Complex Lagrangians, Mirror Symmetry, and Quantization (23w5068)


John Alexander Cruz Morales (Universidad Nacional de Colombia)

(University of North Carolina at Chapel Hill)

Elba Garcia Failde (Sorbonne Université)

(University of California, Davis)



The Banff International Research Station will host the "Complex Lagrangians, Mirror Symmetry, and Quantization" workshop in Banff from October 15 to October 20, 2023.

In 1987, Hitchin discovered a set of simple nonlinear partial differential equations on a Riemann surface, that arises as the reduction} of the 4-dimensional physical equations known as the self-duality equations of Yang-Mills fields. The workshop is concerned with the geometric nature of the space of solutions, the moduli space, of Hitchin's equations. Since Yang-Mills fields represent forces in high-energy physics, they are naturally associated with a group of internal symmetries. Thus a Hitchin moduli space also depends on this group. Three and a half decades later, the interest in Hitchin's equations and moduli spaces is still exponentially growing. This is due to the following reasons. One is its deep connection to the Langlands duality of groups, a representation theoretic concept discovered by Langlands in 1967. The original context of Langlands can be translated into geometry, which then takes a form of questions about Hitchin moduli spaces and subvarieties in them. The other one is the discovery from physics, identifying the Langlands duality with electromagnetic duality, leading to the physical relevance of quantization of Hitchin moduli spaces.

The workshop is conceived in response to recent breakthroughs in physics and mathematics. One is the \emph{real} quantization of Hitchin moduli spaces due to Gaiotto and Witten, and the other is a deformation quantization theory of Kontsevich and Soibelman that has achieved a vast mathematical foundation and generalization of topological recursion of Eynard and Orantin in random matrix theory. The vision of the organizers is that these two theories should be related through mirror symmetry. The workshop is organized to test this vision, and to explore its mathematical consequences.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).