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arxiv: hep-th/0608050 · v2 · submitted 2006-08-07 · ✦ hep-th · math.AG

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

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classification ✦ hep-th math.AG
keywords operatorsgaugetheorycountingfunctiongeneratingmulti-traceplethystics
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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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  1. Fermionic trace relations and supersymmetric indices at finite $N$

    hep-th 2026-05 unverdicted novelty 7.0

    The supersymmetric index in a one-fermion matrix model for N=4 SYM is independent of N due to exact cancellations between bosonic and fermionic trace relations.