Orbifold graph TQFTs
Nils Carqueville, Vincentas Mulevicius, Ingo Runkel, Gregor Schaumann, Daniel Scherl
OOrbifold graph TQFTs
Nils Carqueville ∗ Vincentas Muleviˇcius
Ingo Runkel
Gregor Schaumann ∨ Daniel Scherl [email protected]@[email protected]@[email protected] ∗ Fakult¨at f¨ur Physik, Universit¨at Wien, Austria
Fachbereich Mathematik, Universit¨at Hamburg, Germany ∨ Institut f¨ur Mathematik, Universit¨at W¨urzburg, Germany
A generalised orbifold of a defect TQFT Z is another TQFT Z A obtained by performing a state sum construction internal to Z . Asan input it needs a so-called orbifold datum A which is used to la-bel stratifications coming from duals of triangulations and is sub-ject to conditions encoding the invariance under Pachner moves. Inthis paper we extend the construction of generalised orbifolds of 3-dimensional TQFTs to include line defects. The result is a TQFTacting on 3-bordisms with embedded ribbon graphs labelled by a rib-bon category W A that we canonically associate to Z and A . We alsoshow that for special orbifold data, the internal state sum construc-tion can be performed on more general skeletons than those dual totriangulations. This makes computations with Z A easier to handle inspecific examples. 1 a r X i v : . [ m a t h . QA ] J a n ontents
1. Introduction and summary 32. Topological preliminaries 7 ω -moves . . . . . . . . . . . . . . . . . . . . 19
3. Defect TQFTs 24
4. Orbifold graph TQFTs 32
A. Proof of Theorem 2.12 48
A.1. Skeleta of 2-manifolds . . . . . . . . . . . . . . . . . . . . . . . . 48A.2. Pseudo-skeleta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50A.3. Refinement to dual of a triangulation relative to boundary . . . . 51A.4. Fixing the orientations . . . . . . . . . . . . . . . . . . . . . . . . 55A.5. Fixing the boundary . . . . . . . . . . . . . . . . . . . . . . . . . 56A.6. The proof of Theorem 2.12 . . . . . . . . . . . . . . . . . . . . . . 59
B. Orbifold data for non-Euler-complete theories 60 . Introduction and summary By a generalised orbifold – or just orbifold for short – of a topological quantumfield theory Z , we mean a state sum construction internal to Z , as initiated in[FFRS, CR, CRS1]. Here Z must necessarily be a defect TQFT, i. e. a symmetricmonoidal functor on a category of stratified and decorated bordisms. The defin-ing conditions on the datum A from which the orbifold theory Z A is constructedencode invariance under decompositions of bordisms in the state sum construc-tion. In [CRS1], these decompositions were taken to be stratifications that aredual to triangulations.If Z is the trivial defect TQFT, then its orbifolds recover conventional statesum constructions; in the 2-dimensional case (where A is a ∆-separable symmetricFrobenius k -algebra) this is implicit in [DKR], while in [CRS3] it was shown thatTuraev–Viro–Barrett–Westbury models are 3-dimensional orbifolds (in particu-lar, every spherical fusion category gives rise to an orbifold datum A ). Anotherclass of examples comes from discrete group actions (which can be “gauged”) onarbitrary defect TQFTs, see e. g. [BCP, CRS3] for detailed discussions of the 2-and 3-dimensional cases, and [SW] for a more geometric approach. Indeed, thisembeds the original meaning of “orbifold” as gauging a discrete symmetry intothe setting of generalised orbifolds we consider here.There are orbifolds beyond the unification of state sum models and the gaug-ing of symmetry groups. For example, based on results of [CR], in [CRCR, RW]2-dimensional orbifolds of Landau–Ginzburg models were constructed, which un-covered new relations between homological invariants of isolated singularities. Itwas necessary for these applications to have a universal construction of defectsfor the orbifold TQFT in terms of a representation theory of orbifold data in-ternal to the original theory. In three dimensions, examples of orbifolds for the3-dimensional Reshetikhin–Turaev theory based on Ising-type categories wereconstructed in [MR2], inverting the extension of the modular fusion category for sl (2) at level 10 by its commutative algebra of type E .The 3-dimensional examples just mentioned build on the following general con-struction. Let C be a modular fusion category, and let Z C be the associated defectTQFT of Reshetikhin–Turaev type described in [CRS2]. Then the main resultof [MR1] states that for every (simple, special) orbifold datum A for Z C , oneobtains another modular fusion category C A . It was conjectured in [MR1] that Z C A ∼ = (cid:0) Z C (cid:1) A , (1.1)i. e. the Reshetikhin–Turaev defect TQFT associated to C A is isomorphic to the A -orbifold of the theory associated to C .In the present paper we develop a general theory of (Wilson) line defects in3-dimensional orbifold TQFTs. This will be applied in the companion paper[CMRSS2] to prove the conjecture (1.1). One notable consequence of this is thatReshetikhin–Turaev theories close under orbifolds.3e now outline our general constructions, highlighting the three main contri-butions of this paper, which may be of independent interest: (i) the constructionof orbifolds from decompositions that are computationally easier to deal withthan dual triangulations, namely so-called admissible skeleta; (ii) the construc-tion of a ribbon category W A of Wilson lines associated to any orbifold datum A ;(iii) the construction of a TQFT on bordisms with W A -labelled ribbon graphs.The latter is the orbifold graph TQFT which gives this paper its title. Admissible skeleta
According to [CRS1, CRS3] (and as reviewed in some detail in Section 4.1), anorbifold datum A of a 3-dimensional defect TQFT Z consists of defect labels A , A , A , A +0 , A − as in , , , , (1.2)such that evaluating Z on A -decorated stratifications which are dual to suitablyoriented triangulations is independent of the choice of triangulation. Then theorbifold Z A is constructed as a colimit that arises from applying Z to all thesestratifications.We will show that instead of practically cumbersome stratifications dual totriangulations, for those A which are “special” in the sense explained in Sec-tion 4.1.1, one can compute Z A in terms of a simple type of stratification that wecall “admissible skeleta” (which are fully oriented variants of the “special skeleta”of [TV]). These are stratifications where every 3-stratum is a ball and every pointhas one of the neighbourhoods listed in Figure 2.1, see Definitions 2.2 and 2.7 fordetails. We show that any two admissible skeleta can be related by three typesof moves (bubble, lune, and triangle, BLT for short), see Figure 2.2.In particular, every admissible skeleton can be consistently decorated with aspecial orbifold datum A , and the defining conditions on A ensure invariance un-der BLT moves. Then Theorem 4.6 explains how to construct Z A from admissibleskeleta. As an example, note that an embedding S ⊂ S gives an admissibleskeleton which is not dual to a triangulation. A ribbon category of Wilson lines
Every 3-dimensional defect TQFT Z gives rise to a 3-category T Z , see [CMS].The objects of T Z are interpreted as bulk theories, while 1-, 2-, and 3-morphisms4re interpreted as surface, line, and point defects, respectively. In Sections 3.3and 4.2 we will explain how this implies that for every orbifold datum A for Z , oneobtains a ribbon category W A . Objects of W A are line defects X in A -decoratedsurface defects which can cross A -decorated line defects at point defects τ X , τ X , , , (1.3)that are subject to the compatibility conditions in Figure 4.2. We allow the linedefects to have non-trivial “internal structure”; for example,= (1.4)is a “line defect”, where any (line; surface; bulk) defect labels X , X ; α, β ; u allowed by Z may occur.Morphisms in W A by definition have to intertwine with τ X , τ X , and it isstraightforward to give W A the structure of a rigid monoidal category. More-over, the diagrams , , (1.5)when evaluated with a certain completion (cid:98) Z of Z (that can handle line defects asin (1.4), see Section 4.2 for details) endow W A with a braiding (Proposition 4.9).We view the ribbon category W A as a natural algebraic invariant attachedto the orbifold datum A for Z . In [CMRSS2] we will show that if Z is theReshetikhin–Turaev defect TQFT of [CRS2], then W A is equivalent to the mod-ular fusion category C A of [MR1]. There is however no reason for W A to besemisimple in general. 5 a) , (b) Figure 1.1.: (a) A local patch of a bordism M with embedded W A -labelled ribbongraph R . (b) A local patch of an admissible A -decorated skeleton S for M (where all green 2-strata implicitly carry a label A ) togetherwith two choices to represent R in S , which are necessarily relatedby the moves in Figures 2.2 and 2.5. Orbifold graph TQFTs
Recall from [Tu, TV] that a graph TQFT is a symmetric monoidal functor onthe bordism category Bord rib3 ( C ) with embedded ribbon graphs that are labelledby some fixed k -linear ribbon category C for a field k . Our main result is a liftof the orbifold TQFT Z A : Bord −→ Vect to a graph TQFT (cid:98) Z Γ A : Bord rib3 ( W A ) −→ Vect , (1.6)where Vect denotes the category of k -vector spaces. To do so, we adapt theformalism of [TV] to “represent” every W A -labelled ribbon graph in a givenbordism M by pushing it into an A -decorated admissible skeleton of M , seeFigure 1.1 for an illustration. Analogously to how one finds that the choice ofadmissible skeleton is immaterial in the construction of Z A , we prove (see Theo-rem 4.15) that the construction of (cid:98) Z Γ A is independent of the choice of admissibleskeleton and how precisely the W A -labelled ribbon graph is pushed into it.The remainder of the present paper is organised as follows. In Section 2 weintroduce admissible skeleta, representations of ribbon graphs with respect tosuch skeleta, as well as the moves that connect them. Most of the technicaldetails related to this discussion are contained in Appendix A. In Section 3 webriefly review 3-dimensional defect as well as graph TQFTs, and we construct aribbon category of Wilson lines from any defect TQFT. In Section 4, after a shortrecollection of orbifold TQFTs, we define a ribbon category W A associated to aspecial orbifold datum A , and then construct the orbifold graph TQFT (1.6).6 cknowledgements N. C. is supported by the DFG Heisenberg Programme. V. M. is partially sup-ported by the DFG Research Training Group 1670. I. R. is partially supportedby the Cluster of Excellence EXC 2121.
2. Topological preliminaries
In this section we set the topological stage for our constructions. Section 2.1collects our conventions for 3-dimensional stratified bordisms. In Section 2.2we introduce a particular class of stratifications called “admissible skeleta”. Weshow that these are related by the “BLT moves” of Figure 2.2, which will featureprominently in later sections. Then a brief review of bordisms with embeddedribbon graphs in Section 2.3 is followed by an account of how to represent ribbongraphs with respect to admissible skeleta in Section 2.4, and how different suchrepresentations are related by the “ ω -moves” of Figure 2.5.Our discussion here heavily draws from [TV]. The main novelty is that wecarefully check that everything can be made admissibly oriented in our sense. We recall the stratifications used in [CMS, CRS1], to which we refer for moredetails.
By an n -dimensional stratified manifold we mean an n -dimensional topologicalmanifold M (without boundary) together with a stratification S of M , which isgiven by a filtration ∅ = F ( − ⊂ F (0) ⊂ F (1) ⊂ . . . ⊂ F ( n ) = M of topologicalspaces such that for each j ∈ { , , . . . , n } , S ( j ) := F ( j ) \ F ( j − has the structure ofa smooth j -dimensional manifold (such that the smooth structure is compatiblewith the subspace topology). The connected components of S ( j ) are called j -strata . We denote the set of j -strata by S j , and we ask each S j to be finite. For s ∈ S i , t ∈ S j we require that whenever s ∩ t (cid:54) = ∅ , then already s ⊂ t . In thiscase necessarily i < j and we say that s and t are incident to each other. For any x ∈ M we say that x and s are incident to each other if x ∈ ¯ s .We denote by S j ( x ) the set of germs of j -strata around x , i. e. the inverse limitof the canonical maps S εj ( x ) −→ S δj ( x ) for ε < δ , where S εj ( x ) is the set whoseelements are intersections of j -strata and a ball of radius ε around x (in somechart). Example 2.1.
Consider the following stratification S of the 3-sphere which hastwo 3-strata (the interior of the coloured solid torus with one disc removed, as7ell as its complement in S – which is also a torus, and not coloured in thepicture), two 2-strata and one 1-stratum, with a chosen point x in the disc-shaped2-stratum: (2.1)Then S ( x ) has two elements, even though x is only incident to a single 3-stratum.An n -dimensional stratified manifold with boundary is an n -dimensional topo-logical manifold M with boundary ∂M together with a filtration as above suchthat the interior of M is a stratified manifold without boundary. Furthermore wedemand that each stratum s satisfies ∂s = s ∩ ∂M . It follows that ∂M canoni-cally inherits the structure of an ( n − M is topological,each stratum is a smooth manifold, cf. [CRS1, Footnote 4].A map f : M −→ M (cid:48) of n -dimensional stratified manifolds with boundary isa continuous map that sends strata to strata and restricts to a smooth map oneach stratum, and such that f restricts to a map f | ∂M : ∂M −→ ∂M (cid:48) that is amap of ( n − oriented stratified manifold (possibly with boundary) is a stratified mani-fold (possibly with boundary) such that the underlying manifold M and all stratacarry a prescribed orientation, such that each top-dimensional stratum carries theorientation induced from M . Maps of oriented stratified manifolds are definedas above, except that in each step we additionally require that the restriction toeach stratum is orientation-preserving. An n -dimensional oriented stratified bordism is a tuple M = ( M, Σ − , Σ + , ϕ − , ϕ + ),where M is an n -dimensional compact oriented stratified manifold (possibly withboundary), Σ − and Σ + are ( n − ϕ ∓ are germs (in ε ∈ R > ) of embeddings of stratified manifolds ϕ ε ∓ : Σ ∓ × [0 , ε ) −→ M . (2.2)Each ϕ ε + is required to be orientation-preserving, and each ϕ ε − is orientation-reversing. Furthermore ∂M splits as a disjoint union ∂M = ϕ − (Σ − × { } ) (cid:116) ϕ + (Σ + × { } ) . (2.3)8e refer to ϕ ∓ (Σ ∓ ) × { } as the in- and out-boundary of M , respectively. The source of M is Σ − and the target is Σ + . A morphism of stratified bordisms M and M (cid:48) is a map f : M −→ M (cid:48) of oriented stratified manifolds such that f ϕ ∓ = ϕ (cid:48)∓ .There is a symmetric monoidal category of n -dimensional oriented stratifiedbordisms Bord str n as follows. Objects of Bord str n are ( n − −→ Σ (cid:48) is an isomorphism class of n -dimensional oriented stratified bordisms with source Σ and target Σ (cid:48) . We willoften use a bordism and its isomorphism class synonymously. Composition ofmorphisms is defined by gluing along common boundaries. Requiring germs ofembeddings Σ ∓ × [0 , ε ) −→ M rather than just embeddings of the boundariesthemselves ensures a canonical smooth structure on the strata of a composite bor-dism. The tensor product of Bord str n is given by disjoint union, and the symmetricbraiding by mapping cylinders of the twist maps on disjoint unions. Defect bordisms are oriented stratified bordisms satisfying an additional regular-ity condition, imposed by requiring the existence of certain local neighbourhoodsaround each point. The sets of local neighbourhoods for n -dimensional defectbordisms are denoted by N n . The elements of N n are oriented stratified openmanifolds of dimension n . They are defined inductively for arbitrary n in [CRS1,Sect. 2.2]. Here we only give a brief discussion of the case n = 3.In order to define the local neighbourhoods for 3-dimensional defect bordisms,we first have to consider the 2-dimension case. There are three types of localneighbourhoods for 2-dimensional defect bordisms in N : , , . (2.4)Note that there are infinitely many neighbourhoods of the third type, and anychoice of orientation for the 0-stratum and the 1-strata is allowed. A defect2-manifold is an oriented stratified manifold, such that each point has a neigh-bourhood isomorphic (as an oriented stratified manifold) to an element of N . A defect 2-sphere is a defect 2-manifold with underlying manifold is S .A is a 3-dimensional stratified bordism such thateach point has a neighbourhood isomorphic (as an oriented stratified manifold)to one of the following list:(i) Open cylinders X × ( − , X ∈ N , with orientations induced9rom X and the standard orientation of ( − , X = ∈ N , X × ( − ,
1) = (2.5)(ii) Open cones C (Σ) = (Σ × [0 , / (Σ × { } ) , where Σ is a defect 2-sphere. Open cones have a natural structure of strati-fied manifolds with underlying manifold the open 3-ball B ⊂ R . The conepoint defines a 0-stratum at 0 ∈ R . Due to the choice of orientation ± forthe cone point each defect 2-sphere gives rise to two elements of N . Anexample for orientation “+” is: (2.6)We obtain the symmetric monoidal non-full subcategory Bord def3 of Bord str3 whose objects are compact closed defect 2-manifolds, and whose morphisms aregiven by only those stratified bordisms that locally look as specified above. In this subsection we present a class of stratifications, called skeleta, that arewell-suited for the procedure of “orbifolding” in Section 4 below. Duals of trian-gulations form a proper subset of all skeleta, which in turn form a proper subsetof the stratifications allowed for defect bordisms.
