Smooth Functorial Field Theories from B-Fields and D-Branes

In the Lagrangian approach to 2-dimensional sigma models, B-fields and D-branes contribute topological terms to the action of worldsheets of both open and closed strings. We show that these terms naturally fit into a 2-dimensional, smooth open-closed functorial field theory (FFT) in the sense of Atiyah, Segal, and Stolz-Teichner. We give a detailed construction of this smooth FFT, based on the definition of a suitable smooth bordism category. In this bordism category, all manifolds are equipped with a smooth map to a spacetime target manifold. Further, the object manifolds are allowed to have boundaries; these are the endpoints of open strings stretched between D-branes. The values of our FFT are obtained from the B-field and its D-branes via transgression. Our construction generalises work of Bunke-Turner-Willerton to include open strings. At the same time, it generalises work of Moore-Segal about open-closed TQFTs to include target spaces. We provide a number of further features of our FFT: we show that it depends functorially on the B-field and the D-branes, we show that it is thin homotopy invariant, and we show that it comes equipped with a positive reflection structure in the sense of Freed-Hopkins. Finally, we describe how our construction is related to the classification of open-closed TQFTs obtained by Lauda-Pfeiffer.


Introduction
A topological quantum field theory (TQFT) is a symmetric monoidal functor where Bord d is a suitable category of closed oriented (d−1)-manifolds and d-dimensional bordisms, symmetric monoidal under the disjoint union of manifolds, and where Vect is the category of finitedimensional complex vector spaces, monoidal under the tensor product. The axioms of the symmetric monoidal functor Z implement abstractly the sewing laws of the path integral. The idea of this formalisation goes back to Atiyah [Ati88] and Segal [Seg87,Seg04]. We refer to [Bae06] for more information about the physical perspective. TQFTs are very rich and interesting objects. A crucial feature is that they can be classified in terms of algebraic objects: one chooses a presentation of the category Bord d in terms of generators and relations and then translates these into algebraic data on the target side. In this way, for example, 2-dimensional TQFTs correspond to commutative Frobenius algebras [Dij89,Abr96,Koc04].
There exist many interesting variations of the notion of a TQFT, which arise by including additional structure. In this article, we study four modifications, all at the same time, and show that the 2-dimensional sigma model with B-field and D-branes fits into this framework. Until now, only two special cases of this picture have been worked out. We shall sketch below separately the four modifications of the bordism category we study; all details are fully worked out in the main text.
(1) D-Branes. We consider a set I of brane labels and form a new category OCBord I d , whose objects are compact oriented (d−1)-manifolds Y with boundary, with each boundary component equipped with a brane label i ∈ I. Morphisms Σ : Y 0 → Y 1 are compact oriented d-manifolds with corners that now have an incoming boundary ∂ 0 Σ ∼ = Y 0 , an outgoing boundary ∂ 1 Σ ∼ = Y 1 , as well as additional brane boundary ∂ 2 Σ, whose components carry brane labels compatible with those of Y 0 and Y 1 . Symmetric monoidal functors Z : OCBord I d → Vect are called open-closed TQFTs with D-brane labels I. In two dimensions, open-closed TQFTs have been discussed and classified by Lazaroiu [Laz01], Moore-Segal [MS], and Lauda-Pfeiffer [LP08]; they correspond to so-called I-coloured knowledgeable Frobenius algebras. Our motivation to include brane labels is string theory, where one needs to consider open and closed strings at the same time, and where the end-points of open strings are constrained to D-branes.
(2) Reflection-positivity. The bordism category Bord d admits two canonical involutions, called dual (..) ∨ and opposite (..). The dual implements duals with respect to the symmetric monoidal structure, while the opposite is an a priori different operation that is usually not considered in the TQFT literature. The category Vect has similar involutions: the usual dual V ∨ and the complex conjugate V of a vector space V . While any monoidal functor sends duals to duals, demanding that a functor Z : Bord d → Vect sends opposite bordisms to complex conjugate vector spaces is a constraint, and requires an additional structure called a reflection structure [FH]. Basically, it consists of natural isomorphisms Z(Y ) ∼ = Z(Y ) for all objects Y of Bord d . It turns out that a reflection structure induces an isomorphism between Z(Y ) ∨ and Z(Y ), and hence a non-degenerate hermitean form. A reflection structure on Z is then called positive if that form is positive definite, for every object Y .
(3) Target spaces. Searching for new invariants of manifolds, Turaev considered bordism categories where all manifolds are equipped with a homotopy class of maps into a fixed topological space [Tur]. Stolz-Teichner [ST04] considered an even more refined bordism category Bord d (M ), where all manifolds are endowed with smooth maps to a smooth manifold M , not taken up to homotopy. Symmetric monoidal functors Z : Bord d (M ) → Vect will be called functorial field theories (FFTs) on M . A FFT Z on M may be invariant under changing the maps to M by homotopies and thus reduce to one of Turaev's homotopy invariant FFTs. Often, however, FFTs are only invariant under thin homotopies, i.e. homotopies whose differential has at most rank d; these FFTs will be called thin homotopy invariant FFTs on M . Motivated by our construction of a FFT, we add a further property that has not been considered before: we call a FFT Z superficial if it is thin homotopy invariant and, in addition, the values of Z agree on two morphisms in Bord d (M ) with the same source and the same target whenever they have the same underlying d-manifold Σ and their (not necessarily homotopic) smooth maps σ, σ ′ : Σ → M are thin in the sense that their differential is of rank strictly less than d everywhere.
(4) Smoothness. This becomes relevant upon including target spaces. In physical field theories, it is generally crucial not only to describe the fields themselves, but also how the fields change in space and over time. For instance, in order to derive the classical equations of motion, one analyses how the action functional changes under smooth variations of the fields. Thus, we consider smooth families of manifolds, rather than just individual ones. The formalism we use here has been invented by Stolz-Teichner [ST11] and is based on presheaves of categories. We define a presheaf Bord d (M ) of symmetric monoidal categories on a suitable category of test spaces. Then, we define a smooth FFT on M to be a morphism Z : Bord d (M ) → VBun of presheaves of symmetric monoidal categories, where VBun is the presheaf that assigns to a test space its symmetric monoidal category of vector bundles. Inserting the one-point test space always brings us back to the previous (discrete) setting, but in general this looses information.
Our motivation for passing from TQFTs to smooth FFTs is to think of M as the background spacetime of a classical field theory. The passage comprises a quite drastic change of perspective, since in general smooth FFTs are much richer than TQFTs: in our formalism, TQFTs turn out to be smooth FFTs on M = { * }. In fact, this is a theorem that we prove -to our best knowledge -here for the first time (Theorem 6.3.3): the presheaf formalism of smooth FFTs disappears for M = { * } automatically and reduces the formalism indeed to the one of TQFTs. Consequently, smooth FFTs provide a common framework for classical and quantum theories. The original motivation of TQFTs to represent the sewing laws of a path integral is now enlarged to a more fundamental statement about the integrands under the path integral.
Let us briefly describe known examples of smooth FFTs on a target space M , in small dimensions d. In dimension d = 1, every vector bundle with connection over M defines a 1-dimensional smooth FFT, see [Bar91,SW09,BEP15,LS], and each of these papers proves (in different formalisms) that in fact every 1-dimensional smooth FFT arises this way. Analogously, in dimension d = 2, a bundle gerbe with connection over M defines a 2-dimensional smooth FFT on M . Again in a slightly different setting, this has been demonstrated by Bunke-Turner-Willerton [BTW04]. The same smooth FFT can also be obtained by restricting the FFT we construct in this paper to the closed sector, as we show in Section 6.1, but our construction is functorial, rather than defined on isomorphism classes only. Each of these FFTs turns out to be superficial in our sense. For M a Lie group, many aspects of the relation between gerbes with connection and smooth FFTs have been treated earlier by Gawȩdzki [Gaw90] and Freed [Fre95] in the process of understanding mathematical properties of Wess-Zumino-Witten models. We remark that Bunke-Turner-Willerton [BTW04] also prove that all 2-dimensional, invertible, thin homotopy invariant, smooth FFTs on M arise from a bundle gerbe with connection over M .
Next, we describe in more detail the framework we set up in this article, which provides a unified treatment of all four modifications described above. The target space is a pair (M, Q) of a smooth manifold M and a family Q = {Q i } i∈I of submanifolds Q i ⊂ M ; these submanifolds are supposed to support the D-branes, indexed by brane labels i ∈ I. We define a presheaf The main result of this paper is the functorial construction of a 2-dimensional, invertible, reflectionpositive, superficial, smooth OCFFT on a target space (M, Q), taking as input a target space brane geometry on (M, Q). These are pairs (G, E) consisting of a bundle gerbe G with connection on M (in string theory called a "B-field"), and of a family E = {E i } i∈I of twisted vector bundles with connection over the submanifolds Q i (the "Chan-Paton bundles"). They enter the OCFFT precisely as expected and as proposed by string theory: the bundle gerbe connection contributes a Wess-Zuminoterm [WZ71,Wit83,Gaw88,Gaw05] Let us now outline some important steps in our constructions and describe how the paper is organised.
(a) The complex vector spaces that our OCFFT assigns to 1-manifolds are constructed in Section 2.
A substantial part of the construction has been performed in our previous paper [BW]: the transgression of bundle gerbes and D-branes to spaces of loops and paths in M . We review the relevant parts in Section 2.2 and Section 2.4 of the present article. While in [BW] it sufficed to parameterise paths and loops in M by [0, 1] and S 1 , respectively, for our present purposes we need to extend the formalism to oriented manifolds Y that are only diffeomorphic to either [0, 1] or S 1 . This is achieved using an enriched two-sided simplicial bar construction and descent theory [Bun]; we summarise the construction in Section 2.5. The vector spaces our OCFFT assigns to closed 1-manifolds are obtained from the fibres of the hermitean line bundle over the loop space of M (the transgression of the bundle gerbe) and the vector spaces assigned to open 1-manifolds are fibres of hermitean vector bundles over spaces of paths connecting two D-branes (the transgression of the twisted vector bundles). In particular, all these vector spaces are equipped with hermitean inner products, which are -as OCFFTs take values in bare vector bundles -discarded in this step.
(b) The linear maps that our smooth OCFFTs assign to 2-dimensional bordisms are constructed in Section 3. There, we actually adopt a dual picture and construct instead the scattering amplitude associated to the bordism for given incoming and outgoing states. Our construction extends and conceptually simplifies well-known constructions of [Kap00, GR02,CJM02], while also providing a careful treatment of corners. Moreover, this smooth OCFFT Z G,E has the following properties: • It is invertible.
• It is superficial, and in particular thin homotopy invariant.
• There is a canonical positive reflection structure on Z G,E . It recovers precisely those hermitean inner products on the vector spaces in the image of Z G,E that have been discarded earlier.
• The dependence on the target space brane geometry is functorial. That is, we obtain a functor A number of further properties of the OCFFT Z G,E are investigated in Section 6. In Section 6.1 we look at closed subsectors of the theory; we relate the restriction of Z G,E to these subsectors to previous work, in particular to that of Bunke-Turner-Willerton [BTW04]. In Section 6.2 we prove the result (Theorem 6.2.1) that the OCFFT Z G,E is homotopy invariant if and only if the connection on the bundle gerbe G is flat. In the remaining subsections we concentrate on the reduction of our results to the case of a one-point target space M = { * }, and explore in detail the relation to the work of Lazaroiu [Laz01], Lauda-Pfeiffer [LP08] and Moore-Segal [MS]. The following theorem summarises these relations.
Theorem 2. There is a strictly commutative diagram of equivalences of categories: The categories in the corners of this diagram are the following. TBG(I) is the bicategory of target space brane geometries for the one-point target space (its only information is the index set of brane labels), and h 1 TBG(I) is its homotopy groupoid. RP-KFrob I C is the category of I-coloured knowledgeable Frobenius algebras whose bulk algebra is isomorphic to C, equipped with a version of a reflection-positive structure. The functor frob is described in detail in Section 6.3; it arises from the geometric formalism in [BW]. (RP-OCTQFT I 2 ) × is the category of 2-dimensional, invertible, reflection-positive, open-closed TQFTs, and F is the functor that establishes the classification result of Lauda-Pfeiffer [LP08], enhanced to include reflection structures. Finally, RP-OCFFT 2 (I) × is the category of 2-dimensional, invertible, reflection-positive, smooth OCFFTs on the one point target space. The functor ev * evaluates a morphism of presheaves of categories on the one-point test space. As mentioned above, it is an important statement on its own that this functor is an equivalence, which is the content of Theorem 6.3.3.
In upcoming work we aim to show that our functor is an equivalence of categories, for all target spaces (M, Q). This would yield a complete classification of 2-dimensional, invertible, reflection-positive, superficial, smooth OCFFTs by target space brane geometries.

