Classifying gauge anomalies through symmetry- protected trivial orders and classifying gravitational anomalies through topological orders
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Citation Wen, Xiao-Gang. “Classifying gauge anomalies through symmetry- protected trivial orders and classifying gravitational anomalies through topological orders.” Physical Review D 88, no. 4 (August 2013). © 2013 American Physical Society
As Published http://dx.doi.org/10.1103/PhysRevD.88.045013
Publisher American Physical Society
Version Final published version
Citable link http://hdl.handle.net/1721.1/81402
Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW D 88, 045013 (2013) Classifying gauge anomalies through symmetry-protected trivial orders and classifying gravitational anomalies through topological orders
Xiao-Gang Wen Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5 Canada Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 5 May 2013; published 9 August 2013) In this paper, we systematically study gauge anomalies in bosonic and fermionic weak-coupling gauge theories with gauge group G (which can be continuous or discrete) in d space-time dimensions. We show a very close relation between gauge anomalies for gauge group G and symmetry-protected trivial (SPT) orders (also known as symmetry-protected topological (SPT) orders) with symmetry group G in one- higher dimension. The SPT phases are classified by group cohomology class H dþ1ðG; R=ZÞ. Through a more careful consideration, we argue that the gauge anomalies are described by the elements in dþ1 dþ1 Free½H ðG; R=ZÞ H_ ðBG; R=ZÞ. The well known Adler-Bell-Jackiw anomalies are classified by the free part of H dþ1ðG; R=ZÞ (denoted as Free½H dþ1ðG; R=ZÞ ). We refer to other kinds of gauge anomalies beyond Adler-Bell-Jackiw anomalies as non-ABJ gauge anomalies, which include Witten dþ1 SUð2Þ global gauge anomalies. We introduce a notion of -cohomology group, H_ ðBG; R=ZÞ, for the classifying space BG, which is an Abelian group and include Tor½H dþ1ðG; R=ZÞ and topological dþ1 dþ1 cohomology group H ðBG; R=ZÞ as subgroups. We argue that H_ ðBG; R=ZÞ classifies the bosonic non-ABJ gauge anomalies and partially classifies fermionic non-ABJ anomalies. Using the same approach that shows gauge anomalies to be connected to SPT phases, we can also show that gravitational anomalies are connected to topological orders (i.e., patterns of long-range entanglement) in one-higher dimension.
DOI: 10.1103/PhysRevD.88.045013 PACS numbers: 11.15. q, 11.15.Yc, 02.40.Re, 71.27.+a
anomalies through SPT states, which allow us to under- I. INTRODUCTION stand gauge anomalies for both continuous and discrete Gauge anomaly in a gauge theory is a sign that the theory gauge groups directly. is not well defined. The first known gauge anomaly is the What are SPT states? SPT states are short-range Adler-Bell-Jackiw anomaly [1,2]. The second type of gauge entangled states with an on-site symmetry described by anomaly is the Witten SUð2Þ global anomaly [3]. Some the symmetry group G [35,36]. It was shown that one can recent work on gauge anomaly can be found in Refs. [4–9]. use distinct elements in group cohomology class Those anomalies are for continuous gauge groups. The gauge H dþ1ðG; R=ZÞ to construct distinct SPT states in anomalies can also appear for discrete gauge groups. (d þ 1)-dimensional space-time [37–39]. Previously, the understanding of those discrete-group The SPT states have very special low energy boundary anomalies was obtained by embedding the discrete gauge effective theories, where the symmetry G in the bulk groups into continuous gauge groups [10,11], which only is realized as a non-on-site symmetry on the boundary. If captures part of the gauge anomalies for discrete gauge we try to gauge the non-on-site symmetry, we will get groups. an anomalous gauge theory, as demonstrated in In condensed matter physics, close relations between Refs. [38,40–43] for G ¼ Uð1Þ, SUð2Þ. This relation be- gauge and gravitational anomalies and gapless edge exci- tween SPT states and gauge anomalies on the boundary of tations [12,13] in quantum Hall states [14,15] have been the SPT states is called anomaly inflow (the first example found. Also close relations between gauge and gravitational was discovered in Refs. [44,45], which allows us to obtain anomalies of continuous groups and topological insulators the following result: and superconductors [16–29] have been observed [30–34], which have been used extensively to understand and study one can use different elements in group cohomol- topological insulators and superconductors [30]. ogy class H dþ1ðG; R=ZÞ to construct different In this paper, we will give a systematic understanding of bosonic gauge anomalies for gauge group G in gauge anomalies in weak-coupling gauge theories, where d-dimensional space-time. weakly fluctuating gauge fields are coupled to matter fields. If the matter fields are all bosonic, the corresponding This result applies for both continuous and discrete gauge anomalies are called bosonic gauge anomalies. If gauge groups. The free part of H dþ1ðG; R=ZÞ, some matter fields are fermionic, the corresponding gauge Free½H dþ1ðG; R=ZÞ , classifies the well known Adler- anomalies are called fermionic gauge anomalies. We find Bell-Jackiw anomaly for both bosonic and fermionic sys- that we can gain a systematic understanding of gauge tems. The torsion part of H dþ1ðG; R=ZÞ corresponds to
1550-7998=2013=88(4)=045013(22) 045013-1 Ó 2013 American Physical Society XIAO-GANG WEN PHYSICAL REVIEW D 88, 045013 (2013) dþ1 new types of gauge anomalies beyond the Adler-Bell- We will define H_ ðBG; R=ZÞ later in Sec. VB.In Jackiw anomaly (which will be called non-ABJ gauge the next two sections, we will first give a general picture anomalies). of our approach and present some simple examples of However, in the above systematic description, the the new non-ABJ gauge anomalies. Then we will give a nontrivial gauge anomalies come from the nontrivial general systematic discussion of gauge anomalies, and homological structure of the classifying space BG of the their description or classification in terms of dþ1 dþ1 gauge group G. On the other hand, we know that nontrivial Free½H ðG; R=ZÞ H_ ðBG; R=ZÞ. global anomalies come from nontrivial homotopic struc- Last, we will use the connection between gauge anoma- ture dðGÞ of G, which is the same as the homotopic lies and SPT phases (in one-higher dimension) to construct structure of the classifying space since dþ1ðBGÞ¼ a nonperturbative definition of any anomaly-free chiral dðGÞ. Therefore, the cohomology description of gauge gauge theories. We find that even certain anomalous chiral anomalies may miss some global anomalies which can gauge theories can be defined nonperturbatively. only be captured by the homotopic structure of BG, instead of the homological structure. II. A GENERAL DISCUSSION In an attempt to obtain a more general description OF GAUGE ANOMALIES of gauge anomalies, we introduce a notion of the dþ1 A. Study gauge anomalies in one-higher dimension -cohomology group H_ ðBG; R=ZÞ for the classifying dþ1 and in zero-coupling limit space BG of the gauge group G. H_ ðBG; R=ZÞ is an Abelian group which include the topological cohomology We know that anomalous gauge theories are not well class Hdþ1ðBG; R=ZÞ and group cohomology class defined. But, how can we classify something that are not Tor½H dþ1ðG; R=ZÞ as subgroups (see Appendix D): well defined? We note that if we view a gauge theory with the Adler-Bell-Jackiw anomaly in d-dimensional space- Tor ½H dþ1ðG; R=ZÞ Hdþ1ðBG; R=ZÞ time as the boundary of a theory in (d þ 1)-dimensional space-time, then the combined theory is well defined. The H_ dþ1ðBG; R=ZÞ: (1) gauge noninvariance of the anomalous boundary gauge theory is canceled by the gauge noninvariance of a If G is finite, we further have Chern-Simons term on (d þ 1)-dimensional bulk which is gauge invariant only up to a boundary term. So we define Tor ½H dþ1ðG; R=ZÞ ¼ Hdþ1ðBG; R=ZÞ d-dimensional anomalous gauge theories through defining dþ1 ¼ H_ ðBG; R=ZÞ: (2) a(d þ 1)-dimensional bulk theory. The classification of the (d þ 1)-dimensional bulk theories will leads to a classifi- dþ1 We like to remark that, by definition, H_ ðBG; R=ZÞ is cation of anomalies in d-dimensional gauge theories. more general than Hdþ1ðBG; R=ZÞ. But at the moment, we The (d þ 1)-dimensional bulk theory has the following dþ1 do not know if H_ ðBG; R=ZÞ is strictly larger than generic form Hdþ1ðBG; R=ZÞ. It is still possible that H_ dþ1ðBG; R=ZÞ¼ TrðF F Þ dþ1 R Z L Lmatter c H ðBG; = Þ even for continuous group. dþ1D ¼ dþ1Dð ; ;A Þþ ; (3) We find that we can use the different elements in the g _ dþ1 R Z -cohomology group H ðBG; = Þ to construct differ- where (or c ) are bosonic (or fermionic) matter fields ent non-ABJ gauge anomalies. So, more generally, that couple to a gauge field A of gauge group G. In this paper, we will study gauge anomalies in weak-coupling the bosonic/fermionic gauge anomalies are described gauge theory. So we can take the zero-coupling limit: H dþ1 R Z _ dþ1 R Z by Free½ ðG; = Þ H ðBG; = Þ.It ! 0. In this limit we can treat the gauge field A as H dþ1 R Z g is possible that Free½ ðG; = Þ nondynamical probe field and study only the theory of _ dþ1 R Z H ðBG; = Þ classify all the bosonic gauge the mater fields Lmatter ð ; c ;A Þ, which has an on-site anomalies. H_ dþ1ðBG; R=ZÞ includes dþ1D symmetry with symmetry group G if we set the probe field Hdþ1ðBG; R=ZÞ as a subgroup. A ¼ 0. So we can study d-dimensional gauge anomalies d þ We note that Witten’s SUð2Þ global anomaly is a through ( 1)-dimensional bulk theories with only G fermionic global anomaly with known realization by matter and an on-site symmetry . fermionic systems. Since the -cohomology result dþ1 dþ1 B. Gauge anomalies and SPT states Free½H ðG; R=ZÞ H_ ðBG; R=ZÞ only describes part of fermionic gauge anomalies, it is not clear if it Under the above set up, the problem of gauge anomaly includes Witten’s SUð2Þ global anomaly. On the other becomes the following problem: hand, we know for sure that the group cohomology result Given a low energy theory with a global symmetry Hdþ1ðBG; R=ZÞ does not include the SUð2Þ global anom- G in d-dimensional space-time, is there a nonper- aly since H5ðBSUð2Þ; R=ZÞ¼0. turbatively well-defined theory with on-site
