University of Southampton — September 2026

Conformal defects in neural network field theories

symmetry breaking · solvable nn-dCFTs

Benjamin Suzzoni
Department of Mathematical Sciences, UNIST
arXiv:2512.07946

with

PC P. Capuozzo BR B. Robinson

Outline

Why NN? Why defects CFTs?

physics for ML for physics · conformal symmetry ubiquitous

CFT and dCFT data

conformal primaries · OPE · crossing · ambient and defect channels

Neural-network CFTs

UATs · parameter averages · symmetry in architecture and distribution

Neural-network defect CFTs

general construction · monomial and reciprocal examples

What we learn

results · limits · directions opened by the defect viewpoint

Chapter 1

Why NN? Why defect CFTs?

a lot can be found by combining the two

Strong interplay between ML and physics

ML → physics

Find hidden symmetries in data
Compute numerical approximations
Formulate exact conjectures
Assist theorem proving

physics → ML

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.

Conformal symmetry and defect conformal symmetry

Why CFTs?

Symmetry controls dynamics

The conformal group strongly constrains correlators.
CFTs sit at RG fixed points; exactly marginal deformations can connect families of them.
They arise naturally in critical systems and D-brane constructions.

Why dCFTs?

Extended operators complete the picture

Boundaries, interfaces and defects carry their own degrees of freedom.
Higher-form and generalized symmetries act on extended operators.
Ambient spectrum + defect spectrum + couplings characterize the deformed theory.

The same organization can be transferred to neural-network ensembles.

Chapter 2

CFT and dCFT data

we can bootstrap from building blocks

A CFT is encoded by dimensions and OPE coefficients

correlators reveal the independent CFT data

conformal primary

\(\mathcal O_i:\)( Δᵢ, Jᵢ)

\(\mathcal O_i(\rho x)=\rho^{-\Delta_i}\mathcal O_i(x)\)

symmetry fixes the scalar two-point function

\[ \langle\Phi_i(x_1)\Phi_j(x_2)\rangle =\frac{\delta_{ij}}{|x_{12}|^{2\Delta_i}} \]

the scalar three-point function defines the OPE coefficient

\(\langle\Phi_i(x_1)\Phi_j(x_2)\Phi_k(x_3)\rangle=\) λᵢⱼₖ \(|x_{12}|^{\Delta_i+\Delta_j-\Delta_k}|x_{23}|^{\Delta_j+\Delta_k-\Delta_i}\)
\(|x_{13}|^{\Delta_i+\Delta_k-\Delta_j}\)

Scalar correlators shown; spinning tensor structures are suppressed.

CFT data={ Δᵢ, Jᵢ, λᵢⱼₖ}

Δ_i J_i λ_ijk

OPE data obeys the crossing equations

\[ \phi_1(x)\phi_2(y)\sim\sum_{\mathcal O} f_{\phi_1\phi_2\mathcal O}\, P(x-y,\partial_y)\mathcal O(y) \]

The same four-point function must be reconstructed in either channel.

A defect breaks the symmetry subgroup

symmetry acts differently once the defect is present

\(SO(1,D+1)\)

↓

\(SO(1,p+1)\)

\(SO(q)\)

\(q=D-p\)

schematic 2D slice

An ambient operator expands into defect primaries

bring \(\mathcal O_i\) to the defect

bulk-to-defect OPE (BOE) · transverse-scalar sector \(s_a=0\)

\[ \begin{aligned} \mathcal O_i(x_\parallel,x_\perp) &=\sum_a b_{ia}|x_\perp|^{\widehat\Delta_a-\Delta_i}\\[-1pt] &\quad\times\mathcal C_{ia}(x_\perp^2\partial_\parallel^2) \widehat{\mathcal O}_a(x_\parallel) \end{aligned} \]

ambient
\(\Delta_i,J_i,\lambda_{ijk}\)

defect
\(\widehat\Delta_a,s_a,\widehat\lambda_{abc}\)

couplings
\(a_i,b_{ia}\)

full dCFT data

Crossing: two OPE channels reconstruct one correlator

Two ambient insertions can reach the defect identity in two orders.

