Week 03.06.2024 – 09.06.2024

TITLE: An introduction to Geometric Langlands Duality: from Faraday to Montonen-Olive to Hitchin to Kapustin-Witten ABS: As the first 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.

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.

We consider type IIB string theory with $N$ D3 branes and various configurations of sevenbranes, such that the string coupling $g_s$ is fixed to a constant finite value. These are the simplest realizations of F-theory, and are holographically dual either to a to a rank $N$ gauge for any coupling tau, or to non-Lagrangian CFTs such as Argyres-Douglas and Minahan-Nemeschansky theory. We compute the mass deformed sphere free energy F(m) using localization in the case of the Lagrangian theory, and the Seiberg-Witten curve for the non-Lagrangian theories. We show how F(m) can be used along with the analytic bootstrap to fix the large N expansion of flavor multiplet correlators in these CFTs, which are dual to scattering of gluons on AdS_5 x S^3, and in the flat space limit determine the effective theory of sevenbranes in F-theory. In particular, we compute the log threshold terms for all the theories and the first higher derivative correction F^4 for the Lagrangian theory for finite tau, and find a precise match in the flat space limit in all cases. Finally, we use numerical bootstrap to study the Lagrangian theory at finite N and tau.

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.