A generalization of decomposition in orbifolds
AA generalization of decomposition in orbifolds
Daniel G. Robbins , Eric Sharpe , Thomas Vandermeulen Department of PhysicsUniversity at AlbanyAlbany, NY 12222 Department of Physics MC 0435850 West Campus DriveVirginia TechBlacksburg, VA 24061 [email protected] , [email protected] , [email protected] This paper describes a generalization of decomposition in orbifolds. In general terms,decomposition states that two-dimensional orbifolds and gauge theories whose gauge groupshave trivially-acting subgroups decompose into disjoint unions of theories. However, de-composition can be, at least naively, broken in orbifolds if the orbifold has discrete torsionin the trivially-acting subgroup. (Formally, this breaks finite global one-form symmetries.)Nevertheless, even in such cases, one still sees rudiments of decomposition. In this paper,we generalize decomposition in orbifolds to include such examples of discrete torsion, whichwe check in numerous examples. Our analysis includes as special cases (and in one sensegeneralizes) quantum symmetries of abelian orbifolds.January 2021 1 a r X i v : . [ h e p - t h ] J a n ontents ι ∗ ω (cid:54) = 0 X/ Z (cid:111) Z ] with discrete torsion and trivially-acting Z × Z subgroup . . . 214.3 [ X/S ] with discrete torsion and trivially-acting Z × Z subgroup . . . . . . 224.4 [ X/S ] with discrete torsion and trivially-acting A subgroup . . . . . . . . . 244.5 [ X/D × ( Z ) ] with discrete torsion and trivially-acting ( Z ) subgroup . . . 26 ι ∗ ω = 0 and β ( ω ) (cid:54) = 0 X/ Z × Z ] with discrete torsion and trivially-acting Z . . . . . . . . . . . 285.2 Example with nonabelian K . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.3 [ X/ Z × Z ] with discrete torsion and trivially-acting Z or Z × Z subgroup 315.4 [ X/ Z × Z ] with discrete torsion and trivially-acting Z subgroup . . . . . . 335.5 [ X/D ] with discrete torsion and trivially-acting Z subgroup . . . . . . . . . 345.6 [ X/ Z (cid:111) Z ] with discrete torsion and trivially-acting subgroups . . . . . . . 35 ι ∗ ω = 0 and ω = π ∗ ω X/ Z × Z ] with discrete torsion and trivially-acting Z subgroup . . . . . . 396.2 [ X/ Z (cid:111) Z ] with discrete torsion and trivially-acting Z subgroup . . . . . . 412 Mixed examples 44 Z × Z orbifold with discrete torsion and trivially-acting Z × Z . . . . . . 447.2 Z × Z orbifold with discrete torsion and trivially-acting Z × Z . . . . . . 47 A.1 Group cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50A.2 Projective representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
B Some calculations with cocycles 53C Explicit realization of β
55D Pertinent group theory results 59
D.1 Z × Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59D.2 Z × Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59D.3 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60D.4 Z (cid:111) Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62D.5 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 References 63 Introduction
This paper is motivated by a revival of interest in two old topics, namely • orbifolds, see e.g. [1–7], utilizing new methods and insights into anomalies from topo-logical defect lines and related technologies, and • decomposition, first described in [8], a property of two-dimensional theories with whatare now called one-form symmetries and various generalizations, see e.g. [9–13] forrecent activity in this area.Decomposition was first introduced in [8] to understand orbifolds and two-dimensionalgauge theories in which a finite subgroup of the gauge group acts trivially on the theory.(If the subgroup is abelian, this means that the theory has a finite global one-form sym-metry, in modern language, but decomposition is defined more generally.) See [14–16] fordetailed discussions of such examples, and see also [17–22] for applications to Gromov-Wittentheory, [23–29] for applications to phases of gauged linear sigma models (GLSMs), [30] forapplications in heterotic string compactifications, [12] for applications to elliptic genera oftwo-dimensional pure supersymmetric gauge theories, and [10] for four-dimensional ana-logues, for example.The original work on decomposition [8] studied orbifolds and gauge theories, but didnot consider orbifolds in which discrete torsion was turned on in a way that obstructed theexistence of the one-form symmetry (or its analogues), except to note that the decompositionstory did not apply to such cases. The purpose of this paper is to fill that gap, by generalizingdecomposition in orbifolds to include cases in which discrete torsion is turned on.As one application, we shall see how quantum symmetries are described in this framework.Recall [31, 32] that a Z k orbifold has a Z k quantum symmetry that, when gauged, returnsthe original orbifold. We shall see that the composition of orbifolds can be described as asingle Z k × Z k orbifold with discrete torsion, for which our generalization of decompositionpredicts that the orbifold is equivalent to the original space. In fact, our generalization ofdecomposition will predict analogous results in more general cases, that often the effect ofgauging a trivially-acting subgroup K of the orbifold group G with nonzero discrete torsion isto partly ‘undo’ the underlying G/K orbifold. Indeed, part of our results is a generalization,in a certain direction, of quantum symmetries.We begin in section 2 by reviewing decomposition for the special case of orbifolds, spe-cializing a number of very general (and necessarily abstract) statements in [8] to the simplerconcrete case that the theory is a finite gauge theory, an orbifold by a finite group.In section 3 we then make our conjecture for the generalization of decomposition. Ex-amples naturally fall into three classes. Given an orbifold with orbifold group G , with4rivially-acting subgroup K , where1 −→ K ι −→ G π −→ G/K −→ , (1.1)with discrete torsion ω ∈ H ( G, U (1)), the three cases are as follows:1. ι ∗ ω (cid:54) = 0 as an element of H ( K, U (1)),2. ι ∗ ω = 0, β ( ω ) (cid:54) = 0,3. ι ∗ ω = 0, ω = π ∗ ω, where the maps π ∗ , β arise in the exact sequence H ( G/K, U (1)) π ∗ −→ Ker ι ∗ β −→ H ( G/K, H ( K, U (1))) . (1.2)In the remainder of this paper we check the claim in a number of concrete examples. Insection 4, we discuss examples with ι ∗ ω (cid:54) = 0; in section 5 we discuss examples with ι ∗ ω = 0and β ( ω ) (cid:54) = 0, and relate β to quantum symmetries; in section 6 we discuss examples with ι ∗ ω = 0 and ω = π ∗ ω for some ω ∈ H ( G/K, U (1)). Finally in section 7 we discuss examplesspanning all three categories, depending upon the value of discrete torsion ω .For reference, in appendix A we briefly review some pertinent aspects of group coho-mology and projective representations, in appendix B we give some technical details of ourcocycle computations, in appendix C we make explicit the map β that plays a prominentrole in our analysis, and in appendix D we collect a number of group-theoretic results on thefinite groups appearing in the examples, including explicit representatives of discrete torsioncocycles and genus-one twisted sector phases.Finally, we should comment on additive versus multiplicative notation in cocycles. Atvarious points in this paper, it will greatly improve readability to use one or the other, sowe have adopted both notations, and leave the reader to infer from context which is beingused in any one section. Let G be a finite group acting on a space X , with K ⊂ G a normal subgroup acting trivially.It was argued in [8, section 4.1] that this orbifold is equivalent to a disjoint union, specificallyQFT ([ X/G ]) = QFT (cid:32)(cid:34) X × ˆ KG/K (cid:35) ˆ ω (cid:33) , (2.1)5here ˆ K denotes the set of isomorphism classes of irreducible representations of K , and ˆ ω denotes discrete torsion we shall describe momentarily. This is known as decomposition,referring to the fact that the theory on the right-hand side typically has multiple differentdisjoint components. These different components or summands are sometimes referred toin the literature as universes, given that in a string compactification, they would definelow-energy theories with multiple independent decoupled gravitons.The group G/K acts on ˆ K as follows: pick a section s : G/K → G , so that π ( s ( q )) = q for all q ∈ G/K . If the extension does not split (if G is not isomorphic to a semi-directproduct K (cid:111) G/K ), then s cannot be chosen to be a group homomorphism, but it alwaysexists as a map. For any representation φ : K → GL( V ) of K and any q ∈ G/K , define anew representation L q φ : K → GL( V ) by( L q φ )( k ) = φ ( s ( q ) − ks ( q )) (2.2)(here we are suppressing the map ι and are simply taking K to be a normal subgroup of G ). It’s easy to check that L q φ is a homomorphism and that if φ is irreducible then L q φ isalso irreducible. By verifying that L φ is isomorphic to φ , that L q ( L q φ ) is isomorphic to L q q φ , and that if φ is isomorphic to φ then L q φ is isomorphic to L q φ , one shows thatthis defines a G/K action on ˆ K by q · [ φ ] = [ L q φ ]. Moreover, this action is independent ofthe choice of section s .With this understanding we can interpret (2.1) and determine the discrete torsion ˆ ω .Let { ρ a } be a collection of irreducible representations of K chosen so that the equivalenceclasses [ ρ a ] are representatives of the distinct orbits of the G/K action on ˆ K . For each ρ a ,let H a ⊆ G/K be the stabilizer of [ ρ a ] in ˆ K . Then, decomposition becomes the statementthat QFT ([ X/G ]) = QFT (cid:32)(cid:97) a [ X/H a ] ˆ ω a (cid:33) , (2.3)where ˆ ω a ∈ H ( H a , U(1)) (our notation and conventions for group cohomology are reviewedin appendix A.1) denotes the discrete torsion in the summand [
X/H a ].If K is abelian, then there is a simple way to determine ˆ ω a . First, given q , q ∈ G/K define e ( q , q ) = s ( q ) s ( q ) s ( q q ) − . (2.4)Since this is in the kernel of the map π , it must lie in the subgroup K . Indeed, e turnsout to be a 2-cocycle valued in Z ( G/K, K ), where K is taken to be a G/K -module withaction q · k = s ( q ) ks ( q ) − . Different choices of section s will lead to cocycles that differ fromthis by coboundary terms, but the cohomology class of e in H ( G/K, K ) depends only onthe extension (1.1) and is called the extension class [33, section IV.3]. Since K is finite andabelian, its irreducible representations are all one-dimensional and map K into U(1). Sogiven one of our irreducible representations ρ a , we can apply it to e restricted to H a to get6ur discrete torsion ˆ ω a , ˆ ω a ( h , h ) = ρ a ( s ( h ) s ( h ) s ( h h ) − ) . (2.5)If in addition, G itself is abelian, then the extension class (2.4) is symmetric in q and q ,and hence so is ˆ ω a . But this means that the discrete torsion phases (cid:15) ( g, h ) = ˆ ω a ( g, h ) / ˆ ω a ( h, g )which appear in the partition function are all unity.If K is not abelian, one must work harder to define the discrete torsion ˆ ω a . Beforedescribing the general case, let us describe three examples in which K is abelian. • For one example, if G is the trivial extension K × H , with K abelian, then the extensionclass vanishes, and so QFT ([ X/G ]) = QFT (cid:97) ˆ K [ X/H ] , (2.6)where no copies have any discrete torsion. • For another example, if G = D and K = Z , then using the fact that G/K = Z × Z ,and that D (cid:54)∼ = Z × ( Z × Z ) (so the extension class is nontrivial),QFT ([ X/D ]) = QFT (cid:16) [ X/ Z × Z ] (cid:97) [ X/ Z × Z ] ˆ ω (cid:17) , (2.7)where the second summand has discrete torsion ˆ ω given by the nontrivial irreduciblerepresentation of Z applied to the extension class, which one can check is the nontrivialelement of H ( Z × Z , U (1)). See [8, section 5.2] for a detailed verification that physicsdoes, indeed, obey this equivalence. • For our third example, let G = H , the eight-element group of quaternions, and K = (cid:104) i (cid:105) ∼ = Z . In this case, K has four irreducible representations, two of whichare invariant under the action of G/K = Z , and two of which are exchanged. In thiscase, decomposition predictsQFT ([ X/ H ]) = QFT (cid:16) [ X/ Z ] (cid:97) [ X/ Z ] (cid:97) X (cid:17) . (2.8)See [8, section 5.4] for a detailed verification that physics does, indeed, obey thisequivalence.In passing, ultimately because of the Cartan-Leray spectral sequence, if X/ ( G/K ) issmooth, then the prescription above will determine a B field on that quotient, which is inthe spirit of the original phrasing of decomposition in [8].7ext, we shall describe how the discrete torsion ˆ ω a is determined in the general case,when K need not be abelian.Since H a fixes the isomorphism class [ ρ a ], it means that for each h ∈ H a the represen-tations ρ a : K → GL( V a ) and L h ρ a are isomorphic. Explicitly, this means that for each h ∈ H a we can find an element f a ( h ) ∈ GL( V a ) so that the following diagram commutes forall k ∈ K , V a ρ a ( k ) (cid:47) (cid:47) f a ( h ) (cid:15) (cid:15) V af a ( h ) (cid:15) (cid:15) V a ρ a ( s a ( h ) − ks a ( h )) (cid:47) (cid:47) V a (2.9)The map f a : H a → GL( V a ) is called the intertwiner.Let G a = π − ( H a ), so that we have an exact sequence1 −→ K ι −→ G a π −→ H a −→ . (2.10)The idea of the construction will be that for each ρ a we will define a projective representation (cid:101) ρ a on G a . Projective representations are reviewed in appendix A.2. The associated 2-cocycle (cid:101) ω a will turn out to be the pullback of a 2-cocycle on H a which we’ll identify with ˆ ω − a .Indeed, by restricting our choice of section s to H a , we can write each element of G a uniquely as g = s ( h ) k for some h ∈ H a and k ∈ K . Now define a map (cid:101) ρ a : G a → GL( V a ) by (cid:101) ρ a ( s ( h ) k ) = f a ( h ) − ρ a ( k ) . (2.11)We claim that (cid:101) ρ a is a projective representation on G a . Indeed, for any two elements g , g ∈ G a , define an operator C a ( g , g ) by C a ( g , g ) = (cid:101) ρ a ( g ) (cid:101) ρ a ( g ) (cid:101) ρ a ( g g ) − . (2.12)To show that (cid:101) ρ a is a projective representation, we need to show that C a ( g , g ) is a scalarmultiple of the identity operator ∈ GL( V a ), say C a ( g , g ) = (cid:101) ω a ( g , g ) . It then followsthat (cid:101) ρ a ( g ) (cid:101) ρ a ( g ) = (cid:101) ω a ( g , g ) (cid:101) ρ a ( g g ) . (2.13)Indeed, computing we have (abbreviating s a ( h ) = s , s a ( h h ) = s , f a ( h ) = f , etc.) (cid:101) ρ a ( g g ) = (cid:101) ρ a ( s k s k ) = (cid:101) ρ a ( s ( s − s s ) s − k s k ) = f − ρ a ( s − s s ) ρ a ( s − k s k ) , (2.14)and so C a ( g , g ) = f − ρ a ( k ) f − ρ a ( k ) ρ a ( s − k s k ) − ρ a ( s − s s ) − f , = f − f − ρ a ( s − k s ) ρ a ( k ) ρ a ( k ) − ρ a ( s − k s ) − ρ a ( s − s − s ) f , = f − f − f ρ a ( s s − s − ) . (2.15)8ote that C a ( s ( h ) k , s ( h ) k ) is in fact independent of k and k . To see that C a ( g , g ) isa multiple of the identity, take any k ∈ K and check that C a ( g , g ) commutes with ρ a ( k ), C a ( g , g ) ρ a ( k ) = f − f − f ρ a ( s s − s − k )= ρ a ( ks s − s − ) f − f − f = ρ a ( k ) f − f − f ρ a ( s s − s − ) = ρ a ( k ) C a ( g , g ) , (2.16)where we repeatedly used the intertwiner property ρ a ( k ) f a ( h ) − = f a ( h ) − ρ a ( s ( h ) − ks ( h )) . (2.17)Since ρ a was an irreducible representation of K , Schur’s lemma tells us that anything thatcommutes with all the operators ρ a ( k ) must be a multiple of the identity.Note that associativity of (cid:101) ρ a implies the coclosure of (cid:101) ω a . Specifically, the cocycle conditionfollows by plugging (2.13) into( (cid:101) ρ a ( k ) (cid:101) ρ a ( k )) (cid:101) ρ a ( k ) = (cid:101) ρ a ( k ) ( (cid:101) ρ a ( k ) (cid:101) ρ a ( k )) , (2.18)which clearly holds since the (cid:101) ρ a ( k i ) are simply products of GL( V a ) matrices whose multipli-cation is associative. Alternatively, one can show by direct calculation that C a ( g , g ) C a ( g , g g ) C a ( g g , g ) − = C a ( g , g ) . (2.19)Finally, we define ˆ ω a ( h , h ) as the inverse of (cid:101) ω a , i.e.ˆ ω a ( h , h ) = C a ( s , s ) − = ρ a ( s s s − ) f − f f . (2.20)The fact that (cid:101) ω a is a cocycle implies that ˆ ω a is also coclosed, and thus ˆ ω a defines a class in H ( G/K,
