Construction of the ReshetikhinTuraev TQFT from conformal field theory
Abstract.
In [14] we constructed the vacua modular functor based on the sheaf of vacua theory developed in [44] and the abelian analog in [13]. We here provide an explicit isomorphism from the modular functor underlying the skeintheoretic model for the ReshetikhinTuraev TQFT due to Blanchet, Habbeger, Masbaum and Vogel to the vacua modular functor. This thus provides a geometric construction of the TQFT constructed first by ReshetikhinTuraev from the quantum group .
Contents
 1 Introduction
 2 Modular functors
 3 The Vacua modular functors
 4 The Skein theory construction of modular categories
 5 The representation theory of the Hecke algebra
 6 The ReshetikhinTuraev modular functor via skein theory

7 The genus zero isomorphism
 7.1 The label sets and the action of the Hecke algebra
 7.2 The morphism from the Hecke module to the space of vacua
 7.3 The isomorphism for general boxlabeled object
 7.4 The isomorphism for general labels
 7.5 The isomorphism for arbitrary genus zero marked surfaces
 7.6 Compatibility with glueing in genus zero
 8 The matrices and the higher genus isomorphism
 9 Appendix Basic notations and normalizations for the Lie algebra
1. Introduction
This is the main paper in a series of four papers ([13], [14], [15]), where we provide a geometric construction of modular functors and topological quantum field theories (TQFT’s) from conformal field theory building on the constructions of Tsuchiya, Ueno and Yamada in [44] and [49] and Kawamoto, Namikawa, Tsuchiya and Yamada in [31] and further provide an explicit isomorphism of the modular functors underlying the ReshetikhinTuraev TQFT for and the vacua modular functors coming from conformal field theory for the Lie algebra based on the constructions of in [44] and [49] and [31]. We uses the Skein theory approach to the ReshetikhinTuraev TQFT of Blanchet, Habegger, Masbaum and Vogel [20], [21] and [19] to set up this isomorphism. Since the modular functor determines the TQFT uniquely, this therefore also provides a geometric construction of the ReshetikhinTuraev TQFT. That there should be such an isomorphism is a well established conjecture which is due to Witten, Atiyah and Segal (see e.g. [56], [17] and [42]).
Let us now outline our construction of the above mention isomorphism between the two theories. In [13] we described how to reconstruct the rank one abelian theory first introduced by Kawamoto, Namikawa, Tsuchiya and Yamada in [31] from the the point of view of [44] and [49]. In [14] we described how one combines the work of Tsuchiya, Ueno and Yamada ([44] and [49]) with [13] to construct the vacua modular functor for each simple Lie algebra and a positive integer call the level. Let us here denote the theory we constructed for the the Lie algebra and level by (we briefly give the outline of this construction in section 3). We recall that a modular functors is a functor from a certain category of extended labeled marked surfaces (see section 2) to the category of finite dimensional vector spaces. The functor is required to satisfy Walker’s topological version [52] of Segal’s axioms for a modular functor [42] (see section 2). Note that we do not consider the duality axiom as part of the definition of a modular functor. We consider the duality axioms as extra data. For modular functors which satisfies the duality axiom, we say that it is a modular functor with duality. In [19] Blanchet constructed a modular tensor category which we will here denote (see section 4). It is constructed using skein theory and one can build a modular functor and a TQFT from this category following either the method of [21] or [45]. We denote the resulting modular functor . It is easy to check that these two modular functors have the same label set. In this paper we explicitly construct an isomorphism between these to modular functors
Theorem 1.1.
There is an isomorphism of modular functors
i.e. for each extended labeled marked surface we have an isomorphism of complex vector spaces
which is compatible with all the structures of a modular functor.
