Quantum 4d Yang-Mills Theory and Time-Reversal Symmetric 5d Higher-Gauge Topological Field Theory
Zheyan Wan, Juven Wang, Yunqin Zheng
TL;DR
This paper investigates 4d SU(2) Yang-Mills theories with topological terms as boundaries of 5d topological states, revealing new anomalies, exotic statistics, and links to higher-dimensional topological quantum field theories.
Contribution
It introduces a classification of time-reversal symmetric SU(2)$_{ heta=\pi}$ YM theories with background fields, and connects them to 5d higher-symmetry-protected topological states and topological orders.
Findings
Identifies four classes of symmetry-enriched SU(2)$_{ heta=\pi}$ YM theories with anomalies.
Derives exotic anyonic statistics for extended objects in 5d SETs.
Discovers link invariants related to 5d higher-gauge TQFTs.
Abstract
We explore 4d Yang-Mills gauge theories (YM) living as boundary conditions of 5d gapped short/long-range entangled (SRE/LRE) topological states. Specifically, we explore 4d time-reversal symmetric pure YM of an SU(2) gauge group with a second-Chern-class topological term at (SU(2) YM), by turning on background fields for both the time-reversal (i.e., on unorientable manifolds) and 1-form center global symmetry. We find four classes of time-reversal and Lorentz symmetry-enriched SU(2) YM, labeled by : specifies Kramers singlet/doublet Wilson line and new mixed higher 't Hooft anomalies; specifies boson/fermionic Wilson line and a new Wess-Zumino-Witten-like counterterm. Higher anomalies indicate that to realize all higher -global symmetries locally on -simplices, the 4d theory becomes a boundary of a 5d…
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**Quantum 4d Yang-Mills Theory and
Time-Reversal Symmetric
5d Higher-Gauge Topological Field Theory:
** Anyonic-String/Brane Braiding Statistics to Link Invariants
Zheyan Wan1,2, e-mail: [email protected] March 2019 Juven Wang3,4, e-mail: [email protected] (Corresponding Author) and Yunqin Zheng5 e-mail: [email protected]
Dedicated to Professor Shing-Tung Yau and his 70th Birthday celebration on April 4th, 2019.
1*Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China
2*School of Mathematical Sciences, USTC, Hefei 230026, China
3*School of Natural Sciences, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA
4*Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138, USA
5*Physics Department, Princeton University, Princeton, New Jersey 08544, USA
We explore various 4d Yang-Mills gauge theories (YM) living as boundary conditions of 5d gapped short/long-range entangled (SRE/LRE) topological states. Specifically, we explore 4d time-reversal symmetric pure YM of an SU(2) gauge group with a second-Chern-class topological term at (SU(2)θ=π YM), by turning on the background fields for both the time-reversal (i.e., on unorientable and non-spin manifolds) and the 1-form center global symmetry. We find Four Siblings of time-reversal and Lorentz symmetry-enriched SU(2)θ=π YM with distinct couplings to background fields, labeled by : specifies Kramers singlet/doublet Wilson line and new mixed higher ’t Hooft anomalies; specifies boson/fermionic Wilson line and a new Wess-Zumino-Witten-like counterterm. Higher anomalies indicate that in order to realize all higher -global symmetries locally on -simplices, the 4d theory becomes a boundary of a 5d higher-symmetry-protected topological state (SPTs, as an invertible topological quantum field theory (iTQFT) or a cobordism invariant in math, or as a 5d higher-symmetric interacting topological superconductor in condensed matter). Via Weyl’s gauge principle, by dynamically gauging the 1-form symmetry, we transform a 5d bulk SRE SPTs into an LRE symmetry-enriched topologically ordered state (SETs); thus we obtain the 4d SO(3)θ=π YM-5d LRE-higher-SETs coupled system with dynamical higher-form gauge fields. We further derive new exotic anyonic statistics of extended objects such as 2d worldsheet of strings and 3d worldvolume of branes, physically characterizing the 5d SETs. We discover triple and quadruple link invariants potentially associated with the underlying 5d higher-gauge TQFTs, hinting at a new intrinsic relation between non-supersymmetric 4d pure YM and topological links in 5d. We provide 4d-5d lattice simplicial complex regularizations and bridge to 4d SU(2) and SO(3)-gauged quantum spin liquids as 3+1 dimensional realizations. We constrain YM low-energy gauge dynamics by higher anomalies and a higher symmetry-extension method.
Contents
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2 4d SU(2)θ=π Yang-Mills Gauge Theories coupled to 5d Short Range Entangled SPTs
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2.1 Ordinary and Higher Global Symmetries of Yang-Mills Theory
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2.2 Derivation of New Higher-Anomalies of SU(2) Yang-Mills Theory at on Unorientable Manifolds
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2.3 Proof of Anomaly Matching of 5d-4d Inflow and 5d Cobordism Group Data
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2.3.1 Identifying the 4d anomaly with 5d Cobordism Group Data
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2.5 Consequences and Interpretations of Four Siblings of “Anomalies”
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2.6 5d SPTs/Bordism Invariants Whose Boundary Allows 4d SU(2)θ=π YM
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3.1 Kramers Singlet/Doublet under Time-Reversal and Bosonic/Fermionic Wilson line
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5 Anyonic String/Brane Braiding Statistics and Link Invariants of 5d TQFTs
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5.4.2 Extended 2-Surface/3-Brane Operators and Link Invariants
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6 Anyonic String/Brane Spacetime Braiding Processes and Link Configurations of Extended Operators
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6.1 Quadratic Link (Aharanov-Bohm) Configuration in Any Dimension
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8 Lattice Regularization, UV completion and Symmetric Anomalous TQFT
1 Introduction and Summary
The world where we reside, to our best present understanding, can be described by quantum theory and the underlying long-range entanglement. Quantum field theory (QFT) and in particular quantum gauge field theory, under the spell of Gauge Principle following the insights since Maxwell, Hilbert, Weyl, Pauli, and others (see a historical review [1]), embodies the quantum, special relativity and long-range entanglement into a systematic framework. Yang-Mills (YM) gauge theory [2], generalizing the U(1) gauge group of quantum electrodynamics to a non-abelian Lie group, has been proven to be powerful to describe the Standard Model physics.
The Euclidean partition function of a pure YM theory with an SU(N) gauge group on a 4-dimensional (i.e., 4d)111 We denote d for an -dimensional spacetime. We denote D for an -dimensional space and 1-dimensional time. We denote D for an -dimensional spatial object.
spacetime and a second-Chern-class topological term labeled by , i.e., SU(N)θ-YM, is
[TABLE]
where is the gauge field, is the SU(N) field strength, and is the gauge coupling constant. See the footnote222
is locally a 1-form SU(N) connection obtained from parallel transporting the principal-SU(N) bundle over the spacetime manifold . Locally with is the hermitian generator of su(N) Lie algebra, satisfying the commutator where is a fully anti-symmetric structure constant. Locally is a differential 1-form. is the Lie algebra valued gauge field. The path integral is meant to integrate over all the configurations of modding out the SU(N) gauge transformation . The is the Yang-Mills Lagrangian [2] (a non-abelian generalization of Maxwell U(1) gauge theory) where is the Hodge dual of . Tr stands for the trace as an invariant quadratic form of the Lie algebra su(N). The term is related to the second-Chern-class of the SU(N) gauge bundle via . This path integral is sensible for physicists, but may not be mathematically well-defined. We will also point out how to grasp the meaning of YM path integral on unorientable manifolds in Sec. 2.
for further explanations of the notations. When , the SU(N)θ=0 YM theory is believed to be in the confined phase [3], have an energy gap, and have a single ground state on any manifold. In particular, there is no ’t Hooft anomaly [4]. Recently, Ref. [5] discovered that for SU(N)θ=π- YM with even N, there is a subtle ’t Hooft anomaly involving the time-reversal symmetry and 1-form center global symmetry .333We use the subscript to indicate that the symmetry is a 1-form generalized global symmetry [6], and the superscript to indicate the symmetry is electric as opposed to magnetic (i.e., the charged line operators are the Wilson lines rather than the ’t Hooft lines). When we say symmetry in this article, we always mean global symmetry unless we state otherwise. The ’t Hooft anomaly of a 4d theory is captured by a 5d topological term through the anomaly inflow mechanism [7]. Schematically, Ref. [5] suggested a 5d topological term linear in the time-reversal background field and quadratic in the background field ,
[TABLE]
The 5d topological term characterizes the 5d short-rangle-entangled (SRE) phase. See Sec. 2 for more rigorous definitions of the background fields and the 5d topological term.
Further recently, Ref. [8] suggested that there are additional new higher ’t Hooft anomalies for some 4d SU(N)θ=π theories at even N: From one perspective, Ref. [8] suggested that when , there is a mixed anomaly captured by a 5d topological term which is cubic in and linear in , which is schematically denoted as
[TABLE]
From another perspective, Ref. [8] suggested that the SU(N)θ=π YM at an even integer contains new mixed anomalies involving , and a 0-form charge conjugation (i.e., a outer-automorphism) symmetry. For example, at N , another anomaly can be captured by the 5d topological terms schematically as 444Note that so far YM for N , Ref. [8] finds a new 4d anomaly expressible by a term
The is the -th Stiefel-Whitney (SW) class of spacetime ’s tangent bundle . Although there is still a possibility that another 4d anomaly may exist
More precisely, these two 4d anomalies are captured by 5d invertible topological quantum field theories (iTQFTs) and on a 5d closed manifold respectively. Here is the Bockstein homomorphism associated with the extension where is the group homomorphism given by multiplication by . In particular, . The detailed derivation of YM for will be left for the future work [9].
Here is a 1-form background field for the 0-form charge conjugation symmetry. In the following, we will make the above schematic 5d topological terms Eq. (1.2) and Eq. (1.3) mathematically precise, following the setup in Ref. [8] and Ref. [10].
The above 5d topological terms can be regarded as semi-classical partition functions (definable on closed 5-manifolds with appropriate structures) whose functional values depend on the external values to global symmetry-background probe fields. These 5d topological terms are also known as invertible topological quantum field theories (iTQFTs) in the literatures.555 By iTQFT, physically it means that the absolute value of partition function on any closed manifold. Thus this can only be a complex phase , which can thus be inverted and cancelled by as another iTQFT.
In the present work, we will further dynamically gauge the 1-form symmetry associated to the coupled systems of 4d YM and 5d topological terms above. After gauging , the 5d SRE topological terms become 5d long-range entangled (LRE) topological quantum field theories (TQFT). We further apply the methods developed in Refs. [11, 12, 13] to analytically compute the physical observables of the higher-gauge 5d TQFTs. The physical observables of 5d TQFTs include, for example, (i) the partition functions on closed manifolds , (ii) braiding statistics of anyonic strings and anyonic branes whose spacetime trajectories forming 2d worldsheets and 3d worldvolumes respectively, and link invariants of these 2d worldsheets and 3d worldvolumes in . We uncover new spacetime braiding processes and link invariants, including triple and quadruple linkings analogous to previous works [12, 14, 11, 15, 16]. 666 Here we comment on the physical and mathematical meanings of fractional statistics and non-abelian statistics associated with the spacetime braiding processes involving 0D anyonic particles, 1D anyonic strings, 2D anyonic branes and other extended objects. In the discussions below, we take a generalized definition of anyonic.
In a more restricted definition, anyonic means the self-exchange statistics can go beyond bosons or fermions [17].
In our generalized definition, anyonic means that either self-exchange statistics (of identical objects) or the mutual statistics (of multiple distinguishable objects, may involving more than two objects) can go beyond bosonic or fermionic statistics.
— In 3d (2+1D) spacetime , braiding statistics of particles can be fractional (such as the exchange statistics of two identical particles, or mutual statistics of two different particles) which are called anyonic particles (see an excellent historical overview [17]). As an example, this can be understood from a 3d Chern-Simons action with 1-form gauge field integrated over
which modifies the quantum statistics of particle worldline whose open ends host the anyonic particles.
— In 4d (3+1D) spacetime , braiding statistics of particles cannot be fractional as the two 1-worldlines cannot be linked in 4d. Thus there is no anyonic particle and no fractional particle statistics beyond bosons or fermions in 4d. However, the braiding statistics of strings can be fractional which we dub anyonic strings. As an example, the fractional statistics of strings can be understood from a 4d TQFT with a 1-form gauge field and a 2-form gauge field , as
which modifies the mutual quantum statistics of a 0D particle from 1-worldline linked with a 1D string from 2d worldsheet in .
Since a particle cannot carry a fractional charge in 4d, we can interpret the above theory as the anyonic string carrying a fractional flux in 4d. Another way to interpret the fractional statistics of anyonic strings is through the dimensional reduction from 4d to 3d. Let with the size of much smaller than the size of and let the closed anyonic strings wrap around [18, 19, 20] , then the anyonic strings in reduce to anyonic particles in .
From the field theory side, the 4d TQFTs
can modify the braiding statistics of strings [12, 11, 21, 22, 23, 24, 25, 26]. See the relations between Dijkgraaf-Witten’s group cohomology theory [27] and these TQFTs discussed in [12, 11, 21]. Furthermore, there are 4d gauge invariant topological terms with 2-form gauge field [12, 28, 6]
— In 5d (4+1D) spacetime , for example, there exist self and mutual coupling type of 5d topological terms with 2-form gauge fields , , , etc.,
The self coupling term leads to anyonic strings within the restricted definition, where the self-exchange statistics goes beyond bosonic and fermionic[17]. The mutual coupling term leads to anyonic strings within the generalized definition, where anyonic means that mutual statistics of distinguishable 1D strings can go beyond bosonic or fermionic statistics. Both terms modify the quantum statistics of string worldsheet whose open ends host the 1D anyonic string.
We can have another Aharonov-Bohm like topological term with local 1-form gauge field and 3-form gauge field .
When the above term appears together with other topological terms like , the statistics of 2D brane (attached to the end of 3d worldvolume) can have fractional statistics within the general definition, while the statistics of 0D particle (attached to the end of 1-worldline ) remains non-anyonic. Again the anyonic brane in 5d can reduce to anyonic particles in 3d by compactifying along in [18, 19, 20]. There are many other terms allowed in 5d[21]. For a general dimension , there exists the topological term
where is a -form gauge field. we always take the higher-dimensional object from the -field to have fractional statistics (the analogs of fractional flux), while we take the lower-dimensional object from the -field to have a regular statistics (the analogs of integrally quantized charge).
Now let us take a step back to digest the physical meanings of these 5d topological terms Eq. (1.2)-Eq. (1.3). The dimensional ’t Hooft anomaly of ordinary 0-form global symmetries is known to be captured by a dimensional iTQFT. In the condensed matter literatures, these d iTQFTs describe Symmetry-Protected Topological states (SPTs)777We abbreviate both Symmetry-Protected Topological state(s) as SPTs. We abbreviate Symmetry-Enriched Topologically ordered state(s) as SETs. [29, 30, 31, 32]. The relations between the SPTs and the response probe field-theoretic partition functions have been systematically studied, selectively, in [33, 21, 34, 35, 36, 37, 38] (and References therein), and climaxed to the hint of cobordism classification of SPTs[39, 40]. Recently the iTQFTs and SPTs are found to be systematically classified by a powerful cobordism theory framework of Freed-Hopkins [41], following the earlier work of Thom-Madsen-Tillmann spectra [42, 43].
Further recently, Ref. [10] generalized the Thom-Madsen-Tillmann-Freed-Hopkins cobordism theory [42, 43, 41] to the cases with generalized higher global symmetries [6]. The generalized cobordism group computation [10], which involves the bordism group of higher classifying spaces and their fibrations, e.g. , can capture the dimensional higher ’t Hooft anomaly of generalized global symmetries by dimensional bordism invariants (i.e., generalized symmetric iTQFTs or higher symmetric iTQFTs). In the following, we also call the generalized symmetric iTQFTs as higher-SPTs. The d boundary of d higher-SPTs has d higher-anomalies. Gauging the higher-symmetry of higher-SPTs gives rise to higher-gauge theories. Earlier pursuits on a systematic framework of the generalized symmetric iTQFTs through cobordism theories and cohomology theories include, but not limited to, Ref. [44, 45, 46, 47, 48, 49, 50, 51, 52, 53] and references therein. In this work, we identify the 4d anomalies of SU(N)θ=π YM Eq. (1.2)-Eq. (1.3) with the mathematically precise 5d bordism invariants,888 For the mathematical terminology, we call:
the bordism group generators as the manifolds or manifold generators, which generate finite Abelian groups, e.g., .
the cobordism group generators as the topological terms or iTQFTs, which generate Abelian groups, e.g., or , etc.
the co/bordism invariants (people call bordism invariants as cobordism invariants with the same meaning) mean that they are invariants under the bordism class of manifolds, thus co/bordism invariants mean the topological terms or iTQFTs, which again generate Abelian groups, or , etc. obtained in Refs. [8] and [10].