045013-2 CLASSIFYING GAUGE ANOMALIES THROUGH SYMMETRY- ... PHYSICAL REVIEW D 88, 045013 (2013) symmetry in the same dimension which reproduce characterize d-dimensional gauge anomalies. The topo- the low energy theory. logical invariants for (d þ 1)-dimensional SPT states also give rise to anomaly cancellation conditions: Given a We require the global symmetry G to be an on-site potentially anomalous gauge theory in d-dimensional symmetry in the well-defined theory since we need to space-time, we first construct a well defined (d þ 1)- gauge the global symmetry to recover the gauge theory dimensional theory which produce the d-dimensional with gauge group G. gauge theory. (This step is needed since the potentially It turns out that we may not always be able to find a well- anomalous gauge theory may not be well defined in defined theory with on-site symmetry in the same dimen- d-dimensional space-time.) If all the topological invariants sion to reproduce the low energy theory. Let us assume that for the (d þ 1)-dimensional theory are trivial, then the we can always find a well-defined gapped theory with d-dimensional gauge theory is not anomalous. on-site symmetry in higher dimension to reproduce the low energy theory on a lower dimensional defect sub- manifold, such as a boundary, a defect line, etc. Note that C. Gauge anomalies and gauge topological term d þ1 we can always deform the higher dimensional space into a in ( ) dimensions lower dimensional space so that the defect submanifold In addition to the topological invariants studied in looks like a boundary when viewed from far away (see Refs. [49,50], we can also characterize gauge anomalies gauge Fig. 1). So without loosing generality, we assume that we through the induced gauge topological term Wtop ðA Þ in can always find a well-defined gapped theory with on-site the (d þ 1) dimensional theory, obtained by integrating out symmetry in one-higher dimension to reproduce the low the matter fields. The gauge topological term provide us a energy theory on the boundary. Therefore, powerful tool to study gauge anomalies in one lower dimension. We can understand anomalies through studying theo- The above describes the general strategy that we will ries with on-site symmetry in one-higher dimension. follow in this paper. In the following, we will first use this line of thinking to examine several simple examples of In this paper, we will concentrate on ‘‘pure gauge’’ non-ABJ gauge anomalies. anomalies. We require that the theory is not anomalous if we break the gauge symmetry. Within our set up, this means that we can find a well-defined gapped theory in III. SIMPLE EXAMPLES OF NON-ABJ same dimension to reproduce the low energy theory, if GAUGE ANOMALIES we allow to break the symmetry at high energies. If we A. Bosonic Z2 gauge anomaly in 1 þ1D do not allow to break the symmetry, we still need to go to one-higher dimension. However, the fact that the boundary The simplest example of non-ABJ gauge anomaly is the _ 3 R Z theory can be well defined within the boundary (if we break Z2 gauge anomaly in 1 þ 1D. Since H ðBZ2; = Þ¼ H 3 R Z Z the symmetry) implies that the ground state in one-higher ðZ2; = Þ¼ 2, we find that there is only one type dimensional theory has a trivial (intrinsic) topological of nontrivial bosonic Z2 gauge anomaly in 1 þ 1D. order [46–48]. This way, we conclude that To see a concrete example of Z2 gauge anomaly, let us first give a concrete example of non-on-site Z2 symmetry. We can understand ‘‘pure’’ gauge anomalies Gauging the non-on-site Z2 symmetry will produce the Z2 through studying SPT states [37–39] with on-site gauge anomaly. symmetry in one-higher dimension. Let us consider the following spin-1=2 Ising-like model on a one-dimensional lattice whose sites form a ring and A nontrivial SPT state in (d þ 1)-dimensions will are labeled by i ¼ 1; 2; ...L [37,42,51]: correspond to a ‘‘pure’’ gauge anomaly d-dimensions. XL XL (For more detailed discussions, see Sec. IV). z z x x z x z Hring ¼ Ji;iþ1 i iþ1 hi ð i i 1 i iþ1Þ With such a connection between gauge anomalies and i¼1 i¼1 SPT states, we see that the topological invariants for XL (d þ 1)-dimensional SPT states [49,50] can be used to y y z y z hi ð i þ i 1 i iþ1Þ; (4) i¼1 where x, y, z are 2-by-2 Pauli matrices. The model has a non-on-site (or anomalous) Z2 global symmetry generated by
FIG. 1 (color online). A point defect in two dimensions looks YL YL x like a boundary of an effective one-dimensional system, if we U ¼ i i;iþ1; (5) wrap the two-dimensional space into a cylinder. i¼1 i¼1
045013-3 XIAO-GANG WEN PHYSICAL REVIEW D 88, 045013 (2013) where i;j is a 4-by-4 matrix that acts on two spins at site i conserved. So the full symmetry group is generated by z z 2 2 and site j as ¼ j ""ih"" j j "#ih"# j þ j #"ih#" j þ j ##ih## j. (i 1, i L, Uline, U) which is ½ðZ4 Z4Þ Z4 Z2. Some We say U is a non-on-site symmetry transformation since it of the group elements have the relation cannot be written in the direct product form (i.e., the on- z z site form) [37–39,42] U ¼ iUi, where Ui acts only on ði 1ÞU ¼ Uði 1Þ: (8) site i. Such a non-on-site (or anomalous) symmetry is not So all the representations of the group must be even ‘‘gaugable’’. If we try to gauge the Z2 symmetry, we will dimensional. Such a symmetry causes a two-fold degener- get an anomalous Z2 gauge theory in 1 þ 1D. The anoma- acy for all the eigenvalues of Hline. From a numerical lous Z2 gauge theory is not well defined and we cannot calculation, we find that the two-fold degenerate states even write down its Hamiltonian. However, the anomalous always carry opposite Z2 quantum numbers U ¼ 1. 1 þ 1D Z2 gauge theory can be defined as a boundary of a This is a property that reflects the anomaly in the Z2 2 þ 1D Z2 gauge theory. So we can study the physical symmetry. properties of anomalous 1 þ 1D Z2 gauge theory through Z its corresponding 2 þ 1D Z2 gauge theory. We will do this The two-fold degeneracy induced by the 2 non- in the next section for the more general anomalous 1 þ 1D on-site symmetry implies that there is a Majorana þ Zn gauge theory. zero-energy mode at each end of 1 1D system if In the rest of this section, we will not gauge the Z2 the system lives on an open line. symmetry. We will only study the 1 þ 1D model with the non-on-site (i.e., anomalous) Z2 symmetry. We like to Z 1þ1D understand the special properties of the 1 þ 1D model B. Bosonic n gauge anomalies in that reflect the anomaly (the non-on-site character) in the Now let us discuss more general Zn gauge anomaly in Z2 symmetry. 1 þ 1D bosonic gauge theory, which is classified by 3 3 The most natural way to probe the gauge anomaly is to H_ ðBZn; R=ZÞ¼H ðZn; R=ZÞ¼Zn. So there are measure the gauge charge induced by gauge flux. So to n 1 nontrivial Zn gauge anomalies. To construct the probe the Z2 anomaly, we like to add an unit of Z2 flux examples of those Zn gauge anomalies, we will present through the ring on which the 1 þ 1D system is defined, two approaches here. and then measure the induced Z2 charge. But since the Z2 In the first approach, we start with a bosonic Zn SPT symmetry is non-on-site, we do not know how to add an state in 2 þ 1D. We can realize the Zn SPT state through a unit of Z2 flux through the ring. We can add Z2 flux only if 2 þ 1D bosonic Uð1Þ SPT state, which is described by the we view our anomalous 1 þ 1D system as a boundary of a following Uð1Þ Uð1Þ Chern-Simons theory [40,43]: 2 þ 1D Z2 gauge theory, which will be discussed in the 1 1 next section. So here, we will do the next best thing: we L ¼ KIJaI @ aJ þ qIA @ aI þ ; will study the 1 þ 1D model on a open line. The 1 þ 1D 4 2 model on an open line can be viewed as having a strong (9) fluctuation in the Z2 flux through the ring. The Hamiltonian on an open line, Hline, can be obtained where the nonfluctuating probe field A couples to the from that on a ring (4) by removing all the ‘‘nonlocal current of the global Uð1Þ symmetry. Here the K matrix terms’’ that couple the site-1 and site-L, i.e., by setting and the charge vector q are given by [52–54] x x y y JL;1 ¼ h1 ¼ hL ¼ h1 ¼ hL ¼ 0. Hline still has the anoma- ! ! 1 ¼ 01 1 lous (i.e., non-on-site) Z2 symmetry: UHlineU Hline. K ¼ ; q ¼ ;k2 Z: (10) However, as a symmetry transformation on a line, U con- 10 k tains a nonlocal term L;1. After dropping the nonlocal term L;1, we obtain The even diagonal elements of the K matrix are required by the bosonic nature of the theory. The Hall conductance YL LY 1 1 x for the Uð1Þ charge coupled to A is given by Uline ¼ U L;1 ¼ i i;iþ1: (6) i¼1 i¼1 1 T 1 2k xy ¼ð2 Þ q K q ¼ : (11) We find that Uline is also a symmetry of Hline: UlineHline 2 1 Uline ¼ Hline. From the relation The above 2 þ 1D Uð1Þ SPT state is characterized by 2 4 1 1 H 3 R Z U ¼ Uline ¼ 1;UUlineU ¼ Uline; (7) an integer k 2 ½Uð1Þ; = . We also know that an 2 þ 1D Zn SPT state is characterized by a mod-n integer 3 we find that U and Uline generate a dihedral group m 2 H ðZn; R=ZÞ. If we view the 2 þ 1D Uð1Þ SPT state 2 D4 ¼ Z4 Z2—a symmetry group of Hline. In fact, Hline labeled by k as a 2 þ 1D Zn SPT state, then what is the m z z has even higher symmetries since 1 and L are separately label for such a 2 þ 1D Zn SPT state?
045013-4 CLASSIFYING GAUGE ANOMALIES THROUGH SYMMETRY- ... PHYSICAL REVIEW D 88, 045013 (2013) ! ! The mod-n integer m can be measured through a mn =n 2 1 01 topological invariant constructed by creating n identical K~ ¼ ; K~ ¼ : (14) n 0 1=n 2m=n2 Zn monodromy defects [50]: 2m is the total Zn charge of n identical Z monodromy defects. On the other hand, a Z n n When m ¼ 0, the above 2 þ 1D theory is a standard Zn monodromy defect corresponds to 2 =n flux in the Uð1Þ gauge theory, and its low energy edge theory is a standard gauge field A . From the 2k quantized Hall conductance, n Zn gauge theory in 1 þ 1D with no anomaly. Such an 1 þ ð Þ identical 2 =n-flux of A will induce 2kU1 charge, 1D Zn gauge theory can defined within 1 þ 1D without which is also the Zn charge. So the above bosonic Uð1Þ going through a 2 þ 1D theory. When m 0, the m term SPT state correspond to a m ¼ k mod n bosonic Zn SPT corresponds to a quantized topological term in Zn gauge state [50]. theory discussed in Ref. [55]. Such a quantized topological 3 The low energy effective edge theory for the 2 þ 1D term is classified by a mod-n integer m 2 H ðZn; R=ZÞ. system (9) has an non-on-site Zn symmetry if k 0 mod n. To see the relation between the Uð1Þ Uð1Þ (In fact, the low energy edge effective theory has an non- Chern-Simons theory Eq. (13) and the Zn gauge theory on-site Uð1Þ symmetry.) If we gauge such a non-on-site Zn in 2 þ 1D,[57,58] we note that a unit a1 -charge corre- symmetry on the edge, we will get an anomalous Zn gauge spond to a unit Zn charge. A unit Zn charge always carries a þ theory in 1 1D, which is not well defined. (In other Bose statistics. So the Zn gauge theory is a bosonic Zn words, we cannot gauge non-on-site Zn symmetry within gauge theory. On the other hand, a unit of Zn flux is 1 þ 1D.) v described by a particle with lI aI -charge. We find that However, we can define an anomalous Zn gauge theory v l2 ¼ 1 (so that moving a unit Zn charge around a unit Zn in 1 þ 1D as the edge theory of a 2 þ 1D Zn gauge theory. v flux will induce 2 =n phase). l1 can be any integer and the Such a 2 þ 1D Zn gauge theory can be obtained from lv v ¼ðl1 ; 1Þ aI -charge is not a pure Zn flux (i.e., may carry Eq. (9) by treating A as a dynamical Uð1Þ gauge field some Zn charges). and introduce a charge n-Higgs field to break the Uð1Þ When the 2 þ 1D system (13) has holes (see Fig. 2), the down to Z : n theory lives on the edge of hole is an 1 þ 1D anomalous Zn gauge theory. If we add Zn flux to the hole, we may view 1 1 l L ¼ K a @ a þ q A @ a the hole as a particle with ¼ð0; 1Þ aI -charge. Such a 4 IJ I J 2 I I particle carries a fractional 2m=n Zn charge as discussed 2 2 4 þjð@ þ inA Þ j þ aj j bj j : (12) above. We conclude that, when m 0,
The edge theory of the above Ginzberg-Landau-Chern- a unit of Zn flux through a ring, on which a 1 þ 1D Simons theory contain gapless edge excitations with anomalous Zn gauge theory lives, always induces a central charge c ¼ 1 right-movers and central charge fractional Zn charge 2m=n, which is a consequence c ¼ 1 left-movers. Such a 1 þ 1D edge theory is an ex- of Zn gauge anomaly of the 1 þ 1D system. ample of anomalous 1 þ 1D Zn gauge theory that we are looking for. The anomaly is characterized by a mod-n integer m ¼ 2k. A unit of Zn flux (a 2 =n flux) through the hole (see Fig. 2) will induce a 2m=n Zn charge on the edge. Such a property directly reflects a Zn gauge anomaly. To summarize, in the first approach, we start with a Zn SPT state in 2 þ 1D to produce a 1 þ 1D edge theory with a non-on-site Zn symmetry. We then gauge the non-on-site Zn symmetry to obtain an anomalous Zn gauge theory in 1 þ 1D. In the second approach, we use the Levin-Gu duality relation [49,55,56] between the Zn SPT states and the (twisted) Zn gauge theory in 2 þ 1D. We obtain the anoma- lous 1 þ 1D bosonic Zn gauge theory directly as the edge FIG. 2 (color online). A Z2 gauge configuration with two theory of the (twisted) Zn gauge theory in 2 þ 1D. The identical holes on a torus that contains a unit of Z2 flux in (twisted) Zn gauge theory can be described by the follow- each hole. The Z2 link variables are equal to 1 on the crossed ing 2 þ 1D Uð1Þ Uð1Þ Chern-Simons theory [57–59]: links and 1 on other links. If the 1 þ 1D bosonic Z2 gauge theory on the edge of one hole is anomalous, then such a Z2 gauge configuration induces half unit of total Z2 charge on the edge 1 (representing a Z gauge anomaly). Braiding those holes around L ¼ K~IJaI @ aJ þ ; (13) 2 4 each other reveals the fractional statistics of the holes. The edge states for one hole are degenerate with 1=2 Z2 charge if there is where the K~-matrix is given by a time-reversal symmetry.
