Matrix Factorizations, D-Branes and their Deformations

D-branes (see e.g. ref. [1]) play an important rôle for understanding certain properties of string and field theories, as well as for building semirealistic models. However, practically all literature on string phenomenology deals with weakly coupled theories, where compactification radii are large and notions of classical geometry apply: e.g., supergravity solutions, branes wrapping pdimensional cycles, gauge field configurations on top of branes, etc. All this corresponds just to the boundary of the parameter space, which (presumably) is a subset of measure zero of the full string parameter space. In order to improve the understanding of how string theory behaves in the main part of its parameter space, we thus need to move away from the large radius/weak coupling regime. However, naive geometrical notions, such as a D-brane wrapping some p-dimensional cycle of a CalabiYau manifold, then start to break down. When distances become small or curvatures large, quantum corrections tend to blur notions of classical geometry, such as the dimension of a wrapped submanifold. Various physical phenomena can arise, like branes can become unstable and decay in ways that are not visible classically; orientifold planes can disintegrate; new branches in the moduli space can open up; new, non-perturbative critical points of the effective potential can develop; and contrarily, the moduli space of branes can be obstructed while classically it seems unobstructed (in other words, a non-perturbative superpotential can be generated). Recent examples of string phenomenology deep inside the “bulk” of the moduli space are given e.g., by refs. [2, 3]. In order to enter the bulk of the moduli space and meaningfully describe such phenomena, we need to adopt a suitable language for describing general D-brane configurations that goes beyond the notion of branes wrapping cycles. For topological B-type [4–6] D-branes, the proper mathematical framework is a certain enhanced, bounded derived category of coherent sheaves [7–12]; via homological mirror symmetry this maps to the category of A-type branes, which wrap Lagrangian cycles of the Fukaya category [6, 13] or coisotropic A-type branes [14–16]. This framework treats branes as abstract, not necessarily naive geometrical objects, but even in the geometrical, large radius limit it retains more data than the more familiar characterization of branes in terms of K-theory or cohomology (i.e., RR charges). It thus provides a much sharper description of D-branes. That is, the category also contains the information about the brane locations, and other possible gauge bundle moduli. For instance, a configuration consisting of an antiD0-brane located at some point u1 of the compactification manifold, plus aD0-brane located at some other point u2, is trivial from the K-theory point of view, but is a non-trivial object in the categorical description as long as u1 6= u2. Obviously, this extra information is crucial for under-