NOTES ON QUANTUM TEICHMÜLLER THEORY. 1. Introduction

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1 NOTE ON QUANTUM TEICHMÜLLER THEORY DYLAN GL ALLEGRETTI Abstract We review the Fock-Goncharov formalism for quantum Teichmüller theory By a conjecture of H Verlinde, the Hilbert space of quantum Teichmüller theory can be viewed as the space of conformal blocks for the two-dimensional Liouville conformal field theory The Teichmüller space of a surface also arises in the quantization of L 2 (R)-Chern-imons theory We review these relationships and indicate how quantum Teichmüller theory might provide a tool for rigorously understanding some aspects of these physical theories 1 Introduction 11 Classical and quantum Teichmüller theory In low-dimensional topology and geometry, the Teichmüller space of a surface is a space that parametrizes the hyperbolic structures on More precisely, if is a surface that admits a hyperbolic metric, then the Teichmüller space is defined to be the space of all hyperbolic metrics on modulo diffeomorphisms isotopic to the identity map It has been known since the work of Chekhov-Fock [2] and the independent work of Kashaev [10] that the Teichmüller space of a punctured surface admits a canonical quantization By this, we mean that for certain surfaces with punctures there is a version of the Teichmüller space equipped with a special set of coordinates These coordinates define a commutative algebra of functions on the Teichmüller space, and there is a collection of noncommutative algebras, depending on a parameter q, such that we recover the commutative algebra of functions when q = 1 These algebras can be represented as algebras of operators on an infinite-dimensional Hilbert space In these notes, we will review a particular approach to quantum Teichmüller theory developed by Fock and Goncharov in [5, 6, 7, 8] One of the remarkable features of this approach is that it incorporates ideas from Fomin and Zelevinsky s theory of cluster algebras [9] This theory provides an elegant way of formulating quantum Teichmüller theory, and it has important applications in many other parts of mathematics and theoretical physics In addition to reviewing quantum Teichmüller theory, the goal of these notes is to explain the connections between quantum Teichmüller theory and two other subjects in mathematical physics: quantum Liouville theory and L 2 (R)-Chern-imons theory According to conjectures of physicists, these three theories are related in highly nontrivial ways which we now describe 12 From Teichmüller to Liouville theory The first main topic of these notes after quantum Teichmüller theory is the conformal field theory known as Liouville field theory Last update: August 6,

2 This theory describes the quantum mechanics of a single scalar field on a two-dimensional spacetime manifold It has numerous applications in theoretical physics in the study of non-critical string theories and low-dimensional models of quantum gravity The basic data in the mathematical definition of Liouville theory is the space of conformal blocks This is a vector space whose elements are used to compute expectation values in the quantum theory According to the results of Teschner [12], one can use the Moore-eiberg formalism of conformal field theory to describe the space of Liouville conformal blocks in the case when the surface has genus zero By a conjecture of H Verlinde [13], the space of conformal blocks in quantum Liouville theory can be identified with the Hilbert space of quantum Teichmüller theory Thus the formalism of quantum Teichmüller theory can be used to understand some aspects of quantum Liouville theory in a mathematically rigorous way 13 L 2 (R)-Chern-imons theory The final topic of these notes is the quantum field theory known as Chern-imons theory This theory describes a physical field similar to the electromagnetic field but on a three-dimensional spacetime manifold Chern-imons theory with compact gauge group is well studied because of its applications to knot theory and three-dimensional topology In these notes, we focus on the case of non-compact gauge group and describe a connection to Teichmüller theory For Chern-imons theory on a three-dimensional manifold of the form M = R where is a compact oriented surface, the classical phase space is the space of flat connections on Using the theory of characteristic classes, one can show [11] that this moduli space contains a component which can be identified with the Teichmüller space of Quantum Chern-imons theory is obtained by quantizing the phase space, and therefore quantum Teichmüller theory might be useful as a tool for studying the quantization of L 2 (R)-Chern-imons theory 14 Application to (2+1)-dimensional quantum gravity Although we will not discuss it elsewhere in these notes, there is an interesting application of these theories to (2 + 1)-dimensional quantum gravity In the seminal paper [14], Witten showed that Chern- imons theory can be used to get a quantization of general relativity on a three-dimensional spacetime manifold For gravity with negative cosmological constant, the action functional can be rewritten in terms of the action functionals for two L 2 (R)-Chern-imons fields Using Witten s result, Coussaert, Henneaux, and van Driel argued in [3] that (2 + 1)- dimensional gravity with negative cosmological constant is equivalent to Liouville theory on an auxiliary surface located at the boundary of spacetime in a certain sense Their result gives an approach to (2+1)-dimensional gravity in the spirit of the Ad/CFT correspondence of string theory It would be very interesting to make these results more rigorous and see what role quantum Teichmüller theory plays in (2 + 1)-dimensional quantum gravity with negative cosmological constant 15 Organization We begin in ection 2 by defining a version of the Teichmüller space of surface with punctures and a special set of coordinates on this space We explain how the transition maps between different coordinate systems are given by formulas from the theory 2