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From the second description of the anomalous 1 þ 1D edge theories of those twisted Z2 Z2 Z2 gauge theories bosonic Zn gauge theory, we see that if we view the holes realize the 127 types of bosonic Z2 Z2 Z2 gauge with a unit of Zn flux as particles (see Fig. 2), then such anomalies in 1 þ 1D. The edge theories always have degen- particles will carry a unit of a2 -charge. If we braid the erate ground states or gapless excitations, even after we holes with a unit of Zn flux around each others (see Fig. 2), freeze the Zn gauge fluctuations (i.e., treat the Zn gauge 2m those holes will carry a fractional statistics ¼ 2 (the field as a nondynamical probe field). n As discussed in Refs. [62,63], twisted Z Z Z fractional statistics of unit a2 -charges). 2 2 2 gauge theories can be described by U6ð1Þ Chern-Simons theories (13) with One can use fractional (or non-Abelian) statistics 0 1 of the holes with flux to detect 1 þ 1D gauge 2m1 2 m12 0 m13 0 anomaly [49]. B C B 200000C B C The gapless edge excitations of the 2 þ 1D theory (13) B m12 0 2m2 2 m23 0 C K~ ¼ B C; (18) is described by the following 1 þ 1D effective theory B 002000C B C 1 @ m13 0 m23 0 2m3 2 A L1þ1D ¼ ½K~IJ@t I@x J VIJ@x I@x J 4 X X 000020 ~ þ ½c ilKJI I : : ; J;le H c (15) where m , m ¼ 0, 1. The m terms and the m terms are l J¼1;2 i ij i ij the quantized topological terms, which twist the Z2 where the field Iðx; tÞ is a map from the 1 þ 1D space- Z2 Z2 gauge theory. The other 64 twisted Z2 Z2 R Z time to a circle 2 = , and V is a positive definite real Z2 gauge theories are non-Abelian gauge theories [62,64] 2-by-2 matrix. with gauge groups D4, Q8, etc. (see also Ref. [63]). So A Zn flux (not the pure Zn flux which is not allowed) is some of the anomalous Z2 Z2 Z2 gauge theories in described by an unit a2 -charge. From the equation of 1 þ 1D has to be defined via non-Abelian gauge theories in the motion, we find that, in the bulk, a Zn flux correspond 2 þ 1D. (For details, see Refs. [62,64]). In this case, the 2 to a bound state of 1=n a1 -flux and 2m=n a2 -flux. holes that carry the gauge flux have non-Abelian statistics Thus a unit of Zn flux through the hole is described by (see Fig. 2). the following boundary condition [60,61] D. Fermionic Z2 Z2 gauge anomalies in 1 þ 1D 1ðxÞ¼ 1ðx þ LÞþ2 =n; (16) A fermionic Z2 Z2 anomalous (i.e., non-on-site) 2 2ðxÞ¼ 2ðx þ LÞþ2 ð2m=n Þ: symmetry in 1 þ 1D can be realized on the edge of a 2 þ 1D fermionic Z2 Z2 SPT states. Those fermionic We see that the Zn symmetry of the 1 þ 1D theory is SPT states were discussed in detail in Refs. [50,51,65]. generated by We found that there are 16 different fermionic Z2 Z2 ! þ 2 =n; ! þ 4 m=n2: (17) SPT states in 2 þ 1D (including the trivial one) which form 1 1 2 2 Z a 8 group where the group operation is the stacking of Such a Zn symmetry is anomalous (or non-on-site) if the 2 þ 1D states. m 0 mod n in Eq. (36). When n ¼ 2 and m ¼ 1, One type of the fermionic Z2 Z2 anomalous symmetry Eq. (15)is the low energy effective theory of Hring in Eq. (4). in 1 þ 1D is realized by the following free Majorana field If we gauge the Zn symmetry, we will get an anomalous theory 1 þ 1D Z gauge theory, which has no 1 þ 1D nonpertur- n L ¼ i ð@ @ Þ þ i ð@ þ @ Þ : (19) bative definition. This way, we obtain an example of bosonic 1þ1D R t x R L t x L anomalous Zn gauge theory in 1 þ 1D,Eqs.(15)and(17). The Z2 Z2 symmetry is generated by the following two generators C. Bosonic Z2 Z2 Z2 gauge anomalies in 1 þ 1D ð R; LÞ!ð R; LÞ; ð R; LÞ!ð R; LÞ; (20) The bosonic Z2 Z2 Z2 gauge anomalies in 1 þ 1D _ 3 R Z H 3 are classified by H ðBðZ2 Z2 Z2Þ; = Þ¼ ½Z2 i.e., R carries the first Z2 charge and L the second Z2 R Z Z7 Z2 Z2; = ¼ 2. So there are 127 different types charge. The above fermionic anomalous symmetry is the Z of Z2 Z2 Z2 gauge anomalies in 1 þ 1D. Those 127 generator of 8 types of fermionic Z2 Z2 anomalous gauge anomalies in 1 þ 1D can be constructed by starting symmetries. with a 2 þ 1D Z2 Z2 Z2 gauge theory. We then add the Due to the anomaly in the Z2 Z2 symmetry, the above quantized topological terms [55] to twist the Z2 Z2 Z2 1 þ 1D field theory can only be realized as a boundary of a gauge theory. The quantized topological terms are also 2 þ 1D lattice model if we require the Z2 Z2 symmetry H 3 R Z classified by ½Z2 Z2 Z2; = . The low energy to be an on-site symmetry. (However, it may be possible to
045013-6 CLASSIFYING GAUGE ANOMALIES THROUGH SYMMETRY- ... PHYSICAL REVIEW D 88, 045013 (2013) realize the 1 þ 1D field theory by a 1 þ 1D lattice model if To see the above two properties cannot be realized by a we do not require the Z2 Z2 symmetry to be an on-site well-defined 2 þ 1D bosonic theory (i.e., represent a Uð1Þ symmetry.) One example of 2 þ 1D realization is the gauge anomaly), we first note that the requirement that stacking of a p þ ip and a p ip superconductor (denoted there is no degenerate ground states implies that there are as p þ ip=p ip state) [26,50,65]. no excitations with fractional charges and fractional statis- Since the Z2 Z2 symmetry is anomalous in the above tics (since the state has no intrinsic topological order 1 þ 1D field theory, if we gauge the Z2 Z2 symmetry, the [46,47]). Second, the above Uð1Þ Chern-Simons theory resulting 1 þ 1D fermionic Z2 Z2 gauge theory will be with a fractional coefficient has a special property that a anomalous which can not have a nonperturbative definition unit of Uð1Þ flux (2 flux) induces a Uð1Þ charge (since as a 1 þ 1D model. However, the 1 þ 1D fermionic the Hall conductance is 2 ). The flux-charge bound state Z2 Z2 gauge theory can have a nonperturbative defini- has a statistics ¼ . Since a unit of Uð1Þ flux only tion as the boundary of a 2 þ 1D model. One such model is induce an allowed excitation, so for any well-defined the stacking of a bosonic ¼ 1 Pfaffian quantum Hall state 2 þ 1D model with no ground state degeneracy, the in- [66] and a bosonic ¼ 1 Pfaffian quantum Hall state duced charge must be integer, and the induced statistics (denoted as Pfaff=Pfaff state). Note that the bosonic ¼ 1 must be bosonic (for a bosonic theory): Pfaffian quantum Hall state have edge states which include ¼ integer; ¼ even integer: (22) a c ¼ 1=2 Majorana mode and a c ¼ 1 density mode [67]. However, since we do not require boson number conser- We see that, for 2½0; 2Þ, the above Uð1Þ Chern-Simons vation, the density mode of the ¼ 1 Pfaffian state and the theory (with no ground state degeneracy) cannot appear as density mode of the ¼ 1 Pfaffian state can gap out each the low energy effective theory of any well-defined 2 þ 1D other, and be dropped. model. Thus, it is anomalous. Again, it is interesting to see that a nonperturbative But when ¼ even integer, the above 2 þ 1D model definition of an anomalous 1 þ 1D fermionic Z2 Z2 with even-integer quantized Hall conductance can be real- Abelian gauge theory requires an non-Abelian state ized through a well-defined 2 þ 1D bosonic model with [66,68]in2 þ 1D. trivial topological order, [40–43] and thus not anomalous [69]. This is why only 2½0; 2Þ represents the Uð1Þ E. Bosonic Uð1Þ gauge anomalies in 2 þ 1D anomalies in 2 þ 1D. However, the above anomalous 2 þ 1D theory (with no The bosonic Uð1Þ gauge anomalies in 2 þ 1D are ground state degeneracy) can be realized as the boundary described by H_ 4 ½BUð1Þ; R=Z which contains H4½BUð1Þ; theory of a 3 þ 1D bosonic insulator that does not have the R=Z ¼R=Z as subgroup. So what are those Uð1Þ gauge time-reversal and parity symmetry. The 3 þ 1D bosonic anomalies labeled by a real number =2 2 R=Z ¼½0; 1Þ? insulator contains a topological term First, let us give a more general definition of anomalies (which include gauge anomalies as special cases): We start 2 L ¼ @ A @ A (23) with a description of a set of low energy properties, and 3þ1D 2!ð2 Þ2 then ask if the set of low energy properties can be realized that is allowed by symmetry. A unit of magnetic flux by a well-defined quantum theory in the same dimensions? through the boundary will induce a fractional Uð1Þ charge If not, we say the theory is anomalous. on the boundary. Thus the 3 þ 1D bosonic insulator can So to describe the 2 þ 1D Uð1Þ gauge anomaly, we need reproduces the above two mentioned low energy properties. to first describe a set of low energy properties. The Uð1Þ The above result can be generalized to study Ukð1Þ gauge gauge anomaly is defined by the following low energy anomaly in 2 þ 1D. If after integrating out the matter fields, properties: we obtain the following gauge topological term (1) there are no gapless excitations and no ground state degeneracy. L IJ I J 2þ1D ¼ A @ A þ ; (24) (2) the Uð1Þ gauge theory has a fractional Hall conduc- 4 tance ¼ =2 . xy then the theory is anomalous if IJ is not an integer sym- The above low energy properties implies that, after inte- metric matrix with even diagonal elements. The anomalous grating out the matter field, the 2 þ 1D theory produces the Ukð1Þ gauge theory can be viewed as the boundary of a following gauge topological term 3 þ 1D Ukð1Þ gauge theory with topological term L 2 IJ I J 2þ1D ¼ A @ A þ : (21) L ¼ @ A @ A : (25) 4 3þ1D 2!ð2 Þ2 0 When 2½0; 2Þ, the above two low energy properties Two topological terms described by IJ and IJ are cannot be realized by a well-defined local bosonic quantum regarded as equivalent if theory in 2 þ 1D. In this case, the theory has a Uð1Þ gauge 0 even anomaly. IJ IJ ¼ KIJ ; (26)
045013-7 XIAO-GANG WEN PHYSICAL REVIEW D 88, 045013 (2013) even where K is an integer symmetric matrix with even L ¼ IJ AI @ AJ þ ; (28) diagonal elements [70]. 2þ1D 4
F. Fermionic Uð1Þ gauge anomalies in 2 þ 1D then the theory is anomalous if and only if IJ is not an integer symmetric matrix. The anomalous fermionic Ukð1Þ In this section, we consider 2 þ 1D fermion systems gauge theory can be viewed as the boundary of a 3 þ 1D with a Uð1Þ symmetry where the fermion parity symmetry Ukð1Þ gauge theory with topological term is part of Uð1Þ symmetry. As a result, all fermions carry odd Uð1Þ charges. 2 L ¼ IJ @ AI @ AJ : (29) The gauge anomalies in such fermionic Uð1Þ gauge 3þ1D 2!ð2 Þ2 4 theory are described by H_ ½BUð1Þ; R=Z which includes 4 0 H ½BUð1Þ; R=Z ¼R=Z. Thus, the fermionic Uð1Þ gauge Two topological terms described by IJ and IJ are anomaly can be labeled by 2 R=Z ¼½0; 1Þ. regarded as equivalent if The Uð1Þ gauge anomaly correspond to the following 0 low energy properties: The 2 þ 1D anomalous fermionic IJ IJ ¼ KIJ; (30) Uð1Þ gauge theory has (1) a gapped nondegenerate ground state and where K is an integer symmetric matrix. It is interesting to see that the periodicy of IJ is an even integer matrix for (2) a fractional Hall conductance xy ¼ =2 . After integrating out the matter field, the 2 þ 1D theory bosonic systems while the periodicy is an integer matrix produces the following gauge topological term for fermionic systems [70].
