S-duality of boundary conditions and the Geometric Langlands program

Abstract

Maximally supersymmetric gauge theory in four dimensions admits local boundary conditions which preserve half of the bulk supersymmetries. The S-duality of the bulk gauge theory can be extended in a natural fashion to act on such half-BPS boundary conditions. The purpose of this note is to explain the role these boundary conditions can play in the Geometric Langlands program. In particular, we describe how to obtain pairs of Geometric Langland dual objects from S-dual pairs of half-BPS boundary conditions. arXiv:1609.09030v2 [hep-th] 8 Nov 2016 42 B.3.2 The ADHM system 43 B.3.3 Kähler SQM 44 -ii -B.3.4 Hyper-holomorphic SQM 46 B.3.5 Hypermultiplets on a Riemann surface 47 B.3.6 From N = 4 SQM to branes 48 B.3.7 From branes so SQM 48 C The category of BBB branes 49 D Gauge group reductions 50 D.1 Dirichlet boundary conditions 50 D.2 Nahm pole boundary conditions and interfaces 51manner, by tensor product with a certain reference bundle. 'T Hooft operators act by Hecke correspondences, functors which play a key role in Geometric Langlands. Dirichlet boundary conditions D produce "eigenbranes" for the action of Wilson lines, simply because they fix the value at the boundary of the connection and scalar fields which enter the Wilson line path-ordered exponential. S-duality exchanges Wilson lines and 't Hooft lines. As a consequence, the S-dual boundary condition ∨ D will produce "eigenbranes" for the action of 't Hooft lines, i.e. Hecke eigensheaves.The analysis of [1] did not employ directly the S-dual boundary condition ∨ D, as the action of S-duality on boundary conditions for non-Abelian gauge theory was not understood at the time. This provided an important motivation for the work of [4, 5], which described general half-BPS boundary conditions four-dimensional N = 4 gauge theory and their behaviour under the action of S-duality.One conclusion of that work was that ∨ D can be defined with the help of a very special three-dimensional N = 4 SCFT T [G], which unfortunately lacks a simple UV Lagrangian description. For some classical groups a UV Lagrangian description is available, but breaks symmetries which are crucial for GL applications. On the other hand, the analysis also provided us with a large list of S-dual pairs of boundary conditions and interfaces which do admit an effective UV Lagrangian description.The purpose of this paper is to make explicit the map from half-BPS boundary conditions which admit an effective Lagrangian description to objects in the categories relevant for the Geometric Langlands duality (see [6] for some early work on the subject). The expected payoff is a long list of Geometric Langlands dual pairs of mathematical objects which can be defined uniformly on any Riemann surface.The obvious limitation of the computational techniques employed in this paper is the requirement of an effective UV description of the boundary conditions. This prevents us from, say, constructing generic Hecke eigensheaves directly from ∨ D. In a separate paper, [7], we will discuss alternative computational ap...

Abelian dualities of $$ \mathcal{N} $$ = (0, 4) boundary conditions

“…The action of S-duality beautifully produces dualities of these boundary conditions [5] as well as 3d N = 4 mirror symmetry [6]. These half-BPS boundary conditions can also play a role in the Geometric Langlands Program as they can be mapped to mathematical objects in the categories relevant for the Geometric Langlands duality [7,8]. Moreover, such half-BPS interfaces in four-dimensional N = 2 gauge theories have played a central role in finding new duality, that is 3d-3d relation [10,11,12].…”

Spinors, Lagrangians and rank 2 Higgs bundles

“…The following case (and its outcome) was suggested to the author by D. Gaiotto (see [7,Section 7]). Take the group G c = SL(2, C) × SL(2, C) × SL(2, C) acting on U 1 ⊗ U 2 ⊗ U 3 , the tensor product of the three two-dimensional representations.…”

“…The holomorphic symplectic structure is the common feature here, the fibres of both being Lagrangian submanifolds, the second case forming an algebraically completely integrable system. This paper concerns another construction of Lagrangians due to Gaiotto [7], which we use to investigate the simplest case of rank 2 Higgs bundles. The Lagrangians appear in different ways corresponding to the two viewpoints.…”

Quantum Langlands dualities of boundary conditions, $D$-modules, and conformal blocks

“…In the past twenty years, a great effort has been made to search for general methods and structures that could shed light on how and why quantum Geometric Langlands (qGL) duality (1.1) comes about (see [23,55,54,67,44,46,36,37,16,1] for a partial list of references).…”

“…In this paper, we build on these results, as well as the recent works [36,37,39,16], to present a systematic study of the quantum Geometric Langlands dualities in the framework of boundary conditions in 4d gauge theory and the corresponding junction vertex algebras. Here we list the main ingredients of our approach:…”