KMS Fall Meeting, GIST

Neural-Network Wess-Zumino-Witten,
Duality and Machine Learning

Benjamin Suzzoni
2026/10/31

with

PCPietro CapuozzoBRBrandon Robinson

A neural network is a composable map

Neural network:

\(\Phi_{\Theta}:\mathbb R^D\rightarrow\mathbb R\)

Operations: composition, addition and scalar multiplication.

Its functional form is the architecture; its weights and biases are the parameters.

one hidden layer

\[ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Phi_{\Theta}(x)=\sum_i \Theta_i \varphi_i(x) \]

The Universal Approximation Theorem (UAT)

one width parameter \(L\) controls both pictures

\(L=1\) width parameter; displayed nodes are schematic

coarse approximation schematic error

UAT: expressivity/density in a function space.

NNGP: a suitably normalized iid large-width limit can become Gaussian.

A functional integral through the UAT

Network at initialization: weights \(\Theta\) follow a distribution \(P\).

function space

\[ \int\mathcal D\phi\,\mu[\phi]\,\mathcal F[\phi] \]

integrate over field configurations

↔ pushforward
through \(\Phi_\Theta\)

parameter space

\[ \lim_{L\to\infty}\int_{\Theta_L} d\Theta\,P_L(\Theta)\, \mathcal F[\Phi_\Theta] \]

ordinary integrals over network parameters

Key Idea: replace the functional integral with a statistical average.

But: conformal correlators can already emerge at finite architecture—the UAT limit is not required to define an nn-CFT.

What has NNFT achieved?

Foundation of modern NNFT

[Halverson, et al; 2021]

Fermions and supersymmetry

[Frank, et al; 2025]

Virasoro symmetry using NNs

[Robinson; 2025]

String-theory amplitudes

[Frank, Halverson; 2026]

Topological effects: BKT and T-duality

[Ferko, et al; 2026]

T-duality identified in the bosonic-string setting

momentum↔winding

What about other dualities?

WZW and level-rank duality [Ostrik, Sun; 2014]

A 2d sigma model

\[g:\Sigma\longrightarrow SU(N)\]

worldsheet to group manifold

Affine current algebra

\[J^a(z)J^b(w)\sim\frac{K\delta^{ab}}{2(z-w)^2}+\frac{if^{ab}{}_{c}J^c(w)}{z-w}\]

Direct / coset

\[SU(N)_K\simeq\frac{SU(NK)_1}{SU(K)_N}\]

a common chiral current sector

[5:15–6:50, 95 seconds] WZW is an interacting two-dimensional conformal sigma model with group-valued fields. Show only the map, as requested, without the action. Advance 1: the holomorphic current OPE has the Lie-algebra simple pole and a double pole at level K. The displayed normalization is tr(t^a t^b)=delta^{ab}/2, as in the draft. Advance 2: the conformal embedding of SU(N)_K and SU(K)_N in SU(NK)_1 supplies a direct/coset presentation of the same chiral current sector. This does not identify all the auxiliary Wakimoto fields, finite neural ensembles, or the full SU(N)_K and SU(K)_N theories. The example used next is SU(2)_3 and SU(6)_1/SU(3)_2. Sources: draft.tex, target-current-ope, fw_conformal_embedding, fw_same_sector_coset; APCTP WZW slide. Reference: Ostrik and Sun, arXiv:1208.5131.

Level-rank for neural networks [Wakimoto; 1986]

The Wakimoto representation builds WZW currents from free fields.

neural free fields

\(\phi\) and \((\beta,\gamma)\)

one scalar, one commuting pair

Wakimoto readout

Polynomials and derivatives

with normal ordering

WZW observables

\(J^1,\ J^2,\ J^3\)

and the stress tensor \(T\)

One current, for example:

\[J^3=i\sqrt{\frac{K+2}{2}}\,\partial\phi+:\!\beta\gamma\!:\]

The prescribed readouts reproduce the affine algebra in the ordered continuum limit.

[6:50–8:40, 110 seconds] Ask what level-rank looks like when the fields are neural networks. Use the bosonic Wakimoto representation throughout this comparison. Random Fourier-feature networks provide the scalar and the coupled commuting beta/gamma fields. Advance 1: fixed nonlinear and derivative readouts produce the currents and the stress tensor; show only J3 as a representative formula. Normal ordering, derivative corrections and the level-dependent coefficients are part of the prescribed construction, not learned parameters. Advance 2: the elementary neural contractions and Gaussian Wick limit reproduce the affine OPE after the ordered width and regulator limits. Finite networks retain presentation-dependent corrections. Sources: draft.tex sec:bosonicsubstrate, bosonic-scalar-architecture, bosonic-ghost-architecture, wakimoto-su2-currents and su2-wakimoto-stress-tensor. References: Wakimoto, DOI 10.1007/BF01211068; de Boer and Feher, arXiv:hep-th/9611083.

One current sector, two neural presentations

The same limiting \(SU(2)_3\) sector, at the same feature width \(L\).

	Direct W \(SU(2)_3\)	Coset C \(SU(6)_1/SU(3)_2\)
Scalar fields	1	5
Commuting \(\beta\gamma\) pairs	1	15
Gaussian output amplitudes	\(3L\)	\(35L\)
Real sampled parameters	\(6L\)	\(38L\)
Observables retained	Three currents and the stress tensor

Baseline shared frequencies and scalar phases. Count each complex frequency as two real parameters; fixed hyperparameters are excluded.

