Higher Anomalies, Higher Symmetries, and Cobordisms I: Classification of Higher-Symmetry-Protected Topological States and Their Boundary Fermionic/Bosonic Anomalies via a Generalized Cobordism Theory | Tomesphere
arXiv:1812.11967·hep-th·October 4, 2019
Higher Anomalies, Higher Symmetries, and Cobordisms I: Classification of Higher-Symmetry-Protected Topological States and Their Boundary Fermionic/Bosonic Anomalies via a Generalized Cobordism Theory
This paper develops a generalized cobordism theory to classify higher-symmetry-protected topological states and analyze their boundary anomalies in quantum field theories, incorporating advanced mathematical tools and providing numerous examples.
Contribution
It introduces a new generalized cobordism framework for classifying higher symmetries and anomalies in quantum systems, extending previous theories with higher-group structures and invariants.
Findings
01
Classifies higher-SPTs and their boundary anomalies.
02
Provides examples of bordism groups with higher-group structures.
03
Analyzes perturbative and non-perturbative anomalies in various dimensions.
Abstract
By developing a generalized cobordism theory, we explore the higher global symmetries and higher anomalies of quantum field theories and interacting fermionic/bosonic systems in condensed matter. Our essential math input is a generalization of Thom-Madsen-Tillmann spectra, Adams spectral sequence, and Freed-Hopkins's theorem, to incorporate higher-groups and higher classifying spaces. We provide many examples of bordism groups with a generic H-structure manifold with a higher-group G, and their bordism invariants --- e.g. perturbative anomalies of chiral fermions [originated from Adler-Bell-Jackiw] or bosons with U(1) symmetry in any even spacetime dimensions; non-perturbative global anomalies such as Witten anomaly and the new SU(2) anomaly in 4d and 5d. Suitable H such as SO/Spin/O/Pin± enables the study of quantum vacua of general bosonic or fermionic systems with…
Tables118
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Table 2. Table 9:
Note that one of the ℤ ℤ \mathbb{Z} classes in
TP 3 ( SO × U ( 1 ) ) = ℤ 2 subscript TP 3 SO U 1 superscript ℤ 2 \mathrm{TP}_{3}({\rm SO}\times{\rm U}(1))=\mathbb{Z}^{2}
is given by
eqn. ( 3.22 ).
Similarly, one of the ℤ ℤ \mathbb{Z} classes in
TP d + 1 ( SO × U ( 1 ) ) subscript TP 𝑑 1 SO U 1 \mathrm{TP}_{d+1}({\rm SO}\times{\rm U}(1)) for an even d 𝑑 d is given by
eqn. ( 3.20 ).
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Table 6. Table 10:
Note that one of the ℤ ℤ \mathbb{Z} classes in
TP 3 ( Spin c ) = ℤ 2 subscript TP 3 superscript Spin 𝑐 superscript ℤ 2 \mathrm{TP}_{3}({\rm Spin}^{c})=\mathbb{Z}^{2}
is given by
eqn. ( 3.24 ).
Similarly, one of the ℤ ℤ \mathbb{Z} classes in
TP d + 1 ( Spin c ) subscript TP 𝑑 1 superscript Spin 𝑐 \mathrm{TP}_{d+1}({\rm Spin}^{c}) for an even d 𝑑 d is given by
eqn. ( 3.20 ).
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Table 7. Table 11: Bordism group.
Here σ 𝜎 \sigma is the signature of 4-manifolds, PD ( w 2 ) PD subscript 𝑤 2 \text{PD}(w_{2}) is the submanifold of a 6-manifold which represents the Poincaré dual of w 2 subscript 𝑤 2 w_{2} . Note that PD ( w 2 ) PD subscript 𝑤 2 \text{PD}(w_{2}) is Spin.
The N 0 subscript 𝑁 0 N_{0} is the number of the zero modes of the Dirac operator in 4d. It is defined in [ 25 ] . On oriented 4-manifolds, N 0 = N 0 ′ = N + − N − subscript 𝑁 0 superscript subscript 𝑁 0 ′ subscript 𝑁 subscript 𝑁 N_{0}=N_{0}^{\prime}=N_{+}-N_{-} where N 0 ′ superscript subscript 𝑁 0 ′ N_{0}^{\prime} is also defined in [ 25 ] , and N ± subscript 𝑁 plus-or-minus N_{\pm} are the numbers of zero modes of the Dirac operator with given chirality.
Bordism group
bordism invariants
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(, Arf )
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()
Table 8. Table 12: SPT states in d 𝑑 d -dim spacetime.
TP 5 ( Spin × SU ( 2 ) ℤ 2 F ) = ( ℤ 2 ) 2 subscript TP 5 Spin SU 2 superscript subscript ℤ 2 𝐹 superscript subscript ℤ 2 2 \mathrm{TP}_{5}({\frac{{\rm Spin}\times{\rm SU}(2)}{\mathbb{Z}_{2}^{F}}})=(\mathbb{Z}_{2})^{2} .
whose 5d bordism invariants correspond to the old SU(2) [ 23 ] and the new SU(2) anomalies [ 24 ] in 4d.
A related cobordism group in one higher dimension,
TP 6 ( Spin × SU ( 2 ) ℤ 2 F ) = ( ℤ 2 ) 2 subscript TP 6 Spin SU 2 superscript subscript ℤ 2 𝐹 superscript subscript ℤ 2 2 \mathrm{TP}_{6}({\frac{{\rm Spin}\times{\rm SU}(2)}{\mathbb{Z}_{2}^{F}}})=(\mathbb{Z}_{2})^{2} .
whose 6d bordism invariants correspond to the old SU(2) and the new SU(2) anomalies in 5d [ 24 ] . Since Ω 3 Spin × ℤ 2 SU ( 2 ) = 0 superscript subscript Ω 3 subscript subscript ℤ 2 Spin SU 2 0 \Omega_{3}^{{\rm Spin}\times_{\mathbb{Z}_{2}}{\rm SU}(2)}=0 , for any Spin × ℤ 2 SU ( 2 ) subscript subscript ℤ 2 Spin SU 2 {\rm Spin}\times_{\mathbb{Z}_{2}}{\rm SU}(2) 3-manifold M 3 superscript 𝑀 3 M^{3} , M 3 superscript 𝑀 3 M^{3} is the boundary of a Spin × ℤ 2 SU ( 2 ) subscript subscript ℤ 2 Spin SU 2 {\rm Spin}\times_{\mathbb{Z}_{2}}{\rm SU}(2) 4-manifold M 4 superscript 𝑀 4 M^{4} , then we define the 3d topological term X 𝑋 X on M 3 superscript 𝑀 3 M^{3} to be N 0 subscript 𝑁 0 N_{0} of M 4 superscript 𝑀 4 M^{4} . Note that Dirac operator can be defined for manifolds with boundary. If the bulk manifold has Dirac operator which has a mass m 𝑚 m being gapped, then the boundary manifold can have Dirac operator no mass m = 0 𝑚 0 m=0 being gapless.
Cobordism group
topological terms
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Equations824
\displaystyle\left\{\begin{array}[]{ccc}\text{deformation classes of reflection positive}\\
\text{invertible }n\text{-dimensional extended topological}\\
\text{field theories with a symmetry group }H_{n}\times\mathbb{G}\end{array}\right\}\cong[MT(H\times\mathbb{G}),\Sigma^{n+1}I\mathbb{Z}]_{\text{tors}}.
\displaystyle\left\{\begin{array}[]{ccc}\text{deformation classes of reflection positive}\\
\text{invertible }n\text{-dimensional extended topological}\\
\text{field theories with a symmetry group }H_{n}\times\mathbb{G}\end{array}\right\}\cong[MT(H\times\mathbb{G}),\Sigma^{n+1}I\mathbb{Z}]_{\text{tors}}.
0→Ext1(πnB,Z)→[B,Σn+1IZ]→Hom(πn+1B,Z)→0
0→Ext1(πnB,Z)→[B,Σn+1IZ]→Hom(πn+1B,Z)→0
TPn(H×G)≡[MT(H×G),Σn+1IZ].
TPn(H×G)≡[MT(H×G),Σn+1IZ].
\displaystyle\left\{\begin{array}[]{ccc}\text{deformation classes of reflection positive}\\
\text{invertible }n\text{-dimensional extended topological}\\
\text{field theories with a symmetry group }(\frac{H_{n}\ltimes\mathbb{G}}{N_{\text{shared}}})\end{array}\right\}\cong[MT(\frac{H\ltimes\mathbb{G}}{N_{\text{shared}}}),\Sigma^{n+1}I\mathbb{Z}]_{\text{tors}}.
\displaystyle\left\{\begin{array}[]{ccc}\text{deformation classes of reflection positive}\\
\text{invertible }n\text{-dimensional extended topological}\\
\text{field theories with a symmetry group }(\frac{H_{n}\ltimes\mathbb{G}}{N_{\text{shared}}})\end{array}\right\}\cong[MT(\frac{H\ltimes\mathbb{G}}{N_{\text{shared}}}),\Sigma^{n+1}I\mathbb{Z}]_{\text{tors}}.
TPn(NsharedH⋉G)≡[MT(NsharedH⋉G),Σn+1IZ].
TPn(NsharedH⋉G)≡[MT(NsharedH⋉G),Σn+1IZ].
(p,g)⋅h=(p⋅h,ρ(h)−1g),p∈P,g∈O,h∈H
(p,g)⋅h=(p⋅h,ρ(h)−1g),p∈P,g∈O,h∈H
[M,K(G,n)]=Hn(M,G)
[M,K(G,n)]=Hn(M,G)
ΩnH(X):={(M,f)∣M is a closed
ΩnH(X):={(M,f)∣M is a closed
n-manifold with H-structure, f:M→X is a map}/bordism.
[X,BG]={isomorphism classes of principal G-bundles over X}
[X,BG]={isomorphism classes of principal G-bundles over X}
(p,v)g=(pg,g−1v)
(p,v)g=(pg,g−1v)
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Full text
**Higher Anomalies, Higher Symmetries,
and
Cobordisms I:
Classification of Higher-Symmetry-Protected Topological States and Their
Boundary Fermionic/Bosonic Anomalies via a Generalized Cobordism Theory
1*Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China
2*School of Mathematical Sciences, USTC, Hefei 230026, China
3*School of Natural Sciences, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA
4*Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138, USA
By developing a generalized cobordism theory,
we explore the higher global symmetries
and higher anomalies of quantum field theories
and interacting fermionic/bosonic systems in condensed matter.
Our essential math input is a generalization of
Thom-Madsen-Tillmann spectra, Adams spectral sequence, and Freed-Hopkins’s theorem, to incorporate
higher-groups and higher classifying spaces.
We provide many examples of bordism groups with a generic H-structure manifold with a higher-group G, and their bordism invariants
— e.g. perturbative anomalies of chiral fermions [originated from Adler-Bell-Jackiw] or bosons with U(1) symmetry in any even spacetime dimensions;
non-perturbative global anomalies such as Witten anomaly and the new SU(2) anomaly in 4d and 5d.
Suitable H such as SO/Spin/O/Pin± enables the study
of quantum vacua of general bosonic or fermionic systems
with time-reversal or reflection symmetry on (un)orientable spacetime.
Higher ’t Hooft anomalies of dd live on the boundary of (d+1)d higher-Symmetry-Protected Topological states (SPTs) or
symmetric invertible topological orders (i.e., invertible topological quantum field theories at low energy);
thus our cobordism theory also classifies and characterizes higher-SPTs.
Examples of higher-SPT’s anomalous boundary theories include strongly coupled non-Abelian Yang-Mills (YM) gauge theories and sigma models,
complementary to physics obtained in [arXiv:1810.00844, 1812.11955, 1812.11968, 1904.00994].
This article is a companion with further detailed calculations supporting other shorter articles.
Thom, as the pioneer of bordism theory, studied the criteria when the disjoint union of two closed n-manifolds is the boundary of a compact (n+1)-manifold [1]. Thom found that this relation is an equivalence relation on the set of closed n-manifolds. Moreover, the disjoint union operation defines an abelian group structure on the set of equivalence classes. This group is called the unoriented bordism group, it is denoted by ΩnO. Furthermore, Thom found that the Cartesian product defines a graded ring structure on Ω∗O:=⨁n≥0ΩnO, which is called the unoriented bordism ring.
Thom also found that the bordism invariants of ΩnO are the Stiefel-Whitney numbers. Namely,
two manifolds are unorientedly bordant if and only if they have identical sets of Stiefel-Whitney characteristic numbers. This yields many interesting consequences. For example,
the real projective space RP2 is not a boundary while RP3 is; also
the complex projective space CP2 and RP2×RP2 are unorientedly bordant.
Many generalizations are made to bordism theories so far.
For example, we can consider manifolds which are equipped with an H-structure,
we follow the definition of H-structure given in [2].
Our work is inspired by a cobordism theory from the Madsen-Tillmann spectrum [3] and from Freed-Hopkins [4].
Freed and Hopkins propose a cobordism theory [4] to classify the
Symmetry-Protected Topological states (SPTs) [5] in condensed matter physics [6] with ordinary internal global symmetries of group G
and their classifying space BG. Examples of SPTs include the famous topological insulators and topological superconductors [7, 8].
The major motivation of our work is to generalize the calculations and the cobordism theory of Freed-Hopkins [4] —
such that, instead of the ordinary group G or ordinary classifying space BG, we consider a generalized cobordism theory studying manifolds (i.e., spacetime manifolds)
endorsed with H structure, with additional higher group G (i.e., generalized as principal-G bundles)
and higher classifying spacesBG.
We consider this particular generalized cobordism theory in order to study, characterize and classify:
Generalized higher global symmetries of G [9] in physics 111Generalized higher global symmetries may or may not be
higher-differential form global symmetries. For example,
there exist certain fermionic SPTs
whose higher global symmetries whose
charged objects are not in terms of higher-differential forms, see LABEL:1812.11959 and References therein. and their higher classifying spaces BG.
2. 2.
Higher-Symmetry-Protected Topological states (higher-SPTs), which are nontrivial quantum vacua protected by higher global symmetries of G.
Higher SPTs are characterized by (co)bordism invariants obtained from (co)bordism groups of higher classifying spaces BG.
For example, in (d+1)-dimensional spacetime, denoted (d+1)d,
consider the quantum vacua of internal global symmetry G on a (d+1)d spacetime manifold with H-structure,
we will propose a bordism group Ω(d+1)H(BG) and a related cobordism group TP(d+1)(H×G)
to classify higher-SPTs in (d+1)d. See the earlier pioneer work on higher-SPTs in [11, 12].
3. 3.
Higher quantum anomalies, e.g. higher ’t Hooft anomalies: The ordinary ’t Hooft anomalies [13] of global symmetry G is the anomaly for
QFT of the ordinary global symmetry G.
In comparison, given the internal higher global symmetry G and the dd spacetime manifold with H-structure,
we can ask what are the possible higher quantum anomalies in the dd physical theories?
The associated higher anomalies, given by the data G and H, in the dd spacetime,
via a generalization of the anomaly-inflow picture [14],
turns out to relate to the anomalies of the dd boundary theory (called the boundary anomalies) of (d+1)d higher-SPTs (given by the same data G and H).
So the characterization and classification of (d+1)d higher-SPTs in the previous remark turns out to help on the characterization and classification of
dd higher ’t Hooft anomalies.
Modern examples of higher ’t Hooft anomalies are found
in quantum field theories (QFTs) including Yang-Mills gauge theories [15] and sigma models.
The first example of higher ’t Hooft anomalies is discovered by a remarkable work LABEL:Gaiotto2017TTT1703.00501
for a pure 4d SU(N) Yang-MiIls gauge theory of even integer N
with a second Chern-class topological term (called the θ-term or θTr[F∧F]-term in particle physics.)
Further new higher ’t Hooft anomalies are found in [17, 18, 19, 20].
In summary, as we have said, we aim to study higher global symmetries, characterize and classify higher-SPTs and higher quantum anomalies.
•
By characterization, we mean that given certain physics phenomena or theories (here, higher-SPTs and higher quantum anomalies),
we like to write down their mathematical invariants (here, we mean the bordism invariant) to
fully describe or capture their essences/properties. Hopefully, we can further compute their physical observables.
•
By classification, we mean that given the spacetime dimensions (here d+1d for higher-SPTs or dd for higher quantum anomalies),
their H-structure and the internal higher global symmetry G,
we aim to know how many classes (a number to count them) there are?
Also, we aim to determine the mathematical structures of classes (i.e. here group structure as for (co)bordism groups:
would the classes be a finite group Zn or an infinite group Z or their mixing, etc.).
Another purpose of this article is a companion article with further detailed mathematical calculations in order to support other shorter articles [18, 19, 20].
In this Introduction, we will provide some basic physics preliminaries in Sec. 1.2
and mathematical preliminaries in Sec. 1.3.
Since the concepts of higher symmetries and higher anomalies are crucial, we will also clarify what precisely we mean by
higher symmetries/anomalies in condensed matter, in QFTs and in mathematics, in Sec. 1.4.
After some additional introduction to mathematical background in Sec. 2,
we will provide explicit interpretations of familiar examples (to QFT-ist and physicists) of
perturbative anomalies in Sec. 3.1:
(1):
Perturbative fermionic anomalies from chiral fermions with U(1) symmetry, originated from Adler-Bell-Jackiw (ABJ) anomalies [21, 22].
2. (2):
Perturbative bosonic anomalies from bosonic systems with U(1) symmetry.
We will also provide more exotic non-perturbative global anomalies in Sec. 3.2:
matching the physics results of dd anomalies to mathematical cobordism group calculations in (d+1)d.
We briefly comment the difference between a previous cobordism theory [4, 25] and this work:
In all Adams charts of the computation in [4, 25], there are no nonzero differentials,
while in this paper we encounter nonzero differentials dn due to the (p,pn)-Bockstein homomorphisms in the computation involving B2Zpn and BZpn.
1.2 Physics preliminaries
Freed-Hopkins’s work [4] is motivated by the development of cobordism theory classification [26, 27] of so-called the
Symmetry Protected Topological (SPT) state in condensed matter physics [6].
In a very short summary,
Freed-Hopkins’s work [4] applies the theory of
Thom-Madsen-Tillmann spectra [1, 3],
to prove a theorem relating the
“Topological Phases” (which later will be abbreviated as TP) or certain
deformation classes of reflection positive invertible n-dimensional extended
topological field theories (iTQFT) with
symmetry group (or in short, symmetric iTQFT),
to Madsen-Tillmann spectrum [3] of the symmetry group.
Here an n-dimensional extended
topological field theory is a symmetric monoidal functor F from the (∞,n)-category of extended cobordisms
Bordn(Hn) to a symmetric monoidal (∞,n)-category C
where Bordn(Hn) is defined as follows (all manifolds are equipped with H-structures, see definition 1):
•
objects are 0-manifolds;
•
1-morphisms are 1-cobordisms between objects;
•
2-morphisms are 2-cobordisms between 1-morphisms;
•
…
•
n-morphisms are n-cobordisms between (n−1)-morphisms;
•
(n+1)-morphisms are diffeomorphisms between n-morphisms;
•
(n+2)-morphisms are smooth homotopies between (n+1)-morphisms;
•
…
An n-dimensional extended
topological field theory is called invertible if F factor through the Picard groupoid C×.
By a theorem of Galatius-Madsen-Tillmann-Weiss [3], the classifying space
of Bordn(Hn) is exactly the 0-th space of the Madsen-Tillmann spectrum ΣnMTHn.
In this work, we will consider the generalization of [4]
to include higher symmetries [9], for example, including both 0-form symmetry of group G(0) and 1-form symmetry of group G(1),
or in certain cases, as higher symmetry group of higher n-group.222
For the physics application of our result, please see [18, 19, 20].
Some of these 4d non-Abelian SU(N) Yang-Mills[15]-like gauge theories can be obtained from
gauging the time-reversal symmetric SU(N)-SPT generalization of topological insulator/superconductor (TI/SC) [25].
We can understand their anomalies of 0-form symmetry of group G(0) and 1-form symmetry of group G(1),
as the obstruction to regularize the global symmetries locally in its own dimensions (4d for YM theory).
Instead, in order to regularize the global symmetries locally and onsite,
the 4d gauge theories need to be placed on the boundary of 5d higher SPTs.
The 5d higher SPTs corresponds to the nontrivial generators of cobordism groups of higher classifying spaces.
We write G(0) or Ga to indicate some 0-form symmetry probed by 1-form a field.
We write G(1) or Gb to indicate some 1-form symmetry probed by 2-form b field.
Other physics motivations to study higher group can be found in
[28, 29, 30, 31] and references therein.
We generalize the work of Freed-Hopkins [4]: there is a 1:1 correspondence
[TABLE]
where H is the space time symmetry, G is the internal symmetry which is possibly a higher group,
MT(H×G) is the Madsen-Tillmann spectrum [3] of the group H×G,
Σ is the suspension, IZ is the Anderson dual spectrum, and tors means the torsion part.
Since there is an exact sequence
[TABLE]
for any spectrum B, especially for MT(H×G).
The torsion part [MT(H×G),Σn+1IZ]tors is Ext1((πnMT(H×G))tors,Z)=Hom((πnMT(H×G))tors,U(1)).