Let us rephrase the higher anomalies into a condensed matter language on the lattice: Higher anomaly indicates that in order to realize all higher -global symmetries locally on -simplices, the theory needs to become a boundary of a one-higher dimensional higher-symmetry-protected topological state (higher-SPTs). If a theory has a higher anomaly, then the theory in its own dimension has some higher -global symmetries acting non-locally (on -simplices). Then there is an obstruction to gauge such a non-local symmetry, hence the name of higher (’t Hooft) anomaly [4]. More in Sec. 8.
1.1 The Outline
Here are the outlines of the present work.
Sec. 2 — We identify the 5d bordism invariants with the ’t Hooft anomalies of 4d SU(N)θ=π YM theory (where we focus on ), as a version of higher-anomaly matching.
Sec. 3 — We clarify and enumerate possible distinct classes of 4d SU(2)θ=π YM theories. We take the condensed matter viewpoint, where we regard the SU(2)θ=π YM theories as infrared theories emerging from high-energy ultraviolet (UV) bosonic systems with a lattice cutoff, as opposed to fermionic systems. We thus call the UV system as bosonic YM theories. These bosonic YM theories still allow Wilson line operators as worldlines of particles being (1) either bosonic or fermionic in quantum statistics, (2) either Kramers doublet or Kramers singlet under the time-reversal symmetry. This supplements as a partial classification of 4d SU(2)θ=π bosonic YM theories. We apply the tools in Ref. [38] to understand the relation between gauge bundle constraint and the properties of line/surface operators.
From Sec. 2 and Sec. 3, we will see that there are at least four closely related 4d SU(2)θ=π non-supersymmetric pure YM theories (which we nickname them as Four Siblings of 4d SU(2)θ=π YM theories) with bosonic UV completions. Each of them carries either a distinct 4d ’t Hooft anomaly associated with a 5d higher-SPTs or a distinct 4d counterterm. The distinct 5d higher-SPTs labeled by distinct the 5d bordism invariants are actually the physical analogs of the 5d (4+1D) one-form-center-symmetry-protected interacting topological superconductors in the condensed matter language.
Sec. 4 and Sec. 5 — We dynamically gauge the 1-form center symmetry . This turns the 4d SU(2)θ=π YM/5d-higher-SPTs coupled systems in [8] into 4d SO(3)θ=π YM/5d-higher-SETs coupled systems, where SETs stand for the symmetry-enriched topologically ordered states.999 Symmetry-Protected Topological state (SPTs) is a short-ranged entangled (SRE) gapped quantum state that can be defined on a lattice. (UV complete such as defining the quantum state on a triangulable manifold or a simplicial complex.) If we preserve the global symmetry, SPTs cannot be deformed to a trivial tensor product state under finite steps of local unitary transformations, assuming without closing any energy gap and without level crossing. Once we break the global symmetry, SPTs can be deformed to a trivial tensor product state under finite steps of local unitary transformations, even without closing any energy gap and without energy level crossing. Symmetry-enriched topologically ordered state (SETs) is a long-ranged entangled (LRE) gapped quantum state that can be defined on a lattice. (Here we only discuss the SETs that are anomaly free.) Even if the global symmetry is completely broken, SETs cannot be deformed to a trivial tensor product state under finite steps of local unitary transformations, assuming without closing any energy gap and without level crossing. The SETs have the same LRE nature as topologically ordered states. See recent reviews [29, 30, 31, 32].
We then explore the detailed properties of various 5d higher-SETs. The 5d higher-SETs are described by 5d time-reversal symmetric higher-TQFTs with emergent 2-form dynamical gauge fields. We mainly focus on the Four Siblings of 5d higher-SETs, while also consider other highly relevant exotic 5d higher-SETs. To characterize these 5d higher-SETs, we study the following aspects:
Partition function without extended operator (1-line, 2-surface, 3-submanifold) insertions on 5-manifold . We compute following the techniques and tools built from [12] and [13]. In particular, when , the partition function is the topological ground state degeneracy (topological GSD) on a spatial . This issue is addressed in Sec. 4. 2. 2.
Braiding statistics involving anyonic 1D string/2D branes, and the associated link invariants of the spacetime 2d worldsheet/3d worldvolume. Here we compute the path integral with extended-operator insertions (), following the techniques and tools built from [11, 12, 15, 16]. This issue is addressed in Sec. 5.
Sec. 6 — We provide the exemplary spacetime braiding processes of anyonic string/brane in 5d, and the link configurations of extended operators, which can be detected by the link invariants that we derived in Sec. 5.
Sec. 7 — We re-examine the coupled system of 4d SO(3)θ=π YM theories and 5d-higher-SETs appeared in Sec. 4.
Sec. 8 — We construct the lattice regularization and UV completion of some of our systems. This includes a lattice realization of 5d higher-SPTs and higher-gauge SETs by implementing on 5d simplicial complex spacetime path integral, and a 4+1D “condensed matter” realization on the spatial Hamiltonian operator. We also provide a lattice regularization of (1) higher-symmetry-extended and (2) higher-symmetry-preserving anomalous 3+1D topologically ordered gapped boundaries by generalizing the method of [54]. The higher-symmetry-extension method was also developed in [55].
Sec. 9 — We conclude and make connections to physics and mathematics in other perspectives.
Before we proceed to the detailed discussions in the main text, we first give a quick overview on more colloquial and pedestrian summaries in terms of schematic descriptions and Table 1, in Sec. 1.2. Readers who are not familiar with certain mathematical information or physical motivations may seek for additional helps from Refs. [38] (and its Appendices), [8] and [10].
1.2 Summaries and Tables
As we mentioned, in Sec. 2 and Sec. 3, we will see that there are at least four closely related 4d SU(2)θ=π non-supersymmetric pure YM theories (nicknamed the Four Siblings of 4d SU(2)θ=π YM theories are labeled by ) with a bosonic UV completion. They carry either distinct 4d higher ’t Hooft anomalies101010 Distinct 4d higher ’t Hooft anomalies correspond to distinct 5d higher-SPTs/counterterms labeled by distinct 5d bordism invariants: physical analogs of 5d (4+1D) one-form-center-symmetry-protected interacting “topological superconductors” in a condensed matter language. In condensed matter, topological superconductors refer to electronic systems with time-reversal symmetry but without U(1) electron charge conservation symmetry (see an overview [30, 31]), for example due to the Cooper pairing breaking U(1) charge symmetry down to a discrete subgroup or down to nothing. or distinct 4d counterterms. All these anomalies that we will discuss below are the mod 2 non-perturbative global anomalies, similar to the old and the new SU(2) anomalies [56, 57]; except that instead of an ordinary global symmetry, now we require a higher 1-form symmetry to probe higher anomalies. Here we advertise these results in a colloquial and pedestrian manner.
. The 1st Sibling of 4d SU(2)θ=π with Kramers singlet () bosonic Wilson line has the 4d anomaly/5d bordism invariant schematically as:
[TABLE]
with the -th Stiefel-Whitney (SW) class of spacetime manifold ’s tangent bundle . Here is a degree-2 cohomology class obtained from restricting the 2-form field via and for any closed surface . More rigorously, stands for , explained in Sec. 2, with a twisted SW class , a cup product , and a Pontryagin square . 2. 2.
. The 2nd Sibling of 4d SU(2)θ=π with Kramers doublet () bosonic Wilson line has the 4d anomaly/5d bordism invariant schematically as:
[TABLE]
We note that the the 4d anomaly associated with the 5d term is highly related to the 2d charge conjugation anomaly associated to the 3d cubic term for a -valued 1-cohomology class . See the relevant studies of 2d anomaly from the 3d cubic term in [58, 22, 59, 60, 8] and References therein. 3. 3.
. The 3rd Sibling of 4d SU(2)θ=π with Kramers doublet () fermionic Wilson line has the 4d anomaly/5d bordism invariant schematically as:
[TABLE]
Here is a coboundary operator, sending a -cochain in the cochain group to a -coboundary in the coboundary group . Note that there are maps and , so in the cohomology group can be pulled back to another cohomology group , with O the orthogonal group O for -manifold. In this case, the is a cohomology class in . Meanwhile sends to a cohomology class in . The is mathematically precisely a Steenrod square [61]. 4. 4.
. The 4th Sibling of 4d SU(2)θ=π with Kramers singlet () fermionic Wilson line has the 4d anomaly/5d bordism invariant schematically as:
[TABLE]
We remark that our investigations on Kramers time-reversal properties and bosonic/fermionic statistics of line operators (for non-abelian gauge theories here) give rise to a further refined classification of gauge theories somehow beyond the previous framework of Ref. [62] and [6]. See Ref. [63, 64] for the case of abelian U(1) gauge theories. See also [38], [65] and [66] for other examples of non-abelian gauge theories.
The schematic term in Eq. (1.6) and Eq. (1.7) is written as mathematically precisely on a 5-manifold in Sec. 2. We will see that such a term vanishes (as the 0 mod 2), when is a closed 5-manifold. However, Sec. 2 shows that when has a boundary , transforms nontrivially under where is a 1-cochain. This nontrivial transformation is essential to cancel the noninvariance of the 4d YM theory. This observation indicates a subtle fact that cannot be dropped and should be kept as a certain physical term, since we are studying the physics on a 5d manifold with 4d boundary. To summarize:
- •
vanishes as 0 (mod 2) on a closed 5-manifold . This can be interpreted in many distinct but related ways. It describes a trivial gapped vacuum with no SPT order, or a trivial gapped insulator in condensed matter language, or a trivial iTQFT on .
- •
However, has essential physical effects on a 5-manifold with a nontrivial boundary . Under the background gauge transformation , the gauge variant is non-zero.
- •
on an with boundary may behave like — which is half of a 4d bordism invariant . Twice of this fractional term is a 4d bordism invariant, and quadruple of this fractional term mod 2 is a trivial 4d bordism invariant. Thus cannot be interpreted as a 4d local counter term. Instead, we interpret it as a non-local counter term, a fractional counter term, or a fractional SPTs on . This is analogous to a certain Wess-Zumino-Witten (WZW)-like term111111We thank Ho Tat Lam for an inspiring conversation on this issue. with the following new features:
(i) The standard WZW term [67, 68] is labeled by an integer, but here is labeled by a number.
(ii) The standard WZW term is written in terms of dynamical fields, but the WZW-like term here is written in terms of the background fields of the time-reversal symmetry and a higher symmetry .
- •
Similar to the standard WZW term, our WZW-like term affects the symmetry quantum numbers of physical observables, i.e., the statistics and Kramers degeneracy (i.e. singlet or doublet) of the Wilson lines.
A schematic illustration of 4d SU(2)θ=π YM-5d SRE higher-SPTs coupled system is shown in Fig. 1. See Table 1 for a short summary for the Four Siblings of 4d SU(2)θ=π YM theories and their coupling to the 5d systems, as well as their physical properties. See Table 2 for a summary of the link invariants and link configurations of 5d TQFTs.
2 4d SU(2)θ=π Yang-Mills Gauge Theories coupled to 5d Short Range Entangled SPTs
2.1 Ordinary and Higher Global Symmetries of Yang-Mills Theory
We discuss the global symmetries of SU(N)θ YM theory in Minkowski/Lorentz signature version of Eq. (1.1): \int[{\cal D}{a}]\exp\big{(}\hskip 1.0pt\mathrm{i}\hskip 1.0pt\int\limits_{M^{4}}\frac{1}{g^{2}}\text{Tr}\,(F_{a}\wedge\star F_{a})-\hskip 1.0pt\mathrm{i}\hskip 1.0pt\int\limits_{M^{4}}\frac{\theta}{8\pi^{2}}\text{Tr}\,(F_{a}\wedge F_{a})\big{)} =\int[{\cal D}{a}]\exp\big{(}\hskip 1.0pt\mathrm{i}\hskip 1.0pt\int\limits_{M^{4}}(-\frac{1}{4g^{2}}\,F_{\mu\nu}^{\alpha}F^{\alpha,\mu\nu}{}-\frac{\theta}{64\pi^{2}}\epsilon^{{\mu\nu}{\mu^{\prime}\nu^{\prime}}}\,F_{\mu\nu}^{\alpha}F_{{\mu^{\prime}\nu^{\prime}}}^{\alpha})\hskip 1.0pt\mathrm{d}^{4}x\big{)}, for the convenience of studying the anti-unitary time-reversal symmetry that sends an imaginary number .
We first focus on the discrete time-reversal symmetry and its symmetry transformation acting on the gauge field with the hermitian Lie algebra generator (namely ) and the real-valued . The temporal component is and the spatial component is . The anti-unitary acts on as an active transformation as:
[TABLE]
Here and below, the gauge field associated with a real symmetric Lie algebra generator (namely ) has the upper version of the sign choices. The gauge field associated with an imaginary antisymmetric Lie algebra generator (namely ) has the lower version of the sign choices. The components of the field strength are and . Under , only one of the and flips its overall sign:
[TABLE]
Here is the structure constant of the SU(N) Lie algebra which is real. The reason that this is a good symmetry choice in contrast to the familiar -symmetry of U(1) gauge theory case is explained in the footnote.121212 The familiar U(1) gauge theory’s transformation sends and . If we choose instead and for gauge field, then and are not mapped back to themselves (not even up to a sign); thus this does not define any symmetry of SU(N) YM. However, by including the ’s anti-unitary transformation on the Lie algebra generator T^{\alpha}\to{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}T^{\alpha*}}, our transformation on a non-abelian gauge field overall still sends and .
Given a gauge group , the above discussion is related to the center Z, the automorphism group Aut(), the outer automorphism Out() and the inner automorphism Inn(). They form short exact sequences:
and a combined exact sequence
If is a simply-connected compact Lie group and is its Lie algebra (which would necessarily be semi-simple), then , , and is isomorphic to the automorphism group of the Dynkin diagram of the Lie algebra .
For , we have , , .
For , we have , , , and .
For with N , we have , , and . We also have where acts on by on the Lie algebra alone, and by in Eq. (2), with a minus sign and a complex conjugation .
The validity of the charge conjugation symmetry , with a global symmetry transformation, is based on the validity of the outer automorphism that includes a as a .
It is obvious that the kinetic term is invariant under . The term flips the sign under :131313More explicitly, under : (Using Eq. (2.2))
{\cal T}:\epsilon^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}0}ijk}F^{\alpha}_{0i}(t,x)F^{\alpha}_{jk}(t,x)\to-\epsilon^{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}0}ijk}F^{\alpha}_{0i}(-t,x)F^{\alpha}_{jk}(-t,x).
The time reversal here is chosen as an active transformation that changes the sign of the time coordinate of the field here up to some phase. When we wrote Eq. (1)’s , we mean to say the coordinate assignment to the fields are flipped so to define an active transformation. The chosen does not do a passive transformation here so that the coordinate and integration measure maintains . Also for example, in (1), its time derivative in (2.2) says maps under to \underset{{\Delta t\to 0}}{\lim}\frac{a^{\prime\alpha}_{i}(t+{\Delta t})-a^{\prime\alpha}_{i}(t)}{\Delta t}=\mp\underset{{\Delta t\to 0}}{\lim}\frac{a_{i}^{\alpha}(-t-{\Delta t})-a_{i}^{\alpha}(-t)}{\Delta t}=\mp\underset{{\Delta t\to 0}}{\lim}\frac{-(a_{i}^{\alpha}(-t)-a_{i}^{\alpha}(-t-{\Delta t}))}{\Delta t}=\pm\underset{{\Delta t\to 0}}{\lim}\frac{(a_{i}^{\alpha}(\tilde{t})-a_{i}^{\alpha}(\tilde{t}-{\Delta t}))}{\Delta t}{\big{|}_{({\tilde{t}}=-t)}}=\pm\partial_{\tilde{t}}a_{i}^{\alpha}{\big{|}_{({\tilde{t}}=-t)}} with the time derivative evaluated at .
[TABLE]
The has a periodicity, thus the theories at and are time-reversal invariant. 2. 2.
We can define the symmetry associated with the transformation for an SU(N) gauge theory:
[TABLE]
Here is the complex conjugation. The is anti-unitary so also sends an imaginary number . We also define the charge conjugation symmetry associated with the transformation for an SU(N) gauge theory:141414The active transformation acts on the fields only, not on the Lie algebra generator . But effectively on the full , the transformation is also equivalent to the Lie algebra outer automorphism transformation , , and overall .
[TABLE]
However for N = 2, the SU(2) YM does not have global symmetry because SU(2) does not have a outer automorphism. The transformation is part of the SU(2) gauge transformation. Let be the matrix that provides an isomorphism between fundamental representation of SU(2) and its conjugate, and be the unitary SU(2) transformation on the SU(2)-fundamentals, where are Pauli matrices. Then In other words, and are the same symmetry for the SU(2) YM. See more discussions in the footnote 12, Sec. 2.2 of [38] and Sec. 2 of [8]. 3. 3.
Parity symmetry is another discrete symmetry associated with the transformation :
[TABLE]
is related to via the symmetry:
[TABLE] 4. 4.
