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

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arxiv hep-th/0608050 v2 pith:OPKVNAAP submitted 2006-08-07 hep-th math.AG

Sergio Benvenuti , Bo Feng , Amihay Hanany , Yang-Hui He This is my paper · ORCID
classification hep-thmath.AG
keywords operatorsgaugetheorycountingfunctiongeneratingmulti-traceplethystics
verification ladder T0 review T1 audit T2 compute T3 formal

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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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