L ¼ A @ A þ : (27) G. Uð1Þ ½Uð1Þ 2 Z2 gauge anomalies in 2 þ 1D 2þ1D 4 After understanding the Uð1Þ gauge anomalies in 2 þ 1D 2½ Þ þ When 0; 1 the above 2 1D theory is anomalous. for bosonic and fermionic systems, we are ready to discuss When ¼ integer, the above 2 þ 1D model with inte- 2 a more interesting example—Uð1Þ ½Uð1Þ Z2 gauge ger quantized Hall conductance can be realized through a anomalies in 2 þ 1D. well-defined 2 þ 1D fermionic model – an integer quan- tized Hall state which has no ground state degeneracy. So the 2 þ 1D theory with integer is not anomalous. 1. Cohomology description 2 This is why only 2½0; 1Þ represents the fermionic Uð1Þ The 2 þ 1D Uð1Þ ½Uð1Þ Z2 gauge anomalies are 4 2 R Z anomalies in 2 þ 1D. described by H_ ½BðUð1Þ ½Uð1Þ Z2 Þ; = which con- 4 2 R Z Similarly, the above result can also be generalized to tains H ½BðUð1Þ ½Uð1Þ Z2 Þ; = as a subgroup. study fermionic Ukð1Þ gauge anomaly in 2 þ 1D. If after Using Ku¨nneth formula [see Eq. (E15)], we can compute d 2 Z d 2 Z integrating out the matter fields, we obtain the following H ½BðUð1Þ ½Uð1Þ Z2 Þ; from H ½B½Uð1Þ Z2 ; gauge topological term and Hd½BUð1Þ; Z :
d: 0; 1; 2; 3; 4; 5; 6 d 2 Z Z Z Z Z Z Z 2 H ½B½Uð1Þ Z2 ; : ; 0;Z2; 2; 2; 2; 2 ; (31) Hd½BUð1Þ; Z : Z; 0;Z; 0; Z; 0 Z d 2 Z Z Z 2 Z 2 Z 2 Z 2 Z 4 H ½BðUð1Þ ½Uð1Þ Z2 Þ; : Z; 0;Z Z2 2; 2 ; 2 ; 2 2 where Zn Zn Zn. Then using the universal coefficient theorem (see Appendix E), we find
d: 0; 1; 2; 3; 4; 5 (32) d 2 R Z R Z R Z Z Z 2 R Z 2 Z 2 Z 4: H ½BðUð1Þ ½Uð1Þ Z2 Þ; = : = ;Z2; = 2; 2 ; ð = Þ 2 ; 2
2 2 We see that some of the Uð1Þ ½Uð1Þ Z2 gauge anoma- a Uð1Þ ½Uð1Þ Z2 gauge theory coupled to matter R Z 2 Z 2 lies in 2 þ 1D can be described by ð = Þ 2 fields. If integrating out the matter field produces the 4 2 R Z H_ ½BðUð1Þ ½Uð1Þ Z2 Þ; = . following gauge topological term in 3 þ 1D: 2. Continuous gauge anomalies 2 2 L 1 The gauge anomalies described by ðR=ZÞ can be 3þ1D ¼ 2 @ A1 @ A1 2 2!ð2 Þ labeled by two real numbers ð 1; 2Þ2ðR=ZÞ (for fermions) or ð ; Þ2ðR=2ZÞ 2 (for bosons). An ex- 2 2 1 2 þ @ A2 @ A2 ; (33) ample of such a gauge anomaly can be obtained through 2!ð2 Þ2
045013-8 CLASSIFYING GAUGE ANOMALIES THROUGH SYMMETRY- ... PHYSICAL REVIEW D 88, 045013 (2013) then the 3 þ 1D gauge theory describes the desired To further understand the physical property of such a gauge anomaly. Here A1 is for the first Uð1Þ and A2 discrete gauge anomaly, let us assume that the 3 þ 1D 0 the second Uð1Þ, and A2 changes sign under the Z2 gauge space-time has a topology M2 M2. We also assume 0 transformation. that the A1 gauge field has 2 flux on M2. In the large M2 limit, the Lagrangian (34) reduces to an effective 3. First discrete gauge anomaly Lagrangian on M2 which has a form If integrating out the matter field produces the following L ¼ @ A : (38) gauge topological term in 3 þ 1D [55]: M2 2 2 L 3þ1D ¼ @ A1 @ A2 ; (34) We note that the A1 gauge configuration preserve the ð2 Þ2 2 Uð1Þ ðUð1Þ Z2Þ symmetry. The above Lagrangian is 2 then the 3 þ 1D gauge theory describes a discrete Uð1Þ the effective Lagrangian of the Uð1Þ ½Uð1Þ Z2 sym- 2 Z 2 metric theory on M probed by the A gauge field [50]. ½Uð1Þ Z2 gauge anomaly (which belongs to 2 ). The 2 2 boundary 2 þ 1D theory of the 3 þ 1D system will be a Such an effective Lagrangian implies that the Uð1Þ 2 Uð1Þ ½Uð1Þ 2 Z gauge theory with the discrete Uð1Þ ½Uð1Þ Z2 symmetric theory on M2 describe a nontrivial 2 2 ½Uð1Þ 2 Z2 gauge anomaly. Such an anomalous 2 þ 1D Uð1Þ ½Uð1Þ Z2 SPT state labeled by the nontrivial H 2 2 R Z Z theory must be gapless or have degenerate ground states, if element in ½Uð1Þ ðUð1Þ Z2Þ; = ¼ 2 [59]. 2 we freeze the gauge fluctuations without breaking the The nontrivial 1 þ 1D Uð1Þ ½Uð1Þ Z2 SPT state on 2 Uð1Þ ðUð1Þ Z2Þ symmetry. We suspect that, in our par- M2 has the following property: Let M2 ¼ Rt I, where Rt ticular case, 2 þ 1D boundary theory is actually gapless. is the time and I is a spatial line segment. Then the This is because if the Z2 gauge symmetry is broken on excitations at the end of the line are degenerate, and the the 2 þ 1D boundary, we will have the following effective degenerate end states form a projective representation of 2 2 þ 1D boundary theory: Uð1Þ ½Uð1Þ Z2 [71–74]. The above result has another interpretation. Let the j j = 3 þ 1D space-time have a topology R I M0 . Such L ¼ K~IJAI @ AJ þ ; (35) t 2 4 a space-time has two boundaries. Each boundary has a 0 where is the Higgs field that breaks the Z gauge topology Rt M2, and the theory on the boundary is 2 2 symmetry, and the K~ matrix is given by a Uð1Þ ½Uð1Þ Z2 gauge theory with the first discrete 2 ! Uð1Þ ½Uð1Þ Z2 gauge anomaly. If we freeze the 01=2 2 K~ ¼ : (36) Uð1Þ ½Uð1Þ Z2 gauge fields without break the Uð1Þ 2 1=20 ½Uð1Þ Z2 symmetry, then all the low energy excitations 0 on M2 at one boundary form a linear representation of The above theory has a fractional mutual Hall conductance, 2 0 Uð1Þ ½Uð1Þ Z2 , if the A1 gauge field is zero on M2. K~ However, all the low energy excitations on M0 at one IJ ¼ IJ ; 2 xy j j 2 boundary will form a projective representation of Uð1Þ ½Uð1Þ 2 Z , if the A gauge field has 2 flux on M0 . This 11 ¼ 22 ¼ 0; (37) 2 1 2 xy xy result also implies that 12 21 1=2 xy ¼ xy ¼ : 2 j j the monopole of A1 gauge field in the correspond- 2 ing 3 þ 1D Uð1Þ ½Uð1Þ Z2 SPT state will Such a theory can be realized by a double-layer bosonic carries a projective representation of Uð1Þ fractional quantum Hall state described by K matrix 2 ! ½Uð1Þ Z2 . 02 K ¼ ; Note that the monopole of A gauge field does not j j 1 20 2 break the Uð1Þ ½Uð1Þ Z2 symmetry. where the bosons in the two layers carry unit charges of Again consider only one boundary, we have seen that the two Uð1Þ’s separately. adding 2 flux of A1 gauge field changes the Uð1Þ 2 If the Z2 gauge symmetry does not break, will fluc- ½Uð1Þ Z2 representation of the boundary excitations tuate with equal probability to be ¼ j j. Due to the from linear to projective. If the 2 flux is concentrated separate conservation of the two Uð1Þ charges, the domain within a region of size L, we may assume that the boundary wall between ¼þj j and ¼ j j will support gap- excitations that from a projective representation of Uð1Þ 2 less edge excitations [12,60]. Because there are long do- ½Uð1Þ Z2 is concentrated within the region. When L is main walls in the disordered phase of , this suggests that large, the 2 flux is a weak perturbation. The fact that a 2 the theory is gapless if the Uð1Þ ½Uð1Þ Z2 symmetry is weak perturbation can create an nontrivial excitation in a not broken. projective representation implies that the excitations on the
045013-9 XIAO-GANG WEN PHYSICAL REVIEW D 88, 045013 (2013) 0 2 2 þ 1D boundary Rt M2 is gapless. To summarize, we the Uð1Þ ½Uð1Þ Z2 gauge fields without break the 2 have the following two results: Uð1Þ ½Uð1Þ Z2 symmetry and consider the large S1 0 small S1 limit, then the excitations on S1 are gapped with a 2 The 2 þ 1D Uð1Þ ½Uð1Þ Z2 gauge theory with nondegenerate ground state, if the A2 gauge field is zero 0 the anomaly described by (34) is gapless, if we on S1 S1. However, the excitations on S1 will be gapless 2 freeze the Uð1Þ ½Uð1Þ Z2 gauge fields without or have degenerate ground states, if there is flux of A2 break the Uð1Þ ½Uð1Þ 2 Z symmetry 0 2 gauge field going through S1 [50]. (The gapless or degen- The 3 þ 1D Uð1Þ ½Uð1Þ 2 Z SPT state char- 2 erate ground states on S1 are edge state of nontrivial acterized by the topological term (34) of the probe 0 2 þ 1D Z2 SPT state.) Since adding flux to small S1 is gauge fields [50,55] has gapless boundary excita- not a small perturbation, we cannot conclude that the ex- tions, if the Uð1Þ ½Uð1Þ 2 Z symmetry is not 0 2 citations on the 2 þ 1D boundary Rt S1 S1 are gapless. broken. We also note that the monopole of A2 gauge field in the 3 þ 1D bulk breaks the Z2 symmetry. In this case, we can In other words, the edge of this particular 3 þ 1D only discuss the Uð1Þ Uð1Þ charges of the monopoles 2 Uð1Þ ½Uð1Þ Z2 SPT state cannot be a gapped topologi- (see Ref. [55]). cally ordered state that does not break the symmetry. The first discrete gauge anomaly generates one of the Z 4 2 R Z R Z 2 Z 2 IV. UNDERSTANDING GAUGE ANOMALIES 2 in H ½BðUð1Þ ½Uð1Þ Z2 Þ; = ¼ð = Þ 2 . 4 2 R Z H 4 THROUGH SPT STATES Since Dis½H ½BðUð1Þ ½Uð1Þ Z2 Þ; = ¼Torð 2 R Z Z 2 ½Uð1Þ ½Uð1Þ Z2 ; = Þ ¼ 2 , the first discrete After discussing some examples of gauge anomalies, let Z gauge anomaly also generates one of the 2 in us turn to the task of trying to classify gauge anomalies of H 4 2 R Z Torð ½Uð1Þ ½Uð1Þ Z2 ; = Þ. According to the gauge group G. We will do so by studying a system with Ku¨nneth formula [see Eq. (E15)], on-site symmetry G in one-higher dimension. We have H 4½Uð1Þ ½Uð1Þ 2 Z ; R=Z described the general idea of such an approach in Sec. II. 2 In this section, we will give more details. H 2 H 2 2 R Z ¼ ðUð1Þ; ½Uð1Þ Z2; = Þ H 0 H 4 2 R Z ðUð1Þ; ½Uð1Þ Z2; = Þ; (39) A. The emergence of non-on-site symmetries in bosonic systems where we have only kept the nonzero terms, and Before discussing gauge anomalies, let us introduce the H 2 H 2 2 R Z H 2 Z Z notion of non-on-site symmetries, and discuss the emer- ðUð1Þ; ½Uð1Þ Z2; = Þ ¼ ½Uð1Þ; 2 ¼ 2; gence and a classification of non-on-site symmetries. The (40) non-on-site symmetries appear in the low energy boundary effective theory of a SPT state. So let us first give a brief H 0 H 4 2 R Z ðUð1Þ; ½Uð1Þ Z2; = Þ introduction of SPT state. Recently, it was shown that bosonic short-range en- H 4 2 R Z Z ¼ ½Uð1Þ Z2; = ¼ 2: (41) tangled states [48] that do not break any symmetry can Z be constructed from the elements in group cohomology So the discrete gauge anomaly generates the 2 of dþ1 2 2 class H ðG; R=ZÞ in d spatial dimensions, where G is H ðUð1Þ; H ½Uð1Þ 2 Z2; R=Z Þ, which is a structure that involves both Uð1Þ’s. the symmetry group [37–39]. Such symmetric short-range entangled states are called symmetry-protected trivial (SPT) states or symmetry-protected topological (SPT) 4. Second discrete gauge anomaly states. In this section, we will discuss the second discrete gauge A bosonic SPT state is the ground state of a local Z anomaly that generates the other 2 associated with bosonic system with an on-site symmetry G. A local H 0 H 4 2 R Z H 4 2 R Z ðUð1Þ; ½Uð1Þ Z2; = Þ ¼ ½Uð1Þ Z2; = . bosonic system is a Hamiltonian quantum theory with a The second discrete gauge anomaly is actually a gauge total Hilbert space that has direct-product structure: H H H anomaly of Uð1Þ 2 Z2 described by the nontrivial element ¼ i i where i is the local Hilbert space on site H 4 2 R Z Z in ½Uð1Þ Z2; = ¼ 2. At the moment, we do not i which has a finite dimension. An on-site symmetry is a know how to use a 3 þ 1D gauge topological term to representation UðgÞ of G acting on the total Hilbert space describe such an anomaly. However, we can describe the H that have a product form physical properties (i.e., the topological invariants) of the second discrete gauge anomaly [50]. UðgÞ¼ iUiðgÞ;g2 G; (42) Let the 3 þ 1D space-time has a topology Rt I 0 0 S1 S1. The theory on a boundary Rt S1 S1 has the where UiðgÞ is a representation G acting on the local 2 H second Uð1Þ ½Uð1Þ Z2 gauge anomaly. If we freeze Hilbert space i on site-i.
