Witten Index and Threshold Bound States of D-Branes

TL;DR

This paper calculates the Witten index for SU(2) Super-Yang-Mills quantum mechanics with different supersymmetries, providing insights into the number of bound states of D-branes and testing string theory predictions.

Contribution

It explicitly computes the bulk contribution to the Witten index for N=16, 8, 4 supersymmetries, revealing a common defect term and clarifying the bound state counting.

Findings

Witten index for N=16 is 5/4, indicating a defect contribution of -1/4.

Witten index for N=8 and 4 is 1/4, consistent with theoretical expectations.

The defect contribution of -1/4 is explained through effective dynamics analysis.

Abstract

We consider the Witten index ${\cal I}= Tr (-1)^F$ of SU(2) Super-Yang-Mills quantum mechanics (SYMQ) with N=16, 8, 4 supersymmetries. The theory governs the interactions between a pair of D-branes under various circumstances, and our goal is to count the number of the threshold bound states directly from the low-energy effective theory. The string theory and M theory have predicted that ${\cal I}=1$ for N=16, which in fact forms an underlying hypothesis of the M(atrix) theory formulation. Also the consistency of conifold transitions in type II theories is known to require ${\cal I}=0$ for N=8 and 4. Here, the bulk contribution to ${\cal I}$ is computed explicitly, and for N=16, 8, 4, found to be 5/4, 1/4, 1/4 respectively, suggesting a common defect contribution of -1/4. We illustrate how the defect term of -1/4 may arise in the SU(2) SYMQ by considering the effective dynamics along the asymptotic region.