Orientation twisted homotopy field theories and twisted unoriented Dijkgraaf-Witten theory

Given a finite $\mathbb{Z}_2$-graded group $\hat{\mathsf{G}}$ with ungraded subgroup $\mathsf{G}$ and a twisted cocycle $\hat{\lambda} \in Z^n(B \hat{\mathsf{G}}; \mathsf{U}(1)_{\pi})$ which restricts to $\lambda \in Z^n(B \mathsf{G}; \mathsf{U}(1))$, we construct a lift of $\lambda$-twisted $\mathsf{G}$-Dijkgraaf--Witten theory to an unoriented topological quantum field theory. Our construction uses a new class of homotopy field theories, which we call orientation twisted. We also introduce an orientation twisted variant of the orbifold procedure, which produces an unoriented topological field theory from an orientation twisted $\mathsf{G}$-equivariant topological field theory.


Introduction
Fix n ≥ 1. Let G be a finite group and let λ ∈ Z n (BG; U(1)) be an n-cocycle on the classifying groupoid BG. The associated Dijkgraaf-Witten theory Z λ G is an n-dimensional oriented topological quantum field theory. In this paper we regard Z λ G as a once-extended theory, so that it can be evaluated on oriented manifolds of codimension 0, 1 and 2. The theory Z λ G was first introduced in [8] as a finite version of Chern-Simons theory [51] and has since been developed from a number of different perspectives [14], [10], [33], [20], [40]. One reason for the enduring interest in Dijkgraaf-Witten theory is that it is a topological field theory which is both interesting and amenable to direct study. It is therefore beneficial to develop techniques in this finite setting before approaching more complicated topological field theories, such as Chern-Simons theory. On the other hand, Dijkgraaf-Witten theory itself has found applications in many areas of mathematics and physics, such as low dimensional topology, representation theory, tensor category theory and, more recently, condensed matter physics.
The main result of the present paper, which is stated as Theorem 4.1, is a geometric construction of a class of unoriented lifts of Dijkgraaf-Witten theory. More precisely, for each Real structureĜ on G, that is, a short exact sequence of finite groups 1 → G →Ĝ π − → Z 2 → 1, and lift of λ to a π-twisted n-cocycleλ ∈ Z n (BĜ; U(1) π ), we construct an ndimensional unoriented topological quantum field theory Zλ G whose oriented restriction is Z λ G . The theories Zλ G recover as special cases the previously known unoriented lifts of Dijkgraaf-Witten theory. WhenĜ = G × Z 2 andλ = 1, we recover the unoriented lift determined by the stack Bun G (−) of principal G-bundles (viewed as a functor out of the unoriented cobordism category) [14], while for n = 2, the trivial Real structureĜ = G × Z 2 and particularλ we recover the state sum theories of [46], although in a different realization. On the other hand, both physical and abstract mathematical arguments, the latter involving the cobordism hypothesis [30] or methods from stable homotopy theory, have been used to assert the existence of an unoriented topological field theory attached to a pair (Ĝ,λ) as above [11], [22], [12]. Theorem 4.1 gives a concrete and direct geometric realization of these theories.
Our construction of Zλ G is as a composition Zλ G : Cob n,n−1,n−2 Here Cob n,n−1,n−2 is the n-dimensional unoriented cobordism bicategory, 2Vect C is the bicategory of Kapranov-Voevodsky 2-vector spaces, 2Vect C (Grpd) is the bicategory of 2-vector bundles on spans of groupoids, as constructed in [33], [39], [19], and Par is the linearization functor of [33], [38]. The main problem, whose solution we outline below, is therefore to define Aλ G , which we regard as a classical unoriented topological G-gauge theory with Lagrangianλ. As expected from the Lagrangian point of view, we prove that the values of Aλ G can be computed from the data (Ĝ,λ) via a suitable pushforward procedure; see Theorem 4.2. For example, for a closed unoriented (and possibly nonorientable) (n − 2)-manifold X, the underlying groupoid of Aλ G (X) is the groupoid Bun or G (X) of principalĜ-bundles on X with an or X -twisted reduction of structure group to G, where or X → X is the orientation double cover. The additional cohomological data on Bun or G (X) is constructed from a flat U(1)-gerbe on Bun or G (X), which is in turn obtained as the image ofλ under an orientation twisted transgression map τ or X : Z n (BĜ; U(1) π ) → Z 2 (Bun or G (X); U(1)). This transgression map is introduced in Section 3. 3.
In order to construct Aλ G , we first introduce the notion of orientation twisted homotopy field theory, which is of independent interest. Such a theory depends on a chosen topological double cover Π : T →T . The relevant cobordism bicategorŷ T -Cob Π n,n−1,n−2 is one of unoriented cobordisms with maps toT together with lifts to Z 2 -equivariant maps from the orientation double cover of the cobordism to T . Our definition ofT -Cob Π n,n−1,n−2 is motivated by Atiyah's oriented cobordism categories with coefficients in a double cover [3] and by mathematical approaches to unoriented Wess-Zumino-Witten theory [36] and string theory with orientifolds [9]. In the non-extended setting, various specializations of orientation twisted field theories have been studied under different names; see [45], [44], [23]. A key result of the present paper is Theorem 2.5, which associates to an n-cocycleλ ∈ Z n (T ; U(1) T ), whose coefficients are twisted by the double cover T →T , an n-dimensional orientation twisted homotopy field theory Pλ T . Geometrically, Pλ T can be understood as parallel transport along the Jandl (n − 1)-gerbeλ. The oriented theory which underlies Pλ T is that determined by the pullback λ ∈ Z n (T ; U(1)) ofλ, as constructed by Turaev [47] and Turner [49] in the non-extended setting and by Müller-Woike [34] in the once-extended setting.
The connection between Dijkgraaf-Witten theory and homotopy field theory arises when the double cover T →T is taken to be the (geometric realization of the) map of classifying groupoids BG → BĜ. In this setting, homotopy field theories are called G-equivariant field theories. Our main result about G-equivariant theories is Theorem 3.12, which is a generalization of the oriented orbifold construction of Schweigert-Woike [39], [37]. The theorem allows us to functorially associate to each orientation twisted G-equivariant topological quantum field theory an unoriented topological field theory with target 2Vect C (Grpd). The desired theory Aλ G can then be defined to be the orientation twisted orbifold of the G-equivariant theory Pλ BĜ .
