Online Seminar
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05192020
Hai Dong: Grassmanian via Dual Grassmanian
Abstract: Grassmannian Gr(r,n) is geometrically isomorphic to dual Grassmannian Gr(n-r,n). However, they have very different combinatorial structures, originated from their GIT presentations. It is a mysterious and yet highly nontrivial problem to match their combinatorial structures directly. A famous example is level-rank duality from physics. In this talk, I will examine the relation of I-functions of Grassmannian and its dual in both quantum cohomology and quantum K-theory cases. Furthermore, the twisted I-functions of vector bundles of Grassmannians in quantum cohomology and I-functions with the level structure in quantum K-theory, which is introduced by Yongbin Ruan and Ming Zhang, will also be examined.
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05122020
Albrecht Klemm: Topological strings on genus one fibered Calabi-Yau 3-folds and string dualities
Abstract: We calculate the generating functions of BPS indices using their modular propertiesin Type II and M-theory compactifications on compact genus one fibered CY 3-foldswith singular fibers and additional rational sections or just N-sections, in order tostudy string dualities in four and five dimensions as well as rigid limits in which gravitydecouples. The generating functions are Jacobi-forms of Γ_1(N) with the complexifiedfiber volume as modular parameter. The string coupling λ, or the \epsilon_{±} parameters inthe rigid limit, as well as the masses of charged hypermultiplets and non-Abelian gaugebosons are elliptic parameters. To understand this structure, we show that specificauto-equivalences act on the category of topological B-branes on these geometries andgenerate an action of Γ1(N) on the stringy K¨ahler moduli space. We argue that theseactions can always be expressed in terms of the generic Seidel-Thomas twist with respectto the 6-brane together with shifts of the B-field and are thus monodromies. This impliesthe elliptic transformation law that is satisfied by the generating functions. We useHiggs transitions in F-theory to extend the ansatz for the modular bootstrap to genusone fibrations with N-sections and boundary conditions fix the all genus generatingfunctions for small base degrees completely. This allows us to study in depth a widerange of new, non-perturbative theories, which are Type II theory duals to the CHLZN orbifolds of the heterotic string on K3 × T2. In particular, we compare the BPSdegeneracies in the large base limit to the perturbative heterotic one-loop amplitudewith $R^2_+F_+^{2g−2} insertions for many new Type II geometries. In the rigid limit we canrefine the ansatz and obtain the elliptic genus of superconformal theories in 5d.
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04282020
Rahul Pandharipande: Descendents for stable pairs on 3-folds
Abstract: Descendent classes on moduli spaces of sheaves are defined via the Chern characters of the universal sheaf.I will present several conjectures and results concerning stable pairs descendent invariants for 3-folds: rationality of generating functions, functional equations, cobordism classes, and Virasoro constraints.