Target space brane geometry
In this section we recall the basic definitions of bundle gerbes and D-branes as well as important results on their structure on which our constructions in the later sections will be based. Bundle gerbes have been introduced in [Mur96]. The notion of morphisms that we are going to employ originated in [MS00] and has been generalised in [Wal07b,Wal07a]. For detailed proofs of the statements on the 2-category of bundle gerbes we refer the reader to the above references as well as [BSS18,Bun17]; for a non-technical introduction see also [BS17]. The conventions used in this article are compatible with our previous article [BW]. We denote by Mfd the category of smooth manifolds and smooth maps.
The approach to bundle gerbes we present here is the compact treatment worked out in [NS11]. Let HVBun ∇ (M ) denote the category of hermitean vector bundles with connection on a manifold M . To every manifold M we assign a 2-category TrivGrb ∇ (M ) whose objects are 2-forms B ∈ Ω 2 (M ) and whose morphism categories Hom(B 1 , B 2 ) are the full subcategories on those hermitean vector bundles E over M with (unitary) connection whose curvature satisfies The composition of 1-morphisms and the horizontal composition of 2-morphisms are induced by the tensor product in HVBun ∇ (M ). The assignment M → TrivGrb ∇ (M ) defines a presheaf of symmetric monoidal 2-categories on Mfd.
The sheafification of the presheaf TrivGrb ∇ with respect to the Grothendieck topology of surjective submersions yields a sheaf Grb ∇ of symmetric monoidal 2-categories on Mfd. Its sections are called bundle gerbes with connections. In short, a bundle gerbe G with connection consists of a surjective submersion π : Y → M , a 2-form B ∈ Ω 2 (Y ), a hermitian line bundle L with connection over the 2-fold fibre product Y [2] = Y × M Y , whose curvature satisfies curv(L) = pr * 2 B − pr * 1 B (called the curving of G), and a unitary, connection-preserving line bundle isomorphism µ : pr * 12 L ⊗ pr * 23 L → pr * 13 L over Y [3] which satisfies an associativity condition over Y [4] . A morphism between bundle gerbes G 1 and G 2 with connections consists of a surjective submersion ζ : Z → Y 1 × M Y 2 , a hermitian vector bundle E over Z with connection whose curvature satisfies (2.1.1), and a unitary, connection-preserving vector bundle isomorphism that is compatible with the line bundle morphisms µ 1 and µ 2 . Here, all fibre products are taken over M , the maps pr Y i are the canonical projections Y 1 × M Y 2 → Y i , and the projections pr i are the canonical projections Z [2] → Z onto the i-th factor. In the following, for the sake of legibility we will usually not display these pullbacks explicitly unless it might cause confusion. One can show that a morphism E : G → G ′ of bundle gerbes is invertible if and only if its vector bundle E → Z is of rank one; see [Wal07b]. Concretely, the gerbe I ρ has as its surjective submersion the identity 1 M : M → M , as its line bundle with connection the trivial line bundle M × C → M with the trivial connection, and its structure isomorphism µ acts as µ((x, z 1 ), (x, z 2 )) = (x, z 1 z 2 ) for all x ∈ M and z 1 , z 2 ∈ C. Finally, the curving of I ρ is given by the 2-form ρ ∈ Ω 2 (M ).
A trivialisation of a bundle gerbe G with connection on M is a 1-isomorphism T : G ∼ = −→ I ρ for some ρ ∈ Ω 2 (M ). The bundle gerbe I 0 is the monoidal unit of Grb ∇ (M ). ⊳ Example 2.1.3. Consider a 1-morphism E : I ω → I ρ between two trivial bundle gerbes on M . This is a hermitean vector bundle E → Z with connection over a surjective submersion ζ : Z → M , and a connection-preserving isomorphism α : pr * 1 E → pr * 0 E over Z [2] , satisfying a cocycle condition over Z [3] . In other words, E is a descent datum for a hermitean vector bundle RE with connection on M . Descent thus induces an equivalence of categories An inverse of this functor is given by the canonical inclusion as a full subcategory. ⊳ We briefly recall that for any bundle gerbe with connection G ∈ Grb ∇ (M ) and any 2-forms ω 1 , ω 2 ∈ Ω 2 (M ) there exists a functor ∆ : Hom(G, I ω 2 ) × Hom(G, I ω 1 ) op −→ HVBun ∇ (M ) ω 2 −ω 1 , which derives from the closed module category structure of the morphism categories of Grb ∇ (M ) over the category HVBun ∇ (M ) 0 [BSS18,Bun17]. Explicitly, ∆ can be defined as follows, see [BW]. We denote by the 2-functor that is the identity on objects, that sends a 1-morphism to the morphism defined by the dual vector bundle, and that sends a 2-morphism to the 2-morphism induced by the fibre-wise transpose of the original 2-morphism. The functor (−) * reverses the direction of 1-morphisms and 2-morphisms. Then we define ∆ to be the composition Hom(G, I ω 2 ) ⊗ Hom(G, I ω 1 ) op where R is the descent functor as in Example 2.1.3. That is, we set ; this follows from the Definition (2.1.4) of the functor ∆ and the fact that T * = T −1 for any 1-isomorphism of bundle gerbes [Wal07a]. We further recall from [BW, Remark 2.1.1] that there are canonical morphisms for all 1-morphisms E a : G → I ωa for a = 1, 2, 3 and isomorphisms A : G ′ → G in Grb ∇ (M ). Here, the first morphism is induced by composition, the second is induced by the dual in the category of hermitean vector bundles with connection, and the third is induced from the evaluation isomorphism for A. Moreover, there exists a canonical isomorphism where Γ par,uni is the functor that takes parallel unit-length global sections of a hermitean vector bundle with connection. This can be checked directly from the definition of ∆.
To conclude this section, we recall the definitions of a D-brane and of a target space brane geometry: Definition 2.1.7. Let G ∈ Grb ∇ (M ) be a bundle gerbe over M , and let Q ⊂ M be a submanifold. A D-brane for G supported on Q is a morphism E : G |Q → I ω of bundle gerbes over Q for some 2-form ω ∈ Ω 2 (Q).
We call a pair (M, Q) of a manifold M and a collection Q = {Q i } i∈I of submanifolds in M a target space. Given a target space (M, Q), we define a 2-groupoid TBG(M, Q) of target space brane geometries on (M, Q) as follows. Its objects are pairs (G, E) of a bundle gerbe with connection G ∈ Grb ∇ (M ) and of a family commutes for every D-brane label i ∈ I. We refer to [BW] for a more detailed discussion of target space brane geometry.

The transgression bundles over paths between D-branes
Let (M, Q) be a target space, and let (G, E) ∈ TBG(M, Q) be a target space brane geometry. In this section we review how (G, E) give rise to vector bundles over the spaces of paths between the submanifolds Q i [BW].
Consider two arbitrary brane labels i, j ∈ I, and let P ij M be the diffeological space of smooth paths γ : [0, 1] → M in M with sitting instants that start in Q i and end in Q j . For a path γ ∈ P ij M and a trivialisation T : γ * G → I 0 of the pullback gerbe over the interval, we set For later use, we point out that one can alternatively write The Hilbert space R ij|γ (T ) depends on the choice of a trivialisation T of γ * G ∈ Grb ∇ ([0, 1]). However, changing the trivialisation T to a trivialisation T ′ changes the Hilbert space R ij|γ (T ) only up to a canonical isomorphism. To see this, we note that there always exists a 2-isomorphism ψ : T → T ′ of trivialisations of γ * G -this follows from (2.1.6), for instance. Any such 2-isomorphism induces a canonical isomorphism where we have abbreviated ψ t := ∆(1, ψ |t ) for t ∈ {0, 1}. The following result was proved in [BW, Section 4.2].
Lemma 2.2.3. The following statements hold true: (1) For any other isomorphism ψ ′ : T → T ′ , we have r ψ = r ψ ′ . We can thus denote this isomorphism by r T ,T ′ : (2) Given a third trivialisation T ′′ of γ * G, we have We thus define The finite-dimensional Hilbert space R ij|γ comes with canonical isomorphisms comes with a canonical map π : R ij → P ij M which is a diffeological hermitean vector bundle of rank For later use, we briefly recall the definition of the diffeology on R ij from [BW]. Instead of using open subsets U ⊂ R n to define plots of diffeological spaces, we only use cartesian spaces U ∈ Cart, i.e. embedded submanifolds U ⊂ R m that are diffeomorphic to R n for some n, m ∈ N 0 . Since any of the former test spaces can be covered by the latter, the resulting categories of diffeological spaces are equivalent. A mapĉ : U → R ij from an arbitrary cartesian space U ∈ Cart to R ij is a plot if it has the following properties: we demand that the composition c := π •ĉ : U → P ij M is a plot of P ij M . This means, equivalently, that the adjoint map is a smooth map. Since U ∼ = R n for some n ∈ N 0 , there exists a trivialisation T : c ⊣ * G → I ρ . Define the inclusion maps ι t : U ֒→ U × [0, 1], x → (x, t), for any t ∈ [0, 1]. Over U we obtain the hermitean vector bundles We move on to briefly recall from [BW] the construction of the connection on the bundles Observe that there is a canonical isomorphism which is induced by the isomorphisms r (T |{s}×[0,1] ) from (2.2.5). We then define where on the right-hand side, pt denotes the parallel transport in Hom(V (i,T 0 ) , V (j,T 1 ) ) along the interval. Here we orient the boundary of [0, 1] 2 using an inward-pointing normal vector field to match our later conventions -this explains the different sign as compared to [BW].
is a smooth map.
Lemma 2.3.1. Given a target space (M, Q) and D-brane labels i, j ∈ I, the group Diff + ([0, 1]) acts smoothly on the space P ij M via Proof. This follows by the construction of the mapping space diffeology [IZ13, Paragraph 1.59].
Lemma 2.3.2. The action R lifts to a smooth action This turns R ij into a Diff + ([0, 1])-equivariant hermitean vector bundle on P ij M .
Here we have used that τ is the identity on ∂[0, 1], so that the action on ψ is trivial.
Proof. We let f : U → Diff + ([0, 1]) be a plot, and we consider a plotĉ : U → R ij . In the notation of Section 2.2 that means that there exists a trivialisation T : c ⊣ * G → I ρ (where c := π •ĉ), an open covering {U a } a∈A and morphisms ψ a : F i|Ua → F j|Ua such that We need to show that R which is a plot of P ij M by Lemma 2.3.1. The plot f defines a fibre-wise diffeomorphism Over U a , we then have Thus, the trivialisationf * T and the bundle morphisms ψ a render the map R The equivariant structure R on R ij can be induced from the connection on R ij in the following way.
from γ to γ • F (1). We have that Proof. Since the expressions (2.2.7) and (2.3.3) are well-defined on equivalence classes, it suffices to prove the identity (2.3.5) with respect to any one representative, i.e. with respect to any one trivialisation of the pullback gerbe ( First, we consider the map which satisfies Let T 0 : γ * G → I 0 be a trivialisation. We obtain a trivialisation F ⊣ * T 0 : (R F γ) ⊣ * G → I 0 over [0, 1] 2 , which has the following properties: (1) It has 2-form ρ = 0.
Combining property (3) with the fact that R F γ(s, t) = γ(t) for all s ∈ [0, 1] and t ∈ {0, 1}, we obtain that in this special case the bundles V (i,T 0 ) and V (j,T 1 ) (cf. (2.2.6)) are pullbacks of bundles over the point. Thus, their parallel transport is trivial. Inserting this insight and properties (1) and (2) into the definition (2.2.7) of the parallel transport pt ij readily yields the identity (2.3.5).
Corollary 2.3.6. The lifted action R commutes with the parallel transport on the bundle R ij . Consequently, R ij is Diff + ([0, 1])-equivariant as a hermitean diffeological vector bundle with connection.
Proof. This corollary follows from combining Proposition 2.3.4 with the fact that pt ij is superficial (see [ Here, R ij denotes the complex conjugate vector bundle, and ψ * is the fibre-wise hermitean adjoint of ψ. Equivalently, we have a commutative square of diffeological spaces where a rev denotes conjugation by rev in Diff + ([0, 1]). Finally, we point out that there is an isomorphism where −, − denotes the evaluation pairing. Alternatively, we can write where ♭ h ji : R ji → R ∨ ji is the musical isomorphism induced by the hermitean metric h ji on R ji .

The transgression line bundle over the loop space
Here we recall the construction of the transgression line bundle of a bundle gerbe from [Wal16]. Let G ∈ Grb ∇ (M ) be a bundle gerbe with connection over M . We denote by LM the diffeological free loop space of M . There exists a principal U(1)-bundle LG over LM , whose fibre over a loop γ is the set of isomorphism classes of trivialisations of γ * G. This is a U(1)-torsor, since the groupoid of trivialisations of a bundle gerbe with connection is a torsor groupoid over the groupoid of hermitean line bundles with connection [Wal07b]. A line bundle J ∈ HLBun ∇ (S 1 ) acts on the fibre LG |γ as where T is a trivialisation of γ * G and [T ] denotes its 2-isomorphism class. It has been shown in [Wal16] that LG is a diffeological U(1)-bundle on LM . Further, LG carries a symmetrising fusion product and a superficial, fusive connection [Wal16] (though these are not relevant here). The transgression line bundle L is the associated diffeological hermitean line bundle Thus, the elements of the fibre L |γ over a loop γ ∈ LM are equivalence classes [[T ], z], consisting of an isomorphism class of a trivialization T : γ * G → I 0 and of a complex number z ∈ C. The equivalence relation identifies representatives (T , z) ∼ (T ′ , z ′ ) if there exists a hermitean line bundle J on S 1 with connection such that T ⊗ J ∼ = T ′ and z = z ′ · hol(J).
Let Diff + (S 1 ) denote the diffeological group of orientation-preserving diffeomorphisms of S 1 . This acts on LM by pre-composition, i.e. via the map The map R is smooth by arguments analogous to those in Lemma 2.3.1. Further, it lifts to a map The latter map can be expressed in terms of parallel transport along paths induced by isotopies that connect τ and 1 S 1 , in analogy with Proposition 2.3.4 (using the fact that Diff + (S 1 ) is connected). However, note that there are non-homotopic isotopies of this kind, since S 1 is not simply connected. The map described above is well-defined nevertheless, due to the superficiality of the connection on Finally, again in analogy to Section 2.3, there is a Z 2 -action on LM via R rev : γ → γ • rev, with rev : S 1 → S 1 denoting the orientation-reversing diffeomorphism exp(2π i t) → exp(−2π i t). This descends to a Z 2 -action R rev on LM , which lifts tõ λ : L → R * rev L respectively. As for R ij , this can equivalently be cast as an isomorphism

