Decompositions of moduli spaces of vector bundles and graph potentials

TL;DR

This paper explores advanced decompositions of moduli spaces of vector bundles, linking algebraic geometry, motivic theory, and mirror symmetry, and proposes conjectural structures supported by various evidence.

Contribution

It introduces a conjectural semiorthogonal decomposition for the derived category of certain moduli spaces and relates it to motivic and mirror symmetry decompositions.

Findings

Proposes a semiorthogonal decomposition conjecture for moduli space derived categories.

Connects motivic decompositions in the Grothendieck ring to geometric structures.

Relates mirror symmetry decompositions to Fukaya categories and quantum cohomology.

Abstract

We propose a conjectural semiorthogonal decomposition for the derived category of the moduli space of stable rank 2 bundles with fixed determinant of odd degree, independently formulated by Narasimhan. We discuss some evidence for, and furthermore propose semiorthogonal decompositions with additional structure. We also discuss two other decompositions. One is a decomposition of this moduli space in the Grothendieck ring of varieties, which relates to various known motivic decompositions. The other is the critical value decomposition of a candidate mirror Landau-Ginzburg model given by graph potentials, which in turn is related under mirror symmetry to Munoz's decomposition of quantum cohomology. This corresponds to an orthogonal decomposition of the Fukaya category. We will explain how these decompositions can be seen as evidence for the conjectural semiorthogonal decomposition.