Orientation Twisted Homotopy Field Theories and Twisted Unoriented Dijkgraaf–Witten Theory

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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 [10] as a finite version of Chern-Simons theory [54] and has since been developed from a number of different perspectives [12,17,23,36,43]. 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 1 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 n-dimensional 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) [17], while for n = 2, the trivial Real structureĜ = G × Z 2 and particularλ we recover the state sum theories of [49], although in a different realization. On the other hand, both physical and abstract mathematical arguments, the latter involving the cobordism hypothesis [33] 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 [13,14,25]. Theorem 4.1 gives a concrete and direct geometric realization of these theories. 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 [41], and Par is a linearization functor [41]. The main problem, whose solution we outline below, is therefore to define AλĜ, 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 AλĜ 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 AλĜ(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 This transgression map is introduced in Sect. 3.3.

Our construction of Zλ
In order to construct AλĜ, 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 : T →T . The relevant cobordism bicategorŷ T -Cob 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 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 [39] and string theory with orientifolds [11]. In the nonextended setting, various specializations of orientation twisted field theories have been studied under different names; see [26,47,48,50]. 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 [50] and Turner [52] in the non-extended setting and by Müller-Woike [37] 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 Gequivariant 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 [40,42]. The theorem allows us to functorially associate to each orientation twisted Gequivariant topological quantum field theory an unoriented topological field theory with target 2Vect C (Grpd). The desired theory AλĜ can then be defined to be the orientation twisted orbifold of the G-equivariant theory Pλ 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 [7,14,18,25,26]. In this context, Dijkgraaf-Witten theories play an important role due to their relation to Kitaev's quantum double model [7,30]. 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) [55].
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 onceextended oriented topological field theories [8], 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, AλĜ 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 [7,9]. Finally, our geometric approach admits a natural generalization to allow for defects and boundary conditions, along the lines of the oriented case [21]. 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 [20]. Finally, we expect that discrete torsion in string and M-theory with orientifolds (see [6,44]) 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 Sect. 2 we introduce orientation twisted homotopy field theories and construct the theories Pλ T . In Sect. 3 we study orientation twisted equivariant theories and describe the orientation twisted orbifold construction. Finally, in Sect. 4 we construct twisted unoriented Dijkgraaf-Witten theory, studying in detail the theory in dimensions one and two.

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 : consisting of morphisms f : x → x and g : y → y for which the diagram We will use the following basic result below. Proof. The homotopy fibre R F −1 (y) is the groupoid with objects and morphisms The stated assumptions imply that Obj 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.
− → B be topological spaces over B. The relative mapping 2-groupoid Map ≤2 B (X, Y ) is the homotopy fibre over π X of the pseudofunctor An explicit description of Map ≤2 B (X, Y ) is as follows. 2 (i) Objects are pairs ( f ; m) consisting of a continuous map f : X → Y and a homotopy m : consisting of a homotopy F : X × I → Y from f 1 to f 2 and an equivalence class of homotopies 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 Sect. 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 Recall that a topological space X is called aspherical if π n (X, x) = 0 for all n ≥ 2 and x ∈ X . Lemma 1.2. Suppose that Y and B are aspherical. 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 2-groupoids 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).

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 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 and all automorphism groups of X are 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 [35]. 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 [22]. In the bicategorical setting there are explicit constructions [41,42].
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 [24]. The Deligne product gives 2Vect C the structure of a symmetric monoidal bicategory.
Given a groupoid X , recall that the functor category Hom Cat (X , Vect C ) is called the category of vector bundles on X . Since X is essentially finite, vector bundles on X are necessarily flat. Motivated by this, the pseudofunctor bicategory Hom Bicat (X , 2Vect C ) is called the bicategory of (necessarily flat) 2-vector bundles on X and is denoted by 2Vect C (X ). For an exposition of the role of higher flat vector bundles in topological field theory, the reader is referred to [53,Introduction].
The composition of the 1-morphisms (X 1 , α 1 ) together with the 1-morphism of 2-vector bundles the middle arrow being a coherence 2-morphism for Y × h X 2 Y . Similarly, the vertical composition of the 2-morphisms The reader is referred to [41] 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 post-composing with a suitable functor to a sufficiently linear target higher category [15]. In the present bicategorical setting, following earlier work of Morton [35,36] in the case without local systems, Schweigert-Woike [41,42] 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 [41, §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].

