Corpus ID: 221641031

# Graph potentials and moduli spaces of rank two bundles on a curve

@article{Belmans2020GraphPA,
title={Graph potentials and moduli spaces of rank two bundles on a curve},
author={Pieter Belmans and Sergey Galkin and Swarnava Mukhopadhyay},
journal={arXiv: Algebraic Geometry},
year={2020}
}
• Published 2020
• Mathematics
• arXiv: Algebraic Geometry
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… Expand
4 Citations

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#### References

SHOWING 1-10 OF 105 REFERENCES
Graph Curves *
We study a family of stable curves defined combinatorially from a trivalent graph. Most of our results are related to the conjecture of Green which relates the Clifford index of a smooth curve, anExpand
Moduli spaces of local systems and higher Teichmüller theory
• Mathematics
• 2003
Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of SExpand
Coordinate rings for the moduli stack of SL2(C) quasi-parabolic principal bundles on a curve and toric fiber products
Abstract We continue the program started in Manon (2010) [M1] to understand the combinatorial commutative algebra of the projective coordinate rings of the moduli stack M C , p → ( SL 2 ( C ) ) ofExpand
Generalized Theta Functions, Strange Duality, and Odd Orthogonal Bundles on Curves
• Mathematics
• Communications in Mathematical Physics
• 2019
This paper studies spaces of generalized theta functions for odd orthogonal bundles with nontrivial Stiefel-Whitney class and the associated space of twisted spin bundles. In particular, we prove aExpand
Infinite Grassmannians and moduli spaces ofG-bundles
• Mathematics
• 1994
Let C be a smooth projective irreducible algebraic curve over C of any genus and G a connected simply-connected simple affine algebraic group over C. In this paper we elucidate the relationshipExpand
Quantum periods for 3-dimensional Fano manifolds
• Mathematics
• 2016
The quantum period of a variety X is a generating function for certain Gromov-Witten invariants of X which plays an important role in mirror symmetry. In this paper we compute the quantum periods ofExpand
Desingularizations of the moduli space of rank 2 bundles over a curve
• Mathematics
• 2004
Abstract.Let X be a smooth projective curve of genus g≥3 and M0 be the moduli space of rank 2 semistable bundles over X with trivial determinant. There are three desingularizations of this singularExpand
Gravitational Quantum Cohomology
• Physics
• 1997
We discuss how the theory of quantum cohomology may be generalized to "gravitational quantum cohomology" by studying topological σ models coupled to two-dimensional gravity. We first consider σExpand
The Algebra of Conformal Blocks
For each simply connected, simple complex group $G$ we show that the direct sum of all vector bundles of conformal blocks on the moduli stack $\bar{\mathcal{M}}_{g, n}$ of stable marked curvesExpand
Triangulated categories of singularities and D-branes in Landau-Ginzburg models
In spite of physics terms in the title, this paper is purely mathematical. Its purpose is to introduce triangulated categories related to singularities of algebraic varieties and establish aExpand