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Confluence in quantum K-theory of weak Fano manifolds and q-oscillatory integrals for toric manifolds

Todor Milanov, Alexis Roquefeuil

Abstract

For a smooth projective variety whose anti-canonical bundle is nef, we prove confluence of the small $K$-theoretic $J$-function, i.e., after rescaling appropriately the Novikov variables, the small $K$-theoretic $J$-function has a limit when $q\to 1$, which coincides with the small cohomological $J$-function. Furthermore, in the case of a Fano toric manifold $X$ of Picard rank 2, we prove the $K$-theoretic version of an identity due to Iritani that compares the $I$-function of the toric manifold and the oscillatory integral of the toric mirror. In particular, our confluence result yields a new proof of Iritani's identity in the case of a toric manifold of Picard rank 2.

Confluence in quantum K-theory of weak Fano manifolds and q-oscillatory integrals for toric manifolds

Abstract

For a smooth projective variety whose anti-canonical bundle is nef, we prove confluence of the small -theoretic -function, i.e., after rescaling appropriately the Novikov variables, the small -theoretic -function has a limit when , which coincides with the small cohomological -function. Furthermore, in the case of a Fano toric manifold of Picard rank 2, we prove the -theoretic version of an identity due to Iritani that compares the -function of the toric manifold and the oscillatory integral of the toric mirror. In particular, our confluence result yields a new proof of Iritani's identity in the case of a toric manifold of Picard rank 2.

Paper Structure

This paper contains 39 sections, 38 theorems, 311 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Foreword
  3. Plan of the paper
  4. The ABC of twisted Gromov--Witten theory
  5. Definition of ABC twists
  6. Removing the twists
  7. Fake $K$-theoretic GW invariants
  8. Stem invariants
  9. Confluence of the small $K$-theoretic $J$-function
  10. The Kawasaki--Riemann--Roch formula
  11. Stems as Kawasaki strata
  12. Reconstruction of the small $K$-theoretic $J$-function
  13. Confluence of the small $K$-theoretic $J$-function
  14. Oscillatory integrals in Gromov--Witten theories
  15. Symplectic toric manifolds
  16. ...and 24 more sections

Key Result

Theorem 1

If $X$ is a smooth projective variety, such that, the anti-canonical bundle $K_X^\vee$ is nef, then the limit exists and it coincides with $J^{\rm coh}(1,Q)$.

Figures (1)

  • Figure 1: Kawasaki stratum $I_\zeta X_{0,1,d}$

Theorems & Definitions (92)

  • Definition 1.1.1
  • Definition 1.1.3: IMT, Definition 2.4
  • Theorem : Theorem \ref{['t1']}
  • Theorem : Iritani:gamma_structure, see Theorem \ref{['thm:gamma_cohomology_oscillatory_i_function']}
  • Remark 1.1.5
  • Definition : Definition \ref{['def:qosc_q_gamma_class']}
  • Theorem : Theorem \ref{['thm:gamma_comparison_theorem_in_quantum_k_theory']}
  • Remark 1.2.1
  • Definition 2.1.1
  • Definition 2.1.2
  • ...and 82 more