Mastering the Master Space

TL;DR

Notes the master space geometry as a central organizing object for the vacuum moduli spaces of N=1 quiver gauge theories from D3-branes at Calabi-Yau singularities. It develops a toric-algebraic framework where the F-flat space is toric with a coherent Calabi-Yau component and a symplectic/GLSM description via perfect matchings, enabling Hilbert-series calculations that expose palindromicity and hidden non-Abelian symmetries. The Plethystic program connects the one-brane master space to multi-brane spectra, yielding the N-brane moduli space and Higgsing flows between toric theories. It shows Seiberg-duality invariance for the coherent component and reveals that the refined Hilbert series organizes into representations of hidden global symmetries, with practical implications for counting BPS operators and mapping moduli spaces in string-theory quiver gauge theories.

Abstract

Supersymmetric gauge theories have an important but perhaps under-appreciated notion of a master space, which controls the full moduli space. For world-volume theories of D-branes probing a Calabi-Yau singularity X the situation is particularly illustrative. In the case of one physical brane, the master space F is the space of F-terms and a particular quotient thereof is X itself. We study various properties of F which encode such physical quantities as Higgsing, BPS spectra, hidden global symmetries, etc. Using the plethystic program we also discuss what happens at higher number N of branes. This letter is a summary and some extensions of the key points of a longer companion paper arXiv:0801.1585.