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

Sergio Benvenuti·Bo Feng·Amihay Hanany·Yang-Hui He
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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.

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Sergio Benvenuti, Bo Feng, Amihay Hanany, Yang-Hui He. 2006-08-21. Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics. https://doi.org/10.1088/1126-6708%2F2007%2F11%2F050

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