045013-10 CLASSIFYING GAUGE ANOMALIES THROUGH SYMMETRY- ... PHYSICAL REVIEW D 88, 045013 (2013) A bosonic SPT state is also a short-range entangled state B. Anomalous gauge theories as the boundary that is invariant under UðgÞ. The notion of short-range effective theory of bosonic SPT states entangled state is introduced in Ref. [48] as a state that We can alway generalize an on-site global symmetry can be transformed into a product state via a local unitary transformation into a local gauge transformation by mak- transformation [75–77]. A SPT state is always a gapped ing g to be site dependent, state. It can be smoothly deformed into a gapped product state via a path that may break the symmetry without gap- UgaugeðfgigÞ ¼ iUiðgiÞ; (43) closing and phase transitions. However, a nontrivial SPT Ns state cannot be smoothly deformed into a gapped product which is a representation of G , where Ns is the number of state via any path that does not break the symmetry without sites, phase transitions. UgaugeðfhigÞUgaugeðfgigÞ ¼ UgaugeðfhigigÞ: (44) Since SPT states are short-range entangled, it is relatively easy to understand them systematically. In particular, a sys- So we say that the on-site symmetry (i.e., the anomaly-free tematic construction of the bosonic SPT state in d spatial symmetry) is ‘‘gaugable.’’ dimensions with on-site symmetry G can be obtained through On the other hand, the non-on-site symmetry of the the group cohomology class H dþ1ðG; R=ZÞ [37–39]. boundary effective theory is not ‘‘gaugable.’’ If we try to The SPT states are gapped with no ground state generalize a non-on-site symmetry transformation to a degeneracy when there is no boundary. If we consider a local gauge transformation: Unon-on-siteðgÞ!UgaugeðfgigÞ, Ns d-space-dimensional bosonic SPT state with a boundary, then UgaugeðfgigÞ does not form a representation of G .In then any low energy excitations must be boundary excita- fact, if we do ‘‘gauge’’ the non-on-site symmetry, we will tions. Also since the SPT state is a short-range entangled get an anomalous gauge theory with gauge group G on the state, those low energy boundary excitations can be de- boundary, as demonstrated in Refs. [38,40–43] for G ¼ scribed by a pure local boundary theory [37–39]. However, Uð1Þ, SUð2Þ. Therefore, gauge anomaly non-on-site if the SPT state is nontrivial (i.e., described by a nontrivial symmetry. This is why we also refer the non-on-site sym- element in H dþ1ðG; R=ZÞ), then the symmetry transfor- metry as anomalous symmetry. Gauging anomalous sym- mation G must act as a non-on-site symmetry [37–39,42] metry will lead to an anomalous gauge theory. in the effective boundary theory. The non-on-site symme- Since nonsite symmetries emerge at the boundary of try action UðgÞ does not have a product form UðgÞ¼ SPT states. Thus gauging the symmetry in the SPT state iUiðgÞ. So the SPT phases in d spatial dimensions lead in (d þ 1)-dimensional space-time is a systematic way to to the emergence of non-on-site symmetry in d 1 spatial construct anomalous gauge theory in d-dimensional space- dimensions. As a result, the different types of non-on-site time. Then from the group cohomology description of the symmetry in (d 1) spatial dimensions are described by SPT states, we find that the gauge anomalies in bosonic H dþ1ðG; R=ZÞ. gauge theories with a gauge group G in d space-time The non-on-site symmetry has another very interesting dimensions are described by H dþ1ðG; R=ZÞ (at least (conjectured) property: partially).
the ground states of a system with a non-on-site C. The gauge noninvariance (i.e., the gauge anomaly) symmetry must be degenerate or gapless of non-on-site symmetry and the cocycles [37–39,50,69]. The degeneracy may be due to the in group cohomology symmetry breaking, topological order, [46,47]or both. The standard understanding of gauge anomaly is its ‘‘gauge noninvariance.’’ However, in above, we introduce The above result is proven only in 1 þ 1D [37]. gauge anomaly through SPT state. In this section, we will For certain types of non-on-site symmetries, the ground show that the two approaches are equivalent. We also state may even have to be gapless, if the symmetry is not discuss a direct connection between gauge noninvariance dþ1 broken. and the group cocycles in H ðG; R=ZÞ. For a reason that we will explain later, we will refer non- The SPT state in the (d þ 1)-dimensional space-time on-site symmetry as anomalous symmetry and on-site bulk manifold M can be described by a nonlinear model symmetry as anomaly-free symmetry. We see that a system with G as the target space, Z with an anomalous symmetry cannot have a ground state 1 S ¼ ddþ1x ½@gðx Þ 2 þ iW ðgÞ ; (45) that is nondegenerate. On the other hand a system with an top anomaly-free symmetry can have a ground state that is M s nondegenerate (and symmetric). So the anomaly-free prop- in large s limit. Here we triangulate the (d þ 1)- erty of a global symmetry is a sufficient condition for the dimensional bulk manifold M to make it a (random) lattice existence of a gapped ground state that do not break any or a (d þ 1)-dimensional complex.R The field gðx Þ live on symmetry. the vertices of the complex. So ddþ1x is in fact a sum
045013-11 XIAO-GANG WEN PHYSICAL REVIEW D 88, 045013 (2013) over lattice sites and @ is the lattice difference operator. ðfgigMÞ¼sum of local terms for the cells in M (50) The above action S actually defines a lattice theory. R [i.e., ðfg g Þ¼ ddþ1xWgaugeðg 1@ gÞ]. The second iWtopðgÞ is a lattice topological term which is defined i M M top and classified by the elements in H dþ1ðG; R=ZÞ one is Refs. [38,39,49,55,56,59]. This is why the bosonic SPT ðfg g Þ¼0 mod 2 (51) states are classified by H dþ1ðG; R=ZÞ. i M Since G is an on-site symmetry in the d þ 1D bulk, we if M has no boundary, since the theory is gauge invariant can always gauge the on-site symmetry to obtain a gauge when M has no boundary. Equation (51) is the cocycle theory in the bulk by integrating out gðx Þ condition in group cohomology theory and the function ðfgigMÞ satisfying (51) is a cocycle. Z ð Þ2 dþ1 Tr F gauge When M does has a boundary, the gauge noninvariance S ¼ d x þ iWtop ðA Þ : (46) g ðfgigMÞ only depend on gi’s on the boundary (mod 2 ). So it is a gauge noninvariance (or a gauge anomaly) on the gauge The resulting topological term Wtop ðAÞ in the gauge d-dimensional boundary. Some times, such a gauge non- theory is always a ‘‘quantized’’ topological term discussed invariance ðfgigMÞ can be expressed as the sum of local in Ref. [55]. It is a generalization of the Chern-Simons terms for the cells on the boundary @M [this is potentially term Refs. [55,56,78]. It is also related to the topological possible since ðfgigMÞ only depend on gi’s on the bound- term WtopðgÞ in the nonlinear -model when A is a pure ary mod 2 ], then such a ðfgigMÞ will be called coboun- gauge dary. The associated gauge noninvariance is an artifact of us adding gauge noninvariant boundary terms as we create gauge 1 Wtop ðA Þ¼WtopðgÞ; where A ¼ g @ g: (47) the boundary of the space-time. Such a gauge noninvar- iance is removable. So a coboundary does not represent a (A more detailed description of the two topological gauge gauge anomaly. Only those gauge noninvariance ðfgigMÞ terms WtopðgÞ and Wtop ðA Þ on lattice can be found in gauge that cannot be expressed as the sum of local terms repre- Ref. [55].) So the quantized topological term Wtop ðAÞ in sent real gauge anomalies. After we mod out the coboun- the gauge theory is also described by H dþ1ðG; R=ZÞ. daries from the cocycles, we obtain H dþ1ðG; R=ZÞ. H dþ1 R Z Since WtopðgÞ is a cocycle in ðG; = Þ, we have This way, we see more directly that Refs. [38,39] Z the elements in H dþ1ðG; R=ZÞ describe the gauge dþ1 gauge 1 ½gðx Þ ¼ d xWtop ðg @ gÞ¼0 mod 2 (48) anomalies in d-dimensional space-time for gauge M group G, assuming the gauge transformations are if the space-time M has no boundary. But if the space-time described by fgig on the vertices of the space-time M has a boundary, then complex M. Z dþ1 gauge 1 We also see that a nontrivial gauge anomaly (described ½gðx Þ ¼ d xWtop ðg @ gÞ 0 mod 2 ; (49) M by a nontrivial cocycle) represents a gauge noninvariance in the boundary gauge theory. We believe that the above which represents a gauge noninvariance (or a gauge anom- argument is very general. It applies to both continuous and aly) of the gauged bulk theory in (d þ 1)-dimensional discrete gauge groups, and both bosonic theories and space-time. (This is just like the gauge noninvariance of fermionic theories. (However, fermionic theories may con- the Chern-Simons term, which is a special case of tain extra structures. See Sec. VI.) It turns out that the free gauge Wtop ðA Þ.) Note that the gauge anomaly ½gðx Þ mod part of H dþ1ðG; R=ZÞ, Free½H dþ1ðG; R=ZÞ , gives rise 2 only depend on gðx Þ on the boundary of M. Such a to the well known Adler-Bell-Jackiw anomaly. The torsion gauge anomaly is canceled by the boundary theory which part of H dþ1ðG; R=ZÞ correspond to new types of gauge is an anomalous bosonic gauge theory. Such a point was anomalies called non-ABJ gauge anomalies. discussed in detail for G ¼ Uð1Þ, SUð2Þ in Ref. [12]. From the above discussion, we see that the bulk theory V. MORE GENERAL GAUGE ANOMALIES on the (d þ 1)-dimensional complex M is gauge invariant d if M has no boundary, but may not be gauge invariant if M A. -dimensional gauge anomalies and d þ 1 has a boundary. Since a gauge transformation gðx Þ lives ( )-dimensional gauge topological terms on the vertices, it is described by fgiji labels verticesg. Thus, the gauge noninvariance of the bulk theory is de- In the last section, when we discuss the connection Ns scribed by a mapping from G to phase 2 R=Z: ðfgigMÞ, between gauge noninvariance and the group cocycles, we where Ns is the number of lattice sites (i.e., the number of assume that the gauge transformations on the vertices of d the vertices). Such a mapping has two properties. The first the space-time complex M , fgig, can be arbitrary. one is However, in this paper, we want to understand the gauge
045013-12 CLASSIFYING GAUGE ANOMALIES THROUGH SYMMETRY- ... PHYSICAL REVIEW D 88, 045013 (2013) anomalies in weak-coupling gauge theories in d space-time The Chern-Simons term is an example of this type of dimensions, where gauge field strength is small. In this topological terms. case, gauge transformations fgig on the vertices are not The second type of gauge topological terms has an arbitrary. action amplitude that does not change under any perturba- For finite gauge group G, the gauge transformations fgig tive modifications of the gauge field in a local region (away on the vertices of the space-time complex Md are indeed from the boundary): arbitrary. Therefore, we have the following: R R i ddþ1xWgaugeðA þ A Þ i ddþ1xWgaugeðA Þ e M top ¼ e M top : (53) H dþ1ðG; R=ZÞ classifies the bosonic gauge gauge anomalies in d-dimensional space-time for finite Wtop ðA Þ¼@ A @ A is an example of such kind gauge group G. of topological terms. We will refer the second type of topological terms as locally null topological terms. Some H dþ1ðG; R=ZÞ partially describes the fermionic of the locally null topological terms are described by H dþ1 R Z gauge anomalies in d-dimensional space-time for Tor½ RðG; = Þ [55,78]. i ddþ1xWgaugeðA Þ finite gauge group G. Since e top does not change for any pertur- bative modifications of the gauge field away from the We will discuss the distinction between gauge anomalies boundary, one may naively think that it only depends on in bosonic and fermionic gauge theory in Sec. VI. the fields on the boundary and write it as a pure boundary However, for continuous gauge group G, we further term, require that gauge transformations fg g on the vertices of R R i dþ1 gauge d gauge d i d xW ðA Þ i d xL ðA Þ the space-time complex M are close to smooth functions e M top ¼ e @M top : (54) on the space-time manifold. In this case, there are more general gauge anomalies. Free½H dþ1ðG; R=ZÞ still de- However,R the above is not valid in general since i ddþ1xWgaugeðA Þ scribes all the Adler-Bell-Jackiw anomaly. But there are e top does dependR on the bulk gauge field dþ1 gauge non-ABJ anomalies that are beyond Tor½H ðG; R=ZÞ . i ddþ1xW ðA Þ away from the boundary: e top can change if To understand more general non-ABJ gauge anomalies the modification in the gauge field away from the boundary ½H dþ1ðG; R=ZÞ beyond Tor , let us view gauge anomalies cannot be continuously deformed to zero. In this case, the d in -dimensional space-time