F-theory paints a beautiful picture that relates gauge theories to purely geometric information, whereby the Dynkin diagrams of gauge groups come to life as 2-cycles of internal spaces. The caveat is that, such spaces are necessarily singular, and treating them sensibly requires resolving or deforming the singularities. Recently, my collaborator R. Savelli and I have proposed a new strategy that allows one to deal with singular spaces directly. It is based on Eisenbud’s so-called matrix factorisations. This remarkably simple concept has very deep connections to the mathematics of singularities. In this talk, I will review the basic notions needed to formulate the issues, assuming only standard knowledge of string theory. Then, after introducing our proposal, I will show examples of its applications, such as computations of chiral spectra for bound states of 7-branes.

D-instantons are Euclidean (-1)-brane solutions of type IIB supergravity, which have been used to compute non-perturbative corrections to the effective action of type IIB string theory. I will discuss the most general non-extremal, non-supersymmetric generalization to the D-instanton. By repeating the calculations for arbitrary dimensions and dilaton coupling we will see that these solutions can be viewed as wormholes of non-extremal Reissner-Nordstroem black holes in one higher dimension. I will also discuss the role of D-instantons in the context of AdS/CFT and make a conjecture about the CFT counterpart of these non-extremal solutions.