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Factorization of 4d N=1 superconformal index

Yutaka Yoshida

TL;DR

This paper demonstrates that the four-dimensional $\mathcal{N}=1$ superconformal index for $U(N)$ (and $SU(N)$) SQCD with $N_F$ flavors factorizes into an elliptic uplift of vortex and anti-vortex partition functions, under anomaly-free $R$-charge assignments and, for non-Abelian theories, a traceless vorticity condition. The factorization is shown explicitly for both Abelian and non-Abelian cases, with the non-Abelian result expressed as a sum over vacua involving vortex/anti-vortex contributions and open topological string amplitudes on strip geometries. The work connects these factorized forms to open topological strings and demonstrates that in the $3d$ limit the index reduces to the factorized partition function on the squashed sphere, consistent with Higgs-branch localization. It also outlines future directions toward a 4d analog of holomorphic blocks and extensions to broader geometries and dualities.

Abstract

We study the factorization of four dimensional N=1 superconformal index for U(N) (SU(N)) SQCD with N_F fundamental and anti-fundamental chiral multiplets. When both the anomaly free R-charge assignment and the traceless condition for total vorticities are satisfied, we find that the superconformal index factorizes to a pair of the elliptic uplift of the vortex partition functions. We also study the relation between open topological string and the the elliptic uplift of the vortex partition functions. In the three dimensional limit, we show index for U(N) theory reduces to the factorized form of the partition function on the three dimensional squashed sphere.

Factorization of 4d N=1 superconformal index

TL;DR

This paper demonstrates that the four-dimensional superconformal index for (and ) SQCD with flavors factorizes into an elliptic uplift of vortex and anti-vortex partition functions, under anomaly-free -charge assignments and, for non-Abelian theories, a traceless vorticity condition. The factorization is shown explicitly for both Abelian and non-Abelian cases, with the non-Abelian result expressed as a sum over vacua involving vortex/anti-vortex contributions and open topological string amplitudes on strip geometries. The work connects these factorized forms to open topological strings and demonstrates that in the limit the index reduces to the factorized partition function on the squashed sphere, consistent with Higgs-branch localization. It also outlines future directions toward a 4d analog of holomorphic blocks and extensions to broader geometries and dualities.

Abstract

We study the factorization of four dimensional N=1 superconformal index for U(N) (SU(N)) SQCD with N_F fundamental and anti-fundamental chiral multiplets. When both the anomaly free R-charge assignment and the traceless condition for total vorticities are satisfied, we find that the superconformal index factorizes to a pair of the elliptic uplift of the vortex partition functions. We also study the relation between open topological string and the the elliptic uplift of the vortex partition functions. In the three dimensional limit, we show index for U(N) theory reduces to the factorized form of the partition function on the three dimensional squashed sphere.

Paper Structure

This paper contains 9 sections, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. $\mathcal{N}=1$ superconformal index in four dimensions
  3. Factorization of superconformal index
  4. Abelian case
  5. Non-Abelian case
  6. Relation to open topological string
  7. Three dimensional limit
  8. Summary
  9. Conventions and useful formula

Figures (1)

  • Figure 1: Left: The $(p,q)$-web for the Calabi-Yau 3-folds. Two vertical external legs are identified and becomes an internal line. $Q_a, (a=1,\cdots, 2N_F)$ represents the exponentiated Kahler parameter of $a$-th internal line. Right: a A-brane is inserted at $r_a$-th right horizontal external leg $(a=1, \cdots, N)$ and the representation is $1^{k_a}$ . The other external legs are the trivial representation