ICBS — August 2026
symmetry breaking · solvable nn-dCFTs
Benjamin Suzzoni
Department of Mathematical Sciences, UNIST
arXiv:2512.07946
Chapter 1
a lot can be found by combining the two
Neural Network Field Theories
Then symmetry, correlation functions and defects become tools for organizing families of random networks.
A defect is a controlled way to break symmetry—and probe what survives.
The same organization can be transferred to neural-network ensembles.
Chapter 2
we can bootstrap from building blocks
\(\mathcal O_i(\rho x)=\rho^{-\Delta_i}\mathcal O_i(x)\)
\[ \langle\Phi_i(x_1)\Phi_j(x_2)\rangle =\frac{\delta_{ij}}{|x_{12}|^{2\Delta_i}} \]
Scalar correlators shown; spinning tensor structures are suppressed.
\[ \phi_1(x)\phi_2(y)\sim\sum_{\mathcal O} f_{\phi_1\phi_2\mathcal O}\, P(x-y,\partial_y)\mathcal O(y) \]
The same four-point function must be reconstructed in either channel.
\(q=D-p\)
\[ \begin{aligned} \mathcal O_i(x_\parallel,x_\perp) &=\sum_a b_{ia}|x_\perp|^{\widehat\Delta_a-\Delta_i}\\[-1pt] &\quad\times\mathcal C_{ia}(x_\perp^2\partial_\parallel^2) \widehat{\mathcal O}_a(x_\parallel) \end{aligned} \]
\[ \left\{ \begin{aligned} &\Delta_i,J_i,\lambda_{ijk};\, \widehat\Delta_a,s_a,\widehat\lambda_{abc};\\[-2pt] &a_i,b_{ia} \end{aligned} \right\} \]
\[ \begin{aligned} \chi&=-\frac{2X_1\bullet X_2}{\lVert X_1\rVert_\circ\lVert X_2\rVert_\circ},\\ \cos\psi&=\frac{X_1\circ X_2}{\lVert X_1\rVert_\circ\lVert X_2\rVert_\circ} \end{aligned} \]
Chapter 3
from function approximation to parameter-space field theory
Neural network:
\(f:\mathbb R^m\rightarrow\mathbb R^n\)
Operations: composition, addition and scalar multiplication.
Its functional form is the architecture; its weights and biases are the parameters.
\[ f(x)=f_o\!\left(\sum_i w_i f_i(x)\right) \]
\[ \int\mathcal D\phi\,\mu[\phi]\,\mathcal F[\phi] \]
integrate over field configurations
\[ \lim_{N\to\infty}\int_{\Theta_N} d\Theta\,P_N(\Theta)\, \mathcal F[\Phi_\Theta] \]
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.
| Field-theory language | Neural-network language |
|---|---|
| \(\phi(\lambda x)=\lambda^{-\Delta}\phi(x)\) | \(\Phi_\Theta(\lambda X)=\lambda^{-\Delta}\Phi_\Theta(X)\) |
| \(\displaystyle Z[J]=\int\mathcal D\phi\,e^{-S[\phi]}e^{\int J\phi}\) | \(\displaystyle Z[J]=\int d\Theta\,P(\Theta)e^{\int J\Phi_\Theta}\) |
| \(\langle\phi(x)\phi(y)\rangle\) | \(\mathbb E_{P(\Theta)}[\Phi_\Theta(X)\Phi_\Theta(Y)]\) |
Takeaway: conformal structure is a property of the architecture–distribution pair, not of the \(N\to\infty\) limit alone.
\[ Z[J]=\int\mathcal D\phi\,e^{-S[\phi]} e^{\int d^D x\,J(x)\phi(x)} \]
\[ Z[J]=\int d\Theta\,P(\Theta) e^{\int d^D x\,J(x)\Phi_\Theta(x)} \]
\[ \mathbb E_{P(\Theta)} [\Phi_{\Theta,1}\cdots\Phi_{\Theta,n}] \]
inherit the symmetries respected by both \(P(\Theta)\) and \(\Phi_\Theta\).
Key Idea: Symmetries of \(S[\phi]\) are encoded in the probability measure \(P(\Theta)\).
\(SO(D+2)\)-invariant \(P(\theta)\)
\(\Phi_\Delta(X)=(X\cdot\theta)^{-\Delta}\)
Chapter 4
break the symmetry, choose an invariant distribution, pick an architecture
\[ SO(1,D+1)\rightarrow SO(1,p+1)\times SO(q) \]
with \(q=D-p\)
\[ P(g\theta)=P(\theta),\qquad g\in G_{\rm defect} \]
ambient \(\Phi_\Delta(X)\)
defect \(\widehat\varphi_{\widehat\Delta,s}(X)\)
\[ Z[\widehat J,J]= \int d\theta\,P(\theta)\, e^{\int d^p x_\parallel\,\widehat J(x_\parallel)\widehat\varphi(x_\parallel)} e^{\int d^D x\,J(x)\Phi(x)} \]
\(\widehat P(\hat\theta)\widetilde P(\tilde\theta)\): centered Gaussians.
second moments \(\widehat\mu_2\) and \(\widetilde\mu_2\)
\[ \widehat\varphi_{\widehat n}(X) =(X\bullet\hat\theta)^{\widehat n} \]
\[ \Phi_n(X)= (X\bullet\hat\theta+X\circ\tilde\theta)^n \]
\(\widehat\Delta=-\widehat n\), \(\Delta=-n\)
negative dimensions; \(n,\widehat n\in\mathbb N\)
Convention: hats are tangent/\(\bullet\); tildes are normal/\(\circ\).