U(1)).As a special case, suppose again that K is abelian (but not necessarily in the center of G ). Then all irreducible representations ρ a are one-dimensional, and the intertwiners f a ( h )are all scalars. In this case, the cocycle ˆ ω given in equation (2.20), modulo coboundaries,reduces to ˆ ω ( h , h ) = ρ a (cid:0) s a ( h ) s a ( h ) s a ( h h ) − (cid:1) . (2.21)Now, the product s a ( h ) s a ( h ) s a ( h h ) − (2.22)is precisely the cocycle representing the extension class in H ( H a , K ) in the case that K isabelian [33, section IV.3], [34, exercise VI.10.1], so we see that ˆ ω is the image of the extension Technically, our construction only ensures that (cid:101) ω a and hence ˆ ω a are C × -valued, not U(1)-valued. How-ever, we can always multiply the intertwiners by C × scalars to arrange that their determinants have magni-tude one, and in that case our cocycles are U(1)-valued. G a in H ( H a , K ) under ρ a , as claimed earlier. If K is a subgroup of the center of G ,then each H a = G/K , and then ˆ ω is the image of the extension class of G in H ( G/K, K )under ρ a , also as described earlier.These statements have been checked in many examples in many ways. In orbifolds, theoriginal work [14–16] first checked that physics ‘sees’ trivially-acting groups, through studiesof multiloop factorization (target-space unitarity), as well as massless spectra and partitionfunctions, which confirmed that not only does one get distinct theories, but one encountersphysical contradictions if one tries to ignore them. Decomposition in such theories was testedin [8] by comparing partition functions at all genus, construction of projection operatorsprojecting onto the various universes, and comparisons of massless spectra and correlationfunctions, as well as studies of open string sectors. Decomposition is also defined for gaugetheories with finite trivially-acting subgroups, and there it has been similarly tested in bothsupersymmetric and nonsupersymmetric models, via comparisons of partition functions andelliptic genera (using supersymmetric localization), quantum cohomology rings, and mirrors,to name a few. See for example [9, 12, 13] for some more recent discussions and reviews citedtherein.Sometimes, for K abelian, decomposition can be understood in terms of the existence offinite global one-form symmetries in the theory, see e.g. [9] for a recent discussion. A finiteglobal one-form K symmetry, technically denoted BK , acts in an orbifold by interchangingthe twisted sectors (and in more general gauge theories, the nonperturbative sectors). If wethink of each twisted sector contribution to the partition function as associated with someprincipal bundle, the action of BK is to tensor that principal bundle with a principal K bundle to get another twisted sector contribution. If K acts trivially, then the two twistedsectors make identical contributions to the total partition function.As a quick consistency check when doing computations, given a decomposition of theform QFT ([ X/G ]) = QFT (cid:32)(cid:97) i [ X/G i ] ω i (cid:33) , (2.23)then it should be true that the total number of irreducible representations of G shouldmatch the sum of the number of ω i -twisted irreducible representations of each G i . (Thisis a consequence of the special case that X is a point.) For example, as noted earlier, for G = D , decomposition predictsQFT ([ X/D ]) = QFT (cid:16) [ X/ Z × Z ] (cid:97) [ X/ Z × Z ] ω (cid:17) . (2.24)Now, D has five irreducible representations, Z × Z has four ordinary irreducible represen-tations, and one irreducible projective representation twisted by the nontrivial element of These are not vector bundles, so the term ‘tensor product’ is not completely accurate, but neverthelessthere can exist, under suitable circumstances, a product structure defined in the obvious fashion. ( Z × Z , U (1)), as described in appendix A.2. As expected, 5 = 4 + 1. This observationcan sometimes be handy when double-checking expressions for decomposition.Previous work such as [8] focused on orbifolds (and related theories) in which a subgroupacted trivially, but no discrete torsion was turned on that interacted with that subgroupin any way, aside from making the observation that discrete torsion typically ‘broke’ thedecomposition story, yielding theories with fewer components than decomposition wouldpredict. (For example, discrete torsion typically breaks much of the finite global one-formsymmetry.) The purpose of this paper is to extend to orbifolds with trivially-acting subgroupsin which discrete torsion has been turned on, and understand what is happening in such cases. Now, consider an orbifold [
X/G ] ω by a finite group G as above, where one has includeddiscrete torsion, given by some cocycle ω , so [ ω ] ∈ H ( G, U (1)) (with trivial action on thecoefficients). Suppose as above that a normal subgroup K ⊆ G acts trivially on X , anddescribe G as the extension1 −→ K ι −→ G π −→ G/K −→ . (3.1)As before we will choose a section s : G/K → G which is not a homomorphism in general.At the end of the day, nothing physical will depend on the choice of section.In the previous section, when ω was trivial, we found that the G orbifold decomposedinto a number of disjoint pieces, one for each orbit of the G/K action on ˆ K , the set ofisomorphism classes of irreducible representations of K . Now, the role of ˆ K will be replacedby the set ˆ K ι ∗ ω of isomorphism classes of irreducible projective representations of K withrespect to the cocycle ι ∗ ω (which, if we view K as a normal subgroup of G , is simply therestriction of ω to K ). Some facts about projective representations of finite groups arereviewed in appendix A.2.Our first task is to understand the natural action of G/K on ˆ K ι ∗ ω in this case. We can’tsimply make the naive definition ( L q φ )( k ) = φ ( s ( q ) − ks ( q )), since L q φ defined in this wayturns out to be projective, but not with respect to ι ∗ ω . Instead, the cocycle which arises is ω (cid:48) ( k , k ) = ω ( s ( q ) − k s ( q ) , s ( q ) − k s ( q )) , (3.2)which differs from ι ∗ ω by a coboundary. Fortunately, we can cancel coboundary terms bymultiplying our projective representations by phases, which leads us to instead make the11efinition ( L q φ )( k ) = ω ( s ( q ) − k, s ( q )) ω ( s ( q ) , s ( q ) − k ) φ ( s ( q ) − ks ( q )) . (3.3)To check that L q φ is a projective representation with respect to ι ∗ ω , we compute (abbre-viating g = s ( q ))( L q φ )( k )( L q φ )( k ) = ω ( g − k , g ) ω ( g − k , g ) ω ( g, g − k ) ω ( g, g − k ) φ ( g − k g ) φ ( g − k g ) , = ω ( g − k , g ) ω ( g − k , g ) ω ( g, g − k ) ω ( g, g − k ) ω ( g − k g, g − k g ) φ ( g − k k g ) , = ω ( k , k )( L q φ )( k k ) , (3.4)where we used the fact that ω is coclosed and ω ( g − k , g ) ω ( g − k , g ) ω ( g, g − k ) ω ( g, g − k ) ω ( g − k g, g − k g )= dω ( g, g − k , g ) dω ( g, g − k , g ) dω ( g, g − k g, g − k g ) dω ( k , k , g ) dω ( g, g − k k , g ) dω ( k , g, g − k g ) · ω ( g − k k , g ) ω ( k , k ) ω ( g, g − k k ) . (3.5)This suggests that we define the action of G/K on ˆ K ι ∗ ω by q · [ φ ] = [ L q φ ]. Indeed, wehave that (abbreviating s = s (1) ∈ K )( L φ )( k ) = ω ( s − k, s ) ω ( s, s − k ) φ ( s − ks ) = ω ( s − k, s ) ω ( s, s − k ) φ ( s − ) φ ( ks ) ω ( s − , ks ) , = ω ( s − k, s ) ω ( s, s − k ) ω ( s − , ks ) ω ( s, s − ) φ ( s ) − φ ( k ) φ ( s ) ω ( k, s ) , = dω ( s, s − k, s ) dω ( s, s − , ks ) φ ( s ) − φ ( k ) φ ( s ) = φ ( s ) − φ ( k ) φ ( s ) , (3.6)where we used the cocycle condition on ω (meaning the dω ’s evaluate to 1) and the projectiverepresentation properties (A.7) and (A.10). So L φ is isomorphic to φ and hence 1 · [ φ ] =[ φ ]. Similarly, we can show that L q ( L q φ ) is isomorphic to L q q φ , that if φ and φ areisomorphic, then so are L q φ and L q φ , and finally that if φ is irreducible then so is L q φ .This shows that we do have a good G/K action on ˆ K ι ∗ ω .Note that even when ι ∗ ω is trivial and K is a central subgroup of G (so that the usual G/K actions on K and ˆ K are trivial), this action we have constructed is not necessarilytrivial! In other words, the choice of prefactors required in (3.3) to make the projectivitywork out in general have the side effect that they can lead to a nontrivial action even in the12on-projective case, and this will turn out to be crucial in correctly accommodating someexamples.One example that we will explore in more detail below is to take G = Z × Z = { , a, b, c } , K = { , a } ∼ = Z , so G/K = { K, bK } ∼ = Z . We’ll choose our section to be s ( K ) = 1, s ( bK ) = b . If we turn on the nontrivial discrete torsion (A.15) in G , then ι ∗ ω is trivial(since ω ( a, a ) = 1). In this case ˆ K = { [ ρ + ] , [ ρ − ] } , where ρ ± are one-dimensional irreduciblerepresentations of K defined by their action on the generator, ρ ± ( a ) = ±
1. The usual actionof
G/K on ˆ K , as constructed in the previous section, would be trivial, but instead the actionwe have defined above satisfies( L bK φ )( a ) = ω ( b − a, b ) ω ( b, b − a ) φ ( b − ab ) = ω ( c, b ) ω ( b, c ) φ ( a ) = − φ ( a ) , (3.7)and hence bK · [ ρ ± ] = [ ρ ∓ ]. Thus, in this example, L bK exchanges different representations,and so ˆ K consists of only a single G/K orbit, with representative [ ρ + ] and trivial stabilizersubgroup H + ∼ = 1.More generally, define β ( ω ) : G/K × K −→ U (1) (3.8)by β ( ω )( q, k ) = ω ( ks ( q ) , s ( q ) − ) ω ( s ( q ) − , ks ( q )) . (3.9)(We will discuss this function in greater generality and detail in appendix C, where we willsee that for ι ∗ ω trivial, it represents an element of H ( G/K, H ( K, U (1)).) (When K iscentral, β ( ω ) is invariant under coboundaries and so gives a well-defined U (1) phase.) In allcases, we will see that β also defines a suitable homomorphism in appendix C. We will alsosee in appendix C that the phase factor in L q φ is β ( ω ) − , so that( L q φ )( k ) = β ( ω )( q, k ) − φ ( s ( q ) − ks ( q )) . (3.10)Later we will use β to give a more efficient approach to decomposition with discrete torsion.Now that we have defined our G/K action, we can decompose ˆ K ι ∗ ω into G/K orbits, andchoose representatives [ ρ a ], where ρ a is an irreducible projective representation of K chosento stand in for its isomorphism class. Let H a ⊆ G be the stabilizer subgroup of [ ρ a ]. Ourconjecture is that the G -orbifold with discrete torsion ω decomposes into a disjoint union oftheories, one for each orbit [ ρ a ], with orbifold group H a and discrete torsion ˆ ω a which wewill construct below. In other words, schematically we haveQFT ([ X/G ] ω ) = QFT (cid:32)(cid:34) X × ˆ K ι ∗ ω G/K (cid:35) ˆ ω (cid:33) , (3.11)or with slightly more detail, [ X/G ] ω = (cid:97) a [ X/H a ] ˆ ω a . (3.12)13t remains to construct the cocycles ˆ ω a ∈ Z ( H a , U(1)).We will follow a similar approach to what we did in the previous section. Given theprojective irreps ρ a of K , we will construct projective irreps (cid:101) ρ a for G a = π − ( H a ). The stepswill all be the same, but we will have to be more careful with ω -related phase factors. Thestatement that H a is the stabilizer of [ ρ a ] means that there exist intertwiners f a ( h ) ∈ GL( V a )for each h ∈ H a such that the diagrams V a ρ a ( k ) (cid:47) (cid:47) f a ( h ) (cid:15) (cid:15) V af a ( h ) (cid:15) (cid:15) V a ( L h ρ a )( k ) (cid:47) (cid:47) V a (3.13)commute for each k ∈ K . In other words, the intertwiners satisfy ρ a ( k ) f a ( h ) − = ω ( s ( h ) − k, s ( h )) ω ( s ( h ) , s ( h ) − k ) f a ( h ) − ρ a ( s ( h ) − ks ( h )) . (3.14)As before, each element g of G a can uniquely be written as g = s ( h ) k for some h ∈ H a and k ∈ K , and we can define a map (cid:101) ρ a : G a → GL( V a ) by (cid:101) ρ a ( s ( h ) k ) = ω ( s ( h ) , k ) − f a ( h ) − ρ a ( k ) . (3.15)The phase ω ( s ( h ) , k ) − is chosen to make the formulas below cleaner. A different choicewould lead to a projective representation whose cocycle differed by a coboundary.To check that this is indeed a projective representation and to find the associated cocycle,we first compute (cid:101) ρ a ( g g ) = ω ( s , s − s k s k ) − f − ρ a ( s − s k s k )= ω ( s , s − s k s k ) − ω ( s − s s , s − k s k ) − ω ( s − k s , k ) − × f − ρ a ( s − s s ) ρ a ( s − k s ) ρ a ( k ) , (3.16)where we used (A.7) twice to factorize ρ a ( s − s k s k ). Then we define an operator C a ( g , g ) = (cid:101) ρ a ( g ) (cid:101) ρ a ( g ) (cid:101) ρ a ( g g ) − . (3.17)One can show that this operator is a scalar multiple of the identity, C a ( g , g ) = (cid:101) ω a ( g , g ) , (3.18)specifically, C a ( g , g ) = ω ( s k , s k ) ω ( s s s − , s ) ω ( s , s ) ω ( s s − s − , s s s − ) f − f − f ρ a ( s s − s − ) , (3.19)14ence (cid:101) ρ a is a projective representation of G a with cocycle (cid:101) ω a . Since the details of thecalculation are not particularly enlightening, they are relegated to appendix B.It is convenient to write the cocycle which appears as (cid:101) ω a ( g , g ) = ω ( g , g )ˆ ω a ( h , h ) , (3.20)where ˆ ω a can be defined by combining (3.20) and (3.19) and using (A.10) to getˆ ω a ( h , h ) = ω ( s , s ) ω ( s s s − , s ) ρ a ( s s s − ) f − f f , (3.21)generalizing expression (2.20) for ordinary decomposition.From the definition (3.17) of C a ( g , g ), it now follows that (cid:101) ρ a is a projective representa-tion on G a with respect to the cocycle (cid:101) ω a , (cid:101) ρ a ( g ) (cid:101) ρ a ( g ) = (cid:101) ω a ( g , g ) (cid:101) ρ a ( g g ) . (3.22)The manifest associativity of the matrices (cid:101) ρ a ( g ) implies the cocycle condition for (cid:101) ω a , andcombined with the fact that ω is also coclosed we learn that ˆ ω a is a cocycle and hence definesa class in H ( H a , U(1)). This is the discrete torsion which will appear in the correspoondingfactor of the decomposition.Let’s note some aspects of this result. The factor ω ( s , s ) which appears in ˆ ω a is es-sentially the pullback of ω along the section s , i.e. ω ( s , s ) = ( s ∗ ω )( h , h ). Suppose that ω = π ∗ ¯ ω is the pullback of some cocycle ¯ ω ∈ H ( G/K,