as an obstruction to have a appearance of the locally null gauge topological term in nonperturbative definition (i.e., a well-defined UV com- (d þ 1)-dimensions represents an obstruction to view the pletion) of the gauge theory in the same dimension. To (d þ 1)-dimensional theory as a pure d-dimensional understand such an obstruction, let us consider a theory in boundary theory. This is why we can study non-ABJ gauge (d þ 1)-dimensional space-time where gapped matter anomalies through (d þ 1)-dimensional locally null gauge fields couple to a gauge theory of gauge group G.We topological terms. view of the gauge field as a nondynamical probe field and only consider the excitations of the matter fields. Since the matter fields are gapped in the bulk, the low B. Classifying space and -cohomology classes energy excitations only live on the boundary and are To have a systematic description of the locally null described by a boundary low energy effective theory with topological terms, let us use the notion of the classifying the nondynamical gauge field. We like to ask, can we space BG for group G. The gauge configurations (with define the boundary low energy effective theory as a weak field strength) on the (d þ 1)-dimensional space-time pure boundary theory, instead of defining it as a part of manifold Mdþ1 can be described by the embeddings of dþ1 dþ1 dþ1 (d þ 1)-dimensional theory? M into BG, M ! MBG BG [55,78]. So we can This question can be answered by considering the in- rewrite our quantized topological term as a function of dþ1 duced gauge topological terms (the terms that do not the embeddings MBG : depend on space-time metrics) in the (d þ 1)-dimensional Z theory as we integrate out the gapped mater fields. There dþ1 gauge gauge dþ1 d xWtop ðA Þ¼Stop ðMBG Þ: (55) are two types of the gauge topological terms that can be Mdþ1 induced. The first typeR of gauge topological terms has an One way to construct the topological term is to use the i ddþ1xWgaugeðA Þ M top dþ1 R Z action amplitude e that can change as we topological (d þ 1)-cocycles dþ1 2 H ðBG; = Þ: change the gauge field slightly in a local region: gauge dþ1 dþ1 R R Stop ðMBG Þ¼2 h dþ1;MBG i: (56) i ddþ1xWgaugeðA þ A Þ i ddþ1xWgaugeðA Þ e M top e M top : (52) Note that cocycles are cochains, and cochains are defined dþ1 They are classified by Free½H ðG; R=ZÞ ¼ as linear maps from cell-complices M to R=Z. h dþ1;Mi Free½Hdþ2ðBG; ZÞ [55,78] and correspond to the Adler- denotes such a linear map. As a part of definition, dþ1 Bell-Jackiw anomalies in d-dimensional space-time. h dþ1;MBG i satisfies the locality condition
045013-13 XIAO-GANG WEN PHYSICAL REVIEW D 88, 045013 (2013) dþ1 d-dimensional space-time are classified by h dþ1;MBG i¼sum of local terms for the cells in M; Free½H dþ1ðG; R=ZÞ H_ dþ1ðBG; R=ZÞ. (57) which is similar to Eq. (50). The gauge anomalies in fermionic weak-coupling It turns out that the most general locally null topological gauge theories with gauge group G in terms can be constructed from cocycles. By definition, a d-dimensional space-time are partially described by Free½H dþ1ðG; R=ZÞ H_ dþ1ðBG; R=ZÞ. ðd þ 1Þ -cocycle dþ1 is a (d þ 1)-cochain that satis- fies the condition dþ1 As an Abelian group, H_ ðBG; R=ZÞ may contain dþ1 dþ1 R Z Z Z _ dþ1 R Z h dþ1;MBG i¼h dþ1;NBG i mod 1 (58) = , , and/or n. Dis½H ðBG; = Þ is the discrete dþ1 part of H_ ðBG; R=ZÞ, which is obtained by dropping the dþ1 dþ1 dþ1 dþ1 if MBG and NBG have no boundaries and MBG and NBG R=Z parts. We can show that, for finite group G (see dþ1 dþ1 are homotopic to each other (i.e., MBG and NBG can Appendix D), deform into each other continuously.) As a comparison, a H_ dþ1ðBG; R=ZÞ¼Dis½H_ dþ1ðBG; R=ZÞ ; (d þ 1)-cocycle dþ1 are (d þ 1)-cochains that satisfy a stronger condition, dþ1 dþ1 H_ ðBG; R=ZÞ¼Tor½H ðG; R=ZÞ (63) dþ1 dþ1 ¼ H dþ1ðG; R=ZÞ: h dþ1;MBG i¼h dþ1;NBG i mod 1; (59)
dþ1 dþ1 if MBG NBG is a boundary of a (d þ 2)-dimensional cell complex. VI. BOSONIC GAUGE ANOMALIES AND dþ1 FERMIONIC GAUGE ANOMALIES Let us use Z ðBG; R=ZÞ to denote the set of dþ1 (d þ 1)- -cocycle. Clearly, Z ðBG; R=ZÞ contains the Why does the -cohomology theory developed above fail dþ1 dþ1 set of (d þ 1)-cocycles: Z ðBG; R=ZÞ Z ðBG; to classify all the fermionic gauge anomalies? In this sec- R=ZÞ, which in turn contains the set of (d þ 1)- tion, we will reveal the reason for this failure. Our discussion coboundaries: Bdþ1ðBG; R=ZÞ Zdþ1ðBG; R=ZÞ.The also suggests that the -cohomology theory may provide a dþ1 cohomology class H_ ðBG; R=ZÞ is defined as classification of all bosonic gauge anomalies. We have been studying gauge anomalies in _ dþ1 R Z Zdþ1 R Z dþ1 R Z H ðBG; = Þ¼ ðBG; = Þ=B ðBG; = Þ; d-dimensional space-time through a bulk gapped theory (60) in (d þ 1)-dimensional space-time. The anomalous gauge theory is defined as the theory on the d-dimensional bound- i.e., two cocycles are regard as equivalent if they are differ ary of the (d þ 1)-dimensional bulk. In our discussion, we _ dþ1 R Z by a coboundary. Clearly H ðBG; = Þ contains have made the following assumption. We first view the dþ1 R Z H ðBG; = Þ as a subgroup. gauge field as nondynamical probe field (i.e., take the gauge coupling to zero). When the (d þ 1)-dimensional dþ1 dþ1 dþ1 H ðBG; R=ZÞ Z ðBG; R=ZÞ=B ðBG; R=ZÞ bulk has several disconnected boundaries, we assume that dþ1 the total low energy Hilbert space of the matter fields for all H_ ðBG; R=ZÞ: (61) the boundaries is a direct product of the low energy Hilbert spaces for each connected boundary. So the total low However, although in definition, H_ dþ1ðBG; R=ZÞ is more energy Hilbert space of the matter fields can be described general than Hdþ1ðBG; R=ZÞ, at the moment, we do not by independent matter degrees of freedom on each bound- know if H_ dþ1ðBG; R=ZÞ is strictly larger than Hdþ1ðBG; ary. In this case, when we glue two boundaries together, R=ZÞ. It might be possible that H_ dþ1ðBG; R=ZÞ¼ other boundary will not be affected. This assumption Hdþ1ðBG; R=ZÞ. allows us to use cochains in the classifying space to Using the cocycles 2 H_ dþ1ðBG; R=ZÞ, we can dþ1 describe the low-energy effective theory with boundaries. construct generic locally null topological terms as In the following, we like to argue that the above assump- gauge dþ1 dþ1 tion is valid for bosonic theories. This is because when we Stop ðMBG Þ¼2 h dþ1;MBG i: (62) studied gauge anomalies, we made an important implicit Thus locally null topological terms in weak-coupling assumption: we only study pure gauge anomalies. Had we gauge theories in (d þ 1)-dimensional space-time are broken the gauge symmetry, we would be able to have a dþ1 classified by H_ ðBG; R=ZÞ. Since the non–locally null nonperturbative definition of the theory in the same dimen- topological terms are classified by Free½H dþ1ðG; sion. This implies that the matter degrees of freedom in the R=ZÞ , we obtain the following: (d þ 1)-dimensional bulk form a short-range entangled state [48] with a trivial intrinsic topological order. For The gauge anomalies in bosonic weak-coupling bosonic systems, short-range entangled bulk state implies gauge theories with gauge group G in that the total Hilbert space for all the boundaries is a direct
045013-14 CLASSIFYING GAUGE ANOMALIES THROUGH SYMMETRY- ... PHYSICAL REVIEW D 88, 045013 (2013) product of the Hilbert spaces for each connected boundary, correspond to the same SPT phase. The gauge anomalies for any bulk gauge configurations. This result can be and the SPT phases in one-higher dimension are related by obtained directly from the canonical form of the bosonic an exact sequence (a many-to-one mapping): short-range entangled states suggested in Refs. [37,38]. However, above argument breaks down for fermionic d-dimensional gauge anomalies of gauge group G systems, as demonstrated by the 2 þ 1D p þ ip=p ip !d þ 1-dimensional SPT phases of symmetry group G fermionic superconductor with Z2 Z2 symmetry. The edge state of the p þ ip=p ip superconductor is de- !0: scribed by Eq. (19), which has a Z2 Z2 fermionic gauge anomaly. If we break the Z2 Z2 symmetry down to the Using such a relation between gauge anomalies and SPT þ fermion parity symmetry, the 1 1D theory (19) can phases, we can introduce the notions of gapless gauge þ indeed be defined on 1D lattice. Thus the p ip=p ip anomalies and gapped gauge anomalies. We know that superconductor has no intrinsic topological order. some SPT states must have gapless boundary excitations However, we do not know the canonical form for such if the symmetry is not broken at the boundary. We call short-range entangled fermionic state. The bulk short- those gauge anomalies that map into such SPT states as range entanglement does not imply that the total Hilbert ‘‘gapless gauge anomalies’’. We call the gauge anomalies space for all the boundaries is a direct product of the that map into the SPT states that can have a gapped Hilbert spaces for each connected boundary, for any bulk boundary states without the symmetry breaking ‘‘gapped Z2 Z2 gauge configurations. We believe this is the reason gauge anomalies’’. why the cohomology theory fail to described all the It appears that all the ABJ anomalies are gapless gauge fermionic gauge anomalies. anomalies. The 2 þ 1D continuous Uð1Þ gauge anomalies discussed above (see Sec. III E) are examples of gapped VII. THE PRECISE RELATION BETWEEN GAUGE gauge anomalies, which are non-ABJ anomalies. The first 2 ANOMALIES AND SPT STATES discrete 2 þ 1D Uð1Þ ðUð1Þ Z2Þ gauge anomaly dis- Despite the very close connection between gauge anoma- cussed in Sec. III G 3 is an example of gapless gauge lies and SPT states, different gauge anomalies and different anomaly, which is also a non-ABJ anomaly. All the 1 þ 1D SPT phases do not have a one-to-one correspondence. gauge anomalies are gapless gauge anomalies, since 2 þ 1D Remember that the gauge anomaly is a property of a low SPT state always have gapless edge excitations if the energy weak-coupling gauge theory. It is the obstruction to symmetry is not broken [37]. have a nonperturbative definition (i.e., a well-defined UV completion) of the gauge theory in the same dimension. VIII. NONPERTURBATIVE DEFINITION OF While a SPT phase is a phase of short-range entangled CHIRAL GAUGE THEORIES states with a symmetry. To see the connection between gauge anomalies and In this section, we will discuss an application of the SPT phases, we note that the low energy boundary excita- deeper understanding of gauge anomalies discussed in tions of a SPT state in d þ 1 space-time dimensions can this paper: a lattice nonperturbative definition of any always be described by a pure boundary theory, since the anomaly-free chiral gauge theories. This idea can be used bulk SPT states are short-range entangled. However, the to construct a lattice nonperturbative definition of the on-site symmetry of the bulk state must become a non-on- SOð10Þ grant unification chiral gauge theory [79]. site symmetry on the boundary, if the bulk state has a nontrivial SPT order. If we try to gauge the non-on-site A. Introduction symmetry, it will lead to an anomalous gauge theory in d The Uð1Þ SUð2Þ SUð3Þ standard model space-time dimensions. Refs. [80–85] is the theory which is believed to describe Every gauge anomaly can be understood this way. In all elementary particles (except the gravitons) in nature. other words, every gauge anomaly correspond a SPT state The standard model is a chiral gauge theory where the which give rise to a non-on-site symmetry on the boundary. SUð2Þ gauge fields couple differently to right-/left-hand However, some times, two different gauge anomalies may fermions. For a long time, we only know a perturbative correspond to two SPT states that can be smoothly con- definition of the standard model via the perturbative ex- nected to each other.R For example, 3 þ 1D Uð1Þ gauge pansion of the gauge coupling constant. The perturbative topological terms 2!ð2 Þ2 @ A @ A gives rise to definition is not self consistent since the perturbative ex- different 2 þ 1D Uð1Þ gauge anomalies for different values pansion is known to diverge. In this section, we would like of (see Sec. III E). However, the Uð1Þ gauge topological propose a nonperturbative definition of any anomaly-free terms with different values of correspond to SPT states chiral gauge theories. We will construct well-regulated that can connect to each other without phase transition. Hamiltonian quantum models [86] whose low-energy Thus, the different 2 þ 1D Uð1Þ gauge anomalies effective theory is any anomaly-free chiral gauge theory.