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

(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. A 2-morphism is an n-dimensional oriented collared cobordism with corners with a compatible continuous map to T . For precise definitions, see [50, §I.1] in the setting of cobordism categories and [40, §2.1] in the once-extended setting. When T is a point, T -Cob or n,n−1,n−2 reduces to the oriented cobordism bicategory Cob or n,n−1,n−2 of [38, 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

Orientation twisted cobordism bicategories.
In this section we introduce the cobordism bicategory which underlies orientation twisted homotopy field theory. Let :T → BZ 2 be a topological space over BZ 2 . The map 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 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 is an equivalence class of triples ((Z ; σ • ), ϕ; η) consisting of: commute. The bicategoryT -Cob 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.
Remark. (i) There is a pointed versionT * -Cob n,n−1,n−2 ofT -Cob n,n−1,n−2 in whicĥ T 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 mapT * -Cob n,n−1,n−2 → T -Cob n,n−1,n−2 is a monoidal biequivalence. See [42,Remark 2.6] in the oriented case. (ii) The above definition truncates to define a categoryT -Cob n,n−1 . This category is monoidally equivalent to the category of 1-endomorphisms of the monoidal unit ∅ n−2 ofT -Cob n,n−1,n−2 .
(iii) By construction, the group π 0 (T -Cob n,n−1 ) is isomorphic to the oriented cobordism group M SO n−1 (T,T ) of T with coefficients inT , as introduced by Atiyah We establish some basic properties ofT -Cob n,n−1,n−2 . Proof. The double cover classified by 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 can be defined to be post-composition with π T . More precisely, set can be taken to be the respective identities. There is a canonical lift of to a symmetric monoidal pseudofunctor. Given 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 is essentially surjective. Consider now the functor X 1 ,X 2 . Let (Y, F) be an object of the codomain of 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 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 X 1 ,X 2 is locally full on 2-morphisms.
The next result provides a generalization of the forgetful map (2).

Proposition 2.2. There is a symmetric monoidal pseudofunctor
Proof. Let ν be a null-homotopy of the composition T ρ − →T − → 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 4 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 2isomorphisms and the lift of F to a symmetric monoidal pseudofunctor are canonical.