U(1)). In that case we have s ∗ ω = ¯ ω and the other ω factor in (3.21) equals one, so we see that ˆ ω a is simply ¯ ω multiplied by theresult (2.20) that we obtained in the ordinary decomposition case. It’s interesting to observealso that in this case ω = π ∗ ¯ ω drops out entirely from (cid:101) ω a ; the projective representation (cid:101) ρ a is insensitive to the presence of ¯ ω . As this has been a somewhat long-winded discussion, in this section we summarize the high-lights. We can essentially break this into three cases, determined by the image of the discretetorsion ω ∈ H ( G, U (1)) under various maps defined by the short exact sequence (3.1).1. If ι ∗ ω (cid:54) = 0, then QFT ([ X/G ] ω ) = QFT (cid:32)(cid:34) X × ˆ K ι ∗ ω G/K (cid:35) ˆ ω (cid:33) , (3.23)15here ˆ K ι ∗ ω denotes the set of irreducible projective representations of K , defined withrespect to ι ∗ ω ∈ H ( K, U (1)), and ˆ ω refers to discrete torsion we defined in the previoussubsection.Without the twisting, when the cocycle ι ∗ ω vanishes, the number of irreducible ordinaryrepresentations of a finite group is counted by the number of conjugacy classes. Asreviewed in appendix A.2, the number of irreducible projective representations of afinite group K twisted by some ω ∈ H ( K, U (1)) is equal to the number of conjugacyclasses in K which consist of g ∈ K such that ω ( g, h ) = ω ( h, g ) for all h ∈ K thatcommute with g : | ˆ K ω | = |{ [ g ] | ω ( g, h ) = ω ( h, g ) for all h s.t. hg = gh }| . (3.24)We can slightly simplify the prediction (3.23) as follows. Let { ρ a } be a collection ofrepresentatives of the orbits of G/K on ˆ K ι ∗ ω . For each ρ a , let H a ⊂ G/K be thestabilizer of [ ρ a ] in ˆ K ι ∗ ω . Then, (cid:34) X × ˆ K ι ∗ ω G/K (cid:35) ˆ ω = (cid:97) a [ X/H a ] ˆ ω a , (3.25)where the details of the discrete torsion ˆ ω a was described in the previous subsection.2. Suppose that ω is annihilated by ι ∗ . As we have seen, the G/K action on the irreduciblerepresentations can still involve phases, which can exchange representations. Thosephases are determined by β ( ω ) ∈ H ( G/K, H ( K, U (1)), as described in appendix C,where following [35], and as we describe in detail in appendix C, β itself is the map inthe following exact sequence: H ( G/K, U (1)) π ∗ −→ Ker (cid:0) ι ∗ : H ( G, U (1)) −→ H ( K, U (1)) (cid:1) β −→ H ( G/K, H ( K, U (1))) . (3.28)If β ( ω ) is nontrivial in cohomology, and if K is in the center of G (so that in particularˆ K = H ( K, U (1))), then our prescription can be efficiently summarized asQFT ([
X/G ] ω ) = QFT (cid:18)(cid:20) X × Coker( β ( ω ))Ker( β ( ω )) (cid:21) ˆ ω (cid:19) , (3.29) This is one piece of the seven-term exact sequence (see e.g. [35])0 −→ H ( G/K, U (1)) π ∗ −→ H ( G, U (1)) ι ∗ −→ H ( K, U (1)) d −→ H ( G/K, U (1)) π ∗ −→ Ker ( ι ∗ ) β −→ H ( G/K, H ( K, U (1))) d −→ H ( G/K, U (1)) (3.26)that is a consequence of the Lyndon-Hochschild-Serre spectral sequence [36], slightly generalizing theinflation-restriction exact sequence (see e.g. [36], [37, example 6.8.3], [38, section I.6], [39, section 3.3])0 −→ H ( G/K, U (1)) π ∗ −→ H ( G, U (1)) ι ∗ −→ H ( K, U (1)) d −→ H ( G/K, U (1)) π ∗ −→ H ( G, U (1)) (3.27)(where we have specialized to trivial action on the coefficients). β ( ω ) as a homomorphism G/K → ˆ K ∼ = H ( K, U (1)), the irre-ducible representations of K , and where we gave the discrete torsion ˆ ω in the previoussubsection. (If K is not central then we can still appeal to the methods of the previ-ous subsection to understand decomposition; however, we will still classify examplesaccording to the (non)triviality of β ( ω ) in cohomology when ι ∗ ω is trivial.)We can simplify this expression slightly as follows. Let { ρ a } be a collection of repre-sentatives of the orbits of Ker β ( ω ) on the set of irreducible (honest) representationsof the cokernel. For each ρ a , let H a ⊂ Ker β ( ω ) be the stabilizer of ρ a , then (cid:20) X × Coker( β ( ω ))Ker( β ( ω )) (cid:21) ˆ ω = (cid:97) a [ X/H a ] ˆ ω a . (3.30)We shall argue in section 5 that in fact β defines an analogue of a quantum symmetry.3. Finally, suppose ι ∗ ω = 0 and β ( ω ) = 0. Then, there exists ω ∈ H ( G/K, U (1)) suchthat ω = π ∗ ω . In this case,QFT ([ X/G ] ω ) = QFT (cid:32)(cid:34) X × ˆ KG/K (cid:35) ˆ ω (cid:33) . (3.31)We will describe the discrete torsion ˆ ω later in this section. Briefly, if K is abelian,then ˆ ω = ω + ˆ ω , (3.32)where ˆ ω is the discrete torsion predicted by ordinary decomposition as in section 2,and ω = π ∗ ω .For example, if K is in the center of G , so that G/K acts trivially on ˆ K , then (cid:34) X × ˆ KG/K (cid:35) ω +ˆ ω = (cid:97) ρ ∈ ˆ K [ X/ ( G/K )] ω +ˆ ω ( ρ ) , (3.33)where ˆ ω ( ρ ) is the image under ρ of the extension class of G in H ( G/K, K ): ρ : H ( G/K, K ) −→ H ( G/K, U (1)) . (3.34)In the special case ω = 0, this reduces to the ordinary decomposition story reviewed insection 2.The third case above ( ω = π ∗ ω ) can be viewed more formally as a K -gerbe over [ X/G ] ω ,and so this case can be thought of as another example of decomposition, in the spirit of [8].One of the motivations for the original decomposition conjecture was the open stringsector. A group that acts trivially on bulk states, may act nontrivially on boundary states,17nd so the boundary states decompose into representations of the group. For example, Ktheory on gerbes decomposes, as was reviewed in [8]. Here, we have a slightly more compli-cated situation, in that discrete torsion in open strings ‘twists’ ordinary representations intoprojective representations [40, 41]. An open-string interpretation of that result over gerbesin the second two cases is somewhat beyond the scope of this article, but its interpretation isimmediate in the first case, and is the underlying physics reason for the appearance of ˆ K ι ∗ ω ,the set of irreducible projective representations of the trivially-acting subgroup.A quick consistency check closely akin to that for ordinary decomposition also applies tothese theories. Given a decomposition of the formQFT ([ X/G ] ω ) = QFT (cid:32)(cid:97) i [ X/G i ] ω i (cid:33) , (3.35)then it should be true that the total number of ω -twisted irreducible representations of G should match the sum of the number of ω i -twisted irreducible representations of each G i .(This is a consequence of the special case that X is a point.) This observation can sometimesbe handy when double-checking expressions for decomposition.Many of the resulting theories have multiple disconnected components. This reflects somesubtle “one-form” symmetries of the theory, as can typically be seen in two ways: • An orbifold has a one-form symmetry denoted BH when its genus-one twisted sectorscan be exchanged by tensoring with H bundles, for H abelian. In the presence ofdiscrete torsion, one must be careful, as discrete torsion weights different twisted sectorswith different phases, and so can break such symmetries, but there can be residualsymmetries remaining. • A sigma model on a disjoint union of k identical target spaces has a BH symmetry for | H | = k , as explained in [9].One consistency test of the proposal above is that, in examples, both sides (the originalorbifold and its final simplified description) have the same one-form symmetry. We shalldiscuss this phenomenon in examples as it arises.In the remainder of this paper, we will work through a number of examples, to illustrateand test the various features of the prediction above. Among other things, we will see thatin cases where a choice of normal subgroups K is ambiguous, the resulting prediction for thefield theories for all choices of K will be the same, so that the physics is well-defined. Forexample, the Z × Z orbifold of a point, with discrete torsion, can be considered either as anexample with K = Z × Z and ι ∗ ω (cid:54) = 0, as we discuss in section 4.1, or as an example with ‘Tensoring’ is perhaps not precisely the right term, as these are principal bundles not vector bundles,but nevertheless there is a product operation defined in the obvious way. = Z , ι ∗ ω = 0, and β ( ω ) (cid:54) = 0, as we discuss in section 5.1. From both perspectives, wewill get the same physics, that the theory is equivalent to a sigma model on a single point.The latter perspective ties into quantum symmetries in abelian orbifolds, as we discuss insection 5.1.In previous work such as [8], decomposition in orbifolds was tested extensively, by e.g.computing not just genus-one partition functions, but also partition functions at all genera,as well as projection operators, massless spectra, comparing open string states and K theory,and more. However, as the basic point of decomposition now seems well-established, in thispaper for brevity we will test our claims merely by computing genus-one partition functions. ι ∗ ω (cid:54) = 0 In this section we will consider examples of orbifolds [
X/G ] with discrete torsion ω , where ι ∗ ω (cid:54) = 0. Consider an orbifold of a point by a finite group G with discrete torsion. Since all of G actstrivially, we have K = G , hence ι ∗ ω = ω (cid:54) = 0 (by assumption).From section 3, this theory is predicted to be the same as that of N points, where N isthe number of irreducible projective representations of G with respect to ω . Using [42, equ’n(6.40)] N = 1 | G | (cid:88) gh = hg ω ( g, h ) ω ( h, g ) , (4.1)we see immediately that the genus-one partition function of the G orbifold with discretetorsion is always the same as that of N points.We collect here a number of special cases, to more explicitly check this claim. • G = Z × Z . As discussed in appendix D.1, there is only one irreducible projectiverepresentation of G , and given the cocycle given there, it is also straightforward tocompute that the genus-one partition function is 1. • G = ( Z ) . Assume that the discrete torsion is in one Z × Z factor. Then, the totalnumber of irreducible projective representations is two – the tensor product of one irre-ducible projective representation of Z × Z and two irreducible honest representationsof Z – and it is straightforward to compute that the genus-one partition function is 2.19he reader should also note in this case that because of the Z factor, this theory hasa B Z symmetry, which is consistent with the fact that it decomposes into two equalfactors [9]. • G = Z × Z . As discussed in appendix D.2, there are two irreducible projectiverepresentations of Z × Z with nontrivial discrete torsion, and utilizing the phases intable D.2, it is straightforward to compute that the genus-one partition function is Z = 1 | Z × Z | (cid:88) gh = hg (cid:15) ( g, h ) = 18 (16) = 2 = Z (two points) , (4.2)where (cid:15) ( g, h ) = ω ( g, h ) /ω ( h, g ), in agreement with the prediction.The reader should also note that the discrete torsion phases in table D.2 are periodicunder multiplication of group elements by the subgroup (cid:104) b (cid:105) ∼ = Z . As a result, thistheory has a B Z (one-form) symmetry, interchanging twisted sectors weighted by thesame phase, which is consistent with the fact that this theory describes a disjoint unionof two objects (points) [9]. • G = D . As discussed in appendix D.3, there are two irreducible projective represen-tations of D , and given the phases in table D.4, it is also straightforward to computethe genus-one partition function. Given that there are 28 ordered commuting pairs forwhich (cid:15) ( g, h ) = +1, and 12 ordered commuting pairs for which (cid:15) ( g, h ) = −
1, one finds Z ([point /D ]) = 1 | D | (28 −
12) = 2 = Z (two points) , (4.3)agreeing with the prediction for this case.As the result has two equivalent disconnected components, this theory has a B Z symmetry, just like the previous case. Unlike the previous case, this B Z does not seemto have a group-theoretic origin in D or its discrete torsion itself. Instead, becausethis is an orbifold of a point, the contribution of each twisted sector is determinedsolely by the discrete torsion phase (cid:15) ( g, h ) – so there is room for other permutations oftwisted sectors, unrelated to group theory, which appears to be what is happening inthis case, and one expects, in most cases of orbifolds of points. More to the point, italso seems to be true of the other examples in this list, so we will not explicit discussthese symmetries further in this section. • G = Z (cid:111) Z . As discussed in appendix D.4, there are four irreducible projectiverepresentations of Z (cid:111) Z . Utilizing the phases in table D.6, it is straightforward tocompute that the genus-one partition function is Z = 1 | Z (cid:111) Z | (cid:88) gh = hg (cid:15) ( g, h ) = 4 = Z (four points) , (4.4)in agreement with the prediction. 20 G = S . As discussed in appendix D.5, there are three irreducible projective represen-tations of S . Adding up the contributions, weighted by signs as given in table D.9,one finds Z ([point /S ]) = 1 | S | (72) Z (point) = (3) Z (point) , (4.5)agreeing with the prediction. [ X/ Z (cid:111) Z ] with discrete torsion and trivially-acting Z × Z subgroup In this section we consider an orbifold by the semidirect product of two copies of Z , Z (cid:111) Z .It can be shown (see appendix D.4) that H ( Z (cid:111) Z , U (1)) = Z , so there is one nontrivialvalue ω of discrete torsion, which we turn on in this orbifold. Furthermore, we take theaction of the subgroup K = (cid:104) x , y (cid:105) ∼ = Z × Z (in the notation of appendix D.4) to be trivial.Now, H ( K, U (1)) = Z in this case (see appendix D.2), so in principle, ι ∗ ω could benonzero. To check, we note that, from table D.6, for ι ∗ ω the genus-one phase (cid:15) ( x , y ) = − ι ∗ ω will be nontrivial, and as H ( K, U (1)) = Z , we see that there is only one nontrivialchoice.For completeness, let us also compute the full cocycle. Applying table D.5, the fullcocycle for ι ∗ ω is given in table 4.1.1 x y x y y x y y x y x i − i i − iy − i i − i ix y i − i i − i y x y i − i i − iy − i i − i ix y i − i i − i ι ∗ ω for ω the nontrivial element of H ( Z (cid:111)Z , U (1)), and ι : Z × Z (cid:44) → Z (cid:111) Z .The cocycle for ι ∗ ω matches the cocycle for the nontrivial element of H ( Z × Z , U (1))given in table D.1. (In principle, they only needed to match up to a coboundary; it is areflection of our conventions that they happen to match on the nose.)Thus, we see that ι ∗ ω (cid:54) = 0. Furthermore, G/K = Z . Our conjecture of section 3 then21redicts thatQFT ([ X/ Z (cid:111) Z ] ω ) = QFT (cid:32)(cid:34) X × ˆ K ι ∗ ω G/K (cid:35)(cid:33) = QFT (cid:32)(cid:34) X × ˆ K ι ∗ ω Z (cid:35)(cid:33) . (4.6)As discussed in appendix D.2, there are only two irreducible projective representations of K = Z × Z , corresponding to the conjugacy classes { } , { b = y } . Furthermore, the pref-actors (3.9) which appears in the definition of ( L q φ )( k ) is unity for all q and k . This combinedwith the fact that conjugation by G leaves both conjugacy classes invariant, implies that G/K acts trivially on ˆ K ι ∗ ω . Therefore, we make the prediction thatQFT ([ X/ Z (cid:111) Z ] ω ) = QFT (cid:32)(cid:34) X × ˆ K ι ∗ ω G/K (cid:35)(cid:33) = QFT (cid:16) [ X/ Z ] (cid:97) [ X/ Z ] (cid:17) , (4.7)two copies of the orbifold [ X/ Z ].It is straightforward to check this in genus-one partition functions. If we let x denote thegenerator of G/K = Z , then using the Z (cid:111) Z discrete torsion phases in table D.6, we find Z ([ X/ Z (cid:111) Z ] ω ) = 1 | Z (cid:111) Z | (cid:88) gh = hg (cid:15) ( g, h ) g h , (4.8)= 1616 (cid:20) + x + x + x x (cid:21) , (4.9)= 2 Z ([ X/ Z ]) = Z (cid:16) [ X/ Z ] (cid:97) [ X/ Z ] (cid:17) , (4.10)matching the prediction.As this theory has two equal disconnected components, the two copies of [ X/ Z ], it hasa B Z symmetry, which is reflected in the fact that ι ∗ ω ( g, h ) is invariant under the thesubgroup (cid:104) y (cid:105) ∼ = Z , as is visible in table 4.1. [ X/S ] with discrete torsion and trivially-acting Z × Z sub-group Consider [
