Graph potentials and symplectic geometry of moduli spaces of vector bundles

TL;DR

This paper constructs new examples of Fano manifolds with optimal Lagrangian tori, linking symplectic geometry, algebraic geometry, and mirror symmetry through graph-based models and moduli spaces of vector bundles.

Contribution

It introduces the first examples of Fano manifolds with multiple optimal tori and establishes a mirror symmetry correspondence with graph potentials.

Findings

Construction of monotone Lagrangian tori with maximal holomorphic disc counts.

Association of optimal tori to trivalent graphs on symplectic Fano manifolds.

Confirmation of mirror symmetry between these pairs and graph potential B-models.

Abstract

We give the first examples of Fano manifolds with multiple optimal tori, i.e.~we construct monotone Lagrangian tori $L$, such that the weighted number of holomorphic Maslov index two discs with boundary on $L$ equals the upper bound given by the symplectic invariant $\limsup_n ([m_0(L)^n]_{x^0})^{1/n}$, where $m_0(L)$ is the Floer potential. To every trivalent graph $\gamma$ of genus $g$ we associate an optimal torus $L_\gamma$ on the celebrated symplectic Fano manifold $\mathcal{N}_g$ (of complex dimension $3g-3$) with $\mathrm{T}_{\mathcal{N}_g} = 8g-8$), given by the character variety of rank 2 on a genus $g$ surface with prescribed odd monodromy at a puncture, We moreover show that all pairs $(\mathcal{N}_g,L_\gamma)$ are pairwise non-isotopic. In particular, we confirm a form of mirror symmetry between the A-model of the pairs $(\mathcal{N}_g,L_\gamma)$ (and also spaces $\mathcal{N}_g$ standalone) and B-model of graph potentials, a family of Laurent polynomials we introduced in earlier work. A crucial input from outside of symplectic geometry is an analysis of Manon's toric degenerations of algebro-geometric models $\mathrm{M}_C(2,\mathcal{L})$ for the spaces $\mathcal{N}_g$, as moduli spaces of stable rank $2$ bundles on an algebraic curve with a fixed determinant, constructed using conformal field theory.