Counting Gauge Invariants: the Plethystic Program

TL;DR

This paper introduces a systematic method using plethystic functions to count gauge invariant operators in gauge theories, revealing deep links between geometry and gauge theory, and enabling entropy calculations.

Contribution

It develops a plethystic program for counting gauge invariants, connecting geometric structures with gauge theory operators and extending to entropy computations.

Findings

Effective counting of gauge invariants via plethystic functions

Connections established between geometry, Young Tableaux, and gauge theories

Entropy of gauge theories computed using plethystic exponential

Abstract

We propose a programme for systematically counting the single and multi-trace gauge invariant operators of a gauge theory. Key to this is the plethystic function. We expound in detail the power of this plethystic programme for world-volume quiver gauge theories of D-branes probing Calabi-Yau singularities, an illustrative case to which the programme is not limited, though in which a full intimate web of relations between the geometry and the gauge theory manifests herself. We can also use generalisations of Hardy-Ramanujan to compute the entropy of gauge theories from the plethystic exponential. In due course, we also touch upon fascinating connections to Young Tableaux, Hilbert schemes and the MacMahon Conjecture.