Christian Ferko

A bstract: Neural network field theory (NN-FT) formulates field theory in terms of a network architecture and a density on its parameters. We derive Schwinger–Dyson equations and Ward identities in NN-FT and utilize them to study anomalies. The equations depend on a conserved parameter space current that characterizes symmetries and how they break. It is relevant even in non-local NN-FTs, but can recover local currents in the case of a local Lagrangian by an appropriate fiber-wise average. In machine learning, this formalism is applied to feedforward networks and the attention mechanism. In physics, we use this machinery to study U(1) symmetry for a complex scalar, the scale anomaly in 4 d massless ϕ ⁴ theory, the Weyl anomaly for the bosonic string (including a new computation of the critical dimension), and examples involving discrete topological data, such as winding numbers and T-duality. Since the results are obtained in network parameter space rather than the standard field space, they represent a new way to understand symmetries in quantum field theories.

A bstract: 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 compact and topological settings by allowing mixed continuous/discrete parameter spaces whose discrete labels encode topological sectors. For the Berezinskii–Kosterlitz–Thouless system, starting from the standard spin-wave/Coulomb gas decomposition of the compact boson, we construct a mixed sampler whose Gaussian component matches the spin-wave critical line and whose explicitly supplied vortex component gives finite volume observables compatible with vortex proliferation in the disordered regime. For the bosonic string compactified on S ¹ , once the momentum–winding lattice and its duality action are supplied explicitly, finite width samples match the standard T-duality benchmark identities for states, correlators, constant toroidal backgrounds, enhanced current data at the self-dual radius, and a toy non-geometric T-fold gluing.

Neural network field theory (NN-FT) defines a field theory by a network architecture together with a probability density on its latent variables. For compact theories the local Gaussian sector is only part of the story: one must also sum over discrete topological sectors. We review a mixed continuous/discrete latent-variable construction and apply it to two compact bosons. For the Berezinskii–Kosterlitz–Thouless transition, a random Fourier feature spin-wave sampler supplemented by an explicit Coulomb gas vortex sector reproduces the Gaussian critical line below [Formula: see text], vortex proliferation above [Formula: see text], the essential singularity of the correlation length, and the Nelson–Kosterlitz jump. For the bosonic string, oscillator modes augmented by momentum–winding labels reproduce circle T-duality, Buscher transformations on constant toroidal backgrounds, self-dual current algebra enhancement, and a toy T-fold. The common lesson is that the same local neural sampler, paired with different discrete topological data, yields physically distinct compact theories. These proceedings are based on [1].

We demonstrate that any Euclidean-time quantum mechanical theory may be represented as a neural network, ensured by the Kosambi-Karhunen-Loeve theorem, mean-square path continuity, and finite two-point functions. The additional constraint of reflection positivity, which is related to unitarity, may be achieved by a number of mechanisms, such as imposing neural network parameter space splitting or the Markov property. Non-differentiability of the networks is related to the appearance of non-trivial commutators. Neural networks acting on Markov processes are no longer Markov, but still reflection positive, which facilitates the definition of deep neural network quantum systems. We illustrate these principles in several examples using numerical implementations, recovering classic quantum mechanical results such as Heisenberg uncertainty, non-trivial commutators, and the spectrum.

Entanglement cohomology assigns a graded cohomology ring to a multipartite pure state, providing homological invariants that are stable under local unitaries and characterize inequivalent patterns of entanglement. In this work we derive exact expressions for the dimensions of these cohomology groups in two canonical entanglement classes, generalized GHZ and W states on an arbitrary number of parties and local Hilbert space dimensions, thus proving conjectures of arXiv:1901.02011. Using the additional structure of the Hodge star and wedge product operations, we propose two new classes of local unitary invariants: the spectrum of the natural Laplacian acting on entanglement $k$-forms, and the intersection numbers obtained from wedge products of representatives for cohomology classes. We present numerical experiments which investigate these invariants in particular states, suggesting that they may provide useful quantities for describing multipartite entanglement.