By the generalized Pontryagin-Thom isomorphism (1.20), πnMT(H×G)=ΩnH×G=ΩnH(BG) which is the bordism group defined in definition 2.
Namely, we can classify the deformation classes of symmetric iTQFTs and also symmetric invertible topological orders (iTOs), via
the particular group
[TABLE]
Here TP means the abbreviation of “Topological Phases”
classifying the above symmetric iTQFT,
the torsion part of TPn(H×G) and ΩnH(BG) are the same.
In this work, we compute the (co)bordism groups ΩdH(BG) (TPd(H×G)) for H=O/SO/Spin/Pin± and several G, we also consider ΩdG where BG is the total space of the nontrivial fibration with base space BO and fiber B2Z2 in section 4.1.
If there is a nontrivial group action between H and G (let us denote the action as the semi-direct product ⋉),
or if there is a shared common normal subgroup Nshared or sub-higher-group between H and G,
then we can generalize our above proposal eqn. (1.4) and eqn. (1.6) to
[TABLE]
and
[TABLE]
For readers who wishes to explore other physics stories and some introduction materials, we suggest to look at the introduction of [25]
and other shorter articles [18, 19, 20, 32, 33].
In particular, LABEL:Wen2018zux1812.02517 provides a condensed matter interpretation of higher symmetries.
We also encourage readers to read the Section I to III of [18].
We will explore the generic manifold with H-structure, including the orientable H=SO, Spin, etc.,
or unorientable H=O, Pin±.
In physics, the quantum system that can be put on an unorientable H manifold implies that
there is a time-reversal symmetry or a reflection symmetry (commonly termed the parity symmetry in an odd dimensional space).
Physicists can find the introduction materials on the reflection symmetry and unorientable manifolds in LABEL:Witten2016cio1605.02391,_Barkeshli2016mew1612.07792.
For readers who wishes to explore other mathematical introductory materials, we suggest to look at the [4, 36] and
Appendices of [25].
Readers may be also interested in other recent work along the cobordism theory applications to physics
[37] [38] [39] [40].
1.3 Mathematical preliminaries
In this subsection, we review the basics of bordism theory and possible generalizations.
Definition 1**.**
If H is a group with a group homomorphism ρ:H→O, V is a vector bundle over M with a metric, then an H-structure on V is a principal
H-bundle P over M, together with an isomorphism of bundles P×HO∼BO(V) where P×HO is the quotient (P×O)/H where H acts freely on right of P×O by
[TABLE]
and BO(V) is the orthonormal frame bundle of V.
In particular, if V=TM, then an H-structure on TM is also called a tangential H-structure (or an H-structure) on M.
Here we assume the H-structures are defined on the tangent bundles instead of normal bundles.
Below we consider manifolds with a metric.
Any manifold admits an O-structure,
a manifold M admits an SO-structure if and only if w1(TM)=0, a manifold admits a Spin structure if and only if w1(TM)=w2(TM)=0,
a manifold admits a Pin+ structure if and only if w2(TM)=0, a manifold admits a Pin- structure if and only if w2(TM)+w1(TM)2=0. Here wi(TM) is the i-th Stiefel-Whitney class of the tangent bundle of M.
Moreover, we can consider manifolds equipped with a map to a fixed topological space X,
we are interested in the case when X is an Eilenberg-MacLane space since
[TABLE]
where the left hand side is the group of homotopy classes of maps from M to K(G,n), the right hand side is the n-th cohomology group of M with coefficients in G.
Definition 2**.**
Let H be a group, X be a fixed topological space, we can define an abelian group
[TABLE]
where bordism is an equivalence relation, namely, (M,f) and (M′,f′) are bordant if there exists a compact n+1-manifold N with H-structure and a map h:N→X such that the boundary of N is the disjoint union of M and M′,
the H-structures on M and M′ are induced from the H-structure on N and h∣M=f, h∣M′=f′.
In particular, when X=B2Zn, f:M→B2Zn is a cohomology class in H2(M,Zn).
When X=BG, with G is a Lie group or a finite group (viewed as a Lie group with discrete topology), then f:M→BG is a principal G-bundle over M.
To explain our notation, here BG is a classifying space of G, and B2Zn is a higher classifying space (Eilenberg-MacLane space K(Zn,2)) of Zn.
In the particular case that H=O and X is a point, this definition 2 coincides with Thom’s original definition.
In this article, we study the cases in which H=O/SO/Spin/Pin±, and X is a higher classifying space, or more complicated cases.
If ΩnH(X)=G1×G2×⋯×Gr where Gi are cyclic groups, then
group homomorphisms ϕi:ΩnH(X)→Gi form a complete set of bordism invariants if
ϕ=(ϕ1,ϕ2,…,ϕr):ΩnH(X)→G1×G2×⋯×Gr is a group isomorphism.
Elements of ΩnH(X) are manifold generators if their images in G1×G2×⋯×Gr under ϕ generate G1×G2×⋯×Gr.
We first introduce several concepts which are important for bordism theory:
Thom space: Let V→Y be a real vector bundle, and fix a Euclidean metric. The Thom space Thom(Y;V) is the quotient D(V)/S(V) where D(V) is the unit disk bundle and S(V) is the unit sphere bundle. Thom spaces satisfy
[TABLE]
where V→X and W→Y are real vector bundles, Rn is the trivial real vector bundle of dimension n, Σ is the suspension, X+ is the disjoint union of X and a point.
We follow the definition of Thom spectrum and Madsen-Tillmann spectrum given in [2].
Thom spectrum [1]:
MH is the Thom spectrum of the group H,
it is the spectrification (see 2.2) of the prespectrum whose n-th space is
MH(n)=Thom(BH(n);Vn), and Vn is the induced vector bundle (of dimension n) by the map BH(n)→BO(n).
In other words,
MH=Thom(BH;V), where V is the induced virtual bundle (of dimension [math]) by the map BH→BO.
Madsen-Tillmann spectrum [3]:
MTH is the Madsen-Tillmann spectrum of the group H, it is the colimit of ΣnMTH(n), where
MTH(n)=Thom(BH(n);−Vn), and Vn is the induced vector bundle (of dimension n) by the map BH(n)→BO(n).
The virtual Thom spectrum
MTH(n) is the spectrification (see 2.2) of the prespectrum whose (n+q)-th space is
Thom(BH(n,n+q),Qq)
where BH(n,n+q) is the pullback
[TABLE]
and there is a direct sum Rn+q=Vn⊕Qq of vector bundles over Grn(Rn+q) and, by pullback, over BH(n,n+q) where Rn+q is the trivial real vector bundle of dimension n+q.
In other words,
MTH=Thom(BH;−V), where V is the induced virtual bundle (of dimension [math]) by the map BH→BO.
Here Ω is the loop space, Σ is the suspension.
Note: “T” in MTH denotes that the H-structures are on tangent bundles instead of normal bundles.
(Co)bordism theory is a generalized (co)homology theory which is represented by a spectrum by the Brown representability theorem.
In fact, it is represented by Thom spectrum due to the Pontryagin-Thom isomorphism:
[TABLE]
In the case when tangential H-structure is the same as normal H′-structure, the relevant Thom spectra are weakly equivalent. In particular,
MTO≃MO, MTSO≃MSO, MTSpin≃MSpin, MTPin+≃MPin−, MTPin−≃MPin+.
Pin± cobordism groups are not rings, though they are modules over the Spin cobordism ring.
By the generalized Pontryagin-Thom construction, for X a topological space, then the group of H-bordism classes of H-manifolds in X is isomorphic to the generalized homology of X with coefficients in MTH:
[TABLE]
where
πd(MTH∧X+) is the d-th stable homotopy group of the spectrum MTH∧X+.
The d-th stable homotopy group of a spectrum M is
[TABLE]
So the computation of the bordism group ΩdH(X) is the same as the computation of the stable homotopy group of the spectrum MTH∧X+ which can be computed by Adams spectral sequence method.
Next, we introduce the
Thom isomorphism [1]:
Let p:E→B be a real vector bundle of rank n. Then there is an isomorphism, called Thom isomorphism
[TABLE]
where H~ is the reduced cohomology, T(E)=Thom(E;B) is the Thom space and
[TABLE]
where U is the Thom class.
We can define the i-th Stiefel-Whitney class of the vector bundle p:E→B by
[TABLE]
where Sq is the Steenrod square.
1.4 Basics of Higher Symmetries and Higher Anomalies of Quantum Field Theory for Physicists and Mathematicians
In order to obtain a complete classification of ’t Hooft anomalies of quantum field theories (QFTs), we aim to first identify the relevant (if not all of) global symmetry G
(here we will abuse the notation to have G including the higher symmetry G)
of QFTs.
Then we couple the QFTs to classical background-symmetric gauge field of G. Then we try to detect the possible obstructions of such
coupling [13]. Such obstructions, known as the obstruction of gauging the global symmetry, are termed
“ ’t Hooft anomalies” in QFT.
In the literature, when people refer to “anomalies,” however, they can means several related but different issues.
To fix our terminology, we refer “anomalies” to be one of the followings:
Classical global symmetry is violated at the quantum theory, such that the
classical global symmetry fails to survive as a quantum global symmetry, e.g. the original Adler-Bell-Jackiw (ABJ) anomaly [21, 22].
2. 2.
Quantum global symmetry is well-defined and preserved (for the Hamiltonian or path integral Lagrangian formulation of quantum theory).
Namely, global symmetry is sensible, not only at a classical theory (if there is any classical description), but also for a quantum theory.
However, there is an obstruction to gauge the global symmetry.
Specifically, we can detect a certain obstruction to even weakly gauge
the symmetry or couple the symmetry to a non-dynamical background probed gauge field.
(We may abbreviate this background field as “bgd.field.”)
This is known as “’t Hooft anomaly,” or sometimes
regarded as a “weakly gauged anomaly” in condensed matter.
Namely, the partition function Z does not sum over background gauge connections, but only fix a background gauge connection
and only depend on the background gauge connection as a classical field (as a classical coupling constant).
Say if the background gauge connection is A, the partition function is Z[A] depending on A.
Normally, the Z[A] on a closed manifold in its own dimension is an invertible topological QFT (iTQFT), such that
Z[A]=exp(iθ(A)) is a complex phase (thus physically meaningfully invertible) while its absolute value ∣Z[A]∣=1 for any choice of background A.
3. 3.
Quantum global symmetry is well-defined and preserved (for the Hamiltonian or path integral Lagrangian formulation of quantum theory).
However, once we promote the global symmetry to a gauge symmetry of the dynamical gauge theory,
then the gauge theory becomes inconsistent. Some people call this as
a “dynamical gauge anomaly” which makes a quantum theory inconsistent.
Namely, the partition function Z after summing over dynamical gauge connections becomes inconsistent or ill-defined.
From now on, when we simply refer to “anomalies,” we mean mostly “’t Hooft anomalies,”
which still have several intertwined interpretations:
Interpretation (1): In condensed matter physics, “’t Hooft anomalies” are known as the obstruction to lattice-regularize the global symmetry’s quantum operator in a strictly
local manner. By claiming local on a lattice or on a simplicial complex,
we mean:
∙ on-site (e.g. on 0-simplex) for which 0-form symmetry operator acts on.
∙ on-link (e.g. on 1-simplex) for which 1-form symmetry operator acts on.
∙ on-plaquette (e.g. on 2-simplex) for which 2-form symmetry operator acts on.
…
∙ on n-simplex for which n-form symmetry operator acts on.
This obstruction is due to the symmetry-twists (See [Ref. [41, 42, 43]] for QFT-oriented discussion and references therein).
This obstruction can be detected at high energy lattice scale (known as the ultraviolet [UV] in QFT).
This “non-onsite symmetry” viewpoint is generically applicable to both, perturbative anomalies, and non-perturbative global anomalies:
∙Perturbative anomalies — Characterized and captured by perturbative Feynman diagram calculations. Classified by an infinite integer Z class,
known as the free (sub)group.
∙Non-perturbative or global anomalies —
Examples of global anomalies include the old and the new SU(2) anomalies [23, 24]
(here we mean their ’t Hooft anomaly analogs if we view the SU(2) gauge field as a non-dynamical classical background field)
and the global gravitational anomalies [44].
These are classified by a finite group Zn class, known as the torsion (sub)group.
These anomalies are sensitive to the underlying UV-completion
not only of fermionic systems, but also of bosonic systems [45, 42, 46, 47].
We term the anomalies of QFT whose UV-completion requires only the bosonic degrees of freedom as bosonic anomalies [45].
While we term those must require fermionic degrees of freedom as fermionic anomalies.
Interpretation (2): In QFTs, the obstruction is on the impossibility of adding any counter term
in its own dimension (d-d) in order to absorb a one-higher-dimensional counter term (e.g. (d+1)d topological term) due to background G-field [48].
This is named the “anomaly-inflow [14].”
The (d+1)d topological term is known as the (d+1)d SPTs in condensed matter physics [5, 49].
Interpretation (3): In math, the dd anomalies can be systematically captured by (d+1)d topological invariants [23] known as
bordism invariants [50, 26, 27, 4].
∙ Bosonic anomalies or bosonic SPTs are normally characterized by topological terms detected via manifolds with H=SO (orientable) or O (unorientable) structures.
∙ Fermionic anomalies or fermionic SPTs
are normally characterized by topological terms detected via manifolds with H=Spin (orientable) or Pin± (unorientable) structures.
Below we summarize the higher symmetry G systematically introduced in [9].
(i) Higher symmetries and higher anomalies:
The ordinary 0-form global symmetry has a charged object of 0d measured by the charge operator of (d−1)d.
The generalized q-form global symmetry is introduced by LABEL:Gaiotto2014kfa1412.5148.
A charged object of qd is measured by the charge operator of (d−q−1)d (i.e. codimension-(q+1)). This concept turns out to be powerful to
detect new anomalies, e.g. the pure SU(N)-YM at θ=π
has a mixed anomaly between 0-form time-reversal symmetry Z2T
and 1-form center symmetry ZN,[1] at an even integer N, firstly discovered in a remarkable work [Ref. [16]].
(ii) Relate (higher)-SPTs to (higher)-topological invariants:
In the condensed matter literature, based on the earlier discussion on the symmetry twist,
it has been recognized that the classical background-field partition function under the symmetry twist, called Zsym.twist
in (d+1)d
can be regarded as the partition function of (d+1)d SPTs ZSPTs.
These descriptions are applicable to both low-energy infrared (IR) field theory, but also to the UV-regulated SPTs on a lattice,
see [Ref. [41, 42, 26]]
and References therein. Schematically, we follow the framework of [42],
[TABLE]
In general,
the partition function Zsym.twist=ZSPTs[A1,B2,wi,…]
is a functional containing background gauge fields of 1-form A1, 2-form B2 or higher forms;
and can contain characteristic classes [51] such as the i-th Stiefel-Whitney class (wi)
and other geometric probes such as gravitational background fields, e.g. a gravitational Chern-Simons 3-form CS3(Γ) involving the Levi-Civita connection or
the spin connection Γ.
For our convention, we use the capital letters (A,B,...) to denote non-dynamical background gauge fields (which, however, later they may or may not be dynamically gauged), while
the little letters (a,b,...) to denote dynamical gauge fields.
More generally,
∙ For the ordinary 0-form symmetry, we may couple the charged 0d point operator to 1-form background gauge field (so the symmetry-twist occurs
in the Poincaré dual codimension-1 sub-spacetime [dd] of SPTs).
∙ For the 1-form symmetry, we may couple the charged 1d line operator to 2-form background gauge field (so the symmetry-twist occurs in the Poincaré dual codimension-2 sub-spacetime [(d−1)d] of SPTs).
∙ For the q-form symmetry, we may couple the chargedqd extended operator to (q+1)-form background gauge field.
The chargedqd extended operator can be measured by another charge operator of codimension-(q+1) [i.e. (d−q)d].
In summary, for the q-dimensional symmetry, we use the following terminology:
(♢ 1):
Charged object: The chargedq-dimensional extended operator as the q-dimensional-symmetry generator which is being measured
by a symmetry generator.
2. (♢ 2):
Charge operator: The corresponding charge operator of codimension-(q+1) [i.e. (d−q)-dimension] which measures the q-dimensional-symmetry
charged object.
So the symmetry-twist can be interpreted as the occurrence of the codimension-(q+1)charge operator.
In other words, the symmetry-twist happens at a Poincaré dual codimension-(q+1) sub-spacetime [(d−q)d] of SPTs.
We shall view the measurement of a chargedqd extended object, happening at any q-dimensional intersection
between the (q+1)d form background gauge field
and the codimension-(q+1) symmetry-twist or charge operator of this SPT vacua.
By higher-SPTs, we mean SPTs protected by higher symmetries (for generic q, especially for any SPTs with at least a symmetry of q>0).
So our principle above is applicable to higher-SPTs [12, 30].
In the following of this article, thanks to (1.25), we can interchange the usages and interpretations of
“higher SPTs ZSPTs,” “higher topological terms due to symmetry-twist {\mathbf{Z}}^{\text{(d+1)d}}_{{\text{sym.twist}}},” “higher topological invariants {\mathbf{Z}}^{\text{(d+1)d}}_{{\text{topo.inv}}}” or “bordism invariants
{\mathbf{Z}}^{\text{(d+1)d}}_{{\text{Cobordism.inv}}}” in (d+1)d.
They are all physically equivalent, and can uniquely determine a dd higher anomaly: if we study the anomaly of any
boundary theory of the (d+1)d higher SPTs living on a manifold with dd boundary.
Thus, we regard all of them as physically tightly-related given by (1.25).
By turning on the classical background probed field (denoted as “bgd.field” in (1.26)) coupled to dd QFT, under the symmetry transformation (i.e. symmetry twist),
its partition function {\mathbf{Z}}^{\text{dd}}_{{\text{QFT}}}
can be shifted
[TABLE]
to detect the underlying (d+1)d topological terms/counter term/SPTs, namely the (d+1)d partition function {\mathbf{Z}}^{\text{(d+1)d}}_{{\text{SPTs}}}.
To check whether the underlying (d+1)d SPTs really specifies a true dd ’t Hooft anomaly unremovable from dd counter term,
it means that {\mathbf{Z}}^{\text{(d+1)d}}_{{\text{SPTs}}}(\text{bgd.field}) cannot be absorbed by a lower-dimensional SPTs
{\mathbf{Z}}^{\text{dd}}_{{\text{SPTs}}}(\text{bgd.field}), namely
We explain the convention for our notations and terminology below.
Most of our conventions follow [4] and [25].
•
We denote O the stable orthogonal group,
SO the stable special orthogonal group,
Spin the stable spin group,
and Pin± the two ways of Z2 extension (related to the time reversal symmetry) of Spin group.
•
Zn is the finite cyclic group of order n, n is a positive integer.
•
A map between topological spaces is always assumed to be continuous.
•
For a (pointed) topological space X, Σ denotes a suspension ΣX=S1∧X=(S1×X)/(S1∨X) where ∧ and ∨ are smash product and wedge sum (one point union) of pointed topological spaces respectively.
For a graded algebra A, A=⨁iAi, ΣA is the graded algebra defined by ΣA=⨁i(ΣA)i where (ΣA)i=Ai−1.
•
For a (pointed) topological space X with the base point x0, ΩX is the loop space of X:
[TABLE]
•
Let R be a ring, M a topological space,
H∗(M,R) is the cohomology ring of M with coefficients in R.
•
We will abbreviate the cup product x∪y by xy.
•
If Md (or simply M) is a d-dimensional manifold, then TMd (or simply TM) is the tangent bundle over Md (or M).
•
Rank r real (complex) vector bundle V is a bundle with fibers being real (complex) vector spaces of real (complex) dimension r.
•
wi(V) is the i-th Stiefel-Whitney class of a real vector bundle V (which may be also complex rank r but considered as real rank 2r).
•
pi(V) is the i-th Pontryagin class of a real vector bundle V.
•
ci(V) is the i-th Chern class of a complex vector bundle V.
Pontryagin classes are closely related to Chern classes via complexification:
[TABLE]
where V⊗RC is the complexification of the real vector bundle V.
The relation between Pontryagin classes and Stiefel-Whitney classes is
[TABLE]
•
For a top degree cohomology class
we often suppress explicit integration over the manifold (i.e. pairing with the fundamental class [M]).
If M is orientable, then [M] has coefficients in Z.
If M is non-orientable, then [M] has coefficients in Z2.
•
If x is an element of a graded vector space, ∣x∣ denotes the degree of x.
•
For an odd prime p and a non-negatively and integrally graded vector space V over Zp, let Veven and Vodd be even and odd graded parts of V . The free algebra FZp[V] generated by the graded vector space V is the tensor product of the polynomial algebra on Veven and the exterior algebra on Vodd:
[TABLE]
We sometimes replace the vector space with a set of bases of it.
•
Ap denotes the mod p Steenrod algebra where p is a prime.
•
Sqn is the n-th Steenrod square, it is an element of A2.
•
A2(1) denotes the subalgebra of A2 generated by Sq1 and Sq2.