X/S ] with nontrivial discrete torsion, and with Z × Z ⊂ S acting trivially on X . This is a consequence of the fact that the decomposition of a projective representation into irreducibleprojective representations is determined by the projective characters which are given by tracing over thematrices of the representation. One can show that the projective characters vanish except on ω -trivialconjugacy classes and hence two irreducible projective representations are isomorphic if and only if theircharacters agree on all ω -trivial conjugacy classes. In particular, if β ( ω )( q, k ) = 1 for all q and k and the G/K action preserves all of the ι ∗ ω -trivial conjugacy classes of K , then L q φ and φ will be isomorphic. Z × Z subgroup of S has elements1 , (12)(34) , (13)(24) , (14)(23) , (4.11)and the coset S / Z × Z = S , with elements1 = { , (12)(34) , (13)(24) , (14)(23) } , (4.12) a = { (123) , (134) , (243) , (142) } , (4.13) b = { (132) , (143) , (234) , (124) } , (4.14) c = { (12) , (34) , (1423) , (1324) } , (4.15) d = { (13) , (24) , (1234) , (1432) } , (4.16) e = { (14) , (23) , (1243) , (1342) } . (4.17)As elements of S , b = a − and a = b = 1 , c = d = e = 1 . (4.18)The only distinct elements that commute with one another are a and b .Analyzing this formally, define K = Z × Z , G = S . The restriction of the nontrivialelement of H ( S , U (1)) to K ⊂ S , ι ∗ ω , is the nontrivial element of H ( Z × Z , U (1)).From section 3, since there is only a single projective irreducible representation of K withnontrivial H ( K, U (1)) in this case, we predict thatQFT ([
X/S ] d . t . ) = QFT ([ X/S ]) . (4.19)Since H ( S , U (1)) = 0, there is no possibility of discrete torsion in the [ X/S ] orbifold.We can confirm this at the level of genus-one partition functions. Using table D.8, it isstraightforward to compute that Z ([ X/S ] d . t . ) = 1 | S | (cid:88) gh = hg (cid:15) ( g, h ) g h , (4.20)= 1 | S | (cid:88) gh = hg g h , (4.21)= Z ([ X/S ]) , (4.22)where g, h ∈ S . Thus, in this case, the S orbifold with discrete torsion and trivially-acting Z × Z has the same partition function as the S / Z × Z ∼ = S orbifold.23 .4 [ X/S ] with discrete torsion and trivially-acting A subgroup Consider [
X/S ] where K = A ⊂ S acts trivially and the orbifold has discrete torsion. Thecoset S /A = Z .The elements of A are the even permutations, which are transpositions of the form 1,( ab )( cd ). The odd permutations are of the form ( ab ), ( abcd ).Let us first work out the prediction of section 3, then compare to physics. First, it canbe shown that H ( A , U (1)) = Z , (4.23)(with trivial action on the coefficients,) and the restriction of the nontrivial element of H ( S , U (1)), ι ∗ ω , is the nontrivial element of H ( A , U (1)), as can be seen by restrictingthe genus-one phases in table D.8.Since ι ∗ ω (cid:54) = 0, in general terms, section 3 predictsQFT ([ X/S ] ω ) = QFT (cid:32)(cid:34) X × ˆ K ι ∗ ω G/K (cid:35) ˆ ω (cid:33) . (4.24)In this case, G/K = Z , and as H ( Z , U (1)) = 0, there will not be any discrete torsion con-tributions ˆ ω , but we do need to compute the number of irreducible projective representationsof K = A and the action of G/K on those representations.Now, let us compute the irreducible projective representations of A . The group A hasfour conjugacy classes, which we list below: { } , { (12)(34) , (13)(24) , (14)(23) } , { (123) , (421) , (243) , (341) } , { (132) , (412) , (234) , (314) } , (4.25)and from table D.8, the conjugacy classes consisting of g ∈ A such that ω ( g, h ) = ω ( h, g )for all h ∈ A commuting with g are the first one and the last two, { } , { (123) , (421) , (243) , (341) } , { (132) , (412) , (234) , (314) } . (4.26)For example, ω ((12)(34) , (13)(24)) = − ω ((13)(24) , (12)(34)) . (4.27)Thus, A has three irreducible projective representations with respect to the nontrivial ele-ment of H ( A , U (1)). 24onjugating by odd elements of S exchanges the two nontrivial conjugacy classes, forexample: (12)(123)(12) = (132) . (4.28)Since the prefactors β ( ω )( q, k ) are trivial, logic similar to footnote 5 allows us to concludethat the S /A = Z action will exchange two of those irreducible projective representations.As mentioned previously, section 3 predictsQFT ([ X/S ] d . t . ) = QFT (cid:32)(cid:34) X × ˆ A ,ι ∗ ω Z (cid:35)(cid:33) , (4.29)for ˆ A ,ι ∗ ω the set of irreducible projective representations with respect to the restriction of ω ∈ H ( S , U (1)). As a set, ˆ A ,ι ∗ ω has three elements, but two are interchanged by the Z ,so that QFT ([ X/S ] d . t . ) = QFT (cid:32)(cid:34) X × ˆ A ,ι ∗ ω Z (cid:35)(cid:33) = QFT (cid:16) X (cid:97) [ X/ Z ] (cid:17) . (4.30)Computing the genus-one partition function, we find Z ([ X/S ] d . t . ) = 1 | S | (cid:88) gh = hg (cid:15) ( g, h ) g h , (4.31)= 1 | S | (cid:34) (36) + (12) ξ + (12) ξ + (12) ξ ξ (cid:35) , (4.32)= Z ( X ) + Z ([ X/ Z ]) . (4.33)Thus, at least at the level of partition functions, we see[ X/S ] d . t . = X (cid:97) [ X/ Z ] , (4.34)matching the prediction of section 3.By way of comparison, for the orbifold [ X/S ] with trivially-acting A and no discretetorsion, ordinary decomposition impliesQFT ([ X/S ]) = QFT (cid:16) X (cid:97) [ X/ Z ] (cid:97) [ X/ Z ] (cid:17) , (4.35)i.e. it has one additional copy of [ X/ Z ] relative to the case with discrete torsion.25 .5 [ X/D × ( Z ) ] with discrete torsion and trivially-acting ( Z ) subgroup Consider [
X/G ] ω , where G = D × ( Z ) , and ω = p ∗ ω , where p : D × ( Z ) −→ ( Z ) (4.36)is the projection map onto the second factor, and ω is the nontrivial element of H ( Z × Z , U (1)). Let us assume that the trivially acting subgroup is K = ( Z ) ⊂ G , where one Z factor is the center of D and the remaining ( Z ) factors match those in G .Let p (cid:48) : K → ( Z ) be the projection onto the second two Z factors, so that the diagram G p (cid:47) (cid:47) Z × Z K ι (cid:79) (cid:79) p (cid:48) (cid:58) (cid:58) (cid:118)(cid:118)(cid:118)(cid:118)(cid:118)(cid:118)(cid:118)(cid:118)(cid:118) (4.37)commutes. Then we have ι ∗ ω = ι ∗ p ∗ ω = p (cid:48)∗ ω (cid:54) = 0 . (4.38)Following section 3, next consider the set ˆ K ι ∗ ω . This is essentially a product of twofactors, Z and ( Z ) , with the discrete torsion entirely in the ( Z ) factor. There is only oneirreducible projective representation of ( Z ) , as can be seen by counting conjugacy classes,but the remaining Z factor has two honest representations. Furthermore, since K is in thecenter of G , G/K acts trivially on ˆ K ι ∗ ω .Putting this together, we have the predictionQFT ([ X/G ] ω ) = QFT (cid:16) [ X/ Z × Z ] ˆ ω (cid:97) [ X/ Z × Z ] ˆ ω (cid:17) , (4.39)where ˆ ω , are elements of discrete torsion we determine next. We are given discrete torsion ω as the pullback of the generator of H ( Z × Z , U (1)). From equation (3.21) for ˆ ω , theratio of ω ’s multiplying the image of the extension class is determined by the values of ω on a section s : D / Z → D × ( Z ) . We can choose the section so that p ( s ( q )) = 1 ∈ Z × Z for all q ∈ D / Z . Then the ω ’s in (3.21) are trivial, and the discrete torsionˆ ω is determined solely by the image of the extension class. Since D is not a semidirectproduct Z (cid:111) ( Z ) , the extension class H (( Z ) , Z ) is nontrivial, and applying the twohonest irreducible representations of Z , we get that one of ˆ ω , is trivial, and the other isnontrivial. Thus, we predictQFT ([ X/G ] ω ) = QFT (cid:16) [ X/ Z × Z ] (cid:97) [ X/ Z × Z ] d . t . (cid:17) , (4.40)where exactly one of the [ X/ Z × Z ] summands has discrete torsion.26ext, let us compare to physics. We can view this example as a pair of successive orbifolds,first [ X/D ] without discrete torsion and with trivially-acting Z , and then a trivially-acting( Z ) orbifold with discrete torsion. Applying the ordinary version of decomposition [8], weknow that QFT ([ X/D ]) = QFT (cid:16) [ X/ Z × Z ] (cid:97) [ X/ Z × Z ] d . t . (cid:17) , (4.41)a disjoint union of two ( Z ) orbifolds where one copy has discrete torsion and the other doesnot. As we shall see in section 5.1, a ( Z ) orbifold with discrete torsion in which either Z factor acts trivially is just a realization of orbifolding by a quantum symmetry, and returnsthe original theory. Putting this together, we find thatQFT ([ X/G ]) ω = QFT (cid:16) [ X/ Z × Z ] (cid:97) [ X/ Z × Z ] d . t . (cid:17) . (4.42) ι ∗ ω = 0 and β ( ω ) (cid:54) = 0 We will see in this section that this case corresponds to one generalization of quantumsymmetries of orbifolds. Ordinarily, in quantum symmetries in abelian orbifolds, we havea G/K orbifold,
G/K abelian, which when orbifolded by a further K (abelian), with anaction only on twist fields and not the underlying space, one recovers the original theory.The action of K on G/K can be encoded in a set of phases K × G/K −→ U (1) , (5.1)which are equivalent to a map G/K −→ Hom(
K, U (1)) . (5.2)Taking into account the action of G/K on K and its induced action on H ( K, U (1)), this mapis a crossed homomorphism . This is equivalent to an element of H ( G/K, H ( K, U (1))).The map β sends β : H ( G, U (1)) −→ H (cid:0) G/K, H ( K, U (1)) (cid:1) , (5.3)and so it can be interpreted (at least in abelian cases) as giving us the quantum symmetryaction corresponding to an element of discrete torsion in the extension G .In more general cases, these maps all factor through abelianizations, H ( G/K, H ( K, U (1))) = H (( G/K ) ab , H ( K ab , U (1))) , (5.4) See also [2] for a different generalization of quantum symmetries to nonabelian orbifolds. Suppose a group G acts on an abelian group H . Then a crossed homomorphism φ : G → H is a mapsatisfying φ ( g g ) = g · φ ( g ) + φ ( g ). G orbifold interms of the abelianization of G , G ab = G/ [ G, G ].We shall elaborate on this perspective in the examples. [ X/ Z × Z ] with discrete torsion and trivially-acting Z Consider [ X/ Z × Z ] with nontrivial discrete torsion, and with one Z acting trivially on X .In this case, K = Z , G = Z , and since H ( K, U (1)) = 0, we have that ι ∗ ω = 0 trivially.Furthermore, since H ( G/K, U (1)) = 0 also, we see that β ( ω ) (cid:54) = 0.In principle, to apply the analysis of section 3, we need to compute β . In this example,since the phases are nontrivial, one can get to the result without detailed computation, butto illustrate the method, we work through the details here.From appendix C, recall that for q ∈ G/K , k ∈ K , β ( ω )( q, k ) = ω ( ks ( q ) , s ( q ) − ) ω ( s ( q ) − , ks ( q )) . (5.5)Here, write K = Z = (cid:104) a (cid:105) , G/K = Z = (cid:104) b (cid:105) . Without loss of generality, we take thesection s : Z → Z × Z to be given by, s ( q ) = q . Following the explicit cocycle given inappendix D.1, it is straightforward to compute that β ( ω )( q, k ) = 1 for either q = 1 or k = 1,and its only different value is for k = a and q = b , for which β ( ω )( b, a ) = ω ( ab, b ) ω ( b, ab ) = − . (5.6)Using H ( G/K, H ( K, U (1))) = H ( Z , Z ) = Z , (5.7)we interpret β ( ω ) as a map H ( G, U (1)) → H ( G/K, H ( K, U (1))). From the analysis of β above, we see that β ( ω )( q = 1 , − ) is the trivial map that sends all elements of K to theidentity, and β ( ω )( q = b, − ) is the identity, sending any element of K = Z to itself. Thus,we see that β ( ω ) is an isomorphism H ( G, U (1)) → H ( G/K, H ( K, U (1))), henceKer( β ) = 0 , Coker( β ) = 0 . (5.8)Given the form for β , we predict from our analysis in section 3 thatQFT ([ X/ Z × Z ] ω ) = X. (5.9) Unlike abelian cases, orbifolding by the abelianization does not return the original theory. A fix forthis was recently proposed in [2], in which the abelianization was extended to a unitary fusion category,orbifolding by which would then return the original theory. This is somewhat beyond the scope of thisarticle, however.
28t is straightforward to see this directly in physics. In fact, this example is the same asorbifolding [ X/ Z ] by the quantum symmetry Z [31]:[ X/ ˆ Z × Z ] d . t . = (cid:104) [ X/ Z ] / ˆ Z (cid:105) . (5.10)Following [32, section 8.5], if we let primes denote the twisted sectors of the ˆ Z orbifold, then + (cid:48) + = 12 (cid:18) + + + − + + + − + − − (cid:19) , (5.11) − (cid:48) + = 12 (cid:18) + + + − + − + − − − − (cid:19) , (5.12) + (cid:48)− = 12 (cid:18) + + − − + + + − − − − (cid:19) , (5.13) − (cid:48)− = 12 (cid:18) + + − − + − + − + − − (cid:19) . (5.14)Putting this together, the combination of the Z and ˆ Z orbifolds can be expressed as asingle ˆ Z × Z orbifold with phases (cid:15) ( a, g ; b, h ) where a, b ∈ ˆ Z , g, h ∈ Z , given by (cid:15) ( a, +; b, +) = +1 , (5.15) (cid:15) ( a, − ; b, − ) = ab, (5.16) (cid:15) ( a, +; b, − ) = a, (5.17) (cid:15) ( a, − ; b, +) = b, (5.18)where we have taken a, b ∈ {± } . This means that (cid:15) = − , − ) , ( − , +)) , ((+ , − ) , ( − , − )) , (( − , +) , ( − , − )) , (5.19)and the other three obtained by reversing the order. For all other elements of ˆ Z × Z , (cid:15) = +1. These are precisely the phases assigned by discrete torsion.Thus, we see that the orbifold [[ X/ Z ] / ˆ Z ] is the same as [ X/ ( ˆ Z × Z )], with phasesgiven by discrete torsion. In particular,[[ X/ Z ] / ˆ Z ] = [ X/ ( ˆ Z × Z )] d . t . = X, (5.20)agreeing with our prediction of section 3.As a final consistency check, let us redo the computation of section 3 without appealingto the description of this case in terms of kernels and cokernels of β ( ω ). Briefly, since K iscentral in G , the action of L q for q ∈ G/K is to map φ ( k ) to( L q φ )( k ) = β ( ω )( q, k ) − φ ( k ) . (5.21)29s computed above, and using the same section as there, β ( ω )( b, a ) = −
1, so we find L b φ (1) = + φ (1) , L b φ ( a ) = − φ ( a ) . (5.22)In other words, the effect of L b is to exchange the two irreducible representations of K = Z :the trivial representation becomes nontrivial, and vice-versa. As a result, there is only oneorbit of G/K on ˆ K , consisting of both the elements of ˆ K , and so one recovers the resultabove.In passing, the reader should note the critical role played by the phases encoded in β ( ω ):if those phases were all +1, then L q φ would be isomorphic to φ for all q ∈ G/K , and wewould have predicted instead that the resulting theory be a disjoint union of two copies of[ X/ Z ], instead of one copy of X , which is not what we see in physics. K In this subsection we will consider an example that is closely related to the previous one,essentially adding a nonabelian factor to G .Consider [ X/G ] where G = ˜ G × Z × Z , and K = ˜ G × Z . Let α denote the nontrivialelement of H ( Z × Z , U (1)), and let p : G → Z × Z be the projection, then take ω = p ∗ α .Since K is (potentially) nonabelian, we cannot simply resort to a computation of thekernel and cokernel of β ( ω ), but must work slightly harder. Let a generate the Z factorin K , and b generate the remaining Z in G , and pick a section s : G/K → G such that s (1) = 1, s ( bK ) = b . In general,( L q φ )( k ) = ω ( s ( q ) − , ks ( q )) ω ( ks ( q ) , s ( q ) − ) φ ( s ( q ) − ks ( q )) , (5.23)and the nontrivial case, for which the phase factor (the ratio of ω ’s) is nontrivial is q = bK :( L q φ )( ag ) = ω ( b, agb ) ω ( agb, b ) φ ( ag ) = α ( b, ab ) α ( ab, b ) φ ( ag ) = − φ ( ag ) , (5.24)( L q φ )( g ) = ω ( b, gb ) ω ( gb, b ) φ ( g ) = α ( b, b ) α ( b, b ) φ ( g ) = + φ ( g ) , (5.25)for g ∈ ˜ G . We see that the effect of L q is to exchange two copies of the irreducible represen-tations of ˆ G , indexed by the irreducible representations of Z . Taking that into account, wesee that the prediction for physics is thatQFT ([ X/G ] ω ) = QFT (cid:97) ˆ G X , (5.26)30he same decomposition as for [ X/ ˜ G ] with a trivial action of ˜ G and no discrete torsion – anexample of ordinary decomposition. This is easily verified in partition functions.Before going on, let us quickly walk through the details of the analogous computationin terms of kernels and cokernels of β ( ω ), which is not applicable since K is (potentially)nonabelian, to illustrate the problem. Let the section s : G/K → G be as above. Then it isstraightforward to compute that the only nontrivial elements β ( ω )( q, k ) are β ( ω )( bK, ag ) = − a the generator of the Z factor in K , and g ∈ ˜ G . As a result, β ( ω )(1 , − ) is theidentity element of H ( K, U (1)), and β ( ω )( bK, − ) is the product of the trivial element ofHom( ˜ G, U (1)) and the nontrivial element of H ( Z , U (1)). As a result, β ( ω ) is giving a map G/K → H ( K, U (1)) with zero kernel and cokernel Hom( ˜
