## $$D$$-brane bound state redux.(English)Zbl 0911.53052

Summary: We study the existence of $$D$$-brane bound states at threshold in type II string theories. In a number of situations, we can reduce the question of existence to quadrature, and the study of a particular limit of the propagator for the system of $$D$$-branes. This involves a derivation of an index theorem for a family of non-Fredholm operators. In support of the conjectured relation between compactified eleven-dimensional supergravity and type IIA string theory, we show that a bound state exists for two coincident zero-branes. This result also provides support for the conjectured description of $$M$$-theory as a matrix model. In addition, we provide further evidence that there are no BPS bound states for two and three-branes twice wrapped on Calabi-Yau vanishing cycles.

### MSC:

 53Z05 Applications of differential geometry to physics 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
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