A generalization of decomposition in orbifolds

This paper describes a generalization of decomposition in orbifolds. In general terms, decomposition states that two-dimensional orbifolds and gauge theories whose gauge groups have trivially-acting subgroups decompose into disjoint unions of theories. However, decomposition can be, at least naively, broken in orbifolds if the orbifold has discrete torsion in 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, which we check in numerous examples. Our analysis includes as special cases (and in one sense generalizes) quantum symmetries of abelian orbifolds.


Introduction
This paper is motivated by a revival of interest in two old topics, namely • orbifolds, see e.g. [1][2][3][4][5][6][7], utilizing new methods and insights into anomalies from topological defect lines and related technologies, and • decomposition, first described in [8], an equivalence between two-dimensional theories with what are now called one-form symmetries (and various generalizations) and disjoint unions of other two-dimensional theories, see e.g. [9][10][11][12][13] for recent activity in this area.
Decomposition was first introduced in [8] to understand orbifolds and two-dimensional gauge 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 symmetry, in modern language, but decomposition is defined more generally.) Briefly, decomposition says that such a quantum field theory is equivalent to ('decomposes' into) a disjoint union of related quantum field theories, often constructed from orbifolds and gauge theories by effectively-acting quotients of the original gauge group. See [14][15][16] for detailed discussions of such examples, and see also [17][18][19][20][21][22] for applications to Gromov-Witten theory, [23][24][25][26][27][28][29] for applications to phases of gauged linear sigma models (GLSMs), [30] for applications in heterotic string compactifications, [12] for applications to elliptic genera of two-dimensional pure supersymmetric gauge theories, and [10] for four-dimensional analogues, for example.
The original work on decomposition [8] studied orbifolds and gauge theories, but did not consider orbifolds in which discrete torsion was turned on in a way that obstructed the existence of the one-form symmetry (or its analogues), except to note that the decomposition story did not apply to such cases. The purpose of this paper is to fill that gap, by generalizing decomposition 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, returns the original orbifold. We shall see that the composition of orbifolds can be described as a single Z k × Z k orbifold with discrete torsion, for which our generalization of decomposition predicts that the orbifold is equivalent to the original space. In fact, our generalization of decomposition will predict analogous results in more general cases, that often the effect of gauging a trivially-acting subgroup K of the orbifold group G with nonzero discrete torsion is to 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, specializing a number of very general (and necessarily abstract) statements in [8] to the simpler concrete case that the theory is a finite gauge theory, an orbifold by a finite group. Briefly, decomposition in this case reduces to the statement where K ⊂ G is a trivially-acting subgroup,K the set of isomorphism classes of irreducible representations of K, andω represents discrete torsion on the components. Note that the right-hand size of the expression above has multiple connected components in general. In the case of ordinary decomposition in orbifolds, the G orbifold [X/G] does not have discrete torsion.
In the remainder of this paper we check the conjecture in a number of concrete examples. In section 4, we discuss examples with ι * ω = 0; in section 5 we discuss examples with ι * ω = 0 and β(ω) = 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 2 (G/K, U (1)). Finally in section 7 we discuss examples spanning all three categories, depending upon the value of discrete torsion ω.
For reference, in appendix A we briefly review some pertinent aspects of group cohomology and projective representations, in appendix B we give some technical details of our cocycle computations, in appendix C we make explicit the map β that plays a prominent role in our analysis, and in appendix D we collect a number of group-theoretic results on the finite groups appearing in the examples, including explicit representatives of discrete torsion cocycles and genus-one twisted sector phases.
Finally, we should comment on additive versus multiplicative notation in cocycles. At various points in this paper, it will greatly improve readability to use one or the other, so we have adopted both notations, and leave the reader to infer from context which is being used in any one section.
After this paper appeared on the arXiv, we learned of [34] which also discusses decomposition in the presence of B fields and derives similar conclusions, albeit from a different computational perspective.

