Abstract:We compute the dimension of the moduli space of gauge-inequivalent solutions to the Bogomolny equation on R 3 with prescribed singularities corresponding to the insertion of a finite number of 't Hooft defects. We do this by generalizing the methods of C. Callias and E. Weinberg to the case of R 3 with a finite set of points removed. For a special class of Cartan-valued backgrounds we go further and construct an explicit basis of L 2 -normalizable zero-modes. Finally we exhibit and study a two-parameter family of spherically symmetric singular monopoles, using the dimension formula to provide a physical interpretation of these configurations. This paper is the first in a series of three on singular monopoles, where we also explore the role they play in the contexts of intersecting D-brane systems and four-dimensional N = 2 super Yang-Mills theories.

We provide a semiclassical description of framed BPS states in fourdimensional N = 2 super Yang-Mills theories probed by 't Hooft defects, in terms of a supersymmetric quantum mechanics on the moduli space of singular monopoles. Framed BPS states, like their ordinary counterparts in the theory without defects, are associated with the L 2 kernel of certain Dirac operators on moduli space, or equivalently with the L 2 cohomology of related Dolbeault operators. The Dirac/Dolbeault operators depend on two Cartan-valued Higgs vevs. We conjecture a map between these vevs and the SeibergWitten special coordinates, consistent with a one-loop analysis and checked in examples. The map incorporates all perturbative and nonperturbative corrections that are relevant for the semiclassical construction of BPS states, over a suitably defined weak coupling regime of the Coulomb branch. We use this map to translate wall crossing formulae and the no-exotics theorem to statements about the Dirac/Dolbeault operators. The no-exotics theorem, concerning the absence of nontrivial SU(2) R representations in the BPS spectrum, implies that the kernel of the Dirac operator is chiral, and further translates into a statement that all L 2 cohomology of the Dolbeault operator is concentrated in the middle degree. Wall crossing formulae lead to detailed predictions for where the Dirac operators fail to be Fredholm and how their kernels jump. We explore these predictions in nontrivial examples. This paper explains the background and arguments behind the results announced in the short note [1].

We study a system of N c D3-branes intersecting D7-branes and O7-planes in 1 + 1-dimensions. We use anomaly cancellation and string dualities to argue that there must be chiral fermion zero-modes on the D3-branes which are localized near the O7-planes. Away from the orientifold limit we verify this by using index theory as well as explicit construction of the zero-modes. This system is related to F-theory on K3 and heterotic matrix string theory, and the heterotic strings are related to Alice string defects in N = 4 Super-Yang-Mills. In the limit of large N c we find an AdS 3 dual of the heterotic matrix string CFT.