Three questions in Gromov-Witten theory. (English) Zbl 1047.14043
Li, Ta Tsien (ed.) et al., Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20–28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press; Singapore: World Scientific/distributor (ISBN 7-04-008690-5/3-vol. set). 503-512 (2002).
From the introduction: Let \(X\) be a nonsingular projective variety over \(\mathbb{C}\). Gromov-Witten theory concerns integration over \(\overline M_{g,n}(X,\beta)\), the moduli space of stable maps from genus \(g\), \(n\)-pointed curves to \(X\) representing the class \(\beta\in H_2(X, \mathbb{Z})\). While substantial progress in the mathematical study of Gromov-Witten theory has been made in the past decade several fundamental questions remain open.
The present author describes three conjectural directions: Gorenstein properties of tautological rings, BPS states for threefolds, Virasoro constraints. Each points to basic structures in Gromov-Witten theory which are not yet understood. New ideas in the subject will be required for answers to these questions.
For the entire collection see [Zbl 0993.00022].
The present author describes three conjectural directions: Gorenstein properties of tautological rings, BPS states for threefolds, Virasoro constraints. Each points to basic structures in Gromov-Witten theory which are not yet understood. New ideas in the subject will be required for answers to these questions.
For the entire collection see [Zbl 0993.00022].
MSC:
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 14H10 | Families, moduli of curves (algebraic) |
| 14C15 | (Equivariant) Chow groups and rings; motives |
| 14J30 | \(3\)-folds |
