Decompositions of moduli spaces of vector bundles and graph potentials

14J33 18G80 14D20 14J45; Analysis; Theoretical Computer Science; Algebra and Number Theory; Statistics and Probability; Mathematical Physics; Geometry and Topology; Discrete Mathematics and Combinatorics; Computational Mathematics; 14J33; 18G80; 14D20; 14J45; Mathematics - Algebraic Geometry; Mathematics - Geometric Topology; Mathematics - Quantum Algebra; Mathematics - Symplectic Geometry

Abstract :

[en] 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 Muñoz's decomposition of quantum cohomology. This corresponds to an orthogonal decomposition of the Fukaya category. We discuss how decompositions on different levels (derived category of coherent sheaves, Grothendieck ring of varieties, Fukaya category, quantum cohomology, critical sets of graph potentials) are related and support each other.

Disciplines :

Mathematics

Author, co-author :

BELMANS, Pieter ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)

Galkin, Sergey ; PUC-Rio, Departamento de Matemática, Rio de Janeiro, Brazil

Mukhopadhyay, Swarnava; School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India

External co-authors :

yes

Language :

English

Title :

Decompositions of moduli spaces of vector bundles and graph potentials

Publication date :

07 March 2023

Journal title :

Forum of Mathematics, Sigma

eISSN :

2050-5094

Publisher :

Cambridge University Press

Volume :

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S2050509423000142

Funding text :

The first author was partially supported by the FWO (Research Foundation–Flanders). The third author was partially supported by the Department of Atomic Energy, India, under project no. 12-R&D-TFR-5.01-0500 and also by the Science and Engineering Research Board, India (SRG/2019/000513).

Available on ORBilu :

since 27 November 2023

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