Field theories with defects and the centre functor
aa r X i v : . [ m a t h . QA ] J u l ZMP-HH/11-8Hamburger Beitr¨age zur Mathematik 407
Field theories with defects and the centre functor
Alexei Davydov a , Liang Kong b , Ingo Runkel c , ∗ a Department of Mathematics and StatisticsUniversity of New Hampshire, Durham NH, USA b Institute for Advanced Study (Science Hall)Tsinghua University, Beijing 100084, China c Fachbereich Mathematik, Universit¨at HamburgBundesstraße 55, 20146 Hamburg, Germany
July 2011
Abstract
This note is intended as an introduction to the functorial formulation of quantumfield theories with defects. After some remarks about models in general dimension,we restrict ourselves to two dimensions – the lowest dimension in which interestingfield theories with defects exist.We study in some detail the simplest example of such a model, namely a topo-logical field theory with defects which we describe via lattice TFT. Finally, we givean application in algebra, where the defect TFT provides us with a functorial def-inition of the centre of an algebra. This involves changing the target category ofcommutative algebras into a bicategory.Throughout this paper, we emphasise the role of higher categories – in our casebicategories – in the description of field theories with defects. ∗ Emails: [email protected] , [email protected] , [email protected] ontents One way to think about quantum field theory – motivated by conformal field theory andstring theory [FS, Va] – is as functors from bordisms to vector spaces [Se, At]; here, each ofthe terms ‘functor’, ‘bordism’, ‘vector space’ has to be supplemented with the appropriatequalifiers for the application in mind. In its most basic form, the bordisms for an n -dimensional quantum field theory form a symmetric monoidal category whose objects are( n −
1) dimensional manifolds equipped with ‘collars’ and whose morphisms are equivalenceclasses of n -dimensional manifolds with parametrised boundary.2o study quantum field theories beyond this basic functorial definition, it is oftenappropriate to employ higher categories. There are two natural ways in which such highercategories enter.1. The ( n − n − n − n -morphisms. The resulting field theories are most studied in the case of topologicalfield theories [Lu1].2. One can let the bordisms remain a (1-)category but equip them with extra struc-ture, namely with ‘defects’. These are submanifolds embedded in the ( n − n -dimensional bordisms, decorated with labels which describe different possible ‘de-fect conditions’. A field theory on bordisms with defects equips the set of defectconditions with the structure of a higher category.Here we want to elaborate on the second point. Some other works which also stress theappearance of higher categories in field theories with defects, and which the reader couldconsult for further references, are [SFR, Lu1, BDH, Ka, KK]. In the present paper, wewill concentrate on the simplest interesting class of models, namely two-dimensional fieldtheories with defects. In section 2 we will see how a field theory with a particular type ofdefects – so called topological defects – gives rise to a 2-category defined in terms of theset of defect conditions. In section 3 we use lattice topological field theory to construct avery simple but still non-trivial example of a field theory with defects. This example willmotivate – in section 4 – a nice mathematical construction, namely a method to make theassignment which maps an algebra to its centre functorial. Section 5 contains an outlookon further developments.These three constructions – the 2-category of topological defects (section 2.4), latticetopological field theory with defects (theorem 3.8), and the centre functor (theorem 4.12and remark 4.19) – are the main points of this paper. We hope that they provide someintuition on how to work with field theories containing defects and illustrate their useful-ness. It is beyond the scope of this article (and the present abilities of the authors) to developan all-purpose formalism for field theories with defects. In this subsection we briefly sketchthe basic features of the functorial formulation of field theory with defects. In doing so will omit most details. For those who are familiar with the functorial formulation, some ofthese details are: we should equip our object-( n −
3s usual, a field theory will be a functor from a bordism category to a category of vectorspaces. In the presence of defects, the target category of the functor remains unchanged.However, we do modify the source category. The category of n - dimensional bordisms withdefects contains the following ingredients. • Sets of defect conditions: the bordism category will depend on a choice of n +1(possibly empty) sets, D k , k = 0 , . . . , n . The elements of D k serve as defect conditionsfor k -dimensional defects. • Objects: the objects are ( n − U with emptyboundary, together with a disjoint decomposition into submanifolds. That is, U = S n − i =0 U i , where each U i is an i -dimensional oriented submanifold of U and U k ∩ U l = ∅ for k = l . The orientation of U n − is induced by that of U .For example, U n − = U and U k = ∅ for k < n − U i ( i < n −
1) is a closed submanifold of U , then we can take U n − = U \ U i and U k = ∅ for k = i, n − U k is decorated with a defect condition from D k +1 , i.e. we have a collection of maps d k +1 : π ( U k ) → D k +1 ; the reason for the shiftin k is that the U k will appear as boundaries of ( k + 1)-dimensional submanifolds inthe n -dimensional manifold making up a morphism. • Morphisms: a morphism M : U → V has a structure analogous to objects, exceptin one dimension higher. In more detail, M is an n -dimensional compact orientedmanifold, together with a decomposition M = S ni =0 M i , where each M i is an i -dimensional oriented submanifold, possibly with non-empty boundary ∂M k , and M k ∩ M l = ∅ for k = l (and footnote 2 applies analogously). The orientation of M n isinduced by that of M . Each connected component of M k is labelled by a defectcondition, but this time from D k , that is, we have maps ˆ d k : π ( M k ) → D k .The boundary ∂M is identified via an orientation preserving diffeomorphism (whichis part of the data of a morphism) with the disjoint union − U ⊔ V ; we require that ∂M k ⊂ ∂M , and that the resulting decomposition and labelling of ∂M agrees withthe one induced by − U ⊔ V . operation; objects and morphisms could carry extra geometric data such as a metric, a spin structure, etc.;we should work with families to have a natural notion of continuous or smooth dependence of the functoron the bordism; the functor from bordisms to vector spaces is symmetric monoidal; the target categoryof the functor consists of topological vector spaces with an appropriate tensor product. These issues aretreated carefully in [ST]. We also demand that the partial union S ki =0 U i is a closed subset of U for k = 0 , . . . , n −
1; this ensuresthat ¯ U k \ U k (the difference of U k and its closure) is contained in the union S k − i =0 U i of lower dimensionalpieces. Let us give a non-example in U = S , which we present as the one-point compactification of R . Take U = ∅ , U = ( − , × { (0 , } , U = S ⊂ R , U = U \ ( U ∪ U ). In this case, all U i aresubmanifolds, but U ∪ U is not closed, which is not allowed. (But U ∪ U ∪ U is closed). To turn thisinto an allowed decomposition, take instead U ′ = { ( ± , , } and U ′ = S \ { ( ± , , } . Then U ′ , U , U ′ , U is an allowed decomposition of U . + xs ( x ) t ( x ) b) − xt ( x ) s ( x ) c) x t ( x ) s ( x ) Figure 1:
Figures a)–c) show open subsets of a bordism in dimension n = 1 and n = 2. They giveour orientation convention in the compatibility condition for the assignment of defect conditionsin the case n = 1 (figs. a, b) and n = 2 (fig. c). The arrows represent positively oriented orderedbases. For example, in n = 3 dimensions, a generic morphism would look like a foam, where theinterior of each bubble is ‘coloured’ by an element of D , the walls between two bubblesby elements of D , lines along which the walls between bubbles meet by elements of D ,and points where these lines meet by elements of D (somewhat problematic in the foamanalogy, but nonetheless allowed, are 1- and 0-dimensional submanifolds not attached toany walls).While we have given the overall name ‘defect conditions’ to elements of the sets D k ,more descriptive names in the various dimensions 0 ≤ k ≤ n would be that they areconditions for • D n : domains (or phases of the field theory) • D n − : domain walls (or phase boundaries) • D n − , . . . , D : junctionsThe sets D k are equipped with additional structure describing in which geometric config-urations the domains can occur. This is complicated in general, but it is easy to state fordomain walls. Since the n -dimensional manifold underlying the morphism and the ( n − s, t : D n − −→ D n (2.1)(for ‘source’ and ‘target’), and a domain wall of type x ∈ D n − must have a domain labelledby s ( x ) on its left and t ( x ) on its right. This gives a restriction on the allowed maps d n , d n − in objects and ˆ d n , ˆ d n − in morphisms. In this work, we will only discuss the cases n = 1and n = 2 and our orientation conventions are shown in figure 1. Before passing to the more interesting two-dimensional situation, let us briefly discuss thesimplest one-dimensional field theory with domain walls, namely the case where the fieldtheory is topological. 5e fix two sets D and D , together with two maps s, t : D → D . The objects in thebordism category are finite sets of oriented points U , together with a map d : U → D .The morphisms M : U → V are (diffeomorphism classes of) 1-dimensional manifolds M with a finite set W of marked points in the interior of M . Each connected component of M \ W is labelled by an element of D , and each element of W by an element of D . Onthe boundary ∂M the D -labels have to agree with those of U , resp. V .A symmetric monoidal functor from this bordism category to (necessarily finite dimen-sional) k -vector spaces for some field k is then determined • on objects: by a collection of vector spaces ( V i ) i ∈ D . The value of the functor on apoint with orientation ‘+’ and label i is given by V i , while a point with orientation‘ − ’ gets mapped to V ∗ i . On 0-dimensional manifolds with more than one point thefunctor is fixed by the monoidal structure as usual. • on morphisms: by two collections of linear maps ( L + x ) x ∈ D and ( L − x ) x ∈ D , where L + x : V s ( x ) → V t ( x ) and L − x : V t ( x ) → V s ( x ) . Let ε ∈ {± } . The map L εx is the valueof the functor on the interval [ − ,
1] with standard orientation, together with the0-dimensional submanifold { } with orientation ε and label x ∈ D . If ε = +, thesub-interval [ − ,
0) is labelled by s ( x ) ∈ D and (0 ,
1] by t ( x ) ∈ D , while for ε = − ,the label of [ − ,
0) and (0 ,
1] is t ( x ) and s ( x ), respectively, as in figure 1 a, b). Anarbitrary morphism can be obtained by composing and tensoring the above maps,as well as the cup and cap bordisms, which the functor maps to evaluation andco-evaluation.We can collect this data in a category D , together with a distinguished subset of arrows,as follows. Take D as objects of D . As space of morphisms i → j , for i, j ∈ D , take D := Hom k ( V i , V j ), the linear maps from V i to V j . Finally, fix a map D × {±} → Mor( D ),which assigns to ( x, ± ) the arrow L ± x , with source and target as described above. Let us look in more detail at an instance of a bordism category with defects in two dimen-sions; the exposition essentially follows [RS, Sec. 3]. A note on convention: by manifoldwe mean smooth manifold, and by a map between manifolds we mean a smooth map; afinite or countable disjoint union has an ordering of its factors, so that for two sets A , B the disjoint unions A ⊔ B and B ⊔ A are isomorphic but not equal. Sets of defect conditions
We start with the three sets D , D , and D , which are the sets of world sheet phases,domain wall conditions, and junction conditions, respectively. As above we have two maps s, t : D → D giving the phase to the left and right of a domain wall; our orientationconventions are shown in figure 1 c). For a junction in D we need to specify which domainwalls can meet with which orientations at a junction point.6 u, +) x s ( x ) t ( x ) x s ( x ) t ( x ) x s ( x ) t ( x ) x t ( x ) s ( x ) x s ( x ) t ( x ) x t ( x ) s ( x ) Figure 2:
Illustration of the condition of cyclic composability of domain walls. Given the n -tuple (( x , ε ) , . . . , ( x n , ε n )), the i ’th domain wall (counted anti-clockwise) is labelled by x i andis pointing towards the junction point if ε i = + and away from the junction point if ε i = − . Inthe present example the 6-tuple is (( x , +) , ( x , − ) , ( x , − ) , ( x , +) , ( x , − ) , ( x , +)). The imagesunder the maps s and t to D have to agree as shown, e.g. s ( x ) = s ( x ) and t ( x ) = s ( x ). Thejunction point has orientation ‘+’ and is labelled by u ∈ D . The combinatorial description thereof is a bit lengthy: Let D ( n )1 be the set of tuplesof n cyclically composable domain walls . By this we mean the subset of n -fold cartesianproduct ( D × {±} ) × n selected by the following condition: For (( x , +) , . . . , ( x n , +)) werequire t ( x i +1 ) = s ( x i ) and t ( x ) = s ( x n ). If some of the ‘+’ are changed for ‘ − ’, the roleof s and t is exchanged as in figure 2. The group C n of cyclic permutations acts on the n -tuples in D ( n )1 . The set D is equipped with a map j : D −→ ∞ G n =0 (cid:16) D ( n )1 /C n (cid:17) . (2.2)In words, for each element u of D , the map j determines how many domain walls can endat a junction labelled by u and what their orientations and domain wall conditions are, upto cyclic reordering.The map j is similar in spirit to the relation between D and D . There, we can combinethe ‘source’ and ‘target’ maps into a single map ( s, t ) : D → D × D , which determinesthe world sheet phases that must lie on the two sides of a domain wall labelled by a givenelement of D . There is no need to divide by the symmetric group in two elements, becausethe orientations allow one to distinguish the ‘left’ and ‘right’ side of a domain wall. Objects
In short, an object is a disjoint union of a finite number of unit circles S with markedpoints, together with a germ of a collar. In more detail, for a single S the structure is as follows. Take U = S to be the unitcircle in C , decorated as in section 2.1: a 0-dimensional submanifold U ⊂ S (i.e. a set of This is more restrictive than allowing general one-dimensional manifolds as in section 2.1 but does notloose any generality and has the advantage that objects form a set, and that the connected componentsof an object are already ordered by our convention on disjoint unions. + −− ws ( w ) t ( w ) xs ( x ) t ( x ) y t ( y ) s ( y ) zt ( z ) s ( z ) ReIm abc d Figure 3:
Illustration of a collar which forms part of the data for an object in the bordismcategory. In the notation of the text, the solid (blue) circle is a unit circle U = S , the shadedarea is an open neighbourhood A , the solid (red) short lines form the oriented submanifold A of A which intersects S in U . Our convention for the orientation of A induced by that of U (the signs ‘ ± ’) is as shown. The elements a, b, c, d ∈ D label connected components of U andtheir extension A ; these labels have to agree with the source and target maps of the domain walllabels w, x, y, z ∈ D as shown. E.g. t ( w ) = a = s ( x ). points decorated by signs ± ), a map d : π ( U ) = U → D , and a map d : π ( U ) → D ,where U = S \ U . The maps d , d have to be compatible with s, t as in section 2.1 (cf.figure 1 a, b)).A collar is, in short, an extension of the above structure to an open neighbourhoodof S in C , see figure 3. Let A be an open neighbourhood of S , and let A be a one-dimensional submanifold, closed in A , which intersects S transversally (the tangents to A and S are linearly independent at intersection points). Set A = A \ A . There aremaps ˆ d i : π ( A i ) → D i , i = 1 ,
2, compatible with s, t as in figure 1 c). The restriction of A and A to S has to reproduce U and U with labelling and orientation, with conventionsas in figure 3. Finally, A carries a metric in conformal gauge, i.e. g ( z ) ij = e σ ( z ) δ ij for areal-valued function σ on A .Two collars are equivalent if they agree in some open neighbourhood of S ; an equiva-lence class is called a germ of collars .For a disjoint union U of such S with collars, write U in for the subset obtained bytaking only points | z | ≥ S , and U out when taking only points with | z | ≤ Morphisms
Morphisms M : U → V are equivalence classes of surfaces with extra structure as in section2.1, together with a metric. Thus we have a decomposition M = M ∪ M ∪ M , mapsˆ d i : π ( M i ) → D i , i = 0 , ,
2. The map ˆ d is compatible with s, t as in figure 1 c). Goingbeyond the level of detail in section 2.1, we also require the following:8 + u − x x x x x Figure 4:
Some domain walls and two junctions placed on S ; we only display a fragmentafter projection to the plane. The two junctions are labelled by the same junction condition u ∈ D but with opposite orientation ‘ ± ’. Here, j ( u ) is the cyclic permutation equivalence classof (( x , +) , ( x , +) , ( x , − ) , ( x , +) , ( x , − )). Thus, the junction labelled by u with orientation ‘+’must have domain walls ( x , +) , ( x , +) , ( x , − ) , ( x , +) , ( x , − ) attached in anti-clockwise order,where for ( x i , +) the domain wall is oriented towards the junction and for ( x i , − ) it is orientedaway from the junction. The junction labelled by u with orientation ‘ − ’ must have domain walls( x , +) , ( x , − ) , ( x , +) , ( x , − ) , ( x , − ) attached in anti-clockwise order. compatibility condition for ˆ d : For a point p ∈ M labelled by u ∈ D (i.e. ˆ d ( p ) = u ), let(( x , ε ) , . . . , ( x n , ε n )) denote the domain wall conditions and orientations in anti-clockwiseorder (with arbitrary starting point). We require that j ( u ) is the cyclic permutationequivalence class of (( x , ε ) , . . . , ( x n , ε n )) if the junction point has orientation ‘+’, cf.figure 2, and that it is in the class of (( x n , − ε n ) , . . . , ( x , − ε )) if the junction orientationis ‘ − ’. Junctions with opposite orientation are dual in the following sense (figure 4): ifthe bordism is a 2-sphere with two antipodal junction points both labelled by u but withorientations ‘+’ and ‘ − ’, the domain walls starting at the two junctions can be joined up(intersection-free) by half-circles around the S . boundary parametrisation : A choice U ′ , V ′ of collars representing the germs U , V , to-gether with injective maps f in : U in → M and f out : V out → M which preserve the orienta-tion, metric, boundary, 1-dimensional submanifold (with orientation) and labelling. Theimages of the factors S in U ′ and V ′ are disjoint and cover the boundary of M .Two surfaces are equivalent if they are isometric and the isometry preserves the decom-position M = M ∪ M ∪ M together with orientations and labelling, and commutes withthe boundary parametrisation in some open neighbourhood of ∂M .Composition of morphisms is defined by choosing representatives and gluing via theboundary parametrisation; the collars ensure that this does not introduce ‘corners’ andresults again in a surface as described above. The equivalence class of the glued surface isindependent of the choice of representatives. Identities and symmetric structure
So far there are no identity morphisms. We will add these by hand by extending themorphisms to include permutations of the S factors in the disjoint union of a given object U . If we denote the permutation by σ and the permuted disjoint union by σ ( U ), we addmorphisms σ : U → σ ( U ). Each morphism U → V is either a permutation (only possible9f V = σ ( U )) or a bordism; composing σ − ( U ) σ −→ U M −→ V τ −→ τ ( V ) produces a bordism σ − ( U ) M ′ −→ τ ( V ), where M ′ differs from M only in the boundary parametrisation maps.This endows the bordism category with a symmetric structure (the tensor product isdisjoint union as usual). Topological domain walls and junctions
Denote the symmetric monoidal category described above byBord def2 , ( D , D , D ) , (2.3)or Bord def2 , for short. A two-dimensional quantum field theory with defects can now bedefined as a symmetric monoidal functor Q from Bord def2 , to topological vector spaces,which depends continuously on the moduli (namely, the metric on a morphism M , thedecomposition M = M ∪ M ∪ M , and the boundary parametrisation). Remark 2.1.