3 of cluster algebras, and we explain how the Teichmüller space can be canonically quantized The main references for this section are [5, 7, 8] In ection 3, we review the Moore-eiberg formalism for conformal field theory Using this formalism, we describe the space of conformal blocks in quantum Liouville theory and explain the relationship with quantum Teichmüller theory This section is based on the paper [12] We will not attempt in these notes to make the connection between quantum Teichmüller theory and quantum Liouville theory completely rigorous For a more detailed discussion of this connection, we refer the reader to [12] Finally, in ection 4, we describe Chern-imons theory and its relation to Teichmüller theory We begin with the definition of the Chern-imon action for gauge group L 2 (R) We will not attempt to develop Chern-imons theory rigorously from this point of view, but we will use the formulation in terms of an action to explain why the phase space of Chern-imons theory is identified with the space of flat connections on a surface We then explain how the Teichmüller space arises as a component of the Chern-imons phase space The main reference for this section is [11] 2 Quantum Teichmüller theory 21 The Teichmüller space of a surface In order to quantize the Teichmüller space of a surface, we will need to assume that the surface has punctures More precisely, we will consider a closed oriented surface with finitely many points removed The classical Teichmüller space T of a punctured surface can be viewed as the quotient T = Hom (π 1 (), P L 2 (R))/P L 2 (R) where Hom (π 1 (), P L 2 (R)) is the set of all discrete and faithful representations of π 1 () into P L 2 (R) such that the image of a loop surrounding a puncture is parabolic The group P L 2 (R) acts on this set by conjugation In these notes, we will consider a slight modification of this concept that parametrizes more general surface group representations More precisely, we consider the set Hom (π 1 (), P L 2 (R))/P L 2 (R) where Hom (π 1 (), P L 2 (R)) is the set of all discrete and faithful representations of π 1 () into P L 2 (R) such that the image of a loop surrounding a puncture is either parabolic or hyperbolic uppose we are given a representation ρ : π 1 () P L 2 (R) in the latter set A puncture p in the surface will be called a hole if ρ maps the homotopy class of a loop surrounding p to a hyperbolic transformation Definition 21 Let be a punctured surface The (enhanced) Teichmüller space T + parametrizes pairs (ρ, ) where ρ is an element of the above quotient and is a set of orientations, one for each hole 22 Coordinates on Teichmüller space The first step in studying the space T + define a system of coordinates associated to a certain triangulation of the surface is to Definition 22 Let be a punctured surface An ideal triangulation T of is a triangulation whose vertices are the punctures 3