ambient OPE → identity term of the BOE

two bulk-to-defect OPEs → defect OPE

\[ \boxed{\displaystyle \sum_k \lambda_{ij}{}^{k}a_k\,\mathcal G_k^{\mathrm{amb}}(\chi,\psi) =\sum_a b_{ia}b_{ja}\,\widehat{\mathcal G}_a^{\mathrm{def}}(\chi,\psi)} \]

\(a_k=b_{k\widehat{\mathbf 1}},\quad r_\ell=\lVert X_\ell\rVert_\circ,\quad \chi=-\dfrac{2X_1\bullet X_2}{r_1r_2},\quad \cos\psi=\dfrac{X_1\circ X_2}{r_1r_2}\)

The two-ambient-point function admits an ambient-channel expansion and a defect-channel expansion. The animation follows the identity contribution on each route; the displayed equation restores the full ambient and defect conformal-block sums. Here xi = -2 X_1 dot X_2/(r_1 r_2) = chi - 2 cos(psi). The displayed shorthand uses G_amb = xi^[-(Delta_i+Delta_j)/2] f_(Delta_k,J_k)(xi,psi) and G_def = f-hat_(Delta-hat_a,0,s_a)(chi,psi), matching the standard two-ambient-point crossing convention. The defect two-point basis is taken orthonormal, so the defect identity coefficient is delta_ab. For a nonorthonormal basis, contract with the appropriate defect two-point metric. Degenerate families mean that crossing fixes basis-invariant sums of products rather than individual b coefficients. No positivity assumption is needed. [Sources] - https://arxiv.org/html/1601.02883#S4 - https://arxiv.org/html/1601.02883#S4.SS1 - https://arxiv.org/abs/2512.07946

Chapter 3

Neural-network CFTs

from function approximation to parameter-space field theory

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 \(N\) controls both pictures

\(N=1\) 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

\[ \lim_{N\to\infty}\int_{\Theta_N} d\Theta\,P_N(\Theta)\, \mathcal F[\Phi_\Theta] \]

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.

What is a NN-CFT?[Halverson, et al; 2024]

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.

How to encode symmetry?

field theory

\[ Z[J]=\int\mathcal D\phi\,e^{-S[\phi]} e^{\int d^D x\,J(x)\phi(x)} \]

parameter ensemble

\[ Z[J]=\int d\Theta\,P(\Theta) e^{\int d^D x\,J(x)\Phi_\Theta(x)} \]

parameter ensemble

\(P(\Theta)\)

→

correlators

\[ \mathbb E_{P(\Theta)} [\Phi_{\Theta,1}\cdots\Phi_{\Theta,n}] \]

inherit the symmetries respected by both \(P(\Theta)\) and \(\Phi_\Theta\).

Key Idea: Symmetries of \(S[\phi]\) are encoded in the probability measure \(P(\Theta)\).

Embedding space gives a linear action

\(\mathbb R^{D+2}\) \(\mathbb R^{1,D+1}\)

Wick rotation

\(\mathrm{NC}=\{X\in\mathbb R^{1,D+1}\mid X^2=0\}\) projective identification \(X\sim\lambda X\)

\(\mathbb R^D:\ X^+=1\) \(x^\mu\mapsto(1,x^2,x^\mu)\)

\(\mathbb R^D:\ X^+=-1\) antipodal representative

distribution

\(SO(D+2)\)-invariant \(P(\theta)\)

homogeneous architecture

\(\Phi_\Delta(X)=(X\cdot\theta)^{-\Delta}\)

Toy model: the monomial nn-CFT \(\Delta_n=-n\)