It is also easy to say when such a functor Q describes a conformal field theorywith defects . Namely, the vector space Q ( U ) assigned to an object U has to be independentof the conformal factor e σ ( z ) giving the metric g ( z ) ij = e σ ( z ) δ ij on U , and the linear map Q ( M ) assigned to a morphism M changes by at most a scalar factor if the metric on M is changed by a conformal factor g e f g for some f : M → R . Thus Q would in generalonly give a projective functor if one passes to conformal equivalence classes of manifolds(it would be a true functor if the so-called central charge vanishes).With respect to a chosen Q , we can define an interesting subset of domain walls andjunctions: • topological domain walls are elements x of D such that1. for all objects U , the vector space Q ( U ) is unchanged under isotopies movingcomponents of U labelled by x (and their extension into the collars with them)such that no point of U crosses the point − ∈ S . This condition renders thespace of such isotopies contractible (on germs of collars). In particular, a full 2 π -rotation is excluded, as it would in general induce a non-trivial endomorphismof Q ( U ). The metric on U stays fixed.2. for all morphisms M , Q ( M ) is invariant under isotopies moving componentsof M labelled by x while leaving M fixed and restricting on ∂M to isotopiesrespecting the condition in 1. • topological junctions are elements u of D such that j ( u ) only contains elements of D labelling topological domain walls, and such that Q ( M ) is invariant under isotopiesmoving components of M labelled by topological domain wall conditions and pointsin M labelled by u .From now on we will concentrate on topological domain walls and junctions. We willdenote the corresponding subsets by D top i ( Q ), i = 0 ,
1, or just D top i .10 .4 2-categories of defect conditions Let us fix a two-dimensional field theory with defects as above, i.e. a continuous symmetricmonoidal functor Q from Bord def2 , ( D , D , D ) to an appropriate category of topologicalvector spaces. The construction below will only make use of the sets D and D , but notof D . To emphasise this, we take D = ∅ (i.e. no junctions are allowed).Consider the topological domain walls s, t : D top1 ( Q ) → D . This is a pre-category(which is nothing but a graph, see e.g. [ML, Ch. II.7]), and the aim of this section is toshow that Q turns the free category (with conjugates) generated by this pre-category into a2-category. This 2-category can be thought of as capturing some of the genus-0 informationof the field theory Q . Our conventions for bicategories are collected in appendix A.Recall (e.g. from [ML, Ch. II.7]) that the free category is generated by tuples of com-posable arrows. By the free category with conjugates we mean the category whose objectsare D and whose morphisms a → b (for a, b ∈ D ) are tuples x ≡ (( x , ε ) , . . . , ( x n , ε n )) (2.4)where x i ∈ D top1 , ε i ∈ {±} . As for cyclically composable domain walls, if all signs ε i = +,then we require s ( x i ) = t ( x i +1 ) and s ( x n ) = a , t ( x ) = b . If some ε i = − , the role of s and t changes as in figure 2. Composition of x : a → b and y : b → c is by concatenation, y ◦ x = (( y , ν ) , . . . , ( y m , ν n ) , ( x , ε ) , . . . , ( x n , ε n )) . (2.5)Let us denote this category by D ≡ D [ D , D top1 ]. Morphism spaces a → b are written as D ( a, b ). The conjugation is the involution ∗ : D ( a, b ) → D ( b, a ) given by(( x , ε ) , . . . , ( x n , ε n )) ∗ = (( x n , − ε n ) , . . . , ( x , − ε )) . (2.6)Note that endomorphisms x : a → a in D are precisely the tuples of cyclically composabledomain walls. In particular, for any x, y : a → b , the morphism y ◦ x ∗ is cyclicallycomposable.To define the 2-category structure on domain walls, we need the notion of translationand scale invariant families of states. Their definition will take us a few paragraphs.We will only need to know Q on a subset of bordisms, each of which consists of asingle disc in R from which a number of smaller discs have been removed (if there wereno domain walls, these bordisms would form the little discs operad, see e.g. [Ma]). Themetric on the bordism is the one induced by R and the boundaries are parametrised bylinear maps that are a combination of a translation and a scale transformation x rx + v ,where x, v ∈ R and r ∈ R > .The objects which serve as source and target of these bordisms are described as follows.Let x be cyclically composable and denote by O ( x ; r ) an object in the bordism categoryconsisting of a single S with 0-dimensional submanifold given by n points not containing − ∈ S . These are clockwise cyclically labelled x , . . . , x n , such that x labels the firstpoint in clockwise direction from − ∈ S . The collar around S is obtained by taking11 x ; R = Q ! ( ψ x ; r ) = Q ! ( ψ x ; r ) = Q ! ( ψ x ; r ) Figure 5:
Illustration of the condition for scale and translation invariant family of states: Let ψ x be such a family. The figure shows Q applied to three annuli, understood as bordisms O ( x, r i ) → O ( x, R ), for i = 1 , ,
3, where r i denotes the radius of the inner disc of the i ’th annulus shownabove. All three annuli have the same outer radius R . Applying Q to the bordism and evaluatingthe resulting linear map on ψ x,r i ∈ Q ( O ( x, r i )), i = 1 , ,
3, results always in the vector ψ x ; R ∈ Q ( O ( x, R )) of the same family. concentric copies to fill a small neighbourhood. The conformal factor defining the metricon the collar is e σ = r , so that the parametrising map x rx + v , which takes the S (with radius 1) to a circle of radius r is an isometry. As all x i are in D top1 , by definition thevector space Q ( O ( x ; r )) does not depend on the precise position of the n marked points on S , as long as they are in the prescribed ordering. However, the vector space Q ( O ( x ; r ))may still depend on r . Let D ( R ; r, v ) : O ( x ; r ) → O ( x ; R ) be the bordism given by a disc of radius R in R centred at the origin, from which a smaller disc of radius r and centre v has been removed.The domain walls are straight lines and the boundary parametrisation is given by scalingand translation as above (see figure 5). A scale and translation invariant family ψ x is afamily of vectors { ψ x ; r } r ∈ R > with ψ x ; r ∈ Q ( O ( x ; r )) such that ψ x ; R = Q (cid:0) D ( R ; r, v ) (cid:1) ( ψ x ; r ) for all r, R > , v ∈ R with r + | v | < R . (2.7)This condition is illustrated in figure 5. The space of scale and translation invariantfamilies , H inv ( x ) , (2.8)is defined to be the vector space of all scale and translation invariant families ψ x ≡{ ψ x ; r } r ∈ R > for fixed x . The space H inv ( x ) may be zero-dimensional.Scale and translation invariant families have the following important property: allamplitudes Q ( M ) – with M a disc in R with smaller discs removed – are independentof the position and size of an in-going boundary circle O ( x ; r ) for which ψ x ; r is insertedas the corresponding argument. This can be seen by using functoriality of Q to cut outa disc D ( R ; r, v ) from M containing such an in-going boundary, then moving the in-goingboundary circle using the defining property (2.7), and finally gluing the resulting disc D ( R ; r ′ , v ′ ) back. In many examples (but not always), the spaces Q ( O ( x ; r )) for different values of r are isomorphic,with a preferred isomorphism given by evaluating Q on an annulus with the two radii. But even in thiscase we do not demand that one passes to a formulation of the theory where these state spaces are actually equal . b a ( x , − )( x , +) ( x , − )( x , − )( x , +)( x , − ) ( x , +)( x , +) b) {{ b auvx ∗ zx x p y y q z z r c) { {{ { c b ax ∗ y ∗ x ′ y ′ x x p y q y x ′ x ′ r y ′ s y ′ q p Figure 6:
Bordisms defining the structure maps for the 2-category. a) the identity on a 1-morphism x : a → b ; b) vertical composition of 2-morphisms u ∈ D ( x, y ) and v ∈ D ( y, z ),where x, y, z ∈ D ( a, b ) (drawing the vector inside the cut-out disc means that this vector is to beused as the corresponding argument after applying Q ); c) horizontal composition of 2-morphisms p ∈ D ( x, x ′ ) and q ∈ D ( y, y ′ ), where x, x ′ : a → b and y, y ′ : b → c . Given x, y : a → b , we define the space of 2-morphisms from x to y to be D ( x, y ) := H inv ( y ◦ x ∗ ) . (2.9)The identity 2-morphisms, and the horizontal and vertical composition are defined by thebordisms shown in figure 6. The identity 1-morphism a : a → a , for a ∈ D , is the emptytuple a = (). Remark 2.2. (i) In order to obtain families of states from the bordisms shown in figure 6,one uses that there is an R > -action on metric bordisms given by rescaling the metric. Forthe disc shaped bordisms in R relevant here, this amounts to a rescaling by some R > M and assume thatthe radius of its outer disc is 1. Thus M : O ( z ◦ y ∗ ; r ) ⊔ O ( y ◦ x ∗ ; r ) −→ O ( z ◦ x ∗ ; 1) . (2.10)For each R > RM : O ( z ◦ y ∗ ; Rr ) ⊔ O ( y ◦ x ∗ ; Rr ) → O ( z ◦ x ∗ ; R ).Given two families φ ∈ H inv ( z ◦ y ∗ ) and φ ∈ H inv ( y ◦ x ∗ ), we obtain a family of vectors ψ R := Q ( RM ) (cid:0) φ Rr , φ Rr (cid:1) ∈ Q (cid:0) O ( z ◦ x ∗ ; R ) (cid:1) . (2.11)The family { ψ R } R ∈ R > is again scale and translation invariant. This follows by substitutinginto the defining property (2.7) and using that φ and φ are scale and translation invariantfamilies.(ii) For all r > v ∈ R with r + | v | <
1, the bordism D (1; r, v ) : O ( x ; r ) → O ( x ; 1)induces the identity map on H inv ( x ). This is just a reformulation of the defining property(2.7) using the prescription in (i). In other words, cylinders give the identity map on H inv ,not just idempotents. This is important when verifying that the bordism in figure 6 a) isindeed the unit for the vertical composition in figure 6 b).13) { { x x x m x x ∗ ab b) { { x x x m x x ∗ ba c) { { x x x m x ∗ xba d) { { x x x m x ∗ xab Figure 7:
Bordisms defining the adjunction maps in D [ Q ]: The left and right adjoint of x : a → b is x ∗ : b → a and applying Q to the bordisms shown gives the adjunction maps a) b x : b → x ◦ x ∗ ;b) d x : x ∗ ◦ x → a ; c) ˜ b x : a → x ∗ ◦ x ; d) ˜ d x : x ◦ x ∗ → b . The fact that we are working with topological domain walls and with scale and trans-lation invariant families of states ensures that the properties of a 2-category are satisfied(as composition of 1-morphisms is strictly associative, and the unit 1-morphisms are strict,we indeed have a 2-category and not only a bicategory). Let us denote this 2-category as D [ Q ] ≡ D [ D , D top1 ; Q ] . (2.12)As was to be expected, moving one dimension up from the example in section 2.2 also in-creased the categorial level: in this construction the pre-category D top1 ⇒ D gets extendedto a 2-category. Remark 2.3. (i) Actually, D [ Q ] carries more structure. For example, each 1-morphism x : a → b has a left and a right adjoint, namely x ∗ : b → a , together with adjunction mapsas shown in figure 7 (see [Gr, Sec. I.6] for more on adjunctions in bicategories). Such rigidand related structures on the category of defects were discussed already in [Fr¨o1, MN, CR].Rigid and pivotal structures on the 2-category D [ Q ] were studied in the context of planaralgebras in [Go, DGG].(ii) The 2-category D [ Q ] is an invariant attached to a quantum field theory with defects.Interestingly, even though only ‘topological data’ enters its definition (topological domainwalls and scale and translation invariant families of states), general quantum field theoriesdo produce more general 2-categories D [ Q ] than topological field theories. In a nutshell,the reason is that the rigid structure mentioned in (i) will tend to produce integer quan-tum dimensions for topological field theories, while for example in rational conformal fieldtheories non-integer quantum dimensions occur. A planar algebra [Jo] can be understood as a two-dimensional theory with exactly two world sheetphases D = { a, b } , exactly one topological domain wall type D = { a x −→ b } , and no junctions, D = ∅ .Furthermore, the theory is only defined on genus zero surfaces with exactly one out-going boundary circle,that is, on discs with smaller discs removed, see figures 5–7. A notion of 2d TFT with domain walls was also studied in [KPS] in relation to subfactor planaralgebras. Apart from there being exactly two world sheet phases and one type of domain wall – as isusual in the planar algebra setting – there is one important difference: in [KPS] a bordism is in addition
14n slightly more detail, fix a topological field theory with defects and take k = C .Consider the bordism M : ∅ → ∅ given by a torus (say [0 , × [0 ,
1] with opposite edgesidentified) with a single defect line labelled x ∈ D wrapping a non-contractible cycle (say[0 , × { } ). We label the unique connected domain of the bordism by a ∈ D , so that x : a → a . Then Q ( M ) : C → C is just a number. This number can be computed in twoways. Let M | : O ( x ) → O ( x ) be the annulus obtained by cutting M along { } × [0 , M . Then M | is just the cylinder over O ( x ). Wewill learn later in remark 3.12, that for a topological field theory the space of scale andtranslation invariant states H inv ( x ) can be identified with the image of Q ( M | ) (which is anidempotent in topological field theory) in Q ( O ( x )). Thus, using also functoriality of Q , Q ( M ) = tr Q ( O ( x )) Q ( M | ) = tr H inv ( x ) id = dim( H inv ( x )) ∈ Z ≥ . (2.13)On the other hand, we can consider M − : O ( a ) → O ( a ), which is obtained by cutting M along [0 , × { } , i.e. by not identifying the horizontal edges. The endomorphism Q ( M − ) : Q ( O ( a )) → Q ( O ( a )) is called defect operator for the defect x : a → a , we will return to thisbriefly in section 3.5 below. By the same reasoning as above, Q ( M ) = tr Q ( O ( a )) Q ( M − ).Let C O ( a ) be the cylinder over O ( a ). By functoriality (and again only for topological fieldtheory) we have Q ( M − ) = Q ( C O ( a ) ) ◦ Q ( M − ) ◦ Q ( C O ( a ) ), so that the image of Q ( M − ) lies inthe image of the idempotent Q ( C O ( a ) ) and Q ( M − ) acts trivially on the kernel of Q ( O ( a )).By restriction we obtain an endomorphism Q ( M − ) : H inv ( a ) → H inv ( a ) and the tracecan be computed in this restriction, Q ( M ) = tr Q ( O ( a )) Q ( M − ) = tr H inv ( a ) Q ( M − ) . (2.14)The fact that the traces (2.13) and (2.14) agree is known as the Cardy condition (becauseof the paper [Ca]) and was first investigated for topological defects in [PZ] in the contextof rational conformal field theory.In summary, for topological field theories, the trace of a defect operator over the spaceof scale and translation invariant states is equal to the dimension of a vector space, andis thus a non-negative integer. In rational conformal field theories with non-degeneratevacuum (this means that the space H inv ( a ) is one-dimensional), the defect operator actsby multiplying with a number – the (left or right) quantum dimension of x – and the tracetr H inv ( a ) Q ( M − ) is then equal to this number. In many examples, this quantum dimensionis not an integer (and not even rational, though still algebraic). This is the case for theexamples studied in [PZ] and [FRS1, Fr¨o1]. It is difficult to find functors from Bord def2 , ( D , D , D ) to topological vector spaces whichdepend non-trivially on the metric. On the other hand, it is easy to construct examples equipped with a decomposition into ‘genus 0 components’, and bordisms with different such decompositionsare considered distinct, unless they have a common refinement (see [KPS, Def. 2.7]). This excludes forexample the decomposition of a torus along different cycles to be used below (the ‘Cardy condition’). Thusthe functors constructed in [KPS] are in general not defect TFTs in our sense (cf. eqn. (3.2) below). k . Instead of posing restrictions on the functor one can modify the bordism category accord-ingly. This leads us to define a symmetric monoidal category of smooth bordisms withdefects , which we denote as Bord def , top2 , ( D , D , D ) . (3.1)The modifications relative to the definition in section 2.3 are as follows. • Objects:
The collar around an S no longer carries a metric. • Morphisms:
The manifold does not carry a metric and the parametrising maps areonly required to be smooth (rather than isometric). Two bordisms are equivalent ifthere is a diffeomorphism between them which preserves orientation, decomposition,labelling, as well as the image of the point − ∈ S in each connected component ofthe boundary (rather than commuting with the parametrising maps in some neigh-bourhood of the boundary).The symmetric monoidal structure is as in section 2.3.Note that this definition is different form the standard 2-bordism category for topolog-ical field theories even in the case without domain walls ( D = {∗} and D = D = ∅ )because we have still added the identities (and S n -action) by hand to the space of mor-phisms; in particular, the cylinder over a given object U is not the identity morphism (butit is still an idempotent).Denote by V ect f ( k ) the symmetric monoidal category of finite-dimensional k -vectorspaces. Fix furthermore sets D i , i = 0 , ,
2, of defect labels with maps as required. Theaim of this section is to construct examples of symmetric monoidal functors T : Bord def , top2 , ( D , D , D ) −→ V ect f ( k ) . (3.2)We will do this via a lattice TFT construction which is a straightforward generalisation ofthe original lattice TFT without domain walls [BP, FHK] and of the lattice constructionof homotopy TFTs in [Tu, Sec. 7]. A construction of field theory correlators on arbitraryworld sheets in the presence of domain walls first appeared in [FRS1, FRS2, Fr¨o1], whereit was carried out in the context of two-dimensional rational conformal field theory. Theconstruction for TFTs given in this section can be extracted from this in the special casethat the modular category underlying the CFT is that of vector spaces.Note, however, that this will not give the most general such functor T (as it does noteven in the case without domain walls); a classification of functors (3.2) akin to the one inthe situation without domain walls in [Di, Ab] is at present not known. A classification of 2d16FTs with defects that can be extended down to points has been reported in [SP]; relatedresults are given in [Lu1, Ex. 4.3.23]. Two dimensional homotopy TFTs over X = K ( G, G , have been classified in [Tu, Thm. 4.1]. The classification in the case ofsimply connected X is given in [BT, Thm. 4.1].The construction works in two steps. In the first step, we introduce a larger category,Bord def , top , cw2 , , where objects and morphisms are in addition endowed with a cell decompo-sition (section 3.2). It comes with a forgetful functor F : Bord def , top , cw2 , → Bord def , top2 , , whichis surjective (not just essentially surjective) and full (but not faithful). We then constructa symmetric monoidal functor T cw : Bord def , top , cw2 , → V ect f ( k ) (section 3.5). In the secondstep, we show that T cw is independent of the cell-decomposition in the sense that thereexists a symmetric monoidal functor T which makes the diagramBord def , top , cw2 , F (cid:15) (cid:15) T cw / / V ect f ( k )Bord def , top2 , ∃ ! T lllllll (3.3)commute on the nose (section 3.6). Since F is surjective and full, this diagram defines T uniquely. We will use a class of cell decompositions introduced in [Ki, Def. 5.1], called PLCW decom-positions there. These are less general than CW-complexes, which for example allow oneto decompose S into a 0-cell (a point on S ) and 2-cell ( S minus that point). However,they are more general than regular CW-complexes, where one cannot identify differentfaces of one given cell, or even triangulations, where in addition each cell is a simplex. Itis shown in [Ki] that any two PLCW decompositions are related by a simple collection oflocal moves.Given a compact n -dimensional manifold M , possibly with non-empty boundary, wewill consider decompositions of M into a finite number of mutually disjoint open k -cells, k = 0 , . . . , n , with the following properties. Let B k be the closed unit ball in R k and let ˚ B k By a defect TFT with only invertible defects we mean the case that D = {∗} , D = ∅ and that D = G is a group (more generally we can take s, t : D → D to be a groupoid). We demand that thelinear map assigned by T in (3.2) to a bordism does not change if we replace two parallel defect lines withopposite orientation “ ⇄ ” labelled by the same element of D by the ‘reconnected’ defect lines “ ⊃⊂ ”. Givena bordism M with defect lines labeled by group elements in G , we can construct a principal G -bundleon the bordism by taking the trivial G -bundle over each component of M and choosing the transitionfunctions across M to be multiplication with the group element labelling that component of M . Thisgives a functor from Bord def , top2 , ( {∗} , G, ∅ ) to principal G -bundles with two-dimensional base.
