[Paper Review] Graph potentials and moduli spaces of rank two bundles on a curve

Pieter Belmans, Sergey Galkin+1 more|arXiv (Cornell University)|Sep 11, 2020

Homotopy and Cohomology in Algebraic Topology115 references4 citations

TL;DR

This paper introduces graph potentials—Laurent polynomials associated to colored trivalent graphs—that encode degenerations of the moduli space of rank two vector bundles with fixed determinant on a curve. It establishes that the birational type of these potentials depends only on the homotopy type of the graph, thereby defining a topological quantum field theory, and links them to mirror symmetry invariants and derived categories via toric degenerations and Grothendieck ring analysis.

ABSTRACT

We introduce graph potentials, which are Laurent polynomials associated to (colored) trivalent graphs. These graphs encode degenerations of curves to rational curves, and graph potentials encode degenerations of the moduli space of rank 2 bundles with fixed determinant. We show that the birational type of the graph potential only depends on the homotopy type of the colored graph, and thus define a topological quantum field theory. By analyzing toric degenerations of the moduli spaces we explain how graph potentials are related to these moduli spaces in the setting of mirror symmetry for Fano varieties. On the level of enumerative mirror symmetry this shows how invariants of graph potentials are related to Gromov-Witten invariants of the moduli space. In the context of homological mirror symmetry we formulate a conjecture regarding the shape of semiorthogonal decompositions for the derived category. Studying the properties of graph potentials we provide evidence for this conjecture. Finally, by studying the Grothendieck rings of varieties and categories we will give further geometric evidence.

Motivation & Objective

• To define graph potentials as Laurent polynomials associated to colored trivalent graphs that encode degenerations of moduli spaces of rank two vector bundles with fixed determinant.
• To establish that the birational type of a graph potential depends only on the homotopy type of the underlying colored graph, thus constructing a topological quantum field theory.
• To connect graph potentials to mirror symmetry by relating their enumerative invariants to Gromov-Witten invariants of the moduli space.
• To formulate a conjecture on the shape of semiorthogonal decompositions in the derived category of the moduli space, supported by properties of graph potentials.
• To provide geometric evidence for the conjecture using Grothendieck rings of varieties and categories.

Proposed method

• Construct graph potentials as Laurent polynomials from colored trivalent graphs that model degenerations of algebraic curves to rational curves.
• Use toric degenerations of the moduli space of rank two bundles to relate graph potentials to mirror symmetry structures.
• Analyze the enumerative mirror symmetry correspondence by comparing invariants of graph potentials to Gromov-Witten invariants of the moduli space.
• Formulate a conjecture on the structure of semiorthogonal decompositions in the derived category of the moduli space based on the combinatorial and algebraic properties of graph potentials.
• Apply Grothendieck ring techniques to varieties and categories to derive geometric invariants that support the conjecture.
• Establish that the birational equivalence class of a graph potential is invariant under homotopy of the colored graph, thereby defining a topological quantum field theory.

Experimental results

Research questions

• RQ1 How do graph potentials, defined from colored trivalent graphs, encode degenerations of the moduli space of rank two vector bundles with fixed determinant?
• RQ2 What is the dependence of the birational type of a graph potential on the topological structure of the underlying graph?
• RQ3 How are invariants of graph potentials related to Gromov-Witten invariants of the moduli space in the context of enumerative mirror symmetry?
• RQ4 What is the expected structure of semiorthogonal decompositions in the derived category of the moduli space, and how can graph potentials inform this structure?
• RQ5 What geometric evidence can be derived from Grothendieck rings of varieties and categories to support the conjectured structure of derived categories?

Key findings

• The birational type of a graph potential depends only on the homotopy type of the colored trivalent graph, establishing a topological quantum field theory structure.
• Graph potentials are related to Gromov-Witten invariants of the moduli space through enumerative mirror symmetry, linking their invariants to curve counting invariants.
• The paper formulates a conjecture on the shape of semiorthogonal decompositions in the derived category of the moduli space, supported by structural properties of graph potentials.
• Toric degenerations of the moduli space provide a geometric bridge between graph potentials and mirror symmetry for Fano varieties.
• Grothendieck ring techniques yield additional geometric invariants that support the conjecture on derived categories.
• The construction of graph potentials via trivalent graphs provides a unified framework for studying degenerations of moduli spaces and their mirror duals.