Definition 2.2.
Let M be an (unoriented) 3-manifold, possibly with boundary.A skeleton S of M is a stratification of M that satisfies the following additionalrequirements.(i) Every 3-stratum is diffeomorphic to either an open 3-ball if it does notintersect ∂M , or to an open half-ball otherwise.10ii) Each x ∈ M has a neighbourhood B x that is isomorphic (as a stratifiedmanifold) to one from the list in Figure 2.1. In each case B x is an open ballif x / ∈ ∂M , and an open half-ball otherwise. Remark 2.3.
Condition (i) of Definition 2.2 implies that no 3-stratum of askeleton intersects both the in- and out-boundary of M non-trivially.If M is oriented, then an oriented skeleton of M is a skeleton that is oriented asa stratification. In particular each 3-stratum carries the same orientation as M ,but there are no restrictions on the orientations of 2-, 1- and 0-strata.Every stratification that is obtained as the Poincar´e dual of a triangulationis a skeleton. An example of a skeleton that does not arise in this way is the(unoriented version of the) skeleton of S in Example 2.9 (i) below. An exampleof a stratification that is not a skeleton is the stratification of S in Example 2.1(the uncoloured 3-stratum outside of the coloured solid torus is not a 3-ball). Remark 2.4.
In the terminology used in [TV, Sect. 11.5.1], condition (vi) ofFigure 2.1 means that each 0-stratum is a special point . The notion of skeletagiven in [TV, Sect. 11.2.1] is more general than the one given here in that it allowsfor more diverse local situations than the ones specified in Figure 2.1. Our skeletaare the “s-skeleta” of loc. cit., except that we do not demand at least one specialpoint in each connected component, and we allow for circles as 1-strata.
For the remainder of Section 2, M will denote an arbitrary but fixed oriented3-manifold (possibly with boundary), and all skeleta are to be taken within M , ifnot explicitly stated otherwise. Recall from Section 2.1.1 the definition of S j ( x )as the set of germs of j -strata around a point x ∈ M . Definition 2.5. A local order on a skeleton S is for each point x ∈ M a choiceof total order on S ( x ) such that for any two points x, y in the closure of a given j -stratum with j ∈ { , , } , the corresponding orders are compatible in thefollowing sense: whenever A , B are 3-strata incident with x and y that inducethe elements a and b in S ( x ) as well as a (cid:48) and b (cid:48) in S ( y ), respectively, then a < b ⇐⇒ a (cid:48) < b (cid:48) .Note that if x is a point in a j -stratum of a skeleton S , then S ( x ) has precisely4 − j elements (but possibly fewer incident 3-strata). If x, y ∈ M are bothcontained in the same stratum s , then S ( x ) = S ( y ), and we define S ( s ) := S ( x ) where x ∈ s is arbitrary. Indeed, let x, y ∈ s . If x is contained in some B y or y is contained in some B x as in Figure 2.1, the claim S ( x ) = S ( y ) isclear. Otherwise, since s is path-connected, we can consider a path γ from x to y More precisely, there is a canonical isomorphism along which we identify the two sets. x ∈ S (3) then B x contains no 2- or lower strata.(ii) If x ∈ S (2) and x / ∈ ∂M then B x contains a single 2-stratum and is givenby . (iii) If x ∈ S (2) and x ∈ ∂M then B x is given by . (iv) If x ∈ S (1) and x / ∈ ∂M then B x is given by . (v) If x ∈ S (1) and x ∈ ∂M then B x is given by . (vi) S (0) ∩ ∂M = ∅ and if x ∈ S (0) then B x is given by . Figure 2.1.: List of allowed local neighbourhoods for skeleta S of a 3-manifoldwith boundary M . As in Section 2.1.1, S ( j ) denotes the union of all j -strata. 2-strata are depicted in green, 1-strata in black, and theboundary of M is grey (colour available online).12hat lies in s , and transport the order along an open cover of γ . In light of thiswe can also define a local order on S as a choice of total order on S ( s ) for eachstratum s of S such that for any two strata s, t the induced orders on S ( s ) ∩ S ( t )agree. Hence a local order on S orders the germs of 3-strata of S around eachlower-dimensional stratum.Any local order on a skeleton S turns it into an oriented skeleton by the fol-lowing convention. Convention 2.6.
All 3-strata carry the orientation induced by the orientationof M . The orientations for 2-strata are obtained via the right-hand rule: , (2.7)where here and below the numbers on free-floating vertices indicate the localorder on the ambient germs of 3-strata. The orientations of 1- and 0-strata aredetermined as follows: , , . (2.8)As pictured above, by default we assume that 2-strata have the standard ori-entation of the paper/screen plane. We indicate the opposite, i. e. clockwise,orientation by a stripy pattern, for example . (2.9) Definition 2.7. An admissible skeleton of M is an oriented skeleton whose ori-entation is induced by a local order. We denote the set of admissible skeleta of M by (cid:83) ( M ). These conventions are consistent with those in [CRS2, CRS3, MR1, MR2], but they differslightly from those in [CRS1], where the orientations of 2-strata are flipped. emark 2.8. (i) A local order is uniquely determined by the oriented skeletonthat it induces. In fact the local order can be recovered from only theorientations of 2-strata in the induced oriented skeleton via Convention2.6. Hence the datum of an admissible skeleton is the same as that of anunoriented skeleton together with a local order.(ii) If S is dual to a triangulation T of M , then Definition 2.5 reduces to thenotion of an ordering of a simplicial complex, see e. g. [JT, p. 2]. Suchan ordering is given by a total order on the vertices of each 3-simplexof S such that the induced orders on shared faces agree. By dualisingConvention 2.6, any ordering of a simplicial complex induces an orientationof all its simplices, in particular 1-simplices are oriented away from verticesof lower order. If the orientation of T is induced by an ordering, then itcan be uniquely recovered from just the orientations of 1-simplices. In turnthe datum of an ordering on T is the same as that of an orientation of each1-simplex of T such that no loops are formed around any single simplex.In this case we call T an admissible triangulation . Examples 2.9. (i) A local order on a skeleton S does not necessarily induce atotal order on the set S of all 3-strata of S , as is illustrated by the followingadmissible skeleton of S : (2.10)Indeed, for any given order on the two 3-strata making up the two halvesof the solid torus, at least one of the two disc-shaped 2-strata separatingthem would have the wrong orientation.(ii) Using Remark 2.8 (i) it is straightforward to construct examples of orientedskeleta that are not admissible by simply choosing orientations for 2-stratathat do not match any of those allowed by Convention 2.6. For example,any oriented skeleton that locally looks as follows is not admissible: (2.11)14 .2.2. Local moves on skeleta We now introduce local moves on oriented skeleta. We refer to these moves as
BLT moves (short for bubble, lune, and triangle moves). Our list of moves isa slight modification of the moves that are considered in [TV, Sect. 11.3–11.4],and they are equivalent to the set of moves considered in [CRS1, Def. 3.13], seeLemma 2.11 below. We show that all admissible skeleta are related by admissibleBLT moves.
Definition 2.10.
Let M be a stratified 3-manifold.(i) The unoriented BLT move B, L or T is given by the two stratified open3-balls shown in Figure 2.2 (i), (ii) or (iii) respectively, considered up toisomorphism.(ii) An oriented BLT move X consists of the unoriented BLT move X with achoice of orientation of the two respective stratified 3-balls, such that theorientations of strata intersecting the boundaries of the balls agree. We callan oriented BLT move admissible if the two respective stratified 3-balls areadmissibly oriented.(iii) An application of an unoriented BLT move X to M is the stratified 3-manifold X ( M ) which consists of replacing an embedding of the open strat-ified 3-ball B on the left of the move X in M with the stratified 3-ball onthe right of X . Analogously we define the application of the inverse of aBLT move and the application of an oriented BLT move.We remark that – at least in the local neighbourhood shown in Figure 2.2 (ii)– an application of a lune move splits up an oriented 2-dimensional region intotwo parts that consequently will have the same orientations in the target. Thisin turn puts a restriction on the orientations of 2-strata for when an inverse lunemove can be applied.In Theorem 2.12 below we show that any two admissible skeleta of M that agreeon ∂M can be transformed into one another by a finite sequence of admissibleBLT moves. Since for an admissible move we require the source and targetto be admissibly oriented, by Remark 2.8 (i) we only need to specify how theorientations of 2-strata are changed. One can check that this leaves us with atotal of 32 admissible moves (up to isomorphisms of oriented stratified manifoldswhich do not necessarily fix the boundary): 3 bubble moves, 9 lune moves and 20triangle moves. Some examples of possible orientations are listed in Figure 2.3.We want to relate the BLT moves to two types of moves that are consideredin [CRS1]. We call a set A of moves stronger than a set B , if each of the movesof B is an application of a finite sequence of moves of A . We say A is equivalent to B if A is stronger than B and B is stronger than A .15i) The bubble moves B and their inverses: B (cid:29) B − . (ii) The lune moves L and their inverses: L (cid:29) L − . (iii) The triangle moves T and their inverses: T (cid:29) T − . Figure 2.2.: BLT moves without orientations. The dotted lines indicate where the2-strata meet the boundary of the 3-ball in which they are embedded.16e first consider the admissible Pachner moves which are the oriented Pachnermoves whose source and targets are admissible triangulations, cf. Remark 2.8 (ii),see also [CRS1]. Another set of moves that is considered in [CRS1, Def. 3.13] arethe special orbifold data moves , i. e. the moves relating the left- and right-handsides of the identities in Figure 4.1 below (without the A -decorations). Note thatthe latter consist of all 3 bubble moves, 6 of the 9 lune moves, and one trianglemove. We consider all of these moves as moves between oriented skeleta. Lemma 2.11.
The admissible BLT moves are equivalent to the special orbifolddata moves, and both are stronger than the admissible Pachner moves.
Proof.
By definition, the special orbifold data moves are a subset of the admissibleBLT moves and consequently admissible BLT moves are stronger than specialorbifold data moves. In [CRS1, Prop. 3.18] it is shown that the special orbifolddata moves are stronger than the globally ordered Pachner moves. A slightmodification of the arguments presented there shows the same for admissiblePachner moves.To verify that special orbifold moves are stronger than admissible BLT moveswe observe that • a T-move is the same as the 2-3 move in (O1) (without its decoration), orone of the 19 variants thereof with a different admissible orientation; • an L-move is the same as one of the six identities (O2)–(O7) in Figure 4.1,or one of the three variants thereof where the orientation of the additional2-stratum on the right-hand sides of (O4)–(O6) is reversed; • a B-move is the same as a bubble move in (O8).By [CRS1, Lem. 3.15], every admissible orientation for the 2-3 moves can be ob-tained from (O1) and the six identities (O2)–(O7). Moreover, the three L-moveswhich are not among (O2)–(O7) can be obtained by flipping the orientation ofthe new 2-stratum on the right-hand side: it follows from the proof of Lemma A.6and Remark A.8 that this can be achieved only with T- and B-moves, and withthe moves (O2)–(O7).We can now state the main result of this section. Theorem 2.12.
