QUANTUM MECHANICS AND NON-ABELIAN THETA FUNCTIONS ...

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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
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