Higher AGT Correspondences, W-algebras, and Higher Quantum Geometric Langlands Duality from M-Theory

Abstract

We further explore the implications of our framework in [arXiv:1301.1977, arXiv:1309.4775], and physically derive, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent, (i) a 5d AGT correspondence for any compact Lie group, (ii) a 5d and 6d AGT correspondence on ALE space of type ADE, and (iii) identities between the ordinary, q-deformed and elliptic affine W-algebras associated with the 4d, 5d and 6d AGT correspondence, respectively, which also define a quantum geometric Langlands duality and its higher analogs formulated by Feigin-Frenkel-Reshetikhin in [3,4]. As an offshoot, we are led to the sought-after connection between the gauge-theoretic realization of the geometric Langlands correspondence by Kapustin-Witten [5,6] and its algebraic CFT formulation by Beilinson-Drinfeld [7], where one can also understand Wilson and 't Hooft-Hecke line operators in 4d gauge theory as monodromy loop operators in 2d CFT, for example. In turn, this will allow us to argue that the higher 5d/6d analog of the geometric Langlands correspondence for simply-laced Lie (Kac-Moody) groups G (G_k), ought to relate the quantization of circle (elliptic)-valued G Hitchin systems to circle/elliptic-valued ^LG (^LG_k)-bundles over a complex curve on one hand, and the transfer matrices of a G (G_k)-type XXZ/XYZ spin chain on the other, where ^LG is the Langlands dual of G. Incidentally, the latter relation also serves as an M-theoretic realization of Nekrasov-Pestun-Shatashvili's recent result in [8], which relates the moduli space of 5d/6d supersymmetric G (G_k)-quiver SU(K_i) gauge theories to the representation theory of quantum/elliptic affine (toroidal) G-algebras.

73 pages. Presented at "Strings 2016". Published version

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Cited by in corpus (11)

• $(q,t)$-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces
• Anomaly in RTT relation for DIM algebra and network matrix models
• A slow review of the AGT correspondence
• Triality in Little String Theories
• (q,t)-KZ equation for Ding-Iohara-Miki algebra
• On determinant representation and integrability of Nekrasov functions
• A Macdonald refined topological vertex
• Orthogonal Polynomials in Mathematical Physics
• q-Painleve equation from Virasoro constraints
• Elliptic modular double and 4d partition functions
• Elliptic algebra, Frenkel-Kac construction and root of unity limit