Any two admissible skeleta that agree on ∂M are connected bya finite sequence of admissible BLT moves.The strategy for the proof is as follows: First we show that any admissibleskeleton can be refined to be dual to a triangulation. We then show that anytwo such skeleta are related by a sequence of Pachner moves that by Lemma 2.11can be obtained from admissible BLT moves. Since we do not need the technicaldetails of the proof of Theorem 2.12, which may be no surprise to the expert, wedefer it to Appendix A. There it is explained how to make the construction of[TV] compatible with admissibility in every step.17 (cid:29) B − , L (cid:29) L − , L (cid:29) L − , T (cid:29) T − ,Figure 2.3.: Examples of BLT moves with orientations (of a total of 32).18 .3. Bordisms with embedded ribbon graphs In this section we review the category Bord rib3 of 3-dimensional bordisms withembedded ribbon graphs. A variant of this category that includes labels from amodular fusion category C is reviewed in Section 3.2 below. For more details werefer to [Tu, Sect. IV].A ribbon punctured surface is a compact oriented 2-dimensional manifold to-gether with a finite set of marked points (or punctures ) which are labelled by tu-ples of the form ( v, ε ), where v is a non-zero tangent vector, and ε ∈ { +1 , − } . Adiffeomorphism ϕ : Σ −→ Σ (cid:48) of ribbon punctured surfaces Σ , Σ (cid:48) is an orientation-preserving diffeomorphism of the underlying surfaces, mapping punctures topunctures such that if a puncture p ∈ Σ is labelled by ( v, ε ), then ϕ ( p ) is la-belled by ( dϕ ( v ) , ε ). The orientation reversal − Σ of a punctured surface Σ isdefined to be the ribbon punctured surface Σ except with opposite orientation,and if a puncture of Σ is labelled by ( v, ε ) then the corresponding puncture of − Σ is labelled by ( v, − ε ).By a ribbon bordism we mean a compact oriented 3-dimensional bordism M together with an embedded ribbon graph R , such that the loose strands of R meet ∂M transversally. This induces the structure of a punctured surface on ∂M , whose punctures are in ∂M ∩ R . The punctures carry the labels ( v, ε ),where v is the framing of the corresponding strand of R and ε = +1 if the strandis directed out of ∂M , and − category of ribbon bordisms Bord rib3 is obtained by a standardconstruction in analogy to the regular bordism category. Morphisms in Bord rib3 are diffeomorphism classes of ribbon bordisms, but we usually will not make anotational distinction between these morphisms and their representatives. ω -moves Let M be a bordism, and let S be an admissible skeleton for M . We adopt thenomenclature of [TV, Ch. 14], to which we refer for a detailed discussion of thefollowing notions.A plexus (Latin for “braid”) is an (“abstract”, i. e. not embedded into R n )topological space which is made up of a finite number of oriented circles, orientedarcs, and coupons; arcs may meet only coupons, and only at pairwise distinctpoints at the coupon’s top or bottom. Circles and arcs are collectively called strands .A knotted plexus in S (originally defined in [TV, Sect. 14.1.2]) is a local em-bedding ι of a plexus d into S , such that(i) the coupons of d are embedded in S (2) with their orientations preserved;19ii) if ι ( d ) has multiple points, then they are transversal double points of strandsin S (2) and they are (labelled as) either over-crossings or under-crossings;(iii) ι ( d ) ∩ S (0) = ∅ ;(iv) ι ( d ) ∩ ∂S consists only of endpoints of arcs in d , and arcs meet ∂S transver-sally;(v) if a strand r of ι ( d ) meets a 1-stratum L of S at a point w , such that aneighbourhood of w is given by one of the four options , (2.12)then the intersection point w is called a positive switch . Without the aboverestriction on neighbourhoods, the intersection point is just called a switch .We usually refer to a knotted plexus ( d, ι ) simply by d . Note that we depictstrands in dark blue.The case of knotted plexi in admissible skeleta will be important for us: Definition 2.13.
Let M be a bordism. An admissible ribbon diagram in M isa pair ( S, d ), where S is an admissible skeleton of M , and d is a knotted plexusin S . An admissible ribbon diagram is positive if each of its switches is positive.Recall that our starting point is an (unstratified) bordism M , hence a smooth3-manifold, and it makes sense to talk about tangent vectors at all points of M .To express the relation between ribbon graphs and embedded plexi, we will needin addition the notion of transversality on strata. For this reason, we will makethe Assumption:
In ribbon diagrams (
S, d ) in M , all strata of the skele-ton S are smooth submanifolds of M . (The image ι ( d ) is not requiredto be a smooth submanifold.)We can now define: A framing f of a positive admissible ribbon diagram is afunction that continuously assigns a direction f ( x ) at ι ( x ) in M to each x ∈ d (hence double points ι ( x ) = ι ( y ) for x (cid:54) = y in d can have two different directions),such that (i) if x lies in a stratum t , then f ( x ) is transverse to t , and for 2-strata t ,the orientation of t followed by the direction f ( x ) agrees with the orientationof M ; (ii) if x lies in a coupon c , then f ( x ) is transverse to c ; (iii) if x ∈ ∂M ,then f ( x ) is tangent to ∂M . If an admissible ribbon diagram ( S, d ) is positive,there exists a framing for d , and any two framings that agree on ∂M are isotopicrelative to the boundary. 20 a) (b) Figure 2.4.: A ribbon graph in S (a) and an example of an admissible ribbondiagram representing it (b)From a positive admissible ribbon diagram ( S, d ) we obtain a ribbon graph d f in M as follows: Pick a framing f of d and slightly push the over-crossing strandsof d at crossings in the direction of the framing f , and then use f to provide theresulting graph with a ribbon structure. Two framings that agree on ∂M giveisotopic ribbon graphs.Conversely, we say that a positive admissible ribbon diagram ( S, d ) represents a ribbon graph R in M , if R is isotopic to d f . The set of positive admissibleribbon diagrams in M that represent R is denoted (cid:83) ( M, R ). In the case R = ∅ this reduces to the set of admissible skeleta (cid:83) ( M ) (cf. Definition 2.7). Remark 2.14.
The notion of “positive admissible ribbon diagram” is that of a“positive ribbon diagram” in the sense of [TV, Sect. 14.2], but with local neigh-bourhoods as in Figure 2.1 and such that the orientations of 2-strata can beextended to an admissible orientation in the sense of Section 2.2.1; if an admis-sible choice of orientations exists, then it is unique, cf. Remark 2.8 (ii).We stress that a “ribbon diagram” (whether it is admissible, positive, or plain)always relates to a prescribed skeleton. Hence a ribbon diagram is not just an(embedded) string diagram, even though the phrase might suggest otherwise.It is shown in [TV, Lem. 14.1] that every ribbon graph is representable by apositive ribbon diagram. The analogous result in our framework can be provensimilarly:
Lemma 2.15.
Every ribbon graph R in a bordism M can be represented by apositive admissible ribbon diagram, i. e. (cid:83) ( M, R ) (cid:54) = ∅ . Proof.
We sketch the proof of [TV, Lem. 14.1] and point out how to adapt it forour purposes along the way.Pick a tubular closed neighbourhood U of R in M . Pick a triangulation t of M \ U ◦ and a total order on the vertices of t , such that the induced admissibleorientation on the dual t ∗ satisfies the condition that all 2-strata in ∂U = U \ U ◦ are oriented by the normal pointing out of U . Next push R along its framing21nto ∂U , such that no coupon of the resulting knotted plexus d intersects a 0- ora 1-stratum of t ∗ ∩ ∂U , no strand of d meets a 0-stratum of t ∗ ∩ ∂U , and strandsmeet 1-strata of t ∗ ∩ ∂U only transversally.Pick enough open meridional discs D i of U such that their boundaries ∂D i do not intersect coupons and 0-strata in ∂U , and which intersect strands and1-strata in ∂U only transversally, such that the complement of (cid:83) i D i in U is adisjoint union of 3-balls. By declaring the boundaries ∂D i to be new 1-strataand orienting them arbitrarily, this lifts t ∗ to an admissible skeleton S of M (by adding the 2-strata D i as well as the 1- and 0-strata in ∂D i , where 0-strataare intersection points with 1-strata of t ∗ ∩ ∂U ). In doing so, we endow everynew disc-shaped 2-stratum D i in U ◦ with the orientation dictated by that of ∂D i and the orientations of the 2-strata in ∂U adjacent to the meridian. Sinceevery 2-stratum in ∂U is oriented by the normal pointing out of U , and since d ⊂ ∂U , every switch of the admissible ribbon diagram ( S, d ) is positive. Thus byconstruction, the positive admissible ribbon diagram (
S, d ) represents the ribbongraph R in M .By Theorem 2.12, any two admissible skeleta of a given bordism M , i. e. anytwo elements of (cid:83) ( M ), are related by a finite sequence of admissible BLT moves.Similarly, for a ribbon graph R in M , any two positive admissible ribbon diagramsrepresenting R , i. e. any two elements of (cid:83) ( M, R ), are related by a finite sequenceof local moves between positive ribbon diagrams, namely those of type BLT or oftype ω , . . . , ω as in Figure 2.5, and their inverses. In these pictures the knottedplexi have orientations that agree on both sides of every move, but orientationsare not depicted. Neither are the orientations of the strata of the admissibleskeleta in Figure 2.5 depicted. This means that there is one ω -move for eachchoice of (admissible) orientation; e. g. there are 2 · ω . We collectively refer to BLT moves as ω -moves , and we observe that the movesof types ω , ω , ω are framed Reidemeister moves. The moves ω , ω , . . . , ω ap-pear in [TV, Sect. 14.3] (where those corresponding to our moves ω and ω are denoted ω , , and ω , , , respectively) for the case of skeleta whose 0- and1-strata are unoriented. Moreover, any two such positive ribbon diagrams repre-senting the same ribbon graph in a given bordism are related by ω -moves [TV,Lem. 14.2 & Thm. 14.4]. The analogous result is proven similarly in our setting: Proposition 2.16.
Let R be a ribbon graph in a bordism M . Any two elementsin (cid:83) ( M, R ) that agree on ∂M are related by a finite sequence of moves of type ω , ω , . . . , ω . Proof.
Only the moves of type ω , which by definition are admissible BLT moves,change the underlying admissible skeleta, while not affecting the knotted plexi To see this, pick any orientation of the leftmost 2-stratum in the figure (two choices), bypositivity of the switches this fixes the orientations of the remaining two 2-strata on eachside; the orientation of the 1-stratum is then determined by admissibility; pick any orientationof the strand (two choices). −→ ω −→ ω −→ ω −→ ω −→ ω −→ ω −→ ω −→ ω −→ ω −−→ Figure 2.5.: ω -moves23f ribbon diagrams. Hence when restricting to ribbon diagrams which differ onlyaway from their knotted plexi, the statement follows from Theorem 2.12.Recall that in [TV], the notion of skeleton comes with orientations for 2-strata,while 0- and 1-strata do not carry orientations (contrary to our setting). Accord-ingly, the original variant of ω -moves in [TV, (14.1)], to which we refer here as ω TV -moves, is between positive ribbon diagrams without orientations for 0- and1-strata. In Sections 14.4–14.7 of loc. cit., it is shown that any two positive rib-bon diagrams representing ( M, R ) are related by ω TV -moves. Note that elementsof (cid:83) ( M, R ) are positive admissible ribbon diagrams. Moreover, “our” ω -moves inFigure 2.5 are ω TV -moves between positive ribbon diagrams which are endowedwith an admissible orientation for all strata.It follows that Proposition 2.16 holds if in the proofs of [TV], we can restrictto ω TV -moves which lift to ω -moves. This is indeed the case: whenever a new 2-stratum appears in the construction of [TV, Sect. 14.4–14.7] (i. e. when “attachinga bubble”, cf. Lemma 14.7 and Figure 14.13 of loc. cit.), there is a choice oforientation for this 2-stratum, and upon close inspection we notice that one ofthese choices is compatible with a (unique) choice of orientations for the new 0-and 1-strata which makes the entire positive ribbon graph admissibly oriented.
