QUANTUM MECHANICS AND NON-ABELIAN THETA FUNCTIONS ...

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42RĂZVAN GELCA AND ALEJANDRO URIBEif and only if k = r − 1, in which case we are forced to have α = 0, because−1 is not an eigenvalue of X.In any other situation, by subtracting the equations we obtain(t 4k − t 4 − t −4k+4 + 1)v = 0,that is t 4 (t 4k−4 − 1)(t −4k + 1)v = 0. This can happen only if t 4k = −1,namely if 2k = r.So if k ≠ r 2 , then Y v k = v k+1 +v k−1 with v k+1 and v k−1 being eigenvectorsof X with eigenvalues 2cos (k+1)πrrespectively 2cos (k−1)πr, and Y v k±1 =v k + v k±2 , where v k±2 lie in the eigenspaces of X corresponding to theeigenvalues 2cos (k±2)πr.Let us see what can happen if k = r 2 . One of v k+1 and v k−1 is not zero,say v k+1 . Applying the above considerations to v k+1 we have Y v k+1 =αv k + v k+2 and Y αv k = v k+1 + vk−1 ′ , for some vector v′ k−1in the eigenspaceof X corresponding to the eigenvalue 2cos (k−1)π . Then on the one handY v k = v k+1 + v k−1 and on the other hand αY v k = v k+1 + vk−1 ′ . This showsthat α = 1, and because (α − 1) + (β − 1) = 0, it follows that β = 1 as well.Repeating the argument we conclude that the irreducible representation,which must be the span of X m Y n v k for m,n ≥ 0, has the basisv 1 ,v 2 ,... ,v r−1 , and X and Y act on these vectors byrXv j = 2cos jπ r , Y v j = v j+1 + v j−1 ,with the convention v 0 = v r = 0. And we recognize the representation givenby the Weyl quantization of the moduli space of flat SU(2)-connections onthe torus.□5.4. The Reshetikhin-Turaev representation of the mapping classgroup of the torus. Let us first concentrate on the generators S and Tof the mapping class group of the torus. Because the Weyl quantization onthe pillow case is obtained by doing equivariant Weyl quantization on thetorus, it follows that the values of ρ(S) and ρ(T) on the pillow case can be
QUANTUM MECHANICS AND GENERALIZED THETA FUNCTIONS 43computed from those on the torus as follows:ρ(S)ζ τ j (z) = 4√ rρ(S)(θ τ j (z) − θτ −j (z)) = 4√ rk=1r−12r−1∑k=0= 4√ ∑r−1)r(e − πir kj θk τ πi(z) − e r kj θ−k τ (z)+ 4√ ∑rk=1r−1= 2 4√ ∑rk=1k=1)(e − πir (2r−k)j θ2r−k τ πi(z) − e r (2r−k)j θk τ (z)(e − πir kj − e πir kj) (θ τ j (z) − θ τ −j(z))∑r−1= −4i sin π r kjζτ j (z),which is a scalar multiple of the discrete sine transform, and()e − πir kj θk τ πi(z) − e r kj θ−k τ (z)ρ(T)ζ τ j (z) = 4√ rρ(T)(θ τ j − θτ −j (z)) = 4√ re πir j2 (θ τ j (z) − θτ −j (z))= e πir j2 ζ τ j (z).These coincide, up to multiplication by a constant, with the S and T matricesdefined using quantum groups [24], in which the jk entry of ρ(S) isthe Reshetikhin-Turaev invariant of the Hopf link with components coloredby V j respectively V k , while ρ(T) introduces a positive twist on each basiselement (see Figure 18).S: jk T:V VjVFigure 18The relationship with classical theta functions allows us to adapt a formulaof Kač and Peterson [13] in order to obtain a general formula for theReshetikhin-Turaev representation of the mapping class group of the torus.Theorem 5.3. Leth =(a bc dbe an element of the mapping class group of the torus. Then there is anumber c(2r,h) ∈ C such thatρ(h)ζj τ (z) = c(2r,h)∑ e πi2r (cdk2 +abj 2) [bckj]ζaj+ck τ (z)kwhere the sum is taken over a family of j ∈ Z that give all representativesof the classes cj modulo 2rZ and the square brackets denote a quantizedinteger.),
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