Titlebook: Recent Progress on the Donaldson–Thomas Theory; Wall-Crossing and Re Yukinobu Toda Book 2021 The Editor(s) (if applicable) and The Author(s

创作

y as combatants in order to re-enter civilian life without the stigmas that are attributed to women with a military past. As a result, the lives of women combatants are marked by neglect as they search for recognition and acknowledgement, and by poverty. This chapter is based on in-depth life-histor

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,Donaldson–Thomas Invariants on Calabi–Yau 3-Folds,s on CY 3-folds. More precisely, they are defined as integrations of zero-dimensional virtual fundamental classes on moduli spaces of stable coherent sheaves. In the rank one case, the generating series of DT invariants are conjectured to be related to the generating series of Gromov–Witten invarian

阴郁

,Generalized Donaldson–Thomas Invariants,tion. Although the first condition is not essential, the latter condition is much more essential, and it is much more difficult to define DT invariants when there exist strictly semistable sheaves. In this chapter, we explain the construction of DT invariants without the ss=st condition by Joyce–Son

漫步

,Donaldson–Thomas Invariants for Bridgeland Semistable Objects,semi)stable objects in the derived category of coherent sheaves. For this purpose, we need a suitable notion of stability in the derived category. The notion of stability conditions on derived categories (more generally triangulated categories) was introduced by Bridgeland [27] as a mathematical fra

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Cohomological Donaldson-Thomas Invariants,sely related to topological Euler numbers of the moduli spaces, one may ask a question whether the DT invariants can be refined to some cohomology theory whose Euler characteristics recover the DT invariants. In this chapter, we explain how to establish such a theory via Joyce’s d-critical structure

Narcissist

,Gopakumar–Vafa Invariants,se sheaves on moduli spaces of one-dimensional stable sheaves are also used to define Gopakumar–Vafa invariants in [128]. In this chapter, we give an introduction to Gopakumar–Vafa invariants defined in [128] and discuss the conjecture relating GV invariants with PT invariants.

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Some Future Directions,asize that the topics in this chapter are only a part of future directions chosen from the author’s preference. The topics we discuss here are not entirely out of reach at this moment, but rather ongoing research subjects which are not yet mature. We expect great progress on these topics in the comi

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,Donaldson–Thomas Invariants on Calabi–Yau 3-Folds,sheaves. In the rank one case, the generating series of DT invariants are conjectured to be related to the generating series of Gromov–Witten invariants, which is proved in many cases. In this chapter, we give an overview of DT invariants on CY 3-folds and GW/DT correspondence conjecture.

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Book 2021ounting 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

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2197-1757 DT theory.Contains a mathematical theory of Gopakumar–Vafa 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