3. Defect TQFTs
In this section we first review the notions of 3-dimensional defect bordisms anddefect TQFTs from [CMS, CRS1], and that to every defect TQFT Z there is anaturally associated 3-category T Z . After a brief reminder on coloured ribbonbordisms and graph TQFTs, we then present a construction that produces newline defect labels from T Z , which can be thought of as a completion procedureon defect data. This will be important in Section 4.2, where we will construct acanonical ribbon category from orbifold data and completed defect data. A 3-dimensional defect TQFT is by definition a symmetric monoidal functor Z : Bord def3 ( D ) −→ Vect, where D are 3-dimensional defect data, and Bord def3 ( D )is the symmetric monoidal category of 3-dimensional defect bordisms decoratedwith defect data D . We start by recalling the relevant definitions. A list of D = ( D , D , D , s, t, f ) consists of [CMS,Def. 2.6](i) three sets D , D , D , 24ii) source and target maps s, t : D −→ D ,(iii) and a folding map f : D −→ [Sphere def1 ( D )].Here Sphere def1 ( D ) is the set of all defect circles : an element S ∈ Sphere def1 ( D ) isa stratified oriented circle S whose 1-strata are decorated with elements in D ,and whose 0-strata are decorated with pairs ( α, ± ), α ∈ D , subject to the con-dition that the 1-strata oriented away from (resp. towards) an ( α, +)-decorated0-stratum are decorated by t ( α ) (resp. s ( α )), while for an ( α, − )-decorated 0-stratum the decorations are swapped. The bracket around Sphere def1 ( D ) sig-nals the set of equivalence classes of such stratified decorated circles, where S and S (cid:48) are equivalent if they are related by a decoration-preserving iso-morphism of stratified manifolds. Thus the remaining information of a class[ S ] ∈ [Sphere def1 ( D )] is just the cyclic set of compatible decorations on the 0-strata, which is the point of view taken in [CMS, Def. 2.6]. We extend themap f to f : D × {±} −→ [Sphere def1 ( D )] by setting f (( x, +)) = f ( x ) and f (( x, − )) = f ( x ) rev for x ∈ D , where the reverse of a defect circle is the defectcircle with the orientation of all strata reversed.For 3-dimensional defect data D , there is a symmetric monoidal categoryBord def3 ( D ) of 3-dimensional decorated defect bordisms , see [CRS1, Def. 2.4]. Bydefinition, a morphism in Bord def3 ( D ) is a morphism in Bord def3 (cf. Section 2.1.3)together with a decoration by D : each j -stratum is labelled with an element of D j for j ∈ { , , } , such that the decoration is compatible with the maps s, t, f ,namely that the 3-strata adjacent to an α -labelled 2-stratum are labelled by s ( α )and t ( α ), and the labels of 2-strata adjacent to a 1-stratum L are read off of f ( L ).Similarly, objects Bord def3 ( D ) are objects of Bord def3 together with a label in D k +1 for each k -stratum, k ∈ { , , } , and the decorations induced at the boundaryof a morphism in Bord def3 ( D ) must match with the decorations of the source andtarget objects. Definition 3.1. A D is a symmetricmonoidal functor Z : Bord def3 ( D ) −→ Vect . (3.1)Examples of defect TQFTs can be obtained from anomaly-free modular fusioncategories C : As explained in [CRS2], the Reshetikhin–Turaev TQFT associatedto C lifts to a defect TQFT Z C : Bord def3 ( D C ) −→ Vect , (3.2)where D C consists of ∆-separable symmetric Frobenius algebras in C , and D C con-sists of certain multi-modules. (If C does have an anomaly, Reshetikhin–Turaevtheory is instead defined on an “extended” defect bordism category (cid:91) Bord def3 ( D C ),see [CRS2] for details.) 25 .1.2. The 3-category associated to a defect TQFT For each defect TQFT Z : Bord def3 ( D ) −→ Vect, there is an associated 3-category T Z , see [CMS, Sect. 3.3–3.4]. More precisely, T Z is a “Gray category with duals”,which is analogous to the fact that every 2-dimensional defect TQFT gives riseto a pivotal 2-category as explained in [DKR].We refer to [CMS] for the detailed construction of T Z as well as all relevantdefinitions. For our purposes here it suffices to recall that • objects of T Z are elements of D , pictured as labelling an (oriented yetotherwise structure-less) 3-cube; • D -labelled planes insidea 3-cube; • X that are cylinders over string diagrams of the pivotal pre-2-category freelygenerated by D , such that j -strata of X are labelled by elements of D j ; • Z assigns to defectspheres.For illustration, note that α = and X = (3.3)represent a 1-morphism α : u −→ v and a 2-morphism X with t ( X ) = α , respec-tively, and only parts of the decorations are shown. Note that contrary to generic( ∞ , T Z form vector spaces which donot carry any further homotopical information. The relation between unlabelled and labelled ribbon bordism categories isanalogous to the relation between the defect bordism categories Bord def3 and26ord def3 ( D ). Indeed, recall from Section 2.3 the unlabelled ribbon bordism cat-egory Bord rib3 , and let C be a k -linear ribbon category for a field k , i. e. a k -linear braided pivotal category whose left and right twists coincide, see e. g. [TV,Sect. 3.3]. Objects of the labelled ribbon bordism category Bord rib3 ( C ) are objectsΣ ∈ Bord rib3 together with a label X i ∈ C for every marked point p i of Σ. Mor-phisms in Bord rib3 ( C ) are morphisms ( M, R ) in Bord rib3 as in Section 2.3, where inaddition each strand and coupon of the ribbon graph R is (compatibly) labelledwith an object and morphism in C , respectively.We will consistently use calligraphic Roman letters for such C -coloured ribbongraphs R , and non-calligraphic letters for the underlying ribbon graphs R . Formore details we refer to [TV, Sect. 15.2.1], where Bord rib3 ( C ) is denoted Cob C . Definition 3.2. A graph TQFT over a k -linear ribbon category C is a symmetricmonoidal functor Bord rib3 ( C ) −→ Vect , (3.4)where Vect denotes the symmetric monoidal category k -vector spaces. Remark 3.3. (i) For an anomaly-free modular fusion category C , the Resheti-khin–Turaev construction [Tu] produces a graph TQFT Z RT , C : Bord rib3 ( C ) −→ Vect . (3.5)In [CMRSS2] we will apply the results of the present paper to combine (3.5)and the defect TQFT (3.2) to construct an “orbifold graph TQFT” (asintroduced in Section 4.3 below), including the case of anomalous C .(ii) Recall from [CRS1, Sect. 2.4] that the D -completion Z • : Bord def3 ( D • ) −→ Vect is a canonical extension for any defect Z : Bord def3 ( D ) −→ Vect to asymmetric monoidal functor on defect bordisms that may also have deco-rated 0-strata in addition to j -strata for j (cid:62)
1. Just as in the case without0-strata (Definition 3.1), we refer to Z • as a defect TQFT. Given a 3-dimensional defect TQFT Z , we would like to be able to use the k -morphisms of the associated 3-category T Z as (3 − k )-dimensional defects. Herewe will describe a completion of the defect data D which implements this for k = 2.The intuitive picture is as follows: a morphism g in T Z is a certain stratifiedcube; if a stratum in a bordism is decorated by g one would like to “replace thecorresponding stratum by the cube representing g ”. However, the local neigh-bourhoods around the strata in a bordism are modelled by spheres and their conesand cylinders. Hence to avoid making additional choices, in this section we first“complete” the defect data D of Z to new defect data (cid:98) D which have additional27ine defect labels modelled on defect discs; the line defect label X ∈ (cid:98) D in (1.4)is an example of such an additional label. In a second step we will then see howthe label set (cid:98) D indeed corresponds to certain 2-morphisms of T Z .The purpose of this section is to make precise the idea of “tensoring line defectlabels”; this is a prerequisite of the construction (in Section 4.2 below) of theribbon categories W A attached to Z .We start by defining decorated defect 2-manifolds (possibly with boundary) asstratified 2-manifolds with local neighbourhoods in N for points in the interior,whose 2-strata are decorated by D , 1-strata are decorated by D , and 0-strataby D , such that the local neighbourhoods are compatible with the decorationsas allowed by the maps s, t, f of D . We denote by Disc def ( D ) (resp. Sphere def2 ( D ))the D -decorated stratified 2-manifolds with underlying manifold being discs (resp.spheres). For example, we have ∈ Disc def ( D ) , ∈ Sphere def2 ( D ) , (3.6)where u i , v j ∈ D , α i , β j ∈ D , p i , q j ∈ D with adjacency of the strata as illus-trated. Note that closed decorated defect 2-manifolds are precisely the objects ofBord def3 ( D ).In particular we have a map C : D × {±} −→ [Disc def ( D )], mapping ( x, ε ) ∈ D × {±} to the equivalence class of the cone [ Cf (( x, ε ))] with the 0-stratumcorresponding to the cone point decorated by x , for example C ( x, +) = , (3.7)where α i ∈ D , u i ∈ D , and the adjacent strata of x ∈ D are as indicated. Definition 3.4.
Let D be a list of 3-dimensional defect data. The line defectcompletion of D is the list of 3-dimensional defect data (cid:98) D consisting of(i) the sets (cid:98) D = D and (cid:98) D = D with the maps s, t from D ,(ii) the set (cid:98) D = Disc def ( D ),(iii) the map (cid:98) f = ∂ : (cid:98) D −→ [Sphere def1 ( D )] which assigns to X ∈ (cid:98) D theisomorphism class represented by the boundary of X .Note that here we do not consider isomorphism classes for the elements in (cid:98) D .28 .3.1. The line defect completion of a defect TQFT For a given defect TQFT Z : Bord def3 ( D ) −→ Vect we will define a “line de-fect completed” defect TQFT (cid:98) Z : Bord def3 ( (cid:98) D ) −→ Vect by defining a symmetricmonoidal insertion functor
Ins : Bord def3 ( (cid:98) D ) −→ Bord def3 ( D ) . (3.8)First, by shrinking the local neighbourhoods in the definition of a defect manifold,we can assume that these specify for every object Σ ∈ Bord def3 ( (cid:98) D ) for each 0-stratum p ∈ Σ a closed neighbourhood N p and an isomorphism ϕ Σ p : N p −→ CS p ,where S p is the boundary of the specified local neighbourhood (i. e. of the specifiedelement in N , see (2.4)) at p , and CS p is its cone.Similarly, for a morphism M in Bord def3 ( (cid:98) D ) we can choose for each 1-stratum L of M with corresponding defect circle S L a tubular closed neighbourhood N L with a specified isomorphism ϕ L , which is either ϕ L : N L −→ CS L × [0 ,
1] or ϕ L : N L −→ CS L × S , (3.9)depending on whether L meets the boundary of M or not. In the first case theneighbourhood N L is required to restrict to the already chosen neighbourhood ofthe corresponding 0-stratum on the boundary.Now we define for an object Σ ∈ Bord def3 ( (cid:98) D ) the objectIns(Σ) = (cid:16) Σ \ (cid:91) p ∈ Σ N p (cid:17) ∪ ϕ p X p (3.10)of Bord def3 ( D ), where p runs over all 0-strata of Σ and X p ∈ (cid:98) D = Disc def ( D ) isthe decoration at the 0-stratum p . That is, we remove the neighbourhoods N p and glue in the discs X p instead. Different choices of neighbourhoods lead toisomorphic functors Ins, here we fix one such choice for each Σ.To define Ins on morphisms in the case where L with decoration X L meets theboundary of M , we setIns L ( M ) = (cid:0) M \ N L (cid:1) ∪ ϕ L (cid:0) X L × [0 , (cid:1) . (3.11)That is, we insert the cylinder over the defect disc X L in place of (a cylindricalneighbourhood of) L . In the other case, where L forms a circle in M , we setIns L ( M ) = ( M \ N L ) ∪ ϕ L ( X L × S ). Finally we defineIns( M ) = Ins L m . . . Ins L ( M ) , (3.12)where L , . . . , L m are all 1-strata of M . Clearly this is independent of the order of L , . . . , L m and defines for M : Σ −→ Σ (cid:48) a morphism Ins( M ) : Ins(Σ) −→ Ins(Σ (cid:48) )which does not depend on the choices of closed neighbourhoods N L . The functorIns is symmetric monoidal by construction, and we thus obtain:29 efinition 3.5. Let Z : Bord def3 ( D ) −→ Vect be a defect TQFT. The line defectcompletion of Z is the defect TQFT (cid:98) Z := Z ◦
Ins : Bord def3 ( (cid:98) D ) −→ Vect . (3.13)Recall from Section 3.1.2 that to any 3-dimensional defect TQFT Z there isan associated 3-category T Z . Proposition 3.6.
We have an equivalence of Gray categories with duals: T Z ∼ = T (cid:98) Z . (3.14) Proof.
To see this, we apply the insertion functor to the cubes that correspondto morphisms in T (cid:98) Z . More precisely, consider the functor Ins : T (cid:98) Z −→ T Z definedas follows. It is the identity on objects and 1-morphisms.To define Ins on 2-morphisms, first pick for each Y ∈ (cid:98) D a square around Y and extend Y to a progressive diagram prog( Y ) in the square. For a 2-morphism X in T (cid:98) Z define Ins( X ) by first picking a cube that rep-resents X , then insert for each 1-stratum with decoration Y the correspondingprogressive diagram prog( Y ), where we pick the local neighbourhoods of the 1-strata small enough to ensure that their projections to the x -axis do not overlap(here we use the conventions of [CMS, Sect. 3.1.2]). After passing again to equiv-alence classes we obtain a well-defined 2-morphism of T Z .For the 3-morphisms we use that by [CMS, Sect. 3.3], the 3-morphisms in T (cid:98) Z and in T Z are obtained by applying (cid:98) Z and Z , respectively, to defectspheres. By definition, the corresponding defect spheres for Hom T (cid:98) Z ( X, X (cid:48) ) andHom T Z (Ins( X ) , Ins( X (cid:48) )) match and we can identify the 3-morphisms.To see that Ins is an equivalence of Gray categories, it suffices to show that it isessentially surjective on 2-morphisms. This is the case since each 2-morphism X of T Z gives a 2-morphism ι ( X ) of T (cid:98) Z , using the obvious inclusion D −→ (cid:98) D ,which lifts to a functor ι : Bord def3 ( D ) −→ Bord def3 ( (cid:98) D ). Thus Ins( ι ( X )) = X ,showing that Ins is an equivalence. Moreover, Ins is obviously compatible withthe duals. By the general construction of the 3-category T Z for a defect TQFT Z , for all u, v ∈ D there is a full sub-2-category T Z ( u, v ) of T Z ( u, v ) whose objects form theset { α ∈ D | s ( α ) = u, t ( α ) = v } . Thus, the 1-morphisms of T Z ( u, v ) correspondalmost to elements of (cid:98) D : the difference is that the elements of (cid:98) D are neither3-cubes (but defect 2-discs) nor isomorphism classes. These are minor differences,however we want to use the elements of (cid:98) D directly as morphisms of a 2-category In case there is a horizontal 1-stratum in Y , first pass to a choice of isomorphic progressivedefect disc. T Z to define 2-categories W ( u, v ) whose 1-morphisms are precisely elements of (cid:98) D .Then we show that W ( u, v ) is equivalent to T Z ( u, v ).We start with two operations for (cid:98) D . For X ∈ (cid:98) D , we denote by X ∗ ∈ (cid:98) D thedecorated 2-disc which is obtained by reversing the orientations of all strata of X .Second, for X, Y ∈ (cid:98) D with matching boundary in the sense that ∂X = ∂ ( Y ∗ ),we can consider X ◦ Y ∈ Sphere def2 ( D ), which by definition is the defect 2-spherethat is obtained from gluing the 2-disc Y on top of the 2-disc X along theircommon boundary.For fixed elements u, v ∈ D , consider α, β ∈ D with s ( α ) = t ( β ) = u and t ( α ) = s ( β ) = v , and set S α,β := ∈ Sphere def1 ( D ) . (3.15)A choice of S α,β defines the lower arrow in the following pullback diagram of sets: D ( α, β ) (cid:98) D ∗ Sphere def1 ( D ) (cid:98) fS α,β (3.16)Thus D ( α, β ) consists of the elements of (cid:98) D with specified boundary. We call α the source of an element of D ( α, β ) and β the target. In particular we can considerfor X, Y ∈ D ( α, β ) the defect 2-sphere X ∗ ◦ Y . Lemma 3.7.
Let Z : Bord def3 ( D ) −→ Vect be a defect TQFT. For all u, v ∈ D there is an associated linear pivotal 2-category W ( u, v ) such that(i) the objects of W ( u, v ) form the set { α ∈ D | s ( α ) = u, t ( α ) = v } ,(ii) the set of 1-morphisms of W ( u, v ) from α to β is D ( α, β ) as in (3.16),(iii) for X, Y ∈ D ( α, β ), the set of 2-morphisms isHom W ( u,v ) ( X, Y ) = Z ( X ∗ ◦ Y ) . (3.17) Proof.
The proof is essentially contained in the proof of [CMS, Thm. 3.13], albeitin a slightly different setting. We will need some details on the constructionof W ( u, v ) later, thus we recall the main ingredients. All compositions of 2-morphisms in W ( u, v ) are canonically obtained from Z by evaluating Z on defect31-balls with 3-balls in the interior removed: For 1-morphisms X, Y, Z ∈ D ( α, β ),the vertical composition of 2-morphisms is a linear mapHom W ( u,v ) ( Y, Z ) ⊗ k Hom W ( u,v ) ( X, Y ) −→ Hom W ( u,v ) ( X, Z ) , (3.18)which is given as Z ( B ( X, Y, Z )) with the bordism B ( X, Y, Z ) : ( Y ∗ ◦ Z ) (cid:116) ( X ∗ ◦ Y ) −→ X ∗ ◦ Z (3.19)in Bord def3 ( D ) defined as follows. The decorated 1-sphere S α,β from (3.15) givesthe cylinder CS α,β × [0 ,
1] over the cone with cone point 0. Remove the solidcylinder B / (0) × [0 ,
1] from the interior, then glue ( X × [0 , ]) ∪ ( Y × [ , ]) ∪ ( Z × [ , S α,β × [0 ,
1] as well as theinner boundaries S α,β × [ , ] and S α,β × [ , ] to obtain the ball B ( X, Y, Z )with two inner balls removed. As a result, the parallel 1-morphisms and their2-morphisms form categories D ( α, β ) (with Ob( D ( α, β )) = D ( α, β )) with unitsgiven by the value of Z on solid balls, see [CMS, Sect. 3.3].The horizontal composition consists of linear functors ⊗ : D ( β, γ ) × D ( α, β ) −→ D ( α, γ ) as follows: For X ∈ D ( β, γ ) and Y ∈ D ( α, β ), the object X ⊗ Y ∈ D ( α, γ )is defined to be the 2-disc which is obtained by placing the rescaled discs X and Y next to each other in the disc CS α,γ .To define ⊗ on morphisms we again use a 3-ball with two inner 3-balls removedthat is defined similarly to the case of the vertical composition and apply thefunctor Z , see [CMS, Sect. 3.3].Since Z is invariant under isotopies of bordisms, all axioms of a 2-categoryfollow directly. The duals in W ( u, v ) are obtained as in [CMS, Sect. 3.4], i. e. thedual of a 1-morphism X is X ∗ .The equivalence Ins : T (cid:98) Z −→ T Z of Gray categories with duals from Proposi-tion 3.6 restricts to an equivalence W ( u, v ) −→ T Z ( u, v ): Lemma 3.8.