•
β(n,m):H∗(−,Zm)→H∗+1(−,Zn) is the Bockstein homomorphism associated to the extension Zn→⋅mZnm→Zm, when n=m=p is a prime, it is an element of Ap. If p=2, then β(2,2)=Sq1.
•
Ppn:H∗(−,Zp)→H∗+2n(p−1)(−,Zp) is the n-th Steenrod power, it is an element of Ap where p is an odd prime. For odd primes p, we only consider p=3, so we abbreviate P3n by Pn.
•
P2 is the Pontryagin square operation H2i(M,Z2k)→H4i(M,Z2k+1).
Explicitly,
P2
is given by
[TABLE]
and it satisfies
[TABLE]
Here 1∪ is the higher cup product.
•
Postnikov square
P3:H2(−,Z3k)→H5(−,Z3k+1)
is given by
[TABLE]
where β(3k+1,3k) is the Bockstein homomorphism associated to 0→Z3k+1→Z32k+1→Z3k→0.
•
For a finitely generated abelian group G and a prime p, Gp∧=limnG/pnG is the p-completion of G.
•
For a topological space M, πd(M) is the d-th (ordinary) homotopy group of M.
•
For an abelian group G, the Eilenberg-MacLane space K(G,n) is a space with homotopy groups satisfying
[TABLE]
•
Let X, Y be topological spaces, [X,Y] is the set of homotopy classes of maps from X to Y.
•
Let G be a group, the classifying space of G, BG is a topological space such that
[TABLE]
for any topological space X. In particular, if G is an abelian group, then BG is a group.
•
There is a vector bundle associated to a principal G-bundle PG: PG×GV=(PG×V)/G which is the quotient of PG×V by the right G-action
[TABLE]
where V is the vector space which G acts on.
For characteristic classes of a principal G-bundle, we mean the characteristic classes of the associated vector bundle.
1.6 Tables and Summary of Some Co/Bordism Groups
Below we use the following notations, all cohomology class are pulled back to the d-manifold M along the maps given in the definition of cobordism groups:
∙wi is the Stiefel-Whitney class of the tangent bundle of M,
∙a is the generator of H1(BZ2,Z2),
∙a′ is the generator of H1(BZ3,Z3), b′=β(3,3)a′,
∙x2 is the generator of H2(B2Z2,Z2), x3=Sq1x2, x5=Sq2x3,
∙x2′ is the generator of H2(B2Z3,Z3), x3′=β(3,3)x2′,
∙wi′=wi(PSU(2))∈Hi(BPSU(2),Z2) is the Stiefel-Whitney class of the principal PSU(2) bundle,
∙ci=ci(PSU(3))∈H2i(BPSU(3),Z) is the Chern class of the principal PSU(3) bundle,
∙z2=w2(PSU(3))∈H2(BPSU(3),Z3) is the generalized Stiefel-Whitney class of the principal PSU(3) bundle, z3=β(3,3)z2.
Conventions:
All product between cohomology classes are cup product, product between a cohomology class x and η~ (or Arf, ABK, etc) means the value of η~ (or Arf, ABK, etc) on the submanifold of M which represents the Poincaré dual of x.
In Section 4.1, we compute the topological terms (involving the cohomology classes of B2Z2) of Ω5G where G is a 2-group with Ga=O, Gb=Z2
We find that the term x2w3 (or x3w2) survives only for β=0,w13 (the Postnikov class β∈H3(BO,Z2)=Z23 which is generated by w13,w1w2,w3). This term also appears in eq. 2.57 of [17].
Let X be a topological space, an n-simplex of X is a map σ:Δn→X where
[TABLE]
it is denoted by [v0,…,vn] where vi are vertices of Δn.
n-simplexes of X generates an abelian group Cn(X), the elements of Cn(X) are called n-chains.
Δn−1 embeds in Δn in the canonical way, define ∂:Cn(X)→Cn−1(X) by
[TABLE]
It is easy to verify that ∂2=0, so (C∙(X),∂) is a chain complex.
Let G be an abelian group, let Cn(X,G):=Hom(Cn(X),G), the elements of Cn(X,G) are called n-cochains with coefficients G. Define δ:Cn(X,G)→Cn+1(X,G) by
δ(α)(σ)=α(∂(σ)), then δ2=0, so (C∙(X,G),δ) is a cochain complex.
Hn(X,G) is defined to be Imδ:Cn−1(X,G)→Cn(X,G)Kerδ:Cn(X,G)→Cn+1(X,G).
It is an abelian group, called the n-th cohomology group of X with coefficients G, the elements of the abelian group Zn(X,G):=Kerδ:Cn(X,G)→Cn+1(X,G) are called n-cocycles, the elements of Bn(X,G):=Imδ:Cn−1(X,G)→Cn(X,G) are called n-coboundaries.
By abusing the notation, we also use [v0,…,vn] to denote an n-chain.
If G is additionally a ring R, then we can define a cup product such that H∗(X,R) is a graded ring.
First we define the cup product of two cochains:
[TABLE]
[TABLE]
where ⋅ is the multiplication in R.
The cup product satisfies
[TABLE]
α∪β is a cocycle if both α and β are cocycles.
If both α and β are cocycles, then α∪β is a coboundary if
one of α and β is a coboundary. So the cup product is also an
operation on cohomology groups ∪:Hn(X,R)×Hm(X,R)→Hn+m(X,R). The cup product of two cocycles satisfies
[TABLE]
For the convenience of defining higher cup product, we use the notation i→j for the consecutive sequence from i to j
[TABLE]
We also denote an n-chain by (0→n). We use ⟨α,σ⟩ to denote the value of α(σ) for n-cochain α and n-chain σ.
Let fm be an m-cochain, hn be an n-cochain, we define higher cup product fmk∪hn which yields an (m+n−k)-cochain:
[TABLE]
and fmk∪hn=0 for k>m or n or k<0.
Here
i→j is the sequence i,i+1,⋯,j−1,j, and
p is the number of transpositions (it is not unique but its parity is unique) in the decomposition of the permutation to bring the sequence
[TABLE]
to the sequence
[TABLE]
For example
[TABLE]
We can see that 0∪=∪.
Unlike cup product at k=0, the higher cup product of two
cocycles may not be a cocycle.
Steenrod studied the higher cup product of cochains and found a formula [55, Theorem 5.1]:
[TABLE]
where u is a p-cochain, v is a q-cochain.
Also Steenrod defined Steenrod square using higher cup product:
[TABLE]
2.1.2 Universal coefficient theorem and Künneth formula
If X is a topological space,
R is a principal ideal domain (Z or a field), G is an R-module, then the homology version of universal coefficient theorem is
[TABLE]
The cohomology version of universal coefficient theorem is
[TABLE]
We will abbreviate Tor1Z by Tor, ExtZ1 by Ext.
If X and X′ are topological spaces,
R is a principle ideal domain and G,G′ are R-modules such that
Tor1R(G,G′)=0.
We also require either
(1) Hn(X;Z) and Hn(X′;Z) are finitely generated, or
(2) G′ and Hn(X′;Z) are
finitely generated.
The homology version of Künneth formula is
[TABLE]
The cohomology version of Künneth formula is
[TABLE]
Note that Z and R are principal ideal
domains, while R/Z is not. Also, R and R/Z are not finitely
generate R-modules if R=Z.
Special cases: 1. R=G′=Z.
In this case, the
condition Tor1R(G,G′)=Tor1Z(G,Z)=0 is always
satisfied. G can be R/Z, Z, Zn *etc *. So we have
[TABLE]
Take X to be the space of one point in (2.1.2),
and use
where X′ is renamed as X. This is also called the universal coefficient
theorem which can be used to calculate H∗(X,G) from H∗(X;Z) and the
module G.
Here Tor=Tor1Z.
Homology version of (2.20) is just the universal coefficient theorem for homology with R=Z.
R=G=G′=F is a field, Tor1R(G,G′)=0.
[TABLE]
This is called the Künneth formula.
There is also a relative version of Künneth formula [52, Theorem 3.18]:
[TABLE]
Here X∧X′ is the smash product, H~ is the reduced cohomology.
2.2 Spectra
Definition 3**.**
∙
A prespectrum T∙ is a sequence {Tq}q∈Z≥0 of pointed spaces and maps sq:ΣTq→Tq+1.
∙
An Ω-prespectrum is a prespectrum T∙ such that the adjoints tq:Tq→ΩTq+1 of the structure maps are weak homotopy equivalences.
∙
A spectrum is a prespectrum T∙ such that the adjoints tq:Tq→ΩTq+1 of the structure maps are homeomorphisms.
Example 4**.**
∙ Let X be a pointed space, Tq=ΣqX for q≥0, then T∙ is a prespectrum.
∙Tq=Sq, T∙ is a prespectrum.
∙ Let G be an abelian group, Tq=K(G,q) the Eilenberg-MacLane space, T∙ is an Ω-prespectrum.
Spectrification: Let T∙ be a prespectrum, define
(LT)q to be the colimit of
[TABLE]
Namely,
[TABLE]
then (LT)∙ is a spectrum.
Example 5**.**
∙Tq=Sq, (LT)∙ is a spectrum S.
∙ Let G be an abelian group, Tq=K(G,q) the Eilenberg-MacLane space, (LT)∙ is a spectrum HG (the Eilenberg-MacLane spectrum).
Stable homotopy groups of spectra:
Let M∙ be a spectrum, define
πdM∙ to be the colimit of
[TABLE]
Namely,
[TABLE]
Maps between spectra: If M∙,N∙ are two spectra, then for any integer k, the abelian group of homotopy classes of maps from M∙ to N∙ of degree −k: [M∙,N∙]−k is defined as follows: a map in
[M∙,N∙]−k is a sequence of maps Mn→Nn+k such that the following diagram commutes
[TABLE]
where the columns are the structure maps of the spectra M∙ and N∙.
If in addition the spectrum N∙ is a ring spectrum, then the abelian groups
[M∙,N∙]−k form a graded ring [M∙,N∙]−∗.
Example 6**.**
πdM∙=[S,M∙]d.
Cohomology rings of spectra:
Definition 7**.**
A ring spectrum is a spectrum E along with a unit map η:S→E and a multiplication map μ:E∧E→E.
Example 8**.**
Let R be a ring, then the Eilenberg-MacLane spectrum HR is a ring spectrum.
The cohomology ring of a spectrum M∙ with coefficients in R is defined to be [M∙,HR]−∗.
2.3 Spectral sequences
In this paper, we use three kinds of spectral sequence: Adams spectral sequence, Atiyah-Hirzebruch spectral sequence, and Serre spectral sequence.
2.3.1 Adams spectral sequence
The Adams spectral sequence is a spectral sequence introduced by Adams in [56], it is of the form
[TABLE]
where Y is any spectrum.
We consider Y=MTH∧X+ and focus on p=2 and p=3.
We introduce the notions used in Adams spectral sequence:
p-completion: For any finitely generated abelian group G, Gp∧=limnG/pnG is the p-completion of G. If G is finite, then Gp∧ is the Sylow p-subgroup of G. If G=Z, Gp∧ is the ring of p-adic integers.
Steenrod algebra:
The mod p Steenrod algebra is Ap:=[HZp,HZp]−∗ where HZp is the mod p Eilenberg-MacLane spectrum.
For any spectrum Y, the cohomology ring H∗(Y,Zp)=[Y,HZp]−∗ is an Ap-module.
For p=2, the generators of A2 are Steenrod squares Sqn.
Definition 9** (Axioms).**
For each i≥0, there is a natural transformation
[TABLE]
such that
∙Sq0=Id
∙
If i>∣x∣, then Sqix=0
∙
If i=∣x∣, then Sqix=x2
∙
(Cartan formula)
Sqn(xy)=∑i+j=nSqi(x)Sqj(y)
∙
(Adem relation)
If a<2b, then
[TABLE]
The subalgebra A2(1) of A2 generated by Sq1 and Sq2 looks like Figure 1.
Each dot stands for a Z2, all relations are from Adem relations (2.111).
For odd primes p, the generators of Ap are
the Bockstein homomorphism β(p,p) and Steenrod powers Ppn.
Ext functor: Let R=Ap or A2(1).
ExtRs,t is the internal degree t part of the s-th derived functor of HomR∗.
In general, we can find a projective R-resolution P∙ of L to compute ExtRi(L,Zp), ExtRi(L,Zp)=Hi(HomR(P∙,Zp)) (the i-th cohomology of the chain complex HomR(P∙,Zp)).
In Adams chart, the horizontal axis is degree t−s and the vertical axis is degree s. The differential drs,t:Ers,t→Ers+r,t+r−1 is an arrow starting at the bidegree (t−s,s) with direction (−1,r).
Er+1s,t:=Imdrs−r,t−r+1Kerdrs,t for r≥2. There exists N such that EN+k=EN for k>0, denote E∞:=EN.
We explain how to read the result from the Adams chart: In the E∞ page, one dot indicates a Zp, an vertical line connecting n dots indicates a Zpn, when n=∞, the line indicates a Z.
In the H=O cases, MO is the wedge sum of suspensions of the Eilenberg-MacLane spectrum HZ2, H∗(MO,Z2) is the direct sum of suspensions of the Steenrod algebra A2.
H∗(MO∧X+,Z2)=H∗(MO,Z2)⊗H∗(X,Z2) is also the direct sum of suspensions of the Steenrod algebra A2.
We have used the Künneth formula (2.22).
Let L=H∗(MO∧X+,Z2), then P0=L, Ps=0 for s>0 gives a projective A2-resolution of L.
Since
[TABLE]
all dots are concentrated in s=0 in the Adams chart of ExtA2s,t(H∗(MO∧X+,Z2),Z2), there are no differentials, E2=E∞,
ΩdO(X) is a Z2-vector space.
In the H=SO cases, the localization of MSO at the prime 2 is
[TABLE]
where HZ is the Eilenberg-MacLane spectrum and H∗(HZ,Z2)=A2/A2Sq1.
[TABLE]
is an A2-resolution (denoted by P∙) where the differentials d1 are induced by Sq1.
When X is a point, the Adams chart of ExtA2s,t(H∗(MSO,Z2),Z2) is shown in Figure 2. For general X, P∙⊗H∗(X,Z2) is a projective A2-resolution of H∗(HZ,Z2)⊗H∗(X,Z2) (since P∙ is actually a free A2-resolution), the differentials d1 are induced by Sq1.
The localization of MSO at the prime 3 is the wedge sum of suspensions of the Brown-Peterson spectrum BP (MSO(3)=BP∨Σ8BP∨⋯) and H∗(BP,Z3)=A3/(β(3,3)) where (β(3,3)) is the two-sided ideal generated by β(3,3).
[TABLE]
is an A3-resolution of A3/(β(3,3)) (denoted by P∙′) where the differentials d1 are induced by β(3,3).
When X is a point, the Adams chart of ExtA3s,t(H∗(MSO,Z3),Z3) is shown in Figure 3. For general X, P∙′⊗H∗(X,Z3) is a projective A3-resolution of H∗(BP,Z3)⊗H∗(X,Z3) (since P∙′ is actually a free A3-resolution), the differentials d1 are induced by β(3,3).
There may be differentials dn corresponding to the Bockstein homomorphism β(p,pn) [57] for both p=2 and p=3.
See 2.5 for the definition of Bockstein homomorphisms.
Since MSO(3)=MSpin(3), ΩdSO(X)3∧=ΩdSpin(X)3∧.
In the H=Spin/Pin± cases, since the mod 2 cohomology of the Thom spectrum MSpin is
[TABLE]
where M is a graded A2(1)-module with the degree i homogeneous part Mi=0 for i<8.
Theorem 10** (Change of rings/Frobenius reciprocity).**
[TABLE]
When we compute ΩdH(X)2∧,
we are reduced to compute ExtA2(1)s,t(L,Z2) for t−s<8, where L is some A2(1)-module (our cases are some mod 2 cohomology H∗(−,Z2)).
The only possible differentials are dr(h1)=h0r+1 where h0∈ExtA2(1)1,1(Z2Z2), h1∈ExtA2(1)1,2(Z2Z2). If there were such a differential dr for r≥2, then since h0h1=0, 0=dr(h0h1)=h0r+2 which is not true. Hence E2=E∞.
This is in fact real Bott periodicity (πi+8ko=πiko, H∗(ko,Z2)=A2⊗A2(1)Z2).
Our computation is based on the following fact:
Lemma 11**.**
Given a short exact sequence of A2(1)-modules
[TABLE]
then for any t, there is a long exact sequence
[TABLE]
After using this fact repeatedly, we obtain the E2 page.
Example 3:
[TABLE]
is a short exact sequence where the left dot is L1, the middle part is L2, the right dot is L3.
Given a fibration F→E→B, the Serre spectral sequence is the following:
[TABLE]
This can be used in computing the integral cohomology group of the total space of a nontrivial fibration.
There is also a homology version:
[TABLE]
2.3.3 Atiyah-Hirzebruch spectral sequence
The Atiyah-Hirzebruch spectral sequence can be viewed as a generalization of the Serre spectral sequence. Given a fibration F→E→B, the Atiyah-Hirzebruch spectral sequence is the following:
[TABLE]
where h∗ is an extraordinary homology theory. For example, h∗ can be the bordism theory Ω∗H. In particular, if the fiber F is a point, then the Atiyah-Hirzebruch spectral sequence is of the form:
[TABLE]
2.4 Characteristic classes
2.4.1 Introduction to characteristic classes
Characteristic classes are cohomology classes of the base space of a vector bundle.
Stiefel-Whitney classes are defined for real vector bundles, Chern classes are defined for complex vector bundles, Pontryagin classes are defined for real vector bundles.
All characteristic classes are natural with respect to bundle maps. Characteristic classes of a principal bundle are defined to be the characteristic classes of the associated vector bundle of the principal bundle.
Given a real vector bundle V→M and a complex vector bundle E→M, the i-th Stiefel-Whitney class of V is
wi(V)∈Hi(M,Z2), the i-th Chern class of E is ci(E)∈H2i(M,Z), the i-th Pontryagin class of V is pi(V)∈H4i(M,Z).
Pontryagin classes are closely related to Chern classes via complexification:
[TABLE]
where V⊗RC→M is the complexification of the real vector bundle V→M.
The relation between Pontryagin classes and Stiefel-Whitney classes is
[TABLE]
For a manifold M, the integrals over M of characteristic classes of a vector bundle over M (the pairing of the characteristic classes with the fundamental class of M) are called characteristic numbers.
Let En be the universal n-bundle over BO(n), the colimit of En−n is a virtual bundle E (of dimension 0) over BO, the pullback of E along the map g:M→BO given by the O-structure on M is just TM−d where M is a
d-manifold and TM is the tangent bundle of M. By the naturality of characteristic classes, the pullback of the characteristic classes of E is the characteristic classes of TM.
Chern-Simons form:
By Chern-Weil theory, Chern classes (and Pontryagin classes) can also be defined as a closed differential form (in de Rham cohomology). By Poincaré Lemma, they are exact locally:
[TABLE]
where d is the exterior differential operator, CS2n−1 is called the Chern-Simons 2n−1-form.
Whitney sum formula:
Let w(V)=1+w1(V)+w2(V)+⋯∈H∗(M,Z2) be the total Stiefel-Whitney class,
c(E)=1+c1(E)+c2(E)+⋯∈H∗(M,Z) be the total Chern class,
p(V)=1+p1(V)+p2(V)+⋯∈H∗(M,Z) be the total Pontryagin class,
then
[TABLE]
[TABLE]
[TABLE]
That is, the total Stiefel-Whitney class and the total Chern class are multiplicative with respect to Whitney sum of vector bundles, the total Pontryagin class is multiplicative modulo 2-torsion with respect to Whitney sum of vector bundles.
2.4.2 Wu formulas
The total Stiefel-Whitney class w=1+w1+w2+⋯ is related to the
total Wu class u=1+u1+u2+⋯ through the total Steenrod square:
[TABLE]
Therefore,
wn=∑i=0nSqi(un−i).
The Steenrod squares satisfy:
[TABLE]
for any xj∈Hj(Md;Z2).
Thus
[TABLE]
This allows us to compute un iteratively, using Wu formula
[TABLE]
and the Steenrod relation
[TABLE]
We find
[TABLE]
On the tangent bundle of Md, the corresponding Wu class and the
Steenrod square satisfy
[TABLE]
This is also called Wu formula.
2.5 Bockstein homomorphisms
In general, given a chain complex C∙ and a short exact sequence of abelian groups:
[TABLE]
we have a short exact sequence of cochain complexes:
[TABLE]
Hence we obtain a long exact sequence of cohomology groups:
[TABLE]
the connecting homomorphism ∂ is called Bockstein homomorphism.
For example,
β(n,m):H∗(−,Zm)→H∗+1(−,Zn) is the Bockstein homomorphism associated to the extension Zn→⋅mZnm→Zm.
Let ρ(nm,m):H∗(−,Znm)→H∗(−,Zm) be the mod m reduction map, then β(n,m)ρ(nm,m)=0 by the long exact sequence.
In particular, β(2,2)ρ(4,2)=0.
