QUANTUM MECHANICS AND NON-ABELIAN THETA FUNCTIONS ...

32RĂZVAN GELCA AND ALEJANDRO URIBEconclude that σ ′ is in the cyclic representation generated by e p , and thereforein the cyclic representation generated by σ.Repeating, we eventually descend to the empty link. It remains to showthat the empty link is cyclic. But this is obviously true, since each basiselement can be represented as the image of a collection of nonintersectingsimple closed curves on the boundary. This completes the proof. □4.4. The Reshetikhin-Turaev representation of the mapping classgroup as a Fourier transform for non-abelian theta functions. TheReshetikhin-Turaev projective representation ρ of the mapping class groupof Σ g is constructed as follows. By the Lickorish twist theorem [15], everyelement h of the mapping class group is a product of twists. Each twist canbe represented as surgery with integer coefficients along a link in Σ g ×[0,1].Hence h itself can be represented as surgery with integer coefficients inΣ g × [0,1], so it can be represented by a framed link, the framings of whosecomponents are equal to the surgery coefficients.In order to find ρ(h) it suffices to compute [ρ(h)e p ,e q ] for all basis elementse p and e q , where [·, ·] is the nondegenerate bilinear pairing discussedin the previous section. To do this, place Σ g × [0,1] with the surgery link instandard position in S 3 , then place e p in one handlebody and e q in the other.∑ r−1j=1 [j]V j where [j] isColor each link component of the surgery link by 1 Xthe quantized integer and X is the square root of ∑ r−1j=1 [j]2 . Then computethe Reshetikhin-Turaev invariant of the framed graph obtained by projectingeverything onto a plane to obtain the desired value.On the other hand, there is an action of the mapping class group of thesurface Σ g on the ring F(M g ) of regular functions on the moduli space. Anelement h of the mapping class group acts byh · f(A) = f(h −1∗ A)where h ∗ A denotes the image of the connection A through h. In particularthe Wilson line of a curve γ is mapped to the Wilson line of the curve h(γ).Each elemenent of the mapping class group preserves the Atiyah-Bott symplecticform, so it induces a symplectomorphism of M g . The Reshetikhin-Turaev representation gives a method for quantizing these symplectomorphisms.The action of the mapping class group on F(M g ) induces an action onthe quantum observables byh · op(f(A)) = op(f(h −1∗ A)).In order for the Reshitikhin-Turaev topological quantum field theory to beconsistent, this action must be compatible with the Reshetikhin-Turaev representationof the mapping class group. The compatibility condition isop(W h(γ),n ) = ρ(h)op(W γ,n )ρ(h) −1 ,which holds true for all elements h of the mapping class group, curves γ onthe surface and positive integers n. We recognize the Egorov identity, whichis satisfied exactly.