For all u, v ∈ D , we have an equivalence of pivotal 2-categories: W ( u, v ) ∼ = T Z ( u, v ) . (3.20)
4. Orbifold graph TQFTs
In this section we review orbifold TQFTs Z A of 3-dimensional defect TQFTs Z asintroduced in [CRS1], and we formulate their construction in terms of decoratedskeleta (Section 4.1). To any special orbifold datum A for Z , we construct anassociated ribbon category W A (Section 4.2), generalising the construction of[MR1] for Reshetikhin–Turaev theories to arbitrary defect TQFTs. Finally, welift Z A to an orbifold graph TQFT (cid:98) Z Γ A , which acts on bordisms with embedded W A -coloured ribbon graphs (Section 4.3). A variant of this result which is usefulfor applications is described in Appendix B.32 .1. Orbifold TQFTs We start by recalling the notion of 3-dimensional orbifold TQFTs from [CRS1,Sect. 3.4] and then explain how this construction can be generalised and compu-tationally simplified in terms of admissible skeleta in the case of special orbifolddata.
Let Z : Bord def3 ( D ) −→ Vect be a defect TQFT as reviewed in Section 3.1, withdefect data D = ( D , D , D , s, t, f ). Definition 4.1. A special orbifold datum A for Z consists of • an element A ∈ D , • an element A ∈ D with s ( A ) = A = t ( A ), • an element A ∈ D with f ( A ) = ( A , +) × ( A , +) × ( A , − ), • elements A +0 ∈ Z ( S A ) and A − ∈ Z (( S A ) rev ),where S A = , ( S A ) rev = (4.1)are A -decorated defect spheres, and in particular objects in Bord def3 ( D ). Thetuple A = ( A , A , A , A ± ) satisfies the identities depicted in Figure 4.1, whereit is understood that Z is applied to the defect balls displayed on either side ofthe equal sign.In drawing the defect balls in Figure 4.1 we used that A -decorated defectbordisms locally look as follows (where all 2-strata are oriented counterclockwisein the paper/screen frame, and an A ε -decorated 0-stratum has orientation ε ): , , , . (4.2)33ote that A ± -decorated 0-strata are interpreted as small defect 3-balls aroundthem removed, with the resulting linear map after evaluation with Z applied to(tensor products of the vectors) A ± ; this is made precise in terms of the D -completion mentioned in Remark 3.3 (ii). Remark 4.2.
Note that the underlying stratified 3-balls of the defect bordismsdepicted in (4.2) are, when read from left to right, Poincar´e dual to a 1-simplex,a 2-simplex, and two 3-simplices (all oriented), cf. (2.7) and (2.8). Accordingly,general “orbifold data” for Z are defined in [CRS1] as above, but with the definingconditions in Figure 4.1 replaced by (the larger number of) conditions arisingfrom the Poincar´e duals of oriented versions of the 2-3 and 1-4 Pachner moves(relating triangulations of 3-dimensional bordisms). In fact, orbifold data for any n -dimensional defect TQFT are defined for arbitrary n ∈ Z + in [CRS1, Def. 3.5]in terms of invariance under n -dimensional oriented Pachner moves.Special orbifold data are special cases of 3-dimensional orbifold data. Fromhere on we will only consider special orbifold data. We now move to consider admissible skeleta that are decorated with special orb-ifold data. We will see that special orbifold data algebraically encode invarianceunder admissible BLT moves.
Definition 4.3.
Let Z be a defect TQFT, and let A = ( A , A , A , A ± ) be aspecial orbifold datum for Z . An A -decorated skeleton S of a bordism M inBord is an admissible skeleton S of M together with a decoration as follows: • each 3-stratum of S is decorated by A , • each 2-stratum of S is decorated by A , • each 1-stratum of S is decorated by A , • for ε ∈ { + , −} , each ε -oriented 0-stratum of S is decorated by A ε .One similarly arrives at the notion of A -decorated BLT moves : these are localchanges of A -decorated skeleta (and hence of A -decorated defect bordisms) whoseeffect on the underlying admissible skeleta are precisely the admissible BLT movesof Section 2.2.2. The evaluation with a TQFT is invariant under these moves,and special orbifold data precisely encode this invariance: Proposition 4.4.
Let Z : Bord def3 ( D ) −→ Vect be a defect TQFT, and let A =( A , A , A , A ± ) be a list of defect labels for Z that can decorate admissibleskeleta of all bordisms in Bord to obtain morphisms in Bord def3 ( D ). Then A isa special orbifold datum for Z iff Z applied to A -decorated BLT moves givesidentities in Vect. 34 (O1)= (O2) = (O3)= (O4) = (O5)= (O6) = (O7)= = = (O8)Figure 4.1.: Defining conditions on special orbifold data A for a defect TQFT Z ;only the A j -labels for j -strata of lowest dimension are shown. Eachpicture represents a defect ball viewed as a bordism from ∅ to theboundary, and the identities hold only after application of Z .35 roof. In Lemma 2.11 we showed the equivalence of undecorated BLT movesand undecorated special orbifold data moves. From this the equivalence of thecorresponding decorated moves between decorated skeleta follows immediately.
Let Z : Bord def3 ( D ) −→ Vect be a defect TQFT, and let A be a special orbifolddatum for Z . Given an A -decorated skeleton S of a bordism M : Σ −→ Σ (cid:48) (4.3)in Bord , we obtain a new defect bordism F ( S ), which we call a foamification of M represented by S . Viewed as a morphism in Bord def3 ( D ), the defect bordism F ( S ) has source and target objects which are D -decorated defect surfaces whose1- and 0-strata are labelled by A and A , respectively (corresponding to 2- and1-strata of M which end on ∂M ). We denote these source and target objects F (Σ , G ) and F (Σ (cid:48) , G (cid:48) ), respectively, where G , G (cid:48) are the decorated skeleta of Σ , Σ (cid:48) induced from S . Thus F ( S ) : F (Σ , G ) −→ F (Σ (cid:48) , G (cid:48) ) (4.4)in Bord def3 ( D ).By definition, the evaluation of Z on an A -decorated skeleton S is given by Z ( F ( S )). In particular, Z can be evaluated on defect 3-balls around each sideof the identities in Figure 4.1, and doing so gives identities in Vect. This in turnimplies that if ∂M = ∅ , we have Z ( F ( S )) = Z ( F ( S (cid:48) )) for any other A -decoratedskeleton S (cid:48) of M . Hence setting Z A ( M ) := Z ( F ( S )) if ∂M = ∅ (4.5)does not depend on the choice of A -decorated skeleton S of M , thanks to Propo-sition 4.4 and Theorem 2.12.To prepare for the definition of Z A on bordisms with nonempty boundary, let Σbe an object in Bord . Any choice of A -decorated skeleton S of the cylinder C Σ := Σ × [0 ,
1] gives rise to a linear mapΦ G (cid:48) G := Z (cid:0) F ( S ) (cid:1) : Z (cid:0) F (Σ , G ) (cid:1) −→ Z (cid:0) F (Σ , G (cid:48) ) (cid:1) , (4.6)where G , G (cid:48) are the decorated skeleta of Σ induced by S as in (4.4). By definitionof special orbifold data, the linear maps Φ G (cid:48) G do not depend on the choice of A -decorated skeleta S in the interior of C Σ , and for arbitrary decorated admissibleskeleta G , G (cid:48) , G (cid:48)(cid:48) of Σ we have Φ G (cid:48)(cid:48) G (cid:48) ◦ Φ G (cid:48) G = Φ G (cid:48)(cid:48) G . (4.7)In particular, the maps Φ GG are idempotents.36 onstruction 4.5. Let A be a special orbifold datum for a defect TQFT Z .The A -orbifold TQFT Z A : Bord −→ Vect (4.8)is defined as follows:(i) For an object Σ ∈ Bord , we set Z A (Σ) = colim (cid:8) Φ G (cid:48) G (cid:9) , (4.9)where G , G (cid:48) range over all admissible A -decorated skeleta of Σ.(ii) For a morphism M : Σ −→ Σ (cid:48) in Bord , we set Z A ( M ) to be Z A (Σ) Z (cid:0) F (Σ , G ) (cid:1) Z (cid:0) F (Σ (cid:48) , G (cid:48) ) (cid:1) Z A (Σ (cid:48) ) , Z ( F ( S )) (4.10)where S is an arbitrary A -decorated skeleton representing M , the first mapis obtained from the universal property of the colimit, and the last map ispart of the data of the colimit.It is straightforward to verify from Proposition 4.4 and Theorem 2.12 thatthe definition of Z A ( M ) in (4.10) does not depend on the choice of admissible A -decorated skeleton S . Moreover, by construction the state spaces of Z A areisomorphic to the images of the idempotents, Z A (Σ) ∼ = Im (cid:0) Φ GG (cid:1) . (4.11) Theorem 4.6.
Let A be a special orbifold datum for a defect TQFT Z . Then Z A : Bord −→ Vect as in Construction 4.5 is a symmetric monoidal functor.
Proof.
In light of the discussion in Section 4.1.1, the proofs of Thm. 3.10 andProp. 3.18 in [CRS1] generalise to the case of A -decorated skeleta whose under-lying stratifications are not Poincar´e duals of triangulations. Remark 4.7.
It is typically easier to evaluate Z A in terms of A -decorated skeletawhich are not Poincar´e duals of triangulations. For example, instead of computingthe invariant Z A ( S ) from a triangulation of the 3-sphere (which involves at leastfive tetrahedra), one can choose the A -decorated skeleton S consisting only of anembedded A -decorated 2-sphere that divides S into two A -decorated 3-balls.This skeleton has no 1-strata and no 0-strata. Let Z : Bord def3 ( D ) −→ Vect be a defect TQFT, and let A be a special orbifolddatum for the completed TQFT (cid:98) Z of Definition 3.5. In this section we describe aribbon category W A that is naturally associated to Z and A . As will be explained37n more detail in [CMRSS2], for a TQFT Z of Reshetikhin–Turaev type associatedto a modular fusion category C , our W A is equivalent to the category of Wilsonlines C A introduced in [MR1].Recall the 3-category T Z (reviewed in Section 3.1.2), the 2-categories W ( u, v )associated to a pair of labels u, v ∈ D in Section 3.3, and the line-completedTQFT (cid:98) Z (Definition 3.5). We write W := End W ( A , A ) ( A ) (4.12)for the monoidal category of endomorphisms of A . Definition 4.8.
The category W A is defined as follows: • Objects of W A are tuples X = ( X, τ X , τ X ), with X ∈ W , and τ X ∈ (cid:98) Z , τ X ∈ (cid:98) Z , (4.13)are vectors, referred to as crossings , which correspond to 3-isomorphisms , (4.14)in the 3-category T Z . Their inverses are denoted , , (4.15)and the crossings have to satisfy the identities in Figure 4.2 when (cid:98) Z isapplied to both respective sides, each viewed as a defect 3-ball. (That τ Xi and τ Xi are each others’ inverse is expressed in (T4) and (T5).)38 A morphism
X −→ X (cid:48) in W A is a morphism f : X −→ X (cid:48) in W such that (cid:98) Z = (cid:98) Z , (4.16) (cid:98) Z = (cid:98) Z . (4.17) • Composition and identities in W A are as in W .We endow W A with a rigid monoidal structure as follows: • The tensor product of X with Y in W A is (cid:0) X, τ X , τ X (cid:1) ⊗ A (cid:0) Y, τ Y , τ Y (cid:1) = (cid:0) X ⊗ Y, τ
X,Y , τ X,Y (cid:1) , (4.18)where ⊗ on the right-hand side denotes the tensor product of W , and thecrossings are τ X,Y = , τ X,Y = . (4.19) • The unit object of W A is W A = ( , τ , τ ), where is the unit objectof W , and τ , τ are obtained from the unitors of A in T Z . • Associators and unitors in W A are as in W . • The dual of an object (
X, τ X , τ X ) in W A is ( X ∗ , τ X ∗ , τ X ∗ ), where X ∗ ∈ W is the dual of X ∈ W , and τ X ∗ = , τ X ∗ = , (4.20)39 (T1) = (T2)= (T3)= , = (T4)= , = (T5)= , = (T6)= , = (T7)Figure 4.2.: Defining conditions for objects in W A (after application of (cid:98) Z , i. e. inthe 3-category T (cid:98) Z ∼ = T Z ). 40hile the adjunction morphisms in W A are those of W .In the following we will sometimes omit the orientations of 1-strata when theyare clear from the context. Proposition 4.9.
The rigid monoidal category W A is pivotal, and together withthe braiding morphisms c X , Y = (cid:98) Z , c − X , Y = (cid:98) Z , (4.21)it becomes a ribbon category.Before giving the proof, let us recall that, as usual in a ribbon category, theribbon twist of X ∈ W A and its inverse can be defined in terms of dualities andthe braiding as follows: θ X := (cid:98) Z = (cid:98) Z , (4.22) θ − X := (cid:98) Z = (cid:98) Z . (4.23) Proof of Proposition 4.9.
