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. 4/97

We consider quantum mechanical gauge theories with sixteen supersymmetries. The Hamiltonians or Lagrangians characterizing these theories can contain higher derivative terms. In the operator approach, we show that the free theory is essentially the unique abelian theory with up to four derivatives in the following sense: any small deformation of the free theory, which preserves the supersymmetries, can be gauged away by a unitary conjugation. We also present a method for deriving constraints on terms appearing in an effective Lagrangian. We apply this method to the effective Lagrangian describing the dynamics of two well-separated clusters of D0-branes. As a result, we prove a nonrenormalization theorem for the v 4 interaction. 5/98 1 paban@sns.ias.edu 2 sethi@sns.ias.edu 3 stern@math.duke.edu , the real part of h {0,0} 3

Higher derivative terms in the effective action of certain Yang-Mills theories can be severely constrained by supersymmetry. We show that requiring sixteen supersymmetries in quantum mechanical gauge theory determines the v 6 term in the effective action. Even the numerical coefficient of the v 6 term is fixed in terms of lower derivative terms in the effective action. 6/981 paban@sns.ias.edu

We count the supersymmetric bound states of many distinct BPS monopoles in N = 4 Yang-Mills theories and in pure N = 2 Yang-Mills theories. The novelty here is that we work in generic Coulombic vacua where more than one adjoint Higgs fields are turned on. The number of purely magnetic bound states is again found to be consistent with the electromagnetic duality of the N = 4 SU(n) theory, as expected. We also count dyons of generic electric charges, which correspond to 1/4 BPS dyons in N = 4 theories and 1/2 BPS dyons in N = 2 theories. Surprisingly, the degeneracy of dyons is typically much larger than would be accounted for by a single supermultiplet of appropriate angular momentum, implying many supermutiplets of the same charge and the same mass.

We study the existence of monopole bound states saturating the BPS bound in N=2 supersymmetric Yang-Mills theories. We describe how the existence of such bound states relates to the topology of index bundles over the moduli space of BPS solutions. Using an L 2 index theorem, we prove the existence of certain BPS states predicted by Seiberg and Witten based on their study of the vacuum structure of N=2 Yang-Mills theories. 8/95