QUANTUM MECHANICS AND NON-ABELIAN THETA FUNCTIONS ...

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