Neural and Non-Equilibrium methods for numerical Effective String Theory calculations

Effective String Theory (EST) is a powerful non-perturbative approach used to describe confinement in Yang-Mills theory by modeling the confining flux tube as a thin, vibrating string. EST calculations are typically performed using zeta-function regularization; however, some observables, such as the shape of the flux tube, are too complex to be addressed with this method. This issue can, in principle, be approached by modeling the EST action as a spin system on a discretized two-dimensional lattice, making it amenable to Monte Carlo simulations. However, the strong non-linearity of the model makes conventional algorithms highly inefficient for such computations. Instead, the problem is well-suited for two distinct numerical approaches based on deep learning methods and out-of-equilibrium statistical mechanics. In the first approach, a variational density is learned using a class of deep generative models called Normalizing Flows (NFs), allowing unbiased expectation values over the EST distribution to be computed via a reweighting procedure. In contrast, out-of-equilibrium methods, such as Non- Equilibrium Markov Chain Monte Carlo (NE-MCMC) algorithms, enable exact calculations of vacuum expectation values using Crooks’ theorem and Jarzynski’s equality. Notably, these two approaches can be combined into an efficient sampling method known as Stochastic Normalizing Flows (SNFs). This thesis introduces EST and its lattice regularization, followed by a brief overview of standard MCMC methods. Next, we introduce NFs and present a proof-of-concept using Continuous NFs to demonstrate the feasibility of our method for sampling EST on the lattice. We then introduce non-equilibrium methods and SNFs, showcasing state-of-the-art results achieved with the latter. The combination of these findings enables a quantitative description of the fine details of the confinement mechanism in different lattice gauge theories. The thesis concludes by discussing the main implications of our results for EST and gauge theory, as well as potential directions for future research.