Week 01.06.2024 – 09.06.2024

TITLE: An introduction to Geometric Langlands Duality: from Faraday to Montonen-Olive to Hitchin to Kapustin-Witten ABS: As the first of a 4-part Bragg Lectures, we will furnish a pedestrian introduction to geometric Langlands duality via its manifestation in physical gauge theory. We will cover the underlying physics and math ideas starting with Faraday and electromagnetism, on to Montonen-Olive and the Langlands duality of electric-magnetic charges, then to Hitchin and the mirror symmetry of his moduli spaces, and finally, to Kapustin-Witten and S-duality of 4d N=4 topological gauge theory.

TITLE: An introduction to Geometric Langlands Duality: from Faraday to Montonen-Olive to Hitchin to Kapustin-Witten ABS: We will furnish a pedestrian introduction to geometric Langlands duality via its manifestation in physical gauge theory. We will cover the underlying physics and math ideas starting with Faraday and electromagnetism, on to Montonen-Olive and the Langlands duality of electric-magnetic charges, then to Hitchin and the mirror symmetry of his moduli spaces, and finally, to Kapustin-Witten and S-duality of 4d N=4 topological gauge theory.

TITLE: Braverman-Finkelberg Generalization of Geometric Langlands Duality in String Theory ABS: Braverman-Finkelberg considered a generalization of the geometric Satake isomorphism to involve not Lie but Kac-Moody groups, and in so doing, arrived at a formulation of geometric Langlands duality which involves not complex curves, but complex surfaces. Specifically, the formulation relates the intersection cohomology of the moduli space of G-instantons on orbifold complex surfaces, to modules of a Langlands-dual affine Lie algebra with level determined by the order of the singularity. We will furnish a string/M-theoretic derivation of their mathematical conjecture.

TITLE: The AGT Duality and Beilinson-Drinfeld's Original Formulation of Geometric Langlands Duality via CFT ABS: The original mathematical formulation of geometric Langlands duality by Beilinson-Drinfeld involved CFT, not gauge theory. It was therefore an outstanding question how Kapustin-Witten's gauge-theoretic approach is related to Beilinson-Drinfeld's CFT approach. We will shed light on this question via string/M-theory. In particular, we first consider a modification of our physical setup manifesting the Braverman-Finkelberg geometric Langlands duality to arrive at an AGT duality which relates gauge theory to affine W-algebras. Then, it can be explained that the KW and BD formulations are just string-dual to each other.