Global Anomalies on the Hilbert Space
GGlobal Anomalies on the Hilbert Space
Diego Delmastro, ab Davide Gaiotto, a Jaume Gomis aa Perimeter Institute for Theoretical Physics,Waterloo, Ontario, N2L 2Y5, Canada b Department of Physics, University of Waterloo,Waterloo, ON N2L 3G1, Canada
Abstract
We show that certain global anomalies can be detected in an elementary fashionby analyzing the way the symmetry algebra is realized on the torus Hilbert space ofthe anomalous theory. Distinct anomalous behaviours imprinted in the Hilbert spaceare identified with the distinct cohomology “layers” that appear in the classification ofanomalies in terms of cobordism groups. We illustrate the manifestation of the layers inthe Hilbert for a variety of anomalous symmetries and spacetime dimensions, includingtime-reversal symmetry, and both in systems of fermions and in anomalous topologicalquantum field theories (TQFTs) in 2 + 1 d . We argue that anomalies can imply an exactbose-fermi degeneracy in the Hilbert space, thus revealing a supersymmetric spectrumof states; we provide a sharp characterization of when this phenomenon occurs andgive nontrivial examples in various dimensions, including in strongly coupled QFTs.Unraveling the anomalies of TQFTs leads us to develop the construction of the Hilbertspaces, the action of operators and the modular data in spin TQFTs, material that canbe read on its own. a r X i v : . [ h e p - t h ] J a n ontents Z T in 0+1 dimensions . . . . . . . . . . . . . . . . . . . . . . 113.2 Anomalous Z T in 2+1 dimensions . . . . . . . . . . . . . . . . . . . . . . 143.3 Anomalous Z in 1+1 dimensions . . . . . . . . . . . . . . . . . . . . . . 18 ν = 2 mod 4: Arf layer . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 ν odd: fermion layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 A Spin TQFTs and anyon condensation 40
A.1 Boson anyon condensation . . . . . . . . . . . . . . . . . . . . . . . . . . 44A.2 Fermion anyon condensation . . . . . . . . . . . . . . . . . . . . . . . . . 47
B Examples of anyon condensation 53
B.1 SO ( n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54B.2 SO ( n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56B.3 SO ( n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59B.4 U (1) k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Introduction and Summary
Consider a system with a classical global symmetry group G . Powerful constraints on thedynamics can be derived by coupling the system to a background connection A for thesymmetry G . The system has an ’t Hooft anomaly [1] if the non-invariance of the partitionfunction under background gauge transformations generated by g ∈ GZ [ A ] (cid:55)→ e iα ( g,A ) Z [ A ] (1.1)cannot be cancelled by a local counterterm constructed out of the background fields. This isthe physics that endows anomalies with a cohomological formulation [2–4].The anomaly α ( g, A ) is a local functional of the background connection and of thetransformation g ∈ G . An ’t Hooft anomaly is captured via anomaly inflow [5] from atopological term in one dimension higher. Each such topological term can be thought of asthe effective action characterizing a symmetry protected topological (SPT) phase [6–8] withsymmetry G in one higher dimension. The topological term is gauge invariant on a closedmanifold and reproduces the anomaly on a manifold with a boundary. Being topological,an ’t Hooft anomaly is robust under deformations that preserve the symmetry, includingrenormalization group transformations. ’t Hooft anomalies give physicists some of the veryfew clues into the nonpertubative dynamics of a quantum system.A combination of insights from condensed matter physics, particle physics, quantuminformation and mathematics has culminated in a conjecturally complete answer to theproblem of classifying the possible anomalies in various dimensions [7, 9–15]. This includesanomalies in bosonic as well as fermionic systems, for discrete and continuous internalsymmetry groups as well as discrete spacetime symmetries such as time-reversal and parity. This has led to the topological classification of anomalies in terms of cobordism theory andgeneralized cohomology theories [9, 11, 13–15, 20–27].Consider first a bosonic system, one which can be defined without a choice of spin structureof the underlying manifold. By Wigner’s theorem, symmetries come in two flavours: linearand unitary, or antilinear and antiunitary, with time-reversal being the prototypical exampleof an antiunitary symmetry. Thus, the symmetry data of a bosonic system is specified by thepair (
G, w ) , (1.2)where G is a group and w ∈ H ( G, Z ) a certain cohomology class w : G → Z that encodesthe unitarity/antiunitarity of the group elements in G . The anomalies of a bosonic systemwith symmetry data ( G, w ) in D spacetime dimensions are classified by the twisted cobordismgroup [13] Ω D +1so ( G ; w ) . (1.3) Coupling to a time-reversal background requires defining the system on unoriented manifolds [13, 16–19].
1n low spacetime dimensions, for D ≤
2, the anomaly classification reduces to group co-homology: Ω D +1so ( G ; w ) = H D +1 ( G, U (1)), extending the classic result that anomalies inquantum mechanics (i.e. D = 1) are classified by H ( G, U (1)), that is, by the projectiverepresentations of G . In higher dimensions, Ω D +1so ( G ; w ) can be reconstructed (losing someinformation about the addition law) from the Atiyah-Hirzebruch spectral sequence [28], thatcombines H D +1 ( G, U (1)) with other cohomology groups of lower degrees.Now recall the characterization of symmetries and classification anomalies of a fermionicsystem, which requires the choice of a ( G -twisted) spin structure to be defined. A fermionicsystem has a universal and unbreakable Z F unitary symmetry generated by fermion parity,denoted by ( − F . This symmetry induces a Z -grading in the Hilbert space H of fermionicsystems, which become super-vector spaces. Since (classically) symmetries cannot changethe fermion parity, that is [ g, ( − F ] = 0, the symmetry group G f acting on the localoperators of a fermionic system is necessarily a Z F central extension of a group G , suchthat G = G f / Z F . Also, by virtue of Wigner’s theorem, a symmetry can be either unitaryor antiunitary. Therefore, the symmetries of a fermionic system are characterized by acocycle w ∈ H ( G, Z F ) specifying the Z F central extension and by a cocycle w ∈ H ( G, Z )encoding the unitarity/antiunitarity of group elements. The anomalies of a fermionic systemwith symmetry data ( G ; w , w ) (1.4)in D spacetime dimensions are classified by the twisted cobordism group [14, 24]Ω D +1spin ( G ; w , w ) . (1.5)State-of-the-art mathematical techniques allow for the computation of these twisted cobordismgroups; see [29–32] for many relevant examples together with reviews aimed at physicists. Aparticularly convenient computational tool is again the Atiyah-Hirzebruch spectral sequence.The different ingredients that go into the computation of (1.5) in this spectral sequence canbe given a nice physical interpretation in terms of layers in various dimensions (see below).While the topological classification of anomalies is rather well understood, detecting whether a physical system is anomalous can be a difficult task. Intuitively, one has to keeptrack of all the arbitrary choices required for a sharp definition of the system on a generalbackground and then quantify the topological obstruction to the trivialization of these choices.A concrete calculation may involve hard-to-determine data characterizing the system. While When w is nontrivial the cocycle condition defining H ∗ ( G, K ) is twisted by the action of w , which actsas an involution on K . This action is nontrivial for K = U (1) and K = Z but trivial for K = Z . In order toavoid clutter we do not write the twisting by w explicitly. For example, for G = Z , and taking w and w to be the nontrivial Z = { , } element in H ( Z , Z )and H ( Z , Z F ) yields the symmetry group generated by time-reversal T obeying T = ( − F , sometimesdenoted by Z T . This is the relevant symmetry group of the celebrated topological superconductors. For example, detecting anomalies in bosonic topological quantum field theories (TQFTs) requires knowingthe F -symbols [33]. G is textbook material, the detection of global anomalies, which includes anomalies for alldiscrete symmetries, is more subtle [34, 35]. The approach is often indirect, for exampleby embedding some global anomalies into a perturbative ones (see for example the recentwork [44] and references therein).In this paper we exhibit an elementary method for detecting some anomalies, basedon constructing the Hilbert space of the theory on a flat (spatial) torus T D − as well asdetermining how the algebra of symmetries is realized on the Hilbert space. This can begiven the following physical interpretation. The anomaly of a D -dimensional theory can berepresented by the class α D +1 ∈ Ω D +1spin ( G ; w , w ) . (1.6)Studying the Hilbert space of the D -dimensional anomalous theory on a spatial torus produces upon integration a class ˜ α = (cid:90) T D − α D +1 . (1.7)The class ˜ α can be viewed as the effective anomaly class of a quantum mechanical theory in0 + 1 d , which we recognize from the properties of the Hilbert space. As a result, we expectto be able to detect this way all anomalies whose cobordism class can be recognized fromthe values on manifolds of the form T D − × Σ , equipped with generic flat connections, spinstructures, etc.This perspective also shows that a torus compactification can provide useful anomalyinformation only if the relevant structures – either the background G connection or spinstructure – do not extend to one higher dimension (i.e. if they are not the boundary ofa manifold in one higher dimension). Indeed, if these structures were all bounding suchthat T D − = ∂M D , then (cid:82) T D − α D +1 = (cid:82) M D dα D +1 = 0, and the effective anomaly in 0+1dvanishes. This means that in order to detect the anomaly in the torus Hilbert space wemust either turn on non-trivial holonomies for the symmetry G or we must consider periodicboundary conditions on the torus for fermionic theories – or both. In practice, we find that this method captures a surprisingly large amount of anomalyinformation. This is especially true for fermionic systems.In order to illustrate how various anomalies are manifested in the Hilbert space, it isuseful to recall some ingredients of the (partial) reconstruction of Ω D +1spin ( G ; w , w ) via the In the case of antiunitary symmetries it requires, for example, learning how to define spin TQFTs onunoriented manifolds, which is an open problem. Numerous interesting partial results have been obtained,however [18, 36–43]. One could also study the reduction of the anomaly class on more general manifolds, potentially detectingmore anomalies. Turning on non-trivial holonomies for the symmetry G defines a G -twisted Hilbert spaces, which are theHilbert spaces where to detect anomalies if the spin structures are bounding. (see e.g. [22–24]) ... ν D − ∈ H D − ( G, Z ) p x + ip y layer ν D − ∈ H D − ( G, Z ) Arf layer ν D ∈ H D ( G, Z ) ψ layer ν D +1 ∈ H D +1 ( G, U (1)) Bosonic layer (1.8)with nontrivial differentials connecting the various classes. Each layer has a physical andgeometric interpretation (see section 2 for more details). In particular, the groups whichappear in the second slot of H D − k ( G, · ) are the groups of k -dimensional SPT phases withno symmetries. We summarize them in table 1.0 + 1 d d d SPTs Z Z Z generator ψ Arf p x + ip y aka SO (1) Z S ± (cid:55)→ ± (cid:55)→ ( − Arf(Σ) M (cid:55)→ e i CS grav [ M ] Table 1: The first row gives the classification of SPT phases with no symmetries, the secondthe generators of the SPT classes, and the last the partition functions of the generators. S ± denotes a circle with periodic/antiperiodic (R / NS) boundary conditions; Σ is a compactRiemann surface, and Arf(Σ) is the Arf-invariant of the surface with spin structure, whichevaluates to 1 on even and to − M is a three-manifold, withCS grav = π (cid:82) M tr( ω d ω + ω ).The endpoint of the Atiyah-Hirzebruch spectral sequence calculation is the associatedgraded of a filtration of Ω D +1spin ( G ; w , w ): the addition law on the k -th layer is modified byunknown carry-overs from higher layers, which are somewhat tricky to compute. Physically,that means that even if the non-trivial differentials vanish, we can only really assign a specificvalue to ν D − k if all ν D − k (cid:48) with k (cid:48) > k vanish, or we can only discuss the difference in the ν D − k anomaly of two theories for which all ν D − k (cid:48) with k (cid:48) > k are the same.We now demonstrate the Hilbert space manifestation of the layers in the anomalies of0 + 1 d fermionic systems with an antiunitary time-reversal symmetry T with T = 1, so that G f = Z T × Z F . The anomalies of such a system are classified by Ω ( Z ; 1 ,
0) = Ω − = Z Recall that the action of w is trivial on Z coefficients. ν ∈ H ( Z T , Z ) (cid:39) Z Arf layer ν ∈ H ( Z T , Z ) (cid:39) Z ψ layer ν ∈ H ( Z T , U (1)) (cid:39) Z Bosonic layer (1.9)We can think about these three groups as compiling into Z , corresponding to the binaryexpansion ν = ν + 2 ν + 4 ν mod 8 . (1.10)The simplest 0 + 1 d system with anomaly ν ∈ Z is a set of ν free massless Majorana fermions,with time-reversal acting as T ( ψ ( t )) = ψ ( − t ) on all ν fermions. At the level of operators,this system has symmetries generated by T and ( − F , these two operations commuting andbeing both of order two, i.e., G f = Z T × Z F . At the level of the Hilbert space, the anomaly ν is manifested through the following anomalous pattern: • ν = odd: There is no graded Hilbert space H . This arises from the Arf layer ν . • ν = 2 mod 4: There is a graded Hilbert space H but the symmetry generators on H donot commute. Instead, they anti-commute: { T , ( − F } = 0 . (1.11)This arises from the fermion ψ layer ν . • ν = 4 mod 8: There is a graded Hilbert space H with [ T , ( − F ] = 0 on it, but thesymmetry algebra T = 1 is realized projectively on H , that is T = − H . (1.12)This arises from the bosonic layer ν .As we compactify higher-dimensional systems on tori, we will use this characterization torecognize the image of various anomalies. Let us explain our approach to detecting anomalies in the celebrated example of topologicalsuperconductors. Consider the anomalies of 2 + 1 d fermionic systems with time-reversalsymmetry T obeying T = ( − F . The symmetry group is G f = Z T , with G = G f / Z F = Z T and the symmetry is twisted by the nontrivial Z classes w and w in H ( Z T , Z ) and H ( Z T , Z F ). The anomalies are classified by Ω ( Z , ,
1) = Ω + = Z [14, 46–48]. By An elegant instance of this general idea is Witten’s SU (2) anomaly [34], which is described by a cobordismclass η ∧ c ( F ) [32], that when integrated over a four sphere with background gauge fields with minimalinstanton number, yields the SPT class η in 0 + 1d with no symmetries, which describes the ψ -phase (seetable 1). Therefore the SU (2) global anomaly is detected as an anomaly in ( − F due to a fermion zeromode in the instanton background and arises from the ψ -layer. d .The anomalies are constructed from the following layers ν ∈ H ( Z T , Z ) (cid:39) Z p x + ip y layer ν ∈ H ( Z T , Z ) (cid:39) Z Arf layer ν ∈ H ( Z T , Z )) (cid:39) Z ψ layer ν ∈ H ( Z T , U (1)) (cid:39) Z Bosonic layer (1.13)These four groups compile into Z , corresponding to the binary expansion ν = ν + 2 ν + 4 ν + 8 ν mod 16 . (1.14)Anomalies ν ∈ Z can be detected by studying the Hilbert spaces H XY of the theory onthe two-torus T , which depend on the choice of spin structure on T , where X, Y ∈ { NS , R } .This gives rise to the Hilbert spaces associated to even spin structures H NS-NS , H NS-R , H R-NS ,and to the odd spin structure H R-R . As explained above, anomalies can only appear in H R-R ,as the other three spin structures are bounding.The following anomalies can be detected on the Hilbert space, as we show in both thestudy of spin TQFTs and fermions in 2 + 1 d : • ν odd: In H R-R the classically ( − F -even time-reversal symmetry generator T becomes( − F -odd, thus changing the parity of the states in H R-R . This corresponds to T anticommuting with ( − F instead of commuting in H R-R : { T , ( − F } = 0 . (1.15)This anomalous behaviour is associated with the p x + ip y layer in (1.13). For ν even[ T , ( − F ] = 0 on H XY . • ν = 2 mod 4: The Z T symmetry algebra on H XY is T = ( − F × ( − Arf( T ) on H XY . (1.16)The symmetry algebra is undeformed on the Hilbert spaces with even spin structureand deformed in the Hilbert space with odd spin structure. This anomalous behaviouris associated with the Arf layer in (1.13). For ν = 0 mod 4, T = ( − F on H XY . • The next two layers ν and ν , corresponding to ν = 4 mod 8 and ν = 8 mod 16, arenot visible on the torus Hilbert space and require other observables to detect them.The analysis of anomalies for time-reversal symmetry T = ( − F in the Hilbert spaceof spin TQFTs [49] requires constructing H XY in the first place, and also learning how tocompute the action of the operators (Wilson lines) on H XY . We explain how to do this forarbitrary spin TQFTs. We show that the Hilbert spaces H XY and the action of ( − F , the6atrix elements of operators and the spin modular data can be unambiguously constructedfrom the data of a suitable bosonic shadow/parent TQFT (encapsulated in a unitary modulartensor category). This involves some novel ingredients which require, for example, the use ofsome F -symbols of the bosonic TQFT. This construction not only allows us to study theanomalies of time-reversal symmetry, but it is interesting in its own right and can be read onits own.Another interesting example where the anomaly layers can be detected on the Hilbertspace is in 1 + 1 d fermionic systems with a unitary Z symmetry. The overall symmetryis G f = Z × Z F and the anomalies are classified by Ω ( Z , ,