17) b)
Figure 8:
Allowed configurations of 2-cells and defects: (a) 2-cell containing a point from M ;here only a star-shaped pattern of domain walls is allowed and each edge has to be crossed byprecisely one domain wall. (b) 2-cell containing no point from M but part of M ; here only onesegment of M may lie in the 2-cell, and it must enter and leave the 2-cell via distinct edges. be its interior. For each k -cell C there has to exist a continuous map ϕ : B k → M such that- C is the homeomorphic image of ˚ B k ,- there exists a decomposition of the boundary S k − of B k which gets mapped by ϕ tothe decomposition in M ,- ϕ is a homeomorphism when restricted to the interior of each cell on S k − .For more details we refer to [Ki]. We just note that this last condition excludes thedecomposition of S into a single 0-cell and a single 2-cell mentioned above.By abuse of terminology, we will refer to a decomposition of M as just described simplyas a cell decomposition C ( M ). The set of i -dimensional cells ( i -cells for short) will be called C i ( M ).The category Bord def , top , cw2 , ( D , D , D ) is the same as Bord def , top2 , ( D , D , D ), exceptthat objects and morphisms are equipped with the following extra structure: • Objects:
The 1-dimensional part of an object U (the disjoint union of circles, nottheir collars) are equipped with a cell decomposition C ( U ) such that each point ofthe set U of marked points lies in a 1-cell (recall that a 1-cell is homeomorphic toan open interval), and such that each 1-cell contains at most one point of U . • Morphisms:
By a bordism with cell decomposition we mean a bordism M = M ∪ M ∪ M in Bord def , top2 , , which is equipped with a cell decomposition C ( M ) such that – the 1-dimensional submanifold M only intersects 1-cells and 2-cells, but not0-cells. Each 1-cell intersects M in at most one point. – each point of M lies in a 2-cell, and each 2-cell contains at most one point of M . A 2-cell containing a point of M must be homeomorphic to one of thetype shown in figure 8 a), i.e. it may only contain a ‘star-shaped’ configurationof domain walls, such that each edge of this 2-cell is traversed by exactly onedomain wall. To match [Ki] we should work in the PL setting. We thus impose the additional condition that themanifold M with the decomposition as defined is homeomorphic to a PLCW complex. This is automaticallysatisfied if the manifold and the cell maps are already PL. a 2-cell containing no point from M but a part of M must be homeomorphicto one of the type shown in figure 8 b), i.e. its intersection with M is a singleopen interval.If M has non-empty boundary ∂M , we demand that ∂M is a union of 0-cells and1-cells (and hence does not intersect 2-cells). Furthermore, the decomposition of ∂M has to be the image of the cell decomposition of the source and target object underthe parametrising maps.Morphisms are equivalence classes of bordisms with cell decomposition. Two bor-disms with cell decomposition are equivalent if their underlying bordisms are equiva-lent in Bord def , top2 , and if the two cell decompositions are related by an isotopy (whichat each instance has to give a bordism with cell decomposition).All cells in C ( U ) and C ( M ) are a priori unoriented. However, the 2-cells of C ( M ) have anorientation induced by that of M . By an algebra we shall always mean a unital, associative algebra over k . The centre Z ( A )of an algebra A is the (commutative, unital) subalgebra Z ( A ) = (cid:8) z ∈ A (cid:12)(cid:12) za = az for all a ∈ A (cid:9) . (3.4)Given a right A -module M and a left A -module N , the tensor product M ⊗ A N is definedas the cokernel M ⊗ A ⊗ N l − r −−→ M ⊗ N π ⊗ −→ M ⊗ A N , (3.5)where l ( m ⊗ a ⊗ n ) = m ⊗ ( a.n ) and r ( m ⊗ a ⊗ n ) = ( m.a ) ⊗ n denote the left and rightaction. Given an A - A -bimodule X , we shall also require the ‘cyclic tensor product’ whichidentifies the left and right action of A on X . We denote this tensor product by (cid:9) A X ; itis defined as the cokernel A ⊗ X l − r −−→ X π ⊗ −→ (cid:9) A X , (3.6)where l ( a ⊗ x ) = a.x and r ( a ⊗ x ) = x.a . Note that (cid:9) A X is in general no longer an A - A -bimodule; it does, however, carry a coinciding left and right action of Z ( A ). Denoteby [ A, A ] the linear subspace of A (not the ideal) generated by all elements of the form ab − ba . Then by definition (cid:9) A A = A/ [ A, A ]. A Frobenius algebra is a finite-dimensional algebra A together with a linear map ε : A → k , called the counit , such that the bilinear pairing h a, b i := ε ( ab ) (3.7)on A × A is non-degenerate. We denote by β : k → A ⊗ A the unique linear map suchthat (( ε ◦ m ) ⊗ id A ) ◦ (id ⊗ β ) = id A . In other words, if a i is a basis of A and a ′ i the dual The quotient A/ [ A, A ] is not necessarily isomorphic to Z ( A ) – just take A to be the 3-dimensionalalgebra of upper triangular 2 × Z ( A ) ∼ = k while A/ [ A, A ] ∼ = k ⊕ k . h a i , a ′ j i = δ i,j , then β (1) = P i a ′ i ⊗ a i . The defining property canbe written as X i h x, a ′ i i a i = x for all x ∈ A . (3.8)By associativity of A , the pairing (3.7) is always invariant , i.e. h a, bc i = h ab, c i .For a ∈ A let L a : A → A be the left multiplication by a , L a ( b ) = ab . We say that A isa Frobenius algebra with trace pairing if it is a Frobenius algebra whose counit is given by ε ( a ) = tr A ( L a ). Note that ‘Frobenius’ is extra structure on an algebra, while ‘Frobeniuswith trace pairing’ is a property of an algebra.In the following we list some of the special properties of Frobenius algebras with tracepairing. Firstly, it is automatically symmetric : h a, b i = h b, a i . Consequently also β issymmetric: β (1) = P i a ′ i ⊗ a i = P i a i ⊗ a ′ i . Next, by the definition of the counit we havethe following identity: h x, i = ε ( x ) = tr A ( L x ) = P i a ∗ i ( xa i ) = P i h xa i , a ′ i i = h x, P i a i a ′ i i .Since this holds for all x ∈ A , we conclude X i a i a ′ i = 1 . (3.9)For all a, b ∈ A we have X i ( aa ′ i b ) ⊗ a i = X i a ′ i ⊗ ( ba i a ) ∈ A ⊗ A ; (3.10)this can be proved by pairing both sides with an arbitrary x . On the left hand side, pairingwith x produces P i h x, aa ′ i b i a i = P i h bxa, a ′ i i a i = bxa by (3.8). The right hand side givesthe same result.If we interpret β (1) as an element of the algebra A op ⊗ A , the two equations above leadto the following result: Lemma 3.1.
In the algebra A op ⊗ A we have(i) β (1) · ( a ⊗
1) = β (1) · (1 ⊗ a ) and ( a ⊗ · β (1) = (1 ⊗ a ) · β (1) ,(ii) β (1) · β (1) = β (1) .Proof. By definition, the product of A op ⊗ A is ( a ⊗ a ′ ) · ( b ⊗ b ′ ) = ( ba ) ⊗ ( a ′ b ′ ). Statement(i) is nothing but (3.10), while (ii) follows from (i) and (3.9) via β (1) · β (1) = P i β (1) · ( a ′ i ⊗ · (1 ⊗ a i ) = P i β (1) · (1 ⊗ a ′ i ) · (1 ⊗ a i ) = P i β (1) · (1 ⊗ ( a ′ i a i )) = β (1) · (1 ⊗ A -module M and a left A -module N as above. For Frobenius algebraswith trace pairing, the tensor product M ⊗ A N can be canonically identified with a subspace Let R a ( b ) = ba denote the right multiplication by a and set ε ( a ) = tr A ( L a ), ε ′ ( a ) = tr A ( R a ). Then( A, ε ) is a Frobenius algebra if and only if (
A, ε ′ ) is a Frobenius algebra and in this case ε ( a ) = ε ′ ( a ). For aproof see e.g. [FRS1, Lem. 3.9] in the special case that the tensor category is V ect f ( k ). Note that if ( A, ε )and (
A, ε ′ ) are not Frobenius, the statement may be false; for example, if A is the 3-dimensional algebraof upper triangular 2 × a = (cid:0) (cid:1) , then tr A ( L a ) = 1 and tr A ( R a ) = 2. M ⊗ N as follows. The tensor product M ⊗ N carries an A op ⊗ A -action. Define thelinear map p ⊗ : M ⊗ N → M ⊗ N to be the action of β (1) on M ⊗ N : p ⊗ ( m ⊗ n ) = β (1) . ( m ⊗ n ) = X i ( m.a ′ i ) ⊗ ( a i .n ) . (3.11) Lemma 3.2.
Let A be a Frobenius algebra with trace pairing.(i) p ⊗ ( m.a ⊗ n ) = p ⊗ ( m ⊗ a.n ) holds for all a ∈ A , m ∈ M , n ∈ N .(ii) p ⊗ is idempotent.(iii) im( p ⊗ ) = M ⊗ A N .Proof. Parts (i) and (ii) are immediate consequences of lemma 3.1. For part (iii), denoteby e ⊗ : im( p ⊗ ) −→ M ⊗ N , π ⊗ : M ⊗ N −→ im( p ⊗ ) , (3.12)the embedding of the image of p ⊗ into M ⊗ N and the projection from M ⊗ N to theimage. They satisfy e ⊗ ◦ π ⊗ = p ⊗ , π ⊗ ◦ e ⊗ = id im( p ⊗ ) . (3.13)Consider the diagram M ⊗ A ⊗ N l − r / / M ⊗ N π ⊗ / / f (cid:15) (cid:15) im( p ⊗ ) f ◦ e ⊗ y y r r r r r r V . (3.14)Here V is a vector space and f satisfies f ◦ ( l − r ) = 0. By part (i) the map π ⊗ satisfies π ⊗ ◦ ( l − r ) = 0. From f ( m.a ⊗ n ) = f ( m ⊗ a.n ) it is easy to see that f ◦ p ⊗ = f , and so f = ( f ◦ e ⊗ ) ◦ π ⊗ . Thus the above diagram commutes and we have verified the universalproperty of the cokernel.The construction of p ⊗ , e ⊗ , π ⊗ works similarly for the (cid:9) A tensor product of an A - A -bimodule X . In this case, p ⊗ ( x ) = X i a i .x.a ′ i . (3.15)We use the same notation e ⊗ : (cid:9) A X → X and π ⊗ : X → (cid:9) A X as above. For multipletensor products, the idempotents can be combined. For the state spaces of the TFT withdefects to be constructed below, the maps X ⊗ X ⊗ · · · ⊗ X n π ⊗ / / (cid:9) A n, X ⊗ A , X ⊗ A , · · · ⊗ A n − ,n X ne ⊗ o o (3.16)will be useful. Here the X i are bimodules with left/right actions of algebras as indicated.As above, the maps e ⊗ and π ⊗ satisfy π ⊗ ◦ e ⊗ = id and e ⊗ ◦ π ⊗ = p ⊗ . The projector p ⊗ inthis case is p ⊗ ( x ⊗ x ⊗ · · · ⊗ x n ) = X i ,i ,...,i n ( a i .x .a ′ i ) ⊗ ( a i .x .a ′ i ) ⊗ · · · ⊗ ( a i n .x n .a ′ i ) , (3.17)where the a i k are bases of the corresponding algebras.21 emma 3.3. Let A be a Frobenius algebra with trace pairing. Then Z ( A ) ∼ = (cid:9) A A = A/ [ A, A ] . Proof.
Here p ⊗ : A → A is given by p ⊗ ( x ) = P i a i xa ′ i . By the same reasoning as in (3.14)we conclude that the cokernel A/ [ A, A ] is isomorphic to im( p ⊗ ). It remains to show thatim( p ⊗ ) = Z ( A ). For z ∈ Z ( A ) we have p ⊗ ( z ) = P i a i za ′ i = z ( P i a i a ′ i ) = z , where we used(3.9). Thus Z ( A ) ⊂ im( p ⊗ ). Conversely, if x ∈ im( p ⊗ ) then x = P i a i xa ′ i . Then for all y ∈ A we have xy = X i a i xa ′ i y = m ◦ ( X i a i ⊗ ( xa ′ i y )) = m ◦ ( X i ( ya i x ) ⊗ a ′ i ) = X i ya i xa ′ i = yx , (3.18)where we used (3.10) together with symmetry of β . Thus im( p ⊗ ) ⊂ Z ( A ). Remark 3.4.
Recall that for a unital associative R -algebra (for R a commutative ring),the 0’th Hochschild homology HH ( A ) and cohomology HH ( A ) are given by HH ( A ) = A/ [ A, A ] and HH ( A ) = Z ( A ) , (3.19)see [Lo, Sec. 1.1, 1.5]. Similarly, for an A - A -bimodule X one has H ( A, X ) = (cid:9) A X and H ( A, X ) = { x ∈ X | a.x = x.a for all a ∈ A } . In the situation of lemma 3.3, that is if A is a Frobenius algebra with trace pairing, one finds H ( A, X ) ∼ = H ( A, X ). We will see in(3.49) below that the 0’th Hochschild (co)homology provides the state space which the 2dlattice TFT with defects assigns to a circle with a single marked point.
Fix sets D , D , D with maps s, t : D → D and a map j as in (2.2). The data thatserves as input to the lattice TFT construction is as follows.1. For each a ∈ D a Frobenius algebra A a with trace pairing.2. For each x ∈ D a finite-dimensional A t ( x ) - A s ( x ) -bimodule X x .There is also a piece of data associated to D , but we need a bit of preparation before wepresent it. For an A - B -bimodule X write X + ≡ X and write X − for the B - A -bimodule X ∗ . Recall the free category with conjugates D ≡ D [ D , D ] defined in section 2.4. For x ∈ D ( a, b ) we define the A b - A a -bimodule X x = X ε x ⊗ A , X ε x ⊗ A , · · · ⊗ A n − ,n X ε n x n , (3.20)where x = (( x , ε ) , . . . , ( x n , ε n )) and A i,i +1 denotes the algebra which acts from the righton X ε i i and from the left on X ε i +1 i +1 . Note that X y ◦ x = X y ⊗ A b X x for c y ←− b x ←− a . (3.21)22et χ be an element of D ( n )1 /C n as in (2.2). Let x ∈ D ( n )1 be a representative of χ , i.e. χ = [ x ], and let O χ = C n .x be the C n orbit of x in D ( n )1 (which is independent of the choiceof x ). Let π : J χ → O χ (3.22)be the vector bundle (with discrete base) whose fibres are given by the dual vector space π − ( y ) = Hom k ( (cid:9) A s ( y )= t ( y ) X y , k ) , (3.23)where Hom k ( U, V ) stands for the space of linear maps from U to V . Since y ∈ D ( s ( y ) , t ( y ))is cyclically composable by assumption, we indeed have s ( y ) = t ( y ). An element σ of thecyclic group C n acts on J χ by taking a vector ϕ ∈ π − ( y ) to ϕ ◦ σ − ∈ π − ( σ ( y )). By abuseof notation, here we also denoted by σ the linear isomorphism (cid:9) A s ( y ) X y → (cid:9) A s ( σy ) X σy obtained by shifting tensor factors. Denote by Γ( J χ ) inv the space of C n -invariant sectionsof the bundle J χ . The value of the section at y is then invariant under the action of thestabiliser of y in C n .For example, if χ = [ x ] is such that all ( x i , ε i ) in x are mutually distinct, then the orbithas lengths n , all stabilisers are trivial, and Γ( J χ ) inv is isomorphic to any one of the fibresof J χ . If all ( x i , ε i ) are identical, the orbit has length one and Γ( J χ ) inv consists of the C n invariant vectors in Hom k ( (cid:9) A s ( x ) X x , k ).With this preparation, we can finally state the third piece of data for the lattice TFTconstruction.3. For each u ∈ D , a vector ϕ u ∈ Γ( J j ( u ) ) inv .This complicated construction will later ensure that the junction-condition is unchangedunder ‘rotations which leave the attached domain walls invariant’, or in other words, it hasno preferred ‘starting edge’. Fix D i , i = 0 , , T cw : Bord def , top , cw2 , −→ V ect f ( k ) . (3.24)The action of T cw on objects is as follows. Denote by O an object of the bordism categoryBord def , top , cw2 , consisting of a single S . Recall that C ( O ) is the set of 1-cells (i.e. edges) ofthe cell decomposition of O . To each edge e ∈ C ( O ) we assign the vector space R e = ( A a ; e contains no marked point and carries label a ∈ D , X εx ; e contains a marked point with orientation ε and label x ∈ D , (3.25)23 − + ReIm xzy b ac e e e e e Figure 9:
Assignment of vector spaces R e to edges e ∈ C ( O ): The figure shows a circle withthree domains labelled a, b, c ∈ D and three marked points labelled x, y, z ∈ D . The circle isdecomposed into 5 edges, and the corresponding vector spaces are R e = X x , an A b - A a -bimodule; R e = A a ; R e = X ∗ z , an A a - A c -bimodule (while X z itself is an A c - A a -bimodule); R e = A c ; R e = X y , an A c - A b -bimodule. see figure 9 for an illustration. We set T cw ( O ) = O e ∈ C ( O ) R e . (3.26)For an object U = O ⊔ O ⊔ · · · ⊔ O n we take T cw ( U ) = T cw ( O ) ⊗ · · · ⊗ T cw ( O n ).There are two types of morphisms in the bordism category: permutations of objects,and bordisms. A permutation σ : U → σ ( U ) is mapped by T cw to the correspondingpermutation of tensor factors. The description of T cw for bordisms will take a little while.Given a bordism M : U → V (we assume that a representative of the equivalence class hasbeen chosen and use the same symbol), we will write the functor as a composition of twolinear maps T cw ( M ) : T cw ( U ) id T cw( U ) ⊗ P ( M ) −−−−−−−−−→ T cw ( U ) ⊗ Q ( M ) ⊗ T cw ( V ) E ( M ) ⊗ id T cw( V ) −−−−−−−−−→ T cw ( V ) . (3.27)We will now describe the vector space Q ( M ), and the maps P ( M ) (‘propagator’) and E ( M ) (‘evaluation’).We start with the vector space Q ( M ). Denote by ∂ in M the in-going part of the bound-ary of M , that is, the part parametrised by U , and by ∂ out M the out-going part of theboundary of M , parametrised by V .Consider triples ( p, e, or), where p ∈ C ( M ) is a 2-cell (i.e. an open polygon), e ∈ C ( M )is a 1-cell, and ‘or’ is an orientation of e . We only allow triples which satisfy the following Here, the tensor product stands for the tensor product over k of a family of vector spaces indexedby some set I (here C ( O )). To define this tensor product it is not necessary to choose an ordering of I ,i.e. a preferred way to write out the tensor product in a linear order. The same applies to similar tensorproducts below. Of course we could have taken the definition (3.26) also for a general object U instead of writing T cw ( U ) as a tensor product with implied ordering of the factors. However, in this way it is easier to seethat the functor is symmetric. x t ( x ) = bs ( x ) = a in out b) x a be α e γ e β p Figure 10:
Denote by O ( a ) the circle labelled by a ∈ D ; figure a) shows a bordism A ( x ) : O ( a ) → O ( b ) with a single circular domain wall labelled x . Figure b) gives a possible celldecomposition of A ( x ) into a 2-cell p , 1-cells e α , e β , e γ and two unnamed 0-cells. The edges areoriented for convenience, so that we can describe the two possible orientations simply by ± (theorientation is not part of the data of the cell decomposition); the orientation of the boundaryof p induced by the orientation of A ( x ) is indicated by the arrows placed in p . There are 4allowed triples: ( p, e α , +), ( p, e β , +), ( p, e γ , − ), ( p, e β , − ). The corresponding vector spaces are Q p,e α , + = A a , Q p,e β , + = X ∗ x , Q p,e γ , − = A b , Q p,e β , − = X x . As e α lies on ∂ in A ( x ), we have Q ( A ( x )) = Q p,e β , + ⊗ Q p,e γ , − ⊗ Q p,e β , − . condition: the orientation of M also orients p , and this in turn induces an orientation of theboundary ∂p ; we demand that ( e, or) is part of ∂p as an oriented edge. This is illustratedin figure 10. To each allowed triple ( p, e, or) we assign a vector space: Q p,e, or = A a ; ( e, or) does not intersect M and is in a componentof M labelled a , X x ; ( e, or) intersects a component of M which is labelled x and is oriented into the polygon p , X ∗ x ; ( e, or) intersects a component of M which is labelled x and is oriented out of the polygon p . (3.28)Here, the pair ( e, or) is understood as part of the boundary ∂p ; this is important if thesame edge e occurs twice in ∂p , see the example in figure 10. The vector space Q ( M ) isgiven by Q ( M ) = O ( p,e, or) , e/ ∈ ∂ in M Q p,e, or , (3.29)where the tensor product is taken over all allowed triples ( p, e, or) for which e does not liein ∂ in M .We now turn to the description of the map P ( M ) : k → Q ( M ) ⊗ T cw ( V ). Each edge e ∈ C ( M ) in the interior of M occurs in two allowed triples, let us call them ( p ( e ) , e, or )and ( p ( e ) , e, or ), where or and or are the two possible orientations of e , and p ( e ) i is thepolygon which contains the oriented edge ( e, or i ) in its boundary. Note that it may happen25hat p ( e ) = p ( e ) , as it does in figure 10. For each interior edge e define the linear map P e : k −→ Q p ( e ) ,e, or ⊗ Q p ( e ) ,e, or (3.30)according to the following two cases.1. If M does not intersect e then according to (3.28) we have Q p ( e ) ,e, or = Q p ( e ) ,e, or = A a , where a is the label of the component of M containing e . We take P e = β A a ,with β the dual of the Frobenius pairing as in section 3.3. Since A a has trace paring, β is symmetric and the map P e is independent of the choice of order of ( p ( e ) , e, or )and ( p ( e ) , e, or ).2. Suppose a component of M labelled by x intersects e . Let u i be a basis of X x andlet u ∗ i be the dual basis of X ∗ x . Choose the numbering ‘1’ and ‘2’ of ( p ( e ) , e, or )and ( p ( e ) , e, or ) so that the orientation of the domain wall M is such that it pointsinto the polygon p ( e ) at ( e, or ) and out of the polygon p ( e ) at ( e, or ). Then Q p ( e ) ,e, or = X x and Q p ( e ) ,e, or = X ∗ x (see figure 10) and we set P e ( λ ) = λ P i u i ⊗ u ∗ i .If e is an edge on the out-going boundary ∂ out M , i.e. the boundary component parametrisedby V , then there is exactly one allowed triple which contains e . Let ( p, e, or) be that triple.The parametrisation identifies e with an edge of C ( V ) which we also call e . In this case P e is defined as in (3.30), but with Q p ( e ) ,e, or and Q p ( e ) ,e, or replaced by Q p ( e ) ,e, or and R e .Comparing (3.25) and (3.28) (and using the conventions in figure 3), one checks that cases1 and 2 above still apply.Altogether, the map P ( M ) : k → Q ( M ) ⊗ T cw ( V ) is defined as P ( M ) = O e ∈ C ( M ) , e/ ∈ ∂ in M P e . (3.31)Finally, we need to define E ( M ) : T cw ( U ) ⊗ Q ( M ) → k . Note that T cw ( U ) ⊗ Q ( M )contains one factor Q p,e, or for each p ∈ C ( M ) and ( e, or) ∈ ∂p , even if e ⊂ ∂M , such that T cw ( U ) ⊗ Q ( M ) = O p ∈ C ( M ) , ( e, or) ∈ ∂p Q p,e, or . (3.32)For each polygon p ∈ C ( M ) we define a linear map E p : O ( e, or) ∈ ∂p Q p,e, or −→ k . (3.33)Fix p ∈ C ( M ). There are three cases to distinguish, depending on whether p intersects M and/or M .1. Suppose p intersects neither M nor M . Let a be the label of the component of M containing p . Choose an edge ( e , or ) ∈ ∂p and denote by ( e , or ) , ( e , or ) , . . . , ( e m , or m )all oriented edges of ∂p in anti-clockwise ordering. Let further q ⊗ q ⊗ · · · ⊗ q m ∈ Q p,e , or ⊗ Q p,e , or ⊗ · · · ⊗ Q p,e m , or m . (3.34)26ach Q p,e i , or i is equal to A a and we set E p ( q ⊗ · · · ⊗ q m ) = ε A a ( q · · · q m ), where ε A a is the counit of A a . By symmetry of the pairing of the Frobenius algebra A a , theresult is independent of the choice of starting edge ( e , or ).2. Suppose p intersects M but not M . In this case there is one oriented edge where M leaves p , which we take to be ( e , or ). Then we order the oriented edges of ∂p anti-clockwise as in 1. Let ( e i , or i ) be the edge where M enters p and let x be thelabel of the component of M in p . In the notation from (3.34), we have q ∈ X ∗ x , q i ∈ X x , and q , . . . , q i − ∈ A t ( x ) and q i +1 , . . . , q m ∈ A s ( x ) . We set E p ( q ⊗ · · · ⊗ q m ) = q (cid:0) ( q · · · q i − ) .q i . ( q i +1 · · · q m ) (cid:1) . Unlike case 1., case 2. did not involve an arbitrarychoice, and there is no invariance condition to check.3. Suppose p contains a point u from M of orientation ν u ∈ {±} and with labelˆ d ( u ) = t ∈ D . As in 1., we choose an arbitrary starting edge ( e , or ) ∈ ∂p andorder the remaining edges anti-clockwise. Each edge e i , i = 1 , . . . , m is transversed bya domain wall. For each i = 1 , . . . , m we thereby obtain a pair ( x i , ε i ) where x i ∈ D is the label of the domain wall crossing e i , and ε i = + if this domain wall is orientedinto the polygon at ( e i , or i ) and ε i = − otherwise. Let x = (( x , ε ) , . . . , ( x m , ε m )).If ν u = +, the labelling has to satisfy j ( t ) = [ x ] =: χ . According to the constructionin section 3.4, x ∈ O χ . Evaluating the section ϕ t ∈ Γ( J j ( t ) ) inv at x gives an element ψ ∈ Hom k ( (cid:9) A s ( x )= t ( x ) X x , k ). Precomposing with the projection π ⊗ : X x → (cid:9) A s ( x )= t (( x ) X x we obtain a linear form ψ ◦ π ⊗ : X ε x ⊗ · · · ⊗ X ε m x m → k . We set E p ( q ⊗ · · · ⊗ q m ) = ψ ◦ π ⊗ ( q ⊗ · · · ⊗ q m ). Independence of the choice of ( e , or ) follows since Γ( J j ( t ) ) inv consists of elements invariant under cyclic permutations.If ν u = − , the labelling has to satisfy j ( t ) = [ x ∗ ], and the above construction isrepeated with x ∗ instead of x .Figure 10 gives an example of case 2. There, ( e , or ) = ( e β , +), ( e , or ) = ( e γ , − ),( e , or ) = ( e β , − ), ( e , or ) = ( e α , +), and E p ( q ⊗ q ⊗ q ⊗ q ) = q ( q .q .q ), where q .q .q is the left/right action of q ∈ A b and q ∈ A a on q ∈ X x .Altogether, for E ( M ) we take E ( M ) = O p ∈ C ( M ) E p . (3.35)This completes the definition of T cw .Let us briefly illustrate the construction in two related examples; more examples willbe computed in section 3.7. For the bordism A ( x ) : O ( a ) → O ( b ) considered in figure 10,the composition of maps in (3.27) reads T cw ( A ( x )) : T cw ( O ( a )) id ⊗ P −−−→ R e α ⊗ Q p,e β , + ⊗ Q p,e γ , − ⊗ Q p,e β , − ⊗ R e γ E ⊗ id −−−→ T cw ( O ( b )) , (3.36)27here the edge on O ( a ) is identified with e α via the parametrisation, and the edge on O ( b )with e γ . Substituting the definition of these vector spaces and maps gives T cw ( A ( x )) : A a id ⊗ P −−−→ A a ⊗ X ∗ x ⊗ A b ⊗ X x ⊗ A b E ⊗ id −−−→ A b q P i,j q ⊗ u ∗ i ⊗ b ′ j ⊗ u i ⊗ b j P i,j u ∗ i ( b ′ j .u i .q ) b j . (3.37)This map can be defined for any two Frobenius algebras with trace pairing A, B and afinite-dimensional B - A -bimodule X . One can check that the image of this map lies in Z ( B ), and that the kernel of the projector p ⊗ onto Z ( A ) is contained in the kernel of T ( A ( x )). We therefore lose nothing if we restrict ourselves to Z ( A ) and Z ( B ) from thestart: D ( X ) : Z ( A ) −→ Z ( B ) , z X i,j u ∗ i ( b ′ j .u i .z ) b j . (3.38)This is an example of a defect operator, which we already briefly mentioned in remark2.3 (ii). Such defect operators have some nice properties : if X ∼ = X ′ as bimodules, then D ( X ) = D ( X ′ ); if Y is a C - B -bimodule, then D ( Y ) D ( X ) = D ( Y ⊗ B X ); and for the A - A -bimodule A one has D ( A ) = id Z ( A ) .A related example comes from the annulus as in figure 10, but without the domain wall x , so that necessarily a = b . The map in (3.37) specialises to q P i,j h a ′ i , a ′ j a i q i a j . By(3.8), this is equal to q P i a i qa ′ i = p ⊗ ( q ), cf. lemma 3.3. Thus, T cw maps the cylinderover O ( a ) to the projector onto the centre of A a .It is fairly straightforward to see from the above construction that T cw is compati-ble with composition and tensor products. Since we imposed that permutations of S -components in an object U of the bordism category get mapped to permutations of tensorfactors in T cw ( U ), the functor respects identities and is symmetric. In this subsection we abbreviate Bord cw ≡ Bord def , top , cw2 , and Bord ≡ Bord def , top2 , . Objectsand morphisms in Bord cw will be decorated by a tilde (e.g. ˜ U , ˜ M , . . . ). Recall the forgetfulfunctor F : Bord cw → Bord from section 3.1. We will show that there exists a symmet-ric monoidal functor T making the diagram (3.3) commute (consequently, this functor isunique). This will be done in several steps, the key one being the following lemma. Lemma 3.5.
Let ˜ U , ˜ V ∈ Bord cw and let ˜ M , ˜ M ′ : ˜ U → ˜ V be morphisms. If ˜ M ′ is obtainedfrom ˜ M by one of the local modifications of the cell decomposition shown in figures 11 and12, then T cw ( ˜ M ) = T cw ( ˜ M ′ ) . These properties are all easily checked directly with the methods of section 3.3. They have also beenshown in arbitrary modular categories (instead of just the category V ect f ( k )) in [FRS3, Lem. 2] and [KR1,Lem. 3.1]. ←→ b) ←→ ←→ Figure 11: a) A local modification of the cell decomposition which adds an edge and a vertexturning a 2-gon into two triangles, or conversely. The two exterior vertices are allowed to beidentical. b) The same in the presence of a domain wall; there are two modifications as the vertexcan be added on either side of the domain wall. ←→ ←→ ←→ · · · ←→
Figure 12:
A local modification of the cell decomposition which adds a single edge to the interiorof a 2-cell. The figure shows an exemplary situation. Alternatively, the 2-cell can be any n -gonwith n ≥
2, the domain wall can run between other edges, or there could be no domain wall atall.
Proof.
We will show the equality T cw ( ˜ M ) = T cw ( ˜ M ′ ) in the two cases displayed in figure13, the remaining cases are treated analogously.The part of the cell decomposition in figure 13 a) contributes the factor E p : X ∗ x ⊗ X x → k , ϕ ⊗ x ϕ ( x ) from (3.33) to the map E in (3.35) and (3.27). Figure 13 b) contributesthe factors E p ⊗ E p : ( X ∗ x ⊗ A b ⊗ X x ) ⊗ ( X ∗ x ⊗ A b ⊗ X x ) → k , ϕ ⊗ b ⊗ x ⊗ ϕ ′ ⊗ b ′ ⊗ x ′ ϕ ( b.x ) · ϕ ′ ( b ′ .x ′ ) to E , and to P in (3.31) it contributes the factors P e ⊗ P e : k → ( A b ⊗ A b ) ⊗ ( X x ⊗ X ∗ x ), 1 P i,j b ′ i ⊗ b i ⊗ u j ⊗ u ∗ j . The composition of P e ⊗ P e ⊗ id X ∗ x ⊗ id X x and E p ⊗ E p (with the appropriate permutation of tensor factors) yields ϕ ⊗ x X i,j ϕ ( b ′ i .u j ) · u ∗ j ( b i .x ) = X i ϕ ( b ′ i .b i .x ) = ϕ ( x ) = E p ( ϕ ⊗ x ) . (3.39)Thus if ˜ M and ˜ M ′ differ only in one place as shown in figure 13 a,b), we still have T cw ( ˜ M ) = T cw ( ˜ M ′ ).For figure 13 c,d) the argument is the same. The 2-cell p contributes the map E p : X ∗ x ⊗ A b ⊗ A b ⊗ X x ⊗ A a → k , ϕ ⊗ b ⊗ b ⊗ x ⊗ a ϕ ( b .b .x.a ). The two cells p and p contribute E p ⊗ E p : ( X ∗ x ⊗ A b ⊗ A b ⊗ X x ) ⊗ ( X ∗ x ⊗ X x ⊗ A a ) → k , ϕ ⊗ b ⊗ b ⊗ x ′ ⊗ ϕ ′ ⊗ x ⊗ a ϕ ( b .b .x ′ ) · ϕ ′ ( x.a ). The new edge e gives the map P e : k → X x ⊗ X ∗ x , 1 P j u j ⊗ u ∗ j .Composing the two as ( E p ⊗ E p ) ◦ (id ⊗ P e ⊗ id) results in a map X ∗ x ⊗ A b ⊗ A b ⊗ X x ⊗ A a → k which acts as ϕ ⊗ b ⊗ b ⊗ x ⊗ a P j ( E p ⊗ E p )( ϕ ⊗ b ⊗ b ⊗ u j ⊗ u ∗ j ⊗ x ⊗ a )= P j ϕ ( b .b .u j ) · u ∗ j ( x.a ) = ϕ ( b .b .x.a )= E p ( ϕ ⊗ b ⊗ b ⊗ x ⊗ a ) . (3.40)29) x abe e p X x X ∗ x b) x abe e e e p p X x X ∗ x X x X ∗ x A b A b c) A b A b A a X x X ∗ x x abp e e e e e d) A b A b A a X x X ∗ x x abp p e e e e e X x X ∗ x e Figure 13:
Two examples. Figures a,b: Adding two edges and a vertex; the vector spaces Q p,e, or associated to triples ( p, e, or) as in section 3.5 are also shown, e.g. Q p ,e , − = X ∗ x and Q p ,e , + = X x . Figures c,d: Adding an edge to a 2-cell; also shown are the associated vectorspaces. ←→ fig. 12 ←→ fig. 11 ←→ × fig. 12 Figure 14:
The moves in figures 11 and 12 allow one to split an edge by adding a new vertex.
Hence, if ˜ M and ˜ M ′ differ only in one place as shown in figure 13 c,d), we have T cw ( ˜ M ) = T cw ( ˜ M ′ ).One immediate consequence of the above lemma is that we can insert new vertices onedges which do not belong to the boundary of M via the sequence of moves in figure 14.(The cell-decomposition of the boundary is fixed by the parametrisation in terms of thesource and target objects). In the absence of domain walls, the ‘elementary subdivisions’(and their inverses) of a 2-cell (figure 12) and of a 1-cell (figure 14) have been shown in [Ki]to relate any two cell decompositions (more precisely: two PLCW-decompositions of a com-pact polyhedron, see [Ki] for details). The next lemma extends this to cell decompositionin the presence of domain walls and junctions; we will only sketch its proof. Lemma 3.6.