There are a number of reasons to be interested in unoriented topological field theory in general and unoriented Dijkgraaf-Witten theory in particular. Aside from purely topological applications, unoriented topological field theories are central to the classification of symmetry protected topological phases of matter with time reversal symmetry [22], [12], [5], [23], [15]. In this context, Dijkgraaf-Witten theories play an important role due their relation to Kitaev's quantum double model [27], [5]. As a rather different example, the fully-extended variant of Zλ G can be seen as a physical realization of the Real representation theory of the higher categorical group determined by (G, λ) [52].
The results of this paper suggest a number of follow-up problems. Perhaps the most interesting, which will be the subject of a forthcoming paper, is to study the algebraic structures encoded by Zλ G in dimension three. While there is a classification of once-extended oriented topological field theories [6], there is at present no such unoriented classification. The theories Zλ G provide a simple class of examples which can be used to explicate the additional structure on the modular tensor category determined by the oriented theory. In the same vein, Aλ G can be used to study unoriented equivariant three dimensional theories and the resulting Real generalizations of the G-crossed modular tensor categories. It would be interesting to compare these algebraic structures with the input data of the recently proposed unoriented extension of Turaev-Viro-Barrett-Westbury theory [5], [7]. Finally, our geometric approach admits a natural generalization to allow for defects and boundary conditions, along the lines of the oriented case [18]. Calculations in the case of Dijkgraaf-Witten theory will shed light on the as-of-yet undeveloped theory of defects in unoriented theories. In three dimensions, this will lead to a refinement of the bimodule classification from the oriented case [17]. Finally, we expect that discrete torsion in string and M-theory with orientifolds (see [4], [41]) can be understood in terms of the orientation twisted orbifold construction, by first tensoring the G-equivariant theory describing (mem)branes on a G-space with Zλ G , and then orbifolding.
The structure of this paper is as follows. Section 1 contains relevant background material. In Section 2 we introduce orientation twisted homotopy field theories and construct the theories Pλ T . In Section 3 we study orientation twisted equivariant theories and describe the orientation twisted orbifold construction. Finally, in Section 4 we construct twisted unoriented Dijkgraaf-Witten theory, studying in detail the theory in dimensions one and two.
Acknowledgements. The author would like to thank Jeffrey Morton, Lukas Müller and Mark Penney for discussions related to the content of this paper.
1. Background material 1.1. Homotopy limits. The homotopy limit of a diagram of groupoids of the form This groupoid fits into a homotopy commutative diagram and is characterized as being 2-universal among all such groupoids. We will use the following model for X × h Z Y. Objects are triples (x, y; ϕ) consisting of x ∈ X and y ∈ Y and a morphism ϕ : commutes. The component of η at (x, y; ϕ) ∈ X × h Z Y is ϕ. When F is the inclusion of an object z ∈ Z, the homotopy limit {z} × h Z Y is called the homotopy fibre of F over z and is denoted by RF −1 (z).
We will use the following basic result below.
Lemma 1.1. Let groups G and H act on sets X and Y , respectively, with the latter action being trivial. Let π : G → H be a surjective group homomorphism and let F : X → Y be a π-equivariant map. Then the homotopy fibre over y ∈ Y / /H of the induced functor F : Proof. The homotopy fibre R F −1 (y) is the groupoid with objects and morphisms The stated assumptions imply that Obj(R F −1 (y)) = F −1 (y) × H. The explicit form of morphisms in RF −1 (y) then gives R F −1 (y) ≃ (F −1 (y) × H)/ /G, where G acts on F −1 (y) and H through its action on X and π, respectively. Since π is surjective, every object of R F −1 (y) is isomorphic to one of the form (x ′ , e), where e ∈ H is the identity. The claimed equivalence follows. is the bicategory whose objects are continuous maps X → Y and whose 1-and 2-morphisms are homotopies and equivalence classes of homotopies of homotopies relative X × ∂I, respectively. Two homotopies η, η ′ : X × I × I → Y relative X × ∂I are called equivalent, written η ≃ η ′ , if they are homotopic relative X × ∂(I × I).
Fix a topological space B.
An explicit description of Map ≤2 B (X, Y ) is as follows. 1 (i) Objects are pairs (f ; m) consisting of a continuous map f : X → Y and a homotopy m : are equivalence classes of homotopies ϕ : Here m 2 denotes the extension of m 2 : X × I → B to a map X × I × I → B which is constant along the second factor of I. Equivalently, m 2 * π Y (ϕ) is the whiskering of the 1-morphism m 2 with the 2-morphism π Y (ϕ). Similarly, one defines the relative mapping groupoid Map B (X, Y ). This groupoid is equivalent to the homotopy groupoid of Map ≤2 B (X, Y ). Truncating the explicit description of Map ≤2 B (X, Y ) recovers that of Map B (X, Y ) which arises from the model for homotopy fibres described Section 1.1.
Finally, there is a relative mapping space Map B (X, Y ), defined as the homotopy fibre over π X of the continuous map of topological spaces Proof. It is well-known that asphericity of a topological space Z implies that the canonical map Map ≤2 (X, Z) → Map(X, Z) is a biequivalence. The first statement then follows from the fact that homotopy limits of biequivalent diagrams of 2groupoids are biequivalent.
The second statement follows from the fact that, by asphericity, Map(X, Y ) and Map(X, B) are homotopy equivalent to the geometric realizations of Map(X, Y ) and Map(X, B), respectively, together with the fact that geometric realization commutes with finite homotopy limits.
The final statement follows from the long exact sequence of homotopy groups associated to the fibration and the asphericity of Map(X, Y ) and Map(X, B).

