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

TL;DR

This paper introduces a systematic method for counting BPS operators in gauge theories associated with D-branes, using plethystics and algebraic geometry, applicable to various singular geometries and finite or infinite N.

Contribution

It develops a novel, efficient counting technique for BPS operators that works for general geometries and finite N, extending previous methods.

Findings

The plethystic exponential links Calabi-Yau geometry to operator counting.

Mathematical structures like syzygies reveal deep relations between gauge theories and geometry.

The method applies to non-toric and incomplete intersection geometries.

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 $N \to \infty$ 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.