Counting gauge invariants: The Plethystic program

By Bo Feng, A. Hanany, Yang-Hui He. 2007

335 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/8c80bc4672dc9c292fe8daff8f9db50bd1927d5f

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Citations in the corpus

Branes, anti-branes and Brauer algebras in gauge-gravity duality (2007, background): “For other recent discussions of finite N matrix counting problems see [37, 38, 39].”
Tilings, Chern-Simons Theories and M2 Branes (2008, methodology): “Under the assumption that the moduli space is a symmetric product we can compute the Hilbert series - the partition function for chiral operators - for all values of N using the Plethystic Exponential [33, 34, 35].”
N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals (2008, background): “We can find the protected chiral operators by noticing that these operators are given by considering a certain quantum mechanics on the moduli space, as in [26-28].”
3-Manifolds and 3d Indices (2011, background): “[48]) may help to convert indices IT (−m,−e; q−1) to series in q.”
Low-derivative operators of the Standard Model effective field theory via Hilbert series methods (2015, background): “oups of a generating function known as the 1Provided the ﬁelds transform linearly under the symmetries of the theory, and the symmetry groups are compact Lie groups. – 2 – plethystic exponential (PE) [4, 6, 7, 10, 18]. The input to the PE is the ﬁeld content of the theory, meaning the number of ﬂavo”
Elliptic Hypergeometry of Supersymmetric Dualities (2009, neutral): “, [3,4,24,29,49,57,58,80,95].”
Applications of the superconformal index for protected operators and q-hypergeometric identities to N=1 dual theories (2008, background): “In general to calculate the index it is necessary to identify a supercharge Q, with associated adjoint1 Q+, such that {Q,Q+} = 2H , Q2 = 0 , (1.1) so that H has a positive semi-definite spectrum.”
Hilbert Series for Constructing Lagrangians: expanding the phenomenologist's toolbox (2015, background): “6) is an example of a plethystic exponential [7, 9, 10, 13, 21], a generating function of all symmetric combinations of its argument (see A.”
Diagonal free field matrix correlators, global symmetries and giant gravitons (2008, background): “For developments on the use of the combinatorics related to plethysms in the context of chiral rings of a large class of N = 1 SYM theories see [20].”
On instantons as Kaluza-Klein modes of M5-branes (2011, background): “Perhaps a subtlety in imposing constraints [58, 59] should be properly understood.”
SQCD: A Geometric Apercu (2008, methodology): “We will thus use the plethystic logarithm to deduce the number of generators and constraints at each order of quarks and antiquarks from the generating function [17, 18].”
THE MASTER SPACE OF N=1 GAUGE THEORIES (2008, methodology): “In [9], it was pointed out that it is also key to the problem of counting gauge invariant operators in the quiver gauge theory.”