1.3.
Groupoids of principal bundles. Let M be a compact topological manifold, possibly with boundary. Given a finite group G, denote by Bun G (M) the groupoid whose objects are principal G-bundles P → M and whose morphisms (P → M) → (P ′ → M) are G-equivariant maps P → P ′ which commute with the structure maps to M. The G-action on the total space of a G-bundle is from the right.
The groupoid Bun G (M) has a number of equivalent models. To describe the first, assume for simplicity that M is connected and fix a basepoint m 0 ∈ M, which we henceforth omit from the notation. Then there is an equivalence where G acts on Hom Grp (π 1 (M), G) by conjugation. This equivalence sends a (necessarily flat) G-bundle to its holonomy representation.
The second model is in terms of a classifying space BG of G. Namely, there is an equivalence Bun G (M) ≃ Map(M, BG), a G-bundle P → M being sent to a classifying map f P : M → BG of P . The G-bundle determined by a map f : M → BG will be denoted by P f → M.

Spans of groupoids with local coefficients.
A groupoid X is called essentially finite if π 0 (X ) is finite. The category of essentially finite groupoids is denoted by Grpd. Unless explicitly mentioned otherwise, all groupoids in this paper are assumed to be essentially finite.
The bicategory Span(Grpd) has groupoids as objects, spans of groupoids as 1morphisms and equivalence classes of spans of spans as 2-morphisms. For precise definitions, see [32]. We require a decorated version of Span(Grpd), in which objects and morphisms carry compatible local systems. A general construction in the context of (∞, n)-categories has been developed in [19]. In the bicategorical setting there are explicit constructions [39], [38].
Let 2Vect C be the bicategory of Kapranov-Voevodsky 2-vector spaces, that is, finitely semisimple C-linear additive categories, C-linear functors and natural transformations [21]. The Deligne product ⊠ gives 2Vect C the structure of a symmetric monoidal bicategory.
Given a groupoid X , the pseudofunctor bicategory Hom Bicat (X , 2Vect C ) is called the bicategory of 2-vector bundles on X and is denoted by 2Vect C (X ).
• Objects are pairs (X , α) consisting of a groupoid X and a 2-vector bundle α : X → 2Vect C .
consisting of a span of spans and a 2-morphism γ : . Two such tuples are called equivalent if there is an equivalence of spans of spans which respects the 2-morphisms.
The composition of the 1-morphisms (X 1 , α 1 ) together with the 1-morphism of 2-vector bundles The reader is referred to [38] for the definition of the horizontal composition of 2-morphisms.
Cartesian product of groupoids together with the Deligne product of 2-vector spaces extend to define a symmetric monoidal structure on 2Vect C (Grpd).
1.5. Twisted 2-linearization. Classical topological gauge theories can often be understood as topological field theories valued in (higher) categories of spans of groupoids with local systems. Their quantization can then be approached by postcomposing with a suitable functor to a sufficiently linear target higher category [13]. In the (∞, n)-categorical setting, such linearization functors were constructed by Haugseng [19] while in the present bicategorical setting, following earlier work of Morton [32], [33] in the case without local systems, Schweigert-Woike [39], [38] constructed a symmetric monoidal pseudofunctor The pseudofunctor Par assigns to a 2-vector bundle α : X → 2Vect C its space of flat (or parallel) sections, where Vect C|X is the trivial 2-line bundle on X . The 1-morphism (1) is assigned the pushforward along β: For the definition of Par on 2-morphisms and the verification that Par is a symmetric monoidal pseudofunctor, the reader is referred to [38, §4.2].

Orientation twisted extended homotopy field theories
In this section we define a generalization of unoriented homotopy quantum field theory. The construction is motivated by Atiyah's oriented cobordism groups with coefficients in a double cover [3].
2.1. Orientations. All manifolds are assumed to be smooth and compact. We allow manifolds to have corners. A manifold with empty boundary is called closed. Unless explicitly stated, all manifolds are assumed to be unoriented, and possibly nonorientable.
Let Z 2 be the multiplicative group {±1}. Recall that each manifold M has a canonical orientation double cover or M → M. We fix a classifying map M → BZ 2 of the orientation cover which, if it will not lead to confusion, we also denote by or M . If M is of dimension n, then or M can be constructed as the composition of a classifying map M → BO n of the tangent bundle T M → M with the canonical map w 1 : BO n → BZ 2 . The manifold or M is canonically oriented, and this orientation defines the fundamental class [M] ∈ H n (M, ∂M; Z or M ) of M.

2.2.
(Un)oriented cobordism bicategories. Let T be a topological space. Denote by T -Cob or n,n−1,n−2 the bicategory of n-dimensional oriented (compact) cobordisms with continuous maps to T . An object of T -Cob or n,n−1,n−2 is a closed oriented (n − 2)-manifold with a continuous map to T . A 1-morphism is an (n − 1)dimensional oriented collared cobordism with a continuous map to T which is compatible with the boundaries. There is an unoriented variant T -Cob n,n−1,n−2 of T -Cob or n,n−1,n−2 , defined in the same way as T -Cob or n,n−1,n−2 but with all orientation data omitted. Disjoint union gives T -Cob (or) n,n−1,n−2 the structure of a symmetric monoidal bicategory. Forgetting orientations defines a symmetric monoidal pseudofunctor F : T -Cob or n,n−1,n−2 → T -Cob n,n−1,n−2 .
2.3. Orientation twisted cobordism bicategories. In this section we introduce the cobordism bicategory which underlies orientation twisted homotopy field theory. Let Π :T → BZ 2 be a topological space over BZ 2 . The map Π classifies a double cover, which we denote by ρ : T →T .
While the following definition is rather involved, the basic idea is simple: replace all occurrences of orientations in T -Cob or n,n−1,n−2 with the data of maps of spaces over BZ 2 .
Definition. The orientation twisted cobordism bicategoryT -Cob Π n,n−1,n−2 is defined as follows: • An object is a triple (X, f ; h) consisting of a closed (n − 2)-manifold X, a continuous map f : X →T and an equivalence class of homotopies h : with equivalence classes of homotopies
-A continuous map ϕ : Z →T such that the diagram commutes.
The bicategoryT -Cob Π n,n−1,n−2 is symmetric monoidal under disjoint union. When it will not lead to confusion, we will write Y in place of (Y ; o 1 , o 2 ), and similarly for Z. We will often omit the explicit mention of collars.

Remarks.
(i) There is a pointed versionT * -Cob Π n,n−1,n−2 ofT -Cob Π n,n−1,n−2 in whichT is pointed, objects have a basepoint in each connected component and the map toT is pointed, and similarly for 1-and 2-morphisms. IfT is connected, as will always be the case in this paper, then the forgetful map T * -Cob Π n,n−1,n−2 →T -Cob Π n,n−1,n−2 is a monoidal biequivalence. (ii) The above definition truncates to define a categoryT -Cob Π n,n−1 . This category is monoidally equivalent to the category of 1-endomorphisms of the monoidal unit ∅ n−2 ofT -Cob Π n,n−1,n−2 . (iii) By construction, the group π 0 (T -Cob Π n,n−1 ) is isomorphic to the oriented cobordism group MSO n−1 (T,T ) of T with coefficients inT , as introduced by Atiyah [3, §2].
Proof. The double cover classified by Π is homotopy equivalent to the inclusion T ֒→ T × BZ 2 at a chosen basepoint of BZ 2 . Let π T : T × BZ 2 → T be the projection to the first factor. Then Φ can be defined to be post-composition with π T . More precisely, set Φ(X, f ; h) = (X, π T • f ) on objects. The functor can be taken to be the respective identities. There is a canonical lift of Φ to a symmetric monoidal pseudofunctor.
Given (X,f ) ∈ T -Cob n,n−1,n−2 , we have where we have written or X both for the map X → BZ 2 and for the associated identity homotopy X × I → BZ 2 . This shows that Φ is essentially surjective.
Consider now the functor Φ X 1 ,X 2 . Let (Y, F ) be an object of the codomain of Φ X 1 ,X 2 . Let F be a classifying map of the orientation cover of Y which restricts to and let H be a homotopy from F to or Y . The existence of F is ensured by the fact that (3) classifies or ∂Y while that of H follows from the fact that orientation covers are unique up to equivalence.
, proving that Φ X 1 ,X 2 is essentially full on 1-morphisms. This construction admits an obvious variation in which the (n − 1)-cobordism is replaced with an n-cobordism with corners. This shows that Φ X 1 ,X 2 is locally full on 2-morphisms.
The next result provides a generalization of the forgetful map (2).
There is a symmetric monoidal pseudofunctor Proof. Let ν be a null-homotopy of the composition T ρ − →T Π − → BZ 2 , say to z ∈ BZ 2 . We will interpret an orientation of a manifold M as a homotopy ω M from z to or M . With this notation, the functor F can be defined as follows. On objects set 3 and on 1-morphisms set The lift of i k : X k ֒→ Y to a map over BZ 2 , which has been omitted from the notation, is obtained from the compatible orientations of Y and X 1 ⊔ X 2 . The definition on 2-morphisms is analogous to that on 1-morphisms. The additional compatibility 2-isomorphisms and the lift of F to a symmetric monoidal pseudofunctor are canonical.
Let C be a symmetric monoidal bicategory.

