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# Advances in Theoretical and Mathematical Physics

## Volume 22 (2018)

### Number 2

### Higher AGT correspondences, $\mathcal{W}$-algebras, and higher quantum geometric Langlands duality from M-theory

Pages: 429 – 507

DOI: http://dx.doi.org/10.4310/ATMP.2018.v22.n2.a4

#### Author

#### Abstract

We further explore the implications of our framework in [1, 2], 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 $\mathcal{W}$-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 $G (\widehat{G})$, ought to relate the quantization of circle (elliptic)-valued $G$ Hitchin systems to circle/elliptic-valued ${}^L G (\widehat{{}^L G})$-bundles over a complex curve on one hand, and the transfer matrices of a $G (\widehat{G})$-type XXZ/XYZ spin chain on the other, where ${}^L G$ 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 $G (\widehat{G})$-quiver $SU(K_i)$ gauge theories to the representation theory of quantum/elliptic affine (toroidal) $G$-algebras.

In loving memory of See-Hong Tan