Counting gauge invariants: the plethystic program

Abstract

We propose a programme for systematically counting the single and multitrace 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 worldvolume 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. * b.feng@imperial.ac.uk † ahanany@perimeterinstitute.ca ‡ hey@maths.ox.ac.uk

Low-energy dynamics of 3d N $$ \mathcal{N} $$ = 2 G2 supersymmetric gauge theory

“…In this paper we have shown the existence of the one-dimensional Coulomb branch, so it is possible to calculate a Hilbert seriesà la [58][59][60][61][62]. Hilbert series basically counts the holomorphic (gauge invariant) operators in a theory.…”

BPS states in the Minahan-Nemeschansky $${E_6}$$ theory

“…6 The passage from Q(p) to the coefficients α n (p) is an example of a "plethystic logarithm," as found in [27]; these appear in many counting problems in gauge theory, as discussed in [28].…”

Brane tilings and reflexive polygons

“…The generating function of mesonic gauge invariant operators (GIOs) is known as the mesonic Hilbert series [57][58][59][60][61]. The Hilbert series encodes the generators of the associated moduli space.…”

“…In order to have a complete classification of the mesonic moduli spaces, the moduli space generators for all 30 quiver gauge theories are found by computing a generating function known as the Hilbert series [57][58][59][60][61]. The Hilbert series encodes information about the moduli space generators.…”