Tunnels, Algebras and Geometry

In the tight binding model with multiple degenerate vacua we might treat wave function overlaps as instanton tunnelings between different wells (vacua). An amplitude for such a tunneling process might be constructed as a product of canonical instanton action suppression, and two operators: one annihilates a particle in the one vacuum, whereas another creates a particle in the other vacuum. Adiabatic change of the wells leads to a Berry-phase evolution of the couplings, which is described by the zero-curvature Gauss-Manin connection i.e. by a quantum R-matrix. Zero-curvature is actually a consequence of level repulsion or topological protection, and its implication is the Yang-Baxter relation for the R-matrices. In the simplest case the story is pure Abelian and not very exciting. But when the model becomes more involved, incorporates supersymmetry, gauge and other symmetries, such amplitudes obtain more intricate structures. "Creation/annihilation" operators might also evolve from ordinary Heisenberg operators into a more sophisticated algebraic object – a "tunneling algebra". The result for the tunneling algebra would depend strongly on geometry of the QFT we started with, and, unfortunately, at the moment we are unable to solve the reverse engineering problem. In this talk we revise few successful cases of the aforementioned correspondence: quantum algebras and affine Yangians. For affine Yangians we demonstrate explicitly how instantons "perform" equivariant integrals over associated quiver moduli spaces appearing in the alternative geometric construction. The talk is based on paper https://arxiv.org/abs/2502.11294 with A. Morozov.