Counting Gauge Invariants: the Plethystic Program

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.

51 pages, 2 figures; refs updated, typos corrected

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