G, U (1)), giving as many copies of X as the number of one-dimensional irreducible representations of ˜ G , but the correct answergives as many copies as the number of all irreducible representations of ˜ G , which need notbe one-dimensional in general. [ X/ Z × Z ] with discrete torsion and trivially-acting Z or Z × Z subgroup Consider [ X/ Z × Z ] with discrete torsion. In this section we will consider three closelyrelated examples in which a subgroup K , either Z or Z × Z , acts trivially:1. K = (cid:104) b (cid:105) ∼ = Z (in the presentation in appendix D.2),2. K = (cid:104) ab (cid:105) ∼ = Z ,3. K = (cid:104) a, b (cid:105) ∼ = Z × Z .In each of these three cases, ι ∗ ω = 0. In the first two cases, this is for the trivial reasonthat H ( K, U (1)) = 0. In the third case, H ( K, U (1)) (cid:54) = 0; however, when one computes ι ∗ ω in the third case, from table D.1, one finds it is given by1 a b ab a b ab G/K ∼ = Z , where G = Z × Z . As a result, H ( G/K, U (1)) =0, and so the image of π ∗ vanishes. This means the map β is injective, and so β ( ω ) is anontrivial element of H ( G/K, H ( K, U (1))) = Hom (cid:16) Z , ˆ K (cid:17) . (5.28)In each case, Ker β ( ω ) = 0 , Coker β ( ω ) = Z . (5.29)Let us check this description of β explicitly. As the analysis for all three cases is nearlyidentical, we give the details of β explicitly for only the first case, where K = (cid:104) b (cid:105) ∼ = Z . Wetake the section s : G/K → G to be s ( K ) = 1, s ( aK ) = a . We then compute using β ( ω )( q, k ) = ω ( ks ( q ) , s ( q ) − ) ω ( s ( q ) − , ks ( q )) . (5.30)For q = 1, one has immediately that β = +1, and if q = a , k = b or b , one also computes β = +1. The only two nontrivial cases are as follows: β ( ω )( q = a, k = b ) = ω ( ba, a ) ω ( a, ba ) = − , (5.31) β ( ω )( q = a, k = b ) = ω ( b a, a ) ω ( a, b a ) = − , (5.32)using the cocycle representative in appendix D.2. As a result, β ( ω )( q = 1 , − ) is the element of H ( K, U (1)) that maps all elements of K to the identity, while β ( ω )( q = a, − ) is the elementof H ( K, U (1)) that maps Z onto Z ⊂ U (1). As a homomorphism from G/K = Z to H ( K, U (1)) = Z , we see that β ( ω ) maps injectively into Z , hence the kernel and cokernelare as given above.Putting this together, the conjecture of section 3 then predictsQFT ([ X/G ] ω ) = QFT (cid:32)(cid:34) X × (cid:92) Coker β ( ω )Ker β ( ω ) (cid:35)(cid:33) = QFT (cid:16) X × ˆ Z (cid:17) = QFT (cid:16) X (cid:97) X (cid:17) . (5.33)Next, we shall check this prediction against partition functions. Briefly, in each of thesethree cases, Z ([ X/ Z × Z ] ω ) = 1 | Z × Z | (cid:88) gh = hg (cid:15) ( g, h ) g h , (5.34)= 168 , (5.35)= 2 Z ( X ) = Z (cid:16) X (cid:97) X (cid:17) , (5.36)agreeing with the prediction. 32 .4 [ X/ Z × Z ] with discrete torsion and trivially-acting Z sub-group Consider again [ X/ Z × Z ] with discrete torsion. In this section we will consider two closelyrelated examples in which a subgroup K ∼ = Z acts trivially:1. K = (cid:104) a (cid:105) ,2. K = (cid:104) ab (cid:105) ,(in the presentation of Z × Z described in appendix D.2).In both cases, ι ∗ ω = 0, for the trivial reason that H ( K, U (1)) = 0. Furthermore, in bothcases,
G/K ∼ = Z . As a result, H ( G/K, U (1)) = 0, hence the image of π ∗ vanishes, so themap β is injective. Thus, β ( ω ) is a nontrivial element of H ( G/K, H ( K, U (1))) = Hom( Z , Z ) . (5.37)In each case, Ker β ( ω ) = Z , Coker β ( ω ) = 0 . (5.38)Let us pause for a moment to check this description of β ( ω ) explicitly for the first case,for which K = (cid:104) a (cid:105) . (The second case will be identical.) We take the section s : G/K → G to be s ( b n K ) = b for n <
4, and then compute β ( ω )( q, k ) = ω ( ks ( q ) , s ( q ) − ) ω ( s ( q ) − , ks ( q )) . (5.39)It is straightforward to see that β ( ω )(1 , k ) = 1 for all k ∈ K , and β ( ω )( q,
1) = 1 for all q ∈ G/K . The nontrivial cases are as follows: β ( ω )( bK, a ) = ω ( ab, b ) ω ( b , ab ) = − , (5.40) β ( ω )( b K, a ) = ω ( ab , b ) ω ( b , ab ) = +1 , (5.41) β ( ω )( b K, a ) = ω ( ab , b ) ω ( b, ab ) = − , (5.42)using the explicit expression for the cocycle in appendix D.2. Thus, we see that β ( ω )(1 , − )and β ( b K, − ) both are the trivial element of ˆ K , that maps both elements of K = Z to theidentity, and β ( ω )( bK, − ), β ( ω )( b K, − ) both are the nontrivial element of ˆ K = ˆ Z . As ahomomorphism from G/K = Z to ˆ K ∼ = Z , we see that β ( ω ) maps onto Z , and so has thekernel and cokernel given above. 33utting this together, the conjecture of section 3 then predictsQFT ([ X/G ] ω ) = QFT (cid:32)(cid:34) X × (cid:92) Coker β ( ω )Ker β ( ω ) (cid:35)(cid:33) = QFT ([ X/ Z ]) . (5.43)In the first case, the kernel of β ( ω ) is generated by b , while in the second case, the kernelof β ( ω ) is generated by a , where a = { b, ab } is the coset generating G/K ∼ = Z .Next, comparing to partition functions, it is straightforward to compute in both casesthat Z ([ X/ Z × Z ] ω ) = 1 | Z × Z | (cid:88) gh = hg (cid:15) ( g, h ) g h = Z ([ X/ Z ]) , (5.44)with the effective Z orbifold action predicted above, verifying the conjecture. [ X/D ] with discrete torsion and trivially-acting Z subgroup Consider [
X/D ] with center Z ⊂ D acting trivially, and turn on discrete torsion in the D orbifold.As before, we begin by computing the prediction of section 3 for this case. Here K = Z and G = D , so G/K = Z × Z , and as H ( K, U (1)) = 0, we see immediately that ι ∗ ω = 0.We also compute H ( G/K, H ( K, U (1))) = H ( G/K, Z ) = H ( Z × Z , Z ) , (5.45)= Z × Z , (5.46) H ( G/K, U (1)) = Z , (5.47)The exact sequence (3.28) becomes H ( G/K, U (1)) π ∗ −→ Z β −→ H ( G/K, H ( K, U (1))) , (5.48)From table D.4, since for example (cid:15) ( a, (cid:54) = (cid:15) ( a, z ), we see that ω is not the pullback of anelement of discrete torsion in G/K = Z × Z . Thus, ω must map to a nontrivial elementof H ( G/K, H ( K, U (1)), a nontrivial homomorphism
G/K → Z . From table D.3, we seethat ω is trivial on the subgroup generated by b , but not that generated by a , from whichwe infer that the kernel of the map G/K → Z is the Z generated by b (the image of b in G/K ), and there is no cokernel. Thus, we predictQFT ([
X/D ] ω ) = QFT ([ X/ Z ]) , (5.49)where the Z is generated by b ∈ G/K . 34efore going on to check the physics, let us take a moment to explicitly check the de-scription of β ( ω ) above. In the present case, G/K = Z × Z = (cid:104) a, b (cid:105) , and we pick a section s : G/K → G given by s (1) = 1 , s ( a ) = a, s ( b ) = b, s ( ab ) = ab, (5.50)in the conventions of appendix D.3. As before, β ( ω )( q, k ) = ω ( ks ( q ) , s ( q ) − ) ω ( s ( q ) − , ks ( q )) , (5.51)and it is straightforward to see that β ( ω )(1 , k ) = 1 for all k ∈ K , and β ( ω )( q,
1) = 1 for all q ∈ G/K . The nontrivial cases are as follows: β ( ω )( a, z ) = ω ( za, a ) ω ( a, az ) = − , (5.52) β ( ω )( b, z ) = ω ( zb, bz ) ω ( bz, zb ) = +1 , (5.53) β ( ω )( ab, z ) = ω ( zab, ab ) ω ( ab, zab ) = − , (5.54)using table D.4. From this we read off that β ( ω )(1 , − ) and β ( ω )( b, − ) are the trivial elementsin ˆ K , that map all elements of K to the identity, and β ( ω )( a, − ) and β ( ω )( ab, − ) are thenontrivial elements of ˆ K . Thus, we see that β ( ω ) maps G/K = Z surjectively onto ˆ K = ˆ Z ,as predicted above, and so the kernel and cokernel are as above.Now, let us compare to physics. Using the phases in table D.4, it is straightforward tocompute that Z ([ X/D ] d . t . ) = 1 | D | (cid:88) gh = hg (cid:15) ( g, h ) g h , (5.55)= 12 (cid:34) + b + b + b b (cid:35) , (5.56)= Z ([ X/ Z ]) , (5.57)where the Z appearing is a subgroup of the effectively-acting Z × Z which is generated by b . This matches our prediction. [ X/ Z (cid:111) Z ] with discrete torsion and trivially-acting subgroups In this example we study three closely related examples, orbifolds by Z (cid:111) Z (the semidirectproduct of two copies of Z , discussed in appendix D.4) with discrete torsion, and thefollowing trivially-acting subgroups: 35. K = (cid:104) x (cid:105) ∼ = Z ,2. K = (cid:104) x (cid:105) ∼ = Z ,3. K = (cid:104) y , x (cid:105) ∼ = Z × Z , (a different Z × Z subgroup than was considered in section 4.2)in the presentation of appendix D.4.First, in each case, ι ∗ ω = 0. In the first two cases, this is a trivial consequence of the factthat H ( K, U (1)) = 0. In the third case, H ( Z × Z , U (1)) (cid:54) = 0, so in principle ι ∗ ω couldbe nonzero. To see that in fact ι ∗ ω = 0 in this case as well, one can compute the pullback ofthe genus-one phases, to find that they are all equal to one. (In fact, one can also compute ι ∗ ω directly from the representation of ω in table D.5. They only need to all be one up to acocycle, but in fact, one finds that ι ∗ ω ( g, h ) = 1 on the nose for all g, h ∈ K .)Furthermore, in each case one can also show that ω (cid:54) = π ∗ ω for an ω ∈ H ( G/K, U (1)):1. For K = (cid:104) x (cid:105) , G/K = Z × Z , which does admit discrete torsion. However, fromtable D.6, the genus-one phases are not invariant under x : for example, (cid:15) (1 , y ) (cid:54) = (cid:15) ( x , y ), and so ω cannot be a pullback from H ( Z × Z , U (1)).2. For K = (cid:104) x (cid:105) , G/K = Z , for which H ( G/K, U (1)) = 0, so ω (cid:54) = π ∗ ω for any ω since ω (cid:54) = 0.3. For K = Z × Z , G/K = Z , for which H ( G/K, U (1)) = 0, so again we see that ω (cid:54) = π ∗ ω for an ω ∈ H ( G/K, U (1)).In each of these three cases, β ( ω ) (cid:54) = 0, and determines the prediction.1. K = (cid:104) x (cid:105) is contained in the center of G , so β ( ω ) will be a nontrivial homomorphismfrom G/K ∼ = Z × Z to H ( K, U (1)) ∼ = Z . So we see that Coker β ( ω ) = 0 andthe kernel is either Z or Z × Z . By looking at the cocycles in table D.5, one canexplicitly verify that β ( ω )( yK, x ) = −
1, so yK / ∈ ker β ( ω ), and hence the kernel mustbe (cid:104) xK, y K (cid:105) ∼ = Z × Z .We can see this more explicitly as follows. Let s : G/K → G be the section s ( y n K ) = y n , s ( xy n K ) = xy n , for 0 ≤ n < , (5.58)in the conventions of appendix D.4. As before, β ( ω )( q, k ) = ω ( ks ( q ) , s ( q ) − ) ω ( s ( q ) − , ks ( q )) , (5.59)36nd it is straightforward to compute that the nontrivial elements are β ( ω )( xK, x ) = ω ( x x, x ) ω ( x , x ) = +1 , (5.60) β ( ω )( yK, x ) = ω ( x y, y ) ω ( y , x y ) = − , (5.61) β ( ω )( xyK, x ) = ω ( x xy, xy ) ω ( xy , x y ) = − , (5.62) β ( ω )( y K, x ) = ω ( x y , y ) ω ( y , x y ) = +1 , (5.63) β ( ω )( xy K, x ) = ω ( x xy , x y ) ω ( x y , x y ) = +1 , (5.64) β ( ω )( y K, x ) = ω ( x y , y ) ω ( y, x y ) = − , (5.65) β ( ω )( xy K, x ) = ω ( x xy , xy ) ω ( xy, x y ) = − , (5.66)using table D.5. From this it is straightforward to see that β ( ω )(1 , − ) , β ( ω )( xK, − ) , β ( ω )( y K, − ) , β ( ω )( xy K, − ) (5.67)all are the trivial element of ˆ K = ˆ Z , while β ( ω )( yK, − ) , β ( ω )( xyK, − ) , β ( ω )( y K, − ) , β ( ω )( xy K, − ) (5.68)all are the nontrivial element of ˆ K , so β ( ω ) : G/K → ˆ K is a surjective map. Sincemultiplying by xK leaves the map invariant, the kernel is (cid:104) xK, y K (cid:105) = Z × Z , asanticipated above. The cokernel, trivially, vanishes.We’ll have no discrete torsion ˆ ω in this case, since the cokernel vanishes. Equivalently,the two representations in ˆ K fall into a single G/K orbit, so we can take our represen-tative to be the trivial homomorphism that sends K to 1 ∈ U (1), and the stabilizer is H = ker β ( ω ) = Z × Z . If there was nontrivial discrete torsion, we would be able todetect it by computingˆ (cid:15) ( xK, y K ) = ˆ ω ( xK, y K )ˆ ω ( y K, xK ) = ω ( x, y ) ω ( y , x ) = 1 , (5.69)where we have made the last step using table D.5. So ˆ ω cannot be the nontrivial classin H ( Z × Z , U (1)).In summary, for this case we predictQFT ([ X/ Z (cid:111) Z ] ω ) = QFT ([ X/ Z × Z ]) , (5.70)a Z × Z orbifold without discrete torsion.37. The subgroup K = (cid:104) x (cid:105) is not contained in the center of G , so we can’t really use thecokernel of the map β ( ω ). Instead we will proceed more directly. First of all, we have G/K = (cid:104) yK (cid:105) ∼ = Z . We can choose a section s ( y n K ) = y n . Note that this is actuallya homomorphism, so the extension class is trivial, which is sensible since G is preciselythe semidirect product of K with G/K . Now ˆ K = { [ ρ m ] | ≤ m < } , where each ρ m is a homomorphism from K to U (1) defined by its action on the generator of K , ρ m ( x ) = i m . We compute the action of G/K on ˆ K using (3.3).( L yK ρ m )( x ) = ω ( y − x, y ) ω ( y, y − x ) ρ m ( y − xy ) = ω ( x y , y ) ω ( y, x y ) ρ m ( x ) = i − m − . (5.71)So we have yK · [ ρ ] = [ ρ ] , yK · [ ρ ] = [ ρ ] , yK · [ ρ ] = [ ρ ] , yK · [ ρ ] = [ ρ ] . (5.72)In other words, ˆ K breaks into two orbits under G/K , each with stabilizer (cid:104) y K (cid:105) ∼ = Z .Since H ( Z , U (1)) = 0, we don’t need to compute ˆ ω a .We are left with a prediction ofQFT ([ X/ Z (cid:111) Z ] ω ) = QFT (cid:16) [ X/ Z ] (cid:97) [ X/ Z ] (cid:17) , (5.73)two copies of the [ X/ Z ] orbifold.3. For K = (cid:104) y , x (cid:105) , again we are not in the center. Here we have G/K = (cid:104) yK (cid:105) ∼ = Z , andwe can choose the section s ( K ) = 1, s ( yK ) = y (note that in this case the extensionclass is nontrivial since e ( yK, yK ) = y ). The irreducible representations of K aregiven by ˆ K = { [ ρ m,n ] | ≤ m < , ≤ n < } where the homomorphisms ρ m,n aredefined by ρ m,n ( x ) = i m , ρ m,n ( y ) = ( − n . (5.74)To compute the action of G/K on ˆ K we compute( L yK ρ m,n )( x ) = ω ( y − x, y ) ω ( y, y − x ) ρ m,n ( y − xy ) = i − m − , (5.75)( L yK ρ m,n )( y ) = ω ( y − y , y ) ω ( y, y − y ) ρ ( y − y y ) = ( − n . (5.76)From this we learn that ˆ K breaks into four distinct orbits { [ ρ m,n ] , [ ρ − m,n ] } , whereboth m and n can be 0 or 1. Each orbit has a trivial stabilizer H m,n = 1. Since thestabilizer groups are trivial, there is of course no possible discrete torsion ˆ ω , and soour prediction is QFT ([ X/ Z (cid:111) Z ] ω ) = QFT (cid:32)(cid:97) X (cid:33) , (5.77)a disjoint union of four copies of X . 38n each case, these predictions can be verified by genus-one partition function computa-tions. Briefly, given Z ([ X/ Z (cid:111) Z ] ω ) = 1 | Z (cid:111) Z | (cid:88) gh = hg (cid:15) ( g, h ) g h , (5.78)and using the discrete torsion phases for Z (cid:111) Z given in table D.6, it is straightforward toshow that1. For K = (cid:104) x (cid:105) , Z ([ X/ Z (cid:111) Z ] ω ) = Z ([ X/ Z × Z ]) , (5.79)a Z × Z orbifold without discrete torsion,2. For K = (cid:104) x (cid:105) , Z ([ X/ Z (cid:111) Z ] ω ) = 2 Z ([ X/ Z ]) = Z (cid:16) [ X/ Z ] (cid:97) [ X/ Z ] (cid:17) , (5.80)3. For K = (cid:104) y , x (cid:105) , Z ([ X/ Z (cid:111) Z ] ω ) = 4 Z ( X ) = Z (cid:32)(cid:97) X (cid:33) , (5.81)in each case matching the prediction. ι ∗ ω = 0 and ω = π ∗ ω In this section we will look at examples of G orbifolds in which the discrete torsion is apullback from G/K , where the subgroup K acts trivially. Note that in this case we have β ( ω ) = 0, i.e. the prefactors in (3.3) are trivial. [ X/ Z × Z ] with discrete torsion and trivially-acting Z sub-group In this example we consider [ X/ Z × Z ] where the subgroup (cid:104) b (cid:105) ∼ = Z (in the conventionsof appendix D.2) acts trivially.As given, K = Z = (cid:104) b (cid:105) and G = Z × Z ,, so as H ( K, U (1)) = 0, we have that ι ∗ ω = 0 trivially. Furthermore G/K = Z × Z , and as both H ( G/K, U (1)) = Z and39 ( G, U (1)) = Z , there is a chance that the discrete torsion in Z × Z is a pullback from Z × Z .Write G/K = Z × Z = (cid:104) a, b (cid:105) , so that π ( a ) = π ( ab ) = a, π ( b ) = π ( b ) = b, π ( ab ) = π ( ab ) = ab, π ( b ) = 1 . (6.1)Using the cocycle for the nontrivial element of H ( G, U (1)) given in table D.1, we see that, ω (1 , g ) = ω ( b , g ) , ω ( a, g ) = ω ( ab , g ) , ω ( b, g ) = ω ( b , g ) , ω ( ab, g ) = ω ( ab , g ) (6.2)(and symmetrically for ω ( y, x ) instead of ω ( x, y )) for any element g ∈ G = Z × Z , whichindicates that ω = π ∗ ω for ω ∈ H ( G/K, U (1)), which is given explicitly in the table below:1 a b ab a i − ib − i iab i − i H ( Z × Z , U (1)) (see for exampleappendix D.1), so we see that ω = π ∗ ω for the nontrivial element ω ∈ H ( G/K, U (1)).Since
