Superpotentials, quantum parameter space and phase transitions in $ \mathcal{N} $ = 1 supersymmetric gauge theories

Abstract

We study the superpotentials, quantum parameter space and phase transitions that arise in the study of large N dualities between N = 1 SUSY U (N ) gauge theories and string models on local Calabi-Yau manifolds. The main tool of our analysis is a notion of spectral curve characterized by a set of complex partial 't Hooft parameters and cuts given by projections on the spectral curve of minimal supersymmetric cycles of the underlying Calabi-Yau manifold. We introduce a prepotential functional via a variational problem which determines the complex density as an extremal constrained by the period conditions. This prepotential is shown to satisfy the special geometry relations of the spectral curve. We give a system of equations for the branch points of the spectral curves and determine the appropriate branch cuts as Stokes lines of a suitable set of polynomials. As an application, we use a combination of analytical and numerical methods to study the cubic model, determine the analytic condition satisfied by critical one-cut spectral curves, and characterize the transition curves between the one-cut and two-cut phases both in the space of spectral curves and in the quantum parameter space.

Zero distribution of complex orthogonal polynomials with respect to exponential weights

“…in detail in [1] with a combination of analytical and numerical techniques. See also [2,7,11] for related results.…”

“…This work is motivated by a computational approach to oscillatory integrals based on steepest descent analysis [8]. The cubic model is also studied in detail in [1] with a combination of analytical and numerical techniques. See also [2,7,11] for related results.…”

Spectral curves in gauge/string dualities: integrability, singular sectors and regularization

“…and A j are counterclockwise cycles around q disjoint oriented curves (cuts) γ j with endpoints (β 2 j−1 , β 2 j ). This work is a natural continuation of our study of phase transitions in random matrix models [8] and in the space of vacua in supersymmetric gauge theories [20]. Our aim now is to study the structure of the moduli space of the curves (1) which satisfy a system of period relations of the form (5).…”

“…The regular sectors contain the spectral curves which generically are determined by the constraints (5), so that they depend on q moduli which may be either the 't Hooft parameters or a subset of q deformation parameters. In the language of supersymmetric gauge theory [21], the regular sectors represent the different quantum phases of the vacua of the gauge model and, consequently, the singular sectors describe the regions of phase transitions [20].…”