D-BRANES ON Calabi–Yau MANIFOLDS

Abstract

In this review we study BPS D-branes on Calabi-Yau threefolds. Such D-branes naturally divide into two sets called A-branes and B-branes which are most easily understood from topological field theory. The main aim of this paper is to provide a self-contained guide to the derived category approach to B-branes and the idea of Π-stability. We argue that this mathematical machinery is hard to avoid for a proper understanding of B-branes. A-branes and B-branes are related in a very complicated and interesting way which ties in with the "homological mirror symmetry" conjecture of Kontsevich. We motivate and exploit this form of mirror symmetry. The examples of the quintic 3-fold, flops and orbifolds are discussed at some length. In the latter case we describe the rôle of McKay quivers in the context of D-branes.

D-brane monodromies from a matrix-factorization perspective

“…As a consequence a brane that wraps the compactification space without any lower-dimensional brane charges is massless at the conifold point [43,40]. Such a brane, X, potentially binds to the transported brane probe, P , as follows [30]. The mass of a BPS brane is given by the absolute value of its central charge, Z, which depends holomorphically on the Kähler moduli.…”

“…This shift of the D6-brane grade with respect to conifold monodromy has also been observed in ref. [28,30], where it was traced back to a simple pole in the period of the D6-brane. Then we immediately determine the inverse large radius monodromy of the brane, X 1 , by applying according to eq.…”

Stability conditions and branes at singularities

“…For a CY 3-fold, there are no perturbative corrections and the nonperturbative corrections are given by a power series in q i = exp 2πi C i B + iJ with no constant term [11]. The phase of this central charge is proportional to the grading of the D-brane.…”

“…By solving the PicardFuchs equations, one can then determine how the gradings behave as we move around in moduli space. This has been accomplished in many examples [4,11] but can be difficult in general.…”

Lectures on Bridgeland Stability

“…Another fundamental topic is the connection with counting invariants; for a survey we refer to [Tod12a,Tod12b,Tod14]. Connections to string theory are described for example in [Asp05,Bri06a,GMN13]. Connections to Representation Theory are in [ABM15].…”