---

Higher Algebraic Structures and Quantization

arXiv:hep-th/9212115v2 8 Jun 1993REVISED VERSIONHigher Algebraic Structures and QuantizationDaniel S. FreedDepartment of MathematicsUniversity of Texas at AustinApril 22, 1993Very Abstract. We derive (quasi-)quantum groups in 2+1 dimensional topologicalfield theory directly from the classical action and the path integral.

Detailed com-putations are carried out for the Chern-Simons theory with finite gauge group. Theprinciples behind our computations are presumably more general.

We extend theclassical action in a d + 1 dimensional topological theory to manifolds of dimensionless than d + 1. We then “construct” a generalized path integral which in d + 1 di-mensions reduces to the standard one and in d dimensions reproduces the quantumHilbert space.

In a 2 + 1 dimensional topological theory the path integral over thecircle is the category of representations of a quasi-quantum group. In this paper weonly consider finite theories, in which the generalized path integral reduces to a finitesum.

New ideas are needed to extend beyond the finite theories treated here.Recent work on invariants of low dimensional manifolds utilizes complicated alge-braic structures, for both theory and computation. New invariants of 3-manifolds,and of knots and links in 3-manifolds, are constructed from certain types of Hopfalgebras [RT] or more generally from special sorts of categories [KR].

These in-variants are known to arise from a 2 + 1 dimensional quantum field theory [W].In this paper we derive the algebraic structure from the field theory, starting withthe classical lagrangian, and so express the relationship between the algebra andthe geometry directly. With this understanding the algebra can be put to work tocalculate invariants.

The guiding principle for us is the locality of field theory, asexpressed in gluing laws. The gluing laws resonate well with cut and paste tech-niques in topology.

They are important tools field theory offers for both theoreticalwork and computations. We generalize the standard constructs in a d + 1 dimen-sional field theory—classical action and path integral—to spaces of dimension lessthan d + 1, retaining the essential property of locality.

Whereas the classical actionis always a finite dimensional integral, the path integral over the space of fieldsusually involves infinitely many variables. Our focus here is not on the analyticalThe author is supported by NSF grant DMS-8805684, a Presidential Young Investigators awardDMS-9057144, and by the O’Donnell Foundation.

He warmly thanks the Geometry Center at theUniversity of Minnesota for their hospitality while this work was undertaken.Typeset by AMS-TEX1

difficulties of path integrals over infinite dimensional spaces; we only treat pathintegrals in a “toy model” where they reduce to finite sums.Nevertheless, ourgeneralizations of the classical action and path integral most likely pertain to othertopological field theories.In a d + 1 dimensional field theory the classical action of a field Θ on a (d + 1)-manifold X is usually a real1 number SX(Θ). Often in topological theories only theexponential e2πiSX(Θ) is well-defined.

The simplest example is the holonomy of aconnection: X = S1 is the circle and the field Θ is a connection on a principal circlebundle P →S1. Notice that the action is not as straightforward if X = [0, 1] hasboundary—interpreted as a number the parallel transport of a connection over theinterval depends on boundary conditions.

Rather, the dependence on boundaryconditions is best expressed by regarding the parallel transport as a map P0 →P1 from the fiber of the circle bundle over 0 to the fiber over 1.This is theclassical action over the interval. Our generalization of the classical action assertsthat the classical action of a connection over a point, which is just a principalcircle bundle Q →pt, is the fiber Q.The value of that action is a space onwhich the circle group T acts simply transitively, a so-called T-torsor.

Notice thatthe action of a field (connection) on the interval takes values in the action of therestriction of the field to the boundary. The Chern-Simons invariant in 3 dimensionsis similar—the action in 2 dimensions is a T-torsor—and the story continues to lowerdimensions [F1], [F2].At the crudest level of structure the classical action in d dimensions is a set.

(The classical action in d + 1 dimensions is a number. )The usual path integral in a d + 1 dimensional theory may be written schemati-cally asZCXdµX(Θ) e2πiSX(Θ),where X is a (d+1)-manifold without boundary, CX is the space of fields on X, anddµX is a measure on CX.

Of course, in many examples of interest this is only a for-mal expression since the measure does not exist, or has not been constructed. Thisintegral is a sum of positive numbers (the measure) times complex numbers (theexponentiated action), so is a complex number.

Our generalization to d dimensionsis as follows. The action is now a T-torsor, which we extend to a hermitian line,i.e., a one dimensional complex inner product space.

The original T-torsor is theset of elements of unit norm in the associated hermitian line. The integral is thena sum of positive numbers times hermitian lines.

If L is a hermitian line and µ apositive number, let µ·L be the same underlying one dimensional vector space withinner product multiplied by µ. We sum hermitian lines via direct sum; the sum isa hermitian vector space, or Hilbert space.

Formally, then, this generalized pathintegral is the space of L2 sections of a line bundle over the space of fields. Whenthe space of fields has continuous parameters we can formally reinterpret canonicalquantization, or geometric quantization, as the regularization needed to make senseof the integral.In higher codimensions the classical action and path integral take values in cer-tain generalizations of T-torsors and vector spaces.

The next step after a T-torsoris a T-gerbe [Gi], [Br], [BMc] and the next step after a vector space is a 2-vectorspace [KV], [L]. The underlying structure in both cases is not a set, but rather a1The theories we consider in this paper are unitary.2

category. The continuation to higher codimensions leads to multicategories, andthe foundations become rather murky, at least to this author.

We attempt an ex-position of these “higher algebraic structures” in §1 and §3. Our treatment has nopretensions of rigor.

For this reason throughout this paper we use the term ‘Asser-tion’ as opposed to ‘Theorem’ or ‘Proposition’, except when dealing with ordinarysets and categories. Since we deal with unitary theories our quantum spaces havean inner product, so are Hilbert spaces.

In codimension two we therefore obtain2-inner product spaces or 2-Hilbert spaces. The terminology may be confusing: A2-inner product space is an ordinary category, not a 2-category.The particular model we treat is gauge theory with finite gauge group.

It exists inany dimension. This theory was introduced by Dijkgraaf/Witten [DW] and furtherdeveloped by many authors [S2], [Ko], [Q1], [Q2], [Fg], [Y3], [FQ].

In some ways thispaper is a continuation of [FQ], though it may be read independently. The space offields (up to equivalence) on a compact manifold is a finite set in this model, henceall path integrals reduce to finite sums.

The lagrangian in the d + 1 dimensionaltheory is a singular (d + 1)-cocycle, and the generalized classical action is definedas its integral over compact oriented manifolds of dimension less than or equalto d + 1.Only the cohomology pairing with the fundamental class of a closedoriented (d+1)-manifold is standardly defined. In the appendix we briefly describean integration theory which extends this pairing.

It is the origin of the torsors,gerbes, etc. that we encounter.

We define the generalized classical action in §2 andthe generalized path integral in §4. Our assertions in these sections are formulatedfor all codimensions simultaneously, and we suggest that the reader decipher themstarting in the top dimension, where they reduce to the corresponding theoremsin [FQ].In §5 we explore the structure of the generalized path integral E over a circlein 1 + 1 dimensional theories and in 2 + 1 dimensional theories.

The treatmenthere is based on the generalized axioms of topological field theory2 set out in As-sertion 2.5 and Assertion 4.12, not on any particular features of finite gauge theory.In a 1 + 1 dimensional theory E is an inner product space and we construct acompatible algebra structure and a compatible real structure. The argument hereis standard.

In a 2 + 1 dimensional theory E is a 2-inner product space, which inparticular is a category. The analogue of the real algebra structure, here derivedfrom the generalized path integral, makes this a braided monoidal category withcompatible “balancing” and duality.

Such categories arise in rational conformalfield theory [MS], and have been much discussed in connection with topologicalinvariants and topological field theory. Reconstruction theorems in category the-ory [DM], [Ma1] assert that such a category is the category of representations ofa quasitriangular quasi-Hopf algebra, or quasi-quantum group [Dr].3 In fact, thereconstruction also requires a special functor from the category E to the categoryof vector spaces.

We remark that a different quasi-Hopf algebra related to fieldtheories was proposed in [Ma3].We put the abstract theory of §§1–4 to work in §§6–9, where we carry out thecomputations for the finite gauge theory. We warmup in §6 by discussing some2These axioms are not meant to be complete, and in any case they must be modified in otherexamples to allow for central extensions of diffeomorphism groups.

See [A], [Q2] for a discussionof the general axioms in topological field theory. See [F3] for a discussion of central extensions.3I believe that the quasitriangular quasi-Hopf algebras we obtain will always have a “ribbonelement” [RT] as well.

This certainly holds in the finite gauge theory.3

features of the 1 + 1 dimensional theory. The remainder of the paper treats the2 + 1 dimensional Chern-Simons theory (with finite gauge group).

The quasi-Hopfalgebras we compute via the generalized path integral are the quasi-Hopf alge-bras introduced by Dijkgraaf/Pasquier/Roche [DPR]. They were further studiedby Altschuler/Coste [AC].

The computations are not difficult, but they are nerve-racking! When dealing with categories (and, even worse, multicategories) one mustbe very careful about equality versus isomorphism, at the next level about equal-ity of isomorphisms versus isomorphisms of isomorphisms, and so on.

This sort ofalgebra seems well-adapted to the geometry of cutting and pasting, but as I saidit is nerve-racking. We keep close track of the trivializations we need to introduceat various stages of the computation.Some of these trivializations are used todefine the functor to the category of vector spaces which we need to reconstructthe quasi-Hopf algebra.

Our reconstructions do not follow the procedures in theabstract category theory proofs. Rather, in our examples the algebras are apparentfrom appropriate descriptions of the braided monoidal category.

In §9 we use moresophisticated gluing arguments to choose special bases of the algebras, and so derivethe exact formulas in [DPR]. This involves cutting and pasting manifolds with thesimplest kind of corners.

We formulate a generalized gluing law for the classicalaction in Assertion 9.2. Clearly it generalizes to higher codimensional gluing and tothe quantum theory.

Segal [S1] gives a proof of the “Verlinde diagonalization” [V]using a quantum version of this gluing law. This sort of generalized gluing shouldbe useful in other problems as well.

We also briefly describe at the end of §7 howSegal’s modular functor [S1] fits in with our approach.In gauge theory one usually makes special arguments to account for reducibleconnections. In these finite gauge theories every “connection” is reducible, thatis, every bundle has nontrivial automorphisms, and all of the constructions mustaccount for the automorphism groups.We formulate everything in terms of manifolds, whereas others prefer to workmore directly with knots and links.The relationship is the following (cf.

[W]).Suppose K is a knot in a closed oriented 3-manifold X. Let X′ = X −ν(K) denotethe manifold X with an open tubular neighborhood ν(K) of the knot removed.Then a framing of the normal bundle of K in X determines an isotopy class ofdiffeomorphisms from the standard torus S1 × S1 to ∂X′ = −∂ν(X).

In a 2 +1 dimensional topological field theory this induces an isometry between the quantumHilbert space of ∂X′ and the quantum Hilbert space of the standard torus. So thepath integral over X′ takes values in the Hilbert space of the standard torus.

Aswe explain at the end of §9 this Hilbert space is the “Grothendieck ring” of themonoidal category discussed above, and it has a distinguished basis consisting ofequivalence classes of irreducible representations. These are the “labels” in thetheory, and the coefficients of the path integral over X′ are the knot invariants forlabeled, framed knots.

The generalization to links is immediate.An expository version of some of this material appears in [F3].I warmly thank Larry Breen, Misha Kapranov, Ruth Lawrence, Nicolai Reshetikhin,Jim Stasheff, and David Yetter for informative discussions.4

§1 Higher Algebra IWhereas the classical action in a d + 1 dimensional field theory typically takesvalues in the real numbers, often in topological theories only its exponential withvalues in the circle groupT = {λ ∈C : |λ| = 1}is defined. We remark that nonunitary versions of these theories would replace Tby the group C× of nonzero complex numbers.

For the algebra in this section wecould replace T by any commutative group. The usual action is defined for fields onclosed4 spacetimes of dimension d+1.

In §2 we describe “higher actions” which aredefined for fields on manifolds of dimension less than d + 1 and take their values in“higher groups”. For example, over closed d-manifolds the action takes its values inthe abelian group-like category of T-torsors.

On a closed (d−1)-manifold the actiontakes its values in the abelian group-like 2-category of T-gerbes. And so on.

In thissection we briefly describe these “higher groups”. We also use the term “highertorsors”.

As stated in the introduction we only attempt a heuristic treatment, nota rigorous one. Our goal in this section, then, is to explain a hierarchy:(1.1)T0 = Tcircle groupT1“group” of T-torsors (1-torsors)T2“group” of T-gerbes (2-torsors)etc.Each of these is an abelian group in the sense that there is a commutative associativecomposition law, an identity element, and inverses.

However, only T0 is an honestgroup; in fact, only T0 is a set! The T-torsors T1 form a category,5 the T-gerbesa 2-category,6 etc.

So the group structure must be understood in that framework.Although this will not be relevant for us in this paper, we note that T is a Liegroup and the higher Tn also have some smooth structure.We begin with a definition.4Here ‘closed’ means ‘compact without boundary’. There is also a (relative) action on compactmanifolds with boundary, which we describe below.5We refer to [Mac] for the basics of category theory as well as plenty of examples.

Roughly,a category C is a collection of objects Obj(C) and for every A, B ∈Obj(C) there is a set ofmorphisms Mor(A, B). Morphisms Af−→B and Bg−→C compose to give a morphism Agf−−→C.This composition is associative and there are identity morphisms.Notice that Obj(C) is notnecessarily a set.

We often write ‘A ∈C’ for ‘A ∈Obj(C)’.6A 2-category C has a collection of objects Obj(C) and for each A, B ∈Obj(C) a category ofmorphisms Mor(A, B). In other words, if f, g ∈Mor(A, B), then there is a set of 2-morphismswhich map from f to g. The composition Mor(A, B) × Mor(B, C) →Mor(A, C) is now assumedto be a functor.One obtains different notions depending on whether one assumes that thiscomposition is exactly (strictly) associative or whether one postulates that it is associative upto a given 2-morphism.The former notion generalizes to n-categories.The latter notion wasintroduced by Benabou [Be] for 2-categories (these are called “bicategories”), and apparently acomplete list of axioms for the higher case has not been written down.

(See the lists of axiomsin [KV] to see the complications involved.) Since for three T-torsors A, B, C the torsors (A⊗B)⊗Cand A ⊗(B ⊗C) are strictly speaking different, but isomorphic, the category T1 does not havea strictly associative tensor product.

This propagates through to the higher Tn. Our use of thework ‘n-category’ is in the latter, yet undefined, sense.5

Definition 1.2. A T-torsor T is a manifold with a simply transitive (right) T-action.So ‘T-torsor’ is a short equivalent to ‘principal homogeneous T-space’.

Of course,T itself is a T-torsor, the trivial T-torsor. A nontrivial example, which is of no par-ticular relevance to us, is the nonidentity component of the orthogonal group O(2).An example of more relevance: Let L be any one dimensional complex inner prod-uct space.

Then the set of elements of unit norm is a T-torsor. Any T-torsor takesthis form for some hermitian line L (cf.

(3.2)). Now if T1, T2 are T-torsors, then amorphism h: T1 →T2 is a map which commutes with the T action: h(t·λ) = h(t)·λfor all t ∈T1, λ ∈T.

The collection of all T-torsors and morphisms forms a cate-gory T1. The group of automorphisms Aut(T ) of any T ∈T1 is naturally isomorphicto T: any µ ∈T acts as the automorphism t 7→t · µ.

Also, the set of morphismsMor(T1, T2) is naturally a T-torsor. Finally, every morphism in T1 has an inverse.7So far we have only described the category structure on T1, which is analogousto the set structure on T. The important point is this: Elements of T1 have au-tomorphisms.

We do not identify isomorphic elements which are not equal; thechoice of isomorphism matters. In fact, any two elements of T1 are isomorphic, soall of the information is in the isomorphism.

It does make sense to say that twoisomorphisms are equal, since Mor(T1, T2) is a set for any T1, T2 ∈T1.To describe the abelian group structure we need to introduce new operationswhich serve as the group multiplication and group inverse. These are the productof two torsors and the inverse torsor.

So if T1, T2 ∈T1 are T-torsors, define theproduct T1 · T2 asT1 · T2 = {⟨t1, t2⟩∈T1 × T2}⟨t1 · λ , t2⟩∼⟨t1 , t2 · λ⟩for all λ ∈T. The T action on T1 · T2 is⟨t1, t2⟩· λ = ⟨t1 · λ , t2⟩= ⟨t1 , t2 · λ⟩.The inverse T −1 of a torsor T with T action · has the same underlying set but anew T action ∗given byt ∗λ = t · λ−1.We denote the element in T −1 corresponding to t ∈T as t−1 ∈T −1.

The trivialtorsor T acts as the identity element under the multiplication. One must rememberthe maxim that elements in T1 cannot be declared equal, only isomorphic.

So wedo not have T · T −1 = T, but rather an isomorphismT · T −1 −→T⟨t · λ , t⟩7−→λ.This isomorphism is part of the data describing T1. All other axioms for an abeliangroup, such as commutativity and associativity, must be similarly modified.

Forexample, now the associative law is not an axiom but a piece of the structure—a system of isomorphisms—and these isomorphisms satisfy a higher-order axiomcalled the pentagon diagram.7So T1 is called a groupoid, which is not to be confused with the abelian group-like structurewe introduce below.6

We remark that there is a natural identification(1.3)T2 · T −11∼= Hom(T1, T2)for any T1, T2 ∈T1.Starting with the group T we have outlined the construction of an abelian group-like category T1. Now we want to repeat the construction replacing T with T1.

Inother words, we consider “T1-torsors” and then introduce a product law and inverseso as to obtain what is now an abelian group-like 2-category T2 of the collection ofall “T1-torsors”. The terminology is that a “T1-torsor” is a T-gerbe.The definitions are analogous to those for T-torsors, so we will be brief andincomplete.

