QUANTUM MECHANICS AND NON-ABELIAN THETA FUNCTIONS ...

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