Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordisms
Meng Guo, Kantaro Ohmori, Pavel Putrov, Zheyan Wan, Juven Wang
TL;DR
This paper develops a framework for classifying and computing fermionic topological quantum field theories (TQFTs) with finite group symmetries using cobordism theory, providing explicit formulas and new insights into anomalies and crystalline SPT phases.
Contribution
It introduces a fermionic generalization of Dijkgraaf-Witten TQFTs via cobordism classification, computes relevant groups, and constructs explicit formulas for partition functions and anomalies.
Findings
Classified fermionic SPTs using spin and pin cobordism groups.
Derived explicit formulas for TQFT partition functions on closed manifolds.
Constructed new anomalous boundary spin-TQFTs and explored crystalline SPTs.
Abstract
We formulate a family of spin Topological Quantum Filed Theories (spin-TQFTs) as fermionic generalization of bosonic Dijkgraaf-Witten TQFTs. They are obtained by gauging -equivariant invertible spin-TQFTs, or, in physics language, gauging the interacting fermionic Symmetry Protected Topological states (SPTs) with a finite group symmetry. We use the fact that the latter are classified by Pontryagin duals to spin-bordism groups of the classifying space . We also consider unoriented analogues, that is -equivariant invertible pin-TQFTs (fermionic time-reversal-SPTs) and their gauging. We compute these groups for various examples of abelian using Adams spectral sequence and describe all corresponding TQFTs via certain bordism invariants in dimensions 3, 4, and other. This gives explicit formulas for the partition functions of spin-TQFTs on closed manifolds with…
| Dim | Values | |||
|---|---|---|---|---|
| 3d (2+1D) | ||||
| 3d (2+1D) | ||||
| 4d (3+1D) | ||||
| 4d (3+1D) | ||||
| 4d (3+1D) |
| dim | c-fSPTs classes | c-fSPTs classes | ||
|---|---|---|---|---|
| 1+1D (2d) | ||||
| 2+1D (3d) | ||||
| 3+1D (4d) | ||||
| D (d) | ||||
| dim | c-fSPTs classes | c-fSPTs classes | ||
|---|---|---|---|---|
| 1+1D (2d) | ||||
| 2+1D (3d) | ||||
| 3+1D (4d) | ||||
| D (d) | ||||
| D (d) | ||||
| dim | c-fSPTs classes | c-fSPTs classes | ||
|---|---|---|---|---|
| 1+1D (2d) | ||||
| 2+1D (3d) | ||||
| 3+1D (4d) | ||||
| D (d) | ||||
| dim | c-fSPTs classes | c-fSPTs classes | ||
|---|---|---|---|---|
| 2+1D (3d) | ||||
| 0 | ||||
| 3+1D (4d) | ||||
| 0 | 0 | |||
| 0 | 0 | |||
| 0 | 0 | |||
| 0 | 0 | |||
| D (d) | ||||
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**Fermionic Finite-Group Gauge Theories
and Interacting Symmetric/Crystalline Orders
via Cobordisms **
Meng Guo1,2,3, Kantaro Ohmori4, Pavel Putrov4,5,
Zheyan Wan6, Juven Wang4,7
1*Department of Mathematics, Harvard University, Cambridge, MA 02138, USA
2*Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5, Canada
3*Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada
4*School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA
5* ICTP, Trieste 34151, Italy
6*School of Mathematical Sciences, USTC, Hefei 230026, China
7*Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA, USA
We formulate a family of spin Topological Quantum Filed Theories (spin-TQFTs) as fermionic generalization of bosonic Dijkgraaf-Witten TQFTs. They are obtained by gauging -equivariant invertible spin-TQFTs, or, in physics language, gauging the interacting fermionic Symmetry Protected Topological states (SPTs) with a finite group symmetry. We use the fact that the latter are classified by Pontryagin duals to spin-bordism groups of the classifying space . We also consider unoriented analogues, that is -equivariant invertible pin±-TQFTs (fermionic time-reversal-SPTs) and their gauging. We compute these groups for various examples of abelian using Adams spectral sequence and describe all corresponding TQFTs via certain bordism invariants in dimensions 3, 4, and other. This gives explicit formulas for the partition functions of spin-TQFTs on closed manifolds with possible extended operators inserted. The results also provide explicit classification of ’t Hooft anomalies of fermionic QFTs with finite abelian group symmetries in one dimension lower. We construct new anomalous boundary deconfined spin-TQFTs (surface fermionic topological orders). We explore SPT and SET (symmetry enriched topologically ordered) states, and crystalline SPTs protected by space-group (e.g. translation ) or point-group (e.g. reflection, inversion or rotation ) symmetries, via the layer-stacking construction.
Contents
-
2.2 Relation to Dijkgraaf-Witten gauge theories and bosonic TQFTs
-
2.3 Definitions of invertible spin-TQFTs v.s. short-range entangled fSPTs
-
5 Fermionic topological invariants of links and surface links
-
8 Symmetric anomalous Spin-TQFTs as the boundary state of fermionic SPTs
-
10.10 Fermionic SETs (Symmetry Enriched Topologically ordered states)
-
10.11.1 Fermionic higher global symmetries vs. Higher-form global symmetries
-
10.11.2 Adams spectral sequence vs. Atiyah-Hirzebruch spectral sequence
1 Introduction and summary
Topological quantum field theories (TQFTs) play an important role both in physics and mathematics. In physics, they can be considered as the simplest examples of quantum field theories. Unlike the mathematically poor-defined “physical” QFTs (in particular, the gauge theories in the Standard model, that describe interactions between the quarks, leptons and gauge mediator bosons), TQFTs instead can be mathematically well-defined. Moreover, TQFTs not only play the role of toy models, but can also be used to describe topological phases of condensed matter system. Another application of TQFTs is systematic description of anomalies of general (i.e. not necessarily topological) QFTs in one less dimension.
In mathematics, the TQFTs provide a natural framework for topological invariants of manifolds as well as invariants of embedded submanifolds up to ambient isotopy (i.e. links in 3-manifolds). The TQFT structure allows calculation of invariants on complicated manifolds via surgery.
In this work, we are interested in study of spin-TQFTs, that is TQFTs that provide invariants of manifolds that depend not just on their topology but also a choice of spin structure. From physics point of view this generalization is very natural, since the physical system often contain fermions (in particular, the matter in the real world is composed of fermions). Therefore the spin structure on the space-time manifold is needed to describe how spinors transform under parallel transport.
From mathematics point of view, the spin-structure is also quite natural, since it provides a lift of -principle bundle of orthonormal frames in the tangent bundle to a principle bundle with a simply-connected structure group . Such extra structure often allows construction of more refined invariants.
The structure of the article is the following. In Section 2, we review the notion of spin-TQFT and describe a way to produce a family of spin-TQFTs labeled by elements of a spin-bordism groups of a classifying space of a finite group. In Section 3, we review some invariants of spin-manifolds that will be useful for construction of spin-bordism invariants that appear in expressions for spin-TQFT partition function on a closed manifold. In Section 4, we compute of spin-bordism groups of classifying spaces of some simple abelian groups. In Section 5, we consider the invariants of (surface) links produced by such spin-TQFTs. In Section 6, we calculate bordism groups of manifolds with structure, extending some of the results previously appeared in the literature. In Section 7, we briefly consider unoriented analogues of spin-TQFTs, that is as the pin±-TQFTs. In Section 8 , we construct some symmetry-preserving spin-TQFTs living on the boundary of one-dimension-higher bulk SPTs, explicitly in a TQFT language. In Section 9, we introduce the crystalline-SPTs that correspond to bordism invariants of manifolds with -bundles, where the symmetry involves a spatial translation -symmetry, and then relate them to various previously constructed SPTs with finite abelian symmetries. In Section 10, we comment on interpretation of the results in topological quantum matter.
For notational convention, we abbreviate the -dimensional spacetime as d. However, in certain cases, we may also use the notation from condensed matter community denoting D as an -dimensional spacetime, where . We also use a shorthand notation fSPTs for the fermionic SPTs. In Section 10, other than fSPTs protected by internal symmetries (denoted as a finite group -symmetry in this article), we will also explore crystalline-fSPTs, which means that the SPTs is also protected by space group (such as the translational symmetry) or point group symmetries (e.g. the rotational symmetry). We write the fermion parity symmetry as , where is always implicitly included for any fSPTs. It can also be understood as the center of symmetry that extends Euclidean spacetime symmetry. When we omit writing but only quote -fSPTs, it means fSPTs with and -symmetry, more precisely it means an invertible spin TQFT with -symmetry.
2 Spin-TQFTs and fermionic gauge theories
In this section, for completeness, we briefly review the notions of spin-TQFT and its equivariant version.
An ordinary -dimensional TQFT can be defined as a symmetric monoidal functor from the -dimensional bordism category to the category of complex vector spaces [1]. The objects of are oriented smooth111There are generalization without theses conditions. closed -manifolds. The morphisms are -dimensional bordisms between them modulo diffeomorphisms. A bordism from an -manifold to an -manifold is an oriented smooth -manifold with an isomorphism where bar denotes change of orientation. The fact that the functor is symmetric monoidal means that both categories are treated as symmetric monoidal categories and the functor respects this structure. The tensor product product structure in the bordism category is given by disjoint union, which is symmetric. The role of the unit is played by the empty manifold . This gives a symmetric monoidal structure on . On the symmetric monoidal structure is given by the ordinary tensor product with unit being . The value of the functor on a closed -manifold is then a linear map , which can be identified with an element of itself. The complex number is often referred to as partition function. Physically an objects of the bordism category, an -dimensional manifolds , is a spatial manifolds on which the theory is quantized and the corresponding vector space, that is the value of the functor , is the Hilbert space of the quantum theory222In physics literature such spaces are often denoted by a different symbol, e.g. , while is only used for partition function, that is the value of the functor on a closed -manifold..
This definition has various natural generalizations which involve a choice of some additional structures on manifolds (provided a structure on an -manifold induces a structure on its -dimensional boundary). In particular, one can consider manifolds with spin structure, that is a lift of orthonormal frame bundle to principle bundle, with respect to the extension . A spin structure on a manifold induces a natural spin structure on its boundary (see e.g. [2]), therefore one can define the corresponding bordism category of spin manifolds . The vector spaces associated to -dimensional manifolds can be equipped with -grading: . Physically the even and odd parts of vector spaces are bosonic and fermionic states respectively in the Hilbert space . One can therefore give the following definition (cf. [3, 4, 5]):
Definition 1**.**
An -dimensional spin-TQFT is symmetric monoidal functor
[TABLE]
where is the category of -dimensional spin-bordisms and is the category of -graded vector spaces333The morphisms are grading preserving linear maps., satisfying
[TABLE]
for any closed -dimensional spin-manifold , where is a circle with even/odd-spin structure.
The last condition can be understood as topological spin-statistics constraint. The notion can be also generalized to non-orientable manifolds and bordisms between them. The analog of the spin-structure is pin± structure, the lift of bundle with respect to one of the two non-trivial central extensions . The extensions can be distinguished by the following commutative diagram
[TABLE]
where and , . In this work, we firstly focus on orientable case, and later discuss non-orientable pin*±* cases in Section 7.
Another natural generalization involves a choice of (isomorphism class of) principal -bundle over manifolds, where is topological group. Equivalently, one can consider maps to its classifying space444Which can be defined as a connected topological space (unique up to homotopy) satisfying , up to homotopy. One can consider the corresponding bordism category where objects are pairs . The morphisms between and are pairs such that555As before, the bordism data contains a choice of the isomorphism between and the boundary, however we will usually not indicate it explicitly. , , . Then -equivariant TQFT can be defined as a symmetric monoidal functor . Physically the group has meaning of global symmetry group of the theory.666In the sense that one can couple the theory to a background gauge field. There is no condition that acts faithfully on the operators of the theory.
One can also, of course, combine the notions above and consider -equivariant spin-TQFT which is a symmetric monoidal functor . Even more generally, one can consider the bordism category for any extension (or in non-orientable case) and the corresponding bordism category where manifolds are equipped with tangential structures, that is principle -bundles that are lifts of orthonormal tangent frame bundles. Then one can define TQFT with symmetry as a functor from to the category of complex vector spaces (see [6] for details). The case of -equivariant spin-TQFT is then the particular case with .
In what follows we will also use the notion of an invertible TQFT. This is a TQFT such that the value of the functor on any -manifold is a one dimensional vector space (that is, in general non-canonically, isomorphic to ) and the value on any bordism is an invertible homomorphism. Invertible TQFTs form an abelian group, while all TQFTs only form a monoid. The product of TQFT functors is defined by . Physically taking the product of two quantum field theories means stacking them together without interaction.
2.1 Gauging
In this work, we are interested in obtaining non-trivial spin-TQFTs by gauging -equivariant invertible spin-TQFTs with finite symmetry group . The theories obtained this way can be understood as a direct generalization of Dijkgraaf-Witten TQFTs [7] to the case of spin-TQFTs. From physics point of view an invertible -equivariant TQFT can be understood as a classical field theory, where the partition function is given by the exponentiated action which depends on the background gauge field, while the corresponding gauged TQFT is the quantum theory where the gauge field is dynamical. Formally, the gauging procedure can be understood as the following map
[TABLE]
such that the values of functors on closed -manifolds (i.e. partition function) are related as follows:
[TABLE]
where the sum is performed over , homotopy classes of maps from to , and denotes the automorphism group of a principle -bundle over corresponding to the map . Equivalently, can be understood as the fundamental group of the corresponding connected component of in the space of maps . In the case when is abelian, there is an isomorphism which is the property of the Eilenberg-MacLane space . More explicitly, the function corresponds to an element where is a generator of . In this paper, however, we will use the same symbol (e.g. above) for both a function and the corresponding element in . In the abelian case for any .
