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 $\exp [-S_I/N]=\exp [-8{\pi }^2/g^{2}N]$ 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 $T^3\times S_L^1$ 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 ${\mathbb{CP}}^{N-1},N\ge 3$ on ${\mathbb{R}}^2$.

arXiv comment90 pages, 8 figures, Minor changes, Refs. added

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• Memory efficient finite volume schemes with twisted boundary conditions