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$T$-equivariant disc potential and SYZ mirror construction

Yoosik Kim, Siu-Cheong Lau, Xiao Zheng

TL;DR

This work builds a $G$-equivariant Lagrangian Floer theory by constructing a Morse-model on the Borel construction $L_G$, resulting in a curved $A_50$-algebra whose equivariant disc potential $W^L_G$ encodes open invariants with $H^ullet_G(pt;\Lambda_0)$-coefficients. For toric, semi-Fano contexts, the authors show that the $T$-equivariant disc potential of a regular moment-map fibre reproduces the equivariant toric Landau–Ginzburg mirror $W_{\lambda}$ via the inverse mirror map, connecting to Givental–Hori–Vafa-type data; they also provide a gluing-based computation of the $\mathbb{S}^1$-equivariant disc potential for the immersed pinched torus, yielding $W^{L_0}_{\mathbb{S}^1}=\log(1-uv)$. The framework employs finite-dimensional approximations to $EG\to BG$, an equivariant version of Fukaya–Oh–Ohta–Ono perturbation theory, and a notion of partial units to handle equivariant parameters, enabling explicit open-curve counting for both toric and immersed Lagrangians. Overall, the paper advances equivariant SYZ mirror constructions and provides practical tools for computing equivariant disc potentials across toric and immersed settings, with potential implications for 3D gauge-theoretic mirrors via Teleman’s program.

Abstract

We develop a $G$-equivariant Lagrangian Floer theory and obtain a curved $A_\infty$ algebra, and in particular a $G$-equivariant disc potential. We construct a Morse model, which counts pearly trees in the Borel construction $L_G$. When applied to a smooth moment map fiber of a semi-Fano toric manifold, our construction recovers the $T$-equivariant toric Landau-Ginzburg mirror of Givental. We also study the $\bS^1$-equivariant Floer theory of a typical singular SYZ fiber (i.e. a pinched torus) and compute its $\bS^1$-equivariant disc potential via the gluing technique developed in \cite{CHL18,HKL}.

$T$-equivariant disc potential and SYZ mirror construction

TL;DR

This work builds a -equivariant Lagrangian Floer theory by constructing a Morse-model on the Borel construction , resulting in a curved -algebra whose equivariant disc potential encodes open invariants with -coefficients. For toric, semi-Fano contexts, the authors show that the -equivariant disc potential of a regular moment-map fibre reproduces the equivariant toric Landau–Ginzburg mirror via the inverse mirror map, connecting to Givental–Hori–Vafa-type data; they also provide a gluing-based computation of the -equivariant disc potential for the immersed pinched torus, yielding . The framework employs finite-dimensional approximations to , an equivariant version of Fukaya–Oh–Ohta–Ono perturbation theory, and a notion of partial units to handle equivariant parameters, enabling explicit open-curve counting for both toric and immersed Lagrangians. Overall, the paper advances equivariant SYZ mirror constructions and provides practical tools for computing equivariant disc potentials across toric and immersed settings, with potential implications for 3D gauge-theoretic mirrors via Teleman’s program.

Abstract

We develop a -equivariant Lagrangian Floer theory and obtain a curved algebra, and in particular a -equivariant disc potential. We construct a Morse model, which counts pearly trees in the Borel construction . When applied to a smooth moment map fiber of a semi-Fano toric manifold, our construction recovers the -equivariant toric Landau-Ginzburg mirror of Givental. We also study the -equivariant Floer theory of a typical singular SYZ fiber (i.e. a pinched torus) and compute its -equivariant disc potential via the gluing technique developed in \cite{CHL18,HKL}.

Paper Structure

This paper contains 16 sections, 33 theorems, 274 equations, 6 figures.

Table of Contents

  1. Introduction
  2. A Morse model for Lagrangian Floer theory
  3. The singular chain model
  4. Morse homology $=$ singular homology: an alternative approach
  5. Morse model with a strict unit
  6. Disc potentials in Lagrangian Floer theory
  7. A Morse model for equivariant Lagrangian Floer theory
  8. The equivariant Morse model
  9. Equivariant parameters as homotopy partial units
  10. Disc potentials in equivariant Lagrangian Floer theory
  11. $T$-equivariant disc potentials of toric manifolds
  12. Morse theory on the approximation spaces
  13. Computing $T$-equivariant disc potentials
  14. $\mathbb{S}^1$-equivariant disc potential for the immersed two-sphere
  15. Equivariant Floer theory for Lagrangian immersions
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $X$ be a semi-projective and semi-Fano toric manifold with $\dim_{\mathbb{C}} X=d$, and let $T\subset (\mathbb{S}^1)^d$ be the subtorus determined by the integral basis $u_1,\ldots,u_{\ell} \in \bm{N}\cong \mathbb{Z}^d$. The $T$-equivariant disc potential (def:G_disc_potential) $W^L_{T}$ of a re where $\beta_i$ are the basic disc classes bounded by the toric fiber, $g_i(\check{q}(q))$ is given

Figures (6)

  • Figure 1: On the left is a Morse function on $S^2$. On the right is unstable chain $\Delta_p$ assigned to its maximal point $p$.
  • Figure 2: Pearl trees
  • Figure 3: All possible configurations contributing to $\mathfrak{m}_{2,\beta}(X_1,X_1)$ in the case $\partial \beta\cdot X_1=2$, after requiring $z_0$ to intersect $\mathbf{1}_L$. Here $X^{(i)}_1$ denotes the composition of $X_1$ with the time-$1$ flow of $v_i$.
  • Figure 4: The geometric inputs of $W^{L}_T$. The non-constant discs (with sphere bubbles) contribute to $\bm{1}^{\blacktriangledown}$, and the flow line contributes to $\bm{\lambda}^{\blacktriangledown}$.
  • Figure 5: $L$ can be seen locally as a union of two cleanly intersecting Lagrangians $L_1$ and $L_2$. The pearly tree shown on the right does not exist due to inconsistency of Lagrangian boundary labels.
  • ...and 1 more figures

Theorems & Definitions (77)

  • Theorem 1.1: Corolloary \ref{['cor:tor']}
  • Remark 1.2
  • Theorem 1.3: Theorem \ref{['thm:equiv-imm']}
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Theorem 2.5
  • proof
  • ...and 67 more