The relation (2.5) can be extended to the full functors analogously to Dijkgraaf-Witten theories [8, 9]. In particular, the values of the functor on the objects are related as follows:
[TABLE]
where is the following subset of the set of homotopy classes of maps :
[TABLE]
This subset of has the following meaning. From functoriality of it follows that forms a (one dimensional) representation of . The representation is realized as follows. As was mentioned above, an element correspond to a closed path in starting and ending at , considered up to homotopy. Each such path gives a function , or, equivalently, , such that . Then
[TABLE]
provides the action of on . Then appearing in (2.6) can be also defined as the subset of homotopy classes of functions such that is a trivial representation of . Physically this can be interpreted as the Gauss law constraint. One can easily see that the condition (2.2) is automatically satisfied. In particular,
[TABLE]
where the map is related to the element as above and is the pullback with respect to the projection map . In the last equality we used the fact that is the character of one dimensional representation of group defined above, and applied orthogonality property of characters. Only characters of trivial representations survive after taking the sums over the elements of the group. Finally, since is an invertible spin-TQFT, we have:
[TABLE]
where in the right hand side one of the graded dimensions is zero and the other is . In particular, the total dimension of the Hilbert space of the gauged spin-TQFT on is given by
[TABLE]
The value of the functor on a bordism between and is then a linear map
[TABLE]
given by the following expression:
[TABLE]
where
[TABLE]
The factor is needed so that the functor satisfies composition property (cf. [8]). The formulas (2.12) and (2.12) then can be considered as the definition of the gauging map (2.4).
Invertible TQFTs with symmetry , satisfying certain additional properties: reflection-positivity and being extended, were classified in [6] (also in [10, 11, 12] from more physical perspective). From physics point of view these additional requirements are quite natural and expected in theories describing realistic quantum systems. Namely, reflection-positivity is a Wick-rotated version of unitarity and being extended corresponds to locality. In the case of the result can be formulated as follows777Note that the free part of the classification, conjecturally given by
(2.15)
contains Chern-Simons-like theories, which are not strictly topological.:
[TABLE]
More general formulation of the result will be mentioned and used in Section 6. As was pointed out above, invertible TQFTs form an abelian group (the additional conditions are respected by the product) and the isomorphism above should be understood as an isomorphism between abelian groups. The right hand side is the Pontryagin dual to the torsion subgroup of spin-bordism group of , which is defined as follows:
[TABLE]
where bordisms are understood as morphisms in the category defined above. The set of equivalence classes has a natural abelian group structure under disjoint union operation. This abelian group is always discrete and finitely generated, that is isomorphic to the finite product of finite cyclic groups and copies of .
The correspondence between -equivariant TQFTs and the elements of the abelian group in the right hand side of (2.16) is realized as follows. The embedding of the torsion subgroup into the full spin bordism group induces a surjective map between their Pontryagin duals. Moreover, the connected elements of the group map to the same elements of the Pontryagin dual to the torsion subgroup. Therefore the right hand side of (2.16) can be understood as the group of connected components:
[TABLE]
One can consider a -equivariant spin-TQFT corresponding to an element
[TABLE]
so that TQFTs corresponding to elements in the same connected component can be continuously deformed into each other888Meaning that there are continuous maps from a path connecting points in to all values of the TQFT functor.. Such TQFT can be characterized by its values on closed manifolds as follows:
[TABLE]
where denotes a class in the bordism group (2.17). Applying the gauging map described above one can consider a (generically non-invertible) spin-TQFT labeled by the elements of the same set. Note that non-torsion elements only appear in in dimensions .
2.2 Relation to Dijkgraaf-Witten gauge theories and bosonic TQFTs
The TQFTs labeled by the elements which are constructed above can be understood as generalizations of Dijkgraaf-Witten topological gauge theories [7], for both ungauged (“classical”) and gauged (“quantum”) versions. The Dijkgraaf-Witten theories are labeled by elements of . The explicit relation between two families of TQFTs is the following. The map
[TABLE]
induces
[TABLE]
This maps Dijkgraaf-Witten theories to the theories constructed above. Note that the map is in general neither surjective nor injective. Non-injectivity of the map (2.22) (which is equivalent to non-surjectivity of (2.21)) means that SPTs (the un-gauged Dijkgraaf-Witten theories) labeled by different elements of can become equivalent when considered on smooth spin-manifolds.999 Of course, it is known that Dijkgraaf-Witten theories corresponding to different elements of group cohomology can become equivalent after dynamical gauging (e.g. [13] and References therein). However, here we mean a more surprising statement: There are identical TQFTs even before gauging (i.e. SPTs). After gauging, there might be additional identifications corresponding to field redefinitions. In dimension Dijkgraaf-Witten theories labeled by different elements of can become equivalent even as non-spin TQFTs, cf. [10]. This is because starting from dimension the map
is not injective in general. This is because there are examples in degree 7 when a homology class of cannot be represented by an image of a smooth manifold continuously mapped to . Since SPTs/invertible TQFTs are classified by r.h.s. of the above equation, the TQFTs labeled by the elements of l.h.s. that map to the same element realize equivalent TQFTs (more concretely, the actions constructed via two different classes of will have the same value on any smooth -manifold). However this will not happen in any examples that we consider in this paper. That is, in all examples in this article, the map (2.21) is surjective, and, therefore, (2.22) is injective. In this case one can consider as a subgroup of . The TQFTs and that correspond to elements that are in the image of (2.22) are not proper spin-TQFTs, meaning that the values of the functor do not actually depend on spin structures. Following physics terminology we will call such TQFTs bosonic, while others will be called fermionic. In this work, we are interested in presenting explicit examples of the latter.
Note that invertible -dimensional -equivariant (non-spin) TQFTs are actually classified by , instead of , where is the ordinary oriented bordism group of generated by pairs where is an oriented manifold. The relation to Dijkgraaf-Witten theories is given by the direct analog of (2.21)-(2.22). However, in dimension there is no difference, that is . In general instead of (2.21) one should consider the map
[TABLE]
realized by forgetting spin-structure. The dual map
[TABLE]
provides a relation between invertible -equivariant spin- (fermionic) and non-spin (bosonic) TQFTs.
2.3 Definitions of invertible spin-TQFTs v.s. short-range entangled fSPTs
However, there is a caveat. Some theorists naïvely regard “-equivariant reflection-positive invertible spin-TQFTs” as a definition of SPTs protected by global symmetry . In contrast, other theorists define SPTs as short-range entangled (SRE) states whose existence must be protected by nontrivial global symmetry . The second definition, in some sense, is more physical and suitable for the lattice-regularized condensed matter setting. In quantum system, two distinct condensed matter phases cannot be deformed into each other via local unitary transformations. While all SPTs can be deformed into each other via local unitary transformations if all symmetry is broken, distinct -SPTs cannot be deformed into each other when is preserved. In the second definition, based on the local unitary transformation classification of phases, the invertible spin-TQFTs protected by no symmetry (except the fermion parity ), classified by , are actually long-range entangled (LRE) invertible topologically ordered states (instead of short-range entangled SPTs).101010 See a recent discussion in Ref. [14] and Ref. [15]’s Section 5.4 along this statement, and References therein. For example, the 2d Arf invariant or equivalently the 1+1D Kitaev fermionic chain [16], obtained from the generator of , is actually not a short-range entangled fSPTs, but instead a long-range entangled invertible fermionic topological order in 1+1D (in 2d).
Formally, in the second definition of SPTs, we need to mod out those LRE invertible spin-TQFTs which are invertible topological order states; so we can propose a definition of fermionic -SPTs and their classification, mathematically, by modifying the (2.16) to
[TABLE]
Later in Section 4 and 6, when we show, in various Tables, the data of bordism groups and the classification of fSPTs, we always use the second definition (which is, physically, the precise definition of short-range entangled SPTs), as proposed in (2.25). Since the entangled structure and the locality for local unitary transformation is more sharply defined in spacetime dimensions , we will only classify fSPT for dimensions , focusing on (namely 1+1D, 2+1D, and 3+1D), shown in our Tables.
3 Some useful invariants of spin-manifolds
In this section we review basic invariants and structures that one can consider on closed spin and pin± manifolds in low dimensions (see e.g. [2] for details) and fix their notations. This will be useful later in explicit construction of TQFTs.
3.1 1-manifolds
The only one-dimensional connected closed manifold is . There are two choices of spin-structure that are usually referred to as odd and even. We will denote the corresponding spin circles as and respectively. The circle with even spin structure is a boundary of a disk with the unique spin structure, while the circle with odd spin structure is the generator of . We will denote the corresponding bordism invariant as , its value is determined as follows:
[TABLE]
3.2 2-manifolds
Consider first an oriented 2-manifold . Spin structures then are in one-to-one correspondence with quadratic forms111111Physically the value of corresponds to periodicity condition on spinors along the Poincaré dual 1-cycle: for periodic and [math] for anti-periodic
[TABLE]
such that
[TABLE]
The 2-dimensional spin bordism group is and the corresponding bordism invariant is Arf invariant:
[TABLE]
where is any symplectic basis in .
Consider now non-orientable a 2-manifold . It always admits a pin- structure. Similarly to the spin case, pin- structures are in one-to-one correspondence with quadratic enhancement
[TABLE]
such that
[TABLE]
In particular:
[TABLE]
The pin--bordism group is and the isomorphism is explicitly given by Arf-Brown-Kervaire invariant:
[TABLE]
One of the definitions of the Arf-Brown-Kervaire invariant valued mod 8 is given in terms of the following Gauss sum:
[TABLE]
When surface is orientable this reduces to the previous case with , .
3.3 3-manifolds
Let be an oriented closed 3-manifold. Because of odd dimension there is no usual intersection pairing as in the two dimensional case above. However, one can instead define a linking pairing on the torsion part of the first homology121212There is an analogous definition of linking pairing on for any odd dimensional manifold.:
[TABLE]
where is a 2-chain such that for some (such always exist because is torsion).
A spin-structure on again allows to define a quadratic refinement of the linking pairing:
[TABLE]
The value of can be geometrically defined as follows. Take a smooth embedding representing a torsion element . Framings on , i.e. trivializations of the normal bundle, are in one-to-one correspondence to rational numbers such that (because framings corresponds to a choice of push-off into the boundary of the tabular neighborhood). Given a spin structure on there is a subset of even framings defined as follows. Framing on the normal bundle together with spin structure on fixes a spin structure on . Then one can define a subset of even framings such that , that is is a spin-boundary. Such framings have a fixed value of . Then one defines .
Spin structure on also allows one to define a symmetric function
[TABLE]
which is an enhancement of
[TABLE]
in the sense that
[TABLE]
The values of can be defined as follows. Let be a smooth, possibly non-orientable, surface which represents a class in Poincaré dual to (it always exists). A spin structure on induces canonically a pin*-* structure on . This is follows from the fact that due to orientability of , and from the fact that there is one-to-one correspondence between pin- structures on and spin-structures on . Then
[TABLE]
where is a quadratic enhancement corresponding to the pin- structure on considered above, and is implicitly assumed to be restricted on .
Moreover, there is an enhancement of , given by
[TABLE]
so that
[TABLE]
3.3.1 Abelian spin-Chern-Simons theory
The quadratic enhancement defined above can be used to explicitly write expression for the partition function of abelian spin-Chern-Simons theories, a known simple family of spin-TQFTs. The (“classical”) data needed to define a spin-Chern-Simons TQFT is a symmetric bilinear form
[TABLE]
or, equivalently, a lattice. On the physical level the partition function on a closed oriented spin 3-manifold is given by the path integral
[TABLE]
The expression in the exponent is actually ill defined because connection 1-form is not globally defined for non-trivial bundles. To avoid this problem one has to take a spin 4-manifold such that with induced spin-structure131313We use the usual normalization of connection 1-form/curvature such that the first Chern class is .:
[TABLE]
which is independent of the choice of spin-manifold because of integrality of and the fact that intersection form on spin-manifolds is even. If the spin-structure was not required, would have to be even (i.e. ), which is the quantization condition on the level matrix for the ordinary, non-spin (“bosonic”) Chern-Simons theory. If the spin-structure is required, we have the spin (“fermionic”) Chern-Simons theory.
Different can actually give equivalent spin-TQFTs. The classification of non-equivalent abelian spin-Chern-Simons theories was done in [17].
Even though in general path integral over the space of connections (modulo gauge transformations) is ill-defined, in this case the action is quadratic in connection 1-forms and the path integral can be defined and computed formally141414Alternatively, one can also mathematically define spin-Chern-Simons TQFT via spin-generalization (see e.g. [18, 3, 5]) of Reshetikhin-Turaev [19] construction where the input data is the spin modular tensor category of representations of lattice vertex operator algebra associated to .. The path integral reduces to the sum over critical points of the action, that is flat connections. To simplify the formulas let us assume that is a rational homology sphere, that is . The moduli space of flat connections is then a finite set
[TABLE]
where the last isomorphism is explicitly given by the composition of linking pairing with the exponential map:
[TABLE]
Assuming this correspondence between flat connections and elements of , it is easy to see that the usual Chern-Simons invariant (valued mod 1) of flat connection reads , while its spin-version is . The partition function of level spin-Chern-Simons theory with appropriate normalization151515Such that . is then given by
[TABLE]
As we will see later some of the TQFTs are equivalent to abelian spin-Chern-Simons theories for certain choices of . Consider in particular the case of
[TABLE]
The sum above then can be partially performed explicitly:
[TABLE]
The set over which the sum is performed can be identified with the first cohomology with coefficients:
[TABLE]
The isomorphism is given by the universal coefficient theorem () and the canonical embedding . Let us denote by an element of corresponding . Then we can define a function
[TABLE]
so that
[TABLE]
The function can be understood as a quadratic refinement of the bilinear form
[TABLE]
where is the Bockstein homomorphism corresponding to the following short exact sequence of coefficients:
[TABLE]
So that
[TABLE]
One can argue that is a bordism invariant, that is can be considered as homomorphism
[TABLE]
where we identify an element of with a homotopy class of a map in the usual way. We then arrive at the following relation between the spin-Chern-Simons TQFT and spin-TQFTs associated to elements of the spin-cobordism group of :
[TABLE]
where is considered an element of and is its pullback with respect to the map above.
4 Spin-bordism groups and computations
By the Pontryagin-Thom construction,
[TABLE]
where, as usual, denotes the -th stable homotopy group of a spectrum, is a Thom spectrum associated to stable spin-structure161616Note that in the case of spin structure Thom spectrum is equivalent to Madsen-Tillmann spectrum: .. We use the standard notations in homotopy theory: is a pointed space with a disjoint basepoint added, denotes the smash product. For example, are computed by Anderson-Brown-Peterson ([20]).