Definition.
A once-extended n-dimensional orientation twisted homotopy field theory with targetT valued in C is a symmetric monoidal pseudofunctor 3 For ease of notation, we have written ω X * ν instead of the more accurate ω X * (ν • (f × id I )).
When it will not lead to confusion, we will omit the adjectives 'once-extended' and 'homotopy'. The 2-groupoidT -TFT Π n,n−1,n−2 (C) is defined to be the full subbicategory of Hom Bicat (T -Cob Π n,n−1,n−2 , C) spanned by orientation twisted homotopy field theories. When C = 2Vect C we simply writeT -TFT Π n,n−1,n−2 . Truncating the previous definition defines the groupoidT -TFT Π n,n−1 (A) of non-extended orientation twisted field theories valued in a monoidal category A. Moreover, restriction to EndT -Cob Π n,n−1,n−2 the domain being the homotopy category ofT -TFT Π n,n−1,n−2 (C). Definition. An orientation twisted lift of an oriented homotopy field theory is the data of a map Π :T → BZ 2 for which T is (homotopic to) the total space of the associated double cover and an orientation twisted homotopy field theory Z lift :T -Cob Π n,n−1,n−2 → C which makes the following diagram homotopy commute: The following result is motivated by [3, Proposition 2.3]. Proposition 2.3. LetT = T ×BZ 2 with Π the projection to the second factor. Then restriction along the pseudofunctor Φ from Proposition 2.1 defines a biequivalence Φ * : T -TFT n,n−1,n−2 (C) →T -TFT Π n,n−1,n−2 (C). Proof. Let Z : T -Cob n,n−1,n−2 → C be an unoriented homotopy field theory. Ho- Recall that a pseudofunctor is a biequivalence if and only if it is essentially surjective, essentially full on 1-morphisms and locally fully faithful. The proof of Proposition 2.1 shows that Φ fails to be a biequivalence only because the functors Φ X 1 ,X 2 are not faithful. More precisely, the fibre of Φ X 1 ,X 2 over a 2-morphism (Z, ϕ) consists of all extensions ϕ of ϕ to a classifying map of the orientation cover of Z which restrict to Π • (F 1 ⊔ F 2 ∪ f 1 ∪ f 2 ). All such extensions are homotopic relative ∂Z. In particular, the homotopy invariance axiom implies any Z ∈T -TFT Π n,n−1,n−2 (C) collapses the 2-morphism fibres of Φ X 1 ,X 2 . This ensures that Φ * is a biequivalence.
We will use Proposition 2.3 to identify orientation twisted homotopy field theories with target T × BZ 2 with unoriented homotopy field theories with target T . In the non-extended setting, unoriented homotopy field theories with various targets have been studied by many authors; see [45], [44] and, when T = pt, also [24], [2], [48].
The following result is crucial for the orbifolding of orientation twisted theories.
Proposition 2.4. Let Z :T -Cob Π n,n−1,n−2 → C be an orientation twisted field theory. For each closed (n − 2)-manifold X, there is an induced pseudofunctor BZ 2 (X,T ). Slightly abusively, we denote by H : X × I 2 → BZ 2 a chosen representative of H, which is thus a homotopy relative X × ∂I from h 2 * (Π • F ) to h 1 . We can depict H as Restrictions of H to various faces of X × I 2 = X × I 1 × I 2 have been indicated. Let c 1 : I 2 → I 2 be a continuous map which is homotopic to the identity and which takes the indicated segments to the indicated segments and corner: Then (id X × c 1 ) * H is a map G : X × I 2 → BZ 2 with the following indicated restrictions: Here we regard or X×I as the map Observe that if H and H ′ are homotopic relative X × ∂I, then so too are the associated maps G and G ′ , where we use the same map c 1 to define G and G ′ . From this we conclude that R X Z is well-defined on 1-morphisms. Finally, suppose that we are given a 2-morphism Fix a representative of ϕ and choose a homotopy Q : X × I 3 → BZ 2 relative X × ∂(I × I) realizing this equivalence. Suppressing the X direction, this can be pictured as There exists a mapc 1 : I 3 → I 3 which is homotopic to the identity and which restricts to c 1 on the regions labeled by H 1 and H 2 and for which (id X ×c 1 ) * Q has the form It is now clear that there exists a map c 2 : I 3 → I 3 which is homotopic to the identity and for which R = (id X × c 2 ) * Q has the form can be verified directly. The unit and composition 2-isomorphisms for R X Z are induced by those of Z.

