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We explain how, starting with a stack of D4-branes ending on an NS5-brane intype IIA string theory, one can, via T-duality and the topological-holomorphicnature of the relevant worldvolume theories, relate (i) the lattice modelsrealized by Costello's 4d Chern-Simons theory, (ii) links in 3danalytically-continued Chern-Simons theory, (iii) the quantum geometricLanglands correspondence realized by Kapustin-Witten using 4d N = 4 gaugetheory and its quantum group modification, and (iv) the Gaitsgory-Lurieconjecture relating quantum groups/affine Kac-Moody algebras to WhittakerD-modules/W-algebras. This furnishes, purely physically via branes in stringtheory, a novel bridge between the mathematics of integrable systems, geometrictopology, geometric representation theory, and quantum algebras.

Unifying Lattice Models, Links and Quantum Geometric Langlands via Branes in String Theory