The main idea behind the construction of is to use the GNS construction applied to the infinite Hecke algebra with respect to the relevant Markov traces as was first done by Jones [28] and Wenzl [54]. On the skein theory side, we identify the usual purification construction in terms of the GNS construction. This allows us to show that the resulting representations of the Hecke algebras are isomorphic to Wenzl’s representations. On the vacua side, we know by the results of Kanie [30] (see also [50]) that the space of vacua modular functor gives a geometric construction of Wenzl’s representations. By an inductive limit construction, we build a representation of the infinite Hecke algebra, which we identify with the one coming from Wenzl’s GNS construction. This allows us to establish the important Theorem 7.4, which provides a unique infinite Hecke algebra isomorphism from the inductive limit of the relevant morphism spaces on the Skein side to this inductive limit of spaces of vacua with a certain normalization property. Analyzing the properties of this isomorphism further we find that it determines isomorphism from all relevant morphisms in to the corresponding spaces of vacua (up to a choice od scalar for each label of the theory). By following Turaev’s construction of a modular functor from a modular tensor category, we build the the modular functor from the modular category and the isomorphism from Theorem 7.4 now determines all the needed isomorphism between the vector spaces and the vector spaces associates to all labeled marked surfaces.
We have the following geometric application of our construction.
Theorem 1.2.
The connections constructed in the bundle of vacua for any holomorphic family of labeled marked curve given in [44] preserves projectively a unitary structure which is projectively compatible with morphism of such families.
This theorem is an immediate corollary of our main Theorem 1.1. By definition is the covariant constant sections of the bundle of vacua twisted by a fractional power of a certain abelain theory over Teichmüller space. Using the isomorphism from our main Theorem 1.1, we transfer the unitary structure on to the bundle of vacua over Teichmüller space. Here we have used the preferred section of the abelian theory, to transfer the unitary structure to the bundle of vacua (see [14]). Since the unitary structure on is invariant under the extended mapping class group, the induced unitary structure on the bundle of vacua will be projectively invariant under the action of the mapping class group. But since the bundle of vacua for any holomorphic family naturally is isomorphism to the pull back of the bundle of vacua over Teichmüller space, we get the stated theorem.
Theorem 1.3.
The Hitchin connections constructed in the bundle over Teichmüller space, whose fiber over an algebraic curve, representing a point in Teichmüller space, is the geometric quantization at level of the moduli space of semistable bundles of rank and trivial determinant over the curve, projectively preserves a unitary structure which is projectively preserved by the mapping class group.
This is an immediate corollary of Theorem 1.2 and then the theorem by Laszlo in [35], which provides a projective isomorphism of the two bundle with connections over Teichmüller space from Theorem 1.2 and 1.3. We also get the following corollary
Corollary 1.1.
The projective monodromy of the Hitchin connection is infinite for and .
This is an immediate corollary of Theorem 1.1 and Gregor Masbaum’s corresponding result for the ReshetikhinTuraev Theory [39] (see also [23]). For a purely algebraic geometric proof of this result see [36].
Further the combination of Laszlo’s theorem from [35] allows us to use the geometric quantization of the moduli space of flat connections to study the ReshetikhinTuraev TQFT’s. We have already provided a number of such applications, see e.g. [3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. In fact the result of this paper is used in [11] as an essential component in the first authors proof that the mapping class group does not have Kazhdan’s property T [11].
The paper is organized as follows. In section 2 we briefly recall the theory of modular functors. In section 3 we will recall the vacua modular functor constructed from our previous paper [14]. In section 4 following the description of Blanchet [19], we set up the needed modular tensor category using skein theory. In section 5 we recall Jones and Wenzl work on the representation theory of the Hecke algebra and establish the needed relation between the skein theory representations and Wenzl’s representations of the Hecke algebra. In the following section 6, we define the modular functor which gives the skeintheoretic model for ReshetikhinTuraev TQFT due to Blanchet, Habbeger, Masbaum and Vogel [20], [21]. The genus zero part of the of the isomorphism of the modular functors and is provided in section 7. The extension of the isomorphism to higher genus is provided in section 8. In the appendix we provide some normalizations and notations for the Lie algebra .