4 From now on, we will consider only decorated surfaces that admit an ideal triangulation Note that in general the sides of a triangle in an ideal triangulation may not be distinct In this case, the triangle is said to be self-folded We will write I for the set of all edges of an ideal triangulation T If T is an ideal triangulation with no self-folded triangles, then there is a skew-symmetric matrix ε ij (i, j I) defined by ε ij = t T i, t, j where i, t, j equals +1 (respectively, 1) if i and j are sides of the triangle t and i lies in the counterclockwise (respectively, clockwise) direction from j with respect to their common vertex Otherwise, we set i, t, j = 0 Consider an edge k of the ideal triangulation T A flip at k is the transformation of T that removes the edge k and replaces it by the unique different edge that, together with the remaining edges, forms a new ideal triangulation: A flip will be called regular if none of the triangles above is self-folded It is a fact which we will not prove here that any two ideal triangulations on a surface are related by a sequence of flips Note that there is an natural bijection between the edges of a triangulation and the edges of the triangulation obtained by a flip at some edge If we use this bijection to identify edges of the flipped triangulation with the set I, then it is straightforward to show that a flip at an edge k of an ideal triangulation changes the matrix ε ij to the matrix { ε ε ij if k {i, j} ij = ε ij + ε ik ε kj +ε ik ε kj if k {i, j} 2 Given any ideal triangulation T, there is a system of coordinates on T + such that each coordinate is associated with an edge of T The first step in constructing this system of coordinates is to deform the edges of the triangulation T in a certain canonical way To do this, we first remove a small neighborhood of each hole to get a new surface with provides a choice of orientation for the geodesic holes in, and we can wind each edge of the triangulation infinitely many times in the direction prescribed by this orientation so that the edges of our triangulation spiral into the holes in geodesic boundary Now any point of T + Once we have deformed the edges of our triangulation in this way, we can lift the triangulation to the upper half plane to get a collection of ideal triangles Let k be any edge and consider the two ideal triangles t 1 and t 2 that share this edge By applying an element of 4

5 P L 2 (R), we can map this edge to the ray from 0 to in the upper half plane and map the remaining vertex of t 1 to the point 1 as shown below 1 0 X This element of P L 2 (R) takes the remaining vertex of t 2 to some point X R called the cross ratio There are two ways of arriving at this cross ratio, and one can show that they give the same value X This number X is the coordinate that we associate to the edge k Next, we explain how to reconstruct a point of T + starting from a set of positive coordinates X i (i I) To recover the orientations of the punctures, we use the fact that the number log X i, where the sum is over all edges i incident to a puncture p, encodes the orientation of p This sum is negative (resp positive) precisely when the orientation is induced from the orientation of (resp opposite the orientation of ) To recover the hyperbolic structure on, note that any ideal triangle in the upper half plane can be mapped by an element of P L 2 (R) to the ideal triangle having vertices 1, 0, and We can glue a second triangle along the edge from 0 to in infinitely many ways which are parametrized by the cross ratio defined above Thus, to reconstuct the surface, we take a collection of ideal triangles in the upper half plane and glue them along their sides using the given coordinates X i as gluing parameters This shows that points of T + are completely determined by a collection of positive real coordinates X i (i I) In defining the coordinates corresponding to a point of T +, we worked with a fixed ideal triangulation T If instead of T we had used the ideal triangulation T obtained from T by a regular flip at some edge k, we would get coordinates X i related to the X i by the formula X i = { X 1 k X i (1 + X sgn(ε ik) k ) ε ik if i = k if i k Consider the quadrilateral in the triangulation whose diagonal is the edge k There is a coordinate associated to each edge of this picture, and the change of coordinates is given by the following graphical rule X 1 X 2 X 1 (1 + X 0 ) X 2 (1 + X 1 0 ) 1 X 0 X 1 0 X 4 X 3 X 4 (1 + X 1 0 ) 1 X 3 (1 + X 0 ) 5

