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The tt*-structure for the quantum cohomology of complex Grassmannian

Tadashi Udagawa

TL;DR

This work constructs a global tt*-structure for the quantum cohomology of the Grassmannian $qH^*({\rm Gr}(k,\mathbb{C}^{k+N}))$ by employing a Landau-Ginzburg potential $W_{k,N}$ and the DPW method to obtain radial solutions. The authors show that this Grassmannian tt*-structure is precisely the $k$-th exterior power of the tt*-structure for $qH^*(\mathbb{C}P^{k+N-1})$, establishing a deep link between Grassmannian and projective-space quantum cohomology via exterior powers. They provide explicit holomorphic (DPW) data and asymptotic (Toda) data, and prove a detailed isomorphism between the Grassmannian tt*-structure and wedge$^k$ CP$^n$ tt*-structures, interpreted within a Lie-theoretic principal bundle framework. This yields a unified description of the tt*-equations for Grassmannians, clarifies the role of exterior powers in their solutions, and connects to isomonodromic and Hitchin-type structures, with potential implications for mirror symmetry and quantum cohomology computations.

Abstract

The tt*-equation (topological-anti-topological fusion equation) was introduced by S. Cecotti and C. Vafa for describing massive deformation of supersymmetric conformal field theories. B. Dubrovin formulated the tt*-equation as a flat bundle, called tt*-structure. In this paper, we construct a tt*-structure for the quantum cohomology of the Grassmannian of complex \(k\)-plane and obtain global solutions to the tt*-equation, following the idea of Bourdeau. We give a precise mathematical formulation and a description of the solutions by using p.d.e. theory and the harmonic map theory developed by J. Dorfmeister, F. Pedit and H. Wu (the DPW method). Furthermore, we give an isomorphism between tt*-structure for the \(k\)-th exterior product of tt*-structure for the quantum cohomology of the complex projective space and the tt*-structure for the quantum cohomology of the Grassmannian.

The tt*-structure for the quantum cohomology of complex Grassmannian

TL;DR

This work constructs a global tt*-structure for the quantum cohomology of the Grassmannian by employing a Landau-Ginzburg potential and the DPW method to obtain radial solutions. The authors show that this Grassmannian tt*-structure is precisely the -th exterior power of the tt*-structure for , establishing a deep link between Grassmannian and projective-space quantum cohomology via exterior powers. They provide explicit holomorphic (DPW) data and asymptotic (Toda) data, and prove a detailed isomorphism between the Grassmannian tt*-structure and wedge CP tt*-structures, interpreted within a Lie-theoretic principal bundle framework. This yields a unified description of the tt*-equations for Grassmannians, clarifies the role of exterior powers in their solutions, and connects to isomonodromic and Hitchin-type structures, with potential implications for mirror symmetry and quantum cohomology computations.

Abstract

The tt*-equation (topological-anti-topological fusion equation) was introduced by S. Cecotti and C. Vafa for describing massive deformation of supersymmetric conformal field theories. B. Dubrovin formulated the tt*-equation as a flat bundle, called tt*-structure. In this paper, we construct a tt*-structure for the quantum cohomology of the Grassmannian of complex -plane and obtain global solutions to the tt*-equation, following the idea of Bourdeau. We give a precise mathematical formulation and a description of the solutions by using p.d.e. theory and the harmonic map theory developed by J. Dorfmeister, F. Pedit and H. Wu (the DPW method). Furthermore, we give an isomorphism between tt*-structure for the -th exterior product of tt*-structure for the quantum cohomology of the complex projective space and the tt*-structure for the quantum cohomology of the Grassmannian.
Paper Structure (14 sections, 21 theorems, 131 equations)

This paper contains 14 sections, 21 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. tt*-structures
  4. The DPW method
  5. The tt*-structures for the quantum cohomology ring of the Grassmannian
  6. A tt*-structure for $qH^*(\mathbb{C}P^{k + N -1})$
  7. A tt*-structure for $qH^*({\rm Gr}(k,\mathbb{C}^{k+N}))$
  8. The tt*-equation for $qH^*({\rm Gr}(k,\mathbb{C}^{k+N}))$
  9. The asymptotic data
  10. The holomorphic data
  11. The relation between $qH^*({\rm Gr}(k,\mathbb{C}^{k+N})) and \bigwedge^kqH^*(\mathbb{C}P^n)$
  12. The induce tt*-structure on $\bigwedge^k E^{\mathbb{C}P}_n$
  13. The relation between the tt*-structure on $E^{\rm Gr}_{k,N}$ and the tt*-structure on $\bigwedge^k E^{\rm Gr}_n$
  14. A tt*-structure on a principal $G$-bundle

Key Result

Proposition 3.1

$(E^{\mathbb{C}P}_n,\eta^{\mathbb{C}P},g^{\mathbb{C}P},\Phi^{\mathbb{C}P})$ is a tt*-structure over $\mathbb{C} \backslash (-\infty,0]$ if and only if $\{u_j\}_{j=0}^{n}$ satisfies with the condition $u_j + u_{n-j}=0$ for all $j$.

Theorems & Definitions (51)

  • Definition 2.1
  • Definition 2.2
  • Example 1: The sinh-Gordon equation
  • Example 2: The tt*-Toda equation GIL20152, GIL2020
  • Definition 3.1
  • Proposition 3.1
  • proof
  • Lemma 3.1: Gepner G1991
  • Proposition 3.2
  • proof
  • ...and 41 more