We are interested in the case , a classifying space of a finite abelian group. In this work, we compute some simple examples where is a finite abelian group. At odd torsion . Since we are interested in constructing spin-TQFTs which non-trivially depend on spin-structure (i.e. not “bosonic”), we will only consider some simple examples when itself is 2-torsion. We can then use the Adams spectral sequence for computation:
[TABLE]
where stands for mod 2 cohomology and is for Steenrod algebra. The abelian groups are understood as groups of extensions of length between modules over Steenrod algebra and , where denotes the shift of grading by . Notations stands for 2-completion of the abelian group . We refer to [21] as a brief review of necessary definitions and techniques in stable homotopy theory that requires a minimal prior background.
At 2-torsion and in degree the Thom spectrum is equivalent to , the connective version (stable homotopy groups in negative degree are zero) of the real K-theory . Therefore, if the abelian group has only 2-torsion (which means that is a product of finite abelian groups of the form ), one have
[TABLE]
This relation can be used to immediately give answer for for certain cases of , where were already calculated in the literature ([22, 23]). We consider one of such cases in the next subsection.
We would like to remark that such statement about classification of interacting fermionic SPTs with finite symmetry via the real connective K-theory () should not be confused with a different statement about classification of certain free fermionic SPTs with via real or complex periodic K-theory ( or ) given in [24].
Below we present the calculations of bordism group which classifies the invertible fermionic spin-TQFTs with the fermionic parity symmetry and internal symmetry . The invertible fermionic spin-TQFTs include all the interacting fermionic SPTs (short-ranged entangled states) and some invertible fermionic topological orders (long-ranged entangled states). To recall the definitions, see (2.16) and (2.25). Here the corresponds to the spacetime dimension in physical systems.
4.1
4.1.1 Computation
When one can use the known results in the literature
Theorem 2**.**
[22]** Let denote the -fold smash product of the classifying space of . Then
[TABLE]
where is a -vector space, whose dimension is the coefficient of in the series
[TABLE]
with
[TABLE]
Since we have , from above theorem, we have the following
Theorem 3**.**
[TABLE]
[TABLE]
As described in section 2.2 there is a natural map from spin-bordism groups to integer homology groups (2.21) which provides a relation with Dijkgraaf-Witten theories. Therefore it is instructive to compare (4.5)-(4.6) with:
[TABLE]
[TABLE]
In particular we see that . Therefore in this case we do not get any fermionic TQFTs (according to the terminology explained in section 2.2).
4.1.2 Bordism invariants
Let us explicitly describe the corresponding bordism invariants. In dimension 3, the isomorphism above is explicitly given by:
[TABLE]
where , are functions defined in section 3.3. In the right hand side are treated as the elements of and the multiplication is performed with respect to the usual cup product. The non-underlined bordism invariants, considered as elements of (by embedding a cyclic group in ), belong to the subgroup embedded via the map (2.22)171717Note that via Yoneda lemma there is one-to-one correspondence between cohomology operations
(4.10)
and elements of . .
The underlined bordism invariants, considered as elements of , are fermionic (in the terminology explained in in section 2.2), but provide refinement of bosonic elements from . That is, a certain power of them gives an element from subgroup . For example, from (3.14) it follows that .
In dimension 4 we have:
[TABLE]
where is the first Pontryagin class and the denominator of the quotient contains identically vanishing polynomials in elements of of degree 4. Note that in dimension none of the invariants actually depends on the spin structure. Therefore their gauging does not give proper spin-TQFTs. Moreover, and the gauged TQFTs coincide with Dijkgraaf-Witten TQFTs [25].
Hence we have the following theorem:
Theorem 4**.**
[TABLE]
where in the second column, for comparison, we list the known homology groups. Note that .
4.2
4.2.1 Computation
As was mentioned in the beginning of the section, the computation involves no odd torsion and we can use 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 Steenrod algebra and stands for -algebra generated by and . Thus, for , we can identify the -page with
[TABLE]
The mod 2 cohomology is the following: where , where , , . The differential on the second page are the following: , , . We use the standard notation in the stable homotopy theory where denotes an element of181818There is natural action
(4.16)
realized by corresponding to extension . The -module structure of and the page are depicted in Figures 1 and 2.
Hence we have the following theorem:
Theorem 5**.**
[TABLE]
where in the last column, for comparison, we list the known homology groups.
4.2.2 Bordism invariants and manifold generators
The bordism groups are explicitly realized as follows:
[TABLE]
Equivalently, , . The functions and can be used to pull back the generators of the mod 2 cohomology of classifying spaces to :
[TABLE]
[TABLE]
Consider in particular the dimension . Reading out from the page of Adams spectral sequence, the bordism group is mapped to by sending an element to the invariants and , we can see that
Theorem 6**.**
* generates and generates , where is the Lens space, is the generator of , is the generator of , is the generator of , and is the generator of .*
It follows that complete bordism invariant in dimension read
[TABLE]
where, as before, functions , , are the ones defined in section 3. Below we elaborate on the notations and the definition of the bordism invariants listed above. The expressions written in terms of or assume that the Poincaré duals of and can be represented by an embedding191919In fact, for the purposes of inducing spin structure from the ambient space, as described below, it is enough to have an immersion. of a smooth oriented202020Note that if and a smooth representative of the Poincaré dual of exists, it will be necessarily orientable, because can be obtained by the pushforward of under the embedding. On the other hand, if this implies that is the image of some under the canonical map induced by the non-trivial homomorphism . Such homomorphism also induces a homomorphism and the corresponding bordism invariant for should reduce (i.e. pulled back) to the one of the invariants in Section 4.1.2. A naive argument shows that it should be . manifold of codimension one in . In particular, this is automatically the case when for some element . In this case it is known that the Poincaré dual to can be always represented by a smooth oriented submanifold of codimension one inside and one can take to be this submanifold. An embedding of an oriented , together with the orientation of the ambient space provides a trivialization of the normal bundle. The spin structure on then induces a spin structure on . In the expressions above or are used to denote the spin manifold .
As in Section 4.1.2 non-underlined elements (understood as elements of ) are bosonic (i.e. belong to subgroup), and elements underlined with a single line are fermionic that provide refinement of the bosonic elements from . In particular, from the formulae of Section 3 we have:
[TABLE]
[TABLE]
[TABLE]
The elements in (4.19) underlined with a double line are fermionic and do not refine any elements of .
4.3
4.3.1 Computation
As in the previous case the computation involves no odd torsion and we can use the Adams spectral sequence:
[TABLE]
Using again the expression (4.14) for mod 2 cohomology of , for , we can identify the -page with
[TABLE]
The mod 2 cohomology and the differential are the following: where , , . , for , , , , where and are generators of two copies of .
The -module structure of and the page are presented in Figures 3 and 4 respectively.
Hence we have the following theorem
Theorem 7**.**
[TABLE]
4.3.2 Bordism invariants and manifold generators for
The bordism groups are explicitly realized as follows:
[TABLE]
Equivalently, , . As before, the functions and can be used to pull back the generators of the mod 2 cohomology of the classifying spaces to :
[TABLE]
[TABLE]
Reading out from the page of Adams spectral sequence, the complete bordism invariants in dimension read as follows
[TABLE]
where is the Bockstein homomorphism w.r.t. to the short exact sequence . The codimension one submanifolds and are chosen such that they intersect transversally along a smooth 2-manifold. The spin structure on is induced as follows. The pair of normal vectors to and define a trivialization of the normal bundle to . As described before, together with the spin structure on the ambient space this unambiguously defines a spin-structure on .
It follows that in dimension 4 the generators of the bordism group are given by the following
Theorem 8**.**
* and generates two individually and generates , where is the Lens space, is the generator of , is the generator of , is the generator of , is the generator of , is the generator of the first , is the generator of the second and there are odd spin structures on the last two .*
4.4
4.4.1 Computation
We can again use the Adams spectral sequence, since the computation involves no odd torsion:
[TABLE]
Using (4.14), for , we can identify the -page with
[TABLE]
The cohomology rings of the classifying spaces are the following: where , where , , . The second page differential acts as follows: , , for , where are generators of the two copies of .
The -module structure of and the page are shown in Figures 5 and 6.
Hence we have the following
Theorem 9**.**
[TABLE]
4.4.2 Bordism invariants
The bordism groups are explicitly realized as follows:
[TABLE]
The complete bordism invariant in dimension read as follows
[TABLE]
4.5
4.5.1 Computation
As in the previous sections we use the Adams spectral sequence
[TABLE]
and the fact (4.14). For , we can then identify the -page with
[TABLE]
The cohomology ring of the classifying space is given by: where , where , , . The second page differential acts as follows: , , for . , , , , where and are generators of two copies of .
The -module structure of and the page are shown in Figures 7 and 8 where we use the known result for in Figure 4.
Hence we have the following
Theorem 10**.**
[TABLE]
4.5.2 Bordism invariants
The bordism groups are explicitly realized as follows:
[TABLE]
The complete bordism invariant in dimension read as follows
[TABLE]
4.6
4.6.1 Computation
As before, we can use the Adams spectral sequence
[TABLE]
and, for , we can identify the -page with
[TABLE]
The mod 2 cohomology ring and the second page differential are determined by the following formulas. where , , . , for , , , , . , , , .
The -module structure of is depicted in Figure 9.
The calculation and the result are similar to the previous ones. Here we only present the result in dimension 4:
Theorem 11**.**
[TABLE]
which can be compared with
[TABLE]
4.6.2 Bordism invariants
[TABLE]
[TABLE]
Note that the last invariant can be equivalently written in a following not explicitly symmetric way: .
4.7 for general finite abelian .
We do not present a calculation for general finite abelian , however the result is expected to be qualitatively similar. In particular, the odd torsion part will coincide with the odd torsion part the oriented bordism group and will not provide any fermionic spin-TQFTs (that is, TQFTs depending non-trivially on spin-structure). Moreover, for , . Therefore one can restrict to the case of . Therefore, the result is expected to be of the similar form as in the examples considered above.
5 Fermionic topological invariants of links and surface links
An -dimensional TQFT, when evaluated on a closed -manifold gives a numerical invariant of this manifold, valued in212121As . The TQFT structure also provides invariants of embedded submanifolds with respect to ambient isotopy. Namely, consider a possibly disjoint closed oriented -manifold embedded via a map into a -manifold . Denote the image and its tabular neighborhood as . The complement then can be considered as a bordism from the empty manifold to . The value of a TQFT on it then can be considered as an element in the complex vector space222222As . . If one fixes a basis in this vector space, for each basis element there is a numeric invariant of under ambient isotopy.
A very well known example of this construction is Witten-Reshetikhin-Turaev 3d TQFTs which provide “quantum” invariants of both closed 3-manifolds and links. Similarly, the 3d and 4d spin-TQFTs considered in this article give rise to invariants of links in 3-manifolds and surface links in 4-manifolds respectively.
We will restrict our attention on the codimension 2 oriented submanifolds in where or . In this case the normal bundle to is always trivial. This follows, for example from the triviality of the Euler class (see e.g. Corollary 11.4 in [26]). Let us fix the framing of , that is a trivialization of the normal bundle. Since the normal bundle is two-dimensional, the choice of the framing is equivalent to a choice of non-vanishing section of the normal bundle (up to homotopy), that is a choice of normal vector at each point of . Without loss of generality the end of the vector can lie on the boundary of the tabular neighborhood . The framing thus fixes a homeomorphism by mapping a fixed point on to the end of the framing vector. The framing also provides a spin-structure on induced from the unique spin-structure on . Namely, the trivialization of of normal bundle together with spin structure on , via the decomposition , fixes a spin structure on [2]. Spin structure induced from on the factor in is even (i.e. bounding), since it is identified with a circle bounding a small disk surrounding . Together they give a product spin structure on . Thus a framed oriented codimension 2 submanifold (possible disconnected) embedded in gives a well defined element of in the spin-bordism category and one can consider the evaluation of the spin-TQFT functor on it:
[TABLE]
for different values of .
There is always distinguished choice of the “zero” framing on each connected component of in the following sense. The space of homotopy classes of framings on the normal bundle to is non-canonically isomorphic to . However there is a distinguished isomorphism for which the element corresponding to the given framing is determined by , where is a smooth curve on representing a class and is its push-off in towards the framing vector. For framing corresponding to the zero element in there is always exist oriented Seifert hyper-surface smoothly embedded in such that and the framing vector field on is the normal vector to inside (see e.g. [27]). For such choice of framing , with the spin-structure induced from , represents a trivial element in .
Note that knowledge of invariants of (surface) links (5.1) in allows easy calculation of the invariants on closed manifolds constructed via surgery, by using the functoriality of .
In what follows we consider a few examples of particular choices in more detail, extending and clarifying some of the statements appeared in [25]. All other choices are analogous. Note that the cases when is “bosonic” (i.e. belongs to the subgroup) were already considered in great detail in [25]. We provide a list of examples the invariant of (surface) links produced by 3- and 4-dimensional fermionic finite group gauge theories in Table 1.
5.1 , in dimensions
Consider a framed oriented knot232323The case of the multi-component link is analogous. in , that is an embedding . As described above the unique spin-structure on together with framing defines a spin structure on . The framings of are in one-to-one correspondence with integers numbers. The correspondence is given by the self-linking number of :
[TABLE]
where is a push-off of towards the framing vector. The framing with is often called Seifert framing. In this case the framing vector can be realized as a normal vector to pointing inwards an oriented Seifert surface (which always exist) smoothly embedded in such that . The spin structure induced on is then even, since is the spin-boundary of oriented which has a spin-structure canonically induced from . Any non-zero even framing (i.e. ) can be realized by taking the normal vector pointing inwards an unorientable surface , such that . Such surfaces can be realized by taking band-connected sums of an orientable Seifert surfaces with a Möbius band. Each such sum changes framing by . Any even framing than induces an even spin structure on and, as a spin-manifold where two circles are identified with the meridian and longitude cycles242424Meridian cycle is a small cycle surrounding the knot and longitude is the cycle given by the push-off towards the framing vector.. For odd framings Seifert surface, orientable or not, that is compatible with the framing does not exist. The spin structure on the knot complement is then . This follows, for example, from the realization of spin-structures on as quadratic forms on and the fact that changing framing by one corresponds to the action
[TABLE]
where and are representatives of the median and longitude of respectively.
