Witten index and threshold bound states of D-branes. (English) Zbl 0925.58105

Summary: 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.


58J90 Applications of PDEs on manifolds
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
58J20 Index theory and related fixed-point theorems on manifolds
81T60 Supersymmetric field theories in quantum mechanics
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