Extended orientation twisted homotopy field theories.
For background on (extended) oriented homotopy field theories, the reader is referred to [37,50].
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 which is homotopy invariant in the following sense: if are 2-morphisms inT -Cob n,n−1,n−2 for which there exists a homotopy κ : When it will not lead to confusion, we will omit the adjectives 'once-extended' and 'homotopy'. The 2-groupoidT -TFT n,n−1,n−2 (C) is defined to be the full subbicategory of Hom Bicat (T -Cob n,n−1,n−2 , C) spanned by orientation twisted homotopy field theories. When C = 2Vect C we simply writeT -TFT n,n−1,n−2 . Truncating the previous definition defines the groupoidT -TFT n,n−1 (A) of non-extended orientation twisted field theories valued in a monoidal category A. Moreover, restriction to EndT -Cob n,n−1,n−2 (∅ n−2 ) defines a functor the domain being the homotopy category ofT -TFT n,n−1,n−2 (C).
Definition. An orientation twisted lift of an oriented homotopy field theory is the data of a map :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 n,n−1,n−2 → C which makes the following diagram homotopy commute: Proof. Let Z : T -Cob n,n−1,n−2 → C be an unoriented homotopy field theory. Homotopy invariance of Z implies that Z• is homotopy invariant, whence * is well-defined.
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 fails to be a biequivalence only because the functors X 1 ,X 2 are not faithful. More precisely, the fibre of 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 . All such extensions are homotopic relative ∂ Z . In particular, the homotopy invariance axiom implies any Z ∈T -TFT n,n−1,n−2 (C) collapses the 2-morphism fibres of X 1 ,X 2 . This ensures that * 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 [47,48] and, when T = pt, also [2,27,51].
The following result is crucial for the orbifolding of orientation twisted theories.
Proof. We work with the explicit description of Map ≤2 . Fix a representative X × I 2 → BZ 2 of H . We depict H , with its various restrictions, as Then G := (id X × c 1 ) * H : X × I 2 → BZ 2 is illustrated as follows: . This is welldefined, since if H and H are homotopic relative X × ∂ I , then so are G and G .
Finally, consider a 2-morphism in Map ≤2 BZ 2 (X,T ): There exists a mapc 1 : I 3 → I 3 which is homotopic to the identity, 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 inT -Cob n,n−1,n−2 and we can set R X Z (ϕ) = Z(X × I 2 , ϕ; R). Note that if ϕ is equivalent to ϕ , in that ϕ and ϕ represent the same 2-morphism in Map ≤2 BZ 2 (X,T ), then Z(X × I 2 , ϕ; R) = Z(X × I 2 , ϕ ; R ) by the homotopy invariance of Z.
That R X Z defines a functor can be verified directly. The unit and composition 2-isomorphisms for R X Z are induced by those of Z.

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 :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 ) (resp. C • (Ŝ; A )) the complex of singular chains (resp. cochains) onŜ with coefficients in the local system A = S × Z 2 A →Ŝ. Pullback along S →Ŝ defines a cochain map Let T be a topological space and let λ ∈ Z n (T ; C × ). Independently, Turaev [50, § I.2] and Turner [52] constructed an invertible oriented homotopy field theory Fix a continuous map :T → BZ 2 with double cover T and letλ ∈ Z n (T ; C × ) 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

Pλ
T :T -Cob n,n−1,n−2 → 2Vect C of P λ T . The basic idea is straightforward: replace fundamental chains at all stages of the constructions [37,50] with their orientation twisted variants.
Let (X, f ; h) be an object ofT -Cob 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 Cenriched 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 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 1 . Given fundamental cycles c X 1 and c X 2 of X 1 and X 2 , respectively, let Fund 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 observation We can then define the required functor Pλ Finally, consider a 2-morphism be the natural transformation whose component Pλ with c Y 2 ∈ Fund c X 2 c X 1 (Y 2 ) and set 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. 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 [37] are basic properties of coends, which continue to hold without change in the present setting, and the Glueing Lemma [37, 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 [37, §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 * ( • 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 • 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 is a chain homotopy from (ϕ; η) * to (ϕ ; η ) * . Fix fundamental chains c X k ∈ Fund(X k ), c Y k ∈ Fund c X 2 c X 1 (Y k ) and c Z satisfying equation (4). We compute The third and fifth equalities follow from the fact thatλ is closed and 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  [37]. Hence Pλ T is indeed a lift of P λ T . Consider then the final statement. SinceT -TFT n,n−1,n−2 is a 2-groupoid, it suffices to associate to each cochainν ∈ C n−1 (T ; C × ) a symmetric monoidal pseudonatural transformation This can be done as follows. Define the component of Qλ ,ν T at (X, f ; h) ∈T -Cob n,n−1,n−2 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 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.

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 and, as such, classifies a groupoid double cover D π → BĜ. Explicitly, D π can be realized as the category with objects Z 2 and morphisms ω : 1 → 2 given by elements ω ∈Ĝ which satisfy π(ω) 2 = 1 . See [16,Lemma 10.17] for a construction in the case of arbitrary groupoids. The choice of an element ς ∈Ĝ\G induces an equivalence of D π → BĜ with the morphism Bi : BG → BĜ. In short, Bπ classifies the double cover Bi. Passing to classifying spaces gives a map Bπ : BĜ → BZ 2 with associated (topological) double cover Bi : BG → BĜ.