Let us take to be the generator of :
[TABLE]
Let us choose an even framing. The value of the spin-TQFT functor on is the following complex 3-dimensional space (see general formula (2.6) and the calculation in Section 5.2 of [28])
[TABLE]
where we specify the elements (i.e. the homotopy class of the map ) by their Poincaré duals in . Since is an invertible TQFT, each component in the sum is a one-dimensional vector space. However, in order to obtain numerical invariants of knots one still has to fix a basis, that is provide unit maps
[TABLE]
for each component in (5.5). This can be realized by using the value of the TQFT on the complement of an unknot with trivial framing (that is a solid torus ):
[TABLE]
Note that the value in vanishes because the map cannot be extended to interior of the solid torus and the corresponding term in (2.13) is absent. The choice of the basis in can be done by swapping meridian and the longitude.
On closed 3-manifolds the value of the invertible -equivariant TQFT is given by the right hand side of (5.4). It can be extended to a general element of the bordism in as follows252525The identification of the map between one-dimensional complex spaces with itself requires a choice of basis for each , that is a linear map . Due to the monoidal property and existence of a canonical bordism between disjoint union and connected sum, it is sufficient to consider only for genus one Riemann surfaces. For the case when and is such that represents a zero class in the basis has been fixed above. All other null-bordant pairs can be related by the mapping class group action on . For pairs which represent non-trivial elements in one can first obtain a map by cutting a 3-torus in half. This fixes a basis in up to a sign. Since such only appear in pairs in the boundary of the bordism , different choices do not affect the choice of the isomorphism .:
[TABLE]
where now is a smooth surface (in general non-orientable and with boundary) inside representing an element in the relative homology Poincaré dual to . The condition is equivalent to the requirement that the boundary of is a collection of smooth curves on and representing Poincaré duals to . The spin structure on induces a spin structure on analogously to the closed case discussed in Section 3.3. In the formula above denotes the extension of the valued Arf-Brown-Kervaire invariant to pin- surface with boundary (see e.g. [29]). As in the case of closed discussed in Section 3.2, pin- structures are in one-to-one correspondence with quadratic enhancements of the intersection pairing. The ABK invariant is then defined by a formula similar to (3.9). The functoriality property of (5.8) then follows from the additivity of ABK invariant under gluing surfaces with pin- structure.
The value of the spin-TQFT on the complement of a general oriented knot with even framing inside is given by
[TABLE]
where is a possibly unorientable surface embedded in with and inducing the given framing on . We used the fact that, as in the case of unknot, there is a unique with a fixed . The pin- structure on is induced from the unique spin-structure on . The values of the corresponding quadratic enhancement geometrically can be realized as follows [29]. Take a smooth curve in that represents an element . Then is the number of half-twists mod 4 of the thin band embedded in , that is a tabular neighborhood of the curve.
The mod 8 valued invariant of a knot in (5.9) appeared in [29] and is simply related to more usual Arf invariant of (which is defined using an oriented Seifert surface ):
[TABLE]
where is the same knot but with zero framing. The relation can be shown by changing to an orientable surface by taking a band-connected sum with appropriate number of Möbius bands. A simple example of a knot with is given by the trefoil ( in the classification table).
5.2 , in dimensions
Let us take to be a generator of a subgroup in (see Section 4.1):
[TABLE]
In this case the interesting situation is when is a two component framed oriented link in such that each component has even framing. The case of larger number of components is similar. Denote by possibly unorientable surfaces such that and they induce framings on respectively. We also assume that , and intersect with transversely. Such surfaces exist if and only if the link is proper, that is [30]. The unique spin structure on induces a pin- structure on each which is represented by a quadratic function . The same reasoning as in the previous section can be used to show that:
[TABLE]
where we only listed the projection on the functor on the one-dimensional subspace of in the decomposition (2.6) that gives a non-trivial invariant of the link . Geometrically counts the number of half-twists of the thin band in containing the curve . Since and intersect transversely, . Such mod 4 valued invariant of proper links is known as unoriented Sato-Levine invariant [30]. The examples of 2-component proper links with non-trivial values of unoriented Sato-Levine invariant are shown in Figure 10.
5.3 , in dimensions
Take to be a generator of a subgroup in (see Section 4.1):
[TABLE]
Let be a two-component oriented surface-link in , so that is the image of a closed oriented Riemann surface under the embedding map . Choose zero framing on both components as described in the beginning of the section. Then there exist three-dimensional Seifert volumes such that and each with the induced spin structure is a spin-boundary. Moreover, let us assume that is semiboundary, that is, by definition, there exist Seifert volumes such that and [27]. One can choose to intersect transversally so that is an oriented Riemann surface. The normal bundle to in has a natural framing given by the two normal vectors pointing inward . Given this framing, the spin structure on induces a spin structure on . Then, by an argument similar to the one in the three-dimensional case,
[TABLE]
This -valued invariant of a semi-boundary link is equal to the Sato-Levine invariant of a two-component semi-boundary surface link in [27] (see also [31] for an alternative realization of the same invariant, very close to the one described here) which follows from the canonical isomorphism between and two-dimensional framed bordism group , which is in turn, by Pontryagin-Thom, isomorphic to the second stable homotopy group of spheres. An example of a surface link with non-trivial value of the invariant is shown in Figure 11.
5.4 , in dimensions
Take to be a generator of a subgroup in (see Section 4.6):
[TABLE]
Let be now a three-component oriented surface-link in , where each is the image of a closed oriented Riemann surface under embedding map . Choose again zero framing on each and assume that is semi-boundary, that is, for each component there exist a Seifert volume such that it induces the framing and . One can choose all intersect transversally so that is an oriented Riemann surface and is a smooth curve in it. As in the previous subsection, the spin-structure on induces a spin-structure on described by a certain quadratic form . Moreover, there is a natural framing on the 1-manifold embedded in given by the two three normal vectors pointing inward . Given this framing, the spin structure on induces a spin structure on . Then
[TABLE]
where is the invariant of spin 1-manifolds described in Section 3.1. Note that using the isomorphism , one can define the value as the class of in , similarly to the original Sato-Levine invariant. An example of a surface link with non-trivial value of the invariant is shown in Figure 12.
5.5 Categorification of invariants of links and 3-manifolds
Before we proceed, let us briefly review the general definition of Sato-Levine invariants [27] of higher dimensional links. Let be a pair of disjoint oriented dimension manifolds embedded in (one can replace it with by adding a point at infinity). Each is not necessarily connected. There always exist Seifert volumes such that . By definition the link is semiboundary if one can find such that . Then the -dimensional Sato-Levine invariant is defined as follows262626The more usual notation is , however we are already using this symbol for a different invariant.:
[TABLE]
where, as in Section 5.3, we use vectors tangential to to define a framing on the normal bundle to . Given this dimension submanifold in with a framing of the normal bundle, Pontryagin-Thom construction272727For completeness, let us remind it. Let be a codimension submanifold in with a framing on the normal bundle. The corresponding continuous map then can be explicitly constructed as follows. First let us identify the source with the ambient with added point at infinity, and the target as where is a unit ball. Pick a tabular neighborhood of . The framing on the normal bundle then provides an explicit isomorphism where is the -dimensional unit ball. The map is then given by taking all the points outside of the tabular neighborhood to (collapsed to a single point in ) and the points inside to be projected on . then provides an element in the -homotopy group of .
Similarly, one can define stable Sato-Levine invariant:
[TABLE]
where we replace the ambient with , and with and then take .
As was explained Section 5.3, the 4d spin-TQFT obtained by gauging -equivariant invertible spin-TQFT with action provides a realization of invariant of surface links. By a similar argument one can show that 3d spin-TQFT obtained by gauging invertible -equivariant spin-TQFT with the same and action (where is the invariant of ) provides a realization of invariant of surface links. In both cases one can see this from the fact that for . Namely, if the invariants are understood to be valued in (i.e. multiplicative realization of ):
[TABLE]
[TABLE]
where we used the notations for Seifert surfaces/volumes as before.
Now we would like to claim that, in a certain sense, categorifies . In order to make this statement meaningful one has to extend from invariant of (semiboundary) surface links in to the functor from the category of (semiboundary) link-bordisms to the category of complex -vector spaces :
[TABLE]
The objects in the category are semiboundary links in :
[TABLE]
The morphisms are pairs of 2-manifolds embedded in with boundaries coinciding with the links sitting in and satisfying semiboundary property (see Fig. 13):
[TABLE]
The value of the functor (5.21) on objects is given by
[TABLE]
where denotes a one-dimensional complex vector space with grading , and is the non-trivial (fully extendable) invertible 2d spin-TQFT such that its value on a closed spin-surface is . Such TQFT was considered in detail in [32] (see also [33]). The value of on morphisms (using the conventions in (5.23)) is
[TABLE]
where is a surface in with induced spin-structure and boundary components lying in and . From the definition of semi-boundary link bordisms (5.23) the r.h.s. of (5.25) indeed provides a -linear map between the -vector spaces associated to the objects via (5.24) which is functorial.
Then the functor categorifies link invariant in the sense that
[TABLE]
where, as before, the upper indices denote the components of the vector space with particular grading. This categorification of link invariants can be understood as a toy version of Khovanov homology [34], which categorifies Jones polynomial invariant of links (valued in ), and is also functorial with respect to link bordisms. Moreover, invariant has a close relation to the Jones polynomial invariant evaluated at (see e.g. [29])).
As was described above, can be realized as with being particular elements of . Namely
[TABLE]
It is natural to ask if also categorifies . However, it is easy to see that categorification in the most naive sense, is not possible, because generically the value of on a closed spin 3-manifold is rational (e.g. ), which cannot be interpreted as a (signed) sum of dimensions. That is, it is not possible to have a simple relation of the form for any 4d spin-TQFT . But, as was shown in [28], the 3d and 4d TQFTs and are still closely related. Namely:
[TABLE]
where is the direct sum operation282828The crucial property of the direct sum operation on n-dimensional TQFTs is
(5.29)
The direct operation is then naturally extended to the TQFT values on disjoint -dimensional manifolds and bordisms between them so that functoriality and symmetric monoidal property hold. on TQFT functors and the summands are fermionic topological gauge theories corresponding to different elements of . In particular,
[TABLE]
6 Other bordism groups and computations:
.
Let where the quotient is with respect to the diagonal center subgroup. Similarly to spin manifolds, one can consider manifolds with tangential structure. That is, manifolds equipped with principle bundle which is a lift of the orthonormal tangent frame bundle with respect to the extension .
The Pontryagin-Thom isomorphism provides a relation between the bordism groups of manifolds with (stable292929In the cases when and the stable and unstable structures are equivalent, see [2].) tangential structure and homotopy groups of the Madsen-Tillmann spectrum associated to tangential structure :
[TABLE]
On the other hand, in the work of Freed-Hopkins [6], there is a 1:1 correspondence 303030Note that the free part of the classification contains Chern-Simons-like theories, which are not strictly topological.
[TABLE]
The abelian group is denoted by in [6]. In particular, stands for the torsion part of homotopy classes of maps from spectrum to the -th suspension of spectrum . The Anderson dual is a spectrum that is the fiber of where is the Brown-Comenetz dual spectrum defined by
[TABLE]
[TABLE]
There is an exact sequence
[TABLE]
The torsion part is . This provides relation to the bordism groups in (6.1).
Now let . Write where and is odd, let . Then and , . The 2-torsion part of is which is computed in [35] for , and , . The group has been computed in [36], but here we provide an alternative computation.
For , ,
[TABLE]
For , ,
[TABLE]
For ,
[TABLE]
where the case is new.
Write where are odd primes and . The -torsion part of is .
[TABLE]
where is the mod Steenrod algebra.
[TABLE]
where , is the Bockstein homomorphism associated to , , . The dual of is
[TABLE]
where and , being standard generators of the mod Steenrod algebra. Let , then
[TABLE]
where for . The cohomology is then
[TABLE]
where and is the two-sided ideal of generated by .
If ,
[TABLE]
[TABLE]
is an -resolution of .
Therefore we arrive at the following
Theorem 12**.**
[TABLE]
The topological term of is where is the Postnikov square operation .
If ,
[TABLE]
[TABLE]
is an -resolution of .
Therefore we have
Theorem 13**.**
[TABLE]
for .
The topological terms of are and where is the signature.
7 Time-reversal, pin*±*-TQFTs and non-orientable manifolds
As was already mentioned in Section 2, the notion of spin-TQFT can be generalized to non-orientable manifolds. The analogue of spin structure is pin± structure, that is the lift of principle bundle of orthonormal tangent frames to principle bundle with respect to the extensions (2.3). Physically the subgroup of plays role of the time reversal symmetry and often denoted as . It is extended by fermionic parity to the subgroups and of and respectively, so that the diagram (2.3) commutes. A pin± manifold is a manifold with a chosen pin± structure.
The definition of a pin±-TQFT is the same as the one for a spin-TQFT but with the spin bordism category replaced by the pin± bordism category, that is the category with objects being closed pin± manifolds and morphisms being bordisms between them equipped with pin± structure that is reduced to the pin± structure of manifolds at the boundary. The corresponding bordism group is also defined similarly to (2.17):
[TABLE]
The non-orientable version of (2.16) then reads
[TABLE]
Compared to (2.16) picking the torsion subgroup is not required because the bordism groups are all torsion. Physically, such invertible TQFTs provide description of fSPT with time reversal (TR) symmetry (TR-fSPT). Note that, after Wick rotation from Euclidean to Lorentzian signature, and correspond respectively to and relations between the generators of time-reversal symmetry and fermionic parity.
The gauging operation
[TABLE]
is realized completely analogous the gauging in spin case, described in detail in Section (2.1).
Following the general Pontryagin-Thom isomorphism (cf. (6.1)), the pin*±* bordism groups of a topological space can be related to stable homotopy groups:
[TABLE]
where denotes the Madsen-Tillmann spectrum corresponding to (stable) tangential -structure313131Unlike in the case it is not the same as Thom spectrum . However, ., and, as before, is with a disjoint marked point added. The reduced bordism groups of are given by
[TABLE]
In particular, we have
[TABLE]
In what follows we provide the results of calculations of pin± bordism groups of , for a few simple finite abelian groups .
Since pin± bordism group are 2-torsion, we only need to consider the Adams spectral sequence at prime :
[TABLE]
In particular, if with odd,
[TABLE]
7.1 Pin*+* bordism groups
For calculation we use the fact that [6]
[TABLE]
Together with (4.14) it implies that the second page of the Adams spectral sequence (7.7) for can be identified with
[TABLE]
7.1.1
[TABLE]
The -module structure of is shown in Figure 16.