The continuum relation leaves different finite architectures and readouts.

[8:40–10:10, 90 seconds] Compare direct SU(2)_3 Wakimoto fields with the bosonic SU(6)_1/SU(3)_2 parent-based presentation. Field counts are one versus five scalars and one versus fifteen commuting pairs. There are 3L versus 35L Gaussian amplitudes. With the draft's shared frequency/phase law, add 2L real frequency coordinates and L scalar phases to each: 6L versus 38L real sampled parameters. These counts exclude fixed global hyperparameters and prescribed coefficients, and are not numbers of trained weights. Both retain three currents and a stress tensor. The coset stress is the same-parent subtraction T_P-T_D, with affine corrections retained; the denominator is not a separately sampled network. Advance highlights the different parameter counts and concludes that the continuum identification does not choose a finite architecture. Sources: draft.tex bosonic-scalar-architecture, bosonic-ghost-architecture, fw_parent_neural_fields, fw_coset_stresses and the field/amplitude comparison in sec:finite_width_regulator_design.

Real life neural networks are not infinite...
what happens there?

Finite-width accuracy depends on the presentation

Compare stress-tensor correlations with the common CFT reference.

\(L\): feature width. \(\alpha\): spectral tilt at fixed elementary covariance.

UV damping (R¹)

Direct W

Coset C

IR and UV damping (R³)

Direct W

Coset C

\(\log_{10}E_X^{TT}\)

−115.231.447.663.880

white: resolved minimum
hatched: within 5%

Hover to inspect \(L\), \(\alpha\) and \(E^{TT}\). Same 64 pairs, \(p_-=.01\), \(p_+=1000\). Sampling uncertainty is separate.

[10:25–12:35, 130 seconds] L is sampled feature width; alpha redistributes frequencies with compensated amplitudes, preserving elementary covariances. The displayed color is log10 of the relative population RMS error of the stress two-point function against the common continuum target. The exact formula remains in the bundled data and the draft, rather than on this high-level slide. These are interactive Canvas plots of the active draft's TT numerical protocol. The deleted source grid archives were regenerated with the unchanged CPU evaluator, original saved alpha/width nodes and original 64-pair seed. The recovery matched all 16 retained compact audit values. No values were digitized from images or interpolated. We show R1, then advance to R3, with the draft's common full-range color scale. The fixed bank contains 64 ordered disk pairs, with no minimum-separation cut. Damping is selected once per regulator by minimizing the larger direct/coset Gaussian relative RMS error, then held fixed for all L and alpha. The 91 alpha values span [-1.5,1.5]; 301 distinct integer widths span 1 to 2^18. White curves mark sampled per-width minimizers; hatching is within 5% in RMS. The full common log scale extends from -1 to 80, so the dark region is visually compressed; do not infer equal errors from similar colors. The draft captions give direct alpha*=7/15, coset R1 alpha* in [13/30,1/2] and coset R3 alpha* in [2/5,1/2]. This E compares the population expectation to the continuum target and contains both regulator and finite-width bias. It is not Monte Carlo variance. Finite-width terms can partly cancel regulator bias, so increasing width need not reduce this total error monotonically. Source: fw_finite_width_tt_ratio, fw_finite_width_tt_error, fw_optimal_spectral_exponent, fw_near_optimal_spectral_region, fig:fw_tt_alpha_width and tab:tt-relative-gaussian-main. All plots use the original common color scale. On entry and each regulator change, evaluated cells are revealed in increasing width over one second. Hover gives the cell's L, alpha, population RMS and deterministic numerical sensitivity. The matching cell is highlighted in both presentations. Focus a plot and use arrow keys for neighboring cells, then Escape to return to slide navigation. Reduced-motion preferences disable the reveal animation.

Computational cost depends on the task

Coset / direct median CPU time, at \(L=256\), \(\alpha=0\), \(R^3\).

Ensemble-correlation workloadTime ratio (log scale)

Production + calibration

10.25×

1000 amplitude-dependent calls

72.03×

1000 fixed-amplitude calls

1.27×

131030100

Fixed coefficients allow more reuse. New insertion points require integration again.

A continuum identification does not select the most economical finite network.

These workloads evaluate ensemble correlations. Neither complete task reached the prescribed 1% full-vector accuracy target.

[12:35–14:35, 120 seconds] Equal feature widths do not mean equal resources: the direct and parent presentations have 3L versus 35L Gaussian amplitudes, plus shared frequency/phase parameters. Here the active draft reports ensemble-correlation evaluation costs, not forward milliseconds per sampled network. The baseline direct time is one for each workload. First: complete production and calibration gives 10.2481. Advance 1: retaining amplitude dependence for 1000 repeated calls gives 72.0258. Advance 2: combining fixed-amplitude coefficients in advance reduces the ratio to 1.2743. The three tasks have different construction, integration and reuse costs. Ratios alone do not establish matched-accuracy savings: neither complete task attained the prescribed 1% target for the full correlation vector. The finite-width bias, estimator variance and calibration generalization are separate questions. The practical conclusion is to choose a presentation and prior for a declared observable, error measure and workload. Source: sec:local_calibration_dependence, Dependence on the computational task, fig:calibration_resources. Exact displayed raw ratios: 10.2481, 72.0258, 1.2743.

Thank you

Benjamin Suzzoni

with Pietro Capuozzo and Brandon Robinson