Benjamin Suzzoni
In collaboration with P. Capuozzo and B. Robinson
Neural-Networks can approximate any function in the \(\infty\)-width limit.
The generating functional of correlators can be approximated too:
\[ Z[J] = \int \mathcal{D}\phi\, e^{-S[\phi] + \int d^Dx\, \phi(x)J(x)} \]
\[ Z[J]=\lim_{N\rightarrow\infty} \int_{\Omega} \prod_i^N d\theta_i\, P(\theta_i)\, e^{\int d^Dx\,\phi(x|\theta_i)J(x)} \]
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.
String theory with a group manifold target space \(\longrightarrow\) 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!
\[ \mathbb{E}[T(z)T(w)] = \frac{c/2}{(z-w)^4} + \mathcal{O}\left(\frac{1}{N}\right) \]
Question: what physics do the finite-\(N\) corrections hold?