The -page of Adams spectral sequence is shown in Figure 17.
There are no further differentials due to degree reasons. Hence we have the following theorem:
Theorem 14**.**
[TABLE]
The corresponding bordism invariants of in dimensions and :
[TABLE]
where we used the fact that Poincaré dual to can be represented by an orientable codimension 1 submanifold with spin structure induced from the pin- structure on [2]. The invariant is valued eta-invariant of Dirac operator.
7.1.2
[TABLE]
The -module structure of is shown in Figure 18.
The -page of Adams spectral sequence is shown in Figure 19 .
Hence we have the following theorem:
Theorem 15**.**
[TABLE]
The corresponding bordism invariants of in dimensions and :
[TABLE]
7.2 Pin*-* bordism groups
For calculation we use the fact that [6]
[TABLE]
Together with (4.14) it implies that the second page of the Adams spectral sequence (7.7) for can be identified with
[TABLE]
7.2.1
[TABLE]
The -module structure of is shown in Figure 20.
The -page of Adams spectral sequence is shown in Figure 21.
There is no further differential due to degree reasons. Hence we have the following theorem:
Theorem 16**.**
[TABLE]
The corresponding bordism invariants of in dimensions and :
[TABLE]
7.2.2
[TABLE]
The -module structure of is shown in Figure 22.
The -page of Adams spectral sequence is shown in Figure 23.
Hence we have the following theorem:
Theorem 17**.**
[TABLE]
The corresponding bordism invariants of in dimension :
[TABLE]
Notice that is an isomorphism. The map is a spin 4-manifold with and is sent to an submanifold dual to with .
8 Symmetric anomalous Spin-TQFTs as the boundary state of fermionic SPTs
In this section we construct gapped boundary theories coupled with some of the fSPTs protected by finite group symmetry we have constructed. The construction works for the two cases. The one is where the symmetry group , or its subgroup large enough to trivialize the bordism invariant defining the fSPT, is spontaneously broken at the boundary. Another, more nontrivial, case is when, in 4-dimensions, the symmetry group has the form of and the bordism invariant can be formally written as “”, where is the modulo 2 reduction of the backgrounds , and is a bordism invariant in . In Section 2, we have found three of such invariants. In the latter case no subgroup of is broken on the boundary. The summary of this section can be found in Subsection 8.5.
What we would like to do is in precise the following. A dimensional fSPT with symmetry is an invertible -equivariant TQFT, which is in particular a functor , as stated in section 2. We want to find a topological boundary condition for some of fSPTs we have found. That is, we want to find an enhancement of a given TQFT functor to a functor , where is the bordism category whose objects can have boundaries and morphisms can have corners.323232Here we are not trying to define a full-fledged extended TQFT, which also encodes the set of possible boundary conditions, but we only describe a single boundary condition for a given fSPT. Further, since the fSPT is trivial when its background is ignored (i.e. maps to are trivial), physically we expect that we can obtain the boundary TQFT from the bulk-boundary TQFT . The non-equivariant version of goes through the category of spin null-bordant manifolds,333333That is, and are the same if is spin-diffeomorphic to . This is because we can drill out where is the tubular neighborhood of the boundary, and then fill there with the -dimensional ball without changing the value of , since the bulk theory is trivial. and we demand that this functor enhances to the boundary TQFT . The boundary TQFT cannot be trivial as long as the bulk fSPT is non-trivial.
Physically, it is expected that for any for a finite group there exists a boundary TQFT on which the symmetry is spontaneously broken. In such cases the vector space is dimensional. While for d boundary there is no other option, for higher dimensional boundary we would like to find a more non-trivial boundary TQFT where is one dimensional, which will turns out to be possibly in many cases.
8.1 General strategy
In [11] and more generally in [37], it is stated that a -protected fSPT can be described by attaching intrinsic (i.e. non-equivariant) invertible TQFTs on the -symmetry defects. A good example of the statement appeared in [11] is the 3d fSPT defined by where is the background, since
[TABLE]
Physically, is regarded as the subspace that the symmetry defect occupies, and therefore we regard (8.1) as putting the invertible TQFT defined by the invariant on the symmetry defect.343434This might seem contradicting since, while the invariant is valued, the defect should vanish when two of them are stacked. This is actually not the case because when is oriented has order 2, and when is unorientable it has non-vanishing self-intersection and therefore two of symmetry defect occupying the same unorientable homology class cannot be stacked in a parallel way. The pin- structure on is induced by the spin structure of the total space. See [2]. (It is also briefly explained in [28]). Another, more intricate, example is the 4d fSPT defined by the invariant
[TABLE]
where and are the backgrounds. We regard this invariant as decorating the intersection of defects with the invertible TQFT.
Now we want to put a fSPT on a manifold with boundary. We want to preserve the symmetry on the boundary (though it is spontaneously broken on boundary for a 3d fSPT case). In other words, we take the symmetry background in the cohomology group (and not in the relative cohomology cohomology ), so that its dual is in the relative homology group . A representative of can have its boundary in , which is dimensional. If a symmetry defect (or intersection of them) supports an intrinsic invertible TQFT, the value of the invertible TQFT on the defect (e.g. ) is not well-defined when the defect have a boundary. A naive way to fix this problem is to extend the symmetry defect along to close the boundary of the defect. This operation would define an element of out of . However, the way to close the boundary of the defect is not unique and the ambiguity is captured by , because of the exact sequence
[TABLE]
Therefore, a way to define the bulk-boundary TQFT is to take the sum over (If is not in the image of , simply we set the partition function to be zero). In this way, we arrive at the partition function, for instance, for the fSPT on a spin manifold with boundaries:
[TABLE]
where is the boundary map, and is the class in that is the image of the fundamental class of under the boundary map . The pin- structure on is induced by the spin structure of .
For the invariant (8.2), we would like to propose the partition function on a spin manifold with boundary in a similar fashion to (8.4). The construction is, however, more involved and thus we postpone the discussion till Subsection 8.3.
In the rest of the section, we construct the (1-)functor for the bulk-boundary TQFT for the 3d fSPT , and then also discuss bulk-boundary systems for some of 4d bulk fSPTs.
8.2 Bulk-boundary TQFT for 2+1d fSPT
Let us construct the functor for the bulk fSPT . We can promote the partition function (8.4) into the functor in the following way. The value of on an object , with a (homotopy class of) map , is
[TABLE]
where is an indeterminate vector with degree zero corresponding to an element and is the functor representing the pin- fSPT defined by the invariant. Such (fully extended) 2d TQFT was considered in detail in [32]. The functor evaluated on a morphism is similarly constructed as:
[TABLE]
where is the linear map sending .
By setting , we obtain the boundary TQFT . For example, the functor evaluated on the object gives
[TABLE]
In particular, and .353535The one-dimensional -graded vector space has the odd degree, which can be understood from the partition function of on a torus. The ground state degeneracy of is interpreted as the spontaneous symmetry breaking of the symmetry on the boundary.
The construction here can be easily generalize to arbitrary -fSPT with boundary condition with spontaneously broken symmetry (or its subgroup large enough to trivialize the fSPT anomaly). Next, we would like to construct a more nontrivial boundary condition for a fSPT, that is the boundary condition with which symmetry is not spontaneously broken on boundary.
8.3 Bulk-boundary TQFT for 3+1d fSPT
Partition function
We would like to generalize the above construction to the fSPT defined by the invariant where are the two backgrounds and (the modulo 2 reductions of them). On boundary, we do not want neither symmetry to be broken. In particular, the partition function of the boundary theory on should be 1.
As said in Subsection 8.1, the invariant is interpreted as decorating the intersection of the symmetry defects and . The intersection of the defects intersects with the boundary of the space time at a link . The orientation of induces the fundamental class .
Imitating the previous section, naively one might think that we can define the boundary theory by summing over the preimage of the boundary map . However, for a general element , the pin- structures induced on and are not compatible along . Hence we cannot define the invariant with a general . To avoid this problem, we propose to sum over the surfaces bounded by with a pin- compatible with . This can be done as follows.
The spin structure on is induced form that of using the framing of the normal bundle with framing vectors tangential to and . This also induces the framing of in in the same way. Let be the tubular neighborhood of in . By pushing each component of along one of the framing vectors, we can define a map . We denote the image of under this map by . The homology class does not depends on the choice of the framing vector at each component. We have the long exact sequence
[TABLE]
where all the coefficients of the cohomologies are . For each element , we define the element in by extending to along the framing vector363636Note that because , in principle a smooth representative of ends on a cycle in which represents an element in integral homology that can be different from the one given by the map via pushoff towards a framing vector. However, by gluing the appropriate number of Möbius strips to the boundary components of one can always fix this mismatch. This can be seen from the fact that framings at each component of are (non-canonically) in one-to-one correspondence with integers and gluing a single Möbius strip changes the integer by 2 and that the values of framings mod 2 is fixed by . See [29]. we used to define . By construction, the framing, and hence the spin structure, induced on from and from are the same. Therefore, we can uniquely define the pin- structure on .
Now, we propose that the partition function on a 4-manifold with boundary is
[TABLE]
The overall normalization factor is coming from the gauge redundancy on the boundary theory. When , the partition function is just as desired. If (where is defined in (8.8)), the partition function is zero. Otherwise, the preimage is a -torsor, and thus we can non-canonically map the sum over to the sum over . In particular, when the backgrounds are off, i.e. , we have , and the partition function is
[TABLE]
which is the partition function of the gauge theory with action with on . The boundary theory can be replaced by gauge theory with action with .
The interpretation of the system is that the lines behave the ’t Hooft lines in the boundary theory, and is the framing of the operator. Although there is no naive 1-form symmetry coupled to the operator since the operator requires a framing, the intersection of two 0-from symmetry defects naturally caries a framing as explained, and therefore can couple with the ’t Hooft line of the boundary gauge theory, with anomaly .
Functor
Next, we describe the functor for the bulk fSPT that is compatible with the partition function (8.9). The value of on an object , with maps and , is
[TABLE]
where are points with tubular neighborhood , is the element in induced from , is the boundary map in the exact sequence analogous to (8.8). is the element obtained by extending to .
evaluated on a morphism is:
[TABLE]
Here is the normalization factor coming from the gauge redundancy on boundary, which is essentially the same as the factors appeared in (2.13). Since the restriction of on coincides with appeared in (8.11), we have consistent restriction map from to for each and .
One can directly observe that, as a functor, the boundary TQFT is the gauge theory with the action , namely:
[TABLE]
as expected above.
8.4 Bulk-boundary TQFT for 3+1d fSPT
In the previous two examples, the boundary TQFT depends on the spin-structure on its argument, when all the backgrounds are set to be trivial. This is not always the case even if the fSPT in the bulk depends on the spin-structure. However, in such a case symmetry action on the boundary TQFT depends on the spin-structure on the boundary.
Consider the 3+1d fSPT , where are the backgrounds and is the background. As before, we can construct a bulk-boundary TQFT as
[TABLE]
and evaluated on a morphism is:
[TABLE]
where is the fSPT for the invariant , and . Here is understood to be restricted on the argument of . Other notations are the same as those in the previous subsection. The normalization factor is also the same as that in the previous example.
When all the backgrounds and are turned off, the boundary TQFT is identified with the gauge theory with the trivial action. Further, the boundary theory can be promoted into a equivariant TQFT with background . The value on an object is
[TABLE]
This -graded vector space homogeneously have even degree (i.e. there is no fermionic states), but is non-trivially acted on by the global symmetry, due to the second factor. For example, on the torus with the even spin structure (meaning that a fermion is anti-periodic along at least one of the directions of ), the vector space has 3 states neutral under the global symmetry (not to be confused with the of the grading, which is related to the fermion parity symmetry) and 1 charged state, and on the torus with the odd spin-structure (meaning that a fermion is periodic along both directions of ), the vector space has 1 neutral state and 3 charged states.
8.5 General structure and comments
Here we summarize the results we have found on 3+1d fSPTs so far in this section, and add a several discussions. For a -protected spin-SPT listed in Table 2, we can construct bulk-boundary TQFT as follows. Here, the first column was discussed in Subsection 8.3, the second was in Subsection 8.4, and the third is a generalization of them. In all the cases, the symmetry group has the form of , and the bordism invariant defining the fSPTs formally has the form of ”” with an invariant . For an object with background , the value of is
[TABLE]
The notations here was introduced in Subsection 8.3. Similarly, the value of on a morphism is
[TABLE]
As before, the value of the normalization constant is , where is the set of the connected components of that does not intersect with neither or .
In all the cases listed in Table 2, the boundary TQFT is a gauge theory. Its action is , where is specified in Table 2. When is odd the ground state degeneracy on a torus is 3 (all bosonic for even spin-structure and all fermionic for odd spin-structure), and otherwise it is (all bosonic).
Let us make a comment about a physics interpretation of the construction. As stated in Subsection 8.3, the intersection of the symmetry defects acts like a one-form symmetry defect on the boundary gauge theory. In fact, the gauge theory with the trivial action have electric and magnetic one-form symmetries, with a mixed anomaly between them, where are the backgrounds for the one-form symmetries.373737This means that the can be a boundary theory of the invertible TQFT defined on orientable manifolds with structure maps that have the partition function . Roughly speaking, the equations (8.17) and (8.18) can be regarded as the theory obtained by “substituting ” into the magnetic one-form symmetry background and substituting into the electric one-form symmetry background realizes the anomaly “”, which is the rough structure of the precise invariants in Table 2. In this sense, the construction of the boundary TQFT for the fSPT we presented is an analog of what is done in [10] for bosonic SPTs.
However, it is not very precise to say we substituted into the magnetic one-form symmetry background of , since the ’t Hooft loop of the theory requires a framing, which cannot be specified just by a 1-cycle or 2-cocycle, which is supposed to be the background field for a one-form symmetry. Therefore, in some see, the theory have ”framed one-form symmetry”, meaning that the background is a 1-cycle equipped with framing, and the construction (8.17) and (8.18) amounts to substituting into the background of ”framed one-form symmetry”. It would be interesting to find a precise physics meaning of such a concept.