The 1-form electric center global symmetry: The charged object of the 1-form -symmetry is a gauge-invariant Wilson line
[TABLE]
The gauge field is Lie algebra su(N) valued. The specifies a SU(N) group element where P is the path ordering. Tr is the trace in the representation R of SU(N). For the SU(N) gauge theory, R can be any possible representation. If R is an irreducible representation and let be the number of boxes in the Young diagram of R, then transforms under as
[TABLE]
For the fundamental representation, there is only one box in the Young diagram, hence the Wilson line transforms under as . For N2, the Wilson line in the fundamental representation transforms under by a sign .
The charge operator (i.e., symmetry generator) of the -symmetry is a co-dimension 2 (thus a 2D operator in 4d spacetime) electric surface operator . For SU(2) gauge theory, we will see that
[TABLE]
where as a cohomology class.
One can couple the SU(2) theory to background gauge field . Following [69, 6, 5], we first promote the SU(2) gauge field to a U(2) gauge field ,
[TABLE]
where is a two dimensional identity matrix. The first Chern class of the U(2) bundle is where is a U(2) field strength. Then we couple to by requiring , which can be done via introducing a Lagrangian multiplier (see Eq. (2.46)). This amounts to introducing the following term in the path integral,
[TABLE]
The minimal coupling implies that the symmetry generator (i.e., charge operator) of is precisely . This explains Eq. (2.10). Notice that integrating out the Lagrangian multiplier removes the U(1) degree of freedom, hence the gauge group is SO(3)PSU(2) (rather than SU(2)),
[TABLE]
with the gauge bundle constraint . Here the second Stiefel-Whitney class is the obstruction of promoting the SO(3) bundle to SU(2) bundle, which we explain in detail below. The nontrivial SU(2) gauge bundle on a manifold can be constructed by finding an open cover of and then gluing together trivial bundles from adjacent open patches via the SU(2) transition functions. Suppose is the transition function (which plays the role of gauge transformation) defined on the intersections of two open covers indexed by and . There is a consistency condition
[TABLE]
on the triple overlapping intersections of three open patches indexed by , and . However, the consistency condition of SO(3)-bundle is weaker. Let be the transition function in the SO(3)-bundle, and is the lift of in the SU(2)-bundle. Then
[TABLE]
while
[TABLE]
The is related to evaluated on the simplex .151515The patch is dual to a 0-simplex in the dual cell decomposition of spacetime. The intersection of two patches and is dual to a 1-simplex in the dual cell decomposition of . The intersection of the patches and is dual to a 2-simplex in the dual cell decomposition of . Thus SO(3) bundle can be lifted to an SU(2) bundle only when is trivial, i.e., the background field is trivial. Namely, activating allows us to study the SU(2) gauge theory with nontrivial SO(3)-gauge bundle. In short,
[TABLE]
and we learn that the SU(2) gauge theory coupled to a background field can be regarded as a path integral summing over SO(3) gauge bundle subject to the gauge bundle constraint . We will soon propose a new generalization of gauge bundle constraint of Eq. (2.16) on unorientable or non-spin manifolds. See Eq. (2.26) in Sec. 2.2.
Coupling to background field allows one to say more on various line and surface operators. First, one can use to construct a magnetic 2-surface . When is a surface with boundary, a Wilson line in the fundamental representation (below, we will simply use for simplicity) can be supported on the boundary so that is invariant under the background gauge transformation . Second, when the surface of the electric 2-surface operator , Eq. (2.10), has a boundary , a ’t Hooft line can be supported on . Since is dynamical in the SU(2) gauge theory, the ’t Hooft line is not a genuine line operator, and has to live on the boundary of . Thus ’t Hooft line as the worldline of probe background magnetic monopole must be attached with the dynamical and detectable open Dirac string, which is visible by . The closed 2d worldsheet of detectable Dirac string forms the operator. This can be seen from the correlation function
[TABLE]
where R stands for the fundamental representation. is the linking number between and . The is on the right-hand side of (2.17), because its expectation value depends on a small perturbation and thus is not a topological operator.
From the Hamiltonian point of view, the spatial Wilson line operator and the spatial ’t Hooft operator (as two canonically quantized line operators) in the SU(N) gauge theory satisfy the commutation relation [70]:
[TABLE]
where is the linking number between and in the 3d space. For the SU(2) YM, Eq. (2.18) reduces to
[TABLE]
The non-commutative nature of Eq. (2.18) implies that the and are not mutually local, which is consistent with the fact that is a genuine line operator while is not a genuine line operator as discussed in the last paragraph. 5. 5.
The full symmetry : The full symmetry of SU(2) YM theory relevant in our study is . 161616Since and differ by a SU(2) gauge transformation, we only discuss . The symmetry implies the spacetime symmetry has an orthogonal group O() via a short exact sequence extension where SO() is the spacetime rotation symmetry. Knowing the full relevant global symmetry, , we can classify the ’t Hooft anomalies based on Thom-Madsen-Tillmann-Freed-Hopkins bordism spectra and cobordism theory[42, 43, 41]. In terms of a bordism group (more precisely, we focus on the torsion part ), the classification of 4d ’t Hooft anomalies for 4d SU(2) YM can be written as linear combinations of bordism invariants for [8, 10]. (We leave the details of bordism invariants later in Eq. (2.49) and in Sec. 2.3.)
2.2 Derivation of New Higher-Anomalies of SU(2) Yang-Mills Theory at
on Unorientable Manifolds
We start with the SU(2) Yang-Mills theory (YM) with , denoted SU(2)θ=π. The Euclidean action from (1.1) is
[TABLE]
Since the anomaly is a renormalization group flow invariant, in the following discussion, the kinetic term which is proportional to the running coupling constant will not play a role. Hence we only consider the second term in Eq. (2.19), which we call the theta term. To probe the anomaly, we turn on the background gauge field for the 1-form symmetry. Here is a -valued 2-form gauge field with for any closed surface . The 2-form gauge field is related to the 2-cochain via , and we also convert the wedge product to the cup product when the action is written in terms of cochains. To couple the SU(2) YM theory to the background gauge field , we promote the SU(2) gauge field to a U(2) gauge field , and the theta term at reads171717 The topological term for the Euclidean action in the Euclidean partition function contains a factor of imaginary , namely in Eq. (2.19). However, by converting , we have the following Minkowski in Eq. (2.20).
[TABLE]
where is the U(2) field strength, and is the two dimensional identity matrix. To restore the SU(2) gauge field, the U(2) field strength should satisfy the gauge bundle constraint
[TABLE]
Here is the Stiefel-Whitney class of the associated vector bundle of the (the principal gauge bundle of ).
To activate the background field for the time-reversal symmetry, we formulate Eq. (2.20) on an unorientable manifold . On an unorientable manifold, the top differential form is not well-defined, due to the lack of the volume form whose definition needs an orientation. To make sense of Eq. (2.20) on an unorientable manifold, we reformulate it in terms of the Chern characteristic classes. We denote the th Chern class of the U(N) gauge bundle as . For , we have
[TABLE]
Replacing by , we rewrite Eq. (2.20) as181818 Some of mathematical-oriented readers may wonder how to rigorously define Eq. (2.20)’s \frac{\pi}{8\pi^{2}}\int_{M^{4}}{\mathrm{Tr}}\big{(}(\widehat{F}-{\cal B}\mathbb{I}_{2})\wedge(\widehat{F}-{\cal B}\mathbb{I}_{2})\big{)} to a term \frac{\pi}{8\pi^{2}}\int_{M^{4}}{\mathrm{Tr}}\big{(}(\widehat{F}-\pi B\mathbb{I}_{2})\wedge(\widehat{F}-\pi B\mathbb{I}_{2})\big{)} with the continuous differential form coupling to a discrete cohomology class . In fact, the physics way to interpret this coupling is related to the coupling between QFT to TQFT explained in [69]. More formally, we can also implement mathematical methods [71] to formulate such couplings. JW thanks Shing-Tung Yau for insightful conversations on this method [71].
[TABLE]
Using Eq. (2.22), Eq. (2.23) can be re-interpreted as
[TABLE]
where is the Pontryagin square191919Notice it is crucial to treat as the more precise re-writing for the later purposes. The denotes the Pontryagin square, e.g. see Ref. [10, 8]. of .
Note that Eq. (2.24) is not well-defined even on an orientable manifold. In Sec. 2.4, we resolve this problem for the torsion-free oriented manifolds . Yet, Eq. (2.24) is also not well-defined on an unorientable manifold. In general, if is a -dimensional unorientable manifold and is a -cocycle, is well-defined only when is valued in .202020Using the definition of the fundamental class of an unorientable manifold , i.e., , one has where is the valued pairing between and . Since is integer valued, the first term in Eq. (2.24) makes sense when is unorientable. However, the other terms are fractional, hence the integral of those terms does not make sense if is unorientable. To make sense of Eq. (2.24), we actually need to define it on both the unorientable and an unorientable such that .212121 Note that if is orientable, then must be orientable. Conversely, if is unorientable, must be unorientable. However, if is orientable, can be orientable or unorientable. To proceed, we extend the integer valued cohomology class on to an integer valued cochain on . Note that on does not have to be an element in , i.e, does not have to hold on . The requirement of will be imposed later by the gauge bundle constraint. The extension means, in particular, that when restricting to , it reduces to a -valued cohomology class . We further extend the -valued cohomology class on to a -valued cohomology class on , and for simplicity, we use the same notation on as well. Thus we define Eq. (2.24) as follows:
[TABLE]
with the background field properly extended to . Here is a coboundary operator, such that we apply from (2.24) to (2.25). To make sure that the integral on an unorientable is well-defined, and also independent of the dynamical gauge field, we need to utilize the gauge bundle constraint, which relates with the background gauge fields and . Below, we will see that the 5-dimensional integral does not depend on the dynamical gauge fields due to the gauge bundle constraints. Hence the 5d integral is an invertible TQFT whose partition function is a local function of the background fields. In summary, we find that in order to make sense of the theta term of the SU(2) YM theory with the background fields on an unorientable manifold, one needs to treat the YM theory as a 4d-5d coupled system. This is a manifestation of the mixed ’t Hooft anomaly between the 1-form global symmetry and the time-reversal symmetry .
On an unorientable manifold , the is non-trivial and one can treat it as the background gauge field for the time-reversal symmetry. This allows us to modify the gauge bundle constraint Eq. (2.21) by an additional term , with . Furthermore, we are also allowed to consider the manifold with non-trivial since the underlying manifold does not necessarily allow a Spin/Pin structure, hence we activate the term with . In summary, there are four choices of gauge bundle constraints labeled by the pair as
[TABLE]
This is a nontrivial constraint between the gauge bundle , the spacetime tangent bundle and the background field . The value of has physical consequences: when , the SU(2) gauge charge (in the fundamental representation of SU(2)) is a Kramers singlet () or a Kramers doublet () under time-reversal transformation;222222For an SU(2) gauge theory, one can either use or as the time-reversal transformation because the charge conjugation of SU(2) is an inner automorphism. The Kramers doublet () of Wilson line (in the SU(2) fundamental representation) means that there is a doublet (two-fold) degeneracy associated with the Wilson line. The two states of the Wilson line, say and forming a 2-dimensional Hilbert space, transforms as and under the time-reversal transformation. when , the SU(2) gauge charge is a boson (spin-statistics as an integer spin) or a fermion (spin-statistics as a half-integer spin). More details about the Wilson line properties are derived in Sec. 3.
The gauge bundle constraint Eq. (2.26) is defined on . We would like to promote it to as follows,
[TABLE]
Eq. (2.27) imposes additional constraints on . Since and are cohomology on , the is equivalent to a -valued cohomology mod 2 (although it is not a -valued cohomology), i.e., .
We further apply the gauge bundle constraint Eq. (2.26) to the 5-dimensional integral Eq. (2.25). We should be aware that the 5-manifold has a boundary . Here we summarize some helpful formulas and mathematical definitions in a footnote232323 We clarify the definitions of various fields we used in terms of cochain (), cocycle (), coboundary (), or cohomology ():
\displaystyle\left\{\begin{array}[]{l}c_{1}(V_{{\rm U}(2)})\in\operatorname{H}^{2}(M,\mathbb{Z}),\\ \delta c_{1}(V_{{\rm U}(2)})=0\in B^{3}(M,\mathbb{Z}),\\ {\widetilde{c}_{1}\in C^{2}(M,\mathbb{Z}),\quad(\widetilde{c}_{1}\mod 2)\in\operatorname{H}^{2}(M,\mathbb{Z}_{2}),}\\ c_{2}(V_{{\rm U}(2)})\in\operatorname{H}^{4}(M,\mathbb{Z}),\\ B\in\operatorname{H}^{2}(M,\mathbb{Z}_{2}),\quad{\cal P}(B)\in\operatorname{H}^{4}(M,\mathbb{Z}_{4}),\\ w_{1}(TM)\in\operatorname{H}^{1}(M,\mathbb{Z}_{2}),\\ w_{2}(TM)\in\operatorname{H}^{2}(M,\mathbb{Z}_{2})\\ \lambda\in C^{1}(M,\mathbb{Z}_{2}),\quad\delta\lambda\in B^{2}(M,\mathbb{Z}_{2}),\quad\delta^{2}\lambda=0\mod 2,\quad\mathrm{Sq}^{1}\delta\lambda\in Z^{3}(M,\mathbb{Z}_{2}).\end{array}\right.
(2.36) Here stands for the -th cochain, for the -th cohomology, for the -th cocycle, and for the -th coboundary. When discussing the cup products, there are subtle distinctions between (1) cohomology classes in , (2) cocycles in and (3) cochains in , which we enumerate below:
(1)
The cup product between two cohomology classes are super-commutative, i.e.,
(2.37)
(2)
The cup products between two cocycles are not super-commutative. If and are general -th and -th cocycles, their commutation relation is governed by the Steenrod’s relation [61]
(2.38)
where we have used the cocycle condition , .
(3)
The cup products between two cochains satisfy Steenrod’s relation [61]
(2.39)
(2.40)
where and are general -th and -th cochains.
In this section, all the calculation still go through if we regard the field as a 2-cocycle, because we did not use the super-commutativity. . Since is in , it makes sense to define its Steenrod square . Then the 5d integral in Eq. (2.25) can be written as
[TABLE]
In the first equality, we simply stated the initial definition. In the second equality, we plugged in the coboundary operator . In the third equality, we used Eq. (2.40) and replaced by which is valid for -valued cocycles. We also used the identity since is a -valued 2-cocycle [8].242424For a 2-cocycle , the following equality holds:
See Eq. (124) in [8] for further details.
In the fourth equality, we plug in the gauge bundle constraint Eq. (2.26). Eq. (2.26) also implies . In the fifth equality, we used In the last equality, we used \big{(}(K_{1}w_{1}^{2}+K_{2}w_{2})\cup\mathrm{Sq}^{1}w_{2}+\mathrm{Sq}^{1}w_{2}\cup(K_{1}w_{1}^{2}+K_{2}w_{2})\big{)}=0\mod 2 since the Stiefel-Whitney classes are super-commutative.
Several comments are in order:
As mentioned below Eq. (2.25), is a properly quantized integral of the background field and the Stiefel-Whitney class , which is independent of the dynamical gauge field. Hence is an invertible TQFT. 2. 2.
In Eq. (2.25) and Eq. (2.41), the 5d unorientable manifold has a boundary .
- •
If does not have a boundary, the term vanishes, due to
[TABLE]
In the last step, we have used the Wu-formula , on a closed 5-manifold. Hence Eq. (2.41) simplifies to
[TABLE]
- •
If has a boundary, transforms non-trivially under the background gauge transformation ,
[TABLE]
This compensates the non-invariance of the 4d theory under . Thus although the terms vanish when is a closed manifold, when has a boundary, it is crucial to keep track of this term.
- •
We can show that the term is well-defined in 4d by showing that this terms only depends on the 4d boundary . The triviality of on a closed implies that when the 5d manifold has a boundary, such a term does not depend on the choice of extension, i.e., given two 5d extensions and , we know because is closed, thus we derive . Note that when has a boundary, can be nonzero. This is analogues to the WZW term. See Sec.2.5 for further discussions. 3. 3.
The 4d-5d integral Eq. (2.25) is invariant under a 1-form gauge transformation . We will show this explicitly in Sec. 2.3. 4. 4.
Although only depends on when is closed, we still label it as the 5d anomaly polynomial parameterized by , due to the subtlety that the 5d integral still depends on when has boundary.
To summarize, the partition function of the combined 4d-5d coupled system
[TABLE]
is fully gauge invariant under the transformation of the background gauge field and time-reversal symmetry, the full partition function also makes sense when and are unorientable, where
[TABLE]
and
[TABLE]
The combined 4d-5d system is anomaly free. Equivalently, to couple the background fields of both time-reversal symmetry and the 1-form global symmetry , the YM theory cannot be background gauge invariant by being placed on an unorientable only. Instead, one needs to place on the boundary of an unorientable which supports a 5d invertible TQFT . This is the manifestation of the the YM’s mixed ’t Hooft anomaly between the 1-form global symmetry and the time-reversal symmetry .
2.3 Proof of
Anomaly Matching of 5d-4d Inflow and 5d Cobordism Group Data
In this subsection, we identify the 5d topological terms Eq. (2.41) with the mathematically well-defined 5d bordism invariants, and further explicitly check the invariance of the 4d-5d system Eq. (2.45) under .
