M-Theoretic Derivations of 4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems

Abstract

In Part I, we extend our analysis in [arXiv:0807.1107], and show that a mathematically conjectured geometric Langlands duality for complex surfaces in [1], and its generalizations -- which relate some cohomology of the moduli space of certain ("ramified") G-instantons to the integrable representations of the Langlands dual of certain affine (sub) G-algebras, where G is any compact Lie group -- can be derived, purely physically, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent. In Part II, to the setup in Part I, we introduce Omega-deformation via fluxbranes and add half-BPS boundary defects via M9-branes, and show that the celebrated AGT correspondence in [2, 3], and its generalizations -- which essentially relate, among other things, some equivariant cohomology of the moduli space of certain ("ramified") G-instantons to the integrable representations of the Langlands dual of certain affine W-algebras -- can likewise be derived from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent. In Part III, we consider various limits of our setup in Part II, and connect our story to chiral fermions and integrable systems. Among other things, we derive the Nekrasov-Okounkov conjecture in [4] -- which relates the topological string limit of the dual Nekrasov partition function for pure G to the integrable representations of the Langlands dual of an affine G-algebra -- and also demonstrate that the Nekrasov-Shatashvili limit of the "fully-ramified" Nekrasov instanton partition function for pure G is a simultaneous eigenfunction of the quantum Toda Hamiltonians associated with the Langlands dual of an affine G-algebra. Via the case with matter, we also make contact with Hitchin systems and the "ramified" geometric Langlands correspondence for curves.

214 pages. Minor imprecisions corrected. Added acknowledgment

References in corpus (7)

• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• Instanton counting with a surface operator and the chain-saw quiver
• The Omega Deformation From String and M-Theory
• Instanton partition functions in N=2 SU(N) gauge theories with a general surface operator, and their W-algebra duals
• Open M5-branes
• Compactification on the Ω-background and the AGT correspondence
• N=1 superconformal blocks with Ramond fields from AGT correspondence

Cited by in corpus (25)

• Topological strings and 5d T_N partition functions
• M2-brane surface operators and gauge theory dualities in Toda
• Extended Conformal Symmetry and Recursion Formulae for Nekrasov Partition Function
• Classical torus conformal block, N=2* twisted superpotential and the accessory parameter of Lame equation
• A slow review of the AGT correspondence
• Intersecting Surface Defects and Two-Dimensional CFT
• Orbifolds and Exact Solutions of Strongly-Coupled Matrix Models
• 2d-4d Connection between q-Virasoro/W Block at Root of Unity Limit and Instanton Partition Function on ALE Space
• An M-Theoretic Derivation of a 5d and 6d AGT Correspondence, and Relativistic and Elliptized Integrable Systems
• Exact results for $\mathcal{N}=2$ supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants
• N=1 Geometries via M-theory
• Classical irregular block, N=2 pure gauge theory and Mathieu equation
• The Stringy Instanton Partition Function
• 4d Quantum Geometry from 3d Supersymmetric Gauge Theory and Holomorphic Block
• Coset conformal field theory and instanton counting on C^2/Z_p
• Classical limit of irregular blocks and Mathieu functions
• N=2 gauge theories, instanton moduli spaces and geometric representation theory
• q-Vertex Operator from 5D Nekrasov Function
• Classical irregular blocks, Hill's equation and PT-symmetric periodic complex potentials
• AGT relation in the light asymptotic limit
• Refined toric branes, surface operators and factorization of generalized Macdonald polynomials
• The Euler anomaly and scale factors in Liouville/Toda CFTs
• Generalized Toda Theory from Six Dimensions and the Conifold
• Elliptic algebra, Frenkel-Kac construction and root of unity limit
• Super-Whittaker vector at c=3/2