Let ˜ U , ˜ V ∈ Bord cw and let ˜ M , ˜ M ′ : ˜ U → ˜ V be two morphisms such that F ( ˜ M ) = F ( ˜ M ′ ) . Then T cw ( ˜ M ) = T cw ( ˜ M ′ ) . ketch of proof. A 2-cell containing a junction is by construction the same in all cell de-compositions, so we will remove such 2-cells from ˜ M and ˜ M ′ and treat their boundaryedges as additional boundary components for the remaining cell complexes.Next use the moves in figures 12 and 14 to refine the cell decomposition of ˜ M and ˜ M ′ to a triangulation. The same moves allow us to make the part of the triangulation touchedby the domain walls in ˜ M and ˜ M ′ agree: each component of the domain wall submanifold M defines a string of triangles in ˜ M and ˜ M ′ and one can pass to a common refinement.(To achieve this refinement it is allowed to change the triangulation away from the domainwalls.) We now remove all triangles containing domain walls from ˜ M and ˜ M ′ , giving riseto yet more boundary components.The triangulations of ˜ M and ˜ M ′ still remaining do no longer contain any domain wallsor junctions, and the standard proof of triangulation independence applies (see [Ki] or justcheck that figures 12 and 14 imply invariance under the Pachner moves).The next step is to study the behaviour of T cw on preimages of cylinders under F . For U ∈ Bord denote by C U : U → U the morphism given by the cylinder over U . That is, C U = U × [ − ,
1] with decomposition induced by that of U via C U = ( C U ) ∪ ( C U ) ∪ ( C U ) with ( C U ) i = U i − × [ − , i = 2 , C U ) = ∅ . Orientations and labellings are inducedby U as well. Note that because morphisms of Bord are diffeomorphism classes, we have C U ◦ C U = C U (3.41)as morphisms in Bord. Now pick objects ˜ U , ˜ U ′ with F ( ˜ U ) = F ( ˜ U ′ ) = U and a morphism˜ M : ˜ U → ˜ U ′ with F ( ˜ M ) = C U . Define ζ ˜ U ′ , ˜ U := T cw ( ˜ M ) : T cw ( ˜ U ) −→ T cw ( ˜ U ′ ) . (3.42)By lemma 3.6, ζ ˜ U, ˜ U ′ is independent of ˜ M . As a consequence, given another preimage ˜ U ′′ of U , we have ζ ˜ U ′′ , ˜ U ′ ◦ ζ ˜ U ′ , ˜ U = ζ ˜ U ′′ , ˜ U . (3.43)For an object O ∈ Bord consisting of a single component S write x ( O ) for the list(( x , ε ) , . . . , ( x n , ε n )) of the marked points on O together with their orientation, orderedclockwise starting from the point − ∈ S . If ˜ O is a preimage of O , denote by e , . . . , e m the 1-cells, again ordered clockwise starting from − ∈ S . For example, figure 9 shows apreimage ˜ O with m = 5 1-cells, for O a circle with x ( O ) = (( y, +) , ( x, +) , ( z, − )) so that n = 3. Consider the maps e ( ˜ O ) := (cid:16) (cid:9) A n, X x ( O ) ∼ −−→ (cid:9) A m, R e ⊗ A , R e ⊗ A , · · · ⊗ A m − ,m R e m e ⊗ −→ T cw ( ˜ O ) (cid:17) (3.44)and π ( ˜ O ) := (cid:16) T cw ( ˜ O ) π ⊗ −→ (cid:9) A m, R e ⊗ A , R e ⊗ A , · · · ⊗ A m − ,m R e m ∼ −−→ (cid:9) A n, X x ( O ) (cid:17) (3.45)Here, X x is the notation introduced in (3.20), R e was defined in (3.25) and the intermediatealgebras A i,i +1 are as required by the bimodules. By (3.26), T cw ( ˜ O ) consists precisely ofthe tensor factors R e ⊗ · · · ⊗ R e m , and the maps e ⊗ , π ⊗ are as in (3.16).31 emma 3.7. Let ˜ O , ˜ O ′ in Bord cw be preimages of O . Then π ( ˜ O ) ◦ e ( ˜ O ) = id (cid:9) An, X x ( O ) and e ( ˜ O ′ ) ◦ π ( ˜ O ) = ζ ˜ O ′ , ˜ O .Proof. The first equality is the defining property of the maps π ⊗ and e ⊗ in (3.16).Let ˜ C ˜ U be the cylinder over ˜ U obtained by equipping C U with the cell decompositioninduced by that of ˜ U : each edge e of ˜ U gets extended to the square 2-cell e × [ − , T cw to ˜ C ˜ O , a short calculation starting from the definition of T cw (illustrated in the first example in section 3.7 below) shows that ζ ˜ O, ˜ O = p ⊗ , (3.46)where p ⊗ is the idempotent on X ε x ⊗ · · · ⊗ X ε n x n whose image is (cid:9) A n, X x ( O ) . Below we willfurthermore check that π ( ˜ O ′ ) ◦ ζ ˜ O ′ , ˜ O ◦ e ( ˜ O ) = id (cid:9) An, X x ( O ) . (3.47)Composing this from the left with e ( ˜ O ′ ) and from the right with π ( ˜ O ), and using e ⊗ ◦ π ⊗ = p ⊗ together with (3.46) and (3.43), proves the second equality of the lemma.Let us now sketch the proof of (3.47). We identify (cid:9) A n, X x ( O ) and (cid:9) A m, R e ⊗ A , · · · ⊗ A m − ,m R e m with the images of the corresponding projectors p ⊗ in X ε x ⊗ · · · ⊗ X ε n x n and R e ⊗ · · · ⊗ R e m ≡ T cw ( ˜ O ). Let P i x ( i )1 ⊗ · · · ⊗ x ( i ) n be an element of X ε x ⊗ · · · ⊗ X ε n x n in theimage of p ⊗ . The first arrow in (3.44) is the isomorphism mapping this to the element v = p ⊗ ◦ (cid:16) X i A n, ⊗ · · · ⊗ x ( i )1 ⊗ A , ⊗ · · · ⊗ x ( i )2 ⊗ A , ⊗ · · · ⊗ x ( i ) n ⊗ · · · ⊗ A n, (cid:17) (3.48)of R e ⊗ · · · ⊗ R e m ≡ T cw ( ˜ O ). Here one unit element has been inserted for each factor R e k for which e k does not contain a marked point (in this case R e k = A a for an appropriate a , cf. (3.25)). One can convince oneself that ζ ˜ O ′ , ˜ O maps v to an element v ′ of the sameform in T cw ( ˜ O ) (i.e. v ′ has same factors x ( i ) k but possibly a different number of factors1 A k,k +1 ). We omit the details of this step. The final isomorphism in (3.45) maps v ′ back to P i x ( i )1 ⊗ · · · ⊗ x ( i ) n .In the last step, we define the sought-after functor T . On objects O ∈ Bord with asingle S component, we set T ( O ) := (cid:9) A m, X x ( O ) . (3.49)For U = O ⊔ · · · ⊔ O n , monoidality then requires T ( U ) = T ( O ) ⊗ · · · ⊗ T ( O n ). For abordism M : U → V in Bord pick a preimage ˜ M : ˜ U → ˜ V under the forgetful functor.Extend the definition of e ( ˜ O ) and π ( ˜ O ) to ˜ U by taking tensor products. Define T ( M ) := (cid:16) T ( U ) e ( ˜ U ) −−→ T cw ( ˜ U ) T cw ( ˜ M ) −−−−→ T cw ( ˜ V ) π ( ˜ V ) −−→ T ( V ) (cid:17) . (3.50)The first main result of this paper is: 32 heorem 3.8. (i) T ( M ) is independent of the choice of preimage ˜ M of M .(ii) T ( C U ) = id T ( U ) .(iii) T : Bord def , top2 , → V ect f ( k ) is a symmetric monoidal functor.Proof. (i) Choose another preimage ˜ M ′ : ˜ U ′ → ˜ V ′ in Bord cw of M : U → V , and choosepreimages ˜ C U : ˜ U → ˜ U ′ and ˜ C V : ˜ V → ˜ V ′ of the cylinder C U and C V . Consider thediagram T cw ( ˜ U ) ζ ˜ U ′ , ˜ U (cid:15) (cid:15) T cw ( ˜ M ) / / T cw ( ˜ V ) ζ ˜ V ′ , ˜ V (cid:15) (cid:15) π ( ˜ V ) * * UUUUUU (cid:9) A m, X x ( U ) e ( ˜ U ) iiiiiii e ( ˜ U ′ ) * * UUUUUU (cid:9) A n, X x ( V ) T cw ( ˜ U ′ ) T cw ( ˜ M ′ ) / / T cw ( ˜ V ′ ) π ( ˜ V ′ ) iiiiii . (3.51)To see that the left triangle commutes, substitute ζ ˜ U ′ , ˜ U = e ( ˜ U ′ ) ◦ π ( ˜ U ) and use that π ( ˜ U ) ◦ e ( ˜ U ) = id (lemma 3.7). Commutativity of the right triangle follows analogously.The following chain of equalities shows that also the central square commutes: ζ ˜ V ′ , ˜ V ◦ T cw ( ˜ M ) (1) = T cw ( ˜ C V ) ◦ T cw ( ˜ M ) (2) = T cw ( ˜ C V ◦ ˜ M ) (3) = T cw ( ˜ M ′ ◦ ˜ C U ) (4) = T cw ( ˜ M ′ ) ◦ T cw ( ˜ C U ) (5) = T cw ( ˜ M ′ ) ◦ ζ ˜ U ′ , ˜ U . (3.52)Step (1) is the definition of ζ ˜ V ′ , ˜ V in (3.42); step (2) is functoriality of T cw ; step (3) followsfrom lemma 3.6 since ˜ C V ◦ ˜ M and ˜ M ′ ◦ ˜ C U are just different cell decompositions (butidentical on the boundary) of the same bordism ˜ U → ˜ V ′ ; steps (4) and (5) are the sameas (2) and (1). Thus the diagram (3.52) commutes, establishing (i).(ii) By definition (3.50) and lemma (3.7), T ( C U ) = π ( ˜ U ′ ) ◦ T cw ( ˜ C U ) ◦ e ( ˜ U ) = π ( ˜ U ′ ) ◦ ζ ˜ U ′ , ˜ U ◦ e ( ˜ U ) = π ( ˜ U ′ ) ◦ e ( ˜ U ′ ) ◦ π ( ˜ U ) ◦ e ( ˜ U ) = id.(iii) Let U M −→ V N −→ W be two composable morphisms in Bord and choose a preimage˜ U ˜ M −→ ˜ V ˜ N −→ ˜ W in Bord cw . To check compatibility with composition, we need to show T ( N ◦ M ) = T ( N ) ◦ T ( M ). Inserting the definition, this amounts to π ( ˜ W ) ◦ T cw ( ˜ N ◦ ˜ M ) ◦ e ( ˜ U ) = π ( ˜ W ) ◦ T cw ( ˜ N ) ◦ e ( ˜ V ) ◦ π ( ˜ V ) ◦ T cw ( ˜ M ) ◦ e ( ˜ U ) (3.53)That the two sides are indeed equal can be seen as follows. By lemma 3.7, e ( ˜ V ) ◦ π ( ˜ V ) = ζ ˜ V , ˜ V , and, if ˜ C is a preimage of C V , by functoriality of T cw the rhs is equal to π ( ˜ W ) ◦ T cw ( ˜ N ◦ ˜ C ◦ ˜ M ) ◦ e ( ˜ U ). But F ( ˜ N ◦ ˜ C ◦ ˜ M ) = N ◦ M , so that by lemma 3.6, the rhs isindeed equal to the lhs.Monoidality and symmetry of T are implied by that of T cw .This concludes our construction of an example of a two-dimensional topological fieldtheory with defects. 33) b ayx p p e e e e e e b) xb a p p e e e e e Figure 15:
Two bordisms with defects together with a choice of cell-decomposition used in thesample computation. As in figure 10, the orientations of the edges are chosen for convenienceand are not part of the data of the cell decomposition.
Let us work through two more examples to see how the amplitude of a bordism M : U → V (3.54)in Bord ≡ Bord def , top2 , is computed in lattice TFT. As in section 3.6, we denote by ˜ M : ˜ U → ˜ V a lift to Bord cw ≡ Bord def , top , cw2 , .The first example is shown in figure 15 a). Using the notation of section 2.4, let U = V = O ( y ◦ x ∗ ) be the object of Bord consisting of a single S with two marked points ( x, − )and ( y, +). For ˜ U = ˜ V we choose a decomposition with two 1-cells. The spaces R e are: e e e e e R e X y X ∗ x X y X ∗ x The allowed triples are:( p, e, or) ( p , e , − ) ( p , e , +) ( p , e , +) ( p , e , − ) Q p,e, or X ∗ x A b X x A a ( p, e, or) ( p , e , − ) ( p , e , +) ( p , e , +) ( p , e , − ) Q p,e, or X y A a X ∗ y A b We now evaluate T cw ( ˜ M ) as given in (3.27), which is a linear map from R e ⊗ R e to R e ⊗ R e , both of which spaces are equal to X y ⊗ X ∗ x . The map id ⊗ P in (3.27) maps theelement w ⊗ ϕ ∈ R e ⊗ R e to (not writing all ‘ ⊗ ’-symbols) −→ ( R e R e ) ( Q p ,e , + Q p ,e , + Q p ,e , − Q p ,e , + Q p ,e , + Q p ,e , − ) ( R e R e ) X i,j,k,l ( w ⊗ ϕ ) ⊗ ( b ′ i ⊗ u j ⊗ a ′ k ⊗ a k ⊗ v ∗ l ⊗ b i ) ⊗ ( v l ⊗ u ∗ j )which in turn gets mapped by E ⊗ id to X i,j,k,l ϕ ( b ′ i .u j .a ′ k ) v ∗ l ( b i .w.a k ) v l ⊗ u ∗ j (3.55)34n R e ⊗ R e . This can be simplified by carrying out the sum over the bases u j , v l and theirduals, resulting in T cw ( ˜ M )( w ⊗ ϕ ) = X i,k ( b i .w.a k ) ⊗ ( a ′ k .ϕ.b ′ i ) . (3.56)Comparing to the discussion in section 3.3, we see that this is nothing but the projector p ⊗ on X y ⊗ X ∗ x whose image is (cid:9) A b X y ⊗ A a X ∗ x . Combining this with the definition (3.50)of T , we see that T ( M ) = id on (cid:9) A b X y ⊗ A a X ∗ x . This illustrates point (ii) of theorem 3.8.The following lemma provides a different point of view on this result which will be usefulin understanding the 2-category of defect conditions. Denote by ev V : V ∗ ⊗ V → k theevaluation of a vector space on its dual and write Hom A | B ( X, Y ) for the space of bimodulemaps between two A - B -bimodules X, Y . Lemma 3.9.
Let
A, B be Frobenius algebras with trace pairing, and let
X, Y be finitedimensional A - B -bimodules. The map φ : (cid:9) A Y ⊗ B X ∗ → Hom A | B ( X, Y ) , φ ( γ ) := (cid:16) X γ ⊗ id X −−−−→ ( (cid:9) A Y ⊗ B X ∗ ) ⊗ X e ⊗ ⊗ id X −−−−→ Y ⊗ X ∗ ⊗ X id X ⊗ ev X −−−−−→ Y (cid:17) (3.57) is an isomorphism.Proof. Let γ ∈ (cid:9) A Y ⊗ B X ∗ . That φ ( γ ) is a bimodule map is a straightforward calculationusing ev X ( u ⊗ ( a.v.b )) = ev X (( b.u.a ) ⊗ v ) and ( e ⊗ ◦ γ ) .a = a. ( e ⊗ ◦ γ ). That φ is anisomorphism follows by the standard argument using the corresponding coevaluation mapand the duality properties.Via this lemma we can also think of T ( O ( y ◦ x ∗ )) as Hom A b | A a ( X x , X y ). If we identify X y ⊗ X ∗ x with Hom( X x , X y ) then T cw ( ˜ M ) becomes the projection from general linear mapsto bimodule intertwiners.The second example – figure 15 b) – is again an annulus with a domain wall, butthis time the in-going boundary sits entirely in one domain. Here, the source-object is U = O ( b ), an S with no marked points and labelled b ∈ D , and the target object is V = O ( x ◦ x ∗ ). The lift ˜ U we chose contains a single edge, while ˜ V contains two edges.The spaces R e and Q p,e, or in this case are: e e e e R e A b X x X ∗ x ( p, e, or) ( p , e , − ) ( p , e , +) ( p , e , +) ( p , e , − ) ( p , e , − ) ( p , e , +) ( p , e , +) Q p,e, or A b A b X x X ∗ x A b X x X ∗ x The map id ⊗ P takes w ∈ A b to −→ R e ( Q p ,e , − Q p ,e , + Q p ,e , + Q p ,e , + Q p ,e , − Q p ,e , + ) ( R e R e ) X i,j,k,l w ⊗ ( u ∗ i ⊗ b ′ j ⊗ u k ⊗ u ∗ l ⊗ b j ⊗ u i ) ⊗ ( u l ⊗ u ∗ k ) . P i,j,k,l u ∗ i (( w · b ′ j ) .u k ) u ∗ l ( b j .u i ) · u l ⊗ u ∗ k by E ⊗ id, which simplifiesto T cw ( ˜ M )( w ) = X j,k (cid:0) ( b j wb ′ j ) .u k (cid:1) ⊗ u ∗ k = X k ( p ⊗ ( w ) .u k ) ⊗ u ∗ k , (3.58)where p ⊗ : A b → A b is the projection to the centre of A b , see lemma 3.3. Accordingly, theresulting map for the bordism M is T ( M ) : Z ( A b ) −→ Hom A b | A a ( X x , X x ) z ( q z.q ) , (3.59)where we have identified (cid:9) A b X x ⊗ A a X ∗ x ∼ = Hom A b | A a ( X x , X x ) via lemma 3.9. In the construction in section 3.4–3.6, the domain wall conditions were given by bimodules.Bimodules naturally form a bicategory (see [Be, Sec. 2.5], [Gr, Sec. I.3] or [ML, Ch. XII.7]),and in this subsection we want to compare this bicategory to the 2-category of defectconditions described in section 2.4. Our conventions for bicategories can be found inappendix A.
