Neural-Network Field Theory Correspondence (for Wess-Zumino-Witten Theories)

26xx.xxxxx

Benjamin Suzzoni

In collaboration with P. Capuozzo and B. Robinson

Table of Contents

I Background

    1.1 Neural Networks and UATs

    1.2 NN-FT Correspondence

II A Timeline of NNFT

III CFTs, Bosonic Strings and WZW

    3.1 NN-CFT Correspondence

    3.2 Bosonic Strings and WZW

IV Advertizing

    4.1 Topological effects

    4.2 Superstrings

1.1 Neural-Networks

What do we call a Neural Network (NN)? How do they work?

Neural Network: a map \(f:\mathbb{R}^m\longrightarrow\mathbb{R}^n\)

Operations: composition, addition, scalar multiplication

Functional form of \(f\) is called an architecture

eg. Feed forward network

1.1 Neural-Networks

Initialization: pick parameters from a distribution \(\longrightarrow\) get a random function \(f\)

Eg. \(f(x) = \sigma(\sum_i^3 w_ix_i+b)\), $$ w_1,w_2,w_3 \sim P_w \qquad\qquad\qquad b \sim P_b $$

Many known architectures have a parameter \(N\) s.t. \(N\to\infty\) gives a Gaussian draw \(f\sim\mathcal{N}\).

eg. Feed forward, Convolution, Recurrent, Graph Convolution, etc

This is the Neural-Network-Gaussian-Process (NNGP) correspondence.

1.1 Universal Approximation Theorems

How well do NNs approximate functions?

Ingredients: activation functions/architecture, large-parameter limit

Universal Approximation Theorem: NNs are dense in a space of maps (in the \(N\to\infty\) limit)

1.2 The Neural-Network Field-Theory Correspondence

Textbook method for describing a Quantum Field Theory: path integral

$$ \langle\mathcal{O}(x)\rangle = \int\mathcal{D}\phi\, e^{-S[\phi]}\,\mathcal{O}(x) $$

The measure \(\mathcal{D}\phi\) is generally not well defined.

\(\hookrightarrow\) it is a shorthand for integrate over all fields

We can define it

• rigorously sometimes (eg. TQFTs)
• by discretizing space (eg. lattice methods)
• or using Neural Network and UATs!

1.2 The Neural-Network Field-Theory Correspondence

Key idea: use initialization distributions together with UATs to integrate over all possible neural-network configurations.

We recover the standard path integral in the UAT limit* (\(N\to\infty\)) $$ \int\mathcal{D}\phi\, e^{-S[\phi]}\,\mathcal{O}(\phi(x)) \qquad\longleftrightarrow\qquad \lim_{N\to\infty}\int_{\Omega}\prod_i^N d\theta_i P(\theta_i)\mathcal{O}(\phi_{\theta}(x))$$

* NNGP correspondence \(\rightarrow\) generalized free theory only

Key takeaway:

• Recast ill-defined path integral into limit of well-defined integrals over network parameters
• Converted field integrals into statistical integrals

1.2 The Neural-Network Field-Theory Correspondence

Can compute correlators in the usual way, using generating functional \(Z[J]\) $$ Z[J] = \lim_{N\to\infty}\int_{\Omega}\prod_i^N d\theta_i P(\theta_i)\,e^{\int dx\, J(x)\phi_{\theta}(x)} $$

but those are now standard statistical averages (\(\mathbb{E}\)), $$ K(x,y) = \langle\phi(x)\phi(y)\rangle \qquad\longleftrightarrow\qquad \frac{\delta^2 Z[J]}{\delta J(x)\delta J(y)} = \mathbb{E}_{P(\theta)}[\phi_{\theta}(x)\phi_{\theta}(y)] $$

We get a free scalar boson with propagator \(K\) $$ S[\phi] = -\frac{1}{2}\int dxdy\,\phi(x)K^{-1}(x,y)\phi(y) $$

\(\hookrightarrow\) one can choose \(P,\phi_{\theta}\) (hence \(K(x,y)\)) to engineer different QFTs