The coherent pull-push construction
In Section 5 it will be crucial to not just consider the mapping spaces P ij M of paths between D-branes that are parameterised over the unit interval [0, 1]. We would like to consider, for any oriented 1manifold Y diffeomorphic to [0, 1], the spaces P Y ij M of smooth maps γ : Y → M with sitting instants around the initial point y 0 ∈ Y and the endpoint y 1 ∈ Y as defined by the orientation of Y , and such that γ(y 0 ) ∈ Q i and γ(y 1 ) ∈ Q j . The set P Y ij M of such maps is a subset of the diffeological mapping space M Y , and we endow it with the subspace diffeology. The equivariant structures on the bundles R ij → P ij M from Section 2.3 allow us to transfer the bundles R ij to the spaces P Y ij M in a coherent way. Analogously, we need to transfer the hermitean line bundle L → LM to the diffeological mapping space L Y M of smooth maps Y → M for every oriented manifold Y that is diffeomorphic to S 1 . In this section we make precise what we mean by this and outline a construction that achieves this transfer. The full details of this construction can be found in the separate article [Bun].
For Y 0 , Y 1 oriented manifolds (possibly with boundary), we write D(Y 0 , Y 1 ) := Diff + (Y 0 , Y 1 ) for the diffeological space of orientation-preserving diffeomorphisms from Y 0 to Y 1 . Note that if Y 0 = Y 1 = Y , the space D(Y, Y ) =: D(Y ) canonically has the structure of a diffeological group. For Y ∼ = [0, 1], this group acts on the diffeological space P Y ij M by pre-composition; there is a smooth map which is compatible with the group structure on D(Y ). Similarly, for Y 0 , Y 1 ∼ = [0, 1] there are smooth maps which fit into a commutative square in Dfg for any oriented manifolds Y 0 , Y 1 , Y 2 ∼ = [0, 1]. We also define maps Note that this notation stems from the use of simplicial techniques, which are at work in the background here; for more details on this, see [Bun]. Let M [0,1] be the groupoid of oriented manifolds Y that are isomorphic (as oriented manifolds) to [0, 1] with its standard orientation. The morphisms in M [0,1] are the orientation-preserving diffeomorphisms. Note that Y → P Y ij M defines a functor P Definition 2.5.1. Let (M, Q) be a target space. A coherent hermitean vector bundle on P for any i, j ∈ I, where Desc(HVBun, Ψ Y ) is the category of descent data for hermitean vector bundles with respect to the subduction Ψ Y . We can now use the fact that HVBun admits a functorial descent in order to obtain a hermitean vector bundle on P Y ij M from any equivariant hermitean vector bundle on P ij M . The above constructions and the following theorem will be discussed in more general context in [Bun].
• and descent along Ψ Y naturally assemble into an equivalence of categories

A completely analogous construction yields an equivalence
Let (R ij , R) ∈ HVBun(P ij M ) D([0,1]) and (L, R) ∈ HVBun(LM ) D(S 1 ) be the equivariant hermitean vector bundles from Section 2.2 and Section 2.4, respectively. Applying the coherent pull-push of Theorem 2.5.3 we obtain coherent vector bundles We will write R Y ij → P Y ij M and L Y → L Y M for their components over P Y ij M and over L Y M , respectively.
Remark 2.5.4. Since the D([0, 1])-equivariant structure R on the bundle R ij is compatible with the connection on R ij , we could even construct the extended bundle R Y ij as a coherent diffeological hermitean vector bundle with connection on P (−) ij M (and similarly for L), but for our purposes we will only need the connection on R ij , as given in Section 2.2. ⊳ By construction, there are canonical identifications R of simplicial diffeological spaces (compare also diagram (2.3.9)). We can thus use R rev × a • rev to pull back the equivariant bundle R ji → P ji M to an equivariant bundle R * rev R ji → P ij M . Then, the morphism α ij : R ij → R * rev R ji induces an isomorphism (which we also denote α ij ) of D([0, 1])-equivariant hermitean vector bundles on P ij M . By the functoriality of Ψ * Φ * and the compatibility of descent of vector bundles with taking the complex conjugate vector bundle, we thus obtain an isomorphism ij M . We denote this isomorphism by α ij . We can give yet a different perspective on this isomorphism: observe that the oriented manifold The morphism at the top pre-composes a path γ by rev, while still seeing 0 ∈ [0, 1] as the initial point of the new path γ • rev. The right-hand morphism also pre-composes by rev, but for the resulting path γ • rev : [0, 1] → M we view 1 ∈ [0, 1] as its initial point. Finally, the left-hand map just sends a map γ to itself, but now views 1 as the initial point in the parameterising manifold [0, 1]. Using the coherent structure R on R ij (see Definition 2.5.1) and the morphism α ij , we obtain a commutative diagram of isomorphisms of D([0, 1])-equivariant hermitean vector bundles that covers diagram (2.5.5), where the isomorphism α of coherent hermitean vector bundles on P ij M for all i, j ∈ I. Analogously, the map rev : S 1 → S 1 as in Section 2.4 and the isomorphismλ from the same section in diagram (2.5.6) yield an isomorphism λ : L → r * L of coherent hermitean vector bundles over L (−) M . This also extends the isomorphisms β ij from (2.3.11) and̺ from 2.4.2 to isomorphisms (2.5.8)

Surface amplitudes
In this section we use the coherent bundles R ij and L to extend the usual holonomy of bundle gerbes to amplitudes for surfaces with corners, whose boundary is partly contained in D-branes. In Section 5 we assemble the resulting amplitudes into a smooth functorial field theory which describes the B-fielddependent part of open-closed bosonic string amplitudes.

Scattering diagrams
First, we recall the geometric tools necessary to describe surfaces with corners. Our main reference for this interlude is [Sch]. An m-dimensional manifold with corners N is a topological manifold with (possibly empty) boundary, equipped with a maximal smooth atlas whose charts are continuous maps that are homeomorphisms onto their images, with U ⊂ N open and R + denoting the set of non-negative real numbers. The index of a point x ∈ N is the number of coordinates of ϕ(x) that are zero. Compatible charts yield the same index; thus, each point , while a face of N is a disjoint union of connected faces. A manifold with faces is a manifold with corners such that each point x ∈ N belongs to ind(x) different faces.
A morphism of n -manifolds N → N ′ is a continuous map f : N → N ′ whose representatives in all charts are smooth, and such that f |∂aN : Morphisms of 3 * -manifolds are the same as those of 3 -manifolds.
Note that, in particular, a 3 * -manifold satisfies ∂ 0 N ∩∂ 1 N = ∅. If N is an oriented manifold with corners, all connected faces of N carry an induced orientation, which we define using an inward-pointing normal vector field. We will be concerned with compact (oriented) 2-dimensional 3 * -manifolds N . Each connected face c ⊂ ∂N is a compact 1-dimensional manifold with boundary, and hence either diffeomorphic to S 1 or to [0, 1]. Throughout this paper we call a 1-manifold with corners closed if it is diffeomorphic to S 1 and open if it is diffeomorphic to [0, 1]. If c a is a connected face in ∂ a N and c b is a connected face in ∂ b N with a = b then c a ∩ c b is either empty, one point, or two points.
Consider a target space (M, Q) and a target space brane geometry (G, E) ∈ TBG(M, Q). Let Σ be an oriented, compact, 2-dimensional 3 * -manifold and let σ : Σ → M be a smooth map. By Definition 3.1.1, the boundary of Σ comes with a partition ∂Σ = ∂ 0 Σ ∪ ∂ 1 Σ ∪ ∂ 2 Σ. We then think of ∂ 0 Σ as the incoming string boundary of Σ, of ∂ 1 Σ as the outgoing string boundary of Σ, and of ∂ 2 Σ as the brane boundary of Σ. In order to compute a surface amplitude of G over Σ, we need the following decorations of Σ.
(SD1) Corners lie on D-branes: for every corner (SD2) String endpoints move in D-branes: for each connected face b ⊂ ∂ 2 Σ in the brane boundary, Note that ∂b may be empty.
(SD3) Incoming and outgoing states: per assumption on Σ, the boundaries are disjoint unions of connected faces c 0,u , c 1,u ∼ = S 1 and s 0,v , s 1,v ∼ = [0, 1]. For a less cluttered notation, if i, j ∈ I are the brane labels assigned to the initial and end points, respectively, of the oriented edge s 0,v , we just write R s 0,v := R s 0,v ij for the vector bundle constructed in Section 2.2; the brane labels are then understood from the data of s 0,v . We choose an "incoming state" vector Similarly, we choose an "outgoing state" vector Remark 3.1.4. The reason why we use the notation V 1 (Σ, σ) ∨ here will become evident in Section 5, where the correct vector space to assign to the outgoing boundary of (Σ, σ) is the dual of the vector space considered here; the identification between V 1 (Σ, σ) ∨ and V 1 (Σ, σ) will rely on the isomorphisms (2.5.8). ⊳ Definition 3.1.5. Let (M, Q) be a target space, and let (G, E) ∈ TBG(M, Q) be a target space brane geometry.
(2) We call two scattering diagrams (Σ, σ, ψ ∨ 1 , ψ 0 ) and • t preserves brane labels, • the states ψ a and ψ ′ a agree under the isomorphism V a (Σ, σ) ∼ = V a (Σ ′ , σ ′ ) induced by evaluating the coherence isomorphisms R of R and L on the restrictions of t to the connected components of ∂ a Σ, for a = 0, 1.
We denote the equivalence class of a scattering diagram

Definition of the surface amplitude
In this subsection we define the surface amplitude for scattering diagrams (Σ, σ, ψ ∨ 1 , ψ 0 ) (Definition 3.1.5) and then show that it depends only on the equivalence class [(Σ, σ, ψ ∨ 1 , ψ 0 )]. We first consider the case where the vectors ψ 0 and ψ ∨ 1 in the tensor product vector spaces (3.1.2) and (3.1.3) are pure vectors and use the following auxiliary data, on which the amplitude will not depend: • String boundary parameterisations: for each connected face s ⊂ ∂ 0 Σ or s ⊂ ∂ 1 Σ we fix an orientation-preserving diffeomorphism γ s : • Trivialisation: we fix a trivialisation T : σ * G → I ρ .
The pure incoming state vector ψ 0 is then represented (under the coherence isomorphism R) by a tensor product Here, n 0 is the number of connected components c 0,u of ∂ 0 Σ that are diffeomorphic to S 1 , and m 0 is the number of connected components s 0,v of ∂ 0 Σ that are diffeomorphic to [0, 1]. Further, we have used the explicit form of the bundles R ij and L from Section 2.2 and 2.4. Observe that once the parameterisations γ s and the trivialisation T have been fixed, the tensor factors ψ 0,u and ψ 0,v each have unique representatives as in the above formula by the fact that ( R ij , R) and ( L, R) are coherent. An analogous statement holds true for the outgoing state vector ψ ∨ 1 ; can be written as a linear combination of tensor products of pure states (compare (3.1.3)) We now proceed to define the surface amplitude of a scattering diagram whose states ψ 0 and ψ ∨ which is a hermitean vector bundle over a point and hence a finite-dimensional Hilbert space. For each connected face b ⊂ ∂ 2 Σ let which is a hermitean vector bundle with connection over b. For each connected component c of ∂Σ we produce a number z c ∈ C in the following way. If c is diffeomorphic to S 1 , it is already a connected face in ∂ 0 Σ, ∂ 1 Σ, or ∂ 2 Σ. Otherwise, c is a union of connected faces, each of which is diffeomorphic to [0, 1]. Thus, we have to treat the following cases: (SA1) c ∼ = S 1 and c is some connected face c ⊂ ∂ 0 Σ or c ⊂ ∂ 1 Σ: in this case, we have c = c ǫ,u for some ǫ ∈ {0, 1} and u ∈ {1, . . . , n ǫ } (in the notation of (3.2.1) and (3.2.2)). Further, we have chosen a parameterisation γ cǫ,u of c (for ǫ = 0 or 1) and an element [[γ * cǫ,u T ], z cǫ,u ] ∈ L |γc ǫ,u . We set z c := z cǫ,u .
This is well-defined because the trace of the holonomy of a vector bundle is independent of the choices of a base point and of a parameterisation.
(SA3) c is a union of connected faces c 1 , . . . , c n ⊂ ∂Σ: by definition of a 3 * -manifold we can order these faces in such a way that there are corners x 0 , . . . , x n ∈ c with x n = x 0 and ∂c a = {x a−1 , x a }, where x a−1 is the initial point and x a is the end point of c a . We define the following linear maps λ a : E x a−1 → E xa : (a) If c a ⊂ ∂ 2 Σ is brane boundary, then the hermitean vector bundle E ca over c a comes with a connection, whose parallel transport yields an isomorphism then c a = s ǫ,v for some ǫ = 0, 1 and v ∈ {1, . . . , m ǫ } (in the notation of (3.2.1) and (3.2.2)). It thus comes with the element ψ sǫ,u chosen as part of the incoming and outgoing states (compare (3.2.1) and (3.2.2)). Under the canonical isomorphism we see that λ a := ψ sǫ,u defines a morphism of vector spaces λ a : E x a−1 → E xa .
We thus obtain a linear map and we define By the cyclicity of trace, this expression is invariant under cyclic permutations of the labels c 1 , . . . , c n of the connected faces, compatible with the orientation on ∂Σ.
Example 3.2.4. If ∂Σ = ∅, the surface amplitude A G,E (Σ, σ, 1, 1) recovers the surface holonomy of G over (Σ, σ) and thus the closed WZW amplitude (see e.g. [Mur96]). In the case of only closed brane boundary it coincides with the "holonomy on D-branes" of [CJM02]. ⊳ In the following two lemmata we prove that the surface amplitude A G,E (Σ, σ, ψ ∨ 1 , ψ 0 ) is defined independently of the choice of auxiliary data.
Proof. Suppose T ′ : σ * G → I ρ ′ is another trivialisation. Define a hermitean line bundle with connection J := ∆(T ′ , T ) over Σ. Its curvature satisfies curv(J) = ρ ′ −ρ. All quantities used above will be written with a prime when constructed from T ′ instead of T . The isomorphisms (2.1.5) yield connectionpreserving isomorphisms for every connected face b ⊂ ∂ 2 Σ and for every corner x of Σ. Consequently, for c ⊂ ∂ 2 Σ a closed 1-manifold, i.e. in case (SA2), we obtain that z ′ c = hol(J, c) −1 · z c . For the other cases, we have to represent the states ψ 0 and ψ ∨ 1 with respect to the new trivialisation T ′ (cf. (SD3)). Using the parameterisations γ, chosen as part of the auxiliary data, and the coherent structure R of R and L, as well as the respective formulae for changes of trivialisations of G, we obtain Thus, we infer that z ′ c = hol(J, c) −1 · z c for closed 1-manifolds c ⊂ ∂ 0 Σ or c ⊂ ∂ 1 Σ. From our construction (2.2.2) of the isomorphisms r T ′ ,T we obtain the following commutative diagram: This holds because the 2-isomorphisms ψ used to construct r T ′ ,T are connection-preserving. Therefore, with the above conventions for λ ′ a , we find that Putting everything together while keeping track of orientations, we obtain as stated. The integral of curv(J) comes with a positive sign because Stokes' Theorem holds true for the orientation on the boundary induced by an outward-pointing vector field, but the holonomy is taken around the boundary with the opposite orientation.
Lemma 3.2.6. The surface amplitude is invariant under orientation-preserving reparameterisation of the string boundary components: (1) If s ⊂ ∂ 0 Σ or s ⊂ ∂ 1 Σ is a closed connected face and τ ∈ Diff + (S 1 ), then the surface amplitude is invariant under changing γ s to γ s • τ .
Proof. In case (1), the change of parameterisation leads to a new representation of the state ψ 0 ∈ L s |σ |s as an element of L, using the coherence of L, according to , z s ∈ L |σ•γs•τ , and similarly for ψ ∨ 1 . Analogously, in case (2) the image of the state ψ 0 ∈ R s |σ |s under R changes as •γs•τ , and accordingly for ψ ∨ 1 . Both claims then follow from the fact that the action of reparameterisations leaves the representing elements z s and ψ s unchanged -see (2.4.1) and (2.3.3).
It remains to prove that the surface amplitude is well-defined on equivalence classes of scattering diagrams.
Lemma 3.2.7. The surface amplitude is invariant under diffeomorphisms of scattering diagrams as in Definition 3.1.5(2).
The statement then follows after choosing a trivialisation T : σ * G → I ρ to compute A G,E (Σ, σ, ψ ∨ 1 , ψ 0 ) and using the pullback t * T to trivialise σ ′ * G and to compute A G, Combining Lemmas 3.2.5, 3.2.6, and 3.2.7, we obtain: Proposition 3.2.8. The surface amplitude is well-defined and depends only on the equivalence class of a scattering diagram. That is, we can define