Gaussian parameter ensemble

Euclidean average → Wick rotate

\[ P_E(\Theta)=\frac{e^{-\Theta^2/(2\sigma^2)}}{(2\pi\sigma^2)^{(D+2)/2}} \]

normalized homogeneous primaries

\[ \mathcal O_n(X)=\frac{(\Theta\!\cdot\!X)^n}{\sigma^n\sqrt{n!}}, \qquad \Delta_n=-n,\quad n>0 \]

one-point function

\[ \langle\mathcal{O}_n(X)\rangle= \begin{cases}0,&n\ \mathrm{odd},\\\frac{(n-1)!!}{\sqrt{n!}}(X^2)^{n/2},&n\ \mathrm{even},\end{cases} \]

On the Poincaré section: \(\langle\mathcal O_n(X)\rangle=0\)

orthogonality on the null cone

\[ \boxed{\; \langle\mathcal{O}_n(X_1)\mathcal{O}_m(X_2)\rangle =\delta_{nm}(X_1\!\cdot\!X_2)^n\;}, \qquad X_1^2=X_2^2=0 \]

On the Poincaré section: \(\langle\mathcal{O}_n(X_1)\mathcal{O}_m(X_2)\rangle = \delta_{nm}|x_{12}|^{2n}\)

mixed four-point correlator · \((-n,-m,-n,-m)\)

\[ G_{nmnm}(x_i) =|x_{12}|^{n+m}|x_{34}|^{n+m} \left(\frac{|x_{24}|}{|x_{13}|}\right)^{m-n} g_{nm}(u,v) \]

conformal-block decomposition

\[ g_{nm}(u,v)= \sum_{\mathcal X\in\mathcal O_n\times\mathcal O_m} \lambda_{nm\mathcal X}^{\,2}\, g_{\Delta_{\mathcal X},J_{\mathcal X}}^{(m-n,m-n)}(u,v) \]

\(u=\dfrac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2}\), \(v=\dfrac{x_{14}^2x_{23}^2}{x_{13}^2x_{24}^2}\); \(\Delta_{12}=\Delta_{34}=m-n\).

paper example in \(D=4\) · \(n=1,\ m=2\)

\(\mathcal O_1=\Phi/\sigma,\quad \mathcal O_2=\Phi^2/(\sqrt2\,\sigma^2),\quad \mu_6=15\sigma^6\) ⇒ divide by \(2\sigma^6\)

normalized reduced correlator

\[ \boxed{\; g_{12}(u,v)= 2u^{-1/2}+2v\,u^{-3/2}+u^{-3/2}\;} \]

normalized three-point data → the same \(g_{12}\)

\[ \boxed{\; g_{12}(u,v)= 3\,g_{-3,0}^{(1,1)}(u,v) +\frac52\,g_{-1,0}^{(1,1)}(u,v)\;} \]

\(\widehat{\mathcal O}_{0,3}=\dfrac{(\Theta\cdot X)^3}{\sigma^3\sqrt6}\), \(\lambda_{12,\widehat{\mathcal O}_{0,3}}=\sqrt3\); \(\widehat{\mathcal O}_{1,1}=\dfrac{(\Theta\cdot\Theta)(\Theta\cdot X)}{\sigma^3\sqrt{80}}\), \(\lambda_{12,\widehat{\mathcal O}_{1,1}}=\sqrt{\frac52}\).
Crossing: \(g_{12}(u,v)=\left(\frac vu\right)^{3/2}g_{12}(v,u)\).