045013-15 XIAO-GANG WEN PHYSICAL REVIEW D 88, 045013 (2013) Our approach will apply to the standard model if the 2 H dþ1ðG; R=ZÞ. On the d-dimensional boundary, the standard model is free of all anomalies. low energy effective theory will have a non-on-site sym- There are many previous researches that try to give metry (i.e., an anomalous symmetry) G. Here we will chiral gauge theories a nonperturbative definition. There assume that the d-dimensional boundary excitations are are lattice gauge theory approaches, [87] which fail since gapless and do not break the symmetry G. After ‘‘gauging’’ they cannot reproduce chiral couplings between the gauge the on-site symmetry G in the (d þ 1)-dimensional bulk, field and the fermions. There are domain-wall fermion we get a bosonic chiral gauge theory on the d-dimensional approaches Refs. [88,89]. But the gauge fields in the boundary whose anomaly is described by the cocycle . domain-wall fermion approaches propagate in one-higher Then let us consider a stacking of a few bosonic SPT dimension: 4 þ 1 dimensions. There are also overlap- states in (d þ 1)-dimensional space-time described by co- dþ1 fermion approaches [90–93]. However, the path integral cycles i 2 H ðG; R=ZÞ where the interaction between in overlap-fermion approaches may not describe a the SPTP states are weak [see Fig. 3(b)]. We also assume Hamiltonian quantum theory (for example, the total that i i ¼ 0. Because the stacked system has a trivial Hilbert space in the overlap-fermion approaches, if it ex- SPT order, if we turn on a proper G-symmetric interaction ists, may not have a finite dimension, even for a space- between different layers on one of the two boundaries, we lattice of a finite size). can fully gap the boundary excitations in such a way that Our construction has a similar starting point as the the ground state is not degenerate. (Such a gapping process mirror fermion approach discussed in Refs. [94–97]. also do not break the G symmetry.) Thus the gapping However, later work either fail to demonstrate [98–100] process does not leave behind any low energy degrees of or argue that it is almost impossible [101] to use mirror freedom on the gapped boundary. Now we ‘‘gauge’’ the fermion approach to nonperturbatively define anomaly- on-site symmetry G in the (d þ 1)-dimensional bulk. The free chiral gauge theories. Here, we will argue that the resulting system is a nonperturbative definition of mirror fermion approach actually works. We are able to use anomaly-freeP bosonic chiral gauge theory described by i the defining connection between the chiral gauge theories with i ¼ 0. Since the thickness l of the (d þ 1)- in d-dimensional space-time and the SPT states in (d þ 1)- dimensional bulk is finite (although l can be large so that dimensional space-time to show that, if a chiral gauge the two boundaries are nearly decoupled), the system theory is free of all the anomalies, then we can construct actually has a d-dimensional space-time. In particular, a lattice gauge theory whose low energy effective theory due to the finite l, the gapless gauge bosons of the gauge reproduces the anomaly-free chiral gauge theory. We show group G are gapless excitations on the d-dimensional that lattice gauge theory approaches actually can define space-time. anomaly-free chiral gauge theories nonperturbatively with- The same approach also works for fermionic systems. out going to one-higher dimension, if we include a proper We can start with a few fermionic SPT states in (d þ 1)- direct interactions between lattice fermions. dimensional space-time describedP by super-cocycles i (Ref. [51]) that satisfy i ¼ 0 (i.e., the combined B. A nonperturbative definition of any anomaly-free fermion system is free of all the gauge anomalies). If we chiral gauge theories turn on a proper G-symmetric interaction on one boundary, Let us start with a SPT state in (d þ 1)-dimensional we can fully gap the boundary excitations in such a way space-time with a on-site symmetry G [see Fig. 3(a)]. that the ground state is not degenerate and does break the We assume that the SPT state is described by a cocycle symmetry G. In this case, if we gauge the bulk on-site symmetry, we will get a nonperturbative definition of gapping anomaly-free fermionic chiral gauge theory. SPTν states the mirror 1 chiral SPT anomaly−free ν the mirror of of chiral chiral gauge 2 anomaly−free gauge state gauge ν theory ν theory 3 chiral gauge C. A nonperturbative definition of some anomalous theory l theory chiral gauge theories (a) (b) In the above nonperturbative definition of some FIG. 3 (color online). (a) A SPT state described by a cocycle anomaly-free chiral gauge theories, the lattice gauge 2 H dþ1ðG; R=ZÞ in (d þ 1)-dimensional space-time. After theories reproduce all the low energy properties of the ‘‘gauging’’ the on-site symmetry G, we get a bosonic chiral anomaly-free chiral gauge theories, including all the low gauge theory on one boundary and the ‘‘mirror’’ of the bosonic energy particle-like excitations and degenerate ground chiral gauge theory on the other boundary. (b) A stacking of a states. This is because the gapped mirror sector on the few SPT states in (d þ 1)-dimensional space-time described by P other boundary has a nondegenerate ground state. cocycles i.If i i ¼ 0, then after ‘‘gauging’’ the on-site symmetry G, we get a anomaly-free chiral gauge theory on However, for the application to high energy physics, in one boundary. We also get the ‘‘mirror’’ of the anomaly-free particular, for the application to nonperturbatively define chiral gauge theory on the other boundary, which can be gapped the standard model, we only need the nonperturbatively without breaking the ‘‘gauge symmetry’’. defined theory to reproduce all the low energy particle-like
045013-16 CLASSIFYING GAUGE ANOMALIES THROUGH SYMMETRY- ... PHYSICAL REVIEW D 88, 045013 (2013) excitations. In this case, the gapped mirror sector on the dimension. To see such a connection, we like to point out other boundary can have degenerate ground states and that if a theory cannot be nonperturbatively defined in the nontrivial topological orders. same dimension even after we break all the gauge symme- If we only need the nonperturbatively defined theory to tries, then the theory should have an anomaly that is reproduce all the low energy particle-like excitations, we beyond the gauge anomaly. This more general anomaly can even define certain anomalous chiral gauge theories can be identified as gravitational anomaly. A theory with nonperturbatively, following the method outlined in the gravitational anomaly can only appear as an effective previous section. Using the notions of ‘‘gapless gauge theory on the boundary of a bulk theory in one-higher anomalies’’ and ‘‘gapped gauge anomalies’’ introduced dimension, which has a nontrivial intrinsic topological in the last section, we see that we can use a lattice gauge order [46,47]. This line of thinking suggests that the gravi- theory to give nonperturbative definition of an anomalous tational anomalies are classified by topological orders chiral gauge theory, if the chiral gauge theory has a (i.e., patterns of long-range entanglement [48]) in one- ‘‘gapped gauge anomaly.’’ higher dimension, leading to a new fresh point of view Thus all the chiral gauge theories with the ABJ anoma- on gravitational anomalies. lies do not have a nonperturbative definition. The 2 þ 1D We also like to remark that in Ref. [55], quantized chiral gauge theories with the first discrete 2 þ 1D Uð1Þ topological terms in d-space-time-dimensional weak- 2 ðUð1Þ Z2Þ gauge anomaly discussed in Sec. III G 3 also do coupling gauge theory are systematically constructed not have a nonperturbative definition. However, many using the elements in H dþ1ðG; ZÞ. The study in this paper other anomalous chiral gauge theories have ‘‘gapped gauge shows that more general quantized topological terms anomalies’’ and they do have a nonperturbative definition. can be constructed using the discrete elements in d d The gapped boundary states of those anomalous chiral Free½H ðG; R=ZÞ H_ ðBG; R=ZÞ. gauge theories have nontrivial topological orders and ground state degeneracies. ACKNOWLEDGMENTS I would like to thank Xie Chen, Zheng-Cheng Gu, and IX. SUMMARY Ashvin Vishwanath for many helpful discussions. This research is supported by NSF Grants No. DMR-1005541, In this paper, we introduced a -cohomology No. NSFC 11074140, and No. NSFC 11274192. Research theory to systematically describe gauge anomalies. We at the Perimeter Institute is supported by the Government propose that bosonic gauge anomalies in d-dimensional of Canada through Industry Canada and by the Province of space-time for gauge group G are classified by the ele- dþ1 dþ1 Ontario through the Ministry of Research. ments in Free½H ðG; R=ZÞ H_ ðBG; R=ZÞ, where dþ1 H_ ðBG; R=ZÞ is the -cohomology class of the classi- fying space BG of group G. We show that the APPENDIX A: THE NON-ABJ GAUGE dþ1 -cohomology class H_ ðBG; R=ZÞ contains the topo- ANOMALIES AND THE GLOBAL dþ1 logical cohomology class H ðBG; R=ZÞ as a subgroup. GAUGE ANOMALIES The -cohomology theory also apply to fermion The non-ABJ gauge anomalies described by systems, where Free½H dþ1ðG; R=ZÞ H_ dþ1ðBG; R=ZÞ H_ dþ1ðBG; R=ZÞ is closely related to bosonic global gauge describes some of the fermionic gauge anomalies. The anomalies. The definition of bosonic global gauge anoma- gauge anomalies for both continuous and discrete groups lies is very similar to the definition of the fermionic SUð2Þ are treated at the same footing. global gauge anomaly first introduced by Witten [3]. In this Motivated by the -cohomology theory and the closely section, we will follow Witten’s idea to give a definitions of related group cohomology theory, we studied many ex- bosonic global gauge anomalies for continuous gauge amples of non-ABJ anomalies. Many results are obtained, groups [4]. We then discuss the relation between the non- which are stressed by the framed boxes. ABJ gauge anomalies and newly defined bosonic global The close relation between gauge anomalies and SPT gauge anomalies, for the case of continuous gauge groups. states in one-higher dimension allows us to give a non- We like to point out that the bosonic global gauge perturbative definition of any anomaly-free chiral gauge anomalies defined here are potential global gauge anoma- theory in terms of lattice gauge theories. In this paper, we lies. They may or may not be realizable by boson systems. outline a generic construction to obtain such a nonpertur- bative definition. The close relation between gauge anomalies and 1. A definition of bosonic/fermionic global gauge SPT states also allows us to gain a deeper understanding anomalies for continuous gauge groups for both gauge anomalies and SPT states. Such a deeper We use the gauge noninvariance of the partition function understanding suggests that gravitational anomalies are under the ‘‘large’’ gauge transformations to define the classified by topological orders [46,47] (i.e., patterns of global gauge anomalies. Let us consider a weak-coupling long-range entanglement, see Ref. [48]) in one-higher gauge theory in closed d-dimensional space-time Sd which
045013-17 XIAO-GANG WEN PHYSICAL REVIEW D 88, 045013 (2013) has a spherical topology. We also assume a continuous 2. The non-ABJ gauge anomalies and the bosonic gauge group G.If dðGÞ is nontrivial, it means that there global gauge anomalies are exist nontrivial ‘‘large’’ gauge transformations that We note that ðGÞ also describes the classes of G gauge does not connect to the identity gauge transformation d configurations on Sdþ1 that cannot be continuously (i.e., the trivial gauge transformation). Note that dðGÞ deformed into each others. Those classes of G gauge forms a group. Under a ‘‘large’’ gauge transformation, configurations on S þ correspond to classes of embedding the partition function may change a phase d 1 Sdþ1 ! BG that cannot be continuously deformed into each others. This picture explains a mathematical result 0 i 0 1 1 Z½A ¼e Z½A ;A ¼ g A g ig @ g; (A1) dðGÞ¼ dþ1ðBGÞ. So the potential global gauge anomalies in d-dimensional space-time are defined as H 1 R Z where gðxÞ is a nontrivial map from Md to G. The different one-dimensional representations ½ dþ1ðBGÞ; = . i _ dþ1 R Z choices of the phases e correspond to different one- Each cocycle dþ1 in H ðBG; = Þ induces a dimensional representations of dðGÞ, which are classified one-dimensional representation of dþ1ðBGÞ via by first group cohomology classes H 1½ ðGÞ; R=Z .So d h ;Sdþ1i mod 1; (A3) the potential global gauge anomalies are described by dþ1 BG H 1½ ðGÞ; R=Z . dþ1 d where SBG is an embedding Sdþ1 ! BG. Thus we have a map The potential global gauge anomalies in dþ1 R Z H 1 R Z d-dimensional space-time and for gauge group G H_ ðBG; = Þ! ½ dþ1ðBGÞ; = : (A4) are described by H 1½ ðGÞ; R=Z . d The above map represents the relation between the dþ1 non-ABJ gauge anomalies described by H_ ðBG; R=ZÞ Since dðGÞ is an Abelian group, we have H 1½ ðGÞ; R=Z ¼ ðGÞ. In Table I, we list ðGÞ for and the global gauge anomalies described by d d d H 1ð ð Þ R ZÞ some groups. For a more general discussion of global dþ1 BG ; = . If a one-dimensional representation ð Þ gauge anomalies along this line of thinking, see Ref. [4]. of dþ1 BG cannot be induced by any cocycle, then the We will refer those global gauge anomalies that appear corresponding global gauge anomaly is not realizable by in a pure bosonic systems as bosonic global gauge anoma- local bosonic systems. lies. We will refer those global gauge anomalies that appear _ d d in a fermionic systems as fermionic global gauge anoma- APPENDIX B: H ðBG; R=ZÞ¼H ðG; R=ZÞ lies. Witten’s SUð2Þ global anomaly is a special case of FOR FINITE GROUPS fermionic global gauge anomalies, which exists because ðSUð2ÞÞ ¼ Z . So for a fermionic SUð2Þ gauge theory When G is finite, any closed complex MBG in BG can be 4 2 deformed continuously into a canonical form where all the defined on space-time manifold S4, its partition function Z½A may change sign as we make a large SUð2Þ gauge vertices of MBG is on the same point in BG. All the edges of M is mapped to ðBGÞ¼G. So each edge of M is transformation: BG 1 BG labeled by a group element. All the canonical complex M , with all the vertices on the same point and with fixed 0 0 1 1 BG Z½A ¼ Z½A ;A ¼ g A g ig @ g; the group elements on all the edges, can deform into each gðxÞ2G; (A2) other, since nðBGÞ¼0 for n>1 if G is finite. In this case, an evaluation of a cocycle on MBG is a function of the group elements on the edges. Such a function is a group where gðxÞ is a nontrivial map from S4 to SUð2Þ. This is described by the nontrivial element in cocycle. This way we map a cocycle to a group cocycle. H 1½ ½SUð2Þ ; R=Z . We also note that the group cocycle condition implies 4 that the evaluation on any d-sphere is trivial. So a group cocycle is also a cocycle. The fact that cocycle ¼ TABLE I. A list of homotopy groups dðGÞ which describe group cocycle for finite groups allows us to show the global gauge anomalies in d space-time dimensions. d d H_ ðBG; R=ZÞ¼H ðG; R=ZÞ.
