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Recent Progress on the Donaldson–Thomas Theory

Part of the SpringerBriefs in Mathematical Physics Book Series (volume 43)

DOI: 10.1007/978-981-16-7838-7

ISBN: 978-981-16-7837-0

Publisher: Springer Singapore - Springer Journals

Dec 16, 2021

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Chapter 1Donaldson–Thomas Invariants on Calabi–Yau 3-Folds

Toda, Yukinobu

Pages: 1 –16

[The Donaldson–Thomas invariants are introduced by Thomas [163] as a holomorphic analogue of Casson invariants [162]. They count stable coherent sheaves on CY 3-folds. More precisely, they are defined as integrations of zero-dimensional virtual fundamental classes on moduli spaces of stable...
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Chapter 2Generalized Donaldson–Thomas Invariants

Toda, Yukinobu

Pages: 17 –28

[In the last chapter, we gave a definition of DT invariants on CY 3-folds when the moduli spaces of stable sheaves are fine and satisfy the ss=st condition. Although the first condition is not essential, the latter condition is much more essential, and it is much more difficult to define DT...
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Chapter 3Donaldson–Thomas Invariants for Quivers with Super-Potentials

Toda, Yukinobu

Pages: 29 –38

[The original DT invariants are defined for projective CY 3-folds. However, using the description of the DT invariants via Behrend functions, we can also define DT invariants for quivers with super-potentials. In this chapter, we review the theory of quivers with super-potentials and the...
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Chapter 4Donaldson–Thomas Invariants for Bridgeland Semistable Objects

Toda, Yukinobu

Pages: 39 –51

[In Chaps. 1, 2, the DT invariants are defined as counting (semi)stable sheaves on CY 3-folds. In this chapter, we extend them to invariants counting (semi)stable objects in the derived category of coherent sheaves. For this purpose, we need a suitable notion of stability in the derived category....
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Chapter 5Wall-Crossing Formulas of Donaldson–Thomas Invariants

Toda, Yukinobu

Pages: 53 –69

[In this chapter, we explain wall-crossing formulas of DT invariants and their applications. The applications include DT/PT correspondence, rationality of PT generating series, flop formula of DT invariants, etc.]
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Chapter 6Cohomological Donaldson-Thomas Invariants

Toda, Yukinobu

Pages: 71 –82

[The DT invariants counting stable sheaves are integers and coincide with Behrend-weighted Euler characteristics of the moduli spaces. As these are closely related to topological Euler numbers of the moduli spaces, one may ask a question whether the DT invariants can be refined to some cohomology...
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Chapter 7Gopakumar–Vafa Invariants

Toda, Yukinobu

Pages: 83 –90

[In the last chapter, we explained a cohomological refinement of DT invariants via gluing of perverse sheaves of vanishing cycles. The resulting perverse sheaves on moduli spaces of one-dimensional stable sheaves are also used to define Gopakumar–Vafa invariants in [128]. In this chapter, we give...
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Chapter 8Some Future Directions

Toda, Yukinobu

Pages: 91 –94

[In this final short chapter, we discuss some possible future directions of the study on DT invariants. The DT theory is now a huge subject, and I emphasize 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...

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