2.3.1 Identifying the 4d anomaly with 5d Cobordism Group Data
We compare the 4d anomaly in Eq. (2.41) with the bordism group data given in [8] and [10]. Since the global symmetries of 4d SU(2)θ=π YM theory is , we compute the 5d bordism group252525In addition to [8] and [10], we notice that the oriented version of the bordism group has been studied recently in [46] for different purposes. Here we study instead the unoriented version of the bordism group , new to the literature. See details in Appendix A. 262626 For an ordinary (0-form) global symmetry, we denote as the [math]-form global symmetry group. When gauging a [math]-form symmetry, we introduce a -form flat gauge field with a gauge group , whose classifying space is . For an abelian group and for a higher symmetry: We denote as the -form global symmetry group. When gauging a -form symmetry, we introduce a -form flat gauge field with a higher gauge group, whose classifying space is associated with . Similarly, for an abelian -form global symmetry group , we have the associated classifying space . See [48, 49, 50].
[TABLE]
Hence there are four independent generators of the bordism group ,
[TABLE]
where the equalities hold only on closed 5-manifolds. Clearly, in Eq. (2.41) is a bordism invariant except the term proportional to . Setting , is identified with the sum of first three bordism invariants in Eq. (2.49),
[TABLE]
As explained in Sec. 2.2, the fourth term in is a trivial when does not have a boundary. This is consistent with the fact that there isn’t any bordism invariant of ) of the form .
Notice that the last invariant in Eq. (2.49) 272727The is a bordism invariant in , , and , see [10]. Namely, this is not only a topological term respecting a spacetime symmetry and 1-form -symmetry, but also a topological term respecting a spacetime or symmetry alone, or respecting an enhanced spacetime-internal locked symmetry . Thus the 4d anomaly from is a signature for the new SU(2) anomaly [57]. In fact, the topological term plays an important role as the only possible anomaly of an interacting Spin(10) chiral fermion theory — which is responsible for the anomaly-free of the SO(10) Grand Unification [72, 57].
, i.e. , does not participate in the anomaly of SU(2)θ=π YM. However it is responsible for the new SU(2) anomaly [57]: 4d SU(2) gauge theory with an odd number of fermion multiplets in representations of isospin of the gauge group is inconsistent, for a non-negative integer . The theory is nevertheless consistent on certain manifolds with Spin or Spinc structure. The new SU(2) anomaly [57] is in contrast of the old SU(2) anomaly [56]. The familiar SU(2) anomaly [56] states that a 4d SU(2) gauge theory with an odd number of fermion multiplets in the isospin representation is inconsistent.
In this subsection, we proved that the 4d YM anomaly (derived in Sec. 2.2) given in Eq. (2.47) matches a mathematically well-defined 5d bordism invariant from a bordism group data (given in Appendix A).
2.3.2 Anomaly Matching of 4d-5d Inflow
We first highlight the distinctions between the derivation of anomalies in [5] and in our Sec. 2.2.
- •
Ref. [5] places the SU(2)θ=π YM on an orientable manifold, and turns on the 2-form background field of the 1-form symmetry (or a 2-cochain ). By performing a time-reversal transformation, Ref. [5] detects the anomaly, which is linear in transformation and quadratic to the 2-cochain .
- •
In Sec. 2.2, we have derived the anomaly by first turning on the 2-form gauge field , and further place the theory on an unorientable manifold. We find that to make sense of the 4d theta term on an unorientable manifold, we need to promote the original 4d YM theory to a combined 4d-5d system. The 5d theory is an invertible TQFT. In the following, we reverse the logic:
- (Step 1)
We first formulate the YM on an unorientable manifold before activating .
- (Step 2)
We further match the non-invariance of the 4d SU(2)θ=π YM theory Eq. (2.46) under with the non-invariance of in Eq. (2.41).
- (Step 1)
We first place the Yang-Mills theory on an unorientable manifold without activating the background field . If we limit to case that the gauge bundle constraint Eq. (2.26) as , then the theta term is simplified to
[TABLE]
which is a well-defined 4d term. If we further change the time-reversal property (i.e., Kramers singlet/doublet) and the statistics (i.e., bosonic/fermionic) of the SU(2) gauge charge by modifying the gauge bundle constraint to , the theta term is
[TABLE]
The second term does not make sense for unorientable, and one needs to define it by promoting the integral to a 5d unorientable manifold . Following the discussion around Eq. (2.27), the -valued cohomology class is extended to a cohomology class , along with the gauge bundle constraint, . Then, Eq. (2.52) shall be re-interpreted as
[TABLE]
When does not have a boundary, vanishes. This means that, for a fixed , the second term in Eq. (2.53) does not depend on the choice of . Hence, when is turned off, there is no anomaly for generic . To summarize, there is no pure time-reversal anomaly of Yang-Mills with . 2. (Step 2)
We further turn on the background field . Under the gauge transformation where is a -valued 1-cochain, the field strength transforms as
[TABLE]
Using Eq. (2.22), we determine that
[TABLE]
The only 4d term in Eq. (2.25) is the first term proportional to . Under ,
[TABLE]
In the second equality, we replaced by which is valid for -valued cocycles, and used the identity since is a cocycle [8]. On the other hand, the variation of the bulk invertible TQFT , i.e. the 5d integral in Eq. (2.25), is
[TABLE]
In the second equality, we used the identity since is a cocycle [8], and the formula since and are both cocycles. In the third equality, we replaced by which is valid for -valued cocycles, and used the identity since is a cocycle [8].
Comparing Eq. (2.55) and Eq. (2.56), we find that the non-invariance of the 4d terms Eq. (2.55) precisely cancels the non-invariance of the 5d terms Eq. (2.56). Thus the combined 4d-5d coupled system is symmetric under the background gauge transformation of , thus is anomaly free under the 1-form background gauge transformation.282828On an unorientable manifold, the mixed time-reversal and 1-form anomaly reduces to 1-form anomaly, since time-reversal symmetry is “gauged” on an unorientable manifold and it is too late to break . Furthermore, since both the boundary theory Eq. (2.51) and the bulk invertible TQFT are well-defined on an unorientable manifold and respectively, the full system also respects the time-reversal symmetry. Thus we again arrive at the conclusion that the combined partition function Eq. (2.45) is well-defined and free of the ’t Hooft anomalies of both 1-form symmetry, time-reversal symmetry and their mixed anomaly.
2.4 Topological Term On Torsion-Free Orientable Manifolds
In the previous sections 2.2 and 2.3, we derived the mixed anomaly by first reformulating the theta term in terms of characteristic classes, and then make sense of it on unorientable manifolds by promoting the ill-defined terms on 5-manifolds. However, there is a loop-hole: Eq. (2.24) is not well-defined even on an oriented manifold, because and , as a and valued cohomology respectively, are ill-defined when the coefficients are fractional. In this subsection, we resolve this issue, for certain manifolds, by lifting the class to a class , i.e,
[TABLE]
Here we restrict to the orientable manifolds with torsion-free cohomology class [73] where the lifting makes sense. Hence Eq. (2.24) becomes
[TABLE]
To further formulate Eq. (2.58) on an unorientable manifold, we note that every unorientable manifold contain nontrivial torsion in , and thus the lifting does not exist. This implies that on an unorientable manifold and , it is not possible to promote a cohomology class to a cohomology class. However, the derivation of the 5d anomaly polynomial Eq. (2.41) still goes through.
2.5 Consequences and Interpretations
of Four Siblings of “Anomalies”
In this section, we discuss the two siblings of anomalies labeled by and . We also compare our results with the known mixed - anomaly discussed in [5].
When , the bulk anomaly polynomial is
[TABLE]
which is non-vanishing only on an unorientable . This equality has been explored in Ref. [8] in relating to the 4d YM theory’s anomaly. Furthermore, we find that this equality is also explained in a remarkable mathematical note Ref. [74].
Below let us gain a better understanding based on Ref. [74]: Let be the orientation local system, then . Indeed, this is the group cohomology , where denotes with the sign action. The pullback of the nonzero element of under the map determined by is called the twisted first Stiefel-Whitney class . Its mod 2 reduction is the usual first Stiefel-Whitney class in an untwisted -cohomology. We consider its reduction in a twisted mod 4 cohomology. Here denotes the Pontryagin square . In Eq. (2.59), we use cup and cap products in twisted -cohomology: if denotes the fundamental class in the twisted -cohomology, this means that
[TABLE]
However, since is a twisted coboundary, , is even, hence it makes sense to divide by 2 and obtain an element of . This defines as a mod 2 class in the 5th cohomology group which is also a bordism invariant of 5th bordism group .
There are two options for the boundary : orientable or unorientable.
When is orientable, the time reversal of the theory is not gauged. However, there is still a way to probe the mixed - anomaly, following the approach of [5]. We first couple the Yang-Mills to background gauge field , and then perform a global time-reversal transformation. To determine how the theta term changes under timer reversal, we make use of the fact that shifting by amounts to change the parameter of the counter term by , where the counter term is and , i.e.,
[TABLE]
Under time reversal, both the theta term Eq. (2.20) and the counter term change sign, i.e., . Using the identification Eq. (2.61),
[TABLE]
Equivalently, under time reversal, the theta term is unchanged, but there is a shift of the counter term
[TABLE]
The non-invariance in Eq. (2.63) cannot be zero by properly choosing , which represents an anomaly. The anomaly Eq. (2.63) can be canceled by the ’t Hooft anomaly inflow Eq. (2.59).
So it is important to emphasize that the 4d anomaly from detected by [5] (and Sec. 2 of Ref. [8]) is precisely captured by the bordism invariant in Eq. (2.59) noticed in Ref. [8]. 2. 2.
When is unorientable, the anomaly can be detected as well, as discussed in Sec.2.2 and 2.3.
When , the bulk action is
[TABLE]
which is non-vanishing only when is unorientable.
When is orientable, one cannot probe . This is because for to be non-vanishing mod 2 on , there should be at least two or more orientation reversing cycles in , hence there should be at least one orientation cycle in . Thus if is orientable, even if is unorientable, we still cannot detect a particular 4d anomaly associated to the 5d term . 2. 2.
When is unorientable, the anomaly can be detected, as discussed in Sec.2.2 and 2.3.
When ,
If is a closed 5d manifold (regardless orientable or unorientable), we cannot detect the term .
If is a 5d manifold with a 4d boundary (regardless orientable or unorientable in 5d or in 4d) and is nontrivial on both and (e.g., non-Pin*+* manifolds), we can detect the term on the 4d boundary via the 1-form background gauge transformation. On an with a boundary , we regard schematically as a 4d fractional SPTs, which is characterized by a 4d ill-defined term with a fractional coefficient . Two copies of such 4d fractional SPTs become a well-defined time-reversal and 1-form symmetric 4d SPTs/bordism invariant , with respect to a nontrivial -generator in , see Ref. [8] and Appendix A. Thus, four layers of such 4d fractional SPTs become a trivial SPTs with respect to .
The is similar to Wess-Zumino-Witten (WZW) term [67, 68] in some way but with its own exoticness:
(1) The familiar WZW term is an integer class [67, 68], here this has a fractional discrete class. (In some sense, seems to be a unit generator in respect to a 4d trivial SPTs.)
(2) The familiar WZW term is written as a path integral of dynamical fields, but here depends on the background probe fields and .
(3) Both WZW and govern the 4d physics, but they need to be written in one extra higher dimension. It is tempting to speculate that may be a non-local counter term on , which is 4d in nature but cannot be written in 4d alone. The can access the 5d extra bulk, but it does not depend on how is chosen as long as .
Related interpretations and facts about are also summarized in Sec. 1.2.
When , the interpretation is simply the linear combination of and interpretations above.
We will further comment about the fate of dynamics of Four Siblings of SU(2) YM based on their “anomalies” Eq. (2.47), in Sec. 8 and in Sec. 9.
2.6 5d SPTs/Bordism Invariants Whose Boundary Allows 4d SU(2)θ=π YM
2.6.1 On a closed manifold
We now give various equivalent formulas of the 5d SPTs/bordism invariant in Eq. (2.47) on a closed 5-manifold :
[TABLE]
In the second line, we knew already from the derivation of Eq. (2.42) that on a closed manifold.
In the fourth line, we used where the second equality uses Wu formula on a closed manifold. We also used
[TABLE]
[TABLE]
[TABLE]
by the Wu formula on a closed 5-manifold.292929If we consider instead a different 5d SPTs/bordism invariant as , we have the following equalities on a closed 5-manifold:
\displaystyle{\mathbf{Z}}_{\text{SPT}_{(K_{3})}}^{5\text{d}}[M^{5}_{\text{closed}}]=\exp\Bigg{(}{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}K_{3}w_{2}(TM)\mathrm{Sq}^{1}B}\Bigg{)}=\exp\Bigg{(}{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}K_{3}w_{3}(TM)B}\Bigg{)}.
(2.69)
In the fifth line, Eq. (2.67) is based on Eq. (2.59) and Ref. [8, 74].
In the sixth line, Eq. (2.68) is based on Eq. (124) in [8].
2.6.2 On a manifold with a boundary
We also give various equivalent formulas of the 5d SPTs/bordism invariant in Eq. (2.47) on a 5-manifold with a non-empty 4d boundary :
[TABLE]
In the third line, we followed Ref. [8] to define as the Bockstein homomorphism associated to the extension , where is the group homomorphism given by multiplication by . We can show that [8]. Using the bordism group data and the identities given in Ref. [8] and [10], we rewrite the 4d higher-anomalies and 5d higher-SPTs/bordism invariants/anomaly polynomials.
3 Classification of 4d SU(2)θ=π Yang-Mills Theories and Classification of 4d Time-Reversal Symmetric Bosonic/Fermionic SU(2)-SPTs
In this section, we explore the physical meaning of the gauge bundle constraint in Eq. (2.26), i.e.,
[TABLE]
and discuss their physical consequences.
3.1 Kramers Singlet/Doublet under Time-Reversal and Bosonic/Fermionic Wilson line
Below we provide some physical interpretations of the Four Siblings of 4d SU(2) YM theories in terms of the Wilson line properties.
We introduce the standard 4d SU(2) Yang-Mills path integral {\mathbf{Z}}^{\text{4d}}_{{{\rm SU}(2)}{\text{YM}}}[B] coupled to the background 2-form gauge field . {\mathbf{Z}}^{\text{4d}}_{{{\rm SU}(2)}{\text{YM}}}[B] is obtained by replacing with in {\mathbf{Z}}^{\text{4d}}_{{\text{YM}}} in Eq. (1.1). We also need to impose the gauge bundle constraint , which can be imposed by introducing a Lagrangian multiplier303030We can also introduce an additional Pontryagin square term \exp\big{(}\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{\pi}{2}p\mathcal{P}(B)) with into the path integral, as the pioneer works Ref. [62] and Ref. [6] do. However, this weight factor term only will result in shifting (thus relabeling) of the classification of 4d SU(2)θ=π theories that we are going to reveal. We use the notations in [8],
[TABLE]
Electric 2-surface : Mathematically, integrating out the Lagrange multiplier sets . Physically, plays the role of an electric 2-surface , which measures 1-form -symmetry . The magnetic ’t Hooft line lives on the boundary of an electric 2-surface . Since is dynamical, ’t Hooft line is not genuine thus not in the line spectrum for the SU(2) gauge theory [6].
Magnetic 2-surface is given by . The boundary of supports the Wilson loop . Unlinking a 2-surface and a Wilson loop yields a nontrivial statistical -phase .
Following Sec. 2, we enrich the gauge bundle constraint as Eq. (2.26) by introducing two couplings labeled by , and the partition function is
[TABLE]
As we just deduced, the magnetic 2-surface has its boundary as Wilson loop . We will apply this relation to the Four Siblings with the YM partition function Eq. (3.1) and its constraint Eq. (2.26) and discuss the properties of the Wilson lines.
: The gauge bundle constraint is . The magnetic 2-surface has no decoration other than the 2-form background field. Thus the 1-Wilson line (which can live on the magnetic 2-surface ’s boundary) is Kramer singlet () and bosonic. 2. 2.
: The gauge bundle constraint becomes . The magnetic 2-surface has a decoration other than the 2-form field. But is a topological term in a cohomology group also in bordism group , which is effectively a D Haldane’s anti-ferromagnetic quantum spin-1 chain (Haldane-chain) protected by time-reversal symmetry. It is well-known that there exists two-fold degeneracy due to Kramer doublet () on the boundary of Haldane-chain. Thus due to decoration, the Wilson line is Kramer doublet () and bosonic. 3. 3.