\[ \mathbb E[\widehat\varphi_{\widehat n}(X)] \overset{\mathrm{P.S.}}{=}0,\qquad \widehat n>0 \]
The identity \(\widehat n=0\) has expectation value 1.
\[ \mathbb E[\Phi_n(X)]\overset{\mathrm{P.S.}}{=} \begin{cases} \displaystyle \frac{\Gamma(n+1)} {2^{n/2}\Gamma(\frac n2+1)} \big[(\widetilde\mu_2-\widehat\mu_2)X\circ X\big]^{n/2}, & n\in2\mathbb Z,\\[4pt] 0,&\text{otherwise.} \end{cases} \]
The variance difference measures the defect deformation; it vanishes when the two sectors match.
\[ \mathbb E[\widehat\varphi_{\widehat n_1}(X_1) \widehat\varphi_{\widehat n_2}(X_2)] =\delta_{\widehat n_1,\widehat n_2} \Gamma(\widehat n_1+1) (\widehat\mu_2X_1\bullet X_2)^{\widehat n_1} \]
\[ \begin{aligned} &\mathbb E[\widehat\varphi_{\widehat n_1}(X_1)\Phi_{n_2}(X_2)] =\frac{\Gamma(n_2+1)} {2^{(n_2-\widehat n_1)/2} \Gamma(\frac{n_2-\widehat n_1}{2}+1)} (\widehat\mu_2X_1\bullet X_2)^{\widehat n_1}\\ &\hspace{2.3cm}\times [(\widetilde\mu_2-\widehat\mu_2)X_2\circ X_2]^{ (n_2-\widehat n_1)/2} \end{aligned} \]
nonzero for \(n_2\geq\widehat n_1\) and \(n_2-\widehat n_1\in2\mathbb Z\)
\[ \begin{aligned} \mathbb E[\Phi_{n_1}(X_1)\Phi_{n_2}(X_2)] ={}&\frac{\Gamma(n_2+1)} {\Gamma(\frac{n_2-n_1}{2}+1)} \left(\frac{\alpha_{22}}{2}\right)^{\frac{n_2-n_1}{2}} \alpha_{12}^{n_1}\\ &\times{}_2F_1\!\left( \frac{1-n_1}{2},-\frac{n_1}{2}; \frac{n_2-n_1}{2}+1; \frac{\alpha_{11}\alpha_{22}}{\alpha_{12}^2} \right) \end{aligned} \]
for \(n_1\leq n_2\) with equal parity; exchange \(1\leftrightarrow2\) otherwise
\[ \Phi_n(X)= \sum_{d=0}^{n}\binom nd \underbrace{(X\circ\tilde\theta)^{n-d}}_{\text{normal degree }n-d} \underbrace{(X\bullet\hat\theta)^d}_{\text{tangent degree }d} \]
Each term is a smaller network with a definite scaling dimensions.
For \(n=4\): exactly five network factors ⟾ no infinite defect tower.
\(\widehat P(\hat\theta)\widetilde P(\tilde\theta)\), with moments \(\widehat\mu_2,\widetilde\mu_2\).
\[ \widehat\varphi_{\widehat\Delta}(X) =(X\bullet\hat\theta)^{-\widehat\Delta} \]
\[ \Phi_\Delta(X)= (X\bullet\hat\theta+X\circ\tilde\theta)^{-\Delta} \]
\(\widehat\Delta,\Delta>0\)
positive does not by itself imply unitarity
\[ \Phi_\Delta(X)= \sum_{n=0}^{\infty}\frac{(\Delta)_n}{n!}(-1)^n\, \widetilde\varphi_{-n}(X)\, \widehat\varphi_{\Delta+n}(X) \]
\[ \mathbb E[\widehat\varphi_{\widehat\Delta}(X)]=0 \]
\[ \mathbb E[\Phi_\Delta(X)]=0 \]
With no \(\widehat\Delta=0\) term, there is no coupling to the defect identity.
\[ \begin{aligned} &\mathbb E[ \widehat\varphi_{\widehat\Delta_1}(X_1) \widehat\varphi_{\widehat\Delta_2}(X_2)]\\ &\quad= \frac{\sec(\pi\widehat\Delta_1)} {\Gamma(\widehat\Delta_1)} (\widehat\mu_2X_1\bullet X_2)^{-\widehat\Delta_1} \delta_{\widehat\Delta_1,\widehat\Delta_2}. \end{aligned} \]
For integer \(\widehat\Delta\), \(\sec(\pi\widehat\Delta)=(-1)^{\widehat\Delta}\).
\[ \begin{aligned} &\mathbb E[ \Phi_{\Delta_1}(X_1) \widehat\varphi_{\widehat\Delta_2}(X_2)] = \frac{ 2^{(\Delta_1-\widehat\Delta_2)/2} \sec(\pi\widehat\Delta_2)} {\Gamma(\Delta_1) \Gamma(\frac{\widehat\Delta_2-\Delta_1}{2}+1)} \\[-1pt] &\qquad\times (\widetilde\mu_2X_1\circ X_1)^{ (\widehat\Delta_2-\Delta_1)/2} (\widehat\mu_2X_1\bullet X_2)^{ -\widehat\Delta_2}. \end{aligned} \]
nonzero for \(\widehat\Delta_2\geq\Delta_1\)
The mixed result has exactly the kinematic form required by defect conformal symmetry.
Chapter 5
a solvable laboratory · a new toolset for dCFT data
UATs give a well-defined approximation of the path integral.
Finite neural architectures can already define CFT and dCFT data. A UAT limit is not required.
Large-width Gaussian limits generally yield generalized-free theories
Can combine these nn-CFTs (nn-dCFTs) to generate infinitely many new theories
Questions are welcome — b.suzzoni@benterre.com