A T-gerbe is a category G equipped with a simply transitive actionof T1. The action is a functor G × T1 →G whose action is denoted ⟨G, T ⟩7→G · T .The simple transitivity means that the functorG × T1 −→G × G⟨G, T ⟩7−→⟨G, G · T ⟩is an equivalence, and we are given an “inverse” function and equivalences of thecomposites to the identity.

This amounts to the specification of a torsor T (G1, G2)for G1, G2 ∈G together with natural equivalences G2 ∼= G1 · T (G1, G2) and T ∼=T (G, G · T ). This definition may be more rigid than the standard definition, but itfits our examples.Now if G1 and G2 are T-gerbes, then a morphism G1 →G2 is a functor whichcommutes with the T1 action.

This means that part of the data of the morphismis a natural transformation between the two functors obtained by traveling fromnorthwest to southeast around the squareG1 × T1 −−−−→G1yyG2 × T1 −−−−→G2It is easy to see that the collection of morphisms G1 →G2 forms a category andthat the morphisms Mor(G1, G2) form a T-gerbe. The collection of T-gerbes forms a2-category T2.

One can introduce an abelian group-like structure on this 2-categoryby defining the product of two T-gerbes and the inverse of a T-gerbe, which we leaveto the reader.I hope that at this stage it is in principle clear how I mean to define the seriesof abelian group-like structures listed in (1.1), and that it is clear what their basicproperties are, though the detailed definition promises to be a combinatorial mess.We need one more notion, which is a symmetry of such abelian group-like structures.Suppose A is a finite group. To say that A acts on T by symmetries means thatwe have a homomorphism A →T, i.e., a character of A, and then A acts on Tas multiplication by this character.

If T is a T-torsor, then since Aut(T ) ∼= T, anaction of A on T is again given by a character of A. Note that the characters formthe cohomology group H1(A; T).

Next, an action of A on T1 means that we havea “homomorphism” A →T1. More precisely, for each a ∈A we have a T-torsor Taand for a1, a2 ∈A an isomorphism Ta1 · Ta2 ∼= Ta1a2.

These isomorphisms must7

satisfy an associativity constraint.Such a system of torsors describes a centralextension ˜A = ∪a∈ATa of A:(1.4)1 −→T −→˜Aπ−→A −→1.The fiber of π over a is Ta. Up to isomorphism the central extension is classifiedby an element of the cohomology group H2(A; T).

An action of A on a T-gerbe Galso leads to a cohomology class, since different trivializations of G lead to equiv-alent extensions of A. The continuation of this discussion to higher Tn leads torepresentatives of higher group cohomology (with abelian coefficients).8

§2 Classical TheoryIn this section we describe a classical (gauge) field theory in d+1 dimensions withfinite gauge group Γ. We generalize the classical theory to higher codimensions,that is, to lower dimensional manifolds.

The (exponentiated) action on fields on a(d+1)-manifold takes values in T. For fields on a d-manifold the action takes valuesin T1, i.e., the value of the action is a T-torsor. More generally, over a (d + 1 −n)-manifold the action takes values in Tn.

We construct the action using the integrationtheory of the Appendix. Since this is a straightforward generalization of [FQ,§1],given the algebra in §1 and the integration theory in the Appendix, we defer tothat reference for more details and exposition.Throughout this paper we use a procedure to eliminate the dependence of quan-tities on extra variables or choices.

In [FQ,§1] we call this the invariant sectionconstruction after the special case mentioned in the footnote below. Here, follow-ing MacLane [Mac] (cf.

Quinn [Q1]) we call it an inverse limit of a functor. LetC be a groupoid and F : C →D a functor to a category (or multicategory) D.We define8 an element of the inverse limit to be a collection {v(C) ∈F(C)} suchthat F(C →C′)v(C) = v(C′) for all morphisms C →C′.

The inverse limit is anobject in D. In our applications D is Tn for some n or is the multicategory Vn ofhigher inner product spaces which we introduce in §3. Also, in our applicationsthe groupoid C has only a finite number of components.

For D = Vn the inverselimit always exists. If D = Tn we must also assume that F(C →C) is trivial forall automorphisms C →C, i.e., that “F has no holonomy”.Fix a finite group Γ.

For any manifold M we let CM denote the category ofprincipal Γ bundles over M. This is the collection of fields in the theory. Thereare symmetries as well: A morphism f : P ′ →P is a smooth map which commuteswith the Γ action and induces the identity map on M. Notice that every morphismis invertible.

Define an equivalence relation by setting P ′ ∼= P if there exists amorphism P ′ →P. Let CM denote the space of equivalence classes of fields; it is afinite set if M is compact.

If M is connected there is a natural identificationCM ∼= Homπ1(M, m), Γ Γfor any basepoint m ∈M. Here Γ acts on a homomorphism by conjugation.Let BΓ be a classifying space for Γ, which we fix together with a universal bundleEΓ →BΓ.

If P →M is a principal Γ bundle, then there exists a Γ map P →EΓand any two such classifying maps are homotopic through Γ maps.Fix a singular (d + 1)-cocycle α ∈Cd+1(BΓ; R/Z). This is the lagrangian ofour theory.

The action is constructed as follows. Suppose M is a compact orientedmanifold of dimension at most d + 1.

Let P ∈CM. Then if F : P →EΓ is aclassifying map for P, with quotient F : M →BΓ, consider the integralexp2πiZMF∗α,which is defined via the integration theory of the Appendix.

We need then to de-termine the dependence on F and obtain something independent of F. We treat8Think of the following example.Let C be the category whose objects are the points of amanifold M and whose morphisms are paths on M. Let D be the category of vector spaces andlinear isomorphisms. A vector bundle with connection over M determines a functor F : C →D(the morphisms act by parallel transport), and the inverse limit is the space of flat sections.9

closed manifolds and arbitrary compact manifolds (possibly with boundary) sepa-rately, though the second case clearly includes the first.Suppose first that Y is a closed oriented (d+1−n)-manifold, n > 0, and Q ∈CYis a Γ bundle over Y . Define a category CQ whose objects are classifying mapsf : Q →EΓ and whose morphisms are homotopies fh−→f ′.Define a functorFQ;α : CQ →Tn by(2.1)FQ;α(f) = exp2πiZY¯f ∗α= IY, ¯f ∗α,where ¯f : Y →BΓ is the quotient map determined by f : Q →EΓ.

For a homotopyfh−→f ′, let FQ;α(fh−→f ′) be the morphism(2.2)exp 2πiZ[0,1]×Y¯h∗α! : IY, ¯f ∗α −→IY, ¯f ′∗α.Here the homotopy h: [0, 1]×Q →EΓ has quotient map ¯h: [0, 1]×Y →BΓ.

Since∂([0, 1]×Y ) = {1}×Y ⊔−{0}×Y , the isomorphisms (A.6), (A.8), and (1.3) identifythe integral (2.2) as a map between the spaces shown. The gluing law (A.10) appliedto gluings of cylinders shows that FQ;α is indeed a functor.An automorphismfh−→f determines a classifying map h: S1 ×Q →EΓ, by gluing, and so extends toa classifying map H : D2×Q →EΓ.

Then ¯h: S1×Y →EΓ extends to H : D2×Y →EΓ, and by Stokes’ theorem (A.11) the morphism FQ;α(fh−→f) acts trivially. Sothere is an inverse limit of FQ;α in Tn, which we denote T αY (Q) = TY (Q).

(We omitthe ‘α’ if it is understood from the context.) It should be thought of as the valueof the classical action on Q.Now suppose X is a compact oriented (d+2−n)-manifold, possibly with bound-ary, and P ∈CX is a Γ bundle over X.Let CP be the category of classifyingmaps F : P →EΓ and homotopies FH−→F ′.

Restriction to the boundary definesa functor CP∂−→C∂P . If F ∈CP then by integration we obtain(2.3)exp2πiZXF∗α∈I∂X,∂F∗α = F∂P ;α(∂F).Furthermore, one can check that if FH−→F ′ is a homotopy, then (A.11) impliesthatF∂P ;α(∂F∂H−−→∂F ′) exp2πiZXF∗α= exp2πiZXF′∗α.These equations imply that (2.3) determines an element(2.4)e2πiSX(P ) ∈T∂X(∂P).We state the properties of this action without proof.Assertion 2.5.

Let Γ be a finite group and α ∈Cd+1(BΓ; R/Z) a cocycle. Thenthe assignments9(2.6)Q 7−→TY (Q) ∈Tn,Q ∈CY ,P 7−→e2πiSX(P ) ∈T∂X(∂P),P ∈CX9It is possibly better notation to write e2πiSX (P ) ∈e2πiS∂X (∂P ) for any compact oriented X,or perhaps instead TX(P ) ∈T∂X(∂P ).

We will sometimes use the latter notation, especially in §9.10

defined above for closed oriented (d+1−n)-manifolds Y and compact oriented (d+2−n)-manifoldsXsatisfy:(a) (Functoriality) If ψ: Q′ →Q is a bundle map covering an orientation pre-serving diffeomorphism ψ: Y ′ →Y , then there is an induced isomorphism(2.7)ψ∗: TY (Q′) −→TY (Q)and these compose properly. If ϕ: P ′ →P is a bundle map covering an orientationpreserving diffeomorphism ¯ϕ: X′ →X, then there is an induced isomorphism10(2.8)(∂ϕ)∗e2πiSX′(P ′)−→e2πiSX(P ),where∂ϕ: ∂P ′→∂Pistheinducedmapovertheboundary.

(b) (Orientation) There are natural isomorphisms(2.9)T−Y (Q) ∼=TY (Q)−1,and(2.10)e2πiS−X(P ) ∼=e2πiSX(P )−1. (c) (Additivity) If Y = Y1 ⊔Y2 is a disjoint union, and Qi are bundles over Yi,then there is a natural isomorphismTY (Q1 ⊔Q2) ∼= TY (Q1) · TY (Q2).If X = X1 ⊔X2 is a disjoint union, and Pi are bundles over Xi, then there is anatural isomorphism(2.11)e2πiSX1⊔X2 (P1⊔P2) ∼= e2πiSX1(P1) · e2πiSX2 (P2).

(d) (Gluing) Suppose Y ֒→X is a closed oriented codimension one submanifold andXcut is the manifold obtained by cutting X along Y . Then ∂Xcut = ∂X ⊔Y ⊔−Y .Suppose P is a bundle over X, P cut the induced bundle over Xcut, and Q therestriction of P to Y .

Then there is a natural isomorphism(2.12)TrQe2πiSXcut (P cut)−→e2πiSX(P ),where TrQ is the contractionTrQ : TXcut(∂P cut) ∼= TX(∂P) · TY (Q) · TY (Q)−1 −→TX(∂P).The Functoriality Axiom (a) means in particular that for any Q ∈CY there isan action of the finite group Aut Q on TY (Q). As explained in §1 the isomorphismclass of this action is an element of Hn(Aut Q; T).

For n = 2 this action determinesa central extension of Aut Q by T. We use an additional property of gluing in §9:Iterated gluings commute. As always, we must interpret ‘commute’ appropriatelyin categories.10If n = 1 then (2.8) is an equality of elements in a T-torsor.Similarly for (2.10), (2.11),and (2.12).11

§3 Higher Algebra IIThe quantum integration process is this: We integrate the classical action overthe space of equivalence classes of fields on some manifold. As explained in §2 theclassical action in codimension n takes values in Tn (or in a Tn-torsor for manifoldswith boundary).

For example, in the top dimension it takes values in T. But wecannot add elements of T. Rather, to form the quantum path integral we embedT ֒→C and add up the values of the classical action as complex numbers. In highercodimensions we introduce “higher inner product spaces” where we can perform thesum.11 The collection Vn of all complex n-inner product spaces12 is an n-category,which is in some sense the trivial complex (n + 1)-inner product space, and thereis an embedding Tn ֒→Vn onto the set of elements of “unit norm”.

We view theaction as taking values in Vn and then take sums there to perform the path integral.Our goal in this section, then, is to describe this hierarchy:(3.1)V0 = Cfield of complex numbersV1“ring” of (virtual) finite dimensional complex inner product spacesV2“ring” of (virtual) finite dimensional complex 2-inner product spacesetc.The inner product space notions of dual space (or conjugate space), direct sum,and tensor product generalize to Vn, and this gives it a structure analogous to acommutative ring with involution.The notion of a 2-vector space appears in work of Kapranov and Voevodsky [KV],and also in lectures of Kazhdan and in recent work of Lawrence [L]. We in no wayclaim to have worked out the category theory in detail, and we feel that this sortof “higher linear algebra” merits further development.The terminology is confusing: An n-inner product space is an (n −1)-category.Thus a 2-inner product space is an ordinary category.Recall that an inner product space V is a set with an commutative vector sumV ×V →V , a scalar multiplication C×V →V , and an inner product (·, ·): V ×V →C.

(The conjugate inner product space V is defined below.) We will not review allof the axioms here.

There are two trivial examples: the zero inner product space Oconsisting of one element, and C with its usual inner product (z, w) = z · ¯w. IfV1, V2 are inner product spaces, then a morphism is a linear map V1 →V2 whichpreserves the inner product.

The collection of inner product spaces and linear mapsforms a category V1.Suppose T ∈T1 is a T-torsor. From T we form the one dimensional complexinner product space (hermitian line)(3.2)LT = T ×T C= {⟨t, z⟩∈T × C}⟨t · λ, z⟩∼⟨t, λ · z⟩11Since our basic group is T (as opposed to C×) we obtain complex inner product spaces (asopposed to simply complex vector spaces).

Presumably one can generalize to other base fields orrings.12It is probably better to consider the category of virtual complex n-inner product spaces,that is, formal differences of complex n-inner product spaces. This provides additive inverses andis more closely analogous to a ring.

However, we will only encounter “positive” elements of this“ring” so do not insist on the inclusion of virtual inner product spaces.12

for all λ ∈T. Note that LT ∼= C. The inner product on LT is⟨t, z⟩, ⟨t, w⟩= z · ¯w.If V ∈V1 is an inner product space, we form the dual space V ∗= Hom(V, C)with its usual inner product.

The conjugate inner product space V has the sameunderlying abelian group as V but the conjugate scalar multiplication and thetransposed inner product. There is a natural isometry V ∼= V ∗given by the innerproduct.

If V1, V2 ∈V1 then one can form the direct sum V1 ⊕V2 and the tensorproduct V1 ⊗V2 with the inner products(v1 ⊕v2 , w1 ⊕w2) = (v1, w1) + (v2, w2)(v1 ⊗v2 , w1 ⊗w2) = (v1, w1) (v2, w2).Notice that there are natural isomorphisms O ⊕V ∼= V and C ⊗V ∼= V . Also, ifT1, T2 ∈T1 then LT −1 ∼= L∗T and LT1·T2 ∼= LT1 ⊗LT2.

The direct sum and tensorproduct give V1 a commutative ring-like13 structure with involution, the involutionbeing the conjugation or duality.It is useful to observe that for any inner product space V , the induced innerproduct on V ∗⊗V is(T1, T2) = Tr(T1T ∗2 ),Ti ∈Hom(V ),where we identify V ∗⊗V ∼= Hom(V ) via the canonical isomorphism, and T ∗is thehermitian adjoint of T .Finally, we introduce an “inner product”(·, ·): V1 × V1 −→V1by(V1, V2) = V1 ⊗V2,and the associated “norm” |V |2 = V ⊗V .Notice that the elements of “unitnorm”, that is of norm C, are precisely the hermitian lines, i.e., the image of theembedding T1 ֒→V1. The image is closed under tensor product and the embeddingis a homomorphism.Starting with the field C we have outlined the construction of a commutativering-like category V1 (with involution) consisting of inner product spaces over C.Now we iterate and consider inner product spaces over V1, which we call complex2-inner product spaces.14 So a complex 2-inner product space W is a category withan abelian group law W ×W →W, a “scalar multiplication” V1 ×W →W, and an“inner product” W × W →V1.

There is a zero complex 2-inner product space O.The dual, conjugate, direct sum, and tensor product are defined. The category V1is a 2-inner product space which is an identity element for the tensor product.13As we mentioned above, we should include virtual inner product spaces to have additiveinverses.14Since V1 is analogous to a ring, not a field, we expect that not all of its modules are free.The ones we consider in this paper are sums of one dimensional cyclic modules, so are free.

Aformal development of this concept should probably demand freeness in the definition [KV].13

The collection of all (virtual) complex 2-inner product spaces forms a commutativering-like 2-category V2 with involution.Because a 2-inner product space is a category, and not a set, there is an extralayer of structure (natural transformations) and so additional data as part of thedefinition. We do not claim to have a complete list, but mention some additionalstructure related to the inner product.

Namely, for all W1, W2 ∈W there is aspecified map(W2, W1) · W1 −→W2.The ‘·’ here is the scalar product. We might further assume that Mor(W1, W2) isisomorphic to the vector space (W2, W1); this holds in the examples.

In addition,we postulate a preferred isometry(W1, W2) −→(W2, W1)whose “square” is the identity. In particular, (W, W) has a real structure for all W ∈W, and we assume the existence of compatible maps(3.3)C −→(W, W) −→C.The composition is then multiplication by a real number, which we call dim W.A linear map of complex 2-inner product spaces L: W1 →W2 is a functor whichpreserves the addition and scalar multiplication.

The space of all such linear maps isthe 2-inner product space Hom(W1, W2) ∼= W2 ⊗W∗1. If we assume some freenesscondition on 2-inner product spaces (see previous footnote), then we can clearlygeneralize other standard notions of linear algebra.

For example, we should be ableto define linear independence and bases. Then if P : W →W is a linear operatoron W, a matrix representation relative to a basis of W is a matrix of inner productspaces P ij ∈V1.

The trace Tr(P) = Li P ii is then an inner product space. Thedimension of W is the trace of the identity map, which is dim W = Cn for some n.It makes sense, then, to identify the dimension of W as n.If G is a T-gerbe, then we form the one dimensional complex 2-inner productspace(3.4)WG = G ×T1 V1= {⟨G, V ⟩∈G × V1}⟨G · T, V ⟩∼⟨G, LT ⊗V ⟩for all T-torsors T .