: The gauge bundle constraint becomes . The magnetic 2-surface has a decoration other than the 2-form field. But is associated with a spin structure. The 1d boundary of the 2d theory supports a worldline of particle with fermionic statistics. Thus due to decoration, the Wilson line living on the boundary of the magnetic 2-surface is fermionic. Since specifies the extension of by the fermionic-parity via the short exact sequence or equivalently the induced fiber sequence , specifies a projective representation of the spacetime symmetry [38]. The demands the Euclidean reflection , thus the Wick rotated time-reversal transformation in Lorentz signature [40]. Another way to see is to use the methods of symmetry extension and the pullback trivialization [54, 57]. Defining the Wilson line operator on the boundary of the magnetic 2-surface requires a trivialization of , which amounts to requiring a structure. The structure imposes and fermionic statistics on the line. In summary, due to the decoration, the Wilson line is both Kramer singlet () and fermionic. 4. 4.
: The gauge bundle constraint is . Since specifies the extension of by the fermionic-parity via the short exact sequence or equivalently the induced fiber sequence , so specifies a projective representation of the spacetime symmetry [38]. The demands the Euclidean reflection , thus the Wick rotated time-reversal transformation in Lorentz signature [40]. Another way to see is to use the methods of symmetry extension and the pullback trivialization [54, 57]. Defining Wilson line operator on the boundary of the magnetic 2-surface requires the trivialization of , which amounts to requiring the structure. The structure imposes and fermionic statistics on the line.
The combined effect of decoration means that the 1-Wilson line is Kramer singlet () and fermionic.
In fact, our above discussions are universally applicable to more general SU(N) YM theories!313131 Related studies along this line of analysis have also appeared in [38], [65] and [66]. This way of enumerating gauge theories (based on new gauge bundle constraints) guides us to obtain new classes of gauge theories beyond the frame work of Ref. [62]. The implications are not restricted to merely 4d SU(2)θ=π YM. This phenomenon (also in [38]) can be poetically phrased as Lorentz symmetry fractionalization [75].
3.2 Enumeration of Gauge Theories from Dynamically Gauging 4d SPTs:
View from 4d Cobordism Group Data
We have discussed the Four Siblings of SU(2)θ=π YM theories given by {\mathbf{Z}}^{\text{4d}}_{{{\rm SU}(2)}{\text{YM}}_{(K_{1},K_{2})}}[B] in Eq. (3.1), with four distinct sets of new anomalies derived in Sec. 2, and with Kramer singlet/doublet () or bosonic/fermionic Wilson lines in Sec. 3.1. With these properties shown, we are confident that they are really four distinct classes of SU(2)θ=π YM theories (at least at the UV high energy). The two distinct ’t Hooft anomalies of also shows that SU(2)θ=π YM theories with distinct are distinct.
In this subsection, we would like to construct and enumerate these Four Siblings of SU(2)θ=π YM theories by dynamically gauging the SU(2) symmetry from 4d time-reversal symmetric SU(2)-SPTs. To this end, we follow Freed-Hopkins [41] to consider a suitable group extension from the time-reversal symmetry (where the spacetime -manifold requires the orthogonal group O()-structure) via a SU(2) extension:
[TABLE]
These 4d SPTs can be regarded as 4d co/bordism invariants of
[TABLE]
which is the torsion subgroup of for all the possible under the above group extension. The extension is classified by for , generated by and .
The solution of this extension problem is given in [41] with indeed four choices of , or , or , or .323232The notation is defined as the product group mod out their (’s and ’s) common normal subgroup [41].
Following the similar study in Ref. [38], there is a correspondence between the element and . It will soon become clear that is related to (i.e., the difference of the gauge bundle connection and the background gauge connection ). Then the 4 central extension choices labeled by b are:
After gauging SU(2), we gain the gauge bundle constraint with ,
[TABLE]
We compute the co/bordism group in Table 3 (details given in Appendix A). For , we obtain
[TABLE]
The is the second Chern class of the SU(2) gauge bundle. 2. 2.
333333Here E satisfies the following two short exact sequences:
\left\{\begin{array}[]{l}1\to\mathbb{Z}_{2}\to{\rm E}(d)\to{\rm O}(d)\to 1\\ {1\to{\rm SO}(d)\to{\rm E}(d)\to\mathbb{Z}_{4}\to 1},\end{array}\right.
given that we also accept the well-known fact . Here the above finite groups have physical interpretations: is a bosonic group, is the extended group under . Thus . Another way to define is a specific subgroup of given in [41].
After gauging SU(2), we gain the gauge bundle constraint with and ,
[TABLE]
We compute the co/bordism group in Table 4 (details given in Appendix A). For , we obtain
[TABLE]
E is defined in [41] which is a subgroup of , described by two data such that the .
By a different but more physical understanding (see footnote 33), we can further obtain that
[TABLE]
where the bosonic internal symmetry and the time reversal form the extended group under .
Here the is the second Chern class of the U(2) gauge bundle.343434 Since the constraint is satisfied, let denote the Bockstein homomorphism associated to the extension , then where is the third integral Stiefel-Whitney class of and we have used the fact that , hence lifts to a bundle , here is the second Chern class of . 3. 3.
After gauging SU(2), we gain the gauge bundle constraint with and ,
[TABLE]
The co/bordism group is computed in [41, 38] and in Table 5 (see also Appendix A). For , we obtain
[TABLE]
This is related to the interacting version of CI class topological superconductor in condensed matter physics ([76], [41], and [38]). Details of these topological terms are discussed in [38]. 4. 4.
After gauging SU(2), we gain the gauge bundle constraint with ,
[TABLE]
The co/bordism group is computed in [41, 38] and in Table 6 (see also Appendix A). For , we obtain
[TABLE]
This is related to the interacting version of CII class topological insulator in condensed matter physics ([76], [41], and [38]). Details of these topological terms are discussed in [38].
More information about these (co)bordism group calculations can be read from [41, 38]. See Appendix of [38] for a quick background review. In particular, since the computation involves no odd torsion, we can use Adams spectral sequence to compute :
[TABLE]
Here is the 2-completion of the group . For example,
[TABLE]
The is the disjoint union of and a point, while is the suspension.
Let be an -manifold, and be the associated vector bundle of the SO(3) gauge bundle. Below we compute the Stiefel-Whitney classes of . They are used to express the cobordism invariants of . Below means the -th Stiefel-Whitney class, means the total Stiefel-Whitney class, namely, we have . We denote and . In addition, the means specifically the -th Stiefel-Whitney class of spacetime tangent bundle .
[TABLE]
So , , , etc.
We also use the notation TP for the classification of topological phases defined in [41], such that
[TABLE]
Here are the list of tables summarizing the results in 4d and in 5d: Table 3, 4, 5 and 6.
We conclude this section with a summary. The Four Siblings of 4d SU(2)θ=π YM theories are obtained, specifically, from summing over the SU(2) gauge connections of the following four topological terms (i.e., gauging the SU(2) global symmetry of the following four distinct SPTs):
in Eq. (3.8). 2. 2.
in Eq. (3.13). (See the footnote 34.) 3. 3.
with an odd class of in Eq. (3.18). 4. 4.
in Eq. (3.23).
These four theories exactly map to the enumeration of four gauge theories in Sec. 3.1. Adding other SPTs/bordism invariants such as and (and then dynamically gauging them), do not alter or gain new classes of gauge theories. They only tensor product the gauge theory with 4d SPTs, namely (4d SU(2)θ=π YM) (4d SPTs).353535 Here for the classification of gauge theories, we identify the following phases
\text{(gauge theory) \otimes (SPTs)}\simeq\text{(gauge theory)}.
For the classification of 4d SU(2)θ=π YM, we identify the following phases
\text{(4d SU(2){}{\theta=\pi}\otimes (4d SPTs)}\simeq\text{(4d SU(2){}{\theta=\pi} YM)}.
See more physically motivated discussions in [38] and References therein.
4 Time-Reversal Symmetry-Enriched 5d Higher-Gauge TQFTs
4.1 Partition Function of 5d Higher-Gauge TQFTs
Following the discussions of four classes of 5d time-reversal and 1-form center symmetry higher-SPTs in Sec. 2.6.1 with their partition functions in Eq. (2.70), we proceed to dynamically gauge the 1-form symmetry . Then we obtain the 5d time-reversal symmetric enriched topologically ordered state (SETs) with 2-form -valued dynamical gauge fields. We expect a precise mathematical formulation requires a certain version of higher category theory. Below we instead approach from a higher-gauge TQFT perspective.
We can define the four classes of 5d partition functions as:
[TABLE]
In the last step (under the symbol ), we have converted the 5d higher-cochain TQFT to 5d higher-form gauge field continuum TQFT for . Moreover, we can insert extended operators (say ) into the path integral:
[TABLE]
Note that since is trivial for closed 5-manifolds, the partition function and the correlation function computed from the path integral do not depend on .
4.2 Partition Function and Topological Degeneracy
Below we compute the partition function on closed manifolds . When , we can interpret it as topological ground state degeneracy (GSD) of TQFT. Our computations follow the strategy in [13, 15]. We directly summarize the results in Tables 7, 37, and 9.
4.2.1 5d SPTs as Short-Range Entangled Invertible TQFTs
We evaluate the partition function of various 5d iTQFTs on various manifolds, and enumerate the results in Table 7. Below we denote the 5-dimensional Wu manifold as .
4.2.2 5d SETs, as Long-Range Entangled TQFTs
We evaluate the partition function of various 5d TQFTs (as 5d SETs) on various manifolds, and enumerate the results in Table 37 and Table 9.
4.2.3 Computation
Now we illustrate our computation:
For , let be the generator of and be the generator of . Note that . The cohomology groups have that , , and whose two generators are and . If , then . Hence
[TABLE]
On the other hand, since for , we have
[TABLE]
Since the number of satisfying the constraint is only one:
[TABLE]
also note that changing to doesn’t affect the sum, so
[TABLE] 2. 2.
For , let be the generator of , be the generator of . Note that . , , whose three generators are , and . If , then . Hence
[TABLE]
On the other hand, since for , so
[TABLE]
Since
[TABLE]
so
[TABLE] 3. 3.
For , let be the generator of and be the generator of of the -th factor (). Note that . , , whose five generators are , , , and . If , then . Hence
[TABLE]
On the other hand, since for , so
[TABLE]
Since
[TABLE]
also note that changing to doesn’t affect the sum, so
[TABLE] 4. 4.
For a 5d Wu manifold , with , , note that , which is generated by . .
[TABLE]
so
[TABLE]
In the next section, we will use the anyonic string/brane braiding statistics and the link invariants of 5d TQFTs to characterize and distinguish these 5d SETs.
5 Anyonic String/Brane Braiding Statistics and
Link Invariants of 5d TQFTs
Now we compute the path integral Eq. (4.5) with extended operator insertions. To recall the general definitions, we have
Partition or path integral w/out insertion is
[TABLE]
In physics, the vacuum expectation value (v.e.v) of a theory is defined as
[TABLE]
For example, this includes the link invariant that we will focus on in this section:
[TABLE]
For conventions of our notations, we label the 1d Wilson line as , the 2d surface operator as , etc. We label the 3d membrane operator as and the 4d operator as , etc. We label the d-hyper-surface of general operators that we inserted as , while we label this ’s d-Seifert-hyper-volume as .
In this section 5, we focus on deriving the general link invariants for these 5d TQFTs/SETs.383838For more guidance on the physical interpretations of link invariants, please see [12] and its Introduction. In the next Sec. 6, we will provide explicit examples of the spacetime braiding process as the link configurations that can be detected by these link invariants derived here in Sec. 5. The techniques for computing all these link invariants below are based on Ref. [12]. Below we simply apply the methods and notations introduced in Ref. [12].
Caveat: Note that while in the first section 5.1, we explicitly study the discrete cochain version of TQFT, in the sections below we implement the continuum formulation of TQFT. The reason is related to a fact that the graded non-commutativity of cochain fields is much more complicated to be dealt with than the continuum differential form fields. The subtle fact will be commented further in footnotes 40 and 41. We also note that when we deal with the continuum differential form fields later in Sec. 5.2 to Sec. 5.4, we choose a normalization of differential form fields as its enclosed 2-surface integral with the periodicity (thus more similar to the convention of discrete cochain fields), instead of the more conventional with the periodicity .
5.1 and a Triple Link Invariant
We start with a 5d TQFT obtained from summing over 2-form field of . This amounts to gauging 1-form of this 5d SPTs. The resulting theory is in Eq. (4.5). The topological action and the partition function are
[TABLE]
We consider the gauge transformation:393939One may consider add additional terms on the gauge transformations, such as and , etc. However, terms such as and will need to be constant, which act as the higher-form global symmetry transformation, instead of gauge transformation.
[TABLE]
under which the action transforms as
[TABLE]
The gauge variance of the action is:
[TABLE]
In Eq. (5.9), we have used the formula404040 This is based on Steenrod’s work “Products of Cocycles and Extensions of Mappings [61],” which derives
(5.10) where and .
[TABLE]
and . In Eq. (5.9), we have used integration by part: for a closed 5-manifold without boundary, after integration by part we can drop the boundary term . Since , we drop which has no effect on a closed 5-manifold without boundary. Denote as the second Wu class. We have also used the formula in footnote 40 as
[TABLE]
In Eq. (5.9), we used and we dropped the total derivative term on a closed 5-manifold. The solution of gauge invariance, i.e., , imposes: 414141In general, when we study the action Eq. (5.3), we have made a convenient choice with a term instead of . For a generic 3-cochain , is not true, by Steenrod’s formula in footnote 40 Eq. (5.10), , we can only drop the total derivative terms (i.e. the coboundary terms). In our present case, we consider . So if is a coboundary, then we can also drop it, which results in
If is not a coboundary, we need the extra term
When is not a coboundary, this results in a modified gauge transformation to . By writing the action as in Eq. (5.3), we can avoid additional complications, thus we end up with a simpler gauge transformation Eq. (5.14). The graded non-commutativity of cochain fields is much more complicated than the case for continuum differential form fields. JW thanks Pierre Deligne for a discussion on the related issues.
[TABLE]
The 3-submanifold gauge invariant operator is,
[TABLE]
where . To verify the gauge invariance, we use and on the 4-submanifold Seifert volume .
The 2-submanifold (2-surface) operator gauge invariant is
[TABLE]
where is an integer mod 2. To verify that is gauge invariant, we use on the 3-submanifold Seifert volume .
We insert and into the path integral , and write the correlation function either in the continuum field theory formulation, or in the discrete cochain field theory formulation, interchangeably as
[TABLE]
We compute the correlation functions as follows.
Integrating out yields
[TABLE]
hence as a consequence, . So with the above configuration constraint, we get the double-counting mod 2 cancellation in the exponent of . This boils down to
[TABLE] 2. 2.
Integrating out yields
[TABLE] 3. 3.
We finally integrate out , from Eq. (5.19):
[TABLE]
In Eq. (5.21), we used \tilde{w}_{1}(TM)(w_{2}(TM)+w_{1}(TM)^{2})B=\tilde{w}_{1}(TM)u_{2}B=\mathrm{Sq}^{2}\big{(}\tilde{w}_{1}(TM)B\big{)}. In Eq. (5.22), we rewrote via 424242We use the Cartan formula of the Steenrod square: where .
[TABLE]
We plugged in all the constraints into the path integral Eq. (5.22) to obtain Eq. (5.23).434343 Here are some more explanations to derive Eq. (5.23).
For \int[{\cal D}B]{\rm e}^{-{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi}(\int_{M^{5}}(\tilde{w}_{1}(TM)BB))}\bigg{|}\text{\tiny{{{{\tilde{w}{1}(TM)=k\delta^{\perp}(V^{4}{X}),\quad\quad\quad\quad\quad\quad;;;}\atop{B=\ell_{\bf(i)}\delta^{\perp}(V^{3}{U{\bf(i)}})+\ell_{\bf(ii)}\delta^{\perp}(V^{3}{U{\bf(ii)}}).}}}}}, we get a mutual-quadratic crossing term with a multiple exponent in which does not contribute to the expectation value. There are also two self-quadratic terms for or . These self-quadratic terms contribute, in principle, infinite many intersecting numbers in for or . Since a multiple exponent have zero contribution to the expectation value, therefore either we can design an even but infinite number of points on each of for or , or we can absorb them into the (\text{Self-intersecting # terms}) in Eq. (5.23). In either cases, this term does not have any net contribution in the end at Eq. (5.24).
For \int[{\cal D}B]{\rm e}^{-{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi}(\int_{M^{5}}(\frac{1}{2}\tilde{w}_{1}(TM)BB))}\bigg{|}\text{\tiny{{{{\tilde{w}{1}(TM)=k\delta^{\perp}(V^{4}{X}),\quad\quad\quad\quad\quad\quad;;}\atop{B=\ell_{\bf(i)}\delta^{\perp}(V^{3}{U{\bf(i)}})+\ell_{\bf(ii)}\delta^{\perp}(V^{3}{U{\bf(ii)}}).}}}}}, we get a mutual-quadratic crossing term with a multiple exponent in , which does contribute to the expectation value when this intersecting number is odd, in a 1 mod 2 effect. There are also two self-quadratic terms for or . Again either we can design an quadruple/four-multiplet but infinite number of points for each of , or we can absorb them into the (\text{Self-intersecting # terms}) in Eq. (5.23).
