Counting BPS operators in gauge theories: quivers, syzygies and plethystics

Abstract

We develop a systematic and efficient method of counting single-trace and multi-trace BPS operators with two supercharges, for world-volume gauge theories of N D-brane probes for both N → ∞ and finite N . The techniques are applicable to generic singularities, orbifold, toric, non-toric, complete intersections, et cetera, even to geometries whose precise field theory duals are not yet known. The socalled "Plethystic Exponential" provides a simple bridge between (1) the defining equation of the Calabi-Yau, (2) the generating function of single-trace BPS operators and (3) the generating function of multi-trace operators. Mathematically, fascinating and intricate inter-relations between gauge theory, algebraic geometry, combinatorics and number theory exhibit themselves in the form of plethystics and syzygies.

At the horizon of a supersymmetricAdS5black hole: Isometries and half-BPS giants

“…To make further progress in this direction one has to classify all the BPS objects in the global coordinates with a given set of supersymmetries. Then one should be able to quantise them using methods similar to [13,14] (see also [32,33,34,35]) and count the different configurations with fixed quantum numbers.…”

New results for the SQCD Hilbert series

“…We will review the Hilbert series and its connection to (2) in Section 2. We discuss two techniques that have been applied to the integrals (2), and will hence also apply to the Hilbert series: i) the use of orthogonal polynomials on the unit circle (Schur polynomials), and ii) relating the integrals to partition functions of loggases with external charges, and then using electrostatics for saddle point estimates.…”

“…2 In this paper we derive several new results for the Hilbert series based on standard properties of Schur polynomials. The second technique to be discussed in this paper, the log-gas approach, has been studied by the authors previously in [24,29] as a way to obtain useful approximations to integrals (2). 1 Incidentally, as a precursor of connections to Yang-Mills theory, the appearance of U (N ) matrix integrals inspired [22] to speculate that TBL is related to QCD in two dimensions.…”

“…1 Incidentally, as a precursor of connections to Yang-Mills theory, the appearance of U (N ) matrix integrals inspired [22] to speculate that TBL is related to QCD in two dimensions. 2 It is interesting to note that recent studies of N = 2 dualities utilize Macdonald polynomials, which are generalizations of Schur polynomials [26,27]. Moreover, in some cases the superconformal index can be reduced to the Hilbert series [28].…”

Refined partition functions for open superstrings with 4, 8 and 16 supercharges

“…It has also been extensively applied to the study of moduli spaces of supersymmetric gauge theories [10][11][12][13][14][15][16][17][18]; in such a context the partition function is also known as the Hilbert series.…”