Note that WT1 ∼= V1. If we define the inner product(W1, W2) = W1 ⊗W2on V2, then we see that the image of the embedding T2 ֒→V2 determined by (3.4)consists of complex 2-inner product spaces of “unit norm”.

The image is closedunder tensor product and the embedding is a homomorphism.Here is a more concrete example of a nontrivial 2-inner product space whichis important in what follows. Suppose A is a finite group.

Let (V1)A denote thecategory of finite dimensional unitary representations of A. The morphisms arerequired to commute with the A action.

Then (V1)A is a 2-inner product space asfollows. If W ∈(V1)A and V ∈V1 then we can “scalar multiply” V by W using theordinary vector space tensor product.

We obtain V ⊗W, which is a representation14

of A. The vector sum in (V1)A is the usual direct sum of representations.

The innerproduct on (V1)A is(3.5)(W1, W2) = (W1 ⊗W2)A,where for any representation W ∈(V1)A the inner product space W A ∈V1 is thesubspace of invariants. Note that if W is an irreducible unitary representation of A,then (cf.

[FQ,Appendix A])(W, W) = dim W · C,since dim W is the norm square of the canonical element of W ⊗W. The compo-sition (3.3) is dim W in the usual sense.

The dimension of (V1)A is the number ofisomorphism classes of irreducible representations of A.More generally, suppose that G is a T-gerbe with a nontrivial A action, whichwe denote by ρ. For any G ∈G let(3.6)LG = ⟨G, C⟩∈WG.

(Recall the definition of WG in (3.4).) Note that for any line L ∈V1, the element⟨G, L⟩∈WG is equivalent to LG′ for some G′ ∈G, and so any element of WG isisomorphic to a finite sum LG1 ⊕· · · ⊕LGk.

Let A act on WG by(3.7)a · LG = La·G.Finally, set15(3.8) (WG)A,ρ = span{W = LG1 ⊕· · ·⊕LGk : W is invariant under the A action}.This is our sought-after 2-inner product space. If G = T1 is the trivial T-gerbe,then according to (1.4) the action ρ determines a central extension ˜A of A by T,and to each a ∈A corresponds a T-torsor Ta which is the fiber of ˜A over a.We can describe (WT1)A,ρ = (V1)A,ρ as the category of representations of ˜A suchthat the central T acts by standard scalar multiplication.For then an elementof this category is of the form W = LT1 ⊕· · · ⊕LTk, where for each a ∈A andeach index i there is an index j with Ta · Ti = Tj.Then (3.7) is an isometryLTa ⊗LTi →LTj, and so each ˜a ∈Ta induces an isometry LTi →LTj.Thisdescribes the ˜A action.

The dimension of (WT1)A,ρ as the number of isomorphismclasses of such irreducible representations. Since any T gerbe G is (noncanonically)isomorphic to T1, this is also the dimension of (WG)A,ρ.

If ρ is the trivial A actionon T1, then (WT1)A,ρ = (V1)A,ρ is the 2-inner product space (V1)A we defined inthe previous paragraph.Think of (WT1)A,ρ as the space of A-invariants in WG. We can also considerinvariants in the analogous situation “one dimension down”.

That is, if A acts ona T-torsor T through a character µ: A →T, then A also acts on the hermitianline LT through the same character. We define(LT )A,µ = {ℓ∈LT : ℓis invariant under the A action}.15To make good sense of “invariant” we must identify certain canonically isomorphic elements.For example, we need to identify different permutations of the sum LG1 ⊕· · · ⊕LGk.

Also, thisdefinition is suspicious—the dimension of the invariants is larger than the dimension of WG!15

But this is simple:(LT )A,µ = LT ,if µ is trivial;0,otherwise.We remark that whereas (V1)A has a natural monoidal structure16 given by thetensor product of representations, the category (V1)A,ρ for ρ nontrivial do not: thetensor product of representations of ˜A where T acts as scalar multiplication is arepresentation of ˜A where T acts as the square of scalar multiplication. Also, if G isa nontrivial gerbe, then (WG)A,ρ is not monoidal in a natural way.Finally, by forgetting the A action we obtain an “augmentation” linear map(WG)A,ρ −→WG.If G = T1 is trivial, it takes values in WT1 = V1.Clearly these constructions have analogs in the higher complex inner productspaces (3.1).16A monoidal category is a category equipped with a tensor product and an identity element.In addition, an “associator” and natural transformations related to the identity element must bespecified explicitly.

A monoidal category is the category-theoretic analogue of a monoid, which isa set with an associative composition law and an identity element.16

§4 Quantum TheoryNow we are ready to quantize the classical d+1 dimensional classical field theorydescribed in §2. We carry out the quantization on any compact oriented manifoldof dimension less than or equal to d + 1 by integrating the classical action overthe space of fields.

(We first use the constructions in §3 to convert the values ofthe classical action from an n-torsor to an n-inner product space. )Since thereare symmetries of the fields, we only integrate over equivalence classes of fields.The residual symmetry, that is, the automorphism groups of the fields, must alsobe taken into account.

Since the gauge group is finite, the space of equivalenceclasses of fields on a compact manifold is a finite set, so all we need to perform thepath integral is a measure on this finite set. We also need to define the productof a positive number µ (the measure) by an element W ∈Vn.

This we denoteas µ · W and interpret it as W with the inner product multiplied by µ. The restis a straightforward generalization of [FQ,§2], given the higher algebra of §3 andthe classical theory of §2.

For a closed oriented (d + 1 −n)-manifold Y , n > 0,the resulting quantum invariant is a complex n-inner product space E(Y ) ∈Vn. IfY = ∅is the empty manifold, then E(∅) = Vn−1 is the trivial space.

The quantuminvariant of a compact oriented (d+2−n)-manifold X, possibly with boundary, is anelement ZX ∈E(∂X). For n = 1 we recover the quantum invariants of [FQ,§2]—the ordinary path integral (partition function) and the quantum Hilbert space.For n = 2 the quantum invariant of a closed oriented (d−1)-manifold S is a 2-innerproduct space E(S), and the quantum invariant of a compact oriented d-manifold Yis an object ZY in the category E(∂Y ).

Et cetera.We first introduce a measure µ on the category of principal Γ bundles CM overany manifold M. For P ∈CM set(4.1)µP =1# Aut P .Clearly µP ′ = µP for equivalent bundles P ′ ∼= P, so µ determines a measure on theset of equivalence classes CM. This is the assertion that the measure is invariantunder the symmetries of the fields.If M has a boundary, for each Q ∈C∂M set(4.2)CM(Q) = {⟨P, θ⟩: P ∈CM, θ: ∂P →Q is an isomorphism}.A morphism ϕ: ⟨P ′, θ′⟩→⟨P, θ⟩is an isomorphism ϕ: P ′ →P such that θ′ = θ◦∂ϕ.The morphisms define an equivalence relation on CM(Q), and we denote the set ofequivalence classes by CM(Q).

Equation (4.1) determines a measure on CM(Q).Note that any automorphism of ⟨P, θ⟩∈CM(Q) is the identity on componentsof M with nontrivial boundary. If ψ: Q′ →Q is an isomorphism of Γ bundlesover ∂M, then ψ induces a measure-preserving mapψ∗: CM(Q′) −→CM(Q)by ψ∗(P, θ) = ⟨P, ψθ⟩.

In particular, for Q′ = Q this gives a measure-preservingaction of Aut Q on CM(Q).One important property of µ, which is an ingredient in the proof of the gluinglaw (4.17), is its behavior under cutting and pasting. Suppose N ֒→M is an ori-ented codimension one submanifold and M cut the manifold obtained by cutting M17

along N. For each Q ∈CN, Q′ ∈C∂M, we obtain a gluing mapgQ : CMcut(Q ⊔Q ⊔Q′) −→CM(Q′)⟨P cut; θ1, θ2, θ⟩7−→⟨P cut/(θ1 = θ2) ; θ⟩.We refer to [FQ,§2] for the proof of the following.Lemma4.3. ThegluingmapgQsatisfies:(a) gQ maps onto the set of equivalence classes of bundles over M whose restrictiontoNisisomorphictoQ.

(b) Let φ ∈Aut Q act on ⟨P cut; θ1, θ2, θ⟩∈CMcut(Q ⊔Q) byφ · ⟨P cut; θ1, θ2, θ⟩= ⟨P cut; φ ◦θ1, φ ◦θ2, θ⟩.Then the stabilizer of this action at ⟨P cut; θ1, θ2, θ⟩is the image Aut P →Aut Q de-terminedbytheθi,whereP=gQ(⟨P cut; θ1, θ2, θ⟩). (c) There is an induced action on equivalence classes CMcut(Q⊔Q), and Aut Q actstransitivelyong−1Q ([P])forany[P]∈CM.

(d) For any [P] ∈CM(Q) we have(4.4)µ[P ] = volg−1Q ([P])· µQ.Now we are ready to carry out the quantization.We treat all codimensionssimultaneously, but suggest that the reader first review the top dimensional quan-tizations in [FQ,§2].Again for clarity we first treat closed manifolds and thenarbitrary compact manifolds (possibly with boundary), though the second caseincludes the first.Suppose first that Y is a closed oriented (d+1−n)-manifold, n > 0. The classicalaction defined in §2 is a mapTY : CY −→Tn,which we can think of as a bundle of “n-torsors” over CY .

By Assertion 2.5(a)for each Q ∈CY there is an action ρQ of Aut Q on TY (Q).Use the construc-tion (3.2), (3.4) to replace each TQ by the one dimensional n-inner product space(4.5)WQ = WTY (Q).Assertion 2.5(a) also implies that an isomorphism ψ: Q′ →Q induces an isomor-phism ψ∗: WQ′ →WQ. However, an automorphism ψ ∈Aut Q does not necessarilyact trivially on WQ.

Rather, it only acts trivially on the subspace of invariants un-der the Aut Q action (cf. (3.8)).

More precisely, we construct a “quotient” complexn-inner product space W[Q] associated to the equivalence class [Q] ∈CY as aninverse limit. (The inverse limit picks out the invariants under automorphisms.

)Consider the category C[Q] of bundles Q in the isomorphism class [Q], and letF[Q] : C[Q] →Vn be the functor whose value at Q is WQ. Set W[Q] to be the inverselimit of F[Q].

As [Q] varies we then obtain a mapWY : CY −→Vn.18

The quantum space E(Y ) is the integral of WY over CY , which in this case is afinite sum:(4.6)E(Y ) =ZCYdµ([Q]) WY ([Q]) =M[Q]∈CYµ[Q] · W[Q] ∈Vn.If we think of WY as a bundle of n-inner product spaces over CY , then E(Y ) is thespace of L2 sections of that bundle.Now suppose that X is a compact oriented (d + 2 −n)-manifold, possibly withboundary. The classical action on the boundary ∂X is a bundle of n-torsors T∂X →C∂X, and the classical action e2πiSX on X is a section of the pullback r∗T∂X, wherer is restriction to the boundary:r∗T∂X −−−−→T∂XyyCXr−−−−→C∂XBy Assertion 2.5(a) the action is invariant under the morphisms in CX, that is,under symmetries of the fields.

Now for each P ∈CX we use the construction (3.6)to define an element(4.7)LX(P) = Le2πiSX (P ) ∈W∂P = WT∂X(∂P ).Now LX(P) is not necessarily invariant under Aut P; it transforms under ψ ∈Aut Paccording to the action of the restricted automorphism ∂ψ ∈Aut(∂P) on WT∂X(∂P ).We only obtain invariance after integrating. Thus fix Q ∈C∂X and consider CX(Q)as defined in (4.2).

If ⟨P, θ⟩∈CX(Q) then using θ to identify T∂X(∂P) ∼= T∂X(Q)we have the action e2πiSX(P,θ) ∈T∂X(Q) and the associated LX(P, θ) ∈WQ, asin (4.7). If ⟨P, θ⟩∼= ⟨P ′, θ′⟩then there is an isomorphism between the values of theactions on these fields as elements of T∂X(Q).

By another inverse limit constructionwe define LX([P, θ]) ∈WQ. Set(4.8)ZX(Q) =ZCX(Q)dµ([P, θ]) LX([P, θ]) =M[P,θ]∈CX(Q)µ[P,θ] · LX([P]) ∈WQ.Now we claim that ZX(Q) is invariant under the Aut Q action on WQ, and so(4.9)ZX(Q) ∈(WT∂X(Q))Aut Q,ρQMore generally, we check that for an isomorphism ψ: Q′ →Q we haveψ∗ZX(Q′) =M[⟨P ′,θ′⟩]µ[P ′] · ψ∗LX([P ′, θ′])∼=M[⟨P ′,θ′⟩]µ[P ′] · LX([P ′, ψθ′])= ZX(Q),(4.10)19

since ⟨P ′, ψθ′⟩runs over a set of equivalence classes in CX(Q) as ⟨P ′, θ′⟩runs over aset of equivalence classes in CX(Q′). Using the definition (3.8) of (WT∂X(Q))Aut Q,ρQwe deduce (4.9), and furthermore (4.10) shows that {ZX(Q) : Q ∈[Q]} is a collec-tion of elements in {WQ : Q ∈[Q]} invariant under symmetries.

In other words, itis an element of the inverse limit W[Q]:ZX([Q]) ∈W[Q].Finally, then,(4.11)ZX =M[Q]∈C∂XZX([Q]) ∈M[Q]∈C∂Xµ[Q] · W[Q] = E(∂X)is the desired quantum invariant.The basic properties of these quantum invariants, which we might term “higherquantum Hilbert spaces” and “higher path integrals”, are listed in the following.Assertion 4.12. Let Γ be a finite group and α ∈Cd+1(BΓ; R/Z) a cocycle.

Thenthe assignments17Y 7−→E(Y ) ∈Vn,X 7−→ZX ∈E(∂X),defined above for closed oriented (d+1−n)-manifolds Y and compact oriented (d+2−n)-manifoldsXsatisfy:(a) (Functoriality) Suppose f : Y ′ →Y is an orientation preserving diffeomor-phism. Then there is an induced isometry(4.13)f∗: E(Y ′) −→E(Y )and these compose properly.

If F : X′ →X is an orientation preserving diffeomor-phism, then there is an induced isometry18(4.14)(∂F)∗(ZX′) −→ZX,where∂F : ∂X′→∂Xistheinducedmapovertheboundary. (b) (Orientation) There are natural isometriesE(−Y ) ∼= E(Y ),and(4.15)Z−X ∼= ZX.

(c) (Multiplicativity) If Y = Y1 ⊔Y2 is a disjoint union, then there is a naturalisometryE(Y1 ⊔Y2) ∼= E(Y1) ⊗E(Y2).17Again the notation is awkward, and possibly it is best to use ZX for all X and writeZX ∈Z∂X.18If n = 1 this is an equality, as are (4.15), (4.16), and (4.17).20

If X = X1 ⊔X2 is a disjoint union, then there is a natural isometry(4.16)ZX1⊔X2 ∼= ZX1 ⊗ZX2. (d) (Gluing) Suppose Y ֒→X is a closed oriented codimension one submanifold andXcut is the manifold obtained by cutting X along Y .

Write ∂Xcut = ∂X ⊔Y ⊔−Y .Then there is a natural isometry(4.17)TrY (ZXcut) −→ZX,where TrY is the contraction(4.18)TrY : E(∂Xcut) ∼= E(∂X) ⊗E(Y ) ⊗E(Y ) −→E(∂X)using the inner product on E(Y ).Just as on the classical level, iterated gluings commute.Proof. We only comment on the gluing law (d).

The proof is formally the same asthe one in [FQ,§2], but we repeat it here anyway. Recall that for a field P over acompact oriented (d + 2 −n)-manifold X we have the action e2πiSX(P ) ∈T∂X(∂P)which lives in an n-torsor, and the associated LX(P) ∈WT∂X(∂P ) which lives in ann-vector space (cf.

(2.4) and (4.7).) Fix a bundle Q′ →∂X.

Then for each Q →Yand each P cut ∈CXcut(Q′ ⊔Q ⊔Q) we have an isometry(4.19)LX(gQ(P cut)) ∼= TrQLXcut(P cut)by (2.12), where now T rQ is the contractionTrQ : WT∂Xcut(∂P cut) ∼= WT∂X(∂P ) ⊗WTY (Q) ⊗WTY (Q) −→WT∂X(∂P )using the inner product on WTY (Q), and gQ is the gluing map(4.20)gQ : CXcut(Q′ ⊔Q ⊔Q) −→CX(Q′).Fix [P] ∈CX(Q′) and consider g−1Q ([P]). By Lemma 4.3(c) the group Aut Q actstransitively on g−1Q ([P]).

This means that the invariants in the representation(4.21)M[P cut]∈g−1Q ([P ])LXcut([P cut])of Aut Q by its diagonal action on WTY (Q) × WT−Y (Q) via ρQ × ρQ are the “con-stant functions” under the isomorphism (4.19). Then the inner product (3.5) in(WTY (Q))Aut Q,ρQ applied to (4.21) gives(4.22)M[P cut]∈g−1Q ([P ])LXcut([P cut])Aut Q∼= #g−1Q ([P]) · LX([P]).21

Fix a set of representatives {Q} for CY . Let CX(Q′)Q denote the equivalence classesof bundles over X whose restriction to ∂X is Q′ and to Y is Q (with given iso-morphisms as in (4.2)).

Thus using equation (4.4) on the measure and the isome-try (4.22) we calculateZX(Q′) =ZCX(Q′)dµ ([P]) LX([P])=XQ∈{Q}ZCX(Q′)Qdµ ([P]) LX([P])∼=XQ∈{Q}µQ ·"ZCXcut (Q′⊔Q⊔Q)dµ([P cut]) TrQLXcut([P cut])#Aut Q=XQ∈{Q}µQ · TrQZXcut(Q′ ⊔Q ⊔Q)Aut Q= TrY (ZXcut(Q′)).22

§5 Product StructuresSome form of the following assertion holds: In a d + 1 dimensional topologicalquantum field theory the d-inner product space E(S1) has the structure of a “highercommutative associative algebra with identity and compatible real structure andinner product”. In this section we only discuss the cases d = 1 and d = 2.

