|Speaker:||Motohico Mulase (UCDavis)|
|Title:||Mirror Symmetry of Catalan Numbers and Quantum Curves|
|Date:||Thu, Nov 08, 2012, 15:30 - 17:00|
|Place:||Seminar Room A|
The talk is aimed at presenting a few rigorous mathematical examples of the emerging picture from topological string theory on quantum curves and quantum knot invariants. So far our examples are not directly related to any knots, yet they exhibit deep mathematical structures of the theory.
Let K be a knot. The emerging picture is that the classical knot invariants, called the A-polynomial of K, and the colored Jones polynomials of K, are mirror symmetric one another. This means one can compute one from the other. For example, from the colored Jones polynomial, one can find a holonomic system, and whose Lagrangian (the semi-classical limit) is the A-polynomial (the conjecture of Garoufalidis). The other direction, from the A-polynomial to colored Jones polynomials, is the speculation by Dijkgraaf-Fuji-Manabe and Gukov-Sulkowski, utilizing the B-remodeling idea of Marino et al. and the B-model recursion of Eynard-Orantin.
We exhibit two examples, though unrelated to knots, to illustrate the mathematical roles of the quantum curves and the mirror symmetry in this particular context.
The talk is based on my joint paper with Sulkowski.