Tunnels Under Geometries (or Instantons Know Their Algebras)

TL;DR

This paper explores the algebraic and geometric structures underlying instanton tunneling processes in quantum field theories with multiple vacua, revealing connections to quantum groups, Yang-Baxter relations, and equivariant integrals over moduli spaces.

Contribution

It introduces the concept of a tunneling algebra that encodes instanton effects and demonstrates explicit links between quantum algebras and geometric instanton moduli spaces.

Findings

Tunneling amplitudes relate to quantum R-matrices and Yang-Baxter equations.

Explicit construction of instanton effects in affine Yangians.

Demonstration of equivariant integrals over quiver moduli spaces.

Abstract

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 $\mathsf{T}_{i\to j}\sim e^{-S_{\mathrm{ inst}}}{\mathbf{v}}_j^{+}{\mathbf{v}}_i^{-}$, where there is canonical instanton action suppression, and $\mathbf{v}_i^{-}$ annihilates a particle in the $i^{\mathrm{th}}$ vacuum, whereas $\mathbf{v}_j^{+}$ creates a particle in the $j^{\mathrm{th}}$ 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. Operators $\mathbf{v}_i^{-}$, $\mathbf{v}_j^{+}$ 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 note we revise few successful cases of the aforementioned correspondence: quantum algebras $U_q(\mathfrak{g})$ and affine Yangians $Y(\hat{\mathfrak{g}})$. For affine Yangians we demonstrate explicitly how instantons ``perform'' equivariant integrals over associated quiver moduli spaces appearing in the alternative geometric construction.