QUANTUM MECHANICS AND NON-ABELIAN THETA FUNCTIONS ...

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.