0) = Z [14, 22, 50, 51],constructed from the layers ν ∈ H ( Z , Z ) (cid:39) Z Arf layer ν ∈ H ( Z , Z ) (cid:39) Z ψ layer ν ∈ H ( Z , U (1)) (cid:39) Z Bosonic layer (1.17)These four groups compile into Z , corresponding to the binary expansion ν = ν + 2 ν + 4 ν mod 8 . (1.18)The simplest example of a 1 + 1 d theory with symmetry Z × Z F that realizes the ν ∈ Z anomaly is a system of ν Majorana fermions. The generator of Z is the chiral symmetry g = ( − F L , which acts trivially on the right-moving fermions and negates the left-movingfermions.The anomaly ν ∈ Z can be detected by studying the untwisted Hilbert space H X andthe Z -twisted Hilbert space H gX of the ν Majorana fermions, where X ∈ { NS , R } labels thespin structure on the (spatial) circle. We observe the following pattern: • ν odd: The theory does not have proper graded twisted Hilbert spaces H g NS and H g R .Also, while the untwisted Hilbert spaces H X are well-defined, ( − F L and ( − F do notcommute on H R { ( − F L , ( − F } = 0 on H R . (1.19)For ν even H X and H gX are properly graded and [( − F L , ( − F ] = 0. • The ν = 2 mod 4 and ν = 4 mod 8 layers are not visible on the Hilbert space as ananomalous realization of symmetry or a projective representation. Indeed reducing the H class in (1.17) on the circle produces a trivial class in H ( Z , U (1)), signaling thatthere are no nontrivial projective representations of Z in H X or H gX . We note, however,that the anomaly ν = 4 mod 8 can be detected by measuring the spin of states in thetwisted Hilbert spaces H gX , and we indeed show that the theory with ν = 4 mod 8Majorana fermions has this anomaly by computing the spin of states in H gX .7n interesting application of these results is the following. As explained above, someanomalies imply that the symmetry generator is fermion-odd in the Hilbert space H withthe appropriate (non-bounding) structure: the operator that implements the symmetryanticommutes with ( − F instead of commuting. This immediately implies that the spectrumof the theory is supersymmetric, namely for any state in H there is a partner with the sameenergy and with opposite fermion parity. This property of the theory is rather surprising: thebose-fermi degeneracy is a consequence of an anomaly instead of a conventional supersymmetry.This provides a unified perspective on several observations in the literature, and it leads togeneralizations and new predictions: • Any 0 + 1 d theory with an antiunitary Z T symmetry and an odd number of Diracfermions has exact bose-fermi degeneracy. The supersymmetric spectrum of an oddnumber of free Dirac fermions was described in [52], and has been studied more recentlyin [53, 54]. From our perspective, any theory with a Z T anomaly ν = 2 mod 4 has asupersymmetric spectrum. • Any 1 + 1 d theory with a unitary chiral Z symmetry and an odd number of Majoranafermions has exact bose-fermi degeneracy in the untwisted Ramond Hilbert space H R .This includes the supersymmetric spectrum of SU ( N ) adjoint QCD with N even in1 + 1 d recently discussed in [55] (the spectrum is supersymmetric in spite of the fact thatLagrangian of adjoint QCD is not supersymmetric). The fact that that a Z -symmetrictheory with an odd number of Majorana fermions has { ( − F L , ( − F } = 0 in H R impliesthat any such a theory will have a supersymmetric spectrum. This includes examplesin Yang-Mills with Spin ( N ) gauge group, e.g. in the fundamental representation for N odd and in the traceless symmetric representation for N = 0 , • Any 2 + 1 d theory with antiunitary Z T symmetry and an odd number of Majoranafermions has exact bose-fermi degeneracy in the odd spin structure Hilbert space H R-R .This is a nontrivial prediction for the spectrum of gauge theories in 2 + 1 d theories,which are strongly coupled in the infrared. An instance of a theory that should havea supersymmetric spectrum is SO ( N ) gauge theory (with vanishing Chern-Simonscoupling) with a fermion in the traceless symmetric representation. Time-reversalinvariance requires that N is even and ν odd further requires that N = 0 mod 4. Whilethe Lagrangian of this theory is not supersymmetric, the anomaly implies that thespectrum is nonetheless supersymmetric. We can provide nontrivial evidence for thisclaim. In [56] the infrared dynamics of this theory was proposed to be captured by SO ( N +22 ) N +22 Chern-Simons theory. Using the formulae in [57] for the number of In [57] it was shown that SO (2 n + 1) k +1 Chern-Simons theory has (cid:0) n + k − k − (cid:1) bosons and (cid:0) n + k − k (cid:1) fermionsin the Hilbert space H R-R . The spectrum is supersymmetric for n = k , when the theory is time-reversalinvariant. H R-R of this Chern-Simons theory we find that thespectrum is indeed supersymmetric! Our argument applies to other gauge theories withhigher rank real representations and is a nontrivial prediction of their spectrum.The next layer, measuring the projectivity of the symmetry algebra on H , also hasnontrivial implications, the most famous being Kramers theorem. From our analysis one canconclude that any theory in 2 + 1 d with Z T symmetry and anomaly ν = 2 mod 4 has (atleast) two-fold degeneracy in the fermionic part of the even-spin-structure Hilbert spaces,and in the bosonic part of the odd-spin-structure Hilbert space. When ν = 0 mod 4 there is(at least) two-fold degeneracy for all the fermionic states, in any of the spin structures.Finally, we should stress that our analysis may not yet capture all the information aboutanomalies which is encoded in the torus Hilbert spaces. Isometries of the internal space willact on the Hilbert space of a compactified theory. As a result, one could study anomaliesfor the combination of the original symmetries and the new internal symmetries of thecompactified system. We leave this to future work.The plan for the rest of the paper is as follows. In section 3 we study free fermions invarious dimensions and illustrate how anomalies manifest themselves at the level of theirHilbert space. We consider antiunitary time-reversal symmetry in 0 + 1 and 2 + 1 dimensions,where the algebra is T = 1 and T = ( − F , respectively; and we also consider unitary chiralsymmetry in 1 + 1 dimensions with algebra g = 1. After that, in section 4 we consider thesame problem in 2 + 1 d spin TQFTs. We study how their anomalies are seen by constructingtheir Hilbert spaces. Here we revisit the algebra T = ( − F and find the same behaviour asin the case of free fermions. In appendix A we describe how to construct the Hilbert spacesof fermionic TQFTs, that is TQFTs that depend on the spin structure. We demonstratethis framework through examples in appendix B, where we work out in some detail theconstruction of the Hilbert space for several interesting TQFTs. This last appendix alsoincludes a few remarks about a more exotic time-reversal symmetry with algebra T = C ,where C denotes charge conjugation, a unitary Z symmetry. The classification of SPT phases in terms of generalized cohomology/cobordism and associated“layers” is somewhat forbidding, but has a rather transparent physical meaning. Ultimately,we want to have a procedure to associate a partition function to a manifold equippedwith appropriate structures. First, we can triangulate the manifold, equipping it with adiscretization of the various structures we want to endow it with: a flat connection alongthe edges of the triangulation, some discrete version of the spin structure and orientation,etc. Next, we can take the cell decomposition C dual to the triangulation, and place on thefacets of C some collection of invertible TFTs (meaning here SPTs with no symmetries) of9ppropriate dimension, following some rules which take into account the discrete data weput on the manifold. The partition function is then defined as the partition function of thecollection of invertible TFTs.The “layers” of the cohomology theory are simply a way to encode which rules we use toplace invertible TFTs on facets. The differential in the generalized cohomology theory imposesthe constraint that the final answer should be independent of the choice of triangulation aswell as any other choices made at intermediate steps of the construction. It also identifiespairs of rules which give the same final answer.As an example, consider orientable, spin SPTs for unitary symmetries. A discretized G flat connection is given as a collection of G elements on the edges of the triangulation.1. If we were to include only the bottom layer, we would leave all facets bare and only focuson vertices of C . At each vertex we place some complex phase (aka elements of U (1))determined by the group elements along the edges of the dual simplex. This is literallythe cochain ν D +1 representing an element in the group cohomology H D +1 ( G, U (1)). Thecocycle condition ensures that the partition function defined as the product of all thephases is independent of the choice of triangulation and gauge. Coboundaries givepartition functions which evaluate to 1 in a trivial manner.2. Following [9], the next refinement of the story involves placing a fermionic one-dimensionalHilbert space along some of the edges of C . The choice is the cochain ν D representing anelement in the group cohomology H D ( G, Z ). The cocycle condition ensures that eachvertex is connected to an even number of fermionic edges. At each vertex we now get topick a vector in the (one-dimensional, Grassmann even) tensor product of these vectorspaces. This is roughly the same as a choice of ν D +1 , but not canonically, because of signambiguities in the tensor product. The Grassmann combinatorics needed to rearrangethe tensor products when contracting states at the endpoints of fermionic edges, as wellas the (spin structure dependent) signs arising from fermion loops contribute to theoverall sign of the partition function.3. At the next level of refinement, we can place Arf theories on some two-dimensionalfacets according to some ν D − . The cocycle condition ensures that we have an evennumber of Arf facets impinging on an edge, but the edge must now carry a specificchoice of how to gap the corresponding Majorana modes. The two possible choices haveopposite Grassmann parity, so this choice is similar but not canonically equivalent to achoice of ν D , etc. The evaluation of the partition function will now require a carefulmanipulation of the Majorana modes.4. Next, we can place SO ( n ) ± Chern-Simon theories on three-dimensional facets accordingto some ν D − . The cocycle condition insures that we have the same number of chiraland anti-chiral fermions at two-dimensional facets, but facets must now carry a specificchoice of how to gap these 2 d fermions. Two inequivalent choices differ by a factor of10rf. The evaluation of the partition function must now cope with this extra level ofcomplication.5. In principle, we could continue, selecting some invertible fermionic theories to place onfour-dimensional facets, etc. In practice, no non-trivial invertible theories are expectedto exist up to dimension 7, so we can safely stop here for most physical systems.On general grounds, the differential in the generalized cohomology theory takes a triangularform, with the diagonal being the standard differential for H D +1 − k ( G, T k ), where T k is thegroup of invertible theories we can place on the k -th dimensional facets. The off-diagonalcomponents of the differential are non-trivial and somewhat tricky to compute far from thediagonal. Furthermore, the “stacking” operation on generalized cohomology classes, i.e. thesum of anomalies, is also defined in a triangular manner, with the diagonal being the usualoperation of stacking invertible theories.As one compactifies an SPT on, say, a circle, one can take a triangulation of the D -dimensional manifold M and refine it to a triangulation of M × S in a systematic way.Applying the rules above to M × S and reducing them to some evaluation on the triangulationof M one can figure out the resulting SPT theory in one dimension lower. This was done forthe Gu-Wen layer in [58], but has not been done in full generality. In this section we demonstrate how the Hilbert space on the torus detects a variety ofanomalies in systems of free fermions in various dimensions. Z T in 0+1 dimensions The anomalies of a fermionic system in 0+1 dimensions with an antiunitary time-reversalsymmetry T with T = 1, so that G f = Z T × Z F , are classified by Ω ( Z ; 1 ,
0) = Ω − = Z .These anomalies arise from three layers ν ∈ H ( Z T , Z ) (cid:39) Z ν ∈ H ( Z T , Z ) (cid:39) Z ν ∈ H ( Z T , Z ) (cid:39) Z , (3.1)which generate the Z anomaly.We shall study the Z anomaly in a system of free fermions. Related considerations canbe found in [59, 60].Consider ν Majorana fermions in 0 + 1 dimensions L = ν (cid:88) a =1 i ψ a ∂ t ψ a . (3.2)11he theory has a Z T time-reversal symmetry which acts as T ψ a ( t ) = ψ a ( − t ) T (3.3)and Z F fermion parity { ( − F , ψ a ( t ) } = 0 . (3.4)It is known that a Z T -symmetric quartic interaction that gaps out the fermions can bewritten for ν = 8 [45]. This realizes in the fermion system the Z anomaly expected from thecobordism classification.Canonical quantization of (3.2) leads to a Clifford algebra of rank ν { ψ a , ψ b } = 2 δ ab a, b = 1 , , . . . , ν . (3.5)We now proceed to identifying the anomaly layers (3.1). Each layer is implemented in acharacteristic way in the fashion that symmetries are realized on the Hilbert space H . • ν odd:There is a rather severe anomaly for ν odd as the operator ( − F generating the Z F symmetry does not exist. The theory does not admit a proper graded Hilbert spaceof states. Equivalently stated, the Clifford algebra of odd rank has two irreduciblerepresentations, and ( − F exchanges them, instead of acting within an irreduciblerepresentation. This anomaly is associated with the H ( Z T , Z ) = Z layer, the Arflayer.This anomaly due to the lack of proper Hilbert space can also be detected by studyingpartition functions. Consider the partition function on the circle with antiperiodic(NS) and periodic (R) boundary conditions. The partition function with NS boundaryconditions is Z NS = 2 ν/ . (3.6)Nominally, this partition function should count the number of states in H , that is Z NS = tr H ( ) = dim( H ). The answer (3.6) mirrors the statement that there is noproper Hilbert space for ν odd, as 2 ν/ is not an integer. Likewise, while the partition For the purposes of studying anomalies it suffices to take all fermions to transform with the same signunder T . If a fermion is assigned the transformation T ψ − ( t ) = − ψ − ( − t ) T , we can then write a Z T -invariantmass term iψ + ψ − that couples a pair of fermions which transform with opposite signs under T . This liftsboth fermions and therefore without loss of generality we can focus on a collection of fermions that transformwith the same sign under T . This partition function can be evaluated by taking the square root of partition function of 2 ν Majoranafermions, which has a 2 ν -dimensional Hilbert space. It can also be computed by zeta-regularizing Z ≡ Pf( i∂ t ) ν = (cid:81) n ∈ Z λ νn , where the eigenvalues of the 0 + 1 d Dirac operator are λ n = n + 1 / λ n = n in the R sector. Z R = tr H ( − F = 0, the correlator (cid:104) ψ ψ · · · ψ ν (cid:105) R (3.7)is non-vanishing, as the insertions compensate the zero-modes. This observable (3.7)changes sign under the action of ( − F , signaling that ( − F is anomalous as the Z F Ward identities are violated. • ν = 2 mod 4. For ν even the theory has a well-defined Hilbert space and operator ( − F acting on it. The Clifford algebra of even rank ν has a unique irreducible representationof dimension 2 ν/ , thus all representations are unitarily equivalent, and we can study theimplementation of symmetries in any choice of basis. We can construct H by definingthe creation and annihilation operators ψ A ± = 12 (cid:0) ψ A − ± iψ A (cid:1) A = 1 , . . . , ν/ , (3.8)which obey { ψ A + , ψ B − } = δ AB , { ψ A + , ψ B + } = { ψ A − , ψ B − } = 0 A, B = 1 , . . . , ν/ , (3.9)We define the vacuum by ψ A | (cid:105) = 0 A = 1 , . . . , ν/ , (3.10)and create the whole module by acting with the different ψ A + on it. Time-reversal actsby exchanging the creation and annihilation operators (see (3.3) and recall that T isantilinear) T ψ A ± = ψ A ∓ T . (3.11)This allows us to determine the action of T on the vacuum | (cid:105) by considering the mostgeneral state T | (cid:105) = α | (cid:105) + α A ψ A + | (cid:105) + · · · + α ...ν/ ψ ψ · · · ψ ν/ | (cid:105) , (3.12)for some yet-to-be-fixed coefficients { α } . Acting on both sides with ψ A − and us-ing (3.9), (3.10) and (3.11) we conclude that all but the last coefficient vanish, namely T | (cid:105) = α ...ν/ ψ ψ · · · ψ ν/ | (cid:105) , (3.13)with | α ...ν/ | = 1 so that state is normalized. Note that T adds ν/ ν/ Z T × Z F symmetry generators on H obey { T , ( − F } = 0 for ν = 2 mod 4 . (3.14) This corresponds to what is usually referred to as particle-hole symmetry : time-reversal exchanges ψ + and ψ − , so a state full of ψ + is mapped to a state full of ψ − , and vice-versa. T , ( − F ] = 0 for ν = 0 mod 4 , (3.15)Therefore, the anomaly corresponding to the ν = 2 mod 4 layer is detected by virtueof the operators T and ( − F anticommuting in H . This anomaly is associated withthe H ( Z T , Z ) = Z layer, the ψ layer. • ν = 4 mod 8. For ν = 4 mod 8 the theory has a proper Hilbert space and [ T , ( − F ] =0. We now proceed to study how the Z T symmetry is realized on the Hilbert space.Acting with T again in (3.12) yields T | (cid:105) = | α ...ν/ | ψ − ψ − · · · ψ ν/ − ψ ψ · · · ψ ν/ | (cid:105) = ( − ν/ ν/ − | (cid:105) . (3.16)Therefore, for ν = 4 mod 8 the Z T symmetry is realized projectively on the Hilbertspace, that is T = − . (3.17)Therefore, the anomaly corresponding to the ν = 4 mod 8 layer is detected by virtueof the Z T symmetry being realized projectively on the Hilbert space. This anomaly isassociated with the H ( Z T , U (1)) = Z layer, the bosonic layer. Z T in 2+1 dimensions The anomalies of a fermionic system in 2+1d with an antiunitary time-reversal symmetry T = ( − F are classified by Ω ( Z ; 1 ,
