A journey from the Hitchin section to the oper moduli

2016/12/31 by Dumitrescu, Olivia
#53C07 Secondary: 14D21 #81T13 #Algebraic Geometry (math.AG) #FOS: Mathematics #Primary: 58E15

paper · doi:10.48550/arxiv.1701.00155

Abstract

This paper provides an introduction to the mathematical notion of quantum curves. We start with a concrete example arising from a graph enumeration problem. We then develop a theory of quantum curves associated with Hitchin spectral curves. A conjecture of Gaiotto, which predicts a new construction of opers from a Hitchin spectral curve, is explained. We give a step-by-step detailed description of the proof of the conjecture for the case of rank 2 Higgs bundles. Finally, we identify the two concepts of quantum curve arising from the topological recursion formalism with the limit oper of Gaiotto's conjecture.

Related

• Lectures on the topological recursion for Higgs bundles and quantum curves
• An invitation to 2D TQFT and quantization of Hitchin spectral curves
• Quantization of spectral curves for meromorphic Higgs bundles through topological recursion
• Quantum curves and topological recursion
• In Search of a Hidden Curve