We discuss ensemble averages of two-dimensional conformal field theories associated with an arbitrary indefinite lattice with integral quadratic form Q. We provide evidence that the holographic dual after the ensemble average is the three-dimensional Abelian Chern-Simons theory with kinetic term determined by Q. The resulting partition function can be written as a modular form, expressed as a sum over the partition functions of Chern-Simons theories on lens spaces. For odd lattices, the dual bulk theory is a spin Chern-Simons theory, and we identify several novel phenomena in this case. We also discuss the holographic duality prior to averaging in terms of Maxwell-Chern-Simons theories.
We show that the four-dimensional Chern-Simons theory studied by Costello, Witten and Yamazaki, is, with Nahm pole-type boundary conditions, dual to a boundary theory that is a three-dimensional analogue of Toda theory with a novel 3d W-algebra symmetry. By embedding four-dimensional Chern-Simons theory in a partial twist of the five-dimensional maximally supersymmetric Yang-Mills theory on a manifold with corners, we argue that this three-dimensional Toda theory is dual to a two-dimensional topological sigma model with A-branes on the moduli space of solutions to the Bogomolny equations. This furnishes a novel 3d-2d correspondence, which, among other mathematical implications, also reveals that modules of the 3d W-algebra are modules for the quantized algebra of certain holomorphic functions on the Bogomolny moduli space.
We explain how, starting with a stack of D4-branes ending on an NS5-brane in type IIA string theory, one can, via T-duality and the topological-holomorphic nature of the relevant worldvolume theories, relate (i) the lattice models realized by Costello's 4d Chern-Simons theory, (ii) links in 3d analytically-continued Chern-Simons theory, (iii) the quantum geometric Langlands correspondence realized by Kapustin-Witten using 4d N = 4 gauge theory and its quantum group modification, and (iv) the Gaitsgory-Lurie conjecture relating quantum groups/affine Kac-Moody algebras to Whittaker D-modules/W-algebras. This furnishes, purely physically via branes in string theory, a novel bridge between the mathematics of integrable systems, geometric topology, geometric representation theory, and quantum algebras.
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