tailieunhanh - Đề tài " Asymptotic faithfulness of the quantum SU(n) representations of the mapping class groups "

We prove that the sequence of projective quantum SU(n) representations of the mapping class group of a closed oriented surface, obtained from the projective flat SU(n)-Verlinde bundles over Teichm¨ller space, is asymptotically u faithful. That is, the intersection over all levels of the kernels of these representations is trivial, whenever the genus is at least 3. For the genus 2 case, this intersection is exactly the order 2 subgroup, generated by the hyper-elliptic involution, in the case of even degree and n = 2. Otherwise the intersection is also trivial in the genus 2 case. . | Annals of Mathematics Asymptotic faithfulness of the quantum SU n representations of the mapping class groups By Jorgen Ellegaard Andersen Annals of Mathematics 163 2006 347 368 Asymptotic faithfulness of the quantum SU n representations of the mapping class groups By J0RGEN Ellegaard Andersen Abstract We prove that the sequence of projective quantum SU n representations of the mapping class group of a closed oriented surface obtained from the projective flat SU n -Verlinde bundles over Teichmuller space is asymptotically faithful. That is the intersection over all levels of the kernels of these representations is trivial whenever the genus is at least 3. For the genus 2 case this intersection is exactly the order 2 subgroup generated by the hyper-elliptic involution in the case of even degree and n 2. Otherwise the intersection is also trivial in the genus 2 case. 1. Introduction In this paper we shall study the finite dimensional quantum SU n representations of the mapping class group of a genus g surface. These form the only rigorously constructed part of the gauge-theoretic approach to topological quantum field theories in dimension 3 which Witten proposed in his seminal paper W1 . We discovered the asymptotic faithfulness property for these representations by studying this approach which we will now briefly describe leaving further details to Sections 2 and 3 and the references given there. Let s be a closed oriented surface of genus g 2 and p a point on s. Fix d E Z nZ ZSU n in the center of SU n . Let M be the moduli space of flat SU n -connections on s p with holonomy d around p. By applying geometric quantization to the moduli space M one gets a certain finite rank vector bundle over Teichmuller space T which we will call the Verlinde bundle Vk at level k where k is any positive integer. The fiber of this bundle over a point Ơ eT is Vkyơ H MƠ L g where Mơ is M equipped with a complex structure induced from Ơ and cơ is an ample generator of the Picard group