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Title:
Developing the Quantum q-Langlands Program
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Abstract:
In its simplest incarnation, the geometric Langlands program was defined by Beilinson and Drinfeld in the 90’s as relating, on one side, a flat connection on a Riemann surface, and on the other side, a more sophisticated structure known as a D-module. Since its inception, this conjectured correspondence has been a highly active and fruitful topic of research both for mathematicians and theoretical physicists. In this talk, we will review a generalization of the correspondence known as the quantum q-Langlands program, due to Aganagic-Frenkel-Okounkov, which establishes an isomorphism between q-deformed versions of conformal blocks, for a W-algebra on one side, and a Langlands dual affine Lie algebra on the other side. Next, we will elucidate the meaning of ramification in this program, or adding punctures with data on the Riemann surface. We will present various applications: the construction of the primary vertex operators of deformed W-algebras, and the physics of handsaw quiver gauge theories in three dimensions with N=2 supersymmetry. Finally, we will comment on the conformal limit; for instance, when the Lie algebra is specialized to be sl(2), one obtains a new (dual) perspective on recent results of Nekrasov and Tsymbaliuk.
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