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Seminars at the Harvard Mathematics DepartmentMon, 12 Jul 2021 18:54:11 -0400CMSA QUANTUM MATTER AND QUANTUM FIELD THEORY SEMINAR:Artan Sheshmani (CMSA) speaks on <b>Higher rank flag sheaves and Vafa-Witten invariants</b> on Dec 18, 19, 12:00 pm - 1:00 pm in CMSA, 20 Garden St, G10:
Abstract: We study moduli space of holomorphic triples $f: E_{1} \rightarrow E_{2}$, composed of (possibly rank $>1$) torsion-free sheaves $(E_{1}, E_{2})$ and a holomorphic map between them, over a smooth complex projective surface $S$. The triples are equipped with a Schmitt stability condition. We prove that when the Schmitt stability parameter becomes sufficiently large, the moduli space of triples benefits from having a perfect relative and absolute obstruction theory in some cases (depending on Chern character of $E_{1}$). We further generalize our construction to higher-length flags of higher rank sheaves by gluing triple moduli spaces, and extend earlier work, with Gholampur and Yau, where the obstruction theory of nested Hilbert schemes over the surface was studied. Here we extend the earlier results to the moduli space of flags $E_{1}\rightarrow E_{2}\rightarrow \cdots \rightarrow E_{n}$, where the maps are injective (by stability). There is a connection, by wall-crossing in the master space, developed by Mochizuki, between the theory of such higher rank flags, and the theory of Higgs pairs on the surface, which provides the means to relate the flag invariants to the local DT invariants of any threefold given by a line bundle over the surface, $X :={\rm Tot}(L \rightarrow S)$. The latter DT invariants, when L is the canonical bundle of S, contribute to Vafa-Witten invariants. Joint work with Shing-Tung Yau, arXiv:1911.00124.
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