2.5.
Orientation twisted theories from twisted cohomology. In this section we use cohomology with twisted coefficients to construct a basic class of examples of orientation twisted field theories.
Let Π :Ŝ → BZ 2 be a continuous map with associated double cover S →Ŝ and let A be an abelian group, viewed as a Z 2 -module via inversion. Denote by C • (Ŝ; A Π ) (resp. C • (Ŝ; A Π )) the complex of singular chains (resp. cochains) onŜ with coefficients in the local system Let T be a topological space and let λ ∈ Z n (T ; C × ). Independently, Turaev [47, I.2] and Turner [49] constructed an invertible oriented homotopy field theory P λ T : T -Cob or n,n−1 → Vect C valued in complex vector spaces. The construction of Turaev is direct while that of Turner is in terms of higher gerbes with connection. Turaev's construction was recently extended by Müller and Woike [34,Theorem 3.19] to give a once-extended theory P λ T : T -Cob or n,n−1,n−2 → 2Vect C . Fix a continuous map Π :T → BZ 2 with double cover T and letλ ∈ Z n (T ; C × Π ) be a twisted n-cocycle which restricts to λ. The goal of this section is to modify the constructions of Turaev and Müller-Woike so as to define an orientation twisted lift The basic idea is straightforward: replace fundamental chains at all stages of the constructions [47], [34] with their orientation twisted variants.
Let (X, f ; h) be an object ofT -Cob Π n,n−1,n−2 . Denote by Fund(X) the groupoid of fundamental cycles of X; objects are cycles c X ∈ Z n−2 (X; Z or X ) which represent the fundamental class [X] ∈ H n−2 (X; Z or X ) and morphisms c X,1 → c X,2 are chains d X ∈ C n−1 (X; Z or X ) which satisfy ∂d X = c X,2 − c X,1 . Morphisms are composed using the abelian group structure of C n−1 (X; Z or X ). Define Pλ T (X, f ; h) to be the Vect C -enriched category whose objects are formal sums with W i ∈ Vect C and c X,i ∈ Fund(X). Morphisms in Pλ T (X, f ; h) are defined by Vect C -linearity and the requirement that the quotient of the free vector space on Hom Fund(X) (c X,1 , c X,2 ) by the relations whenever e X ∈ C n (X; Z or X ) satisfies ∂e X = d X,2 − d X,1 . The notation is as follows. The homotopy h induces an isomorphism of local systems C × . Note that if h and h ′ are homotopic relative X × ∂I, then the maps h(−) and h ′ (−) on twisted cochains coincide. Finally, −, − is the canonical pairing between or X -twisted cochains and chains.
Next, suppose that is a 1-morphism inT -Cob Π n,n−1,n−2 . Let Fund(Y ) be the groupoid of fundamental chains of Y ; objects are chains c Y ∈ C n−1 (Y ; Z or Y ) which induce the fundamental class [Y ] ∈ H n−1 (Y, ∂Y ; Z or Y ) and morphisms c Y,1 → c Y,2 are chains d Y ∈ C n (Y ; Z or Y ) which satisfy ∂d Y = c Y,2 −c Y,1 . Given fundamental cycles c X 1 and c X 2 of X 1 and X 2 , respectively, let Fund c X 2 c X 1 (Y ) ⊂ Fund(Y ) be the full subgroupoid spanned by fundamental chains which satisfy ∂c Y = i 2 * c X 2 − i 1 * c X 1 . Here we have omitted from the notation the maps o k which are used to identify the coefficients of i k * c X k with Z or Y ; similar omissions will occur below. Define be the functor which assigns to an object c X 2 the vector space Y (F ;H) (c X 1 , c X 2 ) and which assigns to a morphism d X : c X 2 ,1 → c X 2 ,2 the linear map Let us verify that this is well-defined. That this assignment defines a linear map C[Fund That this assignment respects the equivalence relation ∼ follows from the observa- Continuing, Y (F ;H) (c X 1 , −) can be augmented to a bifunctor by sending an object (c X 2 ,1 , c X 2 ,2 ) to Y (F ;H) (c X 1 , c X 2 ,1 ) · c X 2 ,2 . We can then define the required functor Pλ T ((Y ; o • ), F ; H) on objects to be the coend Finally, consider a 2-morphism For each c X 1 ∈ Fund(X 1 ), let be the natural transformation whose component Pλ T ((Z; σ • ), ϕ; η) c X 2 at c X 2 is defined as follows. Given c Y 1 ∈ Fund The natural transformation Pλ T induces a natural transformation of coends, thereby producing the required 2-morphism We can now state the main result of this section.
Theorem 2.5. For each continuous map Π :T → BZ 2 and twisted n-cocyclê λ ∈ Z n (T ; C × Π ), the above construction defines an invertible orientation twisted lift Pλ T :T -Cob Π n,n−1,n−2 → 2Vect C of P λ T . Moreover, the equivalence class of Pλ T inT -TFT Π n,n−1,n−2 depends only on the cohomology class [λ] ∈ H n (T ; C × Π ). Proof. The verification that Pλ T is indeed a symmetric monoidal pseudofunctor proceeds as in the oriented case. The key points of the proof in the oriented case are basic properties of coends, which continue to hold without change in the present setting, and the Glueing Lemma [34,Lemma 3.3]. The latter admits a straightforward modification in which homology with Z or (−) , instead of Z, coefficients is used. The proof is therefore very similar to that of [34, §3] and we omit the details.
Let us verify the homotopy invariance of Pλ T . The argument again mirrors the oriented case. Let κ : Z × I →T be a homotopy relative ∂Z from ϕ to ϕ ′ which satisfies η ′ * (Π • κ) ≃ η. Fix a homotopy Q realizing this equivalence. After suppressing the Z direction, the map Q can be depicted as As in the proof of Proposition 2.4, we construct from Q a map thereby giving a homotopy from Π • κ to or Z×I . By functoriality of homology with local coefficients, the pair (ϕ; η) determines a chain map and similarly for (ϕ ′ ; η ′ ) and (κ; R). The composed map and c Z satisfying equation (4). We compute The third and fifth equalities follow from the fact thatλ is closed and that κ is a homotopy relative ∂Z, respectively. Here p ∂Z : ∂Z × [0, 1] → ∂Z is the canonical projection and p ∂Z * denotes the associated map on twisted homology. The n-chain p ∂Z * (∂c Z × [0, 1]) is a cycle, as follows from the construction of c Z , and hence is also a boundary, as H n (∂Z; Z or ∂Z ) = 0. In view of the definition (5), this completes the verification of the homotopy axiom.
The uniqueness of fundamental classes in twisted cohomology implies that Pλ T factors through the Picard 2-groupoid of 2Vect C . This shows that the theory Pλ T is invertible. Upon restriction to T -Cob or n,n−1,n−2 , the orientations trivialize all orientation double covers appearing the construction of Pλ T . The definition of Pλ T then reduces to that of [34]. Hence Pλ T is indeed a lift P λ T . Consider then the final statement. SinceT -TFT Π n,n−1,n−2 is a 2-groupoid, it suffices to associate to each cochainν ∈ C n−1 (T ; C × Π ) a symmetric monoidal pseudonatural transformation to be the functor which is the identity on objects and which sends d X ∈ Hom Fund(X) (c X,1 , c X,2 ) to h(f * ν ), d X d X . Given a 1-morphism inT -Cob Π n,n−1,n−2 , define the compatibility 2-morphism to be that induced by the C-linear map The subscripts on ∼ indicate the n-cocycle used to define the equivalence relation on More precisely, these linear maps are the components of a natural transformation which in turn induces the required morphism of coends. The modifications which encode the compatibility of Qλ ,ν T with the monoidal structure can be taken to be the identities.