Relations between the Bockstein homomorphisms:
If we have a chain complex C∙ and a commutative diagram of abelian groups with exact rows:
[TABLE]
then we have a commutative diagram of cochain complexes with exact rows:
[TABLE]
By the naturality of the connecting homomorphism [58, Theorem 6.13], we have a commutative diagram of abelian groups with exact rows:
[TABLE]
There are commutative diagrams:
[TABLE]
[TABLE]
By (2.86), we have the following commutative diagrams:
[TABLE]
[TABLE]
Hence we have
[TABLE]
and
[TABLE]
By definition,
[TABLE]
where δ is the coboundary map.
Moreover, Sq1=β(2,2).
By (2.108), β(2,4)=ρ(4,2)β(4,4), thus
β(2,2)β(2,4)=β(2,2)ρ(4,2)β(4,4)=0.
Similarly, β(2,8)=ρ(4,2)β(4,8), thus
β(2,2)β(2,8)=β(2,2)ρ(4,2)β(4,8)=0, etc.
Combining this with the Adem relation Sq1Sq1=0, we obtain the important formula:
[TABLE]
2.6 Useful fomulas
Adem relations:
[TABLE]
for 0<a<2b.
In particular, we have Sq1Sq1=0, Sq1Sq2Sq1=Sq2Sq2.
Recall that
[TABLE]
where a is the generator of H1(BZ2,Z2).
[TABLE]
where x2 is the generator of H2(B2Z2,Z2), x3=Sq1x2, x5=Sq2x3, x9=Sq4x5, etc.
[TABLE]
where wi′ is the i-th Stiefel-Whitney class of the universal PSU(2)=SO(3) bundle.
Combining (2.68) and (2.66), we have the following useful formulas in the presentation of cobordsim invariants:
[TABLE]
where wi is the i-th Stiefel-Whitney class of the tangent bundle of M, all cohomology classes are pulled back to M along the maps given in the definition of cobordism groups.
3 Warm-Up Examples
3.1 Perturbative chiral anomalies in even dd
and associated Chern-Simons (d+1)-form theories — SO- and Spin- Cobordism groups of BU(1)
3.1.1 Perturbative bosonic/fermionic anomaly in an even dd and U(1) SPTs in an odd (d+1)d
We start from a warming-up example familiar to most physicists and quantum field theorists: the perturbative anomalies that can be captured by
a 1-loop calculation via Feynman-Dyson diagrams involved with a U(1) group.
Of course, our discussion on the U(1) group can be generalized to any compact semi-simple Lie group such as SU(N), although we focus mostly on U(1) in this section.
We will consider a Dirac fermion theory in any even dimensional spacetime, denoted as dd with d as the even integer (say d=2,4,6,8,10,…).
The Dirac fermion Ψ (or a complex Dirac spinor) is in a 2[d/2]-dimensional spinor representation of Spin(1,d−1) (or Spin(d) in the Euclidean signature,
where Spin(d)/Z2F=SO(d), with the continuous spacetime rotational symmetry SO(d) and
the fermion parity Z2F symmetry acts on any fermion Ψ→−Ψ).
The Dirac fermion Ψ can be coupled to non-dynamical U(1) or dynamical U(1) gauge fields, as Model (1) and Model (2) below respectively.
Model (1):
The U(1) is treated as a U(1) global internal symmetry for the ’t Hooft anomaly.
The so-called path integral or partition function Z of this Dirac fermion theory is defined as a functional integral (here in Minkowski signature):
[TABLE]
where DA is the Dirac operator endorsed with the Feynman slash notation, defined as:
[TABLE]
The g is the coupling constant for the non-dynamical 1-form U(1) gauge field A:=Aμdxμ.
The γμ with μ=0,1,…,d−1 are so-called
gamma matrices, satisfying the Clifford algebra Cℓ1,d−1(R) under the anti-commutator constraint:
[TABLE]
The standard Dirac matrices correspond to d=2[d/2]=4.
The ημν is the Minkowski metric
[TABLE]
with one + sign and (d−1)− sign along the diagonal.
The hermitian chiral matrix γChiral≡γFIVE can be defined for even d dimensions
[TABLE]
2. Model (2):
The U(1) is treated as a U(1) gauge group for the dynamical gauge anomaly.
The so-called path integral or partition function Z of this Dirac fermion coupled to a dynamicsl U(1) gauge field
theory is defined as a functional integral (here in Minkowski signature):
[TABLE]
Here the dynamical 1-form U(1) gauge field A is integrated over in the path integral measure ∫[DA] as a dynamical gauge variable. For the
quantum electrodynamics (QED) as a Dirac-U(1) gauge theory, it is commonly defined as
DA:=γμDμ=γμ(∂μ+ieAμ) where e is the electric charge constant.
For the spacetime index μ=0,1,…,d−1,
the left-moving current Jμ,L is defined as:
[TABLE]
The right-moving current Jμ,R is defined as:
[TABLE]
The vector current Jμ≡Jμ,V is defined as:
[TABLE]
The axial chiral current Jμ≡Jμ,A≡Jμ≡Jμ,Chiral is defined as:
[TABLE]
We also define the left and right-handed Weyl fermions, ΨL and ΨR, projected from the Dirac fermion via:
[TABLE]
In the classical theory (without doing the path integral), the classical Dirac theory has both the continuous vector symmetry U(1)V and
the continuous axial symmetry (or the so-called chiral symmetry) U(1)A, given by the following symmetry transformation:
[TABLE]
Under the Noether theorem, the corresponding continuous currents are
Jμ,V and Jμ,A respect to the U(1)V and U(1)A symmetry respectively.
In a classical theory, both U(1)V and U(1)A symmetries are global symmetries.
However, in the quantum theory, we need do the path integral to get the partition function Z[A] for a quantum theory in eqn. (3.1).
Now under the continuous axial (or chiral) symmetry transformation labeled by a U(1) parameter
αA∈[0,2π), the partition function Z[A] shifts to
[TABLE]
(In terms of quantum electrodynamics notation, people set the g=−e.)
This means the axial (chiral) current is not conserved ∂μJμ,Chiral=0:
If the classical gauge field A has a nontrivial background for F∧…F term in Model (1) where F=dA.
The above calculation can be done based on the Fujikawa’s path integral method[59].
The non-conservation of the axial (chiral) current has the form:
[TABLE]
In the differential form in terms of the top form paired with the fundamental class of the spacetime manifold, we can rewrite the above formula eqn. (3.15) as:
[TABLE]
The above formula is for the ’t Hooft anomaly associated with the probed background Abelian gauge fields (here U(1)).
If we instead consider the background non-Abelian gauge fields (like SU(n)),
then the (F∧⋯∧F) with F=dA is replaced to
a non-abelian field strength F=dA+A∧A;
while (F∧⋯∧F) is replaced by Tr(Fd/2):
[TABLE]
The ωd−1 is the Chern-Simons (d−1)-form [60] as the secondary characteristic classes:
[TABLE]
Other than Fujikawa’s path integral method[59], we can also capture the perturbative anomaly via a 1-loop Feynman-Dyson diagram calculation.
The vertex term means gΨˉγμAμΨ:=gΨˉAΨ
How do we obtain an integer Z class for perturbative anomalies in an even d-dimensional spacetime?
Say from the formulas of eqn. (3.15) and eqn. (3.16)?
The answer is that we can consider the
modified axial symmetry transformation U(1)A,k labeled by a integer charge
k∈Z class, such that the ΨL and ΨR transformed differently,
[TABLE]
The chiral symmetry transformation is labeled by a U(1) parameter αA,k∈[0,2π).
Below we give
a physical explanation of a Z class in
Sec. 3.1.2 for 2d bosonic anomaly, and also of a Z class Sec. 3.1.3 for 2d fermionic anomaly.
Below our derivation is in a similar spirit of 2d anomaly and 3d Chern-Simons theory [61, 42],
but ours is generalizable to arbitary dimensions analogs to [42].
In general, we suggest that the for generic even dd U(1) bosonic or fermionic anomalies (on non-spin or spin manifolds respectively)
can be captured by a
partition function depending on the probed U(1) background gauge field A:
[TABLE]
where the precise normalization c depends on the dimensions d, and non-spin or spin manifolds,
with the integer k∈Z class.
3.1.2 2d anomaly and 3d bosonic-U(1) SPTs:
Integer Z class ∈TP3(SO×U(1))=Z2
Below we explicitly derive the analogous eqn. (3.20) for d=2 bosonic anomaly (on non-spin manifolds).
The physics idea is that we write down the internal field theory with dynamical gauge fields a coupled to background non-dynamical
gauge fields A, and integrate out those
internal degrees of freedom to get a response theory depending on A.
The symmetric bilinear form K matrix Chern-Simons theory with a U(1)2 gauge group
of internal dynamical a gauge field, and the charge q vector coupling to the background U(1) gauge fields A are the following:
[TABLE]
and qT=(1,1).
In other words, the path integral, written by dynamical gauge fields a and background fields A, is
[TABLE]
Under GL(2,Z) or SL(2,Z) redefinition of gauge fields, the K can be changed to
K=(0110),
and qT=(1,k).
In other words, the path integral is
[TABLE]
If we integrate out the dynamical internal gauge field a (“emergent” from the gapped matter field) of SPTs, we obtain the partition function of probed background field
[TABLE]
This Chern-Simons field theory characterizes the low energy physics of a quantum Hall state and its response function.
So the effective bulk quantized Hall conductance is labeled by 2k in 2Z, as
[TABLE]
The boundary theory has a Z class of perturbative Adler-Bell-Jackiw type of U(1)-axial-background gauge anomaly.
(Due to the L and R chiral fermion carrying imbalanced U(1) charges, in
K=(0110),
and q=(1,k).)
The above physics derivation coincides with the mathematical cobordism group calculation,
matching one of the integer Z class ∈TP3(SO×U(1))=Z2 shown later in our Theorem
9.
The reason we require the (co)bordism group of SO×U(1) is due to that the bosonic system has a continuous spacetime SO(d) symmetry (in the
dd Euclidean signature), while the boson has an internal U(1) symmetry.
Similarly, the above analysis can be generalized to any even dd by writing down a one-higher dimensional bosonic
(on manifolds with SO(d+1) or non-spin structures)
Chern-Simons theory given by a certain Chern-Simons (d+1)d form.
3.1.3 2d anomaly and 3d fermionic-U(1) SPTs:
Integer Z class ∈TP3(Z2Spin×U(1))=Z2
Similar to Sec. 3.1.2,
for a fermionic theory, we should have a SPT invariant of U(1) background gauge field,
[TABLE]
This may be obtained from integrating out the fermionic SPTs
K=(100−1),
and qT=(1,−k+1) or simply qT=(0,−k).
In other words, the path integral is
[TABLE]
This Chern-Simons field theory characterizes the low energy physics of another fermionic quantum Hall state and its response function.
So the effective bulk quantized Hall conductance is labeled by k in Z, as
[TABLE]
The above physics derivation coincides with the mathematical cobordism group calculation,
matching one of the integer Z class ∈TP3(Z2Spin×U(1))=TP3(Spinc)=Z2 shown later in our Theorem
10.
The reason we require the (co)bordism group of (Z2Spin×U(1))≡Spinc is due to that this
fermionic system has a continuous spacetime Spin(d) symmetry (in the
dd Euclidean signature) under, the extension of SO(d) via 1→Z2F→Spin(d)→SO(d)→1,
while the fermion has an internal U(1) ⊃Z2F containing the fermion parity symmetry.
A common normal subgroup
Z2F is mod out due to the fact that rotating a fermion by 2π in the
spacetime (i.e., the spin statistics) gives rise to the same fermion parity
minus sign for the fermion field Ψ→−Ψ.
Similarly, the above analysis can be generalized to any even dd by writing down a one-higher dimensional fermionic
(on manifolds with Spin(d+1) thus spin structures)
Chern-Simons theory given by a certain Chern-Simons (d+1)d form.
3.1.4 ΩdSO(BU(1))
Since the computation involves no odd torsion, we can use the Adams spectral sequence
[TABLE]
The mod 2 cohomology of Thom spectrum MSO is
[TABLE]
[TABLE]
is an A2-resolution where the differentials d1 are induced by Sq1.
We also have
[TABLE]
where c1 is the first Chern class of the universal U(1) bundle.
The bordism invariants of Ω4SO(BU(1)) are σ,c12.
Here σ is the signature of a 4-manifold.
The bordism invariant of Ω5SO(BU(1)) is w2w3.
The bordism invariants of Ω6SO(BU(1)) are σc1,c13.
Here σc1=σ(PD(c1)) where PD(c1) is the submanifold of the 6-manifold which represents the Poincaé dual of c1.
The bordism invariant of Ω7SO(BU(1)) is c1w2w3.
Theorem 13**.**
The 1d topological term is the Chern-Simons 1-form CS1(U(1)) of the U(1) bundle.
The 3d topological terms are 31CS3(TM) and CS1(U(1))c1 where CS3(TM) is the Chern-Simons 3-form of the tangent bundle.
The 5d topological terms are c131CS1(TM), CS1(U(1))c12 and w2w3.
3.1.5 ΩdSpin(BU(1))
Since the computation involves no odd torsion, we can use the Adams spectral sequence
[TABLE]
The mod 2 cohomology of Thom spectrum MSpin is
[TABLE]
where M is a graded A2(1)-module with the degree i homogeneous part Mi=0 for i<8. Here A2(1) stands for the subalgebra of A2 generated by Sq1
and Sq2.
For t−s<8, we can identify the E2-page with
[TABLE]
The A2(1)-module structure of H∗(BU(1),Z2) is shown in Figure 9.
Here η~ is the “mod 2 index” of the 1d Dirac operator (#zero eigenvalues mod 2, no contribution from spectral asymmetry).
The bordism invariants of Ω2Spin(BU(1)) are c1 and Arf (the Arf invariant).
The bordism invariants of Ω4Spin(BU(1)) are 16σ and 2c12.
Here c12 is divided by 2 since c12=Sq2c1=(w2(TM)+w1(TM)2)c1=0mod2 on Spin 4-manifolds.
The bordism invariants of Ω6Spin(BU(1)) are c13 and 8c1(σ−F⋅F).
Here 8c1(σ−F⋅F) is defined to be 81(σ(PD(c1))−F⋅F) where PD(c1) is the submanifold of a Spin 6-manifold which represents the Poincaré dual of c1, σ(PD(c1)) is the signature of PD(c1), and F is a characteristic surface of PD(c1). By Rokhlin’s theorem, σ(PD(c1))−F⋅F is a multiple of 8 and 81(σ(PD(c1))−F⋅F)=Arf(PD(c1),F)mod2. See [62]’s Lecture 10 for more details.
Theorem 15**.**
The 1d topological terms are CS1(U(1)) and η~.
The 2d topological term is Arf.
The 3d topological terms are 481CS3(TM) and 21CS1(U(1))c1.
The 5d topological terms are CS1(U(1))c12 and μ(PD(c1)).
Here μ(PD(c1)) is the Rokhlin invariant (see [62]’s Lecture 11) of PD(c1) where PD(c1) is the submanifold of a Spin 5-manifold which represents the Poincaré dual of c1
3.1.6 Ωd(Z2Spin×U(1))=ΩdSpinc
Since the computation involves no odd torsion, we can use the Adams spectral sequence
[TABLE]
By 1801.07530, we have MTSpinc=MSpin∧Σ−2MU(1).
The mod 2 cohomology of Thom spectrum MSpin is
[TABLE]
where M is a graded A2(1)-module with the degree i homogeneous part Mi=0 for i<8. Here A2(1) stands for the subalgebra of A2 generated by Sq1
and Sq2.
For t−s<8, we can identify the E2-page with
[TABLE]
By Thom’s isomorphism, H∗+2(MU(1),Z2)=Z2[c1]U where U is the Thom class of the universal U(1) bundle and c1 is the first Chern class of the universal U(1) bundle.
The A2(1)-module structure of H∗+2(MU(1),Z2) is shown in Figure 11.
Here c1 is divided by 2 since c1mod2=w2(TM) while w2(TM)=0 on Spinc 2-manifolds.
The bordism invariants of Ω4Spinc are c12 and 8σ−F⋅F.
Here F is a characteristic surface of the Spinc 4-manifold M. By Rokhlin’s theorem, σ−F⋅F is a multiple of 8 and 81(σ−F⋅F)=Arf(M,F)mod2. See [62]’s Lecture 10 for more details.
The bordism invariants of Ω6Spinc are 2c13 and c116σ.
Here c116σ=16σ(PD(c1)) where PD(c1) is the submanifold of the Spinc 6-manifold which represents the Poincaré dual of c1. Note that PD(c1) is Spin.
Theorem 17**.**
The 1d topological term is 21CS1(U(1)).
The 3d topological terms are CS1(U(1))c1 and μ.
Here μ is the Rokhlin invariant (see [62]’s Lecture 11 ) of the Spinc 3-manifold.
The 5d topological terms are 21CS1(U(1))c12 and c1481CS1(TM).
3.2 Non-perturbative global anomalies: Witten’s SU(2) Anomaly and A New SU(2) Anomaly in 4d and 5d
We now provide another warm-up example,
a (d+1)-th cobordism group calculation associated with dd non-perturbative global anomalies
for the SU(2) anomaly of Witten [23] and
the new SU(2) anomaly [24].
Here these SU(2) anomalies will be interpreted as the ’t Hooft anomalies of the internal SU(2) global symmetries in
the QFT, whose fermion multiplets are only in half-integer isospin (say, 1/2, 3/2, 5/2, …)-representation of SU(2);
while whose bosons are only in an integer isospin (say, 0, 1, 2, …)-representation of SU(2).
Similar to the Spinc≡(Z2Spin×U(1)) of Sec. 3.1.3,
in the following subsections, we will study the
(co)bordism group of (Z2Spin×SU(2)).
Here the fermionic system has a continuous spacetime Spin(d) symmetry (in the
dd Euclidean signature) under, the extension of SO(d) via 1→Z2F→Spin(d)→SO(d)→1,
while the fermion has an internal SU(2) ⊃Z2F containing the fermion parity symmetry at SU(2)’s center.
A common normal subgroup Z2F is mod out due to the fact that rotating a fermion by 2π in the
spacetime (i.e., the spin statistics) gives rise to the same fermion parity minus sign for the fermion field Ψ→−Ψ.
Thus we need to study the (Z2Spin×SU(2))-structure.
We will see that
TP5(Z2FSpin×SU(2))=(Z2)2.
These 5d bordism invariants generates (Z2)2, they correspond to the old SU(2) [23] and the new SU(2) anomalies [24] in 4d,
shown in Table 12.
We will see that
TP6(Z2FSpin×SU(2))=(Z2)2.
These 6d bordism invariants generates (Z2)2, they correspond to the old SU(2) and the new SU(2) anomalies in 5d [24],
shown in Table 12.
3.2.1
ΩdZ2FSpin×SU(2)=ΩdZ2FSpin×Spin(3)
Let H=Z2FSpin×Spin(3), we have a homotopy pullback square
[TABLE]
There is a homotopy equivalence
f:BSO×BSO(3)∼BSO×BSO(3) by (V,W)↦(V−W+3,W).
Note that f∗(w2)=w2(V−W)=w2(V)+w1(V)w1(W)+w2(W)=w2(TM)+w2′.
Then we have the following homotopy pullback
[TABLE]
This implies that BH∼BSpin×BSO(3).
MTH=Thom(BH;−V), where V is the induced virtual bundle (of dimension [math]) by the map BH→BO.
We can identify BH→BO with
BSpin×BSO(3)V−V3+3BSO↪BO.
The spectrum MTH is homotopy equivalent to Thom(BSpin×BSO(3);−(V−V3+3)), which is MSpin∧Σ−3MSO(3).
For t−s<8,
[TABLE]
By Thom’s isomorphism, H∗+3(MSO(3),Z2)=Z2[w2′,w3′]U where wi′ is the Stiefel-Whitney class of the universal SO(3) bundle and U is the Thom class of the universal SO(3) bundle.
Since w1(TM)=w1′=0, and w2(TM)=w2′ by the gauge bundle constraint, we have w3(TM)=w3′. Below we use wi to denote both wi(TM) and wi′ for i≤3.
The A2(1)-module structure of H∗+3(MSO(3),Z2) and the E2 page are shown in Figure 13, 14.
4 Higher Group Cobordisms and Non-trivial Fibrations
In this section, we use the Serre spectral sequence method explored in appendix of [11] and the Adams spectral sequence method to derive the 5d topological terms for the higher group cobordism Ω5G with non-trivial fibration where G is defined as follows.
If Ga is a group, Gb is an abelian group, then it is well-known that BGb is a group. Consider the group extension
[TABLE]
we have a fibration
[TABLE]
which is classified by the Postnikov class β∈H3(BGa,Gb).
4.1 (BGa,B2Gb):(BO,B2Z2)
We consider the simplest case: Ga=O and Gb=Z2. Note that there is also a group action α:Ga→AutGb in [11], since AutZ2 is trivial, so α is trivial in this special case.
For the fibration
[TABLE]
there is a Serre spectral sequence
[TABLE]
where Hp(BO,Hq(B2Z2,Z)) actually should be the α-equivariant cohomology, but since α is trivial, Hp(BO,Hq(B2Z2,Z)) is the ordinary cohomology.
Note that Hn(B2Z2,Z) is computed in Appendix C of [63].