We thank Christian Blanchet, Yukihiro Kanie, Gregor Masbaum, Akihiro Tsuchiya and Yasuhiko Yamada for valuable discussions.
2. Modular functors
2.1. The axioms for a modular functor
We shall in this section give the axioms for a modular functor. These are due to G. Segal and appeared first in [42]. We present them here in a topological form, which is due to K. Walker [52]. See also [25]. We note that similar, but different, axioms for a modular functor are given in [45] and in [18]. It is however not clear if these definitions of a modular functor is equivalent to ours.
Let us start by fixing a bit of notation. By a closed surface we mean a smooth real two dimensional manifold. For a closed oriented surface of genus we have the nondegenerate skewsymmetric intersection pairing
Suppose is connected. In this case a Lagrangian subspace is by definition a subspace, which is maximally isotropic with respect to the intersection pairing.  A basis for is called a symplectic basis if
for all .
If is not connected, then , where are the connected components of . By definition a Lagrangian subspace is in this paper a subspace of the form , where is Lagrangian. Likewise a symplectic basis for is a basis of the form , where is a symplectic basis for .
For any real vector space , we define
Definition 2.1.
A pointed surface is an oriented closed surface with a finite set of points. A pointed surface is called stable if the Euler characteristic of each component of the complement of the points is negative. A pointed surface is called saturated if each component of contains at least one point from .
Definition 2.2.
A morphism of pointed surfaces is an isotopy class of orientation preserving diffeomorphisms which maps to . Here the isotopy is required not to change the induced map of the first order Jet at to the first order Jet at .
Definition 2.3.
A marked surface is an oriented closed smooth surface with a finite subset of points with projective tangent vectors and a Lagrangian subspace .
Remark 2.1.
The notions of stable and saturated marked surfaces are defined just like for pointed surfaces.
Definition 2.4.
A morphism of marked surfaces is an isotopy class of orientation preserving diffeomorphisms that maps to together with an integer . Hence we write .
Remark 2.2.
Any marked surface has an underlying pointed surface, but a morphism of marked surfaces does not quit induce a morphism of pointed surfaces, since we only require that the isotopies preserve the induced maps on the projective tangent spaces.
Let be Wall’s signature cocycle for triples of Lagrangian subspaces of (See [53]).
Definition 2.5.
Let and be morphisms of marked surfaces then the composition of and is
With the objects being marked surfaces and the morphism and their composition being defined as in the above definition, we have constructed the category of marked surfaces.
The mapping class group of a marked surface is the group of automorphisms of . One can prove that is a central extension of the mapping class group of the surface defined by the 2cocycle , . One can also prove that this cocycle is equivalent to the cocycle obtained by considering twoframings on mapping cylinders (see [16]).
Notice also that for any morphism , one can factor
In particular is .
Definition 2.6.
The operation of disjoint union of marked surfaces is
Morphisms on disjoint unions are accordingly .
We see that disjoint union is an operation on the category of marked surfaces.
Definition 2.7.
Let be a marked surface. We denote by the marked surface obtained from by the operation of reversal of the orientation. For a morphism we let the orientation reversed morphism be given by .
We also see that orientation reversal is an operation on the category of marked surfaces. Let us now consider glueing of marked surfaces.
Let be a marked surface, where we have selected an ordered pair of marked points with projective tangent vectors , at which we will perform the glueing.
Let be an orientation reversing projective linear isomorphism such that . Such a is called a glueing map for . Let be the oriented surface with boundary obtained from by blowing up and , i.e.
with the natural smooth structure induced from . Let now be the closed oriented surface obtained from by using to glue the boundary components of . We call the glueing of at the ordered pair with respect to .
Let now be the topological space obtained from by identifying and . We then have natural continuous maps and . On the first homology group induces an injection and a surjection, so we can define a Lagrangian subspace by . We note that the image of (with the orientation induced from ) induces naturally an element in and as such it is contained in .
