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Vanishing superconformal indices and the chiral symmetry breaking

V. P. Spiridonov, G. S. Vartanov

TL;DR

The paper reveals that superconformal indices for 4d $ obreak N=1$ theories with gauge groups $SU(N)$ and $SP(2N)$ exhibit delta-function singularities on special fugacity submanifolds, corresponding to chiral symmetry breaking and quantum-modified moduli spaces. By analyzing contour integrals and residues, the authors show that the usual SCI equality between electric and magnetic duals holds only in a distributional sense, with delta constraints enforcing symmetry breaking patterns such as $SU(N)_l imes SU(N)_r o SU(N)_d$ or $U(1)_B$-breaking, and they extend the analysis to $3d$ reductions where 3d partition functions display the same phenomenon. The work provides explicit treatments for $SU(2)$ with $N_f=2$, general $N>2$ cases, and $SP(2N)$ with $N_f=N+1$, highlighting the need to modify the standard SCI construction in nonconformal regimes. It also demonstrates consistent reduction to 3d dualities, reinforcing the physical relevance of delta-function structures as markers of chiral symmetry breaking in both 4d and 3d contexts.

Abstract

Superconformal indices of 4d \N=1 SYM theories with SU(N) and SP(2N) gauge groups are investigated for N_f=N and N_f=N+1 flavors, respectively. These indices vanish for generic values of the flavor fugacities. However, for a singular submanifold of fugacities they behave like the Dirac delta functions and describe the chiral symmetry breaking phenomenon. Similar picture holds for partition functions of 3d supersymmetric field theories with the chiral symmetry breaking.

Vanishing superconformal indices and the chiral symmetry breaking

TL;DR

The paper reveals that superconformal indices for 4d theories with gauge groups and exhibit delta-function singularities on special fugacity submanifolds, corresponding to chiral symmetry breaking and quantum-modified moduli spaces. By analyzing contour integrals and residues, the authors show that the usual SCI equality between electric and magnetic duals holds only in a distributional sense, with delta constraints enforcing symmetry breaking patterns such as or -breaking, and they extend the analysis to reductions where 3d partition functions display the same phenomenon. The work provides explicit treatments for with , general cases, and with , highlighting the need to modify the standard SCI construction in nonconformal regimes. It also demonstrates consistent reduction to 3d dualities, reinforcing the physical relevance of delta-function structures as markers of chiral symmetry breaking in both 4d and 3d contexts.

Abstract

Superconformal indices of 4d \N=1 SYM theories with SU(N) and SP(2N) gauge groups are investigated for N_f=N and N_f=N+1 flavors, respectively. These indices vanish for generic values of the flavor fugacities. However, for a singular submanifold of fugacities they behave like the Dirac delta functions and describe the chiral symmetry breaking phenomenon. Similar picture holds for partition functions of 3d supersymmetric field theories with the chiral symmetry breaking.

Paper Structure

This paper contains 13 sections, 92 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Superconformal index
  3. Chiral symmetry breaking for $G_c=SU(2)$
  4. $SU({N}),\; {N}>2$, gauge group case
  5. Breaking to the diagonal subgroup: $SU({N})_l \times SU({N})_r \rightarrow SU({N})_d$.
  6. Breaking of $U(1)_B$.
  7. Proofs
  8. $N_f={N}=2$ case
  9. The higher rank case, $N_f={N}>2$
  10. The case $N_f<{N}$
  11. Chiral symmetry breaking for $G_c=SP(2{N})$
  12. $3d$ theories with chiral symmetry breaking
  13. Conclusion