Witten index and threshold bound states of D-branes

Abstract

We consider the Witten index 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. String theory and M-theory have predicted that 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 I = 0 for N = 8 and 4. Here, the bulk contribution to 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.