Properties of the surface amplitude
In this section we investigate further the properties of the surface amplitude of Definition 3.2.3. For a smooth map h ∈ Mfd(N, M ), denote its differential by h * : T N → T M .
Lemma 3.3.4. Let G ∈ Grb ∇ (M ) be a bundle gerbe with connection, defined over a surjective submersion π : Y → M and with curving B ∈ Ω 2 (Y ).
(1) Consider a morphism E : G → I ω , with an hermitean vector bundle E with connection over a surjective submersion ζ : Z → Y . The 2-form curv(E) + B ∈ Ω 2 (Z, End(E)) descends to a 2-form (2) Let X be a manifold with corners, and let f : X → M be a smooth map such that there exists a trivialisation T : f * G → I ρ for some ρ ∈ Ω 2 (X). We have the identity . We let α be the vector bundle morphism over Z [2] which is part of E. Since it is connection-preserving, the curvature of E and the curving 2-forms satisfy the identity Ad (2): Let f : X → M be a smooth map such that there exists a trivialisation T : f * G → I ρ for some ρ ∈ Ω 2 (X). The map f pulls back the sequence of surjective submersions We compute the curvature of the bundle ∆(f * E, T ): the morphism E • T * is defined over the surjective submersion ξ : Since the functor R (see Example 2.1.3) is built from the descent functor for hermitean vector bundles with connection [Wal07b,Wal07a], this yields as claimed.
Proof of Proposition 3.3.2. First, we choose orientation-preserving parameterisations γ of the connected components of the string boundary of Σ as in Section 3.2. Consider the pullback bundle Under the coherence isomorphism R, a pure vector ψ 0 then corresponds to a tensor product Here, x is the initial point of s 0,v and y is its endpoint. Further, we have set , and similarly for E y . With these choices of z ′ c 0,u and ψ ′ s 0,v , the identity (3.3.6) holds true because the vectors in the second and the third lines are related by parallel transports in L and R ij along constant paths (compare (2.2.7) and [Wal16,BW]), which are identity maps.
Its contribution to the the amplitude at t = 1 is the parallel transport λ b := pt E b {1}×b , which enters in the trace term (cf. (SA3)(i) and Definition 3.2.3). Let x and y denote the initial and endpoint of b, respectively. We have where we have used equation (3.3.5) and that h |[0,1]×b has rank one. By an analogous argument, if b ⊂ ∂ 2 Σ is a closed connected face, its contribution to the surface amplitude at t = 1 differs from that at t = 0 by the factor exp( [0,1]×b −ρ).
Inserting these findings into the expression for the surface amplitude and applying Stokes' Theorem, we obtain the identity (3.3.3).
Summarising Propositions 3.3.2 and 3.3.7, the surface amplitude A G,E has the following properties: if σ 0 and σ 1 are thin homotopic rel boundary.
whenever σ 0 and σ 1 are thin maps that agree on ∂Σ. Eventually, we refer to these properties by saying that the surface amplitude A G,E is superficial.

Smooth open-closed functorial field theories
We turn to the construction of a smooth open-closed bordism category suitable for our purposes and to the notion of smooth open-closed FFTs. Subsequently, we assemble the amplitudes defined in Section 3 into a smooth FFT in Section 5.

Smooth open-closed bordisms on a target space
Let (M, Q) be a target space. In this section, we define a smooth version of the open-closed bordism category in dimension d ≥ 2. Roughly speaking, its objects are smooth families of compact (d−1)dimensional manifolds Y , possibly with boundary, and its morphisms are smooth families of compact d-dimensional bordisms with corners. Additionally, objects and bordisms carry smooth maps to M that map brane boundaries to the submanifolds Q i . Our model for smooth bordism categories borrows from that in [ST11], modified by adding boundaries, smooth maps to a target space, and collars.
Let Cart denote the category of cartesian spaces, whose objects are submanifolds U of some R n that are diffeomorphic to some R m , and whose morphisms are smooth maps between these submanifolds. Our goal is to assemble bordisms into a presheaf of symmetric monoidal categories on Cart, which we will denote by OCBord d (M, Q).
Let U ∈ Cart. If Y is a (d−1)-manifold with boundary and if f : U × Y → M is a smooth map, we say that f has fibre-wise sitting instants if there exists an open neighbourhood V of ∂Y in Y and a diffeomorphism g : V → ∂Y × R ≥0 such that the following diagram commutes: • Y is a compact (d−1)-manifold with an orientation or T of its stabilised tangent bundle R ⊕ T Y and a map or R : π 0 (Y ) → {+, −}, • b : π 0 (∂Y ) → I is a map to the set I of D-brane labels, • f : U × Y → M is a smooth map with fibre-wise sitting instants and such that for each y ∈ ∂Y and x ∈ U we have f (x, y) ∈ Q b[y] , where [y] ∈ π 0 (∂Y ) denotes the connected component of y in ∂Y .
Here we regard the symbols {+, −} as the two possible orientations on R, so that the map or R assigns to each connected component of Y an orientation on R (with + amounting to the standard orientation). We induce an orientation on Y by saying that a local frame (e 1 , . . . , e d−1 ) on Y at a point y ∈ Y is oriented if the tuple (or R [y], e 1 , . . . , e d−1 ) is oriented with respect to or T as a local frame for the stabilised tangent bundle R ⊕ T Y , where [y] ∈ π 0 (Y ) is the connected component containing y.
An important difference between our bordism category and other versions is that the orientations of the manifolds Y are not part of the data of the object; the orientation of Y is determined from the orientations or R and or T . This is essential for the definition of reflection structures in Section 4.4.
We will generally omit the map b and the orientations or T , or R from the notation and abbreviate . This is endowed with the orientation induced from or T under the canonical identification of a frame (r, e 1 , . . . , e d−1 ) of C ± Y at (t, y) ∈ C ± Y with a frame of R ⊕ T Y at y ∈ Y . Note that the datum or R does not enter in the definition of the orientation of C ± Y ; it only enters in the definition of the underlying manifold. For non-connected Y , we define C ± Y as the disjoint union of the component-wise construction.
Next, we define the morphisms of OCBord d (M, Q)(U ) over U ∈ Cart. To that end, we consider tuples of the following data: • The pairs (Y a , f a ) are objects of OCBord d (M, Q)(U ) for a = 0, 1.
We equip W 0 and W 1 with the orientations induced from the orientations on CY 0 and CY 1 .
• The w a are smooth, orientation-preserving embeddings w a : W a → Σ. These maps must have disjoint images, satisfy that for all (t, y) ∈ W a with y ∈ ∂Y a . Furthermore, they must restrict to embeddings • We demand that the following diagram commutes: One can view this diagram as a sitting instant condition on the map σ in a direction normal to the image of U × Y a in U × Σ.
We then define an equivalence relation on the set of tuples: consider two tuples with the same objects (Y 0 , f 0 ) and (Y 1 , f 1 ). We say that (Σ, σ) and (Σ ′ , σ ′ ) are equivalent and write if there exists a triple (V 0 , V 1 , Ψ) consisting of a diffeomorphism Ψ : U × Σ → U × Σ ′ that commutes with the projections to U and that preserves brane labels and the orientation on each fibre, as well as common refinements V a ⊂ W a ∩ W ′ a of the open neighbourhoods of U × {0} × Y a in C + Y 0 and C − Y 1 , respectively. These data have to satisfy for a = 0, 1 .

Composition of two morphisms
is defined by gluing the manifolds Σ and Σ ′ using the collars (W 1 , w 1 ) and (W ′ 1 , w ′ 1 ) of (Y 1 , f 1 ), just as in the ordinary bordism category. Note that the maps σ : Σ → M and σ ′ : Σ ′ → M glue smoothly due to the sitting instant condition (4.1.3). As we are using collars as part of the data of a bordism, gluing of representatives is associative up to a canonical diffeomorphism (see e.g. [Koc04]). Moreover, this diffeomorphism induces an equivalence of tuples: the preservation of brane labels and orientations is immediate, and the compatibility with the maps to M follows from the fact that on the level of the sets underlying the bordisms, gluing amounts to forming a pushout; the universal property of the pushout then guarantees compatibility with maps out of the glued manifolds.
The identity morphism of an object (Y, f, b, or T , or R = +) reads as Remark 4.1.7. We have chosen a particularly simple dependence on the test space U ∈ Cart here, in that we only allow the maps σ to the target manifold M to vary over U . The full picture would be to consider bundles of 3 * -manifolds Σ → U with maps ς : Σ → M , which can be done by a straightforward adaptation of [ST11]. However, since we work over contractible test spaces, any such bundle is trivialisable, and the trivial bundles Σ = U × Σ capture all possible smooth families in this sense. Another reason to not work with the bordism category from [ST11] is that we wish to make contact with earlier literature on open-closed field theories such as [LP08,MS,Laz01]. The bordism categories used in these references differ in some important features from the model of [ST11]; most strikingly, the identity bordisms in [ST11] are degenerate, and it is somewhat tedious to work out a precise relation between both versions of the bordism category. ⊳ Remark 4.1.8. The fact that we do not allow Σ to vary over U prohibits us from allowing its collars to vary over U : otherwise, composition of families of bordisms would yield families of bordisms that do vary over U . Thus, we keep the collars (W a , w a ) constant over U , but that requires restrictions on the smooth maps σ : U × Σ → M : all maps σ |{x}×Σ : Σ → M , for x ∈ U , must have sitting instants on these collars, as we impose in (4.1.3). However, in our setting this is not actually restrictive: consider some U ∈ Cart, some 3 * -manifold Σ, and suppose we are given any smooth map σ : U × Σ → M . Suppose further, we have a (d−1)-manifold Y with a orientations or T and or R as in the definition of an object of OCBord d (M, Q)(U ), as well as an orientation-preserving embedding w 0 : Let us assume that Y is connected and that or R = +; the following argument extends to the general case in a straightforward way. Then, we can choose a δ > 0 such that [0, δ) × Y ⊂ W 0 and we can choose a smooth function ε : [0, δ) → [0, δ) that is constantly zero on [0, δ/3) and which is the identity on (2δ/3, δ). Then, we can replace σ by a smooth map σ ′ such that The new map σ ′ now has sitting instants on the collar neighbourhood [0, δ/3) × Y and for each x ∈ U the map σ ′ |{x}×Σ is thin homotopic to σ |{x}×Σ . That is, up to fibre-wise thin homotopy, we still capture all families of smooth maps U × Σ → M . This is analogous to the treatment of concatenation of smooth paths in [IZ13, Paragraph 5.5]. ⊳