[TABLE]
The E2 page of the Serre spectral sequence is Hp(BO,Hq(B2Z2,Z)). The shape of the relevant piece is shown in Figure 15.
Note that p labels the columns and q labels the rows.
The bottom row is Hp(BO,Z).
The universal coefficient theorem (2.15) tells us that H3(B2Z2,Z)=H2(B2Z2,R/Z)=Hom(H2(B2Z2,Z),R/Z)=Hom(π2(B2Z2),R/Z)=Hom(Z2,R/Z)=Z^2, so the q=3 row is Hp(BO,Z^2).
It is also known that H5(B2Z2,Z)=H4(B2Z2,R/Z) is the group of quadratic functions q:Z2→R/Z [65]. The isomorphism is discussed in detail in [66].
The first possibly non-zero differential is on the E3 page:
[TABLE]
Following the appendix of [11], this map sends a
quadratic form q:Z2→R/Z to ⟨β,−⟩q, where the bracket denotes the bilinear pairing ⟨x,y⟩q=q(x+y)−q(x)−q(y).
The next possibly non-zero differentials are on the E4 page:
[TABLE]
The first map is contraction with β.
The second map comes from the long exact sequence
[TABLE]
If Hn(BO,R)=Hn+1(BO,R)=0, then Hn(BO,R/Z)=Hn+1(BO,Z).
Since Hn(BO,R)=Hn(BO,Z)⊗R and Hn(BO,Z) is finite if n is not divisible by 4, Hn(BO,R)=0 if n is not divisible by 4, thus Hn(BO,R/Z)=Hn+1(BO,Z) for n=1,2mod4.
The last relevant possibly non-zero differential is on the E6 page:
[TABLE]
Following the appendix of [11], this differential is actually zero.
So the only possible differentials in Figure 15 below degree 5 are d3 from (0,5) to (3,3) and d4 from the third row to the zeroth row.
The Madsen-Tillmann spectrum MTG=Thom(BG;−V) where V is the induced virtual bundle over BG (of dimension [math]) from BG→BO.
By Thom isomorphism (1.22), H∗(MTG,Z2)=H∗(BG,Z2)U where U is the Thom class of −V with SqiU=wˉiU where wˉi is the Stiefel-Whitney class of −V such that (1+wˉ1+wˉ2+⋯)(1+w1+w2+⋯)=1 where wi is the Stiefel-Whitney class of V, i.e., wˉ1=w1, wˉ2=w2+w12, etc. Here the U on the right means the cup product with U.
We have the Adams spectral sequence
[TABLE]
where A2 is the mod 2 Steenrod algebra. The last equality is Pontryagin-Thom isomorphism.
The A2-module structure of H∗(MTG,Z2) below degree 5 is shown in Figure 16 where we intentionally omit terms that don’t involve the cohomology classes of B2Z2.
Note that the position (0,3) in Figure 15 contributes to both H2(B2Z2,Z2) which is generated by x2 and H3(B2Z2,Z2) which is generated by x3, Sq1x2=x3. The position (2,3) corresponds to xw12,xw2, the position (3,3) corresponds to xw13,xw1w2,xw3 for both x=x2 and x=x3.
Since ⟨β,β⟩q=−2q(β), 4q(β)=q(2β)=0, there are 2 among the 4 choices of q(β) such that q→⟨β,−⟩q maps to the dual linear function of β, if we identify Z^2 with Z2, then the nonzero element in the image of q→⟨β,−⟩q is just β. So
Imd3(0,5) is spanned by xβ.
The differential d4(2,3):H2(BO,Z^2)→H5(BO,R/Z)=H6(BO,Z) is defined by
[TABLE]
Let β=a1w13+a2w1w2+a3w3, if we identify Z^2 with Z2,
then d4(2,3)(γ)=γ∪β which is in H5(BO,Z2),
while H5(BO,R/Z)=H6(BO,Z).
Let γ=b1w12+b2w2, then γ∪β=a1b1w15+(a1b2+a2b1)w13w2+a2b2w1w22+a3b1w12w3+a3b2w2w3.
The differential d4(3,3):H3(BO,Z^2)→H6(BO,R/Z)=H7(BO,Z) is defined by
[TABLE]
Let β=a1w13+a2w1w2+a3w3, if we identify Z^2 with Z2,
then d4(3,3)(ζ)=ζ∪β which is in H6(BO,Z2),
while H6(BO,R/Z)=H7(BO,Z).
Let ζ=c1w13+c2w1w2+c3w3, then ζ∪β=a1c1w16+a2c2w12w22+a3c3w32+(a1c2+c1a2)w14w2+(a1c3+c1a3)w13w3+(a2c3+c2a3)w1w2w3.
Note that by the Universal Coefficient Theorem (2.20),
where we list the generators below degree 7.
Among the linear combinations of w15,w13w2,w1w22,w12w3,w2w3, only w13w2+w12w3 is in H5(BO,Z)⊗Z2
While among the linear combinations of w16,w12w22,w32,w14w2,w13w3,w1w2w3, only w12w22, w13w3, w16, and w12w22+w32 are in H6(BO,Z)⊗Z2.
So we claim that
Kerd4(2,3) is spanned by xγ where γ=b1w12+b2w2 with
a1b1=0, a1b2+a2b1=a3b1, a2b2=0, a3b2=0.
While Kerd4(3,3) is spanned by xζ where ζ=c1w13+c2w1w2+c3w3 with
a1c2+c1a2=0, a2c3+c2a3=0.
In the following cases, we only consider the topological terms involving the cohomology classes of B2Z2.
Case 1:β=0, there is no differential in Figure 15, the 5d topological terms are x3w2 (or x2w3), x3w12 (or x2w13) and x2x3 (see Figure 16).
Note that x3w2=x2w3, x3w12=x2w13 and x5=x3(w12+w2) by Wu formula (2.68).
In this case BG=BO×B2Z2, MTG=MO∧(B2Z2)+, πdMTG=ΩdO(B2Z2). This case will be discussed later in another way.
Case 2:β=w13, a1=1, a2=a3=0. Kerd4(2,3) is spanned by xγ where γ=b1w12+b2w2 with b1=b2=0.
Kerd4(3,3) is spanned by xζ where ζ=c1w13+c2w1w2+c3w3 with c2=0.
At the position (3,3), x2w13 is killed in the E3 page, x2w3 survives to the E∞ page, so the 5d topological terms are x2w3 and x2x3.
Case 3:β=w1w2, a1=0, a2=1, a3=0. Kerd4(2,3) is spanned by xγ where γ=b1w12+b2w2 with b1=b2=0. Kerd4(3,3) is spanned by xζ where ζ=c1w13+c2w1w2+c3w3 with c1=c3=0.
At the position (3,3),
x2w3 and x2w13 are killed in the E4 page, so the 5d topological term is x2x3.
Case 4:β=w3, a1=a2=0, a3=1. Kerd4(2,3) is spanned by xγ where γ=b1w12+b2w2 with b1=b2=0. Kerd4(3,3) is spanned by xζ where ζ=c1w13+c2w1w2+c3w3 with c2=0.
At the position (3,3),
x2w3 is killed in the E3 page, x2w13 survives to the E∞ page, so the 5d topological terms are x2w13 and x2x3.
Case 5:β=w13+w1w2, a1=a2=1, a3=0.
Kerd4(2,3) is spanned by xγ where γ=b1w12+b2w2 with b1=b2=0. Kerd4(3,3) is spanned by xζ where ζ=c1w13+c2w1w2+c3w3 with c1+c2=0, c3=0.
At the position (3,3),
x2(w13+w1w2) is killed in the E3 page, x2w3 is killed in the E4 page, but since x2w1w2=Sq3x2=0 by Wu formula (2.68), so the 5d topological term is x2x3.
Case 6:β=w1w2+w3, a1=0, a2=a3=1.
Kerd4(2,3) is spanned by xγ where γ=b1w12+b2w2 with b2=0. Kerd4(3,3) is spanned by xζ where ζ=c1w13+c2w1w2+c3w3 with c1=0, c2+c3=0.
At the position (3,3),
x2(w1w2+w3) is killed in the E3 page, x2w13 is killed in the E4 page, but since x2w1w2=Sq3x2=0 by Wu formula (2.68), so the 5d topological terms is x2x3.
Case 7:β=w13+w3, a1=1, a2=0, a3=1.
Kerd4(2,3) is spanned by xγ where γ=b1w12+b2w2 with b1=b2=0. Kerd4(3,3) is spanned by xζ where ζ=c1w13+c2w1w2+c3w3 with c2=0.
At the position (3,3),
x2(w13+w3) is killed in the E3 page, so the 5d topological terms are x2w13=x2w3 and x2x3.
Case 8:β=w13+w1w2+w3, a1=a2=a3=1.
Kerd4(2,3) is spanned by xγ where γ=b1w12+b2w2 with b1=b2=0. Kerd4(3,3) is spanned by xζ where ζ=c1w13+c2w1w2+c3w3 with c1+c2=0, c2+c3=0.
At the position (3,3),
x2(w13+w1w2+w3) is killed in the E3 page, but since x2w1w2=Sq3x2=0 by Wu formula (2.68), so the 5d topological term are x2w13=x2w3 and x2x3.
5 O/SO/Spin/Pin± bordism groups of classifying spaces
In this section, we compute the O/SO/Spin/Pin± bordism groups of the classifying space of the group G=Ga×BGb: BG=BGa×B2Gb.
Here BGb is a group since Gb is abelian.
We briefly comment the difference between a previous cobordism theory [25] and this work:
In all Adams charts of the computation in [25], there are no nonzero differentials,
while in this paper we encounter nonzero differentials dn due to the (p,pn)-Bockstein homomorphisms in the computation involving B2Zpn and BZpn.
5.1 Introduction
For H=O/SO/Spin/Pin± and the group H×G, define
[TABLE]
where V is the induced virtual bundle over B(H×G) by the composition B(H×G)→BH→BO
where the first map is the projection, the second map is the natural homomorphism.
By the Pontryagin-Thom isomorphism (1.20) and the property of Thom space (1.3), ΩdH(BG)=πd(MTH∧BG+)=πd(MT(H×G)).
Hence we can define
[TABLE]
[TABLE]
Here X+ is the disjoint union of X and a point. MTO=MO, MTSO=MSO, MTSpin=MSpin, MTPin+=MPin−, MTPin−=MPin+.
πd(B) is the d-th stable homotopy group of the spectrum B.
[B,Σn+1IZ] stands for the homotopy classes of maps from spectrum B to the (n+1)-th suspension of spectrum IZ. The Anderson dual IZ is a spectrum that is the fiber of IC→IC×
where IC(IC×) is the Brown-Comenetz dual spectrum defined by
[TABLE]
[TABLE]
By the work of Freed-Hopkins [4], there is a 1:1 correspondence
[TABLE]
There is an exact sequence
[TABLE]
for any spectrum B, especially for MT(H×G).
The torsion part [MT(H×G),Σn+1IZ]tors is Ext1((πnMT(H×G))tors,Z)=Hom((πnMT(H×G))tors,U(1)).
Here β(3,3) is the Bockstein homomorphism in A3.
[TABLE]
where ∣x2′∣=2 and Qi is defined inductively by Q0=β(3,3), Qi=P3i−1Qi−1−Qi−1P3i−1 for i≥1.
Let x3′=β(3,3)x2′, x2⋅3i+1′=Qix2′, x2⋅3i+2′=β(3,3)Qix2′ for i≥1.
Here Pn is the n-th Steenrod power in A3.
[TABLE]
Here wi′ is the i-th Stiefel-Whitney class wi(PSU(2)) of the universal principal PSU(2)-bundle over BPSU(2).
Let pi′ be the i-th Pontryagin class pi(PSU(2)) of the universal principal PSU(2)-bundle over BPSU(2), then p1′(mod2)=w2′2.
ci is the i-th Chern class ci(PSU(3)) of the universal principal PSU(3)-bundle over BPSU(3).
where ∣zi∣=i, J=(z2z3,z2z7,z2z8+z3z7) is the ideal generated by z2z3,z2z7,z2z8+z3z7 and z3=β(3,3)z2, z7=P1z3, z8=β(3,3)z7.
Note that c2(mod3)=z22, c3(mod3)=z23.
In the following subsections, all bordism invariants are the pullback of cohomology classes along classifying maps f:M→X and g:M→BH.
5.2 Point
5.2.1 ΩdO
Since the computation involves no odd torsion, we can use the Adams spectral sequence
[TABLE]
Here πt−s(MO)2∧ is the 2-completion of the group πt−s(MO).
The mod 2 cohomology of Thom spectrum MO is
[TABLE]
where Ω=Z2[y2,y4,y5,y6,y8,…] is the unoriented bordism ring, Ω∗ is the Z2-linear dual of Ω.
On the other hand, H∗(MO,Z2)=Z2[w1,w2,w3,…]U where U is the Thom class of the virtual bundle (of dimension 0) over BO which is the colimit of En−n and En is the universal n-bundle over BO(n), wi is the i-th Stiefel-Whitney class of the virtual bundle (of dimension 0) over BO. Note that the pullback of the virtual bundle (of dimension 0) over BO along the map g:M→BO is just TM−d where M is a d-dimensional manifold and TM is the tangent bundle of M, g is given by the O-structure on M. We will not distinguish wi and wi(TM).
Here yi are manifold generators, for example, y2=RP2, y4=RP4, y5 is Wu manifold SU(3)/SO(3).
By Thom’s result [1], two manifolds are unorientedly bordant if and only if they have identical sets of Stiefel-Whitney characteristic numbers. The nonvanishing Stiefel-Whitney numbers of y2=RP2 are w2 and w12, the nonvanishing Stiefel-Whitney numbers of y22=RP2×RP2 are w22 and w4,
the nonvanishing Stiefel-Whitney numbers of y4=RP4 are w14 and w4, the only nonvanishing Stiefel-Whitney number of Wu manifold SU(3)/SO(3) is w2w3.
So y2∗=w12 or w2, (y22)∗=w22, y4∗=w14, y5∗=w2w3, etc, where yi∗ is the Z2-linear dual of
yi∈Ω.
Below we choose y2∗=w12 by default, this is reasonable since Sq2(xd−2)=(w2+w12)xd−2 on d-manifold by Wu formula (2.68).
Hence we have the following theorem
Theorem 21**.**
The bordism invariant of Ω2O is w12.
The bordism invariants of Ω4O are w14,w22.
The bordism invariant of Ω5O is w2w3.
Theorem 22**.**
The 2d topological term is w12.
The 4d topological terms are w14,w22.
The 5d topological term is w2w3.
5.2.2 ΩdSO
Since the computation involves no odd torsion, we can use the Adams spectral sequence
[TABLE]
The mod 2 cohomology of Thom spectrum MSO is
[TABLE]
[TABLE]
is an A2-resolution where the differentials d1 are induced by Sq1.
Since σ=3p1(TM), p1(TM)=dCS3(TM), the 3d topological term is 31CS3(TM).
The 5d topological term is w2w3.
5.2.3 ΩdSpin
Since the computation involves no odd torsion, we can use the Adams spectral sequence
[TABLE]
The mod 2 cohomology of Thom spectrum MSpin is
[TABLE]
where M is a graded A2(1)-module with the degree i homogeneous part Mi=0 for i<8. Here A2(1) stands for the subalgebra of A2 generated by Sq1
and Sq2.
For t−s<8, we can identify the E2-page with
Here η~ is the “mod 2 index” of the 1d Dirac operator (#zero eigenvalues mod 2, no contribution from spectral asymmetry).
The bordism invariant of Ω2Spin is Arf (the Arf invariant).
The bordism invariant of Ω4Spin is 16σ.
Theorem 26**.**
The 1d topological term is η~.
The 2d topological term is Arf.
The 3d topological term is 481CS3(TM).
5.2.4 ΩdPin+
Since the computation involves no odd torsion, we can use the Adams spectral sequence
[TABLE]
MPin−=MTPin+∼MSpin∧S1∧MTO(1).
For t−s<8, we can identify the E2-page with
[TABLE]
By Thom’s isomorphism,
[TABLE]
where U is the Thom class of the virtual bundle −E1 over BO(1), E1 is the universal 1-bundle over BO(1) and w1 is the 1st Stiefel-Whitney class of E1 over BO(1).
The A2(1)-module structure of H∗−1(MTO(1),Z2) and the E2 page are shown in Figure 19, 20.
Hence we have the following theorem
Theorem 27**.**
The bordism invariant of Ω2Pin+ is w1∪η~.
The bordism invariant of Ω3Pin+ is w1∪Arf.
The bordism invariant of Ω4Pin+ is η.
Here η is the usual Atiyah-Patodi-Singer eta-invariant of the 4d Dirac operator (=“#zero eigenvalues + spectral asymmetry”).
Theorem 28**.**
The 2d topological term is w1∪η~.
The 3d topological term is w1∪Arf.
The 4d topological term is η.
5.2.5 ΩdPin−
Since the computation involves no odd torsion, we can use the Adams spectral sequence
[TABLE]
MPin+=MTPin−∼MSpin∧S−1∧MO(1).
For t−s<8, we can identify the E2-page with
[TABLE]
By Thom’s isomorphism,
[TABLE]
where U is the Thom class of the universal 1-bundle E1 over BO(1) and w1 is the 1st Stiefel-Whitney class of E1 over BO(1).
The A2(1)-module structure of H∗+1(MO(1),Z2) and the E2 page are shown in Figure 21, 22.
Hence we have the following theorem
Theorem 29**.**
The bordism invariant of Ω1Pin− is η~.
The bordism invariant of Ω2Pin− is ABK (the Arf-Brown-Kervaire invariant).
Theorem 30**.**
The 1d topological term is η~.
The 2d topological term is ABK.
5.3 Atiyah-Hirzebruch spectral sequence
If H=O/SO/Spin/Pin±, by the Atiyah-Hirzebruch spectral sequence, we have
[TABLE]
If H=O/Pin±,
since ΩdH are finite, ΩdH×G=ΩdH(BG) are also finite, so
TPd(H×G)=ΩdH×G for H=O/Pin±.
If H=SO/Spin,
[TABLE]
[TABLE]
If Hp(BG,Z) are finite for p>0, then Ω6H(BG) is finite and
TP5(H×G)=Ω5H(BG) for H=SO/Spin.
If G=PSU(2)=SO(3), since
H2(BSO(3),Z) and H6(BSO(3),Z) are finite, Ω6H(BG) is also finite and
TP5(H×G)=Ω5H(BG) for H=SO/Spin.
If G=PSU(3), then H6(BPSU(3),Z) contains a Z while H2(BPSU(3),Z) does not, so Ω6H(BG) contains a Z and
TP5(H×G)=Ω5H(BG)×Z for H=SO/Spin.
5.4 B2Gb:B2Z2,B2Z3
5.4.1 ΩdO(B2Z2)
Since the computation involves no odd torsion, we can use the Adams spectral sequence
[TABLE]
Theorem 31** (Thom).**
∙π∗MO=Ω=Z2[y2,y4,y5,y6,y8,…].
∙H∗(MO,Z2)=A2⊗Ω∗ where Ω∗ is the Z2-linear dual of Ω.
y2=RP2, y4=RP4, y5 is the Wu manifold W=SU(3)/SO(3), …
y2∗=w2(TM) or w1(TM)2, y2∗=w2(TM)2, y4∗=w1(TM)4, y5∗=w2(TM)w3(TM), …
Theorem 32** (Serre).**
[TABLE]
where x3=Sq1x2, x5=Sq2x3, x9=Sq4x5, …
By Künneth theorem,
[TABLE]
Here ΣnA2 is the n-th iterated shift of the graded algebra A2.
Since
[TABLE]
we have the following theorem
Theorem 33**.**
The bordism invariants of Ω2O(B2Z2) are x2,w12.
The bordism invariant of Ω3O(B2Z2) is x3=w1x2.
The bordism invariants of Ω4O(B2Z2) are x22,w14,w12x2,w22.
The bordism invariants of Ω5O(B2Z2) are x2x3,x5,w12x3,w2w3.
Here wi is the i-th Stiefel-Whitney class of the tangent bundle of M, x2 is the generator of H2(B2Z2,Z2), x3=Sq1x2, x5=Sq2Sq1x2, since there is a map f:M→B2Z2 in the definition of cobordism group, we identify x2 with f∗(x2)=f. By Wu formula, Sq2xd−2=(w2+w12)xd−2 on d-manifolds.
Theorem 34**.**
The 2d topological terms are x2,w12.
The 3d topological term is x3=w1x2.
The 4d topological terms are x22,w14,w12x2,w22.
The 5d topological terms are x2x3,x5,w12x3,w2w3.
5.4.2 ΩdSO(B2Z2)
Since the computation involves no odd torsion, we can use the Adams spectral sequence
[TABLE]
Since
H∗(B2Z2,Z2)=Z2[x2,x3,x5,x9,…] where
x2 is the generator of H2(B2Z2,Z2), x3=Sq1x2, x5=Sq2Sq1x2, x9=Sq4Sq2Sq1x2, etc,
Sq1x2=x3, Sq1x3=0, Sq1(x22)=0, Sq1(x2x3)=Sq1(x5)=x32.
We have used (2.66) and the Adem relations (2.111).
