Complex Path Integrals and Saddles in Two-Dimensional Gauge Theory.

Area of Science:

• Theoretical Physics
• Quantum Field Theory
• Computational Physics

Background:

• Two-dimensional lattice gauge theories are fundamental models in quantum field theory.
• Understanding nonperturbative effects is crucial for describing phenomena like phase transitions.
• The Gross-Witten-Wadia model serves as a key example for studying these theories.

Purpose of the Study:

• To numerically explore the complex saddle point structure of the two-dimensional lattice gauge theory.
• To investigate the nature of nonperturbative effects in both weak and strong coupling phases.
• To elucidate the mechanism driving the phase transition in this model.

Main Methods:

• Numerical analysis of saddle points in the Gross-Witten-Wadia unitary matrix model.
• Examination of complex-valued saddle points beyond the real integration variables.
• Confirmation of established structures and identification of new interpretations.

Main Results:

• Saddle points are generally complex-valued, deviating from real integration variables and action.
• Trans-series and instanton gas structures are confirmed in the weak-coupling phase.
• A novel complex-saddle interpretation for nonperturbative effects in the strong-coupling phase is identified.
• Eigenvalue tunneling into the complex plane is observed in both phases.
• Saddle condensation is identified as the driving mechanism for the weak-to-strong coupling phase transition.

Conclusions:

• The study reveals the critical role of complex saddle points in understanding lattice gauge theories.
• Nonperturbative effects and phase transitions are intricately linked to eigenvalue dynamics in the complex plane.
• The findings offer a new perspective on the mathematical structure underlying these physical phenomena.