Quantization of moduli spaces and TQFT
Platica dada por William Mistegaard (IST, Austria) en Geometry and Quantization of Moduli Spaces — 5,6 and 7 July, 2021
Abstract: The Reshetikhin-Turaev topological quantum field theory (TQFT) was motivated from physics by Witten’s work on quantum Chern-Simons. In Witten’s work quantization of moduli spaces of flat connections and conformal field theory (CFT) was presented as two equivalent approaches to construct the Hilbert space associated to an oriented two-manifold. Both approaches depend a priori on a choice of complex structure on the two-manifold, although the topological nature of the theory suggests that the Hilbert space should be independent of this choice, and support a projective linear action of the mapping class group. On the CFT side this topological invariance and the existence of a mapping class group action was proven by Tsuchia, Ueno and Yamada. On the quantization side it was proven for some two-manifolds independently by Hitchin and Axelrod, Della Pietra and Witten. Laszlo proved mathematically that the CFT approach and the quantization approach of Hitchin are equivalent. Finally, Andersen and Ueno have established that the CFT representations of the mapping class groups are isomorphic to the Reshetikhin-Turaev TQFT mapping class group action. In this talk, we will; 1) partly review the above story, 2) review how quantization was used to prove important results in quantum topology, and 3) present work in progress joint with Andersen, which constructs the TQFT-representations of the mapping class groups from the quantization approach in some of the remaining (parabolic cases), not previously dealt with by Hitchin or Axelrod-Della Pietra and Witten.
Pagina de VBAC – Vector Bundles on Algebraic Curves