6 The above formula for change of coordinates also appears in the theory of cluster algebras As we will see, we can use ideas from the theory of cluster algebras to give an abstract combinatorial description of the enhanced Teichmüller space T + 23 The cluster point of view In order to describe the combinatorics of an ideal triangulation, we introduced the skew-symmetric matrix ε ij Abstracting from this situation, we arrive at the important notion of a seed from the theory of cluster algebras Here we present a simplified version of the definition of a seed without frozen variables Definition 23 A seed i = (I, ε ij ) consists of a finite set I, and a skew-symmetric integer matrix ε ij (i, j I) If i = (I, ε ij ) is a seed, then we can consider an associated lattice Λ = Z[I] This lattice has a basis {e i } given by e i = {i} for i I and a Z-valued skew-symmetric bilinear form (, ) given on basis elements by (e i, e j ) = ε ij Thus we have the following equivalent definition of a seed Definition 24 A seed i = (Λ, {e i } i I, (, )) consists of a lattice Λ with basis {e i } i I and a Z-valued skew-symmetric bilinear form (, ) on Λ We have seen how the matrix ε ij transforms under a flip of the triangulation This transformation rule leads to the notion of seed mutation Definition 25 Let i = (I, ε ij ) be a seed and k I Then we define a new seed µ k (i) = i = (I, ε ij) called the seed obtained by mutation in the direction k by setting I = I and defining ε ij by the formula ε ij = { ε ij if k {i, j} ε ij + ε ik ε kj +ε ik ε kj 2 if k {i, j} Naturally, there is also a notion of mutation for seeds in the sense of Definition 24 For any integer n, let us write [n] + = max(0, n) Definition 26 Let i = (Λ, {e i } i I, (, )) be a seed and e k (k I) a basis vector Then we define a new seed i = (Λ, {e i} i I, (, ) ) called the seed obtained by mutation in the direction of e k It is given by Λ = Λ, (, ) = (, ), and { e e k if i = k i = e i + [ε ik ] + e k if i k 6

7 24 Quantization of the space T + In the next step of our construction, we employ the following special function Definition 27 The quantum dilogarithm is the formal power series Ψ q (x) = (1 + q 2k 1 x) 1 k=1 The quantum dilogarithm power series first appeared in the 19th century under the names q-exponential and infinite Pochhammer symbol It was rediscovered in the 1990s by Faddeev and Kashaev, who showed that it satisfies a quantum version of a certain functional equation for the classical dilogarithm function [4] In subsequent works, the quantum dilogarithm was used to quantize the Teichmüller space of a surface [2, 10] In this section, we will study the quantum dilogarithm as a function on the following algebra of q-commuting variables Definition 28 Let Λ be a lattice equipped with a Z-valued skew-symmetric bilinear form (, ) Then the quantum torus algebra is the noncommutative algebra over Z[q, q 1 ] generated by variables Y v (v Λ) subject to the relations q (v 1,v 2 ) Y v1 Y v2 = Y v1 +v 2 This definition allows us to associate to any seed i = (Λ, {e i } i I, (, )), a quantum torus algebra T i The basis {e i } provides a set of generators X i = Y ei for this algebra T i which obey the commutation relations X i X j = q 2ε ji X j X i This algebra T i satisfies the Ore condition from ring theory, so we can form its noncommutative fraction field T i In addition to associating a quantum torus algebra to every seed, we use the quantum dilogarithm to construct a natural map T i T i whenever two seeds i and i are related by a mutation Definition 29 (1) The automorphism µ k : T i T i is given by conjugation with Ψ q (X k ): µ k = Ad Ψ q (X k ) (2) The isomorphism µ k : T i T i is induced by the natural lattice map Λ Λ (3) The mutation map µ q k : T i T i is the composition µ q k = µ k µ k Although conjugation by Ψ q (X k ) produces a priori a formal power series, this construction in fact provides a map T i T i of skew fields The proof of this fact is rather tedious, and the interested reader is referred to [8] Theorem 210 In the classical limit q = 1, the map µ q k is given on generators by { X 1 µ 1 k(x i) k if i = k = X i (1 + X sgn(ε ik) k ) ε ik if i k 7