Definition 3.10. (i) The category A lg ( k ) has associative unital algebras over k as objectsand (unital) algebra homomorphisms as morphisms.(ii) The bicategory Alg ( k ) has associative unital algebras over k as objects. The mor-phism category Alg ( k )( A, B ) is given by the category of B - A -bimodules and bimoduleintertwiners. The composition functor Alg ( k )( B, C ) × Alg ( k )( A, B ) → Alg ( k )( A, C ) is( − ) ⊗ B ( − ).We will start with a small digression which is not restricted to Frobenius algebras withtrace pairing. Namely, we will look at some properties of Alg ( k ).Given a 1-category C , we denote the bicategory obtained from C by adding only identity2-morphisms again by C . When comparing A lg ( k ) and Alg ( k ), we understand A lg ( k ) asa bicategory in this sense. For an algebra map f : A → B and a right B -module M ,we denote by M f the right A -module with action ( m, a ) m.f ( a ). In particular, B f isa B - A -bimodule. The next lemma (following [Be, Sec. 5.7]) makes precise the idea that Alg ( k ) contains more 1- and 2-morphisms than A lg ( k ). Lemma 3.11. (i) Let A f −→ B g −→ C be algebra maps. The following map is well-definedand an isomorphism of C - A -bimodules: m g,f : C g ⊗ B B f −→ C g ◦ f , c ⊗ B b c · g ( b ) . (3.60) Note that, while A lg ( k )( A, B ) is not additive (since f + g is never an algebra homomorphism if f and g are), the category Alg ( k )( A, B ) has direct sums of 1-morphisms, so that we have added enoughmorphisms to ‘linearise’ A lg ( k ). ii) The assignment i : A lg ( k ) −→ Alg ( k ) , (3.61) which is the identity on objects and which maps A f −→ B to B f , is a (non-lax) functor.The unit transformations are identities and the multiplication transformations are givenby m g,f .(iii) Let f, g : A → B be algebra maps. Then i ( f ) and i ( g ) are 2-isomorphic in Alg ( k ) ifand only if f ( − ) = u · g ( − ) · u − for some u ∈ B × .Proof. Abbreviate m ≡ m g,f .(i) To see that m is well-defined, consider the map ¯ m : C g ⊗ B f → C g ◦ f given by u ⊗ v ug ( v ). We verify the cokernel condition: for b ∈ B we have ¯ m (( u.b ) ⊗ v ) = ¯ m (( ug ( b )) ⊗ v ) = ug ( b ) g ( v ) = ug ( bv ) = ¯ m ( u ⊗ ( bv )). Therefore, ¯ m induces a map C g ⊗ B B f → C g ◦ f , whichis precisely m . Since c c ⊗ B B is an isomorphism from C g ◦ f to C g ⊗ B B f , and since bycomposing with m one obtains the identity on C g , it follows that m is an isomorphism. Itis straightforward to check that m intertwines the C - A -bimodule structures.(ii) We have to verify associativity and unit properties of the functor. We start withassociativity. Given algebra maps A f −→ B g −→ C h −→ D , we must show commutativity of thediagram ( D h ⊗ C C g ) ⊗ B B f ∼ / / m h,g ⊗ B id (cid:15) (cid:15) D h ⊗ C ( C g ⊗ B B f ) id ⊗ C m g,f (cid:15) (cid:15) D h ◦ g ⊗ B B fm h ◦ g,f (cid:15) (cid:15) D h ⊗ C C g ◦ fm h,g ◦ f (cid:15) (cid:15) D h ◦ g ◦ f = / / D h ◦ g ◦ f (3.62)Acting on an element d ⊗ C c ⊗ B b , the left brach gives d · h ( c ) · h ( g ( b )) and the right branchgives d · h ( c · g ( b )). These are equal as h is an algebra map. The unit properties in turnamount to commutativity of the following two diagrams: B ⊗ B B f ∼ / / id (cid:15) (cid:15) B f B id ⊗ B B f m id ,f / / B id ◦ f id O O , B f ⊗ A A ∼ / / id (cid:15) (cid:15) B f B f ⊗ A A id m f, id / / B id ◦ f id O O (3.63)In the left diagram, both branches give the map b ⊗ B b ′ b · b ′ , and in the right diagram,both branches give b ⊗ B a b · f ( a ).Since by part (i) the m ’s are isomorphisms, we do indeed obtain a functor, not just alax functor.(iii) ‘ ⇒ ’: Suppose that ψ : B f → B g is an isomorphism of B - A -bimodules. Then forall x, b ∈ B and a ∈ A we have ψ ( b · x · f ( a )) = b · ψ ( x ) · g ( a ). From this we concludethat f ( a ) · ψ (1) = ψ ( f ( a )) = ψ (1) · g ( a ). Since ψ is invertible, ψ (1) ∈ B × , and so f ( a ) = ψ (1) · g ( a ) · ψ (1) − . 37 ⇐ ’: The isomorphism is given by b b · u .Recall the construction of the 2-category D [ D , D ; T ] in (2.12), the assignment ofalgebras and bimodules to elements of D and D in the beginning of section 3.4, and thedefinition of the defect TFT T in theorem 3.8. We want to define a functor∆ : D [ D , D ; T ] −→ Alg ( k ) , (3.64)which on objects a ∈ D acts ∆( a ) = A a and on 1-morphisms x ∈ D ( a, b ) as ∆( x ) = X x ,using the notation (3.20). The action on 2-morphisms will be described after the followingremark. Remark 3.12. (i) By (2.9), the 2-morphism spaces of D [ D , D ; T ] are given by D ( x, y ) := H inv ( y ◦ x ∗ ). One may think that in a TFT all states are scale and translation invariant,and this is true but for one detail. Let x : a → a and let C O ( x ) be the cylinder over O ( x ).The defining property (2.7) of a scale and translation invariant family implies that all vec-tors ψ x ; r lie in the image of the idempotent T ( C O ( x ) ) : T ( O ( x )) → T ( O ( x )). Conversely,each vector in the image of T ( C O ( x ) ) gives rise to a scale and translation invariant family.Indeed, for TFTs, T ( O ( x )) ≡ T ( O ( x ; r )) is independent of r , and so is the family ψ x ; r . Wecan therefore identify H inv ( x ) with the image of the idempotent T ( C O ( x ) ). For our latticeTFT construction, by theorem 3.8 (iii) this does not make a difference, but for a generalTFT, T ( C O ( x ) ) may be different from the identity map on T ( O ( x )).(ii) Given a TFT for which the idempotents T ( C U ) for objects U ∈ Bord def , top2 , are notalways identity maps, one can define a new TFT T ′ in which one replaces all state spaces T ( U ) by the image of the corresponding idempotent T ( C U ). The embedding of the imageof T ( C U ) into T ( U ) provides a monoidal natural transformation from T ′ to T . One canthink of T ′ as the ‘non-degenerate subtheory’ of T , because an amplitude T ( U M −→ V )vanishes if its argument comes from the kernel of T ( C U ). In principle, one can alwayswork with non-degenerate TFTs, but in some situations degenerate TFTs are useful as anintermediate step (such as in the orbifold construction of [Fr¨o2], or in a sense also the con-struction in section 3.6, where T was defined precisely as the restriction of T cw to imagesof idempotents).According to part (i) of the above remark, in our lattice TFT example we have H inv ( x ) = T ( O ( x )). Substituting the definition of T on objects in (3.49), we see that for x, y : a → b , D ( x, y ) = (cid:9) A b X y ⊗ A a X x ∗ . (3.65)Using this and lemma 3.9, we can finally state the action of the functor ∆ on morphisms.Namely for u ∈ D ( x, y ) we set ∆( u ) = φ ( u ) : X x → Y y .We should now proceed to show that ∆ thus defined is indeed a functor between bicat-egories, which in addition is locally fully faithful (since φ is an isomorphism). However, wewill not go through these details and instead turn to the next topic, the relation betweenlattice TFT with defects and the centre of an algebra.38 The centre of an algebra
The map which assigns to an algebra A its centre Z ( A ) is not functorial, at least notin the obvious sense. Namely, given A ∈ A lg ( k ), then also Z ( A ) ∈ A lg ( k ), but for analgebra homomorphism f : A → B it is in general not true that f | Z ( A ) lands in Z ( B ). Forexample, if A is the algebra of diagonal 2 × B is all 2 × f isthe embedding map, then Z ( A ) = A , but Z ( B ) = k id which does not contain f ( Z ( A )).For Frobenius algebras with trace pairing one could use the maps e ⊗ and π ⊗ between A and Z ( A ) = (cid:9) A A (cf. lemma 3.3 – not true for general algebras) to map f to π ⊗ ◦ f ◦ e ⊗ ,but this would in general not be compatible with composition and multiplication.The main point of this section is to define a functorial version of the centre. This isdone by first constructing a bicategory – or rather two versions thereof – whose objects arecommutative algebras. The centre is then a lax functor into this bicategory; this functorwill also be given in two versions (theorem 4.12 and remark 4.19). These constructions aremotivated by 2-dimensional TFT with defects, so we begin the discussion by highlightingthe relevant algebras and maps in the defect TFT. Let T : Bord def , top2 , ( D , D , D ) → V ect f ( k ) be a defect TFT (not necessarily obtained vialattice TFT). The functor T encodes an infinite number of state spaces and linear mapsbetween them. In this subsection we will pick out some of the more fundamental ones andinvestigate their properties.By remark 3.12 (ii) we are entitled to assume that all idempotents T ( C U ) are in factidentity maps, and we will make this assumption for the rest of this subsection. Recallfrom section 2.4 the 2-category D [ D , D ; T ] associated to a field theory with defects. Byremark 3.12 (i) and because of our assumption that T ( C U ) = id T ( U ) , definition (2.9) of the2-morphism spaces becomes D ( x, y ) = T ( O ( y ◦ x ∗ )) . (4.1)Recall that the identity 1-morphism a : a → a , for a ∈ D , is the empty tuple a = (). Consider the space of 2-endomorphisms of a , D ( a , a ) = T ( O ( a )). This isan associative, commutative, unital algebra; the bordisms which give the multiplicationand unit morphisms are those in figure 6 a,b), but without domain walls. Commutativityfollows since precomposing the multiplication bordism with a transposition σ : O ( a ) ⊔ O ( a ) → O ( a ) ⊔ O ( a ) gives a diffeomorphic bordism. In fact, by the usual arguments, it iseven a Frobenius algebra, and this Frobenius algebra defines the defect-free TFT given by D = { a } and D = D = ∅ .For an arbitrary 1-morphism x : a → b , the 2-endomorphisms D ( x, x ) do still forman associative, unital algebra (even a Frobenius algebra), but this algebra need not becommutative. The horizontal composition functors for ( a a −→ a x −→ b ) = a x −→ b and39 b axxx !(cid:16) L ( q ) , w (cid:17) = T b axx !(cid:16) q, w (cid:17) = T xxb a !(cid:16) q, w (cid:17) = T b axxx !(cid:16) L ( q ) , w (cid:17) Figure 16:
Manipulation of bordisms showing that L ≡ ˆ L ( − ⊗ id x ) : D ( b , b ) → D ( x, x )maps to the centre of D ( x, x ). Here q ∈ D ( b , b ) and w ∈ D ( x, x ). ( a x −→ b b −→ b ) = a x −→ b give linear mapsˆ R : D ( x, x ) ⊗ D ( a , a ) −→ D ( x, x ) , ˆ L : D ( b , b ) ⊗ D ( x, x ) −→ D ( x, x ) . (4.2)The bordisms for the maps ˆ L and ˆ R are as in figure 6 c), provided we specialise the latter tothe case where only one of the two in-going boundary circles has domain walls attached toit. If we insert the identity 2-morphism id x , we obtain maps R := ˆ R (id x ⊗− ) : D ( a , a ) → D ( x, x ) and L := ˆ L ( − ⊗ id x ) : D ( b , b ) → D ( x, x ). The corresponding bordisms areobtained by gluing a disc as in figure 6 a) into the hole which has the domain walls attached.Figure 15 b) shows a bordism obtained in this way.With the help of bordisms, it is easy to see that R and L are algebra homomorphismswhose image lies in the centre of D ( x, x ). The bordism manipulations showing that theimage of L lies in the centre are given in figure 16. Next, consider the space of 2-morphisms D ( x, y ) between two 1-morphisms x, y : a → b . This is the TFT state space for a circle with sequence of marked points y ◦ x ∗ . By verticalcomposition, D ( x, y ) is a right D ( x, x )-module and a left D ( y, y )-module. Using R tomap D ( a , a ) into D ( x, x ) and D ( y, y ), we see that D ( x, y ) is also a bimodule for D ( a , a ). However, by an argument analogous to that in figure 16 it is easy to checkthat the left and right action agree. Equally, L turns it into an D ( b , b ) bimodule withidentical left and right action.Let f ∈ D ( x, y ) be a 2-morphism. Pre- and post-composing with f defines maps f ◦ ( − ) : D ( x, x ) → D ( x, y ) , ( − ) ◦ f : D ( y, y ) → D ( x, y ) (4.3)Again by manipulating bordisms, one checks that f ◦ ( − ) intertwines the right D ( x, x )action and ( − ) ◦ f intertwines the left D ( y, y )-action. All these maps are collected infigure 17. Manipulating such disk-shaped bordisms reminds one of the string-diagram notation for 2-categories[St]. Indeed a string-diagram identity implies an identity for defect correlators on disks, but the converseis not true – the 2-category D [ D , D ; T ] satisfies more conditions than a generic 2-category. (cid:16) xx + − ab (cid:17) T b ayxxf ! (cid:15) (cid:15) T (cid:16) b (cid:17) T b ax ! T b ay ! , , T (cid:16) yx + − ab (cid:17) T (cid:16) a (cid:17) T b ax ! l l T b ay ! r r T (cid:16) yy + − ab (cid:17) T b ayyxf ! O O Figure 17:
Summary of the state spaces and maps between them as described in section 4.1.Here only the case that x = (( x, +)) and y = (( y, +)) is shown. For tuples with more elements,the bordisms involve the corresponding sequences of parallel lines. Remark 4.1.
In conformal (and thus in particular in topological) field theory, one hasthe state-field correspondence, which says that the space of fields associated to a point onthe world sheet is the same as the space of states on a small circle obtained by cuttingout a small disc around this point; in fact, one can take this as a definition of what onemeans by a field. Then D ( a , a ) is the space of ‘bulk fields’ and D ( x, x ) is the spaceof ‘defect fields’ supported on the defect x . An important notion in quantum field theoryis the short distance expansion or operator product expansion (OPE). In TFT, of course,the distance between insertion points is immaterial. The above considerations isolate thethree most important OPEs: the OPE of two bulk fields; the OPE of two defect fields; theexpansion of a bulk field close to a defect line in terms of defect fields. In this subsection we will use some of the structure seen in 2d TFT with defects in theprevious subsection to define a bicategory of commutative algebras in terms of cospans.All algebras will be unital, associative algebras over a field k . Definition 4.2. A cospan between commutative algebras , or cospan for short, is a tuple( A, α, T, β, B ), where
A, B are commutative algebras, T is an algebra, and α, β are algebrahomomorphisms TA α ; ; vvvvv B β c c HHHHH (4.4)41uch that the images of α and β lie in the centre Z ( T ) of T .The definition has no preferred ‘direction’, but we will pick one anyway: we will thinkof (4.4) as going from B to A . The reason for this choice is that we will use the maps α, β to turn T into an A - B -bimodule and the composition of cospans (to be defined in moredetail below) will be the tensor product of bimodules, just as in the bicategory Alg ( k ). Inthe latter, A - B -bimodules serve as 1-morphisms B → A . We will write T : B → A , or just T , to abbreviate the data in (4.4). Two different cospans T, T ′ : B → A can be comparedvia algebra homomorphisms T → T ′ . This leads first to a category of cospans from B to A , and then to a bicategory CAlg ( k ) of cospans between commutative algebras. Theconstruction is almost identical to the standard construction of the bicategory of spansfor a given category with pullbacks (see [Be, Sec. 2.6] or [Gr, ML]), with the exceptionthat not all three objects in (4.4) are taken from the same category (we use commutativealgebras for the starting points of the cospan and not necessarily commutative algebras forthe middle term). Definition 4.3.
The category C osp ( A, B ) of cospans between commutative algebras from A to B is defined as follows. • objects T ∈ C osp ( A, B ) are cospans T : A → B . • morphisms f ∈ C osp ( A, B )( T, T ′ ) from T : A → B to T ′ : A → B are algebra maps f : T → T ′ such that the following diagram commutes: T f (cid:15) (cid:15) B β : : uuuuu β ′ $ $ HHHHH A α d d IIIII α ′ z z vvvvv T ′ (4.5) • the unit morphism in C osp ( A, B )( T, T ) is the identity map id T , and composition ofmorphisms is composition of algebra maps.The composition of two cospans A T −→ B S −→ C is defined by the usual pushout square, S ⊗ B TS id ⊗ B T : : uuuuuuu T S ⊗ B id d d IIIIIII C γ ? ? ~~~~~ B β d d IIIIIIII β ′ : : uuuuuuuu A α _ _ @@@@@ (4.6)The algebra homomorphism β turns S into a right B -module, and β ′ turns T into a left B -module; these module structures are implied when writing S ⊗ B T . We still need to turn S ⊗ B T into an algebra and show that A and C get mapped to the centre of this algebra.The multiplication on S ⊗ B T is given by( s ⊗ B t ) · ( s ′ ⊗ B t ′ ) := ( ss ′ ) ⊗ B ( tt ′ ) . (4.7)42o check that this is well-defined, we start with the map ¯ m : ( S ⊗ T ) ⊗ ( S ⊗ T ) → S ⊗ B T ,which takes ( s ⊗ t ) ⊗ ( s ′ ⊗ t ′ ) to ( ss ′ ) ⊗ B ( tt ′ ) and verify the cokernel condition. We presentthe calculation for the first factor, the one for the second factor is similar. With b ∈ B ,¯ m ( s.b ⊗ t ⊗ s ′ ⊗ t ′ ) = ( sβ ( b ) s ′ ) ⊗ B ( tt ′ ) ( ∗ ) = ( ss ′ β ( b )) ⊗ B ( tt ′ )= ( ss ′ ) .b ⊗ B ( tt ′ ) = ( ss ′ ) ⊗ B b. ( tt ′ ) = ( ss ′ ) ⊗ B ( β ′ ( b ) tt ′ ) = ¯ m ( s ⊗ b.t ⊗ s ′ ⊗ t ′ ) (4.8)The step marked ‘( ∗ )’ uses that the image of the algebra homomorphism β is in the centreof S . This, by the way, is the reason not to allow general algebra homomorphisms indefinition 4.2: we want the tensor product over B to carry an induced algebra structure.Finally, it is straightforward to check that for all a ∈ A , c ∈ C , the elements 1 S ⊗ B α ( a )and γ ( c ) ⊗ B T are in the centre of S ⊗ B T .The composition of cospans defined above forms part of a functor: Lemma 4.4.
The assignment ⊚ C,B,A : C osp ( B, C ) × C osp ( A, B ) −→ C osp ( A, C ) T g (cid:15) (cid:15) C γ : : uuuuu γ ′ $ $ HHHHH B β d d IIIII β ′ z z vvvvv T ′ , S f (cid:15) (cid:15) B β : : uuuuu β ′ $ $ HHHHH A α d d IIIII α ′ z z vvvvv S ′ ! T ⊗ B S g ⊗ B f (cid:15) (cid:15) C γ ⊗ B oooooo γ ′ ⊗ B ' ' OOOOOO A ⊗ B α g g OOOOOO ⊗ B α ′ w w oooooo T ′ ⊗ B S ′ (4.9) defines a functor.Proof. Note that the algebra map g is automatically a C - B -bimodule map. For example, g ( t.b ) = g ( t · β ( b )) = g ( t ) · g ( β ( b )) = g ( t ) · β ′ ( b ) = g ( t ) .b . Similarly, f is a B - A -bimodulemap. Thus g ⊗ B f is well-defined. That the two triangles on the rhs of (4.9) commute isimmediate. Finally, functoriality of ⊚ C,B,A amounts to the statement that( g ◦ g ) ⊗ B ( f ◦ f ) = ( g ⊗ B f ) ◦ ( g ⊗ B f ) , (4.10)which is a property of the tensor product over B .We have now gathered the ingredients to define the first version of the bicategory ofcommutative algebras, which we denote by CAlg ( k ) . (4.11)Its objects are commutative algebras over k . Given two such algebras A, B , the categoryof morphisms from A to B is C osp ( A, B ). The identity in C osp ( A, A ) is AA id ; ; vvvvv A id c c HHHHH . (4.12)The composition functor is ⊚ C,B,A from lemma 4.4. The associativity and unit iso-morphisms are just the natural isomorphisms T ⊗ C ( S ⊗ B R ) ∼ = ( T ⊗ C S ) ⊗ B R and R ⊗ A A ∼ = R ∼ = B ⊗ B R , which we will not write out in the following. It is then clear thatthe coherence conditions of a bicategory – as listed in appendix A – are satisfied.43 emark 4.5. For a category C , we denote by C , the subcategory containing only invert-ible morphisms. Similarly, given a bicategory B , denote by B , the bicategory obtainedfrom B by restricting to invertible 2-morphisms, and by B , the bicategory consisting onlyof invertible 1- and 2-morphisms (and B ≡ B , ; see [Lu1] for more on ( m, n )-categories).If B is a k -linear bicategory, we obtain a lax functor E : B , −→ CAlg ( k ) , (4.13)where ‘ E ’ stands for endomorphism. We will illustrate this functor in the case of the 2-category B ≡ D [ D , D ; T ] for a fixed 2d TFT T (which need not come from the latticeconstruction). On objects and 1-morphisms we set E ( a ) = B ( a , a ) , E ( b x ←− a ) = B ( x, x ) B ( b , b ) L lllll B ( a , a ) R h h RRRRR ! ; (4.14)the maps L and R have been given in section 4.1. To an invertible 2-morphism u : x → y we assign the algebra map E ( u ) : E ( x ) → E ( y ) given by conjugation with u . That is, f : x → x gets mapped to E ( u )( f ) = (cid:16) y u − −−→ x f −−→ x u −−→ y (cid:17) . (4.15)We will omit the details of the proof that E is a lax functor. Note that, because (4.15)involves an inverse, E is only defined on B , . Nonetheless, B , is not enough to define E , instead one requires all of B so that B ( a , a ), etc., are indeed k -algebras. Also, eventhough the image E ( u ) of a 2-morphism is always invertible, it is not true that E is afunctor to CAlg ( k ) , , because the associativity 2-morphism E ( y ) ⊚ E ( x ) → E ( y ◦ x ) isnot necessarily invertible.Denote by A lg ( k ) com the full subcategory of commutative algebras in A lg ( k ). In theremainder of this subsection we illustrate that CAlg ( k ) enlarges the morphism spaces of A lg ( k ) com , but that it does not add new invertible morphisms. Lemma 4.6.
The assignment I : A lg ( k ) com −→ CAlg ( k ) , B f ←− A BB id : : vvvvv A f d d HHHHH (4.16) defines a (non-lax) functor.Proof.
Clearly, the identity gets mapped to the identity. Given two algebra homomor-phisms A f −→ B g −→ C , one verifies that the map m g,f from lemma 3.11 defines an isomor-phism of cospans C ⊗ B B m g,f (cid:15) (cid:15) C id ⊗ B oooooo id ' ' OOOOOOO A ⊗ B f g g OOOOOO g ◦ f w w ooooooo C . (4.17)44he verification of the associativity condition works along the same lines as the proof oflemma 3.11 (ii).A 2-morphism between the cospans I ( f ) and I ( g ) would necessarily have to be theidentity map in oder to make the left triangle in the condition (4.5) commute. This thenimplies f = g . In particular, different algebra maps get mapped to non-2-isomorphiccospans. In this sense, I is faithful. Lemma 4.7.
A cospan TB β ; ; vvvvv A α c c HHHHH (4.18) is invertible in
CAlg ( k ) if and only if α and β are isomorphisms.Proof. ‘ ⇐ ’: Suppose α and β are isomorphisms. Then T β − (cid:15) (cid:15) B β : : vvvvv id $ $ HHHHH A α d d HHHHH β − ◦ α { { vvvvv B (4.19)is an isomorphism of cospans, i.e. T ∼ = I ( β − ◦ α ). The latter cospan has inverse I ( α − ◦ β )by lemma 4.6. Thus also T is invertible.‘ ⇒ ’: Suppose ( A, α ′ , S, β ′ , B ) is a two-sided inverse of T . This means that there areisomorphisms f, g of cospans S ⊗ B T f (cid:15) (cid:15) A α ′ ⊗ B pppppp id ' ' OOOOOOO A ⊗ B α g g NNNNNN id w w ooooooo A and T ⊗ A S g (cid:15) (cid:15) B β ⊗ A pppppp id ' ' OOOOOOO B ⊗ A β ′ g g NNNNNN id w w ooooooo B . (4.20)Consider the left diagram. Since f is an isomorphism, it implies that also f − = 1 S ⊗ B α is an isomorphism. Thus we have the identities f ◦ (1 S ⊗ B α ) = id A , (1 S ⊗ B α ) ◦ f = id S ⊗ B T . (4.21)The first of these can be rewritten as f ◦ (1 S ⊗ B id T ) ◦ α = id A , showing that α hasleft-inverse ˆ f := f ◦ (1 S ⊗ B id T ) : T → A . An analogous argument gives the left inverseˆ g := g ◦ (1 T ⊗ A id S ) : S → B of β ′ .Since β ′ : B → S is an algebra map and an intertwiner of right B -modules (by definitionof the right B -action on S ), we can write 1 S ⊗ B α : A → S ⊗ B T as ( β ′ ⊗ B id T ) ◦ (1 B ⊗ B α ).Inserting this into the second identity in (4.21) givesid S ⊗ B T = (cid:16) S ⊗ B T f −→ A α −−→ T B ⊗ B id T −−−−−→ B ⊗ B T β ′ ⊗ B id T −−−−−→ S ⊗ B T (cid:17) (4.22)45e compose both sides with ˆ g ⊗ B id T and use that ˆ g is left-inverse to β ′ . This resultsin ˆ g ⊗ B id T = (1 B ⊗ B id T ) ◦ α ◦ f . Finally, composing with 1 S ⊗ B id T from the left andusing that ˆ g (1 S ) = g (1 T ⊗ A S ) = 1 B , shows that 1 B ⊗ B id T = (1 B ⊗ B id T ) ◦ α ◦ ˆ f . Since1 B ⊗ B id T is an isomorphism, we see that ˆ f is also a right-inverse for α , and hence α is anisomorphism. That β is an isomorphism follows along the same lines.From the proof we see that the cospan T in (4.18) is 2-isomorphic to I ( β − ◦ α ). Thusevery 1-isomorphism lies in the essential image of I . Remark 4.8.
