[Paper Review] The strange duality conjecture for generic curves

Prakash Belkale|ArXiv.org|Feb 1, 2006

Algebraic Geometry and Number Theory13 references4 citations

TL;DR

This paper proves the strange duality conjecture for generic curves of genus $ g \geq 1 $, establishing that sections $ \Theta_F $ associated to semistable vector bundles $ F \in U_X^*(k) $ span the space $ H^0(SU_X(r), \mathcal{L}^k) $. The proof combines factorization of conformal blocks, Gromov-Witten invariants on Grassmannians, and degeneration techniques to construct a geometric enumerative problem whose solutions yield the required theta sections, confirming the duality for generic curves.

ABSTRACT

For X a compact Riemann surface of positive genus, the strange duality conjecture predicts that the space of sections of certain theta bundle on moduli of bundles of rank r and level k is naturally dual to a similar space of sections of rank k and level r. We prove this conjecture for X generic in the moduli space of curves of a given genus.

Motivation & Objective

• To establish the strange duality conjecture for generic curves in the moduli space $ M_g $, which relates theta functions of rank $ r $, level $ k $ to those of rank $ k $, level $ r $.
• To show that the sections $ \Theta_F $ for $ F \in U_X^*(k) $ span $ H^0(SU_X(r), \mathcal{L}^k) $, which is equivalent to the conjecture being an isomorphism.
• To construct a geometric enumerative problem in higher genus that mirrors the structure of quantum cohomology and Schubert calculus on Grassmannians.
• To use factorization of conformal blocks and the Verlinde formula to compute the dimension $ M(r,k,g) $ of $ H^0(SU_X(r), \mathcal{L}^k) $, linking it to Gromov-Witten invariants.

Proposed method

• Use the factorization formula of Tsuchiya-Ueno-Yamada to reduce the dimension $ M(r,k,g) $ of $ H^0(SU_X(r), \mathcal{L}^k) $ to conformal blocks on $ \mathbb{P}^1 $.
• Apply the Verlinde formula to compute dimensions of conformal blocks, linking them to the rank of $ H^0(SU_X(r), \mathcal{L}^k) $.
• Use Witten’s relation between conformal blocks on $ \mathbb{P}^1 $ and small quantum cohomology of Grassmannians to express $ M(r,k,g) $ as a sum of twisted Gromov-Witten invariants.
• Introduce a new enumerative problem in higher genus by constructing a partial degeneration of the diagonal in $ \operatorname{Gr}(r,n) \times \operatorname{Gr}(r,n) $, modeled on the Gromov-Witten sum in Equation (\ddagger).
• Show that the linearly independent sections arising from transversal intersections in this enumerative problem are of the form $ \Theta_F $, thus spanning $ H^0(SU_X(r), \mathcal{L}^k) $.
• Use formal deformation theory and Artin approximation to lift vector bundles over generic curves, preserving trivial determinant and semistability.

Experimental results

Research questions

• RQ1 Does the strange duality isomorphism hold for generic curves of genus $ g \geq 1 $, i.e., is the map $ H^0(U_X^*(k), \mathcal{M}^r)^* \to H^0(SU_X(r), \mathcal{L}^k) $ an isomorphism for generic $ X \in M_g $?
• RQ2 Can the dimension $ M(r,k,g) $ of $ H^0(SU_X(r), \mathcal{L}^k) $ be expressed as a sum of twisted Gromov-Witten invariants on Grassmannians?
• RQ3 Is there a geometric enumerative problem in higher genus whose solutions naturally produce the theta sections $ \Theta_F $ for $ F \in U_X^*(k) $?
• RQ4 How does the structure of Schubert varieties and their duals in Grassmannians relate to the factorization of conformal blocks in higher genus?
• RQ5 Can the duality be proven via degeneration techniques that mimic the shift operations in Gromov-Witten theory?

Key findings

• The strange duality conjecture holds for generic curves $ X \in M_g $, meaning the sections $ \Theta_F $ for $ F \in U_X^*(k) $ span $ H^0(SU_X(r), \mathcal{L}^k) $, confirming the isomorphism in the conjecture.
• The dimension $ M(r,k,g) $ of $ H^0(SU_X(r), \mathcal{L}^k) $ is given by a sum of twisted Gromov-Witten invariants: $ M(r,k,g) = \sum' \langle \omega_{I^1}, \dots, \omega_{I^g}, \omega_{(I^1)'}^\prime, \dots, \omega_{(I^g)'}^\prime \rangle_{0,-k(g-1)} $.
• The construction of the enumerative problem in higher genus is modeled on a partial degeneration of the diagonal in $ \operatorname{Gr}(r,n) \times \operatorname{Gr}(r,n) $, with the sum restricted to subsets $ I^j \subset [r+k] $ of size $ r $ containing 1.
• The linear independence of the sections $ \Theta_F $ arises from transversal intersections in this enumerative problem, which geometrically realize the Gromov-Witten invariants in the sum.
• The shift operation in Gromov-Witten theory (Proposition A.3) is used to relate invariants under degeneration, enabling the construction of the full sum via recursive reduction.
• The proof relies on the fact that $ h^0(U_X^*(k), \mathcal{M}^r) = h^0(SU_X(r), \mathcal{L}^k) $, so spanning implies isomorphism, and this is achieved via the geometric construction of the $ \Theta_F $ sections.