An Exact Elliptic Superpotential for N=1^* Deformations of Finite N=2 Gauge Theories

TL;DR

This work analyzes the ${\cal N}=1^{*}$ deformations of ${\cal N}=2$ quiver theories arising from $N$ D3-branes at elliptic $A_{k-1}$ singularities, establishing the vacuum structure and holomorphic observables for arbitrary $k,N$. The authors employ two complementary routes: lifting to M-theory where the Coulomb branch is encoded by a Seiberg-Witten curve $\Sigma$ and studying its maximally degenerate genus-one limits, and compactifying to three dimensions where mirror symmetry yields an exact superpotential identified with a linear combination of the quadratic Hamiltonians of the elliptic spin Calogero-Moser system. They derive the degeneracy pattern of massive vacua as $N^{k-1}\sum_{p|N} p$ and construct duality actions (including tilde- S-duality) that permute vacua, with chiral condensates encoded in modular covariant quantities $H_i$, $H_0$, and $H^*$. The results connect holomorphic data, integrable systems, and duality symmetries, providing exact formulas for condensates and the 3D superpotential, and paving the way for applications to domain walls, gluino condensates, and string duals of orbifold backgrounds.

Abstract

We study relevant deformations of the N=2 superconformal theory on the world-volume of N D3 branes at an A_{k-1} singularity. In particular, we determine the vacuum structure of the mass-deformed theory with N=1 supersymmetry and show how the different vacua are permuted by an extended duality symmetry. We then obtain exact, modular covariant formulae (for all k, N and arbitrary gauge couplings) for the holomorphic observables in the massive vacua in two different ways: by lifting to M-theory, and by compactification to three dimensions and subsequent use of mirror symmetry. In the latter case, we find an exact superpotential for the model which coincides with a certain combination of the quadratic Hamiltonians of the spin generalization of the elliptic Calogero-Moser integrable system.

Figures (5)

• Figure 1: The Type IIA setup of the elliptic model consisting of four-branes suspended between five-branes arranged on a compact direction.
• Figure 2: ${\cal N}=2^*$ SUSY Yang-Mills realized in Type IIA theory.
• Figure 3: (a) The Coulomb branch singularity of ${\cal N}=2^*$ SUSY Yang-Mills which yields the Higgs vacuum of ${\cal N}=1^*$ theory. (b) The singular point with $\text{SU}(p)^q\times \text{U}(1)^{q-1}$ gauge symmetry which descends to the $\text{SU}(p)$ vacuum of ${\cal N}=1^*$ theory.
• Figure 4: A Coulomb branch singularity of the elliptic model with $\text{SU}(p)^{kq}\times \text{U}(1)^{k(q-1)}$ gauge symmetry. Upon soft breaking to ${\cal N}=1$ SUSY it yields $p^k$ massive vacua.
• Figure 5: The $N$-fold cover of $E_\tau$ with ${\tilde{\tau}}=q\tau/p$. The $\times$'s represent the positions of the $k$ NS5-branes.