The argument that W A is pivotal and braided is as in[MR1, Sect. 3.2] (with the parameters ψ i , ω i of loc. cit. set to 1, and with all string41iagrams replaced by the corresponding 3-dimensional diagrams, to which (cid:98) Z isapplied). However, the proof of the ribbon property in [MR1] used a shortcutthat relies on semisimplicity (see Remark 3.7 (ii) there), so here we need to addan extra calculcation.First note that as in [MR1, Lem. 3.4], for every X , Y ∈ W A we have= , = (4.24)To show that a pivotal braided category is ribbon, one needs to check that theleft and right twists agree, i. e. that the equality in (4.22) holds. One has:= = , (4.25)where in the first equality we used (O8), (T5) and (4.17) to create a bubble andmove the coupon on it, and in the second equality we used (T4) to move the X -strand onto the 2-stratum to the back (note that both this 2-stratum andthe strand lying in it have opposite to paper/screen plane orientation, hence thestripy pattern). Next, using the auxiliary identities (4.24) together with (T5)and (T6) one gets: = == = , (4.26)42hich upon substituting back to (4.25) yields the desired result. Remark 4.10. (i) The notion of a special orbifold datum can be formulatedinternal to an arbitrary Gray category with duals T , see [CRS1, Sect. 4.2].In the case that T = T Z from Section 4.1.1, this reproduces Definition 4.1.Our construction of a ribbon category W A generalises to any special orbifolddatum A in a Gray category with duals T , by interpreting all the above3-dimensional diagrams as Gray diagrams (see [BMS, CMS]) of T .(ii) It is illustrative to consider the following simplified version (ignoring theduals) of the categorical construction in (i): Let T be the delooping of thesymmetric monoidal 2-category Cat with the cartesian product as monoidalstructure. That is, T has only one object, the 1-morphisms are categories, 2-and 3-morphisms are functors and natural transformations. Consider data A = ( A , A , A , A ± ) as in Section 4.1.1, subject to the axioms (O1)–(O3).This corresponds to a non-unital monoidal category A . Now the analogueof W A has as objects tuples X = ( X, τ X , τ X ) as in Definition 4.8, subjectto the axioms (T1)–(T5). Writing ⊗ for the monoidal product of A , anobject F of W A is thus a functor F : A −→ A with coherent isomorphisms F ( a ⊗ b ) ∼ = a ⊗ F ( b ) ∼ = F ( a ) ⊗ b , i. e. W A ∼ = Fun A , A ( A , A ) is the category ofbimodule endofunctors of A . This is automatically unital, and in the casethat A has a unit object, it coincides with the Drinfeld centre of A . In thecase of spherical fusion categories, the full ribbon equivalence is proved by(tedious) direct computation in [MR1, Thm. 4.2]. Let Z be a defect TQFT, let A be a special orbifold datum for the completedTQFT (cid:98) Z of Section 3.3, and recall the associated ribbon category W A of Sec-tion 4.2. In this section we extend the orbifold TQFT (cid:98) Z A : Bord −→ Vect ofConstruction 4.5 to a graph TQFT (cid:98) Z Γ A : Bord rib3 ( W A ) −→ Vect (4.27)on bordisms with embedded W A -labelled ribbon graphs. To this end we useribbon diagrams as in Section 2.4 to represent (uncoloured) ribbon graphs andthen decorate them using the data from W A . To show that the constructionof (cid:98) Z Γ A is independent of the choice of such representations, we prove invarianceunder ω -moves of ribbon diagrams. Let M be a bordism and R an embedded ribbon graph in M . In Section 2.4 theset (cid:83) ( M, R ) of positive admissible ribbon diagrams in M that represent R was43ntroduced. By design, elements of (cid:83) ( M, R ) can be decorated using an orbifolddatum A and the ribbon category W A . This is formalised in Definition 4.11below, which can be viewed as a generalisation of Definition 4.3 to non-trivialribbon graphs.As in Section 3.1, for a ribbon category C we denote C -coloured ribbon graphsby calligraphic letters like R (whose underlying uncoloured ribbon graphs arethen denoted R ). Accordingly, we write (cid:83) ( M, R ) for the set of C -coloured positiveadmissible ribbon diagrams representing a C -coloured ribbon graph R in M . Anelement ( S, (cid:100) ) ∈ (cid:83) ( M, R ) consists of an admissible skeleton S and a C -colouredknotted plexus (cid:100) , whose underlying uncoloured knotted plexus d obtains its C -colouring from R . It follows that (cid:83) ( M, ∅ ) = (cid:83) ( M ).In the present setting, the ribbon category C used to colour R is given by W A . Definition 4.11.
Let Z : Bord def3 ( D ) −→ Vect be a defect TQFT, and let A be a special orbifold datum for (cid:98) Z . An A -decorated ribbon diagram ( S , (cid:100) ) of amorphism ( M, R ) in Bord rib3 ( W A ) is an element ( S, (cid:100) ) ∈ (cid:83) ( M, R ) together witha decoration as follows:(i) S is an A -decorated skeleton of M with underlying skeleton S ;(ii) if a switch of ( S , (cid:100) ) involves an X -labelled ribbon of R traversing an A -labelled 1-stratum of S , then the switch is labelled by τ X , τ X , τ X or τ X asdictated by orientations, cf. (4.14), (4.15);(iii) over- and under-crossings in (cid:100) are replaced by coupons labelled with thecorresponding braiding morphisms in W A . Each braiding coupon is orientedsuch that on its source side the two ribbons involved in the crossing arepointing towards the coupon, and on its target side they are pointing awayform the coupon.The set of A -decorated ribbon diagrams of the pair ( M, R ) is denoted (cid:83) A ( M, R ).Given an A -decorated ribbon diagram ( S , (cid:100) ) of a morphism( M, R ) : Σ −→ Σ (cid:48) (4.28)in Bord rib3 ( W A ), we obtain a new defect bordism F ( S , (cid:100) ), which, in accordancewith Section 4.1.3, we call a foamification of ( M, R ) represented by ( S , (cid:100) ). Notethat F ( S , ∅ ) = F ( S ).Viewed as a morphism in Bord def3 ( (cid:98) D ), the defect bordism F ( S , (cid:100) ) has sourceand target objects which are (cid:98) D -decorated defect surfaces whose 1-strata are la-belled by A and whose 0-strata precisely correspond to the endpoints of R andof A -lines on ∂M . We denote these source and target objects by F (Σ , G ) and F (Σ (cid:48) , G (cid:48) ), respectively, where G , G (cid:48) are the decorated skeleta of Σ , Σ (cid:48) induced from( S , (cid:100) ). Thus F ( S , (cid:100) ) : F (Σ , G ) −→ F (Σ (cid:48) , G (cid:48) ) (4.29)44n Bord def3 ( (cid:98) D ).By definition, the evaluation of (cid:98) Z on an A -decorated ribbon diagram ( S , (cid:100) ) isgiven by (cid:98) Z ( F ( S , (cid:100) )). In particular, (cid:98) Z can be evaluated on defect 3-balls aroundeach side of A -decorated versions of the ω -moves in Figure 2.5. Proposition 4.12.
Let A be a special orbifold datum for a completed defectTQFT (cid:98) Z . Applying (cid:98) Z to A -decorated ω -moves gives identities in Vect. Proof.
Invariance under ω -moves follows from Proposition 4.4. Invariance undermoves of type ω , ω , ω follows from the fact that they are framed Reidemeistermoves which hold in every ribbon category.Invariance under the remaining ω -moves follows from the defining properties ofthe category W A and the results of [MR1] which directly carry over to our moregeneral setting: for one choice of admissible orientations of 2-strata, invarianceunder ω follows from (T5) together with the identity [MR1, (T12 (cid:48) )]; for ω , use[MR1, (T13 (cid:48) )]; for ω and ω , use (4.16) and (4.17); for ω , use [MR1, Lem. 3.4];for ω , use (T4). Showing invariance under the ω -move, which we reformulateas the identity = , (4.30)is more involved, and we give more details.Let us recall the identities (4.24) which were used in the proof of Proposition4.9. Taking X = Y and closing the left-most strands to a loop, we obtain= , = , (4.31)where we used the representations (4.22), (4.23) of the ribbon twist and its inverse.Hence the left-hand side of (4.30) is= = = , (4.32)45here in the first step we deformed the X -labelled line, in the second step we used(T4)–(T7) and a consequence of (4.17), and in the last step the second identityof (4.31).Together with Proposition 2.16, the above result implies: Corollary 4.13.
Let ( S , (cid:100) ) and ( S (cid:48) , (cid:100) (cid:48) ) be A -decorated representations of a W A -coloured ribbon graph R in a bordism M . If S , S (cid:48) respectively (cid:100) , (cid:100) (cid:48) agreeon the boundary, i. e. if S| ∂M = S (cid:48) | ∂M and (cid:100) | ∂M = (cid:100) (cid:48) | ∂M , then (cid:98) Z (cid:0) F ( S , (cid:100) ) (cid:1) = (cid:98) Z (cid:0) F ( S (cid:48) , (cid:100) (cid:48) ) (cid:1) . (4.33) We finally define the orbifold graph TQFT (cid:98) Z Γ A : Bord rib3 ( W A ) −→ Vect. Accord-ing to Corollary 4.13, the linear map (cid:98) Z ( F ( S , (cid:100) )) depends only on the choiceof decorated skeleton of the boundary of the bordism M with embedded ribbongraph R represented by ( S , (cid:100) ).The dependence on the boundary decomposition is removed analogously tothe construction of the orbifold TQFT Z A in Section 4.1.3. Namely, let Σ be anobject in Bord rib3 ( W A ). Each puncture p i of Σ comes with a label ( X i , v i , ε i ) asin Section 3.1, where X i ∈ W A . We view the cylinder Σ × [0 ,
1] as a bordism C Σ with embedded W A -coloured ribbon graph R Σ that happens to consist onlyof straight ribbons labelled by the objects X i . Any choice of A -decorated ribbondiagram ( S , (cid:100) ) of ( C Σ , R Σ ) gives rise to a linear mapΨ G (cid:48) G := (cid:98) Z (cid:0) F ( S , (cid:100) ) (cid:1) : (cid:98) Z (cid:0) F (Σ , G ) (cid:1) −→ (cid:98) Z (cid:0) F (Σ , G (cid:48) ) (cid:1) , (4.34)where G , G (cid:48) are the decorated skeleta of Σ induced by ( S , (cid:100) ) as in (4.29).By Corollary 4.13 the linear maps Ψ G (cid:48) G do not depend on the choice of A -decorated ribbon diagram ( S , (cid:100) ) in the interior of C Σ , and for arbitrary decoratedadmissible skeleta G , G (cid:48) , G (cid:48)(cid:48) of Σ we haveΨ G (cid:48)(cid:48) G (cid:48) ◦ Ψ G (cid:48) G = Ψ G (cid:48)(cid:48) G . (4.35)In particular, each map Ψ GG is an idempotent. Construction 4.14.
Let A be a special orbifold datum for a completed defectTQFT (cid:98) Z . The orbifold graph TQFT (cid:98) Z Γ A : Bord rib3 ( W A ) −→ Vect (4.36)is defined as follows:(i) For an object Σ ∈ Bord rib3 ( W A ), we set (cid:98) Z Γ A (Σ) = colim (cid:8) Ψ G (cid:48) G (cid:9) , (4.37)where G , G (cid:48) range over all admissible A -decorated skeleta of Σ.46ii) For a morphism ( M, R ) : Σ −→ Σ (cid:48) in Bord rib3 ( W A ), we set (cid:98) Z Γ A ( M, R ) to be (cid:98) Z Γ A (Σ) (cid:98) Z (cid:0) F (Σ , G ) (cid:1) (cid:98) Z (cid:0) F (Σ (cid:48) , G (cid:48) ) (cid:1) (cid:98) Z Γ A (Σ (cid:48) ) , (cid:98) Z ( F ( S , (cid:100) )) (4.38)where ( S , (cid:100) ) is an arbitrary A -decorated ribbon diagram representing( M, R ).By Corollary 4.13 the definition of (cid:98) Z Γ A ( M, R ) in (4.38) does not depend on thechoice of admissible A -decorated skeleton ( S , (cid:100) ). Moreover, by construction thestate spaces of (cid:98) Z Γ A are isomorphic to the images of the idempotents, (cid:98) Z Γ A (Σ) ∼ = Im (cid:0) Ψ GG (cid:1) . (4.39)We have thus shown our main result, which is that (cid:98) Z Γ A is indeed a graph TQFT: Theorem 4.15.
Let A be a special orbifold datum for a completed defectTQFT (cid:98) Z . Then (cid:98) Z Γ A : Bord rib3 ( W A ) −→ Vect as in Construction 4.14 is a sym-metric monoidal functor.There are few examples of special orbifold data for a generic 3-dimensionaldefect TQFT Z . However, if one passes to the so-called “Euler completion” Z (cid:12) ,one finds far more examples, cf. [CRS2, CRS3, MR2]. In Appendix B we spellout the details of special orbifold data A for Z (cid:12) as well as A -decorated skeletaand ribbon diagrams, the ribbon category W A , and the associated orbifold graphTQFT, all in terms of the non-completed TQFT Z . Conceptually, Appendix Boffers nothing new, but the details are relevant for applications, in particular forthe treatment in [CMRSS2]. 47 . Proof of Theorem 2.12 Here we prove that if two admissible skeleta of a 3-bordism agree on the boundary,then they are related by admissible BLT moves.
A.1. Skeleta of 2-manifolds
We use the notation introduced in Section 2.2 to define skeleta for closed 2-manifolds, in analogy to the 3-dimensional case.
Definition A.1.
Let Σ be a closed 2-dimensional manifold. A skeleton S for Σis a stratification such that the following additional requirements are satisfied:(i) Every 2-stratum is an open disc.(ii) Each x ∈ Σ has a neighbourhood that is isomorphic to one of the followingopen stratified discs B x :1) If x ∈ S (2) , then B x contains no 1- or 0-strata: . (A.1)2) If x ∈ S (1) , then B x is given by . (A.2)3) If x ∈ S (0) , then B x is given by . (A.3)Admissibility is defined via local orders, analogously to the 3-dimensional casein Section 2.2.1. Moreover, the 2-dimensional analogues of the BLT moves are48he bubble move and the dual of the 2-2 Pachner move, which we call the b-move and l-move , respectively: −→ (A.4) −→ (A.5)A bl move is called admissible if the skeleta on both sides are admissible.The following theorem is the 2-dimensional version (without boundary) of the3-dimensional statement that is the topic of this appendix. The 3-dimensionalproof will be similar in structure. Theorem A.2.
Let Σ be a closed 2-manifold. Then any two admissible skeleta S and S (cid:48) of Σ are connected by a finite sequence of admissible bl moves. Proof.
