Deformations of the moduli space and superpotential flows in 3D SUSY QCD

Yuri Shirman,Michael Waterbury

doi:10.1103/physrevd.99.125002

Yuri Shirman, Michael Waterbury

Open Access

https://doi.org/10.1103/physrevd.99.125002

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Abstract

We study the moduli space of three-dimensional $\mathcal{N}=2$ SQCD with $SU(N)$ gauge group and $F<N$ massless flavors. In the case of an $SU(2)$ theory with a single massless flavor, we explicitly calculate the quantum constraint $YM=1$ and generalize the calculation to models with arbitrary $N$ and $F=N-1$ flavors. In theories with $F<N-1$ flavors, we find that analogous constraints exist in locally defined coordinate charts of the moduli space. The existence of such constraints allows us to show that the Coulomb branch superpotential generated by single monopole effects is equivalent to the superpotential generated by multi-monopole contributions on the mixed Higgs-Coulomb branch. As a check for our result, we implement the local constraints as Lagrange multiplier terms in the superpotential and verify that deformations of a theory by a large holomorphic mass term for the matter fields results in a flow of the superpotential from the $F$-flavor model to the superpotential of an $(F-1)$-flavor model.

Highlights

The study of nonperturbative dynamics of SUSY gauge theories in three and four dimensions leads to invaluable insights in understanding quantum field theories
The existence of local constraints allowed us to construct a set of coordinate charts that covers the entire moduli space and show that the superpotential calculations in different charts are equivalent
The existence of local constraints ensures the agreement between the superpotentials generated by fundamental monopoles on the pure Coulomb branch of an SUðNÞ theory [11] with the superpotentials arising from fundamental monopoles and multimonopole contributions on the mixed Higgs-Coulomb branch of the theory [20]

Summary

INTRODUCTION

The study of nonperturbative dynamics of SUSY gauge theories in three and four dimensions leads to invaluable insights in understanding quantum field theories. Recall that this term in the superpotential can be interpreted as a two-monopole contribution generated on the mixed Higgs-Coulomb branch accessible from the boundary between the (k þ 1)st or (k þ 2)nd subwedges This procedure can be used to recursively generate the sets of coordinate charts and transition functions required to cover the entire quantum moduli space of the theory and to define it as a smooth, locally connected manifold. This local constraint given by is easiest to see by performing a calculation on the mixed Higgs-Coulomb branch where the rank F − 1 meson VEV is allowed This is the region VEVs, where vi 1⁄4 0 for i 1⁄4 the multimonopole Y. into a single fundamental monopole of the low energy SUðN − F þ 1Þ theory Y1L 1⁄4 Yð11;FÞ det0 M=g2ðF−1Þ where det0 M denotes a determinant taken over F − 1 flavors with large VEV.

Y ð12Þ

CONCLUSIONS

Monopole-instanton measure

Gaugino zero modes

Quark zero modes

Evaluating the integral