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Virasoro Constraints for Toric Bundles

Tom Coates, Alexander Givental, Hsian-Hua Tseng

Abstract

We show that the Virasoro conjecture in Gromov--Witten theory holds for the the total space of a toric bundle $E \to B$ if and only if it holds for the base $B$. The main steps are: (i) we establish a localization formula that expresses Gromov--Witten invariants of $E$, equivariant with respect to the fiberwise torus action, in terms of genus-zero invariants of the toric fiber and all-genus invariants of $B$; and (ii) we pass to the non-equivariant limit in this formula, using Brown's mirror theorem for toric bundles.

Virasoro Constraints for Toric Bundles

Abstract

We show that the Virasoro conjecture in Gromov--Witten theory holds for the the total space of a toric bundle if and only if it holds for the base . The main steps are: (i) we establish a localization formula that expresses Gromov--Witten invariants of , equivariant with respect to the fiberwise torus action, in terms of genus-zero invariants of the toric fiber and all-genus invariants of ; and (ii) we pass to the non-equivariant limit in this formula, using Brown's mirror theorem for toric bundles.

Paper Structure

This paper contains 22 sections, 10 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Genus-Zero Gromov--Witten Theory
  3. Grading
  4. Virasoro Constraints in Genus Zero
  5. Virasoro Constraints in Higher Genus
  6. Loop Group Covariance
  7. Toric Bundles
  8. The Proof of Theorem \ref{['thm:main']}
  9. Descendant-Ancestor Correspondence
  10. Fixed-Point Localization
  11. Why Does The Limit Exist?
  12. Fixed-Point Localization
  13. The $T$-Action on $E$
  14. The $T$-Action on the Moduli Space of Stable Maps
  15. Virtual Localization
  16. ...and 7 more sections

Key Result

Proposition 1.1

If the linear vector field on ${\mathcal{H}}$ defined by $l_0$ is tangent to the overruled Lagrangian cone ${\mathcal{L}}_X \subset {\mathcal{H}}$, then the linear vector fields defined by the operators $l_m$, $m\geq -1$, are all tangent to ${\mathcal{L}}_X$.

Theorems & Definitions (22)

  • Proposition 1.1: see Giv_Frob
  • proof
  • Conjecture 1.2: Virasoro Conjecture
  • Proposition 1.3: Loop Group Covariance
  • proof
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Claim
  • Proposition 3.1
  • ...and 12 more