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Orbifold Kodaira-Spencer maps and closed-string mirror symmetry for punctured Riemann surfaces

Hansol Hong, Hyeongjun Jin, Sangwook Lee

TL;DR

The paper develops a comprehensive framework for closed-string mirror symmetry in the presence of finite abelian group actions on noncompact symplectic manifolds, by constructing mirrors as $\hat G$-orbifolds of quotients and by using a Kodaira-Spencer-type closed-open map. It works out an explicit, computable correspondence between the symplectic cohomology of abelian covers of the pair-of-pants and orbifold Koszul cohomology of the mirror potential $W=xyz$, via equivariant Lagrangian Floer theory and weighted holomorphic-curve counts. The key technical contributions include a concrete equivariant Floer theory, a robust description of the Seidel Lagrangian in the base and its Maurer-Cartan deformation, and a detailed analysis of twisted sectors in orbifold Koszul algebras. The results provide a first-principles, Kodaira-Spencer-type realization of closed-string mirror symmetry for certain noncompact settings and yield explicit orbifold LG mirrors for punctured Riemann surfaces, with transparent examples such as sphere with four punctures and a tri-punctured torus.

Abstract

When a Weinstein manifold admits an action of a finite abelian group, we propose its mirror construction following the equivariant TQFT-type construction, and obtain as a mirror the orbifolding of the mirror of the quotient with respect to the induced dual group action. As an application, we construct an orbifold Landau-Ginzburg mirror of a punctured Riemann surface given as an abelian cover of the pair-of-pants, and prove its closed-string mirror symmetry using the (part of) closed-open map twisted by the dual group action.

Orbifold Kodaira-Spencer maps and closed-string mirror symmetry for punctured Riemann surfaces

TL;DR

The paper develops a comprehensive framework for closed-string mirror symmetry in the presence of finite abelian group actions on noncompact symplectic manifolds, by constructing mirrors as -orbifolds of quotients and by using a Kodaira-Spencer-type closed-open map. It works out an explicit, computable correspondence between the symplectic cohomology of abelian covers of the pair-of-pants and orbifold Koszul cohomology of the mirror potential , via equivariant Lagrangian Floer theory and weighted holomorphic-curve counts. The key technical contributions include a concrete equivariant Floer theory, a robust description of the Seidel Lagrangian in the base and its Maurer-Cartan deformation, and a detailed analysis of twisted sectors in orbifold Koszul algebras. The results provide a first-principles, Kodaira-Spencer-type realization of closed-string mirror symmetry for certain noncompact settings and yield explicit orbifold LG mirrors for punctured Riemann surfaces, with transparent examples such as sphere with four punctures and a tri-punctured torus.

Abstract

When a Weinstein manifold admits an action of a finite abelian group, we propose its mirror construction following the equivariant TQFT-type construction, and obtain as a mirror the orbifolding of the mirror of the quotient with respect to the induced dual group action. As an application, we construct an orbifold Landau-Ginzburg mirror of a punctured Riemann surface given as an abelian cover of the pair-of-pants, and prove its closed-string mirror symmetry using the (part of) closed-open map twisted by the dual group action.
Paper Structure (22 sections, 26 theorems, 163 equations, 13 figures)

This paper contains 22 sections, 26 theorems, 163 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symplectic cohomology
  4. Symplectic cohomology of pair-of-pants
  5. Orbifold LG B-models
  6. Finite group action on Lagrangian Floer theory and semidirect product
  7. The Seidel Lagrangian in the pair-of-pants $P$
  8. Orbifold Koszul cohomology of $(xyz,{\widehat{G}})$
  9. Equivariant constructions
  10. Trivializing the principal $G$-bundle $X \to B$
  11. $\hat{G}$-equivariant symplectic chain complex
  12. $\hat{G}$-equivariant Lagrangian Floer theory
  13. Example: Seidel Lagrangian in the pair-of-pants
  14. Orbifold mirror LG models for some punctured Riemann surfaces
  15. Kodaira-Spencer map
  16. ...and 7 more sections

Key Result

Theorem 1.3

Let $G$ be a finite abelian group. For a principal $G$-bundle $X \to P$ of the pair-of-pants $P$, we have an isomorphism obtained by a sequence of quasi-isomorphisms where $K^\ast (W,\hat{G})$ is a cochain complex underlying $Kos(W,\hat{G})$. In particular, if we transfer the algebra structure from $CF_{\hat{G}} ( (\mathbb{L},b),(\mathbb{L},b))^{\hat{G}}$, then we get a new product structure on

Figures (13)

  • Figure 1: Mirrors of the four-punctured sphere
  • Figure 2: Mirrors of the tri-punctured torus
  • Figure 3: Choice of the Hamiltonian $H$ on $P$ and its critical points in $P^{in}$
  • Figure 4: The Seidel Lagrangian $\mathbb{L}$
  • Figure 5: $\Theta$ whose associated $CF_{\hat{G}} (\mathbb{L},\mathbb{L})$ coincides with $CF(\mathbb{L},\mathbb{L};\hat{G})$ as algebras. (This is valid only for an abelian $G$. One of labelings in the bottom triangle should be replaced by its conjugate when $G$ is not abelian.)
  • ...and 8 more figures

Theorems & Definitions (54)

  • Theorem 1.3: Corollary \ref{['cor:final']}
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5: CLe
  • Proposition 2.6
  • proof
  • Definition 2.7
  • ...and 44 more