For d =1 we obtain an ordinary algebra structure on the vector space E(S1), togetherwith a compatible real structure. The inner product on E(S1) is compatible withall of these structures.

This is a standard argument, which we repeat here as awarmup. For d = 2 the quantum space E(S1) is a 2-inner product space, whichin particular is a category.

The algebra structure we discuss gives it the structureof a braided monoidal category [JS].19 Here the commutativity and associativityconditions give additional data (rather than being conditions on the multiplication,as in an ordinary algebra), and there is an additional piece of data coming fromnontrivial loops of diffeomorphisms of the circle (a balancing). All of the argumentsin this section proceed directly from the axioms in Assertion 4.12.

So they hold forany theory which obeys these axioms, not just for a gauge theory with finite gaugegroup.We begin with some standard deductions about arbitrary d+1 dimensional the-ories. First, a deduction about the classical theory.

Suppose Y is a closed orientedmanifold and Q ∈CY a Γ bundle. Consider the product [0, 1] × Q ∈C[0,1]×Y , whichis a bundle over the “cylinder” [0, 1] × Y .

The classical action20 T[0,1]×Y ([0, 1] × Q)is an automorphism of TY (Q). Now glue two copies of [0, 1] × Q end to end andapply the gluing law (2.12) to construct an isomorphism(5.1)T[0,1]×Y ([0, 1] × Q) · T[0,1]×Y ([0, 1] × Q) 7−→T[0,1]×Y ([0, 1] × Q).This implies that there is a canonical element(5.2)t ∈T[0,1]×Y ([0, 1] × Q)which satisfies t · t = t. In other words, the classical action of a product field istrivialized.

If dim Y = d the classical action is the identity map of TY (Q). Thequantum version of (5.1), obtained from the quantum gluing law (4.17), assertsthat(5.3)Z[0,1]×Y : E(Y ) −→E(Y )is an idempotent.

In other words, there is an isometry(5.4)(Z[0,1]×Y )2 −→Z[0,1]×Y .We may as well assume that Z[0,1]×Y is isometric to the identity, since in anycase we can replace E(Y ) by the image of (5.3) to obtain a new theory with thisproperty. Similarly, gluing the ends of [0, 1] × Y together we deduce the existenceof an isometry(5.5)ZS1×Y ∼= dim E(Y ).19In fact, we obtain what some refer to as a tortile category.

See [Y1,§1], [Y2] for a precisedefinition and more thorough discussion. The notion of a tortile category is due to Shum [Sh].20We use the notation TX(P ) instead of e2πiSX (P ), even though X = [0, 1] × Y is not closed.23

Here the dimension of an n-inner product space is an (n −1)-inner product space,as discussed in §3. More generally, if f : Y →Y is an orientation preserving diffeo-morphism, we can glue with a twist by f to form the mapping torus S1 ×f Y .

Theaxioms now imply the existence of an isometryZS1×f Y ∼= TrE(Y )(f∗),where f∗: E(Y ) →E(Y ) is the isometry (4.13).Another easily deduced property also relates to the functoriality (4.13). Supposethat f0, f1 : Y ′ →Y are isotopic orientation preserving diffeomorphisms, and thatft : Y ′ →Y is an isotopy.

Form the mapF : [0, 1] × Y ′ −→[0, 1] × Y⟨t, y′⟩7−→⟨t, ft(y′)⟩. (More generally, our considerations apply to pseudoisotopies F, that is, to arbitrarydiffeomorphisms F which restrict on the ends to f0 and f1.) Now apply the func-toriality axiom (4.14) as follows.

The partition functions Z[0,1]×Y ′ and Z[0,1]×Y arethe identity, according to (5.3). The boundary maps f0 and f1 induce isometries(fi)∗: E(Y ′) →E(Y ).

The functoriality axiom asserts that F induces an isometrybetween (f1)∗◦(f0)−1∗and idE(Y ), or equivalently that(5.6)F induces an isometry F∗: (f0)∗→(f1)∗.The proper interpretation of (5.6) depends on the dimension of Y . For example, ifdim Y = d then E(Y ) is an ordinary inner product space and (5.6) asserts an equal-ity (f0)∗= (f1)∗.

This implies in particular that the action of Diff+(Y ) on E(Y )factors through an action of orientation-preserving diffeomorphisms π0 Diff+(Y )on E(Y ).If dim Y = d −1, then E(Y ) is a 2-inner product space, which is acategory, and (5.6) asserts that F induces a natural transformation F∗between thefunctors (f0)∗and (f1)∗. A further argument shows that isotopic maps F inducethe same natural transformation.

In the particular case where f0 = f1 = id, thisshows that π1 Diff+(Y ) acts on E(Y ) by automorphisms of the identity functor.21This discussion generalizes to higher codimensions.Now fix a 1 + 1 dimensional theory and denoteE = E(S1).Since any orientation-preserving diffeomorphism of S1 is isotopic to the identity,(5.6) implies that we can uniquely identify E(S) with E for any connected closedoriented 1-manifold S. Also, any two orientation-reversing diffeomorphisms of S1are isotopic, so there is a well-determined isometryc: E −→E.21An automorphism of the identity functor (i.e., a natural transformation from the identityfunctor to itself) on a category C is for each object W ∈C a choice of morphism θW : W →Wsuch that if Wf−→W ′ is any morphism in C, thenf ◦θW = θW ′ ◦f.24

Since the composite of two orientation-reversing diffeomorphisms is orientation-preserving, ¯cc = id. Thus c defines a real structure on E:(5.7)ER = {e ∈E : c(e) = e}.Since c is an isometry, ER is a real inner product space.

The inner product identifiesER ∼= E∗R as usual.Since any compact oriented 2-manifold has an orientation-reversing diffeomorphism, the generalized partition function of any such manifoldis real, by (4.14).Next, we observe that the generalized partition function of the disk1 = ZD2 ∈ERis a special element of ER.The partition function of the “pair of pants” P, which is a disk with two smallerdisks removed (Figure 2), is an element(5.8)ZP ∈ER ⊗ER ⊗ER.Equation (4.14) applied to diffeomorphisms of P which permute the boundary cir-cles (as in Figure 5) implies that ZP lives in the symmetric triple tensor productof ER. Identifying ER ∼= E∗R with the inner product, this defines a commutativemultiplication ER ⊗ER →ER.

In fact, the trilinear formx ⊗y ⊗z 7−→x · y, zER,x, y, z ∈ER,dual to (5.8) is totally symmetric. This symmetry is a compatibility condition be-tween the inner product and the multiplication.

For the complex vector space E =E(S1) we have the analogous statement that(5.9)x ⊗y ⊗z 7−→(x · y, c(z))E,x, y, z ∈E,is totally symmetric. Gluing a disk D2 onto P and applying (4.17) and (5.3) wededuce that 1 acts as the identity map for the multiplication.Finally, a stan-dard gluing argument that we do not repeat here shows that the multiplication isassociative.We summarize this discussion in the following.Proposition 5.10.

In a 1+1 dimensional topological quantum field theory (whichsatisfies the axioms of Assertion 4.12) the inner product space E(S1) has a compat-ible real algebra structure which is commutative, associative, and has an identity.In addition, the map (5.9) is totally symmetric.It is not too hard to see that E = E(S1) contains no nilpotents. For if x ̸= 0, thensince (xc(x), 1) = (x, x) ̸= 0, we see that xc(x) ̸= 0.

Iterating we find x2nc(x)2n ̸= 0and (x2nc(x)2n, 1) = (x2n, x2n) ̸= 0 for all n. Standard theorems in algebra implythat E contains a basis of idempotents e1, . .

. , eN, unique up to permutation, witheiej = 0 for i ̸= j, and that E is a product of one dimensional algebras.22 It is easy22We need the complex algebra since there exists a nontrivial commutative algebra over R,namely C. Note too that the conjugation ⟨z, w⟩7→⟨¯w, ¯z⟩on E = C × C produces ER ∼= C as analgebra over R. So it is not true in general that the idempotents belong to ER.25

to express the partition function of a closed oriented surface Σg of genus g in termsof the norms λ2i = |ei|2:ZΣg =Xi(λ2i )1−g.Now consider a 2 + 1 dimensional theory, and as before denote E = E(S1).Here E is a 2-inner product space, so in particular is a category. If f : S →S1 isan orientation-preserving diffeomorphism, then there is an induced linear isometryf∗: E(S) →E.

Furthermore, any two such f0, f1 : S →S1 are homotopic, and ahomotopy F : f0 →f1 induces an isometry F∗: (f0)∗→(f1)∗, as in (5.6), but nowF∗depends on the choice of F. (In the 1 + 1 dimensional theory F∗is an equality. )In fact, the positive generator of π1 Diff+(S1) ∼= Z induces an automorphism of theidentity functor on E, that is, a morphism(5.11)θW : W −→Wfor each object W ∈Obj(E).

So we cannot assert that E(S) and E are uniquelyisomorphic.We do need, however, to identify the spaces E(S) for different circles S to derivethe “algebra” structure on E, so we resort to the following device in what follows.We use circles S which lie in C. There is a unique composition of translations andhomotheties which maps any such circle S to the standard circle S1 = T ⊂C. Weuse this to uniquely identify E(S) ∼= E for any such S.As for the automorphism of the identity θ, we can compute it from the diffeo-morphism of the cylinder(5.12)τ : [0, 1] × S1 −→[0, 1] × S1⟨t, s⟩7−→⟨t, s + t⟩,where here we write S1 = R/Z additively.

This glues to a diffeomorphism of thetorus S1 × S1 described by the matrix(5.13)T =1011.By (5.5) we have an isomorphismE(S1 × S1) ∼= dim E,where dim E is understood as an inner product space, and in some sense the actionof (5.13) on E(S1 × S1) is the action of θ on the identity endomorphism of E.The reflection s 7→−s of the circle S1 = R/Z induces an isometry(5.14)c: E −→E.On the underlying category E determines an involution on the objects. Denotec(W) = W ∗,W ∈Obj(E).This is the definition of ‘∗’.

As in (5.7) we can consider the invariants ER. For any 2-manifold Y there is an isometry ZY ∼= Z∗Y determined by any orientation-reversing26

diffeomorphism of Y which restricts to r on ∂Y . Of course, this isometry dependson the choice of diffeomorphism, which we will standardize in what follows.

Namely,our figures will sit in C, symmetrically about the real axis, and the boundary circleswill have centers on that axis. Then reflection about the real axis is our standardorientation-reversing diffeomorphism.To compute the relationship between c and θ, consider the cylinder C as shown inFigure 1.

The cylinder sits in C, the boundary circles have centers on the real axis,and C is symmetric about the real axis. Now the diffeomorphism (5.12) does notcommute with reflection in the real axis, but rather the reflection conjugates it tothe diffeomorphism ⟨t, s⟩7→⟨t, s−t⟩.

However, since the orientation of the boundarycircles are reversed under reflection, this conjugated diffeomorphism represents thepositive generator of π1 Diff+ S1 for the reflected circle. Thus we conclude that forany W ∈Obj(E),(5.15)θW ∗= θ∗W .Here θ∗W denotes the image of θW under the functor (5.14).Figure 1: The cylinder CLet D2 be the unit disk in C. Then(5.16)1 = ZD2 ∈Eis a distinguished element of E, and reflection in the real axis determines an isometry(5.17)1 ∼= 1∗.Fix a standard pair of pants P as shown in Figure 2.

(The ordering of theboundary circles is motivated by Figure 8.) As with all of our figures it is symmetricabout the real axis and the boundary circles have centers on that axis.

Any otherP ′ with the same properties is isotopic to P by an isotopy which moves the boundarycircles only by translations along the real axis and by homotheties. Furthermore,any two such isotopies are isotopic, since any self-diffeomorphism of P which is27

the identity on ∂P is isotopic to the identity. This means that there is a uniquelydefined isotopy ZP ′ ∼= ZP .

Now the partition function isZP ∈E ⊗E ⊗E,and reflection about the real axis determines an isometry(5.18)ZP ∼= Z∗P .By duality ZP determines a map(5.19)m: E ⊗E −→E.In particular, m is a functor E × E →E, but it has linearity properties as well.Denotem(W1, W2) = W1 ⊙W2,W1, W2 ∈Obj(E).This is the definition of ‘⊙’. The isometry (5.18) translates into a natural isometry(5.20)(W1 ⊙W2)∗∼= W ∗1 ⊙W ∗2 ,W1, W2 ∈Obj(E).Glue a disk to the inner boundary circles in P to obtain natural isometries(5.21)1 ⊙W ∼= W,W ⊙1 ∼= W,for all W ∈Obj(E).23213Figure 2: The pair of pants P23There should also be natural transformations W ⊙W ∗→1 and 1 →W ⊙W ∗which we didnot succeed in finding.28

32321144Figure 3: AssociativityIt remains to discuss associativity and commutativity. Whereas in the 1 + 1 di-mensional theory these are constraints on the multiplication, here they are newstructures which satisfy “higher order” constraints.

The associative law is a natu-ral isometry(5.22)ϕW1,W2,W3 : (W1 ⊙W2) ⊙W3 −→W1 ⊙(W2 ⊙W3),W1, W2, W3 ∈Obj(E),obtained from the obvious diffeomorphism indicated in Figure 3. This figure indi-cates an isometry between two different contractions of ZP ⊗ZP , which is equivalentto (5.22).

One can think of (5.22) as obtained by gluing and ungluing accordingto the dashed lines in Figure 3. Performing such gluings and ungluings in Figure 4makes obvious the commutativity of the usual pentagon diagram(5.23)((W1 ⊙W2) ⊙W3) ⊙W4 −−−−→(W1 ⊙W2) ⊙(W3 ⊙W4) −−−−→W1 ⊙(W2 ⊙(W3 ⊙W4))y(W1 ⊙(W2 ⊙W3)) ⊙W4 −−−−→W1 ⊙((W2 ⊙W3) ⊙W4)A similar check shows that(5.24)(W1 ⊙1) ⊙W2 −−−−→W1 ⊙(1 ⊙W2))yW1 ⊙W2commutes.29

Figure 4: Gluings and ungluings of pieces of this surface prove the pentagonβ321312Figure 5: The braiding diffeomorphism βIt does not make sense to say that the multiplication (5.19) is commutative.Rather, there is a natural braiding isometry(5.25)RW1,W2 : W1 ⊙W2 −→W2 ⊙W130

obtained from the self-diffeomorphism β : P →P indicated in Figure 5. The aux-iliary dashed lines indicate the motion of the boundary circle labeled 2 over thatlabeled 1.

There is a compatibility between the braiding R and the automorphism θ:the diagram(5.26)W1 ⊙W2RW1,W2−−−−−→W2 ⊙W1θW1⊙W2yyθW2 ⊙θW1W1 ⊙W2R−1W2,W1−−−−−→W2 ⊙W1commutes for W1, W2 ∈Obj(E). (Thus θ is termed “balanced”.) This follows froman equation in Diff+(P).

Namely, let τi denote a positive Dehn twist around theboundary labeled i. Then the desired equation isτ2τ1β = β−1τ3,which is easily checked using pictures like those in Figure 5.

Similar computationsusing Figure 6 show that the hexagon diagrams(5.27)(W1 ⊙W2) ⊙W3RW1,W2 ⊙id−−−−−−−→(W2 ⊙W1) ⊙W3ϕW2,W1,W3−−−−−−−→W2 ⊙(W1 ⊙W3)ϕW1,W2,W3yyid ⊙RW1,W3W1 ⊙(W2 ⊙W3)RW1,W2⊙W3−−−−−−−−→(W2 ⊙W3) ⊙W1ϕW2,W3,W1−−−−−−−→W2 ⊙(W3 ⊙W1)and(5.28)W1 ⊙(W2 ⊙W3)id ⊙RW2,W3−−−−−−−−→W1 ⊙(W3 ⊙W2)ϕ−1W1,W3,W2−−−−−−−→(W1 ⊙W3) ⊙W2ϕ−1W1,W2,W3yyRW1,W3 ⊙id(W1 ⊙W2) ⊙W3RW1⊙W2,W3−−−−−−−−→W3 ⊙(W1 ⊙W2)ϕ−1W3,W1,W2−−−−−−−→(W3 ⊙W1) ⊙W2commute. Each of (5.27) and (5.28) follows from an equation in the diffeomorphismgroup of the surface pictured in Figure 6.

The diffeomorphisms are formed fromthe braiding β shown in Figure 5. The associators are formed from gluings andungluings, so do not enter.We summarize this discussion in the following.Proposition 5.29.

In a 2+1 dimensional topological quantum field theory (whichsatisfies the axioms of Assertion 4.12) the 2-inner product space E(S1) is a braidedmonoidal category with a compatible balanced automorphism of the identity andcompatible duality.24There is a notion of semisimplicity for such categories [Y2], and it is desirableto prove that E is semisimple using the inner product, as we indicated for the1+1 dimensional case after Proposition 5.10. Surely one should think of the 2-inner24As mentioned earlier, this is sometimes termed a tortile category.

Also, there is a gap herein that we did not find the natural transformations mentioned in the footnote following (5.21).31

3421Figure 6: Surface used to prove hexagon diagrams (5.27) and (5.28)product space structure together with the monoidal structure. In other words, oneshould think of E as a higher version of the algebra encountered in Proposition 5.10.There are reconstruction theorems in category theory which recover certain al-gebraic objects from certain types of categories.

For example, in [DM] it is shownhow to recover a group from its category of representations.The structure inProposition 5.29 is almost enough to reconstruct a quasitriangular quasi-Hopf al-gebra [Ma1]. (This is often termed a quasi-quantum group.

Probably there is aribbon element as well [RT], [AC] corresponding to the automorphism of the iden-tity.) Missing is a functor from E to the category of vector spaces, though moreabstract reconstructions are possible [Ma2].

We remark that there are exampleswhere no such “fiber functor” exists; the simplest is V1 × V1. (This arises from athree dimensional σ-model into a space consisting of two points.) But it seems thatwe can always decompose into a product of spaces where reconstruction is possi-ble.

