QUANTUM MECHANICS AND NON-ABELIAN THETA FUNCTIONS ...

QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 3produces the classical theory of theta functions, an action of the finiteHeisenberg group, and the well known Hermite-Jacobi action of SL(2, Z)on theta functions. Later it has been observed that classical theta functionsare holomorphic sections of a line bundle over the moduli space of flatU(1)-connections on the torus. By analogy, the holomorphic sections of thesimilar line bundle over the moduli space of flat G-connections over a surface(where G is a compact simple Lie group) were called non-abelian thetafunctions. Edward Witten [29] established a relationship between conformalfield theory and the theory of non-abelian theta functions. This had atremendous impact on the latter theory, in the guise of the Verlinde formulawhich computes the dimension of the space of non-abelian theta functions.In what follows we will concentrate on the Lie group SU(2) and explainthe analogy between André Weil’s quantum mechanical approach to thetafunctions and the quantum mechanics of the quantum group quantizationof the moduli space of flat SU(2)-connections on a surface.In order to better illustrate this analogy, we briefly recall the Schrödingerand metaplectic representations, then proceed with a detailed discussion ofclassical theta functions. What we bring new to this theory is a descriptionof the Schrödinger representation of the finite Heisenberg group and of thefinite Fourier transform from an entirely topological perspective. We dothis using the language of skein modules - algebraic structures defined byknots and links and combinatorial relations among them, which were firstintroduced by Turaev [25] for cylinders over surfaces, and by Przytycki [19]for general 3-dimensional manifolds. The skein modules used here are thosecorresponding to the linking number. The fact that skein modules can beused to describe classical theta functions is a corollary of Witten’s Feynmanintegral approach to the Chern-Simons theory with gauge group U(1).We proceed with a description of the quantum group quantization of themoduli space of flat SU(2)-connections on a surface, then rephrase this intothe language of skein modules. We obtain that this quantization is a representationgiven as the left action of a skein algebra on a quotient of itself.We prove that this representation is irreducible (as required by a postulateof quantum mechanics). Then, we recall the Reshetikhin-Turaev representationof the mapping class group and observe that, as a consequence ofSchur’s lemma, it is the only such representation that intertwines the operatorsof the quantum group quantization. As a corollary, we give a simplerdescription of the Reshetikhin-Turaev representation, which establishes itas an analogue of the Fourier-Mukai transform.Next, we discuss the case of the torus. Here the moduli space of flatSU(2)-connections is the quotient of the torus by the antipodal map, so wecan do equivariant Weyl quantization on the torus. We recall our previousresult that this quantization coincides with the quantum group quantization.We explain this coincidence through a Stone-von Neumann theorem. Whilewe are able to prove a Stone-von Neumann theorem for the quantization ofthe moduli space of flat SU(2)-connections on the torus, we don’t know ifsuch a theorem is true for higher genus surfaces. We suspect that such a

QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 7one arising from the quantization in a complex polarization, and one definedby the right action on equivariant functions on the Heisenberg group givenby a choice of a Lagrangian subspace of R 2 .2.2. The metaplectic representation. A corollary of the Stone-von Neumanntheorem is that if we compose the Schrödinger representation withan automorphism of the Heisenberg group and then quantize, we obtaina unitarily-equivalent representation. A linear symplectic transformationof R 2 induces an automorphism of the Heisenberg group. Therefore, theStone-von Neumann theorem implies that linear symplectomorphisms canbe quantized, giving rise to unitary operators. A corollary of the Stone-vonNeumann theorem is, by Schur’s lemma, that these unitary operators areunique up to a multiplication by a constant, hence we have a projectiverepresentation of the linear symplectic group, which in this case is SL(2, R).We thus have a projective representation ρ of SL(2, R) on L 2 (R). Thiscan be made into a true representation if we pass to the double cover ofSL(2, R), namely to the metaplectic group Mp(2, R). The representationof the metaplectic group is known as the metaplectic representation or theSegal-Shale-Weil representation.The fundamental symmetry that Weyl quantization has is that, if h ∈Mp(2, R), thenop(f ◦ h) = ρ(h)op(f)ρ(h) −1 ,for every observable f ∈ C ∞ (R 2 ), where op(f) is the operator associated tof through Weyl quantization. For other quantization models this relationholds only mod O(), (Egorov’s theorem). When it is satisfied with equality,as it is in our case, it is called the exact Egorov identity.An elegant way to define the metaplectic representation is to use the thirdversion of the Schrödinger representation, which identifies the metaplecticrepresentation as a Fourier transform (see [16]). Let h be a linear symplectomorphismof the plane, then let L 1 be a Lagrangian subspace of RP +RQand L 2 = h(L 1 ). Define the quantization of h as ρ(h) : H(L 1 ) → H(L 2 ),∫(ρ(h)φ)(u) =φ(uu 2 )χ L2 (u 2 )dµ(u 2 ),exp L 2 /exp(L 1 ∩L 2 )where dµ is the measure induced on the space of equivalence classes by theHaar measure on H(R). This formula outlines the fundamental principlewhich governs the Fourier-Mukai transform: in order to map a function ofvariable x to a function of variable y you lift the first function to one thatdepends on both x and y, then average over x.To write explicit formulas for ρ(h) one needs to choose the algebraic complementsL ′ 1 and L′ 2 of L 1 and L 2 and unfold the isomorphism L 2 (L ′ ) ∼ =L 2 (R). For example, forS =(0 1−1 0),

8RĂZVAN GELCA AND ALEJANDRO URIBEif we set L 1 = RP with variable y and L 2 = S(L 1 ) = RQ with variable xand L ′ 1 = L 2 and L ′ 2 = S(L′ 1 ) = L 1, then∫ρ(S)f(x) = f(y)e −2πixy dy,which is the usual Fourier transform. Similarly, for( )1 aT a = ,0 1if we set L 1 = L 2 = RP =, L ′ 1 = RQ, and L′ 2 = R(P + Q), thenRρ(T a )f(x) = e 2πix2a f(x)which is the multiplication by the exponential of a quadratic function. Theseare the well known formulas that define the action of the metaplectic groupon L 2 (R,dx).One should also note that in the first situation L 1 ∩ L 2 = {0} so theintegration takes place over the entire exp L 2 . In the second situation L 1 ∩L 2 ≠ {0}, so the integration takes place over a quotient of exp L 2 ; in thissense it is a partial Fourier transform.The cocycle of the projective representation of the symplectic group isc L (h ′ ,h) = e − iπ 4 τ(L,h(L),h′ ◦h(L))where τ is the Maslov index. This means thatfor h,h ′ ∈ SL(2, R).ρ(h ′ h) = c L (h ′ ,h)ρ(h ′ )ρ(h)3. Classical theta functions3.1. Classical theta functions from the quantization of the torus.For an extensive treatement of theta functions, the reader can consult [17],[16], [18]. From the point of view of this paper, we are quantizing a phasespace which is the 2-dimensional torus T 2 in the holomorphic polarizationof an arbitrary complex structure. The moduli space of complex structureson the torus is the upper half-plane factored by the group PSL(2, Z). Twocomplex numbers in the upper half-plane yield the same complex structureif they correspond through a Möbius transformation. To a complex numberτ = a + bi with positive imaginary part one associates the complex torusobtained as the quotient of C by the lattice Z + Zτ. The complex variableis z = x + τy, where (x,y) are the coordinates in the basis (1,τ). The torusis endowed with the symplectic form ω = π bdz ∧ d¯z, which is 2πi times thegenerator of H 2 (T 2 , Z).To obtain the Hilbert space of the quantization we apply the procedure ofgeometric quantization. We start by setting = 1 N. Moreover, for technicalreasons such as the fact that quantizations of exponentials have the samespectra, the requirement that SL θ (2, Z) = SL(2, Z), and the possibility toestablish a relationship with Chern-Simons theory, we restrict N to be aneven integer, say N = 2r.

QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 9The Hilbert space consists of the holomorphic sections of a line bundlewhich is the tensor product of a line bundle of curvature Nω and a halfdensity(the necessity of which is explained in [23]). For the half-density wechoose the trivial line bundle defined by the 1-form dz, and simply ignoreit.A holomorphic line bundle with curvature Nω can be obtained by factoringthe trivial line bundle on R 2 by a cocycle Λ : R 2 × Z 2 → C ∗ of theformΛ((x,y),(m,n)) = α(m,n)e 2πiN( τ 2 n2 −n(x+τy)) = α(m,n)e 2πiN( τ 2 n2 −nz) .where α is a character of Z 2 which parametrizes a flat line bundle on thetorus, i.e. α(m,n) = e 2πi(mµ+nν) for some real numbers µ and ν. Theholomorphic sections of this line bundle are the classical theta functions.They can be identified with holomorphic functions on the plane that satisfythe periodicity conditionf(z + m + nτ) = α(m,n)e 2πiN(τn2 −2nz) f(z).To simplify the discussion, we restrict ourselves to the case µ = ν = 0, whenα ≡ 1.An orthonormal basis of the Hilbert space consists of the so called thetaseries. To specify them one needs a rigid structure on the torus. Thisrigid structure consists of the images on the torus of two generators of itsfundamental group. The first curve cuts the torus open into an annulus,while the second keeps track of twistings (see Figure 1, the dotted line keepstrack of the twistings). The group of transformations of rigid structuresis PSL(2, Z), therefore the complex tori endowed with rigid structures areparametrized by the upper half-plane. They form the Teichmüller space ofthe torus. For a point τ in the upper half-plane, the corresponding thetaseries areθ τ j (z) = ∑ n∈Ze 2πiN h 12( j N +n)2 (a+bi)−(x+ay+biy)( j N +n)i= ∑ n∈Ze 2πiN h τ2( j N +n)2 −z( j N +n)i , j = 0,1,2,... ,N − 1.We extend this definition of theta series to all indices j by the periodicitycondition θj+N τ (z) = θτ j (z), namely we work with indices modulo N.Figure 1Let us now turn to the operators. The only exponentials on the planethat are double periodic, and therefore give rise to functions on the torus,

10RĂZVAN GELCA AND ALEJANDRO URIBEare of the formf(x,y) = exp 2πi(mx + ny), m,n ∈ Z.We quantize them using Weyl quantization. In the holomorphic polarizationWeyl quantization is defined as follows (see [5]): A fundamental domain ofthe torus is the unit square [0,1]×[0,1] (this is done in the (x,y) coordinates,in the complex plane it is actually a parallelogram). The value of a thetafunction is completely determined by its values on this unit square. TheHilbert space of classical theta functions can be isometrically embeddedinto L 2 ([0,1] × [0,1]) with the inner product√ ∫ 2 1∫ 1〈f,g〉 = f(x,y)g(x,y)e −2Nπy2 dxdyN00(the constant in front of the integral is chosen so that the specified thetaseries have norm 1). The operator associated to a function f on the torus isthe Toeplitz operator with symbol e − ∆4N f, where ∆ is the Laplacian definedas∆f = 1 ( ∂ 2 )f2π ∂x 2 + ∂2 f∂y 2 .This operator maps a classical theta function g to the orthogonal projectiononto the Hilbert space of classical theta functions of (e − ∆4N f)g.The Weyl quantization of the exponentials gives rise to the Schrödingerrepresentation of the Heisenberg group with integer entries H(Z) onto thespace of theta functions. We define this Heisenberg group aswith multiplicationH(Z) = {(p,q,k), p,q,k ∈ Z}(p,q,k)(p ′ ,q ′ ,k ′ ) = (p + p ′ ,q + q ′ ,k + k ′ + (pq ′ − qp ′ )).As before, we denote exp(pP +qQ+kE) = (p,q,k). Then exp(pP +qQ+kE)is the Weyl quantization of the function e πiN k exp 2πi(qx+py). A few computationsthat can be found in [9] show that the quantization of exponentialsyields the action of this Heisenberg group on theta functions given byIn particularexp(pP)θ τ j = θ τ j+p,exp(pP + qQ + kE)θ τ jπi= e− N pq−2πi Nexp(qQ)θ τ j = e −2πi N qj θ τ j ,qj+πiN k θ τ j+p .exp(kE)θ τ j = e πiN k θ τ j .Here the indices of theta functions are taken mod N. Note that the thetaseries are the eigenvectors of exp(Q) and they can be generated from θ0 τ byθj τ = exp(jP)θ0 τ for all j = 1,2,... ,N − 1. The rigid structure specifiesP and Q, once they are known, the theta series are characterized by theseproperties (up to a multiplication of all of them by the same constant). Thisis how the rigid structure determines the theta series.We want to factor out by the exponentials that act as identity operators.

QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 11Proposition 3.1. The set of elements in H(Z) that act on theta functionsas identity operators is a normal subgroup consisting of the Nth powers ofelements of the form exp(pP + qQ + kE) with k even. The quotient groupis a finite Heisenberg group.Proof. Ifexp(pP + qQ + kE)θ τ jπi= e− N pq−2πi πiqj+ N N k θj+pτequals θj τ for all j, then p is a multiple of N. Setting j = N 2, we concludethat k is a multiple of 2N. Finally, because −2q(j + N 2) must be an multipleof 2N for all j, it follows that q must be a multiple of N. It is not hard tosee that any such element is the Nth power of an element with the thirdentry even. The computation(exp(pP + qQ + kE)) N θ τ jN(N−1)= e−2πijq−2πi 2Npq−N πipq πik+N N N θj+Nτ= e −2πijq−πiNpq+πik θ τ j = e πik θ τ jshows that the Nth powers of elements with third entry even act as theidentity operator. This proves the first part of the statement.Next, define the finite Heisenberg groupby the multiplication ruleH(Z N ) = {(p,q,k), p,q ∈ Z N ,k ∈ Z 2N }(p,q,k)(p ′ ,q ′ ,k ′ ) = (p + p ′ ,q + q ′ ,k + k ′ + pq ′ ).It is straightforward to check that the mapφ((p,q,k)) = (p,q,k + pq)is a group homomorphism whose kernel is the set of elements of the formexp(pP +qQ+kE) with p and q divisible by N and k divisible by 2N. Hencethe desired quotient group is isomorphic to a finite Heisenberg group. □From now on we will denote by H(Z N ) the quotient of the Heisenberg withinteger entries defined in the previous proposition. Whenever we mentionthe finite Heisenberg group, we have in mind this group. We have seenthat indeed, this group is isomorphic to an actual finite Heisenberg group,but we do not work with the latter, because for example the bilinear form((p,q),(p ′ ,q ′ )) → pq ′ which defines multiplication is not invariant undersymplectomorphisms. Thus whenever we work with an exponential of theform exp(pP +qQ+kE), we think of it as the equivalence class of the element(p,q,k) ∈ H(Z).Theorem (Stone-von Neumann) The Schrödinger representation of H(Z N )is the unique irreducible unitary representation of this group with the propertythat exp(kE) acts as e πiN k Id for all k ∈ Z.

QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 133.2. The Schrödinger representation in an abstract setting. We willformulate the Schrödinger representation of quantized exponentials on thespace of theta functions in an abstract setting by rephrasing the standardconstruction based on a lattice and a Lagrangian subspace of the plane (see[18]), which is similar to the third construction in 2.1. The analogue of theLagrangian subspace of the plane is a one-dimensional free Z-submoduleof H 1 (T 2 , Z) generated by a simple closed curve on the torus. Consider ahomeomorphism h 0 of the torus to the boundary of the solid torus S 1 × Dthat maps this curve to a homologically trivial one. Then h 0 defines a grouphomomorphism H 1 (T 2 , Z) → H 1 (S 1 × D, Z) (that is Z ⊕ Z → Z), which wealso denote by h 0 , whose kernel is L.Viewing the torus as the quotient of the plane by the lattice Z + Zτ, thecurve that generates L is the curve of slope p/q, for some coprime integers pand q. We denote this curve by (p,q). To L we associate the maximal abeliansubgroup of the finite Heisenberg group exp(L+Z 2N E), whose elements areequivalence classes of elements of the form exp(npP + nqQ + kE), withn,k ∈ Z. Also, we define the character χ L : exp(L+Z 2N E) → C, χ L (npP +nqQ + kE) = e πiN k . The Hilbert space of the quantization consists of thefunctions φ : H(Z N ) → C satisfying the equivariance conditionφ(uu ′ ) = χ L (u ′ ) −1 φ(u) for all u ′ ∈ exp(L + Z 2N E).The action of the finite Heisenberg group on this space isu 0 φ(u) = φ(u −10 u).In order to show that this gives rise to the same Schrödinger representation,we need to explicate an algebraic complement L ′ of L in H 1 (T 2 , Z).Then L ′ is also spanned by a simple closed curve on the torus, which intersectsthe generating curve of L at one point. These two curves determinea rigid structure on the torus; the requirement that this rigid structure ismapped to the one shown in Figure 1 completely determines the homeomorphismh 0 .To make specific computations, let L and L ′ be spanned by the curves(p,q) respectively (p ′ ,q ′ ) on the torus, with pq ′ − p ′ q = 1, meaning thatL ′ = Z(p ′ ,q ′ ) and L = Z(p,q). The finite Heisenberg group is generated byexp L := exp(Z(p,q)) and exp L ′ := exp(Z(p ′ ,q ′ )). The Hilbert space canbe identified with the functions on exp(Z N (p ′ ,q ′ )), and the action of theHeisenberg group is given byexp(x 0 p ′ ,x 0 q ′ )φ(exp(xp ′ ,xq ′ )) = φ(exp(x 0 p ′ ,x 0 q ′ ) −1 exp(xp ′ ,xq ′ ))= φ(exp((x − x 0 )p ′ ,(x − x 0 )q ′ )),exp(x 0 p,x 0 q)φ(exp(xp ′ ,xq ′ )) = φ(exp(x 0 p,x 0 q) −1 exp(xp ′ ,xq ′ ))= φ(e 2πi(xx 0pq ′ −xx 0 p ′ q) exp(xp ′ ,xq ′ )exp(x 0 p,x 0 q) −1 )= e −2πi N xx 0φ(exp(xp ′ ,xq ′ )exp(x 0 p,x 0 q) −1 ) = e −2πi N xx 0φ(exp(xp ′ ,xq ′ )),

14RĂZVAN GELCA AND ALEJANDRO URIBEandexp(kE)φ(exp(xp ′ ,xq ′ )) = e πiN k φ(exp(xp ′ ,xq ′ ))for all x,x 0 ∈ Z. Identifying further exp(Z N (p ′ ,q ′ )) with Z N we obtain theactionexp(x 0 p ′ ,x 0 q ′ )φ(x) = φ(x − x 0 )exp(x 0 p,x 0 q)φ(x) = e −2πi N xx 0φ(x)exp(kE)φ(x) = e πiN k φ(x),and we recognize the Schrödinger representation. The Schrödinger representationis given here in the real polarization; one should note that a leftshift in the variable is a right shift in the index of the basis elements.This action of the Heisenberg group extends linearly to the group ring ofthe Heisenberg group with coefficients in C. Moreover, because of finiteness,the space of functions on the finite Heisenberg group can be identified, asa vector space, with the group ring of the finite Heisenberg group withcoefficients in C. In conclusion, the Schrödinger representation is the actionof the group ring of the finite Heisenberg group on a quotient of itself. Thuswe have:Proposition 3.3. Given a submodule L of H 1 (T 2 , Z) generated by a simpleclosed curve, there is an isomorphism that intertwines the Schrödingerrepresentation of the finite Heisenberg group on the space of theta functionsand the representation given by the left action of the finite Heisenberg groupon its group ring with coefficients in C taken modulo the equivalence relationu ∼ u ′ whenever u = v + w and u ′ = v + w ′ with w = χ(w ′′ )w ′ w ′′ for somew ′′ ∈ exp(L + Z 2N E).The above result can be rephrased in the framework of the noncommutativetorus. Again, for p,q coprime integers, we let (p,q) be the curve on thetorus with slope p/q. We factor the noncommutative torus ˜C t [U ±1 ,V ±1 ]by this curve on the torus, by setting p 1 (U,V ) ∼ p 2 (U,V ) if and only ifp 1 (U,V ) = q(U,V ) + r 1 (U,V ) and p 2 (U,V ) = q(U,V ) + r 2 (U,V ), such thatr 2 (U,V ) = r 1 (U,V )(t −pq U p V q ) k for some integer number k. We denotethe space obtained this way by ˜C t [U ±1 ,V ±1 ]/ (p,q) . The multiplication inthe noncommutative torus turns ˜C t [U ±1 ,V ±1 ]/ (p,q) into a left ˜C t [U ±1 ,V ±1 ]-module.Proposition 3.4. The Schrödinger representation of the group ring with coefficientsin C of the finite Heisenberg group factored by the relation exp(kE) =e πiN k coincides with the left action of ˜C t [U ±1 ,V ±1 ] on ˜C t [U ±1 ,V ±1 ]/ (p,q) .Thus we can model the Schrödinger representation of the finite Heisenberggroup as the left action of the noncommutative torus on a quotient of itself.3.3. Classical theta functions from a topological perspective. Inorder to stress out the analogy with non-abelian theta functions, we will

QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 15rephrase the theory of theta functions in terms of the skein modules of thelinking number.Let M be an orientable 3-dimensional manifold (which in our considerationswill be either the cylinder over a torus or a solid torus). A link inM is a smooth embedding of a disjoint union of finitely many circles, whilea framed link is a smooth embedding of a disjoint union of finitely manyannuli. We consider framed oriented links. By viewing each link componentas an embedded S 1 × [0,1], an orientation is defined by a smooth field of2-dimensional frames on the annulus with the property that the first vectorof the frame at any point (x,y) is tangent to S 1 × {y}.Consider the free C[t,t −1 ]-module with basis the set of isotopy classes offramed oriented links in M, including the empty link ∅. Factor this moduleby all equalities of the form depicted in Figure 2. In each of these diagrams,the two links are identical except for an embedded ball in M, inside ofwhich they look as shown. This means that in each link we are allowed tosmoothen a crossing provided that we add a coefficient of t or t −1 , and anytrivial link component can be ignored. These are called skein relations. Theskein relations are considered for all possible embeddings of a ball. We callthe module obtained this way the linking skein module of M and denote itby L t (M). If M is a 3-dimensional sphere, then each link L is, as an elementof L t (S 3 ), equivalent to the empty link with the coefficient equal to t raisedto the sum of the linking numbers of ordered pairs of components and thewrithes of the components, hence the name.t 1 Figure 2;tφLet us now restrict ourselves to the case M = T 2 × [0,1]. Then thetopological operation of gluing one cylinder on top of another induces amultiplication on L t (T 2 × [0,1]) which turns L t (T 2 × [0,1]) into an algebra,the linking skein algebra of the cylinder over the torus. We want to explicateits structure.For p and q coprime integers, view the torus as the quotient of the planeby the integer lattice and denote by (p,q) the curve of slope p/q, oriented bythe vector that joins the origin to the point (p,q). Frame this curve so thatthe annulus is parallel to the torus. In general, for p and q not necessarilycoprime, let n be their greatest common divisor, and denote by (p,q) theframed link consisting of n parallel copies of (p/n,q/n). Finally, let (0,0)be the empty link. Aditionally, denote by ˜L t (T 2 × [0,1]) the ring obtained