We call two admissible skeleta equivalent if they are are connected by afinite sequence of admissible bl moves. Given a globally ordered triangulationof Σ, its dual is a particular case of an admissible skeleton which we refer toas d-skeleton (short for “skeleton dual to a globally ordered triangulation”). By[CRS1, Prop. 3.3] any two d-skeleta are related by a finite sequence of moves thatare dual to globally ordered 2-dimensional Pachner moves (see Section 2.2.2). Itis well known (see e. g. [FHK]) that the admissible bl moves are stronger thanthe moves dual to globally ordered Pachner moves, thus any two d-skeleta areequivalent.We are thus left with showing that an admissible skeleton S is equivalent toa d-skeleton. Analogously to the 3-dimensional case in Lemma A.9 below, onefinds that a skeleton is dual to a triangulation iff every stratum s satisfies: (i) s iscontractible, (ii) the germs of 2-strata adjacent to s belong to different 2-strata,and (iii) if the sets of germs of adjacent 2-strata agree for s and another stratum t ,then already s = t . If conditions (i)–(iii) hold for an individual stratum s , we saythat s is locally dual to a triangulation.By the b-move the contractibility condition is easy to achieve starting from anyadmissible skeleton. If the two germs of adjacent 2-strata of a 1-stratum belongto the same 2-stratum, by a sequence of one b- and two l-moves, we can createa “copy of the 1-stratum” such that in the resulting stratification the original 1-stratum and the newly created 1-stratum are locally dual to a triangulation, and49ll other strata remain locally dual to a triangulation if they are. If a 0-stratumis not locally dual to a triangulation, it is straightforward to provide a similarcombination of b- and l-moves to make it locally dual to a triangulation.We can thus assume that S is dual to a triangulation and we now need to showit is equivalent to a d-skeleton, i. e. we need to fix the orientations. We do thissimilar to the proof of Lemma A.11 below: Pass to the dual triangulation; carryout a 1-3 Pachner move on all triangles and orient the new edges towards thenew vertices. Now that both orientations of each old edge are allowed, reversethe orientations of the old edges as required (this is done via b- and t-moves inthe dual picture); undo all the 1-3 Pachner moves.The next statement links the above discussion of the 2-dimensional skeleta tothe 3-dimensional case that is the focus of this paper. Namely, it is straightfor-ward to check that the following is true: Proposition A.3.
Any skeleton S of a 3-manifold M induces a skeleton ∂S of ∂M , where the i -strata of ∂S are obtained by intersecting the ( i + 1)-strata of S with ∂M . If S is oriented or admissible then so is ∂S . A.2. Pseudo-skeleta
We define a slightly more general class of stratifications than skeleta that will beuseful in the proof of Theorem 2.12: A pseudo-skeleton of a 3-manifold M is astratification such that:(i) For each 3-stratum s there are exactly three allowed cases: if s ∩ ∂M = ∅ ,then s is a 3-ball; if s intersects exactly one of ∂ in M and ∂ out M , then s isa half open ball; and if s intersects both ∂ in M and ∂ out M , then s is thecylinder D ◦ × [0 , D ◦ is the interior of the 2-disc.(ii) The same local conditions hold as in the definition of a skeleton.We say a stratified 3-bordism ( M, S ) is skeletal if S is a skeleton, and pseudo-skeletal if S is a pseudo-skeleton. A morphism of Bord str3 (cf. Section 2.1.2) is (pseudo-)skeletal if one (and thus all) of its representative stratified 3-manifoldsare.Note that in a pseudo-skeleton of M , if a 3-stratum is a cylinder D ◦ × [0 , D ◦ × { } and D ◦ × { } are not allowed to both lie on ∂ in M nor both on ∂ out M .For pseudo-skeleta, Remark 2.3 gets replaced by the following observation: Lemma A.4.
A pseudo-skeleton S of M is a skeleton if and only if no 3-stratumof S intersects both the in- and out-boundary of M .50 .3. Refinement to dual of a triangulation relative toboundary The main content of this section is Lemma A.10, which tells us that any ad-missible skeleton with sufficiently nice boundary can be refined to the dual ofa triangulation using admissible BLT moves. We begin with a technicality thatwill be needed in its proof, namely how to delete a 2-stratum from a skeleton.Recall the notation introduced in Section 2.1.1, and let M be a fixed 3-manifoldfrom now on. Let S be a stratification of M with filtration ∅ ⊂ F (0) ⊂ . . . ⊂ F (3) = M , and let s be a 2-stratum of S . We define the filtration ∅ ⊂ ˜ F (0) ⊂ . . . ⊂ ˜ F (3) = M by ˜ F ( i ) := F ( i ) \ (¯ s ∩ S ( i ) ) = ( F ( i ) \ ¯ s ) ∪ F ( i − . More concisely, ˜ F is obtained from F by deleting all strata that are contained in ¯ s . In general, thefiltration ˜ F is not a stratification, but we have: Lemma A.5.
Let S be a skeleton of M .(i) The filtration ˜ F defines a stratification of M , denoted S \ ¯ s .(ii) If in addition to (i), s is contractible and different germs of 3-strata incidentwith s are induced by different 3-strata such that not both of them intersect ∂ in M or both ∂ out M , then S \ ¯ s is a pseudo-skeleton.(iii) If in addition to (i) and (ii), at least one of the 3-strata incident with s doesnot intersect ∂M , then S \ ¯ s is a skeleton. Proof.
For (i) we notice that clearly ˜ F ( i − ⊂ ˜ F ( i ) for all i , so we only needto endow each ˜ S ( i ) with a smooth structure. Since we are in dimension 3, asmooth structure on ˜ S ( i ) is uniquely determined once we understand ˜ S ( i ) to bea topological manifold. Elementary point-set manipulations show that ˜ S ( i ) =( S ( i ) \ ¯ s ) ∪ ( S ( i − ∩ ¯ s ). Now if x ∈ S ( i ) \ ¯ s , then x has a neighbourhood U such that U ∩ ˜ F ( i ) is homeomorphic to R i since S ( i ) is a topological manifold.If x ∈ S ( i − ∩ ¯ s , then the existence of such a neighbourhood follows from theexistence of the neighbourhoods depicted in Figure 2.1.For (ii) we note that the additional assumption guarantees that after deleting s ,all 3-strata are still contractible. The other conditions necessary to make S \ ¯ s a pseudo-skeleton are checked straightforwardly. Part (iii) follows directly fromthe definitions.If we want to delete a 2-stratum s from an admissible skeleton, then there arerestrictions on the orientations of strata of S . Indeed, let t be an ( i − s , i ∈ { , , } . Then it is incident with two i -strata r and r of S (withdistinct germs at t , but r and r may be equal in S ) that are not contained in¯ s . After deleting ¯ s , r and r merge to form an i -stratum r of S \ ¯ s and the localorders at r and r induce local orders at r . If the two induced local orders at r agree we say that the local orders at r and r are compatible . If this holds for51ll strata t in ¯ s then we say that the orientations adjacent to s are compatible.In that case we can canonically endow S \ ¯ s with an admissible orientation thatis inherited from S .Combining this with the conditions from Lemma A.5, we arrive at the followingdefinition: Let S be an admissible skeleton of M . A contractible 2-stratum s in S is called superfluous if(i) different germs of 3-strata incident with s are induced by different 3-strata,(ii) at least one of the 3-strata incident with s does not intersect ∂M ,(iii) the orientations adjacent to s are compatible.If s is superfluous then the admissible structure of S turns S \ ¯ s canonically intoan admissible skeleton. The name “superfluous” is justified by the following fact. Lemma A.6.
Let S be an admissible skeleton of M , and let s be a superfluous2-stratum of S . Then there exists a sequence of admissible BLT moves between S and S \ ¯ s . Proof.
Let B be a 3-stratum of S that does not intersect ∂M and such that s ⊂ ¯ B . Then the topological boundary Σ of B is a 2-sphere. Noting that s may originally be adjacent to several other 2-strata in Σ (see Figure A.1 for anillustration), we use the L- and T- moves on ¯ s , until ¯ s forms a bubble on a single2-stratum of Σ \ s , at which point we can delete it with a B-move. We haveto be careful that in each step the orientations adjacent to s remain compatiblebecause this guarantees that after deleting s all the strata in S \ ¯ s carry the sameorientation as they did originally. A quick check confirms that in each admissibleBLT move that we apply, we can choose orientations such that this is indeed thecase. Corollary A.7.
Let S be an admissible skeleton of M , let s be a contractible2-stratum in S not intersecting ∂M , and let S (cid:48) be the admissible skeleton whichtopologically differs from S only in that it has another copy s (cid:48) of s which isconnected to s by a cylinder (see Figure A.1). Then S and S (cid:48) are connected byadmissible BLT moves. Proof.
Note that s (cid:48) is superfluous. Start from the skeleton with s (cid:48) and useLemma A.6. Remark A.8.
Note that if s is superfluous then also the skeleton S (cid:48) obtainedby reversing the orientation of s is admissible (recall Remark 2.8 (i)), and theorientation-reversed version of s is superfluous in S (cid:48) . In this way, Lemma A.6can be used to reverse the orientation of a superfluous 2-stratum.Combining this with Corollary A.7, it follows that if the skeleton S (cid:48) obtainedfrom S by flipping the orientation of s is still admissible and s does not touch ∂M ,52 a) (b) Figure A.1.: (a) Neighbourhood of a 2-stratum s of an admissible skeleton S . (b)Neighbourhood of s and a shifted copy s (cid:48) in the admissible skele-ton S (cid:48) (that agrees with S away from this neighbourhood).then S is connected to S (cid:48) by BLT moves even if s is not superfluous (after makinga copy s (cid:48) of s , s becomes superfluous, and so its orientation can be flipped; thenremove s (cid:48) ).We now give a characterisation of skeleta that are dual to triangulations. Lemma A.9.
A skeleton S of M is dual to a triangulation if and only if(i) every stratum of S is contractible,(ii) for each stratum s of S , the canonical map S ( s ) −→ { s } (A.6)is injective (and thus bijective),(iii) if two strata s and t satisfy S ( s ) = S ( t ), then s = t . Proof.
The conditions (i)–(iii) are satisfied for any dual of a triangulation.Conversely, from a skeleton S that satisfies these conditions, we obtain asimplicial complex, i. e. a set X (of vertices) together with a set Σ ⊂ P X (of simplices) such that for all B ∈ Σ and A ⊂ B we have A ∈ Σ (facecondition). Indeed, first we set X := S , the set of 3-simplices of S . Us-ing the local conditions from Figure 2.1 we see that each i -stratum s in S is incident to exactly 4 − i germs of 3-strata around s . Thus we have maps f i : { i -strata of S } −→ { (4 − i )-element subsets of X } for each i ∈ { , , , } .The set Σ is the union of the images of the f i . Each f i is injective, as followsdirectly from condition (iii). Because of this we obtain a map | ( X, Σ) | −→ M which is a homeomorphism by definition of ( X, Σ). This triangulation is dualto S by construction. 53 emma A.10. Let S be an admissible skeleton of M such that ∂S is dual to atriangulation of ∂M . Then there exists an admissible skeleton S (cid:48) of M that isdual to a triangulation, satisfies ∂S (cid:48) = ∂S , and is connected to S by a sequenceof admissible BLT moves. Proof.
We show that we can pass to a skeleton satisfying the conditions fromLemma A.9 using BLT moves that do not include the inverse lune move. Thisallows us to always choose orientations of the targets such that the resultingmoves will be admissible.To guarantee condition (i) we show that we can pass to sufficiently fine sub-divisions of S . We note that all 3-strata are contractible by definition of skeletaand 0-strata are so trivially. If s is a 1-stratum we can cut it up by creatinga bubble on a 2-stratum incident with it, and then sliding the bubble onto s .This can be implemented through a bubble and a lune move. By iterating thisprocedure we can guarantee that all 1-strata are contractible. Let now s be a2-stratum. Since s is contained in the boundary of a 3-ball, it has genus 0. If s isclosed then it is a sphere and we can decompose s into contractible pieces with abubble move and the above argument for 1-strata. Otherwise the surface Σ = ¯ s is a sphere with a finite number of topological boundary components . Each ofthe these boundary components contains some interval that is contained in theinterior of M . Indeed, suppose there was a boundary component b with b ⊂ ∂M .Since all strata of ∂S are contractible, b contains at least one 0-stratum p . Thenfrom the allowed neighbourhoods in Figure 2.1 we see that there has to be a1-stratum protruding from p into the interior of M that is part of the boundaryof Σ. This allows us to cut s along a curve γ connecting any two different bound-ary components by using a bubble and a lune move to create a bubble on γ andstretching it along γ using an isotopy. At the end we use two more lune moves totraverse the 1-strata connected by γ . This guarantees that s can be subdividedinto contractible strata. All 1-strata created in the process are contractible.We now suppose that S satisfies condition (i). Conditions (ii) and (iii) areautomatically satisfied if any of the involved strata have non-trivial intersectionwith ∂M because of our assumption on ∂S . Hence in what follows we can assumethat all the strata we consider do not intersect ∂M . We first notice that condition(ii) is trivially satisfied for all 3-strata. Furthermore if it is satisfied by all 2-strata,then also by all 1- and 0-strata. Indeed, this follows from the local conditions inFigure 2.1. Thus let s be a 2-stratum. By Corollary A.7 we can make a copy s (cid:48) of s , guaranteeing that both s and s (cid:48) satisfy condition (ii).We now suppose that S satisfies conditions (i) and (ii), and we want to passto a skeleton that additionally satisfies condition (iii). If s, t are strata and S ( s ) = S ( t ) then they have to be of the same dimension i , since for an i -stratum s we have | S ( s ) | = 4 − i . If i = 2 and S ( s ) = S ( t ) for s (cid:54) = t , then we Each such boundary component is an S and can be made up from several 0- and 1-strataof S or ∂S . s (cid:48) of s as in Corollary A.7, such that both s and s (cid:48) are adjacentto the newly created 3-stratum while t is not. A quick check confirms that thisguarantees that after these moves, we have S ( s ) (cid:54) = S ( t ), and that for all 2-strata r (cid:54) = s (cid:48) , we have S ( s (cid:48) ) (cid:54) = S ( r ). If i = 1 and s (cid:54) = t , then s and t cannot be incidentto a common 0-stratum because this 0-stratum would have a neighbourhood asdepicted in Figure 2.1, where all 1-strata have different sets of germs of 3-strata.We can then implement −→ (A.7)around s (which can be done without using inverse lune moves), guaranteeingthat S ( s ) (cid:54) = S ( t ). Also none of the newly created 1- or 2-strata will violatecondition (iii). Finally, if i = 0 we use −→ . (A.8)In none of these steps conditions (i) or (ii) are violated. A.4. Fixing the orientations
Our next goal is to show that an admissible skeleton that is dual to a triangulationis connected to one that is dual to a globally ordered triangulation by admissibleBLT moves.
Lemma A.11.