Review of decomposition in orbifolds
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, specifically whereK denotes the set of isomorphism classes of irreducible representations of K, andω 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 different disjoint components. These different components or summands are sometimes referred to in the literature as universes, given that in a string compactification, they would define low-energy theories with multiple independent decoupled gravitons.
The group G/K acts onK 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-direct product K G/K), then s cannot be chosen to be a group homomorphism, but it always exists as a map. For any representation φ : K → GL(V ) of K and any q ∈ G/K, define a new representation L q φ : K → GL(V ) by (L q φ)(k) = φ(s(q) −1 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 is also irreducible. By verifying that L 1 φ is isomorphic to φ, that L q 1 (L q 2 φ) is isomorphic to L q 1 q 2 φ, and that if φ 1 is isomorphic to φ 2 then L q φ 1 is isomorphic to L q φ 2 , one shows that this defines a G/K action onK by q · [φ] = [L q φ]. Moreover, this action is independent of the choice of section s.
With this understanding we can interpret (2.1) and determine the discrete torsionω. Let {ρ a } be a collection of irreducible representations of K chosen so that the equivalence classes [ρ a ] are representatives of the distinct orbits of the G/K action onK. For each ρ a , let H a ⊆ G/K be the stabilizer of [ρ a ] inK. Then, decomposition becomes the statement that whereω a ∈ H 2 (H a , U(1)) (our notation and conventions for group cohomology are reviewed in appendix A.1) denotes the discrete torsion in the summand [X/H a ].
If K is abelian, then there is a simple way to determineω a . First, given q 1 , q 2 ∈ G/K define e(q 1 , q 2 ) = s(q 1 )s(q 2 )s(q 1 q 2 ) −1 . (2.4) Since this is in the kernel of the map π, it must lie in the subgroup K. Indeed, e turns out to be a 2-cocycle valued in Z 2 (G/K, K), where K is taken to be a G/K-module with action q · k = s(q)ks(q) −1 . Different choices of section s will lead to cocycles that differ from this by coboundary terms, but the cohomology class of e in H 2 (G/K, K) depends only on the extension (1.2) and is called the extension class [35,section IV.3]. Since K is finite and abelian, its irreducible representations are all one-dimensional and map K into U(1). So given one of our irreducible representations ρ a , we can apply it to e restricted to H a to get our discrete torsionω a ,ω a (h 1 , h 2 ) = ρ a (s(h 1 )s(h 2 )s(h 1 h 2 ) −1 ). (2.5) If in addition, G itself is abelian, then the extension class (2.4) is symmetric in q 1 and q 2 , and hence so isω a . But this means that the discrete torsion phases (g, h) =ω a (g, h)/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 torsionω a . Before describing 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 extension class vanishes, and so 6) where no copies have any discrete torsion.
• For another example, if G = D 4 and K = Z 2 , then using the fact that G/K = Z 2 × Z 2 , and that D 4 ∼ = Z 2 × (Z 2 × Z 2 ) (so the extension class is nontrivial), where the second summand has discrete torsionω given by the nontrivial irreducible representation of Z 2 applied to the extension class, which one can check is the nontrivial element of H 2 (Z 2 ×Z 2 , U (1) In passing, ultimately because of the Cartan-Leray spectral sequence, if X/(G/K) is smooth, then the prescription above will determine a B field on that quotient, which is in the spirit of the original phrasing of decomposition in [8].
Next, we shall describe how the discrete torsionω 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 representations 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 for all k ∈ K, The map f a : H a → GL(V a ) is called the intertwiner.
Let G a = π −1 (H a ), so that we have an exact sequence The idea of the construction will be that for each ρ a we will define a projective representation ρ a on G a . Projective representations are reviewed in appendix A.2. The associated 2-cocycle ω a will turn out to be the pullback of a 2-cocycle on H a which we'll identify withω −1 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 ρ a : G a → GL(V a ) by (2.11) We claim that ρ a is a projective representation on G a . Indeed, for any two elements g 1 , g 2 ∈ G a , define an operator C a (g 1 , g 2 ) by To show that ρ a is a projective representation, we need to show that C a (g 1 , g 2 ) is a scalar multiple of the identity operator 1 ∈ GL(V a ), say C a (g 1 , g 2 ) = ω a (g 1 , g 2 )1. It then follows that ρ a (g 1 ) ρ a (g 2 ) = ω a (g 1 , g 2 ) ρ a (g 1 g 2 ). (2.13) Indeed, computing we have (abbreviating s a (h 1 ) = s 1 , s a (h 1 h 2 ) = s 12 , f a (h 2 ) = f 2 , etc.) ρ a (g 1 g 2 ) = ρ a (s 1 k 1 s 2 k 2 ) = ρ a (s 12 (s −1 12 s 1 s 2 )s −1 2 k 1 s 2 k 2 ) = f −1 12 ρ a (s −1 12 s 1 s 2 )ρ a (s −1 2 k 1 s 2 k 2 ), (2.14) and so is in fact independent of k 1 and k 2 . To see that C a (g 1 , g 2 ) is a multiple of the identity, take any k ∈ K and check that C a (g 1 , g 2 ) commutes with ρ a (k), where we repeatedly used the intertwiner property Since ρ a was an irreducible representation of K, Schur's lemma tells us that anything that commutes with all the operators ρ a (k) must be a multiple of the identity.
Note that associativity of ρ a implies the coclosure of ω a . Specifically, the cocycle condition follows by plugging (2.13) into which clearly holds since the ρ a (k i ) are simply products of GL(V a ) matrices whose multiplication is associative. Alternatively, one can show by direct calculation that Finally, we defineω a (h 1 , h 2 ) as the inverse of ω a , i.e.
The fact that ω a is a cocycle implies thatω a is also coclosed, and thusω a defines a class in 2 H 2 (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 cocycleω given in equation (2.20), modulo coboundaries, reduces toω is precisely the cocycle representing the extension class in H 2 (H a , K) in the case that K is abelian [35,section IV.3], [36,exercise VI.10.1], so we see thatω is the image of the extension class of G a in H 2 (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 thenω is the image of the extension class of G in H 2 (G/K, K) under ρ a , also as described earlier.
These statements have been checked in many examples in many ways. In orbifolds, the original work [14][15][16] first checked that physics 'sees' trivially-acting groups, through studies of multiloop factorization (target-space unitarity), as well as massless spectra and partition functions, which confirmed that not only does one get distinct theories, but one encounters physical contradictions if one tries to ignore them. Decomposition in such theories was tested in [8] by comparing partition functions at all genus, construction of projection operators projecting onto the various universes, and comparisons of massless spectra and correlation functions, as well as studies of open string sectors. Decomposition is also defined for gauge theories with finite trivially-acting subgroups, and there it has been similarly tested in both supersymmetric and nonsupersymmetric models, via comparisons of partition functions and elliptic 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 cited therein.
Sometimes, for K abelian, decomposition can be understood in terms of the existence of finite global one-form symmetries in the theory, see e.g. [9] for a recent discussion. A finite global one-form K symmetry, technically denoted BK, acts in an orbifold by interchanging the twisted sectors (and in more general gauge theories, the nonperturbative sectors). If we think of each twisted sector contribution to the partition function as associated with some principal bundle, the action of BK is to tensor 3 that principal bundle with a principal K bundle to get another twisted sector contribution. If K acts trivially, then the two twisted sectors make identical contributions to the total partition function.
As a quick consistency check when doing computations, given a decomposition of the form then it should be true that the total number of 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.) For example, as noted earlier, for G = D 4 , decomposition predicts Now, D 4 has five irreducible representations, Z 2 × Z 2 has four ordinary irreducible representations, and one irreducible projective representation twisted by the nontrivial element of H 2 (Z 2 × Z 2 , U (1)), as described in appendix A.2. As expected, 5 = 4 + 1. This observation can sometimes be handy when double-checking expressions for decomposition.
Previous work such as [8] focused on orbifolds (and related theories) in which a subgroup acted trivially, but no discrete torsion was turned on that interacted with that subgroup in any way, aside from making the observation that discrete torsion typically 'broke' the decomposition story, yielding theories with fewer components than decomposition would predict. (For example, discrete torsion typically breaks much of the finite global one-form symmetry.) The purpose of this paper is to extend to orbifolds with trivially-acting subgroups in which discrete torsion has been turned on, and understand what is happening in such cases.
3 General conjecture 3.1 Complete argument Now, consider an orbifold [X/G] ω by a finite group G as above, where one has included discrete torsion, given by some cocycle ω, so [ω] ∈ H 2 (G, U (1)) (with trivial action on the coefficients). Suppose as above that a normal subgroup K ⊆ G acts trivially on X, and describe G as the extension 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 decomposed into a number of disjoint pieces, one for each orbit of the G/K action onK, the set of isomorphism classes of irreducible representations of K. Now, the role ofK will be replaced by the setK ι * ω of isomorphism classes of irreducible projective representations of K with respect to the cocycle ι * ω (which, if we view K as a normal subgroup of G, is simply the restriction of ω to K). Some facts about projective representations of finite groups are reviewed in appendix A.2.
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 andK are trivial), this action we have constructed is not necessarily trivial! In other words, the choice of prefactors required in (3.3) to make the projectivity work out in general have the side effect that they can lead to a nontrivial action even in the non-projective case, and this will turn out to be crucial in correctly accommodating some examples.
One example that we will explore in more detail below is to take 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 caseK = {[ρ + ], [ρ − ]}, where ρ ± are one-dimensional irreducible representations of K defined by their action on the generator, ρ ± (a) = ±1. The usual action of G/K onK, as constructed in the previous section, would be trivial, but instead the action we have defined above satisfies and hence bK · [ρ ± ] = [ρ ∓ ]. Thus, in this example, L bK exchanges different representations, and soK consists of only a single G/K orbit, with representative [ρ + ] and trivial stabilizer subgroup H + ∼ = 1.
More generally, define β(ω) : (3.9) (We will discuss this function in greater generality and detail in appendix C, where we will see that for ι * ω trivial, it represents an element of H 1 (G/K, H 1 (K, U (1)).) (When K is central, β(ω) is invariant under coboundaries and so gives a well-defined U (1) phase.) In all cases, we will see that β also defines a suitable homomorphism in appendix C. We will also see in appendix C that the phase factor in L q φ is β(ω) −1 , so that 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 decomposeK ι * ω into G/K orbits, and choose representatives [ρ a ], where ρ a is an irreducible projective representation of K chosen to stand in for its isomorphism class. Let H a ⊆ G be the stabilizer subgroup of [ρ a ]. Our conjecture is that the G-orbifold with discrete torsion ω decomposes into a disjoint union of theories, one for each orbit [ρ a ], with orbifold group H a and discrete torsionω a which we will construct below. In other words, schematically we have 11) or with slightly more detail, It remains to construct the cocyclesω a ∈ Z 2 (H a , U(1)).
We will follow a similar approach to what we did in the previous section. Given the projective irreps ρ a of K, we will construct projective irreps ρ a for G a = π −1 (H a ). The steps will all be the same, but we will have to be more careful with ω-related phase factors. The statement 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 commute for each k ∈ K. In other words, the intertwiners satisfy (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 ρ a : G a → GL(V a ) by The phase ω(s(h), k) −1 is chosen to make the formulas below cleaner. A different choice would 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 where we used (A.7) twice to factorize ρ a (s −1 12 s 1 k 1 s 2 k 2 ). Then we define an operator C a (g 1 , g 2 ) = ρ a (g 1 ) ρ a (g 2 ) ρ a (g 1 g 2 ) −1 . (3.17) One can show that this operator is a scalar multiple of the identity, C a (g 1 , g 2 ) = ω a (g 1 , g 2 )1, (3.18) specifically, hence ρ a is a projective representation of G a with cocycle ω a . Since the details of the calculation are not particularly enlightening, they are relegated to appendix B.
From the definition (3.17) of C a (g 1 , g 2 ), it now follows that ρ a is a projective representation on G a with respect to the cocycle ω a , ρ a (g 1 ) ρ a (g 2 ) = ω a (g 1 , g 2 ) ρ a (g 1 g 2 ). (3.22) The manifest associativity of the matrices ρ a (g) implies the cocycle condition for ω a , and combined with the fact that ω is also coclosed we learn thatω a is a cocycle and hence defines a class in H 2 (H a , U(1)). This is the discrete torsion which will appear in the correspoonding factor of the decomposition.
Let's note some aspects of this result. The factor ω(s 1 , s 2 ) which appears inω a is essentially the pullback of ω along the section s, i.e. ω(s 1 , s 2 ) = (s * ω)(h 1 , h 2 ). Suppose that ω = π * ω is the pullback of some cocycleω ∈ H 2 (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 thatω a is simplyω multiplied by the result (2.20) that we obtained in the ordinary decomposition case. It's interesting to observe also that in this case ω = π * ω drops out entirely from ω a ; the projective representation ρ a is insensitive to the presence ofω.

