Supersymmetric Index and s-rule for Type IIB Branes

TL;DR

The paper links the three-dimensional $N=2,3$ $SU(n)$ Yang–Mills–Chern–Simons index at level $k$ to Type IIB brane dynamics with $(p,q)$5-branes, showing that SUSY breaking occurs when $k<n$ as dictated by the $s$-rule and that the nonzero index counts SUSY brane configurations. It derives the index microscopically via Born–Oppenheimer quantization on ${\mathbb R}\times T^2$ and via Serre duality, obtaining $I_{N=2,3}(k)=0$ for $0<k<n$ and $I_{N=2,3}(k)=\binom{k-1}{n-1}$ for $k\ge n$. The analysis is complemented by a brane/M-theory picture, including a Landau-problem dual description on ${\mathbb{CP}}^{n-1}$ and a duality web (Hanany–Witten moves and S-duality) that preserves the index, yielding a family of theories with identical indices. Together, these results provide a coherent brane-based understanding of nonperturbative SUSY dynamics in 3d CS theories and establish a concrete connection between the index, s-rule, and dualities, with potential generalizations to other gauge groups and orientifold setups.

Abstract

We investigate the supersymmetric index of N=2,3 SU(n) supersymmetric Yang-Mills Chern Simons theories at level k by using the brane configuration with a (p,q)5-brane. We can explain that the supersymmetry breaking occurs when k<n in terms of the s-rule for Type IIB branes. The supersymmetric index coincides with the number of the possible supersymmetric brane configurations. We also discuss a construction of a family of theories which have the same supersymmetric index.

Figures (2)

• Figure 1: The brane configuration in M-theory for $n\leq k$. $n$ M2-branes can be attached one by one only on $k$ positions.
• Figure 2: The Landau system on the D3-D5 bound state.