Orientation twisted equivariant field theories and orbifolding
In this section we study the simplest class of orientation twisted field theories, that in which the targetT is aspherical. Concretely, we takeT to be the classifying space of a finite Z 2 -graded group. This leads to an interpretation in terms of equivariant field theories.
3.1. Finite Z 2 -graded groups. Let Grp be the category of finite groups. The slice category Grp /Z 2 is the category of finite Z 2 -graded groups. The identity map Z 2 id − → Z 2 is a terminal object of Grp /Z 2 . Objects of Grp /Z 2 will be denoted by πĜ :Ĝ → Z 2 . We write π for πĜ if it will not cause confusion. If π is non-trivial, which we will assume to be the case unless explicitly mentioned otherwise, thenĜ is an extension The map π induces a morphism of classifying groupoids Bπ : BĜ → BZ 2 . The associated groupoid double cover is equivalent to Bi : BG → BĜ. Passing to classifying spaces gives a map Bπ : BĜ → BZ 2 with associated double cover Bi : BG → BĜ. Objects of Bun or G (M) are thus pairs (P, ǫ) consisting of a principalĜ-bundle P → M and an isomorphism ǫ : P × π G Z 2 ∼ − → or M of double covers, which we call an orientation framing of P . A morphism (P, ǫ) → (P ′ , ǫ ′ ) is a morphism f : P → P ′ of principalĜ-bundles which satisfies ǫ ′ • Ind π (f ) = ǫ.
defines an orientation framing of k * P , where the unnamed isomorphism is canonical. This defines (k; h) * on objects. On morphisms (k; h) * acts as pullback by k. Proof. Given (P, ǫ) ∈ Bun or G (M), letφ * (P, ǫ) = (P ×ĜĤ, ǫ) where, by a slight abuse of notation, the map ǫ on the right hand side denotes the composition At the level of morphisms letφ * (f ) = f ×φ G idĤ.
The following three propositions give alternative models of Bun or G (M). The claimed equivalence now follows by applying Lemma 1.1 to the map π•(−). Up to natural isomorphism, these equivalences intertwine the functors Ind π and Bπ • (−). The claimed equivalence now follows from the fact that homotopy limits of equivalent diagrams of groupoids are equivalent.
Proposition 3.5. The groupoid Bun or G (M) is equivalent to the category of principal G-bundles P → M with a section of Ind π (P ) ⊗ Z 2 or M → M and their section preserving morphisms.
Proof. The notation ⊗ Z 2 indicates the monoidal structure on Bun Z 2 (M) induced by the abelian group structure of Z 2 . After fixing an identification of or M ⊗ Z 2 or M with the trivial double cover M × Z 2 , an orientation framing of P can be used to construct a map which determines the required section. In this way we obtain a functor from Bun or G (M) to the categoryĜ-bundles together with a section of Ind π (P ) ⊗ Z 2 or M . Reversing the above construction defines a quasi-inverse.
Recall that sections of Ind π (P ) → M are equivalent to reductions of structure group of P fromĜ to G. From this perspective, Proposition 3.5 gives an interpretation of Bun or G (M) as a groupoid of principalĜ-bundles together with an or M -twisted reduction of structure group to G. This explains their naming.
We now describe two situations in which Bun or G (M) reduces to a familiar groupoid. Then ω M determines a homotopy commutative diagram A quasi-inverse of ϕ ω M is constructed as follows. Let (P, ǫ) ∈ Bun or G (M). The orientation framing ǫ and orientation ω M determine a section s(ǫ, ω M ) of Ind π (P ) → M through the composition The assignment (P, ǫ) → s(ǫ, ω M ) * (P → P/G) extends to a functor Bun or G (M) → Bun G (M) which is a quasi-inverse of ϕ ω M . Proof. SinceĜ = G × Z 2 , aĜ-bundle P is the data of a G-bundle Q → M and a double cover E → M. The assignment (P, ǫ) → Q extends to a functor Bun or G (M) → Bun G (M). On the other hand, the 2-universal property of Bun or G (M) implies that the commutative diagram Indπ induces a functor Bun G (M) → Bun or G (M). It is straightforward to verify that this functor is quasi-inverse to that constructed above.

Remarks.
(i) The mapping space interpretation of Bun or G (M) given in Proposition 3.4 can also be used to prove of Propositions 3.6 and 3. Example. We use Proposition 3.3 to give explicit models of groupoids of orientation twisted G-bundles in simple cases. We fix basepoints and orientations where necessary without comment.
(i) Let T n ≃ (S 1 ) n be an n-dimension torus. There is an equivalence (iii) As π 1 (RP 2 ) ≃ Z 2 and the holonomy representation of or RP 2 sends the generator to −1, there is an equivalence (iv) Let K be the Klein bottle. The double cover or K is the torus T 2 . Writing π 1 (T 2 ) = A, B|ABA −1 B −1 and π 1 (K) ≃ a, b | abab −1 , the covering or K → K induces the homomorphism It follows that there is an equivalence (v) Below we will consider T 2 and K as comprising the one loop sector of the theory. By parts (i) and (iv), the one loop moduli space There is a double coverĜ (2) / /G →Ĝ (2) / /Ĝ, whereĜ acts by Real conjugation and conjugation on G andĜ, respectively. The groupoidĜ (2) / /Ĝ is equivalent to ΛΛ ref π BĜ, the loop groupoid of the unoriented loop groupoid of BG, which plays an important role in the Real (categorical) representation theory of G [52]. Bi ev or G Proof. As the double cover Bi is classified by Bπ : BĜ → BZ 2 , the homotopy limit under consideration is the double cover classified by Bπ • ev or G . This composition fits into the following diagram: The left hand square homotopy commutes by the definition of Bun or G (M). The upper triangle commutes by definition and the right hand square commutes by inspection. It follows that the outside square homotopy commutes, proving the desired statement.
Pullback along ev or G gives a map (ev or G ) * : . The equivalence of Proposition 3.8 yields an isomorphism of twisted cochain complexes: is by definition the standard (oriented) transgression map τ M along M, as defined in [50], for example. The proposition follows.
Because the boundary ∂M is not assumed to be empty, the transgressed cochain τ or M (λ) need not be closed, even ifλ is so. A precise statement is as follows.
The penultimate equality follows from the unoriented form of Stokes' Theorem, as in [1,Theorem 7.2.15]. The final equality follows from the assumption thatλ is a twisted cocycle. This completes the proof.
We record the following immediate corollary of Theorem 2.5.