Summary
As this has been a somewhat long-winded discussion, in this section we summarize the highlights. We can essentially break this into three cases, determined by the image of the discrete torsion ω ∈ H 2 (G, U (1)) under various maps defined by the short exact sequence (3.1).
Without the twisting, when the cocycle ι * ω vanishes, the number of irreducible ordinary representations of a finite group is counted by the number of conjugacy classes. As reviewed in appendix A.2, the number of irreducible projective representations of a finite group K twisted by some ω ∈ H 2 (K, U (1)) is equal to the number of conjugacy classes in K which consist of g ∈ K such that ω(g, h) = ω(h, g) for all h ∈ K that commute with g: We can slightly simplify the prediction (3.23) as follows. Let {ρ a } be a collection of representatives of the orbits of G/K onK ι * ω . For each ρ a , let H a ⊂ G/K be the stabilizer of [ρ a ] inK ι * ω . Then, where the details of the discrete torsionω 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 irreducible representations can still involve phases, which can exchange representations. Those phases are determined by β(ω) ∈ H 1 (G/K, H 1 (K, U (1)), as described in appendix C, where following [33], and as we describe in detail in appendix C, β itself is the map in the following 4 exact sequence: If β(ω) is nontrivial in cohomology, and if K is in the center of G (so that in particular K = H 1 (K, U (1))), then our prescription can be efficiently summarized as where we interpret β(ω) as a homomorphism G/K →K ∼ = H 1 (K, U (1)), the irreducible representations of K, and where we gave the discrete torsionω in the previous subsection. (If K is not central then we can still appeal to the methods of the previous subsection to understand decomposition; however, we will still classify examples according 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 representatives of the orbits of Ker β(ω) on the set of irreducible (honest) representations of the cokernel. For each ρ a , let H a ⊂ Ker β(ω) be the stabilizer of ρ a , then that is a consequence of the Lyndon-Hochschild-Serre spectral sequence [37], slightly generalizing the inflation-restriction exact sequence (see e.g. [37], [ (where we have specialized to trivial action on the coefficients).
We shall argue in section 5 that in fact β defines an analogue of a quantum symmetry.
For example, if K is in the center of G, so that G/K acts trivially onK, then In the special case ω = 0, this reduces to the ordinary decomposition story reviewed in section 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 string sector. A group that acts trivially on bulk states, may act nontrivially on boundary states, and so the boundary states decompose into representations of the group. For example, K theory on gerbes decomposes, as was reviewed in [8]. Here, we have a slightly more complicated situation, in that discrete torsion in open strings 'twists' ordinary representations into projective representations [41,42]. An open-string interpretation of that result over gerbes in the second two cases is somewhat beyond the scope of this article, but its interpretation is immediate in the first case, and is the underlying physics reason for the appearance ofK 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 to these theories. Given a decomposition of the form 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 sometimes be handy when double-checking expressions for decomposition.
Many of the resulting theories have multiple disconnected components. This reflects some subtle "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 sectors can be exchanged by tensoring 5 with H bundles, for H abelian. In the presence of discrete torsion, one must be careful, as discrete torsion weights different twisted sectors with different phases, and so can break such symmetries, but there can be residual symmetries 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 original orbifold and its final simplified description) have the same one-form symmetry. We shall discuss this phenomenon in examples as it arises.
In the remainder of this paper, we will work through a number of examples, to illustrate and test the various features of the prediction above. Among other things, we will see that in cases where a choice of normal subgroups K is ambiguous, the resulting prediction for the field theories for all choices of K will be the same, so that the physics is well-defined. For example, the Z 2 × Z 2 orbifold of a point, with discrete torsion, can be considered either as an example with K = Z 2 × Z 2 and ι * ω = 0, as we discuss in section 4.1, or as an example with K = Z 2 , ι * ω = 0, and β(ω) = 0, as we discuss in section 5.1. From both perspectives, we will 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 in section 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 this paper for brevity we will test our claims merely by computing genus-one partition functions.