1.2 The Neural-Network Field-Theory Correspondence

Going beyond the Gaussian limit

There are two ways of introducing interactions

• Finite-\(N\) corrections
• Break UAT requirements (eg. statistical dependence of parameters)

eg. \(\phi^4\)-theory can be defined via a deformation of the distribution of parameters $$ P(\theta) \to P(\theta)\exp\left(-\frac{\lambda}{4!}\int dx\,\phi^4_{\theta}(x)\right) $$

1.2 Neural-Network-Field-Theory - What to remember?

• Convert functional integral into average over parameters of a NN
• Large NN parameter limit gives (generalized) free theory
• Integration measure \(\int d\theta\, P(\theta)\) is synonymous to \(\int D\phi e^{-S[\phi]}\)
• Interactions appear by moving away from the UAT limit
• \(\hookrightarrow\) finite NN parameter corrections
• \(\hookrightarrow\) other choices of parameter distributions



NNFT: good playground for analytics but also strong numerical convergence!
\(\hookrightarrow\) good for simulations

Table of Contents

I Background

    1.1 Neural Networks and UATs

    1.2 NN-FT Correspondence

II A Timeline of NNFT

III CFTs, Bosonic Strings and WZW

    3.1 NN-CFT Correspondence

    3.2 Bosonic Strings and WZW

IV Advertizing

    4.1 Topological effects

    4.2 Superstrings

II The NNFT Timeline

II The NNFT Timeline

Table of Contents

I Background

    1.1 Neural Networks and UATs

    1.2 NN-FT Correspondence

II A Timeline of NNFT

III CFTs, Bosonic Strings and WZW

    3.1 NN-CFT Correspondence

    3.2 Bosonic Strings and WZW

IV Advertizing

    4.1 Topological effects

    4.2 Superstrings

3.1 Neural-Network Conformal-Field-Theory

General philosophy in NNFT: global symmetries of the action should appear as global symmetries of the parameter distribution.

Conformal symmetry in \(D\) dimensions \(\longrightarrow\) distribution of parameters with \(SO(D+2)\) isometry*.

* actually \(SO(1,D+1)\) but should consider Wick rotation for well-defined probability distribution.

NN architecture for conformal primary should obey $$ \phi_{\theta}(\lambda x) \longrightarrow \lambda^{-\Delta}\phi_{\theta}(x) $$ with \(x \in \mathbb{R}^{D+2}\) (i.e. embedding space)

eg. \(\phi_{\theta}(x) = (x\cdot\theta)^{-\Delta}\)

3.1 Neural-Network Conformal-Field-Theory

We have a NN with one node.

Architecture: $$\phi_{\theta}(x) = (x\cdot\theta)^{-\Delta}$$

Distribution: $$ P(\theta) = f(\theta\cdot\theta) $$

Compute correlators $$ Z[J] = \int_{\mathbb{R}^{D+2}}d\theta\,P(\theta)\, e^{\int dx\, J(x)\phi_{\theta}(x)} $$

One single NN node \(\longrightarrow\) UAT doesn't apply
\(\hookrightarrow\) don't expect a CFT

3.1 Neural-Network Conformal-Field-Theory

One single NN node \(\longrightarrow\) don't expect a CFT

Correlators obey crossing symmetry and have standard CFT structure!

eg. $$ \mathbb{E}[\phi_{\theta,\Delta_1}(x_1)\phi_{\theta,\Delta_2}(x_2)] = \delta_{\Delta_1,\Delta_2}\frac{c_{12}}{(x_1\cdot x_2)^{\Delta_1+\Delta_2}} $$

Choose architecture and distribution to get any CFT you want*.

Can combine NN nodes together to make new CFTs.

Formalism also works with conformal defects [Capuozzo, Robinson, Suzzoni; 2025].

What about CFTs in 2d?

3.2 Bosonic Strings

Neural-Networks can approximate any function in the \(\infty\)-width limit.
The generating functional of correlators can be approximated too:

Distribution \(P(\theta_i)\) usually has global symmetries of the action \(S[\phi]\).

Full Virasoro symmetry achieved in the limit\({}^\ast\) using cos-net architecture and log-kernel [Robinson; 2025], [Frank and Halverson; 2026],

\[ \phi(z|\{a_i,W_i,\gamma_i\}) = \frac{1}{N}\sum_{i=1}^N a_i\cos(zW_i+\bar{z}\overline{W}_i + \gamma_i) \]

\[ \mathbb{E}_{a_iW_i,\gamma_i}[\phi(z|\{a_i,W_i,\gamma_i\})\phi(w|\{a_i,W_i,\gamma_i\})] \overset{N\to\infty}{\longrightarrow} -\alpha^\prime\ln(|z-w) \]

Key idea: Finite-\(N\) corrections are deformations away from the free boson theory.

