QUANTUM MECHANICS AND NON-ABELIAN THETA FUNCTIONS ...

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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
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