# MS Seminar (Mathematics - String Theory)

Speaker: Miroslav Rapcak (Perimeter) The Vertex Algebra Vertex Tue, Feb 27, 2018, 13:15 - 14:30 Seminar Room A It has been known for a long time that BPS counting associated to D6-D4-D2-D0 branes wraping holomorphic cycles in toric Calabi-Yau threefolds is done by topological vertex-like techniques. We categorify this construction by promoting the gluing of the topological vertex to the gluing of vertex operator algebras. The elementary building block of the D6-D2-D0 counting (trivalent vertex of the corresponding toric diagram) can be identified with \mathcal{W}_{1+\infty}$algebra. The gluing procedure can be understood in terms of conformal extension of the product of$\mathcal{W}_{1+\infty}\$ associated to each trivalent vertex by fusions of bimodules associated to each internal line in the toric diagram. On the other hand, D4-D2-D0 counting is categorified by truncations of such glued infinite algebras. The standard generating functions from the D6-D4-D2-D0 counting are recovered from the vacuum characters of the glued algebras. The construction introduces new, intuitive techniques to study vertex operator algebras and provides us with a large class of BPS/CFT pairs.