Groupoids of orientation twisted principal bundles.
In this section we introduce the groupoid of orientation twisted principal G-bundles on a manifold. These groupoids are central to our construction of unoriented Dijkgraaf-Witten theory.
Let M be a manifold with classifying map or M : M → BZ 2 . Fix a Z 2 -graded group π :Ĝ → Z 2 . The homomorphism π induces a functor defines an orientation framing of k * P. This defines (k; h) * on objects. On morphisms (k; h) * acts as pullback by k.  Proof. Fix a trivialization of or M ⊗ Z 2 or M . Then an orientation framing of P determines the required section through the composition This defines a functor from Bun or G (M) to the desired category. Reversing the above construction defines a quasi-inverse. Proof. The orientation determines a homotopy commutative diagram The assignment (P, ) → s( , ω M ) * (P → P/G) extends to a quasi-inverse. 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|AB A −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 is equivalent toĜ (2) There is a double coverĜ (2) / /G →Ĝ (2) / /Ĝ, whereĜ acts by Real conjugation and conjugation on G andĜ, respectively. The groupoidĜ (2)   As the orientation twist of the map pr 1 is the double cover or M → M, pushforward along pr 1 is a map These considerations lead to the following definition.
Definition. The orientation twisted transgression map along M, denoted by is defined to be the composition Note that τ or M restricts to a map on twisted cochains with U(1) coefficients. As the following result shows, τ or M reduces to the standard transgression map τ M when M is oriented. Bπ is by definition the standard (oriented) transgression map τ M along M, as defined in [53], 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. 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 1.2 to identify 5 the domain Map ≤2 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 where we have implicitly used˜ to identify F |X k and H |X k with f k and h k , respectively. A 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 To verify the homotopy invariance of Zφ, suppose that κ : Z × I → BĜ is a homotopy relative ∂ Z from ϕ to ϕ which satisfies η * ( • κ) η. The map κ induces a functor under which σ and τ and pullback to σ and τ , respectively. The spans of spans of groupoids which underlie RInd −1 φ (Z , ϕ; η) and RInd −1 φ (Z , ϕ ; η ) are therefore equivalent. Under this identification, the homotopy invariance of Z then ensures that the resulting maps of intertwiners are equivalent.
The functor Zφ is the change to equivariant coefficients of Z alongφ. Motivated by [15] and [36], we interpret Zφ as a classical topological H-gauge theory. The quantization of Zφ is then given by the composition with Par as in Sect. 1.5. The compositionφ * Z is called the orientation twisted pushforward of Z alongφ.
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 . For ease of notation, we write Z π,orb for the orientation twisted pushforward πĜ * Z. We can now 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 [34].
Oriented and orientation twisted pushforward 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 change to equivariant coefficients map and that of the oriented version in [41].
3.6. Non-extended theories in two dimensions. As a first application of the orbifold construction, we discuss non-extended orientation twisted equivariant topological field theories in two dimensions. This leads to a new perspective on the algebraic classification of such theories by Kapustin-Turzillo [26].
We begin by recalling the classification of two dimensional (un)oriented topological field theories. Let (A, ⊗, 1) be a symmetric monoidal category.
Definition. [2,51] and A morphism of unoriented Frobenius algebras is a morphism 6 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. Proof. The first statement is proved in [31,Theorem 3.6.19], for example. When A = Vect C the second statement appears as [51,Proposition 2.9], [2,Theorem 4.4]. 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.
