Topological Effects in Neural Network Field Theory

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.

Virasoro Symmetry in Neural Network Field Theories

Neural Network Field Theories (NN-FTs) can realize global conformal symmetries via embedding space architectures. These models describe Generalized Free Fields (GFFs) in the infinite width limit. However, they typically lack a local stress-energy tensor satisfying conformal Ward identities. This presents an obstruction to realizing infinite-dimensional, local conformal symmetry typifying 2d Conformal Field Theories (CFTs). We present the first construction of an NN-FT that encodes the full Virasoro symmetry of a 2d CFT. We formulate a neural free boson theory with a local stress tensor $T(z)$ by properly choosing the architecture and prior distribution of network parameters. We verify the analytical results through numerical simulation; computing the central charge and the scaling dimensions of vertex operators. We then construct an NN realization of a Majorana Fermion and an $\mathcal{N}=(1,1)$ scalar multiplet, which then enables an extension of the formalism to include super-Virasoro symmetry. Finally, we extend the framework by constructing boundary NN-FTs that preserve (super-)conformal symmetry via the method of images.

Conformal Defects in Neural Network Field Theories

Neural Network Field Theories (NN-FTs) represent a novel construction of arbitrary field theories, including those of conformal fields, through the specification of the network architecture and prior distribution for the network parameters. In this work, we present a formalism for the construction of conformally invariant defects in these NN-FTs. We demonstrate this new formalism in two toy models of NN scalar field theories. We develop an NN interpretation of an expansion akin to the defect OPE in two-point correlation functions in these models.