An algebra isomorphism f : A → B , when restricted to Z ( A ), provides anisomorphism f | Z ( A ) : Z ( A ) → Z ( B ). This gives a functor from A lg ( k ) , to A lg ( k ) , ,which, when composed with I , gives a functor A lg ( k ) , Z −−→ A lg ( k ) , I −−→ CAlg ( k ) , . (4.23)In theorem 4.12 below, we will extend this beyond the groupoid case to a lax functor Z : A lg ( k ) → CAlg ( k ). As an aside, note that the composed functor in (4.23) is neitherfull nor faithful (isomorphic centres do not imply isomorphic algebras, and different algebraisomorphisms may restrict to the same map on the centre). However, I : A lg ( k ) , → CAlg ( k ) , is an equivalence since for invertible 1- and 2-morphisms, I is an equivalenceon the morphism categories (as those morphism categories which lie in the image of I onlycontain identity 2-morphisms), and on objects it is just the identity. Given two not necessarily commutative algebras A , B and an algebra homomorphism f : A → B , we define the centraliser Z A,B ( f ) to be the centraliser of the image of f in B , Z A,B ( f ) = (cid:8) b ∈ B (cid:12)(cid:12) f ( a ) b = b f ( a ) for all a ∈ A (cid:9) . (4.24)Let ι : Z ( B ) → B be the embedding map and denote the restriction of f to Z ( A ) also by f . Lemma 4.9.
Let
A, B be algebras and f : A → B an algebra homomorphism. Then Z A,B ( f ) Z ( B ) ι nnnnn Z ( A ) f g g PPPPP (4.25) is a cospan of commutative algebras.Proof.
Since Z ( B ) ⊂ Z A,B ( f ), ι is an algebra map which maps to the centre of Z A,B ( f ).Next we check that the image of Z ( A ) under f : A → B lies in Z A,B ( f ). Given z ∈ Z ( A ) set b = f ( z ). For all a ∈ A we have f ( a ) b = f ( a ) f ( z ) = f ( az ) = f ( za ) = f ( z ) f ( a ) = bf ( a ).Thus f ( z ) ∈ Z A,B ( f ). It is then immediate that f ( Z ( A )) lies in the centre of Z A,B ( f ).46 emark 4.10. In the more restrictive setting of Frobenius algebras with trace pairing, thecospan (4.25) has actually already appeared in disguise in the lattice TFT construction.Consider the cospan forming the top of the diamond of maps in figure 17. By definition(3.49) and lemma 3.3, T ( O ( a )) = (cid:9) A a A a = Z ( A a ) and T ( O ( b )) = Z ( A b ). For the top entrywe have T ( O ( x ∗ ◦ x )) = (cid:9) A b X x ⊗ A a X ∗ x ∼ = Hom A b | A a ( X x , X x ), where we used lemma 3.9.To make the connection to (4.25), take A = A a , B = A b and let f : A → B be an algebramap. For X x take the bimodule B f defined in section 3.8. The map φ in Z A,B ( f ) φ (cid:15) (cid:15) Z ( B ) ι jjjjjjjj act ) ) TTTTTTTT Z ( A ) f i i TTTTTTTT act u u jjjjjjjj Hom B | A ( B f , B f ) , (4.26)defined as φ ( b ) := ( u u · b ), provides an isomorphism of cospans. Here ‘act’ refers tothe map that takes b ∈ Z ( B ) to the bimodule map u b.u ; u ∈ B f (resp. a ∈ Z ( A ) to u u.a ). We omit the details. Lemma 4.11.
Let A f −→ B g −→ C be algebra maps. The map m g,f : Z B,C ( g ) ⊗ Z ( B ) Z A,B ( f ) → Z A,C ( g ◦ f ) given by u ⊗ Z ( B ) v u · g ( v ) is a morphism of cospans Z B,C ( g ) ⊗ Z ( B ) Z A,B ( f ) m g,f (cid:15) (cid:15) Z ( C ) ι ⊗ B hhhhhhhhhh ι * * VVVVVVVVVV Z ( A ) ⊗ B f j j VVVVVVVVVV g ◦ f t t hhhhhhhhhh Z A,C ( g ◦ f ) (4.27) Proof.
Abbreviate m ≡ m g,f and Y ≡ Z ( B ). m is well-defined: That m gives a well-defined map to C is the same argument as in theproof of lemma 3.11 (i). That the image of m lies in the centraliser Z A,C ( g ◦ f ) amountsto, for all a ∈ A , g ( f ( a )) · m ( u ⊗ Y v ) = g ( f ( a )) · u · g ( v ) (1) = u · g ( f ( a )) · g ( v ) = u · g ( f ( a ) v ) (2) = u · g ( vf ( a )) = u · g ( v ) · g ( f ( a )) = m ( u ⊗ Y v ) · g ( f ( a )) , (4.28)where (1) follows as u ∈ Z B,C ( g ) commutes with anything in the image of g , and (2) followsanalogously from v ∈ Z A,B ( f ). m is an algebra map: We have m (cid:0) ( u ⊗ Y v ) · ( u ′ ⊗ Y v ′ ) (cid:1) = m (cid:0) ( uu ′ ) ⊗ Y ( vv ′ ) (cid:1) = uu ′ g ( vv ′ ) = uu ′ g ( v ) g ( v ′ ) ( ∗ ) = ug ( v ) u ′ g ( v ′ ) = m ( u ⊗ Y v ) · m ( u ′ ⊗ Y v ′ ) . (4.29)The only perhaps not immediately obvious step is ( ∗ ), which follows since by definition forall u ′ ∈ Z B,C ( g ) and v ∈ B we have u ′ g ( v ) = g ( v ) u ′ .47 he triangles commute: Acting on arbitrary elements c ∈ Z ( C ) and a ∈ Z ( A ), com-mutativity of the two triangles amounts to the identities c = m ( c ⊗ Y B ) and g ( f ( a )) = m (1 B ⊗ Y f ( a )), both of which are immediate upon substituting the definition of m .We have now collected the ingredients to state the second main result of this note. Theorem 4.12.
The assignment Z : A lg ( k ) −→ CAlg ( k ) B f ←− A Z A,B ( f ) Z ( B ) ι nnnnn Z ( A ) f g g PPPPP (4.30) defines a lax functor. The unit transformations are identities and the multiplication trans-formations are given by m g,f . In other words, on objects the lax functor acts as A Z ( A ), on 1-morphisms A → B as f Z A,B ( f ), and all 2-morphisms in A lg ( k ) are identities, which get mapped to identity2-morphisms in CAlg ( k ). Proof.
It remains to verify the associativity and unit properties. The argument is identicalto that in the proof of lemma 3.11 (ii).
Remark 4.13. (i) The map m g,f in lemma 4.11 is typically not an isomorphism. Forexample, take A = C = k ⊕ k and B to be upper triangular 2 × f we take the diagonal embedding and for g the projection onto the diagonal part. Bycommutativity of the underlying algebra we see Z ( A ) = Z ( C ) = Z B,C ( g ) = Z A,C ( g ◦ f ) = k ⊕ k . The remaining algebras are Z ( B ) ∼ = k (multiples of the identity matrix) and Z A,B ( f ) ∼ = k ⊕ k , the diagonal 2 × Z B,C ( g ) ⊗ Z ( B ) Z A,B ( f ) ∼ = ( k ⊕ k ) ⊗ k ( k ⊕ k )while Z A,C ( g ◦ f ) = k ⊕ k . Therefore, we only have a lax functor.(ii) If we restrict Z to commutative algebras A lg ( k ) com we obtain the functor I from lemma4.6 (since then Z ( B ) = B , Z ( A ) = A and Z A,B ( f ) = B for all f ). In this sense, Z is anextension of I to all algebras; the price to pay is that we have to work with bicategoriesand the functor becomes lax. Figure 17 suggests that there is an enlargement of
CAlg ( k ), where the 2-morphisms arealso replaced by cospans. The lattice TFT construction suggests that this enlargementbecomes relevant if one wants to extend the centre functor from A lg ( k ) to Alg ( k ). Thisis the topic of the present subsection, as well as of the next one.48 efinition 4.14. (i) A from a cospan S : A → B to T : A → B is a triple( g, M, f ), where M is a T - S -bimodule, f is a right S -module map and g is a left T -modulemap such that the two squares in the diagram S f (cid:15) (cid:15) B β β & & M A α f f α x x T g O O (4.31)commute, and such that the induced left and right action of A on M agree, and those of B on M agree, i.e. that for all a ∈ A , b ∈ B , m ∈ Mα ( a ) .m = m.α ( a ) , β ( b ) .m = m.β ( b ) . (4.32)We will also abbreviate M : S → T .(ii) A between two 2-diagrams ( g, M, f ) and ( g ′ , M ′ , f ′ ) is a T - S -bimodule map δ : M → M ′ such that the following diagram commutes: S f (cid:1) (cid:1) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3) f ′ (cid:30) (cid:30) ===== M δ / / M ′ T g ] ] ;;;;; g ′ @ @ (cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (4.33)(iii) The category of 2-diagrams D iag AB ( S, T ) has 2-diagrams as objects and 3-cells asmorphisms. The identity 3-cell for the object ( g, M, f ) is the identity map on M , thecomposition of 3-cells is given by composition of bimodule maps. Remark 4.15. (i) The category D iag AB ( S, T ) can be used to define a bicategory
Cosp ( A, B )whose objects are cospans of commutative algebras from A to B and whose morphism cat-egories are D iag AB ( S, T ); this will be done in detail in [DKR2]. Conjecturally, there is atricategory CALG( k ) whose objects are commutative algebras and whose morphism bicat-egories are given by Cosp ( A, B ); we hope to return to this in the future.(ii) The conjectural tricategory CALG( k ) of part (i) is similar (but not equal) in structureto the tricategory of conformal nets described in [BDH]. In [BDH], objects are conformalnets. An object in CALG( k ), i.e. a commutative algebra, could be thought of as a ‘topolog-ical net’ which assigns the same algebra to every interval; this algebra is then necessarilycommutative (but it is not a conformal net: the algebra does not have to be von Neumannand in general it violates the split property). According to [BDH, Def. 3], 1-morphismsare ‘defects’, i.e. conformal nets for bicoloured intervals. In the present language, thiscorresponds to the data of a cospan ( A, α, T, β, B ): evaluating the net on mono-colouredsubintervals produces the two commutative algebras
A, B , for a bicoloured interval oneobtains T , and the inclusion of a mono– into a bicoloured interval provides the two maps49 , β . In [BDH, Def. 4], 2-morphisms are sectors between the defect nets – this means aHilbert space with compatible actions of all four conformal nets involved: the two defectnets and the two conformal nets which the defects go between. This provides all the dataand constraints of a 2-diagram as in (4.31) except for the maps f and g , which are not partof the setting of [BDH]. We will need these two maps for the centre functor, see lemma4.18 below. The third level of categorical structure is constructed in [BDH] by makingconformal nets a bicategory internal to symmetric monoidal categories.Part (i) of the above remark motivates the notation Cosp ( A, B ) for the category whoseobjects are cospans from A to B and whose morphisms are isomorphism classes of 2-diagrams. Composition in Cosp ( A, B ) is given by ⊙ : Cosp ( A, B )( S, T ) × Cosp ( A, B )( R, S ) −→ Cosp ( A, B )( R, T ) S g (cid:15) (cid:15) B β β & & N A α f f α x x T g ′ O O , R f (cid:15) (cid:15) B β β & & M A α f f α x x S f ′ O O ! R g (1) ⊗ S f (cid:15) (cid:15) B β β . . N ⊗ S M A α n n α p p T g ′ ⊗ S f ′ (1) O O . (4.34)The right hand side is again a 2-diagram. Let us check explicitly the left square in (4.31)and the first condition in (4.32). For b ∈ B , g (1) ⊗ S f ( β ( b )) (1) = g (1) ⊗ S f ′ ( β ( b )) (2) = g (1) ⊗ S β ( b ) .f ′ (1) (3) = g (1) .β ( b ) ⊗ S f ′ (1) (4) = g ( β ( b )) ⊗ S f ′ (1) (5) = g ′ ( β ( b )) ⊗ S f ′ (1) . (4.35)Here, (1) is commutativity of the left square in the 2-diagram ( f ′ , M, f ), (2) the fact that f ′ is left S -module map, (3) the property of ⊗ S , (4) follows since g is a right S -modulemap, and finally (5) is commutativity of the left square in the 2-diagram ( g ′ , N, g ). Next,that the left action of B on N ⊗ S M agrees with the right action of B follows from b. ( n ⊗ S m ) = ( β ( b ) .n ) ⊗ S m = ( n.β ( b )) ⊗ S m = n ⊗ S ( β ( b ) .m ) = n ⊗ S ( m.β ( b )) = ( n ⊗ S m ) .b (4.36)The unit morphism in Cosp ( A, B )( T, T ) is T id (cid:15) (cid:15) B β β & & T A α e e α y y T id O O (4.37) Lemma 4.16.
A 2-diagram ( g, M, f ) between S, T : A → B is invertible if and only ifboth f and g are invertible. Lemma 4.17.
The assignment ⊚ C,B,A : Cosp ( B, C ) × Cosp ( A, B ) −→ Cosp ( A, C ) T g (cid:15) (cid:15) C γ γ ′ % % N B β f f β ′ y y T ′ g ′ O O , S f (cid:15) (cid:15) B β β ′ & & M A α f f α ′ y y S ′ f ′ O O ! T ⊗ B S g ⊗ B f (cid:15) (cid:15) C γ ⊗ B γ ′ ⊗ B & & N ⊗ B M A ⊗ B α g g ⊗ B α ′ x x T ′ ⊗ B S ′ g ′ ⊗ B f ′ O O (4.38) defines a functor.Proof. It is evident that the functor maps a pair of identity cospans (4.37) to the identitycospan. To verify functoriality, choose another pair of 2-diagrams N ′ : T ′ → T ′′ and M ′ : S ′ → S ′′ in the product category. First composing in the product with ⊙ × ⊙ givesthe pair ( N ′ ⊗ T ′ N, M ′ ⊗ S ′ M ). Applying ⊚ yields X := N ′ ⊗ T ′ N ⊗ B M ′ ⊗ S ′ M . In the otherorder, by first applying ⊚ , we obtain the two 2-diagrams N ′ ⊗ B M ′ : T ′ ⊗ B S ′ → T ′′ ⊗ B S ′′ and N ⊗ B M : T ⊗ B S → T ′ ⊗ B S ′ . Applying ⊙ to this gives Y := ( N ′ ⊗ B M ′ ) ⊗ T ′ ⊗ B S ′ ( N ⊗ B M ).We claim that the isomorphism N ′ ⊗ N ⊗ M ′ ⊗ M → N ′ ⊗ M ′ ⊗ N ⊗ M given by permutingfactors induces maps X → Y and Y → X ; these are then automatically inverse to eachother. For example, the condition that the map φ : N ′ ⊗ N ⊗ M ′ ⊗ M → Y respects thetensor product over T ′ amounts to, for t ′ ∈ T , φ (cid:0) ( n ′ .t ′ ) ⊗ n ⊗ m ′ ⊗ m (cid:1) = (cid:0) ( n ′ .t ′ ) ⊗ B m ′ (cid:1) ⊗ T ′ ⊗ B S ′ ( n ⊗ B m )= (cid:0) ( n ′ ⊗ B m ′ ) . ( t ′ ⊗ B (cid:1) ⊗ T ′ ⊗ B S ′ (cid:0) n ⊗ B m (cid:1) = (cid:0) n ′ ⊗ B m ′ (cid:1) ⊗ T ′ ⊗ B S ′ (cid:0) ( t ′ ⊗ B . ( n ⊗ B m ) (cid:1) = (cid:0) n ′ ⊗ B m ′ (cid:1) ⊗ T ′ ⊗ B S ′ (cid:0) ( t ′ .n ) ⊗ B m (cid:1) = φ (cid:0) n ′ ⊗ ( t ′ .n ) ⊗ m ′ ⊗ m (cid:1) . (4.39)The other tensor product cokernel conditions are checked similarly. That the inducedisomorphism X → Y is a 3-cell is equally straightforward.We can now define the second version of the bicategory of commutative algebras, whichwe denote by CALG( k ) ; (4.40)Objects and 1-morphisms are as in CAlg ( k ), but for 2-morphisms we take equivalenceclasses of 2-diagrams. In other words, the morphism category A → B is Cosp ( A, B ).The composition functor is given in lemma 4.17. The notation CALG( k ) is motivated bythe conjectural tricategory of remark 4.15 (i). The associativity and unit isomorphisms ofCALG( k ) are just those of bimodules, and the required coherence conditions are satisfiedfor the same reason. 51s compared to CAlg ( k ), the category CALG( k ) has more 2-morphisms. This is madeprecise by the observation that I : CAlg ( k ) −→ CALG( k ) S f (cid:15) (cid:15) B β ; ; vvvvv β ′ HHHHH A α c c HHHHH α ′ { { vvvvv T S f (cid:15) (cid:15) B β β ′ & & T A α f f α ′ y y T id O O (4.41)is a locally faithful functor from CAlg ( k ) to CALG( k ); we skip the details. By lemma4.16, each invertible morphism in Cosp ( A, B )( S, T ) lies in the image of I (we again skipthe details). Therefore, the restriction I : CAlg ( k ) , ∼ −−−→ CALG( k ) , (4.42)is an equivalence of bicategories. In this sense, passing from CAlg ( k ) to CALG( k ) addsmore non-invertible 2-morphisms. In this subsection we will try to extend the centre functor to
Alg ( k ). We will see thatCALG( k ) is not quite good enough as a target category, and we have to restrict ourselvesto appropriate subcategories of Alg ( k ).Let A, B be algebras and let X be a B - A -bimodule. Then Hom B | A ( X, X ) Z ( B ) act kkkkkkk Z ( A ) act i i SSSSSSS (4.43)is a cospan of commutative algebras. As in remark 4.10, ‘act’ refers to the map that takes b ∈ Z ( B ) to the bimodule map x b.x (resp. a ∈ Z ( A ) to x x.a ). Lemma 4.18.