Remark 2.3.
If we have two glueing maps we note that there is a diffeomorphism of inducing the identity on which is isotopic to the identity among such maps, such that . In particular induces a diffeomorphism compatible with , which maps to . Any two such diffeomorphims of induces isotopic diffeomorphims from to .
Definition 2.8.
Let be a marked surface. Let
be a glueing map and the glueing of at the ordered pair with respect to . Let be the Lagrangian subspace constructed above from . Then the marked surface is defined to be the glueing of at the ordered pair with respect to .
We observe that glueing also extends to morphisms of marked surfaces which preserves the ordered pair , by using glueing maps which are compatible with the morphism in question.
We can now give the axioms for a 2 dimensional modular functor.
Definition 2.9.
A label set is a finite set furnished with an involution and a trivial element such that .
Definition 2.10.
Let be a label set. The category of labeled marked surfaces consists of marked surfaces with an element of assigned to each of the marked point and morphisms of labeled marked surfaces are required to preserve the labelings. An assignment of elements of to the marked points of is called a labeling of and we denote the labeled marked surface by , where is the labeling.
We define a labeled pointed surface similarly.
Remark 2.4.
The operation of disjoint union clearly extends to labeled marked surfaces. When we extend the operation of orientation reversal to labeled marked surfaces, we also apply the involution to all the labels.
Definition 2.11.
A modular functor based on the label set is a functor from the category of labeled marked surfaces to the category of finite dimensional complex vector spaces satisfying the axioms MF1 to MF5 below.
Mf1
Disjoint union axiom: The operation of disjoint union of labeled marked surfaces is taken to the operation of tensor product, i.e. for any pair of labeled marked surfaces there is an isomorphism
The identification is associative.
Mf2
Glueing axiom: Let and be marked surfaces such that is obtained from by glueing at an ordered pair of points and projective tangent vectors with respect to a glueing map . Then there is an isomorphism
which is associative, compatible with glueing of morphisms, disjoint unions and it is independent of the choice of the glueing map in the obvious way (see remark 2.3). This isomorphism is called the glueing isomorphism and its inverse is called the factorization isomorphism.
Mf3
Empty surface axiom: Let denote the empty labeled marked surface. Then
Mf4
Once punctured sphere axiom: Let be a marked sphere with one marked point. Then
Mf5
Twice punctured sphere axiom: Let be a marked sphere with two marked points. Then
In addition to the above axioms one may has extra properties, namely
MfD
Orientation reversal axiom: The operation of orientation reversal of labeled marked surfaces is taken to the operation of taking the dual vector space, i.e for any labeled marked surface there is a pairring
compatible with disjoint unions, glueings and orientation reversals (in the sense that the induced isomorphisms and are adjoints).
and
MfU
Unitarity axiom
Every vector space is furnished with a hermitian inner product
so that morphisms induces unitary transformation. The hermitian structure must be compatiple with disjoint union and glueing. If we have the orientation reversal property, then compatibility with the unitary structure means that we have a commutative diagrams
where the vertical identifications come from the hermitian structure and the horizontal from the duality.
3. The Vacua modular functors
In this section we recall the construction of the vacua modular functor given in [14].
3.1. Affine Lie Algebra and Sheaf of Vacua
Let be a simple Lie algebra over the complex numbers . The normalized CartanKilling form is defined to be a constant multiple of the CartanKilling form such that
for the longest root . By and we mean the ring of formal power series in and the field of formal Laurent power series in , respectively. The affine Lie algebra over associated with is defined to be
where belongs to the center of and the Lie algebra structure is given by
for
Let us fix a positive integer and put
(1) 
where is the set of dominant integral weights. The involution
is defined by
(2) 
where is the longest element of the Weyl group of the simple Lie algebra . Then for each there exists a unique left module (called the integrable highest weight module of level ) characterized by the following properties (see [29]):

module with highest weight where .
is the irreducible left 
The central element acts on as .