1) = Ω + = Z . These anomalies arise from fourlayers (1.13) ν ∈ H ( Z T , Z ) (cid:39) Z ν ∈ H ( Z T , Z ) (cid:39) Z ν ∈ H ( Z T , Z )) (cid:39) Z ν ∈ H ( Z T , U (1)) (cid:39) Z , (3.18)which compile into Z .In this section we study this anomaly in a system of free Majorana fermions, and insection 4 we will do the same in time-reversal symmetric spin TQFTs. We note that T = − ν = 4 , T = 1 for ν = 0 , T − γ a T = γ a , or equivalently U γ a U − = ( γ a ) ∗ ,where we have written T = U K , with K denoting complex conjugation and U a unitary. Therefore, T = U ∗ U = ± T = 1 corresponding to a real involution and T = − ν = 0 , ν = 4 , ν Majorana fermions ψ . We shall work in the Majorana basis wherethe gamma matrices are real, γ = iσ , γ = σ , γ = σ . (3.19)In this basis the Majorana condition is simply ψ ∗ = ψ so ψ is a real two-componentGrassmann-odd spinor. We can without loss of generality take time-reversal to act as T ( ψ ( t )) = ± γ ψ ( − t ) . (3.20)Given a pair of fermions transforming with opposite signs, we can write down a T -invariantmass term, which means that such a pair does not contribute to anomalies. Therefore, as faras anomalies is concerned, we can take all fermions to transform with the same sign, say +1.It is known that a T -invariant interaction exists with 16 fermions that lifts all of them [11,46–48, 61–63].We now construct the torus Hilbert space of the system and study how the time-reversalanomaly manifests itself on it. A subtle but important difference in 2 + 1 d as opposed to theexamples in 0 + 1 d and 1 + 1 d is that Ω spin4 = Z contains a free part: in 2 + 1 d there existsa purely gravitational SPT. This invertible theory is intertwined with time-reversal in aninteresting way, which we review next. The generator of SPTs with no symmetry in 2+1d isgiven by the spin TQFT denoted by SO (1) , corresponding to the super Ising category [18,64–66]. The partition function of this theory is e − i CS grav , where locallyCS grav = 14 π (cid:90) M tr (cid:18) ω d ω + 23 ω (cid:19) , (3.21)where ω is the spin connection for the gravitational background of M . An arbitrary SPT withno symmetry is given by a number n ∈ Z of copies of the generator, namely SO ( n ) := SO (1) n ,whose partition function is e − in CS grav . As a spin TQFT, SO ( n ) can be obtained by condensinga certain fermion in the bosonic TQFT Spin ( n ) , that is by gauging a certain Z one-formsymmetry (see section 4). Note that the Chern-Simons form CS grav is a volume form, so it isodd under time-reversal.If the manifold is non-trivial, the fermions automatically couple to the Chern-Simonsterm for the background gravitational field, because the Dirac operator contains a pieceproportional to the spin connection. In 2 + 1 d time-reversal acts both on the fermions andon the Chern-Simons interactions, and the combined system is only time-reversal invariantif the coefficient of the latter is adjusted appropriately. This behaviour should be thoughtof as a mixed time-reversal-gravitational anomaly, and it can be ascribed to a controllednon-invariance of the fermion path-integral measure Dψ . This non-invariance is a topologicalphase, the eta invariant η , and we can summarize the anomaly as the statement that eachmassless Majorana fermion ψ transforms as T : Dψ (cid:55)→ e − iπη/ Dψ . (3.22)15n absence of other background fields, the eta invariant is precisely the gravitational Chern-Simons term, 12 πη = CS grav mod 2 π Z . (3.23)In this sense, time-reversal does not map the QFT of a single massless fermion into itself,but rather into itself tensored with a copy of the SPT SO (1) ; schematically T (cid:0) massless ψ (cid:1) = massless ψ × SO (1) . (3.24)In order to compensate for the anomalous phase e − iπη/ , we formally need to attach toeach massless Majorana fermion a copy of CS grav , i.e., to a copy of a “square root” of SO (1) .The combined object e CS grav Dψ is now time-reversal invariant. In the notation of (3.24),we formally need to move “half” of SO (1) to the left, so as to have T mapping a QFT intoitself instead of into a second QFT.The discussion above is equivalent to the statement that a massless Majorana fermioncarries chiral central charge c = 1 / framing anomaly [67]; recallthat c measures the coupling of the theory to CS grav ). As c is odd under time-reversal, asystem with c (cid:54) = 0 is not invariant by itself, but must be coupled to a suitable SPT, whosecentral charge is − c , in order to make the total central charge zero. The generator of SPTs SO (1) has c = 1 /
2, so in order to compensate for the c = 1 / SO (1) . More generally, given an arbitrary number ν of massless Majorana fermions, the system ψ ν is not actually time-reversal invariant, butthe combined system ψ ν × SO ( ν/ − is. Naturally, if the number of fermions ν is odd, thecoefficient of CS grav is not properly normalized, and the system does not make sense as apurely 2 + 1 d object: we either give up time-reversal invariance and drop the gravitationalcounterterm, or we keep the symmetry and regard the system as the boundary of a 3 + 1 d theory. For ν even, we can maintain time-reversal invariance and still have a conventional2 + 1 d theory, but only after coupling the fermions to SO ( ν/ − . For now, we will considerthe ν fermions alone, and later on we will study the effect of turning on SO ( ν/ − for ν even.With this in mind, let us go back to studying the system of 2 + 1 d ν massless Majoranafermions on the torus T . Anomalies, being renormalization-group invariant, always arise inthe realization of the symmetry on the low energy states; therefore, in order to detect theanomalies, it suffices to look at the vacuum sector. For even spin structure on T there areno zero modes and no anomalies; this agrees with the general discussion in section 1 wherewe argued that anomalies can only be detected on manifolds that do not bound.For odd spin structure there are zero modes and potential anomalies. Roughly, the systemwith odd spin structure on T behaves as 2 ν copies of the 0 + 1 d system of Majoranas weanalyzed earlier, the factor of 2 being due to the fact that each ψ has two real componentsinstead of one. In this sense, the analogous to the first layer in 0 + 1 d is never activated in16 + 1 d , because the number of Majorana components is always even. In other words, theHilbert space H XY of 2 + 1 d Majorana fermions is always well-defined, regardless of theparity of the number of fermions. But the other two layers, those measured by the fermionparity of T and the sign in T = ±
1, are potentially activated. The first one is measuredby the parity of ν , and the second one by the parity of ν/
2. We will exhibit the followinganomalous behavior in the Hilbert space: • ν odd: In H R-R time-reversal is fermion-odd, it anticommutes with ( − F { T , ( − F } = 0 . (3.25)This anomaly is associated to the p x + ip y layer ν . For ν even [ T , ( − F ] = 0. • ν = 2 mod 4: In the even spin structure Hilbert spaces H NS-NS , H NS-R , H R-NS time-reversal satisfies the standard algebra T = ( − F , but in the odd spin structure Hilbertspace H R-R this algebra is realized projectively, namely T = − ( − F . In other words,the time-reversal symmetry on H XY satisfies T = ( − F × ( − Arf( T ) . (3.26)This anomaly is associated to the Arf layer ν .The next two layers, ν , ν , which measure ν mod 8 and ν mod 16, respectively, are invisibleon the torus Hilbert spaces.The discussion regarding the first two layers is essentially identical to the 0 + 1 d case, sowe only highlight the differences. The fermions now depend on both time t and the spatialcoordinate x , which we take to coordinatize a torus T . The Hilbert space associated to thisspatial slice is built by acting with the spatial modes on the vacuum sector. If Arf( T ) = 0,then there are no zero-modes, and the vacuum Hilbert space is trivial: there is a uniquevacuum state | (cid:105) . Therefore, here time-reversal acts quite trivially: T is fermion-even andsatisfies T = ( − F on the nose: neither layer is activated. In order to detect the anomalywe have to look at the non-bounding torus, i.e., where both boundary conditions are periodicsuch that Arf( T ) = 1. Here there is a single zero-mode for each Majorana fermion, which isspatially constant. In what follows we shall study this vacuum module generated by thesezero-modes in H R-R .First of all, since T ( ψ ) = γ ψ where γ = iσ , time-reversal acts on the two componentsof the Majorana fermion as T ( ψ ) = + ψ , T ( ψ ) = − ψ . (3.27)In terms of the complex spinor Ψ = √ ( ψ + iψ ) this becomes T (Ψ) = i Ψ ∗ . (3.28)17he Hilbert space is built by declaring that Ψ i | (cid:105) = 0 for all i = 1 , , . . . , ν , and by repeatedlyacting with Ψ ∗ i on | (cid:105) . The action of time-reversal on the whole vacuum Hilbert space isuniquely fixed in terms of its action on | (cid:105) , which again reads T | (cid:105) = Ψ ∗ Ψ ∗ · · · Ψ ∗ ν | (cid:105) (3.29)up to an inconsequential phase. We thus see that, indeed, if ν is odd T anticommutes with( − F . Now, if we act with T twice we get T | (cid:105) = T Ψ ∗ Ψ ∗ · · · Ψ ∗ ν | (cid:105) = ( − i ) ν Ψ Ψ · · · Ψ ν T | (cid:105) = ( − i ) ν Ψ Ψ · · · Ψ ν Ψ ∗ Ψ ∗ · · · Ψ ∗ ν | (cid:105) = i − ν | (cid:105) . (3.30)When ν is even we get T | (cid:105) = + | (cid:105) . More generally, as T = ( − F when acting on thecreation operators, the relation T | (cid:105) = + | (cid:105) lifts to T = ( − F on the whole Hilbert space.We now return to the effect of the gravitational SPT SO ( ν/ − for ν even that isneeded in order to have a time-reversal symmetric theory. This SPT has a unique stateon any spin structure Hilbert space H XY . The fermion parity of this state is known to be( − F = ( − Arf( T ) ν/ (see [21, 57] and appendix B.2). Therefore, the T -invariant combinedsystem ψ ν × SO ( ν/ − has a time-reversal algebra T = ( − F × ( − Arf( T ) ν/ . (3.31)This means that the operator algebra T = ( − F is undeformed for ν = 0 mod 4, while itgets deformed by the Arf theory for ν = 2 mod 4 in H R-R , as claimed. Z in 1+1 dimensions The anomalies of a fermionic system in 1 + 1 d with a unitary Z symmetry such that G f = Z × Z F are classified by Ω ( Z ; 0 ,
0) = Z . These anomalies arise from three layers ν ∈ H ( Z , Z ) (cid:39) Z ν ∈ H ( Z , Z ) (cid:39) Z ν ∈ H ( Z , Z ) (cid:39) Z , (3.32)which generate the Z anomaly.Consider ν Majorana fermions in 1 + 1 d L = ν (cid:88) a =1 iψ aL ∂ + ψ aL + iψ aR ∂ − ψ aR , (3.33) The anomalies in 1 + 1 d are actually Z × Z , the second factor being the gravitational anomaly. We take ν L = ν R = ν to cancel this gravitational anomaly and focus directly on the Z factor. ∂ ± = ∂ t ± ∂ x . This system has a chiral Z unitary symmetry generated by g = ( − F L which combines with the nonchiral Z F symmetry generated by ( − F to yield the symmetrygroup G f = Z × Z F . These symmetries act on the fermions as { ( − F L , ψ aL } = [( − F L , ψ aR ] = 0 { ( − F , ψ aL } = { ( − F , ψ aR } = 0 . (3.34)It is known that a Z -symmetric interaction that gaps out the fermions can be written for ν = 8 [50, 68–70]. This realizes in the fermion system the Z anomaly expected from thecobordism classification.We now analyze the anomaly layers that can be detected in the Hilbert space. We discussin turn the Hilbert space H X and the Z -twisted Hilbert space H gX , where X ∈ { NS , R } denotes the spin structure on the spatial circle. The twisted Hilbert space H gX is defined byquantizing in the presence of a nontrivial Z (flat) connection around the circle for the Z symmetry.In order to detect the anomalies we proceed to study the implementation of symmetrieson the zero-mode operators in H X and H gX in turn.Anomalies in H X Since in the NS sector there are no fermion zero-modes, there is a unique, trivial vacuumand symmetries are realized on H NS in an non-anomalous fashion. In the R sector there arefermion zero-modes which upon quantization furnish a Clifford algebra of rank 2 ν { ψ aL , ψ bL } = { ψ aR , ψ bR } = 2 δ ab , { ψ aL , ψ bR } = 0 a, b = 1 , , . . . , ν . (3.35)This Clifford algebra has a unique irreducible representation of dimension 2 ν , thus allrepresentations are unitarily equivalent, and we can study the implementation of symmetriesin any choice of basis. We can construct H R by defining the creation and annihilationoperators ψ a + = ( ψ aR + iψ aL ) and ψ a − = ( ψ aR − iψ aL ), such that ψ a − | (cid:105) = 0. It followsfrom (3.34) that ( − F L ψ a + = ψ a − ( − F L . (3.36)The Z symmetry generator thus maps the empty vacuum to the completely filled state( − F L | (cid:105) = αψ ψ · · · ψ ν + | (cid:105) . (3.37)for some phase α . This implies that the Z × Z F symmetry generators on H R obey { ( − F L , ( − F } = 0 for ν odd , (3.38)and [( − F L , ( − F ] = 0 for ν even . (3.39)19herefore, the anomaly corresponding to the ν odd layer is detected by virtue of the operators( − F L and ( − F anticommuting in H R . This has also been noticed in [24, 71].The anomaly associated to the ν odd layer can also be detected in the torus partitionfunction with periodic boundary conditions around both the spatial circle and temporal circle,that is with (R , R) boundary conditions along the two cycles of the torus. The zero-modesin H R imply that the partition function vanishes, but the partition function with fermionzero-modes saturated is nonvanishing: (cid:104) ψ L ψ L · · · ψ νL ψ R ψ R · · · ψ νR (cid:105) (cid:54) = 0 . (3.40)This implies that the ( − F L Ward identities are violated for ν odd, that is, there is ananomaly for the chiral Z symmetry. Anomalies in H gX This Hilbert space is constructed by imposing boundary conditions twisted by ( − F L when fermions are transported around the spatial circle. This yields different boundaryconditions for the left-moving and right-moving fermions, which we we will denote by [ X L , X R ],where X L/R ∈ { NS , R } . Let us consider H gX in turn: H g NS . This corresponds to [R , NS] boundary conditions on the fermions. There are ν zeromodes from the left movers and none from the right movers. The zero mode algebra is thus aClifford algebra of rank ν { ψ aL , ψ bL } = 2 δ ab a, b = 1 , , . . . , ν . (3.41) • ν odd. There is a rather severe anomaly for ν odd as the operator ( − F generating Z F obeying { ( − F , ψ aL } = 0 does not exist. The theory does not admit a proper gradedHilbert space of states. Equivalently stated, the Clifford algebra of odd rank has twoirreducible representations, and ( − F exchanges them, instead of acting within anirreducible representation. This anomaly is associated with the H ( Z , Z ) = Z layer,the Arf layer.The Z anomaly associated to the Arf layer and the corresponding lack of a Hilbertspace can also be detected in the torus partition function with antiperiodic boundaryconditions around both the spatial circle and the temporal circle, that is, with (NS , NS)boundary conditions along the two cycles of the torus. This partition function is givenby Z ( g, , NS) = ( √ χ σ ) ν ( χ + χ (cid:15) ) ν , (3.42)where χ , χ σ and χ (cid:15) are the Virasoro characters with weight 0 , /
16 and 1 /
2. For ν odd,the partition function indeed does not have an integral expansion, and thus there is no This is to be contrasted with the boundary condition in the untwisted Hilbert space, where X L = X R . √ (cid:96) (1), whose dimensionis formally √ • ν even. For ν even the theory has a well-defined Hilbert space H g NS and well defined( − F and ( − F L operators obeying [( − F L , ( − F ] = 0 in the Hilbert space. The Z symmetry generator maps the empty vacuum in H gX to itself up to phase ( − F L | (cid:105) = α | (cid:105) .It is worth mentioning that the theory with ν = 4 mod 8 has a bosonic Z anomaly,measured by H ( Z , U (1)) = Z . While this anomaly is not visible in the way thesymmetry is realized in the Hilbert space, it can be detected by the presence of stateswith anomalous spin in the Hilbert space (see e.g. [72] for a similar discussion for bosonicsystems). This anomaly can be detected from the torus partition function with (NS , NS)boundary conditions (3.42). Under the modular transformation T all spin structuresin the torus are invariant, which implies that in a non-anomalous theory T = 1 andthe spin of states in H g NS should be Z /
2. Instead, in an anomalous theory, T = − H g NS can be 1 / Z /
2. From (3.42) we see that indeed for ν = 4 mod 8 fermions, the states in H g NS have spins 1 / Z /
2. This diagnoses that thesystem with ν = 4 mod 8 fermions has a Z anomaly arising from the bosonic layer. H g R . This corresponds to [NS , R] boundary conditions on the fermions. There are ν zeromodes from the right-movers and none from the left-movers. The zero mode algebra is thus aClifford algebra of rank ν { ψ aR , ψ bR } = 2 δ ab a, b = 1 , , . . . , ν . (3.43) • ν odd. Verbatim our discussion above: this system has a Z anomaly associated to theArf layer, diagnosed by the lack of a proper graded Hilbert space of states. This canalso be seen from the lack of integral expansion of the torus partition function with(R , NS) boundary conditions along the two cycles of the torus Z ( g, , NS) = ( χ + χ (cid:15) ) ν ( √ χ σ ) ν (3.44)which again does not have a properly quantized expansion. • ν even. For ν even the theory has a well-defined Hilbert space H g R and well defined( − F and ( − F L operators obeying [( − F L , ( − F ] = 0 in the Hilbert space.The partition function Z ( g, , NS) also exhibits a bosonic Z anomaly ν = 4 mod 8 as H g R has states with spin 1 / Z / H X , ν = 1 mod 2 means that g anticommutes with( − F . For twisted Hilbert spaces H gX , ν = 1 mod 2 means lack of graded Hilbert spaces.21 Anomalies in spin TQFT Hilbert space