G/K acts trivially on ˆ K , the conjecture of section 3 is going to predict two compo-nents, two copies of [ X/ Z × Z ]. We need to compute the discrete torsion in each component.One contribution to that discrete torsion will be from ω . There is another potential contri-bution, from the image of the extension class of G , an element of H ( G/K, K ), under each ρ ∈ ˆ K . Since G is abelian, the argument below (2.5) shows that this additional contribu-tion vanishes. As a result, the only contribution to discrete torsion on each [ X/ Z × Z ]component is from ω .Putting this together, from the conjecture of section 3, we predict thatQFT ([ X/G ] ω ) = QFT (cid:32)(cid:34) X × ˆ KG/K (cid:35) ω (cid:33) = QFT (cid:16) [ X/ Z × Z ] d . t . (cid:97) [ X/ Z × Z ] d . t . (cid:17) , (6.3)or more simply, two copies of the Z × Z orbifold with discrete torsion ω ∈ H ( Z × Z , U (1)).Next, we shall check this prediction at the level of partition functions. Using table D.2,it is straightforward to compute that the genus-one partition function of the Z × Z orbifold40ith discrete torsion ω is given by Z ([ X/ Z × Z ] ω )= 1 | Z × Z | (cid:88) gh = hg (cid:15) ( g, h ) g h , (6.4)= 12 (cid:34) + a + b + ab + a + a a + b + b b + ab + ab ab − a b − a ab − b a − b ab − ab a − ab b (cid:35) , (6.5)= 2 Z ([ X/ Z × Z ] d . t . ) , (6.6)= Z (cid:16) [ X/ Z × Z ] d . t . (cid:97) [ X/ Z × Z ] d . t . (cid:17) , (6.7)precisely matching the prediction for this case.By way of comparison, ordinary decomposition ( ω = 0) says something very similar inthis case: Z ([ X/ Z × Z ]) = Z (cid:32)(cid:97) [ X/ Z × Z ] (cid:33) , (6.8)with no discrete torsion on either side. This is consistent with our computation that theimage of the extension class of Z × Z , an element of H ( Z × Z , Z ), in H ( Z × Z , U (1)),necessarily vanishes for all ρ ∈ ˆ K . Thus, in both this case and in the example above, thetwo copies of [ X/ Z × Z ] have the same discrete torsion. [ X/ Z (cid:111) Z ] with discrete torsion and trivially-acting Z sub-group In this section we consider a Z (cid:111) Z orbifold (the semidirect product of two copies of Z )with discrete torsion in which a subgroup K = (cid:104) y (cid:105) ∼ = Z acts trivially, where Z (cid:111) Z isgenerated by x and y in the notation of appendix D.4. There is only one nontrivial valueof discrete torsion since H ( Z (cid:111) Z , U (1)) = Z . Also, it will be useful to observe that K = (cid:104) y (cid:105) is a subgroup of the center of Z (cid:111) Z .First, note that G/K = D . Relating to the notation of appendix D.3, in which D hasgenerators a , b , we identify a = { y, y } , b = { x, xy } . It is straightforward to check, forexample, that b = { x , x y } generates the center, and that a = 1 = b , ba = { xy, xy } = ab , (6.9)as expected for D . 41ext, since H ( K, U (1)) = 0, we trivially have ι ∗ ω = 0.Since H ( G/K, U (1)) = Z , the same as H ( G, U (1)), it is possible that the element ofdiscrete torsion in the Z (cid:111) Z orbifold is a pullback from D . Indeed, the phases in table D.6are symmetric under K : for example, (cid:15) ( g, h ) = (cid:15) ( gy , h ) (6.10)and symmetrically, hence they pull back from G/K = D . If we let ω be the nontrivialelement of discrete torsion in D , represented by a cocycle given in table D.3, then thepullback π ∗ ω is given in table 6.1.1 b = x b = x b = x a = y ab = x y ab = x y ab = xy b = x ξ − ξ − ξ − ξb = x i + i − i − ib = x − ξ ∗ − ξ ∗ − ξ ∗ + ξ ∗ a = y − ξ − i ξ ∗ − ξ ∗ + i ξab = x y − ξ − i − ξ ∗ − ξ − ξ ∗ + iab = x y − ξ + i − ξ ∗ − i − ξ − ξ ∗ ab = xy ξ + i − ξ ∗ ξ ∗ − i − ξ π ∗ ω for ω the nontrivial element of D discrete torsion and π : Z (cid:111) Z → D . We define ξ = exp(+6 πi/ ξ of table D.5.Taking into account the y periodicities, it is straightforward to verify that table 6.1,describing π ∗ ω , matches the cocycle for the nontrivial element of H ( Z (cid:111) Z , U (1)) givenin table D.5. (Of course, they need only match up to coboundaries, but our conventions aresuch that they match on the nose.) Therefore, we see explicitly that the Z (cid:111) Z discretetorsion ω = π ∗ ω .Applying section 3, we can now see that the [ X/ Z (cid:111) Z ] orbifold with discrete torsionshould be equivalent to a disjoint union of two [ X/D ] orbifolds, using the fact that in thiscase, D = G/K acts trivially on ˆ K .It remains to compute the discrete torsion on each [ X/D ] summand. Part of that discretetorsion will be ω , but there could also be another contribution, the image of the extensionclass H ( D , Z ) under each of the irreducible representations of K = Z . Since Z (cid:111) Z isnot Z (cid:111) D (in fact, Z × D , since the Z is central), the extension class in H ( D , Z ) isnontrivial. However, we claim that any map induced by a representation into H ( D , U (1))vanishes. This is because the two conjugacy classes corresponding to elements of ˆ K , namely { } and { y } , areinvariant under conjugation by elements of G . s : G/K → G (so π ( s ( q )) = q for all q ∈ G/K ), and then define the extension class by e ( q , q ) = s ( q ) s ( q ) s ( q q ) − ∈ K, for all q , q ∈ G/K. (6.11)Different choices of section will give extension classes that differ by something exact. Forthe current example, we could take the section s ( a n b m ) = y n x m , where 0 ≤ n < ≤ m <
4. Then we compute e ( b k , b m ) = e ( ab k , b m ) = e ( b k , ab m ) = 1 , e ( ab k , ab m ) = y . (6.12)Now there are two possible irreducible representations of K = Z , a trivial one ρ which sends y to 1, and a nontrivial one ρ which sends y to −
1. Applying these to the extension class e gives us a pair of 2-cocycles, ˆ ω a ( q , q ) = ρ a ( e ( q , q )). Clearly ˆ ω is the trivial 2-cocycle.The other possibility ˆ ω is not trivial, but it is exact, and in particular it is symmetric inits arguments, ˆ ω ( q , q ) = ˆ ω ( q , q ), which implies that the corresponding discrete torsioncontribution vanishes.Putting these pieces together, we can now make the prediction thatQFT ([ X/ Z (cid:111) Z ] ω ) = QFT (cid:32)(cid:34) X × ˆ KD (cid:35) ω (cid:33) = QFT (cid:16) [ X/D ] ω (cid:97) [ X/D ] ω (cid:17) . (6.13)The fact that the QFT decomposes into two identical summands (universes), and so hasa B Z symmetry, reflects the group-theoretic fact that the discrete torsion ω in Z (cid:111) Z isinvariant under (cid:104) y (cid:105) ∼ = Z : ω ( g, h ) = ω ( gy , h ) = ω ( g, hy ) = ω ( gy , hy ) , (6.14)as can be seen in table D.5. (Of course, such a relation need only hold up to cocycles, butin our conventions it holds on the nose.)Next, we shall compare this prediction to physics results. Using table D.6 of phases fordiscrete torsion in Z (cid:111) Z orbifolds, it is straightforward to compute that the genus-oneorbifold partition function is Z ([ X/ Z (cid:111) Z ] ω ) = 1 | Z (cid:111) Z | (cid:88) gh = hg (cid:15) ( g, h ) g h , (6.15)= 2 Z ([ X/D ] ω ) = Z (cid:16) [ X/D ] ω (cid:97) [ X/D ] ω (cid:17) , (6.16)or two copies of the D orbifold, each with discrete torsion, confirming the prediction.By way of comparison, ordinary decomposition ( ω = 0) says something very similar inthis case: Z ([ X/ Z (cid:111) Z ]) = Z (cid:32)(cid:97) [ X/D ] (cid:33) , (6.17)with no discrete torsion on either side. 43 Mixed examples
In this section we will describe examples that encompass further aspects of the conjecture,depending upon the value of discrete torsion.Each of these examples will be a Z k × Z k orbifold. To that end, recall that the phaseassigned by discrete torsion ω ∈ Z k in a Z k × Z k orbifold is determined as follows. Let g ∈ Z k × Z k be determined by two integers ( a, b ), a, b ∈ { , · · · , k − } . Then, associated tothe twisted sector ( g, h ) = ( a, b ; a (cid:48) , b (cid:48) ) is the phase [43, equ’n (2.2)] (cid:15) ( a, b ; a (cid:48) , b (cid:48) ) = ξ ab (cid:48) − ba (cid:48) , (7.1)where ξ is a k th root of unity corresponding to the value of discrete torsion. Z × Z orbifold with discrete torsion and trivially-acting Z × Z Consider the case G = Z × Z , with trivially-acting K = Z × Z , so that G/K = Z × Z .Suppose the orbifold has discrete torsion, ω ∈ H ( Z × Z , U (1)) = Z , which we encode ina fourth root of unity labelled ξ .Let us first compute our predictions from section 3 for the various possible values ofdiscrete torsion. Let ξ be a fourth root of unity which encodes the value of discrete torsion.Note that for any value of ξ , ι ∗ ω = 0. To see this, it suffices to consider the genus-onephases (7.1). If ( a, b ) ∈ Z × Z , with a, b ∈ { , } , then the inclusion ι is given by ι ( a, b ) = (2 a, b ) , (7.2)and so as a result, ι ∗ (cid:15) ( a, b ; a (cid:48) , b (cid:48) ) = (cid:15) (2 a, b ; 2 a (cid:48) , b (cid:48) ) = ξ ab (cid:48) − ba (cid:48) ) . (7.3)Since ab (cid:48) − ba (cid:48) ∈ {− , , +1 } , (7.4)and ξ = 1, we see that all the phases ι ∗ (cid:15) ( a, b ; a (cid:48) , b (cid:48) ) = 1, so the cocycle ι ∗ ω must be trivial.Next, let us consider the four possible values of discrete torsion. • First, consider the case of vanishing discrete torsion. Then, decomposition [8] applies,and we haveQFT ([ X/ Z × Z ]) = QFT (cid:32)(cid:34) X × ˆ K Z × Z (cid:35)(cid:33) = QFT (cid:32)(cid:97) [ X/ Z × Z ] (cid:33) , (7.5)44 disjoint union of four copies of [ X/ Z × Z ], where here ˆ K denotes the set of irreduciblehonest representations of K . • ξ = −
1. In this case, using the phases in equation (7.1), it is straightforward to seethat the discrete torsion in Z × Z is a pullback of discrete torsion in Z × Z . (Since ξ = +1, the phases (7.1) are invariant under shifting any of a , b , a (cid:48) , b (cid:48) by 2, or moresimply, invariant under K , reflecting an underlying BK one-form symmetry.) As aresult, decomposition [8] applies again, and we haveQFT ([ X/ Z × Z ] ω ) = QFT (cid:32)(cid:34) X × ˆ K Z × Z (cid:35) d . t . (cid:33) = QFT (cid:32)(cid:97) [ X/ Z × Z ] d . t . (cid:33) , (7.6)a disjoint union of four copies of [ X/ Z × Z ] d . t . , where here ˆ K denotes the set ofirreducible honest representations of K . • ξ = −
1. In these cases, the phases (7.1) are not invariant under K , since shifting anyof a , b , a (cid:48) , b (cid:48) changes the genus-one phase (7.1) by a sign. For this reason, the cocycle ω (cid:54) = π ∗ ω for any ω ∈ H ( Z × Z , U (1)). Instead, in these cases, β ( ω ) (cid:54) = 0. Now, β ( ω )is an element of H ( G/K, H ( K, U (1))) = Hom ( Z × Z , Z × Z ) , (7.7)so enumerating possibilities, if β ( ω ) is nontrivial, one quickly deduces thatKer β ( ω ) = Coker β ( ω ) ∈ { , Z , Z × Z } . (7.8)Given that the genus-one phases (7.1) are entirely nontrivial, they appear to defineisomorphisms, hence Ker β ( ω ) = Coker β ( ω ) = 0 , (7.9)and so for these cases, we predictQFT ([ X/ Z × Z ] ω ) = QFT (cid:32)(cid:34) X × (cid:92) Coker β ( ω )Ker β ( ω ) (cid:35)(cid:33) = QFT ( X ) . (7.10)Next, we compute the genus-one partition functions in the different cases, to confirm thepredictions above. In the current example, since the Z × Z ⊂ Z × Z acts trivially, weshould compute the numerical factors multiplying each effective Z × Z orbifold twistedsector. These are listed in table 7.1, where we use the notation ( a, b ; a (cid:48) , b (cid:48) ) for a, b, a (cid:48) , b (cid:48) ∈{ , } to indicate an effective Z × Z twisted sector, and where ξ is a fourth root of unity,corresponding to the choice of Z × Z discrete torsion.Our results for genus-one partition functions are as follows.450 ,
0; 0 ,
0) 16 (1 ,
0; 1 ,
0) 8 ξ (1 + ξ )(1 ,
0; 0 ,
0) 8(1 + ξ ) (1 ,
1; 0 ,
1) 8 ξ (1 + ξ )(0 ,
1; 0 ,
0) 8(1 + ξ ) (1 ,
0; 1 ,
1) 8 ξ (1 + ξ )(0 ,
0; 1 ,
0) 8(1 + ξ ) (0 ,
1; 1 ,
1) 8 ξ (1 + ξ )(0 ,
0; 0 ,
1) 8(1 + ξ ) (1 ,
1; 1 ,
1) 8(1 + ξ )(1 ,
1; 0 ,
0) 8(1 + ξ )(1 ,
0; 1 ,
0) 8(1 + ξ )(1 ,
0; 0 ,
1) 8 ξ (1 + ξ )(0 ,
1; 1 ,
0) 8 ξ (1 + ξ )(0 ,
1; 0 ,
1) 8(1 + ξ )(0 ,
0; 1 ,
1) 8(1 + ξ )Table 7.1: Multiplicities and phases of effective Z × Z twisted sectors in Z × Z orbifoldwith discrete torsion. • Case ξ = 1. In this case, there is no discrete torsion, and using the results in table 7.1,it is straightforward to show that Z ([ X/ Z × Z ] d . t . ) = 1 | Z × Z | (cid:88) a,b,a (cid:48) ,b (cid:48) ∈{ , ··· , } (cid:15) ( a, b ; a (cid:48) , b (cid:48) ) ( a,b ) a (cid:48) ,b (cid:48) , (7.11)= (4) Z ([ X/ Z × Z ]) , (7.12)consistent with the prediction of decomposition [8] that this orbifold be equivalent toa disjoint union of four copies of an effective [ X/ Z × Z ] orbifold. • Case ξ = −
1. In this case, using the results in table 7.1, it is straightforward to showthat Z ([ X/ Z × Z ] d . t . ) = 1 | Z × Z | (cid:88) a,b,a (cid:48) ,b (cid:48) ∈{ , ··· , } (cid:15) ( a, b ; a (cid:48) , b (cid:48) ) ( a,b ) a (cid:48) ,b (cid:48) , (7.13)= (4) Z ([ X/ Z × Z ] d . t . ) , (7.14)consistent with the statement that in this case, the Z × Z orbifold with discretetorsion is equivalent to a disjoint union of four copies of an effective Z × Z orbifoldwith discrete torsion. • Cases ξ = −
1. In these cases, using the results in table 7.1, it is straightforward toshow that Z ([ X/ Z × Z ] d . t . ) = 1 | Z × Z | (cid:88) a,b,a (cid:48) ,b (cid:48) ∈{ , ··· , } (cid:15) ( a, b ; a (cid:48) , b (cid:48) ) ( a,b ) a (cid:48) ,b (cid:48) , (7.15)= 116 (cid:34) (16) (0 ,
