Recent Progress on the Donaldson–Thomas Theory
Wall-Crossing and Refined Invariants
Seiten
2021
|
1st ed. 2021
Springer Verlag, Singapore
978-981-16-7837-0 (ISBN)
Springer Verlag, Singapore
978-981-16-7837-0 (ISBN)
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 firstproposed 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.
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 firstproposed 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.
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.
1Donaldson–Thomas invariants on Calabi–Yau 3-folds.- 2Generalized Donaldson–Thomas invariants.- 3Donaldson–Thomas invariants for quivers with super-potentials.- 4Donaldson–Thomas invariants for Bridgeland semistable objects.- 5Wall-crossing formulas of Donaldson–Thomas invariants.- 6Cohomological Donaldson–Thomas invariants.- 7Gopakumar–Vafa invariants.- 8Some future directions.
| Erscheinungsdatum | 29.12.2021 |
|---|---|
| Reihe/Serie | SpringerBriefs in Mathematical Physics ; 43 |
| Zusatzinfo | 3 Illustrations, black and white; VIII, 104 p. 3 illus. |
| Verlagsort | Singapore |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika |
| Mathematik / Informatik ► Mathematik ► Algebra | |
| Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
| Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik | |
| Schlagworte | Bridgeland stability conditions • Derived algebraic geometry • Donaldson-Thomas invariants • Gopakumar-Vafa invariants • Wall-crossing formula |
| ISBN-10 | 981-16-7837-5 / 9811678375 |
| ISBN-13 | 978-981-16-7837-0 / 9789811678370 |
| Zustand | Neuware |
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