In this section we demonstrate the existence of anomalies in fermionic TQFTs by lookingdirectly at their Hilbert space. We follow the construction of the Hilbert space of a fermionicTQFT in [57]; see [21, 73–77] for related work. Here we summarize the main ingredients,leaving most details to appendix A.Given an arbitrary spin TQFT, one may sum over all spin structures in order to yield abosonic TQFT. We refer to this theory as the bosonic parent/shadow of the original spinTQFT. This process of summing over spin structures corresponds to gauging the zero-formsymmetry generated by fermion parity Z F = (cid:104) ( − F (cid:105) . Given such bosonic theory, one mayundo the gauging, i.e., we can recover the original spin TQFT by gauging a dual Z symmetry,this time a one-form symmetry [78]. This symmetry is generated by a certain fermionic lineoperator ψ , i.e., Z (1)2 = (cid:104) ψ (cid:105) . Gauging this one-form symmetry is also known as condensing the anyon ψ .With this in mind, the Hilbert space of the spin TQFT is easily obtained in terms ofthe Hilbert space of the bosonic parent, by means of the standard procedure of gauginga symmetry. The Hilbert space of the bosonic parent, being a standard TQFT, is well-understood. Specifically, the torus Hilbert space has a basis of states labelled by theanyons [67]: H ( T ) = Span β ∈A (cid:20) × β (cid:21) (4.1)Here A denotes the set of all anyons in the bosonic parent – a finite set – and the loopdenotes a Wilson line labeled by β wrapped along the b -cycle of the torus (see figure 1). β α g = × g αβ Figure 1: Schematic notation for an arbitrary configuration of anyons on the torus, in thepresence of a puncture g . The green line represents the vertical (time) direction, orthogonalto the torus. We insert an anyon g along this direction, which from the point of view of thetorus becomes a puncture (a marked point). The red line represents the a -cycle, and the blueone the b -cycle. We insert Wilson lines with anyons α, β along these cycles, respectively. Thecross × represents the hole. The states in the Hilbert space H are created by wrapping anyonsaround the b -cycle. The states in the twisted Hilbert space H g are created by wrappinganyons around the b -cycle, in presence of a vertical anyon g .The set A comes equipped with extra structure; for example, we have the modular matrix22 : A × A → C that implements the large diffeomorphism ( a , b ) (cid:55)→ ( b , − a ). Similarly, wealso have the modular matrix T : A × A → C that implements the large diffeomorphism( a , b ) (cid:55)→ ( a , a + b ); these two transformations S, T generate the set of all large diffeomorphisms,the modular group SL ( Z ). By our choice of basis (4.1) where the states are wrapped aroundthe b -cycle, the T -matrix is diagonal, with T = diag( e π ( h − c/ ), where c is the central chargeof the system and h : A → Q / Z denotes the topological spin of the lines.The Hilbert space of the fermionic theory, let us call it ˆ H , depends on the spin structure ofthe torus. We denote these tori as T s a ,s b , where s a , s b = ± a - and b -cycles, respectively. We also denote the s = − s = +1 boundary condition by R. We claim that the corresponding Hilbertspaces are spanned by the following bases:ˆ H ( T ) : × a + × a × ψ ˆ H ( T ) : × a − × a × ψ ˆ H ( T ) : × x + × x × ψ × m ˆ H ( T ) : × x − × x × ψ × mψ (4.2)Here, a takes values in the subset of lines of A with the property that h α ≡ h α × ψ mod 1. Onthe other hand, both x and m denote the lines in A such that h α ≡ h α × ψ + 1 / x lines and m lines is that the former satisfy x × ψ (cid:54) = x while the lattersatisfy m × ψ = m . Finally, the state with an open line denotes the anyon β around thespatial torus and the anyon ψ running along the time direction, cf. figure 1. (From the pointof view of the spatial torus, the line operator ψ looks like a puncture, i.e., a local operator; itis essentially a constant spinor, a zero-mode, which explains why it only exists in the oddspin structure).The rationale behind the structure above is the following. Given the bosonic theory,gauging the one-form symmetry generated by ψ means inserting this operator in all possibleways; summing over insertions along the spatial cycles projects the spectrum into the invariant23tates, and summing over insertions along the time circle introduces twisted sectors. Onecan check that the states in (4.2) are indeed invariant under insertion of ψ along any of thespatial cycles. The twisted sectors are precisely the states with a puncture, which do notlive in H but in the defect Hilbert space H ψ instead. The details of this construction areelaborated upon in appendix A.The explicit geometric structure of states in (4.2) also allows us find how the operators ofthe spin TQFT act on the different states. For example, the Wilson lines act by inserting ananyon around the cycle they are supported on, and therefore they act as W ( a ) ( α ) : × β (cid:55)→ × α β ≡ S α,β S ,β × β W ( b ) ( α ) : × β (cid:55)→ × αβ ≡ × α × β (4.3)on states without puncture, and as [79] W ( b ) ( α ) : × ψ β (cid:55)→ × αψ β ≡ F β,α × β (cid:20) ψ α × ββ α (cid:21) × α × βψ , (4.4)on states with a puncture. Here S : A × A denotes the modular matrix of the bosonic parent,and F its F -symbols. From these expressions it is trivial to write down the Wilson operatorsas matrices acting on ˆ H (with respect to the basis (4.2)).Similarly, one can also write down how modular transformations map the different Hilbertspaces ˆ H ( T XY ) into each other. For example, an S -transformation acts as S : × β (cid:55)→ (cid:88) α ∈A S α,β × α (4.5)on states without puncture, and as S : × ψ β (cid:55)→ (cid:88) α ∈A S α,β ( ψ ) × α × βψ , (4.6)on states with a puncture. Here S ( ψ ) is the S -matrix of the bosonic parent in the once-punctured torus (cf. (A.18)). Given these two expressions one can easily check that S -transformations reshuffle the different spin structures as expected, namely ( X, Y ) (cid:55)→ ( Y, X ).Identical considerations hold for T -transformations (these are actually simpler because theydo not see the puncture, so the formula T = diag( e π ( h − c/ ) is still valid for states in H ψ ).24he final important remark concerning the fermionic Hilbert space ˆ H is that it is asuper-vector space, i.e., its states are either bosons or fermions. Given that ( − F is, bydefinition, dual to the gauged one-form symmetry Z (1)2 , it is clear that the states chargedunder the former correspond to the states coming from the twisted sector, i.e., the fermionsin ˆ H are precisely those that include a ψ -puncture. In this sense, ( − F is trivial in the evenspin structure Hilbert spaces, and it equals +1 on x -lines and − m -lines. In a fermionictheory the modular group is no longer SL ( Z ), but rather a Z F extension, known as themetaplectic group M p ( Z ), defined by the relations S = ( ST ) , S = ( − F . (4.7)The torus Hilbert space of spin TQFTs realize a unitary representation of this group.We next illustrate all these considerations by explicitly working out several specificexamples of spin TQFTs. We show that time-reversal invariant theories with T = ( − F with Z anomalies ν = 2 mod 4 have time-reversal in the Hilbert space realized as T = ( − F × ( − Arf( T ) , (4.8)thus exhibiting the anomalies associated with the Arf layer. We then show that spin TQFTswith ν odd have a time-reversal symmetry that anticommutes with ( − F on H R-R { T , ( − F } = 0 , (4.9)thus exhibiting the anomalies associated with the ψ layer ν = 2 mod 4 : Arf layer In this section we consider time-reversal invariant theories with ν = 2 mod 4 and show thatthey realize the expected behavior associated with the Arf layer. We work out in detail herethe example of the semion-fermion theory which has ν = 2, although the same behaviouris observed in any theory with ν = 2 mod 4. Other common examples of time-reversalinvariant spin TQFTs are Sp ( n ) n and SO ( n ) n , which have ν = 2 n and ν = n , respectively.One can check that e.g. Sp (3) and SO (6) , which have ν = 6, exhibit the same behaviour.We do not reproduce the explicit computation here because the matrices are very large andthe details do not contain any new ingredients.The semion-fermion theory is a fermionic TQFT with four anyons: the vacuum , asemion s , a transparent fermion ψ , and the composite s × ψ . A Chern-Simons realization ofthis theory is U (1) × { , ψ } , where U (1) = { , s } is the TQFT of a single semion, and ψ denotes a transparent fermion. The invertible factor can be written as { , ψ } = SO ( n ) forany n ∈ Z ; the most convenient choice is n = −
2, so that the theory has vanishing central25harge (as required by time-reversal). In other words, we shall consider U (1) × U (1) − ,whose Lagrangian reads L = 14 π (2 a d a − b d b ) , (4.10)where a, b are U (1) connections. The time-reversal symmetry of this Lagrangian acts asfollows [64, 80]: T ( a ) = a − b, T ( b ) = 2 a − b . (4.11)One way of constructing the Hilbert space of this theory is to take U (1) × U (1) − , whichis bosonic, and condense the line ψ = (0 , ψ becomes transparent.We begin by constructing the Hilbert space of the bosonic parent, U (1) × U (1) − . Thisis a 2 × | α, β (cid:105) = W ( b ) ( α, β ) | (cid:105) , ( α, β ) ∈ Z × Z , (4.12)where | (cid:105) denotes the vacuum state – the empty torus – and W denotes a Wilson line: W ( c ) ( α, β ) := exp i (cid:73) c ( αa + βb ) , (4.13)for any c ∈ H ( T , Z ) = Z [ a ] ⊕ Z [ b ].The Wilson lines along the b -cycle act on a generic state as follows: W ( b ) ( α, β ) | α (cid:48) , β (cid:48) (cid:105) = | α + α (cid:48) mod 2 , β + β (cid:48) mod 4 (cid:105) . (4.14)The action of the Wilson lines associated to other cycles can be obtained using the modularoperations. For example, on the a -cycle, one has W ( a ) ( α ) | α (cid:48) (cid:105) = S αα (cid:48) S α (cid:48) | α (cid:48) (cid:105) , (4.15)where S denotes the S -matrix of the system. In the semion-fermion theory, this matrix reads S ( α,β ) , ( α (cid:48) ,β (cid:48) ) = e πi ( ββ (cid:48) / − αα (cid:48) ) / ψ = (0 , α, β ) withrespect to the fermion is B (( α, β ) , ψ ) = e − iπβ , and so the NS- and R-lines are as follows:NS : A NS = { ( α, β ) : β = 0 , } R : A R = { ( α, β ) : β = 1 , } (4.16)Furthermore, there are no fixed-points under fusion with ψ . Indeed, the lines are all intwo-dimensional orbits, paired as follows:NS : ( α, × ψ ←→ ( α, α, × ψ ←→ ( α, . (4.17)The lack of fixed-points indicates that there are no Majorana lines in the theory, i.e., allstates are bosonic. In the terminology of section A.2, all lines ( α, β ) are a -type or x -type,depending on whether β is even or odd; and there are no m -lines.26 ilbert space. With these preliminaries in mind, we now construct the torus Hilbertspace(s) of the fermionic theory. As in section A.2, the states of the condensed phase areexpressed as linear combinations of those of the bosonic parent, and the specific combinationsare determined by the spin structure (cf. (A.33)): • If we take
NS-NS boundary conditions, the two states are |
0; NS-NS (cid:105) = 1 √ | , (cid:105) + | , (cid:105) ) |
1; NS-NS (cid:105) = 1 √ | , (cid:105) + | , (cid:105) ) . (4.18) • If we take
NS-R boundary conditions, the two states are |
0; NS-R (cid:105) = 1 √ | , (cid:105) − | , (cid:105) ) |
1; NS-R (cid:105) = 1 √ | , (cid:105) − | , (cid:105) ) . (4.19) • If we take
R-NS boundary conditions, the two states are |
0; R-NS (cid:105) = 1 √ | , (cid:105) + | , (cid:105) ) |
1; R-NS (cid:105) = 1 √ | , (cid:105) + | , (cid:105) ) . (4.20) • If we take
R-R boundary conditions, the two states are |
0; R-R (cid:105) = 1 √ | , (cid:105) − | , (cid:105) ) |
1; R-R (cid:105) = 1 √ | , (cid:105) − | , (cid:105) ) . (4.21) Modularity.
As a consistency check, we can study how modular transformations move usaround these four Hilbert spaces. Take for example the S -transformation. In the bosonicparent, this operation acts as S | α, β (cid:105) = (cid:88) α (cid:48) ∈ Z β (cid:48) ∈ Z S ( α,β ) , ( α (cid:48) ,β (cid:48) ) | α (cid:48) , β (cid:48) (cid:105) (4.22)27ith S ( α,β ) , ( α (cid:48) ,β (cid:48) ) = e iπ ( ββ (cid:48) / − αα (cid:48) ) /
4. Using this we obtain the action of S on the fermionictheory. For example, it acts on the NS-NS states as follows: S |
0; NS-NS (cid:105) = 1 √ (cid:20) S | , (cid:105) + S | , (cid:105) (cid:21) = 1 √ (cid:20) | , (cid:105) + | , (cid:105) + | , (cid:105) + | , (cid:105) + | , (cid:105) + | , (cid:105) + | , (cid:105) + | , (cid:105) + | , (cid:105) − | , (cid:105) + | , (cid:105) − | , (cid:105) + | , (cid:105) − | , (cid:105) + | , (cid:105) − | , (cid:105) (cid:21) = 1 √ | , (cid:105) + | , (cid:105) + | , (cid:105) + | , (cid:105) ) . (4.23)We recognize this state as ( |
0; NS-NS (cid:105) + |
1; NS-NS (cid:105) ). Through an identical computation onecan show that S maps |
1; NS-NS (cid:105) into ( |
0; NS-NS (cid:105) − |
1; NS-NS (cid:105) ). In both cases we see thatan S -transformation maps states in ˆ H NS-NS into ˆ H NS-NS , precisely as expected (cf. (A.26));and, moreover, the specific matrix that realizes this transformation isˆ S NS-NS → NS-NS = 12 (cid:18) − (cid:19) . (4.24)By performing S -transformations on the other three Hilbert spaces we see that they arepermuted exactly as they should, namely S : ˆ H s a ,s b → ˆ H s b ,s a ; and that they act as thefollowing matrices:ˆ S NS-R → R-NS = ˆ S R-NS → NS-R = − i ˆ S R-R → R-R = 12 (cid:18) − (cid:19) . (4.25)A T -transformation, on the other hand, acts in the bosonic parent as T | α, β (cid:105) = e iπ ( α / − β / | α, β (cid:105) , (4.26)which induces the following transformation in the fermionic quotient: T : ˆ H s a ,s b → ˆ H s a ,s a s b ,with matricesˆ T NS-NS → NS-R = ˆ T NS-R → NS-NS = ˆ T R-NS → R-R = ˆ T R-R → NS-R = (cid:18) i (cid:19) . (4.27)(The semion-fermion theory is rather degenerate, not least due to the fact that it factorizesinto a bosonic TQFT and a trivial spin TQFT; in the general case, the matrices ˆ S s a ,s b , ˆ T s a ,s b are all typically distinct.)A final ingredient as regards modularity is the charge-conjugation operation, which actsin homology as C : ( a , b ) (cid:55)→ ( − a , − b ). Unlike the S - and T -operations, charge-conjugationfixes all spin structures: C : ˆ H s a ,s b → ˆ H s a ,s b . This operation acts on the U (1) connection as28 (cid:55)→ − a or, equivalently, on the anyons as α (cid:55)→ ¯ α = − α . The semion and the fermion areboth self-conjugate (cf. − C acts trivially on all the anyonsof the theory. That being said, this operator need not act trivially on the Hilbert space. Itsaction is easily computed given the expression of the fermionic Hilbert space in terms of thebosonic parent, namely C | α, β (cid:105) = | − α mod 2 , − β mod 4 (cid:105) . (4.28)For example, this action induces the following action on the quotient theory: C |
0; NS-NS (cid:105) = 1 √ C | , (cid:105) + C | , (cid:105) )= 1 √ | , (cid:105) + | , (cid:105) )= |
0; NS-NS (cid:105) . (4.29)Repeating this operation for the rest of basis vectors, we arrive atˆ C NS-NS → NS-NS = ˆ C NS-R → NS-R = ˆ C R-NS → R-NS = − ˆ C R-R → NS-R = (4.30)or, more succinctly, ˆ C = ( − Arf( s ) .Given the explicit expressions for the ˆ S, ˆ T , ˆ C matrices, we can check that they realize aunitary representation of the modular group. In this case, the lack of Majorana lines meansthat ( − F is trivial, which means that the modular algebra is just that of the regular toruswith no punctures. In other words, the modular transformations satisfy S = ( ST ) = C ,with C = 1. The matrices ˆ S, ˆ T , ˆ C calculated above indeed satisfy this algebra, as expected.In checking this one must keep in mind that ˆ S, ˆ T do not live in End( ˆ H s ) (unlike in thebosonic case) but rather in Hom( ˆ H s , ˆ H s (cid:48) ) (cf. (A.41)). Wilson lines.
An arbitrary element | v (cid:105) ∈ ˆ H s a ,s b ∼ = C | can be written as | v (cid:105) = c | s a s b (cid:105) + c | s a s b (cid:105) for some coefficients c , c ∈ C . Furthermore, all operators O ∈
End( H s a ,s b ) canbe represented as 2 × × W ( a ) (1 , |
1; NS-NS (cid:105) = 1 √ W ( a ) (1 , | , (cid:105) + | , (cid:105) )= 1 √ e − iπ ( | , (cid:105) + | , (cid:105) )= −|
0; NS-NS (cid:105) , (4.31)29nd W ( b ) (1 , |
1; NS-NS (cid:105) = 1 √ W ( b ) (1 , | , (cid:105) + | , (cid:105) )= 1 √ | , (cid:105) + | , (cid:105) )= + |
0; NS-NS (cid:105) . (4.32)The rest of matrix elements are computed using the same idea. Denoting the semion by ς = (1 , ψ = (0 , W ( a ) s a ,s b ( ς ) = σ , W ( b ) s a ,s b ( ς ) = σ , W ( c ) s a ,s b ( ψ ) = − s c . (4.33)(As before, the fact that W ( c ) ( ς ) is independent of s a , s b is rather particular to this simplesystem; in generic spin TQFTs these matrices depend non-trivially on the spin structure.)One can easily check that the Wilson lines satisfy the fusion rules of the theory, namely ς = ψ = . Time-reversal.
Finally, we implement time-reversal as an explicit operator in ˆ H s a ,s b . Wewrite T = τ K , where K denotes complex conjugation and τ ∈ C × C ; this factorisation is notcanonical, in the sense that τ and K are separately convention-dependent – only their productis meaningful. We shall fix them by declaring that our basis is real, K | α ; s a s b (cid:105) = | α ; s a s b (cid:105) ,where α = 0 ,
1. In other words, K acts by complex-conjugating the coefficients: K ( c | s a s b (cid:105) + c | s a s b (cid:105) ) ≡ c ∗ | s a s b (cid:105) + c ∗ | s a s b (cid:105) . (4.34)Naturally, this definition is not basis-independent. But T , which is the object we care about,is, so this is enough for our purposes.We recall that T acts on the U (1) fields as T ( a ) = a − b, T ( b ) = 2 a − b (cf. (4.11)). Thisinduces the following transformation on the Wilson lines: W ( c ) ( ς ) = exp i (cid:73) c a (cid:55)→ exp i (cid:73) c ( a − b ) ≡ W ( c ) ( ς ) W ( c ) ( ψ ) W ( c ) ( ψ ) = exp i (cid:73) c b (cid:55)→ exp i (cid:73) c (2 a − b ) ≡ W ( c ) ( ψ ) (4.35)where we have used W ( ς ) = W ( ς ) = . More to the point, time-reversal acts on theanyons by fixing the vacuum and the fermion, and by exchanging ς ↔ ς × ψ .With this, time-reversal acts on the Hilbert space as follows: T W ( c ) s a ,s b ( α ) T − = W ( c ) s a ,s b ( T ( α )) c ∈ { a , b } , (4.36)where α ∈ { , ς, ψ, ψ × ς } . As W ( ψ ) ∝ , the only non-trivial equation corresponds to thesemion, W ( ς ), which in our basis reads τ s a ,s b ( W ( c ) s a ,s b ( ς )) ∗ τ − s a ,s b = − s c W ( c ) s a ,s b ( ς ) c ∈ { a , b } . (4.37)30his is nothing but a set of linear equations in the components of τ , with solution( τ s a ,s b ) α,β = ( − s b ) α δ α + β + ( s a +1) (4.38)up to an inconsequential global phase, and where α, β = 0 , δ x = 1 if x is even, and δ x = 0 if odd.We next note that ( τ τ ∗ ) α,β = (cid:88) γ τ α,γ τ ∗ γ,β = (cid:88) γ ( − s b ) α + γ δ α + γ + ( s a +1) δ β + γ + ( s a +1) = ( − s b ) ( s a +1) δ α + β . (4.39)Finally, observe that the expression above can also be written as τ τ ∗ = ( − Arf( s ) , whichmeans that time-reversal satisfies T = τ τ ∗ ≡ ( − Arf( s ) . (4.40)In this theory, fermion parity is trivial, ( − F ≡
1. Therefore, the equation above meansthat the time-reversal algebra T = ( − F is deformed when acting on the Hilbert space, inthe form T = ( − F × ( − Arf( s ) (4.41)which is precisely what we expected, given that the theory has ν = 2 mod 4. ν odd: fermion layer In this section we consider time-reversal invariant theories with ν odd and show that theyrealize the expected behavior associated with the fermion layer. We work out in detail herethe example of SO (3) Chern-Simons theory that has ν = 3. One can repeat the exercisefor other theories with ν odd, such as SO (5) , which has ν = 5; the main conclusions areidentical.With this in mind, we next study the spin TQFT SO (3) . This is the smallest intrinsicallyfermionic topological theory, in the sense that it supports both bosonic and fermionic states,unlike the previous section, where all states were bosonic. The presence of fermionic statesis directly related to the presence of Majorana lines, i.e., of fixed-points of the condensingfermion. These will introduce new ingredients into the picture.The bosonic parent of the theory is Spin (3) = SU (2) , which becomes SO (3) upongauging the Z center. So we first construct the bosonic theory. This theory has seven states,31abelled by their isospin: | j (cid:105) , where j = 0 , , , . . . ,
3. Under modular transformations, thesestates transform as S | j (cid:105) = (cid:88) j (cid:48) S j,j (cid:48) | j (cid:48) (cid:105) , S j,j (cid:48) = 12 sin π j + 1)(2 j (cid:48) + 1) T | j (cid:105) = e πi (2 j (2 j +2) − / | j (cid:105) . (4.42)The condensing fermion ψ corresponds to the line j = 3. The braiding phase of a genericline j with respect to ψ is B ( j, ψ ) = ( − j , which means that the NS-lines are those withintegral isospin, and the R-lines are those with half-integral isospin:NS : A NS = { j = 0 , , , } R : A R = { j = 12 , , } . (4.43)The NS-lines are all in two-dimensional orbits, paired up as follows:0 × ψ ←→ , × ψ ←→ . (4.44)On the other hand, in the R sector there is one two-dimensional orbit, and one fixed-point:12 × ψ ←→ , (cid:120) × ψ . (4.45)In other words, 0 , , , a -type; , are both x -type; and is m -type. Hilbert space.
