Witten index for weak supersymmetric systems: invariance under deformations

Abstract

When a $4D$ supersymmetric theory is placed on $S^3\times \mathbb{R}$, the supersymmetric algebra is necessarily modified to $su(2|1)$ and we are dealing with a weak supersymmetric system. For such systems, the excited states of the Hamiltonian are not all paired. As a result, the Witten index Tr$\{(-1{)}^F{e}^{-\beta H}\}$ is no longer an integer number, but a $\beta$-dependent function. However, this function stays invariant under deformations of the theory that keep the supersymmetry algebra intact. Based on the Hilbert space analysis, we give a simple general proof of this fact. We then show how this invariance works for two simplest weak supersymmetric quantum mechanical systems involving a real or a complex bosonic degree of freedom.

A bug in Eq. (3.9) fixed. 11 pages, 1 figure

References in corpus (2)

• Calculating the Superconformal Index and Seiberg Duality
• Classical and Quantum Dynamics of Higher-Derivative Systems

Cited by in corpus (3)

• Weak supersymmetry and superconformal indices
• Variations of supersymmetric quantum mechanics and superconformal indices
• Weak supersymmetric $su(N|1)$ quantum systems