• reinforcement learning
• machine learning
• soft actor critic
• algorithms
• conformal field theories
• crossing equation
• Ising
• Yang-Mills theory

Conformal Field Theories (CFT) are special types of Quantum Field Theories (QFTs) which are invariant under conformal transformations. This requirement poses strong constraints on the observables of CFT. Two and three-point correlation functions between operators are completely fixed up to certain number called OPE coefficients, while four-point correlators are determined up to an arbitrary function of two variables, called conformal cross-ratios. The operator product expansion (OPE) allows to formulate a constraint on this function called crossing equation. This equation must be satisfied for all values of the cross ratios and, therefore, it poses strong constraints on the unknowns of the theory, known as CFT data: the OPE coefficients, together with the spectrum of local operators, parametrized by their conformal dimension and spin.

Given their power and flexibility, Machine Learning and Reinforcement Learning models recently begun to be applied in the CFT framework. Results indicate that it is possible to find an approximate solution to the conformal bootstrap equation using the Soft Actor-Critic (SAC) algorithm. To reformulate the problem in a RL framework an agent selects the CFT data, while the reward and the information received are given by evaluations of the crossing equation on the CFT data and a set of points in the complex plane as cross-ratios. We validate the approach using a well-known CFT, the Ising 2D model, for which an analytical solution is available. We look at the results on the search with all 22 unknowns and with an increasing number of scaling dimensions fixed, pushing the algorithm towards some optimal minima.

After experimenting with the toy model, we switch to an interesting theory that has not yet been fully solved: a 1D defect CFT defined by a straight ½-BPS Wilson line in the 4D N=4 super Yang-Mills theory. In this case, the only labels for the operators are their conformal dimensions, which have been computed numerically using a technique known as integrability. Moreover, the four-point function we are interested in is also required to satisfy certain integrated constraints. We develop a method to incorporate these constraints in our algorithm to improve the precision on the predictions for the only unknowns of this problem: the OPE coefficients. We study how the precision on the results varies with the different values of the coupling constant g.