In this paper, the authors could not find a construction of a bulk-boundary TQFT for the bulk 4d fSPT . This invariant involves, unlike other invariants we have discussed in this section, the background itself, which is not modulo 2 reduction. This indicates we need a gauge theory on boundary. Indeed, in [38], a boundary gauge theory is proposed. It would be interesting to try to reconstruct the system in the language we have used in this section. In addition, the boundary TQFTs for pin+ bordism invariants and for the global symmetries and are also remained to be constructed. Although the former only involves the symmetry defects, the method developed in this section is not applicable since there is no known way to give a pin- structure inside a codimension-1 submanifold in the 3-dimensional pin+ boundary manifold.
In this paper we have only considered fSPTs with finite group . On the other hand, in [39] it is proposed that a particular 9d fSPT should have boundary TQFT. It would be intriguing if the method in this section can be generalized into the continuous group case.
9 Crystalline fSPTs (fermionic Symmetry Protected Topological states) and bordism groups :
Dimensional extension
In the rest of the paper, for the sake of brevity, we denote the group that classifies fSPTs with symmetry as
[TABLE]
where is a tangential structure (e.g. ), and refer to as the cobordism group. Now we would like to re-organize spin-TQFT data obtained previously in Section 4 and explore their relation to a different group, . As explained in Figure 24, there is a physical application of a part of , namely the layer-stacking of lower ()d -fSPTs to ()d crystalline -SPTs along an extra dimension, where is interpreted as an internal onsite symmetry of ()d fSPTs (associated to , where the on-site internal symmetry represents the symmetry group considered previously in Section 4). Mathematically, we use the fact that a -th bordism group associated to d invertible spin-TQFTs with -symmetry
[TABLE]
contains explicitly the subgroup associated to d invertible spin-TQFTs with -symmetry. The (canonical) isomorphism (9.2) immediately follows from 383838Note that the classifying space . The formula (9.4) can be understood as a generalization of the particular case of the Künneth formula for integral homology: to the spin-bordism generalized homology. The isomorphism (9.4) can be derived, for example, from the Adams spectral sequence for the bordism groups considered earlier in the paper. It follows from the fact that
(9.3)
and that the are no non-trivial differentials in the Adams spectral sequence coming from and terms.
[TABLE]
that can be geometrically realized as follows:
[TABLE]
where is a smooth oriented codimension 1 submanifold (which always exist) of representing a Poincaré dual to . The spin structure on is induced from the spin structure on as previously described.
Thus, when we interpret as a translation symmetry along an extra dimension, all the spin-TQFTs associated to survive and can be stacked into crystalline fSPTs in one higher dimension. If the crystalline symmetry is more general as instead of simply as , and if commutes with the internal symmetry , then we can evaluate instead . More generally, one can consider and to be non-commutative, e.g. , but we will not need this in this article, since already provides a non-trivial relations between fSPTs in d and d, see Section 9.2 for the details. In the next section, we will focus on and relate topological terms between d and ()d associated to .
9.1 Computations of
for a finite abelian
Using the analogue of Künneth formula (9.2), we can obtain the cobordism groups , and associate this data to crystalline fermionic SPTs classes. In the data below, the crystalline fermionic SPTs classes in d are directly given by again thanks to (9.2).
We remark that the 2d Arf invariant spin-TQFT (1+1D Kitaev chain) is not an SRE fSPTs but is an LRE invertible fermionic topological order (see Section 2.3, eqs.(2.16) and (2.25)). However, stacking LRE 1+1D Kitaev chains into a 2+1D system protected by -translational symmetry, it becomes an SRE 2+1D crystalline fSPTs.393939Thanks to the local unitary transformation, this 2+1D crystalline fSPTs can be deformed to a trivial tensor product state once we break the -translational symmetry. Therefore, all 2d invertible spin-TQFTs with symmetry can contribute to 3d crystalline -fSPTs.
We only list crystalline fSPTs classes for dimensions and , since we have more clear definitions of fSPTs with internal onsite symmetry in 1+1D (2d) or above, which then can be stacked along one extra dimension to crystalline fSPTs in 2+1D (3d) and 3+1D (4d). See below.
Theorem 18**.**
[TABLE]
Theorem 19**.**
[TABLE]
Theorem 20**.**
[TABLE]
Theorem 21**.**
[TABLE]
Theorem 22**.**
[TABLE]
9.2 Relations between fSPTs and crystalline-fSPTs:
and
We now relate the crystalline--fSPTs in d associated to (within the bordism group thanks to (9.2)) to the fSPTs protected by an internal onsite symmetry , obtained from . Namely, we study the map
[TABLE]
dual to the map between the corresponding bordism groups:
[TABLE]
We also use the map
[TABLE]
obtained by composing the embedding via (9.2) with (9.4).
In Table 3, we consider the map (9.11) in the case of spacetime dimension and for or , while or . In Table 4, we consider the map of (9.11) for or , while or . We denote the topological terms/bordism invariants in terms of the notations introduced in Section 4, and also the more informal notations used in our previous work [28]. Furthermore, we denote , , and are the maps from the manifold to the classifying space defining background gauge fields for the corresponding groups.
Fermionic topological terms. Stacking vs. Non-stacking: In Table 3 and 4, we list down the classifications associated to or . The boxed classification groups are such that their generators are intrinsically fermionic (i.e. they are not generated by bosonic Dijkgraaf-Witten topological term). There we only list down their intrinsically fSPT invariants. Since , those fermionic topological terms that occur in the former (the right hand side of (9.11)) but not in the latter (the left hand side of (9.13)’s) cannot be obtained from a 3d-layer-stacking construction as a crystalline 4d-SPT — We brace-labeled those topological terms with “non-stacking” label in Table 4. The non-stacking terms lack the dependence on the (equivalently ) gauge field.
In Tables 3 and 4, “-spin-CS” means an invertible (that is, depending on background gauge fields) spin-CS theory with minimal non-zero level (i.e. level 1, in a proper normalization) with broken to subgroup, see Sec. 10.1.1 for detailed discussions.
Interestingly, as shown in Table 3 and 4, it is not possible to always map injectively from , see the arrows labeled with “No” or “Partially.”404040 For example, it is not possible to obtain the classes generated by in from the classes generated by in , via obtaining a new the gauge field from the gauge field by mod 2 reduction. We can prove it is impossible by contradiction. Suppose it is possible, then
(9.14)
for any , and any spin . But then one can take , to be the generator of , and to be in . Then the right hand side of the above formula (9.14) becomes (where is Bockstein morphism) and in general is not zero, while the left hand side is identically zero in this setup (because self-intersection of PD() inside PD() is trivial), so we have a contradiction. The formula (9.14) is false. However, it is always allowed to map injectively from the . One formal way to understand this fact is because Poincaré duals (which physically correspond to domain walls) for elements of are in general non-orientable manifolds, while for both and are oriented (-orientation is equivalent to -orientation). This is why one can reproduce fSPTs with , but not , symmetry from the fSPTs with symmetry.
In Table 3 and 4, we put labels [Ab] (abelian) or [NAb] (non-abelian) on the right-hand-side of some invertible spin TQFTs with -symmetry (or -fSPTs). What we mean there is that the corresponding spin-TQFTs with dynamically gauged , are [Ab] (abelian) or [NAb] (non-abelian). The criteria for determining, whether spin TQFTs with dynamically gauged symmetry in 3d, 4d or above is [Ab] or [NAb], have been given in Ref. [28]’s Sec. 1.2. We briefly remind the readers our definition in Section 9.3.
9.3 Abelian vs. Non-Abelian gauged spin-TQFTs: Criteria
In this work, the criteria we define for determining if the topological gauge theories, of finite gauge group , are non-abelian [NAb] instead of abelian [Ab] for spacetime dimensions , are the following:
Criteria: If (and only if, in our work), the partition function of the gauged spin-TQFT defined in eqns. (2.5) computed on a -torus () is reduced to a smaller value from a particular power of the order finite gauge group, namely , then the gauged spin-TQFT is non-abelian [NAb].
Physically, the partition function represents the dimensions of Hilbert space, or equivalently the ground state degeneracy (GSD) on a -torus (see various examples computed in [28]). If is reduced, this implies that the certain extended surface operator (physically, the open ends of a surface have anyonic extended/string excitations attached) have the quantum dimension (associated to the local or non-local Hilbert space of such anyonic extended/string excitations ) larger than 1, namely . This non-abelian property can be seen, for example, from the quantum representation of the mapping class group MCG()=SL(), in which case the modular -matrix (a generator of SL()) will have at least one entry that satisfies .414141See for example, Ref. [13] on the computation of modular -matrix (a generator of modular SL() data) that shows this non-abelian property with for certain non-abelian TQFTs and a certain anyonic string excitation from a surface operator labeled by . More examples of non-abelian TQFTs are given in Ref. [28] and in Table 3 and 4. The non-abelian property also implies that a certain 3-loop braiding process will have non-abelian unitary matrix acting on the eigenstate-vector of the Hilbert space after the completion of adiabatic evolution of braiding process.
From Tables 3 and 4, we learn, example by example, that the fact that the dynamically gauged spin-TQFTs defined by become non-abelian spin-TQFTs, when the original un-gauged invertible -spin-TQFT theories (with abelian ) involve either of the following:
an odd multiple of Arf invariant (generating ), 2. 2.
an odd multiple of ABK (generating ), 3. 3.
an odd multiple of (generating class) 4. 4.
an odd multiple of (generating class, see Ref. [28]’s Sec. 5.)
In physics, this means that the above non-abelian TQFTs must induce lower dimensional 2d spin-TQFT (such as based on dimensional reduction [28] or compactification) that involve the Kitaev’s Majorana fermionic chain [16], which corresponds to 2d Arf or ABK invariants. We give more accounts on this phenomena in the next Section 10.
To summarize Section 9, the physical meaning of global symmetry is a discrete translation symmetry in one extra spatial dimension. Applications to realistic systems of -crystalline fSPTs make sense when we consider the spin bordism group of up to dimension . In Section 10, we would like to explain all of the above in a more down to earth way and in a setting suitable for condensed matter community.
10 Interpretation of the results
in quantum matter and more
Now we reorganize our results into an alternative understanding: in terms of the setting and the language of current developments of condensed matter and topological quantum matter, and also in comparison to the tools developed by physics setting (versus the mathematical settings).
We will relate various fSPTs protected by internal onsite symmetry, to crystalline-fSPTs protected by the translational symmetry of infinite integer symmetry (i.e. space group symmetry). We had presented the summary of new cobordism calculations involving the and the classifications of crystalline-fSPTs, altogether in Sec. 9.1. Note that we explain the first example in more details in Sec. 10.1, while we go through later examples rather quickly based on the similar strategy.
Then we will discuss more general crystalline-fSPTs (c-fSPT). We first clarify the differences and the meanings of the crystalline symmetry and the internal symmetry in Sec. 10.6. Then we include various examples of crystalline symmetries and their complete classifications via cobordism approach:
- •
Reflection/Mirror c-fSPT in Sec. 10.7.
- •
Inversion c-fSPT in Sec. 10.8.
- •
Rotation c-fSPT in Sec. 10.9.
Then we discuss the partially gauging of fSPTs to obtain fermionic SETs (Symmetry Enriched Topologically ordered states) in Sec. 10.10. These fermionic SETs are fermionic gauge theories but enriched by the additional global symmetry protection.
10.1 4d and crystalline--fSPTs
Here we particularly aim to understand the 4d cobordism/SPT invariant given in eqn. (4.19) and in Table 3, where we have the spin -manifolds with maps , (equivalently, and ). Following the informal physical notation in [28], this corresponds to the partition function424242As discussed in Section 4, in such expression we assume that there exists a representation of Poincaré dual of by an immersed manifold .
[TABLE]
where and we focus on the minimal generator with below.
10.1.1 Construction from 2d to 3d to 4d
How do we construct this 4d -fSPTs starting from lower dimensions, say from 2d and 3d?
Decorated domain wall construction: We can start from the 2d -fSPTs (with symmetry) classified by , whose generator is the ABK invariant (see a recent exploration [40] and reference therein) — Physically this is the so-called 2d Kitaev’s fermionic chain [16, 41] with each isolated 0+1D edge on an open chain has a Majorana zero mode. Now we can place the ABK invariants (i.e. Kitaev chains) on all the 2d -symmetry-breaking domain walls of 3 -fSPTs (with symmetry) classified by Pontryagin dual group to . Then we can restore the full symmetry by, in condensed matter language, proliferating or condensing the 2d domain wall. This is known as the decorated domain walls proliferation construction of SPTs, proposed by [42].434343See a field theory derivation of decorated domain walls proliferation construction of SPTs and topological terms related to bosonic TQFTs and Dijkgraaf-Witten gauge theory in Ref. [43]. Similar construction for fermionic SPTs is discussed in Ref. [37] Although the 3 -fSPTs is placed on an oriented spin 3-manifold, the 2d domain wall can have an induced pin- structure thanks to Smith isomorphism [2, 11]: . This construction matches the 3d partition function with [25].
Continuing from 3d -fSPTs, we can further construct both the 4d crystalline--fSPTs and 4d -fSPTs by stacking 3d systems layer by layer. In that case, we need to have as 2-copies of ABK or Kitaev chain proliferated-domain walls in order to construct of 3d -fSPTs. Physically, a single 3d -fSPTs has two copies of and superconductors,444444 Hereby the chiral and anti-chiral--wave superconductors, or the superconductors, we mean the Cooper pairing of two fermions are in the -orbital of -orbital pairing states ( in terms of the angular momentum in the spherical harmonics, or the in the , etc. of atomic orbitals). The pairing function results in the superconductor order parameter , where are spatial momentum in the -directions.
so called the chiral and anti-chiral--wave superconductors. The 2d boundary of each chiral--wave TSC (living on a 3-manifold) hosts a conformal field theory (CFT) described by a single free left moving the real Majorana-Weyl fermion with chiral central charge . So for a single 3d -fSPTs boundary, there are left-chiral central charge and right-chiral central charge , combining into a 1+1D Dirac spinor fermion (with a total ). The corresponding mathematical 3d bulk spin-TQFTs (as dynamically gauged fSPTs) and 2d boundary theories are given in Section 8 and Table 2 of [25]. Although a total central charge of 2d edge states is non-chiral, but the global -symmetry assignment is actually chiral, anomalous and non-onsite on the 2d edge modes.