8 Thus we see that in the classical limit µ q k reduces to the transformation that we encountered previously for coordinates on the enhanced Teichmüller space The quantum torus algebras T i, together with the maps µ q k, form the structure that we call quantum Teichmüller space The quantum torus algebra T i used in this construction has a natural representation as an algebra of unbounded operators on a Hilbert space Indeed, consider the Hilbert space H i = L 2 (R I ), and let a i (i I) be the natural coordinate functions on R I Consider the differential operator x p on H given by x p = 2πi a p α + p where α + k = j I [ε kj] + a j It is straightforward to check that the operators defined in this way satisfy the relations of the Heisenberg -algebra, namely [ x p, x q ] = 2πi ε pq, x p = x p for all p, q I If we set X p = exp( x p ), then it follows that the operators X p obey the commutation relations of a quantum torus algebra where q = e πi In fact, Fock and Goncharov take this construction further and constuct intertwining operators H i H i using a version of the quantum dilogarithm [8] 3 Quantum Liouville theory 31 The Moore-eiberg groupoid Let 0,n denote the standard sphere with n holes given by 0,n = CP 1 {D 1,, D n }, where D k = {z : z z k < ε} are disks with z 1 < < z n, and ε > 0 is small enough that the disks do not intersect In addition, let us mark on each surface a point p k = z k εi Then we can draw a graph on the standard sphere 0,n consisting of a single n-valent vertex in the interior of 0,n and an edge connecting it to each of the marked points on 0,n The diagram below shows this graph in the case n = 3 By gluing together copies of 0,2 and 0,3 and identifying the univalent vertices of the embedded graphs, we can get a graph on an arbitrary surface uch a graph will be called a Moore-eiberg graph In the approach to two-dimensional conformal field theory developed by Moore and eiberg, one considers for a given surface, a groupoid called the Moore-eiberg groupoid The objects of this groupoid are all of the possible Moore-eiberg graphs on, and the morphisms are transitions between different Moore-eiberg graphs The A-, B-, and -moves illustrated below are generators for the Moore-eiberg groupoid 8

9 A-move: A B-move: B -move: These generators satisfy a number of relations which we will not explain here For more information, see [1] 32 Liouville conformal blocks As we explained in the introduction, Liouville theory is a conformal field theory describing a scalar field on a two-dimensional spacetime manifold The state space of the theory can be represented as an integral dαv α V α where the spaces V α and V α are unitary highest weight representations of the Virasoro algebra parametrized by the set = Q/2 + ir + where Q is given in terms of a quantization parameter b by Q = b+b 1 These representations V α and V α have central charge c = 1+6Q 2 and highest weight parametrized as α = α(q α) In quantum Liouville theory, one computes quantities of physical interest using mathematical objects called conformal blocks To specify a conformal block in quantum Liouville theory, one must first specify a list E = (α 1,, α s ) s of parameters associated to the 9

10 punctures in These parameters are called external parameters Given such external parameters, a conformal block is a functional F E : V α1 V αs C satisfying a physical condition called the conformal Ward identity Below we will describe a gluing procedure for constructing a large class of solutions to this conformal Ward identity If our surface is a copy of 0,3, then it is known that the space of conformal blocks is one-dimensional We will denote by C 0,3 E the conformal block given by C 0,3 E (v α 3 v α2 v α1 ) = N(α 3, α 2, α 1 ) where N is the function from [12] On the other hand, if our surface is a copy of 0,2, we denote by, α a bilinear form V α V α C satisfying L n w, v α = w, L n v α where L n and L n are generators of the Virasoro algebra Finally, suppose i (i = 1, 2) are surfaces with m i + 1 punctures and G 1 E 1 and H 2 E 2 are conformal blocks associated to 1 and 2 and labeled by the external parameters E 1 = (a 1,, a m1, a) and E 2 = (a, a 1,, a m 2 ), respectively Let p i (i = 1, 2) be punctures on i labeled by the parameter a, and choose a local coordinate z i around p i so that z i = 0 parametrizes the point p i Let A i be the annuli r < z i < R and D i the disks z i r where R is small so that the disks do not intersect Then we can form the surface Σ 1 Σ 2 = ((Σ 1 D 1 ) (Σ 2 D 2 )) / where denotes the equivalence relation that identifies the point on A 1 with coordinate z 1 and the point on A 2 with coordinate z 2 = rr/z 1 Thus Σ 1 Σ 2 is a surface obtained by removing disks around p 1 and p 2 and gluing the resulting surfaces Given the external parameters E 12 = (a 1,, a m1, a 1,, a m 2 ) and the parameter a, we define a conformal block given by F Σ 1 Σ 2 a,e 12 : V a1 V am1 V a 1 V a m2 C F Σ 1 Σ 2 a,e 12 (v 1 v m1 w 1 w m2 ) = i,j I G 1 E 1 (v 1 v m1 v i ) v i, e tl 0 v j a H 2 E 2 (v j w 1 w m2 ) where {v i } i I and {v i } i I are bases for V a such that v i, v j a = δ ij Using this gluing procedure, we can construct conformal blocks on an arbitrary surface Indeed, suppose the surface is obtained by gluing copies of 0,2 and 0,3 along their boundaries so that the resulting surface carries a Moore-eiberg graph Γ Associate an element of to each hole in and to each internal edge of Γ Then one can use these the gluing rule described above to construct a conformal block on The conformal blocks obtained in this way span a vector space which we denote by H L (, Γ) 33 Representation of the Moore-eiberg groupoid We will now describe a representation of the Moore-eiberg groupoid on the space of conformal blocks of quantum Liouville theory To do this, it suffices to associate an operator to every morphism in the Moore- eiberg groupoid joining a graph Γ to another graph Γ which is related to Γ by an A-, B-, or -move 10