For \int[{\cal D}B]{\rm e}^{(-{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi}(\int_{M^{5}}(\frac{1}{8}\delta\tilde{w}_{1}(TM)\delta B)))}\bigg{|}\text{\tiny{{\delta\tilde{w}{1}(TM)=k\delta^{\perp}(\Sigma^{3}{X});;\quad\quad\quad\quad;;}\atop{\delta B=\ell_{\bf(i)}\delta^{\perp}(\Sigma^{2}{U{\bf(i)}})+\ell_{\bf(ii)}\delta^{\perp}(\Sigma^{2}{U{\bf(ii)}})}}}\;={\rm e}^{-{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi}\int_{M^{5}}(\frac{1}{8}(k\delta^{\perp}(\Sigma^{3}_{X}))\Big{(}\ell_{\bf(i)}\delta^{\perp}(\Sigma^{2}_{U_{\bf(i)}})+\ell_{\bf(ii)}\delta^{\perp}(\Sigma^{2}_{U_{\bf(ii)}})\Big{)})}, we find the exponent depends on the intersecting number for or , between 3-surface and 2-surface in a 5 manifold — although generically this number is finite but can be nonzero, we design by default that there is no intersection between any of our insertions of 3-surface and 2-surface into the path integral. Thus we set by default.
We propose a set-up to remove or renormalize the (Self-intersecting terms) appeared in Eq. (5.24), described in the footnote 43. The second exponent in Eq. (5.23) shows that \int_{M^{5}}\delta^{\perp}(\Sigma^{3}_{W})\big{(}\delta^{\perp}(\Sigma^{2}_{U_{\bf(i)}})+\delta^{\perp}(\Sigma^{2}_{U_{\bf(ii)}}))=\#(\Sigma^{3}_{X}\cap\Sigma^{2}_{U_{\bf(i)}})+\#(\Sigma^{3}_{X}\cap\Sigma^{2}_{U_{\bf(ii)}}), which counts the number of intersections between our insertions of 3-surface and 2-surface. However, we choose by default that our insertions of 3-surface and 2-surface have no intersections (to avoid unnecessary singularities). Namely, we set for or , and by default. Overall, under the default assumption and the clarifications in footnote 43, we obtain a final relation between Eq. (5.23) and our final effective answer Eq. (5.24). We use the congruence symbol () to express that other unwanted terms that can be removed by design.
In summary, we have derived the link invariant for the 5d TQFT in Eq. (5.24):
[TABLE]
The path integral, with appropriate insertions of extended operators, becomes Eq. (5.24) which provides the above link invariant.
5.2
5.2.1 Version I: and a Quartic Link Invariant Qlk
As a test example, we consider a 5d TQFT obtained from summing over 2-form field with the topological action (i.e., gauging the 1-form symmetry of this 5d higher-SPTs). For simplicity, we convert the cochain TQFT to a differential-form continuum TQFT. The partition function and the topological action of the gauged theory (see footnote 17) are:
[TABLE]
This 5d TQFT is distinct from any of four classes of , but it still serves as a useful toy model.
We first specify the gauge transformations of various fields. Let us assume the gauge transformations take the following form (see footnote 39):
[TABLE]
The variation of action under the gauge transformations is
[TABLE]
where we have used integration by part. For a closed 5-manifold without boundary, after integration by part we drop the total derivative terms which have no effect on a closed 5-manifold without boundary. The gauge variance of the action, i.e. , requires
[TABLE]
The gauge invariant 3-submanifold operator is
[TABLE]
and the gauge invariant 2-surface operator is
[TABLE]
where . To verify the gauge invariance, we need to use on the 2-surfaces and 3-submanifolds.
To compute the link invariants, we insert into the path integral . In the continuum field theory formulation, the link invariant is
[TABLE]
We compute as follows.
Integrating out , we get
[TABLE]
With the above configuration constraint, we get the double-counting mod 2 cancellation in the exponent of . Consequently the link invariant boils down to
[TABLE] 2. 2.
Integrating out further yields the constraint
[TABLE] 3. 3.
We finally integrate out as follows
[TABLE]
We propose a set-up to remove or renormalize the (Self-intersecting terms) appeared in Eq. (5.39), following the same strategy in footnote 43.
For , we derive the link invariant for the 5d TQFT in Eq. (5.39) and Eq. (5.40):
[TABLE]
The path integral with appropriate extended operator insertions become Eq. (5.40) which provides the above link invariant. However, note that the factorial trivializes the complex phase to . It may be possible to take into account (see footnote 41) from the subtle graded non-commutativity of cochain field effect. Thus one may need to go beyond the continuum differential form TQFT formulation by using the cochain TQFT formulation in order to see the subleading effect.
5.2.2 Version II: and a Triple Link Invariant Tlk
As another test example, we consider a 5d TQFT obtained from summing over 2-form field with the topological action . We again use the continuum version of the TQFT.444444Even though is a rewriting of on a closed 5-manifold, it turns out that we still gain new insights about an additional link invariant. Its partition function and the topological action (see footnote 17) are:
[TABLE]
We assume the gauge transformations have the following ansatz
[TABLE]
Under the gauge transformations, the action transforms as
[TABLE]
where we have used integration by part. requires
[TABLE]
The gauge invariant 3-submanifold operator is
[TABLE]
and the gauge invariant 2-surface operator is
[TABLE]
where .
We proceed to compute the link invariants by inserting into the path integral ,
[TABLE]
To evaluate , we integrate out various fields step by step.
Integrating out , we get
[TABLE]
The link invariant thus boils down to
[TABLE] 2. 2.
Integrate out , we get the constraint
[TABLE] 3. 3.
We finally integrate out in Eq. (LABEL:eq:triple2-step1):
[TABLE]
We propose a set-up to remove or renormalize the (Self-intersecting terms) appeared in Eq. (5.57), following the same strategy as footnote 43.
For , we derive the link invariant for the 5d TQFT in Eq. (5.57) and Eq. (5.58):
[TABLE]
The path integral with appropriate extended operators insertions become Eq. (5.58) which provides the above link invariant.
However, note that the two terms on the exponent of Eq. (5.58) are the same, which trivializes the complex to . It may be possible to take into account (see footnote 41) from the subtle graded non-commutativity of cochain field effect. Thus one may need to go beyond the continuum differential form TQFT formulation by using the cochain TQFT formulation in order to see the subleading effect.
5.3 and a Quadratic Link Invariant
We further consider a 5d TQFT obtained from summing over 2-form field of . We again adopt the continuum version of TQFT. The partition function and action (see footnote 17) are:
[TABLE]
We assume the gauge transformations have the following ansatz
[TABLE]
Under the gauge transformation, the action transforms as
[TABLE]
Thus requires
[TABLE]
There are two types of gauge invariant 2-surface operators,
[TABLE]
and
[TABLE]
where .
We define the link invariant by inserting into the path integral
[TABLE]
Below we evaluate by integrating out various fields.
Integrating out , we get
[TABLE]
Plugging the above constraints into the partition function, we find the double-counting mod 2 cancellation in the exponent of . Thus the link invariant boils down to
[TABLE] 2. 2.
Integrating out , we get the constraint
[TABLE] 3. 3.
We finally integrate out in Eq. (LABEL:eq:link2-step1):
[TABLE]
We derive the link invariant for the 5d TQFT for in Eq. (5.75):
[TABLE]
The path integral with appropriate extended operators insertions become Eq. (5.75) which provides the above link invariant.
5.4
and More Link Invariants:
, and
We finally consider the generic form including the four classes of in Eq. (4.5) by gauging in Eq. (2.65), with labeling the Four Siblings. Below, we find it convenient to introduce via mod 2.
The partition function and action (see footnote 17) are:
[TABLE]
5.4.1 Gauge Invariance
We assume the following ansatz of the gauge transformations
[TABLE]
The gauge variation of the action is
[TABLE]
where we have used integration by part. Gauge invariance, i.e. , requires
[TABLE]
5.4.2 Extended 2-Surface/3-Brane Operators and Link Invariants
The gauge invariant 3-manifold operator is
[TABLE]
is trivial when mod 2.
There are two types of gauge invariant 2-surface operators,
[TABLE]
and
[TABLE]
where .
Inserting into the path integral , we define the link invariant as
[TABLE]
We evaluate the path integral below.
Integrating out , we get
[TABLE]
We keep on both sides because when mod 2 we have this constraint; while when mod 2 the constraint is trivial. Using the above constraints, the path integral boils down to
[TABLE] 2. 2.
Integrating out , we get
[TABLE]
Substituting these into the path integral, we find
[TABLE] 3. 3.
Integrate out , we get the constraint
[TABLE] 4. 4.
We finally integrate out in Eq. (5.89):
[TABLE]
We propose a set-up to remove or renormalize the (Self-intersecting terms) appeared in Eq. (5.94), following the same strategy as footnote 43.
For , we derive the link invariant for the 5d TQFT in Eq. (5.94) and Eq. (5.95):
(5.96)
The path integral with appropriate extended operators insertions becomes Eq. (5.95) which provides the above link invariant.
5.4.3 or : 1st and 3rd Sibling
The gives rise to a 5d triple link invariant:
- •
in Sec. 5.1’s Eq. (5.25). We present an exemplary link configuration later in (Sec. 6.2) that can be detected by this link invariant.
In another equivalent expression, in Eq. (4.5) gives rise to other link invariants in Eq. (5.96) including:
- •
, a second type of triple link in 5d (although seemly undetectable due to an exponent factor in the expectation value). We present an exemplary link configuration later in (Sec. 6.3) that can be detected by this link invariant.
- •
, a quadratic link of 2-surfaces in 5d. We present an exemplary link configuration later in (Sec. 6.5) that can be detected by this link invariant.
- •
, another quadratic link of 2-surfaces in 5d. We present an exemplary link configuration later in (Sec. 6.6) that can be detected by this link invariant.
Physically, these link invariants may be related to each other by re-arranging the spacetime braiding process of strings/branes. It will be interesting to find a precise mathematical equality to relate these link invariants.
5.4.4 or : 2nd and 4th Sibling
in Eq. (4.5) gives rise to link invariants in Eq. (5.96) including:
- •
, a quadratic link of 2-surfaces in 5d. We present an exemplary link configuration later in (Sec. 6.5) that can be detected by this link invariant.
- •
, another quadratic link of 2-surfaces in 5d. We present an exemplary link configuration later in (Sec. 6.6) that can be detected by this link invariant.
Similar to our comments above in Sec. 5.4.3, it will be interesting to find a precise mathematical equality to relate these link invariants.
6 Anyonic String/Brane Spacetime Braiding Processes and Link Configurations of Extended Operators
We provide the exemplary spacetime braiding processes of anyonic strings and branes in general dimensions (with an emphasis on d), and the link configurations of extended operators, which can be detected by the link invariants that we derived in Sec. 5.
6.1 Quadratic Link (Aharanov-Bohm) Configuration in Any Dimension
To warm up, we first discuss the quadratic link, associated with the Aharanov-Bohm statistics in d due to the linking of 1-worldline of the charged particle and the d-worldsheet of the fractional flux. In 3d spacetime, we have the following presentation
[TABLE]
where gluing two solid tori gives rise to a 3-sphere: . We represent the two solid tori as a blue solid tori and a red solid tori, ({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}D^{2}_{\rm L}\times S^{1}_{\rm R}})\cup({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}S^{1}_{\rm L}\times D^{2}_{\rm R}})=S^{3}. The quadratic link invariant detecting this Aharanov-Bohm configuration is given by (Ref. [12] and References therein): , which we also express as
[TABLE]
based on the color labeling of the inclusion of two circles belonging to which of the two solid tori. This link invariant can be computed from the intersection number,
[TABLE]
where
[TABLE]
means the point now is attached to a line. The intersection number \#(({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0_{\rm pt})_{\rm L}\times S^{1}_{\rm R}})\cap({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}D^{2}_{\rm L}\times(0_{\rm pt-})_{\rm R}}))=1 precisely corresponds to the black dot .
In d spacetime, can be obtained by
[TABLE]
which can be graphically represented as
[TABLE]
The associated link invariant is
[TABLE]
with the color prescription explained earlier. This link invariant can be computed from the intersection number,
[TABLE]
[TABLE]
Here, means the point now is attached with a line. We see the intersection number \#(({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0_{\rm pt})_{\rm L}\times S^{1}_{\rm R}})\cap({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}D^{d-1}_{\rm L}\times(0_{\rm pt-})_{\rm R}}))=1 precisely corresponds to the black dot .
6.2 The 1st Triple Link
Configuration in 5d
We proceed to discuss the triple link configuration for derived in Sec. 5.1.454545Effectively, can be also regarded as where is other 1-form gauge field.
We propose that this link invariant derived in Sec. 5.1 can detect the link configuration in Fig. 2.
To explain, we start by constructing the 5-sphere via . More explicitly, we color the different components as S^{5}=({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}S^{3}_{\rm L}\times D^{2}_{\rm R}})\cup({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}D^{4}_{\rm L}\times S^{1}_{\rm R}}) and S^{5}=({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}S^{3}_{\rm L}\times D^{2}_{\rm R}})\cup({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}D^{2}_{\rm L}\times D^{2}_{\rm L}\times S^{1}_{\rm R}}).
Consider the link invariant defined by , we see that the link configuration in Fig. 2 gives the intersection number 1 in Fig. 3. Again in Fig. 3 associated with the intersection number , means the point now is attached to a line. We see the intersection number precisely corresponds to the black dot .
6.3
The 2nd Triple Link Configuration in 5d
We now discuss , or schematically . This link invariant is derived in Sec. 5.2.2.
Let us consider the link invariant defined by . We see that the link configuration in Fig. 4 gives the intersection number 1 in Fig. 5.
6.4 Quadruple Link
Configuration in 5d
We now discuss , or schematically . This link invariant is derived in Sec. 5.2.1.
Let us consider the link invariant defined by
[TABLE]
. We see that the link configuration in Fig. 6 gives the intersection number 1 in Fig. 7.
6.5 Quadratic Link
Now we discuss . This link invariant is derived in Sec. 5.4.
Let us consider the link invariant defined by
[TABLE]
. We see that the link configuration in Fig. 8 gives the intersection number 1 in Fig. 9.
6.6 Quadratic Link
Now we discuss or . This link invariant is derived in Sec. 5.3.
Let us consider the link invariant defined by . We see that the link configuration in Fig. 10 gives the intersection number 1 in Fig. 11.
6.7 The 3rd Triple Link
Configuration in 5d
Finally, we discuss a third triple link invariant . We have not derived these from 4d YM-5d SET coupled systems. However, to get this, we need a topological term . This is possible however from type of TQFTs.
Let us consider the link invariant defined by . We see that the link configuration in Fig. 12 gives the intersection number 1 in Fig. 13.
7 4d SO(3)θ=π Yang-Mills Gauge Theories coupled to the Boundary of 5d SETs/Long-Range Entangled TQFTs
In Sec. 2, we have shown that that the Yang-Mills theory with , with the gauge bundle constraint , has two distinct ’t Hooft anomalies as shown in Eq. (2.47). In this section, we further comment on gauging the 1-form center symmetry of the Four Siblings of YM to obtain YM theories. Since the ’t Hooft anomalies involve the 1-form center symmetry and the spacetime symmetries (whose background fields are the Stiefel-Whitney classes ), depending on which manifold we formulate the Yang-Mills, one obtains different theories.
7.1 From SU(2) to SO(3) Gauge Theory
To illustrate, we start by gauging the 1-form symmetry [6, 77] of the YM theories which is time-reversal symmetric and anomaly free. There are still four choices of gauge bundle constraints labeled by , i.e. Eq. (3.1). Let {\mathbf{Z}}^{\text{4d}}_{{{\rm SU}(2)}{\text{YM}}}[B] be the path integral without specifying the gauge bundle constraint, the partition function with the gauge bundle constraint is
[TABLE]
More generally, we can add the counter term labeled by an integer , which modifies the partition function as
[TABLE]
Below we would like to obtain YM by gauging 1-form center symmetry. The theta angle of the resulting theory is . If is nontrivial, the resulting theory is time-reversal symmetric only when and . When is trivial, the resulting theory is time-reversal symmetric for and . In the following, we always restrict to the time-reversal symmetric case. Gauging 1-form center symmetry amounts to summing over the background gauge field (promoting to a dynamical gauge field),
[TABLE]
Integrating out enforces the relation between SO(3)-gauge bundles and 2-form dynamical gauge field . This outputs the SO(3)-gauge theory {\mathbf{Z}}^{\text{4d}}_{{{\rm SO}(3)}{\text{YM}}_{(K_{1},K_{2})}} with .
7.2 Gauging 1-form -symmetry of Gauge Theory with
We proceed to discuss gauging the 1-form symmetry of Yang-Mills with .
If one formulates the Yang-Mills on an orientable and spin manifold, i.e., (hence as well), for spacetime tangent bundle , there is the freedom to ignore the time reversal as a symmetry of the theory. The only symmetry of interest is the 1-form symmetry, which does not have anomaly with itself. Hence one can gauge the 1-form symmetry and the resulting theory is Yang-Mills with . Indeed, Yang-Mills with does not respect time reversal, which maps to due to the identification on a spin manifold.