Layer-stacking: Now we can, along the spatial -direction, stack integral layers of of 3d -fSPTs placed at the spatial --direction, following the idea of [38]. In this way, we can construct a 4d crystalline--fSPTs placed in the spatial -- space, protected by the internal -symmetry and the spatial lattice translational symmetry along the -direction. How do we classify the interacting 4d crystalline--fSPTs for this particular construction? Mathematically this question can be answered by computing , which we show in Theorem 18. Physically this question can be answered by asking how many (say, a number of ) layers of this of 3d -fSPTs do we need in order to fully gap out the boundary gapless modes without breaking any global symmetries? Namely, how many layers we need to add with non-perturbative interactions among these copies of of 2d boundary CFTs, in order to fully generate the energy gap for a symmetric topological gapped boundary? Our answer is . Because the , since 8 layers of 3d -fSPTs indeed can be fully gapped, thanks to their classification (physically, derived from gapping the boundary Majorana modes). By adding the interactions among the neighbor 4 layers () along -direction, we actually break the -translation down to -translation symmetry, which again is still a lattice translation symmetry (redefined the as a new integer by rescaling the translational lattice constant by 4 times). So physically we predict that there must be at least -class for 4d crystalline--fSPTs thus also for . Indeed the answer agrees between our math result (Theorem 18’s 4d classification454545Which is generated by with the gauge field and the gauge field of along the -direction. contains the normal subgroup) and physics arguments.
Moreover, we can view this 4d crystalline--fSPTs for the -class as another 4d fSPTs protected only by internal symmetries, without the need of translational symmetry.464646This means we can also re-interpret the role of crystalline- lattice translation as a new internal symmetry instead. Which additional internal symmetry is required? The answer is constrained by what the internal symmetry can still be preserved when we add non-perturbative interactions among the -directional neighbors of 4d crystalline--fSPTs (4 layers of of 3d -fSPTs). For 8 Majorana zero modes in 1d, or 8 left-moving + 8 right-moving (8L+8R) chiral Majorana-Weyl fermions in 2d, Ref. [41] shows that, under quartic-interaction fermion gapping terms, there is an internal symmetry of rotating between Majorana modes that can still be preserved (also there is the obvious fermion parity -symmetry preserved). Apparently, the rotates between 8L+8R Majorana-Weyl fermions in 2d, while we wish to keep the subgroup of acting only among each 2L+2R Majorana-Weyl fermions, as a proper internal symmetry. Thus, we can take the intersection between the 2-flavor symmetry of left and right and the . The finite group -internal symmetry of 4d -fSPTs is within the overlapped subgroup. Therefore, we derive physically the relation of constructions between 4d crystalline--fSPTs and 4d -fSPTs. Formally, Table 3’s map labeled “Yes” for is given by
[TABLE]
with and , . The correspondence is that is mapped to . Namely, the map is
[TABLE]
For this reason, we see that from 2d-to-3d-to-4d construction starts from an even number of Kitaev chains. The corresponding gauged 4d spin-TQFT will be abelian [Ab].
Although 4d crystalline--fSPTs contains a classification subgroup (Theorem 19), it is distinct from the subgroup of crystalline--fSPTs. The reason is the following. The former -classes are associated to the 3d -fSPTs, which can be described by ungauged (i.e. depending on a background gauge field) -spin-CS theory given in (3.19) with a single broken down to subgroup and , . The corresponding -classes of dynamically gauged 3d -fSPTs become -classes of gauged spin-CS theory with gauge group and -matrix with , potentially with an additional fully gapped fermionic sector (a fermionic trivial tensor product state). This spin-CS theory is therefore distinct from the ungauged and gauged theory of (10.1). Thus, it turns out that we cannot relate the constructions between 4d crystalline--fSPTs and 4d -fSPTs. This explains the fact that in Table 3’s the map is labeled “No” for .
10.1.2 Symmetric anomalous gapped boundary TQFT
We would like to comment the symmetric gapped boundary topological orders (TQFT) of SPTs (see a systematic study on this subject in [44]). Recently, Ref. [38] derives a 3d anomalous -gauge theory preserving the full global symmetry and living on the boundary of 4d -fSPTs. Namely Ref. [38]’s 3d anomalous -gauge theory can capture all the ’t Hooft anomalies of the boundary of 4d -fSPTs.
Symmetry Extension and Dimensional Reconstruction from 2d to 3d to 4d: We first apply some simple arguments to construct a potential symmetry-preserving gapped boundary state. Our strategy is to firstly follow the so-called symmetry extension construction developed in Ref. [44]. It is known that for any bosonic types of ’t Hooft anomalies of ordinary 0-form finite group global symmetries, there always exists a symmetry-extension constructed gapped boundary when spacetime dimensions , proven in [44], also later confirmed in a more mathematical setting in [45]. However, we are facing now the fermionic types of ’t Hooft anomalies. It is not guaranteed that symmetry extension [44] necessarily apply to all these cases. Following Sec. 10.1.1, we can reduce the problem from 4d to 2d, by asking how to obtain symmetry extended gapped boundary for 2 layers of Kitaev chains ( class in ). The attempt was made recently in Ref. [15] (Section 4.3), which finds that the finite symmetry group of a finite extension, say a group (usually a normal subgroup), must not commute with the fermion parity symmetry . Thus does not commute with the -dimensional spacetime symmetry group [15]! Similar outcome happens to 2 layers of 3d -fSPTs ( class in ), where the finite symmetry group of a finite extension must not commute with the fermion parity symmetry . Going from 2d to 3d back to 4d, for 4d -fSPTs, if symmetry extension gapped boundary construction applies, its finite extension still cannot commute with the fermion parity symmetry . This indicates that in putative gauge theory (after gauging the extension ), the anyon (at the ends of open line operators) can be permuted under the -symmetry transformation. This 2d-to-3d-to-4d argument leads to agreement with the anyon-permuting anomalous global symmetry, on the 3d boundary of 4d SPTs, emphasized in [38]!
10.2 4d and crystalline--fSPTs
Construction from 2d to 3d to 4d: To obtain 4d crystalline--fSPTs in Table 3, we can stack the 3d -fSPTs which have -classes. The -sub-classes are can be realized as the level 1 -spin-CS theory given in (3.19) with a symmetric bilinear form and broken to subgroup (See the construction in the last paragraph of Sec. 10.1.1.) The generator of the subgroup can be obtained from the decoration and proliferation of 2d Kitaev chain into the -symmetry breaking domain wall of 3d -fSPTs. Thus, in the above construction, we can gain all 4d crystalline--fSPTs from lower dimensions via the stacking and proliferation of condensed matter methods.
In fact, the -classes in are directly mapped to the -classes in , and then to the -classes in , which are also the image of the -classes in w.r.t. the map (9.11). All these sub-classifications are related to the fact that two layers of 1+1D Kitaev chains (i.e. 2 Arf) becomes trivial. Physically, it means that the two nearest-neighbored layers of Kitaev chains (in 2d, 3d or 4d) can be fully gapped by adding interactions but without breaking any symmetry.
10.3 4d , crystalline- and -fSPTs
Construction from 2d to 3d to 4d for -fSPTs: We first construct the intrinsically fermionic 4d -fSPTs, which have classes, shown in Table 4. Both subgroups obviously can be constructed from the -classes of Kitaev chain (ABK invariants) from 2d to 3d to 4d via stacking and proliferation as shown in Sec. 10.1.1, which we will not repeat again.
The generator of classes has an interesting 4d topological term , which can be constructed by stacking (see ) along the -direction a generator (from ). Each 3d slice of the gauged is a 3d non-abelian spin-TQFT, due to the ground state degeneracy (GSD) reduction on the 3-torus. (See Sec. 9.3 and Ref.[28]’s Sec. 5’s calculation on this GSD.) Thus, the gauged version of this minimal generator of class of 4d fSPT is also a non-abelian [NAb] spin-TQFT. Therefore we construct above all fermionic states of classes of 4d -fSPTs from the 2d-to-3d-to-4d procedure.
Next, given 4d -fSPTs, we construct 4d -fSPTs. Namely, we would like to know the fermionic states mapped in Table 3 from the -classes of to the -classes of . The classes in the subgroups are obviously and directly related to the 3d theory described earlier associated to . The remaining classes are related by to copies of the map , the same as (10.2), where in (10.2) can be replaced to either or to get two different subgroups. Then for and as in (10.2), the correspondence again is mapped to . The -classes of spin-TQFTs obtained from dynamical gauged fSPTs, are again abelian [Ab].
Construction from 2d to 3d to 4d for -fSPTs:
We first construct the intrinsically fermionic 4d -fSPTs that are crystalline fSPTs, which has classes (from , box labeled), shown in Table 4.
- •
One classification subgroup (involving ABK invariant) obviously can be constructed from the -class generated by Kitaev chain from 2d to 3d to 4d via stacking and proliferation. The generator is explicitly obtained from stacking the generator of -classes of 3d -fSPTs (given by a topological term ) along the direction.
- •
The generator of another classification subgroup (involving a spin-CS) can be constructed from the class of a ungauged -spin-CS theory, given in (3.19), with broken to as shown in Sec. 10.1.1, which we will not repeat again.
- •
The non-trivial -class can be constructed via the 2d-3d-4d procedure: from the 2d Kitaev chain via Arf invariant generating , we can map it to the generator of , then to the generator of subclass in . This fSPTs is already discussed in Sec. 10.1.1.
- •
Another non-trivial class involves all -, - and -gauge fields. Consider the 1d intersection of the - - and -symmetry-breaking domain walls with trapped term (i.e. fermionic mode trapped at the 1d (=0+1D) intersection of domain wall decorated by the induced spin structure [43]) and then proliferate it to restore the full symmetry. This gives rise to the desired 4d fSPTs.
Next we would like to relate the above classes of intrinsically fermionic 4d -crystalline fSPTs (l.h.s.) to the classes of 4d -fSPTs (r.h.s., within ). Depend on the meaning of gauge field (treated as or ), we can use the same map as (10.2), taking to . Moreover, we can map between one of the of r.h.s. and the of r.h.s. is given by:
[TABLE]
with and , . We find that , so the map is given by mod 2 reduction. Alternatively, we can say that the topological term maps to . In summary, we have the r.h.s. fSPTs map to the l.h.s. fSPTs:
[TABLE]
This explains the map (labeled “Partially”) from 4d - fSPTs to the 4d -crystalline-fSPTs.
10.4 4d , crystalline- and -fSPTs
We had constructed the intrinsically fermionic 4d -fSPTs that are crystalline fSPTs with classes in Sec. 10.3 from the 2d-to-3d-to-4d stacking procedure. By writing down the topological terms, in Table 4, we can see explicitly the injective map (labeled “Yes”) between the 4d -fSPTs and 4d -crystalline fSPTs via
[TABLE]
Construction from 2d to 3d to 4d for -fSPTs: We now construct the intrinsically fermionic 4d -fSPTs, which form classification, shown in Table 4. The -classes of 4d fSPTs involve either the 2d Arf invariant or the 1d invariant. All these can be obtained from the decoration and proliferation of the domain-wall trapping 1+1D Kitaev chain or the 0+1D fermionic mode (with an induced spin structure along an ). We shall not repeat these constructions.
Next we construct the map between the above classes of intrinsically fermionic 4d -crystalline fSPTs (l.h.s.) and the classes of 4d -fSPTs (r.h.s., within ). In Table 4, we can see explicitly the map (labeled “Partially”) between the 4d -fSPTs and 4d -crystalline fSPTs via
[TABLE]
10.5 4d and crystalline--fSPTs
We had constructed the intrinsically fermionic 4d -fSPTs that are crystalline fSPTs with classes in Sec. 10.4. By writing down the topological terms, in Table 4, we see explicitly the surjective map (labeled “Yes”) between the 4d -fSPTs and 4d -crystalline fSPTs via
[TABLE]
Strictly speaking the first , the coefficient multiplying , is from a non-stacking 4d SPTs, since it does not involve the -gauge field. This generator, however, maps to the generator of the first and the same group in the 4d -fSPTs. Note that here all dynamically gauged theories become non-abelian [NAb].
10.6 Crystalline Symmetry v.s. Internal Symmetry
In previous sections, we mainly consider our spin TQFTs as either the fermionic SPTs (fSPT) protected by onsite internal global symmetry , or considering their fermionic finite group gauged theory. In this section 10.6 and later on, we will map our results and classifications obtained from cobordism theory to other new types of fSPTs protected by crystalline global symmetry . These crystalline fSPTs (c-fSPT) not only can include the translational symmetry of the space group (discussed in Sec. 9 and Sec. 10.1 to 10.5), but also other crystalline symmetries. Crystalline symmetries, of great interest and applicable to condensed matter system and topological phases [46], include:
- •
Space group, in crystallography, which means the symmetry of the underlying lattice and crystal. In 3 dimensional spatial lattice, there are 230 possible space groups (e.g. [47, 48] and references therein). Each element contains a collection of symmetry operations, like translation, glide, skew axis, etc.
- •
Point group (e.g. [49, 50] and references therein on their SPTs) which corresponds to the symmetries or isometries that keep at least one point fixed on the underlying lattice and crystal. Point group is the quotient group of the space group by the translational symmetry group. Here we will focus on the reflection/mirror c-fSPT in Sec. 10.7, inversion c-fSPT in Sec. 10.8, and the rotation c-fSPT in Sec. 10.9.
Part of our work is inspired by Ref. [47]’s Crystalline Equivalence Principle: “Euclidean crystalline-SPTs with symmetry group are in 1-to-1 correspondence with SPTs protected by the internal symmetry , where if has an orientation-reversing, it is mapped to an anti-unitary symmetry in the internal symmetry.” We would like to apply their principles to include reflection/mirror, inversion and rotation SPTs, and compute explicit classifications from cobordism theory approach — the cobordism theory will also suggest the appropriate relations between the topological terms of c-fSPTs and internal symmetric fSPTs. Moreover, we can compare with another independent approach on lattice models from the recent Ref. [51] on interacting rotational fSPTs, which we find in agreement. Additional comments on other recent works can be found Sec. 10.11.3.
10.7 Reflection/Mirror SPTs
Reflection symmetry transformation means one of spatial coordinates (say ) is sent to its negative value (Fig. 25), say with respect to the origin:
[TABLE]
while others on the mirror hyperplane (say , ) remain the same.
