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

In Part I, we extend our analysis in [<a href="/abs/0807.1107">arXiv:0807.1107</a>], 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.