0) (0 , (cid:35) = Z ( X ) , (7.16)46onsistent with the statement that in these two cases, the Z × Z orbifold with discretetorsion is equivalent to no orbifold at all.This confirms our predictions.In the first two cases, in which ξ = +1, the Z × Z orbifold has a natural B ( Z × Z )symmetry, as the discrete torsion phases (cid:15) ( a, b ; a (cid:48) , b (cid:48) ) are invariant under incrementing anyof a, b, a (cid:48) , b (cid:48) by 2. This is a symmetry-based reason why there is a decomposition [8] in thesecases.Thus, depending upon the value of discrete torsion, we see that the Z × Z orbifold isequivalent to one of the following three possibilities: (cid:97) [ X/ Z × Z ] , (cid:97) [ X/ Z × Z ] d . t . , X. (7.17) Z × Z orbifold with discrete torsion and trivially-acting Z × Z In this section, we will consider a potentially more complex case of a Z k × Z k orbifold.Here, there are more possible values of discrete torsion in the effectively-acting orbifoldsummands/universes than in the previous (set of) examples, which will enable us to conductmore thorough tests of predictions for discrete torsion in summands/universes.Much as in the last section, here ι ∗ ω = 0 in all cases. We can see this as follows. Themap ι embeds Z × Z in Z × Z as ι ( a, b ) = (4 a, b ) , (7.18)hence from equation (7.1), we see that the pullback of the genus-one phase is( ι ∗ (cid:15) )( a, b ; a (cid:48) , b (cid:48) ) = (cid:15) (4 a, b ; 4 a (cid:48) , b (cid:48) ) = ξ ab (cid:48) − a (cid:48) b ) = 1 , (7.19)as in this case ξ = 1 in all cases. Since the phases are trivial, we see that ι ∗ ω = 0.Next, let us (briefly) predict the decomposition for various cases, following section 3:1. ξ = +1. In this case, as for example (cid:15) ( a + 4 , b ; a (cid:48) , b (cid:48) ) = ξ ( a +4) b (cid:48) − a (cid:48) b = ξ ab (cid:48) − a (cid:48) b = (cid:15) ( a, b ; a (cid:48) , b (cid:48) ) , (7.20)we see that ω = π ∗ ω , where ω is discrete torsion in the effectively-acting Z × Z orbifold.In principle, the discrete torsion is then given by ω + ˆ ω , where ˆ ω is that predicted bythe original decomposition. In this case, the group G is abelian, hence ˆ ω = 0 on each47omponent, as observed in section 2. Mathematically, the same result can be obtainedby observing that since in this case the extension class is symmetric, as can be mademanifest by picking matching sections, one hasˆ ω ( a, b ) = ˆ ω ( b, a ) , (7.21)and so the corresponding genus-one phases are trivial.Therefore, we predict that for ξ = +1,QFT ([ X/ Z × Z ] ω ) = QFT (cid:32)(cid:97) [ X/ Z × Z ] ω (cid:33) . (7.22)2. ξ = −
1. For similar reasons as the case above, here ω (cid:54) = π ∗ ω , so instead β ( ω ) (cid:54) = 0.Judging from the phases, one expects that β describes a map of the generators of each Z to the generator of each Z , so that Coker β ( ω ) = 1, and Ker β ( ω ) = Z × Z .Thus, we predict QFT ([ X/ Z × Z ] ω ) = QFT ([ X/ Z × Z ] ˆ ω ) , (7.23)a single copy of [ X/ Z × Z ].To complete the prediction, we need to compute the discrete torsion, following equa-tion (3.21). What really matters are the phases ˆ (cid:15) ( q , q ) = ˆ ω ( q , q ) / ˆ ω ( q , q ), where q , q ∈ ker β ( ω ). But since G is abelian, it’s easy to see that the only part of theexpression (3.21) that doesn’t drop out of ˆ (cid:15) is the contribution of the ω ( s , s ) factor.In other words, we have ˆ (cid:15) ( q , q ) = ω ( s , s ) ω ( s , s ) . (7.24)Now Ker β ( ω ) here consists of ( a, b ) = (2 c, d ), where c, d ∈ { , } , and from (7.1)we then have ˆ (cid:15) ( q , q ) = ( − c d − c d . In other words, ˆ ω is in the nontrivial class of H ( Z × Z , U (1)).Next, we compute partition functions, to compare to the results above. To that end, itis helpful to first write the genus-one partition function of the Z × Z orbifold in the form Z = 1 | Z × Z | (cid:88) a,b,a (cid:48) ,b (cid:48) =0 ξ ab (cid:48) − a (cid:48) b ( a,b ) ( a (cid:48) ,b (cid:48) ) , (7.25)= 164 (cid:88) a,b,a (cid:48) ,b (cid:48) =0 (cid:16) ξ ab (cid:48) − a (cid:48) b (cid:17) (cid:16) ξ ) b (cid:48) + ( ξ ) a + ( ξ ) b + ( ξ ) a (cid:48) + ( ξ ) a + b (cid:48) + ( ξ ) a + a (cid:48) + ( ξ ) a + b + ( ξ ) b (cid:48) + a (cid:48) + ( ξ ) b (cid:48) + b + ( ξ ) a (cid:48) + b + ( ξ ) a + b (cid:48) + a (cid:48) + ( ξ ) a + b (cid:48) + b + ( ξ ) a + a (cid:48) + b + ( ξ ) b (cid:48) + a (cid:48) + b + ( ξ ) a + b (cid:48) + a (cid:48) + b (cid:17)(cid:32) ( a,b ) ( a (cid:48) ,b (cid:48) ) (cid:33) . (7.26)48n table 7.2 we collect multiplicities of a few sectors of the effective Z × Z orbifold. (Asthe total number of twisted sectors in the effective Z × Z orbifold is 16 = 256, we onlylist some representative examples, instead of trying to list every case.)(0 ,
0; 0 ,
0) 16 (2 ,
0; 0 ,
0) 16(1 ,
0; 0 ,
0) 8(1 + ξ ) (3 ,
0; 0 ,
0) 8(1 + ξ )(1 ,
1; 0 ,
0) 8(1 + ξ ) (2 ,
1; 0 ,
0) 8(1 + ξ )(3 ,
1; 0 ,
0) 8(1 + ξ ) (2 ,
2; 0 ,
0) 16(1 ,
0; 0 ,
1) 8 ξ (1 + ξ ) (2 ,
0; 0 ,
2) 16 ξ (2 ,
0; 0 ,
1) 8 ξ (1 + ξ ) (2 ,
0; 0 ,
3) 8 ξ (1 + ξ )(3 ,
0; 0 ,
1) 8 ξ (1 + ξ ) (3 ,
0; 0 ,
2) 8 ξ (1 + ξ )(3 ,
0; 0 ,
3) 8 ξ (1 + ξ )(0 ,
1; 1 ,
0) 8 ξ − (1 + ξ ) (0 ,
2; 2 ,
0) 16 ξ (0 ,
2; 1 ,
0) 8 ξ − (1 + ξ )Table 7.2: Multiplicities and phases of effective Z × Z twisted sectors in Z × Z orbifoldwith discrete torsion for some representative sectors.Assembling these results, we can now compute genus-one partition functions.1. ξ = +1. In this case, Z ([ X/ Z × Z ] ω ) = 1 | Z × Z | (cid:88) a,b,a (cid:48) ,b (cid:48) =0 ξ ab (cid:48) − a (cid:48) b (cid:32) ( a,b ) ( a (cid:48) ,b (cid:48) ) (cid:33) , (7.27)= 4 (cid:88) a,b,a (cid:48) ,b (cid:48) =0 ξ ab (cid:48) − a (cid:48) b (cid:32) ( a,b ) ( a (cid:48) ,b (cid:48) ) (cid:33) , (7.28)= 4 Z ([ X/ Z × Z ] ω ) = Z (cid:32)(cid:97) [ X/ Z × Z ] ω (cid:33) , (7.29)where the discrete torsion ω = π ∗ ω , confirming our earlier prediction.2. ξ = −
1. In this case, from e.g. table 7.2, the only effective genus-one Z × Z twistedsectors that survive are of the form(0 ,
0; 0 , , (2 ,
0; 0 , , (2 ,
2; 0 , , · · · (7.30)of weight 16, and (2 ,
0; 0 , , (0 ,
2; 2 , , · · · (7.31)of weight 16 ξ . Thus, the only genus-one sectors that survive are those of a Z × Z orbifold, weighted with sums as appropriate for discrete torsion. Taking into account49ultiplicities, we find Z ([ X/ Z × Z ] ω ) = 1 | Z × Z | (cid:88) a,b,a (cid:48) ,b (cid:48) =0 ξ ab (cid:48) − a (cid:48) b (cid:32) ( a,b ) ( a (cid:48) ,b (cid:48) ) (cid:33) , (7.32)= 1664 (4) Z ([ X/ Z × Z ] d . t . ) , (7.33)= Z ([ X/ Z × Z ] d . t . ) , (7.34)confirming our prediction. In this paper we have generalized decomposition [8] in orbifolds with trivially-acting sub-groups to include orbifolds with discrete torsion. Although the discrete torsion breaks (muchof) any original one-form symmetry, there is nevertheless a decomposition-like story. We havedescribed a general prediction for all cases, which we have checked in numerous examples.
Acknowledgements
We would like to thank R. Donagi, T. Pantev, and Y. Tachikawa for useful conversations.D.R. was partially supported by NSF grant PHY-1820867. E.S. was partially supported byNSF grants PHY-1720321 and PHY-2014086.
A Group cohomology and pro jective representations
In this appendix we give a very brief review of group cohomology and projective represen-tations for finite groups, primarily to establish notation and nomenclature.
A.1 Group cohomology
Suppose we have a finite group G and a G -module M . Then for n ≥ C n ( G, M ) of M -valued n -cochains on G as maps from the direct product of n copies of G to M . Of course since these cochains are M -valued, C n ( G, M ) also forms a G -module, whosezero is simply the map sending every element of G n to 0 ∈ M . Note that when talking aboutthe general case of arbitrary G -modules it is standard convention to use additive notation50n M and multiplicative notation for G . This helps clarify the relative structures, so we doso here at the beginning of this appendix. However, in the rest of the paper our module M is always either U(1) (most of the time) or an abelian subgroup of G , and in eithercase multiplicative notation is more natural, so these formulae should be retranscribed usingmultiplicative notation for M . For the case of M = U(1) with trivial G -action, we do so atthe end of this section.We can construct a coboundary map d n : C n ( G, M ) → C n +1 ( G, M ) by taking, for ω ∈ C n ( G, M ),( d n ω )( g , · · · , g n +1 ) = g · ω ( g , · · · , g n +1 ) + ( − n +1 ω ( g , · · · , g n )+ n (cid:88) i =1 ( − i ω ( g , · · · , g i − , g i g i +1 , g i +2 , · · · , g n +1 ) . (A.1)When the context is clear we will often drop the n subscript on d n .A cochain ω satisfying dω = 0 is said to be coclosed, and is called a cocycle (the equation dω = 0 is often called the cocycle condition on ω ). The space of M -valued n -cocycles of G isthus defined to be Z n ( G, M ) = ker( d n ). It can be verified that the coboundary maps satisfy d n +1 d n = 0, and hence the image d n − ( C n − ( G, M )) (known as the space of coboundaries) isa submodule of Z n ( G, M ), and so we can define cohomology groups by taking the quotient, H n ( G, M ) = Z n ( G, M ) /d n − ( C n − ( G, M )) . (A.2)We say that an n -cochain ω is normalized if ω ( g , · · · , g n ) vanishes whenever any one ofits arguments is the identity element of G . It can be easily verified that if ω is normalized,then so is dω and with a bit more work one can show that every cohomology class contains anormalized representative. For instance, consider the case n = 2. Then the cocycle conditionis ( dω )( g , g , g ) = g · ω ( g , g ) − ω ( g g , g ) + ω ( g , g g ) − ω ( g , g ) . (A.3)By setting g = g = 1, we learn that a cocycle satisfies ω (1 , g ) = ω (1 ,
1) for all g , and setting g = g = 1 tells us that ω ( g,
1) = g · ω (1 , µ : G → M satisfying µ (1) = − ω (1 ,
1) and define ω (cid:48) = ω + dµ . Then this new cocycle satisfies ω (cid:48) (1 ,
1) = ω (1 ,
1) + ( dµ )(1 ,
1) = ω (1 ,
1) + 1 · µ (1) − µ (1) + µ (1) = 0 , (A.4)and from the previous argument we also have ω (cid:48) (1 , g ) = ω (cid:48) ( g,
1) = 0, showing that thecohomology class of ω contains a normalized representative ω (cid:48) . Throughout the paper wewill assume that our cochains and cocycles are normalized.The case of greatest interest in this paper involves taking M to be U(1) with trivial G -action, and n = 2. Switching now to multiplicative notation for U(1), a 2-cochain ω isnormalized if ω (1 , g ) = ω ( g,
1) = 1 for all g ∈ G , and the cocycle condition is1 = ( dω )( g , g , g ) = ω ( g , g ) ω ( g , g g ) ω ( g g , g ) ω ( g , g ) . (A.5)51hifting a 2-cocycle ω by a coboundary dµ gives a new 2-cocycle ω (cid:48) , ω (cid:48) ( g , g ) = ω ( g , g ) µ ( g ) µ ( g ) µ ( g g ) − , (A.6)where µ : G → U(1) is any map.The finite cyclic groups have no 2-cohomology, H ( Z N , U(1)) ∼ = 1. Direct productsof cyclic groups satisfy H ( Z N × Z M , U(1)) ∼ = Z gcd( N,M ) . Some other examples include H ( S , U(1)) ∼ = 1, H ( S , U(1)) ∼ = Z , H ( D , U(1)) ∼ = Z , and H ( Z (cid:111) Z , U(1)) ∼ = Z . A.2 Projective representations
Again let G be a group. A projective representation of G with respect to a U(1)-valued2-cocycle ω is a vector space V and a map φ : G → GL( V ) satisfying φ ( g ) φ ( g ) = ω ( g , g ) φ ( g g ) . (A.7)Of course the case when ω is trivial corresponds to an ordinary representation of G . Asso-ciativity of φ implies the cocycle condition on ω since( φ ( g ) φ ( g )) φ ( g ) = ω ( g , g ) φ ( g g ) φ ( g ) = ω ( g , g ) ω ( g g , g ) φ ( g g g ) , (A.8) φ ( g ) ( φ ( g ) φ ( g )) = ω ( g , g ) ω ( g , g g ) φ ( g g g ) , (A.9)but since φ ( g i ) are simply matrices in GL( V ) whose multiplication is associative, these twoexpressions must be equal, and multiplying by φ ( g g g ) − gives us the cocycle condition.Note that if ω is normalized, then any projective representation φ necessarily sends theidentity element to the identity matrix, φ (1) = . Also note that as a consequence of (A.7),the relation between inversion in G and inversion of GL( V ) picks up a phase φ ( g − ) = ω ( g, g − ) φ ( g ) − . (A.10)As long as ω is coclosed, there will exist projective representations with respect to ω .Indeed, we can define a regular projective representation by taking a vector space V r with abasis { v g | g ∈ G } and defining a map φ r : G → GL( V r ) by its action on the basis, φ r ( g )( v h ) = ω ( g, h ) v gh . (A.11)Then ( φ r ( g ) φ r ( g ))( v h ) = ω ( g , g h ) ω ( g , h ) v g g h , (A.12) For any class in H ( G, U(1)) it is always possible to pick a representative ω that is both normalized andalso satisfies ω ( g, g − ) = 1 for all g ∈ G . If we insisted on this property (essentially a sort of gauge choice),then this expression and some others in the paper would simplify, but since we are able to keep it moregeneral we will only ask that our cocycles be normalized in the usual sense. φ r ( g g )( v h ) = ω ( g g , h ) v g g h . (A.13)Thus φ r ( g ) φ r ( g ) = ω ( g , g h ) ω ( g , h ) ω ( g g , h ) φ r ( g g ) = dω ( g , g , h ) ω ( g , g ) φ r ( g g ) , (A.14)so if ω is coclosed then we see that φ r is a projective representation with respect to ω .A subspace U of V is said to be invariant if φ ( g )( U ) ⊆ U for all g ∈ G . If the onlyinvariant subspaces of V are 0 and V itself, then we say that φ is irreducible. We say thattwo projective representations φ : G → GL( V ) and φ : G → GL( V ) with respect tothe same ω are isomorphic if there exists a vector space isomorphism f : V → V so that φ ( g ) = f ◦ φ ( g ) ◦ f − for all g ∈ G . Of course irreducibility is preserved by isomorphism,so given a cocycle ω ∈ Z ( G, U(1)) we can talk about the set ˆ G ω of isomorphism classes ofirreducible projective representations of G with respect to ω .Just as there is a (not necessarily canonical) one-to-one correspondence between theisomorphism classes of ordinary irreducible representations, ˆ G , and conjugacy classes of G [44, section 2.5], there is also a one-to-one correspondence between ˆ G ω and conjugacyclasses [ g ] of G which additionally satisfy [45, prop. 2.6], [46–50], for any element g in theconjugacy class, that ω ( g, g (cid:48) ) = ω ( g (cid:48) , g ) for all g (cid:48) ∈ G such that gg (cid:48) = g (cid:48) g . We will call these ω -trivial conjugacy classes, and their elements will be ω -trivial elements.As an example, consider G = Z × Z = { , a, b, c } . Then H ( G, U(1)) ∼ = Z , and arepresentative ω for the nontrivial class can be defined by ω (1 , g ) = ω ( g,
1) = ω ( g, g ) = 1 , g ∈ G,ω ( a, b ) = ω ( b, c ) = ω ( c, a ) = i, ω ( a, c ) = ω ( b, a ) = ω ( c, b ) = − i. (A.15)An example of a projective representation is φ (1) = (cid:18) (cid:19) , φ ( a ) = (cid:18) (cid:19) , φ ( b ) = (cid:18) − ii (cid:19) , φ ( c ) = (cid:18) − (cid:19) . (A.16)Up to isomorphism, this is the only irreducible projective representation with respect to ω .This is consistent with the observation that the only ω -trivial element of G is the identityelement. If one works out the regular representation for this G and ω , the resulting four-dimensional projective representation can be shown to be isomorphic to the direct sum oftwo copies of the irreducible representation. B Some calculations with cocycles
