October 13, 2025Open Access

A T2 × R2 roadmap to confinement in SU(2) Yang-Mills theory

Key Points

Monte Carlo simulations confirm semiclassical predictions, highlighting the transition in string tension dynamics.
As the torus size increases, the density of fractional instantons grows, influencing the long-distance structure significantly.
The standard string tension observed at low densities matches the expected value for a two-dimensional center-vortex gas.
Preliminary studies extend findings to SU(N) theories and explore large torus sizes where boundary conditions become negligible.

Abstract

A bstract We study the behaviour of SU(2) Yang-Mills fields on a T 2 × R 2 geometry where the two-torus is equipped with twisted boundary conditions. We monitor the evolution of the dynamics of the system as a function of the torus size l s . For small sizes the behaviour of the system is well understood in terms of semiclassical predictions. In our case, the long distance structure is that of a two-dimensional gas of vortex-like fractional instantons with size and density growing with l s . Our lattice Monte Carlo simulations confirm the semiclassical predictions and allow the determination of the relevant scale signalling the transition to the non-dilute situation. At low densities the string tension takes the standard value of a 2D center-vortex gas, growing with the density and approaching the value measured at infinite volume. Our work includes preliminary studies of the extension to SU(N) and to the region of large sizes in which boundary conditions are irrelevant, and all physical scales are determined by the Λ parameter.

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