Required background: Algebra and Combinotorics (at least some interest) for math students; Field theory (at least classical) for physics students
Gauge/YBE correspondence is an exciting ongoing research area. This correspondence, roughly speaking, relates an exact results in supersymmetric gauge theories to exactly calculable quantities in lattice spin models of statistical mechanics.
Supersymmetry is a powerful idea in theoretical physics, which provides a non-trivial extension of the Poincare algebra. It transforms bosons into fermions and vice versa. Nowadays, supersymmetry is one of the key tools for high energy physics research beyond the Standard Model of particle physics.
Supersymmetry cannot be an exact symmetry of Nature. No one has ever seen a supersymmetric particle and so far no experimental evidence for supersymmetry has been discovered at the Large Hadron Collider and Tevatron. However the possibility that the world is supersymmetric at high energies attracts attention of scientists.
In this project we will develop supersymmetry in the framework of quantum mechanics.
Recently, an important progress has been made in studying the question of the relation between the perturbative and non-perturbative contributions in quantum theories. Many of this progress is due to resurgence theory, developed by Ecalle in the early 1980s. Rather than the usual perturbative expansion for an energy eigenvalue resurgence analysis connects perturbative contributions with non-perturbative effects through ``resurgent trans-series'' such that the imaginary terms with ambiguities coming from the Borel summation cancels systematically each other to all orders.
The resurgent expansion of the energy can be obtained by a systematic small g-expansion of exact quantization condition in some quantum mechanical systems. In this approach, the quantization condition includes two functions, one of them is related to the perturbative expansion of the energy eigenvalues and another to instanton contributions to the system. Recently, Dunne and Ünsal have rather revealed a remarkable simple relation between these two functions connecting perturbative and non-perturbative sectors by a differential equation.
Lately, there have been observed strong similarities between Dunne-Ünsal relation and Whitham equations. In particular, the Dunne-Ünsal relation coincides with the Whitham equations for genus one hyperelliptic curves. This has opened a new geometrical interpretation of this relation and much work remains to be done in this direction. In this project we will try to better understand this relationship.