Strongly coupled QFT dynamics via TQFT coupling

Abstract

We consider a class of quantum field theories and quantum mechanics, which we couple to $\mathbb Z_N$ topological QFTs, in order to classify non-perturbative effects in the original theory. The $\mathbb Z_N$ TQFT structure arises naturally from turning on a classical background field for a $\mathbb Z_N$ 0- or 1-form global symmetry. In $SU(N)$ Yang-Mills theory coupled to $\mathbb Z_N$ TQFT, the non-perturbative expansion parameter is $\exp[-S_I/N]= \exp[-{8 \pi^2}/{g^2N}]$ both in the semi-classical weak coupling domain and strong coupling domain, corresponding to a fractional topological charge configurations. To classify the non-perturbative effects in original $SU(N)$ theory, we must use $PSU(N)$ bundle and lift configurations (critical points at infinity) for which there is no obstruction back to $SU(N)$. These provide a refinement of instanton sums: integer topological charge, but crucially fractional action configurations contribute, providing a TQFT protected generalization of resurgent semi-classical expansion to strong coupling. Monopole-instantons (or fractional instantons) on $T^3 \times S^1_L$ can be interpreted as tunneling events in the 't Hooft flux background in the $PSU(N)$ bundle. The construction provides a new perspective to the strong coupling regime of QFTs and resolves a number of old standing issues, especially, fixes the conflicts between the large-$N$ and instanton analysis. We derive the mass gap at $\theta=0$ and gaplessness at $\theta=\pi$ in $\mathbb{CP}^{1}$ model, and mass gap for arbitrary $\theta$ in $\mathbb{CP}^{N-1}, N \geq 3$ on $\mathbb R^2$.