Smooth open-closed functorial field theories
Let VBun denote the sheaf of complex finite-dimensional vector bundles; regarded as a sheaf of symmetric monoidal categories on Cart. The smooth OCFFTs we will construct in Section 5.1 have certain properties that we describe next. The first is related to the rank of the vector bundles in its image. We define three new equivalence relations on the set of tuples (4.1.2), for every test space U ∈ Cart. First, two tuples (Σ, σ) and (Σ, σ ′ ) with the same underlying surface Σ are homotopy equivalent if there exists a homotopy between σ and σ ′ rel boundary that restricts to a constant homotopy on the images of W a in Σ for a = 0, 1 and on ∂ 2 Σ. Secondly, the tuples (Σ, σ) and (Σ, σ ′ ) are called thin homotopy equivalent if they are homotopy equivalent via a homotopy whose restriction to {x} × Σ is thin for all x ∈ U . Thirdly, the tuples (Σ, σ) and (Σ, σ ′ ) are superficially equivalent if σ and σ ′ agree on U × ∂Σ and if for every x ∈ U the maps σ and σ ′ restrict to thin maps on the fibre {x} × Σ.  We have the following obvious result. Remark 4.2.5. A similar treatment of superficial OCFFTs is not easy to obtain. The reason is that two superficially equivalent bordisms may be composed separately with a fixed third bordism, whose target space map σ is not thin. Then, the two separate composites will in general not be superficially equivalent; superficial equivalence is not preserved under composition.
In principle, it is possible to construct a presheaf of symmetric monoidal categories OCBord sf d (M, Q) by starting with the free symmetric monoidal categories OCBord th d (M, Q)(U ) and imposing the relation that superficially equivalent morphisms be identified, somewhat in the spirit of Gabriel-Zisman localisation. Superficial smooth OCFFTs would then equivalently be morphisms OCBord sf d (M, Q) → VBun. However, we feel that such a construction would take us to far away from geometrical and physical intuition. Therefore, we decided to treat superficiality, and then, for the sake of consistency, also (thin) homotopy invariance as additional conditions on functors defined on OCBord d (M, Q), rather than working with functors from OCBord th The presheaf OCBord th d (M, Q) is interesting for another reason as well, though, namely for the study of path bordisms. Consider two objects (Y, f a ) ∈ OCBord d (M, Q)(U ), a = 0, 1, with the same underlying connected manifold Y . We assume that or R (Y ) = + for both objects and consider a morphism (Y, f 0 ) → (Y, f 1 ) represented by for some smooth map σ : U × [0, 1] × Y → M . As in the construction of the identity bordism (4.1.6), the bordism [0, 1] × Y is endowed with the orientation induced from or T under the inclusion [0, 1] × Y ֒→ R × Y . We call such a bordism a path bordism since it represents a smooth family of paths in the diffeological mapping space M Y with sitting instants. This is easily extended to objects with non-connected Y . The following is essentially the statement that that smooth paths in M are invertible only up to thin homotopy. Proof. We can restrict ourselves to connected Y . We provide an explicit inverse path bordism in the case where or R = +; in the case of or R = −, the construction adapts straightforwardly. The inverse of the bordism in (4.2.6) is given by where r s : R → R, t → 2s − t is the reflection of R at s ∈ R. This is the reversal of the path in the mapping space M Y that is defined by the bordism Note that we need to employ a thin homotopy of paths in the mapping space M Y in order to obtain the identity [

Duals, opposites, and hermitean structures for bordisms
In this section we work over a fixed test space U ∈ Cart, and thus with the symmetric monoidal category OCBord d (M, Q)(U ). We note that this monoidal category has (left) duals; moreover, there is a canonical choice of duality data for every object in OCBord d (M, Q)(U ), turning OCBord d (M, Q) into a symmetric monoidal category with fixed duals; see Appendix A. Indeed, if (Y, f ) ∈ OCBord d (M, Q)(U ) is an object, duality data ((Y, f ) ∨ , ev Y , coev Y ) is fixed as follows: Observe that the orientation on R ⊕ T Y is unchanged, so that the orientation induced on Y is reversed, as a consequence of reversing the orientation or R (Y ). The evaluation ev Y and coevaluation coev Y are given by the standard evaluation and coevaluation bordisms represented by the manifold Σ = [0, 1] × Y and its standard collars, multiplied by U and decorated with the maps σ = f • pr U ×Y . In the following we treat this choice of fixed duals in terms of (twisted) involutions.
Definition 4.3.1. Let C be a symmetric monoidal category. An involution on C is a pair (d, δ) of a symmetric monoidal functor d : C → C and a monoidal natural isomorphism δ : d • d → 1 C satisfying In general, in a symmetric monoidal category with fixed duals, the assignment x → x ∨ extends canonically to a symmetric monoidal functor d C : C op → C and further to a twisted involution (d C , δ C ), which we then call the duality involution of C. In the case of the symmetric monoidal category OCBord d (M, Q)(U ) the duality involution d OCBord is in fact strict, and it is straightforward to check that its action on morphisms is explicitly given by There is another interesting operation on OCBord d (M, Q)(U ), given on objects and morphisms by the following definitions: (Y 0 , f 0 ), (Y 1 , f 1 ), Σ, W 0 , W 1 , w 0 , w 1 , σ, ℓ := (Y 0 , f 0 ), (Y 1 , f 1 ), Σ, W 0 , W 1 , w 0 , w 1 , σ, ℓ .
In our shorthand notation, we call (Y, f ) the opposed bordism of (Y, f ). It is straightforward to check that opposition defines a strict involution The duality involution d OCBord and the involution op OCBord are in fact related. This relation is described by a so-called hermitian structure. We first provide the relevant definitions in a more general context, following [FH].
Let C be a symmetric monoidal category with fixed duals, whose duality data we denote by (x ∨ , ev x , coev x ), and with corresponding duality involution (d C , δ). We suppose that (op C , γ) is some involution on C. If x ∈ C is an object, we will also write x as a shorthand for op C (x).

Definition 4.3.2.
A pre-hermitean structure on an object x ∈ C is an isomorphism ♭ x : x → x ∨ . We denote the inverse of ♭ x by ♯ x := ♭ −1 x : x ∨ → x.
A pre-hermitean structure ♭ x on an object x ∈ C defines a pre-hermitean pairing on x, by which we mean the morphism where u is the monoidal unit in C. This pairing is non-degenerate in the sense that we have a prehermitean co-pairingȟ • coev x , so that the usual triangle identities are satisfied.
Definition 4.3.4. A pre-hermitean structure ♭ x on an object x ∈ C is called hermitean structure if its pre-hermitian pairing h x is symmetric in the sense that the following diagram commutes Remark 4.3.5. We remark that a hermitean structure relates the values of functors with different domain categories; therefore, one cannot regard it as a natural transformation. Instead, compatibility with morphisms will be treated as a condition on the morphisms, see Definition 4.3.6. We also remark that a property "positive-definite" cannot be defined in this general context; this will be done by hand for the category of vector bundles, see Definition 4.4.6. ⊳ Definition 4.3.6. Let f : x → y be a morphism between objects with hermitian structures ♭ x and ♭ y , respectively, and induced hermitian pairings h x and h y , respectively. We call f is isometric if it satisfies the identity We say that f is unitary if it is an isometric isomorphism. The adjoint of f is the composition The name adjoint is justified by the following lemma.
Proof. We compute where we have used Lemma A.5 in the third step. Proof. We need to show that f is isometric if and only if f * = f −1 . To that end, we compute where τ x x,u is part of the adjunction of Proposition A.4. On the other hand, using this adjunction, we may write (4.3.3) as h x = τ x −1 x,u (♭ x ). Since τ x −1 x,u is bijective, we thus have that With the definition of the adjoint morphism f * this proves the assertion.
Example 4.3.9. The symmetric monoidal category Vect of finite-dimensional complex vector spaces has its usual fixed duals: for a vector space V ∈ Vect, we set We consider the monoidal strict involution op Vect sending V to the complex conjugate vector space V . Then, a hermitean structure on V ∈ Vect in the sense of (4.3.10) Here we have made use of the involution r s : R → R, t → 2s − t that reflects the real line at an element s ∈ R. Observe that because of this reflection, the inclusion w 0 in ♭ (Y,f ) is indeed orientation-preserving if the product [0, 1] × Y is endowed with the orientation induced from or T . It is straightforward to check that the pre-hermitean structure ♭ (Y,f ) of (4.3.10) is hermitean.
For the following argument we consider the quotient category OCBord th d (M, Q)(U ) where thin homotopy equivalent morphisms are identified. It is easy to see that the involutions d OCBord and op OCBord , as well as the hermitean structures ♭ (Y,f ) descend to this quotient category. Proof. By Lemma 4.2.7, path bordisms are invertible in OCBord th d (M, Q)(U ). Recall that the bordism [0, 1] × Y in a path bordism has its orientation induced from or T . We calculate: Here, the third identity arises from the orientation-preserving diffeomorphism Observe that this change of orientation on [0, 1] × Y is accounted for by changing the embeddings w 0 and w 1 accordingly. The following identity arises from another diffeomorphism and a thin homotopy to express the composition

Positive reflection structures on OCFFTs
In this section we describe how a smooth OCFFT Z relates the involutions d OCBord and op OCBord on the presheaf OCBord d (M, Q) to the involutions d VBun and op VBun on the presheaf VBun of vector bundles.
Again, we first discuss a more general setting. Let C and D be symmetric monoidal categories with fixed duality data and associated duality involutions (d C , δ C ) and (d D , δ D ). Then, for any symmetric monoidal functor F : C → D, there exists a unique natural isomorphism that is compatible with evaluation and coevaluation, and which makes the diagram commutative, see Proposition A.6. For arbitrary other (non-twisted) involutions (op C , γ C ) and (op D , γ D ), such a structure is not automatic; Proposition A.6 relies crucially on the triangle identities for duality data.
Definition 4.4.2. Let C and D be symmetric monoidal categories endowed with involutions (op C , γ C ) and (op D , γ D ), respectively. Let F : C → D be a symmetric monoidal functor. A monoidal natural isomorphism commutes is called a homotopy fixed point structure on F .
Given a homotopy fixed point structure α on a functor F and an object of x of C is equipped with a (pre-)hermitean structure ♭ x : defines a (pre-)hermitean structure on the image F (x). Here β is the canonical natural isomorphism (4.4.1). We have the following result.
Lemma 4.4.4. Suppose f : x → y is an isometric morphism between objects with hermitean structures ♭ x and ♭ y , respectively. Then, F (f ) is isometric with respect to the hermitean structures ♭ F,x and ♭ F,y defined by (4.4.3).
Proof. For x, y ∈ C, let f x,y : F x ⊗ F y → F (x ⊗ y) denote the isomorphism that renders F a symmetric monoidal functor. For simplicity, we denote again the involutions op C and op D by x → x. Then, we calculate Here we have used the naturality of α, the fact that (F, f ) is a monoidal functor, the compatibility of β x with the evaluations ev x and ev F x , see Proposition A.6. It is again straightforward to generalise the definition of a homotopy fixed point structure to morphisms of presheaves of symmetric monoidal categories. Applying this to OCFFTs we obtain the following definition, which is an adaption of the formalism of [FH] from FFTs to smooth FFTs, see [FH,Def. 4.17].
Definition 4.4.5. A reflection structure on a smooth OCFFT Z : OCBord d (M, Q) → VBun is a homotopy fixed point structure α on Z with respect to the involutions op OCBord and op VBun .
We recall that every object (Y, f ) of OCBord d (M, Q)(U ), for every U ∈ Cart, carries a canonical hermitean structure (4.3.10). Hence, by (4.4.3) we obtain hermitean structures ♭ Z,(Y,f ) on the vector bundles Z(Y, f ) over U . As explained in Example 4.3.9, these are nothing but hermitean bundle metrics in the ordinary sense. Now the following definition makes sense:

Smooth OCFFTs from B-fields and D-branes
In this section, we explicitly construct a 2-dimensional smooth OCFFT over a target space (M, Q) from a target space brane geometry (G, E) ∈ TBG(M, Q) as defined in Section 2.1. This construction employs the the coherent transgression vector bundles L and R ij from Section 2 and the surface amplitude A G,E from Section 3. We equip our smooth OCFFT with a positive reflection structure, and show that it depends functorially on the target space brane geometry.

