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Quantum cohomology of projective bundles

Hiroshi Iritani, Yuki Koto

TL;DR

This work advances the genus-zero theory of projective bundles by proving a non-split mirror theorem: the $I$-function for $\mathbb{P}(V)$ is obtained as a Fourier transform of the $S^1$-equivariant $J$-function of $V$ and lies on the Givental cone, enabling the quantum $D$-module of $\mathbb{P}(V)$ to decompose into a direct sum of $QDM(B)$ pieces after localization. It establishes a precise Fourier duality between the equivariant quantum $D$-module of $V$ and the (non-equivariant) quantum $D$-module of $\mathbb{P}(V)$, mediated by a mirror map $\hat{\tau}$ and a functorial isomorphism that exchanges shift and Novikov variables. Building on this, the paper proves a formal decomposition of $QDM(\mathbb{P}(V))$ into $r$ copies of $QDM(B)$, together with explicit projections induced by stationary-phase Fourier analysis, which implies that the big quantum cohomology of $\mathbb{P}(V)$ is generically semisimple iff that of $B$ is. The results yield reconstructive tools for genus-zero Gromov-Witten invariants of $\mathbb{P}(V)$ from those of $B$ and the Chern data of $V$, with concrete implications for classical spaces such as Grassmannians and isotropic Grassmannians via semisimplicity. Overall, the paper provides a coherent framework linking equivariant and non-equivariant theories through Fourier duality, QRR, and stationary-phase methods, yielding new structural insights into quantum cohomology for projective bundles.

Abstract

We construct an I-function of the projective bundle P(V) associated with a not necessarily split vector bundle V\to B as a Fourier transform of the S^1-equivariant J-function of the total space of V and show that it lies on the Givental Lagrangian cone of P(V). Using this result, we show that the quantum cohomology D-module of P(V) splits into the direct sum of the quantum cohomology D-modules of the base space B. This has applications to the semisimplicity of big quantum cohomology.

Quantum cohomology of projective bundles

TL;DR

This work advances the genus-zero theory of projective bundles by proving a non-split mirror theorem: the -function for is obtained as a Fourier transform of the -equivariant -function of and lies on the Givental cone, enabling the quantum -module of to decompose into a direct sum of pieces after localization. It establishes a precise Fourier duality between the equivariant quantum -module of and the (non-equivariant) quantum -module of , mediated by a mirror map and a functorial isomorphism that exchanges shift and Novikov variables. Building on this, the paper proves a formal decomposition of into copies of , together with explicit projections induced by stationary-phase Fourier analysis, which implies that the big quantum cohomology of is generically semisimple iff that of is. The results yield reconstructive tools for genus-zero Gromov-Witten invariants of from those of and the Chern data of , with concrete implications for classical spaces such as Grassmannians and isotropic Grassmannians via semisimplicity. Overall, the paper provides a coherent framework linking equivariant and non-equivariant theories through Fourier duality, QRR, and stationary-phase methods, yielding new structural insights into quantum cohomology for projective bundles.

Abstract

We construct an I-function of the projective bundle P(V) associated with a not necessarily split vector bundle V\to B as a Fourier transform of the S^1-equivariant J-function of the total space of V and show that it lies on the Givental Lagrangian cone of P(V). Using this result, we show that the quantum cohomology D-module of P(V) splits into the direct sum of the quantum cohomology D-modules of the base space B. This has applications to the semisimplicity of big quantum cohomology.
Paper Structure (32 sections, 17 theorems, 175 equations, 2 figures)

This paper contains 32 sections, 17 theorems, 175 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mirror theorem for non-split projective bundles
  3. Decomposition of quantum $D$-modules
  4. Equivariant quantum cohomology and Fourier transformation
  5. Plan of the paper
  6. Preliminaries
  7. Gromov-Witten invariants and quantum cohomology
  8. Quantum connection and fundamental solution
  9. The vector bundle case
  10. Givental Lagrangian cones
  11. The vector bundle case
  12. Twisted Gromov-Witten invariants and quantum Riemann-Roch theorem
  13. Moving points on the Givental cone via differential operators
  14. Shift operators
  15. Proof of the mirror theorem
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $B$ be a smooth projective variety and let $V \to B$ be a vector bundle of rank $r\ge 2$. We assume that the dual bundle $V^\vee$ is generated by global sections. Consider the $S^1$-action on $V$ scaling fibres and let $J_V^\lambda(\tau)$ denote the $S^1$-equivariant $J$-function of the total sp Then $z I_{\mathbb{P}(V)}(\tau,t)|_{z\to -z}$ lies in the Givental cone of $\mathbb{P}(V)$. Here $q

Figures (2)

  • Figure 1: Decomposition pictured over the global Kähler moduli space: $\operatorname{QDM}(\mathbb{P}(V))$ splits into a direct sum of $\operatorname{QDM}(B)$ near the infinity divisor $(q^{-1/r'}=0)$.
  • Figure 2: Decomposition of eigenvalues of the Euler multiplication of $\mathbb{P}(V)$ into $r$ groups ($r=6$ in this picture).

Theorems & Definitions (54)

  • Theorem 1.1: Theorem \ref{['thm:mirrorthm']}
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7: Theorem \ref{['thm:decomp_QDM']}
  • Corollary 1.8: see §\ref{['subsec:semisimplicity']} for the proof
  • Remark 1.9
  • Remark 1.10
  • ...and 44 more