The Algebra & Geometry of Modern Physics
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Quantization of the spectral curves for Hitchin fibrations via Eynard-Orantin theory
Quantum curves are a magical object. It is conjectured that they capture the information of quantum topological invariants in an effective and compact way. The relation between quantum curves and the Eynard-Orantin theory was first exemplified in an influential paper of Gukov and Sułkowski. In the first part of this talk, for introduction, I will present the simplest and mathematically elegant example of quantum curves and the Eynard-Orantin formalism based on the Catalan numbers. (This part is based on my joint paper with P. Sułkowski and an earlier paper with Dumitrescu et al.) In the second part I will explain the construction of quantization of the spectral curves appearing in the theory of Hitchin fibrations. The main theorem is that the Eynard-Orantin theory indeed provides a mechanism of constructing the canonical generator of a D-module on an arbitrary compact Riemann surface. The semi-classical approximation of this D-module coincides with the Hitchin spectral curve. (This part is based on my joint work with O. Dumitrescu of Hanover.) The talk will be given in an elementary and pedagogical language.
A brief introduction to the AdS/CFT correspondence and its applications
One of the most profound development in the contemporary high energy physics is the discovery of the AdS/CFT correspondence (or more generally the gauge-gravity duality), which states exact equivalence between a class of string theory vacua and certain quantum field theories. I will present the original Maldacena's argument for the existence of the AdS/CFT correspondence, as well as various qualitative and quantitative indications why this conjecture seems to be correct. If time permits, I will discuss recent research trends in this field.
Categories & all that ...
While the path integral is rarely mathematically well-defined, it is usually assumed to have some useful properties like sewing laws (relating the integral over a domain which decomposes into two subdomains to path integrals over the subdomains). These were included by Atiyah into the definition of the topological quantum field theory as a functor on the category of cobordisms. This is one of the many ways the modern mathematical language of categories and functors becomes relevant to physicists. The lecture will be an introduction to categories and functors. We will show how various notions from different areas of mathematics get unified in the categorial framework. We will discuss categories with multiplication and some geometric functors (like homology or K-theory) arising whenever global effects of the spacetime play a role.