Orbifold of a point with discrete torsion
Consider an orbifold of a point by a finite group G with discrete torsion. Since all of G acts trivially, we have K = G, hence ι * ω = ω = 0 (by assumption).
From section 3, this theory is predicted to be the same as that of N points, where N is the number of irreducible projective representations of G with respect to ω. Using [43, equ'n (6.40)] we see immediately that the genus-one partition function of the G orbifold with discrete torsion 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 2 × Z 2 . As discussed in appendix D.1, there is only one irreducible projective representation of G, and given the cocycle given there, it is also straightforward to compute that the genus-one partition function is 1.
Assume that the discrete torsion is in one Z 2 × Z 2 factor. Then, the total number of irreducible projective representations is two -the tensor product of one irreducible projective representation of Z 2 × Z 2 and two irreducible honest representations of Z 2 -and it is straightforward to compute that the genus-one partition function is 2.
The reader should also note in this case that because of the Z 2 factor, this theory has a BZ 2 symmetry, which is consistent with the fact that it decomposes into two equal factors [9].
• G = Z 2 × Z 4 . As discussed in appendix D.2, there are two irreducible projective representations of Z 2 × Z 4 with nontrivial discrete torsion, and utilizing the phases in table D.2, it is straightforward to compute that the genus-one partition function is where (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 periodic under multiplication of group elements by the subgroup b 2 ∼ = Z 2 . As a result, this theory has a BZ 2 (one-form) symmetry, interchanging twisted sectors weighted by the same phase, which is consistent with the fact that this theory describes a disjoint union of two objects (points) [9].
• G = D 4 . As discussed in appendix D.3, there are two irreducible projective representations of D 4 , and given the phases in table D.4, it is also straightforward to compute the genus-one partition function. Given that there are 28 ordered commuting pairs for which (g, h) = +1, and 12 ordered commuting pairs for which (g, h) = −1, one finds agreeing with the prediction for this case.
As the result has two equivalent disconnected components, this theory has a BZ 2 symmetry, just like the previous case. Unlike the previous case, this BZ 2 does not seem to have a group-theoretic origin in D 4 or its discrete torsion itself. Instead, because this is an orbifold of a point, the contribution of each twisted sector is determined solely by the discrete torsion phase (g, h) -so there is room for other permutations of twisted sectors, unrelated to group theory, which appears to be what is happening in this case, and one expects, in most cases of orbifolds of points. More to the point, it also seems to be true of the other examples in this list, so we will not explicit discuss these symmetries further in this section.
• G = Z 4 Z 4 . As discussed in appendix D.4, there are four irreducible projective representations of Z 4 Z 4 . Utilizing the phases in table D.6, it is straightforward to compute that the genus-one partition function is in agreement with the prediction.
• G = S 4 . As discussed in appendix D.5, there are three irreducible projective representations of S 4 . Adding up the contributions, weighted by signs as given in table D.9, one finds agreeing with the prediction.
with discrete torsion and trivially-acting Z 2 × Z 4 subgroup In this section we consider an orbifold by the semidirect product of two copies of Z 4 , Z 4 Z 4 . It can be shown (see appendix D.4) that H 2 (Z 4 Z 4 , U (1)) = Z 2 , so there is one nontrivial value ω of discrete torsion, which we turn on in this orbifold. Furthermore, we take the action of the subgroup K = x 2 , y ∼ = Z 2 × Z 4 (in the notation of appendix D.4) to be trivial.
For completeness, let us also compute the full cocycle. Applying table D.5, the full cocycle for ι * ω is given in table 4.1. Table 4.1: Cocycle for ι * ω for ω the nontrivial element of H 2 (Z 4 Z 4 , U (1)), and ι : The cocycle for ι * ω matches the cocycle for the nontrivial element of H 2 (Z 2 × Z 4 , U (1)) given in table D.1. (In principle, they only needed to match up to a coboundary; it is a reflection of our conventions that they happen to match on the nose.) Thus, we see that ι * ω = 0. Furthermore, G/K = Z 2 . Our conjecture of section 3 then predicts that As discussed in appendix D.2, there are only two irreducible projective representations of K = Z 2 × Z 4 , corresponding to the conjugacy classes {1}, {b 2 = y 2 }. Furthermore, the prefactors (3.9) which appears in the definition of (L q φ)(k) is unity for all q and k. This combined with the fact that conjugation by G leaves both conjugacy classes invariant, implies 6 that G/K acts trivially onK ι * ω . Therefore, we make the prediction that two copies of the orbifold [X/Z 2 ].
It is straightforward to check this in genus-one partition functions. If we let x denote the generator of G/K = Z 2 , then using the Z 4 Z 4 discrete torsion phases in table D.6, we find (4.8) matching the prediction.
As this theory has two equal disconnected components, the two copies of [X/Z 2 ], it has a BZ 2 symmetry, which is reflected in the fact that ι * ω(g, h) is invariant under the subgroup y 2 ∼ = Z 2 , as is visible in table 4.1.

[X/S 4 ]
with discrete torsion and trivially-acting Z 2 × Z 2 subgroup Consider [X/S 4 ] with nontrivial discrete torsion, and with Z 2 × Z 2 ⊂ S 4 acting trivially on X.
The normal Z 2 × Z 2 subgroup of S 4 has elements 1, (12) As elements of S 3 , b = a −1 and The only distinct elements that commute with one another are a and b.
Analyzing this formally, define From section 3, since there is only a single projective irreducible representation of K with nontrivial H 2 (K, U (1)) in this case, we predict that Since H 2 (S 3 , U (1)) = 0, there is no possibility of discrete torsion in the [X/S 3 ] orbifold.
We can confirm this at the level of genus-one partition functions. Using table D.8, it is straightforward to compute that where g, h ∈ S 3 . Thus, in this case, the S 4 orbifold with discrete torsion and trivially-acting Z 2 × Z 2 has the same partition function as the S 4 /Z 2 × Z 2 ∼ = S 3 orbifold.

[X/S 4 ] with discrete torsion and trivially-acting A 4 subgroup
Consider [X/S 4 ] where K = A 4 ⊂ S 4 acts trivially and the orbifold has discrete torsion. The coset The elements of A 4 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 can be shown that (with trivial action on the coefficients,) and the restriction of the nontrivial element of H 2 (S 4 , U (1)), ι * ω, is the nontrivial element of H 2 (A 4 , U (1)), as can be seen by restricting the genus-one phases in table D.8.
Since ι * ω = 0, in general terms, section 3 predicts In this case, G/K = Z 2 , and as H 2 (Z 2 , U (1)) = 0, there will not be any discrete torsion contributionsω, but we do need to compute the number of irreducible projective representations of K = A 4 and the action of G/K on those representations. Now, let us compute the irreducible projective representations of A 4 . The group A 4 has four conjugacy classes, which we list below:  Thus, A 4 has three irreducible projective representations with respect to the nontrivial element of H 2 (A 4 , U (1)).
As mentioned previously, section 3 predicts forÂ 4,ι * ω the set of irreducible projective representations with respect to the restriction of ω ∈ H 2 (S 4 , U (1)). As a set,Â 4,ι * ω has three elements, but two are interchanged by the Z 2 , so that Computing the genus-one partition function, we find Thus, at least at the level of partition functions, we see matching the prediction of section 3.
By way of comparison, for the orbifold [X/S 4 ] with trivially-acting A 4 and no discrete torsion, ordinary decomposition implies i.e. it has one additional copy of [X/Z 2 ] relative to the case with discrete torsion.
with discrete torsion and trivially-acting (Z 2 ) 3 subgroup is the projection map onto the second factor, and ω 0 is the nontrivial element of H 2 (Z 2 × Z 2 , U (1)). Let us assume that the trivially acting subgroup is K = (Z 2 ) 3 ⊂ G, where one Z 2 factor is the center of D 4 and the remaining (Z 2 ) 2 factors match those in G.
Let p 2 : K → (Z 2 ) 2 be the projection onto the second two Z 2 factors, so that the diagram commutes. Then we have Following section 3, next consider the setK ι * ω . This is essentially a product of two factors, Z 2 and (Z 2 ) 2 , with the discrete torsion entirely in the (Z 2 ) 2 factor. There is only one irreducible projective representation of (Z 2 ) 2 , as can be seen by counting conjugacy classes, but the remaining Z 2 factor has two honest representations. Furthermore, since K is in the center of G, G/K acts trivially onK ι * ω .
Putting this together, we have the prediction whereω 1,2 are elements of discrete torsion we determine next. We are given discrete torsion ω as the pullback of the generator of H 2 (Z 2 × Z 2 , U (1)). From equation (3.21) forω, the ratio of ω's multiplying the image of the extension class is determined by the values of ω on a section s : We can choose the section so that p 2 (s(q)) = 1 ∈ Z 2 × Z 2 for all q ∈ D 4 /Z 2 . Then the ω's in (3.21) are trivial, and the discrete torsion ω is determined solely by the image of the extension class. Since D 4 is not a semidirect product Z 2 (Z 2 ) 2 , the extension class H 2 ((Z 2 ) 2 , Z 2 ) is nontrivial, and applying the two honest irreducible representations of Z 2 , we get that one ofω 1,2 is trivial, and the other is nontrivial. Thus, we predict where exactly one of the [X/Z 2 × Z 2 ] summands has discrete torsion.
Next, let us compare to physics. We can view this example as a pair of successive orbifolds, first [X/D 4 ] without discrete torsion and with trivially-acting Z 2 , and then a trivially-acting (Z 2 ) 2 orbifold with discrete torsion. Applying the ordinary version of decomposition [8], we know that a disjoint union of two (Z 2 ) 2 orbifolds where one copy has discrete torsion and the other does not. As we shall see in section 5.1, a (Z 2 ) 2 orbifold with discrete torsion in which either Z 2 factor acts trivially is just a realization of orbifolding by a quantum symmetry, and returns the original theory. Putting this together, we find that 5 Examples in which ι * ω = 0 and β(ω) = 0 We will see in this section that this case corresponds to one generalization 7 of quantum symmetries of orbifolds. Ordinarily, in quantum symmetries in abelian orbifolds, we have a G/K orbifold, G/K abelian, which when orbifolded by a further K (abelian), with an action 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 which are equivalent to a map Taking into account the action of G/K on K and its induced action on H 1 (K, U (1)), this map is a crossed homomorphism 8 . This is equivalent to an element of H 1 (G/K, H 1 (K, U (1))).
The map β sends and so it can be interpreted (at least in abelian cases) as giving us the quantum symmetry action corresponding to an element of discrete torsion in the extension G.
In more general cases, these maps all factor through abelianizations, We shall elaborate on this perspective in the examples. 7 See also [2] for a different generalization of quantum symmetries to nonabelian orbifolds. 8 Suppose a group G acts on an abelian group H. Then a crossed homomorphism φ : G → H is a map satisfying φ(g 1 g 2 ) = g 1 · φ(g 2 ) + φ(g 1 ). 9 Unlike abelian cases, orbifolding by the abelianization does not return the original theory. A fix for this 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 this article, however.
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, but to 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) −1 ) ω(s(q) −1 , ks(q)) . (5.5) Here, write K = Z 2 = a , G/K = Z 2 = b . Without loss of generality, we take the section s : Z 2 → Z 2 × Z 2 to be given by, s(q) = q. Following the explicit cocycle given in appendix 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 Using we interpret β(ω) as a map H 2 (G, U (1)) → H 1 (G/K, H 1 (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 the identity, and β(ω)(q = b, −) is the identity, sending any element of K = Z 2 to itself. Thus, we see that β(ω) is an isomorphism H 2 (G, U (1)) → H 1 (G/K, H 1 (K, U (1))), hence Given the form for β, we predict from our analysis in section 3 that It is straightforward to see this directly in physics. In fact, this example is the same as orbifolding [X/Z 2 ] by the quantum symmetry Z 2 [31]: (5.14) Putting this together, the combination of the Z 2 andẐ 2 orbifolds can be expressed as a singleẐ 2 × Z 2 orbifold with phases (a, g; b, h) where a, b ∈Ẑ 2 , g, h ∈ Z 2 , given by and the other three obtained by reversing the order. For all other elements ofẐ 2 × Z 2 , = +1. These are precisely the phases assigned by discrete torsion.
Thus, we see that the orbifold [[X/Z 2 ]/Ẑ 2 ] is the same as [X/(Ẑ 2 × Z 2 )], with phases given by discrete torsion. In particular, agreeing with our prediction of section 3.
As a final consistency check, let us redo the computation of section 3 without appealing to the description of this case in terms of kernels and cokernels of β(ω). Briefly, since K is central in G, the action of L q for q ∈ G/K is to map φ(k) to (5.21) As computed above, and using the same section as there, β(ω)(b, a) = −1, so we find In other words, the effect of L b is to exchange the two irreducible representations of K = Z 2 : the trivial representation becomes nontrivial, and vice-versa. As a result, there is only one orbit of G/K onK, consisting of both the elements ofK, and so one recovers the result above.
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 we would have predicted instead that the resulting theory be a disjoint union of two copies of [X/Z 2 ], instead of one copy of X, which is not what we see in physics.

