Cluster Expansion and Resurgence in the Polyakov Model

Area of Science:

• Theoretical Physics
• Quantum Field Theory
• Mathematical Physics

Background:

• The Polyakov model exhibits a nonperturbative mass gap generated by instanton effects at leading-order semiclassics.
• Higher-order semiclassical effects introduce ambiguities, specifically an imaginary contribution to the mass gap, challenging its real and unambiguous nature.

Purpose of the Study:

• To address the problematic imaginary ambiguity arising in higher-order semiclassical calculations of the Polyakov model's mass gap.
• To develop a method for resolving these ambiguities within the quantum field theory (QFT) framework.

Main Methods:

• Utilized critical points at infinity, cluster expansion, and Lefschetz thimbles for semiclassical analysis.
• Introduced a novel compactification of the Polyakov model to quantum mechanics using a background 't Hooft flux.
• Employed the Born-Oppenheimer approximation to retain monopole instanton information in the quantum mechanical limit.

Main Results:

• Demonstrated that a third-order semiclassical effect introduces an imaginary, ambiguous contribution to the mass gap.
• Proved the resurgent cancellation of ambiguity in the three-instanton sector against the Borel resummation ambiguity of the one-instanton sector in the quantum mechanical limit.
• Derived large-order asymptotics for perturbation theory around both the perturbative vacuum and instanton, assuming the cancellation holds in QFT.

Conclusions:

• The novel compactification successfully preserves crucial instanton information lost in periodic compactifications.
• The resurgent cancellation provides a potential resolution to the mass gap ambiguity in the Polyakov model.
• The derived large-order asymptotics offer new insights into the behavior of perturbation theory in this model.