If one formulates the Yang-Mills on an orientable and non-spin manifold, one still has the freedom to ignore the time reversal as a symmetry of the theory. However, in this case, there is a counter term
[TABLE]
which is a WZW-like term of background fields (i.e., probe fields in condensed matter language). Denoting the partition function of the Yang-Mills coupled to as , after promoting to a dynamical field, the partition function of the entire 4d-5d system is
[TABLE]
If , the 4d-5d system reduces to an intrinsic 4d system. Physically, this corresponds to the case where the gauge charge is a boson. It makes sense to gauge the 1-form symmetry which again gives raise to the time-reversal broken Yang-Mills theory. If , the theory is still an intrinsic 4d system. Physically, this corresponds to the case where the gauge charge is a fermion.
If one formulates the Yang-Mills on an unorientable manifold, the time-reversal symmetry is built in, so time-reversal symmetry is too late to be abandoned. Promoting to a dynamical gauge field, the partition function for the entire 4d-5d system is
[TABLE]
Since is unorientable, for all four choices of , the 5d terms do not vanish (because is always non-vanishing on unorientable manifold). Hence one can only discuss the 4d-5d system rather than discussing the 4d system alone. We summarize all the above cases in Table 10.
8 Lattice Regularization, UV completion and Symmetric Anomalous TQFT
In this section, we formulate the partition function of the 5d higher-SPT on a simplicial complex spacetime. This provides a lattice regularization of the 5d SPT. We also provide lattice realization of (i) 4d higher-symmetry-extended boundary theory or (ii) 4d higher-symmetry-enriched anomalous topologically ordered boundary theory. We will generalize the approach in [54] and follow the Section IX of [55]. In condensed matter physics, this (ii) phenomenon is known as the anomalous surface topological order (firstly noticed in [78]) typically for the 2+1D boundary of 3+1D SPTs, see a review [30].
8.1 Lattice Realization of 4d Higher-SPTs and Higher-Gauge TQFT: 4d Simplicial Complex and 3+1D Condensed Matter Realization
We warm up by considering a lattice realization of 4d Higher-SPTs given by a probe-field partition function
[TABLE]
The path integral can be regularized on a triangulated 4-manifold . The building blocks of are 4-simplices. Without loss of generality, we consider an arbitrary 4-simplex which we denote as where each number labels one vertex. See Fig. 14 for a graphical representation of a 4-simplex. We denote as restricting the 2-cochain on the 2-simplex . We label the path integral amplitude on as , i.e.,
[TABLE]
It is straightforward to verify that satisfies the cocycle condition:
[TABLE]
8.2 Lattice Realization of 5d Higher-SPTs and Higher-Gauge SETs: 5d Simplicial Complex and 4+1D Condensed Matter Realization
The 5d partition function with is
[TABLE]
We start by triangulating the 5d closed spacetime manifold (without boundary) into 5-simplicial complex. There are some useful identities:
[TABLE]
Note that in the second equality of the first line, we have used the cocycle condition that . One can express the SPT action Eq. (8.4) in terms of the sum of cup-products of cochains over 5-simplices
[TABLE]
Without loss of generality, we consider an arbitrary 5-simplex which we denote as where each number labels one vertex. See Fig. 15 for a graphical representation of a 5-simplex. We will label the path integral amplitude on the simplex as , i.e.,
[TABLE]
so that the partition function can be simplified as . Using the definition of the cup products on simplices and the identities Eq. (8.5), we have
[TABLE]
Hence the path integral amplitude on the simplex is
[TABLE]
It is straightforward to verify that satisfies the cocycle condition:
[TABLE]
We emphasize that is a cocycle only when is a cocycle, i.e., . If is a cochain rather than a cocycle, Eq. (8.4) is not a cocycle, hence cannot be a partition function of a topological field theory. 464646The cocycle condition is crucial in proving the partition function to be invariant under re-triangulating the spacetime manifold .
We further comment on the lattice regularization of theory with various choices of .
When , as we derived above, there is a lattice regularization of the 5d SPT partition function. 2. 2.
When , the path integral amplitude depends on the first Stiefel-Whitney class . Using the method of [29], one can write down the simplicial form of using the twisted cocycle, with the coefficient in due to anti-unitary nature of time-reversal symmetry (in the Hamiltonian formalism of [29]). We will not write down the explicit expression for the cocycle. 3. 3.
has the same anomaly polynomial as respectively.
Other than the treating as the background gauge field, we can also sum over to get the the topologically ordered 5d SET .
Given that the 5d SPT and 5d SET path integral can be regularized on a lattice, following [29], one can write down the quantum wavefunction via the spacetime path integral. It is also possible to construct a lattice quantum Hamiltonian on the 4D space (on a constant time slice), for both SPTs and SETs, similar to the formulations of [79, 29, 80, 81]. For the topologically ordered 5d SET, we implement the method of [80, 81]:
[TABLE]
where is an operator acting on the plaquettes (2-simplex) adjacent to the 1-link , and is an operator acting on the boundary of a given 3-simplex which again are plaquettes (2-simplex). The has its effect on imposing the time evolution constraint as the same as the path integral formulation: lifting the state vector to a next time slice locally around the 1-link . The imposes the zero flux condition enclosed by the 3-simplex (which is a 2-sphere in topology). We will not give the explicit expression of the quantum Hamiltonian in this work.
8.3 Higher-Symmetry-Extended 3+1D Gapped Boundaries
One option to saturate the anomaly inflow from the bulk 5d (4+1D) SPT is to extend the global symmetry on the 4d (3+1D) boundary, based on the symmetry-extension method of [54]. Note that and theories have the same 4d anomaly, they are differed by a 4d WZW-like counter term written on a 5d ,
[TABLE]
The 4d counter term is shown to be vanished on a closed in 5d in Eq. (• ‣ 2). Due to Eq. (• ‣ 2), this -dependent term has no consequences via any anomaly consideration on 4d dynamics. Thus we only discuss the D gapped boundary for the two siblings . The 5d partition function is
[TABLE]
In the second line, we have used . Note that when is closed, the total derivative vanishes and we have the standard Wu formula . When has a boundary, the standard Wu formula may no longer hold, and they differ at most by a co-boundary term.474747By the Wu formula for -manifolds with boundary, where is the relative Wu class, and . The total relative Wu class is related to the total Stiefel-Whitney class of M as . So , , etc., still hold but may not be in for a 5-manifold with boundary. Therefore, may not hold for a 5-manifold with boundary. See for instance Ref. [82]. We denote the co-boundary term as in the second line. Since is a well-defined term of background gauge fields, it is a 4d invertible TQFT, which does not contribute to the 4d dynamics. Hence in the construction of 4d symmetric boundary TQFT below, we only focus on the first part in the second line of Eq. (8.12). Using the systematic way in [54] and its generalized higher-symmetry extension [55], we find that the boundary of 5d SPT can support a 4d TQFT via the higher-symmetry extension from a 1-form to a 1-form symmetry. Schematically, let be the 5-cocycle whose product over the 5d manifold gives the 5d SPT partition function Eq. (8.12). Let be a 4-cochain which trivializes the 5d cocycle, i.e.,
[TABLE]
We find that the following satisfies Eq. (8.13):
[TABLE]
where is a valued 2-cochain satisfying , and is a function which maps the 2-cochain to a 2-cochain:
[TABLE]
In summary, the in (8.14) is the partition function of the higher-symmetry-extended 3+1D gapped boundary, while the original 1-form anomalous global symmetry is extended to the anomaly-free 1-form global symmetry . We can also rephrase that the higher-anomaly associated to the bordism group with a higher classifying space can be pulled back and trivialized as fully anomaly-free in the bordism group with an extended higher classifying space :
[TABLE]
This higher-symmetry-extended 3+1D gapped boundary, described by (8.14), has no long-range entanglement and no intrinsic topological order. This higher-symmetry-extended 3+1D gapped boundary is known as the System (i) in Section 7 of Ref. [13] as a short-range entangled state, both in the bulk and on the boundary (denoted as SRE/SRE in Section 7 of Ref. [13]). In fact, this whole SRE/SRE bulk-boundary theory is still an invertible TQFT with a partition function on an with a 4d gapped boundary .
8.4 Higher-Symmetry Anomalous 3+1D Topologically Ordered Gapped Boundaries: Spontaneous Higher-Symmetry Breaking
There also exists another boundary theory of the 5d SPTs in Eq. (8.12), with 2-cochain summed over as dynamical fields on the boundary. The boundary theory is a dynamical gauge theory, which can be obtained from gauging the normal subgroup of in the symmetry extended gapped boundary from the previous section. Schematically, we promote the 2-form (or 2-cochain) gauge field coupling to 1-form -symmetry to dynamical, in a normal subgroup of . The resulting boundary theory has long range entanglement in contrast to the short range entanglement of the symmetric extended boundary theory in the previous section Sec. 8.3. This can be summarized as an induced fiber sequence of their higher classifying space :
[TABLE]
Or we may denote the above as , implementing the notations of [54, 13].
However, as noticed in Ref. [54] and in section 7 of Ref. [13], when the boundary theory is long-range entangled (after gauging a normal subgroup), it is possible that the new fate of low energy dynamics may not preserve the global symmetry. Specifically, Ref. [54] finds that, under the exact sequence , even for a successful -symmetry extended construction of gapped boundary, in certain cases, dynamically gauging the normal subgroup may still result in spontaneously broken. In short, we should question:
*Is it possible that global symmetry in the long-range entangled boundary theory happens to be spontaneously broken? *
Namely, following the notations in Sec. 7 of Ref. [13] on the bulk/boundary (denoted bulk/bdry) systems of the SRE/SRE and SRE/LRE types, the symmetry-extension construction [54, 13] under the exact sequence , may result in different dynamical fates:
[TABLE]
Thus below what we aim to examine is whether a proposal of Case 2 associated with Eq. (8.17) is in fact the Case 3 in disguise, when is dynamically gauged, as
[TABLE]
Concretely, we propose a gauge theory (endorsed with a 2-cochain field and the dual 1-cochain field, as a candidate IR theory of the UV SU(2) YM) which has global symmetry and saturates the ’t Hooft anomaly Eq. (8.12). We will find that the global symmetry is spontaneously broken.
We consider the 4d -gauge TQFT
[TABLE]
Here means we sum over the valued 2-cochain and valued 1-cochain . The first term is the standard BF term of discrete gauge theory, while the second term is the 4d analogue of the Dijkgraaf-Witten type action[12]. We also refer it as the twisting term. Integrating out enforces to be a valued cocycle, hence the twisting term mod 2 is time-reversal symmetric. To match the anomaly from 5d SPTs in Eq. (8.12), we couple to background fields and , as follows,
[TABLE]
Summing over the valued 2-cochain enforces the gauge bundle constraint:
[TABLE]
Notice that under the gauge transformation of the background field , we demand . It is obvioius that the action Eq. (8.20) is gauge invariant. However, due to the gauge bundle constraint Eq. (8.21), the twisting term is no longer [math] or , hence it is not time-reversal invariant, and Eq. (8.20) is not well defined on an unorientable manifold. To make sense of the theory on an unorientable manifold, we use the same idea in Sec. 2 where we promote the twisting term to a 5d integral,
[TABLE]
where we have used and the gauge bundle constraint Eq. (8.21). Thus when we couple theory to the background field and formulate it on an unorientable manifold, only the 4d-5d coupled system is well defined
[TABLE]
Indeed, the 5d integral implies that the gauge theory saturates the anomaly Eq. (8.12). Let us comment on the dynamics of the above 4d gauge theory. The fact that transforms as under 1-form background gauge transformation suggests that the Wilson line
[TABLE]
has a charge 1 under . In a TQFT, any genuine line operators are tensionless, thus the expectation does not satisfy area law. Since is a genuine line operator, it obeys a perimeter law, and it spontaneously breaks the 1-form symmetry.
We comment that in Section 7.1 of Ref. [13], similar arguments have been used to show that the 0-form global symmetry of a 2d TQFT is spontaneously broken. In its 2d action , there is a scalar with charge 1 under 0-form global symmetry whose background field is . (The and are dynamical 0-cochain and 1-cochain fields.) In the 2d TQFT, has a nontrivial expectational value . Thus the 0-form symmetry is spontaneously broken. See Ref. [13] for more details.
The fact that the dynamical gauge theory we constructed spontaneously breaks the 1-form -symmetry is consistent with the conclusion pointed out by Cordova and Ohmori (see [83] and [84]) where the authors showed the impossibility of having any symmetric anomalous gapped 4d TQFT saturating the 4d higher-anomaly of 5d SPTs Eq. (8.12).
9 Conclusions, Discussions and Dynamics
Summary: In this work, we proved (physically via the quantum field theory method) that 4d time-reversal symmetric pure Yang-Mills of an SU(2) gauge group with a second-Chern-class topological term at (i.e., SU(2)θ=π YM) have new higher ’t Hooft anomalies in 4d, given by a 5d topological term Eq. (2.47) and Eq. (2.70):
[TABLE]
The 5d term is a 5d bordism invariant (or equivalently 5d iTQFT/SPTs/counter term) specifies the 4d ’t Hooft anomaly. However, the 5d term is not a 5d bordism invariant but only a 4d WZW-like counterterm, thus strictly speaking it does not indicate any 4d ’t Hooft anomaly. We found that there are at least Four Siblings of SU(2)θ=π YM with bosonic UV completion, labeled by . Their higher ’t Hooft anomalies of generalized global symmetries indicate that 4d SU(2)θ=π YM, in order to realize all global symmetries locally, necessarily couple to 5d higher symmetry-protected topological states (SPTs, as invertible TQFTs [iTQFTs], as 5d 1-form-center-symmetry-protected interacting “topological superconductors” in condensed matter).
We explored various 4d YM gauge theories living as boundary conditions of 5d gapped short/long-range entangled (SRE/LRE) topological states. We revisited 4d SU(2)θ=π YM-5d SRE-higher-SPTs coupled systems [5, 8]. Follow Weyl’s gauge principle, by dynamically gauging the 1-form center symmetry, we transformed a 5d bulk SRE SPTs into an LRE symmetry-enriched topologically ordered state (SETs); thus we obtained the 4d SO(3)θ=π YM-5d LRE-higher-SETs coupled system with dynamical higher-form gauge fields. We illustrate such 4d-5d systems schematically in Fig. 1 and Fig. 16.
The 4d SO(3) YM has a periodicity on a spin manifold (due to that the instanton number becomes fractional as half integer ), and on a non-spin manifold (due to that the instanton number becomes quarter integer as ). Since time-reversal symmetry is preserved if and only if is identified, thus SO(3)θ=π YM explicitly breaks the time-reversal symmetry at . In the right-hand side (b) of Fig. 1 and Fig. 16, we actually have a 5d SETs whose 4d boundary explicitly breaks time-reversal symmetry.
Apply the tool introduced in [12], we derive new exotic anyonic statistics of extended objects such as 2d worldsheet of strings and 3d worldvolume of branes, which physically characterize the 5d SETs. We discover new triple and quadruple link invariants associated with the underlying 5d higher-gauge TQFT, hinting a new intrinsic relation between non-supersymmetric 4d pure YM and topological links in 5d. 2. 2.
Appearances of mod 2 anomalies: We note that the anomaly associated to the 5d term has also appeared in the context of an adjoint QCD4 theory ([85, 86, 87] and [55]). The has also appeared as a new SU(2) anomaly in the SU(2) gauge theory [57]. All these anomalies and all our anomalies in Eq. (2.70) are mod 2 non-perturbative global anomalies, like the SU(2) anomalies [56, 57]. 3. 3.
Mathematical relation between 5d and 4d bordism groups: Mathematically there seems to be an amusing relation between (1) gauging the SU(2) gauge bundle/connection under the coupling of 4d YM with 4d SPTs (4d bordism invariants of ) with derived from a group extension Eq. (3.2):
[TABLE]
and (2) some of the 5d bordism invariants given by . We provide the computations of bordism groups and topological invariants in Appendix A. It will be illuminating to explore this relation further in the future.
Classes of 4d SU(2)θ=π YM: In Ref. [38], it was noted that the Pin+ and Pin- version of the above group extensions and provide two different SPTs vacua after dynamically gauging the SU(2) symmetry give rise to two distinct 4d SU(2)θ=π YM theories. Although Ref. [38] suggested that the Pin+ and Pin- of 4d SU(2)θ=π YM are secretly indistinguishable by correlators of local operators on orientable spacetimes nor by gapped SPT states, can be distinguished on non-orientable spacetimes or potentially by correlators of extended operators.
In this work, we have shown that Pin+ and Pin- of 4d SU(2)θ=π YM indeed have distinct new higher ’t Hooft anomalies, given by Eq. (2.47) and Eq. (2.70), with and respectively. Thus we confirm that Pin+ and Pin- of 4d SU(2)θ=π YM live in distinct Hilbert spaces, thus they are indeed distinct vacua.