3.2 Wess-Zumino-Witten

String theory with group manifold target space: \(S^2\rightarrow G\).

$$ S = \frac{1}{4\lambda^2}\int_{S^2} \operatorname{tr}\left(g^{-1}\partial_{\mu}gg^{-1}\partial^{\mu}g\right) - \frac{ik}{2\pi}\int_{B_3}\operatorname{tr}\left(g^{-1}dg\wedge g^{-1}dg\wedge g^{-1}dg\right) $$

The quantum theory unearths the affine Kac-Moody algebra \(\widehat{\mathfrak{g}}_k\) as its spectrum-generating algebra.

With \(J(z)=-k\partial gg^{-1}\), $$ J^a(z)J^b(z) \sim \frac{k\delta^{ab}}{(z-w)^2} + \frac{if^{ab}{}_cJ^c(w)}{z-w} $$ and \(T(z) = \gamma:J^aJ^a:(z)\), $$ T(z)T(w) \sim \frac{c/2}{(z-w)^4} + \frac{2T(w)}{(z-w)^2} + \frac{\partial T(w)}{z-w} $$

3.2 Wess-Zumino-Witten

Use the Wakimoto representation: \(\beta\gamma\)-ghosts and free boson \(\phi(z)\).
Approximate those using the cos-net architecture.

Result: limit is well behaved and neural-network approximation is numerically powerful!

eg. \(\widehat{\mathfrak{su}(2)}_k\) has three base fields

\[ \begin{align} e(z) &= \beta(z)\\ h(z) &= i\sqrt{2k+4}\partial\phi(z)+2\gamma(z)\beta(z)\\ f(z) &= -i\sqrt{2k+4}\partial\phi(z)\gamma(z)-k\partial\gamma(z)-\beta(z)\gamma^2(z) \end{align} \] \[ T(z) = \frac{1}{2k+4}\left(\frac{1}{2}h^2(z)+2e(z)f(z)\right) \]

WZW Kac-Moody algebra with central charge \(c=\frac{3k}{k+2}\) is recovered in the \(\infty\)-width limit!

\[ \mathbb{E}[T(z)T(w)] = \frac{c/2}{(z-w)^4} + \mathcal{O}\left(\frac{1}{N}\right) \]

Neural-Network Field Theory: Key takeaways (aka conclusion)

• NNFTs recast functional integrals into statistical averages of network parameters
• Can engineer new free theories and add perturbative interactions (eg. finite-\(N\) effects)
• NN-CFTs are CFT for all \(N\)
• \(\hookrightarrow\) new playground for generating CFT data
• NNFTs are interesting both analytically and numerically
• \(\hookrightarrow\) can reproduce bosonic strings amplitudes and WZW correlators

Limitations

• Hard to engineer specific theories (or even local theories)
• Formalism works well for spin-\(0\) and spin-\(\frac{1}{2}\)... unknown for higher spin
• NN-CFT only known for spin-\(0\) conformal primaries
• Gauge field?

Table of Contents

I Background

    1.1 Neural Networks and UATs

    1.2 NN-FT Correspondence

II A Timeline of NNFT

III CFTs, Bosonic Strings and WZW

    3.1 NN-CFT Correspondence

    3.2 Bosonic Strings and WZW

IV Advertizing

    4.1 Topological effects

    4.2 Superstrings

4.1 Topological effects

Added a discrete/topological sector to the distribution of parameters.

The NNFT reproduces known phase transitions in the \(XY\) model.

NNFT have strong numerical capabilities!

C. Ferko

J. Halverson

V. Jejjala

B. Robinson

4.2 Superstrings

Developing a program to reconstruct all superstrings from NNs!

T-duality was already shown to work in NNFT, can NN-superstring give new predictions?

NNFT allow us to efficiently simulate the superstrings.

J. Halverson

B. Robinson

S. Frank

Thank you

P. Capuozzo

B. Robinson