Excited String States and D-branes from Infinite Width Neural Networks

TL;DR

The paper extends the neural-network field theory approach to represent worldsheet string path integrals by including excited closed-string insertions and boundary (D$p$-brane) sectors. It introduces a fixed-feature Gaussian normal-ordering prescription to renormalize derivative composites and uses an image-based neural construction to realize mixed Neumann/Dirichlet boundary conditions, yielding sphere and disk amplitudes that reproduce the standard Koba–Nielsen factors and momentum-conservation structures after renormalization in the large-width limit. The main contributions are the renormalized level-(1,1) insertion on the sphere and the neural D$p$-brane disk amplitudes, together with explicit demonstrations that the parameter-space integrals reproduce the correct string-theoretic factors and limits. This neural-ensemble formulation provides a concrete, controllable framework to study finite-width corrections and potential extensions to open–closed sectors and higher-genus amplitudes within string perturbation theory.

Abstract

We explore recent proposal to represent worldsheet string path integrals by integrating over parameters of a wide random-feature neural network whose output is identified with the embedding field $X^μ$. In this paper we extend it focusing on scattering with excited states insertions and for worldsheets with boundaries introducing fixed-feature Gaussian normal-ordering prescription for derivative composites (removing the neural contact term at finite width), and propose realization of mixed Neumann/Dirichlet boundary conditions interpreted as a neural D$p$-brane. As concrete outputs, we derive the sphere four-point integrand with a single $(1,1)$ insertion and the disk four-tachyon amplitude on a D$p$-brane, recovering the expected derivative prefactors, boundary exponents, and momentum-conservation limits after renormalization.