Counting Gauge Invariants: the Plethystic Program

TL;DR

The paper develops a systematic plethystic framework for counting gauge-invariant operators in D-brane quiver gauge theories on Calabi–Yau singularities, tying Hilbert series to the geometry of the moduli space and using plethystic exponential/inverse to generate single- and multi-trace spectra. It delivers explicit constructions and asymptotic analyses for broad classes of geometries, including ADE and SU(3) orbifolds, along with refined generating functions and connections to Young tableaux, MacMahon/Birational partition problems, and Hilbert schemes. The work demonstrates how entropy and operator growth in gauge theories can be extracted from the geometry via the Meinardus/Haselgrove–Temperley machinery, and it extends the program to discrete torsion, toric $Y^{p,q}$ spaces, and symmetric products, suggesting wide applicability beyond the original D-brane context. Overall, the plethystic programme provides a concrete, geometry-driven method to count and understand operator spectra and their asymptotics in a broad landscape of gauge theories.

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.

Figures (2)

• Figure 1: Examples of Young Tableaux with partition $p=\{p_1,p_2,p_3,...,p_k,...\}$ and $N$. The constraint is that $N = \sum\limits_{k=1} p_k k$.
• Figure 2: The toric diagram for the space $Y^{p,q}$.