11 To define the operator associated with an A-move, we will need the special function e b (z) defined in the strip Iz < Ic b, c b = i(b + b 1 )/2, by the formula ( ) 1 e 2izw dw e b (z) = exp 4 sinh(bw) sinh(b 1 w) w Ω where Ω is a contour that goes along the real axis outside of a small neighborhood of the origin and deviates into the upper half plane near the origin This function e b (z) is a version of the quantum dilogarithm function In terms of e b (z), Teschner gives in [12] the following rules for assigning operators to the A- and B-moves A-move: Consider the sphere 0,4 with four holes and parameters E = (a 4,, a 1 ) associated to the holes Let F E,as and G E,at denote the conformal blocks associated to the left and right hand sides, respectively, of the figure defining the A-move Then the desired operator is an integral operator given by F E,as = dµ(a t )FE L (a s a t )G E,at with kernel where FE L (a s a t ) = s b(u 1 ) s b (w 1 ) s b (u 2 ) s b (w 2 ) and the coefficients are determined by R dt 4 i=1 s b (x) = e πi 2 x2 e πi 24 (2 Q2) e b (x), s b (t r i ) s b (t s i ) r 1 = p 2 p 1, s 1 = c b p 4 + p 2 p t, u 1 = p s + p 2 p 1 r 2 = p 2 + p 1, s 2 = c b p 4 + p 2 + p t, u 2 = p s + p 3 + p 4 r 3 = p 4 p 3, s 3 = c b + p s, w 1 = p t + p 1 + p 4 r 4 = p 4 + p 3, s 4 = c b p s, w 2 = p t + p 2 p 3 where c b = i Q and a 2 = Q + ip 2 for {1, 2, 3, 4, s, t} If we set a t = Q + ip 2 t, then the measure can be written dµ(a t ) = 4 sinh 2πbp t sinh 2πb 1 p t dp t B-move: Here the desired operator is given by multiplication by the function B L (a 3, a 2, a 1 ) = e iπ( a 3 a 2 a 1 ), where the ai are the conformal dimensions a = a(q a) According to [12], it is not known how to represent the -move on the space of conformal blocks We will therefore focus in these notes on quantum Liouville theory defined on a genus zero surface 11