For the finite gauge theory we carry out the reconstruction in §§7–9. Therewe choose various trivializations to construct a functor from E to the category ofvector spaces, and this allows the reconstruction of the quasi-quantum group.Finally, we remark that we can take products with any closed oriented Y inall of these constructions to obtain a higher algebra structure on E(S1 × Y ).

Inparticular, the generalized quantum Hilbert space of any torus S1 × · · · × S1 has ahigher algebra structure.32

§6 The 1 + 1 Dimensional TheoryWe resume our discussion of the finite group gauge theory of §2 and §4. In thissection we examine the d = 1 case.

We know from Assertion 5.10 that E(S1) isan algebra, the algebra of central functions Fcent(Γ) under convolution, as wascomputed in [FQ,§5]. The new point is to compute E(pt) and Z[0,1].

The resultsare fairly trivial, but they illustrate the definitions and constructions of the previoussections and are a good warmup to the d = 2 case we discuss in §§7–9.Recall that the lagrangian is specified by a cocycle α ∈C2(BΓ; R/Z).Wefirst consider the simplest case (the “untwisted theory”) where α = 0. Obviously,Cpt has a single element, the equivalence class of the trivial bundle Qtriv = pt × Γ.The value of the classical action Tpt(Qtriv) ∈T2 is the trivial T-gerbe T1.Weidentify the automorphism group of Qtriv with Γ, acting by left multiplication, andit acts trivially on Tpt(Qtriv).

Hence the associated 2-vector space WQtriv in (4.5)is (V1)Γ, the category of representations of Γ. Now E(pt) is computed by the pathintegral (4.6) as an inverse limit over the category of trivial bundles Q →pt.

Theautomorphism groups Aut Q which enter (4.5) are not canonically isomorphic to Γ.Rather, we use the distinguished bundle Qtriv →pt to trivialize the inverse limit:(6.1)E(pt) ∼=1#Γ · (V1)Γ. (Recall that the prefactor is 1/(# Aut Qtriv).) We use this trivialization in whatfollows.According to (5.4) the generalized partition function Z[0,1] is isometric to theidentity operator on E(pt).

It is instructive to compute this isometry directly fromthe definition of the path integral (4.8). There is a bijection(6.2)C[0,1](Qtriv ⊔Qtriv) ←→Γby comparing the trivializations of a bundle P →[0, 1] over the the two end-points of [0, 1].

More explicitly, fix a basepoint in Qtriv and let p0 ∈P0, p1 ∈P1be the corresponding basepoints in P using the trivializations. Parallel transportalong [0, 1] is an isomorphism ψ: P0 →P1.

Define g ∈Γ by ψ(p0) = p1 · g. Theng is the element of Γ corresponding to P under the correspondence (6.2).Theaction of ⟨h0, h1⟩∈Γ × Γ ∼= Aut(Qtriv) × Aut(Qtriv) on the left hand side of (6.2)corresponds to the action⟨h0, h1⟩· g = h1gh−10 ,g ∈Γ,on the right hand side. The classical action (2.4) is trivial, so in (4.7) we obtainL[0,1](g) = C for all g in (6.2).

Since [0, 1] has nonempty boundary the measure µin (4.1) is identically equal to 1. Hence the path integral (4.8) gives(6.3)Z[0,1] =Mg∈ΓC.We identify this as the set of complex-valued functions F(Γ) on Γ, with Γ×Γ actingas(6.4)⟨h0, h1⟩· f(g) = f(h1gh−10 ),f ∈F(Γ),g ∈Γ,33

with the standard inner product(6.5)(f1, f2) =Xg∈Γf1(g)f2(g),f1, f2 ∈F(Γ).View F(Γ) as an element in E(pt) ⊗E(pt), or using the inner product on E(pt) asan element in E(pt)∗⊗E(pt) ∼= HomE(pt). Call this endomorphism K. Supposethat W ∈E(pt) is a unitary representation of Γ with action ρ: Γ →Aut(W).According to the inner product (3.5) and the factor 1/#Γ in (6.1), the action of Kon W isK(W) =1#Γ ·F(Γ) ⊗WΓHere we take Γ-invariants under the action of h ∈Γ by ⟨h, 1⟩on F(Γ) and ρ(h)on W; then h ∈Γ acts onF(Γ) ⊗WΓ through the action of ⟨1, h⟩on F(Γ).Now by (5.4) we can derive from the gluing law an isometry K2 →K.

(Theunderlying map of categories is a natural transformation.) We compute it by an-alyzing the gluing map (4.20) for the gluing of two intervals.

We find that thedesired isometry is(6.6)1#Γ ·F(Γ) ⊗F(Γ)Γ ∼=1#Γ · F(Γ × Γ)Γ −→F(Γ)f(·, ·) 7−→f(e, ·). (The Γ invariance in (6.6) refers to the action (h·f)(g1, g2) = f(g1h, h−1g2) for h ∈Γ.) This yields the desired isometry K2 →K which on W ∈E(pt) isK2(W) =1(#Γ)2 ·F(Γ × Γ) ⊗WΓ×Γ −→1#Γ ·F(Γ) ⊗WΓ = K(W)f i ⊗wi7−→g 7→f i(e, g)wi.

(These expressions are summed over i.) This is an isometry K →id on the imageof K, and is compatible with the isometry K →id which on W ∈E(pt) is(6.7)1#Γ ·F(Γ) ⊗WΓ −→Wf i ⊗wi 7−→f i(e)wi.We can also check the gluing which leads to (5.5).

That is to say we can checkthe gluing law (4.17) when we glue the two ends of [0, 1] together. Now CS1 canbe identified with the set of conjugacy classes in Γ, and the gluing map (4.20)with Q = Qtriv sends an element in Γ to its equivalence class.The map Trptin (4.18) is 1/#Γ times the Γ-invariants under the diagonal action in (6.4), andapplied to Z[0,1] = F(Γ) this gives(6.8)TrptF(Γ)=1#Γ · Fcent(Γ)where Fcent(Γ) is the space of central functions with inner product (6.5).Thisis E(S1), as follows easily from (4.6) (cf.

[FQ,§5]).34

If the lagrangian α ∈C2(BΓ; R/Z) is nonzero (the “twisted theory”), then theclassical action also enters in a nontrivial way. We compute the classical action onthe trivial bundle Qtriv →pt.

Since there is a unique cycle in C0(pt) which repre-sents the fundamental class [pt] ∈H0(pt), the integration theory in the appendixgivesexp(2πiZpt¯f ∗α) = T1for any ¯f : pt →BΓ. (This is what we must compute in (2.1).) In other words, wecan think of α as defining the trivial T-gerbe bundle over BΓ, which then lifts to thetrivial T-gerbe bundle over EΓ.

The nontrivial part comes from homotopies betweenclassifying maps of Qtriv, which we identify with paths in EΓ. The integral in (2.2)is then a T-torsor.

The classical action T (α)pt (Qtriv) is a nontrivial T-gerbe computedby an inverse limit over the “path category” of EΓ. The value of the classical actionon a field P →[0, 1], whose boundary we assume trivialized by an isomorphism∂P ∼= Qtriv × Qtriv, is then an automorphism of T (α)pt (Qtriv).

By (6.2) we identifythe equivalence class of P with an element g ∈Γ, and by (2.8) the classical action iswell-defined on the equivalence class. Taking an inverse limit over all such bundlesin the equivalence class we obtain for each g ∈Γ an automorphism Tg of T (α)pt (Qtriv).Furthermore, there are isomorphismsTgh −→Tg · Thfrom the gluing law (2.12) applied to the gluing of intervals.As in the α = 0 case we compute the quantum space E(α)(pt) by taking aninverse limit over the category of all trivial bundles.We use the distinguishedobject Qtriv to trivialize the inverse limit (cf.

(4.5) and (3.8)):(6.9)E(α)(pt) ∼=1#Γ · (WT (α)pt (Qtriv))Γ,ρ.Here ρ is the action of Γ on T (α)pt (Qtriv) via the torsors Tg. If we trivialize theT-gerbe T (α)pt (Qtriv), for example by choosing a basepoint in EΓ, then we obtainan isomorphism T (α)pt (Qtriv) ∼= T1, and so the Tg are identified with T-torsors.

Asin (1.4) these torsors define a central extension1 −→T −→˜Γ −→Γ −→1.Incidentally, they are isomorphic to the torsors and central extension which comefrom the action of Aut(Qtriv) ∼= Γ on T (α)pt (Qtriv) ∼= T1. This assertion follows fromthe fact (5.2) that the classical action of the product bundle [0, 1] × Qtriv →[0, 1]is trivial.

With the trivialization of T (α)pt (Qtriv) the isometry (6.9) becomes(6.10)E(α)(pt) ∼=1#Γ · (V1)Γ,ρ.Recall from the paragraph following (3.8) that (V1)Γ,ρ is the category of represen-tations of ˜Γ where the central T acts by scalar multiplication. We emphasize that(6.10) requires two choices of trivialization (of two inverse limits).35

Computing with the trivialization (6.10) we find analogous to (6.3) that(6.11)Z[0,1] ∼=Mg∈ΓLg,where Lg is the hermitian line obtained from the torsor Tg as in (3.2). We leave thereader to modify the verification of (6.7) above to show that (6.11) acts isometricallyto the identity map.

The twisted version of (6.8) is also easy to check.36

§7 The 2 + 1 dimensional Chern-Simons theoryand quasi-quantum groups: Untwisted CaseWe turn to the 2 + 1 dimensional case of gauge theory with finite gauge group,which can be considered as a Chern-Simons theory.Our goal is to derive thequasi-Hopf algebras of [DPR] directly from the path integral (4.6). We alreadyinvestigated several features of this theory in [FQ].

The new point is an investigationof the 2-inner product space E(S1), which according to Assertion 5.29 is a certaintype of braided monoidal category. With suitable trivializations we claim that it isisomorphic to the category of representations25 of the quasitriangular quasi-Hopfalgebra constructed in [DPR].

We also recover the results of [FQ,§§3–4], includingSegal’s modular functor [S1], from our approach here. In this section we treat theuntwisted case where the lagrangian α ∈C3(BΓ; R/Z) vanishes.

In sections §§8–9we generalize to the twisted case α ̸= 0.The holonomy of a bundle around the circle induces a bijection(7.1)CS1 ←→conjugacy classes in Γ.The classical action in the α = 0 theory is trivial. Choose a bundle Q[x] →S1representing each conjugacy class [x] in Γ under the correspondence (7.1).

Thenfollowing the same steps as in (6.1), this choice of bundles leads to an isometry(7.2)E = E(S1) ∼=M[x]1# Aut Q[x]· (V1)Aut Q[x].It is convenient to use a more concrete description of E directly in terms of thegroup Γ, and this requires a choice of some basepoints. (Compare with the choiceof basepoints in [FQ,§3].) Fix a conjugacy class [x] and consider the fiber F[x] ofQ[x] →S1 over the basepoint 1 ∈S1 = T. A point in F[x] determines a particularvalue of the holonomy of Q[x], which is an element of the conjugacy class [x].

Choosea (base)point fx in the fiber of the holonomy map F[x] →[x] for each x ∈[x]. Thenfx induces an isomorphism(7.3)Aut Q[x] −→Cxby assigning to ψ ∈Aut Q[x] the element g ∈Cx which satisfies ψ(fx) = fx · g.Thus if W is a representation of Aut Q[x], then under this isomorphism W is also arepresentation of Cx.

Let W denote the trivial vector bundle over [x] whose fiber ateach x ∈[x] is W. Now Γ acts on [x] on the left by conjugation (g: x 7→gxg−1), andwe want to lift this action to W. For each x ∈[x] the stabilizer Cx already acts onthe fiber Wx = W. For x, x′ ∈[x] there is a unique gx,x′ ∈Γ with fx = fx′ · gx,x′.Then x′ = gx,x′xg−1x,x′. Lift gx,x′ : x 7→x′ to the identity map id: Wx →Wx′.There is then a unique extension of the Cx action and the action of the gx,x′ on Wto a Γ action on W which lifts the conjugation action on [x].25It is probably more natural to use corepresentations here, but in any case we have enoughfiniteness to switch back and forth between representations and corepresentations.Also, thiswill reconstruct the algebras in [DPR] rather than their duals.Our convention here differsfrom [FQ,§3], where we use corepresentations.Note also that in [FQ,§3] we use right comod-ules whereas here we use left modules.

Thus the groupoid (7.5) is opposite that in [FQ,§3].37

Summarizing, the choice of basepoints in the bundles Q[x] leads to an isometry(7.4)E ∼=1#Γ · VectΓ(Γ),where VectΓ(Γ) is the 2-inner product space of hermitian vector bundles over Γwith a unitary lift of the left Γ action on Γ by conjugation. We write an elementof VectΓ(Γ) as W = ⊕x∈ΓWx.

If W1, W2 ∈VectΓ(Γ), then the inner product isdefined as(W1, W2)VectΓ(Γ) = Mx(W1)x ⊗(W2)x!Γ.It is easy to check that 1/#Γ times this inner product is the inner product in (7.2).There is another description of E which is useful. Let G denote the groupoidwhich is the set G × G with the composition law(7.5)⟨x2, g2⟩◦⟨x1, g1⟩= ⟨x1, g2g1⟩,if x2 = g1x1g−11 .Composition is not defined if x2 ̸= g1x1g−11 .

Then(7.6)E ∼=1#Γ · (V1)G,where (V1)G is the 2-inner product space of finite dimensional unitary representa-tions of G. What we mean by a representation of the groupoid G amounts exactlyto a Γ-bundle over Γ, so (7.6) is essentially identical to (7.4). More precisely, theseare representations (left modules) of the “groupoid algebra”(7.7)C[G] =Mx,gC⟨x, g⟩,with multiplication⟨x2, g2⟩· ⟨x1, g1⟩= ⟨x1, g2g1⟩,x2 = g1x1g−11 ;0,otherwise.The unit element is1 =Xx⟨x, e⟩.If W ∈VectΓ(Γ) = (V1)G we use the notationAWg= Ag : Wx −→Wgxg−1for the action of ⟨x, g⟩∈G.

In terms of the G action we have(7.8)Wx = ⟨x, e⟩· W.We use the trivialization (7.4), or equivalently (7.6), in what follows.An irreducible element W ∈E is supported on some equivalence class [x], andthe fiber Wx is an irreducible representation ρ of Cx. Since the various Cx, x ∈[x]38

are identified up to inner automorphisms, the equivalence class [ρ] of the repre-sentation is well-defined.Up to isomorphism the irreducible elements of E arelabelled by the pair ⟨[x], [ρ]⟩. These labels appear in all treatments of this the-ory [DVVV], [DPR], [DW], [FQ].It is convenient to use the isomorphism (7.6) to identify the path integral (4.11)over a compact oriented 2-manifold X, which is an element of E(∂X), as an elementin tensor products of E. Recall our convention stated after (5.11) for identifying ∂Xas a disjoint union of copies of the standard circle S1.For this we restrict tosurfaces X which are subsets of C. Under these identifications each componentof ∂X has a basepoint corresponding to the standard basepoint 1 ∈S1.

Let C′Xdenote the category of principal Γ bundles P →X endowed with a basepoint in thefiber over each basepoint in ∂X. Morphisms are required to preserve the basepoints.Let C′X denote the set of equivalence classes.

For the cylinder C the holonomy andparallel transport define a bijection(7.9)C′C ←→Gas illustrated in Figure 7. (Compare with (6.2).) Now for a surface X we can glue Cto any component of ∂X using the basepoints.

This induces a G action on C′X foreach component of ∂X.gxFigure 7: The bundle over C corresponding to ⟨x, g⟩∈GProposition 7.10. Let X ⊂C be a compact oriented 2-manifold.26 Then underthe isomorphism (7.6) the path integral over X is(7.11)ZX ∼= L2(C′X)with the G actions induced by gluing cylinders onto components of ∂X.Proof.

Let P ∈C′X and fix a component S of ∂X. The basepoint determines anisomorphism PS→Q[x] for some [x].

If the holonomy around S is x, then thebasepoint maps to fx. Apply this to a pointed bundle P ∈C′C over the cylinder Cwhich corresponds under (7.9) to an element ⟨x, g⟩∈G.

Using parallel transportalong the axis of C, this bundle also determines an element of Aut Q[x]. If g ∈Cxthen the correspondence between the automorphism of Q[x] and g agrees with (7.3).26The same arguments apply to arbitrary surfaces with parametrized boundary.

If the surfacehas closed components, then we must modify the inner product in (7.11).39

(This follows from (5.2).) Also, the bundle labeled by ⟨x, gx,x′⟩corresponds to theidentity in Aut Q[x] for all x′ ∈[x].

Thus the action of G ≈C′C on the quantiza-tion (7.11) induced by gluing is the action described in the text leading to (7.4)and (7.6).The 2-inner product space E has extra structure determined by the path integralover special surfaces and special diffeomorphisms, as described in §5.Proposition 7.12. The finite gauge theory described in Assertion 4.12 with α = 0determines the following structure on E.(a) (Automorphism of the identity (5.11)) For W ∈E we have(7.13)θWWx= Ax : Wx −→Wx.

(b) (Involution (5.14)) For W ∈E the dual W ∗∈E is defined by (W ∗)x = W ∗x−1and AW ∗g= (AWg−1)∗. (c) (Identity (5.16)) The identity 1 is(7.14)1x = C,x = e;0,x ̸= e,with Ce = Γ acting trivially on 1e.

(d) (Multiplication (5.19)) The tensor product of W1, W2 ∈E is(7.15)(W1 ⊙W2)x =Mx1x2=x(W1)x1 ⊗(W2)x2with the Γ action(7.16)AW1⊙W2g= AW1g⊗AW2g. (e) (Associator (5.22)) The associator ϕ is induced from the standard associator oftensor products of vector spaces.

(f ) (R-matrix (5.25)) For W1, W2 ∈E we have(7.17)RW1,W2 : (W1)x1 ⊗(W2)x2 −→(W2)x1x2x−11⊗(W1)x1w1 ⊗w2 7−→AW2x1 (w2) ⊗w1and all other components are zero.A few remarks are in order. First, since x is a central element of Cx, the transforma-tion (7.13) is a scalar on each irreducible component of Wx.

