For the first part of the project, I proposed a neural network architecture to compute the Hodge numbers (some topological data describing the shape of the surface, and related to the number of fermions in the low-energy theory) of complete intersection Calabi-Yau 4-folds (which are the manifolds needed to describe compactifications in F-theory), taking inspiration from my previous work on 3-folds (used for compactifications in string theory). In another work, I used this architecture to study again the 3-folds and add new analyses, such that the robustness of the predictions under extrapolation. In both cases, we achieved state-of-the-art accuracy. Moreover, this exemplified for the first time that neural networks can handle data from algebraic topology, which is also interesting for the mathematical and machine learning communities.
For the second part, I provided an algorithm incorporating deep learning to construct string field interactions, and we showed that it reproduces known results at the lowest order. The main building blocks correspond to functions on and subspaces of the moduli spaces of Riemann surfaces: our approach consists in parametrizing these functions by neural networks, which are trained by solving some mathematical constraint. Hence, my results are also useful for mathematicians and the method can be applied to more general problems. In a second work, I explored the Hubbard-Stratonovich method to reduce the maximum interaction order (how many particles/strings can interact at the same time) in a Lagrangian. In particular, we demonstrated that open string field theory with stubs, which is originally non-polynomial (i.e. it has an infinite number of interactions), can be written in a cubic form with a single auxiliary field. This is a major result which opens new possibilities for closed string field theory, whose action is particularly difficult to manipulate. We have also applied these ideas to a scalar field theory, and studied the relation with renormalization. We have proved that particle field theories enjoy a geometric BV algebra similar to the one found in string theory. We have also explored these ideas in the context of the neural network / field theory, where the same structures appear. Finally, we have explored how to formulate closed string field theory with a string field without any constraint: indeed, there are some reasons indicating that the action could be made cubic in this case.
These results have been published as 3 preprints (all submitted to journals, plus 1 to appear), 4 journal articles, 1 book chapter, 2 proceedings, 1 habilitation thesis. They have also been presented at several conferences and institute seminars.
