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Register a new userOn Duality in $mathcal{N}=2$ supersymmetric Liouville Theory
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Yu Nakayama
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Similarly to the bosonic Liouville theory, the $mathcal{N}=2$ supersymmetric Liouville theory was conjectured to be equipped with the duality that exchanges the superpotential and the Kahler potential. The conjectured duality, however, seems to suffer from a mismatch of the preserved symmetries. More than fifteen years ago, when I was a student, my supervisor Tohru Eguchi gave a beautiful resolution of the puzzle when the supersymmetry is enhanced to $mathcal{N}=4$ based on his insight into the underlying geometric structure of the $A_1$ singularity. I will review his unpublished but insightful idea and present our attempts to extend it to more general cases.
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A solution to the infinite coupling problem for N=2 conformal supersymmetric gauge theories in four dimensions is presented. The infinitely-coupled theories are argued to be interacting superconformal field theories (SCFTs) with weakly gauged flavor groups. Consistency checks of this proposal are found by examining some low-rank examples. As part of these checks, we show how to compute new exact quantities in these SCFTs: the central charges of their flavor current algebras. Also, the isolated rank 1 E_6 and E_7 SCFTs are found as limits of Lagrangian field theories.
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The maximal extension of supersymmetric Chern-Simons theory coupled to fundamental matter has $mathcal{N} = 3$ supersymmetry. In this short note, we provide the explicit form of the action for the mass-deformed $mathcal{N} = 3$ supersymmetric $U(N)$ Chern-Simons-Matter theory. The theory admits a unique triplet mass deformation term consistent with supersymmetry. We explicitly construct the mass-deformed $mathcal{N} = 3$ theory in $mathcal{N} = 1$ superspace using a fundamental and an anti-fundamental superfield.
Strings in $mathcal{N}=2$ supersymmetric ${rm U}(1)^N$ gauge theories with $N$ hypermultiplets are studied in the generic setting of an arbitrary Fayet-Iliopoulos triplet of parameters for each gauge group and an invertible charge matrix. Although the string tension is generically of a square-root form, it turns out that all existing BPS (Bogomolnyi-Prasad-Sommerfield) solutions have a tension which is linear in the magnetic fluxes, which in turn are linearly related to the winding numbers. The main result is a series of theorems establishing three different kinds of solutions of the so-called constraint equations, which can be pictured as orthogonal directions to the magnetic flux in ${rm SU}(2)_R$ space. We further prove for all cases, that a seemingly vanishing Bogomolnyi bound cannot have solutions. Finally, we write down the most general vortex equations in both master form and Taubes-like form. Remarkably, the final vortex equations essentially look Abelian in the sense that there is no trace of the ${rm SU}(2)_R$ symmetry in the equations, after the constraint equations have been solved.
We study $mathcal{N} = 3$ supersymmetric Chern-Simons-matter theory coupled to matter in the fundamental representation of $SU(N)$. In the t Hooft large $N$ limit, we compute the exact $2 to 2$ scattering amplitudes of the fundamental scalar superfields to all orders in the t Hooft coupling $lambda$. Our computations are presented in $mathcal{N} = 1$ superspace and make significant use of the residual $SO(2)_R$ symmetry in order to solve for the exact four-point correlator of the scalar superfields. By taking the on-shell limit, we are able to extract the exact $2 to 2$ scattering amplitudes of bosons/fermions in the symmetric, anti-symmetric and adjoint channels of scattering. We find that the scattering amplitude of the $mathcal{N} = 3$ theory in the planar limit is tree-level exact to all orders in the t Hooft coupling $lambda$. The result is consistent with the conjectured bosonization duality and is expected to have enhanced symmetry structures such as dual superconformal symmetry and Yangian symmetry.
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