SQCD: a geometric aperçu

Abstract

This is the unspecified version of the paper.This version of the publication may differ from the final published version. Abstract: We take new algebraic and geometric perspectives on the old subject of SQCD. We count chiral gauge invariant operators using generating functions, or Hilbert series, derived from the plethystic programme and the Molien-Weyl formula. Using the character expansion technique, we also see how the global symmetries are encoded in the generating functions. Equipped with these methods and techniques of algorithmic algebraic geometry, we obtain the character expansions for theories with arbitrary numbers of colours and flavours. Moreover, computational algebraic geometry allows us to systematically study the classical vacuum moduli space of SQCD and investigate such structures as its irreducible components, degree and syzygies. We find the vacuum manifolds of SQCD to be affine Calabi-Yau cones over weighted projective varieties. Permanent repository link

New results for the SQCD Hilbert series

“…Then we would not need to introduce the O(1) functions C which depend in a complicated way on λ i /s and will cancel in all results. 13 The result (105) covers all nontrivial cases, i.e., it holds for k = 0 or N f ≥ N c . For k = 0 and N f < N c the result is not applicable since we used the formula for d λ+k(M ) instead of d λ+k(Nc) , but in this case actually G(s, k) = 0.…”

“…We also notice that s κ+1 (x) = ( ix i )s κ (x) = s κ (x) by Eqs. (7) and (13). Collecting these observations, we get…”

“…In this paper we focus on a narrowly defined calculational problem: computing explicit results for the Hilbert series of N = 1 supersymmetric QCD with U (N c ) or SU (N c ) color symmetry and N f flavors in the fundamental representation of U (N f ) flavor symmetry. The standard method to calculate the Hilbert series is based on the plethystic programme [2][3][4][5][6][7][8][9][10][11][12][13] and the Molien-Weyl formula [13][14][15]. This method typically leads to expressions with a large number of contour integrals, which makes it difficult to obtain explicit exact results or approximations for the Hilbert series.…”

“…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. Schur polynomials have appeared in the context of Hilbert series as the characteristics of SU (N ) representations [13,16] and they have also been used extensively in the analysis of the N = 4 super Yang-Mills [25]. 2 In this paper we derive several new results for the Hilbert series based on standard properties of Schur polynomials.…”

SO and USp Kähler and hyper-Kähler quotients and lumps

“…This method has been widely used in the studies of N = 1 supersymmetric gauge theories [23,24,25]. See [26,27] for recent developments. In particular in…”