With the information above we have all we need in order to construct theHilbert space of the fermionic theory. As usual, the states of the condensed phase SO (3) are expressed in terms of those of its parent, the specific expression being determined by thechoice of spin structure: • If we take
NS-NS boundary conditions, the two states are |
0; NS-NS (cid:105) = 1 √ | (cid:105) + | (cid:105) ) |
1; NS-NS (cid:105) = 1 √ | (cid:105) + | (cid:105) ) . (4.46) • If we take
NS-R boundary conditions, the two states are |
0; NS-R (cid:105) = 1 √ | (cid:105) − | (cid:105) ) |
1; NS-R (cid:105) = 1 √ | (cid:105) − | (cid:105) ) . (4.47)32 If we take
R-NS boundary conditions, the two states are |
0; R-NS (cid:105) = 1 √ | / (cid:105) + | / (cid:105) ) |
1; R-NS (cid:105) = | / (cid:105) . (4.48) • If we take
R-R boundary conditions, the two states are |
0; R-R (cid:105) = 1 √ | / (cid:105) − | / (cid:105) ) |
1; R-R (cid:105) = | / ψ (cid:105) , (4.49)where, we remind the reader, | α ; β (cid:105) denotes the anyon α in presence of a β puncture(cf. (A.6)). Modularity.
As a check of the formalism so far, let us construct the modular data associatedto these states, and check that it behaves as expected from general considerations. Theeven-spin-structure Hilbert spaces do not contain punctures, which means their modular datais computed in exactly the same way as in the previous section. For example, performingan S -transformation on an NS-R state we expect to obtain an R-NS state, which is easilyconfirmed: S |
0; NS-R (cid:105) = 1 √ S | (cid:105) − S | (cid:105) )= 14 (cid:20) + (cid:112) − ξ | (cid:105) + | / (cid:105) + (cid:112) ξ | (cid:105) + √ | / (cid:105) + (cid:112) ξ | (cid:105) + | / (cid:105) + (cid:112) − ξ | (cid:105)− (cid:112) − ξ | (cid:105) + | / (cid:105) − (cid:112) ξ | (cid:105) + √ | / (cid:105) − (cid:112) ξ | (cid:105) + | / (cid:105) − (cid:112) − ξ | (cid:105) (cid:21) = 12 ( | / (cid:105) + √ | / (cid:105) + | / (cid:105) )= 1 √ |
0; R-NS (cid:105) + |
1; R-NS (cid:105) ) , (4.50)where in the second line we have denoted ξ = sin π/ / √ S on the rest of even-spin-structure Hilbert spaces we see that they areindeed permuted as S : ˆ H s a ,s b → ˆ H s b ,s a ; and, moreover, this action is effected by the following33atrices: ˆ S NS-NS → NS-NS = 12 (cid:32) + (cid:112) − √ (cid:112) √ (cid:112) √ − (cid:112) − √ (cid:33) ˆ S NS-R → R-NS = ˆ S R-NS → NS-R = 1 √ (cid:18) +1 +1+1 − (cid:19) . (4.51) T -transformations work similarly. For example, they should map states in the NS-NSsector into the NS-R sector, which is indeed what happens: T |
0; NS-NS (cid:105) = 1 √ T | (cid:105) + T | (cid:105) )= e − πi/ √ | (cid:105) − | (cid:105) )= e − πi/ |
0; NS-R (cid:105) . (4.52)Acting with T on the rest of basis vectors, one confirms that T -transformations map T : ˆ H s a ,s b → ˆ H s a ,s a s b , through the following matrices:ˆ T NS-NS → NS-R = ˆ T NS-R → NS-NS = e − πi/ (cid:18) i (cid:19) ˆ T R-NS → R-NS = (cid:18) e πi/ (cid:19) . (4.53)One can easily check that all the expected properties of the modular group (i.e., (A.41))are satisfied, where ( − F = + , as all states are bosonic.The odd-spin-structure Hilbert space ˆ H R-R is much more interesting. The state |
0; R-R (cid:105) ∼| / (cid:105) − | / (cid:105) contains no punctures, so it is a boson, whereas the state |
1; R-R (cid:105) = | / ψ (cid:105) has a ψ -puncture, so it is a fermion. In other words, in the R-R sector the fermion parityoperator is non-trivial, ( − F = σ z . This makes the analysis of modular transformationsmore involved. In particular, these transformations should not mix these two states, andthey should not take us outside ˆ H R-R (as the s = (+1 , +1) spin structure is fixed by all ofthe MCG). These expectations are confirmed by direct computation. For example, acting on |
0; R-R (cid:105) with an S -transformation we get S |
0; R-R (cid:105) = 1 √ S | / (cid:105) − S | / (cid:105) )= 1 √ | / (cid:105) − | / (cid:105) )= |
0; R-R (cid:105) (4.54)which is indeed in ˆ H R-R (and has not mixed with the fermion, as it never could: modulartransformations do not mix configurations with different punctures).34he action of S on |
1; R-R (cid:105) = | / ψ (cid:105) is more subtle, because the state contains apuncture, so we need the S -matrix in the once-punctured torus. This matrix is givenby (A.18) S / , / ( ψ ) = (cid:88) j =0 θ j θ / S ,j F / , / (cid:20) ψ / / j (cid:21) = (cid:88) j =0 e πi (2 j ( j +1) − / ×
12 sin π j + 1) × ( − j = ( − / , (4.55)which means that, altogether, ˆ S R-R → R-R = (cid:18) − / (cid:19) . (4.56) T -transformations, on the other hand, do not care about the puncture, so they are justgiven by the spin of the states: ˆ T R-R → R-R = (cid:18) − / (cid:19) . (4.57)These two matrices are also easily seen to satisfy the expected modular properties, namelythey are unitary and obey the algebra of M p ( Z ), to wit, ˆ S = ( ˆ S ˆ T ) , ˆ S = ( − F . Wilson lines.
Given the choice of basis for the different Hilbert spaces as above, one canexpress the operators of the theory – the Wilson lines – as 2 × W ( b ) ( j ) | j (cid:48) (cid:105) = | j × j (cid:48) (cid:105) ≡ min( j + j (cid:48) , − j − j (cid:48) ) (cid:88) j (cid:48)(cid:48) = | j − j (cid:48) | | j (cid:48)(cid:48) (cid:105) , (4.58)which means that, for example, W ( b ) (1) |
0; NS-NS (cid:105) = 1 √ W ( b ) (1) | (cid:105) + W ( b ) (1) | (cid:105) )= 1 √ | (cid:105) + | (cid:105) ) , (4.59)which we identify as |
1; NS-NS (cid:105) . Similarly, W ( b ) (1) |
1; NS-NS (cid:105) = |
0; NS-NS (cid:105) + 2 |
1; NS-NS (cid:105) .The a -cycle computation is analogous: in the bosonic parent Wilson lines act as W ( a ) ( j ) | j (cid:48) (cid:105) = S j,j (cid:48) S ,j (cid:48) | j (cid:48) (cid:105) , (4.60)35hich implies that W ( a ) (1) |
0; NS-NS (cid:105) = 1 √ W ( a ) (1) | (cid:105) + W ( a ) (1) | (cid:105) )= 1 √ √ | (cid:105) + | (cid:105) ) , (4.61)which equals (1 + √ |
0; NS-NS (cid:105) . Repeating this calculation on all even-spin-structure Hilbertspaces, we obtain the following collection of matrices: W ( a )NS-NS (1) = (cid:18) √ − √ (cid:19) W ( a )NS-R (1) = (cid:18) √ − √ (cid:19) W ( a )R-NS (1) = (cid:18) − (cid:19) W ( b )NS-NS (1) = (cid:18) (cid:19) W ( b )NS-R (1) = (cid:18) (cid:19) W ( b )R-NS (1) = (cid:18) √ √ (cid:19) , (4.62)The same computation for the rest of NS-lines yields W ( c ) s a ,s b (3) = − s c for the transparentfermion, and W ( c ) s a ,s b (2) = W ( c ) s a ,s b (1) W ( c ) s a ,s b (3) (which is the expected relation given the fusionrule 2 = 1 ×
3, i.e., that the lines j = 1 , j = 3).One can also check that these matrices satisfy the algebra required by the fusion rule1 × W ( c ) s a ,s b (1) = + (1 − s c ) W ( c ) s a ,s b (1). Finally, it is also checkedthat, under modular transformations, these matrices are permuted as they should, e.g. S s a ,s b W ( a ) s a ,s b ( α )( S s a ,s b ) † = W ( b ) s b ,s a ( ¯ α ).We now move on to the odd-spin-structure sector, the R-R Hilbert space. The bosonicstate |
0; R-R (cid:105) ∼ | / (cid:105) − | / (cid:105) contains no punctures, so it behaves in the same manner asthe states in the even-spin-structure sector, for example W ( b ) (1) |
0; R-R (cid:105) = 1 √ W ( b ) (1) | / (cid:105) − W ( b ) (1) | / (cid:105) )= 1 √ | / (cid:105) + | / (cid:105) − | / (cid:105) − | / (cid:105) )= |
0; R-R (cid:105) . (4.63)The fermionic state |
1; R-R (cid:105) = | / ψ (cid:105) , on the other hand, requires using the data of theonce-punctured torus, cf. (A.11): W ( b ) (1) |
1; R-R (cid:105) = W ( b ) (1) | / ψ (cid:105) = F / , / (cid:20) / / (cid:21) | / ψ (cid:105) (4.64)which evaluates to −| / ψ (cid:105) . 36he a -cycle does not see the puncture (cf. (A.10)), and so its evaluation is straightforward.All in all, the Wilson lines in the R-R sector read W ( a )R-R (1) = W ( b )R-R (1) = (cid:18) − (cid:19) , (4.65)together with W ( c )R-R (3) = − for the transparent fermion, and W ( c )R-R (2) = − W ( c )R-R (1) (asexpected from the fusion rule 2 = 1 × S R-R W ( a )R-R ( α )( S R-R ) † = W ( b )R-R ( ¯ α ). Time-reversal.
Finally, we discuss the behaviour of the theory under time-reversal. Recallthat SO (3) is time-reversal symmetric thanks to the level-rank duality [65] SO (3) ←→ SO (3) − × SO (9) (4.66)A key aspect of this duality is that time-reversal is not really a symmetry of SO (3) , butrather a map T : SO (3) (cid:55)→ SO (3) × SO (9) − . The factor SO (9) is invertible, so we shouldthink of SO (3) being time-reversal invariant only if we mod out by SPTs. In the strict sense,it is not.In the U (1) k case this obstruction was easily circumvented: the duality U (1) k ↔ U (1) − k × SO (4) could be rewritten as U (1) k × U (1) − ↔ U (1) − k × U (1) +1 , i.e., we could break up theinvertible factor into two, and split them symmetrically into the two theories. In this situation,we would say time-reversal is not a symmetry of U (1) k , but rather of U (1) k × U (1) − : thistheory is identical to its conjugate, even taking into account SPTs.In the SO (3) case, no such solution is possible: the SPT SO (9) cannot be split into twoequal factors; such splitting would require fractional levels SO (9 / , which is not well-definedas a 3 d theory.An equivalent way to phrase this discussion is by thinking of the central charge – indeed,this number is what classifies 3 d -SPTs with no symmetry. Time-reversal always maps c into − c , which means a theory can only be time-reversal invariant, in the strict sense, if c = 0. If c (cid:54) = 0, we may be able to correct this by multiplying by a suitable SPT, but this is not alwayspossible. Indeed, the SPT SO ( n ) has c = n/
2, which means we can only correct the centralcharge in multiples of 1 /
2. In other words, the minimal SPT has c = 1 /
2, corresponding to asingle edge Majorana fermion. Any other SPT will consist of an integral number of copies ofthis system.In the U (1) k case, the central charge takes value c = 1, so this obstruction is avoidable: wejust have to tensor the theory with two copies of the Majorana fermion, i.e., SO (1) = U (1) .This makes the central charge of the product theory, U (1) k × U (1) − , vanish, making it avalid candidate for a time-reversal invariant theory. In the SO (3) case, c = 9 /
4, which isnot a multiple of 1 /
2, which means no redefinition can correct the central charge. The theory37s not, and cannot be made, time-reversal invariant in the strict sense. Only in the relativesense when we think of QFTs as absolute theories, modulo invertible ones.More generally, if a given theory A is known to be time-reversal invariant in the relativesense, then necessarily c ∝ /
4. Indeed, time-reversal maps A into ¯ A , modulo some SPT, andso A ↔ ¯ A × SO ( n ) for some n . The central charge of A therefore satisfies c ( A ) = − c ( A )+ n/ c ( A ) = n/
4, as claimed. If c = 0 mod 1 /
4, i.e., if c ∝ /
2, then the theorycan be made time-reversal invariant in the strict sense, by considering A × SO (2 c ( A )) − ,whose central charge vanishes. If c (cid:54) = 0 mod 1 /
4, this is not possible. In other words, c mod 1 / d theory. This is the ν layer.The discussion above is set up in the framework of spin TQFTs, but an analogous situationhappens in other families of theories. For example, the minimal bosonic SPT is ( E ) , whichhas c = 8, which means that bosonic time-reversal invariant theories always have c ∝
4, andthat c mod 4 is the first layer in the anomaly of time-reversal invariance.Going back to our example of SO (3) , let us see what we can say about time-reversal,neglecting the fact that it is not an operator acting on SO (3) , but rather a map from thistheory into SO (3) × SO (9) − . Time-reversal acts on the lines of SO (3) as 1 ↔ ≡ × ψ .If we ignore the SPT, this descends to the Hilbert space action τ ( W ( c ) s a ,s b (1)) ∗ τ − = W ( c ) s a ,s b (2)= − s c W ( c ) s a ,s b (1) (4.67)where T = τ K . The solution to this matrix equation is τ NS-NS ∝ τ NS-R ∝ σ z τ R-NS ∝ σ x (4.68)for the even spin structures, and τ R-R = (cid:18) z z (cid:19) (4.69)for the odd spin structure, where z , ∈ C are some arbitrary coefficients.Finally, recall that ( − F = for even spin structure, and ( − F = σ z for the oddspin structure. It is clear from these expressions that τ commutes with ( − F for even spinstructures, and anti-commutes for the odd spin structure { T , ( − F } = 0 , (4.70)precisely as expected from a system with ν odd and associated with the ψ layer. We havealso established that this behavior is present in other time-reversal invariant theories with ν odd. 38 cknowledgments We would like to thank Theo Johnson-Freyd, Nathan Seiberg, and Ryan Thorngren for usefuldiscussions. Research at Perimeter Institute is supported in part by the Government ofCanada through the Department of Innovation, Science and Economic Development Canadaand by the Province of Ontario through the Ministry of Colleges and Universities. Anyopinions, findings, and conclusions or recommendations expressed in this material are thoseof the authors and do not necessarily reflect the views of the funding agencies.39
Spin TQFTs and anyon condensation