(We decompose Wx un-der the Cx action.) If W is an irreducible element of E labelled by ⟨[x], [ρ]⟩, then thescalar transformation Ax is independent of x ∈[x].

The conformal weight h⟨[x],[ρ]⟩is defined up to an integer by the equation(7.18)Ax = e2πih⟨[x],[ρ]⟩.40

This agrees with the results of [FQ,§5], where we calculated the conformal weightfrom the action of (5.13) on the torus. Notice that θW can also be described as theaction of(7.19)v =Xx⟨x, x⟩on W, where v is a special element27 of C[G].

The identity element 1 correspondsto the label ⟨[e], trivial⟩. Another description of the multiplication (7.15), (7.16) isW1 ⊙W2 = µ∗(W1 ⊠W2),where µ: Γ × Γ →Γ is group multiplication and W1 ⊠W2 →Γ × Γ is the exter-nal tensor product.

Finally, we invite the reader to verify (5.15), (5.17), (5.20),(5.21), (5.23), (5.24), (5.26), (5.27), and (5.28) directly from the data listed inProposition 7.12.28Proof. We use Proposition 7.10 to compute the path integrals over the varioussurfaces.

(a) We compute the action of the diffeomorphism (5.12) on the cylinder C.From (7.9) and (7.11) we obtain an isomorphism(7.20)ZC ∼= F(G).An argument similar to that in §6 (see (6.7)) shows that ZC acts isometrically tothe identity on E via the isometry(7.21)1#Γ ·F(G) ⊗WG −→Wf i ⊗wi 7−→f i⟨π(wi), e⟩wi,where π: W →Γ is an element of VectΓ(Γ). On the left hand side of (7.21) we takeG-invariants under the action a: f i(·) ⊗wi 7→f i(a−1·) ⊗awi, and then a ∈G actson the invariants by a: f i(·) ⊗wi 7→f i(·a) ⊗wi.

Here ‘·’ indicates the argument ofthe function. Now the diffeomorphism τ in (5.12) induces by pullback the map(7.22)τ ∗⟨x, g⟩= ⟨x, gx⟩,⟨x, g⟩∈Gon fields (7.9), and so the mapτ∗f⟨x, g⟩= f⟨x, gx⟩,f ∈F(G),on the quantization (7.20).

In terms of the element v ∈C[G] in (7.19), this is(τ∗f)(·) = f(· v).27This element plays the role of the inverse of the ribbon element of Reshetikhin/Turaev [RT].The quasitriangular quasi-Hopf algebras we encounter have a ribbon structure (cf. [AC]).28The natural transformations W ⊙W ∗→1 and 1 →W ⊙W ∗mentioned in the footnotefollowing (5.21) are evidently the duality pairing Lx Wx ⊗W ∗x →C and its dual.41

Thus on the left hand side of (7.21) the diffeomorphism τ induces the actionf i(·) ⊗wi 7−→(τ∗f i)(·) ⊗wi = f i(· v) ⊗wi,which corresponds to the action w 7→vw on the right hand side of (7.21). Thisis (7.13).

(b) We first calculate that the reflection of S1 induces the map Q[x] 7→Q[x−1] onfields by pullback. (Actually, this is the map on equivalence classes of fields writtenusing our distinguished representatives.) Since the reflection reverses orientation,this induces a map T1 7→T −11on the classical action, and in the quantization leadsus to use the dual space.

Under the identification (7.4) this gives (W ∗)x = W ∗x−1.Then the induced representation of Aut Q[x] ∼= Aut Q[x−1] is AW ∗g= (AWg−1)∗. (c) It is easy to see that C′D2 consists of one element, and the restriciton of anyrepresentative bundle to ∂D2 = S1 is Q[e].

Furthermore, Aut Q[e] ∼= Γ acts trivially.x2x1g2g1g1-1g2g1x1g1-1g2x2g2-1Figure 8: The bundle over P corresponding to ⟨x1, g1⟩× ⟨x2, g2⟩∈G × G(d) For the pair of pants P we identify(7.23)C′P ←→G × Gusing the parallel transports and holonomies indicated in Figure 8. This leads toan isometry(7.24)ZP ∼= F(G × G).The actions of ⟨x, g⟩∈G corresponding to the two inner components of ∂P are(7.25)f(·, ·) 7−→f(·⟨x, g⟩−1, ·),f(·, ·) 7−→f(·, ·⟨x, g⟩−1).42

The action of ⟨x, g⟩∈G corresponding to the outer component is(7.26)⟨x, g⟩· f⟨x1, g1⟩, ⟨x2, g2⟩= f⟨x1, gg1⟩, ⟨x2, gg2⟩,if x = g1x1g−11 g2x2g−12 ;0,otherwise.Using the inner product (7.6) on E we see that the multiplication (5.19) is the mapW1 ⊗W2 7−→1(#Γ)2 ·F(G × G) ⊗W1 ⊗W2G×G,where G × G acts on F(G × G) via (7.25). The G action on the right hand side isvia (7.26).

Then a routine check shows that(7.27)1(#Γ)2 ·F(G × G) ⊗W1 ⊗W2G×G −→W1 ⊙W2f ij ⊗w(1)i⊗w(2)j7−→f ij⟨π(w1), e⟩, ⟨π(w2), e⟩w(1)i⊗w(2)jis an isometry, where W1 ⊗W2 is defined by (7.15) and (7.16). (e) This is immediate from the definition of the associator.

(f) We compute the action of the braiding diffeomorphism β (Figure 5) on thefields (7.23) by pullback as(7.28)⟨x1, g1⟩× ⟨x2, g2⟩7−→⟨x2, g1x1g−11 g2⟩× ⟨x1, g1⟩.So the action on the quantization (7.24) by pushforward isβ∗f⟨x1, g1⟩, ⟨x2, g2⟩= f⟨x2, g1x1g−11 g2⟩, ⟨x1, g1⟩.Under the isometry (7.27) this corresponds to (7.17), as desired.Reconstruction theorems in category theory assert that E is (equivalent to) thecategory of representations of a Hopf algebra H. In fact, since E is braided H isa quasitriangular Hopf algebra [Dr]. We do not need the general arguments fromcategory theory to carry out the reconstruction, as the Hopf algebra H is apparentfrom our explicit descriptions of E in (7.4) and (7.6), and from the formulas inProposition 7.12.Indeed, as an algebra H is the “groupoid algebra” H = C[G] defined in (7.7).We have already seen in (7.6) that E is isomorphic to the category of representa-tions of the algebra H. Explicitly, if ρ: H →End(W) is a representation of H,set Wx = ρ(⟨x, e⟩)(W) as in (7.8) and set Ag : Wx →Wgxg−1 equal to ρ(⟨x, g⟩).The quasitriangular Hopf structure on H is easily deduced from Proposition 7.12.From (7.15) and (7.16) we see that the coproduct ∆: H →H ⊗H is∆⟨x, g⟩=Xx1x2=x⟨x1, g⟩⊗⟨x2, g⟩.The counit ǫ: H →C isǫ⟨x, g⟩= 1,x = e;0,otherwise,43

as we see from the action of H on 1 (7.14). The antipode S : H →H is implementedon the dual (Proposition 7.12(b)), so isS⟨x, g⟩= ⟨gx−1g−1, g−1⟩.The quasitriangular structure is an element R ∈H ⊗H such that for every pair ofrepresentations (W1, ρ1), (W2, ρ2) of H, we haveRW1,W2 = τW1,W2 ◦(ρ1 ⊗ρ2)(R),where τW1,W2 : W1 ⊗W2 →W2 ⊗W1 is the transposition.

Hence from (7.17) wededuceR =Xx1,x2⟨x1, e⟩⊗⟨x2, x1⟩.Since the associator ϕ is the standard associator on vector spaces (Proposition 7.12(e)),we obtain a Hopf algebra (as opposed to a quasi-Hopf algebra). Finally, we havealready observed that the automorphism of the identity θ in (7.13) is implementedby the element v in (7.19):v =Xx⟨x, x⟩.This special element in H is the inverse of the ribbon element of Reshetikhin/Turaev [RT].We interpret it here in terms of the “balancing” of the category of representations.The quasitriangular Hopf algebra H is identified in [DPR] as the “quantumdouble” of F(Γ).Finally, we indicate how to recover the “modular functor” [S1], [FQ,§4].

Onceand for all fix a basis {Wλ} of the 2-inner product space E = E(S1). Here λ runsover the labeling set Φ mentioned earlier.

Now suppose X is a compact oriented2-manifold with each boundary component parametrized. The parametrizationsidentify E(∂X) with a tensor product of copies of E and E. Thus we can decom-pose ZX according to the chosen basis for E:ZX ∼=MλE(X, λ) ⊗Wλ,where λ = ⟨λ1, .

. .

, λk⟩runs over labelings of the boundary components andWλ = W ±1λ1 ⊗· · · ⊗W ±1λk ,the signs chosen according to the orientation. The inner product spaces E(X, λ)define the modular functor.

The gluing law for the modular functor follows directlyfrom Assertion 4.12(d).44

§8 The 2 + 1 dimensional Chern-Simons theoryand quasi-quantum groups: Twisted CaseIn this section we extend the results of §7 to the 2 + 1 dimensional finite gaugetheory with nontrivial lagrangian α ∈C3(BΓ; R/Z). The classical theory is non-trivial, and this leads to corresponding modifications of the quantum theory.

Wemust choose additional trivializations (of gerbes) to express the theory in terms offamiliar objects, and in particular to construct a quasi-Hopf algebra. (Recall theremarks following Proposition 5.29.) Such trivializations appear more naturallyin §9, where we cut open the circle and make calculations on the interval.

We relyhere on the exposition in §7 and only indicate the necessary modifications. Thecocycle α ∈C3(BΓ; R/Z) is fixed throughout.

We often omit it from the notation.We use the choices made in §7 of representative bundles Q[x] →S1 and base-points fx. The classical action T (α)S1 (Q[x]) is a T-gerbe, which we denote G[x].

Theautomorphism group Aut Q[x] acts on this gerbe, and the action is a homomorphism(8.1)ρ(α)[x] = ρ[x] : Aut Q[x] −→Aut(G[x]).Fix a trivializing element(8.2)G[x] ∈G[x] = T (α)S1 (Q[x]),and so an isomorphism G[x] ∼= T1.This can be done as in [FQ,§3] by fixing arepresentative cycle s ∈C1(S1) for the fundamental class [S1] ∈H1(S1), and byfixing classifying maps Q[x] →EΓ.With these trivializations the action (8.1)determines a central extension of Aut Q[x] by T, as in (1.4). There is an inducedisometry (see (4.6), (6.10))E(α) = E(α)(S1) ∼=M[x]1# Aut Q[x]· (V1)Aut Q[x],ρ(α)[x] .We want to express this directly in terms of Γ, using the basepoints fx as in §7.The central extensions of Aut(Q[x]) lead via the isomorphism (7.3) to central ex-tensions ˜Cx of the centralizer subgroup of any x ∈Γ.

That is, for each g ∈Cx wehave a T-torsor T (x, g) together with appropriate isomorphisms under composition.Note that there are trivializations(8.3)T (x, e) ∼= Tsince T (x, e) · T (x, e) ∼= T (x, e). (This is (5.2).) Extend to a central extension ofthe groupoid G in (7.5) as follows.

First, for any two elements x, x′ in the sameconjugacy class let(8.4)T (x, gx,x′) = T,x′ ∈[x]. (The element gx,x′ ∈Γ was defined following (7.3).) Then for any x, g ∈Γ we have⟨x, g⟩= ⟨x, gx,gxg−1⟩◦⟨x, h⟩for some unique h ∈Cx.

Set T (x, g) = T (x, gx,gxg−1) ·T (x, h). This determines the desired central extension1 −→T −→˜G(α) −→G −→1,45

where for ⟨x, g⟩∈G the T-torsor T (x, g) is the preimage of ⟨x, g⟩in ˜G(α). Thereare appropriate isomorphisms under composition.

Let L(x, g) be the hermitian linecorresponding to the T-torsor T (x, g), and(8.5)ℓ(x, e) ∈L(x, e)the trivializing element derived from (8.3).We ignore the trivializations (8.4),which are artifacts of our definitions.With this understood an element of E(α) corresponds to a vector bundle W =Lx∈Γ Wx over Γ with isomorphismsAWg= Ag : L(x, g) ⊗Wx −→Wgxg−1which compose properly. Set(8.6)H(α) =Mx,gL(x, g).Define an algebra structure29 using the multiplication in ˜G:(8.7)L(x2, g2) ⊗L(x1, g1) −→ L(x1, g2g1),x2 = g1x1g−11 ;0,otherwise.The identity element in H(α) is(8.8)1 =Xxℓ(x, e).We can view E(α) as the 2-inner product space of representations of H(α), with thenatural inner product multiplied by 1/#Γ.

Or, by analogy with (7.6), we write(8.9)E(α) ∼=1#Γ · (V1)˜G,where we only take representations in which the central circles T (x, e) ∼= T act asscalar multiplication.Now suppose X is a compact oriented surface, either with a given parametriza-tion of the components of ∂X, or with an embedding X ⊂C which induces suchparametrizations according to our conventions.Suppose P ∈C′X is a Γ bun-dle over X with basepoints on the boundary.Let Y be a component of ∂Xand suppose the holonomy of PY is x.Then the basepoint in PY and theparametrization of Y determine an isomorphism PY ∼= Q[x], and so an isomor-phism T (α)∂X (∂P) ∼= T (α)S1 (Q[x]) = G[x].This T-gerbe is trivialized by our choicein (8.2). Hence the classical action (2.6) of P can be identified with a T-torsorT (α)X (P), using this trivialization.

As in (4.7) this T-torsor determines a hermit-ian line, and by taking an inverse limit we obtain a line L(α)X ([P]) depending onlyon the equivalence class of P. (This line could degenerate to 0 if X has a closedcomponent.) LetL(α)X−→C′Xdenote the resulting line bundle over the finite set C′X.

The following generalizesProposition 7.10.29In [FQ] we defined a coalgebra structure instead.46

Proposition 8.10. Let X ⊂C be a compact oriented 2-manifold.30 Then underthe isomorphism (8.9) the path integral over X is space of L2 sections(8.11)Z(α)X∼= L2(C′X, L(α)X ),with the ˜G action induced by gluing cylinders onto components of ∂X.Proof.

The only new point is an isometry(8.12)LC(x, g) ∼= L(x, g),where LC(x, g) = LC([P⟨x,g⟩]) for P⟨x,g⟩→C a pointed bundle over the cylindercorresponding to ⟨x, g⟩∈G under (7.9). Recall the proof of Proposition 7.10, wherewe show that the basepoints determine an isomorphism ∂P⟨x,g⟩∼= Q[x] ⊔Q[x], andso P⟨x,g⟩determines an element of Aut Q[x].

The classical action of P⟨x,g⟩is thenan element of Aut(G[x]) ∼= T1. But by (5.2) the classical action TC([0, 1] × Q[x]) ofa product bundle is trivial, and then the desired isometry (8.12) follows easily.We adopt the notationℓC(x, e) = ℓ(x, e)for the element in (8.5).We need a few preliminaries to generalize Proposition 7.12.

For any x ∈Γ thereis a trivialization(8.13)ℓC(x, x) ∈LC(x, x)as follows.By (7.22) the diffeomorphism τ : C →C satisfies τ∗⟨x, e⟩= ⟨x, x⟩.Notice that τ is the identity on ∂C, so it respects the trivializations (8.2). Bythe functoriality of the classical action (2.7) the diffeomorphism τ induces an iso-morphism TC(x, x) ∼= TC(x, e), and so an isometry LC(x, x) ∼= LC(x, e).ThenℓC(x, x) corresponds to ℓC(x, e) ∈LC(x, e) (cf.

(8.5)).Next, consider the diffeomorphism of the cylinder Cι: [0, 1] × S1 −→[0, 1] × S1⟨t, s⟩7−→⟨−t, −s⟩.It is not the identity on ∂C.Rather, ∂ι swaps the two boundary components,and if we identify them in the obvious way, ∂ι is the reflection s 7→−s.Bythe functoriality (2.7) and the orientation axiom (2.9) this reflection induces anisomorphism TS1Q[x]−1 →TS1Q[x−1], and so we can compare the trivializationsin (8.2). Use this isomorphism to define the T-torsor(8.14)T[x] = G[x] · G[x−1].Let L[x] be the hermitian line corresponding to the T-torsor T[x].

Then since ι in-duces the map ι∗⟨x, g⟩= ⟨gx−1g−1, g−1⟩on fields, the induced isometry on theclassical action is(8.15)ι∗: LC(gx−1g−1, g−1) ⊗L[x] −→LC(x, g) ⊗L[gxg−1].30The same arguments apply to arbitrary surfaces with parametrized boundary. If the surfacehas closed components, then we must modify the inner product in (8.11).47

x1x2x3eeeFigure 9: Field used in the proof of (8.16)x1x2x1-1x1eex1x2Figure 10: The isometry (8.19)Of course, L[gxg−1] = L[x], so we can cancel these terms from (8.15).We use (7.23) to identify an equivalence class of pointed bundles over the pairof pants P with an element in G × G (see Figure 8). LetLP (x1 | x2) = LP (x1, e; x2, e)denote the hermitian line obtained from the classical action on the equivalence class48

corresponding to ⟨x1, e⟩× ⟨x2, e⟩. We claim that for any x1, x2, x3, g ∈Γ there areisometriesφx1,x2,x3 : LP (x1x2 | x3) ⊗LP (x1 | x2) −→LP (x1 | x2x3) ⊗LP (x2 | x3),(8.16)σx1,x2 : LP (x1 | x2) −→LP(x1x2x−11| x1) ⊗LC(x2, x1),(8.17)and(8.18)γx1,x2,g : LC(x1x2, g)⊗LP(x1 | x2) −→LP (gx1g−1 | gx2g−1)⊗LC(x1, g)⊗LC(x2, g).For (8.16) we use the gluings in Figure 3 to see that both sides are isomorphic to thebundle L(x1, e; x2, e; x3, e) indicated in Figure 9.