More generally, in this work, we propose a classification 4d SU(2)θ=π YM with a bosonic UV completion (e.g., on a lattice with bosonic degrees of freedom) and without fermionic parity symmetry . We propose that a classification can be all obtained from dynamically gauging the SU(2) normal subgroup of 4d -SPTs where is given by Eq. (3.2): i.e., follow [38] by coupling the 4d -SPTS to a pure 4d SU(2) YM theory and dynamically gauge their , see Sec. 3.484848Although we mainly focus on 4d SU(2)θ=π YM here, this gauge principle works for more general 4d SU(2) YM, or equivalently 4d SU(2)-gauge quantum spin liquids in condensed matter Other than the Pin+ and Pin- cases, this extension Eq. (3.2) includes two more cases: and . In summary, in terms of the un-gauged 4d SPTs, we have the symmetry group [41, 38]:
[TABLE]
The global symmetries of SPTs for the first two cases are purely bosonic since they do not contain ; the later two cases are fermionic since they do contain (the is shared by Pin and by the center of SU(2)).
- •
Before gauging, see Fig. 17, the 4d SPTs are well-defined 4d topological terms/bordism invariants that can live on 4d with fully local onsite symmetry without the need of an extra 5d bulk.
- •
After gauging SU(2) normal subgroup of symmetry of these SPTs, see Fig. 16(a), there is an emergent 1-form center -symmetry. After gauging SU(2), all four systems become bosonic without symmetry. We obtain new theories: 4d SU(2)θ=π YM as boundary conditions of 5d iTQFT in order to realize all global symmetries locally.
- •
Furthermore, after gauging this -symmetry of the 4d SU(2)θ=π YM and 5d iTQFT coupled system, see Fig. 16(b), we obtain new theories: 4d SO(3)θ=π YM as boundary conditions of 5d TQFT. 5. 5.
Quantum spin liquids in condensed matter: Strong coupled gauge theories have condensed matter implications as quantum spin liquids. Time-reversal symmetric U(1) gauge theories as quantum spin liquids [30] have been explored and classified based on the quantum numbers of gapped electric and magnetic excitations (Wilson and ’t Hooft line operators) in Ref. [63, 64], see also recent works [88, 89]. Time-reversal symmetric SU(N) gauge theories as quantum spin liquids have been explored in [38]. We will leave additional interpretations of our results of non-abelian SU(2) gauge theories in the context of quantum spin liquids for a future work. 6. 6.
Relations of link invariants and braiding statistics in various dimensions: We have applied the tools developed in [12] to compute link invariants of 5d higher-gauge TQFTs in Sec. 4. We anticipate a precise mathematical formulation of this 5d higher-gauge TQFT requires a certain higher category theory. We remark that several link invariants that we find here in 5d have dimensionally reduction analogy to 4d and 3d, such that the dimensional reduced link configurations in 4d and 3d are related to what had been studied in [11], [12] and References therein. 7. 7.
Fate of IR dynamics of gauge theories, UV completion and lattice regularizations at : For the 4d-5d systems that we explore (schematically in Fig. 1 and Fig. 16), we mainly focus on their Four Siblings as the UV theories. We do not yet know the IR fate of their dynamics of these strongly coupled gauge theories. However, given the potentially complete ’t Hooft anomalies in Eq. (2.47) and Eq. (2.70) (at zero temperature), we can constrain the IR dynamics by UV-IR anomaly matching.
The consequence of anomaly matching implies that the IR theories must be at least one of the following scenarios:
- (i)
Time-reversal symmetry broken (spontaneously or explicitly): The conventional standard lore suggests the two fold -breaking degenerate ground states [5]. 2. (ii)
1-form center symmetry broken (spontaneously or explicitly) as deconfinement:
- (2-i)
1-form -breaking and deconfined TQFTs, i.e., topological order in condensed matter. A gauge theory with the 1-form symmetry spontaneously broken has been proposed in Sec. 8.4. 2. (2-ii)
1-form -breaking and deconfined gapless theories or deconfined CFTs. 3. (iii)
1-form symmetry unbroken as confinement:
- (3-i)
1-form symmetry-extended invertible TQFT: This exotic scenario is discussed in Sec. 8.3. In Sec. 8.3, 1-form -symmetry is extended to in order to trivialize, thus saturate, the anomaly. 2. (3-ii)
1-form symmetry-preserving TQFT: Cordova and Ohmori [83, 84] have proved the nonexistence of TQFTs preserving both the 1-form symmetry and time-reversal symmetry while saturating the 4d SU(N)θ=π YM’s anomaly. This is consistent with Sec. 8.4’s analysis, which results in the previous phase of (2-i). 4. (iv)
Full symmetry-preserving gapless theory (CFT): This is a fairly exotic case which seems to be less likely to happen.
Let us comment more on the recent Cordova-Ohmori’s result [83, 84] which rules out the 1-form and time-reversal symmetry-preserving gapped phases for 4d SU(N)θ=π YM, namely the phase of (3-ii). To recall, although in Sec. 8.3, we show the phase of (3-i) 1-form symmetry-extended invertible TQFT can be constructed, but in reality, such phases are unlikely to happen [54] and should be only regarded as an intermediate step to construct the phase of (3-ii) 1-form symmetry-preserving TQFT. Furthermore, we show in Sec. 8.4 that dynamically gauging the extended symmetry of symmetry-extended invertible TQFT (of Sec. 8.3) results in the spontaneous symmetry breaking phases, instead of the symmetry-preserving gapped phases. Thus our Sec. 8 is consistent with [83, 84].
The fate of any of the four scenarios of IR phases above is meant to match the anomaly (or match Lieb-Schultz-Mattis type of theorem in condensed matter physics). Since we have the Four Siblings of 4d SU(2)θ=π YM at UV, labeled by , we can discuss each of their IR dynamics. We leave more systematic discussions of the IR dynamics for a future work.
4d SU(2)θ=π YM with and :
For or with , we see that (or schematically ) vanishes on 5d orientable manifolds. In other words, if -symmetry is spontaneously or explicitly broken, we can match the anomaly at IR. This means that when , the and versions of 4d SU(2)θ=π YM in Eq. (9.1) can indeed flow to the -breaking Scenario (1) at IR.
4d SU(2)θ=π YM with and :
For or with , we see that is nonzero if has a boundary, and this term does not vanish even if we restrict to orientable manifolds. In other words, even if we break -symmetry (spontaneously or explicitly), the still can suffer from background gauge variance. In this case, we should define the physical 4d theory to include not only given in Eq. (2.46) but also the 4d counter term , which combine to
[TABLE]
The intrinsic 5d theory thus contains only the 5d bordism invariant/SPTs/iTQFT: 494949Alternatively, if we instead interpret the background gauge variance of as a 4d higher ’t Hooft anomaly (rather than just a 4d counter term), then it has a consequence on 4d SU(2)θ=π YM dynamics. This means that when , the and versions of 4d SU(2)θ=π YM in Eq. (9.1) cannot flow merely to the -breaking Scenario (1) at IR. However, Scenario (2), (3), and (4) are still possible IR fates. It seems that the Scenario (2) with -breaking with deconfinement (due to the perimeter law of Wilson loop) can be the most likely outcome. Scenario (2-ii) for and versions of 4d SU(2)θ=π YM, are proposed in [38] as two distinct versions of deconfined gapless theories or deconfined CFTs: In this work, we show that these two siblings are indeed distinct theories with different Hilbert spaces at UV, due to their distinct 4d anomalies differed by a 5d invariant .
However, we will see that interpreting as 4d higher ’t Hooft anomaly will lead to a rather bizarre and strong constraint on the dynamics of 4d SU(2)θ=0 YM, see the footnote 50
[TABLE]
In fact, in Sec. 8, we construct the 4d boundary based of the Scenario (3) above as a boundary TQFT with a lattice spacetime path integral or a lattice Hamiltonian regularization; in this case, the full spacetime partition function of 4d-5d system can be explicitly computed as a number (by following Sec. 9 of [54]).
We will revisit other issues of dynamics in the future. 8. 8.
Refinement and modification of Yang-Mills existence and mass gap problem at : The original statement of the Clay Math Millennium Prize Problem [3] is “Prove that for any compact simple gauge group , a non-trivial quantum Yang-Mills theory exists on and has a mass gap .” It seems that in addition to the specification of gauge group and topological term with , we may also need to specify the data for Kramers single/doublet and bosonic/fermionic statistics for quantum number of Wilson lines , as we did in Eq. (2.26) and Eq. (2.46), say for 4d SU(2)θ=0 YM. The data may have been ignored in the past.
Here are possible outcomes for Four Siblings of 4d SU(2)θ=0 YM. Notice that the transition from the vacua of 4d SU(2)θ=π YM to 4d SU(2)θ=0 YM must break in between .
4d SU(2)θ=0 YM and :
Since the anomalies associated to 4d SU(2)θ=π YM with and can be removed by -breaking, therefore 4d SU(2)θ=0 YM and can have no ’t Hooft anomaly, thus it can be trivially gapped as a trivial vacuum.
4d SU(2)θ=0 YM with and :
These two siblings only differ from the trivially gapped vacuum of the previous two siblings, and , by the 4d counter term .505050 Follow the earlier footnote 49, if we instead interpret the background gauge variance of as a 4d higher ’t Hooft anomaly (rather than just a 4d counter term), then it has a consequence on 4d SU(2)θ=0 YM dynamics also. Notice that survive without -protection, therefore if there is an “anomaly” at , then it remains for all including at . The only way to saturate the if-anomaly of for SU(2)θ=0 YM is breaking the 1-form symmetry. If so, this means that SU(2)θ=0 YM with fermionic Wilson line (i.e., ) has 1-form symmetry spontaneously broken thus deconfined, which cannot be trivially gapped nor a trivial vacuum! In this case, if 4d SU(2)θ=0 YM with is still gapped as the conventional wisdom goes, they belong to the scenarios:
(2-i) 1-form -breaking and deconfined TQFTs, i.e., topological order.
This deconfined scenario seems to be too exotic for SU(2)θ=0 YM merely with fermionic Wilson line.
Therefore, our canonical interpretation with being a 4d counter term in 4d YM (see the main text around Eq. (9.2)) avoids leading to this bizarre deconfinement scenario for SU(2)θ=0 YM.
It will be enlightening to contemplate more consequences of their IR dynamics for these Four Siblings of 4d SU(2) YM.
10 Acknowledgements
The authors are listed in the alphabetical order by the standard convention. JW thanks Pierre Deligne, Juan Maldacena, Max Metlitski, Kantaro Ohmori, Nathan Seiberg, Edward Witten and Shing-Tung Yau for encouraging conversations, thanks Robert Gompf and Pavel Putrov for research collaborations in Ref. [90], and in Ref. [12, 38] respectively. We also thank Clay Cordova, Zohar Komargodski and Ho Tat Lam for conversations. JW thanks Harvard CMSA for the seminar invitation (March 11, April 10 and September 10, 2019) on presenting this specific work [91] and thanks the audience for the feedback at the IAS Princeton seminar [92]. ZW acknowledges support from NSFC grants 11431010 and 11571329. ZW is supported by the Shuimu Tsinghua Scholar Program. JW is supported by NSF Grant PHY-1606531. YZ thanks the support from Physics Department of Princeton University. This work is also supported by NSF Grant DMS-1607871 “Analysis, Geometry and Mathematical Physics” and Center for Mathematical Sciences and Applications at Harvard University.
Appendix A Computation of bordism groups
In this Appendix, we compute the bordism groups where is a solution of all possible extensions of
[TABLE]
given by Eq. (9.1)
[TABLE]
The bordism groups and their bordism invariants (topological invariants and SPTs) are used in the main text, for example, in Sec. 3. We also compute , used in Sec. 2, in Appendix A.5.
In the two subsections Appendix A.3-A.4, we will encounter the module structure due to the appearance of in the decomposition of .
Readers can find the introduction to this computation in Ref. [41, 38, 10]. For a short summary of the used concepts and terminology here, the readers may consult a succinct summary in the Appendix B of [38]. For readers who are not familiar with the details of mathematical calculations, we will help by stating the results explicitly.
A.1 Bordism group of :
We first notice that , where is the smash product and is the disjoint union of the topological space and a point. is the Madsen-Tillmann spectrum of the group , is the Thom spectrum of the group .
By the Adams spectral sequence, we have
[TABLE]
The Thom spectrum is the wedge sum of suspensions of the mod 2 Eilenberg-MacLane spectrum , is the direct sum of suspensions of the mod 2 Steenrod algebra , actually Thom proved that
[TABLE]
where the generators are in each degree other than .
So and .
Since
[TABLE]
by the Künneth formula, we get
[TABLE]
Here is the -th iterated shift of the graded algebra .
Since
[TABLE]
we have , .
The bordism invariants of are , , and . Namely, in physics terms, the topological invariants/SPTs from are , , and .
The bordism invariant of is Namely, in physics terms, the topological invariants/SPTs from is
A.2 Bordism group of :
Recall that is defined to be the subgroup of consisting of the pairs such that , there is a fibration .
We can also think of the space as the fiber of , where is the generator of .
Note that , we can think of the space as the fiber of , where is the generator of . Take to be the rank 3 vector bundle on determined by .
Define a map by , with inverse . Observe , so that {\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\mathrm{B}({\rm E}\times_{\mathbb{Z}_{2}}{\rm SU}(2))} is homotopy equivalent to . The canonical bundle {\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\mathrm{B}({\rm E}\times_{\mathbb{Z}_{2}}{\rm SU}(2))}\to\mathrm{B}{\rm O} corresponds to on . So MT({\rm E}\times_{\mathbb{Z}_{2}}{\rm SU}(2))=MT{\rm SO}\wedge\text{Thom}(\mathrm{B}{\rm Pin}^{+}(3),{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}W-3})=M{\rm SO}\wedge\Sigma^{-3}M{\rm Pin}^{+}(3).
Note that .
So , here is the smash product, is the suspension, is the Madsen-Tillmann spectrum of the group and is the Thom spectrum of the group .
By the Adams spectral sequence,
[TABLE]
Since
[TABLE]
where is the Chern class of the bundle and is the Thom class.
By the Künneth formula,
[TABLE]
Hence we have , .
The bordism invariants of are , , and . Namely, in physics terms, the topological invariants/SPTs from are , , and . Since the constraint is satisfied, let denote the Bockstein homomorphism associated to the extension , then where is the third integral Stiefel-Whitney class of and we have used the fact that , hence lifts to a bundle , here is the second Chern class of .
The bordism invariants of is Namely, in physics terms, the topological invariants/SPTs from is
A.3 Bordism group of :
Since there is a homotopy pullback square
{\mathrm{B}H}$${\mathrm{B}{\rm Pin}^{+}\times\mathrm{B}{\rm SO}(3)}$${\mathrm{B}{\rm O}\times\mathrm{B}{\rm SO}(3)}$${\mathrm{B}{\rm O}\times\mathrm{B}{\rm SO}(3)}$${\mathrm{B}^{2}\mathbb{Z}_{2}}$${\mathrm{B}{\rm O}}$$\scriptstyle{\sim}$$\scriptstyle{f}$$\scriptstyle{pr_{1},V}$$\scriptstyle{w_{2}+w^{\prime}_{2}}$$\scriptstyle{w_{2}+0}$$\scriptstyle{V+W-3}
where maps to , we have . By the Adams spectral sequence,
[TABLE]
The mod 2 cohomology of Thom spectrum is
[TABLE]
where is a graded -module with the degree homogeneous part for . Here stands for the sub-algebra of generated by and .
For , we can identify the page with
[TABLE]
For other details and the computation of module structure and Adams chart, please consult Ref. [41, 38]. We can extract the bordism group and their bordism invariants from [38].
A.4 Bordism group of :
Since there is a homotopy pullback square
{\mathrm{B}H}$${\mathrm{B}{\rm Pin}^{-}\times\mathrm{B}{\rm SO}(3)}$${\mathrm{B}{\rm O}\times\mathrm{B}{\rm SO}(3)}$${\mathrm{B}{\rm O}\times\mathrm{B}{\rm SO}(3)}$${\mathrm{B}^{2}\mathbb{Z}_{2}}$${\mathrm{B}{\rm O}}$$\scriptstyle{\sim}$$\scriptstyle{f}$$\scriptstyle{pr_{1},V}$$\scriptstyle{w_{2}+w_{1}^{2}+w^{\prime}_{2}}$$\scriptstyle{w_{2}+w_{1}^{2}+0}$$\scriptstyle{V+W-3}
where maps to , we have . By the Adams spectral sequence,
[TABLE]
For , we can identify the page with
[TABLE]
For other details and the computation of module structure and Adams chart, please consult Ref. [41, 38]. We can extract the bordism group and their bordism invariants from [38].
A.5 Bordism group of :
By the Adams spectral sequence
[TABLE]
Since where is the generator of , , , , etc, so by the Künneth formula,
[TABLE]
Hence we have , .
The bordism invariants of are , , , and . Namely, in physics terms, the topological invariants/SPTs from are , , , and .
The bordism invariants of are , , , and Namely, in physics terms, the topological invariants/SPTs from are , , , and . Readers can find more detailed discussions and calculations of the cobordism theory of higher symmetries in Ref. [10].
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