12 34 Verlinde s conjecture A conjecture of H Verlinde originating in [13] states that the Hilbert space of quantum Teichmüller theory can be identified with the space of conformal blocks in quantum Liouville theory We will now explain the meaning of this conjecture in more detail following Teschner [12] Recall that quantum Teichmüller theory associates a Hilbert space H T to a surface This Hilbert space naturally provides a projective representation (H T, ρ T ) of the mapping class group Γ of According to Teschner, the representation of the Moore-eiberg groupoid described above leads in the same way to a projective representation (H L, ρ L ) of Γ on the space of conformal blocks in quantum Liouville theory Verlinde s conjecture says that these representations are isomorphic: (H T, ρ T ) = (H L, ρ L ) For a more detailed discussion of Verlinde s conjecture in the genus zero case, we refer the reader to [12] 4 L 2 (R)-Chern-imons theory 41 The Chern-imons action In this section, we discuss a quantum field theory called L 2 (R)-Chern-imons theory This theory describes a three-dimensional spacetime manifold M with a field A that one can think of as a three-dimensional analog of an electromagnetic vector potential In general, the space of fields in Chern-imons theory is equipped with the action of a group of symmetries called the gauge group The case of compact gauge group is particularly well studied as it provides an intuitive way of understanding the Jones polynomial of knot theory Here we will be interested in the gauge group L 2 (R) Like most quantum field theories, Chern-imons theory can be defined by means of a function on the space of fields called the action To define the Chern-imons action, let M be a compact three-dimensional manifold and let P M be a principal L 2 (R)-bundle over M Automorphisms of this principal bundle are called gauge transformations The fields in Chern-imons theory are represented by connections A on P Let us give the definition first in the special case where the bundle is trivial, P = M L 2 (R) In this case, we can identify any gauge field with an sl 2 (R)-valued 1-form A on M Then the action is defined by the famous expression (A) = k 2π M Tr(A da A A A) where k is a constant called the level of the theory Here Tr denotes a bilinear form on the Lie algebra sl 2 (R) which is invariant under the adjoint action of L 2 (R) More generally, suppose P M is an arbitrary principal L 2 (R)-bundle In this case, it follows from some results of cobordism theory that there exists a compact oriented 4- manifold Y such that M = Y We can choose this manifold Y in such a way that the bundle P M extends to an L 2 (R)-bundle P Y Then any connection A on P can be extended to a connection  on P Denote by F the curvature of the connection  Then 12

13 the Chern-imons action is defined by the formula (A) = k Tr(F F ) 2π One can show that this definition is independent of all choices made This more general definition specializes to the one above when P M is the trivial bundle Indeed, this bundle extends to the trivial bundle P on Y Our claim then follows from the well known fact that Tr(F F ) = d Tr(A da + 2A A A) 3 In quantum Chern-imons theory, one studies the Feynman integrals f = f(a)e i(a) DA A where A is the space of L 2 (R)-connections modulo gauge transformations, DA denotes a formal measure on A, and f is a function on A representing an observable quantity When we normalize this expression by dividing by the integral Z = e i(a) DA, A which is called the partition function, we obtain the expectation value for the observable represented by f 42 Moduli space of flat connections Although Feynman integrals are extremely useful for doing calculations in physics, they are unfortunately very difficult to define in a mathematically precise way In these notes, we will therefore study quantum Chern-imons theory in a different way, by studying the phase space of the theory Consider Chern-imons theory on an manifold of the form M = R where is a compact oriented surface The L 2 (R)-gauge bundle over M defines an L 2 (R)-bundle V We choose the gauge in which the component A 0 of the connection in the R direction vanishes One can then show, using the calculus of variations, that the condition δ/δa 0 = 0 is satisfied if and only if Y F = 0 where F denotes the curvature of the connection on The phase space of L 2 (R)-Chern- imons theory is defined to be the subspace of the space of all connections on V modulo gauge transformations which satisfy this equation In other words, it is the space of all flat L 2 (R)-connections modulo gauge transformations Let M denote the phase space of L 2 (R)-Chern-imons theory on the manifold M = R In order to relate M to the Teichmüller space of, we will need an alternative description of M The map that sends each loop γ π 1 () to its monodromy defines a representation ρ : π 1 () L 2 (R) of the fundamental group of our surface In fact, one can show that this construction provides a bijection between the set of principal bundles P with flat connection and the set of representations ρ : π 1 () L 2 (R) Let F denote the set of flat connections on an L 2 (R)-bundle V, and let G denote the group of gauge 13

