[Paper Review] Three questions in Gromov-Witten theory

Rahul Pandharipande|ArXiv.org|Feb 7, 2003

Geometric and Algebraic TopologyMathematics24 references92 citations

TL;DR

This paper formulates three central conjectures in Gromov-Witten theory: the Gorenstein property of tautological rings in moduli spaces of curves, the existence of BPS states in threefold Gromov-Witten invariants, and the Virasoro constraints for all nonsingular projective varieties. It proposes that these conjectures reveal deep algebraic and geometric structures underlying Gromov-Witten theory, with the Virasoro constraints being particularly significant as they may link Gromov-Witten theory to integrable systems via explicit operator formulations and connections to matrix models and vertex operator algebras.

ABSTRACT

This article accompanies my ICM talk in August 2002. Three conjectural directions in Gromov-Witten theory are discussed: Gorenstein properties, BPS states, and Virasoro constraints. Each points to basic structures in the subject which are not yet understood.

Motivation & Objective

• To identify and articulate three unresolved foundational problems in Gromov-Witten theory that reveal deeper underlying structures.
• To propose that the tautological rings of moduli spaces of curves and their strata are finite-dimensional Gorenstein algebras, generalizing Faber’s conjecture.
• To formulate a universal conjecture for BPS states in Gromov-Witten theory of threefolds, extending the notion of BPS invariants to all nonsingular projective threefolds.
• To extend the Virasoro constraints beyond known cases (e.g., point, P^1, P^n) to all nonsingular projective varieties, using cohomological data and quantum cohomology.
• To explore the potential of these conjectures in connecting Gromov-Witten theory to integrable systems and algebraic geometry through operator formalism and relative theories.

Proposed method

• Formalizing the tautological ring of moduli spaces of stable maps via closure under pushforwards along forgetful and gluing maps.
• Defining a filtration on M_{g,n} via strata of compact type, rational tails, and fixed stable curves to study Gorenstein properties.
• Introducing a universal Virasoro operator algebra acting on the Gromov-Witten potential, constructed from intersection pairing, Hodge decomposition, and anticanonical class action.
• Deriving explicit formulas for Virasoro generators L_k using symmetric functions, quantum cup product, and derivatives with respect to deformation parameters.
• Using the virtual fundamental class to define Gromov-Witten invariants as integrals over moduli of stable maps to X.
• Extending the Virasoro constraints to relative Gromov-Witten theory, particularly for 1-dimensional targets, to support absolute theory proofs.

Experimental results

Research questions

• RQ1 Are the tautological rings of the moduli spaces of curves and their strata (e.g., C_{g,n}) finite-dimensional Gorenstein algebras?
• RQ2 Do BPS states exist in Gromov-Witten theory for all nonsingular projective threefolds, and how are they encoded in the Gromov-Witten potential?
• RQ3 Do the Virasoro constraints L_k(Z^X) = 0 hold universally for all nonsingular projective varieties X, based on cohomological data?
• RQ4 Can the Virasoro constraints be extended to relative Gromov-Witten theory, and do they play a foundational role in proving absolute constraints?
• RQ5 Is there a deeper connection between Gromov-Witten theory and integrable systems, as suggested by the Virasoro algebra and matrix model/vertex operator realizations?

Key findings

• The tautological ring of M_g is Gorenstein with socle in degree g−2, as proven by Faber for low genus, and the general conjecture extends this to all strata of M_{g,n}.
• The Gorenstein property of tautological rings is conjectured to hold for all strata in the filtration M_{g,n} ⊃ M^c_{g,n} ⊃ M^{rt}_{g,n} ⊃ C_{g,n}, with C_{g,n} requiring a fixed curve C_g.
• The Virasoro constraints are conjectured to hold universally for all nonsingular projective varieties X, with explicit formulas for L_k involving intersection pairing, Hodge decomposition, and anticanonical class.
• For X = point, the Virasoro constraints reduce to Witten’s conjecture, and for X = P^n, they have been proven by Givental, confirming the conjecture in these cases.
• The Virasoro constraints are shown to be compatible with relative theory, and their proof for curves C_g has been established, with implications for absolute constraints.
• The Virasoro algebra's structure, particularly the bracket [L_1, L_{-1}] = 2L_0, depends on Chern number formulas in terms of Hodge numbers, indicating a need for algebraic structure beyond symplectic geometry.