Abstract
We give a microscopic two dimensional ${\cal N}=(2,2)$ gauge theory description of arbitrary M2-branes ending on $N_f$ M5-branes wrapping a punctured Riemann surface. These realize surface operators in four dimensional ${\cal N}=2$ field theories. We show that the expectation value of these surface operators on the sphere is captured by a Toda CFT correlation function in the presence of an additional degenerate vertex operator labelled by a representation ${\cal R}$ of $SU(N_f)$, which also labels M2-branes ending on M5-branes. We prove that symmetries of Toda CFT correlators provide a geometric realization of dualities between two dimensional gauge theories, including ${\cal N}=(2,2)$ analogues of Seiberg and Kutasov--Schwimmer dualities. As a bonus, we find new explicit conformal blocks, braiding matrices, and fusion rules in Toda CFT.
Highlights
Introduction and conclusionsThe traditional order parameters for the phases of four dimensional gauge theories are the Wilson [1] and ’t Hooft [2] operators
We show that the expectation value of these surface operators on the sphere is captured by a Toda CFT correlation function in the presence of an additional degenerate vertex operator labelled by a representation R of SU(Nf ), which labels M2-branes ending on M5-branes
We prove that symmetries of Toda CFT correlators provide a geometric realization of dualities between two dimensional gauge theories, including N = (2, 2) analogues of Seiberg and Kutasov-Schwimmer dualities
Summary
The traditional order parameters for the phases of four dimensional gauge theories are the Wilson [1] and ’t Hooft [2] operators. We review the case of SQED in some detail in section 2.1: this U(1) gauge theory corresponds to the insertion of the simplest Toda CFT degenerate vertex operator, labeled by the fundamental representation of ANf −1 [18]. Conjugating all Toda CFT momenta yields a different set of degenerate operators of the same type, and it turns out that the corresponding dual gauge theories are related by a sequence of Seiberg and N = (2, 2)∗ dualities on all nodes. A particular case is the quiver (1.1) which corresponds to a single degenerate vertex operator labeled by an arbitrary representation R: applying the same sequence of Seiberg and N = (2, 2)∗ dualities corresponds to conjugating R and all Toda CFT momenta This result concludes the description of dualities of two dimensional N = (2, 2) gauge theories which correspond to manifest symmetries of the Toda CFT. Appendix B features analytic proofs that vortex partition of dual theories are equal, for Seiberg duality (appendix B.1), and for dualities of SQCD with an adjoint (appendix B.2)
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