14 transformations Then we have F = Hom(π 1 (), L 2 (R)) and M = F/G = Hom(π 1(), L 2 (R)) L 2 (R) This is the desired description of the phase space 43 The Teichmüller component Finally, we would like to relate the phase space of Chern-imons theory to the Teichmüller space of the surface To do this, we will classify bundles over using characteristic classes Let ρ be a representative for an element of the phase space M, and let P ρ be the corresponding flat principal L 2 (R)-bundle over The group L 2 (R) acts by matrix multiplication on the space R 2, and thus we can form the associated 2-plane bundle E ρ = P ρ L2 (R) R 2 over We associate to this bundle the cohomology class e(e ρ ) H 2 (, Z) = Z known as the Euler class A theorem on Euler classes states that e(e ρ ) χ() = 2g 2 where g is the genus of, and therefore we have a map e : M {2 2g, 3 2g,, 2g 3, 2g 2} The preimage of an integer r under this map is a connected component of M which we will denote by M r = e 1 (r) We claim that the maximal component M 2g 2 can be identified with the Teichmúller space of Indeed, the Teichmüller space consists of all classes [ρ] in the quotient M = Hom(π 1(), L 2 (R)) L 2 (R) where ρ is a faithful representation with discrete image It is a theorem that a representation ρ Hom(π 1 (), L 2 (R)) is faithful with discrete image if and only if e(e ρ ) = 2g 2 This proves our claim that the Teichmüller space is identifiable with M 2g 2 As a consequence of this result, we can quantize the component M 2g 2 by quantizing the Teichmüller space of Although we will not attempt to do it here, it would be very interesting to develop a more rigorous formulation of L 2 (R)-Chern-imons theory using the Fock-Goncharov theory described in ection 2 For a more detailed discussion of the known relationship between Teichmüller theory and L 2 (R)-Chern-imons theory, we refer the reader to [11] and the references cited there Acknowledgments These notes were written as the final project for the course General Relativity in Arbitrary Dimensions taught by Vincint Moncreif at Yale University in the pring semester of 2013 I learned some of the material presented here in a seminar taught by Igor Frenkel in pring of the previous year I thank everyone involved in these courses for helping me learn the material References [1] Bakalov, B and Kirillov, A Lectures on Tensor Categories and Modular Functors American Mathematical ociety, 2000 [2] Chekhov, L and Fock, VV Quantum Teichmüller space arxiv:math/ [mathqa] 14

15 [3] Coussaert, O, Henneaux, M, and van Driel, P The asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant arxiv:gr-qc/ [4] Faddeev, L and Kashaev, RM Quantum dilogarithm arxiv:hep-th/ [5] Fock, VV and Goncharov, AB Dual Teichmüller and lamination spaces arxiv:math/ [mathdg] [6] Fock, VV and Goncharov, AB Cluster ensembles, quantization and the dilogarithm arxiv:math/ v7 [mathag] [7] Fock, VV and Goncharov, AB Cluster ensembles, quantization and the dilogarithm II: The intertwiner arxiv:math/ [mathqa] [8] Fock, VV and Goncharov, AB The quantum dilogarithm and representations of quantum cluster varieties arxiv:math/ [mathqa] [9] Fomin, and Zelevinsky, A Cluster algebras I: Foundations arxiv:math/ [mathrt] [10] Kashaev, RM Quantization of Teichmüller spaces and the quantum dilogarithm arxiv:q-alg/ [11] Killingback, T P Quantization of L(2, R) Chern-imons theory Communications in mathematical physics 1451 (1992): 1 16 [12] Teschner, J On the relation between quantum Liouville theory and the quantized Teichmüller spaces arxiv:hep-th/ v2 [13] Verlinde, H Conformal field theory, two-dimensional quantum gravity and quantization of Teichmüller space Nuclear Physics B 3373 (1990): [14] Witten, E dimensional gravity as an exactly soluble system Nuclear Physics B 3111 (1988):