is generated by over with only one relation
where , is the element corresponding to the maximal root and is the highest weight vector.
The dual space of is defined to be
For put
where is the complete tensor product.
An pointed Riemann surface consists of a compact Riemann surface of genus with distinct points on it. For each positive integer an th order neighbourhood of at a point is a algebra isomorphism
where is the stalk at the point of the sheaf of germs of holomorphic functions on and is the maximal ideal of consisting of germs of holomorphic functions vanishing at . A formal neighbourhood of at a point is a algebra isomorphism
A family of pointed Riemann surfaces with formal coordinates consists of the following data:

Complex manifolds and with and is a proper holomorphic map of maximal rank at each point.

Holomorphic maps such that is the identity map.

Each fiber , is a Riemann surface of genus and
is an pointed Riemann surface.

Maps , which are algebra isomorphisms
where is the ideal sheaf consisting of germs of holomorphic functions vanishing along .
The sheaf of affine Lie algebra over is a sheaf of module
with the following commutation relation, which is bilinear.
where and we require to be central.
Put
where and is the inductive limit of and is the direct image sheaf of by the holomorphic map . Here is a sheaf of germs of meromorphic functions on having poles of order at most along and holomorphic outside .
The Laurent expansions with respect to the formal neighbourhoods ’s gives an module inclusion:
and we may regard as a Lie subalgebra of . For any , put
The sheaf of affine Lie algebra acts on by
where means that acts on the th component of from the right. Similarly acts on from the left.
Definition 3.1.
The sheaves of vacua and covacua attached to the family is defined as
If the base space is a point , that is , and covacua are called the space of vacua and the space of covacua, respectively and written as , where .
Theorem 3.1 ([44]).
The sheaves of vacua and covacua are locally free modules of finite rank and dual to each other. Moreover for any point we have
where and .
3.2. Teichmüller space and the bundle of Vacua
Let us first review some basic Teichmüller theory.
Definition 3.2.
A marked Riemann surface is a Riemann surface with a finite set of marked points and nonzero tangent vectors .
Note that a nonzero tangent vector determines uniquely the first order neighbourhood at the point by
and conversely a first order neighbourhood determines uniquely the tangent vector by the above formula.
Definition 3.3.
A morphism between marked Riemann surface is a biholomorphism of the underlying Riemann surface which induces a bijection between the two sets of marked points and tangent vectors at the marked points.
The notions of stable and saturated is defined just like for pointed surfaces (see Definition 2.1).
Let be a closed oriented smooth surface and let be finite set of points on .
Definition 3.4.
A complex structure on is a marked Riemann surface together with an orientation preserving diffeomorphism mapping the points onto the points . Two such complex structures are equivalent if there exists a morphism of marked Riemann surfaces
such that is isotopic to the identity through maps inducing the identity on the first order neighbourhood of .
We shall often in our notation suppress the diffeomorphism, when we denote a complex structure on a surface.
Definition 3.5.
The Teichmüller space of the pointed surface is by definition the set of equivalence classes of complex structures on .
We note there is a natural projection map from to which we call .
Theorem 3.3 (Bers).
There is a natural structure of a finite dimensional complex analytic manifold on Teichmüller space . Associated to any morphism of pointed surfaces there is a biholomorphism which is induced by mapping a complex structure , to . Moreover, compositions of morphisms go to compositions of induced biholomorphisms.
There is an action of on given by scaling the tangent vectors. This action is free and the quotient is a smooth manifold, which we call the reduced Teichmüller space of the pointed surface . Moreover the projection map descend to a smooth projection map from to , which we denote . We denote the fiber of this map over by . Teichmüller space of a marked surface is by definition , which we call the Teichmüller space of the marked surface. Morphisms of marked surfaces induce diffeomorphism of the corresponding Teichmüller spaces of marked surfaces, which also behaves well under composition. We observe that the selfmorphism of a marked surface acts trivially on the associated Teichmüller space for all integers . General Teichmüller theory implies that
Theorem 3.4.
