A new realization of quantum algebras in gauge theory and Ramification in the Langlands program

We realize the fundamental representations of quantum algebras via the supersymmetric Higgs mechanism in gauge theories with 8 supercharges on an $\Omega$-background. We test our proposal for quantum affine algebras, by probing the Higgs phase of a 5d quiver gauge theory on a circle. We show that our construction implies the existence of tame ramification in the Aganagic-Frenkel-Okounkov formulation of the geometric Langlands program, a correspondence which identifies $q$-conformal blocks of the quantum affine algebra with those of a Langlands dual deformed ${\cal W}$-algebra. The new feature of ramified blocks is their definition in terms of Drinfeld polynomials for a set of quantum affine weights. In enumerative geometry, the blocks are vertex functions counting quasimaps to quiver varieties describing moduli spaces of vortices. Physically, the vortices admit a description as a 3d ${\cal N}=2$ quiver gauge theory on the Higgs branch of the 5d gauge theory, uniquely determined from the Drinfeld polynomial data; the blocks are supersymmetric indices for the vortex theory supported on a 3-manifold with distinguished BPS boundary conditions. The top-down explanation of our results is found in the 6d $(2,0)$ little string theory, where tame ramification is provided by certain D-branes. When the string mass is taken to be large, we make contact with various physical aspects of the point particle superconformal limit: the Gukov-Witten description of ramification via monodromy defects in 4d Super Yang-Mills (and their S-duality), the Nekrasov-Tsymbaliuk solution to the Knizhnik-Zamolodchikov equations, and the classification of massive deformations of tamely ramified Hitchin systems. In a companion paper, we will show that our construction implies a solution to the local Alday-Gaiotto-Tachikawa conjecture.Comment: 246 pages, 18 figures; v.2: added references, fixed typos and figure

Aspects of Supersymmetric Surface Defects

Starting from type IIB string theory on an $ADE$ singularity, the \cN=(2,0) little string arises when one takes the string coupling $g_s$ to 0. We compactify the little string on the cylinder with punctures, which we fully characterize, for any simple Lie algebra \fg. Geometrically, these punctures are codimension two defects that are D5 branes wrapping 2-cycles of the singularity. Equivalently, the defects are specified by a certain set of weights of ^L \fg, the Langlands dual of \fg. As a first application of our formalism, we show that at low energies, the defects have a description as a \fg-type quiver gauge theory. We compute its partition function, and prove that it is equal to a conformal block of \fg-type $q$-deformed Toda theory on the cylinder, in the Coulomb gas formalism.After taking the string scale limit $m_s\rightarrow\infty$, the little string becomes a $(2,0)$ superconformal field theory (SCFT). As a second application, we study how this limit affects the codimension two defects of the SCFT:we show that the Coulomb branch of a given defect flows to a nilpotent orbit of \fg, and that all nilpotent orbits of \fg arise in this way. We give a physical realization of the Bala--Carter labels that classify nilpotent orbits of simple Lie algebras, and we interpret our results in the context of \fg-type Toda. Finally, after compactifying our setup on a torus $T^2$, we make contact with the description of surface defects of 4d $\mathcal{N}=4$ Super Yang-Mills theory due to Gukov and Witten \cite{Gukov:2006jk}

Little string origin of surface defects

Abstract We derive a large class of codimension-two defects of 4d N = 4 $$\mathcal{N}=4$$ Super Yang-Mills (SYM) theory from the (2, 0) little string. The origin of the little string is type IIB theory compactified on an ADE singularity. The defects are D-branes wrapping the 2-cycles of the singularity. We use this construction to make contact with the description of SYM defects due to Gukov and Witten [1]. Furthermore, we provide a geometric perspective on the nilpotent orbit classification of codimension-two defects, and the connection to ADE-type Toda CFT. The only data needed to specify the defects is a set of weights of the algebra obeying certain constraints, which we give explicitly. We highlight the differences between the defect classification in the little string theory and its (2, 0) CFT limit