We shift Figure 2 the same times as the dimension of H∗(B2Z2,Z2) at each degree as a Z2-vector space. We obtain the E1 page for Ω∗SO(B2Z2), the differentials d1 are induced by Sq1, as shown in Figure 23.
There is a differential d2 corresponding to the Bockstein homomorphism β(2,4):H∗(−,Z4)→H∗+1(−,Z2) associated to 0→Z2→Z8→Z4→0 [57]. See 2.5 for the definition of Bockstein homomorphisms.
Note that
[TABLE]
We have used β(2,4)=41δmod2, the Steenrod’s formula (2.12), Sq1=β(2,2)=21δmod2, and the definition Sqkxn=xnn−k∪xn.
So there is a differential such that
d2(x2x3+x5)=x22h02.
Then take the differentials d2 into account, we obtain the E2 page for Ω∗SO(B2Z2), as shown in Figure 24.
Hence we have the following theorem
Theorem 35**.**
The bordism invariant of Ω2SO(B2Z2) is x2.
The bordism invariants of Ω4SO(B2Z2) are σ and P2(x2).
The bordism invariants of Ω5SO(B2Z2) are x5=x2x3 and w2w3.
Here P2(x2) is the Pontryagin square of x2.
σ is the signature of a 4-manifold M.
x2x3+x5=21w~1P2(x2) [69] where w~1 is the twisted first Stiefel-Whitney class of the tangent bundle, in particular, w1=0 implies w~1=0, so x2x3=x5 on oriented 5-manifolds.
Theorem 36**.**
The 2d topological term is x2.
The 3d topological term is 31CS3(TM).
The 4d topological term is P2(x2).
The 5d topological terms are x5=x2x3 and w2w3.
5.4.3 ΩdSpin(B2Z2)
Since the computation involves no odd torsion, we can use the Adams spectral sequence
[TABLE]
For t−s<8, we can identify the E2-page with
[TABLE]
H∗(B2Z2,Z2)=Z2[x2,x3,x5,x9,…] where
x2 is the generator of H2(B2Z2,Z2), x3=Sq1x2, x5=Sq2Sq1x2, x9=Sq4Sq2Sq1x2, etc,
Sq1x2=x3, Sq2x2=x22, Sq1x3=0, Sq2x3=x5, Sq1(x22)=0, Sq1(x2x3)=x32, Sq1x5=Sq2x22=x32, Sq2x5=0. Sq2(x2x3)=x22x3+x2x5.
We have used (2.66) and the Adem relations (2.111).
There is a differential d2 corresponding to the Bockstein homomorphism β(2,4):H∗(−,Z4)→H∗+1(−,Z2) associated to 0→Z2→Z8→Z4→0 [57]. See 2.5 for the definition of Bockstein homomorphisms.
By (5.51), there is a differential such that
d2(x2x3+x5)=x22h02.
The A2(1)-module structure of H∗(B2Z2,Z2) is shown in Figure 25.
The dot at the bottom is a Z2 which has been discussed before. Now we consider the part above the bottom dot. We will use Lemma 11 several times. Two steps are shown in Figure 26, 27.
We will proceed in the reversed order.
First, we apply Lemma 11 to the short exact sequence of A2(1)-modules:
0→P→N→Q→0 in
the second step (as shown in Figure 27),
the Adams chart of ExtA2(1)s,t(N,Z2) is shown in Figure 28.
Next, we apply Lemma 11 to the short exact sequence of A2(1)-modules:
0→L→M→N→0 in
the first step (as shown in Figure 26),
the Adams chart of ExtA2(1)s,t(M,Z2) is shown in Figure 29.
Then take the differentials d2 into account, we obtain the E2 page for Ω∗Spin(B2Z2), as shown in Figure 30.
Hence we have the following theorem
Theorem 37**.**
The bordism invariants of Ω2Spin(B2Z2) are x2 and Arf.
The bordism invariants of Ω4Spin(B2Z2) are 16σ and 2P2(x2).
By Wu formula, x22=Sq2(x2)=(w2(TM)+w1(TM)2)x2=0 on Spin 4-manifolds, x5=Sq2(x3)=(w2(TM)+w1(TM)2)x3=0 on Spin 5-manifolds, P2(x2)=x22=0mod2 on Spin 4-manifolds.
Here Arf is the Arf invariant.
σ is the signature of Spin 4-manifold, it is a multiple of 16 by Rokhlin’s theorem.
Theorem 38**.**
The 2d topological terms are x2 and Arf.
The 3d topological term is 481CS3(TM).
The 4d topological term is 2P2(x2).
5.4.4 ΩdPin+(B2Z2)
Since the computation involves no odd torsion, we can use the Adams spectral sequence
[TABLE]
MPin−=MTPin+∼MSpin∧S1∧MTO(1).
For t−s<8, we can identify the E2-page with
[TABLE]
The A2(1)-module structure of H∗−1(MTO(1),Z2)⊗H∗(B2Z2,Z2) and the E2 page are shown in Figure 31, 32.
Hence we have the following theorem
Theorem 39**.**
The bordism invariants of Ω2Pin+(B2Z2) are x2 and w1η~.
The bordism invariants of Ω3Pin+(B2Z2) are w1x2=x3 and w1Arf.
The bordism invariants of Ω4Pin+(B2Z2) are qs(x2) and η.
The bordism invariants of Ω5Pin+(B2Z2) are x2x3 and w12x3(=x5).
Here η~ is the “mod 2
index” of the 1d Dirac operator (#zero eigenvalues mod 2, no
contribution from spectral asymmetry).
The bordism invariants of Ω4SO(B2Z3) are σ and x2′2.
The bordism invariant of Ω5SO(B2Z3) is w2w3.
Theorem 46**.**
The 2d topological term is x2′.
The 3d topological term is 31CS3(TM).
The 4d topological term is x2′2.
The 5d topological term is w2w3.
5.4.8 ΩdSpin(B2Z3)
[TABLE]
Since H∗(B2Z3,Z2)=Z2, we have ΩdSpin(B2Z3)2∧=ΩdSpin.
[TABLE]
Since there is a short exact sequence of groups
[TABLE]
we have a fibration
[TABLE]
Take the localization at prime 3, we have a homotopy equivalence BSpin(3)∼BSO(3) since the localization of BZ2 at 3 is trivial. Take the Thom spectra, we have a homotopy equivalence MSpin(3)∼MSO(3). Hence
[TABLE]
We have the following
Theorem 47**.**
The bordism invariants of Ω2Spin(B2Z3) are Arf and x2′.
The bordism invariants of Ω4Spin(B2Z3) are 16σ and x2′2.
Theorem 48**.**
The 2d topological terms are Arf and x2′.
The 3d topological term is 481CS3(TM).
The 4d topological term is x2′2.
5.4.9 ΩdPin+(B2Z3)
[TABLE]
[TABLE]
Since MTPin+=MPin−∼MSpin∧S1∧MTO(1) and H∗(MTO(1),Z3)=0,
we have H∗(MPin−,Z3)=0, thus ΩdPin+(B2Z3)3∧=0.
Since H∗(B2Z3,Z2)=Z2, we have ΩdPin+(B2Z3)2∧=ΩdPin+.
Hence ΩdPin+(B2Z3)=ΩdPin+.
Theorem 49**.**
The bordism invariant of Ω2Pin+(B2Z3) is w1η~.
The bordism invariant of Ω3Pin+(B2Z3) is w1Arf.
The bordism invariant of Ω4Pin+(B2Z3) is η.
Theorem 50**.**
The 2d topological term is w1η~.
The 3d topological term is w1Arf.
The 4d topological term is η.
5.4.10 ΩdPin−(B2Z3)
[TABLE]
[TABLE]
Since MTPin−=MPin+∼MSpin∧S−1∧MO(1) and H∗(MO(1),Z3)=0,
we have H∗(MPin+,Z3)=0, thus ΩdPin−(B2Z3)3∧=0.
Since H∗(B2Z3,Z2)=Z2, we have ΩdPin−(B2Z3)2∧=ΩdPin−.
Hence ΩdPin−(B2Z3)=ΩdPin−.
Theorem 51**.**
The bordism invariant of Ω2Pin−(B2Z3) is ABK.
Theorem 52**.**
The 2d topological term is ABK.
5.5 BGa:BPSU(2),BPSU(3)
5.5.1 ΩdO(BPSU(2))
[TABLE]
[TABLE]
Theorem 53**.**
The bordism invariants of Ω2O(BPSU(2)) are w2′,w12.
The bordism invariant of Ω3O(BPSU(2)) is w3′=w1w2′.
The bordism invariants of Ω4O(BPSU(2)) are w2′2,w14,w12w2′,w22.
The bordism invariants of Ω5O(BPSU(2)) are w2w3,w12w3′,w2′w3′.
Theorem 54**.**
The 2d topological terms are w2′,w12.
The 3d topological term is w3′=w1w2′.
The 4d topological terms are w2′2,w14,w12w2′,w22.
The 5d topological terms are w2w3,w12w3′,w2′w3′.
The bordism invariants of Ω4SO(BPSU(2)) are σ,p1′.
The bordism invariants of Ω5SO(BPSU(2)) are w2w3,w2′w3′.
The manifold generators of Ω4SO(BPSU(2)) are (CP2,3) and (CP2,LC+1) where n is the trivial real n-plane bundle and LC is the tautological complex line bundle over CP2.
Note that the principal SO(3)-bundle P associated to LC+1 is the induce bundle P′×SO(2)SO(3) from P′
[TABLE]
by the group homomorphism ϕ:SO(2)→SO(3) which is the inclusion map, that means P=P′×SO(2)SO(3)=(P′×SO(3))/SO(2) which is the quotient of P′×SO(3) by the right SO(2) action
[TABLE]
[TABLE]
So
[TABLE]
Theorem 56**.**
The 2d topological term is w2′.
Since p1′=dCS3(SO(3)), the 3d topological terms are 31CS3(TM) and CS3(SO(3)).
The 5d topological terms are w2w3,w2′w3′.
5.5.3 ΩdSpin(BPSU(2))
[TABLE]
For t−s<8,
[TABLE]
The A2(1)-module structure of H∗(BPSU(2),Z2) and the E2 page are shown in Figure 37, 38.
Theorem 57**.**
The bordism invariants of Ω2Spin(BPSU(2)) are w2′ and Arf.
By Wu formula (2.68), w2′2=Sq2(w2′)=(w2(TM)+w1(TM)2)w2′=0 on Spin 4-manifolds, p1′=w2′2=0mod2 on Spin 4-manifolds.
The bordism invariants of Ω4Spin(BPSU(2)) are 16σ and 2p1′.
Theorem 58**.**
The 2d topological terms are w2′ and Arf.
The 3d topological terms are 481CS3(TM) and 21CS3(SO(3)).
5.5.4 ΩdPin+(BPSU(2))
[TABLE]
MTPin+=MSpin∧S1∧MTO(1).
For t−s<8,
[TABLE]
The A2(1)-module structure of H∗−1(MTO(1),Z2)⊗H∗(BPSU(2),Z2) and the E2 page are shown in Figure 39, 40.
Theorem 59**.**
The bordism invariants of Ω2Pin+(BPSU(2)) are w2′ and w1η~.
The bordism invariants of Ω3Pin+(BPSU(2)) are w1w2′=w3′ and w1Arf.
The bordism invariants of Ω4Pin+(BPSU(2)) are qs(w2′) (this invariant has another form, see the footnotes of Table 3) and η.
The bordism invariant of Ω5Pin+(BPSU(2)) is w12w3′(=w2′w3′).
Theorem 60**.**
The 2d topological terms are w2′ and w1η~.
The 3d topological terms are w1w2′=w3′ and w1Arf.
The 4d topological terms are qs(w2′) and η.
The 5d topological term is w12w3′(=w2′w3′).
5.5.5 ΩdPin−(BPSU(2))
[TABLE]
MTPin−=MSpin∧S−1∧MO(1).
For t−s<8,
[TABLE]
The A2(1)-module structure of H∗+1(MO(1),Z2)⊗H∗(BPSU(2),Z2) and the E2 page are shown in Figure 41, 42.
Theorem 61**.**
The bordism invariants of Ω2Pin−(BPSU(2)) are w2′ and ABK.
The bordism invariant of Ω3Pin−(BPSU(2)) is w1w2′=w3′.
The bordism invariant of Ω4Pin−(BPSU(2)) is w12w2′.
Theorem 62**.**
The 2d topological terms are w2′ and ABK.
The 3d topological term is w1w2′=w3′.
The 4d topological term is w12w2′.
5.5.6 ΩdO(BPSU(3))
[TABLE]
Since H∗(MO,Z3)=0, ΩdO(BPSU(3))3∧=0.
[TABLE]
[TABLE]
Theorem 63**.**
The bordism invariant of Ω2O(BPSU(3)) is w12.
The bordism invariants of Ω4O(BPSU(3)) are w14,w22,c2(mod2).
The bordism invariants of Ω4SO(BPSU(3)) are σ,c2.
The bordism invariant of Ω5SO(BPSU(3)) is w2w3.
The bordism invariant of Ω6SO(BPSU(3)) is c3.
The manifold generators of Ω4SO(BPSU(3)) are (CP2,CP2×PSU(3)) and (S4,H) where H is the induced bundle from the Hopf fibration H′
[TABLE]
by the group homomorphism ρ:SU(2)→PSU(3) which is the composition of the inclusion map SU(2)→SU(3) and the quotient map SU(3)→PSU(3), that means H=H′×SU(2)PSU(3)=(H′×PSU(3))/SU(2) which is the quotient of H′×PSU(3) by the right SU(2) action
[TABLE]
Theorem 66**.**
The 2d topological term is z2.
Since c2=dCS3(PSU(3)), the 3d topological terms are 31CS3(TM) and CS3(PSU(3)).
Since c3=dCS5(PSU(3)), the 5d topological term are CS5(PSU(3)) and w2w3.
Since H∗(MSpin,Z3)=H∗(MSO,Z3), the E2 page is shown in Figure 46.
Theorem 67**.**
The bordism invariants of Ω2Spin(BPSU(3)) are Arf and z2.
The bordism invariants of Ω4Spin(BPSU(3)) are 16σ and c2.
By Wu formula (2.68), c3=Sq2c2=(w2(TM)+w12(TM))c2=0mod2 on Spin 6-manifolds.
The bordism invariant of Ω6Spin(BPSU(3)) is 2c3.
Theorem 68**.**
The 2d topological terms are Arf and z2.
The 3d topological terms are 481CS3(TM) and CS3(PSU(3)).
The 5d topological term is 21CS5(PSU(3)).
5.5.9 ΩdPin+(BPSU(3))
[TABLE]
Since H∗(MPin−,Z3)=H∗(MO,Z3)=0, Ωt−sPin+(BPSU(3))3∧=0.
[TABLE]
For t−s<8,
[TABLE]
The A2(1)-module structure of H∗−1(MTO(1),Z2)⊗H∗(BPSU(3),Z2) and the E2 page are shown in Figure 47, 48.
Theorem 69**.**
The bordism invariant of Ω2Pin+(BPSU(3)) is w1η~.
The bordism invariant of Ω3Pin+(BPSU(3)) is w1Arf.
The bordism invariants of Ω4Pin+(BPSU(3)) are c2(mod2) and η.
Theorem 70**.**
The 2d topological term is w1η~.
The 3d topological term is w1Arf.
The 4d topological terms are c2(mod2) and η.
5.5.10 ΩdPin−(BPSU(3))
[TABLE]
Since H∗(MPin+,Z3)=H∗(MO,Z3)=0, Ωt−sPin−(BPSU(3))3∧=0.
[TABLE]
For t−s<8,
[TABLE]
The A2(1)-module structure of H∗+1(MO(1),Z2)⊗H∗(BPSU(3),Z2) and the E2 page are shown in Figure 49, 50.
Theorem 71**.**
The bordism invariant of Ω2Pin−(BPSU(3)) is ABK.
The bordism invariant of Ω4Pin−(BPSU(3)) is c2(mod2).
Theorem 72**.**
The 2d topological term is ABK.
The 4d topological term is c2(mod2).
5.6 (BGa,B2Gb):(BZ2,B2Z2),(BZ3,B2Z3)
5.6.1 ΩdO(BZ2×B2Z2)
Since the computation involves no odd torsion, we can use the Adams spectral sequence
[TABLE]
[TABLE]
Theorem 73**.**
The bordism invariants of Ω2O(BZ2×B2Z2) are a2,x2,w12.
The bordism invariants of Ω3O(BZ2×B2Z2) are x3=w1x2,ax2,aw12,a3.
The bordism invariants of Ω4O(BZ2×B2Z2) are w14,w22,a4,a2x2,ax3,x22,w12a2,w12x2.
The bordism invariants of Ω5O(BZ2×B2Z2) are
[TABLE]
Theorem 74**.**
The 2d topological terms are a2,x2,w12.
The 3d topological terms are x3=w1x2,ax2,aw12,a3.
The 4d topological terms are w14,w22,a4,a2x2,ax3,x22,w12a2,w12x2.
The 5d topological terms are
[TABLE]
5.6.2 ΩdSO(BZ2×B2Z2)
Since the computation involves no odd torsion, we can use the Adams spectral sequence
[TABLE]
There is a differential d2 corresponding to the Bockstein homomorphism β(2,4):H∗(−,Z4)→H∗+1(−,Z2) associated to 0→Z2→Z8→Z4→0 [57]. See 2.5 for the definition of Bockstein homomorphisms.
By (5.51), there is a differential such that
d2(x2x3+x5)=x22h02.
The bordism invariant of Ω2SO(BZ2×B2Z2) is x2.
The bordism invariants of Ω3SO(BZ2×B2Z2) are ax2,a3.
The bordism invariants of Ω4SO(BZ2×B2Z2) are σ, ax3(=a2x2) and P2(x2).
The bordism invariants of Ω5SO(BZ2×B2Z2) are ax22,a5,x5,a3x2,w2w3,aw22.
Theorem 76**.**
The 2d topological term is x2.
The 3d topological terms are 31CS3(TM),ax2,a3.
The 4d topological terms are ax3(=a2x2) and P2(x2).
The 5d topological terms are ax22,a5,x5,a3x2,w2w3,aw22.
5.6.3 ΩdSpin(BZ2×B2Z2)
Since the computation involves no odd torsion, we can use the Adams spectral sequence
[TABLE]
For t−s<8,
[TABLE]
H∗(BZ2×B2Z2,Z2)=Z2[a,x2,x3,x5,x9,…] where Sq1x2=x3, Sq2x2=x22, Sq1x3=0, Sq2x3=x5, Sq1x5=Sq2x22=x32, Sq2x5=0.
There is a differential d2 corresponding to the Bockstein homomorphism β(2,4):H∗(−,Z4)→H∗+1(−,Z2) associated to 0→Z2→Z8→Z4→0 [57]. See 2.5 for the definition of Bockstein homomorphisms.
By (5.51), there is a differential such that
d2(x2x3+x5)=x22h02.
The A2(1)-module structure of H∗(BZ2×B2Z2,Z2) and the E2 page are shown in Figure 52, 53.
Theorem 77**.**
The bordism invariants of Ω2Spin(BZ2×B2Z2) are x2,Arf,aη~.
The bordism invariants of Ω3Spin(BZ2×B2Z2) are ax2,aABK.
The bordism invariants of Ω4Spin(BZ2×B2Z2) are 16σ, ax3(=a2x2) and 2P2(x2).
The bordism invariant of Ω5Spin(BZ2×B2Z2) is a3x2.
Theorem 78**.**
The 2d topological terms are x2,Arf,aη~.
The 3d topological terms are 481CS3(TM),ax2,aABK.
The 4d topological terms are ax3(=a2x2) and 2P2(x2).
The 5d topological term is a3x2.
5.6.4 ΩdPin+(BZ2×B2Z2)
Since the computation involves no odd torsion, we can use the Adams spectral sequence
[TABLE]
MPin−=MTPin+∼MSpin∧S1∧MTO(1).
For t−s<8,
[TABLE]
The A2(1)-module structure of H∗−1(MTO(1),Z2)⊗H∗(BZ2×B2Z2,Z2) and the E2 page are shown in Figure 54, 55.
Theorem 79**.**
The bordism invariants of Ω2Pin+(BZ2×B2Z2) are w1a=a2,x2,w1η~.
The bordism invariants of Ω3Pin+(BZ2×B2Z2) are a3,w1x2=x3,ax2,w1aη~,w1Arf.
The bordism invariants of Ω4Pin+(BZ2×B2Z2) are ax3,w1ax2(=a2x2+ax3),qs(x2),w1a(ABK),η.
The bordism invariants of Ω5Pin+(BZ2×B2Z2) are
[TABLE]
Theorem 80**.**
The 2d topological terms are w1a=a2,x2,w1η~.
The 3d topological terms are a3,w1x2=x3,ax2,w1aη~,w1Arf.
The 4d topological terms are ax3,w1ax2(=a2x2+ax3),qs(x2),w1a(ABK),η.
The 5d topological terms are
[TABLE]
5.6.5 ΩdPin−(BZ2×B2Z2)
Since the computation involves no odd torsion, we can use the Adams spectral sequence
[TABLE]
MPin+=MTPin−∼MSpin∧S−1∧MO(1).