Example with nonabelian K
In this subsection we will consider an example that is closely related to the previous one, essentially adding a nonabelian factor to G.
Since K is (potentially) nonabelian, we cannot simply resort to a computation of the kernel and cokernel of β(ω), but must work slightly harder. Let a generate the Z 2 factor in K, and b generate the remaining Z 2 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) −1 , ks(q)) ω(ks(q), s(q) −1 ) φ(s(q) −1 ks(q)), (5.23) and the nontrivial case, for which the phase factor (the ratio of ω's) is nontrivial is q = bK: for g ∈G. We see that the effect of L q is to exchange two copies of the irreducible representations ofĜ, indexed by the irreducible representations of Z 2 . Taking that into account, we see that the prediction for physics is that the same decomposition as for [X/G] with a trivial action ofG and no discrete torsion -an example of ordinary decomposition. This is easily verified in partition functions.
Before going on, let us quickly walk through the details of the analogous computation in 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 is straightforward to compute that the only nontrivial elements β(ω)(q, k) are for a the generator of the Z 2 factor in K, and g ∈G. As a result, β(ω)(1, −) is the identity element of H 1 (K, U (1)), and β(ω)(bK, −) is the product of the trivial element of Hom(G, U (1)) and the nontrivial element of H 1 (Z 2 , U (1)). As a result, β(ω) is giving a map G/K → H 1 (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 ofG, but the correct answer gives as many copies as the number of all irreducible representations ofG, which need not be one-dimensional in general.

[X/Z 2 × Z 4 ] with discrete torsion and trivially-acting
Consider [X/Z 2 × Z 4 ] with discrete torsion. In this section we will consider three closely related examples in which a subgroup K, either Z 4 or Z 2 × Z 2 , acts trivially: In each of these three cases, ι * ω = 0. In the first two cases, this is for the trivial reason that H 2 (K, U (1)) = 0. In the third case, H 2 (K, U (1)) = 0; however, when one computes ι * ω in the third case, from table D.1, one finds it is given by 1 a b 2 ab 2 1 1 1 1 1 a 1 1 1 1 b 2 1 1 1 1 ab 2 1 1 1 1 and so is trivial.
In each of these three cases, G/K ∼ = Z 2 , where G = Z 2 ×Z 4 . As a result, H 2 (G/K, U (1)) = 0, and so the image of π * vanishes. This means the map β is injective, and so β(ω) is a nontrivial element of H 1 (G/K, H 1 (K, U (1))) = Hom Z 2 ,K . Let us check this description of β explicitly. As the analysis for all three cases is nearly identical, we give the details of β explicitly for only the first case, where K = b ∼ = Z 4 . We take 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) −1 ) ω(s(q) −1 , ks(q)) .

(5.30)
For q = 1, one has immediately that β = +1, and if q = a, k = b or b 2 , one also computes β = +1. The only two nontrivial cases are as follows: using the cocycle representative in appendix D.2. As a result, β(ω)(q = 1, −) is the element of H 1 (K, U (1)) that maps all elements of K to the identity, while β(ω)(q = a, −) is the element of H 1 (K, U (1)) that maps Z 4 onto Z 2 ⊂ U (1). As a homomorphism from G/K = Z 2 to H 1 (K, U (1)) = Z 4 , we see that β(ω) maps injectively into Z 4 , hence the kernel and cokernel are as given above.

(5.33)
Next, we shall check this prediction against partition functions. Briefly, in each of these three cases,

[X/Z 2 × Z 4 ] with discrete torsion and trivially-acting Z 2 subgroup
Consider again [X/Z 2 × Z 4 ] with discrete torsion. In this section we will consider two closely related examples in which a subgroup K ∼ = Z 2 acts trivially: (in the presentation of Z 2 × Z 4 described in appendix D.2).
Putting this together, the conjecture of section 3 then predicts In the first case, the kernel of β(ω) is generated by b 2 , while in the second case, the kernel of β(ω) is generated by a 2 , where a = {b, ab 3 } is the coset generating G/K ∼ = Z 4 .
Next, comparing to partition functions, it is straightforward to compute in both cases that with the effective Z 2 orbifold action predicted above, verifying the conjecture.