Average with the positive Euclidean Gaussian first, continue scalar products to embedding signature, and only then restrict to the projective null cone. A literal Gaussian in Lorentzian signature is not normalizable. The usual factor -2 relating X_i dot X_j to x_ij squared is suppressed consistently with the rest of the deck. The operators shown here are normalized: O_n=(Theta dot X)^n/(sigma^n sqrt(n!)). On the null cone, every Wick contraction left within a single insertion produces X_i squared and vanishes. For n=m, the n! fully cross-contracted terms cancel the normalization; for n not equal to m, at least one same-point contraction is unavoidable. Hence the exact result delta_nm (X_1 dot X_2)^n. The four-point prefactor and block expansion are equations (2.14), (2.15), and (2.27) of arXiv:2409.12222 after setting Delta_1=Delta_3=-n and Delta_2=Delta_4=-m. For the explicit mixed example, the paper uses unnormalized Phi and Phi squared. Our fields are O_1=Phi/sigma and O_2=Phi squared/(sqrt(2) sigma squared). Since a centered Gaussian has mu_6=15 sigma^6, equations (3.47) and (3.49) must both be divided by 2 sigma^6. This gives the displayed normalized reduced correlator and block coefficients. The exchanged scalar primaries must also be normalized. In D=4, O-hat_(0,3)=O_(0,3)/(sigma^3 sqrt(6)) and O-hat_(1,1)=O_(1,1)/(sigma^3 sqrt(80)). Equations (3.50)-(3.52) then give lambda_(12,O03)=sqrt(3) and lambda_(12,O11)=sqrt(5/2), whose squares are exactly the coefficients of the two blocks. The displayed crossing identity exchanges the two identical O_1 legs, or equivalently the two identical O_2 legs, so it compares the two O_1 times O_2 channel orderings without claiming that individual crossed blocks are invariant. This D=4 scalar decomposition is the paper's explicit n=1, m=2 example; it is not a claim that every mixed n, m correlator has a scalar-only spectrum. [Sources] - https://arxiv.org/html/2409.12222#S2.SS2 - https://arxiv.org/html/2409.12222#S3.SS1 - https://arxiv.org/html/2409.12222#S3.SS2

Chapter 4

Neural-network defect CFTs

break the symmetry, choose an invariant distribution, pick an architecture

Ingredients for nn-dCFTs

1 · preserved symmetry

\[ SO(1,D+1)\rightarrow SO(1,p+1)\times SO(q) \]

with \(q=D-p\)

→

2 · invariant distribution

\[ P(g\theta)=P(\theta),\qquad g\in G_{\rm defect} \]

→

3 · covariant architectures

ambient \(\Phi_\Delta(X)\)
defect \(\widehat\varphi_{\widehat\Delta,s}(X)\)

\[ Z[\widehat J,J]= \int d\theta\,P(\theta)\, e^{\int d^p x_\parallel\,\widehat J(x_\parallel)\widehat\varphi(x_\parallel)} e^{\int d^D x\,J(x)\Phi(x)} \]

Example 1: a monomial nn-dCFT

Factorized distribution

\(\widehat P(\hat\theta)\widetilde P(\tilde\theta)\): centered Gaussians.

second moments \(\widehat\mu_2\) and \(\widetilde\mu_2\)

Defect network

\[ \widehat\varphi_{\widehat n}(X) =(X\bullet\hat\theta)^{\widehat n} \]

Ambient network

\[ \Phi_n(X)= (X\bullet\hat\theta+X\circ\tilde\theta)^n \]

Scaling weights

\(\widehat\Delta=-\widehat n\), \(\Delta=-n\)

negative dimensions; \(n,\widehat n\in\mathbb N\)

Convention: hats are tangent/\(\bullet\); tildes are normal/\(\circ\).

Example 1: Finite defect-network expansion

exact OPE-like decomposition

\[ \Phi_n(X)= \sum_{d=0}^{n}\binom nd \underbrace{(X\circ\tilde\theta)^{n-d}}_{\text{normal degree }n-d} \underbrace{(X\bullet\hat\theta)^d}_{\text{tangent degree }d} \]

Each term is a smaller network with a definite scaling dimensions.

For \(n=4\): exactly five network factors ⟾ no infinite defect tower.

example \(n=4\)

Example 1: One-point functions

selection by the defect

defect insertion ambient insertion

defect

\[ \mathbb E[\widehat\varphi_{\widehat n}(X)] \overset{\mathrm{P.S.}}{=}0,\qquad \widehat n>0 \]

The identity \(\widehat n=0\) has expectation value 1.

ambient

\[ \mathbb E[\Phi_n(X)]\overset{\mathrm{P.S.}}{=} \begin{cases} \displaystyle \frac{\Gamma(n+1)} {2^{n/2}\Gamma(\frac n2+1)} \big[(\widetilde\mu_2-\widehat\mu_2)X\circ X\big]^{n/2}, & n\in2\mathbb Z,\\[4pt] 0,&\text{otherwise.} \end{cases} \]

The variance difference measures the defect deformation; it vanishes when the two sectors match.

