Hilbert Schemes, Donaldson-Thomas theory, Vafa-Witten and Seiberg-Witten theories

Citation:

Artan Sheshmani. 2019. “Hilbert Schemes, Donaldson-Thomas theory, Vafa-Witten and Seiberg-Witten theories.” Notices of International Congress of Chinese Mathematicians (2019), 7, 2, Pp. 25-31. Copy at http://j.mp/2U7qd01

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Abstract:

This article provides the summary of arXiv:1701.08899 and arXiv:1701.08902 where the authors studied the enumerative geometry of “nested Hilbert schemes” of points and curves on algebraic surfaces and their connections to threefold theories, and in particular relevant Donaldson-Thomas, Vafa-Witten and Seiberg-Witten theories.

Last updated on 08/12/2019

Enumerative geometry of surfaces and threefolds

In gauge theory, the prominent examples of enumerative invariants are Donaldson polynomials and Seiberg-Witten invariants, which help to distinguish different smooth structures on 4 dimensional manifolds. In recent years, other 4-manifold invariants have been introduced by changing the gauge theory (the PDE’s and the “counting” problem) or, by changing the dimension, similar gauge theory invariants were defined on higher-dimensional manifolds. Notable examples include Donaldson-Thomas (DT) invariants for six-dimensional, Calabi-Yau, manifolds. The study of enumerative geometry (counting of algebraic subspaces) of complex surfaces and threefolds has proven to be deeply related to physical structures, e.g. around Gopakumar-Vafa invariants (GV); Gromov-Witten invariants, DT, as well as Pandharipande-Thomas (PT) invariants; and their “motivic lifts". On the other hand, physical dualities in Gauge and String theory, such as Montonen-Olive duality and heterotic/Type II duality have also been a rich source of spectacular predictions about these counting invariants. An example of this is the modularity properties of GW or DT invariants which is proved mathematically in some cases, as suggested by the heterotic/Type II duality.

Artan Sheshmani, Ph.D.

Associate Professor of Mathematics (CMSA /QGM)

Hilbert Schemes, Donaldson-Thomas theory, Vafa-Witten and Seiberg-Witten theories

Citation:

Abstract:

Enumerative geometry of surfaces and threefolds