Unifying lattice models, links and quantum geometric Langlands via branes in string theory

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 $\mathcal{N}=4$ 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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Published 8 September 2021