Let
A, B be algebras, let
X, Y be A - B -bimodules and let f : X → Y be abimodule homomorphism. Then Hom A | B ( X, X ) f ◦ ( − ) (cid:15) (cid:15) Z ( A ) act act ( ( Hom A | B ( X, Y ) Z ( B ) act h h act v v Hom A | B ( Y, Y ) ( − ) ◦ f O O (4.44) is a 2-diagram. roof. We need to verify the conditions in definition 4.14. The composition of bimodulemaps turns Hom A | B ( X, Y ) into a right module over Hom A | B ( X, X ) and a left module overHom A | B ( Y, Y ). The map h f ◦ h from Hom A | B ( X, X ) to Hom A | B ( X, Y ) is a right modulemap (this translates into f ◦ ( h ◦ h ′ ) = ( f ◦ h ) ◦ h ′ ). Similarly, ( − ) ◦ f is a left modulemap. Commutativity of the left square amounts to equality of the two maps x f ( a.x )and x a.f ( x ) for all a ∈ Z ( A ), which follows since f is a bimodule map. That the rightsquare commutes follows analogously. Finally, consider the two conditions in (4.32). Thefirst condition amounts to equality of the two maps x g ( x ) .b and x g ( x.b ) for all b ∈ Z ( B ) and g ∈ Hom A | B ( X, Y ), which holds since g is a bimodule map. The secondcondition can be checked similarly.As the constructions will now get somewhat technical, let us just outline in the remarkbelow how the discussion continues from here, leaving the details to [DKR2]. Remark 4.19. (i) The 2-diagram in (4.44) provides a lax functor Z A,B : Alg ( k )( A, B ) −→ Cosp ( Z ( A ) , Z ( B )) . (4.45)This functor is indeed lax for the following reason: The vertical composition (4.34) of two2-diagrams of the form (4.44) belonging to bimodule maps f : X → Y and g : Y → Z yields a 2-diagram with central termHom A | B ( Y, Z ) ⊗ H Hom A | B ( X, Y ) where H ≡ Hom A | B ( Y, Y ) . (4.46)This space is in general not isomorphic to Hom A | B ( X, Z ). So we cannot obtain a functor
Alg ( k )( A, B ) −→ Cosp ( Z ( A ) , Z ( B )) in this way, and consequently not a – lax or other-wise – functor from Alg ( k ) to CALG( k ). However, we conjecture that the 2-diagram (4.44)does give rise to a lax functor Z from Alg ( k ) to the (conjectural) tricategory CALG( k ).(ii) If the maps f, g above are isomorphisms, the space (4.46) is isomorphic to Hom A | B ( X, Z ).In this way, we at least obtain a functor Z A,B : Alg ( k )( A, B ) , −→ Cosp ( A, B ) and withthis also a lax functor Z : Alg ( k ) , −→ CALG( k ) . (4.47)(iii) Denote by F the subcategory of Alg ( k ) consisting of Frobenius algebras with trace-pairing and finite-dimensional bimodules. One can show [DKR2] that the restriction Z : F , −→ CALG( k ) , ∼ = CAlg ( k ) , ∼ = A lg ( k ) , (4.48)is locally fully faithful. This has the interpretation that all isomorphisms of lattice TFTswithout defects (i.e. isomorphisms of Frobenius algebras with trace pairing) are imple-mented by invertible domain walls (i.e. bimodules inducing Morita equivalences). Remark 4.20. (i) There is a close link between the lattice TFTs with defects and thecentre functor just defined. Let T : Bord def , top2 , ( D , D , D ) → V ect f ( k ) be a lattice TFT53ith defects as in theorem 3.8, and let D ≡ D [ D , D ; T ] be the 2-category of defectconditions defined in section 2.4. Then we have the commuting square D , (cid:15) (cid:15) E / / CAlg ( k ) I (cid:15) (cid:15) Alg ( k ) , Z / / CALG( k ) (4.49)where the functor ∆ was given in (3.64), E in (4.13), I in (4.41), and Z in (4.47). Indeed,evaluating the diagram on an invertible 2-morphism f : x → y for x, y : a → b gives forthe upper path and lower path, in this order, Hom A | B ( X, X ) f ◦ ( − ) ◦ f − (cid:15) (cid:15) Z ( A ) act act ( ( Hom A | B ( Y, Y ) Z ( B ) act h h act v v Hom A | B ( Y, Y ) id O O , Hom A | B ( X, X ) f ◦ ( − ) (cid:15) (cid:15) Z ( A ) act act ( ( Hom A | B ( X, Y ) Z ( B ) act h h act v v Hom A | B ( Y, Y ) ( − ) ◦ f O O , (4.50) These are isomorphic 2-diagrams, and thus equal in CALG( k ).(ii) The commuting square (4.49) shows that the lattice construction of defect TFTs isan implementation of the centre functor. Conjecturally, the restriction to invertible 2-morphisms can be dropped if one replaces both bicategories on the right hand side withthe (equally conjectural) tricategory CALG( k ). Rational conformal field theories can be build in two steps. In the first step one starts froma rational vertex operator algebra V and finds its modules and the corresponding spaces ofconformal blocks. The category R ep ( V ) of V -modules is a modular category in this case[HL, Hu].The second step is combinatorial and consists of assigning a correlator to each worldsheet, i.e. choosing a particular vector in the space of conformal blocks corresponding tothe world sheet, such that the factorisation and locality constraints are satisfied. In thecontext of vertex operator algebras and for world sheets of genus zero and certain worldsheets of genus one, such correlators were constructed in [HK] – see also the overview[KR2].The second step can also be solved elegantly for world sheets of arbitrary genus withthe help of three-dimensional topological field theory – provided one assumes that this 3dTFT correctly encodes the factorisation and monodromy properties of conformal blocksat arbitrary genus. The 3d TFT in question is the Reshitikhin-Turaev 3d TFT obtainedfrom the modular category R ep ( V ). This combinatorial construction of CFT correlatorsin terms of 3d TFT was carried out in [FRS1, Fj1] – see also the overview [RFFS] –54nd in particular allows for a description of CFT correlators for world sheets with defects[FRS1, Fr¨o1].Generalising the considerations in section 4.1 from 2d TFT to 2d CFT suggests aninteresting generalisation of the centre construction, which we now sketch.Let us start with the (bi)categories A lg ( k ) and Alg ( k ). Instead of working with al-gebras and bimodules over a field k , that is, with algebras in the symmetric monoidalcategory of k -vector spaces, one considers algebras and bimodules in a general monoidalcategory C (in the CFT-context, this is the category R ep ( V )). In particular, we do notdemand that C is symmetric or braided (though in the CFT context it is braided).To generalise CAlg ( k ) and CALG( k ), we need to be able to talk about commutativealgebras, so here we consider cospans of commutative algebras in a braided monoidalcategory.There is one major new ingredient when passing from vector spaces to more generalcategories, which is based on the following observation. For an algebra A in a generalmonoidal category C it makes no sense to talk about its centre as a subalgebra commutingwith the entire algebra, because the formulation of this condition needs a braiding. Anatural candidate to take the role of the centre in the case of general monoidal categoriesis the so-called full centre Z ( A ) of A [Fj2, Da]. This is a commutative algebra whichlives in the monoidal centre Z ( C ) (see [JS]) of the category C . Since Z ( C ) is braided, wecan talk about commutative algebras there. If C is the category of k -vector spaces, onehas the degenerate situation that Z ( C ) ∼ = C , and so many of the subtleties of the centre-construction are not visible. (In the context of rational CFT, and for modular categoriesin general, one has Z ( C ) = C ⊠ ¯ C , see [M¨u2, Thm. 7.10].)The constructions and results of sections 4.2–4.5 all have analogues in the more gen-eral setting of algebras in monoidal categories. For example, an instance of the equiva-lence (4.48), with the corresponding interpretation in terms of domain walls implementingequivalences of CFTs, has been found in [DKR1, Thm. 3.14]. More details will appear in[DKR2]. Remark 4.21.
As an aside, let us recall an observation from [SFR] which illustrates theusefulness of the 2-category of defect conditions defined in section 2.4 in the context ofrational CFT. Namely, consider a fixed rational CFT (i.e. restrict your attention to onlyone world sheet phase) and consider only topological defects from this world sheet phaseto itself, which in addition commute with the holomorphic and anti-holomorphic copy ofthe rational vertex operator algebra V . Then the 2-category D from section 2.4 has onlyone object (and so is a monoidal category). It turns out that D is Morita equivalent (inthe sense of [M¨u1, Def. 4.2]) to R ep ( V ); this follows since D is monoidally equivalent tothe category of A - A -bimodules in R ep ( V ) for an appropriate Frobenius algebra A withtrace pairing [Fr¨o1, Sec. 2]. It also follows (from [Sch, Thm. 3.3]) that the monoidal centre Z ( D ) is braided monoidally equivalent to the monoidal centre Z ( R ep ( V )) = R ep ( V ) ⊠ R ep ( V ). Thus, quite remarkably, if one knows the one-object 2-category of chiral symmetrypreserving topological defects in a rational CFT, one obtains for free the braided monoidalcategory of representations of its chiral symmetry V ⊗ ¯ V .55 Outlook
In this final section we would like to show some further directions that we find interestingand point out some open questions. From the perspective of this article, there are twoevident problems which we left untouched:1. In the introduction we claimed that there are two natural ways in which highercategories arise in field theory: by demanding that the functor defining the fieldtheory assigns data to manifolds of codimension larger than one, or by working withdefects of various dimensions. Clearly, one should study these two constructions inunison. We are aware of three works in this direction: one in 2d TFT [SP], and twoin arbitrary dimension – [Lu1, Sec. 4.3] and [MW, Sec. 6.7] – both ‘extended downto points’. A better understanding of the relation between the two appearances ofhigher categories should allow one to make precise the idea that n -dimensional TFTextended down to points is in some sense dual to n -dimensional TFT which hasdefects in all dimensions.2. A symmetric monoidal functor defining a 2d TFT or 2d CFT without defects has awell-known presentation in terms of generators and relations which provides a linkwith Frobenius algebras [So, Di, Ab, FRS1, HK]. This connection has been usefulin the construction of examples and in classification questions. For field theorieswith defects in 2d (let alone higher dimensions) such a generators and relationspresentation is presently not known. Nonetheless, progress has been made in relatedquestions: an algebraic description of 2d TFT with defects which extends down topoints was presented in [SP], and for planar algebras, generators are given in [KS] anda construction in terms of a 1-morphism in a pivotal strict 2-category is presentedin [Go]. For 2d homotopy TFTs over spaces with at least one of π or π trivial, aclassification in terms of Frobenius algebras with extra structure is given in [Tu, BT].Apart from these two points, let us list some further miscellaneous points to complementthe material presented in this note.One nice application of quantum field theories with topological defects is the orbifoldconstruction. Here, one introduces a domain wall which implements the ‘averaging overthe orbifold group’, together with a selection of lower-dimensional junctions which allowone to glue these domain walls together. The orbifold theory is then defined in terms ofa cell-decomposition of the original theory with the ‘averaging domain wall’ placed on thecodimension-1 cells. The advantage of this point of view is that the ‘averaging domain wall’need not actually be given by a sum over group-symmetries, giving rise to a generalisationof the orbifold construction. In the case of 2d rational CFTs, this is described in [Fr¨o2].It is proved there that any two rational CFTs over the same left/right chiral symmetryalgebra can be written as a generalised orbifold of one another.In the application of field theories to questions in cohomology one considers field theories‘over a space X ’, see e.g. [Tu, BT, ST]. This means that objects and morphisms of the56ordism category are in addition equipped with continuous maps to X . For each point x ∈ X , a field theory over X gives a field theory for undecorated bordisms by choosingthese continuous maps to be constant with value x . The role of X is reminiscent of our D n , the set labelling the top-dimensional domains M n for an n -dimensional field theorywith defects. However, in our setting the D n label attached to a point in M n is locallyconstant and may change only across M n − , and each such change has to be accompa-nied by specifying a domain wall which mediates this change. It would be interesting tohave a continuous formulation of the framework presented here to be able to incorporatecontinuously changing domain conditions via ‘smeared-out’ domain walls and junctions.While the general setup in section 2.1 allows for non-topological defects, in this note weonly studied the topological case. Theories with non-topological defects are much harderto treat and are much less studied. We mention here four examples in 2d CFT:- There are only two 2d CFTs in which all conformally invariant domain walls (thisincludes the topological ones) from the CFT to itself are known : the Lee-Yangmodel and the Ising model, see [OA, QRW].- In [QRW], a transmission coefficient was introduced which measures the ‘non-topo-logicality’ of a domain wall.- In [BB], the fusion product of certain non-topological domain walls in the free bosonCFT (found in [BBDO]) was computed, showing that at least in these theories thenotion of fusion makes sense for non-topological domain walls despite the short-distance singularities.- In the operator-algebraic approach of [BDH], non-topological domain walls are in-cluded from their start and also their fusion is defined. The definition is via Connes’fusion of bimodules and does not involve a short-distance limit.The centre of an algebra can be interpreted as a ‘boundary-bulk map’ in the follow-ing sense. When considering 2d TFT on surfaces with (unparametrised) boundaries, inaddition to ‘closed states’ associated to circles, there are ‘open states’ associated to inter-vals. The open states form a non-commutative Frobenius algebra and the closed statesform a commutative Frobenius algebra, which one can take to be the centre of A (see e.g.[Laz, MS]). Thus, the centre defines a theory in one dimension higher (here in dimen-sion two) for which the starting theory is a boundary theory (and the boundary is one-dimensional). The construction in section 4 can be understood as turning this boundary-bulk map into a functor. There are a number of situations in which such a boundary-bulkmap occurs: In 2d rational CFT one finds that the boundary theory determines a uniquebulk theory [Fj2]; algebraically this amounts to the construction of the full centre of analgebra in a monoidal category as briefly mentioned in section 4.6. In [DKR2] we will Here ‘known’ means that one has a list of defect operators satisfying a selection of consistencyconditions. Conjecturally, this uniquely specifies all conformally invariant defects, at least in ‘semi-simpletheories’. Only defects whose field content (the space Q ( O ( x ◦ x ∗ ))) has discrete ( L + ¯ L )-spectrum areconsidered. E [ k ]-algebras (related to algebrasover the little-discs operad in k -dimensions). Namely, in [Lu2] a construction is presentedwhich assigns to an E [ k ]-algebra (in a symmetric monoidal ∞ -category) its centre, whichis an E [ k +1]-algebra in the same category, see [Lu2, Cor. 2.5.13]. It would be interestingto understand the precise relation to the constructions presented here.
Acknowledgements : The authors would like to thank Nils Carqueville, Jens Fjelstad,J¨urgen Fuchs, Andr´e Henriques, Chris Schommer-Pries, and especially Sebastian Novakfor helpful discussions and comments on a draft of this article. AD would like to thankTsinghua University, and IR the Beijing International Center for Mathematical Research,for hospitality while part of this work was completed. LK is supported by the Basic Re-search Young Scholars Program of Tsinghua University, Tsinghua University independentresearch Grant No. 20101081762 and by NSFC Grant No. 20101301479. IR is supportedin part by the SFB 676 ‘Particles, Strings and the Early Universe’ of the DFG.
A Appendix: Bicategories and lax functors
In this appendix we recall the definition of bicategories and related notions, see [Be] or[Gr, Le].
Definition A.1.
A bicategory S consists of a set of objects (in a given universe) and acategory of morphisms M or ( A, B ) for each pair of objects A and B together with1. identity morphism: A : → M or ( A, A ) for all A ∈ S , where is a category withonly one object and only the identity morphism. We will abbreviate A ≡ A ( ) ∈M or ( A, A ),2. composition functor: ⊚ C,B,A : M or ( B, C ) × M or ( A, B ) −→ M or ( A, C ) , ( T, S ) T ◦ S , associativity isomorphisms: for A, B, C, D ∈ S , there is a natural isomorphism be-tween functors M or ( C, D ) × M or ( B, C ) × M or ( A, B ) → M or ( A, D ): α : ⊚ D,B,A ◦ ( ⊚ D,C,B × id) −→ ⊚ D,C,A ◦ (id × ⊚ C,B,A ) , left and right unit isomorphisms: for A, B ∈ S there are natural transformationsbetween functors × M or ( A, B ) → M or ( A, B ) and M or ( A, B ) × → M or ( A, B ): l : ⊚ B,B,A ◦ ( B × id) −→ id , r : ⊚ B,A,A ◦ (id × A ) −→ id , IR would like to thank Owen Gwilliam for discussions on this point. associativity coherence: (( S ◦ T ) ◦ U ) ◦ V α ( S,T,U ) ◦ id V / / α ( S ◦ T,U,V ) (cid:15) (cid:15) ( S ◦ ( T ◦ U )) ◦ V α ( S,T ◦ U,V ) (cid:15) (cid:15) ( S ◦ T ) ◦ ( U ◦ V ) α ( S,T,U ◦ V ) ) ) TTTTTTTTTTTTTTT S ◦ (( T ◦ U ) ◦ V ) id S ◦ α ( T,U,V ) u u jjjjjjjjjjjjjjj S ◦ ( T ◦ ( U ◦ V )) (A.1)2. identity coherence: ( S ◦ B ) ◦ T r ( S ) ◦ id T & & NNNNNNNNNNN α ( S, B ,T ) / / S ◦ ( B ◦ T ) id S ◦ l ( T ) x x ppppppppppp S ◦ T (A.2) Definition A.2.
Let C and D be two bicategories. A lax functor F : C → D is aquadruple F = ( F, { F ( A,B ) } A,B ∈ C , i, m ) where1. F is a map of objects X F ( X ) for each object X in C ,2. F ( A,B ) : M or C ( A, B ) → M or D ( F ( A ) , F ( B )) is a functor for each pair of objects A, B ∈ C ,3. unit transformation: natural transformations i A : F ( A ) → F ( A,A ) ◦ A between twofunctors → M or D ( F ( A ) , F ( A )) for all A ,4. multiplication transformation: m : ⊚ D ◦ ( F ( B,C ) × F ( A,B ) ) → F ( A,C ) ◦ ⊚ C , i.e. acollection of morphisms m S,T : F ( B,C ) ( S ) ◦ F ( A,B ) ( T ) → F ( A,C ) ( S ◦ T ) natural in S ∈ M or C ( B, C ) , T ∈ M or C ( A, B ),satisfying the following commutative diagrams:1. associativity: for S ∈ M or C ( C, D ) , T ∈ M or C ( B, C ) , U ∈ M or C ( A, B ),( F ( C,D ) ( S ) ◦ F ( B,C ) ( T )) ◦ F ( A,B ) ( U ) α D / / m ◦ id (cid:15) (cid:15) F ( C,D ) ( S ) ◦ ( F ( B,C ) ( T ) ◦ F ( A,B ) ( U )) id ◦ m (cid:15) (cid:15) F ( B,D ) ( S ◦ T ) ◦ F ( A,B ) ( U ) m (cid:15) (cid:15) F ( C,D ) ( S ) ◦ F ( A,C ) ( T ◦ U ) m (cid:15) (cid:15) F ( A,D ) (( S ◦ T ) ◦ U ) F ( A,D ) ( α C ) / / F ( A,D ) ( S ◦ ( T ◦ U )) , unit properties: for S ∈ M or C ( A, B ), F ( B ) ◦ F ( A,B ) ( S ) l ( F ( S )) / / i B ◦ id (cid:15) (cid:15) F ( A,B ) ( S ) F ( B,B ) ( B ) ◦ F ( A,B ) ( S ) m / / F ( A,B ) ( B ◦ S ) , F ( A,B ) ( l ( S )) O O F ( A,B ) ( S ) ◦ F ( A ) r ( F ( S )) / / id ◦ i B (cid:15) (cid:15) F ( A,B ) ( S ) F ( A,B ) ( S ) ◦ F ( A,A ) ( A ) m / / F ( A,B ) ( S ◦ A ) . F ( A,B ) ( r ( S )) O O If we reverse all arrows, we obtain the notion of oplax functor . Given a lax functor F , ifthe natural transformations i and m are actually isomorphisms, then F is called a functor.Let P be a property of a functor between 1-categories like full, faithful, essentiallysurjective, etc. We say that a (lax, oplax or neither) functor is locally P , if for all objects A, B the functors F ( A,B ) have property P . References [Ab] L.S. Abrams,
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