- •
The local internal symmetry is applied to all local sites, each within a local region around the unfilled circles .
- •
The spatial crystalline symmetry (like space group or point group) is applied to as the symmetry transformations between local sites, implemented between the unfilled circles .
In Table 5, we list down the complete classification of reflection/mirror c-fSPTs. The is the general of reflection/mirror symmetry (up to the fermion parity, it is a group). For fermionic system, there is a choice of or . The latter means that is locked with the fermion parity symmetry. The is +1 or -1 for an even or odd number of fermions. The is obviously related to the fermion statistics, rotating a fermion by gives rise to a sign.
Since the reflection () and the time-reversal symmetry () is the same for the reflection symmetry of Euclidean field theory, we can map the “” or “” to or Euclidean field theory. This leads to the result summarized in Table 5.
10.8 Inversion SPTs
Inversion symmetry transformation means all spatial coordinates (, ) are inverted (Fig. 26), say respect to the origin (the black dot at ):
[TABLE]
In Table 6, we list down the complete classification of inversion c-fSPTs. Let denote the inversion symmetry (up to the fermion parity, it is a group). Again, for fermionic system, there is a choice of or locked with the fermion parity symmetry, as already explained in Sec. 10.7.
In an even- dimensional spacetime, both the inversion (, inverting all the odd-dimensional spatial coordinates, which gives rise to an overall sign ) and the time-reversal symmetry () are exactly the same as the reflection symmetry of Euclidean field theory. We can again map the “” or “” to or spacetime symmetries of Euclidean field theory, the same as for reflection/mirror symmetry in Sec. 10.7. 2. 2.
In an odd- dimensional spacetime, the inversion (, inverting all gives rise to an overall sign ) is rather distinct from the reflection symmetry. We can map the inversion instead to or spacetime symmetries of Euclidean field theory.474747For even , the internal symmetry corresponds to extensions
where () for , while () for . For odd , the internal symmetry corresponds to extension
where (-structure in the cobordism approach) for , while (-structure in the cobordism approach) for .
This leads to the result summarized in Table 6.
10.9 Rotation SPTs
Here we only consider a rotation within a 2-plane, say with coordinates and , such that rotation by a -angle on this 2-plane with respect to the origin is given by
[TABLE]
When , we have, and .
On a lattice or crystal, we have the rotation group ( for cyclic) group, which is the finite group. The generator is the rotation by angle , with , due to the constraint of the lattice periodicity. See for example in Fig. 27.
In an 1+1D (2d) spacetime, the -rotation is the same as the inversion and the reflection. So the first row of data in 1+1D (2d) of Table 5, 6 and 7 exactly coincide and give the same fSPTs. 2. 2.
In an 2+1D (3d) spacetime, the -rotation is the same as the spatial inversion, but not the same as the reflection. So the first row of data in 2+1D (3d) of Table 6 and 7 exactly coincide and give the same fSPTs. 3. 3.
In any dimension larger than 1+1D (d where ), the rotation symmetry acts on the 2-dimensional spatial subspace. We can map the -rotation instead to or spacetime symmetry of Euclidean field theory.484848For all -dimensions where , the internal symmetry corresponds to the extension
where (in terms of bordism of -structure) for , while (in terms of bordism of -structure) for . This is a natural generalization from of -symmetry in 2+1D to generic in other dimensions. Note that is the generator of crystalline rotation symmetry group, not the internal symmetry . The difference between two (factor in the -th power of the generator) has been discussed e.g. in [51].
This leads to the result summarized in Table 7 and 8.
10.10 Fermionic SETs (Symmetry Enriched Topologically ordered states)
We can partially gauge the symmetry group of -fSPT of an internal symmetry . This becomes the so-called Symmetry Enriched Topologically ordered states (SETs):
- •
Fermionic SETs (fSETs), if we only dynamically gauge a subgroup and leave fermion parity ungauged.
- •
Bosonic SETs (bSETs), if we dynamically gauge a subgroup and also dynamically gauge fermion parity. The gauging process is known as a higher dimensional bosonization [52].
We can define the fSET partition function via generalizing the gauging discussion in Sec. 2.1 . Gauging subgroup produces the fSET with symmetry (i.e. -equivariant spin-TQFT), where is the centralizer of in , i.e. . Then, for a given , the Hilbert space of the resulting -dimensional fSET on is given by a generalization of eqn. (2.6) as:
[TABLE]
where is the map induced by the multiplication map (note that here, to define , we use the fact that and are commuting subgroups in so that the map is a group homomorphism) and is the following subset of the set of homotopy classes of maps :
[TABLE]
Similarly, we can define bSETs via combining the formalism in Sec. 2.2 to obtain bosonic theory and the partial gauging in eqn. (10.12). A related gauging procedure from 4d fSPTs to 4d bSETs and to 4d fSETs was recently given in [53]. A worthwhile remark is that Ref. [53] gives a strong conjecture, stating that gauging 4d fSPTs with finite group unitary internal symmetry, may give a large subclass of all 4d bSETs of finite gauge group, and likely also a complete classification of all 4d fSETs of finite gauge group. If so, our formulation provides a systematic study of all 4d bSETs and fSETs of finite gauge group.
10.11 Other aspects
In this subsection, we make some remarks on how our work can be related to the previously existed literature.
10.11.1 Fermionic higher global symmetries vs. Higher-form global symmetries
Ref. [54] proposes the generalized global symmetry for QFTs. For TQFTs, the generalized global symmetry is intrinsically related to the link invariants between two set of extended operators; one is regarded as a “charged” object (being measured), the other is regarded as a “charge” operator (measuring the “charged” object).
First let us note that the usual fermionic parity in many aspects behaves as a 0-form global symmetry. In particular, one can insert a Wilson loop that measures both holonomy of a gauge field for an ordinary 0-form symmetry and the action of invertible spin-TQFT coupled to the spin-structure induced from the ambient space via framing on the normal bundle.
Similarly, in many aspects, one can treat Arf-TQFT as a connection for fermionic 1-form symmetry. For our fermionic spin-TQFTs, there are fermionic loops that are not only charged under 1-form global symmetry, but also “charged under 1-form symmetry generated by Arf invertible spin-TQFT.” So when we parallel transport them along some surface connecting loop with loop (i.e. ), they obtain a phase given by
[TABLE]
where is a background 2-form field for 1-form symmetry ( or , with charge or respectively) and the spin-structure on is induced from the spin structure of the ambient space using framing of the normal bundle to .
In the context of generalized global symmetry, we not only have [54]’s higher-form global symmetries, but also additional fermionic higher global symmetries which may not be written as differential forms.
10.11.2 Adams spectral sequence vs. Atiyah-Hirzebruch spectral sequence
In this section we compare between our classifications and [55, 56]’s classification scheme for fermionic SPTs (fSPTs). Our classification relies on Adams spectral sequence. To our best knowledge, [55, 56]’s structure is similar to Atiyah-Hirzebruch spectral sequence (see also [57, 58]).
Based on a fermionic lattice model for fSPTs, Ref. [55, 56] derives a generalized cohomology group theory, such that there are two layers of short exact sequences and other constraints upon their set of data.
For 3d (2+1D) fSPTs with a total symmetry , with a bosonic internal symmetry and fermion parity symmetry, Ref. [59, 55, 56] (and references therein) summarize three sets of group cohomology data of the symmetry group , namely
[TABLE]
is the obstruction-free subgroup of , generated by that satisfy in , where is the Steenrod square.494949 is the notation indicating that the coefficient of the cohomology group is non-trivially acted by the antiunitary time-reversal symmetry if exists. Here we do not pay attention to the antiunitary symmetry, and assume that all the (considered) symmetries are unitary. is the classification of bosonic SPTs. Physically, the layer can be constructed by decorating a 1+1D Kitaev fermionic chain [16], which is a 2d invertible spin TQFT onto the -symmetry domain walls. The layer is constructed by decorating complex fermions, which are 1d invertible spin TQFT onto the -symmetry domain walls.
For 4d (3+1D) fSPTs with , with a bosonic symmetry , Ref. [55, 56] propose three sets of group cohomology data of the symmetry group , namely
[TABLE]
to classify these 4d fSPT. For the meaning of the first two entries and , see the original reference [55, 56]. The third entry is an obstruction-free subgroup of , generated by that obey in and in . Here is a certain cohomology operation that maps satisfying in into an element in . As far as we are concerned, the “mysterious” discussed in [55] is actually, in Atiyah-Hirzebruch spectral sequence, the dual of the secondary cohomology operation based on the relation , where is the induced coefficient mod 2 reduction from . In general, there is a bijection between stable homology operations and stable cohomology operations . So the dual of is . Apply the functor , we get , here is the classifying space of . Other details of cohomology group notations are explained in [55, 56].
Below we fill in some background knowledge of Atiyah-Hirzebruch spectral sequence for comparison. Here
[TABLE]
where .
There is a filtration
[TABLE]
and an isomorphism
[TABLE]
The first layer is the extension
[TABLE]
Their second layer is the extension
[TABLE]
Following Ref. [55, 56]’s notation, we find the following correspondence for (where , denoted as previously, is the bosonic internal symmetry group). If :
[TABLE]
If :
[TABLE]
On the other hand, our work uses Adams spectral sequence. Our “layer” structure has physical and mathematical interpretations (To recall, the topological terms underlined with a single or double lines are our notations introduced in Sec. 4, such as eq. (4.19)):
- •
Non-underlined topological terms are bosonic (i.e. belong to subgroup), they corresponds to the elements in of Atiyah-Hirzebruch spectral sequence.
- •
Topological terms underlined with a single line are fermionic that provide refinement of the bosonic elements from .
- •
Topological terms underlined with double lines are fermionic and do not refine any elements of . Here “refine” means the fermionic topological term is a nontrivial extension of the elements of .
Examples by examples, our fSPT classification results show general agreements with Ref. [55, 56]. It is worth noticing that similar explicit layer structures of the spin bordism group have been explored in [57, 58]. However, we remain direct comparison of our construction to [57, 58] to be studied.
10.11.3 Relations to other recent works
Here are some final remarks related to the literature and recent works. We also highlight potential future directions.
We have discussed the topological invariants and link invariants of fermionic spin -TQFTs. In condensed matter physics, the topological invariants corresponds to SPTs partition function; the link invariants corresponds the braiding statistical Berry phases of time-evolution trajectory of world-line/world-volume of anyonic particles of anyonic strings (either weakly gauged as probed field defect, or dynamically gauged as topological orders). For the physical meanings of link invariants, one can refer to systematic work [13, 60, 61, 62, 63, 25, 64, 65, 66]. It will be interesting to explore the link invariants in terms of geometric-topology aspects like the surgery theory on submanifolds, along the ideas of [67, 62] in 3 and 4 dimensions. 2. 2.
The SPTs protected by crystalline lattice symmetry has generated broad interests recently, ranging from the earlier work on crystalline topological insulator [46] to the recent work on intrinsic interacting crystalline insulator/superconductor [68], and recent reviews [69] and references therein.
Apart from our discussion and Ref. [47]’s Crystalline Equivalence Principle, other development of general theory of cSPTs include general real-space recipe construction of topological crystalline states [70] and field theory approaches [71] (and references therein).
It is worthwhile to mention that recent Ref. [72] applies generalized homology and Atiyah-Hirzebruch spectral sequence (AHSS) in crystalline SPTs. Therefore, it will be interesting to compare our understanding in Sec. 10.11.2 based on our work (Adams spectral sequence) not only to [55, 56]’s work (AHSS) on fSPT with internal symmetry, but also to [72]’s work (AHSS) on crystalline symmetry. This may guide us to construct analogous lattice models, from our models, for internal [55, 56] and crystalline symmetry [72]. Another approach to construct our models on a lattice or condensed mater systems can be fermion decoration construction [73], similar to our discussion in Sec. 10.1.1.
11 Acknowledgments
The authorship is listed in the alphabetical order. PP would like to thank Anna Beliakova, Ryan Thorngren for useful discussions and the hospitality of SCGP, Weizmann Institute and KITP where parts of the work were done. JW warmly thanks the participants and organizers of the PCTS workshop on Fracton Phases of Matter and Topological Crystalline Order (December 3-5, 2018) for many inspiring conversations near the completion of this work. JW also thanks Meng Cheng, Dominic Else, Sheng-Jie Huang, and Hao Song, for clarifying their own works. MG thanks the support from U.S.-Israel Binational Science Foundation. KO gratefully acknowledges the support from NSF Grant PHY-1606531 and Paul Dirac fund. PP gratefully acknowledges the support from Marvin L. Goldberger Fellowship and the DOE Grant 51 DE-SC0009988 during his appointment at IAS. ZW gratefully acknowledges support from NSFC grants 11431010, 11571329. JW gratefully acknowledges the Corning Glass Works Foundation Fellowship and NSF Grant PHY-1606531. This work was also supported in part by NSF Grant DMS-1607871 “Analysis, Geometry and Mathematical Physics”, Center for Mathematical Sciences and Applications at Harvard University, and the National Science Foundation under Grant No. NSF PHY-1748958.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] M. Atiyah, Topological quantum field theories , Publications Mathématiques de l’Institut des Hautes Études Scientifiques 68 175–186 (1988).
- 2[2] R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds , Geometry of low-dimensional manifolds 2 177–242 (1990).
- 3[3] C. Blanchet, Invariants on three-manifolds with spin structure , Commentarii Mathematici Helvetici 67 406–427 (1992).
- 4[4] C. Blanchet, G. Masbaum et al., Topological quantum field theories for surfaces with spin structure , Duke Mathematical Journal 82 229–268 (1996).
- 5[5] A. Beliakova, C. Blanchet and E. Contreras, Spin modular categories , ar Xiv preprint ar Xiv:1411.4232 (2014).
- 6[6] D. S. Freed and M. J. Hopkins, Reflection positivity and invertible topological phases , ar Xiv preprint ar Xiv:1604.06527 (2016).
- 7[7] R. Dijkgraaf and E. Witten, Topological Gauge Theories and Group Cohomology , Commun. Math. Phys. 129 393 (1990) . · doi ↗
- 8[8] D. S. Freed and F. Quinn, Chern-Simons theory with finite gauge group , Commun. Math. Phys. 156 435–472 (1993) , [ ar Xiv:hep-th/9111004 ]. · doi ↗