For later use, we 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 : We will now explain how Theorem 3.14, which a priori concerns only non-equivariant theories, can be used to understand equivariant theories. We begin by recalling the oriented case, which is discussed in [42,Example 3.11]. Let Z ∈ G-TFT or 2,1 . According to Theorem 3.14, the theory Z G→pt ∈ TFT or 2,1 (Vect C (Grpd)) is determined by a commutative Frobenius algebra in Vect C (Grpd). By the definition of Z G→pt , the underlying object of this Frobenius algebra is In concrete terms, this is the data of a G-graded vector space with a graded G-action. The data of the Frobenius algebra structure is a G-graded algebra structure on A for which the G-action defines 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 : (iii) For each x ∈ A h and y ∈ A g , the equality a g (x y) = yx holds.
The theory Z G→pt has additional structure, however, arising from its construction as a change to equivariant coefficients. Namely, it also satisfies the following conditions (see [50,§II.3.2]): (iv) For each g ∈ G, the restriction of a g to A g is the identity.
(v) For all g, h ∈ G, the equality are the twisted sector comultiplications.
The image of the functor (−) G→pt can be characterized as follows. is an equivalence onto the non-full subcategory of TFT or 2,1 (Vect C (Grpd)) spanned by G-Turaev algebras.
Consider now the unoriented case. Let Z ∈Ĝ-TFT π 2,1 . By Theorem 3.14, the theory Z π ∈ TFT 2,1 (Vect C (Grpd)) is determined by an unoriented Frobenius algebra structure on the functor (11). In addition to the underlying commutative Frobenius algebra, this unpacks to the data of an extension of a to a Z 2 -graded group homomorphismâ :Ĝ → Aut gen alg (A), where Aut gen alg (A) is the Z 2 -graded group of automorphisms and anti-automorphisms of A, and elements Q ς ∈ A ς 2 , ς ∈Ĝ\G, which satisfy the following conditions: (vi) For each g ∈ G and ω ∈Ĝ, the linear mapâ ω : A → A restricts to a map A g → A ωg π(ω) ω −1 .
We briefly explain how some of these conditions arise from the unoriented Frobenius 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 (1) A short argument using the Seifert-van Kampen theorem and Proposition 3.3 shows that where g and ω are the holonomies around 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).
The unorientedĜ-Turaev axioms were discovered by Kapustin-Turzillo [26], in a way analogous to Turaev's derivation the G-Turaev axioms. See [47,48] for earlier work from the same point of view. The above discussion gives a different perspective, showing that the unorientedĜ-Turaev axioms are simply those of an unoriented Frobenius algebra on (G/ /G → Vect C ) ∈ Vect C (Grpd), together with axioms (iv) and (v) of an oriented G-Turaev algebra. In particular, in contrast to the oriented case, we do not need to add any additional axioms by hand, as the next result indicates. As emphasized in [45], a G-Turaev algebra A defines a flat vector bundle on the loop groupoid BG, with fibre A g over g ∈ BG and parallel transport a h along a morphism h. We observe that an unorientedĜ-Turaev algebra defines an extension of A to a flat vector bundle on the unoriented loop groupoid 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 BG (A), whose involution p is induced by the extension of A to 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 BG (A) was proved in [42,Theorem 4.9]; see also [28, Proposition 2.1]. Keeping the notation of diagram (12), the component of the charge of the crosscap at g ∈ BG is Here 1 is the unit section of the trivial line bundle and denotes integration with respect to groupoid cardinality; see [53, §2.1]. The first equality follows from the explicit description of Par given in [42,Remarks 3.18(a)]. For the second equality we have used that a k is the value of Z G→pt (S 1 ) on the morphism (−) → k(−)k −1 in BG.
A similar calculation shows that the involution of BG (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 [10,12,15,17,36,37].
Definition. LetĜ be a finite Z 2 -graded group and letλ ∈ Z n (BĜ; U(1) π ). The unoriented topological quantum field theory We can now state our main result, which justifies our naming of Zλ G . Theorem 4.1. Letλ be a lift of λ ∈ Z n (BG, U(1)) to Z n (BĜ, U(1) π ). Then the theory Zλ Proof. The oriented theory Z λ G was constructed in [37] 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.