In this appendix we provide the details of our calculation of (cid:101) ω a and ˆ ω a in section 3.53aking the definitions (3.15) and (3.17), we have C a ( g , g ) = ω ( s , s − s k s k ) ω ( s − s s , s − k s k ) ω ( s − k s , k ) ω ( s , k ) ω ( s , k ) f − ρ a ( k ) f − ρ a ( k ) × (cid:2) f − ρ a ( s − s s ) ρ a ( s − k s ) ρ a ( k ) (cid:3) − = ω ( s , s − s k s k ) ω ( s − s s , s − k s k ) ω ( s − k s , k ) ω ( s , k ) ω ( s , k ) ω ( s − k , s ) ω ( s , s − k ) × f − f − ρ a ( s − k s ) ρ a ( k ) ρ a ( k ) − ρ a ( s − k s ) − ρ a ( s − s s ) − f = ω ( s , s − s k s k ) ω ( s − s s , s − k s k ) ω ( s − k s , k ) ω ( s − k , s ) ω ( s , k ) ω ( s , k ) ω ( s , s − k ) × f − f − ρ a ( s − s − s ) ω ( s − s − s , s − s s ) f = Ω f − f − f ρ a ( s s − s − ) , (B.1)whereΩ = ω ( s , s − s k s k ) ω ( s − s s , s − k s k ) ω ( s − k s , k ) ω ( s − k , s ) ω ( s , s − s − ) ω ( s , k ) ω ( s , k ) ω ( s , s − k ) ω ( s − s s , s − s − s ) ω ( s − s − , s ) . (B.2)Fortunately, Ω can be greatly simplified using cocycle conditions,Ω = dω ( s , s − s s , s − k s k ) dω ( s s , s − k s , k ) dω ( s , s , s − k ) dω ( s k , s , k ) × dω ( s s , s − k , s ) dω ( s − s − , s , s − s s ) dω ( s , s − s − , s s ) dω ( s s − s − , s s s − , s ) × ω ( s k , s k ) ω ( s s s − , s ) ω ( s , s ) ω ( s s − s − , s s s − ) , (B.3)so we have C a ( g , g ) = ω ( s k , s k ) ω ( s s s − , s ) ω ( s , s ) ω ( s s − s − , s s s − ) f − f − f ρ a ( s s − s − ) . (B.4)To check that C a ( g , g ) is proportional to the identtiy, we first show that it commuteswith all ρ a ( k ), C a ( g , g ) ρ a ( k ) = Ω ω ( s s − s − , k ) f − f − f ρ a ( s s − s − k )= Ω ω ( s − s − k, s ) ω ( s , s − s − ks s − ) ω ( s , s − ks s − s − ) ω ( s , s − s − k ) ω ( s − s − ks s − , s ) ω ( s − ks s − s − , s ) × ω ( s s − s − , k ) ρ a ( ks s − s − ) f − f − f = Ω ω ( s s − s − , k ) ω ( s − s − k, s ) ω ( s , s − s − ks s − ) ω ( k, s s − s − ) ω ( s , s − s − k ) ω ( s − s − ks s − , s ) × ω ( s , s − ks s − s − ) ω ( s − ks s − s − , s ) ρ a ( k ) ρ a ( s s − s − ) f − f − f = Ω (cid:48) ρ a ( k ) C a ( g , g ) , (B.5)54hereΩ (cid:48) = ω ( s s − s − , k ) ω ( s − s − k, s ) ω ( s , s − s − ks s − ) ω ( s , s − ks s − s − ) ω ( k, s s − s − ) ω ( s , s − s − k ) ω ( s − s − ks s − , s ) ω ( s − ks s − s − , s ) × ω ( s − s s − s − , s ) ω ( s − s − s s − , s ) ω ( s , s − s − ) ω ( s , s − s s − s − ) ω ( s , s − s − s s − ) ω ( s − s − , s )= dω ( s s − s − , s s , s − s − k ) dω ( s − s − k, s s − , s ) dω ( s , s − s − ks s − s − , s ) × dω ( s , s − k, s s − s − ) dω ( s , s , s − s − k ) dω ( s , s − s − k, s s − s − ) × dω ( s − s − k, s s − s − , s ) dω ( s s − s − , s , s ) × dω ( s − s s − s − , s , s − s − s s − ) dω ( s , s − s − s s − , s − s s − s − ) × dω ( s − s − s s − , s − s s − s − , s )= 1 . (B.6)Hence C a ( g , g ) commutes with all ρ a ( k ). Then the irreducibility of ρ a and an applicationof Schur’s lemma tell us that C a ( g , g ) is proportional to the identity matrix, C a ( g , g ) = (cid:101) ω a ( g , g ) . (B.7) C Explicit realization of β The map β : Ker ι ∗ −→ H ( G/K, H ( K, U (1)) (C.1)that we have utilized is described explicitly in [35, section 7] as the ‘reduction’ map r , butas the description there is in somewhat different language, in this appendix we will unrollthat definition to understand its properties more explicitly.First, instead of working directly with 2-cocycles ω , [35] manipulates extensions. For ourpurposes, following [33, section IV.3], ω ∈ H ( G, U (1)) corresponds to an extension E of G by U (1), 1 −→ U (1) −→ E σ −→ G −→ , (C.2)where we can think of E as V × G , V = U (1), with product defined by( v , g ) · ( v , g ) = ( v + v + ω ( g , g ) , g g ) , (C.3)which is associative so long as the 2-cocycle condition ω ( g, h ) + ω ( gh, k ) = ω ( h, k ) + ω ( g, hk ) (C.4)is obeyed. The projection map σ : E → G simply maps ( v, g ) (cid:55)→ g , and we assume ω isnormalized so that ω (1 , g ) = ω ( g,
1) = 0 (C.5)55in additive notation).For this appendix, we specialize to the case that ι ∗ ω is trivial, where ι : K (cid:44) → G . Forthis case, [35] defines V · K ⊂ E , which is naturally isomorphic to σ − ( K ).The reference [35] defines two further pertinent maps. First, ρ : G → Aut( V · K ), whichis given by ρ ( g )( v, k ) = ( g · v, gkg − ) = ( v, gkg − ) (C.6)since the group action on the coefficients is trivial here. The reference also defines γ V · K,E : E → Aut( V · K ), by γ V · K,E ( e )( v, k ) = e · ( v, k ) · e − = (cid:0) v + ω ( g, k ) − ω ( g − , g ) + ω ( gk, g − ) , gkg − (cid:1) , (C.7)for all e = (˜ v, g ) ∈ E .Finally, the map β is derived from the homomorphism f : E → Aut( V · K ) given by f ( e ) = γ V · K,E ( e ) ◦ ρ ( σ ( e − )) . (C.8)Explicitly, it is straightforward to compute that f ( e )( v, k ) = (cid:0) v + ω ( kg, g − ) − ω ( g − , kg ) , k (cid:1) , (C.9)and the reader will recognize ω ( kg, g − ) − ω ( g − , kg ) , (C.10)or in multiplicative notation, ω ( kg, g − ) ω ( g − , kg ) (C.11)as defining the map β used in the text.Before going on, let us illustrate some properties. One of the essential properties of f used in [35] is that f ( e e ) = f ( e ) ◦ ρ ( σ ( e )) − ◦ f ( e ) ◦ ρ ( σ ( e )) , (C.12)which is a consequence of the identity ω ( kg g , g − g − ) − ω ( g − g − , kg g ) (C.13)= ω ( g − kg g , g − ) + ω ( kg , g − ) − ω ( g − , kg ) − ω ( g − , g − kg g ) (C.14)+ ( dω )( g − , g − , kg g ) + ( dω )( kg g , g − , g − ) + ( dω )( g − , kg g , g − ) . In multiplicative notation, if we define, for g ∈ G , k ∈ K ,˜ β ( ω )( g, k ) = ω ( kg, g − ) ω ( g − , kg ) , (C.15)56hen this means that ˜ β ( ω )( g g , k ) = ˜ β ( ω )( g , k ) · ˜ β ( ω )( g , g − kg ) . (C.16)Next, we will define a related cocycle in H ( G/K, H ( K, U(1))). Specifically, for a section s : G/K → G , and for q ∈ G/K , k ∈ K , define β ( ω )( q, k ) = ω ( ks ( q ) , s ( q ) − ) ω ( s ( q ) − , ks ( q )) . (C.17)We will check in detail that this is indeed a cocycle.First, since the action on the U(1) coefficients is trivial, H ( K, U(1)) consists of maps φ : K → U(1) such that 1 = dφ ( k , k ) = φ ( k ) φ ( k ) φ ( k , k ) − . (C.18)In other words, H ( K, U(1)) consists of group homomorphisms from K to U(1).The cochains C ( G/K, H ( K, U(1))) are maps from
G/K to H ( K, U(1)). Alternatively,we can think of these cochains as maps ψ : G/K × K → U(1) such that for fixed q ∈ G/K , ψ ( q, − ) acts as a homomorphism from K to U(1). Also, G/K acts on H ( K, U(1)) via( q · φ ) ( k ) = φ ( q − · k ) = φ ( s ( q ) − ks ( q )) , (C.19)which is induced by the standard action q · k = s ( q ) ks ( q ) − of G/K on K , where s : G/K → G is some choice of section. Because of this nontrivial action on the coefficients, the cocyclecondition for ψ ∈ C ( G/K, H ( K, U(1))) is (abbreviating s = s ( q ))1 = ( dψ )( q , q , k ) = ψ ( q , s − ks ) ψ ( q , k ) ψ ( q q , k ) − . (C.20)Moreover, if φ ∈ C ( G/K, H ( K, U(1))) ∼ = H ( K, U(1)) is a 0-cochain, then( dφ )( q, k ) = ( q · φ )( k ) φ ( k ) − = φ ( s ( q ) − ks ( q )) φ ( k ) − . (C.21)Then H ( G/K, H ( K, U(1))) is of course the space of cocycles, i.e. cochains ψ satisfying(C.20), modulo coboundaries (C.21).For the particular choice of 1-cochain (C.17), let’s check some of these properties. First ofall, we check that for fixed q , β ( ω )( q, k ) is a homomorphism in its second argument. Indeed,abbreviating s ( q ) = g − , β ( ω )( q, k ) β ( ω )( q, k ) β ( ω )( q, k k ) = ω ( k g, g − ) ω ( k g, g − ) ω ( g − , k k g ) ω ( g − , k g ) ω ( g − , k g ) ω ( k k g, g − )= ( dω )( g − , k g, g − )( dω )( g − , k g, g − )( dω )( g − k g, g − k g, g − )( dω )( g − , k , k )( dω )( g − , k k g, g − )( dω )( g − k g, g − , k ) × ω ( g − k g, g − k g ) ω ( k , k ) . (C.22)57s long as ι ∗ ω is trivial in cohomology, the right-hand side is trivial in cohomology, and wehave a homomorphism on cohomology β ( ω )( q, k k ) = β ( ω )( q, k ) · β ( ω )( q, k ) . (C.23)(In more detail, if ω ( k, k (cid:48) ) = µ ( k ) µ ( k (cid:48) ) /µ ( kk (cid:48) ) for some µ : K → U(1), then the factors of µ can be absorbed into β , as β (cid:48) ( ω )( q, k ) = β ( ω )( q, k ) µ ( s ( q ) − ks ( q )) µ ( k ) − , which defines thesame cohomology class in H ( G/K, H ( K, U(1))).)Next we check that it is actually a cocycle satisfying (C.20). β ( ω )( q , s − ks ) β ( ω )( q , k ) β ( ω )( q q , k ) = ω ( s − ks s , s − ) ω ( ks , s − ) ω ( s − , ks ) ω ( s − , s − ks s ) ω ( s − , ks ) ω ( ks , s − )= ( dω )( ks s , s − s − s , s − )( dω )( s − s − , ks s , s − s − s )( dω )( s − s − s , s − , ks )( dω )( ks s , s − , s − )( dω )( s − , s − , ks s )( dω )( s − , ks s , s − ) × ω ( s − s − ks s , s − s − s ) ω ( s − s − s , s − ks ) . (C.24)As before, if ι ∗ ω is trivial in cohomology, then in cohomology, β ( ω )( q q , k ) = β ( ω )( q , k ) · β ( ω )( q , s − ks ) . (C.25)If we shift ω by a coboundary to ω (cid:48) ( g , g ) = ω ( g , g ) λ ( g ) λ ( g ) λ ( g g ) − for some λ ,then β ( ω ) shifts to β ( ω (cid:48) ) given by β ( ω (cid:48) )( q, k ) = β ( ω )( q, k ) λ ( ks ( q )) λ ( s ( q ) − ) λ ( k ) λ ( s ( q ) − ks ( q )) λ ( s ( q ) − ) λ ( ks ( q )) , (C.26)= β ( ω )( q, k ) λ ( s ( q ) − ks ( q )) λ ( k ) . (C.27)But this is precisely a coboundary shift, so β ( ω ) and β ( ω (cid:48) ) define the same class in thecohomology group H ( G/K, H ( K, U(1))).Finally, let us compare β ( ω ) above to a phase factor appearing in the definition of L q φ in (3.3), namely ω ( s ( q ) − k, s ( q )) ω ( s ( q ) , s ( q ) − k ) . (C.28)Using cocycle conditions, it is straightforward to check that ω ( s − k, s ) ω ( s, s − k ) = ( dω )( s, s − k, s )( dω )( s, s − , s )( dω )( ks, s − , s )( dω )( s, s − , ks ) ω ( s − , ks ) ω ( ks, s − ) = β ( ω )( q, k ) − , (C.29)hence we see that the phase factor (C.28) in L q φ is identical to β ( ω ) − .58 Pertinent group theory results
In this paper we perform rather detailed manipulations of and computations utilizing rep-resentative cocycles of elements of discrete torsion. To that end, in this section we will giveexplicit presentations and discrete torsion cocycles for several finite groups which we use inthis paper.
D.1 Z × Z As is well-known, H ( Z × Z , U (1)) = Z .Denoting the two generators of Z × Z by a , b , then the group 2-cocycle explicitly is ω ( a, b ) = ω ( b, ab ) = ω ( ab, a ) = + i, (D.1) ω ( b, a ) = ω ( ab, b ) = ω ( a, ab ) = − i, (D.2)with ω ( g, h ) = +1 for other g, h . Given this cocycle, it is straightforward to see that theonly conjugacy class obeying the condition (3.24) is { } . D.2 Z × Z We present the group Z × Z as generated by a , b , where a = 1 = b . It can be shown that H ( Z × Z , U (1)) = Z , and the cocycle and genus-one phases of the nontrivial element aregiven in tables D.1, D.2. 1 b b b a ab ab ab b − i i − i ib b − i i − i ia i i − i − iab − i − i i i ab i i − i − iab − i − i i i ω ( g, h ) for a cocycle representing the nontrivial element of H ( Z × Z , U (1)). 59 b b b a ab ab ab b − − − − b b − − − − a − − − − ab − − − − ab − − − − ab − − − − (cid:15) ( g, h ) = ω ( g, h ) /ω ( h, g ) for a cocycle representing the non-trivial element of H ( Z × Z , U (1)).Since Z × Z is abelian, each element corresponds to its own separate conjugacy class.Of these, from table D.2, only the conjugacy classes { } , { b } satisfy the condition (3.24),and so are associated with irreducible projective representations. D.3 D We present the eight-element dihedral group D as generated by z, a, b , where z generatesthe center, z = 1, a = 1, b = z , so that the elements are described as D = { , z, a, b, az, bz, ab, ba = abz } . (D.3)It can be shown that H ( D , U (1)) = Z . A group cochain representing the nontrivialelement is given as ω ( g, h ) = exp(2 πin ( g, h ) / , (D.4)where the n ( g, h ) are given in table D.3. Note that in the group multiplication, b = z , b = bz , and ba = abz .Using this, the phases weighting a given genus-one twisted sector, associated to a com-muting pair ( g, h ), are given as ratios (cid:15) ( g, h ) = ω ( g, h ) ω ( h, g ) . (D.5)We list the phases in table D.4.Note in passing that this is not the pullback from D / Z = Z × Z , as for example thediscrete torsion there generates phases solely in sectors that do not lift to D . (In particular,for the pullback of H ( Z × Z , U (1)), the discrete torsion phases are trivial, so for that60 b z bz a ba az ab b − − − z − − bz − a − − − ba − − − az − − − ab − − − n ( g, h ), appearing in the cocycle representing the nontrivialelement of H ( D , U (1)). 1 b z bz a ba az ab b z − − − − bz a − − ba − − az − − ab − − T twisted sector phases in a D orbifold with discrete torsion.Non-commuting pairs are indicated with 0.element of H ( D , U (1)), the D orbifold is the same as a D orbifold with no discretetorsion at all.The conjugacy classes of D are { } , { z } , { a, az } , { b, bz } , { ab, ba } . (D.6)Of these, only the conjugacy classes { } , { b, bz } are associated with projective representa-tions. We describe issues with the others below: • ω ( a, z ) (cid:54) = ω ( z, a ), so { z } is not a pertinent conjugacy class. • ω ( a, az ) (cid:54) = ω ( az, a ), so { a, az } is not a pertinent conjugacy class. • ω ( ab, ba ) (cid:54) = ω ( ba, ab ), so { ab, ba } is not a pertinent conjugacy class.61hus, we see that there are two irreducible projective representations of D with the non-trivial element of H ( D , U (1)). (This result is also given in [45, example 3.12].) D.4 Z (cid:111) Z We describe the group Z (cid:111) Z , the semidirect product of two copies of Z , as generated by x , y subject to the conditions x = y = 1 and y = xyx .It can be shown that H ( Z (cid:111) Z , U (1)) = Z , with cocycles as given in table D.5.The group Z (cid:111) Z has ten conjugacy classes, namely { } , { x } , { x, x } , { y, x y } , { y } , { y , x y } , { xy, x y } , { xy , x y } , { x y } , { xy , x y } . (D.7)Of these, the only ones that satisfy condition (3.24) with respect to the cocycle in table D.5are { } , { x, x } , { y } , { xy , x y } . (D.8)As a result, there are four irreducible projective representations of Z (cid:111) Z . D.5 S The group S is the symmetric group on four objects. Its group elements can be presentedas transpositions, of the form 1, ( ab ), ( abc ), ( ab )( cd ), and ( abcd ), where for example ( abc )indicates that a maps to b , b maps to c , and c maps to a , so that, for example, ( abc ) =( bca ) = ( cab ).It can be shown that H ( S , U (1)) = Z . In this appendix we collect an explicit repre-sentative of the nontrivial cocycle and corresponding twisted sector phases.The group S has five conjugacy classes: { } , { (12) , (13) , (14) , (23) , (24) , (34) } , { (12)(34) , (13)(24) , (14)(23) } , { (123) , (132) , (234) , (243) , (341) , (314) , (412) , (421) } , { (1234) , (1243) , (1324) , (1342) , (1423) , (1432) } . 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11 41 i − i ii − i − i − ii − iiii − i − i iiii − i − i − i iii − ii ii ii
11 61 i − i − i − i − ii − i − ii − i − ii − i − i − i − i
11 71 − ii − i − i − i − ii − i i − i − i − i − − − − i − i − ii −
11 81 − i − ii − ii − i − i − i − i i − − i − i − i − ii − i −
11 91 − i − i − i − i − i − i − iiii − i − i i − i − i − i − iiii − i iiii − i − i − i i − i − i − i i − iiii − i i − iii − i − ii i − i i ii − ii i − iii − ii i i − i i i i ii − ii i − ii − − i − ii − i − − i − i − i − i − iii − i − iiii i − i i ii − − i − i − i − i − − i − ii i − i − i − i − i − i − i
11 161 − iii − i i i − i − ii − i i − i − ii
11 171 − − i − ii − i − i − i − iiii − i i − i − i − i − ii − − i − − ii i i − i − i − i ii ii − ii − i − i − i
11 201 − − i − − i − i − − i − i − i i − i − − i − ii − iii − i − ii − i i − − i − − i − i − i − i − iiiii − i − ii i i
111 231 − i − i − i − − − i − i − − i i − i − − i − − i i i T a b l e D . : C o c y c l e f o r n o n tr i v i a l e l e m e n t o f H ( S , U ( )) . T h e tr a n s p o s i t i o n s a r e nu m b e r e d , a s g i v e n i n t a b l e D . . − − − −
111 31 − − − − − − − − − − − − − − − − − −
100 221 − − T a b l e D . : Tw i s t e d s ec t o r ph a s e s f o r d i s c r e t e t o r s i o n i n S o r b i f o l d , d e r i v e d f r o m c o c y c l e i n t a b l e D . . A ze r o e n tr y i nd i c a t e s a n o n - c o mm u t i n g p a i r o f g r o up e l e m e n t s . T r a n s p o s i t i o n s c o rr e s p o nd i n g t og r o up e l e m e n t nu m b e r s a r e li s t e d i n t a b l e D . .
70 1 9 (243) 17 (23)2 (13)(24) 10 (134) 18 (1342)3 (14)(23) 11 (142) 19 (14)4 (12)(34) 12 (123) 20 (1243)5 (234) 13 (34) 21 (24)6 (132) 14 (1324) 22 (13)7 (143) 15 (1423) 23 (1432)8 (124) 16 (12) 24 (1234)Table D.9: Assignments of transpositions to S4