Tunnels Under Geometries (or Instantons Know Their Algebras)

TL;DR

The paper proposes a tunneling-algebra framework that identifies multi-well instanton amplitudes with operators ${\bf v}_i^{\pm}$, yielding $T_{i\to j}\sim e^{-S_{\mathrm{inst}}}\mathbf{v}_j^{+}\mathbf{v}_i^{-}$. Adiabatic variations induce Berry/ Gauss–Manin transport captured by a quantum $R$-matrix whose zero-curvature condition leads to Yang–Baxter relations. In supersymmetric and gauge-theoretic settings these amplitudes acquire richer algebraic structures and can realize quantum algebras $U_q(\mathfrak{g})$ and affine Yangians $Y(\hat{\mathfrak{g}})$ as explicit tunneling algebras. For affine Yangians the construction provides a mechanism where instantons perform equivariant integrals over quiver moduli spaces, linking geometric representation theory to nonperturbative quantum mechanics. The work discusses localization, caveats, and open questions about universality and extensions to singular quivers and DIM algebras.

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.