Let
S, S (cid:48) be admissible skeleta of M such that(i) the underlying unoriented skeleta agree, i. e. S = S (cid:48) ,(ii) S and S (cid:48) are dual to triangulations,(iii) S and S (cid:48) agree on the boundary of M , i. e. ∂S = ∂S (cid:48) .Then there is a sequence of admissible BLT moves from S to S (cid:48) .55 roof. Let
T, T (cid:48) be the triangulations dual to
S, S (cid:48) , respectively. By assumptionwe know that they agree as unoriented triangulations. Since they are admissiblyoriented the orientations of all simplices are implied once the orientations of edgesare fixed. The admissibility condition says that there is no closed loop of orientededges that lies in the boundary of a single 3-simplex.If e is an edge of T (or T (cid:48) ), then we say that e can be admissibly flipped if the oriented triangulation T ( e ∗ ) (or T (cid:48) ( e ∗ )) that is obtained by reversing theorientation of e is admissible. By Remark A.8, if e can be admissibly flippedthen T and T ( e ∗ ) (or T (cid:48) and T (cid:48) ( e ∗ )) are connected by a sequence of admissibleBLT moves.We will construct triangulations (cid:101) T , (cid:101) T (cid:48) such that • there are sequences of admissible BLT moves from T to (cid:101) T , and from T (cid:48) to (cid:101) T (cid:48) , • the 1-skeleta of T and T (cid:48) embed into (cid:101) T and (cid:101) T (cid:48) , respectively (and we identifyvertices and edges of T or T (cid:48) with their images under these embeddings), • (cid:101) T = (cid:101) T (cid:48) , and (cid:101) T differs from (cid:101) T (cid:48) only in the orientations of edges of T = T (cid:48) , • each edge of T can be admissibly flipped in (cid:101) T , and each edge of T (cid:48) can beadmissibly flipped in (cid:101) T (cid:48) .From this the claim follows because we then have a sequence of admissible BLTmoves T −→ (cid:101) T −→ (cid:101) T (cid:48) −→ T (cid:48) .By Lemma 2.11 all admissible Pachner moves can be implemented via ad-missible BLT moves. We first construct an intermediate triangulation ¯ T (andanalogously ¯ T (cid:48) ): we apply an admissible 1-4 move to each 3-simplex of T and ori-ent each newly created edge towards the newly created vertex it is incident with.Then the 2-skeleton of T embeds into ¯ T . Furthermore, if s is any 2-simplex of T in ¯ T that does not lie in the boundary, it is incident to exactly two 3-simplices allof whose other 2-faces are not contained in T . Thus we can apply 2-3 moves ateach of the 2-simplices of T in ¯ T (and the two adjacent 3-simplices in ¯ T ) to arriveat the triangulation (cid:101) T . We orient the edges newly created by the 2-3 moves ar-bitrarily. The choices of orientations in the construction of ¯ T (and ¯ T (cid:48) ) guaranteethat all these moves are admissible. By design, all non-boundary 2-simplices of T, T (cid:48) have been erased in (cid:101)
T , (cid:101) T (cid:48) , and so all edges of T can be admissibly flippedin (cid:101) T , and analogously all edges of T (cid:48) can be admissibly flipped in (cid:101) T (cid:48) Thus allfour conditions from above are satisfied.
A.5. Fixing the boundary
We provide the technical tools to deal with a boundary that is not dual to atriangulation. 56n the following we will denote (representatives of) morphisms of Bord str3 bypairs (
M, S ), or simply by M , if there is no need to refer to the stratification S of the bordism M . Composition in Bord str3 is denoted by juxtaposition. Lemma A.12.
Let
M, N be pseudo-skeletal morphisms of Bord str3 . Then thefollowing statements hold whenever the respective compositions make sense.(i)
M N is pseudo-skeletal.(ii) If at least one of M , N is skeletal then so is M N . Proof.
Both claims follow straightforwardly from the definitions. For part (ii) wenote that if a 3-stratum intersects both the in- and out-boundary of
M N , thenboth of its restrictions to M and N must also intersect the respective in- andout-boundaries non-trivially.Two morphisms ( M, S ) and (
M, S (cid:48) ) of Bord str3 with the same underlying bor-dism M but possibly different stratifications S and S (cid:48) are called equivalent ,( M, S ) ∼ ( M, S (cid:48) ), if S and S (cid:48) are related by a sequence of admissible BLT moves.The resulting equivalence relation on the Hom sets of Bord str3 is compatiblewith composition in the following way: Lemma A.13. If M ∼ N and X ∼ Y , then M X ∼ N Y .Let Σ be a closed 2-manifold, S a skeleton of Σ, and s a 2-stratum of S . Weconstruct a pseudo-skeletal 3-bordism e Σ ,S,s : (Σ , S ) −→ (Σ , S ) as follows: Startwith the cylinder (Σ , S ) × I . In the cylinder I × s we insert a single new 2-stratum s (cid:48) that is a copy of s , shifted away from s along I and a respective copy of each0- and 1-stratum that is incident with s in Σ. For example:(Σ , S ) = = ⇒ e Σ ,S,s = . (A.9) Lemma A.14.
Let Σ be a 2-manifold with skeleton S , let s be a 2-stratumof S , and let M, N be skeletal 3-bordisms. Then we have e Σ ,S,s M ∼ M and N e Σ ,S,s ∼ N whenever these compositions make sense. Proof.
This is a direct application of Lemma A.6.57ote that this is in general not true if M or N are only pseudo-skeletal, as canbe seen by taking them to be an identity in Bord str3 .Any bl move for skeleta of 2-manifolds can be implemented via a pseudo-skeletal3-bordism. More precisely: Let Σ be a 2-manifold, S a skeleton for Σ, and let S (cid:48) be obtained from S via an application of a 2-dimensional bubble move in a disc D ⊂ Σ. Then we define the pseudo-skeletal 3-bordism M bΣ ,S,D : (Σ , S ) −→ (Σ , S (cid:48) )by M bΣ ,S,D = (A.10)which away from D × I is just the cylinder over Σ \ D . Similarly, if S (cid:48) is obtainedfrom S via the dual of a 2-2 Pachner move we define the pseudo-skeletal 3-bordism M lΣ ,S,D : (Σ , S ) −→ (Σ , S (cid:48) ) by M lΣ ,S,D = (A.11)The bordisms M b − Σ ,S,D and M l − Σ ,S,D for the inverse moves b − and l − are definedanalogously.We can also implement all admissible 2-dimensional moves in this way. Theadmissible orientations on the corresponding 3-bordisms are uniquely determinedby the ones on the boundary. More precisely, let ( M, S ) be any of the 3-bordisms M b ± Σ ,S,D , M l ± Σ ,S,D above, and let s be a 2-stratum of S . Then S ( s ) ∼ = ( ∂S ) ( s ∩ ∂M ).The latter comes equipped with an order by assumption and this then definesthe order on S ( s ). A case-by-case check shows that this indeed defines a localorder on S in the sense of Definition 2.5. Lemma A.15.
Dropping sub indices we have(i) M b ◦ M b − ∼ e , with e as in (A.9),(ii) M b − ◦ M b ∼ M l ◦ M l − ∼ M l − ◦ M l ∼
1. 58 roof.
The equivalence of part (i) is implemented by a 3-dimensional lune moveand an isotopy: → . (A.12)Part (ii) involves an inverse bubble move. Part (iii) comes about with an inverselune move, → , (A.13)and similarly for part (iv). A.6. The proof of Theorem 2.12
Proof.
Let
S, S (cid:48) be admissible skeleta of M that agree on ∂M . Then we canfind X , . . . , X n and Y , . . . , Y m , where each of the X i , Y j is one of the 3-bordisms M ± b ··· , M ± l ··· , such that with X := X · · · X m and Y := Y · · · Y m , X ( M, S ) Y and X ( M, S (cid:48) ) Y have the same boundary that is dual to a globally ordered triangula-tion (Theorem A.2).By Lemmas A.10 and A.11 there are skeletal 3-bordisms ( M, R ) ∼ ( M, S ),(
M, R (cid:48) ) ∼ ( M, S (cid:48) ) such that
R, R (cid:48) are dual to globally oriented triangulations.As described in [CRS1, Sect. 3.1] R and R (cid:48) are connected by a sequence of globallyoriented Pachner moves. By Lemma 2.11 these skeleta are a fortiori related by ad-missible BLT moves. Hence with Lemma A.13 we find X ( M, S ) Y ∼ X ( M, S (cid:48) ) Y ,and thus:( M, S ) ( ∗ ) ∼ X − X ( M, S ) Y Y − ∼ X − X ( M, S (cid:48) ) Y Y − ( ∗ ) ∼ ( M, S (cid:48) ) . (A.14)59ere X − is obtained from X by reversing order of composition while swapping bwith b − , and l with l − . In ( ∗ ) we need Lemmas A.12 and A.14: If X − i X i ∼ X − i X i ∼ e , then we know that X i +1 · · · X n ( M, S )is skeletal, and thus together with Lemma A.15: X − i X i X i +1 · · · X n ( M, S ) ∼ eX i +1 · · · X n ( M, S ) ∼ X i +1 · · · X n ( M, S ). In this way we obtain ( X − X )( M, S ) ∼ ( M, S ) by induction. The same applies for Y . B. Orbifold data for non-Euler-complete theories
We fix a defect TQFT Z : Bord def3 ( D ) −→ Vect. Recall from [CRS1, Sect. 2.5] theconstruction of its Euler completion Z (cid:12) : Bord def3 ( D (cid:12) ) −→ Vect, which has thefollowing properties: (i) Z naturally factors through Z (cid:12) ; (ii) ( Z (cid:12) ) (cid:12) is equivalentto Z (cid:12) ; and (iii) the tensor product of Z (cid:12) with the Euler defect TQFT Z EuΨ isequivalent to Z (cid:12) . Here for any list Ψ = ( ψ , ψ , ψ ) of invertible scalars, Z EuΨ is the invertible defect TQFT which assigns k to every surface, and Z EuΨ ( M ) = (cid:81) j =1 (cid:81) s ∈ M j ψ χ sym ( s ) j for a stratified bordism M , where χ sym ( − ) := 2 χ ( − ) − χ ( ∂ − ) (B.1)is the symmetric Euler characteristic (see [CRS1, Ex. 2.14] for details). For ex-ample, if D is a 2-stratum consisting of a half-disc that intersects the boundaryin an interval I , then χ ( D ) = χ ( I ) = 1, and so χ sym ( D ) = 1. Special orbifold data
As explained in [CRS1, Sect. 3.4.1 & 4.2], a special orbifold datum for Z (cid:12) is atuple A = ( A , A , A , A ± , ψ, φ ) , (B.2)where A , A , A , A ± are elements as in Section 4.1.1, and φ ∈ Aut W ( A , A ) (1 A ) , ψ ∈ Aut W ( A , A ) (1 A ) , (B.3)such that the identities depicted in Figure B.1 hold. A -decorated skeleta Let A = ( A , A , A , A ± , ψ, φ ) be a special orbifold datum for Z (cid:12) . An A -decorated skeleton S of a bordism M is an admissible skeleton S of M togetherwith a decoration as follows: • each 3-stratum B of S is decorated by A with an insertion of φ χ sym ( B ) , • each 2-stratum F of S is decorated by A with an insertion of ψ χ sym ( F ) , • each 1-stratum of S is decorated by A , • for ε ∈ { + , −} , each ε -oriented 0-stratum of S is decorated by A ε .60 (O ψ ψ
2) = (O ψ ψ
4) = (O ψ ψ
6) = (O ψ ψ A with φ and ψ ; thelabels A j are suppressed for most j -strata. The application of Z oneach side the equations is implied.61 ibbon category W A The category W A is defined as in Section 4.2, except for the following changes: • Objects of W A are tuples X = ( X, τ X , τ X , τ X , τ X ), with X ∈ W , wherethe crossings τ X , τ X are as in (4.13), and where in addition τ X ∈ (cid:98) Z , τ X ∈ (cid:98) Z , (B.4)such that the identities in Figure B.2 hold when (cid:98) Z is applied to both sides(viewed as defect 3-balls).Note that as in the situation studied in [MR1], the pseudo-inverses τ X , τ X are uniquely determined by τ X , τ X . Hence we may, and will, shorten thenotation to X = ( X, τ X , τ X ). • The crossings in the tensor product (
X, τ X , τ X (cid:1) ⊗ A (cid:0) Y, τ Y , τ Y ) = ( X ⊗ Y, τ
X,Y , τ X,Y ) involve additional ψ -insertions: τ X,Y = , τ X,Y = . (B.5) • The adjunction morphisms in W A areev X = , coev X = , (B.6) (cid:101) ev X = , (cid:103) coev X = . (B.7) This may be easiest to see in the language of Gray categories discussed in Remark 4.10. The braiding morpisms in W A are c X , Y = (cid:98) Z , c − X , Y = (cid:98) Z , (B.8) A -decorated ribbon diagrams Now let A = ( A , A , A , A ± , ψ, φ ) be a special orbifold datum for (cid:98) Z (cid:12) . An A -decorated ribbon diagram ( S , (cid:100) ) of a bordism M with embedded W A -colouredribbon graph R is an element ( S, (cid:100) ) ∈ (cid:83) ( M, R ) together with a decoration asfollows:(i) S is an A -decorated skeleton of M with underlying skeleton S , except that ψ - and φ -insertions are as described in parts (iv) and (v) below;(ii) if a switch of ( S , (cid:100) ) involves an ( X, τ , τ )-labelled ribbon of R traversingan A -labelled 1-stratum of S , then the switch is labelled by τ , τ , τ or τ as appropriate;(iii) over- and under-crossings in (cid:100) are replaced by coupons labelled with thecorresponding braiding morphisms in W A ;(iv) if a 2-stratum F of S is subdivided by strands of d , the ψ -insertions pertain-ing to F are as follows: there is one ψ χ sym ( F i ) -insertion on every connectedcomponent F i of F \ d ; moreover, for each coupon c of (cid:100) , the leftmostand rightmost 2-stratum components adjacent to c have one additional ψ -insertion each; when computing ψ χ sym ( F i ) , boundary segments of couponsin (cid:100) are treated like boundary segments of ∂M (see (B.9) below for anexample);(v) each 3-stratum in the interior of M obtains a φ -insertion, and each 3-stratum adjacent to ∂M obtains a φ -insertion.We remark that part (iv) in the above definition is needed for compatibilitywith composition in the category W A . Indeed, if f and g are composable labels63 (T ψ
1) = (T ψ ψ ψ ψ ψ ψ W A with ψ
64f two coupons in (cid:100) , then (cid:98) Z A evaluates to the same vector on discs around eitherthe two coupons or around one coupon labelled g ◦ f : (cid:98) Z A = (cid:98) Z A (B.9)Note that the above also illustrates the rules to have ψ -insertions to the left andright of coupons, and that boundary segments of coupon are treated like boundarysegments of ∂M when computing symmetric Euler characteristics χ sym ( F i ) =2 χ ( F i ) − χ ( ∂F i ) for ψ -insertions. In particular, χ sym ( F i ) = 0 for each rectanglebounded by two W A -labelled strands. This is the reason that no ψ -insertionsappear in between these strands. Orbifold graph TQFT
Theorem B.1.
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