Virasoro Symmetry in Neural Network Field Theories
This work addresses a foundational gap in theoretical machine learning by enabling local conformal symmetries in neural networks, which is incremental but novel for bridging CFTs and NN-FTs.
The authors tackled the problem of Neural Network Field Theories lacking local stress-energy tensors and infinite-dimensional conformal symmetry, and they constructed the first NN-FT encoding full Virasoro symmetry for a 2d CFT, verifying it numerically by computing central charge and scaling dimensions. They extended this to include super-Virasoro symmetry and boundary theories with preserved conformal symmetry.
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.