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 Sect. 3.3. where H adj is the adjoint of H . Fix a fundamental cycle c Y of Y , thereby identifying Pλ Example. Consider the trivial Real structure,Ĝ = G × Z 2 , and the trivial cocycle, λ = 1. By combining Proposition 3.7 and Theorem 4.2, we find, for example, In fact, inspection of the definitions shows that applying the orientation twisted change to equivariant coefficients functor to Pλ =1 G=G×Z 2 gives Bun G (−), viewed as a symmetric monoidal pseudofunctor Cob n,n−1,n−2 → Span(Grpd). We therefore recover the unoriented untwisted Dijkgraaf-Witten theory from [17,18].
More generally, the pushforward of Pλ G along a morphismφ :Ĝ →Ĥ is an orientation twisted lift of the H-equivariant twisted oriented Dijkgraaf-Witten theory studied in [34,37]. We leave the study of these theories for later work.

One dimension.
We begin with the rather simple one dimensional case. For different perspectives on the oriented theory, see [17, §5], [15, §1]. Letλ ∈ Z 1 (BĜ; U(1) π ). The change to equivariant coefficients A λ G of P λ G is determined by its value on the positively oriented point, which is the one dimensional representation ρ λ of G in which ρ λ (g) : C → C is multiplication by λ([g]).
Consider then the unoriented theory. The value of AλĜ on the incompatibly oriented interval is the diagram BG BG BG Here ς is an arbitrary element ofĜ\G and Ad ς is the weak involution given by conjugation by ς . Up to equivalence, the underlying span of groupoids depends only onĜ, and not on the particular choice of ς . The functor Ad ς arises by first applying Proposition 3.1, and then identifying Bun or G (I ) and Bun or G (pt) with their skeleta consisting of the trivialĜ-bundle with the identity orientation framing. That AλĜ(I −+ ) is indeed equal to (15) now follows from the observation that gauge transformations coming fromĜ\G reverse orientation framings. General principles of topological field theory imply that A λ G (pt − ) is isomorphic to A λ G (pt + ) ∨ , the dual of A λ G (pt + ) considered as an object of the monoidal category Vect C (Grpd). This coincides with the dual representation ρ ∨ λ . The natural transformation AλĜ(I −+ ) is therefore determined by a linear map i : C ∨ → C which satisfies The definition of Pλ G shows that i is multiplication byλ([ς −1 ]). In this way, AλĜ lifts ρ λ to the Real representation of G determined byλ, in the sense of [55, § 2.2]. Passing to quantum theories, that is, applying the functor Par, amounts to taking G-invariants. In the oriented case we get 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 .
Passing to quantum theories amounts to taking bicategories of G-equivariant objects. Hence, the oriented theory assigns to the point the category of finite dimensional λ-twisted representations of G endowed with its standard Calabi-Yau structure. See [37,Proposition 5.4]. The unoriented lift Zλ G determines is a G-torsor of exact duality structures on Rep λ C (G). Concretely, given an element ς ∈Ĝ\G, the associated duality functor where ev φ is the evaluation isomorphism from a finite dimensional vector space to its double dual. For each g ∈ G, the natural transformation F ς g : P ς ⇒ P gς whose component at φ ∈ Rep λ C (G) isλ([g|ς ])φ(g) −∨ defines a non-singular form functor (1 Rep λ C (G) , F ς g ) : (P ς , ς ) → (P gς , gς ).
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 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 ). Turning to surfaces, it is well-known (see, for example, [53,Corollary 12]) that for the sphere and torus we have Zλ G (S 2 ) = 1, Zλ G (T 2 ) = 1 |G| (g 1 ,g 2 )∈G (2)  is equal to τ T 2 τ or K . The latter observation follows from the explicit expressions for these 2cocycles.
Remark. For comparison, [53,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 is given by the Verlinde-type formula where χ(−) is the topological Euler characteristic. This formula can be proved by decomposing 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 [19]; a proof using topological field theory was given in [46]. 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 [49,Theorem 4.1].