In this section we outline the construction of TQFTs that depend on the spin structure ofthe underlying manifold. The strategy we will pursue is the following. Given one such theory,one may sum over all spin structures to yield a bosonic TQFT. This corresponds to gaugingthe zero-form symmetry generated by fermion parity Z = (cid:104) ( − F (cid:105) . This gauging generatesa dual Z ( d − d − and we reduce the problem of constructing spin TQFTs to the morefamiliar problem of gauging a higher-form symmetry in regular (bosonic) TQFTs. We shallfollow this strategy in d = 3 spacetime dimensions, where one can be quite explicit, thanksto the powerful formalism of modular tensor categories and two-dimensional chiral algebras.With this in mind, we begin by reviewing known facts about 3 d TQFTs, and the gaugingof one-form symmetries. From the 2 d point of view this corresponds to extending thechiral algebra by a simple current, and in the condensed-matter language to (abelian) anyoncondensation.Consider a 3 d bosonic TQFT. The most basic observable of the theory is the partitionfunction Z ( M ), where M is a compact 3-manifold. For example, if the manifold takes theform M = S × Σ, with S a circle representing the time direction, and Σ a compact surface,then the partition function computes the dimension of the Hilbert space assigned, by canonicalquantization, to the spatial slice: Z ( S × Σ) = dim( H (Σ)) . (A.1)The observables of the TQFT depend only on the topology of M , and therefore diffeomor-phisms of Σ must act unitarily in H (Σ). Transformations that are continuously connected tothe identity act trivially, so effectively we get a unitary representation of the mapping classgroup, the group of (equivalence classes of) large diffeomorphisms. If one understands theHilbert space H (Σ), and the action of the MCG on it, one can compute – via surgery – thepartition function on an arbitrary 3-manifold M .With this in mind, our main task is to understand the Hilbert space assigned by a TQFTto a compact Riemann surface Σ, and how Dehn twists act on it. The basic data of theTQFT that determines this information is the following: • The set of anyons A , a finite set. This set contains a distinguished anyon, the vacuum . • The modular matrix S : A × A → C . • The topological spin θ = e πih : A → U (1). For example, in d = 2 this corresponds to a zero-form symmetry. This has been studied recently, seee.g. [71, 81, 82]. In d = 4 one gauges a two-form symmetry, cf. e.g. [83]. One should keep in mind that,potentially, an anomaly could make these gaugings ill-defined, e.g. if summing over spin structures leads toan identically vanishing partition function. This subtlety shall play no role in this work. F - and R -symbols. These will play a role later on; for now, the S -matrix is enough. By a key resultof Verlinde, the dimension of H (Σ) is determined by S as follows [84, 85]:dim( H (Σ)) = (cid:88) α ∈A S χ (Σ) ,α , (A.2)where χ denotes the Euler characteristic ( χ (Σ g ) = 2 − g for a genus g surface Σ g ). Inparticular, the torus has χ (Σ ) = 0, which means that H (Σ ) ∼ = C [ A ], i.e., a basis of statesof the torus Hilbert space is labelled by the anyons of the TQFT. The MCG of the torus, SL ( Z ) = (cid:104) S, T (cid:105) , is generated by S and T := e − πic/ diag( θ ).The theory also admits line defect operators, also labelled by A . Namely, we can wrapan anyon α ∈ A around the time circle S , which produces a defect Hilbert space H (Σ α ).From the point of view of the spatial surface, the anyon α looks like a marked point. Given afamily of such punctures α , . . . , α n , the generalization of the Verlinde formula is [85]dim( H (Σ α ··· α n )) = (cid:88) α ∈A S χ (Σ) ,α n (cid:89) i =1 S α i ,α , (A.3)where χ (Σ α ··· α n g ) = 2 − g − n for a surface with g handles and n boundary components.The most fundamental surface is the so-called trinion , i.e., a sphere with three punctures.This surface defines the fusion coefficients : N α ,α ,α := dim( H (Σ α α α )) ≡ (cid:88) β ∈A S α ,β S α ,β S α ,β S ,β , (A.4)which endows A with a product structure, leading to the fusion ring of the TQFT. Thepartition function on an arbitrary surface can be computed by gluing trinions (cf. the “pantsdecomposition”). Using unitarity of S one can recover the general case (A.3) from thetrinion (A.4).An explicit basis of states on H (Σ α ··· α n g ) can be written as follows: α α n × × × · · · × , (A.5)where each cross × represents a handle of Σ g . Each segment carries an orientation and ananyon label (which we omit to simplify the notation). Each trivalent vertex with incominganyons α, β, γ carries an internal vector index, taking values in 1 , , . . . , N α,β,γ , which wealways leave implicit.In short, the states of H (Σ α ··· α n g ) can be represented as labelled oriented graphs with g cycles; leaves labelled by the punctures α , . . . , α n ; edges labelled by anyons α ∈ A ; and41rivalent vertices labelled by internal vector indices taking N α,β,γ values, as determined bythe incident edges α, β, γ .Different bases of the Hilbert space are related to the one above by the F - and R -moves,effected by the aforementioned F - and R -symbols. These are in correspondence with thedifferent pants decompositions of the surface.Of particular relevance is the case of the torus with a single puncture, whose states welabel as | β ; α (cid:105) ∈ H (Σ α ), corresponding to the configuration | β ; α (cid:105) = × βα . (A.6)There is one such state for each possible vertex, i.e., the degeneracy of | β ; α (cid:105) is given by thefusion coefficient N α,β, ¯ β . In particular, the diagram is allowed only if α × β ∝ β + · · · , i.e., if β may “absorb” the puncture α . The case of no punctures corresponds to the vacuum anyon α = , so that all β ∈ A are allowed, and they all carry degeneracy N ,β, ¯ β = 1. For non-trivial α , some β ∈ A may not contribute to H (Σ α ), and some other β ∈ A may contribute morethan one state.Given a basis of states for H (Σ ) one can write down the operators acting on this spaceas matrices. In particular, the Wilson loop operators admit such a representation. Let W ( c ) ( α ) denote the Wilson loop labelled by the anyon α ∈ A running through the cycle c ∈ H (Σ , Z ) = Z [ a ] ⊕ Z [ b ], where a , b are the standard homology cycles. These operatorsact by inserting α along the given cycle, e.g. W ( a ) ( α ) | β (cid:105) = × α β = S α,β S ,β × β W ( b ) ( α ) | β (cid:105) = × αβ = × α × β (A.7)whence (cid:104) β (cid:48) | W ( a ) ( α ) | β (cid:105) = δ ββ (cid:48) S αβ S ,β (cid:104) β (cid:48) | W ( b ) ( α ) | β (cid:105) = N β (cid:48) αβ (A.8)Naturally, given that S interchanges the cycles a and b (up to a sign), one has W ( a ) ( α ) = SW ( b ) ( α ) S † , W ( b ) ( α ) = SW ( a ) ( ¯ α ) S † (A.9)which is just the statement that the characters S αβ /S ,β diagonalize the fusion rules.The higher-genus case is analogous. Given a basis of H (Σ α ...α n g ) one can express theWilson lines as matrices. As above, a Wilson loop W ( c ) ( α ) inserts the anyon α along the42ycle c ∈ H (Σ α ...α n g , Z ). For example, the a -cycles are identical to the torus, inasmuch aswrapping an anyon in the orthogonal cycle is a local operation: one can shrink it to a point.The value of W ( a i ) acting on a given state only depends on the line running through thesegment orthogonal to a i , irrespective of what the rest of the state is doing: α β = S α,β S ,β β (A.10)The b -cycles, on the other hand, cannot be shrunk, and so depend on the entire configu-ration around such cycle: the lines running therein, and the punctures going in and out. Forexample, the once-punctured torus has [79] W ( b ) ( α ) | β ; γ (cid:105) = × αγ β = F β,α × β (cid:20) γ α × ββ α (cid:21) × α × βγ . (A.11)Configurations with more punctures carry more factors of F . The matrix elements of arbitraryconfigurations of Wilson lines, on surfaces with arbitrary genus and arbitrary punctures, isentirely determined in terms of the TQFT data of the theory. Generalizing (A.9), the linesaround the different cycles are unitarily related through the MCG of the surface.The invertible defects – the abelian punctures – correspond to group symmetries of thetheory. These are line operators, so the symmetry is a higher-form symmetry, in this casea one-form symmetry. Gauging the symmetry corresponds to summing over all possibleinsertions of the defect. This produces a new TQFT, whose set of anyons ˆ A and modulardata ˆ S are fixed in terms of the data of the ungauged theory. Making this procedure preciseis the goal of the rest of this appendix.The one-form symmetry group is always a finite abelian group, i.e., a product of cyclicgroups. Each abelian anyon in A generates a cyclic subgroup; condensing this anyon meansgauging this subgroup. If we are interested in gauging a product of cyclic groups, we canalways condense a single generator at a time, iteratively. We can therefore assume withoutloss of generality that the group to be gauged is cyclic, say, Z (1) n = (cid:104) g (cid:105) , with g ∈ A a certainabelian anyon. The Z (1) n symmetry partitions the spectrum A into n equivalence classes,according to their braiding with respect to g : A = n − (cid:71) q =0 A q , A q := { α ∈ A | B ( g , α ) = e πiq/n } , (A.12)where B ( g , α ) := S g ,α /S ,α ≡ θ ( g × α ) /θ ( g ) θ ( α ) is the braiding phase with respect to g . Themodular data of the theory behaves properly with respect to this grading, e.g. [86] S g i × α, g j × β = B ( g , g ) ij B ( g , α ) j B ( g , β ) i S α,β (A.13)43he ’t Hooft anomaly of Z (1) n is given by B ( g , g ), which must equal 1 if the symmetry isto be gauged. In this situation, one can prove that θ ( g ) = ±
1, i.e., the generator is either aboson or a fermion. In the former case, the gauging yields another bosonic TQFT, while inthe latter case the theory acquires a dependence on the spin structure, i.e., it becomes a spinTQFT. For now, we assume that g is a boson, and return to the more interesting case offermionic quotients later on. A.1 Boson anyon condensation
We begin with some bosonic TQFT with anyons A and modular matrix S , and wish tocondense some boson g ∈ A , to produce some other bosonic TQFT, with anyons ˆ A andnew modular matrix ˆ S . The standard lore of this procedure is as follows. First, in order toconstruct ˆ A , one performs the following three steps [87]:1. Select the set of neutral lines, A (cf. (A.12)), i.e., those with trivial braiding withrespect to g .2. Identify any two lines that are in the same Z (1) n -orbit, i.e., if they differ by the action of g j for some j ∈ Z n .3. If a given Z (1) n -orbit has less than n elements, it splits into several different anyons in ˆ A .Specifically, if the length is (cid:96) , then the orbit descends to n/(cid:96) copies in the condensedtheory.In what follows we shall describe the geometric interpretation of these rules, which willallow us to compute the modular data of the condensed theory from first principles, withno need to introduce any ans¨atze. It also admits a natural extension to spin TQFTs whichshines a new light – and goes beyond – what is currently understood about such theories.The main idea to obtain ˆ A is to find the torus Hilbert space of the condensed theory,from which one can read off the set of anyons by writing down a basis of vectors (recall that H (Σ ) ∼ = C [ A ]). The condensed theory is obtained by gauging the Z n one-form symmetry,which means we are to insert the generator g in all possible ways. Summing over all insertions g j , j = 0 , , . . . , n −
1, along the spatial cycles project the Hilbert space into the invariantstates. Insertions along the temporal cycles introduce twisted sectors. We will next see thatinsertions along the three cycles in M = S × Σ indeed reproduce the three steps above.Let us begin with the time circle. Inserting g j along the time direction means takingthe torus with a puncture labelled by g j . Therefore, the states of the condensed theoryare generically of the form | α ; g j (cid:105) for some α ∈ A . In other words, the Hilbert space ofthe condensed theory must be a subspace of the Hilbert space of the original theory, in thepresence of an arbitrary puncture: ˆ H (Σ ) ⊆ n − (cid:77) j =0 H (Σ g j ) . (A.14)44he states of H (Σ g j ) are labelled by anyons α ∈ A with the property g j × α = α . Inparticular, j must be proportional to the length of the Z (1) n -orbit of α . We shall denote thisorbit by [ α ], and its length (cid:96) α := | [ α ] | equals the minimal integer such that g (cid:96) α × α = α . Thisinteger divides n , and any other integer j with g j × α = α is of the form j = (cid:96) α k , for someinteger k = 0 , , . . . , n/(cid:96) α −
1. This reproduces the third condition above, namely if a givenorbit is shorter than (cid:96) α = n , it descends to n/(cid:96) α copies in the condensed theory. The copiesjust label the number of insertions of the symmetry defect we use to create the state.Let us now move on to the spatial circles; insertions of the symmetry elements alongthese circles shall project into the invariant subspace. In this case, the meaning of invariant depends on which cycle we insert the symmetry element on; a symmetry along the a -cycleacts via braiding, and along the b -cycle via fusion. In the end, we must have states that areinvariant under g , both with respect to braiding and to fusion. Let us discuss these two casesin turn. • Take first the a -cycle, which is the circle that is orthogonal to the one we use to createstates. Given a state created by a line α ∈ A running along the b -cycle, the configurationwe obtain by inserting g is B ( α, g ) | α ; g (cid:96) α k (cid:105) (cf. (A.10)). The phase B ( α, g ) equals e πiq/n for α ∈ A q (cf. (A.12)). Summing over all insertions g j produces the phase n − (cid:88) j =0 B ( α, g j ) = n − (cid:88) j =1 e πiqj/n = nδ q, (A.15)which indeed projects to the states with q = 0, i.e., to α ∈ A . We thus reproduce thefirst condition, namely the states in the condensed theory must be neutral under Z (1) n ,i.e., taken from A . • Finally, if we now consider the second spatial circle, the b -cycle, and insert g j , we obtain | g j × α ; g (cid:96) α k (cid:105) . Summing over all j (and normalizing to have unit norm) leads to | [ α ] , k (cid:105) := 1 √ (cid:96) α (cid:96) α − (cid:88) j =0 | g j × α ; g k(cid:96) α (cid:105) , [ α ] ∈ A / ∼ , k ∈ Z n/(cid:96) α . (A.16)which is indeed invariant under g , where now this symmetry acts via fusion (i.e., α (cid:55)→ g × α ). We thus reproduce the second condition, namely the fact that the anyonsof the condensed theory ˆ A are labelled by Z (1) n orbits.We thus see that, as expected, the insertions along the three circles indeed reproduce thethree conditions we are used to. All in all, a basis of states is labelled by a pair of indices:[ α ], denoting a Z (1) n orbit of neutral lines α ∈ A , plus a degeneracy label taking values in k = 0 , , . . . , n/(cid:96) α . This degeneracy label, arguably the most subtle ingredient so far, is infact quite natural from the point of view of gauging Z (1) n : k(cid:96) α just denotes how many copies45f the g -puncture we insert in order to create the state, i.e., from which twisted Hilbert spacethe state comes from.The presentation of ˆ H (Σ ) above also gives us a natural way to compute the modulardata of the condensed theory, in particular, the modular matrix ˆ S . Specifically, a modulartransformation acting on a state | [ α ]; k (cid:105) is nothing but the S -matrix of the uncondensedtheory, in the presence of a puncture g k(cid:96) α : (cid:104) [ α ]; k | ˆ S | [ α (cid:48) ]; k (cid:48) (cid:105) = δ k(cid:96) α ,k (cid:48) (cid:96) α (cid:48) (cid:112) (cid:96) α (cid:96) α (cid:48) S α,α (cid:48) ( g k(cid:96) α ) . (A.17)The modular matrix in the once-punctured torus can be expressed in terms of the F -symbols of the parent theory, namely [79] S α,α (cid:48) ( g j ) = (cid:88) β ∈ α × α (cid:48) θ ( β ) θ ( α ) θ ( α (cid:48) ) S ,β F α,α (cid:48) (cid:20) g j α (cid:48) α β (cid:21) . (A.18)The basis (A.16) of ˆ H (Σ ) is natural because it makes ˆ S block-diagonal, but it doesnot correspond to the anyon basis. The most obvious way to see this is that the would-bequantum dimension d [ α ]; k = ˆ S [ α ]; k, [ ];0 / ˆ S [ ];0 , [ ];0 vanishes for k (cid:54) = 0.In order to identify the anyon basis we can look at the dual Z (0) n symmetry. The chargedstates are those with the puncture. More precisely, in the diagonal basis the states transformas Z (0) n : | [ α ]; k (cid:105) (cid:55)→ e πik/(cid:96) α | [ α ]; k (cid:105) . In the anyon basis, this symmetry should act as a cyclicpermutation of the anyons, that is, as Z (0) n : | [ α ]; ˆ k (cid:105) (cid:55)→ | [ α ]; ˆ k + 1 (cid:105) for some label ˆ k . Weconclude that the anyon basis is in fact nothing but the Fourier transform (Pontryagin dual)of the diagonal basis (A.16): | [ α ] , ˆ k (cid:105) : = 1 (cid:112) n/(cid:96) α n/(cid:96) α − (cid:88) k =0 e πi ˆ kk(cid:96) α /n | [ α ]; k (cid:105) = 1 √ n n/(cid:96) α − (cid:88) k =0 (cid:96) α − (cid:88) j =0 e πi ˆ kk(cid:96) α /n × g j × α g k(cid:96) α , (A.19)where ˆ k ∈ Z ∗ n/(cid:96) α . In this basis, the quantum dimension takes the expected value d [ α ];ˆ k =( (cid:96) α /n ) S α, /S , = ( (cid:96) α /n ) d α . The anyons of the quotient ˆ A create the states | [ α ] , ˆ k (cid:105) by actingon the vacuum.Given the matrix ˆ S in the fusion basis, one can use the Verlinde formula to compute thedimension of the Hilbert space of the condensed theory, for an arbitrary Riemann surface,with an arbitrary number of punctures. In the particular case of no external punctures,the formula only involves matrix elements with the vacuum, in which case the matrix withpunctures S ( g j ) does not contribute except for the vacuum insertion, that is, the regular S S matrix of the parent theory, without the need to know the F -symbols:dim( ˆ H (Σ g )) = (cid:88) α ∈ ˆ A ˆ S χ (Σ) ,α ≡ (cid:88) α ∈A n(cid:96) gα S χ (Σ) ,α . (A.20)It is possible to generalize the expressions above to the case where there is a non-trivialbackground flux for the Z (0) n magnetic symmetry dual to the gauged Z (1) n symmetry. Forexample, if the flux of such background is q ∈ Z n , then the states are created from the subset A q instead of A . Summing over all such backgrounds, i.e., gauging the Z (0) n symmetry, takesus back to the original ungauged theory A . We shall not need this generalization in thispaper. A.2 Fermion anyon condensation