From surface amplitudes to smooth OCFFTs
To begin with, we consider a cartesian space U ∈ Cart and an object (Y, f ) ∈ OCBord 2 (M, Q)(U ). For simplicity, let us first assume that Y ∼ = [0, 1], with brane labels i, j ∈ I assigned to its initial and end point, respectively. The map f : Let U : HVBun ∇ → VBun be the morphism of sheaves of symmetric monoidal categories that forgets hermitean metrics and connections. We set (5.1.1) We extend this assignment to empty Y by setting whenever (Y, f ) ∈ OCBord 2 (M, Q)(U ) with Y = ∅, and to non-connected Y by mapping families of disjoint unions of manifolds to tensor products of vector bundles.
We now use the amplitudes defined in Section 3 in order to define the field theory Z G,E on mor- , f a ) over U , for a = 0, 1, via (5.1.1). Let Ψ 0 ∈ Γ(U, Z G,E (Y 0 , f 0 )) and be arbitrary smooth sections. We think of Ψ 0 as a smoothly parameterised family of incoming states and of Ψ ∨ 1 as a smoothly parameterised family of duals of outgoing states.
We will use this identification throughout. ⊳ The following lemma is an immediate consequence of our definition of scattering diagrams (Definition 3.1.5) and morphisms in OCBord d (M, Q) (Definition 4.1.5).
Since the surface amplitude is well-defined on equivalence classes of scattering diagrams by Proposition 3.2.8, we obtain a well-defined number for every x ∈ U . We prove first that this number depends smoothly on x.
Lemma 5.1.4. The function is smooth.
Proof. Since U is diffeomorphic to R n for some n ∈ N 0 and since Σ is 2-dimensional, we find a trivialisation T : σ * G → I ρ . By the coherence of L and R ij , after choosing parameterisations of the connected components of Y a (i.e. orientation-preserving diffeomorphisms to either [0, 1] or S 1 ), any smooth section Ψ a of Z G,E (Y a , f a ) over U can be written in a unique way as a C ∞ (U, C)-linear combination of tensor products of sections of the form [[T ], z cla ] for smooth functions z cla : U → C, where cl a runs over the connected components of Y a that are diffeomorphic to S 1 , and sections of the form [T , ψ ea ], where ψ ea ∈ Hom ∆(E i(e a,0 )|U ×e a,0 , T |U ×e a,0 ), ∆(E i(e a,1 )|U ×e a,1 , T |U ×e a,1 ) .
Here, e a runs over the connected components of Y a that are diffeomorphic to [0, 1]. For the sake of legibility we have omitted pullbacks, and we have written e a,0 for the initial point of the oriented component of Y a labelled by e a and e a,1 for its end point. The indices i(e a,0 ) and i(e a,1 ) ∈ I are the D-brane indices of the respective corners of Σ (see Definition 4.1.1). The reason that tensor products of these sections generate all smooth sections over U under C ∞ (U, C)-linear combinations is that, by construction of the bundles R ij and L, these sections generate each fibre of their respective pullbacks (cf. Section 2.2) that appear in Z G,E (Y 0 , f 0 ) and Z E ,G (Y 1 , f 1 ) (cf. (5.1.1)). From the explicit construction of the surface amplitude A G,E in Definition 3.2.3, one can now see the claimed smoothness. Now, we define a morphism of vector bundles over U by requiring that for all x ∈ U and for all smooth Since the evaluation pairing between a finite-dimensional vector space and its dual is non-degenerate, the expression (5.1.5) defines a bundle morphism Z G,E [Σ, σ] which is smooth by Lemma 5.1.4.
Theorem 5.1.6. Let (M, Q) be a target space.
We will decompose the proof of Theorem 5.1.6 into a series of smaller assertions. The compatibility with symmetric monoidal structures has been built into the definition of Z G,E on objects. On morphisms it follows from the fact that the surface amplitude A G,E factorises accordingly for disjoint unions of surfaces, as can readily be seen from its definition (see Section 3.2). The main part is now to prove that Z G,E is a morphism of presheaves of categories on Cart. By construction, the pullbacks of objects in OCBord 2 (M, Q)(U ) along morphisms V → U of cartesian spaces get mapped to pullbacks of vector bundles, and the pullback of morphisms pulls back due to its fibre-wise definition (5.1.5). Thus, it suffices to show the functoriality of Z G,E pointwise, i.e. we only have to check that Proof. Since Z G,E is respects the monoidal structure by construction, we can restrict ourselves to connected Y . Assume further that (Y, f ) is an object with or R = +; the case for or R = − is analogous. Recall from (4.1.6) that, for any object (Y, f ) ∈ OCBord 2 (M, Q)( * ), we have 1 , ψ 0 )) is trivial for any pair of incoming states ψ 0 and duals of outgoing states ψ ∨ 1 . Further, the incoming and outgoing string boundaries of 1 (Y,f ) agree as manifolds with maps to the target space M . Consequently, the amplitude is just the evaluation pairing between the finite-dimensional vector space Z G,E | * (Y, f ) and its dual, and thus defines the identity operator is then induced by decomposing the parallel transports in E b at points where c ′′ intersects the image of Y 1 . Consequently, we can rewrite the amplitude (5.1.9) as Here we omitted labelling the identities by the vector spaces which they act on in order to avoid unnecessarily heavy notation. Let x ∈ c ′′ ∩ Y 1 , and let ι x : * ֒→ Σ ′′ denote its inclusion into Σ ′′ . Then, the vector spaces in question are , ι * x T , or their duals, depending on orientations. Note that these are precisely those vector spaces which constitute Z G,E (Y 1 , f 1 ), after forgetting the inner product structure. Choosing bases (e x,µx ) µx=1,...,mx in each of these vector spaces E x , with dual basis elements denoted e ∨ x,µx , we can rewrite each of the identities as Thus, the trace is broken up into a sum over products whose factors are of the form e ∨ x,µx , λ cn (e y,µy ) Ex , where x and y are the initial and endpoint of c n , and accordingly for paths c ′ n . Now we reorder these products: we combine the factors involving the maps λ c i , and we separately combine the factors containing the maps λ c ′ i . The two ways of grouping the factors at a connected component s ⊂ Y 1 are illustrated in the following diagram: after reorganising before reorganising For x ∈ c ′′ ∩Y 1 , let z be the unique point in ∂Σ ′′ such that x and z are joined by a connected component of the image of Y 1 in Σ ′′ . Using the fact that the collection (e z,µz ⊗ e ∨ x,µx ) µx=1,...,mx, µz=1,...,mz forms a basis for E z ⊗ E ∨ x ∼ = R s|f •φs (T |s ), we thus arrive at where (ψ 1,k ) is the basis for Z G,E (Y 1 , f 1 ) formed by using the bases (e z,µz ⊗ e ∨ x,µx ) in the individual tensor factors comprising Z G,E (Y 1 , f 1 ). Now, by the non-degeneracy of the evaluation pairing, it follows that for all ψ 0 ∈ Z G,E (Y 0 , f 0 ) and ψ ∨ 2 ∈ (Z G,E (Y 2 , f 2 )) ∨ , which implies the claim.
To complete the proof of Theorem 5.1.6 we recall that the amplitude A G,E is superficial (Proposition 3.3.2 and Proposition 3.3.7) which readily implies that Z G,E is a superficial smooth OCFFT. Finally, invertibility is immediately build in into the definition (5.1.1).

Reflection positivity
Here we show that the field theory Z G,E carries a canonical positive reflection structure in the sense of Section 4.4. It turns out that the hermitean metrics on the bundles Z G,E (Y, f ) induced from the reflection structures are precisely the hermitean metrics on the bundles R ij and L that have been forgotten in (5.1.1).
Consider an object (Y, f ) ∈ OCBord 2 (M, Q)(U ) over a test space U ∈ Cart. For simplicity, let us first assume that Y is connected and open, i.e. that Y ∼ = [0, 1]. We recall from (5.1.1) that ij ; here, we omit displaying the forgetful functor U. Consequently, we have On the other hand, we have over U , with r ij as in (2.5.6) and α ij as in (2.5.7); see Section 2.5 for details. We show that this yields a natural isomorphism be a smooth family of incoming states, and let Ψ ∨ 1 ∈ Γ(U, Z G,E (Y 1 , f 1 ) ∨ ) be a smooth family of outgoing dual states. We consider the evaluation In the first step we have used nothing but the definition of the transpose of a morphism of vector bundles. The second step is seen from the explicit construction of the amplitude A G,E in Section 3.2: using the coherent structure R on L and R ij to pass to the case where the connected components of Y are given by [0, 1] and S 1 (i.e. by choosing parameterisations), the amplitude in the second line has all faces decorated with the hermitean adjoints of the linear maps that occur in the amplitude in the third line, and the integral is taken over Σ. The last identity is just making use of the relation between evaluation on a complex vector space V and on its complex conjugate, This proves that α is natural in (Y, f ). Furthermore, we see that α satisfies the coherence axiom from Definition 4.4.2 since op 2 OCBord = 1 OCBord and op 2 VBun = 1 VBun , and since with respect to any choice of parameterisations of Y the morphism α acts on tensor factors as the morphism α ij from (2.3.8); one can readily see that α ji • α ij = 1. This proves Proposition 5.2.2. For any target space brane geometry (G, E) ∈ TBG(M, Q), Equation (5.2.1) defines a reflection structure α on Z G,E .
We recall from Section 4.4 that for any test space U ∈ Cart, we have given choices of fixed duals on both OCBord d (M, Q)(U ) and on VBun(U ), which are compatible with pullbacks along smooth maps V → U of cartesian spaces. Such a choice of fixed duality data canonically defines duality functors (see Appendix A). Further, Proposition A.6 implies that for any morphism Z : OCBord d (M, Q) → VBun the fixed duals induce a canonical monoidal natural isomorphism defined in (2.3.10) and (2.3.11). If Y ∼ = S 1 instead, β (Y,f ) is given by the isomorphism̺ defined in (2.4.2). We extend β to general manifolds Y by sending disjoint unions to tensor products. Using the canonical identification of a finite-dimensional vector space with its double dual, we have Here, the first identity is just transposing Proof. Recall the definition of ♭ (Y,f ) form (4.3.10). By (4.4.3) we have that Observe that because of σ = f •pr Y we have σ * curv(G) = 0. Since U ×[0, 1]×Y is homotopy equivalent to Y , we have H 2 dR (U × [0, 1] × Y ) = 0, and we may therefore choose a trivialisation T : σ * G → I 0 . We let Ψ a ∈ Γ(U, Z G,E (Y, f )), for a = 0, 1, be smooth sections and consider the amplitude where h (Y,f ) is the hermitean metric on the vector bundle Z G,E (Y, f ) induced by the hermitean metrics on the transgression bundles L c and R c ij , for the respective connected components c of Y . On the other hand, the left-hand side of (5.
Since the hermitian metrics on R ij and L are positive definite, we obtain the following.
Corollary 5.2.5. The reflection structure α on Z G,E is positive.

Functorial dependence on the target space brane geometry
We fix a target space (M, Q) and investigate the dependence of the functorial field theory Z G,E on the target space brane geometry (G, E) ∈ TBG(M, Q). For a 2-category C, let h 1 C denote its homotopy category. It has the same objects as C, and its morphisms are the 2-isomorphism classes of 1-morphisms in C. We carry out the proof of this theorem in the remainder of this section. Let us first define Z on 1morphisms (A, ξ) : (G, E) → (G ′ , E ′ ) in TBG(M, Q) (see Section 2.1 for the definition). We denote the coherent transgression bundles obtained from (G ′ , E ′ ) by R ′ and L ′ . Consider an arbitrary test space U ∈ Cart, and let (Y, f ) ∈ OCBord 2 (M, Q)(U ). Since Z G,E is symmetric monoidal, we may restrict ourselves to connected manifolds Y .
We first consider the case where Y ∼ = [0, 1]. The field theory Z G,E sends the object (Y, f ) to a vector bundle Z G,E (Y, f ) over U . We choose a parameterisation of Y and writef for the composition t) for t ∈ [0, 1], and write ∂ af :=f • ι a : U → M for a = 0, 1. We choose a trivialisation T :f * G → I ρ over U × [0, 1]. Then there are canonical isomorphisms (cf. (2.2.1) and (5.1.1)) In this representation of the bundle Z G,E (Y, f ), we define isomorphisms and we define ξ A,j,1 analogously. For ψ ∈ Z G,E (Y, f ) we then define smooth isomorphisms We claim that these morphisms are compatible with changes of trivialisations and with reparameterisations of Y , so that they induce a smooth bundle isomorphism Indeed, the compatibility with reparameterisations is seen readily from the construction; acting with an orientation-preserving diffeomorphism g : Thus, if we can show the compatibility with changes of trivialisations T , then, by the functoriality of the coherent pull-push construction Ψ * Φ * , this isomorphism induces an isomorphism We check the compatibility with changes of trivialisations fibre-wise over points x ∈ U : we set , t), and we set Let S :f * G → I ρ ′ be another trivialisation and set We obtain the diagram The right-hand triangle commutes by the construction of the fibre of R ′ ij|γ as a colimit of the R ij|γ (T ′ ) (cf. Equation (2.2.4)). The same argument applies to the left-hand triangle. We are thus left to show that the upper square in (5.3.4) commutes. For every x ∈ U there exists a 2-isomorphism ζ x :  * x T →  * x S, since such 2-isomorphisms are in one-to-one correspondence with parallel unit-length section of  * x ∆(S, T ) ∈ HLBun ∇ ([0, 1]). We also introduce Then we can use ζ x and ζ ′ x to make the canonical isomorphisms r S,T and r S ′ ,T ′ explicit (recall (2.2.2)). For ψ ∈ Z G,E (Y, f ) and omitting pullbacks to {x} × [0, 1], we obtain the commutative diagram Here we denote the horizontal inverse 1 of a (not necessarily invertible) 2-morphism ψ by ψ (−1) . Observe that by (2.2.2) the commutativity of the outer square is the desired identity First of all, the inner square is just the definition of r S,T , so it commutes. The top and bottom squares are the definitions of Z A,T and of Z A,S • r S,T (ψ), respectively, whence they commute as well. Finally, the left and right squares commute as a consequence of the interchange law in a 2-category and the fact that the 2-morphisms on the vertical and horizontal arrows act on different factors in the respective compositions of 1-morphisms. Thus, the representatives Z A,T , which depend on choices of trivialisations, induce a well-defined isomorphism of vector bundles on U .
Proof. In order to see that Z (A,ξ) is natural, we consider an arbitrary smooth family of bordisms [Σ, σ] : (Y 0 , f 0 ) → (Y 1 , f 1 ), parameterised over a cartesian space U ∈ Cart. Let Ψ 0 ∈ Γ(U, Z G,E (Y 0 , f 0 )) and Ψ ∨ 1 ∈ Γ(U, Z G,E (Y 1 , f 1 ) ∨ ) be smooth sections. Choose parameterisations for ∂ 0 Σ and ∂ 1 Σ, let T : σ * G → I ρ be a trivialisation, and set T ′ := T • σ * A −1 : σ * G ′ → I ρ . The value of the scattering amplitude at a point x ∈ U reads as where −, − is the evaluation pairing on Z G,E (Y 1 , f 1 ). We compare the amplitude (5.3.6) to the amplitude where we use the evaluation pairing on Z G ′ ,E ′ (Y 1 , f 1 ). Observe that because the target bundle gerbes of the trivialisations T and T ′ agree, the exponential factors in (5.3.6) and (5.3.7) coincide. We now go through the list (SA1) to (SA3) in Section 3.2 in order determine how the factors z c differ from the factors z ′ c . Regarding (SA1), observe that since on objects c for boundary components c of this type. For (SA2), i.e. if c is a closed loop in the brane boundary ∂ 2 Σ, getting mapped to a D-brane Q i , the 2-isomorphism ξ i induces an isomorphism of hermitean vector bundles with connection as in (5.3.2). Since this isomorphism is connection-preserving, the traces of the holonomies of the source and target bundles around c are equal; hence, z c = z ′ c in this case. Finally, for (SA3) consider an object (Y, f ) with Y ∼ = [0, 1], such that U × ∂ 0 Y gets mapped to some Q i and such that U × ∂ 1 Y gets mapped to some Q j . From (5.3.3) we see that the trace in the expression for z c in (SA3) stays unaffected; the additional terms ξ A,i,0 and ξ A,j,1 from the individual factors cancel each other under the trace.
Thus, we see that the two amplitudes (5.3.6) and (5.3.7) agree factorwise, and hence that Z (A,ξ) is natural. Further, it is symmetric monoidal by construction. Proof. It suffices to prove the statement for the case where Y 0 , Y 1 are disjoint unions of copies of [0, 1] and S 1 , since we have constructed Z (A,ξ) coherently with respect to parameterisations of Y 0 and Y 1 . Using the same notation as above, we additionally define T ′′ := T •f * A ′−1 . First, we consider the case Y = [0, 1], with initial point mapped to Q i and end point mapped to Q j , and show that the diagram The commutativity is seen as follows. The connection-preserving 2-isomorphism ϕ : A → A ′ that defines the 2-isomorphism (A, ξ) → (A ′ , ξ ′ ) induces a connection-preserving isomorphismφ of trivialisations off * G. We use this connection-preserving isomorphism in order to represent the isomorphism r T ′′ ,T ′ (cf. (2.2.2)). For any The top and bottom squares commute by definition of Z A,T and Z A ′ ,T , respectively, while the left and right triangles commute by definition of 2-morphisms in TBG(M, Q). Consequently, we have that for every choice of trivialisation T . The morphisms r T ′ •Z A,T for different choices of trivialisations S of f * G are compatible as we have already checked in diagram (5.3.4). This implies that the compositions r T ′ • Z A,T induce a morphism of the colimits In fact, if we did not know the connection on the transgression bundles, this would reproduce it, and Proposition 4.3.11 would imply that this connection is unitary with respect to the canonically induced hermitean structures ♭ Z G,E (Y,f ) . ⊳

