Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics

2006/08/21 by Sergio Benvenuti, Bo Feng, Amihay Hanany +1 · 1 voice · 2 citations
Physics and Astronomy · Mathematics · #hep-th #math.AG

paper · pdf · doi:10.1088/1126-6708/2007/11/050

published as JHEP 0711:050,2007 · 59+1 pages, 7 Figures

Abstract

We develop a systematic and efficient method of counting single-trace and multi-trace BPS operators with two supercharges, for world-volume gauge theories of N D-brane probes for both N → ∞ and finite N. The techniques are applicable to generic singularities, orbifold, toric, non-toric, complete intersections, et cetera, even to geometries whose precise field theory duals are not yet known. The so-called ``Plethystic Exponential'' provides a simple bridge between (1) the defining equation of the Calabi-Yau, (2) the generating function of single-trace BPS operators and (3) the generating function of multi-trace operators. Mathematically, fascinating and intricate inter-relations between gauge theory, algebraic geometry, combinatorics and number theory exhibit themselves in the form of plethystics and syzygies.

Cited by

• Integrable Systems for Generalized Toric Polygons and Higgsed 5d N=1 Theories
• GOOFy-compatible 3HDMs and beyond

Discussions

• TL;DR: At the level of Hilbert spaces, the plethystic exponential is a map from a 1-particle Hilbert space ${\cal H}_1$ to the corresponding multi-particle Fock-space $$\begin{align} {\cal H}^B~=~F^B( [stackexchange, 1 points]