We now move on to the more interesting case of fermion condensation: we have some bosonicTQFT, and we wish to condense a certain abelian fermion, which we denote by ψ ∈ A . Wecan assume without loss of generality that this line generates a Z (1)2 symmetry, i.e. ψ = ,for otherwise we can first condense the boson g = ψ (as in the previous section), and thencondense the resulting fermion, which will satisfy ψ = .The philosophy underlying fermion condensation is essentially the same as in bosoncondensation: we construct the Hilbert space of the condensed theory from the states in theparent theory, perhaps in presence of ψ -punctures. Roughly speaking, the configurationswith non-trivial background flux can be thought of as the different spin structures on Σ.Before actually constructing the spin TQFT by condensing a fermion in a bosonic TQFT,let us discuss what we are to expect from this condensation in the first place. A spin TQFTshould assign to manifolds of the form S × Σ a super-vector space ˆ H (Σ), which depends onlyon the topology of Σ, together with its spin structure s . (We use a hat to denote the Hilbertspace of the condensed theory, and reserve the notation H (Σ) for that of the bosonic parent).Depending on the spin structure on the time circle S , the partition function computes eitherthe regular trace over ˆ H (Σ), or the super-trace (i.e., the trace weighted by fermion parity).Specifically, the spin generalization of (A.1) is Z ( S × Σ) = tr ˆ H (Σ) (id) Z ( S × Σ) = tr ˆ H (Σ) ( − F (A.21)where S denotes the circle with anti-periodic boundary conditions, and S the circle withperiodic boundary conditions. Therefore, if the super-vector space ˆ H (Σ) is C b | f , then Neveu-47chwarz boundary conditions compute b + f , and Ramond boundary conditions compute b − f .We shall denote a compact surface with genus g and spin structure s by Σ g ; s . As in thebosonic case, large diffeomorphisms act unitarily in ˆ H (Σ g ; s ). The MCG as a spin surface isa subgroup of the MCG as a surface, MCG(Σ g ; s ) ⊆ MCG(Σ g ). The reason for this is thatsome diffeomorphisms that leave Σ invariant as a topological space, actually change the spinstructure s (cid:55)→ s (cid:48) , and so do not constitute elements of the MCG as a spin surface. Thecanonical example is the T -transformation on the torus, which performs a Dehn twist aroundthe a -cycle. As such, it maps ( s a , s b ) (cid:55)→ ( s a , s a s b ). This is an element of the spin MCG if s a = +1, but it is not if s a = −
1. On the other hand, T is in the spin MCG for any spinstructure.Elements of the spin MCG act unitarily in the Hilbert space, namely,MCG(Σ g ; s ) : ˆ H (Σ g ; s ) → ˆ H (Σ g ; s ) . (A.22)On the other hand, elements of the regular MCG induce isomorphisms of (generically distinct)super-vector spaces, MCG(Σ g ) : ˆ H (Σ g ; s ) → ˆ H (Σ g ; s (cid:48) ) . (A.23)This means, for example, that the partition function Z ( S × Σ g ; s ) is invariant under MCG(Σ g );and, more generally, observables only depend on the equivalence class of s under the regularMCG. It is known that there are only two equivalence classes of spin structures modulo MCG,the so-called even and odd spin structures. These are distinguished by the Arf invariant [88].If two spin structures have the same Arf parity, then there exists some MCG element thatmaps one into the other. If they have different Arf parity, no such MCG element exists. Inconclusion, observables of spin TQFTs depend on s only through Arf( s ).For fixed spin structure, MCG(Σ g ; s ) is represented by a unitary operator in ˆ H (Σ g ; s ). Thatbeing said, due to the Z grading of this vector space, this action typically gets extended.Namely, the Hilbert space of spin TQFTs realize a unitary representation of a certain non-trivial Z extension of the spin MCG. To be explicit, (modding out by Torelli, i.e., workingin homology) the MCG of Σ g is the integral symplectic group Sp g ( Z ), and the spin MCGis some subgroup thereof. The Hilbert space of the theory realizes a unitary representationof the so-called metaplectic group M p g ( Z ), which is defined as the (essentially unique) Z extension of the symplectic group Z (cid:44) → M p g ( Z ) (cid:16) Sp g ( Z ) . (A.24)This extension corresponds to the fact that a 2 π rotation is represented by the trivial elementin MCG(Σ g ; s ), while it lifts to ( − F in ˆ H (Σ g ; s ).In order to illustrate these ideas it proves useful to focus on the torus, Σ . There are four48pin tori, depending on the boundary conditions on the two spatial circles Σ s a ,s b = S s a × S s b :Arf( s a , s b ) = +1 : S × S S × S S × S Arf( s a , s b ) = − S × S (A.25)The MCG of the torus is the modular group SL ( Z ) = (cid:104) ˆ S, ˆ T (cid:105) , acting asˆ S : Σ s a ,s b → Σ s b ,s a ˆ T : Σ s a ,s b → Σ s a ,s a s b (A.26)Therefore, the subgroup that fixes each spin structure isMCG(Σ −− ) = (cid:104) ˆ S, ˆ T (cid:105) MCG(Σ − + ) = (cid:104) ˆ S ˆ T ˆ S, ˆ T (cid:105) MCG(Σ − ) = (cid:104) ˆ S ˆ T ˆ S, ˆ T (cid:105) MCG(Σ ) = (cid:104) ˆ S, ˆ T (cid:105) (A.27)Needless to say, the first three groups are all isomorphic, as they are related through SL ( Z )conjugation: (cid:104) ˆ S ˆ T ˆ S, ˆ T (cid:105) = ˆ T (cid:104) ˆ S, ˆ T (cid:105) ˆ T − , (cid:104) ˆ S ˆ T ˆ S, ˆ T (cid:105) = ( ˆ S ˆ T ) (cid:104) ˆ S, ˆ T (cid:105) ( ˆ S ˆ T ) − . (A.28)This group is a congruence subgroup of SL ( Z ) of index 3, usually denoted by Γ (2). Thefourth group, on the other hand, is SL ( Z ) itself.The diffeomorphism ˆ S corresponds to a 2 π rotation and, as such, acts trivially in abosonic theory and so is represented by the identity element in SL ( Z ); conversely, in afermionic theory it is represented by ( − F . Thus, modular transformations in spin theoriessatisfy ˆ S = ( ˆ S ˆ T ) , ˆ S = ( − F , (A.29)with ( − F an order 2 central element. These relations define the group M p ( Z ). The Hilbert spaces ˆ H (Σ g ; s ) are best understood by giving an explicit basis for them. Asin the previous section, the Hilbert space of the fermionic theory can be constructed by One should keep in mind that
M p ( Z ) does not act faithfully in ˆ H (Σ s ) (in fact, the metaplectic groupis not a matrix group; it does not admit faithful finite-dimensional representations). This fact is most drasticwhen the theory, for whatever reason, has no fermionic states at all: in such cases, the actions of Sp ( Z ) and M p ( Z ) are indistinguishable, inasmuch as ( − F is trivial. For example, the theory lacks fermionic states if s is an even spin structure, or if the theory is secretly bosonic (through some non-trivial duality). In suchcases, one can think of the modular group as being SL ( Z ) instead of M p ( Z ): their difference is invisible inthe Hilbert space anyway. Similar considerations hold in higher genus. ψ -puncture, ˆ H (Σ g ; s ) ⊆ H (Σ g ) ⊕ H (Σ ψg ) . (A.30)Let us write down a basis for ˆ H (Σ s ) in terms of the states of the bosonic parent.Recall that the states on the torus in the bosonic theory are labelled by the anyons A .The Z (1)2 symmetry generated by ψ partitions the spectrum A into two equivalence classes,distinguished by the braiding phase B ( ψ, · ) = ±
1. In the case of boson condensation wedenoted these two equivalence classes by A and A ; in the present context it is more naturalto denote them by A NS and A R : A = A NS (cid:116) A R , (cid:40) A NS := { α ∈ A | B ( α, ψ ) = +1 }A R := { α ∈ A | B ( α, ψ ) = − } (A.31)The two equivalence classes are further partitioned according to the length of the orbits.For a generic Z (1) n symmetry, the orbits come in lengths that divide n ; for n = 2, we havetwo-dimensional orbits and one-dimensional ones. We refer to the latter as Majorana lines .It is easy to convince oneself that these can only appear in A R . We shall use the label α to denote generic lines of A ; on the other hand, lines of A NS will be denoted by the morespecific label a , while two-dimensional orbits of A R by x and one-dimensional ones by m . Wewill say that α is a -type, x -type, or m -type, according to this classification: a ∈ A NS x ∈ A R & | x | = 2 m ∈ A R & | m | = 1 . (A.32)In other words, x -lines satisfy x × ψ (cid:54) = x , while m -lines satisfy m × ψ = m .As in the previous section, we take the bosonic theory, and condense a fermion ψ . Inthe condensed theory, ψ becomes almost trivial: it should be represented by the identityoperator, up to a sign, depending on the spin structure around the cycle it is supported on.This determines how the states in the condensed phase are obtained in terms of those of the50ncondensed one. We claim that the basis of ˆ H (Σ s ) can be taken asˆ H (Σ −− ) : √ (cid:18) a + (cid:19) a × ψ ˆ H (Σ − + ) : √ (cid:18) a − (cid:19) a × ψ ˆ H (Σ − ) : √ (cid:18) x + (cid:19) x × ψm ˆ H (Σ ) : √ (cid:18) x − (cid:19) x × ψmψ (A.33)The reasoning behind the construction of this basis is the same as in the case of bosoncondensation. Namely, the gauged theory is obtained by inserting ψ j in all possible ways.Here ψ is of order two, so there are only two possible blocks: j = 0 or j = 1, i.e., noinsertion, or a single ψ -insertion. Furthermore, the specific linear combination of states isdecided by the spin structure. For example, inserting ψ along the a -cycle inserts the phase | α ; ψ j (cid:105) (cid:55)→ B ( α, ψ ) | α ; ψ j (cid:105) . This should reproduce the sign s a , which means that A NS linescreate states in s a = − A R lines create states in s a = +1 boundaryconditions. This explains why the basis is constructed using a -type lines in ˆ H (Σ − , • ), and x -and m -type lines in ˆ H (Σ , • ).Similarly, inserting ψ along the b -cycle fuses the state into | α ; ψ j (cid:105) (cid:55)→ ( − j | ψ × α ; ψ j (cid:105) .If this is to reproduce the boundary condition s b , we are required to consider the linearcombination | α (cid:105) − s b | ψ × α (cid:105) for two-dimensional orbits; and, for Majorana lines, the punctureshould be present if and only if s b = +1.Note that, unlike in the case of bosonic condensation, here the multiple copies associatedto the short orbits live in different spaces. Moreover, there are no short orbits in the NSsector, so the observables of the theories (the Wilson lines associated to the NS anyons) donot require fixed-point resolution. In this sense, the fusion rules of the condensed theory areinherited from those of the parent in a straightforward manner, without the need of knowingthe once-punctured S -matrix. In the bosonic case, the fusion rules of the short orbits dorequire this extra structure.As a consistency check for the basis above, we can easily show that modular transformationsmap the different Hilbert spaces as expected. For example, take ˆ H (Σ −− ), and apply an51 -transformation: 1 √ | a (cid:105) + | ψ × a (cid:105) ) S (cid:55)→ (cid:88) α (cid:48) ∈A √ S a,α (cid:48) + S ψ × a,α (cid:48) ) | α (cid:48) (cid:105) (A.34)As in equation (A.13), we have S ψ × a,α (cid:48) = B ( ψ, α (cid:48) ) S a,α (cid:48) , which means that we may restrictthe sum over α (cid:48) to a -type lines, for the Ramond ones do not contribute – they cancel outpairwise. With this,1 √ | a (cid:105) + | ψ × a (cid:105) ) S (cid:55)→ (cid:88) a (cid:48) ∈A NS / ∼ S a,a (cid:48) √ | a (cid:48) (cid:105) + | ψ × a (cid:48) (cid:105) ) (A.35)which shows that S maps ˆ H (Σ −− ) to itself, as expected (cf. (A.26)). Similarly, T -transformations map1 √ | a (cid:105) + | ψ × a (cid:105) ) T (cid:55)→ e − πic/ √ θ ( a ) | a (cid:105) + θ ( ψ × a ) | ψ × a (cid:105) ) (A.36)Noting that θ ( ψ × a ) = − θ ( a ), this becomes1 √ | a (cid:105) + | ψ × a (cid:105) ) T (cid:55)→ e − πic/ θ ( a ) 1 √ | a (cid:105) − | ψ × a (cid:105) ) (A.37)which shows that T maps ˆ H (Σ −− ) into ˆ H (Σ − + ), again as expected (cf. (A.26)).The other three Hilbert spaces can also be seen to transform into each other in theexpected manner. Not only that, but the exercise gives us the explicit expression for the ˆ S and ˆ T matrices of the condensed theory:ˆ S : ˆ H (Σ −− ) → ˆ H (Σ −− ) = ⇒ (cid:110) ˆ S a,a (cid:48) = 2 S a,a (cid:48) ˆ S : ˆ H (Σ − + ) → ˆ H (Σ − ) = ⇒ (cid:40) ˆ S a,x = 2 S a,x ˆ S a,m = √ S a,m ˆ S : ˆ H (Σ − ) → ˆ H (Σ − + ) = ⇒ (cid:40) ˆ S x,a = 2 S x,a ˆ S m,a = √ S m,a ˆ S : ˆ H (Σ ) → ˆ H (Σ ) = ⇒ ˆ S x,x (cid:48) = 2 S x,x (cid:48) ˆ S x,m = 0ˆ S m,m (cid:48) = S m,m (cid:48) ( ψ ) (A.38)where S m,m (cid:48) ( ψ ) denotes the S -matrix of the bosonic parent, in the once-punctured torus52cf. (A.18)). The ˆ T -matrix is given by a similar expression:ˆ T : ˆ H (Σ −− ) → ˆ H (Σ − + ) = ⇒ (cid:110) ˆ T a,a (cid:48) = e − πic/ θ ( a )( δ a,a (cid:48) − δ a,ψ × a (cid:48) )ˆ T : ˆ H (Σ − + ) → ˆ H (Σ −− ) = ⇒ (cid:110) ˆ T a,a (cid:48) = e − πic/ θ ( a )( δ a,a (cid:48) + δ a,ψ × a (cid:48) )ˆ T : ˆ H (Σ − ) → ˆ H (Σ − ) = ⇒ ˆ T x,x (cid:48) = e − πic/ θ ( a )( δ x,x (cid:48) + δ x,ψ × x (cid:48) )ˆ T x,m = 0ˆ T m,m (cid:48) = e − πic/ θ ( m ) δ m,m (cid:48) ˆ T : ˆ H (Σ ) → ˆ H (Σ ) = ⇒ ˆ T x,x (cid:48) = e − πic/ θ ( a )( δ x,x (cid:48) − δ x,ψ × x (cid:48) )ˆ T x,m = 0ˆ T m,m (cid:48) = e − πic/ θ ( m ) δ m,m (cid:48) (A.39)Finally, we discuss the third generator of the spin modular group, fermion parity. Thiszero-form symmetry is the dual symmetry to the gauged Z (1)2 , which means that the stateswith odd fermion parity are those that carry the puncture. This means that ( − F = 1 in alleven spin structures, while in the R-R sector one has(( − F ) x,x (cid:48) = + δ x,x (cid:48) (( − F ) m,m (cid:48) = − δ m,m (cid:48) (A.40)These matrices are unitary, symmetric, and satisfy ˆ S = ( − F and ˆ S = ( ˆ S ˆ T ) . It isimportant to remark that these properties are understood in the Z -graded sense, i.e., takinginto account (A.26). In other words, the precise relations areˆ S † s a ,s b = ˆ S − s a ,s b ˆ T † s a ,s b = ˆ T − s a ,s b ˆ S ts b ,s a = ˆ S s a ,s b ( ˆ S s b ,s a ˆ S s a ,s b ) = ( − Fs a ,s b ˆ S s b ,s a ˆ S s a ,s b = ˆ S s b ,s a ˆ T s b ,s a s b ˆ S s a s b ,s b ˆ T s a s b ,s a ˆ S s a ,s a s b ˆ T s a ,s b , (A.41)where O s a ,s b denotes the operator O ∈ { ˆ S, ˆ T , ( − F } when acting on ˆ H (Σ s a ,s b ).In the appendix B we construct several examples of quotient TQFTs, namely SO ( N ) k with k = 1 , ,