3.5.
Orientation twisted orbifolding. The orbifold construction is a well-known physical procedure for passing from an equivariant to a non-equivariant field theory. In the setting of oriented topological field theory, the orbifold construction has been studied by many authors; see [25] and [26] for early algebraic discussions in dimensions two and three, respectively. In this section we adapt the functorial formulation of Schweigert-Woike [39], [37] to the orientation twisted setting.
Letφ :Ĝ →Ĥ be a morphism of finite Z 2 -graded groups. Let Z :Ĝ-Cob πĜ n,n−1,n−2 → 2Vect C be an orientation twisted G-equivariant topological field theory. The primary goal of this section is to modify the oriented pushforward construction of [39, §3.3], [37, §3.1] so as to obtain from Z an orientation twistedĤ-equivariant topological field theory Zφ :Ĥ-Cob πĤ n,n−1,n−2 → 2Vect C (Grpd). To begin, suppose that we are given an object (X, f ; h) ∈Ĥ-Cob πĤ n,n−1,n−2 . Define Zφ(X, f ; h) ∈ 2Vect C (Grpd) to be the pseudofunctor The first functor is canonical and R X Z is the pseudofunctor defined in Proposition 2.4. More precisely, we have used Lemma 1. of R X Z with the groupoid Bun or G (X). To a 1-morphism inĤ-Cob πĤ n,n−1,n−2 the pseudofunctor Zφ assigns a 1-morphism whose underlying span of groupoids is To define s and t, observe that there is a homotopy commutative diagram The 2-universal property of RInd −1 φ (X k , f k ; h k ) then determines s and t. The required pseudonatural transformation where we have implicitly usedǫ to identify F |X k and H |X k with f k and h k , respectively. A morphism (( is an equivalence class of homotopies η : Y × I → BĜ from F 1 to F 2 which respects the orientation framings. Completely analogously to [37, Theorem 3.1], a representative of the homotopy η can be used to construct the coherence 2-isomorphisms of R Y,φ Z . It remains to define the value of Zφ on a 2-morphism The underlying span of spans of groupoids is the functors σ and τ being constructed in the same way as s and t above. The component of the map of intertwinerŝ Proof. The proof that Zφ is symmetric monoidal pseudofunctor is a direct modification of the oriented case [37, §3]. The key point there is the gluing property of the stack Bun G (−), which ensures that Bun G (−) is compatible with the various compositions of cobordisms. The corresponding property of Bun or G (−) follows from its construction as a homotopy fibre of π • (−) : BunĜ(−) → Bun Z 2 (−).
The functor Zφ is called theφ-pushforward of Z. Motivated by [13], [33], we interpret Zφ as a classical topological H-gauge theory. The quantization of Zφ is then given by the composition with Par as in Section 1.5. The composition Zφ ,orb is called theφ-orbifold of Z.
In particular, for any Z 2 -graded groupĜ we can apply the orientation twisted pushforward construction to the mapφ = πĜ, regarded as a morphism fromĜ to the terminal object (Z 2 id − → Z 2 ) ∈ Grp /Z 2 . This allows us to formulate the following definition, which makes implicit use of Proposition 2.3.
Note that we orbifold by the group G, as opposed toĜ. This is a key difference between the orientation twisted perspective taken in this paper and the equivariant perspective of [31].
Oriented and orientation twisted orbifolding are compatible in the following sense. Proof. We omit the construction of the pseudofunctor (−)φ, which is a straightforward, but tedious, exercise. The second statement follows from a direct comparison between our construction of the orientation twisted pushforward map and that of the oriented pushforward from [38].
3.6. Non-extended theories in two dimensions. We study two dimensional non-extended orientation twisted equivariant topological field theories and their orbifolds. This leads to a different perspective on the algebraic classification of such theories given by Kapustin-Turzillo [23]. We begin by recalling the classification of two dimensional (un)oriented topological field theories. Let (A, ⊗, 1) be a symmetric monoidal category.
Definition ( [48], [2]). An unoriented Frobenius algebra in A is the data of (i) a commutative Frobenius algebra object A in A, with multiplication m : A ⊗ A → A and comultiplication ∆ : and A morphism of unoriented Frobenius algebras is a morphism 5 of the underlying commutative Frobenbius algebras which intertwines the involutions and respects the charges of the crosscap. Unoriented Frobenius algebras assemble to a category, which is in fact a groupoid.
(i) The groupoid TFT or 2,1 (A) is equivalent to the groupoid of Frobenius algebras in A.
(ii) The groupoid TFT 2,1 (A) is equivalent to the groupoid of unoriented Frobenius algebras in A.
These proofs can be interpreted as giving a generators and relations presentation of the symmetric monoidal category Cob 2,1 and so also apply to a general target category A.
We briefly indicate the construction of an unoriented Frobenius algebra from an unoriented topological field theory. The morphisms of the category Cob 2,1 are generated by the image of the forgetful functor F : Cob or 2,1 → Cob 2,1 together with the incompatibly oriented cylinder S 1 → S 1 and the crosscap M : ∅ 1 → S 1 , which we draw as and , respectively. Here the arrows indicate the embeddings of the boundary circles. Under the equivalence of Theorem 3.14, these cobordisms determine the involution p and the morphism Q. This explains the naming of Q. The constraint (9), for example, is imposed by the following equality of composed cobordisms in Cob 2,1 : Consider now the G-equivariant setting. We begin with the oriented case. Let Z ∈ G-TFT or 2,1 . According to Theorem 3.14 (see also [39,Example 3.11]), the pushforward Z G→pt ∈ TFT or 2,1 (Vect C (Grpd)) is determined by a commutative Frobenius algebra structure on considered as an object of Vect C (Grpd). Concretely, this is the data of a G-graded algebra A = g∈G A g together with a group homomorphism a : G → Aut alg (A) and a trace − e : A e → C which satisfy the following conditions: (i) For all g, h ∈ G, the map a g : A → A restricts to a map A h → A ghg −1 .
(ii) The trace − e is G-invariant and induces a nondegenerate bilinear form −, − g : For each x ∈ A h and y ∈ A g , the equality a g (xy) = yx holds. The theory Z G→pt has additional structure, however, arising from its construction as a pushforward. Namely, it also satisfies the following conditions (see [47,§II.3.2]): (iv) For each g ∈ G, the restriction of a g to A g is the identity.
We briefly explain the origin of some of these conditions. The value of Z π on the crosscap is a morphism which, in terms of the equivalence (6), has underlying span The functor t is given on objects by t(ς) = ς 2 and on morphisms by the identity. We must also give a morphism from the trivial line bundle on Bun or G (M) to t * R S 1 Z . The latter is the data of linear maps henceforth identified with their values on 1 ∈ C, which satisfy a g (Q ς ) = Q gςg −1 . This explains the G-sector of condition (viii); theĜ\G-sector is obtained by capping the constraint (9) from the left. The constraint (9) itself implies condition (ix). To see this, consider the enhancement of equation (10) to an equation inĜ-Cob π 2,1 by equipping the cobordisms with orientation twisted G-bundles whose holonomies along the indicated loops are 3 = ς 2 g and ℓ 4 = ς 2 g. A short argument using the Seifert-van Kampen theorem and Proposition 3.3 shows that Bun or where g ′ and ω ′ are the holonomies around ℓ The functors are the identity on morphisms and are given on objects by s(g, ω) = g, t 1 (g, ω) = ω 2 g and t 2 (g, ω) = ς(ω 2 g) −1 ς −1 . The equality of these decorated spans, k = 1, 2, recovers condition (ix).

Definition.
A tuple (A,â, − e , Q) for which (A, a, − e ) is a G-Turaev algebra and which satisfies conditions (vi)-(x) is called an unorientedĜ-Turaev algebra.
Proof. For specialĜ, the proposition was proved at the level of objects in [ [45], [44] for earlier work in this direction. The above discussion shows that unoriented G-Turaev algebras are simply unoriented Frobenius algebras in Vect C (Grpd) whose underlying Frobenius algebra is G-Turaev. It is surprising that there are no additional conditions, analogous to (iv) and (v) in the oriented case, which must be added by hand to arrive at the definition of an unorientedĜ-Turaev algebra.
A G-Turaev algebra A defines, in particular, a flat vector bundle on the loop groupoid ΛBG, the fibre over g ∈ ΛBG being A g and the parallel transport along h being induced by a h . This perspective is emphasized in [42]. In the same vein, an unorientedĜ-Turaev algebra defines an extension of A to a flat vector bundle on the unoriented loop groupoid Λ ref π BĜ. This geometric perspective makes the orientation twisted orbifold procedure particularly natural.
Proposition 3.17. The orbifold of Z ∈Ĝ-TFT π 2,1 is the unoriented Frobenius algebra whose underlying commutative Frobenius algebra is the space of flat sections Γ ΛBG (A), whose involution p is induced by the extension of A to Λ ref π BĜ and whose crosscap is the flat section given by the assignment Proof. That the underlying commutative Frobenius algebra of Z π,orb is Γ ΛBG (A) was proved in [39,Theorem 4.9]; see also [25, Proposition 2.1]. Keeping the notation of diagram (12), we find that the component of the charge of the crosscap at g ∈ ΛBG is Par(Z π (M))(1)(g) = (ς,k)∈Rt −1 (g) A similar calculation shows that the involution of Γ ΛBG (A) is given by the restriction of the anti-automorphismâ ς , for any choice of ς ∈Ĝ\G.

