Topological Effects in Neural Network Field Theory

TL;DR

This paper extends neural network field theory to include topological aspects, demonstrating phenomena like the Berezinskii--Kosterlitz--Thouless transition and T-duality in string theory.

Contribution

It introduces topological parameters into neural network field theory, revealing complex phase transitions and dualities akin to those in condensed matter and string theories.

Findings

Recovered the Berezinskii--Kosterlitz--Thouless transition.

Verified T-duality invariance in string models.

Demonstrated topological effects in neural network field theory.

Abstract

Neural network field theory formulates field theory as a statistical ensemble of fields defined by a network architecture and a density on its parameters. We extend the construction to topological settings via the inclusion of discrete parameters that label the topological quantum number. We recover the Berezinskii--Kosterlitz--Thouless transition, including the spin-wave critical line and the proliferation of vortices at high temperatures. We also verify the T-duality of the bosonic string, showing invariance under the exchange of momentum and winding on $S^1$, the transformation of the sigma model couplings according to the Buscher rules on constant toroidal backgrounds, the enhancement of the current algebra at self-dual radius, and non-geometric T-fold transition functions.