The isometry (8.17) is constructedfrom the braiding diffeomorphism β, which by (7.28) induces an isometryβ∗: LP (x1, e; x2, e) −→LP(x2, x1; x1, e),and from the gluing in Figure 10, which induces an isometry(8.19)LP (x2, x1; x1, e) −→LP (x1x2x−11 , e; x1, e) ⊗LC(x2, x1).The isometry (8.18) is constructed from the gluing in Figure 11 and the duality(8.20)LC(xi, g) ⊗LC(xi, g−1) −→LC(xi, e) ∼= C,which follows from Figure 12 and (8.5).x1x2eegx1g-1gg-1g-1gx2g-1Figure 11: The isometry (8.18)49

xixig-1gFigure 12: The duality (8.20)Proposition 8.21. Consider the finite gauge theory described in Assertion 4.12with lagrangian α ∈C3(BΓ; R/Z).

This field theory and the trivializations chosenin (8.2) determine the following structure on E(α). (a) (Automorphism of the identity (5.11)) For W ∈E we haveθWWx= Ax(ℓC(x, x)): Wx −→Wx.

(b) (Involution (5.14)) For W ∈E the dual W ∗∈E is defined by (W ∗)x = W ∗x−1 ⊗L∗[x] and AW ∗g= (AWg−1)∗. (c) (Identity (5.16)) The identity 1 is(8.22)1x = LD2([Ptriv]),x = e;0,x ̸= e,with the action of the central extension ˜Ce on 1e determined by gluing a cylinder Cto a disk D2.

(d) (Multiplication (5.19)) The tensor product of W1, W2 ∈E is(8.23)(W1 ⊙W2)x =Mx1x2=xLP (x1 | x2) ⊗(W1)x1 ⊗(W2)x2with the Γ action(8.24)AW1⊙W2g=id ⊗AW1g⊗AW2g◦(γx1,x2,g ⊗id)on LP (x1 | x2) ⊗(W1)x1 ⊗(W2)x2. (e) (Associator (5.22)) For W1, W2, W3 ∈E the associator isϕW1,W2,W3 = φx1,x2,x3 ⊗id50

on LP (x1 | x2) ⊗LP (x1x2 | x3) ⊗(W1)x1 ⊗(W2)x2 ⊗(W3)x3. (f ) (R-matrix (5.25)) For W1, W2 ∈E we have(8.25)RW1,W2 : LP (x1 | x2) ⊗(W1)x1 ⊗(W2)x2 −→LP (x1x2x−11| x1) ⊗(W2)x1x2x−11⊗(W1)x1ℓ⊗w1 ⊗w2 7−→(id ⊗AW2x1 )(σx1,x2(ℓ) ⊗(w2)) ⊗w1and all other components are zero.A few remarks.First, we omitted transposition of ordinary tensor products ofvector spaces from the notation in (8.24) and (8.25).

Also, the conformal weight isdefined by (7.18) with Ax(ℓC(x, x)) replacing Ax on the left hand side. The special(inverse ribbon) element of H(α) replacing (7.19) is(8.26)v(α) =XxℓC(x, x).In (b) the isometry (8.15) is implicit in the equation AW ∗g= (AWg−1)∗.

In (8.22),[Ptriv] is the equivalence class of the trivial bundle over the disk, and gluing acylinder gives isometries(8.27)LC(e, g) ⊗LD2([Ptriv]) −→LD2([Ptriv]),which is the required action of ˜Ce. Of course, (8.27) is equivalent to a linear map(8.28)ǫ:MgLC(e, g) −→C.The verifications of (5.15), (5.17), (5.20), (5.21), (5.23), (5.24), (5.26), (5.27),and (5.28) directly from the data listed in Proposition 8.21 require some addi-tional identities in the classical theory easily derived from simple gluings of thetype already considered.The proof of Proposition 8.21 is a straightforward extension of the proof ofProposition 7.12, so we omit it.It remains to deduce a quasi-Hopf algebra structure on H(α).

For this we needto choose trivializing elements31(8.29)ℓP(x1 | x2) ∈LP (x1 | x2).Define(8.30)ω(x1, x2, x3) =ℓP (x1 | x2x3) ⊗ℓP (x2 | x3)φx1,x2,x3ℓP (x1x2 | x3) ⊗ℓP (x1 | x2) ∈T.An argument with gluings and ungluings of the four times punctured disk showsthat ω satisfies the cocycle identity(8.31)ω(x1 , x2 , x3) ω(x1 , x2x3 , x4) ω(x2 , x3 , x4)ω(x1 , x2 , x3x4) ω(x1x2 , x3 , x4)= 1,x1, x2, x3, x4 ∈Γ.In a sense this is the classical analog of the pentagon diagram(5.23). So ω defines aclass [ω] ∈H3(Γ; R/Z) in group cohomology.

The following proposition is analogousto [FQ,Proposition 3.14]. We state it without proof.31From the point of view of the reconstruction theorems, the reason we need to choose theseelements is to obtain a functor from E to the category of vector spaces which preserves the tensorproduct.

Hence the line which appears in (8.23) must be trivialized.51

Proposition 8.32. Under the isomorphism H•(Γ) ∼= H•(BΓ) the group cohomol-ogy class [ω] corresponds to the singular cohomology class [α].Now we write the quasitriangular quasi-Hopf structure on H(α) induced fromthe data in Proposition 8.21.

The coproduct is(8.33)∆(α)(ℓ) =Xx1x2=xγx1,x2,gℓ⊗ℓP (x1 | x2)ℓP (gx1g−1 | gx2g−1),ℓ∈LC(x, g).The counit is the linear map defined in (8.28); it maps LC(x, g) to 0 if x ̸= 0. Theantipode is computed from Proposition 8.21(b) as the inverse(8.34)S : LC(x, g) −→LC(gx−1g−1, g−1)of (8.15).

The quasitriangular element R(α) ∈H(α) ⊗H(α) is(8.35)R(α) =Xx1,x2ℓC(x1, e) ⊗σx1,x2ℓP (x1 | x2)ℓP (x1x2x−11| x1).Finally, there is an invertible element ϕ(α) ∈H(α) ⊗H(α) ⊗H(α) which implementsthe quasiassociativity condition(id ⊗∆(α))∆(α)(ℓ) =ϕ(α)(∆(α) ⊗id)∆(α)(ℓ)ϕ(α)−1,ℓ∈H(α).This is the element(8.36)ϕ(α) =Xx1,x2,x3ω(x1, x2, x3)−1 ℓC(x1, e) ⊗ℓC(x2, e) ⊗ℓC(x3, e).A routine check shows that the modular tensor category described in Proposi-tion 8.21 is the category of representations of the quasitriangular quasi-Hopf alge-bra H(α).The quasi-Hopf algebra in [DPR,§3.2] looks similar to H(α), but is expressed interms of a basis. We will choose this basis geometrically in the next section, andso construct an isomorphism between H(α) and the algebra in [DPR,§3.2].52

§9 Higher Gluing and Good TrivializationsIn this section we introduce a “higher order gluing law” for gluing manifoldswith corners. The corners we use are in codimension two; clearly there are gen-eralizations of this gluing law to higher codimension.

Also the gluing law we usehere pertains to the classical theory; there are quantum versions as well. Whilethe formulation of this gluing law is rather abstract, the computations which followshould make its meaning clear.

We study the classical theory over the inteval [0, 1].We choose trivializations (9.4) which replace the trivializations (8.2) we chose inthe last section. The procedure here is more natural than that of §8.

Furthermore,the trivializations (9.4) induce trivializations of the lines LP(x1 | x2) which we pre-viously chose separately in (8.29), and they also induce trivializations of the linesL(x, g) ∼= LC(x, g). The latter amount to a basis of the algebra H(α) in (8.6).

Interms of this basis the quasitriangular quasi-Hopf structure we computed in §8 isexactly the one constructed in [DPR,§3.2], as we verify. The reader may wish toconsider analogous, but simpler, computations in the 1 + 1 dimensional theory.YXXcutFigure 13: Gluing manifolds with cornersWe begin with a statement of the gluing law which should hold in any classicalfield theory, but for our purposes we consider the classical d+1 dimensional theoryof Assertion 2.5.

Suppose X is a compact oriented (d+2−n)-manifold and Y ֒→Xa neat oriented codimension one submanifold (Figure 13), that is, ∂Y = Y ∩∂Xand Y intersects ∂X transversely. Then ∂Y ֒→∂X is a closed oriented codimensionone submanifold, and∂Xcut = Y ∪∂Y (∂X)cut ∪−∂Y −Y,∂(∂X)cut = −∂Y ⊔∂Y.Suppose P →X is a Γ bundle and Q →Y its restriction to Y .

Then the usualgluing law Assertion 2.5(d) implies that there is an isomorphism32(9.1)Tr12,34 : TY (Q) · T(∂X)cut(∂P)cut· TY (Q)−1 −→T∂Xcut(∂P cut).32We use the notation TY (Q) for the classical action, even though Y is not closed.53

Note that the left hand side of (9.1) is an element ofT∂Y (∂Q) · T∂Y (∂Q)−1 · T∂Y (∂Q) · T∂Y (∂Q)−1.Also, there is an isomorphismTr14,23 : TY (Q) · T(∂X)cut(∂P)cut· TY (Q)−1 −→T∂X(∂P),and so finally an isomorphismTrQ = Tr14,23 ◦Tr−112,34 : T∂Xcut(∂P cut) −→T∂X(∂P).Assertion 9.2. In the situation described, there is a natural isomorphism(9.3)TrQe2πiSXcut (P cut)−→e2πiSX(P ).Now we resume our work from §8, retaining the notations there.

As in §6 fix atrivial bundle Rtriv = pt × Γ over a point. Use the correspondence (6.2) to identifyequivalence classes of fields over [0, 1] trivialized over the endpoints with elementsof Γ.

Then the classical action of the equivalence class [Qx] corresponding to x ∈Γis a T-gerbe Gx = T (α)[0,1]([Qx]). Choose trivializing elements(9.4)Gx ∈Gx = T (α)[0,1]([Qx]),x ∈Γ.Now for x1, x2 ∈Γ we glue [Qx2] and [Qx1] to obtain [Qx1x2].

Hence the isomor-phism (2.12) implies that there is an isomorphism(9.5)Gx1 · Gx2 −→Gx1x2.In particular, (9.5) implies that Ge has a trivialization compatible with gluing, andwe assume that Ge is that trivialization. In other words,(9.6)Ge · Ge = Ge.Define the T-torsor Tx1,x2 by the equation(9.7)Gx1 · Gx2 = Gx1x2 · Tx1,x2,x1, x2 ∈Γ,where we implicitly use the isomorphism (9.5) to compare the two sides.

Equa-tion (9.6) implies that Te,e = T.Three intervals can be glued together in twodifferent ways to obtain a single interval. The behavior of the classical action underiterated gluings, which we did not explicitly state in Assertion 2.5(d), implies thatfor any x1, x2, x3 ∈Γ the diagram(Gx1 · Gx2) · Gx3 −−−−→Gx1x2 · Gx3 −−−−→Gx1x2x3yGx1 · (Gx2 · Gx3) −−−−→Gx1 · Gx2x354

commutes up to a natural transformation. Using (9.7) this natural transformationamounts to an isomorphism(9.8)Tx1,x2 · Tx1x2,x3 ∼= Tx2,x3 · Tx1,x2x3,x1, x2, x3 ∈Γ.In particular, taking two of x1, x2, x3 to be e we deduce isomorphisms(9.9)Tx,e ∼= Te,x ∼= T,x ∈Γ.Now we explain the relationship of the choices (9.4) to the choices (8.2) made inthe last section.

Fix x ∈Γ and consider the bundle Q[x] →S1 with basepoint fx,as chosen in §§7–8. Cutting the circle at its basepoint, and using the basepoint fxto identify ∂Qcut[x] with Rtriv ⊔Rtriv, we obtain from the gluing law (2.12) an iso-morphism(9.10)Gx −→G[x].It is not necessarily true that the trivializations of Gx′ in (9.4) for different x′ ∈[x]lead to the same trivialization of G[x].33Now let X be a compact oriented 2-manifold with parametrized boundary and P ∈C′X a bundle with basepoints onthe boundary.

Suppose Y is a component of ∂X and PY has holonomy x. Thebasepoint and parametrization induce an identification PY ∼= Q[x], and so by (9.10)an isomorphism T (α)Y(PY ) ∼= Gx. We trivialize this T-gerbe using (9.4).

Then asin the argument preceding Proposition 8.10 the classical action of P is a T-torsor.It is not the same T-torsor obtained in §8, since we use different trivializations.None of the subsequent arguments are affected by this change, and we use thesenew trivializations in what follows.As a first application of Assertion 9.2 we claim that the classical action of thetrivial bundle over the disk is(9.11)TD2([Ptriv]) = Ge.This can be deduced from the gluing in Figure 13 and (9.6).Next, choose trivializing elements(9.12)tx1,x2 ∈Tx1,x2,x1, x2 ∈Γ.We assume that(9.13)tx,e = te,x = 1,x ∈Γ,under the isomorphism (9.9). Define ω(x1, x2, x3) ∈T by the equation(9.14)tx1x2,x3 · tx1,x2 · ω(x1, x2, x3) = tx1,x2x3 · tx2,x3,x1, x2, x3 ∈Γ,33In this connection notice that whereas T(x, gx,x′) was chosen to be T in (8.4), this torsor isnontrivial with our current set of choices (cf.

(9.16)).55

ggxgxg-1Figure 14: The isomorphism (9.16)where the equality refers to the isomorphism (9.8). The behavior of the classicalaction under iterated gluings of four intervals shows that ω satisfies the cocycleidentity (8.31).Let Lx1,x2 be the hermitian line corresponding to the T-torsor Tx1,x2 and(9.15)ℓx1,x2 ∈Lx1,x2the element of unit norm corresponding to tx1,x2.

We claim that with the choicesof trivializations we have made, the higher gluing law (9.3) constructs isometriesLC(x, g) ∼=Lg,xLgxg−1,g,x, g ∈Γ,(9.16)LP (x1 | x2) ∼= Lx1,x2,x1, x2 ∈Γ. (9.17)The isomorphism (9.16) is derived from the gluing in Figure 14, where we obtainthe cylinder C by gluing a disk D2 along part of its boundary.

The usual gluinglaw (2.12) applied to ∂D2 yields an isomorphismGe ∼= Ggxg−1 · Gg · G−1x· G−1g ,and a short computation with (9.7) shows that under this isomorphism we haveGe = Ggxg−1 · Gg · G−1x· G−1g·Tg,xTgxg−1,g.Now (9.16) follows from (9.11) and the gluing law.The isomorphism (9.17) isderived in a similar manner from Figure 15. In that figureGe ∼= Gx1x2 · G−1x2 · G−1x1 ,56

x2x1x1x2eeeeeFigure 15: The isomorphism (9.17)and under this isomorphismGe = Gx1x2 · G−1x2 · G−1x1 · Tx1,x2.The gluing law and (9.11) imply (9.17).We use (9.15) to trivialize the lines LC(x, g) and LP (x1 | x2). Namely, set(9.18)ℓC(x, g) =ℓg,xℓgxg−1,gand(9.19)ℓP (x1 | x2) = ℓx1,x2.The elements in (9.19) replace the arbitrary choice (8.29) we made in §8.

We now de-fine the quasi-Hopf quasitriangular structure on H(α) in terms of the choices (9.19).The elements in (9.18) form a basis of H(α), and our last task is to computethe quasi-Hopf quasitriangular structure in terms of this basis. Observe also that(9.18) agrees with the special trivializations (8.5) and (8.13).First, we compute the isomorphisms (8.16)–(8.18) in terms of (9.18) and (9.19).We make the obvious computations and leave the justification to the reader.

(Thisinvolves the compatibility of various gluings and diffeomorphisms.) The isomor-phism φx1,x2,x3 is still expressed by (8.30), which follows directly from (9.14).For σx1,x2 we computeσx1,x2ℓP (x1 | x2)ℓP (x1x2x−11| x1) ⊗ℓC(x2, x1)=ℓx1,x2ℓx1x2x−11,x1 ⊗ℓx1,x2ℓx1x2x−11,x1= 1.57

A direct computation yieldsγx1,x2,gℓC(x1x2, g) ⊗ℓP (x1 | x2)ℓP (gx1g−1 | gx2g−1) ⊗ℓC(x1, g) ⊗ℓC(x2, g) = ω(g , x1 , x2) ω(gx1g−1 , gx2g−1, g)ω(gx1g−1, g , x2).Now for the structure on H(α). A short computation shows that the multiplica-tion (8.7) is(9.20)ℓC(g1xg−11 , g2) · ℓC(x, g1) = ω(g2 , g1 , x) ω(g2g1xg−11 g−12 , g2 , g1)ω(g2 , g1xg−11 , g1)ℓC(x, g2g1).The identity element is (8.8):(9.21)1 =XxℓC(x, e).The coproduct (8.33) is(9.22)∆(α)(ℓC(x, g)) =Xx1x2=xω(g , x1 , x2) ω(gx1g−1, gx2g−1 , g)ω(gx1g−1, g , x2)ℓC(x1, g) ⊗ℓC(x2, g).The counit (8.28) is(9.23)ǫℓC(x, g)= 1,if x = e;0,otherwise.The quasitriangular element (8.35) is(9.24)R(α) =Xx1,x2ℓC(x1, e) ⊗ℓC(x2, x1).The element ϕ(α) which measures the deviation from coassociativity is (8.36):(9.25)ϕ(α) =Xx1,x2,x3ω(x1, x2, x3)−1 ℓC(x1, e) ⊗ℓC(x2, e) ⊗ℓC(x3, e).Recall that the antipode (8.34) is the inverse of (8.15).

