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.

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: An introduction to Geometric Langlands Duality: from Faraday to Montonen-Olive to Hitchin to Kapustin-Witten ABS: As the first two of a four-part Bragg Lecture Series, we will furnish a pedestrian introduction to geometric Langlands duality via its manifestation in physical gauge theory. To this end, we will first explain what the arithmetic Langlands duality is, and its geometric version. Then, 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: As the first two of a four-part Bragg Lecture Series, we will furnish a pedestrian introduction to geometric Langlands duality via its manifestation in physical gauge theory. To this end, we will first explain what the arithmetic Langlands duality is, and its geometric version. Then, 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.