[X/D 4 ] with discrete torsion and trivially-acting Z 2 subgroup
Consider [X/D 4 ] with center Z 2 ⊂ D 4 acting trivially, and turn on discrete torsion in the D 4 orbifold.
As before, we begin by computing the prediction of section 3 for this case. Here K = Z 2 and G = D 4 , so G/K = Z 2 × Z 2 , and as H 2 (K, U (1)) = 0, we see immediately that ι * ω = 0. We also compute H 1 (G/K, H 1 (K, U (1))) = H 1 (G/K, The exact sequence (3.28) becomes From table D.4, since for example (a, 1) = (a, z), we see that ω is not the pullback of an element of discrete torsion in G/K = Z 2 × Z 2 . Thus, ω must map to a nontrivial element of H 1 (G/K, H 1 (K, U (1)), a nontrivial homomorphism G/K → Z 2 . From table D.3, we see that ω is trivial on the subgroup generated by b, but not that generated by a, from which we infer that the kernel of the map G/K → Z 2 is the Z 2 generated by b (the image of b in G/K), and there is no cokernel. Thus, we predict where the Z 2 is generated by b ∈ G/K.
Before going on to check the physics, let us take a moment to explicitly check the description of β(ω) above. In the present case, G/K = Z 2 × Z 2 = a, b , 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) −1 ) ω(s(q) −1 , 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: where the Z 2 appearing is a subgroup of the effectively-acting Z 2 × Z 2 which is generated by b. This matches our prediction.

[X/Z 4 Z 4 ] with discrete torsion and trivially-acting subgroups
In this example we study three closely related examples, orbifolds by Z 4 Z 4 (the semidirect product of two copies of Z 4 , discussed in appendix D.4) with discrete torsion, and the following trivially-acting subgroups: 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 fact that H 2 (K, U (1)) = 0. In the third case, H 2 (Z 2 × Z 4 , U (1)) = 0, so in principle ι * ω could be nonzero. To see that in fact ι * ω = 0 in this case as well, one can compute the pullback of the 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 a cocycle, 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 = π * ω for an ω ∈ H 2 (G/K, U (1)): 1. For K = x 2 , G/K = Z 2 × Z 4 , which does admit discrete torsion. However, from table D.6, the genus-one phases are not invariant under x 2 : for example, (1, y) = (x 2 , y), and so ω cannot be a pullback from H 2 (Z 2 × Z 4 , U (1)).
In each of these three cases, β(ω) = 0, and determines the prediction.
1. K = x 2 is contained in the center of G, so β(ω) will be a nontrivial homomorphism from G/K ∼ = Z 2 × Z 4 to H 1 (K, U (1)) ∼ = Z 2 . So we see that Coker β(ω) = 0 and the kernel is either Z 4 or Z 2 × Z 2 . By looking at the cocycles in table D.5, one can explicitly verify that β(ω)(yK, x 2 ) = −1, so yK / ∈ ker β(ω), and hence the kernel must be xK, 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 < 4, (5.58) in the conventions of appendix D.4. As before, β(ω)(q, k) = ω(ks(q), s(q) −1 ) ω(s(q) −1 , ks(q)) , (5.59) and it is straightforward to compute that the nontrivial elements are all are the nontrivial element ofK, so β(ω) : G/K →K is a surjective map. Since multiplying by xK leaves the map invariant, the kernel is xK, y 2 K = Z 2 × Z 2 , as anticipated above. The cokernel, trivially, vanishes.
We'll have no discrete torsionω in this case, since the cokernel vanishes. Equivalently, the two representations inK fall into a single G/K orbit, so we can take our representative to be the trivial homomorphism that sends K to 1 ∈ U (1), and the stabilizer is If there was nontrivial discrete torsion, we would be able to detect it by computinĝ where we have made the last step using table D.5. Soω 0 cannot be the nontrivial class in H 2 (Z 2 × Z 2 , U (1)).
In summary, for this case we predict a Z 2 × Z 2 orbifold without discrete torsion.
2. The subgroup K = x is not contained in the center of G, so we can't really use the cokernel of the map β(ω). Instead we will proceed more directly. First of all, we have G/K = yK ∼ = Z 4 . We can choose a section s(y n K) = y n . Note that this is actually a homomorphism, so the extension class is trivial, which is sensible since G is precisely the semidirect product of K with G/K. NowK = {[ρ m ]|0 ≤ m < 4}, 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 onK using (3.3).
In other words,K breaks into two orbits under G/K, each with stabilizer y 2 K ∼ = Z 2 .
We are left with a prediction of In each case, these predictions can be verified by genus-one partition function computations. Briefly, given (5.78) and using the discrete torsion phases for Z 4 Z 4 given in table D.6, it is straightforward to show that in each case matching the prediction.
6 Examples in which ι * ω = 0 and ω = π * ω In this section we will look at examples of G orbifolds in which the discrete torsion is a pullback 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.
with discrete torsion and trivially-acting Z 2 subgroup In this example we consider [X/Z 2 × Z 4 ] where the subgroup b 2 ∼ = Z 2 (in the conventions of appendix D.2) acts trivially.
Since G/K acts trivially onK, the conjecture of section 3 is going to predict two components, two copies of [X/Z 2 ×Z 2 ]. We need to compute the discrete torsion in each component. One contribution to that discrete torsion will be from ω. There is another potential contribution, from the image of the extension class of G, an element of H 2 (G/K, K), under each ρ ∈K. Since G is abelian, the argument below (2.5) shows that this additional contribution vanishes. As a result, the only contribution to discrete torsion on each [X/Z 2 × Z 2 ] component is from ω.
Putting this together, from the conjecture of section 3, we predict that or more simply, two copies of the Z 2 ×Z 2 orbifold with discrete torsion ω ∈ H 2 (Z 2 ×Z 2 , 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 2 × Z 4 orbifold with discrete torsion ω is given by precisely matching the prediction for this case.
By way of comparison, ordinary decomposition (ω = 0) says something very similar in this case: 8) with no discrete torsion on either side. This is consistent with our computation that the image of the extension class of Z 2 × Z 4 , an element of H 2 (Z 2 × Z 2 , Z 2 ), in H 2 (Z 2 × Z 2 , U (1)), necessarily vanishes for all ρ ∈K. Thus, in both this case and in the example above, the two copies of [X/Z 2 × Z 2 ] have the same discrete torsion.