With the trivializations ofthis section equation (8.14) is replaced by the equationGx · Gx−1 = Ge · Tx,x−1,and so (8.15) by a mapι∗: LC(gx−1g−1, g−1) ⊗Lx,x−1 −→LC(x, g) ⊗Lgxg−1,gx−1g−1.The ratioℓC(x, g) ⊗ℓgxg−1,gx−1g−1ι∗ℓC(gx−1g−1, g−1) ⊗ℓx,x−158

is the numerical factor in the expression(9.26)SℓC(x, g)=ω(g−1, gx−1g−1, g) ω(gxg−1, g , x−1)ω(g−1, g , x−1) ω(x−1, g−1, g)ω(g , x , x−1) ω(gxg−1, gx−1g−1, g) ℓC(gx−1g−1, g−1)for the antipode. The inverse ribbon element is (8.26):v(α) =XxℓC(x, x).Equations (9.20)–(9.26) are exactly the equations in [DPR,§3.2], up to somechanges in notation.Suppose we replace the trivializations tx1,x2 in (9.12) with β(x1, x2)tx1,x2 forsome β(x1, x2) ∈T.

We assume that β(x, e) = β(e, x) = 1 for all x ∈Γ so that(9.13) is respected. Then this change of basis has the effect of twisting (cf.

[Dr])the formulas (9.20)–(9.26) by the elementXx1,x2β(x1, x2) ℓC(x1, e) ⊗ℓC(x2, e).We conclude with some brief general remarks about gluing. The first shouldbe valid for arbitrary topological theories in any dimension.

Consider Y ֒→X aclosed oriented codimension one submanifold and Xcut the cut manifold as in As-sertion 2.5(d). For a new manifold W by identifying the two pieces in the boundaryof [0, 1] × Xcut which correspond to [ 12, 1] × Y , as illustrated in Figure 16.

Then∂W = X ⊔−Xcut ⊔[0, 12] × Y ⊔−[0, 12] × Y.In the classical theory we also are given a field P on X and the corresponding P cuton Xcut. We claim that the gluing (2.12) of the classical action (resp.

the glu-ing (4.17) of the path integral) is computed by the classical action (resp. path in-tegral) over W. For this we trivialize the classical action (resp.

path integral) over[0, 12] using (5.2) (resp. (5.4)).

Such pictures help compute the gluing isometries.XXcut[1/2,1] × YFigure 16: Gluing along a closed submanifold59

Figure 16 is a schematic for arbitrary dimensions as well as an exact picture ofthe gluing of two intervals. The reader may wish to contemplate various gluings ofthis figure and relate the computations in §8 to those in §9.There should also be refined gluing laws of the following sort.

Recall from Propo-sition 5.29 that in a 2 + 1 dimensional theory E(S1) is a “higher commutative as-sociative algebra with compatible real structure” which presumably is semisimple(in a unitary theory). In particular, it is a braided monoidal category, or bettera tortile category.

For such categories one can apparently define a “Grothendieckring” GrothE(S1)(see [Y2,Prop. 26]).

If E(S1) is the category of representationsof a quasi-Hopf algebra H, then the Grothendieck ring is the ring of equivalenceclasses of representations, the multiplication given by the tensor product. Equa-tion (5.5) is a gluing law on the level of inner product spaces, and in this case surelythere is an extension to an isomorphismE(S1 × S1) ∼= GrothE(S1)of algebras.

(E(S1×S1) is an algebra by the remark at the end of §5. It is commonlycalled the Verlinde algebra.) The Grothendieck ring is the “dimension” of E(S1)from the point of view of (5.5).

Notice that GrothE(S1)has a distinguished basisof irreducible representations. These are the “labels” mentioned in §7.60

Appendix: Integration of Singular Cocycles RevisitedIn [FQ,Appendix B] we describe some elements of an integration theory for sin-gular cocycles with coefficients in R/Z. Here we describe an extension of that theoryto higher codimensions in terms of the higher algebra discussed in §1.

Notice thatwe do not introduce any basepoints or special choices, as in [FQ,Proposition B.5].Instead, we extend the integration theory in a more intrinsic manner to all codi-mensions. The higher algebra of §1 is a prerequisite to this appendix.Our goal is to integrate a singular (d+1)-cocycle α over compact oriented mani-folds of any dimension less than or equal to (d+1).

In [FQ] we described the integralof α over closed oriented (d+1)-manifolds, compact oriented (d+1)-manifolds (pos-sibly with boundary), and closed oriented d-manifolds. In the easiest case α is a(d + 1)-cocycle on a closed oriented (d + 1)-manifold X.

Then if x ∈Cd+1(X) is anoriented cycle which represents the fundamental class [X] ∈Hd+1(X), we form thepairing e2πiα(x) ∈R/Z. If x′ is another representative, then x′ −x = ∂w for somew ∈Cd+1(X).

Hence α(x′) −α(x) = α(∂w) = δα(w) = 0 since α is a cocycle. Thisis the usual argument which shows that the integral(A.1)exp(2πiZXα) ∈T0 = Tis well-defined.

In fact, (A.1) can be viewed as the pairing between the cohomologyclass [α] ∈Hd+1(X; R/Z) and the homology class [X] ∈Hd+1(X). This is the onlyone of the integrations we discuss which has cohomological meaning.Now suppose α is a (d + 1)-cocycle on a closed oriented d-manifold Y .

Then weclaim that there is a well-defined integral(A.2)IY,α = exp(2πiZYα) ∈T1which is a T-torsor. The following is a slight modification of what appears in [FQ,Appendix B].The justification for terming this an ‘integral’ are the properties listed in Asser-tion A.4.

Let CY be the category whose objects are oriented cycles y ∈Cd(Y ) whichrepresent the fundamental class [Y ] ∈Hd(Y ), and with a unique morphism y →y′for all y, y′ ∈CY . Define a functor FY ;α : CY →T1 by FY ;α(y) = T for each yand FY ;α(y →y′) acts as multiplication by e2πiα(x), where x is any (d + 1)-chainwith y′ = y + ∂x.

An easy argument shows that α(x) = α(x′) for any two choicesof such a chain.Define IY,α as the inverse limit of FY ;α.34That is, an ele-ment of IY,α is a function i(y) ∈FY ;α(y) = T on the objects in CY such thati(y′) = FY ;α(y →y′) i(y) for all morphisms y →y′.It is easy to check thatIY,α exists.Next, suppose α is a (d + 1)-cocycle on a closed oriented (d −1)-manifold S.Then we claim that the integralIS,α = exp(2πiZSα) ∈T2now makes sense as a T-gerbe. The construction is entirely analogous to the pre-vious one except there is one more layer of argument.

So consider the category CS34See the beginning of §2 for a discussion of inverse limits.61

whose objects are oriented cycles s ∈Cd−1(S) which represent the fundamentalclass [S] ∈Hd−1(S), and with a unique morphism between any two objects. Nowif s, s′ ∈CS, construct a category Cs,s′ whose objects are d-chains y which sat-isfy s′ = s + ∂y, and with a unique morphism between any two objects.

Define afunctor Fs,s′;α : Cs,s′ →T1 by Fs,s′;α(y) = T for each y and Fs,s′;α(y →y′) actsas multiplication by e2πiα(x), where x is any (d + 1)-chain with y′ = y + ∂x. Aneasy argument shows that α(x) = α(x′) for any two choices of such a chain.

De-fine the T-torsor Is,s′;α to be the inverse limit of Fs,s′;α. Now define a functorFS;α : CS →T2 by FS;α(s) = T1 for each s and FS;α(s →s′) acts as multiplicationby Is,s′;α.

The T-gerbe IS,α is defined to be the inverse limit of FS;α.It is clear how to continue to higher codimensions. Now we turn to manifoldswith boundary.If α is a (d+1)-cocycle on a compact oriented (d+1)-manifold X, then in [FQ,Proposition B.1]we describe the integralexp(2πiZXα) ∈I∂X,i∗α,where i: ∂X ֒→X is the inclusion of the boundary, and I∂X,i∗α is the T-torsordescribed previously.

We will not review that here, but rather go on to the next case.Namely, suppose that α is a (d + 1)-cocycle on a compact oriented d-manifold Y .The we claim that the integralexp(2πiZYα) ∈I∂Y,i∗αmakes sense, where now I∂Y,i∗α is the T-gerbe described previously. Call S = ∂Yand let s ∈Cd−1(S) represent the fundamental class, i.e., s ∈CS.

By the definitionof I∂Y,i∗α above we must construct a torsor IY,s;α ∈T1 and for any s, s′ ∈CS anisomorphism(A.3)IY,s;α ⊗Is,s′;α −→IY,s′;α.To construct IY,s;α let CY,s be the category whose objects are d-chains y ∈Cd(Y )such that y represents the fundamental class [Y, ∂Y ] ∈Hd(Y, ∂Y ) and ∂y = i∗s.We postulate a unique morphism y →y′ between any two objects of CY,s. Definea functor FY,s;α : CY,s →T1 by FY,s;α(y) = T for each y and FY,s;α(y →y′) ismultiplication by e2πiα(x), where x is any (d+1)-chain with y′ = y +∂x.

As before,this is independent of the choice of x. Set IY,s;α to be the inverse limit of FY,s;α.To construct the isomorphism (A.3), suppose that y ∈CY,s and a ∈Cs,s′, i.e.,y ∈Cd(Y ) represents [Y, ∂Y ] with ∂y = s, and a ∈Cd(S) with ∂a = s′ −s.

Theny + a ∈CY,s′. The isomorphism (A.3) is defined to be the identity relative to thetrivializations of the torsors determined by y, a, and y + a.This discussion indicates the constructions contained in the following assertion,which we boldly state for arbitrary codimension.Assertion A.4.

Let Y be a closed oriented (d + 1 −n)-manifold (n > 0) andα ∈Cd+1(Y ; R/Z) a singular cocycle. Then there is an element IY,α ∈Tn defined.If X is a compact oriented (d + 2 −n)-manifold, i: ∂X ֒→X the inclusion of theboundary, and α ∈Cd+1(X; R/Z) a cocycle, thenexp2πiZXα∈I∂X,i∗α62

isdefined.These“higherT-torsors”andintegralssatisfy:(a) (Functoriality) If f : Y ′ →Y is an orientation preserving diffeomorphism, thenthere is an induced isomorphismf∗: IY ′,f ∗α −→IY,αand these compose properly. If F : X′ →X is an orientation preserving diffeomor-phism, then there is an induced isomorphism35(A.5)(∂F)∗exp2πiZX′ F ∗α−→exp2πiZXα.

(b) (Orientation) There are natural isomorphisms(A.6)I−Y,α ∼= (IY,α)−1,and(A.7)exp2πiZ−Xα∼=exp2πiZXα−1. (c) (Additivity) If Y = Y1 ⊔Y2 is a disjoint union, then there is a natural isomor-phism(A.8)IY1⊔Y2,α1⊔α2 ∼= IY1,α1 · IY2,α2.If X = X1 ⊔X2 is a disjoint union, then there is a natural isomorphism(A.9)exp2πiZX1⊔X2α1 ⊔α2∼= exp2πiZX1α1· exp2πiZX2α2.

(d) (Gluing) Suppose j : Y ֒→X is a closed oriented codimension one submanifoldand Xcut is the manifold obtained by cutting X along Y . Then ∂Xcut = ∂X ⊔Y ⊔−Y .Suppose α ∈Cd+1(X; R/Z) is a singular (d + 1)-cocycle on S, andαcut ∈Cd+1(Xcut; R/Z) the induced cocycle on Xcut.Then there is a naturalisomorphism(A.10)TrY,j∗αexp2πiZXcut αcut−→exp2πiZXα,where TrY,j∗α is the contractionTrY,j∗α : I∂Xcut,αcut ∼= I∂X,i∗α ⊗IY,j∗α ⊗IY,j∗α−1 −→I∂X,i∗α.

(e) (Stokes’ Theorem I) Let α ∈Cd+1(W; R/Z) be a singular cocycle on a compactoriented (d + 3 −n)-manifold W. Then there is a natural isomorphism36(A.11)exp2πiZ∂Wα∼= Tn−2.35If n = 1 then (A.5) is an equality of elements in a Z-torsor. For n > 1 it is an isomorphismbetween elements in a “higher Z-torsor”.

A similar remark holds for (A.7), (A.9), and (A.10).36Note that Tn−2 is the identity element in Tn−1. If n = 1, then (A.11) should be interpretedasexp2πiZ∂Wα= 1.A similar remark applies to (A.12) below.63

(f ) (Stokes’ Theorem II) A singular d-cochain β ∈Cd(Y ; R/Z) on a closed oriented(d + 1 −n)-manifold Y determines a trivializationIY,δβ ∼= Tn−1.A singular d-cochain β ∈Cd(X; R/Z) on a compact oriented (d+2−n)-manifold Xsatisfies(A.12)exp2πiZXδβ∼= Tn−2under this isomorphism.The assertion in (e) only has real content for n = 1.If n > 1, then I∂W,α istrivialized by exp2πiRW α.We leave the reader to contemplate higher order gluing laws analogous to [FQ,Proposition B.10]and those discussed in §9.64

References[AC]D. Altschuler, A. Coste, Quasi-quantum groups, knots, three-manifolds, and topologicalfield theory, Commun. Math.

Phys. 150 (1992), 83–107.[A]M.

F. Atiyah, Topological quantum field theory, Publ. Math.

Inst. Hautes Etudes Sci.

(Paris) 68 (1989), 175–186.[Be]J. Benabou, Introduction to bicategories, Lec.

Notes in Math., vol. 47, Springer-Verlag,1968, pp.

1–71.[Br]L. Breen, Th´eorie de Schreier sup´erieure, Ann.

Sci. Ecole Norm.

Sup. (4) 25 (1992),465–514.[BMc]J.-L.

Brylinski, D. A. McLaughlin, The geometry of degree four characteristic classesand of line bundles on loop spaces I (preprint, 1992).[DM]P. Deligne, J. S. Milne, Tannakian categories, Lec.

Notes in Math., vol. 900, Springer-Verlag, 1982, pp.

101–228.[DPR]R. Dijkgraaf, V. Pasquier, P. Roche, Quasi-quantum groups related to orbifold models,Nuclear Phys.

B. Proc. Suppl.

18B (1990), 60–72. [DVVV] R. Dijkgraaf, C. Vafa, E. Verlinde, H. Verlinde, Operator algebra of orbifold models,Commun.

Math. Phys.

123 (1989), 485–526.[DW]R. Dijkgraaf, E. Witten, Topological gauge theories and group cohomology, Commun.Math.

Phys. 129 (1990), 393–429.[Dr]V.

G. Drinfeld, Quasi-Hopf algebras and Knizhnik-Zamolodchikov equations, Problemsof modern quantum field theory (Alushta, 1989) (A. Belavin et al., eds. ), Springer-Verlag,1989, pp.

1–13.[Fg]K. Ferguson, Link invariants associated to TQFT’s with finite gauge group (preprint,1992).[F1]D.

S. Freed, Classical Chern-Simons Theory, Part 1, Adv. Math.

(to appear).[F2]D. S. Freed, Classical Chern-Simons Theory, Part 2 (in preparation).[F3]D.

S. Freed, Extended structures in topological quantum field theory (preprint, 1993).[FQ]D. S. Freed, F. Quinn, Chern-Simons theory with finite gauge group, Commun.

Math.Phys. (to appear).[Gi]J.

Giraud, Cohomologie non-ab´elienne, Ergeb. der Math., vol.

64, Springer-Verlag, 1971.[JS]A. Joyal, R. Street, The geometry of tensor calculus, I, Adv.

Math. 88 (1991), 55–112.[KV]M.

M. Kapranov, V. A. Voevodsky, 2-Categories and Zamolodchikov tetrahedra equations(preprint, 1992).[KR]D. Kazhdan, N. Y. Reshetikhin (in preparation).[Ko]M.

Kontsevich, Rational conformal field theory and invariants of 3-dimensional mani-folds (preprint).[L]R. Lawrence (in preparation).[Mac]S.

MacLane, Categories for the Working Mathematician, Graduate Texts in Mathemat-ics, Volume 5, Springer Verlag, 1971.[Ma1]S. Majid, Tannaka-Krein theorem for quasi-Hopf algebras and other results, DeformationTheory and Quantum Groups with Appplications to Mathematical Physics, Contempo-rary Mathematics, vol.

134, Amer. Math.

Soc., 1992, pp. 219–232.[Ma2]S.

Majid, Braided groups (preprint, 1990).[Ma3]S. Majid, Quasi-quantum groups as internal symmetries of topological quantum fieldtheories, Lett.

Math. Phys.

22 (1991), 83–90.[MS]G. Moore, N. Seiberg, Classical and quantum conformal field theory, Commun.

Math.Phys. 123 (1989), 177–254.[Q1]F.

Quinn, Lectures on axiomatic topological quantum field theory (preprint, 1992).[Q2]F. Quinn, Topological foundations of topological quantum field theory (preprint, 1991).[RT]N.

Y. Reshetikhin, V. G. Turaev, Invariants of 3-manifolds via link polynomials andquantum groups, Invent. Math.

103 (1991), 547–97.[S1]G. Segal, The definition of conformal field theory (preprint).[S2]G.

Segal, private communication. [Sh]Shum, Tortile tensor categories, J.

Pure Appl. Alg.

(to appear).[V]E. Verlinde, Fusion rules and modular transformations in 2d conformal field theory,Nucl.

Phys. B300 (1988), 360–376.[W]E.

Witten, Quantum field theory and the Jones polynomial, Commun. Math.

Phys. 121(1989), 351–399.65

[Y1]D. N. Yetter, Topology ’90 (B. Apanasov, W. D. Neumann, A. W. Reid, L. Siebenmann,eds.

), Walter de Gruyter, 1992, pp. 399–444.[Y2]D.

N. Yetter, State-sum invariants of 3-manifolds associated to artinian semisimpletortile categories (preprint).[Y3]D. N. Yetter, Topological quantum field theories associated to finite groups and crossedG-sets, J.

Knot Theory and its Ramifications 1 (1992), 1–20.Department of Mathematics, University of Texas, Austin, TX 78712E-mail address: dafr@math.utexas.edu66

---

출처: arXiv:9212.115 • 원문 보기