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Dubrovin conjecture and the second structure connection

John Alexander Cruz Morales, Todor Milanov

Abstract

We give a reformulation of the Dubrovin conjecture about the semisimplicity of quantum cohomology in terms of the so-called second structure connection of quantum cohomology. The key ingredient in our work is the notion of a twisted reflection vector which allows us to give an elegant description of the monodromy data of the quantum connection in terms of the monodromy data of its Laplace transform.

Dubrovin conjecture and the second structure connection

Abstract

We give a reformulation of the Dubrovin conjecture about the semisimplicity of quantum cohomology in terms of the so-called second structure connection of quantum cohomology. The key ingredient in our work is the notion of a twisted reflection vector which allows us to give an elegant description of the monodromy data of the quantum connection in terms of the monodromy data of its Laplace transform.

Paper Structure

This paper contains 19 sections, 20 theorems, 168 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Semi-simple quantum cohomology
  3. Quantum differential equations
  4. Reflection vectors
  5. Monodromy data and reflection vectors
  6. Acknowledgements
  7. Twisted periods of a Frobenius manifold
  8. Frobenius manifolds
  9. Twisted periods
  10. Local monodromy
  11. Asymptotic expansions and Stokes matrices
  12. Stokes matrices and the intersection pairing
  13. The central connection matrix
  14. Dubrovin conjecture
  15. Quantum cohomology
  16. ...and 4 more sections

Key Result

Theorem 1.4.1

Let $\eta$ be an admissible direction and assume that the eigenvalues $u_1,\dots,u_N$ of the operator $E\bullet$ are enumerated according to the lexicographical order corresponding to $\eta$. Then the following statements hold.

Figures (5)

  • Figure 1: Reference paths and admissible directions
  • Figure 2: Domains of analyticity in $z$
  • Figure 3: $\eta_\nu$-sequence
  • Figure 4: Contour deformation
  • Figure 5: Points in the $\lambda$-plane and vectors in the $z$-plane

Theorems & Definitions (48)

  • Theorem 1.4.1
  • Theorem 1.4.2
  • Corollary 1.4.1
  • Definition 2.1.1
  • Remark 2.1.1
  • Definition 2.1.2
  • Definition 2.2.1
  • Remark 2.2.1
  • Proposition 2.3.1
  • Remark 2.3.1
  • ...and 38 more