Twisted unoriented Dijkgraaf-Witten theory
We use the results of the previous sections to construct unoriented lifts of twisted Dijkgraaf-Witten theory. We study this theory in detail in dimensions one and two. For related discussions in the oriented case, see [8], [14], [10], [13], [33], [34] Definition. LetĜ be a finite Z 2 -graded group and letλ ∈ Z n (BĜ; U(1) π ). The unoriented topological quantum field theory Zλ G : Cob n,n−1,n−2 → 2Vect C given by the orbifold of the orientation twisted theory Pλ G of Corollary 3.11 is called the twisted unoriented Dijkgraaf-Witten theory associated to the pair (Ĝ,λ).
We can now state our main result, which justifies our naming of Zλ G .
Theorem 4.1. The theory Zλ G is an unoriented lift of Z λ G . Proof. The oriented theory Z λ G was constructed in [34] as the G-orbifold of P λ G . That Zλ G is an unoriented lift of Z λ G therefore follows by applying Corollary 3.11 and the object-level statement of Proposition 3.13.

4.2.
Relation with orientation twisted transgression. As expected from the point of view of path integral (or cohomological) quantization, the values of Zλ G can be expressed in terms of appropriate pushforwards ofλ. In the present setting, the following result shows that the relevant pushforward is the orientation twisted transgression map of Section 3.3.
where H adj is the adjoint of H. Fix a fundamental cycle c Y of Y , thereby identifying Pλ G (Y, f k ; h k ) with C, k = 1, 2. With this notation, we compute (cf. [34, §4.3

])
which is non-trivial if and only if λ is trivial. In the unoriented case C (ρ λ ,G) inherits an orthogonal structure from the Real structure on ρ λ . 4.4. Two dimensions. Discussions of two dimensional oriented Dijkgraaf-Witten theory can be found in [10, §6], [13, §2.3].
A direct calculation shows that these natural transformations satisfy F g 1 ς g 2 • F ς g 1 = F ς g 2 g 1 . Alternatively, we can regard Zλ G as taking values in the Morita bicategory Alg C of associative algebras, bimodules and intertwiners. In this case, Zλ G (pt + ) is the twisted group algebra C λ [G], considered as a symmetric Frobenius algebra. Elements of C λ [G] will be written as g∈G a g l g with a g ∈ C. Replacing the duality structure P ς is the anti-automorphism The natural isomorphism Θ ς becomes an isomorphism from the Morita context determined by p ς to its opposite. Up to a scalar, this isomorphism is simply left multiplication by l ς 2 .
Continuing, in dimension one we have identified with either K 0 (Rep λ C (G)) ⊗ Z C or the centre of C λ [G]. The G-torsor of duality structures or anti-automorphisms induces a single isometric involution p of Zλ G (S 1 ).
Consider then the crosscap M. Using the equivalence (6), we find This is the (Ĝ,λ)-twisted Frobenius element, in the sense that the assignment V → tr V Q restricts to the (Ĝ,λ)-Frobenius-Schur indicator ν (Ĝ,λ) : Irr λ C (G) → {−1, 0, +1} on the set of irreducible λ-twisted representations of G. Using this observation, if we denote by p V ∈ Zλ G (S 1 ) the primitive orthogonal idempotent associated to V ∈ Irr λ C (G), then we can write Consider now the projective plane. In terms of the equivalence (7) In particular, Zλ G (RP 2 ) vanishes ifĜ is not a split Real structure. On the other hand, Zλ G (RP 2 ) can be computed via the equality Zλ G (RP 2 ) = Q e . Doing so and using equation (16) gives In this way, we obtain a formula for the signed 6 number of odd square roots of e in terms of the Real twisted representation theory of G.
As a final example, in terms of the equivalence (8), we compute for the Klein bottle  holds, where KR 0+λ (BG) is theλ-twisted KR-theory of the groupoid BG, considered as a double cover of BĜ.
Proof. It was proven in [53] that KR 0+λ (BG) is of rank The statement then follows from the fact that Bun or,1-loop G is a double cover of ΛΛ ref π BĜ together with the observation that the pullback of τ S 1 τ ref π to Bun or,1-loop G is equal to τ T 2 ⊔ τ or K . The latter observation follows from the explicit expressions for these 2-cocycles.
Remark. For comparison, [50,Corollary 13] gives the equality Z λ G (T 2 ) = rk K 0+λ (BG). In general, the partition function of a closed nonorientable surface Σ is given by the Verlinde-type formula where χ(−) is the topological Euler characteristic. This formula can be proved by decomposing Σ into cups, caps, genus adding operators , as in the oriented case, and additionally crosscaps. Writing the genus adding operators in terms of {p V } V ∈Irr λ C (G) and the crosscap as equation (16) yields the claimed formula. Combined with equation (13), we obtain the following result. Theorem 4.4 generalizes a number of earlier results. When both the Real structure and twisting are trivial, this is a theorem of Frobenius and Schur [16]; a proof using topological field theory was given in [43]. More generally, whenλ is in the image of π * G : Z 2 (BG; Z 2 ) → Z 2 (B(G × Z 2 ); U(1) π ), the above formula was proved in [46,Theorem 4.1].

Remarks.
(i) By the cobordism hypothesis for two dimensional unoriented theories, in the form of [35,Theorem 3.5.4], the triple (C λ [G], p ς , Θ ς ) completely determines the theory Zλ G : Cob 2,1,0 → Alg C . (ii) Via Theorem 3.14, the triple (Γ ΛBG (τ(λ) −1 C ), p, Q) determines the non-extended truncation of Zλ G . From this perspective, when the Real structure and twisting are trivial the non-extended theory was constructed in [29,Theorem 4]. (iii) The theory Zλ G×Z 2 , withλ in the image of π * G , was constructed using state sums in [46]. (iv) There is a generalization of the theories Zλ G which clarifies the geometric meaning of the charge of the crosscap. Instead of BĜ, consider the action groupoid X/ /Ĝ associated to a finiteĜ-set X. A cocycleλ ∈ Z 2 (X/ /Ĝ; U(1) π ) determines an unoriented topological field theory Zλ X/ /Ĝ ; this can be shown either by generalizing the above arguments or by a direct calculation using part (i). In any case, attached to the circle is the commutative Frobenius algebra Zλ X/ /Ĝ (S 1 ) ≃ Γ Λ(X/ /G) (τ S 1 (λ) −1 C ). The involution p is defined using τ ref π (λ). The charge of the crosscap is Q = where X ς = {x ∈ X | ς · x = x}, and so is a sum over the fixed point set (X/ /G)Ĝ /G = ς∈Ĝ\G X ς .
This fixed point set plays the role of the orientifold plane of X/ /G.