Prove the 4+1 and 5+1 dimensional SYM–many-body correspondence under Ω-deformation

Establish the conjectured correspondence that relates 4+1 and 5+1 dimensional super-Yang–Mills theories with eight supercharges, subject to the Ω-deformation, to integrable many-body systems, by proving the precise non-perturbative mapping between gauge-theoretic observables and the dynamics of the associated many-body models, analogous to the established 3+1 dimensional case.

Background

The lectures present two correspondences between gauge theories and integrable many-body systems: one via Hamiltonian reduction and another via dualities that map quantization parameters to geometric parameters. For supersymmetric theories with eight supercharges, subject to the Ω-deformation, non-local observables obey non-perturbative Dyson–Schwinger equations; in 3+1 dimensions these lead to a Schrödinger equation of a many-body system.

While the 3+1 dimensional case is derived, the extension to higher dimensions is not fully established. The authors explicitly state that in 4+1 and 5+1 dimensions the correspondence is conjectural, highlighting an open problem to formulate and prove the full gauge theory–integrable system mapping in those cases, including the identification of parameters and the equivalence of observables.