The Teichmüller space of any marked surface is contractible.
Since there are many ways to extend the universal family of marked Riemann surfaces over the Teichmüller space of a marked surface to the family of pointed Riemann surfaces with formal coordinates, careful consideration is necessary to construct the bundle of vacua over the Teichmüller space . This was done in our previous paper [14] §5.3. We recall the crucial parts of the construction.
Let be a family of pointed Riemann surfaces with formal neighbourhoods and assume we have a smooth fiber preserving diffeomorphism from to taking the marked points to the sections and inducing the identity on . This data induces a unique holomorphic map from to the Teichmüller space of the surface by the universal property of Teichmüller space.
Definition 3.6.
If a family of pointed Riemann surfaces with formal neighbourhoods on , as above, has the properties, that the base is biholomorphic to an open ball and that the induced map is a biholomorphism onto an open subset of Teichmüller space then the family is said to be good.
Note that if a family of pointed Riemann surfaces with formal neighbourhoods on is versal around some point , in the sense of Definition 1.24 in [50], then there is an open ball around in , such that the restriction of the family to this neighbourhood is good.
Proposition 3.1.
For a stable and saturated pointed surface the Teichmüller space can be covered by images of such good families.
This follows from Theorem 1.28 in [50].
Using these good families we can construct the bundle of vacua over the Teichmüller space (see Definition 5.2 [14] and the arguments after the definition).
Theorem 3.5.
There exists a unique holomorphic vector bundle over Teichmüller space which is specified to be the bundle over for any good families of complex structures on , where is the canonical mapping.
3.3. The bundle of Abelian Vacua
To a family of pointed Riemann surfaces with formal neighbourhoods we can associate the line bundle of abelian vacua which is isomorphic to the determinant bundle of the relative canonical sheaf (Theorem 3.2, [13]).
Theorem 3.6 ([13], Theorem 4.2).
The line bundle carries a family of projectively flat connection parametrized by bidifferentials on .
To compare the curvatures of the connections in the abelian vacua with that for the nonabelain ones, we needed an explicit description of the curvature form, hence we constructed the line bundle of abelian vacua by using the fermion Fock space and fermion operators in [13].
Also by the similar method to the one for nonabelian vacua we can construct the line bundle of abelian vacua over Teichmüller space of a marked surface (section 8 in [14]).
The relation of curvature forms of projectively flat connections of the bundle of vacua and the line bundle of abelian vacua is given the following theorem. The theorem plays the crucial role to construct our modular functor.
Theorem 3.7 ([14], Corollary 9.1).
Let be a family of stable and saturated pointed Riemann surfaces with formal neighbourhoods on . If we use the same bidifferential to define the connections on the bundle of vacua and the line bundle of abelian vacua on , then we have
(3) 
where is the curvature form of the connection of the bundle of vacua, is the curvature form of the connection of the line bundle of abelian vacua constructed by using the bidifferential and
(4) 
where is the dual Coxeter number of the simple Lie algebra ( for example, for ).
To define th root of line bundle , we need a nonvanishing holomorphic section of this bundle (see [14] and the following section). To this end we have the following theorem.
Theorem 3.8 ([14], Theorem 10.8).
Let be a closed oriented surface and let be a symplectic basis of . Then there is a unique nonvanishing holomorphic section in the bundle over which behaves well under an orientation preserving diffeomorphism of surfaces which maps the symplectic basis of for some surface to the symplectic basis of for some other surface .
3.4. The geometric construction of the modular functor.
For the bundle of vacua over the Teichmüller space we summarize its properties in the following theorem.
Theorem 3.9.
Let be a stable and saturated labeled pointed surface.

There exists a vector bundle over the Teichmüller space of whose fiber at a complex structure