Abstract
We consider a class of quantum field theories and quantum mechanics, which we couple to ℤN topological QFTs, in order to classify non-perturbative effects in the original theory. The ℤN TQFT structure arises naturally from turning on a classical background field for a ℤN 0- or 1-form global symmetry. In SU(N) Yang-Mills theory coupled to ℤN TQFT, the non-perturbative expansion parameter is exp[−SI/N] = exp[−8π2/g2N] 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 T3 × {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 θ = 0 and gaplessness at θ = π in mathbbm{CP} 1 model, and mass gap for arbitrary θ in mathbbm{CP} N−1, N ≥ 3 on ℝ2.
Highlights
The applications of coupling a topological quantum field theory (TQFT) to quantum field theory (QFT) received recent interest in the discussion of mixed anomalies [1,2,3]
We consider a class of quantum field theories and quantum mechanics, which we couple to ZN topological QFTs, in order to classify non-perturbative effects in the original theory
We would like to understand the role of topological defects such as instantons, monopole instantons and fractional instantons in the dynamics much more precisely both in weak coupling semi-classical domain and in strong coupling domain by using the restrictions that follows from TQFT
Summary
The applications of coupling a topological quantum field theory (TQFT) to quantum field theory (QFT) received recent interest in the discussion of mixed anomalies [1,2,3]. The simple observation, that saddles with fractional action, but integer topological charge contributes to the path integral (see figure 3), is at the heart of all reliable semiclassical analysis of gauge theories, and sigma models, such as N = 1 SYM, Yang-Mills with double-trace deformations, and CP N−1 models, and is responsible for the multi-branched structure of the vacuum energy. This structure naturally arise in the context of resurgence and sometimes called graded resurgence triangle [11, 50]. This fact will help us in gauge theories on R4 and R3 × S1
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