For t−s<8,
[TABLE]
The A2(1)-module structure of H∗+1(MO(1),Z2)⊗H∗(BZ2×B2Z2,Z2) and the E2 page are shown in Figure 56, 57.
Theorem 81**.**
The bordism invariants of Ω2Pin−(BZ2×B2Z2) are x2,q(a),ABK. (q(a) is explained in the footnotes of Table 1.)
The bordism invariants of Ω3Pin−(BZ2×B2Z2) are a3,w12a,x3=w1x2,ax2.
The bordism invariants of Ω4Pin−(BZ2×B2Z2) are w12x2,w1ax2(=a2x2+ax3),ax3.
The bordism invariants of Ω5Pin−(BZ2×B2Z2) are w12a3,x2x3,w12ax2,w1ax3(=a2x3),a3x2.
Theorem 82**.**
The 2d topological terms are x2,q(a),ABK.
The 3d topological terms are a3,w12a,x3=w1x2,ax2.
The 4d topological terms are w12x2,w1ax2(=a2x2+ax3),ax3.
The 5d topological terms are w12a3,x2x3,w12ax2,w1ax3(=a2x3),a3x2.
5.6.6 ΩdO(BZ3×B2Z3)
[TABLE]
[TABLE]
Since H∗(MO,Z3)=0, we have ΩdO(BZ3×B2Z3)3∧=0.
Since H∗(BZ3×B2Z3,Z2)=Z2, we have ΩdO(BZ3×B2Z3)2∧=ΩdO.
Hence ΩdO(BZ3×B2Z3)=ΩdO.
Theorem 83**.**
The bordism invariant of Ω2O(BZ3×B2Z3) is w12.
The bordism invariants of Ω4O(BZ3×B2Z3) are w14,w22.
The bordism invariant of Ω5O(BZ3×B2Z3) is w2w3.
Theorem 84**.**
The 2d topological term is w12.
The 4d topological terms are w14,w22.
The 5d topological term is w2w3.
5.6.7 ΩdSO(BZ3×B2Z3)
[TABLE]
Since H∗(BZ3×B2Z3,Z2)=Z2, we have ΩdSO(BZ3×B2Z3)2∧=ΩdSO.
The bordism invariant of Ω2SO(BZ3×B2Z3) is x2′.
The bordism invariants of Ω3SO(BZ3×B2Z3) are a′b′,a′x2′.
The bordism invariants of Ω4SO(BZ3×B2Z3) are σ, a′x3′(=b′x2′) and x2′2.
The bordism invariants of Ω5SO(BZ3×B2Z3) are w2w3,a′b′x2′,a′x2′2,P3(b′).
Here P3 is the Postnikov square.
Theorem 86**.**
The 2d topological term is x2′.
The 3d topological terms are 31CS3(TM),a′b′,a′x2′.
The 4d topological terms are a′x3′(=b′x2′) and x2′2.
The 5d topological terms are w2w3,a′b′x2′,a′x2′2,P3(b′).
5.6.8 ΩdSpin(BZ3×B2Z3)
[TABLE]
Since H∗(BZ3×B2Z3,Z2)=Z2, we have ΩdSpin(BZ3×B2Z3)2∧=ΩdSpin.
[TABLE]
Since
[TABLE]
we have the following
Theorem 87**.**
The bordism invariants of Ω2Spin(BZ3×B2Z3) are Arf and x2′.
The bordism invariants of Ω3Spin(BZ3×B2Z3) are a′b′,a′x2′.
The bordism invariants of Ω4Spin(BZ3×B2Z3) are 16σ, a′x3′(=b′x2′) and x2′2.
The bordism invariants of Ω5Spin(BZ3×B2Z3) are a′b′x2′,a′x2′2,P3(b′).
Theorem 88**.**
The 2d topological terms are Arf and x2′.
The 3d topological terms are 481CS3(TM),a′b′,a′x2′.
The 4d topological terms are a′x3′(=b′x2′) and x2′2.
The 5d topological terms are a′b′x2′,a′x2′2,P3(b′).
5.6.9 ΩdPin+(BZ3×B2Z3)
[TABLE]
[TABLE]
Since H∗(MPin−,Z3)=0, we have ΩdPin+(BZ3×B2Z3)3∧=0.
Since H∗(BZ3×B2Z3,Z2)=Z2, we have ΩdPin+(BZ3×B2Z3)2∧=ΩdPin+.
Hence ΩdPin+(BZ3×B2Z3)=ΩdPin+.
Theorem 89**.**
The bordism invariant of Ω2Pin+(BZ3×B2Z3) is w1η~.
The bordism invariant of Ω3Pin+(BZ3×B2Z3) is w1Arf.
The bordism invariant of Ω4Pin+(BZ3×B2Z3) is η.
Theorem 90**.**
The 2d topological term is w1η~.
The 3d topological term is w1Arf.
The 4d topological term is η.
5.6.10 ΩdPin−(BZ3×B2Z3)
[TABLE]
[TABLE]
Since H∗(MPin+,Z3)=0, we have ΩdPin−(BZ3×B2Z3)3∧=0.
Since H∗(BZ3×B2Z3,Z2)=Z2, we have ΩdPin−(BZ3×B2Z3)2∧=ΩdPin−.
Hence ΩdPin−(BZ3×B2Z3)=ΩdPin−.
Theorem 91**.**
The bordism invariant of Ω2Pin−(BZ3×B2Z3) is ABK.
Theorem 92**.**
The 2d topological term is ABK.
5.7 (BGa,B2Gb):(BPSU(2),B2Z2),(BPSU(3),B2Z3)
5.7.1 ΩdO(BPSU(2)×B2Z2)
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Theorem 93**.**
The bordism invariants of Ω2O(BPSU(2)×B2Z2) are w2′,x2,w12.
The bordism invariants of Ω3O(BPSU(2)×B2Z2) are x3=w1x2,w3′=w1w2′.
The bordism invariants of Ω4O(BPSU(2)×B2Z2) are w14,w22,x22,w2′2,x2w12,w2′w12,w2′x2.
The bordism invariants of Ω5O(BPSU(2)×B2Z2) are w2′w3′,x2w3′,w12w3′,w2′x3,x2x3,w12x3,x5,w2w3.
Theorem 94**.**
The 2d topological terms are w2′,x2,w12.
The 3d topological terms are x3=w1x2,w3′=w1w2′.
The 4d topological terms are w14,w22,x22,w2′2,x2w12,w2′w12,w2′x2.
The 5d topological terms are w2′w3′,x2w3′,w12w3′,w2′x3,x2x3,w12x3,x5,w2w3.
5.7.2 ΩdSO(BPSU(2)×B2Z2)
[TABLE]
There is a differential d2 corresponding to the Bockstein homomorphism β(2,4):H∗(−,Z4)→H∗+1(−,Z2) associated to 0→Z2→Z8→Z4→0 [57]. See 2.5 for the definition of Bockstein homomorphisms.
By (5.51), there is a differential such that
d2(x2x3+x5)=x22h02.
The bordism invariants of Ω2SO(BPSU(2)×B2Z2) are w2′,x2.
The bordism invariants of Ω4SO(BPSU(2)×B2Z2) are σ, p1′, w2′x2 and P2(x2).
The bordism invariants of Ω5SO(BPSU(2)×B2Z2) are w2′w3′,x5,w3′x2(=w2′x3),w2w3.
Theorem 96**.**
The 2d topological terms are w2′,x2.
The 3d topological terms are 31CS3(TM), CS3(SO(3)).
The 4d topological terms are w2′x2 and P2(x2).
The 5d topological terms are w2′w3′,x5,w3′x2(=w2′x3),w2w3.
5.7.3 ΩdSpin(BPSU(2)×B2Z2)
For t−s<8,
[TABLE]
There is a differential d2 corresponding to the Bockstein homomorphism β(2,4):H∗(−,Z4)→H∗+1(−,Z2) associated to 0→Z2→Z8→Z4→0 [57]. See 2.5 for the definition of Bockstein homomorphisms.
By (5.51), there is a differential such that
d2(x2x3+x5)=x22h02.
The A2(1)-module structure of H∗(BPSU(2)×B2Z2,Z2) and the E2 page is shown in Figure 60, 61.
Theorem 97**.**
The bordism invariants of Ω2Spin(BPSU(2)×B2Z2) are w2′,x2,Arf.
The bordism invariants of Ω4Spin(BPSU(2)×B2Z2) are 16σ, 2p1′, w2′x2 and 2P2(x2).
The bordism invariant of Ω5Spin(BPSU(2)×B2Z2) is w3′x2(=w2′x3).
Theorem 98**.**
The 2d topological terms are w2′,x2,Arf.
The 3d topological terms are 481CS3(TM), 21CS3(SO(3)).
The 4d topological terms are w2′x2 and 2P2(x2).
The 5d topological term is w3′x2(=w2′x3).
5.7.4 ΩdPin+(BPSU(2)×B2Z2)
For t−s<8,
[TABLE]
The A2(1)-module structure of H∗−1(MTO(1),Z2)⊗H∗(BPSU(2)×B2Z2,Z2) and the E2 page are shown in Figure 62, 63.
Theorem 99**.**
The bodism invariants of Ω2Pin+(BPSU(2)×B2Z2) are w2′,x2,w1η~.
The bodism invariants of Ω3Pin+(BPSU(2)×B2Z2) are w1w2′=w3′,w1x2=x3,w1Arf.
The bodism invariants of Ω4Pin+(BPSU(2)×B2Z2) are qs(w2′),qs(x2),η,w2′x2.
The bodism invariants of Ω5Pin+(BPSU(2)×B2Z2) are
[TABLE]
Theorem 100**.**
The 2d topological terms are w2′,x2,w1η~.
The 3d topological terms are w1w2′=w3′,w1x2=x3,w1Arf.
The 4d topological terms are qs(w2′),qs(x2),η,w2′x2.
The 5d topological terms are
[TABLE]
5.7.5 ΩdPin−(BPSU(2)×B2Z2)
For t−s<8,
[TABLE]
The A2(1)-module structure of H∗+1(MO(1).Z2)⊗H∗(BPSU(2)×B2Z2,Z2) and the E2 page are shown in Figure 64, 65.
Theorem 101**.**
The bordism invariants of Ω2Pin−(BPSU(2)×B2Z2) are w2′,x2,ABK.
The bordism invariants of Ω3Pin−(BPSU(2)×B2Z2) are w1w2′=w3′,w1x2=x3.
The bordism invariants of Ω4Pin−(BPSU(2)×B2Z2) are w12w2′,w12x2,w2′x2.
The bordism invariants of Ω5Pin−(BPSU(2)×B2Z2) are x2x3,w3′x2,w1w2′x2(=w2′x3+w3′x2).
Theorem 102**.**
The 2d topological terms are w2′,x2,ABK.
The 3d topological terms are w1w2′=w3′,w1x2=x3.
The 4d topological terms are w12w2′,w12x2,w2′x2.
The 5d topological terms are x2x3,w3′x2,w1w2′x2(=w2′x3+w3′x2).
5.7.6 ΩdO(BPSU(3)×B2Z3)
[TABLE]
Since H∗(MO,Z3)=0, ΩdO(BPSU(3)×B2Z3)3∧=0.
[TABLE]
Since H∗(BPSU(3)×B2Z3,Z2)=H∗(BPSU(3),Z2), ΩdO(BPSU(3)×B2Z3)2∧=ΩdO(BPSU(3))2∧.
Theorem 103**.**
The bordism invariant of Ω2O(BPSU(3)×B2Z3) is w12.
The bordism invariants of Ω4O(BPSU(3)×B2Z3) are w14,w22,c2(mod2).
The bordism invariant of Ω5O(BPSU(3)×B2Z3) is w2w3.
Theorem 104**.**
The 2d topological term is w12.
The 4d topological terms are w14,w22,c2(mod2).
The 5d topological term is w2w3.
5.7.7 ΩdSO(BPSU(3)×B2Z3)
[TABLE]
Since H∗(BPSU(3)×B2Z3,Z2)=H∗(BPSU(3),Z2), ΩdSO(BPSU(3)×B2Z3)2∧=ΩdSO(BPSU(3))2∧.
The bordism invariants of Ω2SO(BPSU(3)×B2Z3) are x2′,z2.
The bordism invariants of Ω4SO(BPSU(3)×B2Z3) are σ, c2, x2′2 and x2′z2.
The bordism invariants of Ω5SO(BPSU(3)×B2Z3) are w2w3,z2x3′(=−z3x2′).
Theorem 106**.**
The 2d topological terms are x2′,z2.
The 3d topological terms are 31CS3(TM), CS3(PSU(3)).
The 4d topological terms are x2′2 and x2′z2.
The 5d topological terms are CS5(PSU(3)),w2w3,z2x3′(=−z3x2′).
5.7.8 ΩdSpin(BPSU(3)×B2Z3)
[TABLE]
Since H∗(BPSU(3)×B2Z3,Z2)=H∗(BPSU(3),Z2), ΩdSpin(BPSU(3)×B2Z3)2∧=ΩdSpin(BPSU(3))2∧.
[TABLE]
Since H∗(MSO,Z3)=H∗(MSpin,Z3), ΩdSpin(BPSU(3)×B2Z3)3∧=ΩdSO(BPSU(3)×B2Z3)3∧.
Theorem 107**.**
The bordism invariants of Ω2Spin(BPSU(3)×B2Z3) are Arf,x2′,z2.
The bordism invariants of Ω4Spin(BPSU(3)×B2Z3) are 16σ, c2, x2′2 and x2′z2.
The bordism invariant of Ω5Spin(BPSU(3)×B2Z3) is z2x3′(=−z3x2′).
Theorem 108**.**
The 2d topological terms are Arf,x2′,z2.
The 3d topological terms are 481CS3(TM), CS3(PSU(3)).
The 4d topological terms are x2′2 and x2′z2.
The 5d topological terms are 21CS5(PSU(3)),z2x3′(=−z3x2′).
5.7.9 ΩdPin+(BPSU(3)×B2Z3)
[TABLE]
Since H∗(MPin−,Z3)=0, ΩdPin+(BPSU(3)×B2Z3)3∧=0.
[TABLE]
Since H∗(BPSU(3)×B2Z3,Z2)=H∗(BPSU(3),Z2), ΩdPin+(BPSU(3)×B2Z3)2∧=ΩdPin+(BPSU(3))2∧.
Theorem 109**.**
The bordism invariant of Ω2Pin+(BPSU(3)×B2Z3) is w1η~.
The bordism invariant of Ω3Pin+(BPSU(3)×B2Z3) is w1Arf.
The bordism invariants of Ω4Pin+(BPSU(3)×B2Z3) are c2(mod2) and η.
Theorem 110**.**
The 2d topological term is w1η~.
The 3d topological term is w1Arf.
The 4d topological terms are c2(mod2) and η.
5.7.10 ΩdPin−(BPSU(3)×B2Z3)
[TABLE]
Since H∗(MPin+,Z3)=0, ΩdPin−(BPSU(3)×B2Z3)3∧=0.
[TABLE]
Since H∗(BPSU(3)×B2Z3,Z2)=H∗(BPSU(3),Z2), ΩdPin−(BPSU(3)×B2Z3)2∧=ΩdPin−(BPSU(3))2∧.
Theorem 111**.**
The bordism invariant of Ω2Pin−(BPSU(3)×B2Z3) is ABK.
The bordism invariant of Ω4Pin−(BPSU(3)×B2Z3) is c2(mod2).
Theorem 112**.**
The 2d topological term is ABK.
The 4d topological term is c2(mod2).
6 More computation of O/SO bordism groups
6.1 Summary
Below we use the following notations, all cohomology class are pulled back to the d-manifold M along the maps given in the definition of cobordism groups:
∙wi is the Stiefel-Whitney class of the tangent bundle of M,
∙a is the generator of H1(BZ2,Z2),
∙a′ is the generator of H1(BZ3,Z3), b′=β(3,3)a′,
∙x2 is the generator of H2(B2Z2,Z2), x3=Sq1x2, x5=Sq2x3,
∙x2′ is the generator of H2(B2Z3,Z3), x3′=β(3,3)x2′,
∙x2′′ is the generator of H2(B2Z4,Z4), x3′′=β(2,4)x2′′, x5′′=Sq2x3′′,
∙wi′=wi(O(n))∈Hi(BO(n),Z2) is the Stiefel-Whitney class of the principal O(n) bundle,
∙p1′=p1(O(n))∈H4(BO(n),Z2) is the first Pontryagin class of the principal O(n) bundle,
∙z2=w2(PSU(3))∈H2(BPSU(3),Z3) is the generalized Stiefel-Whitney class of the principal PSU(3) bundle, z3=β(3,3)z2.
∙z2′=w2(PSU(4))∈H2(BPSU(4),Z4) is the generalized Stiefel-Whitney class of the principal PSU(4) bundle, z3′=β(2,4)z2′.
∙ For n>1, we also use the notation a for the generator of H1(BZ2n,Z2n), a~=amod2, b is the generator of H2(BZ2n,Z2n), b~=bmod2 and b~=β(2,2n)a.
∙ For n>1, we also use the notation a′ for the generator of H1(BZ3n,Z3n), a~′=a′mod3, b′ is the generator of H2(BZ3n,Z3n), b~′=b′mod3 and b~′=β(3,3n)a′.
Convention:
All product between cohomology classes are cup product.
6.2 B2Z4
6.2.1 ΩdO(B2Z4)
[TABLE]
[TABLE]
where y2∗=w12, (y22)∗=w22, y4∗=w14, y5∗=w2w3, etc.
[TABLE]
where x~2′′=x2′′mod2, x2′′∈H2(B2Z4,Z4), x3′′=β(2,4)x2′′, x5′′=Sq2x3′′, x9′′=Sq4x5′′, etc.
[TABLE]
Hence we have the following theorem
Theorem 113**.**
The 2d bordism invariants are w12,x~2′′.
The 3d bordism invariant is x3′′.
The 4d bordism invariants are w14,w22,w12x~2′′,x~2′′2.
The 5d bordism invariants are w2w3,w12x3′′,x~2′′x3′′,x5′′.
6.2.2 ΩdSO(B2Z4)
[TABLE]
[TABLE]
Note that
Sq1x~2′′=2β(2,4)x2′′=0, β(2,4)(x2′′)=41δx2′′=x3′′, β(2,4)(x2′′2)=2x2′′x3′′=2x~2′′x3′′=0, Sq1(x~2′′2)=2β(2,4)(x2′′2)=0,
Sq1x3′′=0, Sq1(x~2′′x3′′)=0,
Sq1x5′′=Sq1Sq2β(2,4)x2′′=Sq3β(2,4)x2′′=(β(2,4)x2′′)2=x3′′2.
We have used the properties of Bockstein homomorphisms, (2.66) and the Adem relations (2.111).
Also note that
[TABLE]
We have used β(2,8)=81δmod2, the Steenrod’s formula (2.12), β(2,4)=41δmod2, and the definition Sqkxn=xnn−k∪xn.
There is a differential dn corresponding to the Bockstein homomorphism β(2,2n):H∗(−,Z2n)→H∗+1(−,Z2) associated to 0→Z2→Z2n+1→Z2n→0 [57]. See 2.5 for the definition of Bockstein homomorphisms.
So there are differentials such that
d2(x3′′)=x~2′′h02, d3(x~2′′x3′′)=x~2′′2h03.
The 3d bordism invariants are w1′3,w1′w2′=w3′.
The 4d bordism invariants are σ,p1′,w1′2w2′,w4′.
The 5d bordism invariants are w2w3,w22w1′,w2′w3′,w1′w2′2,w1′2w3′=w1′3w2′,w1′5,w1′w4′=w5′.
6.6 BZ2n×B2Zn
6.6.1 ΩdO(BZ4×B2Z2)
[TABLE]
where a∈H1(BZ4,Z4), b∈H2(BZ4,Z4).
[TABLE]
where a~=amod2∈H1(BZ4,Z2), b~=bmod2∈H2(BZ4,Z2).
[TABLE]
[TABLE]
[TABLE]
where y2∗=w12, (y22)∗=w22, y4∗=w14, y5∗=w2w3, etc.
[TABLE]
Hence we have the following theorem
Theorem 121**.**
The bordism groups are
The 2d bordism invariants are b~,x2,w12.
The 3d bordism invariants are a~b~,x3,a~x2,a~w12.
The 4d bordism invariants are a~x3,b~x2,b~2,x22,w14,w22,b~w12,x2w12.
The 5d bordism invariants are \tilde{a}x_{2}^{2},\tilde{b}x_{3},x_{2}x_{3},\tilde{a}\tilde{b}^{2},x_{5},\tilde{a}\tilde{b}x_{2},w_{2}w_{3},\tilde{a}w_{2}^{2},\tilde{a}w_{1}^{4},\tilde{a}\tilde{b}w_{1}^{2},x_{3}w_{1}^{2}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}=w_{1}^{3}x_{2}},\tilde{a}x_{2}w_{1}^{2}.
There is a differential d2 corresponding to the Bockstein homomorphism β(2,4):H∗(−,Z4)→H∗+1(−,Z2) associated to 0→Z2→Z8→Z4→0 [57]. See 2.5 for the definition of Bockstein homomorphisms.