Subsectors, flat gerbes, and TQFTs
In this section, we investigate several variants of the field theories Z G,E constructed in Section 5. We consider its closed subsectors with and without branes, and we treat the special case of flat gerbes.
We conclude by showing that in the case where M is a single point our formalism is compatible with the classification of open-closed topological quantum field theories (TQFTs) in terms of coloured, knowledgeable Frobenius algebras.

Closed subsectors
We consider an arbitrary target space (M, Q). We let CBord d (M, Q) denote the full sub-presheaf of • Its values on bordisms (Σ, σ) without boundary agree with the usual surface holonomy of the bundle gerbe G around σ : Σ → M .
• Its values on bordisms without ingoing or outgoing string boundary agree with the surface holonomy in the presence of D-branes defined in [CJM02,Wal07a].
In other words, our construction extends these surface holonomies to a fully fledged smooth FFT.
Proof. The statement about the values on bordisms is a straightforward comparison of our definition of Section 3 with the literature.
We define the presheaf We conclude this subsection with remarks on the comparison of the smooth field theories considered here and those considered in [BTW04]. The field theories of Bunke-Turner-Willerton are symmetric monoidal functors from a thin-homotopy surface category T M of M to the category Vect × = LBun( * ) of complex lines. From the definitions in [BTW04] we observe that there is an inclusion of symmetric monoidal categories ι : T M ֒→ Bord th 2 (M )( * ) which is an equivalence of categories. Proposition 6.1.6. Given a gerbe G with connection on M , the field theory E G : T M → Vect × constructed in [BTW04]  Proof. Indeed, the value of both functors on simple objects (S 1 , f ) ∈ Bord th 2 (M )( * ) is the fibre T G |f of the transgression line bundle of G at the loop f : S 1 → M . The value on morphisms is essentially the surface holonomy of the gerbe.
Apart from the above relation between both constructions, the categories of field theories used in [BTW04] and here are very different. However, let TIFT(M ) be the set of isomorphism classes of thin-invariant smooth two-dimensional field theories as defined in [BTW04,Prop. 3.3]. We expect that there is an isomorphism of groups TIFT(M ) ∼ = π 0 FFT sf 2 (M ) × between TIFT(M ) and the group of isomorphism classes of two-dimensional superficial, invertible smooth FFTs. Indeed, Bunke-Turner-Willerton show that TIFT(M ) is in bijection with isomorphism classes of gerbes with connection on M [BTW04, Thm. 6.3], and we conjecture that our construction induces a bijection between π 0 (FFT sf 2 (M ) × ) and isomorphism classes of gerbes with connection on M as well (see also the end of Section 1).
Let us comment on why we find that our definition of a smooth field theory is more closely aligned with direct physical intuition (see Section 1). In [BTW04] a rather strict notion of smoothness is used for field theories, which requires the existence of a certain closed 3-form H on M such that for any 3-manifold V with smooth map v : V → M the field theory E : T M → Vect × satisfies This smoothness condition is powerful, but it does not encode the intuition that the values of the field theory functor should depend smoothly on the input data. Further, the smoothness property in [BTW04] implies additional properties such as invariance under thin 3-bordism (see [BTW04,Thm. 3.5]) which a priori are not related to the smoothness of a field theory. In our formalism, any additional properties of field theories are treated separately from smoothness and are implemented in the explicit notion of superficiality. For instance, this implies the invariance under thin three-bordism; we sketch this for the torus: given a solid torus V with a smooth rank-two map v : V → M , we can interpret these data as a thin homotopy between the bordism [∂V, v |∂v ] : ∅ → ∅ and a bordism The presheaf formalism we are using here encodes directly the smooth behaviour of the field theory on both objects and morphisms, and it fits with the modern treatment of functorial field theories [ST11]. Further, our formalism easily extends to field theories valued in targets other than VBun, whereas the implementation of smoothness in [BTW04] makes specific use of the target category (in particular the fact that it is linear).
Finally, we work with the full category of smooth FFTs rather than with their isomorphism classes, and our construction depends functorially on the input gerbe. We further believe that this is why in [BTW04] there was no need to consider reflection structures; we refer to an upcoming paper for details on this.

Flat gerbes and homotopy field theories
Recall the definition of a homotopy invariant smooth OCFFT from Definition 4.2.3.
Theorem 6.2.1. Let (G, E) ∈ TBG(M, Q) be a target space brane geometry on a target space (M, Q). The following are equivalent: (1) The smooth OCFFT Z G,E is homotopy invariant.
(2) The connection on the bundle gerbe G is flat; that is, curv(G) = 0.
Proof. If G is flat, Proposition 3.3.2 implies that Z G,E [Σ, σ] is invariant under homotopies of σ that are constant homotopies on U × ∂ 2 Σ and the collar neighbourhoods of ∂ a Σ for a = 0, 1.
In the other direction, let Σ = D 3 be the three-dimensional closed disc, and let h : D 3 → M be any smooth map. We may regard h as a smooth homotopy t → h t of maps h t : S 2 → M for t ∈ [0, 1], with h 0 a constant map. Proposition 3.3.2 and the homotopy invariance of Z G,E imply that Since h was arbitrary, this implies that curv(G) = 0.
Thus, field theories arising from flat gerbes are homotopy invariant FFTs, similar to those studied by Turaev [Tur10], and for topological (rather than smooth) models of gerbes, in particular, in [MW20]. Our smooth OCFFTs refine these results in that they are smooth, work in the non-flat case, and include D-branes.

Open-closed TQFTs and smooth OCFFTs on the point
In [ST04] the paradigm has been introduced that a functorial field theory with a target space M should be viewed as a classical field theory on M , whereas a field theory over the point (M = * ) should be understood as a quantum field theory.
In this section we explain how our definition of a smooth OCFFT reduces to the common definition of an open-closed topological quantum field theory (OCTQFT). To this end, we consider the target space (M, Q) where M is a single point and where the collection Q = {Q i } i∈I is a family of copies of the point indexed by I. We write I as a shortcut for this target space. We consider the symmetric monoidal category An OCTQFT is called invertible if the vector spaces assigned to closed objects Y ∼ = S 1 are 1-dimensional and all bordisms with empty brane boundary are sent to isomorphisms. Reflection structures for OCTQFTs and their positivity are defined analogously to the ones for smooth OCFFTs. Morphisms of (reflection-positive) OCTQFTs are symmetric monoidal transformations (intertwining the reflection structures). The category of d-dimensional OCTQFTs with D-brane labels I is denoted by OCTQFT I d . We use the prefix "RP-" for reflection-positivity, and the notation (..) × for invertibility. The relation between smooth OCFFTs over I and OCTQFTs with D-brane labels I is the evaluation of a morphism of presheaves on * ∈ Cart, which is a functor We remark that every smooth OCFFT over I is automatically thin homotopy invariant, superficial, and homotopy invariant, so that these properties need not to be discussed here. We describe now the surprising result hat the functor (6.3.2) is an equivalences of categories. This is remarkable, since it may seem that smooth FFTs on the point have vastly more structure than TQFTs, as TQFTs do not encode any smooth families of bordisms. However, functoriality in the test space, combined with our careful construction of the smooth bordism categories OCBord d , leads to the this result: Theorem 6.3.3. The functor ev * is an equivalence of symmetric monoidal categories, between the category of d-dimensional, smooth OCFFTs on I and the category of d-dimensional OC-TQFTs with D-brane labels I. The functor ev * and this equivalence result extends to the full subcategories of reflection-positive and invertible field theories.
We remark that Theorem 6.3.3 holds for an empty index set I = ∅, and hence in particular for smooth FFTs and TQFTs. For the proof of Theorem 6.3.3 we require the following lemma. Let To see this, let Ψ : U × Σ → U × Σ ′ be a fibre-wise diffeomorphism that establishes the equivalence Σ ∼ Σ ′ in OCBord 2 (I)(U ) (compare (4.1.4)). Then, restricting Ψ to the fibre over any point x ∈ U establishes the equivalence Σ * ∼ Σ ′ * in OCBord I 2 . Conversely, given any diffeomorphism Ψ * : Σ * → Σ ′ * that establishes Proof of Theorem 6.3.3. We consider the evaluation functor that evaluates smooth field theories (and their morphisms) on the cartesian space * ∈ Cart. The functor ev * is surjective on objects: let Z ∈ OCTQFT I d be arbitrary. We construct a smooth FFT Z ∈ OCFFT d (I) with ev * (Z) = Z. First, we define Z on objects. Let U ∈ Cart be arbitrary. By Lemma 6.3.4, any object Y ∈ OCBord d (I)(U ) is actually constant, i.e. it arises as a pullback of a unique object Y * ∈ OCBord I d along the collapse map U → * . We set The assignment Z is functorial and symmetric monoidal since Z is so, and by construction it satisfies ev * Z = Z. Hence, ev * is surjective on objects.
We now show that ev * is fully faithful. To that end, let Z, Z ′ : OCBord d (I) → VBun be OCFFTs, and let ζ : Z → Z ′ be a natural transformation of presheaves of categories. The naturality of ζ in U ∈ Cart implies that we have commutative diagrams where the horizontal vector bundle isomorphisms over U are part of the structure that makes Z and Z ′ morphisms of presheaves of categories. Combining diagram (6.3.5) with Lemma 6.3.4 now shows that any morphism ζ is entirely determined by its value on the terminal object * ∈ Cart. The consideration of reflection structures and invertibility is straightforward.
Thus, in our formalism, smooth OCFFTs on the one-point target space I are equivalent to ordinary OCTQFTs with D-brane labels I.

