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El. knyga: Recent Progress on the Donaldson-Thomas Theory: Wall-Crossing and Refined Invariants

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Recent Progress on the Donaldson-Thomas Theory: Wall-Crossing and Refined InvariantsDidesnis paveikslėlis
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This book is an exposition of recent progress on the Donaldson–Thomas (DT) theory. The DT invariant was introduced by R. Thomas in 1998 as a virtual counting of stable coherent sheaves on Calabi–Yau 3-folds. Later, it turned out that the DT invariants have many interesting properties and appear in several contexts such as the Gromov–Witten/Donaldson–Thomas conjecture on curve-counting theories, wall-crossing in derived categories with respect to Bridgeland stability conditions, BPS state counting in string theory, and others. 

Recently, a deeper structure of the moduli spaces of coherent sheaves on Calabi–Yau 3-folds was found through derived algebraic geometry. These moduli spaces admit shifted symplectic structures and the associated d-critical structures, which lead to refined versions of DT invariants such as cohomological DT invariants. The idea of cohomological DT invariants led to a mathematical definition of the Gopakumar–Vafa invariant, which was first proposed by Gopakumar–Vafa in 1998, but its precise mathematical definition has not been available until recently.

This book surveys the recent progress on DT invariants and related topics, with a focus on applications to curve-counting theories.

Recenzijos

The book is directed at readers with a solid foundation in algebraic geometry. the main definitions and theorems are nicely illustrated by examples. The book will serve as a guide to further reading for those wishing to learn more details about the theory. (Matthew B. Young, Mathematical Reviews, March, 2023)

1DonaldsonThomas invariants on CalabiYau 3-folds.- 2Generalized
DonaldsonThomas invariants.- 3DonaldsonThomas invariants for quivers with
super-potentials.- 4DonaldsonThomas invariants for Bridgeland semistable
objects.- 5Wall-crossing formulas of DonaldsonThomas invariants.-
6Cohomological DonaldsonThomas invariants.- 7GopakumarVafa invariants.-
8Some future directions.
The author is currently Professor and Principal investigator at Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo. He was an invited speaker at the ICM 2014.
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