Homological Mirror Symmetry, originating from Kontsevich decades ago, has so far not been much reflected in physics. This is partly due its very abstract concepts, and the focus being different in mathematics and in physics. In mathematics much of the work concentrates on rederiving closed string enumerative invariants from categorical ones. In physics one is interested in concrete computations of correlation functions, which count open string enumerative invariants for intersecting branes. Very little work has been done for such genuinely open string invariants in general, despite this sector of topological open strings is infinitely much richer than the one of closed strings. The exception is the elliptic curve, which has been studied well, and where the open string invariants count polygonal instantons with arbitrary many edges.
Concretely we are interested in computing invariants in the topological A-model via mirror symmetry of the B-model. The proper formulation of the latter is in terms of matrix factorizations. The main problem consists in finding an appropriate connection which allows to determine flat coordinates and operator bases.
In the talk I will present, from a physicist’s perspective, some progress in doing so for simplest possible case, namely reproducing known results for the elliptic curve.