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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• Branes, Anti-Branes and Brauer Algebras in Gauge-Gravity duality
• Brane Tilings and Their Applications
• SQCD: A Geometric Apercu
• Superconformal indices for N=1 theories with multiple duals
• Calculating the Superconformal Index and Seiberg Duality
• Counting Chiral Operators in Quiver Gauge Theories
• Towards M2-brane Theories for Generic Toric Singularities
• Counting BPS operators in N=4 SYM
• Comments on 1/16 BPS Quantum States and Classical Configurations
• The Hilbert Series of Adjoint SQCD
• Dimers and Orientifolds
• Phases of M2-brane Theories
• Mastering the Master Space
• M2-Branes and Quiver Chern-Simons: A Taxonomic Study
• Master Space, Hilbert Series and Seiberg Duality
• Supergravitons from one loop perturbative N=4 SYM
• E3-brane instantons and baryonic operators for D3-branes on toric singularities
• BPS Partition Functions for Quiver Gauge Theories: Counting Fermionic Operators
• Counting Wobbling Dual-Giants
• Black Hole Giants
• Finite N Index and Angular Momentum Bound from Gravity
• Master Space and Hilbert Series for N=1 Field Theories
• Geometric aspects of dibaryon operators