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Superconformal index for $\mathcal{N} = 4$ Super Yang-Mills and Elliptic Macdonald Polynomials

Gao-fu Ren, Min-xin Huang

Abstract

We establish a connection between the superconformal index of $\mathcal{N}=4$ $U(N)$ SYM and the elliptic Ruijsenaars-Schneider integrable system. The index admits an expression in terms of elliptic Macdonald polynomials, which leads to a compact summation over generalized partitions involving the structure constants $B_λ(p,q,t)$ and normalization constants $\mathcal{N}_λ(p,q,t)$. By solving the elliptic Ruijsenaars-Schneider model perturbatively in the elliptic parameter $p$, a systematic expansion of the index in powers of $p$ is obtained. We check that in various limits, namely a deformed 1/2 BPS limit and the large $N$ limit, our formalism reduces to previously known results.

Superconformal index for $\mathcal{N} = 4$ Super Yang-Mills and Elliptic Macdonald Polynomials

Abstract

We establish a connection between the superconformal index of SYM and the elliptic Ruijsenaars-Schneider integrable system. The index admits an expression in terms of elliptic Macdonald polynomials, which leads to a compact summation over generalized partitions involving the structure constants and normalization constants . By solving the elliptic Ruijsenaars-Schneider model perturbatively in the elliptic parameter , a systematic expansion of the index in powers of is obtained. We check that in various limits, namely a deformed 1/2 BPS limit and the large limit, our formalism reduces to previously known results.

Paper Structure

This paper contains 10 sections, 2 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Elliptic Ruijsenaars Schneider model and Elliptic Macdonald Polynomials
  3. Superconformal index of $\mathcal{N}=4$ $U(N)$ SYM
  4. Perturbative expansion of the SCI in elliptic parameter $p$
  5. Superconformal index in perturbation
  6. A deformed 1/2 BPS limit $q=t$
  7. Large $N$ limit
  8. Computing SCI of $\mathcal{N}=4$ SYM via solving eRS model
  9. Conclusion
  10. Appendix

Key Result

Theorem 1

Suppose that $|t|< 1$ and $t^ k \notin q^{\mathbb{Z}>0}$,$(k = 1,\dots,n - 1)$. Then, the normalized joint eigenfunctions $\mathcal{P}_\lambda(\mathbf{x}; p,q,t)$ of the elliptic Ruijsenaars operators, attached to the dominant vectors $\lambda$ in $\mathbb{Z}^N$, are orthogonal with respect to the w $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2