Donaldson-Thomas and Gromov-Witten theories
The study of the moduli spaces of algebraic curves and coherent sheaves, and their induced invariants over ambient complex kahler varieties of complex dimension 2, 3 and higher, has been a central source of focus for mathematicians in the past 50 years, due to their profound connections to geometry, topology, number theory as well as fruitful contributions to superstring theory. The course aims at introducing these topics and provides discussion of computations of Gromov-Witten and Donaldson-Thomas invariants of complex algebraic varieties.
Lecturer
Date
2nd March ~ 13th July, 2023
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Thursday | 09:00 - 10:30 | A3-1-101 | ZOOM 08 | 787 662 9899 | BIMSA |
Syllabus
Part 1: GW invariants
1.1 Stable curves
1.2 Families and moduli spaces of smooth and stale pointed curves
1.3 Gromov-Witten classes
1.4 Gromov-Witten invariant
1.5 Virtualfundamentalclasses
1.6 Formal construction of virtual fundamental class
1.7 Behrend-Pantechi Construction of virtual fundamental class
1.8 Computation of virtual fundamental class in some examples
1.9 Propertiesof GW classes
1.10 Tree-level classes and invariants
1.11 Small quantum cohomology
1.12 Further properties of small quantum product
1.13 Small quantum cohomology of CY manifolds
Part 2: Donaldson-Thomas theory and sheaf theory
2.1 IdealsheavesandDonaldson-Thomastheory
2.2 Deformationtheory
2.3 Primaryinvariants
2.4 Correspondence bewteen DT invariants and GW invariants
2.5 Virtual localizationin DT theory
2.6 Some examples of Torus equivariant localization of DT, for C3 and local P1
2.7 Relative DT Theory(JunLi, BaoseauWu)
2.8 Torsion DT theories and modular forms
Part 3: Physics behind study of torsion sheaves
3.1 Partition function of solutions to SYM theory
3.2 Modularity of torsion sheaf DT theory for K3-fibered 3folds
Part 4: Vafa-Witten theory over algebraic projective surfaces
4.1 Vafa-Witten invariants
4.2 Fixed loci of torsion sheaf moduli space, i.e., C∗-equivariant torsion sheaves
Part 5 4 dimensional DT theories
5.1 Calabi-Yau 4folds and Kapustin-Witten invariants
5.2 Atiyah class
5.3 Universal truncated relative Atiyah class.
5.4 Categorical DT theory for local surfaces
5.5 Work of Yukinobu Toda
5.6 Singular support of coherent sheaves
1.1 Stable curves
1.2 Families and moduli spaces of smooth and stale pointed curves
1.3 Gromov-Witten classes
1.4 Gromov-Witten invariant
1.5 Virtualfundamentalclasses
1.6 Formal construction of virtual fundamental class
1.7 Behrend-Pantechi Construction of virtual fundamental class
1.8 Computation of virtual fundamental class in some examples
1.9 Propertiesof GW classes
1.10 Tree-level classes and invariants
1.11 Small quantum cohomology
1.12 Further properties of small quantum product
1.13 Small quantum cohomology of CY manifolds
Part 2: Donaldson-Thomas theory and sheaf theory
2.1 IdealsheavesandDonaldson-Thomastheory
2.2 Deformationtheory
2.3 Primaryinvariants
2.4 Correspondence bewteen DT invariants and GW invariants
2.5 Virtual localizationin DT theory
2.6 Some examples of Torus equivariant localization of DT, for C3 and local P1
2.7 Relative DT Theory(JunLi, BaoseauWu)
2.8 Torsion DT theories and modular forms
Part 3: Physics behind study of torsion sheaves
3.1 Partition function of solutions to SYM theory
3.2 Modularity of torsion sheaf DT theory for K3-fibered 3folds
Part 4: Vafa-Witten theory over algebraic projective surfaces
4.1 Vafa-Witten invariants
4.2 Fixed loci of torsion sheaf moduli space, i.e., C∗-equivariant torsion sheaves
Part 5 4 dimensional DT theories
5.1 Calabi-Yau 4folds and Kapustin-Witten invariants
5.2 Atiyah class
5.3 Universal truncated relative Atiyah class.
5.4 Categorical DT theory for local surfaces
5.5 Work of Yukinobu Toda
5.6 Singular support of coherent sheaves
Audience
Graduate
Video Public
Yes
Notes Public
No
Language
English
Lecturer Intro
Artan Sheshmani is a Professor of pure Mathematics, specialized in Algebraic geometry, Enumerative and Derived Geometry, and Mathematics of String Theory. He is a Professor at Beijing Institute of Mathematical Sciences and Applications in Beijing, and a senior personnel (Professor) at Simons Collaboration Program on Homological Mirror Symmetry ( Harvard University Center for Mathematical Sciences and Applications), and an Affiliate Faculty Member at Harvard University- MIT IAiFi (Institute for Artificial Intelligence and Fundamental Interactions). Between 2020 and 2023, he jointly held the visiting professor position at Institute for the Mathematical Sciences of the Americas at University of Miami, where he was part of the research collaboration program on "Hodge theory and its applications". During the past 5 years while at Harvard CMSA he was also a visiting professor at Harvard Physics department (2020-2022), and an Associate Professor of Mathematics at Institut for Mathematik (formerly the Center for Quantum Geometry of Moduli Spaces) at Aarhus University in Denmark (2016-2022). His work is mainly focused on Gromov Witten theory, Donaldson Thomas theory, Calabi-Yau geometries, and mathematical aspects of String theory. He studies geometry of moduli spaces of sheaves and curves on Calabi Yau spaces, some of which arise in the study of mathematics of string theory. In his research he has worked on understanding dualities between geometry of such moduli spaces over complex varieties of dimension 2,3,4 and currently he is working on extension of these projects from derived geometry and geometric representation theory point of view. In joint work with Shing-Tung Yau (BIMSA, YMSC, Tsinghua, Harvard Math, Harvard CMSA, and Harvard Physics departments), Cody Long (Harvard Physics), and Cumrun Vafa (Harvard Math and Physics departments) he worked on geometry moduli spaces of sheaves with non-homolomorphic support and their associated non-BPS (non-holomorphic) counting invariants. In 2019 he was one of the 30 winners of the IRFD "Research Leader" grant (approx 1M USD) on his project "Embedded surfaces, dualities and quantum number theory". The project was additionally co-financed by Harvard University CMSA and Aarhus University (Approx total. 400K USD). Detail of IRFD "Research Leader" grant: https://dff.dk/en/grants/research-leaders-2018.