16RĂZVAN GELCA AND ALEJANDRO URIBEby setting (Np,Nq) = ∅ for every p,q ∈ Z, and t = e πiN . Note that here ∅ isthe multiplicative identity of the ring.Theorem 3.5. The ring L t (T 2 ×[0,1]) is isomorphic to the group ring withcoefficients in C of H(Z), with the isomorphism given byt k (p,q) → exp(pP + qQ + kE).This map descends to an isomorphism between ˜L t (T 2 ×[0,1]) and the algebraof Weyl quantizations of trigonometric polynomials.Proof. Let us consider the free C[t,t −1 ]-module with basis the isotopy classesof links in T 2 × [0,1]. Any element of this module of the form t m L, where Lis framed oriented link, can be transformed using the skein relation into aframed oriented link whose projection onto the torus has no crossings, henceinto a link of the form t k (p,q). Moreover, (p,q) is the homology class of Lin H 1 (T 2 × [0,1]) = H 1 (T 2 ) = Z ⊕ Z in the basis given by the curves (1,0)and (0,1), hence is uniquely determined by the link. In particular (p,q) isthe same for all elements equivalent to L modulo the skein relation.Let us show that k is also uniquely determined. We claim that k equalsm plus the number of positive crossings minus the number of negative crossingsin a projection of the link L onto the torus. Denote this last numberby k(t m L). Then k = k(t m L) if we resolve exactly the crossings in oneparticular diagram of the link. The number k(t m L) is invariant under theReidemeister moves II and III, hence it is an isotopy invariant of the framedoriented link. And, for each ball that is used for the skein relation one canfind the ambient isotopy that maps it so that the crossing appears in theprojection of the link on the torus, namely in the link diagram. If t m′ L ′ isthe skein obtained after applying the skein relation, then k(t m L) = k(t m′ L ′ ),so k(t m L) is invariant under skein relations. Hence k(t m L) is invariant aswe resolve the link to produce one that consists of simple closed curves onthe torus, and we conclude that k(t m L) = k, as desired.By further mapping t k (p,q) to exp(pP +qQ+kE) we obtain a well definedhomomorphism of complex vector spaces from the free C[t,t −1 ]-module withbasis the isotopy classes of links to the group ring of H(Z). Because of theinvariance of t k (p,q) under the skein relation applied to the original link,this homomorphism factors to a homomorphism of complex vector spacesfrom L t (T 2 × [0,1]) to the group ring of H(Z). This latter homomorphismis clearly onto. It is also one-to-one, because if t k 1L 1 and t k 2L 2 are bothmapped to exp(pP + qQ + kE), then they can both be transformed via theskein relation into t k (p,q), and hence are equivalent to each other. It followsthat L t (T 2 × [0,1]) and the group ring of H(Z) are isomorphic as complexvector spaces.It is not hard to check that the multiplication in L t (T 2 × [0,1]) is givenby(p,q) · (p ′ ,q ′ ) = t˛p qp ′ q˛˛˛˛˛(p ′ + p ′ ,q + q ′ ).

QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 17Therefore L t (T 2 ×[0,1]) and the group ring of H(Z) are isomorphic as algebras.The skein algebra ˜L t (T 2 ×[0,1]) was defined so that this isomorphismdescends to an isomorphism between ˜L t (T 2 ×[0,1]) and the noncommutativetorus ˜C t [U ±1 ,V ±1 ].□Remark 3.6. The determinant is the sum of the algebraic intersection numbersof the curves in (p,q,k) with the curves in (p ′ ,q ′ ,k ′ ), so the multiplicationrule of the Heisenberg group is defined using the algebraic intersectionnumber of curves on the torus.Identifying the group ring of the Heisenberg group with integer entrieswith C t [U ±1 ,V ±1 ], we obtain the followingCorollary 3.7. The linking skein algebra of the torus is isomorphic to thering of trigonometric polynomials in the noncommutative torus.Now let us turn to the skein module of the solid torus L t (D 2 × S 1 ). Letα be the curve that is the core of the solid torus, with a certain choice oforientation and framing. It is not hard to see that L t (D 2 × S 1 ) is a freemodule with basis α j , j ∈ Z, where α j stands for |j| parallel copies of α,with the same orientation as α if j > 0 and with opposite orientation ifj < 0, and α 0 = ∅. We consider the module obtained by setting α j+N = α jand t = e πiN , and denote it by ˜L t (D 2 × S 1 ).Let h 0 be a homeomorphism of the torus to the boundary of the solidtorus that maps (1,0) to a framed curve isotopic to α. The operation ofgluing T 2 × [0,1] to the boundary of D 2 × S 1 via h 0 induces a left action ofL t (T 2 ×[0,1]) onto L t (D 2 ×S 1 ). This descends to a left action of ˜L t (T 2 ×[0,1])onto ˜L t (D 2 × S 1 ).Theorem 3.8. There is an isomorphism that intertwines the action of thealgebra of Weyl quantizations of trigonometric polynomials on the space oftheta functions and the representation of ˜L t (T 2 × [0,1]) onto ˜L t (D 2 × S 1 ),and which maps the theta series θ τ j (z) to αj for all j = 0,1,... ,N − 1.Proof. The argument of the previous theorem can be repeated mutatis mutandisto show that module L t (D 2 × S 1 ) is free with basis α j , j ∈ Z. Then,one can observe either that the equivalence relation defined in Proposition3.3 is exactly the condition that the framed links are isotopic inside the solidtorus, or that the left action of L t (T 2 × [0,1]) onto L t (D 2 × S 1 ) is given byRemark 3.9. Note that(p,q) · α j = t −qj α j+p .−qj =∣ p qj 0 ∣is the sum of the linking numbers of the curves in the system (p,q) andthose in the system α k . Therefore the Schrödinger representation is definedin terms of the linking number of curves.□

QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 19Alternatively, in the real polarization, S and T act on L 2 (Z 2N ) by thediscrete Fourier transform (also known as finite Fourier transform)(Sf)(k) = ∑f(j)e −2πi N jkj∈Z 2Nand by the multiplication by the exponential of a quadratic function(Tf)(j) = e 2πiN j2 f(j).The latter should be interpreted as a partial Fourier transform.Like in the case of the quantization of the plane, the quantization of theexponentials, and therefore the Weyl quantization on the torus, satisfies withthe Hermite-Jacobi action the exact Egorov identityop(f ◦ h) = ρ(h)op(f)ρ(h) −1 .To fit with our general scheme, we give two more descriptions of theHermite-Jacobi action. Fix an element h of the mapping class group. LetL 1 be a submodule of H 1 (T 2 , Z 2N ) spanned by a simple closed curve (p 1 ,q 1 )and let L 2 = h(L 1 ).Recall that the space of theta functions can be identified with the quotientof the group ring with complex coefficients of the finite Heisenberg groupby the equivalence relation u ≡ u ′ whenever u = v + w, u ′ = v + w ′ withw = χ L1 (w ′′ )w ′ w ′′ for some w ′′ ∈ exp(L 1 + Z 2N E). The representation ρ is,up to multiplication by a constant, given by the formulaρ(h)(umod(exp(L 1 + Z 2N E))) = ∑ χ L1 (u 1 ) −1 uu 1 mod(exp(L 2 + Z 2N E)),where u ∈ H(Z 2N ) the sum is taken over u 1 ∈ exp(L 1 + Z 2N E)/exp(L 1 ∩L 2 + Z 2N E). This is a general discrete Fourier transform, which is againobtained by a procedure of lifting and averaging, and is another instance ofthe Fourier-Mukai transform [18].We translate this formula into the topological language of skein modules.Consider h 1 and h 2 two homeomorphisms of the torus onto the boundary ofthe solid torus such that h 2 = h◦h 1 . They extend to embeddings of T 2 ×[0,1]into D 2 × S 1 which we denote by h 1 and h 2 as well. The homeomorphismsh 1 and h 2 allow us to identify the space of classical theta functions with theskein module of the solid torus in two different ways.Theorem 3.11. Given a link γ in ˜L t (D 2 ×S 1 ), lift γ in all possible nonequivalentways to ˜L t (T 2 × [0,1]) using h 1 , then map all these liftings to ˜L t (D 2 ×S 1 ) using h 2 and take the average of the images. The map defined this wayis, up to multiplication by a constant, the discrete Fourier transform ρ(h).Proof. First let us note that despite the fact that the link can be liftedin infinitely many nonisotopic ways to the cylinder over the torus, afterthe reduction only finitely many distinct liftings remain. This shows thatthe construction is well defined. The map obtained this way interpolatesbetween the Schrödinger representation defined by h 1 and the one definedby h 2 , so the theorem is a consequence of Schur’s lemma.□

20RĂZVAN GELCA AND ALEJANDRO URIBELike in the case of the metaplectic representation, the Hermite-Jacobirepresentation can be made into a true representation by passing to anextension of the mapping class group of the torus. While a Z 2 -extensionwould suffice, we will consider a Z-extension instead, in order to exhibit thesimilarity with the non-abelian theta functions.Fix a Lagrangian subspace L of H 1 (T 2 ). Define the Z-extension of themapping class group of the torus by the multiplication rule on SL(2, Z) ×Z,(h ′ ,n ′ ) ◦ (h,n) = (h ′ ◦ h,n + n ′ + τ(L,h(L),h ′ ◦ h(L)).where τ is the Maslov index. Standard results in the theory of theta functionsshow that the Hermite-Jacobi action lifts to a representation of thisgroup.We give this group itself an entirely topological description, following [27].Let us consider two elements of the group (h,n) and (h ′ ,n ′ ). Let h 1 ,h 2 ,h 3be the homeomorphisms of T 2 onto the bondary of the solid torus suchthat L = ker(h 1 : H 1 (T 2 ) → H 1 (D 2 × S 1 )), h(L) = ker(h 2 : H 1 (T 2 ) →H 1 (D 2 × S 1 )), and h ′ ◦ h(L) = ker(h 3 : H 1 (T 2 ) → H 1 (D 2 × S 1 )). Considerthe lens spacesM = (D 2 × S 1 ) ∪ h1 (T 2 × [0,1]) ∪ h2 (D 2 × S 1 )M ′ = (D 2 × S 1 ) ∪ h2 (T 2 × [0,1]) ∪ h3 (D 2 × S 1 ).Let also W and W ′ be 4-manifolds such that ∂W = M, sign(W) = n, ∂W ′ =M ′ , sign(W ′ ) = n ′ , where sign stands for the signature of the intersectionform in 2-dimensional cohomology. With this construction we see that theelements of the group can be put in the form (h,sign(W)). We can alsodefine(h ′ ,sign(W ′ )) ◦ (h,sign(W)) = (h ′ ◦ h,sign(W ′ ∪ W)),where W ′ ∪ W is obtained by gluing the first copy of the solid torus in M ′to the second copy of the solid torus in M via the identity map. That thisis the same multiplication as the one defined above using the Maslov indexfollows from Wall’s non-additivity formula for the signature (see [28]):sign(W ′ ∪ W) = sign(W) + sign(W ′ ) + τ(L,h(L),h ′ ◦ h(L)).4. Non-abelian theta functions4.1. Non-abelian theta functions from geometric quantization. LetG be a compact simple Lie group and g its Lie algebra. Let also Σ g be aclosed oriented surface of genus g ≥ 1. The non-abelian theta functions arethe elements of the Hilbert space of a quantization of the space of all fieldson Σ g with symmetry group G modulo gauge transformations. In genus oneand for G = U(1) these are the same as the classical theta functions. Let usexplain the construction in detail.Consider the moduli space of g-connections on Σ g , which is the quotient ofthe affine space of all g-connections on Σ g (or rather on the trivial principalG-bundle P on Σ g ) by the group G of gauge transformations A → φ −1 Aφ +

QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 21φ −1 dφ with φ : Σ g → G a smooth function. The space of all connectionshas a symplectic 2-form given by∫ω(A,B) = − tr (A ∧ B),Σ gwhere A and B are connection forms in its tangent space. According to[2], this induces a symplectic form, denoted also by ω, on the moduli space,which further defines a Poisson bracket. By symplectic reduction, the problemof quantizing this moduli space in the direction of the Poisson bracket isreduced to the problem of quantizing the moduli space of flat g-connectionsM g = {A |A : flat g − connection}/G.This space is the same as the moduli space of semi-stable G-bundles on Σ g ,and also the character variety of G-representations of the fundamental groupof Σ g . This is the space of interest to us. One should note that it is an affinealgebraic set over the real numbers.Each curve γ on the surface and each irreducible representation V n ofSU(2) define a classical observable on this spaceW γ,n : A → tr V nhol γ (A),called Wilson line, by taking the trace of the holonomy of the connectionin the n-dimensional irreducible representation of SU(2). Wilson lines areregular functions on the moduli space. When n = 2, we simply denote W γ,2by W γ . The W γ ’s span the algebra F(M g ) of regular functions on M g .The form ω induces a Poisson bracket, which in the case of the gaugegroup SU(2) was found by Goldman [10] to be{W α ,W β } = 1 2∑x∈α∩βsgn(x)(W αβ−1x− W αβx )where αβ x and αβx−1 are computed as elements of the fundamental groupwith base point x (see Figure 3), and sgn(x) is the signature of the crossingx; it is positive if the frame given by the tangent vectors to α respectivelyβ is positively oriented (with respect to the orientation of Σ g ), and negativeotherwise.αβαβFigure 3The moduli space M g , or rather the smooth part of it, can be quantizedin the direction of Goldman’s Poisson bracket as follows. First, set Planck’sconstant = 1 N, where N is a positive integer.αβ−1

22RĂZVAN GELCA AND ALEJANDRO URIBEThe Hilbert space can be obtained using the method of geometric quantizationas the space of sections of a line bundle over M g . The generalprocedure is to obtain the line bundle as the tensor product of a line bundleL with curvature Nω and a half-density [23]. In our case the half-densityis the square root of the canonical line bundle. Because the moduli spacehas a natural complex structure, it is customary to work in the holomorphicpolarization, in which case the Hilbert space consists of the holomorphicsections of the line bundle.Let us briefly recall how each complex structure on the surface inducesa complex structure on the moduli space. The tangent space to M g ata nonsingular point A is the first cohomology group H 1 A (Σ g,ad P) of thecomplex of g-valued formsΩ 0 (Σ g ,ad P) dA → Ω 1 (Σ g ,ad P) dA → Ω 2 (Σ g ,ad P).Here P denotes the trivial principal G-bundle over Σ g . Each complex structureon Σ g induces a Hodge ∗-operator on the space of connections on Σ g ,hence a ∗-operator on HA 1(Σ g,ad P). The complex structure on M g isI : HA 1(Σ g,ad P) → HA 1(Σ g,ad P), IB = − ∗ B. For more details we referthe reader to [12]. This complex structure turns the smooth part of M g intoa complex manifold. It is important to point out that the complex structureis compatible with the symplectic form ω, in the sense that ω(B,IB) ≥ 0for all B.As said, the Hilbert space consists of the holomorphic sections of the linebundle of the quantization. These are the non-abelian theta functions. Thereason for the name is that in the simplest case, when G = U(1) and thesurface has genus one, the Hilbert space has dimension N and its elementsare the classical theta functions. To specify a basis of this vector space,i.e. to obtain the analogues of the theta series, one has to consider a rigidstructure on the original surface and use the relationship of the Hilbert spaceto conformal field theory established in [29].The analogue of the group ring of the finite Heisenberg group is the algebraof operators that are the quantizations of Wilson lines. They arise in thetheory of the Jones polynomial [11] as outlined by Witten [29], being definedheuristically in the framework of quantum field theory. They are integraloperators with kernel∫< A 1 |op(W γ,n )|A 2 >= e iNL(A) W γ,n (A)DA,M A1 A 2where A 1 ,A 2 are conjugacy classes of flat connections on Σ g modulo thegauge group, A is a conjugacy class under the action of the gauge group onΣ g × [0,1] such that A Σg×{0} = A 1 and A Σg×{1} = A 2 , andL(A) = 14π∫Σ g×[0,1]tr(A ∧ dA + 2 )3 A ∧ A ∧ Ais the Chern-Simons Lagrangian. As seen above, the operator quantizinga Wilson line is defined by Feynman path integrals, which does not have

QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 23a rigorous mathematical formulation. It is thought as an average of theWilson line computed over all connections that interpolate between A 1 andA 2 .It is the framework of Chern-Simons theory that motivates the skein theoreticapproach to classical theta functions outlined in Section 3.2. Let usexplain this in more detail.The moduli space of flat U(1)-connections on the torus is itself a torus,the Jacobian. The moduli space is canonically homeomorphic to the originaltorus in the following way. Let p 0 be the point on the original torus that isthe image of the origin under the quotient map C → C/Z + Zτ. Thinkingof the Jacobian as the moduli space of holomorphic topologically trivial linebundles, the homeomorphism in question maps a point p on the originaltorus to the line bundle O(p − p 0 ). Or, when viewing the Jacobian as acharacter variety, this homeomorphism maps the point (x,y) on the originaltorus to the U(1)-representation of the fundamental group that maps (1,0)to e 2πix and (0,1) to e 2πiy .The tangent space to this torus is two dimensional; viewing it as givenby cohomology classes of u(1)-valued 1-forms, it has a basis consisting ofdx and dy, which can be identified with the tangent vectors to the torus∂∂x and ∂ ∂y. Under this identification, we see that a complex structure onthe original torus induces exactly the same complex structure on the modulispace.A rigid structure on the original torus gives rise to a decomposition of thetangent space to the moduli space into a sum of two Lagrangian subspaces.To fit in the framework of section 3.2, we can exponentiate these Lagrangiansubspaces to obtain a rigid structure on the moduli space, which happensto be the same as the rigid structure on the original torus.The Chern-Simons line bundle over the moduli space is just the one fromSection 3.1, so in this case the theta functions are the classical theta functions.The Wilson lines can be quantized either by using one of the classicalmethods for quantizing the torus, or they can be quantized using the Feynmanpath integrals as above. The Feyman path integral approach allows thelocalization of the computations to small balls, in which a single crossingshows up. Witten [29] has explained that in each such ball skein relationshold, in this case the skein relations from Figure 2, which compute the linkingnumber. As such the path integral quantization gives rise to the skeintheoretic model.On the other hand, Witten’s quantization is symmetric under the actionof the mapping class group of the torus, a property shared by Weyl quantization.And indeed, we have seen in section 3.2 that Weyl quantizationand the skein theoretic quantization are the same. For the group U(1), thecase of higher genus surfaces can be done in the same framework, and willbe explained in future work.4.2. Non-abelian theta functions from quantum groups. More thanjust the quantization of moduli spaces of connections on surfaces, Witten

24RĂZVAN GELCA AND ALEJANDRO URIBEdescribed a topological quantum field theory which associates to each surfacea finite dimensional vector space, and to each 3-dimensional cobordism alinear operator between vector spaces. This was all done using Feynmanpath integrals.For the gauge group SU(2), Reshetikhin and Turaev [20] constructedrigorously, by using quantum groups, a topological quantum field theorythat fulfills Witten’s predictions. Within the data of this theory, there is foreach surface a projective finite-dimensional representation of its mappingclass group, which is called the Reshetikhin-Turaev representation.Like Witten’s construction, the Reshetikhin-Turaev theory gives rise toa quantization of the moduli space of flat SU(2)-connections on a closedsurface. The quantum group quantization has the advantage over moreclassical quantization models (e.g. geometric quantization) that it does notdepend on any additional structure, such as a polarization. We describe itbelow for genus g > 1, the case of the torus being discussed in detail inSection 5.We need to restrict Planck’s constant to be the reciprocal of an evenpositive integer = 1 N = 1 2r, and furthermore r > 1. We also set t = eiπand for an integer n let [n] = t2n −t −2nt 2 −t −2= sin nπr /sin π r, which is called thequantized integer.The quantum group associated to SU(2) is U (sl(2, C)), obtained by passingto the complexification SL(2, C) of this group, taking the universal envelopingalgebra of its Lie algebra, and the deforming it with respect to. This quantum group has finitely many irreducible representations amongwhich we distinguish a certain family V 1 ,V 2 ,... ,V r−1 , with the superscriptdenoting the dimension (for more details see [20] or [14]). For further use, weextend the definition of V j to all integer indices j by V j+2r = V j , V 0 = 0,and V r+j = −V r−j (the minus sign does not mean dual, it just means thatwe take the negative when performing computations with this representation).A Clebsch-Gordan theorem holdsV m ⊗ V n = ⊕ pV p ⊕ B,where p runs among all indices that satisfy |m − n| + 1 ≤ p ≤ min(m + n +1,r − 2 − m − n) and B is a representation that is ignored because it has noeffect on computations.The definition of the quantization uses ribbon graphs and framed links embeddedin 3-dimensional manifolds. A ribbon graph consists of the embeddingsin the 3-dimensional manifold of finitely many connected components,each of which is homeomorphic to either an annulus or to an ǫ-neighborhoodin the plane of a planar trivalent graph for a small ǫ > 0. When embeddingthe ribbon graph in a 3- dimensional manifold, the ribbons keep track ofthe twistings of edges. A framed link is a particular case of a ribbon graph.The link components and the edges of ribbon graphs should be oriented. Anedge should be viewed as a product of two intervals, and the orientation is

QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 25given by a smooth field of 2-dimensional frames whose first vector is alwaystangent to the first interval.Unless otherwise specified, all the ribbon graphs and framed links used beloware taken with the “blackboard framing”, meaning that the ǫ-neighborhoodis in the plane of the paper. To simplify the exposition, we will callthem graphs, the framing being self-understood. From the frames that definethe orientation we draw only the first vector, being understood that thesecond vector is chosen so that the orientation agrees with the orientationof the plane of the paper in the usual convention of the x- and y-axes. Wewill be allowed to change locally the orientation of an edge, and we place acoupon (blank box) on the edge to separate the two orientations.Figure 4With this data at hand, let us quantize M g . The Hilbert space H r (Σ g ) isdefined by specifying a basis, the analogue of the theta series. For that weneed a rigid structure on the surface, which is a decomposition of the surfaceinto pairs of pants, together with seems that keep track of the twistings. Mapthe surface to the boundary of a handlebody such that the decompositioncircles bound disks in the handlebody. These disks cut the handlebodyinto balls. Consider the trivalent graph that is the core of the handlebody,whose vertices are these balls and whose edges correspond to the disks. Theframing of the edges should be parallel to the seams (more precisely, to theregion of the surface that lies between the seems). Pick any orientation ofthis graph, this orientation is relevant only in that it decides the sign of thebasis element.The vectors that form an orthonormal basis of H r (Σ g ) consist of all thepossible colorings of this framed oriented trivalent graph by V j ’s such thatat each vertex the indices satisfy the double inequality from the Clebsch-Gordan theorem (note that the double inequality is invariant under cyclicpermutations of m,n,p). For genus 3, and for the rigid structure in Figure 4a basis element is described in Figure 5. The inner product 〈·, ·〉 is definedso that these basis elements are orthonormal.This is a nice combinatorial description of the non-abelian theta functionsfor the Lie group SU(2), which unfortunately obscures their geometricproperties and the origin of the name.The matrix of the operator op(W γ,n ) associated to the Wilson lineW γ,n : A → tr V nhol γ (A)is computed as follows. Place the surface Σ g in standard position in the3-dimensional sphere so that it bounds a genus g handlebody on each side.Draw the curve γ on the surface, give it the framing parallel to the surface,

26RĂZVAN GELCA AND ALEJANDRO URIBEVkVlVmVpVqVnFigure 5then color it by the representation V n of U (sl(2, C)). Add two basis elementse p and e q , viewed as colorings by irreducible representation of thecores of the interior, respectively exterior handlebodies (see Figure 6).VbiV VjV nVkVcaVFigure 6Forget the surface to obtain an oriented tangled ribbon graph in S 3 whoseedges are decorated by irreducible representations of U (sl(2, C) (Figure 7).By projecting this graph on a plane while keeping track of the crossings(which is over, which is under), one can think of the edges as the trajectoriesof 1-dimensional quantum particles with the y-axis as the time direction.The diagram then yields a unitary evolution operator from the Hilbert spaceof a system without particles to the Hilbert space of a system without particles,hence a linear map from C to C. This linear map is the multiplicationby a complex number. The Reshetikhin-Turaev theory [20] gives a way ofcomputing this number, which is independent of the particular projectionand is called the Reshetikhin-Turaev invariant of the framed graph.tbVnVVaViV jVkV cxFigure 7In short, the Reshetikhin-Turaev invariant is computed as follows. Theribbon graph should be mapped by an isotopy to one which, when projected,

QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 27can be cut by finitely many horizontal lines into slices, each of which containingone of the phenomena from Figure 8 and some vertical strands. Toeach of the horizontal lines that slice the graph one associates the tensorproduct of the representations that color the crossing strands, when pointingupwards, or their duals, when pointing downwards. To the phenomenafrom Figure 9 one associates, in order, the operators which are: the quasitriangularR-matrix R : V m ⊗ V n → V n ⊗ V m , its inverse R −1 , the projectionof the tensor product of V m ⊗V n onto V p , the inclusion of V p into V m ⊗V n ,the contraction V n ⊗ V n∗ , its dual, the isomorphism D : V n → V n∗ , and itsdual. One then composes these operators from the bottom of the diagramto the top, to obtain the desired linear map from C to C. The coupons, i.e.the maps D, might be required in order to change the orientations of thethree edges that meet at a vertex, to make them look as depicted.VpVmVnV nVm n m n mVVVV VnVpFigure 9VnVnVnReturning to the quantization, the Reshetikhin-Turaev invariant of thegraph is equal to [op(W γ,n )e p ,e q ], where [·, ·] is the nondegenerate bilinearpairing on H r (Σ g ) defined in Figure 10. One can think of this as beingthe p,q-entry of the matrix of the operator, although this is not quite truebecause the bilinear pairing is not the inner product. But because the pairingis nondegenerate, the above formula completely determines the operatorassociated to the Wilson line. It can be shown that the quantization definedthis way is in the direction of Goldman’s Poisson bracket [1].bV[ep,e q]=VaViV jVkV cFigure 104.3. Non-abelian theta functions from skein modules. We will rephrasethe construction from Section 4.2 in the language of skein modules.The goal is to describe the quantum group quantization as an action of askein algebra on a quotient of itself in the same way as the Schrödinger

28RĂZVAN GELCA AND ALEJANDRO URIBErepresentation of the finite Heisenberg group has been written in 3.2 as theaction of the group ring of this group on a quotient of itself.We first replace the oriented framed graphs and links colored by irreduciblerepresentations of U (sl(2, C)) by formal sums of oriented framedlinks colored by the 2-dimensional irreducible representation. To this endwe use the Clebsch-Gordan theorem for U (sl(2, C)) to writen/2∑( ) n − jV n = (−1) j (V 2 ) n−2j = S n−1 (V 2 ), for n = 1,2, · · · ,r − 1jj=0where S n (x) is, as seen, the Chebyshev polynomial of second kind definedrecursively by S 0 (x) = 1, S 1 (x) = x, S n+1 (x) = xS n (x) − S n−1 (x). Thenwe replace a framed simple closed curve γ on the surface colored by V n byS n (γ) with components colored by V 2 . Here the sum is formal, while a kthpower means k parallel copies of the curve.For an oriented framed graph we first replace the edges using the recursiverelation described in Figure 11 (recall from the previous section that thecoupons stand for the isomorphisms between a representation and its dual).After doing this, at each vertex colored by V m ,V n ,V p (that stands for theprojection of V m ⊗V n onto V p ) there are p strands entering from above andtwo groups of m respectively n strands exiting below. Connect these strandsas shown in Figure 12, where x+y = p, x+z = m, y+z = n. Do the similarthing for the vertices corresponding to the inclusion of V p into V m ⊗V n . Thelink obtained this way has and even number of coupons on each component.Cancel the coupons on each link component in pairs, adding a factor of −1each time the two coupons are separated by an odd number of maxima onthe link component (for those familiar with the subject, note that the linkcomponent is colored by an even-dimensional representation).n n−1 2V V VVn−1V2V3==[n−2]V[n−1]VVV 2 V 2 V V1[2]n−2n−122 2Figure 11V2V2With these transformations, the computation of the matrices of operatorsof the quantum group quantization reduces to computations with links whosecomponents are colored by V 2 . Theorem 4.3 in [14] allows us to perform

QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 29px yzm nFigure 12this computation using skein relations. Specifically, if three framed linksL,H,V in S 3 colored by V 2 coincide except in a ball where they look likein Figure 13, then their Reshetikhin-Turaev invariants, denoted by J L ,J H ,and J V satisfyJ L = tJ H + t −1 J Vif the two crossing strands come from different components, andJ L = ǫ(tJ H − t −1 J V )if the crossing strands come from the same component. Here ǫ is the signof the crossing.Additionally if a link component bounds a disk inside a balldisjoint from the rest of the link, then it is replaced by a factor of t 2 + t −2 .L H VFigure 13Using these skein relations we introduce a different type of skein module.For this, let M be an orientable 3-dimensional manifold. Consider the freeC[t,t −1 ]-module with basis the isotopy classes of framed oriented links inM, then factor it by the skein relationsL = tH + t −1 Vif the two crossing strands come from different components, andL = ǫ(tH − t −1 V )if the crossing strands come from the same component (with the same conventionfor ǫ as above) where the links L,H,V are the same except in anembedded ball in M and inside this ball they look like in Figure 13. Also,impose that any trivial link component that lies inside a ball disjoint fromthe rest of the link is replaced by a factor of t 2 + t −2 . We call the moduleobtained this way the Reshetikhin-Turaev skein module and denote it byRT t (M).Like before, if M = Σ g × [0,1] then the homeomorphismΣ g × [0,1] ∪ Σg Σ g × [0,1] = Σ g × [0,1]

30RĂZVAN GELCA AND ALEJANDRO URIBEinduces a multiplicative structure on the skein module of the cylinder overthe genus g surface. Also the operation of gluing Σ g ×[0,1] to the boundaryof a genus g handlebody H g induces an RT t (Σ g × [0,1])-module structureon the skein module of the handlebody. Moreover, by gluing H g with theempty skein inside to Σ g × [0,1] we see that RT t (H g ) is the quotient ofRT t (Σ g ×[0,1]) obtained by identifying skeins in Σ g ×[0,1]) that are isotopicin H g .We further define the reduced Reshetikhin-Turaev skein module ˜RT t (M)and letting all skeins that contain an S r−1 (γ) for somecurve γ be equal to zero. Note that S r−1 (γ) = 0 stands for the fact thatS r−1 (V 2 ) is a representation that has no effect on computations. By theabove considerations the reduced skein module of the genus g handlebodyis isomorphic to the Hilbert space H r (Σ g ) and the skein algebra of thecylinder over a surface is isomorphic to the algebra of quantized Wilsonlines. Moreover, the action of ˜RT t (Σ g × [0,1]) on ˜RT t (H g ) coincides withthe action of the algebra of operators on the vector space. Hence we haveby setting t = e iπ2rProposition 4.1. The quantum group quantization of the moduli space offlat SU(2)-connections on a surface can be represented as the left multiplicationof the reduced Reshetikhin-Turaev skein algebra of that surface on thereduced Reshetikhin-Turaev skein module of the handlebody.Remark 4.2. The action of the algebra of quantized Wilson lines on theHilbert space in the quantum group quantization is a representation of thisalgebra on a quotient of itself. The skein modules RT t (Σ g × [0,1]) and˜RT t (Σ g × [0,1]) are the analogues, for the gauge group SU(2), of the grouprings of H(Z) and H(Z 2N ).The translation into the language of skein modules allows an easy proof ofthe irreducibility of the representation, which is required by the postulatesof quantum mechanics.Theorem 4.3. For each r > 1, the quantum group quantization of themoduli space of flat SU(2)-connections on the torus is an irreducible representation.Proof. We discuss the case g > 1, since for the case of the torus we have thestronger result of Theorem 5.3. To this end we will show that every nonzerovector in the Hilbert space H r (Σ g ) is a cyclic vector for the representation.The smooth part of M g has real dimension 6g − 6. This smooth partis a completely integrable manifold in the Liouville sense. Indeed, theWilson lines W αi , where α i , i = 1,2,... ,3g − 3 are the curves in Figure14, form a maximal set of Poisson commuting functions (meaning that{W αi ,W αj } = 0). The quantum group quantization of the moduli space offlat SU(2)-connections is thus a quantum integrable system, with the operatorsop(W α1 ), op(W α2 ), ..., op(W α3g−3 ) satisfying the integrability condition.The spectral decomposition of the commuting (3g − 3)-tuple of self-adjointoperators(op(W α1 ),op(W α2 ),...,op(W α3g−3 ))

QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 31has only 1-dimensional eigenspaces: Ce 1 , Ce 2 , Ce 3 ,..., where the e i ’s are thebasis elements described in the previous section.α1 α2α3α4Figure 14α3g−3Given a knot in the handlebody, we can talk about the linking numberof this knot with one of the curves α i ; just embed the handlebody in S 3 instandard position. We agree to take this with a positive sign. The linkingnumber of a link L in H g with the curve α i is the sum of the linking numbersof the components. Associate to L the number d(L) obtained by summingthese for all i = 1,2,... ,3g − 3. Finally, for a skein σ = ∑ c j L j , where L jare links and c j ∈ C, let d(σ) = max j d(L j ). We claim that for each skeinσ which is not a multiple of the empty link, there is a skein σ ′ such thatd(σ ′ ) < d(σ) and σ ′ is in the cyclic representation generated by σ.To this end write σ in the basis as σ = ∑ c j e j . Because the spectraldecomposition of the system(op(W α1 ),op(W α2 ),... ,op(W α3g−3 ))has only 1-dimensional eigenspaces, each of the e j that has a nonzero coefficientis in the cyclic representation generated by σ. Note that for eachsuch e j , d(e j ) ≤ d(σ). If one of these inequalities is sharp, then the claimis proved. If not, we show that if e p is not the empty link (i.e. the trivalentgraph with all edges colored by V 1 ), then in the cyclic representationgenerated by e p there is a skein σ ′ with d(σ ′ ) < d(e p ).After deleting all edges of e p colored by the trival representation V 1 ,the not necessarily connected graph obtained this way has an edge whoseendpoints coincide, which is colored by some nontrivial representation V n .Let β be a framed simple closed curve on σ g that is isotopic to this edge andchoose an α i that intersects β as shown in Figure 15.βV nαiFigure 15The recursive formula in Figure 11 shows that op(W β )e p is the sum oftwo skeins, σ ′ that has the edge linking α i colored by V n−1 and σ ′′ thathas the edge linking α i colored by V n+1 . It is a standard fact that σ ′ is aneigenvector of op(W αi ) with eigenvalue [2n − 2], while, if it is nonzero, thenσ ′′ is an eigenvector of op(W αi ) with eigenvalue [2n + 2]. We can therefore

32RĂZVAN GELCA AND ALEJANDRO URIBEconclude that σ ′ is in the cyclic representation generated by e p , and thereforein the cyclic representation generated by σ.Repeating, we eventually descend to the empty link. It remains to showthat the empty link is cyclic. But this is obviously true, since each basiselement can be represented as the image of a collection of nonintersectingsimple closed curves on the boundary. This completes the proof. □4.4. The Reshetikhin-Turaev representation of the mapping classgroup as a Fourier transform for non-abelian theta functions. TheReshetikhin-Turaev projective representation ρ of the mapping class groupof Σ g is constructed as follows. By the Lickorish twist theorem [15], everyelement h of the mapping class group is a product of twists. Each twist canbe represented as surgery with integer coefficients along a link in Σ g ×[0,1].Hence h itself can be represented as surgery with integer coefficients inΣ g × [0,1], so it can be represented by a framed link, the framings of whosecomponents are equal to the surgery coefficients.In order to find ρ(h) it suffices to compute [ρ(h)e p ,e q ] for all basis elementse p and e q , where [·, ·] is the nondegenerate bilinear pairing discussedin the previous section. To do this, place Σ g × [0,1] with the surgery link instandard position in S 3 , then place e p in one handlebody and e q in the other.∑ r−1j=1 [j]V j where [j] isColor each link component of the surgery link by 1 Xthe quantized integer and X is the square root of ∑ r−1j=1 [j]2 . Then computethe Reshetikhin-Turaev invariant of the framed graph obtained by projectingeverything onto a plane to obtain the desired value.On the other hand, there is an action of the mapping class group of thesurface Σ g on the ring F(M g ) of regular functions on the moduli space. Anelement h of the mapping class group acts byh · f(A) = f(h −1∗ A)where h ∗ A denotes the image of the connection A through h. In particularthe Wilson line of a curve γ is mapped to the Wilson line of the curve h(γ).Each elemenent of the mapping class group preserves the Atiyah-Bott symplecticform, so it induces a symplectomorphism of M g . The Reshetikhin-Turaev representation gives a method for quantizing these symplectomorphisms.The action of the mapping class group on F(M g ) induces an action onthe quantum observables byh · op(f(A)) = op(f(h −1∗ A)).In order for the Reshitikhin-Turaev topological quantum field theory to beconsistent, this action must be compatible with the Reshetikhin-Turaev representationof the mapping class group. The compatibility condition isop(W h(γ),n ) = ρ(h)op(W γ,n )ρ(h) −1 ,which holds true for all elements h of the mapping class group, curves γ onthe surface and positive integers n. We recognize the Egorov identity, whichis satisfied exactly.

34RĂZVAN GELCA AND ALEJANDRO URIBEThen L is a Lagrangian subspace of H 1 (Σ g , R) with respect to the intersectionform. The composition of extended homeomorphisms is defined by(h ′ ,n ′ ) ◦ (h,n) = (h ′ ◦ h,n + n ′ + τ(L,h(L),h ′ ◦ h(L)),where τ is the Maslov index with respect to the intersection pairing. Byusing Wall’s formula for the non-additivity of the signature, this can betranslated into a completely topological language.Let h 1 ,h 2 ,h 3 be the homeomorphisms of Σ g onto the bondaries of the3-manifolds N 1 ,N 2 , respectively N 3 such that L, h(L), respectively h ′ ◦h(L)are the kernels of the morphisms induced in first homology. Consider theclosed 3-manifoldsM = N 1 ∪ h1 (Σ g × [0,1]) ∪ h2 N 2 and M ′ = N 2 ∪ h2 (Σ g × [0,1]) ∪ h3 N 3 .Let also W and W ′ be 4-manifolds such that ∂W = M, sign(W) = n,∂W ′ = M ′ , sign(W ′ ) = n ′ , where sign is the signature of the intersectionform in dimension 2. The elements of the Z-extension of the mapping classgroup can be put in the form (h,sign(W)). Because of Wall’s formula forthe non-addititivity of the signature, the multiplication rule is given by(h ′ ,sign(W ′ )) ◦ (h,sign(W)) = (h ′ ◦ h,sign(W ′ ∪ N2 W)).5. Non-abelian theta functions on the torus5.1. The Weyl quantization of the moduli space of flat SU(2)-connectionson the torus. The moduli space M 1 of flat SU(2)-connectionson the torus is the same as the character variety of SU(2)-representations ofthe fundamental group of the torus. It is, therefore, parametrized by the setof pairs of matrices (A,B) ∈ SU(2) × SU(2) satisfying AB = BA, moduloconjugation. Since commuting matrices can be diagonalized simultaneously,and the two diagonal entries can be permuted simultaneously, the modulispace can be identified with the quotient of the torus {(e 2πix ,e 2πiy ), x,y ∈R} by the “antipodal” map x → −y, y → −y. This space is called the pillowcase.The pillow case is the quotient of R 2 by horizontal and vertical integertranslations and by the symmetry σ with respect to the origin. Alternatively,it is the algebraic set in R 3 defined by the equation X 2 +Y 2 +Z 2 −XY Z =4, shown in Figure 16. Except for four singularities, M 1 is a symplecticmanifold, with symplectic form ω = 2πidx ∧ dy. The associated Poissonbracket is{f,g} = 1 ( ∂f ∂g2πi ∂x ∂y − ∂f∂y)∂g.∂xLet us quantize this space in the holomorphic polarization. Like in thecase of the Reshetikhin-Turaev theory, Planck’s constant will be taken thereciprocal of an even integer = 1 N = 1 2r .The tangent space at an arbitrary point on the pillow case is spannedby the vectors ∂∂x and ∂∂y. In the formalism of Section 4.1, these vectors

QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 3521z 0-1-2-2-10y122-20 xFigure 16are identified respectively with the cohomology classes of the su(2)-valued1-forms( ) ( )1 01 0dx and dy.0 −10 −1It follows that a complex structure on the original torus induces exactly thesame complex structure on the pillow case. So we can think of the pillowcase as the quotient of the complex plane by the lattice Z + Zτ and thesymmetry σ with respect to the origin. As before, we denote by (x,y) thecoordinates in the basis (1,τ) and by z = x + τy the complex variable.As seen in [9], a holomorphic line bundle L 1 with curvature 2πiNdx∧dy =4πirdx ∧ dy on the pillow case is defined by the cocycle Λ 1 : R 2 × Z 2 → C ∗ ,Λ 1 ((x,y),(m,n)) = e −πNiaΛ 1 ((x,y),σ) = 1.(τn 2−2n(x+τy)) = e −πNi (τn 2 −2nz)aSince the form dz is not globally defined on the pillow case, we cannotuse the trivial line bundle as the half-density. The obstruction for dz to beglobally defined given by the symmetry with respect to the origin σ can beincorporated in a line bundle L 2 defined by the cocycle Λ 2 : R 2 × Z 2 → C ∗ ,Λ 2 ((x,y),(m,n)) = 1Λ 2 ((x,y),σ) = −1.The line bundle of the quantization is then L 1 ⊗ L 2 , defined by the cocycleΛ 1 Λ 2 . The Hilbert space of non-abelian theta functions consists of theholomorphic sections of this line bundle. To specify a basis of this vectorspace, we need a rigid structure on the torus. This complex torus with rigidstructure is again specified by a number τ in the upper half-plane. The

36RĂZVAN GELCA AND ALEJANDRO URIBEorthonormal basis of the Hilbert space isζ τ j (z) = 4√ r(θ τ j (z) − θ τ −j(z)), j = 1,2,... ,r − 1,where θ τ j (z) are the theta series from Section 3.1. The definition of ζτ j (z)should be extended to all indices by the conditions ζ τ j+2r (z) = ζτ j (z), ζτ 0 (z) =0, and ζ τ r−j (z) = −ζτ r+j (z).Let us turn our attention to the Weyl quantization of Wilson lines. If pand q are coprime integers, then the Wilson line for the curve with slopep/q on the torus and for the 2-dimensional irreducible representation is justW p/q (x,y) = f(x,y) =sin4π(px + qy)= 2cos 2π(px + qy),sin2π(px + qy)when viewing the pillow case as a quotient of the plane. This is becausethe character of the 2-dimensional irreducible representation is sin2x/sin x.Moreover, if p and q are arbitrary integers, the function f(x,y) = 2cos 2π(px+qy) is a linear combination of Wilson lines. Indeed, if n = gcd(p,q), p = np ′ ,q = nq ′ , then2cos 2π(px + qy) = sin[2π(n + 1)(p′ x + q ′ y)]sin2π(p ′ x + q ′ y)− sin[2π(n − 1)(p′ x + q ′ y)]sin 2π(p ′ x + q ′ ,y)so 2cos 2π(px+qy) = W γ,n+1 −W γ,n−1 where γ is the curve of slope p ′ /q ′ onthe torus. This formula also shows that Wilson lines are linear combinationsof cosines, so it suffices to understand the quantization of the cosines.Because the pillow case is the quotient of the torus by the antipodalmap, the Weyl quantization on the pillow case can be obtained by doingequivariant Weyl quantiztion on the torus. Moreover, since2cos 2π(px + qy) = e 2πi(px+qy) + e −2πi(px+qy) ,we find that the Weyl quantization of the Wilson lines can be obtained byextending linearly the Schrödinger representation to the group ring of thefinite torus, restricting it to the subring invariant under the map exp P →exp(−P) and exp Q → exp(−Q), and then letting it act on odd theta functions.For the quantization of the cosines ( we obtain the formula)op(2cos 2π(px + qy))ζj τ πi(z) = e− 2r pq e πir qj ζj−p τ πi(z) + e− r qj ζj+p τ (z) .In particular the ζj τ ’s are the eigenvectors of op(2cos 2πy), correspondingto the holonomy along the curve which cuts the torus into an annulus. Thisshows that they are correctly identified as the analogues of the theta series.5.2. The quantum group quantization of the moduli space of flatSU(2)-connections on the torus. The quantum group quantization ofM 1 is the particular case of the construction performed in Section 4.2. Setagain = 1 2r. The Hilbert space has an orthonormal basis consisting of ther − 1 vectors V 1 (α),V 2 (α),... ,V r−1 (α), obtained by coloring the core α ofthe torus by the irreducible representations V 1 ,V 2 ,... ,V r−1 of U (sl(2, C))(see Figure 17). These are the quantum group analogues of the ζj τ’s.

QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 37jVFigure 17The operator associated to the function f(x,y) = 2cos 2π(px + qy) iscomputed in a similar fashion as that for higher genus surfaces described inSection 4.2. The required bilinear form on the Hilbert space is now known tobe [V j (α),V k (α)] = [jk], j,k = 1,2,... ,r−1. The value of [op(2cos 2π(px+qy))V j (α),V k (α)] is equal to the Reshetikhin-Turaev invariant of the threecomponentlink obtained by placing the curve of slope p/q on the torusembedded in the standard position in the 3-dimensional space, colored byV n+1 − V n−1 where n is the greatest common divisor of p and q, the coreof the solid torus that lies on one side of the torus colored by V j , and thecore of the solid torus that lies on the other side colored by V k . It has beenshown in [9] that this operator acts on the Hilbert space byop(2cos 2π(px + qy))V j (α) = e − πi2r pq ( e πir qj V j−p (α) + e − πir qj V j+p (α)with the conventions made before if the indices of the irreducible representationsleave the range 1,2,... ,r−1. This is of course the formula from theprevious section, and we haveTheorem 5.1. [9] The Weyl quantization and the quantum group quantizationof the moduli space of flat SU(2)-connections on the torus are unitaryequivalent.In the next section we will show that this theorem is a consequence of aStone–von Neumann theorem. It is important to note thatC(p,0)V k (α) = V k−p (α) + V k+p (α), C(0,q)V k (α) = 2cos qkπr V k (α).These are the analogues of translation and multiplication by a character inthe case of the Heisenberg group.Let us now place ourselves in the framework of Section 4.3. The Reshetikhin-Turaev skein algebra of the cylinder over the torus is isomorphic to thesubalgebra of the noncommutative torusC[U,V,U −1 ,V −1 ]/ UV =e 2πi V Uinvariant under the map U → U −1 , V → V −1 . The isomorphism is definedin the following way. For (p,q) ∈ Z 2 , let n = gcd(p,q), also letT n (x) be the Chebyshev polynomial of the first kind defined by T 0 (x) = 2,T 1 (x) = x, and T n+1 (x) = xT n (x) − T n−1 (x), for n ≥ 1. We consider theframed oriented link (p/n,q/n) on the torus with the convention from Section3.2. The isomorphism maps the skein T n ((p/n,q/n)) ∈ RT t (T 2 × [0,1])to e −πipq (U p V q + U −p V −q ). Since the skeins of the form T n ((p ′ ,q ′ )) span),

38RĂZVAN GELCA AND ALEJANDRO URIBEthe skein module, the map defined above can be extended linearly to theentire skein module.It should be noted that RT t (T 2 × [0,1]) is isomorphic to the Kauffmanbracket skein algebra of the torus (see [6] and [9]). However this is not thecase for higher genus surfaces, as it can be checked by looking at the productof a separating curve with a nonseparating curve.5.3. A Stone–von Neumann theorem on the pillow case. Weyl quantizationyields an irreducible representation of the Reshetikhin-Turaev skeinalgebra of the cylinder over the torus subject to the following conditions:1. t is mapped to the 4rth root of unity e iπ2r ,2. each simple closed curve on the torus γ is mapped to a self-adjointoperator,3. the skein S r−1 (γ) is mapped to zero, where S n (x) denotes the Chebyshevpolynomial of the second kind.This factors to an irreducible representation of the reduced Reshetikhin-Turaev skein algebra ˜RT t (T × [0,1]), which is the non-abelian analogue ofthe group ring of the finite Heisenberg group. Like for the finite Heisenberggroup, a Stone–von Neumann theorem holds.Theorem 5.2. The representation given by the Weyl quantization of themoduli space of flat SU(2)-connections on the torus is the unique irreduciblerepresentation of ˜RT t (T 2 × [0,1]) which maps simple closed curves to selfadjointoperators and t to multiplication by e2r.πiProof. Let us show that each vector is cyclic. Because the eigenspaces of eachquantized Wilson line are 1-dimensional, in particular those of op(2cos 2πx),it suffices to check this for the eigenvectors of this operator, namely for ζ τ j ,j = 1,2,... ,r − 1. And becauseop(2cos 2πy)ζ τ j = ζ τ j+1 + ζ τ j−1op(2cos 2π(x + y))ζ τ j = t −1 (t 2 ζ τ j−1 + t −2 ζ τ j+1),by taking linear combinations we see that from ζj τ we can generate both ζτ j+1and ζj−1 τ . Repeating, we can generate the entire basis. This shows that ζτ jis cyclic for each j = 1,2,... ,r − 1, hence the representation is irreducible.To prove uniqueness, consider an irreducible representation of ˜RT t (T 2 ×[0,1]) that has the properties from the statement. It is first important to notethat the condition S r−1 (γ) = 0 for all γ implies, by the spectral mappingtheorem, that the eigenvalues of the operator associated to γ are among thenumbers 2cos kπ r, k = 1,2,... ,r − 1.Inspired by the structure of the Kauffman bracket skein algebra of thetorus exhibited by Bullock and Przytycki in [3], we can write for the generatorsX = (1,0), Y = (0,1), and Z = (1,1) of the algebra ˜RT t (T 2 × [0,1])

the relationsQUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 39tXY − t −1 Y X = (t 2 − t −2 )ZtY Z − t −1 ZY = (t 2 − t −2 )XtZX − t −1 XZ = (t 2 − t −2 )Yt 2 X 2 + t −2 Y 2 + t 2 Z 2 − tXY Z − 2t 2 − 2t −2 = 0.Substituting Z from the first relation we obtain the equivalent presentation(t 2 + t −2 )Y XY − (Y 2 X + XY 2 ) = (t 4 + t −4 − 2)X(t 2 + t −2 )XY X − (Y X 2 + X 2 Y ) = (t 4 + t −4 − 2)Y(t 6 + t −2 − 2t 2 )X 2 + (t −6 + t 2 − 2t −2 )Y 2 + XY XY + Y XY X−t 2 Y X 2 Y − t −2 XY 2 X = 2(t 6 + t −6 − t 2 − t −2 ).Setting t = e iπ2r the second relation becomes2cos π r XY X − (Y X2 + X 2 Y ) = 4sin 2 π r Y.Now consider some eigenvector v k of X. Its eigenvalue must be of theform 2cos kπ rwhere k is one of the numbers 1,2,... ,r − 1. Exactly as inthe case of the finite Heisenberg group, we wish to generate a basis of therepresentation by letting Y act repeatedly on v k . To this end, set Y v k = w.The above relation yields2cos π r · 2cos kπ r Xw − 4cos2 kπ r w − X2 w = 4sin 2 π r w.Rewrite this as(X 2 − 4cos kπ r cos π r X − 4 (sin 2 π r + cos2 kπ r))w = 0.It follows that either w = 0 or w is in the kernel of the operatorX 2 − 4cos kπ r cos π (r X − 4 sin 2 π r + kπ )cos2 Id.rNow let us use the first relation to deduce that if Y v k = w = 0, thenXv k = 0. This is impossible because of the third relation in the presentation.This shows that w ≠ 0, and so this vector lies in the kernel of the abovementioned operator. Note that if λ is an eigenvalue of X which satisfiesλ 2 − 4cos kπ r cos π (r λ − 4 sin 2 π r + kπ )4cos2 = 0,4then necessarily λ = 2cos (k±1)πr. It follows thatY v k = v k+1 + v k−1 ,where Xv k±1 = 2cos (k±1)π4v k±1 , and v k+1 and v k−1 are not simultaneouslyequal to zero. By applying X repeatedly and taking linear combinations,we see that the vectors v k ,v k+1 ,v k−1 are in the vector space of the representation.We wish to make them elements of a basis of this vector space.

40RĂZVAN GELCA AND ALEJANDRO URIBEFor that we need to make sure that v k+1 and v k−1 are nonzero, and we alsoneed to understand the action of Y on them.Set Y v k+1 = αv k + v k+2 and Y v k−1 = βv k + v k−2 , where Xv k±2 =v k±2 . It might be possible that the scalars α and β are zero. Thevectors v k+2 , v k−2 might as well be zero; if they are not zero, then they areeigenvectors of X, and their respective eigenvalues are as specified becauseof the above argument.Using again the relations satisfied by the three generators of ˜RT t (T 2 ×[0,1]) we write2cos (k±2)πr2cos π r Y XY − (Y 2 X + XY 2 ) = 2cos 2π r X,which impliescos π rThis is equivalent to((k + 2)πcos + cos kπ r rthat is(k + 1)πcos α + cos π r rsin(k − 1)πcos β − cos kπ r r(α + β)= cos 2π r cos kπ r − cos kπ r .) ((k − 2)π(α − 1) + cos + cos kπ r r(k + 1)π(α − 1) + sinrFor further use, we write this as(k − 1)π(β − 1) = 0.r(t 4k+4 − 1)(α − 1) + (t 4k − t 4 )(β − 1) = 0.)(β − 1) = 0Reasoning similarly with the last of the three relations in ˜RT t (T 2 ×[0,1])we obtain the equality(t −6 + t 2 − 2t −2 (2k + 1)π)(α + β) + 4cos α + 4cos π r r α(2k − 1)π+4cos β + 4cos π r r β − (2k + 2)π2t2 cos α − 2t 2 αr−2t 2 (2k − 2)πcos β − 2t 2 β − 2t −2 cos 2kπrr α − 2t−2 cos 2kπr β−2t −2 α − 2t −2 β = 2(t 6 + t −6 − t 2 − t −2 ) − 2(t 6 + t −2 − 2t 2 )−(t 6 + t −2 − 2t 2 )(t 4k + t −4k ).

QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 41Using the fact that t = cos kπ2r+ isinkπ2rwe can transform this into(t −6 + t 2 − 2t −2 (2k + 1)π)((α − 1) + (β − 1)) + 4cos α + 4cos π r r α(2k − 1)π+4cos β + 4cos π (2k + 3)π (2k + 1)πβ − cos α − cos αr r rr(2k + 3)π (2k + 1)π−isin α + isin α − 2cos π rr r α − 2isin π r α(2k − 1)π (2k − 3)π (2k − 1)π (2k − 3)π− cos β − cos β − isin β − isin βrrrr−2cos π r β − 2isin π (2k − 1)π (2k + 1)πβ − cos α − cos αr rr(2k − 1)π (2k + 1)π (2k − 1)π (2k + 1)π−isin α + isin α − cos β − cos βrrrr(2k − 1)π (2k + 1)π−isin β + isin β − 2cos π rr r α + 2isin π r α − 2cos π r β+2isin π r β = −(t6 + t −2 − 2t 2 )(t 4k + t −4k ).After cancellations we obtain()(t −6 + t 2 − 2t −2 (2k + 3)π (2k + 3)π)((α − 1) + (β − 1)) − cos + isin αrr() ()(2k − 3)π (2k − 3)π (2k + 1)π (2k + 1)π− cos − isin β + 2 cos + isin αrrrr() ()(2k − 1)π (2k − 1)π (2k − 1)π (2k − 1)π+2 cos − isin β − cos + isin αrrrr()(2k + 1)π (2k + 1)π− cos − isin βrr= −t 4k+6 + t −4k+6 + t 4k−2 + t −4k−2 − 2t 4k+2 − 2t −4k+2 .This is the same as(t 4k+6 + t 4k−2 + 2t −2 − 2t 4k+2 − t −6 − t 2 )(α − 1)+(t −4k+6 + t −4k−2 + 2t −2 − 2t −4k+2 − t −6 − t 2 )(β − 1) = 0.Dividing through by t −6 + t 2 − 2t −2 we obtain(t 4k+4 − 1)(α − 1) + (t −4k+4 − 1)(β − 1) = 0.Combining this with the relation obtained before, we obtain the system oftwo equations(t 4k+4 − 1)u + (t 4k − t 4 )v = 0(t 4k+4 − 1)u + (t −4k+4 − 1)v = 0in the unknowns u = α − 1 and v = β − 1. Recall that t = e iπ2r .The coefficient of v in one of the equations is equal to zero if and only ifk = 1, in which case we are forced to have β = 0, because zero is not aneigenvalue of X. The coefficient of u in one of the equations is equal to zero

42RĂZVAN GELCA AND ALEJANDRO URIBEif and only if k = r − 1, in which case we are forced to have α = 0, because−1 is not an eigenvalue of X.In any other situation, by subtracting the equations we obtain(t 4k − t 4 − t −4k+4 + 1)v = 0,that is t 4 (t 4k−4 − 1)(t −4k + 1)v = 0. This can happen only if t 4k = −1,namely if 2k = r.So if k ≠ r 2 , then Y v k = v k+1 +v k−1 with v k+1 and v k−1 being eigenvectorsof X with eigenvalues 2cos (k+1)πrrespectively 2cos (k−1)πr, and Y v k±1 =v k + v k±2 , where v k±2 lie in the eigenspaces of X corresponding to theeigenvalues 2cos (k±2)πr.Let us see what can happen if k = r 2 . One of v k+1 and v k−1 is not zero,say v k+1 . Applying the above considerations to v k+1 we have Y v k+1 =αv k + v k+2 and Y αv k = v k+1 + vk−1 ′ , for some vector v′ k−1in the eigenspaceof X corresponding to the eigenvalue 2cos (k−1)π . Then on the one handY v k = v k+1 + v k−1 and on the other hand αY v k = v k+1 + vk−1 ′ . This showsthat α = 1, and because (α − 1) + (β − 1) = 0, it follows that β = 1 as well.Repeating the argument we conclude that the irreducible representation,which must be the span of X m Y n v k for m,n ≥ 0, has the basisv 1 ,v 2 ,... ,v r−1 , and X and Y act on these vectors byrXv j = 2cos jπ r , Y v j = v j+1 + v j−1 ,with the convention v 0 = v r = 0. And we recognize the representation givenby the Weyl quantization of the moduli space of flat SU(2)-connections onthe torus.□5.4. The Reshetikhin-Turaev representation of the mapping classgroup of the torus. Let us first concentrate on the generators S and Tof the mapping class group of the torus. Because the Weyl quantization onthe pillow case is obtained by doing equivariant Weyl quantization on thetorus, it follows that the values of ρ(S) and ρ(T) on the pillow case can be

QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 43computed from those on the torus as follows:ρ(S)ζ τ j (z) = 4√ rρ(S)(θ τ j (z) − θτ −j (z)) = 4√ rk=1r−12r−1∑k=0= 4√ ∑r−1)r(e − πir kj θk τ πi(z) − e r kj θ−k τ (z)+ 4√ ∑rk=1r−1= 2 4√ ∑rk=1k=1)(e − πir (2r−k)j θ2r−k τ πi(z) − e r (2r−k)j θk τ (z)(e − πir kj − e πir kj) (θ τ j (z) − θ τ −j(z))∑r−1= −4i sin π r kjζτ j (z),which is a scalar multiple of the discrete sine transform, and()e − πir kj θk τ πi(z) − e r kj θ−k τ (z)ρ(T)ζ τ j (z) = 4√ rρ(T)(θ τ j − θτ −j (z)) = 4√ re πir j2 (θ τ j (z) − θτ −j (z))= e πir j2 ζ τ j (z).These coincide, up to multiplication by a constant, with the S and T matricesdefined using quantum groups [24], in which the jk entry of ρ(S) isthe Reshetikhin-Turaev invariant of the Hopf link with components coloredby V j respectively V k , while ρ(T) introduces a positive twist on each basiselement (see Figure 18).S: jk T:V VjVFigure 18The relationship with classical theta functions allows us to adapt a formulaof Kač and Peterson [13] in order to obtain a general formula for theReshetikhin-Turaev representation of the mapping class group of the torus.Theorem 5.3. Leth =(a bc dbe an element of the mapping class group of the torus. Then there is anumber c(2r,h) ∈ C such thatρ(h)ζj τ (z) = c(2r,h)∑ e πi2r (cdk2 +abj 2) [bckj]ζaj+ck τ (z)kwhere the sum is taken over a family of j ∈ Z that give all representativesof the classes cj modulo 2rZ and the square brackets denote a quantizedinteger.),

44RĂZVAN GELCA AND ALEJANDRO URIBEProof. Because 2r is an even integer, the group SL θ (2, Z) is the wholeSL(2, Z). By Proposition 3.17 in [13], there is a constant ν(2r,h) suchthatθ τ ′j( zcτ + d)) ∑= ν(2r,h)exp(− 4πircz22(cτ + d)ke iπ2r (cdk2 +2bckj+abj 2) θ τ aj+ck (z)with the same summation convention as in the statement of the theorem,and with τ ′ = aτ+bcτ+d. The mapθ τ j (z) → ∑ ke iπ2r (cdk2 +2bckj+abj 2) θ τ aj+ck (z)is up to multiplication by a constant, the unique map that satisfies theEgorov condition with the representation of the Heisenberg group. It followsthatζ τ j (z) → ∑ ke πi2r (cdk2 +abj 2) [bcjk]ζ τ aj+ck (z)satisfies the Egorov condition with the Weyl quantization of the pillow case.By Theorem 5.2 there is a unique map with this property, up to multiplicationby a constant. Hence the conclusion.□References[1] A.Yu. Alexeev, V. Schomerus, Representation theory of Chern-Simons observables,Duke Math. Journal, 85 (1996), No.2, 447–510.[2] M.F. Atiyah, R. Bott, The Yang-Mills equations over a Riemann surface, Phil.Trans. Royal. Soc. A, 308 (1982), 523–615.[3] D. Bullock, J.H. Przytycki, Multiplicative structure of Kauffman bracket skein modulequantizations, Proc. Amer. Math. Soc., 128 (2000), 923–931.[4] A. Connes, Noncommutative Geometry, Academic Press Inc., San Diego, CA, 1994.[5] G. Folland, Harmonic Analysis in Phase Space, Princeton University Press, Princeton1989.[6] Ch. Frohman, R. Gelca, Skein modules and the noncommutative torus, Trans. Amer.Math. Soc., 352 (2000), 4877–4888.[7] R. Gelca, Topological SL(2, C) quantum field theory with corners, J. Knot TheoryRamif., 7(1998), 893–906.[8] R. Gelca, On the holomorphic point of view in the theory of quantum knot invariants,J. Geom. Phys, 56 (2006), 2163–2176.[9] R. Gelca, A. Uribe, The Weyl quantization and the quantum group quantization ofthe moduli space of flat SU(2)-connections on the torus are the same, Commun.Math. Phys., 233(2003), 493–512.[10] W. Goldman, Invariant functions on Lie groups and Hamiltonian flow of surfacegroup representations, Inventiones Math., 85 (1986), 263–302.[11] V.F.R. Jones, Polynomial invariants of knots via von Neumann algebras, Bull. Amer.Math. Soc., 12(1995), 103–111.[12] N. Hitchin, Flat connections and geometric quantization, Commun. Math. Phys.,131 (1990), 347–380.[13] V. Kač, D.H. Peterson, Infinite-dimensional Lie algebras, theta functions and modularforms, Adv. in Math., 53 (1984), 125–264.[14] R. Kirby, P. Melvin, The 3-manifold invariants of Witten and Reshetikhin-Turaevfor sl(2, C), Inventiones Math., 105 (1991), 473–545.

QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 45[15] W.B.R. Lickorish, A representation of orientable combinatorial 3-manifolds, Ann. ofMath., 76(1962), 531–538.[16] G. Lion, M. Vergne, The Weil Representation, Maslov Index, and Theta Series,Birkhäuser, 1980.[17] D. Mumford, Tata Lectures on Theta, Birkhäuser, 1983.[18] A. Polishchuk, Abelian Tarieties, Theta Functions and the Fourier Transform, CambridgeUniv. Press, 2003.[19] J.H. Przytycki, Skein modules of 3-manifolds, Bull. Pol. Acad. Sci. 39(1-2) (1991)91–100.[20] N. Reshetikhin, V. Turaev, Invariants of 3-manifolds via link polynomials and quantumgroups, Inventiones Math., 103 (1991), 547–597.[21] M. Rieffel, Deformation quantization of Heisenberg manifolds, Commun. Math.Phys. 122(1989), 531–562.[22] J. Roberts, Irreducibility of some quantum representations of mapping class groups,in Knots in Hellas ’98, World Scientific, 2000.[23] J. Sniatycki, Geometric Quantization and Quantum Mechanics, Springer, 1980.[24] Turaev, V.G., Quantum Invariants of Knots and 3-Manifolds, de Gruyter Studiesin Mathematics, de Gruyter, Berlin–New York, 1994.[25] Turaev, V.G., Algebras of loops on surfaces, algebras of knots, and quantization,Adv. Ser. in Math. Phys., 9 (1989), eds. C.N. Yang, M.L. Ge, 59–95.[26] A. Weil, Sur certains groupes d’operateurs unitaires, Acta Math. 111 (1976), 143–211.[27] K. Walker, On Witten’s 3-manifold invariants, preprint 1991.[28] C.T.C. Wall, Non-additivity of the signature, Invent. Math. 7 (1969), 269–274.[29] E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys.,121 (1989), 351–399.Department of Mathematics and Statistics, Texas Tech University, Lubbock,TX 79409 and Institute of Mathematics of the Romanian Academy,Bucharest, RomaniaE-mail address: rgelca@gmail.comDepartment of Mathematics, University of Michigan, Ann Arbor, MI 48109E-mail address: uribe@math.lsa.umich.edu