3. Some of these illustrate bosonic anyon condensation, and some othersfermionic anyon condensation.
B Examples of anyon condensation
Here we collect some extra examples of boson and fermion condensation, using theories ofthe form SO ( n ) k for small values of k . In particular, k = 1 and k = 3 exemplify fermion53ondensation, and k = 2 boson condensation. The case k = 1, i.e., SO ( n ) , is the generatorof fermionic SPTs with no symmetry, and so is a key theory in the study of fermionicTQFTs. The case k = 2 will be related to U (1) theories, through the level-rank duality SO ( n ) ↔ SO (2) − n ≡ U (1) − n . Finally, the case k = 3 will be constructed through the SU (2)theory, thanks to the level-rank duality SO ( n ) ↔ SO (3) − n ≡ SU (2) − n / Z . We also includethe case of U (1) k separately, this time focusing on its time-reversal invariance. B.1 SO ( n ) This is the minimal spin TQFT, and it has central charge n/
2, so corresponds to n boundaryMajorana fermions. A single fermion, SO (1) , is the generator of the group of fermionicSPTs with no extra symmetries, Ω = Z . In other words, any invertible fermionic phase isequivalent to SO ( n ) for some n . The theory can also be written as n copies of (the inverseof) the gravitational Chern-Simons theory.The bosonic parent of this theory is Spin ( n ) . The details of this theory depend on theparity of n . n = 2 m + 1. One can construct SO ( n ) by condensing the fermion in the Ising category.The modular data for the parent theory is that of Ising m = Spin (2 m + 1) , which has threeanyons: = [1 , , , . . . , , σ = [0 , , , . . . , , ψ = [0 , , , . . . , ,
0] (B.1)where [ λ , λ , . . . , λ m ] denote the extended Dynkin labels of the representation. These linesfuse according to ψ × σ = σ and σ = + ψ , and transform under modular transformationsas follows: S | (cid:105) = 12 | (cid:105) + 1 √ | σ (cid:105) + 12 | ψ (cid:105) S | σ (cid:105) = 1 √ | (cid:105) − √ | ψ (cid:105) S | ψ (cid:105) = 12 | (cid:105) − √ | σ (cid:105) + 12 | ψ (cid:105) T | (cid:105) = e πi (cid:0) − n (cid:1) | (cid:105) T | σ (cid:105) = e πi (cid:0) n − n (cid:1) | σ (cid:105) T | ψ (cid:105) = e πi (cid:0) − n (cid:1) | ψ (cid:105) (B.2)The lines are partitioned according to their braiding with respect to ψ asNS : A NS = { , ψ } R : A R = { σ } (B.3)and they are paired-up under fusion as × ψ ←→ ψ, σ (cid:120) × ψ . (B.4)54herefore, the four Hilbert spaces of the theory are • If we take
NS-NS boundary conditions, the state is |
0; NS-NS (cid:105) = 1 √ | (cid:105) + | ψ (cid:105) ) . (B.5) • If we take
NS-R boundary conditions, the state is |
0; NS-R (cid:105) = 1 √ | (cid:105) − | ψ (cid:105) ) . (B.6) • If we take
R-NS boundary conditions, the state is |
0; R-NS (cid:105) = | σ (cid:105) . (B.7) • If we take
R-R boundary conditions, the state is |
0; R-R (cid:105) = | σ ; ψ (cid:105) , (B.8)where, we remind the reader, | α ; β (cid:105) denotes the anyon α in presence of a β puncture(cf. (A.6)).We see that these spaces are all one-dimensional, as expected from an invertible theory.Furthermore, all states are bosonic, except for the one with a puncture, | σ ; ψ (cid:105) , which meansthat ( − F = ( − Arf( s ) .The modular data of the quotient can be computed in a straightforward manner. The onlynon-trivial case is the S matrix in the R-R sector, which has a puncture. We can computethis matrix element using the general formula (A.18), namely S σ,σ ( ψ ) = (cid:88) α = ,ψ θ ( α ) θ ( σ ) S ,α F σ,σ (cid:20) ψ σσ α (cid:21) (B.9)The F -symbols of the Ising category are well-known, cf. F ( α = ) ≡ +1 and F ( α = ψ ) ≡ − S R-R = 12 ( − / ( − i ) m ( F ( ψ ) − F ( )) ≡ e iπ (6 m +7) / (B.10)This result, together with ˆ T R-R = e πi (2 m +1) / , confirms that the theory satisfies theexpected modularity relations, S = ( ST ) and S = ( − F . n = 2 m . Here the bosonic parent
Spin (2 m ) has four lines, the trivial representation,the vector representation, and the two spinor representations. The quotient is obtained bycondensing the vector. The Lie algebra is simply-laced, which automatically implies that thefusion rules are abelian, and so there are no fixed-points under fusion. Therefore, all states55re bosonic. The four lines are split into two NS-lines (the trivial and the vector) and twoR-lines (the two spinors), and each pair belongs to a two-dimensional orbit. This means thateach Hilbert space is one-dimensional, as expected from an invertible theory, and moreoverall states have ( − F = +1. The modular data is trivially computed, given that there are noshort orbits. B.2 SO ( n ) Here we illustrate the construction of the bosonic theory SO ( n ) , by condensing an abelianboson in Spin ( n ) . We focus in particular on the odd- n case, where all the modular data –especially the F -symbols – is fully known [89]. We follow the notation therein.Consider the algebra so n +1 = B n . Its comarks are a ∨ = 1 , , , . . . , ,
1, which means thatthe theory
Spin (2 n + 1) has n + 4 lines. We denote them as , (cid:15), φ i , ψ ± , with i = 1 , , . . . , n .The corresponding affine Dynkin labels are as follows: = [2 , , , . . . , (cid:15) = [0 , , , . . . , φ = [1 , , , . . . , φ i = [0 , . . . , , , , . . . ,
0] at position i + 1 φ n = [0 , , . . . , , ψ + = [1 , , , . . . , , ψ − = [0 , , , . . . , ,
1] (B.11)The S -matrix reads S , = S ,(cid:15) = S (cid:15),(cid:15) = 12 √ n + 1 S ,ψ ± = + 12 S (cid:15),ψ ± = − S ψ s ,ψ s (cid:48) = 12 ss (cid:48) S ,φ i = S (cid:15),φ i = 1 √ n + 1 S ψ ± ,φ i = 0 S φ i ,φ j = 2 √ n + 1 cos 2 πij n + 1 (B.12)56nd the spins are h = 0 h (cid:15) = 1 h φ i = 12 i (2 n + 1 − i )2 n + 1 h ψ + = 18 nh ψ − = 18 n + 12 (B.13)From this one derives the the quantum dimensions d = 1 d (cid:15) = 1 d φ i = 2 d ψ ± = √ n + 1 (B.14)and fusion rules, (cid:15) × (cid:15) = , ψ ± × ψ ± = + n (cid:88) j =1 φ j , ψ ± × ψ ∓ = (cid:15) + n (cid:88) j =1 φ j (cid:15) × φ i = φ i , (cid:15) × ψ ± = ψ ∓ , φ i × ψ ± = ψ ± + ψ ∓ φ i × φ i = + (cid:15) + φ g (2 i ) , φ i × φ j = φ g ( i − j ) × φ g ( i + j ) , i > j (B.15)where g ( i ) = i if 1 ≤ i ≤ n and g ( i ) = 2 n + 1 − i otherwise.We see that there are no multiplicities, and all anyons are self-conjugate. We also notethat (cid:15) is condensable, which leads to the bosonic theory SO (2 n + 1) . Let us analyse thequotient explicitly.By looking at the braiding phase B ( α, (cid:15) ) one learns that the unscreened anyons are , (cid:15), φ i ,while the screened anyons are ψ ± . Moreover, and (cid:15) are in the same orbit, while all the φ i are fixed points. Thus, a basis for the condensed Hilbert space is as follows: | (cid:105) = 1 √ | (cid:105) + | (cid:15) (cid:105) ) | i (cid:105) = | φ i (cid:105)| n + i (cid:105) = | φ i ; (cid:15) (cid:105) (B.16)for i = 1 , , . . . , n , and where | · ; (cid:15) (cid:105) denotes a state in the once-punctured torus. The57ondensed S -matrix is ˆ S , = 2 S , = 1 √ n + 1ˆ S ,i = √ S ,φ i = (cid:114) n + 1ˆ S ,n + i = 0ˆ S i,j = S φ i ,φ j = 2 √ n + 1 cos 2 πij n + 1ˆ S i,j + n = 0ˆ S i + n,j + n = S φ i ,φ j ( (cid:15) ) (B.17)where S φ i ,φ j ( (cid:15) ) is the S -matrix of Spin (2 n + 1) in the presence of a puncture. This matrixelement can be obtain as in (A.18). For example, we compute S φ i ,φ i ( (cid:15) ) = (cid:88) β ∈ φ i × φ i θ ( β ) θ ( φ i ) S ,β F φ i ,φ i (cid:20) (cid:15) φ i φ i β (cid:21) = e − πi i (2 n +1 − i )2 n +1 √ n + 1 × (cid:18) F φ i ,φ i (cid:20) (cid:15) φ i φ i (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) +1 + 12 F φ i ,φ i (cid:20) (cid:15) φ i φ i (cid:15) (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) +1 + e πi (
12 2 i (2 n +1 − i )2 n +1 ) F φ i ,φ i (cid:20) (cid:15) φ i φ i φ g (2 i ) (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) − (cid:19) = 2 i √ n + 1 sin 2 πi n + 1 (B.18)while for i (cid:54) = j , S φ i ,φ j ( (cid:15) ) = (cid:88) β ∈ φ i × φ j θ ( β ) θ ( φ i ) θ ( φ j ) S ,β F φ i ,φ j (cid:20) (cid:15) φ j φ i β (cid:21) = 1 √ n + 1 (cid:18) e iπij n +1 F φ i ,φ j (cid:20) (cid:15) φ j φ i φ g ( i − j ) (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) +1 + e − iπij n +1 F φ i ,φ j (cid:20) (cid:15) φ j φ i φ g ( i + j ) (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) − (cid:19) = 2 i √ n + 1 sin 2 πij n + 1 (B.19)All in all, the S -matrix of the quotient takes the formˆ S = 1 √ n + 1 √ √ πij n +1
00 0 2 i sin πij n +1 ← | (cid:105)← | i (cid:105)← | i ; (cid:15) (cid:105) (B.20)58ne can easily check that this matrix is unitary, and satisfies the algebra of the (bosonic)modular group, S = ( ST ) , S = 1.In order to write down the fusion rules of the quotient we have to switch into the fusionbasis, namely | φ i , ±(cid:105) = 1 √ | φ i (cid:105) ± | φ i ; (cid:15) (cid:105) ) . (B.21)In this basis, the S -matrix becomes ˆ S ij ∼ √ n +1 e πi ij n +1 . This is in agreement with the level-rank duality SO ( n ) ∼ SO (2) − n = U (1) − n , where the S -matrix of U (1) k is e − πiαβ/k / √ k .(Here ∼ denotes duality modulo { , ψ } , since U (1) k is spin for odd k .) B.3 SO ( n ) We construct the theory using level-rank duality SO ( n ) = SO (3) − n = SU (2) − n / Z . So weconsider SU (2) k first.There are k + 1 lines, which we label as j = 0 , , , . . . , k . The S -matrix reads S ij = (cid:114) k + 2 sin π (2 i + 1)(2 j + 1) k + 2 (B.22)and the spins are h j = j ( j +1) k +2 . The fusion rules read j × j = min( J,k − J ) (cid:88) j = | j − j | j, J = j + j (B.23)The quantum dimensions are d j = sin π (2 j +1) k +2 sin πk +2 , and so j = k/ Z symmetry acts as j (cid:55)→ | k − j | . The spin of this line is k/
4, and so the symmetry iscondensable if and only if k is even, k = 2 n . The quotient theory is P SU (2) n = SO (3) n ; itis spin if n is odd.The only fixed-point is j = n/
2, whose S -matrix element is given by (A.18) S n/ ,n/ ( n ) = (cid:88) β ∈ n/ × n/ θ ( β ) θ ( n/ S ,β F n/ ,n/ (cid:20) n n/ n/ β (cid:21) = θ ( n/ − √ n + 1 n (cid:88) j =0 θ ( j ) sin π j + 1 n + 1 F n/ ,n/ (cid:20) n n/ n/ j (cid:21) (B.24)which, using F = ( − j , becomes S n/ ,n/ ( n ) = e − iπn/ . One may check that modularity issatisfied, S = ( ST ) and S = ( − F ≡ ( − n . This is indeed consistent with the quotientbeing bosonic if n is even, and fermionic if odd.59et us consider the case of even n , where the quotient is bosonic. The unscreened linesare those with integer j , and the only fixed point is j = n/
2. Thus, a basis for the Hilbertspace is | j (cid:105) = 1 √ | j (cid:105) + | n − j (cid:105) ) , j = 0 , , . . . , n/ − | a (cid:105) = | n/ (cid:105)| a (cid:105) = | n/ n (cid:105) (B.25)where | · ; α (cid:105) denotes the corresponding state with an α -puncture.The S -matrix of the quotient isˆ S i,j = 2 S i,j = 2 (cid:114) n + 1 sin π (2 i + 1)(2 j + 1)2 n + 2ˆ S i,a = √ S i,n/ = ( − i (cid:114) n + 1ˆ S a ,a = S n/ ,n/ = ( − n/ (cid:114) n + 1ˆ S a ,a = 0ˆ S a ,a = S n/ ,n/ ( n ) = i n/ . (B.26)The fusion basis is defined by | a ± (cid:105) = 1 √ | a (cid:105) ± | a (cid:105) ) . (B.27)One can easily compute the S -matrix in this basis, from where one can compute, for example,the fusion rules of the theory.Consider now the case of odd n , where the quotient is spin. The NS lines are those withintegral isospin, and the R lines with half-integral isospin. A basis of the quotient Hilbertspace is • If we take
NS-NS boundary conditions, the states are | j ; NS-NS (cid:105) = 1 √ | j (cid:105) + | n − j (cid:105) ) , j = 0 , , . . . , n . (B.28) • If we take
NS-R boundary conditions, the states are | j ; NS-R (cid:105) = 1 √ | j (cid:105) − | n − j (cid:105) ) , j = 0 , , . . . , n . (B.29) • If we take
R-NS boundary conditions, the states are | j ; R-NS (cid:105) = 1 √ | j (cid:105) + | n − j (cid:105) ) , j = 12 , , . . . , n , . . . , n | n/
2; R-NS (cid:105) = | n/ (cid:105) . (B.30)60 If we take
R-R boundary conditions, the states are | j ; R-R (cid:105) = 1 √ | j (cid:105) − | n − j (cid:105) ) , j = 12 , , . . . , n , . . . , n | n/
2; R-R (cid:105) = | n/ n (cid:105) . (B.31)The modular data easily follows from this decomposition, and the once-punctured torusmatrix element (B.24). B.4 U (1) k In this section we construct the Hilbert space of the spin TQFT U (1) k . This generalizes theconstruction of the semion-fermion theory of section 4.1. For some special values of k thistheory is time-reversal invariant. The semion-fermion theory is recovered by taking k = 2.For k >
2, the time-reversal symmetry (when present) satisfies a more exotic algebra [80],namely T = C , where C denotes an order-2 unitary symmetry (charge conjugation).The construction of U (1) k is slightly different depending on whether k is even or odd.Indeed, for k odd the theory is naturally spin; but, for k even, it is bosonic, and so it has tobe multiplied by the trivial factor { , ψ } if we are interested in its spin version. The lattercase is rather similar to the semion-fermion theory, so here we will focus on the k odd casehere, and sketch the main differences for k even at the end.Consider the theory U (1) k with k odd. Its bosonic parent is U (1) k , whose anyons arelabelled as α ∈ Z k . The spin theory is obtained by condensing the fermion ψ = 2 k . Thebraiding phase of an arbitrary line α with respect to the fermion is B ( α, ψ ) = e πiα , whichmeans that the anyons are split asNS : 2 α, α = 0 , , . . . , k −
1R : 2 α + 1 , α = 0 , , . . . , k − . (B.32)These are all in two-dimensional orbits, paired up as α × ψ ←→ α + 2 k . (B.33)As there are no fixed-points, all states are bosonic. Hilbert space and modularity.
Given the knowledge of the Hilbert space of the bosonicparent, and the action of the modular group on it, we easily construct the same objects in thequotient theory. In particular, the Hilbert space is
H ∼ = C k , with states | α (cid:105) , and modulartransformations act as S | α (cid:105) = (cid:88) α (cid:48) ∈ Z k S α,α (cid:48) | α (cid:48) (cid:105) T | α (cid:105) = e πi ( α / k − / | α (cid:105) C | α (cid:105) = | − α mod 4 k (cid:105) , (B.34)61here S α,α (cid:48) = e − πiαα (cid:48) / k / √ k , and the term − /
24 in the T -transformation refers to thecentral charge of the theory.The quotient space is as follows: • If we take
NS-NS boundary conditions, the states are | α ; NS-NS (cid:105) = 1 √ | α (cid:105) + | α + 2 k (cid:105) ] , α = 0 , . . . , k − S | α ; NS-NS (cid:105) = 1 √ k k − (cid:88) α (cid:48) =0 e πiαα (cid:48) /k | α (cid:48) ; NS-NS (cid:105) ˆ T | α ; NS-NS (cid:105) = e πi (cid:0) α k − (cid:1) | α ; NS-R (cid:105) ˆ C | α ; NS-NS (cid:105) = k − (cid:88) α (cid:48) =0 ( δ α + α (cid:48) + δ α + α (cid:48) − k ) | α (cid:48) ; NS-NS (cid:105) (B.36)where δ x = 1 if x ≡ k , and δ x = 0 otherwise. • If we take
NS-R boundary conditions, the states are | α ; NS-R (cid:105) = 1 √ | α (cid:105) − | α + 2 k (cid:105) ] , α = 0 , . . . , k − S | α ; NS-R (cid:105) = 1 √ k k − (cid:88) α (cid:48) =0 e πi (2 α +1) α (cid:48) / k | α (cid:48) ; R-NS (cid:105) ˆ T | α ; NS-R (cid:105) = e πi (cid:0) α k − (cid:1) | α ; NS-NS (cid:105) ˆ C | α ; NS-R (cid:105) = k − (cid:88) α (cid:48) =0 ( δ α + α (cid:48) − δ α + α (cid:48) − k ) | α (cid:48) ; NS-R (cid:105) (B.38) • If we take
R-NS boundary conditions, the states are | α ; R-NS (cid:105) = 1 √ | α + 1 (cid:105) + | α + 1 + 2 k (cid:105) ] , α = 0 , . . . , k − S | α ; R-NS (cid:105) = 1 √ k k − (cid:88) α (cid:48) =0 e πiα (2 α (cid:48) +1) / k | α (cid:48) ; NS-R (cid:105) ˆ T | α ; R-NS (cid:105) = e πi (cid:0) (2 α +1)28 k − (cid:1) | α ; R-NS (cid:105) ˆ C | α ; R-NS (cid:105) = k − (cid:88) α (cid:48) =0 ( δ α + α (cid:48) +1 + δ α + α (cid:48) +1 − k ) | α (cid:48) ; R-NS (cid:105) (B.40)62 If we take
R-R boundary conditions, the states are | α ; R-R (cid:105) = 1 √ | α + 1 (cid:105) − | α + 1 + 2 k (cid:105) ] , α = 0 , . . . , k − S | α ; R-R (cid:105) = 1 √ k k − (cid:88) α (cid:48) =0 e πi (2 α +1)(2 α (cid:48) +1) / k | α (cid:48) ; R-R (cid:105) ˆ T | α ; R-R (cid:105) = e πi (cid:0) (2 α +1)28 k − (cid:1) | α ; R-R (cid:105) ˆ C | α ; R-R (cid:105) = k − (cid:88) α (cid:48) =0 ( δ α + α (cid:48) +1 − δ α + α (cid:48) +1 − k ) | α (cid:48) ; R-R (cid:105) (B.42)It is reassuring to see that these modular transformations map the different Hilbert spacesprecisely as they should (cf. (A.26)). Moreover, these matrices are unitary, ˆ S is symmetric( ˆ S ts a ,s b = ˆ S s b ,s a ), and they satisfy the modular algebra ( ˆ S ˆ T ) = ˆ S = ˆ C with ˆ C = 1. Wilson lines.
The Wilson lines are given by W ( a ) ( α ) | γ ; s a s b (cid:105) = e − πiα ( γ +(1+ s a ) / /k | γ ; s a s b (cid:105) W ( b ) ( α ) | γ ; s a s b (cid:105) = | α + γ ; s a s b (cid:105) − s b | α + γ + k ; s a s b (cid:105) (B.43)where α ∈ Z k . They satisfy the expected properties, e.g., W ( c ) s a ,s b ( ψ ) = W ( c ) s a ,s b ( k ) = − s c k W ( c ) s a ,s b ( α × α (cid:48) ) = W ( c ) s a ,s b ( α ) W ( c ) s a ,s b ( α (cid:48) ) S s a ,s b W ( a ) s a ,s b ( α )( S s a ,s b ) † = W ( b ) s b ,s a ( ¯ α ) S s a ,s b W ( b ) s a ,s b ( α )( S s a ,s b ) † = W ( a ) s b ,s a ( α ) (B.44) Time-reversal.
We now implement time-reversal invariance. Recall that U (1) k is time-reversal invariant if and only if q = − k is solvable for some q ∈ Z , in which casetime-reversal acts as α (cid:55)→ qα [80]. This means that, given T = τ K , we require τ ( W ( c )2 ) ∗ τ − = W ( c )2 q = ( W ( c )2 ) q (B.45)with solution τ α,β = ( − s b ) α + β δ αq +2 β + ( s a +1)( q +1) (B.46)up to a global phase. One can check that τ τ ∗ = ( − Arf( s ) (cid:0) δ ( s a +1)+ α + β − s b δ ( s a +1)+ α + β − k (cid:1) (B.47)63nd so T = ( − Arf( s ) ˆ C .We see that the time-reversal algebra is deformed by Arf, signaling an anomaly. In thiscase, the source of the anomaly is clear: the theory has non-vanishing central charge, c = 1,so it is not time-reversal invariant in the strict sense. We have to multiply by a suitable SPTin order to subtract off the central charge. In this case, U (1) − does the trick, as this SPThas c = − U (1) − is straightforward: it suffices to take k = 1 in the discussionabove. Looking at the action of ˆ C on the (one-dimensional) Hilbert space of U (1) − welearn that ˆ C = ( − Arf( s ) . Therefore, multiplying a given theory by U (1) ± has the effect ofredefining ˆ C → ˆ C × ( − Arf( s ) , which means that the theory U (1) k × U (1) − has undeformedalgebra, namely T = ˆ C . The Arf deformation in the case of U (1) k was just signaling that wehad not corrected the central charge down to zero; after doing so, the deformation disappearsfrom the time-reversal algebra.Finally, we make a few remarks concerning the k even case. Now the theory is naturallybosonic, and can be made spin by tensoring with an invertible spin TQFT. If we are interestedin time-reversal invariance, the natural choice is U (1) k × U (1) − , so as to have vanishingcentral charge. As the theory is a tensor product, one factor being bosonic, the total Hilbertspace is straightforward:ˆ H s ( U (1) k × U (1) − ) = H ( U (1) k ) ⊗ ˆ H s ( U (1) − ) (B.48)where H ( U (1) k ) is the space of the bosonic theory U (1) k , and ˆ H s ( U (1) − ) is the space ofthe fermionic theory U (1) − . The Hilbert space of U (1) − was discussed above, and that of U (1) k is well-known, being bosonic. In this sense, no new computation is required in thecase of U (1) k with k even. One can easily check through straightforward computation thatthe main conclusions are identical to those of the k odd case, in particular, time-reversalsatisfies T = ˆ C , with no deformation. (If we reintroduce a non-zero value of the centralcharge, by multiplying by an extra factor of U (1) ± , the deformation reappears, and we get T = ( − Arf( s ) ˆ C , again signalling the anomaly due to c ).64 eferences [1] Gerard ’t Hooft. “Naturalness, chiral symmetry, and spontaneous chiral symmetrybreaking”. In: NATO Sci. Ser. B
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