[X/Z 4 Z 4 ] with discrete torsion and trivially-acting Z 2 subgroup
In this section we consider a Z 4 Z 4 orbifold (the semidirect product of two copies of Z 4 ) with discrete torsion in which a subgroup K = y 2 ∼ = Z 2 acts trivially, where Z 4 Z 4 is generated by x and y in the notation of appendix D.4. There is only one nontrivial value of discrete torsion since H 2 (Z 4 Z 4 , U (1)) = Z 2 . Also, it will be useful to observe that K = y 2 is a subgroup of the center of Z 4 Z 4 .
First, note that G/K = D 4 . Relating to the notation of appendix D.3, in which D 4 has generators a, b, we identify a = {y, y 3 }, b = {x, xy 2 }. It is straightforward to check, for example, that b 2 = {x 2 , x 2 y 2 } generates the center, and that as expected for D 4 .
Since H 2 (G/K, U (1)) = Z 2 , the same as H 2 (G, U (1)), it is possible that the element of discrete torsion in the Z 4 Z 4 orbifold is a pullback from D 4 . Indeed, the phases in table D.6 are symmetric under K: for example, (g, h) = (gy 2 , h) (6.10) and symmetrically, hence they pull back from G/K = D 4 . If we let ω be the nontrivial element of discrete torsion in D 4 , represented by a cocycle given in table D.3, then the pullback π * ω is given in table 6.1. Table 6.1: Cocycle π * ω for ω the nontrivial element of D 4 discrete torsion and π : Z 4 Z 4 → D 4 . We define ξ = exp(+6πi/8), matching the ξ of table D.5.
Taking into account the y 2 periodicities, it is straightforward to verify that table 6.1, describing π * ω, matches the cocycle for the nontrivial element of H 2 (Z 4 Z 4 , U (1)) given in table D.5. (Of course, they need only match up to coboundaries, but our conventions are such that they match on the nose.) Therefore, we see explicitly that the Z 4 Z 4 discrete torsion ω = π * ω.
Applying section 3, we can now see that the [X/Z 4 Z 4 ] orbifold with discrete torsion should be equivalent to a disjoint union of two [X/D 4 ] orbifolds, using the fact that in this case, D 4 = G/K acts trivially 10 onK.
It remains to compute the discrete torsion on each [X/D 4 ] summand. Part of that discrete torsion will be ω, but there could also be another contribution, the image of the extension class H 2 (D 4 , Z 2 ) under each of the irreducible representations of K = Z 2 . Since Z 4 Z 4 is not Z 2 D 4 (in fact, Z 2 × D 4 , since the Z 2 is central), the extension class in H 2 (D 4 , Z 2 ) is nontrivial. However, we claim that any map induced by a representation into H 2 (D 4 , U (1)) vanishes.
Indeed, to compute the extension class, we must first pick a section s : G/K → G (so π(s(q)) = q for all q ∈ G/K), and then define the extension class by e(q 1 , q 2 ) = s(q 1 )s(q 2 )s(q 1 q 2 ) −1 ∈ K, for all q 1 , q 2 ∈ G/K. Now there are two possible irreducible representations of K = Z 2 , a trivial one ρ 0 which sends y 2 to 1, and a nontrivial one ρ 1 which sends y 2 to −1. Applying these to the extension class e gives us a pair of 2-cocycles,ω a (q 1 , q 2 ) = ρ a (e(q 1 , q 2 )). Clearlyω 0 is the trivial 2-cocycle. The other possibilityω 1 is not trivial, but it is exact, and in particular it is symmetric in its arguments,ω 1 (q 1 , q 2 ) =ω 1 (q 2 , q 1 ), which implies that the corresponding discrete torsion contribution vanishes.
Putting these pieces together, we can now make the prediction that The fact that the QFT decomposes into two identical summands (universes), and so has a BZ 2 symmetry, reflects the group-theoretic fact that the discrete torsion ω in Z 4 Z 4 is invariant under y 2 ∼ = Z 2 : ω(g, h) = ω(gy 2 , h) = ω(g, hy 2 ) = ω(gy 2 , hy 2 ), (6.14) as can be seen in table D.5. (Of course, such a relation need only hold up to cocycles, but in our conventions it holds on the nose.) Next, we shall compare this prediction to physics results. Using table D.6 of phases for discrete torsion in Z 4 Z 4 orbifolds, it is straightforward to compute that the genus-one orbifold partition function is 16) or two copies of the D 4 orbifold, each with discrete torsion, confirming the prediction.
By way of comparison, ordinary decomposition (ω = 0) says something very similar in this case: 17) with no discrete torsion on either side.

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 phase assigned 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 ∈ {0, · · · , k − 1}. Then, associated to the twisted sector (g, h) = (a, b; a , b ) is the phase [44, equ'n (2.2)] (a, b; a , b ) = ξ ab −ba , where ξ is a kth root of unity corresponding to the value of discrete torsion.
Let us first compute our predictions from section 3 for the various possible values of discrete torsion. Let ξ be a fourth root of unity which encodes the value of discrete torsion.
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 have a disjoint union of four copies of [X/Z 2 ×Z 2 ], where hereK denotes the set of irreducible honest representations of K.
• ξ = −1. In this case, using the phases in equation (7.1), it is straightforward to see that the discrete torsion in Z 4 × Z 4 is a pullback of discrete torsion in Z 2 × Z 2 . (Since ξ 2 = +1, the phases (7.1) are invariant under shifting any of a, b, a , b by 2, or more simply, invariant under K, reflecting an underlying BK one-form symmetry.) As a result, decomposition [8] applies again, and we have a disjoint union of four copies of [X/Z 2 × Z 2 ] d.t. , where hereK denotes the set of irreducible honest representations of K.
• ξ 2 = −1. In these cases, the phases (7.1) are not invariant under K, since shifting any of a, b, a , b changes the genus-one phase (7.1) by a sign. For this reason, the cocycle ω = π * ω for any ω ∈ H 2 (Z 2 × Z 2 , U (1)). Instead, in these cases, β(ω) = 0. Now, β(ω) is an element of so enumerating possibilities, if β(ω) is nontrivial, one quickly deduces that Given that the genus-one phases (7.1) are entirely nontrivial, they appear to define isomorphisms, hence Ker β(ω) = Coker β(ω) = 0, (7.9) and so for these cases, we predict Next, we compute the genus-one partition functions in the different cases, to confirm the predictions above. In the current example, since the Z 2 × Z 2 ⊂ Z 4 × Z 4 acts trivially, we should compute the numerical factors multiplying each effective Z 2 × Z 2 orbifold twisted sector. These are listed in table 7.1, where we use the notation (a, b; a , b ) for a, b, a , b ∈ {0, 1} to indicate an effective Z 2 × Z 2 twisted sector, and where ξ is a fourth root of unity, corresponding to the choice of Z 4 × Z 4 discrete torsion.
Our results for genus-one partition functions are as follows. Table 7.1: Multiplicities and phases of effective Z 2 × Z 2 twisted sectors in Z 4 × Z 4 orbifold with 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 consistent with the prediction of decomposition [8] that this orbifold be equivalent to a disjoint union of four copies of an effective [X/Z 2 × Z 2 ] orbifold.
• Case ξ = −1. In this case, using the results in table 7.1, it is straightforward to show that consistent with the statement that in this case, the Z 4 × Z 4 orbifold with discrete torsion is equivalent to a disjoint union of four copies of an effective Z 2 × Z 2 orbifold with discrete torsion.
• Cases ξ 2 = −1. In these cases, using the results in  = Z(X), (7.16) consistent with the statement that in these two cases, the Z 4 ×Z 4 orbifold with discrete torsion is equivalent to no orbifold at all.
This confirms our predictions.
In the first two cases, in which ξ 2 = +1, the Z 4 × Z 4 orbifold has a natural B(Z 2 × Z 2 ) symmetry, as the discrete torsion phases (a, b; a , b ) are invariant under incrementing any of a, b, a , b by 2. This is a symmetry-based reason why there is a decomposition [8] in these cases.
Thus, depending upon the value of discrete torsion, we see that the Z 4 × Z 4 orbifold is equivalent to one of the following three possibilities: (7.17) 7.2 Z 8 ×Z 8 orbifold with discrete torsion and trivially-acting Z 2 ×Z 2 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 orbifold summands/universes than in the previous (set of) examples, which will enable us to conduct more thorough tests of predictions for discrete torsion in summands/universes.
In principle, the discrete torsion is then given by ω +ω 0 , whereω 0 is that predicted by the original decomposition. In this case, the group G is abelian, henceω = 0 on each component, as observed in section 2. Mathematically, the same result can be obtained by observing that since in this case the extension class is symmetric, as can be made manifest by picking matching sections, one haŝ and so the corresponding genus-one phases are trivial.
Next, we compute partition functions, to compare to the results above. To that end, it is helpful to first write the genus-one partition function of the Z 8 × Z 8 orbifold in the form  Assembling these results, we can now compute genus-one partition functions.
confirming our prediction.