So there are differentials such that
d2(b~)=a~h02, d2(b~2)=a~b~h02, d2(x2x3+x5)=x22h02, d2(b~x22+a~(x2x3+x5))=a~x22h02, d2(b~3)=a~b~2h02, d2(b~w22)=a~w22h02.
Since H∗(BZ6×B2Z3,Z3)=H∗(BZ3×B2Z3,Z3), we have ΩdSO(BZ6×B2Z3)3∧=ΩdSO(BZ3×B2Z3)3∧.
Hence we have the following theorem
Theorem 124**.**
The bordism groups are
The 2d bordism invariant is x2′.
The 3d bordism invariants are a′b′,a′x2′,a3.
The 4d bordism invariants are σ, a′x3′(=b′x2′) and x2′2.
The 5d bordism invariants are a5,aw22,w2w3,a′b′x2′,a′x2′2,P3(b′).
Here P3 is the Postnikov square.
6.7 BZ2n2×B2Zn
6.7.1 ΩdO(BZ8×B2Z2)
[TABLE]
where a∈H1(BZ8,Z8), b∈H2(BZ8,Z8).
[TABLE]
where a~=amod2∈H1(BZ8,Z2), b~=bmod2∈H2(BZ8,Z2).
[TABLE]
[TABLE]
[TABLE]
where y2∗=w12, (y22)∗=w22, y4∗=w14, y5∗=w2w3, etc.
[TABLE]
Hence we have the following theorem
Theorem 125**.**
The bordism groups are
The 2d bordism invariants are b~,x2,w12.
The 3d bordism invariants are a~b~,x3,a~x2,a~w12.
The 4d bordism invariants are a~x3,b~x2,b~2,x22,w14,w22,b~w12,x2w12.
The 5d bordism invariants are \tilde{a}x_{2}^{2},\tilde{b}x_{3},x_{2}x_{3},\tilde{a}\tilde{b}^{2},x_{5},\tilde{a}\tilde{b}x_{2},w_{2}w_{3},\tilde{a}w_{2}^{2},\tilde{a}w_{1}^{4},\tilde{a}\tilde{b}w_{1}^{2},x_{3}w_{1}^{2}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}=w_{1}^{3}x_{2}},\tilde{a}x_{2}w_{1}^{2}.
There is a differential dn corresponding to the Bockstein homomorphism β(2,2n):H∗(−,Z2n)→H∗+1(−,Z2) associated to 0→Z2→Z2n+1→Z2n→0 [57]. See 2.5 for the definition of Bockstein homomorphisms.
So there are differentials such that
d3(b~)=a~h02, d3(b~2)=a~b~h02, d2(x2x3+x5)=x22h02, d2(a~(x2x3+x5))=a~x22h02, d3(b~3)=a~b~2h02, d3(b~w22)=a~w22h02.
The 4d bordism invariants are σ, P2(x2) and b~x2.
The 5d bordism invariants are {\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}(a\mod 4)\mathcal{P}_{2}(x_{2})}, ab2, a(σmod8), x5=x2x3, a~b~x2 and w2w3.
6.7.3 ΩdO(BZ18×B2Z3)
[TABLE]
where a∈H1(BZ2,Z2).
[TABLE]
Since H∗(MO,Z3)=0, we have ΩdO(BZ18×B2Z3)3∧=0.
[TABLE]
[TABLE]
where y2∗=w12, (y22)∗=w22, y4∗=w14, y5∗=w2w3, etc.
[TABLE]
Hence we have the following theorem
Theorem 127**.**
The bordism groups are
The 2d bordism invariants are w12,a2.
The 3d bordism invariants are a3,aw12.
The 4d bordism invariants are a4,a2w12,w14,w22.
The 5d bordism invariants are a5,a3w12,aw14,aw22,w2w3.
6.7.4 ΩdSO(BZ18×B2Z3)
[TABLE]
Since H∗(BZ18×B2Z3,Z2)=H∗(BZ2,Z2), we have ΩdSO(BZ18×B2Z3)2∧=ΩdSO(BZ2).
[TABLE]
Since H∗(BZ18×B2Z3,Z3)=H∗(BZ9×B2Z3,Z3), we have ΩdSO(BZ18×B2Z3)3∧=ΩdSO(BZ9×B2Z3)3∧.
The 3d bordism invariants are a3,z3′,az~2′,aw12.
Here z2′=w2(PSU(4))∈H2(BPSU(4),Z4) is the generalized Stiefel-Whitney class of the principal PSU(4) bundle, z~2′=z2′mod2, z3′=β(2,4)z2′.
6.8.4 Ω3SO(B(Z2⋉PSU(4)))
[TABLE]
There is a differential d2 corresponding to the Bockstein homomorphism β(2,4):H∗(−,Z4)→H∗+1(−,Z2) associated to 0→Z2→Z8→Z4→0 [57]. See 2.5 for the definition of Bockstein homomorphisms.
Since β(2,4)(z2′)=z3′, there is a differential such that
d2(z3′)=z~2′h02.
7.1 Relations to Non-Abelian Gauge Theories and Sigma Models
As we mentioned, this article is a companion Reference with further detailed calculations supporting other shorter articles [18, 19, 20].
Now we make final comments and remarks on how our cobordism group calculations in the preceding Sec. 5 and Sec. 6 are applied in these works
[18, 19, 20].
Pure SU(2) Yang-Mills theory’s higher anomaly: LABEL:Gaiotto2014kfa1412.5148 introduces the generalized global symmetries include higher symmetries
(See a brief review in Sec. 1.4, Items (♢ 1) and (♢ 2)). The pure SU(N) Yang-Mills (YM) gauge theory has a higher-1-dimensional (1-form)
electric symmetry, denoted as ZN,[1]e (previously known as the ZN-center symmetry).
The pure SU(N) YM theory in 4d has the corresponding 1-form electric ZN,[1]e symmetry charged object: the 1-dimensional gauge-invariant Wilson line We:
[TABLE]
and the 2-dimensional charge operator: the 2-dimensional charge surface operator Ue.
[TABLE]
The spacetime path integral formulation of SU(N) YM higher symmetry becomes a relation:
[TABLE]
with R in fundamental representation.
The remarkable LABEL:Gaiotto2017TTT1703.00501 discovers the mixed higher ’t Hooft anomaly of pure SU(N) YM theory at an even integer N
with a second Chern class topological term (πM4∫c2≃M4∫8π2θTrFa∧Fa at θ=π with
the YM field strength curvature Fa)
between time-reversal Z2T symmetry (with a schematic background field T or w1(TM))
and the 1-form electric ZN,[1]e symmetry (with a schematic 2-form background field B), via a schematic 5d topological term:
[TABLE]
LABEL:Wan:2018zql,Wan:2019oyr
shows that this precise 5d topological term written as a 5d bordism invariant (at N = 2)
of a mod 2 class term
is:
[TABLE]
based on the notation introduced earlier in
Sec. 5.4.1 and Sec. 5.4.1, it can be written as:
[TABLE]
where x2=B is the generator of H2(B2Z2,Z2).
Other than LABEL:Wan:2018zql,Wan:2019oyr, the derivation of the relation of the topological invariant x2x3+x5=21w~1P2(x2) has also been examined
in an excellent note of Debray [69].
For eqn. (7.6), our relevant cobordism theory includes
the unoriented bordism group
ΩdO(B2Z2) in
Sec. 5.4.1
and the oriented bordism group
ΩdSO(B2Z2) in
Sec. 5.4.2.
2. 2.
Pure SU(N) Yang-Mills theory’s higher anomaly:
The above formulas (7.5) and (7.6), are 5d topological invariants characterizing the 4d SU(2) YM at θ=π’s higher anomaly.
For a generic 4d SU(N) YM at θ=π of even integer N=2n,
LABEL:Wan:2018zql proposes a precise 5d topological term written as a 5d bordism invariant (at N=2n) which includes at least a mod 2 class term:
[TABLE]
characterizing (part of) the 4d SU(2) YM at θ=π’s higher anomaly.
Pontryagin square is defined as
P:H2(−,Z2n)→H4(−,Z2n+1).
For example, at N=2, we get eqn. (7.7)
coincides the same formula as eqn. (7.6).
At N=4, we get the formula
Bβ(2,N=4)B=41w~1(TM)P(B).
Our corresponding cobordism group calculations are presented in Sec. 6.2.1 and Sec. 6.2.2.
3. 3.
More discrete symmetries (e.g. charge conjugation) and more higher anomalies:
SU(N) YM theory has charge conjugation symmetry Z2C when N>2. Therefore,
LABEL:Wan:2018zql
presents additional higher ’t Hooft anomalies associated to the
charge conjugation Z2C background field AC.
The relevant cobordism group calculation involving additional Z2C symmetry requires
adding a new BZ2 sector into the previous classifying space.
Relevant cobordism group calculations are presented in LABEL:Wan:2018zql, and also some trial toy-model examples
in Sec. 5.6,
Sec. 6.6,
and Sec. 6.7 involving the classifying space BZm and higher-classifying space B2Zn.
The combined higher-classifying space includes the forms of BZm×B2Zn or
BZm⋉B2Zn (in LABEL:Wan:2018zql).
4. 4.
Non-linear sigma models and their anomalies: Non-linear sigma models such as the CPN−1-sigma models (with the
target space CPN−1) have a global symmetry of PSU(N). Therefore, the relevant
cobordism group calculations presented in LABEL:Wan:2018zql include the classifying space BPSU(N).
We include the pertinent cobordism group calculations also for BPSU(N) in Sec. 5.5,
BPSU(2)=BO(3) in Sec. 6.3,
and B(Z2⋉PSU(N)) in Sec. 6.8.
The time reversal symmetry Z2T of bosonic or fermionic version of sigma models corresponds to
O or Pin*±* structure respectively.
The charge conjugation symmetry Z2C corresponds to the
BZ2 in B(Z2⋉PSU(N)) in Sec. 6.8
5. 5.
Higher-symmetry extension, and the fate of gapped and gapless-ness of quantum phases:
An SU(N) YM gauge theory coupled to SU(N) fundamental fermions break explicitly the 1-form ZN,[1]e-symmetry
(thus does not have the 1-form ZN,[1]e-symmetry).
An SU(N) YM gauge theory coupled to SU(N) adjoint fermions can still possess a 1-form ZN,[1]e-symmetry.
The SU(N) adjoint fermion YM gauge theory is known as the adjoint QCD of SU(N) gauge group.
The relevant global symmetries of this adjoint QCD thus includes
ZN,[1]e and a SU(m) flavor chiral symmetry (say, if there is an m-flavor of Wely fermions in the adjoint representation of SU(N)).
Some trial toy-model examples of cobordism groups, involving these classifying spaces
BSU(m), BPSU(m) and B2ZN, are presented
in Sec. 5.7.
For example, for the adjoint QCD with an SU(2) gauge group and Nf=2 adjoint Weyl fermions,
the pertinent symmetry groups are
{{\rm Spin}\times_{{\mathbb{Z}_{2}^{F}}}\big{(}\frac{{\rm SU}(2)\times\mathbb{Z}_{8,\rm{A}}}{\mathbb{Z}_{2}^{F}}\big{)}\times\mathbb{Z}_{2,[1]}^{e}}
or
{{\rm Pin}^{-}\times_{{\mathbb{Z}_{2}^{F}}}\big{(}\frac{{\rm SU}(2)\times\mathbb{Z}_{8,\rm{A}}}{\mathbb{Z}_{2}^{F}}\big{)}\times\mathbb{Z}_{2,[1]}^{e}} (including a time-reversal symmetry),
see their cobordism groups and higher-anomalies in [18].
Along this development, the fate of relevant theories of the adjoint QCD
is explored recently using the modern language of higher-symmetries and higher-anomalies in
various other
LABEL:Anber2018tcj1805.12290,_2018arXiv180609592C,_Bi:2018xvr,_SWW,_Poppitz2019fnp1904.11640,_Anber2019nfu1906.10315, other than [18],
and References therein.
LABEL:Wan:2018zql employs a generalization of a symmetry-extension method of [43] to
a higher-symmetry-extension method, as a tool of constructing a fully-symmetry-preserving gapped phase saturating the higher ’t Hooft anomalies.
It turns out that:
•
Certain higher ’t Hooft anomalies cannot be saturated by a fully-symmetry-preserving gapped phase (e.g. TQFT); which implies either the symmetry-breaking or gapless-ness
of the dynamical fate of the theories. Examples include P(B) in H4(M,Z4)
and AP(B) in H5(M,Z4) where M is the spacetime manifold [18].
This higher-symmetry-extension approach [18] thus rules out some candidate low-energy infrared phases (as a dual phase of a high-energy QFT) proposed in [71].
•
Certain higher ’t Hooft anomalies can be saturated by a fully-symmetry-preserving gapped phase (e.g. TQFT); which implies a possible exotic dynamical fate
as the confinement with no chiral symmetry nor 1-form center symmetry breaking.
Various examples of pure SU(N) YM gauge theories with θ=π-topological term
indeed afford such an exotic confinement without any (ordinary or higher) symmetry-breaking, see
LABEL:Wan:2018zql,Wan:2019oyr.
7.2 Relations to Bosonic/Fermionic Higher-Symmetry-Protected Topological states:
Bosonic higher-symmetry protected topological states (b-higher-SPTs):
(1) Bosonic symmetry-protected topological states (bSPTs) in d+1d of an internal (ordinary 0-form) global symmetry G(0)
is proposed firstly in Chen-Gu-Liu-Wen LABEL:1106.4772 to be classified by a cohomology group
[TABLE]
or the topological cohomology of
classifying space BG0 as
[TABLE]
(2) It is later proposed by Kapustin in
LABEL:Kapustin2014tfa1403.1467, for bosonic SPTs of G(0) and for bosonic symmetric invertible topological order (denoted as b-iTO) of G(0),
they are classify by a cobordism group classification, which is beyond the group cohomology framework. The torsion (finite group ∏jZnj) part of cobordism group classification
contains:
[TABLE]
where H is an oriented H= SO or an unoriented H= O for the (co)bordism group.
To include the free part (the non-torsion part, infinite integer ∏jZ classes), we need to include additional contribution: In physics,
this is related to the nontrivial thermal Hall response and gravitational Chern-Simons terms.
(3) LABEL:Wen2014zga1410.8477 of Wen proposes the SO(∞) version of bosonic cohomology group to classify the bSPTs beyond Chen-Gu-Liu-Wen’s LABEL:1106.4772 via
[TABLE]
(4) LABEL:Freed2016 of Freed-Hopkins introduces this classification of topological phases (TP),
including the torsion and the free parts, defined as a suitable new cobordism group denoted:
[TABLE]
(5) In our work, we generalize the result of LABEL:Freed2016 of bosonic SPTs to bosonic higher-SPTs including the higher-symmetries (e.g. G(1)),
such as
Fermionic higher-symmetry protected topological states (f-higher-SPTs):
Fermionic symmetry-protected topological states (fSPTs) in d+1d of an internal (ordinary 0-form) global symmetry G(0)
is proposed in LABEL:Gu2012ib1201.2648 by Gu-Wen to be classified by a super-cohomology group.
A corrected modification of Gu-Wen model is presented by Gaiotto-Kapustin in
LABEL:Gaiotto2015zta1505.05856. The Gu-Wen model and Gaiotto-Kapustin model is more or less complete for
the 3d (2+1D) fSPTs with a global symmetry of finite group G(0). They also provide lattice Hamiltonian or wavefunction model constructions.
The relation between the full fermionic symmetry group GF and G(0) is based on a short exact sequence, extended by a normal subgroup
fermionic parity Z2F:
[TABLE]
However, neither Gu-Wen nor Gaiotto-Kapustin models obtain a complete classification for 4d (3+1D) fSPTs, even for a global symmetry of finite group G(0).
Improvements are made via several different approaches:
(1) Kitaev’s in LABEL:Kitaev2015 proposes a homotopy-theoretic approach to SPT phases in action. This gives rise a correct Z16 classification of 3+1D topological superconductors, matching to the cobordism group classification. Kitaev’s LABEL:Kitaev2015
can be regarded as the interaction version of SPT classification, improved from his previous K-theory approach for the topological phase classification of free-fermion systems [78].
Kitaev’s approach is reviewed, for example, in LABEL:Xiong2016deb1701.00004,_Gaiotto2017zba1712.07950.
(2) Kapustin-Thorngren-Turzillo-Wang [27] approaches is based on the H=Spin or Pin± versions of cobordism group
Hom(Ωd+1,torsH(BG(0)),U(1)).
(3) Freed-Hopkins [4] introduces a cobordism group TPd+1(H×G(0)) whose effective computation is based on the Adams spectral sequence, with
H=Spin or Pin± for a fermionic theory.
(4) Kapustin-Thorngren in LABEL:Kapustin2017jrc1701.08264 introduces the higher-dimensional bosonization to construct higher-dimensional fSPTs,
mostly focusing on a finite symmetry group GF.
(5) Wang-Gu in LABEL:WangGu2017moj1703.10937,_Wang2018pdc1811.00536 introduces a generalized group super-cohomology theory with multi-layers of group extension structures
of super-cohomology group,
mostly focusing on a finite symmetry group GF.
The computation of fSPTs classification based on the generalized super-cohomology group is similar to the Atiyah-Hirzebruch spectral sequence method. See related discussions in
LABEL:ShiozakiKen2018yyj1810.00801,_1812.11959 on Atiyah-Hirzebruch spectral sequence for classifying fSPTs.
(6) LABEL:Montero2018arXiv180800009G,_Hsieh2018ifc1808.02881 use a mixture of Dai-Freed theorem [50] and
Atiyah-Hirzebruch-like spectral sequence to determine fSPTs and their discrete anomalies on the boundaries.
(7) LABEL:1812.11959 computes various finite-group fSPTs via Adams spectral sequence. Their methods and their derived fSPT terms can be regarded
as the complementary approach to those derived via Atiyah-Hirzebruch-like spectral sequence [81, 82, 83].
(8) In our work, we generalize the result of LABEL:Freed2016 of fermionic SPTs to fermionic higher-SPTs including the higher-symmetries (e.g. G(1)),
such as
[TABLE]
with H=Spin or Pin±. Or slightly more generally, consider the classification of fermionic higher-SPTs via:
[TABLE]
such that the H, G
and H satisfy the following exact sequences:
[TABLE]
or
[TABLE]
Even more general constructions are explored in Sec. 4.
3. 3.
Braiding statistics and link invariants approach to characterize bosonic/fermionic SPTs and higher-SPTs:
Another useful approach to classify SPTs is based on gauging the global symmetry group of SPTs, such that we obtain a gauge theory or TQFT at the end.
The braiding statistics of the fractionalized excitations of gauged SPTs can characterize the pre-gauged SPTs, the explicit method of 3d (2+1D) SPTs is outlined by Levin-Gu [85].
Here we focus on the case of continuum field theory formulation of braiding statistics and link invariants approach to characterize these higher-dimensional SPTs.
(3) 5d (4+1D) SPTs or higher dimensions: LABEL:Wan:2019oyr.
4. 4.
A Generalized Cobordism Theory of higher-symmetry groups — beyond
Higher-Group Super-Cohomology Theories:
It is known that the cobordism theory approach of
Kapustin et al. [26, 27] and Freed-Hopkins [4]
obtain the classification of fSPTs and bSPTs beyond Chen-Gu-Liu-Wen’s group cohomology [5] or Gu-Wen’s group super-cohomology [75].
A more refined version of generalized group super-cohomology [81, 82]
can obtain some missing classes of [75] to match the cobordism classification.
Therefore, we expect that the our approach, on a generalized cobordism theory including the higher-symmetry groups,
can classify higher-SPTs (including fSPTs and bSPTs) that may or may not be captured by
higher-group super-cohomology theories.
For future work, it will be illuminating to understand the distinctions between the generalized higher-group cobordism theory approach and the generalized higher-group super-cohomology theories.
We expect the comparison between two approaches can be rephrased as
a certain version of Adams spectral sequence method in contrast to
a certain version of Atiyah-Hirzebruch spectral sequence method.
It will also be important to figure the possible lattice-regularization (e.g. lattice Hamiltonian on simplicial complex) of those higher-SPTs classified by our generalized cobordism theory.
8 Acknowledgements
We thank Daniel Freed, Meng Guo, Michael Hopkins, Anton Kapustin, Pavel Putrov, and Edward Witten for conversations.
JW thanks the collaborators for a previous collaboration on Ref. [25].
JW thanks the participants of Developments in Quantum Field Theory and Condensed Matter Physics (November 5-7, 2018)
at Simons Center for Geometry and Physics at SUNY Stony Brook University
for giving valuable feedback where this work is reported [89].
JW thanks the feedback from the attendants of IAS seminar [90].
ZW gratefully acknowledges the support from NSFC grants 11431010, 11571329.
JW gratefully acknowledges the Corning Glass Works Foundation Fellowship and NSF Grant PHY-1606531. This work is also supported by NSF Grant DMS-1607871 “Analysis, Geometry and Mathematical
Physics” and Center for Mathematical Sciences and Applications at Harvard University.
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