d: Gnd 1234 5 6 Uð1Þ Z 00 0 0 0 APPENDIX C: RELATION BETWEEN Hdþ1 BG; Z H d G; R=Z SUð2Þ 00ZZ2 Z2 Z12 ð Þ AND Bð Þ SUð3Þ 00Z 0 ZZ 6 We can show that the topological cohomology of the SUð5Þ 00Z 0 Z 0 Hdþ1ðBG; ZÞ SOð3Þ Z 0 ZZ Z Z classifying space, , and the Borel-group 2 2 2 12 H d R Z Z Z cohomology, BðG; = Þ, are directly related: SOð10Þ 2 0 00 0 Spinð10Þ 00Z 00 0 dþ1 Z H d R Z H ðBG; Þ’ BðG; = Þ: (C1)
045013-18 CLASSIFYING GAUGE ANOMALIES THROUGH SYMMETRY- ... PHYSICAL REVIEW D 88, 045013 (2013) This result is obtained from Ref. [102], p. 16, where it is For d ¼ odd, we also have mentioned in Remark IV.16(3) that H d ðG; RÞ¼Z B 1 d Z H d 1 R Z H d H d Free½H ðBG; Þ ¼ Free½ B ðG; = Þ ¼ 0; [there, BðG; MÞ is denoted as MooreðG; MÞ which is H d d R Z dþ1 Z equal to SMðG; MÞ]. It is also shown in Remark IV.16(1) H ðBG; = Þ¼Tor½H ðBG; Þ : H d Z d Z and in Remark IV.16(3) that SMðG; Þ¼H ðBG; Þ H d R Z H d R Z dþ1 Z ¼ Tor½ BðG; = Þ : (D4) and SMðG; = Þ¼H ðBG; Þ, (where G can have a nontrivial action on R=Z and Z, and Hdþ1ðBG; ZÞ is the For finite group G and any d,wehave usual topological cohomology on the classifying space d Z H d 1 R Z BG of G). Therefore, we have Free½H ðBG; Þ ¼ Free½ B ðG; = Þ ¼ 0; d R Z dþ1 Z H d R Z H dþ1 Z dþ1 Z H ðBG; = Þ¼Tor½H ðBG; Þ ; BðG; = Þ¼ B ðG; Þ¼H ðBG; Þ; H d R Z H d R Z (C2) ¼ Tor½ BðG; = Þ : (D5) BðG; Þ¼ 1;d>0:
These results are valid for both continuous groups and APPENDIX E: THE KU¨ NNETH FORMULA discrete groups, as well as for G having a nontrivial action on the modules R=Z and Z. The Ku¨nneth formula is a very helpful formula that allows us to calculate the cohomology of chain complex X X0 in terms of the cohomology of chain complex X APPENDIX D: GROUP COHOMOLOGY H BðG; MÞ and chain complex X0. The Ku¨nneth formula is given by AND TOPOLOGICAL COHOMOLOGY H ðBG; MÞ (see Ref. [103], p. 247) ON THE CLASSIFYING SPACE d 0 0 H ðX X ; M R M Þ First, we can show that d k M d k 0 M0 ’½ k¼0H ðX; Þ R H ðX ; Þ dþ1 Z H d R Z H ðBG; Þ’ BðG; = Þ; (D1) dþ1 R k M d kþ1 0 M0 ½ k¼0 Tor1 ðH ðX; Þ;H ðX ; ÞÞ : (E1) H d R Z where BðG; = Þ is the Borel group cohomology Here R is a principle ideal domain and M, M0 are classes. In the main text of this paper, we drop the subscript R M M0 R-modules such that Tor1 ð ; Þ¼0. We also require B. This result is obtained from Ref. [102], p. 16, where it is that M0 and HdðX0; ZÞ are finitely generated, such as H d ð RÞ¼ mentioned in Remark IV.16(3) that B G; 0 (there, M0 ¼ Z Z Z Z . H d ð MÞ H d ð MÞ n m B G; is denoted as Moore G; which is equal to A R-module is like a vector space over R (i.e., we can H d M SMðG; Þ). It is also shown in Remark IV.16(1) and in ‘‘multiply’’ a vector by an element of R.) For more details H d Z d Z Remark IV.16(3) that SMðG; Þ¼H ðBG; Þ and on principal ideal domain and R-module, see the corre- H d R Z dþ1 Z SMðG; = Þ¼H ðBG; Þ, (where G can have a non- sponding Wiki articles. Note that Z and R are principal dþ1 trivial action on R=Z and Z, and H ðBG; ZÞ is the usual ideal domains, while R=Z is not. Also, R and R=Z are not topological cohomology on the classifying space BG of finitely generate R-modules if R ¼ Z. The Ku¨nneth for- G). Therefore, we have mula works for topological cohomology where X and X0 are treated as topological spaces. The Ku¨nneth formula H d ðG; R=ZÞ¼H dþ1ðG; ZÞ¼Hdþ1ðBG; ZÞ; B B also works for group cohomology, where X and X0 are H d R 0 0 0 BðG; Þ¼0;d>0: (D2) treated as groups, X ¼ G and X ¼ G , provided that G is a finite group. However, the above Ku¨nneth formula does These results are valid for both continuous groups and not apply for Borel-group cohomology when X0 ¼ G0 is a discrete groups, as well as for G having a nontrivial action H d 0 Z continuous group, since in that case BðG ; Þ is not on the modules R=Z and Z. We see that, for integer finitely generated. coefficient, H d ðG; ZÞ and HdðBG; ZÞ are the same. B The tensor-product operation R and the torsion-product To see how H d ðG; R=ZÞ and HdðBG; R=ZÞ are related, R B operation Tor1 have the following properties: we can use the universal coefficient theorem (E10)to compute HdðBG; R=ZÞ: A Z B ’ B Z A;
Z Z M ’ M Z Z ¼ M; HdðBG; R=ZÞ¼Con½HdðBG; ZÞ Tor½Hdþ1ðBG; ZÞ Z M M Z M M H d Z H dþ1 Z n Z ’ Z n ¼ =n ; ¼ Con½ BðG; Þ Tor½ B ðG; Þ (E2) Z R Z R Z Z H d 1 R Z n Z = ’ = Z n ¼ 0; ¼ Con½ B ðG; = Þ Z Z Z H d R Z m Z n ¼ hm;ni; Tor½ BðG; = Þ ; (D3) ðA BÞ M ¼ðA MÞ ðB MÞ; Z R Z Z M R R R where Con½ ¼ = , Con½ n ¼0, and Con½ 1 M M M M R ðA BÞ¼ðM R AÞ ðM R BÞ; 2 ¼Con½ 1 Con½ 2 .
045013-19 XIAO-GANG WEN PHYSICAL REVIEW D 88, 045013 (2013) ( and M; if d ¼ 0; HdðX; MÞÞ ¼ (E9) R R 0; if d>0; Tor1 ðA; BÞ’Tor1 ðB; AÞ; Z Z M Z M Z Tor1 ð ; Þ¼Tor1 ð ; Þ¼0; to reduce Eq. (E4)to Z Z M M d M M d Z Z M dþ1 Z Tor1 ð n; Þ¼fm 2 jnm ¼ 0g; H ðX; Þ’ Z H ðX; Þ Tor1 ð ;H ðX; ÞÞ; Z Z R Z Z Tor1 ð n; = Þ¼ n; (E3) (E10) Z Z Z Z 0 Tor1 ð m; nÞ¼ hm;ni; where X is renamed as X. The above is a form of the TorRðA B; MÞ¼TorRðA; MÞ TorRðB; MÞ; universal coefficient theorem which can be used to calcu- 1 1 1 late H ðX; MÞ from H ðX; ZÞ and the module M. R M R M R M Tor1 ð ;A BÞ¼Tor1 ð ;AÞ Tor1 ð ;BÞ; The universal coefficient theorem works for topological cohomology where X is a topological space. The universal where hm; ni is the greatest common divisor of m and n. coefficient theorem also works for group cohomology These expressions allow us to compute the tensor-product where X is a finite group. and the torsion-product TorR. R 1 Using the universal coefficient theorem, we can rewrite As the first application of Ku¨nneth formula, we like to Eq. (E4)as use it to calculate H ðX0; MÞ from H ðX0; ZÞ, by choosing M0 Z R M M0 d 0 M d k d k 0 M R ¼ ¼ . In this case, the condition Tor1 ð ; Þ¼ H ðX X ; Þ’ k¼0H ½X; H ðX ; Þ : (E11) TorZðM; ZÞ¼0 is always satisfied. So we have 1 The above is valid for topological cohomology. It is also d 0 M d k M d k 0 Z H ðX X ; Þ’½ k¼0H ðX; Þ Z H ðX ; Þ valid for group cohomology, ½ dþ1 ZðHkðX;MÞ;Hd kþ1ðX0;ZÞÞ : H d 0 M d H k H d k 0 M k¼0Tor1 ðG G ; Þ’ k¼0 ½G; ðG ; Þ ; (E12) (E4) provided that both G and G0 are finite groups. We may apply the above to the classifying spaces of The above is valid for topological cohomology. It is also group G and G0. Using BðG G0Þ¼BG BG0, we find valid for group cohomology: d 0 M d k d k 0 M H d 0 M d H k M H d k 0 Z H ½BðG G Þ; ’ k¼0H ½BG; H ðBG ; Þ : ðG G ; Þ’½ k¼0 ðG; Þ Z ðG ; Þ dþ1 Z H k M Choosing M ¼ R=Z and using Eq. (D2), we have ½ k¼0Tor1 ð ðG; Þ; H d 0 R Z dþ1 0 Z H d kþ1ðG0; ZÞÞ ; (E5) BðG G ; = Þ¼H ½BðG G Þ; ¼ dþ1Hk½BG; Hdþ1 kðBG0; ZÞ provided that G0 is a finite group. Using Eq. (D2), we can k¼0 H d R Z H d 0 R Z rewrite the above as ¼ BðG; = Þ BðG ; = Þ H d 0 M H d M d 2H k M d 1Hk½BG; H d kðG0; R=ZÞ ; ðG G ; Þ’ ðG; Þ ½ k¼0 ðG; Þ k¼1 B d k 1 0 Z H ðG ; R=ZÞ (E13) d 1 Z H k M 1 0 Z ½ k¼0Tor1 ð ðG; Þ; where we have used H ðBG ; Þ¼0. Using H d k 0 R Z d Z H d Z ðG ; = ÞÞ ; (E6) H ðBG; Þ¼ BðG; Þ; d Z H d Z (E14) where we have used H ðBG; nÞ¼ BðG; nÞ; H 1ðG0; ZÞ¼0: (E7) we can rewrite the above as H d R Z d H k H d k R Z If we further choose M ¼ R=Z, we obtain ðGG SG; = Þ¼ k¼0 ½SG; ðGG; = Þ ¼ d H k½GG; H d kðSG; R=ZÞ : H dðG G0;R=ZÞ k¼0 (E15) ’ H dðG;R=ZÞ H dðG0;R=ZÞ 0 d 2H k R Z H d k 1 0 R Z Equation (E15) is valid for any groups G and G . ½ k¼1 ðG; = Þ Z ðG ; = Þ d 1 Z H k R Z H d k 0 R Z ½ k¼1Tor1 ð ðG; = Þ; ðG ; = ÞÞ ; (E8) APPENDIX F: LYNDON-HOCHSCHILD-SERRE SPECTRAL SEQUENCE where G0 is a finite group. We can further choose X to be the space of one point The Lyndon-Hochschild-Serre spectral sequence (or the trivial group of one element) in Eq. (E4) or Eq. (E5) [104,105] allows us to understand the structure of and use H dðGG⋌SG; R=ZÞ to a certain degree. (Here GG⋌SG
045013-20 CLASSIFYING GAUGE ANOMALIES THROUGH SYMMETRY- ... PHYSICAL REVIEW D 88, 045013 (2013) is a group extension of SG by GG: SG ¼ðGG⋌SGÞ=GG.) In other words, all the elements in H dðGG⋌SG; R=ZÞ can H d ⋌SG R Z We find that ðGG ; = Þ, when viewed as an be one-to-one labeled by ðx0;x1; ...;xdÞ with Abelian group, contains a chain of subgroups,
f0g¼Hdþ1 Hd H1 H0 k d k k xk 2 Hk=Hkþ1 H ½SG; H ðGG; R=ZÞ = : (F4) ¼ H dðGG⋌SG; R=ZÞ; (F1) such that Hk=Hkþ1 is a subgroup of a factor The above discussion implies that we can also use group of H k½SG; H d kðGG; R=ZÞ , i.e., H k½SG; ðm0;m1; ...;mdÞ with H d kðGG; R=ZÞ contains a subgroup k, such that H k H d k R Z k Hk=Hkþ1 ½SG; ðGG; = Þ = ; k d k mk 2 H ½SG; H ðGG; R=ZÞ (F5) k ¼ 0; ;d: (F2) H d kð R ZÞ Note that SG has a nontrivial action on GG; = to label all the elements in H dðG; R=ZÞ. However, such a ! ! ⋌SG ! as determined by the structure 1 GG GG labeling scheme may not be one to one, and it may happen SG ! 1. We also have that only some of ðm0;m1; ...;mdÞ correspond to the ele- H 0 H d R Z ments in H dðG; R=ZÞ. But, on the other hand, for every H0=H1 ½SG; ðGG; = Þ ; element in H dðG; R=ZÞ, we can find a ðm ;m ; ...;m Þ H d R Z d 0 1 d Hd=Hdþ1 ¼ Hd ¼ ðSG; = Þ= : (F3) that corresponds to it.
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