Conclusions
In this paper we have generalized decomposition [8] in orbifolds with trivially-acting subgroups to include orbifolds with discrete torsion. Although the discrete torsion breaks (much of) any original one-form symmetry, there is nevertheless a decomposition-like story. We have described a general prediction for all cases, which we have checked in numerous examples. in M and multiplicative notation for G. This helps clarify the relative structures, so we do so 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 either case multiplicative notation is more natural, so these formulae should be retranscribed using multiplicative notation for M . For the case of M = U(1) with trivial G-action, we do so at the end of this section.
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 is thus 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−1 (C n−1 (G, M )) (known as the space of coboundaries) is a submodule of Z n (G, M ), and so we can define cohomology groups by taking the quotient, We say that an n-cochain ω is normalized if ω(g 1 , · · · , g n ) vanishes whenever any one of its 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 a normalized representative. For instance, consider the case n = 2. Then the cocycle condition is (dω)(g 1 , g 2 , g 3 ) = g 1 · ω(g 2 , g 3 ) − ω(g 1 g 2 , g 3 ) + ω(g 1 , g 2 g 3 ) − ω(g 1 , g 2 ). (A.3) By setting g 1 = g 2 = 1, we learn that a cocycle satisfies ω(1, g) = ω(1, 1) for all g, and setting g 2 = g 3 = 1 tells us that ω(g, 1) = g · ω(1, 1). Now pick any map µ : G → M satisfying µ(1) = −ω(1, 1) and define ω = ω + dµ. Then this new cocycle satisfies and from the previous argument we also have ω (1, g) = ω (g, 1) = 0, showing that the cohomology class of ω contains a normalized representative ω . Throughout the paper we will assume that our cochains and cocycles are normalized.
Note that if ω is normalized, then any projective representation φ necessarily sends the identity element to the identity matrix, φ(1) = 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 11 φ(g −1 ) = ω(g, g −1 )φ(g) −1 . (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 a basis {v g |g ∈ G} and defining a map φ r : G → GL(V r ) by its action on the basis, (A.11) Then (φ r (g 1 )φ r (g 2 ))(v h ) = ω(g 1 , g 2 h)ω(g 2 , h)v g 1 g 2 h , (A.12) and φ r (g 1 g 2 )(v h ) = ω(g 1 g 2 , h)v g 1 g 2 h .
If the only invariant subspaces of V are 0 and V itself, then we say that φ is irreducible. We say that two projective representations φ 1 : G → GL(V 1 ) and φ 2 : G → GL(V 2 ) with respect to the same ω are isomorphic if there exists a vector space isomorphism f : Of course irreducibility is preserved by isomorphism, so given a cocycle ω ∈ Z 2 (G, U(1)) we can talk about the setĜ ω of isomorphism classes of irreducible projective representations of G with respect to ω.
Just as there is a (not necessarily canonical) one-to-one correspondence between the isomorphism classes of ordinary irreducible representations,Ĝ, and conjugacy classes of G [45, section 2.5], there is also a one-to-one correspondence betweenĜ ω and conjugacy classes [g] of G which additionally satisfy [46, prop. 2.6], [47][48][49][50][51], for any element g in the conjugacy class, that ω(g, g ) = ω(g , g) for all g ∈ G such that gg = g g. We will call these ω-trivial conjugacy classes, and their elements will be ω-trivial elements.
As an example, consider G = Z 2 × Z 2 = {1, a, b, c}. Then H 2 (G, U(1)) ∼ = Z 2 , and a representative ω for the nontrivial class can be defined by ω(1, g) = ω(g, 1) = ω(g, g) = 1, g ∈ G, An example of a projective representation is 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 identity element. If one works out the regular representation for this G and ω, the resulting fourdimensional projective representation can be shown to be isomorphic to the direct sum of two copies of the irreducible representation.

B Some calculations with cocycles
In this appendix we provide the details of our calculation of ω a andω a in section 3.
Hence C a (g 1 , g 2 ) commutes with all ρ a (k). Then the irreducibility of ρ a and an application of Schur's lemma tell us that C a (g 1 , g 2 ) is proportional to the identity matrix, C Explicit realization of β The map β : Ker ι * −→ H 1 (G/K, H 1 (K, U (1)) (C.1) that we have utilized is described explicitly in [33, section 7] as the 'reduction' map r, but as the description there is in somewhat different language, in this appendix we will unroll that definition to understand its properties more explicitly.
For this appendix, we specialize to the case that ι * ω is trivial, where ι : K → G. For this case, [33] defines V · K ⊂ E, which is naturally isomorphic to σ −1 (K).
The reference [33] defines two further pertinent maps. First, ρ : G → Aut(V · K), which is given by since the group action on the coefficients is trivial here. The reference also defines γ V ·K,E : for all e = (ṽ, g) ∈ E.
First, since the action on the U(1) coefficients is trivial, H 1 (K, U(1)) consists of maps φ : K → U(1) such that In other words, H 1 (K, U(1)) consists of group homomorphisms from K to U(1).

D Pertinent group theory results
In this paper we perform rather detailed manipulations of and computations utilizing representative cocycles of elements of discrete torsion. To that end, in this section we will give explicit presentations and discrete torsion cocycles for several finite groups which we use in this paper.
Denoting the two generators of Z 2 × Z 2 by a, b, then the group 2-cocycle explicitly is with ω(g, h) = +1 for other g, h. Given this cocycle, it is straightforward to see that the only conjugacy class obeying the condition (3.24) is {1}.
We present the group Z 2 × Z 4 as generated by a, b, where a 2 = 1 = b 4 . It can be shown that H 2 (Z 2 × Z 4 , U (1)) = Z 2 , and the cocycle and genus-one phases of the nontrivial element are given in tables D.1, D.2.
Since Z 2 × Z 4 is abelian, each element corresponds to its own separate conjugacy class. Of these, from table D.2, only the conjugacy classes {1}, {b 2 } satisfy the condition (3.24), and so are associated with irreducible projective representations.

D.3 D 4
We present the eight-element dihedral group D 4 as generated by z, a, b, where z generates the center, z 2 = 1, a 2 = 1, b 2 = z, so that the elements are described as where the n(g, h) are given in table D.3. Note that in the group multiplication, b 2 = z, b 3 = bz, and ba = abz.
Using this, the phases weighting a given genus-one twisted sector, associated to a commuting pair (g, h), are given as ratios (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 4 /Z 2 = Z 2 × Z 2 , as for example the discrete torsion there generates phases solely in sectors that do not lift to D 4 . (In particular, for the pullback of H 2 (Z 2 × Z 2 , U (1)), the discrete torsion phases are trivial, so for that Of these, only the conjugacy classes {1}, {b, bz} are associated with projective representations. We describe issues with the others below: • ω(a, z) = ω(z, a), so {z} is not a pertinent conjugacy class.
Thus, we see that there are two irreducible projective representations of D 4 with the nontrivial element of H 2 (D 4 , U (1)). (This result is also given in [46, example 3.12].)

D.4 Z 4 Z 4
We describe the group Z 4 Z 4 , the semidirect product of two copies of Z 4 , as generated by x, y subject to the conditions x 4 = y 4 = 1 and y = xyx.
It can be shown that H 2 (Z 4 Z 4 , U (1)) = Z 2 , with cocycles as given in table D.5.
The group Z 4 Z 4 has ten conjugacy classes, namely

D.5 S 4
The group S 4 is the symmetric group on four objects. Its group elements can be presented as 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 2 (S 4 , U (1)) = Z 2 . In this appendix we collect an explicit representative of the nontrivial cocycle and corresponding twisted sector phases.