Classification of 2-dimensional open-closed TQFTs
Two-dimensional open-closed OCTQFTs have been investigated in [MS,LP08,Laz01], for example. Our definition of the category OCTQFT I 2 coincides precisely with the one given in [LP08]. There, and also in [MS,Laz01], it has been shown that 2-dimensional OCTQFTs with D-brane labels I are in equivalence to so-called I-coloured knowledgeable Frobenius algebras, as defined in [LP08, Section 2], see Theorem 6.4.2 below. We first recall the definition here for convenience. Definition 6.4.1. Let I be a set. An I-coloured knowledgeable Frobenius algebra is a septuple (L, R, χ, ǫ, θ, ι, ι * ) of the following data: (1) L is a commutative Frobenius algebra over C, with trace denoted by ϑ.
(2) R = {R ij } i,j∈I is a family of finite-dimensional complex vector spaces.
(3) χ = {χ ijk } i,j,k∈I is a collection χ ijk : R jk ⊗R ij → R ik of linear maps, which satisfies an associativity condition for quadruples of elements in I.
(4) ǫ = {ǫ i } i∈I is a family of elements ǫ i ∈ R ii that is neutral with respect to χ iii . In particular, (R ii , χ iii , ǫ i ) is an algebra for every i ∈ I.
(5) θ = {θ i } i∈I is a collection of linear maps θ i : R ii → C.
(6) ι = {ι} i∈I is a family of linear maps ι i : L → R ii , central in the sense that for all ℓ ∈ L and v ∈ R ij .
(7) ι * = {ι * i } i∈I is a family of linear maps ι * i : R ii → L, adjoint to ι in the sense that for all v ∈ R ii and ℓ ∈ L.
For i, j ∈ I, let σ ij denote the pairing The above data are subject to the following conditions: • σ ij is non-degenerate; that is, it induces an isomorphism Φ ij : R ij → R ∨ ji for every i, j ∈ I.
A morphism (ϕ, ξ) : (L, R, χ, ǫ, θ, ι, ι * ) → (L ′ , R ′ , χ ′ , ǫ ′ , θ ′ , ι ′ , ι ′ * ) of I-coloured knowledgeable Frobenius algebras consists of a linear map ϕ : L → L ′ and a family ξ = {ξ ij } i,j∈I of linear maps ξ ij : R ij → R ′ ij that respect the products, units and traces, and satisfy This defines a category of I-coloured knowledgeable Frobenius algebras, which we denote by We briefly recall the definition of this functor. If Z is a 2-dimensional OCTQFT with brane labels I, then the Frobenius algebra L of F(Z) has the underlying vector space Z(S 1 ), the product is given by the value of Z on a closed pair of pants, and the trace ϑ is obtained from the cap bordism S 1 → ∅.  Definition 6.4.3. A reflection structure on an I-coloured knowledgeable Frobenius algebra (L, R, χ, ǫ, θ, ι, ι * ) is a pair (λ,α) of an involutive algebra isomorphismλ : L → L and a familỹ α = {α ij } i,j∈I of involutive (meaningα ji •α ij = 1 for all i, j ∈ I), anti-multiplicative isomorphisms α ij : R ij → R ji . These have to satisfy the conditions for all i ∈ I, v ∈ R ii and ℓ ∈ L. The reflection structure (λ,α) is called positive if the pairings are positive definite for all i, j ∈ I. An I-coloured knowledgeable Frobenius algebra with a positive reflection structure is called reflection-positive. A morphism (ϕ, ξ) of I-coloured knowledgable Frobenius algebras respects reflection structures (λ,α) and (λ ′ ,α ′ ) if Definition 6.4.3 gives rise to a category of reflection-positive I-coloured knowledgeable Frobenius algebras, which we denote by RP-KFrob I . The following extension of Theorem 6.4.2 is straightforward to deduce. We shall briefly recall the definition of the functor frob. It assigns to an object (G, E) ∈ TBG(I) an object (L, R, χ, ǫ, θ, ι, ι * ,λ,α) := frob(G, E) ∈ RP-KFrob I C , whose complex line L is the fibre of the transgression line bundle L over the loop space L( * ) = * . For i, j ∈ I, the vector space R ij is the fibre of the transgression bundle over the path space P ij ( * ) ∼ = * as constructed in Section 2.2. The product on the C-algebra L is the fusion product on the transgression line bundle L. In general, it can be computed from compatible trivialisations of pullbacks of G along the loops, as laid out in [BW,Section 4.1], but see also [Wal16] for more details. In the present special case of M = * , with respect to a trivialisationT : G → I 0 on M = * , it simply reads as where γ ′ * γ denotes concatenation. The trace on L is given as ϑ [γ * T ], z = z .
The unit ǫ i ∈ R ii is the element represented by the image of the identity 2-morphism 1 E i . In other words, with respect to the trivialisation γ * iiT we have that On representatives with respect to γ * T , the morphism ι i : L → R ii reads as In the other direction, the morphism ι * i : R ii → L acts on representatives as taking the trace. The last piece of data in [BW,Def. 3.3.3, Def. 3.3.8] are the antilinear mapsλ : L → L andα ij : R ij → R ji for every i, j ∈ I, which define a positive reflection structure on the I-coloured knowledgeable Frobenius algebra (L, R, χ, ǫ, θ, ι, ι * ). With respect to a trivialisation γ * ijT , they act as where ψ * is the hermitean adjoint of ψ. On morphisms, frob acts simply as the transgression functor.
The following is the main result of this section. Proof. The evaluation functor on the right hand side of the diagram is an equivalence by Theorem 6.3.3. Propositions 6.4.5 and 6.4.4 show that F and frob are equivalences. The two-out-of-three property of equivalences of categories implies that Z is an equivalence, provided we prove that the diagram is commutative, which we do in the following.
Consider an arbitrary object (G, E) ∈ TBG(I). We will spell out its image (L ′ , R ′ , χ ′ , ǫ ′ , θ ′ , ι ′ , ι ′ * ,λ ′ ,α ′ ) := F(ev * (Z G,E )) under the functor F • ev * • Z and compare it to its image (L, R, χ, ǫ, θ, ι, ι * ,λ,α) := frob(G, E) ∈ RP-KFrob I C under frob, which we described above. By construction, L ′ is the vector space that Z G,E assigns to the unique loop in M = * , and by construction of Z G,E , this agrees with L. The vector space R ′ ij , for i, j ∈ I, is the vector space that Z G,E assigns to the unique path in M = * from Q i = * to Q j = * ; it is precisely the vector space R ij . That is, we have L = L ′ and R = R ′ .
Letting γ : S 1 → * denote the unique loop in M = * andT a trivialisation of G over * , the fusion product on the transgression line bundle L takes the simple form On the other hand, the product λ ′ on L ′ is obtained as the value of Z G,E on the closed pair of pants. Using the pullback ofT to compute this amplitude and by inspection of the definitions in Section 3.2, we see that The morphism χ ′ is the value of Z G,E on an open (i.e. flat) pair of pants [Σ, σ] with two incoming intervals and one outgoing interval. We can compute the amplitude of this bordism decorated with the states [γ * ijT , ψ ij ] and [γ * jkT , ψ jk ] on its incoming string boundary and with the dual state [γ * ikT , β ki (ψ ki )] on its outgoing string boundary. The contributions of the string boundaries to that amplitude are obvious from these decorations, but for the brane boundary we need parallel transports in V i := ∆(b * E i , b * T ), where b is the restriction of σ to a connected component of the brane boundary of Σ. However, since this bundle is pulled back from a bundle over the point, it is trivialisable as a vector bundle with connection, and so all the parallel transports in the amplitude are trivial. Consequently, the amplitude reads as tr(ψ ki • ψ jk • ψ ij ), and we conclude that In the open-closed TQFT defined by (G, E), the element ǫ ′ i ∈ R ′ ii = R ii is obtained as follows. Under Z G,E , the cap bordism [Σ cap,i , σ] : ∅ → ([0, 1], γ ii ) yields a linear map C → R ii , and ǫ ′ i is the evaluation of this linear map on 1 ∈ C. From the formalism in Section 3.2 and the above arguments regarding the triviality of the parallel transport along the brane boundary of bordisms in the case M = * we infer that β ii (ψ), Z G,E [Σ cap,i , σ](z) = A G,E Σ cap,i , σ * T , {z, β ii (ψ)} = z tr(ψ) for every incoming state z ∈ C and outgoing dual state ψ ∈ R ii ∼ = R ∨ ii . This implies that Z G,E [Σ cap,i , σ](z) = z 1 E i , and thus The trace ϑ ′ on the algebra (L ′ , λ ′ ) is obtained as the evaluation of Z G,E on the cap bordism S 1 → ∅. This acts as [γ * T ], z → z, and thus agrees with the trace ϑ on L, i.e. ϑ = ϑ ′ .
Therefore, the rank-one, commutative Frobenius algebras (L, λ, θ) and (L ′ , λ ′ , θ ′ ) are equal. Recalling that the hermitean metric on R ij is defined on representatives as we see from an analogous argument using the cap bordism [0, 1] → ∅ that also the traces on the algebras R ii and R ′ ii agree for every D-brane label, i.e. θ = θ ′ .
In the field theory, ι ′ i is the morphism induced by the unzip bordism S 1 → [0, 1]. Similarly to the arguments used for the cap bordisms ∅ → [0, 1], we see that this morphism coincides with ι i . That is, In the other direction, we have to consider the zip bordism [0, 1] → S 1 , which on representatives also maps to the trace under Z G,E . This yields ι * = ι ′ * .
Finally, we turn to the reflection structures. On the field theory side a reflection structure α on an OCTQFT naturally induces antilinear morphismsλ andα on the coloured Frobenius algebra that it defines, and these antilinear morphisms behave exactly as in [BW,Definition 3.3.8]. In the case of the field theory Z G,E we directly read off from (5.2.1) and the following paragraph that λ =λ ′ andα =α ′ .
To summarise, we have shown that frob(G, E) = F(ev * (Z G,E )) for all objects (G, E) ∈ TBG(I). That is, the diagram (6.4.6) strictly commutes on objects.
On morphisms in TBG(I), we only need to compare the induced linear maps on the vector spaces underlying the I-coloured Frobenius algebras. However, comparing the definition of the action of frob on morphisms in [BW,Section 4.8] with the action (5.3.3) of Z on morphisms we readily see that the induced linear maps agree.
We point out two consequences of Theorem 6.4.6. The fact that the diagonal of the diagram in Theorem 6.4.6 is an equivalence leads to the following result.
Corollary 6.4.7. The 2-dimensional, invertible, reflection-positive OCTQFTs are precisely those that can be obtained from target space brane geometry over a point, i.e., those that are restrictions of classical field theories to the point.
Secondly, we emphasise that our construction of smooth OCFFTs from target space brane geometry is an equivalence, in the special case of a one-point target space.
Corollary 6.4.8. The construction Z of a smooth OCFFT from a target space brane geometry is an equivalence, TBG(I) ∼ = RP-OCFFT 2 (I) × .
We mention this because we believe that the functor Z is an equivalence for any target space (M, Q), i.e., we conjecture that it is an equivalence TBG(M, Q) ∼ = RP-OCFFT sf 2 (M, Q) × , We plan to prove this equivalence in forthcoming work.

A Monoidal categories with fixed duals
We briefly recall some basic facts about categorical duals, mostly based on [EGNO15].
Definition A.1 ([EGNO15, Definition 2.10.1]). Let C be a monoidal category with monoidal structure ⊗ and unit object u ∈ C. Let x ∈ C be an object. An object x ∨ ∈ C is called a (left) dual of x ∨ if there exist morphisms ev x : x ∨ ⊗ x → u and coev x : u → x ⊗ x ∨ such that the diagrams (where we are omitting the structural isomorphisms) We remark that the inverse of τ x y,z is given by τ x −1 y,z (g) = (1 z ⊗ ev x ) • (g ⊗ 1 x ) .
In a monoidal category with fixed duals, duality extends automatically to the morphisms. For a morphism f : x → y one defines the dual morphism f ∨ : y ∨ → x ∨ as the composition One can show that for g : y → z one has (g • f ) ∨ = f ∨ • g ∨ . We note the following simple fact, of which we could not find an explicit reference.
Lemma A.5. Let f : x → y be a morphism in a monoidal category with fixed duals; then, we have Proof. The identity follows from the commutativity of the diagram The left-hand triangle commutes because of the first triangle identity (A.2). The upper central rectangle is the definition of f ∨ and hence commutes. The lower central rectangle and the right-hand triangle commute by monoidality; morphisms applied to different factors of a tensor product commute in any monoidal category.
If C is a monoidal category with fixed duals, then the assignments x → x ∨ and f → f ∨ define a functor d C : C op → C. Since x ∨∨ and x are both dual to x ∨ , there exists a unique isomorphism δ x : x ∨∨ → x by Proposition A.3. These define a monoidal natural isomorphism δ C : d C • d op C → 1 C , which additionally satisfies In the terminology used in Section 4.3, the pair (d C , δ C ) forms a twisted involution, and is called the duality involution of the monoidal category C with fixed duals.
Proposition A.6. Let C, D be monoidal categories with fixed duals and duality involutions (d C , δ C ) and (d D , δ D ), respectively. Suppose F : C → D is a monoidal functor, i.e. it comes with coherent natural isomorphism f x,y : F x ⊗ F y → F (x ⊗ y). Then, there is a unique monoidal natural isomorphism that is compatible with evaluations and coevaluations in the sense that for all x ∈ C. Moreover, it is compatible with the natural isomorphisms δ C and δ D in the sense that the diagram is commutative.
Proof. Let x ∈ C be an object with fixed left duality data (x ∨ , ev x , coev x ). Let ((F x) ∨ , ev F x , coev F x ) be the fixed left duality data on F x ∈ D. Since F is monoidal, the triple is another set of left duality data for F x. By Proposition A.3 there exists a unique isomorphism β x : F (x ∨ ) → (F x) ∨ compatible with the two sets of left duality data on F x; the compatibility is expressed precisely by the two equations in (A.6). Explicitly, β x is given by the composition [EGNO15, Proposition 2.10.5] We have to show that β is natural. To that end, suppose ψ : x → y is a morphism in C. We have to prove that the diagram commutes. Inserting the definition of the dual morphism, we expand this diagram to Here, the left-hand and right-hand squares commute because of the compatibility of β with the duality data, and the centre square commutes simply by monoidality.
To see that β is monoidal, first observe that for any two objects x, y ∈ C the triple y ∨ ⊗ x ∨ , ev y • (1 ⊗ ev x ⊗ 1), (1 ⊗ coev y ⊗ 1) • coev x is left duality data for x ⊗ y. Thus, there exists a unique isomorphism δ x,y : y ∨ ⊗ x ∨ → (x ⊗ y) ∨ that is compatible with the duality data. Then consider the diagram All of the objects in this diagram come with choices of left duality data for F (x⊗y), and the morphisms in the diagram are each compatible with these left duality data (this uses the coherence of the natural isomorphism f and Lemma A.5). In particular, the diagram yields two possible ways of going from F ((x ⊗ y) ∨ ) to (F y) ∨ ⊗ (F x) ∨ , both of which are compatible with the left duality data on these two objects. Thus, Proposition A.3 implies that the diagram (A.8) commutes, which shows that β respects the tensor product.
The commutativity of the diagram (A.7) can be shown by similar methods; since it will not be used in this article we leave this as an exercise to the reader.