Example 1: Two-point functions

defect–defect

defect–ambient

ambient–defect

ambient–ambient

defect–defect

\[ \mathbb E[\widehat\varphi_{\widehat n_1}(X_1) \widehat\varphi_{\widehat n_2}(X_2)] =\delta_{\widehat n_1,\widehat n_2} \Gamma(\widehat n_1+1) (\widehat\mu_2X_1\bullet X_2)^{\widehat n_1} \]

defect–ambient

\[ \begin{aligned} &\mathbb E[\widehat\varphi_{\widehat n_1}(X_1)\Phi_{n_2}(X_2)] =\frac{\Gamma(n_2+1)} {2^{(n_2-\widehat n_1)/2} \Gamma(\frac{n_2-\widehat n_1}{2}+1)} (\widehat\mu_2X_1\bullet X_2)^{\widehat n_1}\\ &\hspace{2.3cm}\times [(\widetilde\mu_2-\widehat\mu_2)X_2\circ X_2]^{ (n_2-\widehat n_1)/2} \end{aligned} \]

nonzero for \(n_2\geq\widehat n_1\) and \(n_2-\widehat n_1\in2\mathbb Z\)

ambient–ambient

\[ \begin{aligned} \mathbb E[\Phi_{n_1}(X_1)\Phi_{n_2}(X_2)] ={}&\frac{\Gamma(n_2+1)} {\Gamma(\frac{n_2-n_1}{2}+1)} \left(\frac{\alpha_{22}}{2}\right)^{\frac{n_2-n_1}{2}} \alpha_{12}^{n_1}\\ &\times{}_2F_1\!\left( \frac{1-n_1}{2},-\frac{n_1}{2}; \frac{n_2-n_1}{2}+1; \frac{\alpha_{11}\alpha_{22}}{\alpha_{12}^2} \right) \end{aligned} \]

for \(n_1\leq n_2\) with equal parity; exchange \(1\leftrightarrow2\) otherwise

Example 1: exact scalar one-/two-point data

unit-normalized basis

\[ \rho=\frac{\widetilde\mu_2-\widehat\mu_2}{\widehat\mu_2},\qquad \mathcal O_n=\frac{\Phi_n}{\sqrt{n!\,\widehat\mu_2^{\,n}}},\qquad \widehat{\mathcal O}_m=\frac{\widehat\varphi_m}{\sqrt{m!\,\widehat\mu_2^{\,m}}}. \]

scalar spectrum

\[ (\Delta_n,J_n)=(-n,0),\qquad (\widehat\Delta_m,s_m)=(-m,0) \]

For a fixed \(\mathcal O_n\), only

\[ m=n,n-2,n-4,\ldots\geq0, \qquad r_{nm}\equiv\frac{n-m}{2}\in\mathbb N \]

one-point and scalar BOE coefficients

\[ \boxed{\displaystyle a_n=b_{n0}=\delta_{n\,\mathrm{even}} \frac{\sqrt{n!}}{2^{n/2}(n/2)!}\,\rho^{n/2}} \]

\[ \boxed{\displaystyle b_{nm}=\delta_{r_{nm}\in\mathbb N} \frac{\sqrt{n!/m!}}{2^{r_{nm}}r_{nm}!}\,\rho^{r_{nm}}} \]

\[ \langle\mathcal O_n(X)\widehat{\mathcal O}_m(Y)\rangle =b_{nm}(X\circ X)^{r_{nm}}(X\bullet Y)^m, \qquad \langle\widehat{\mathcal O}_m\rangle=\delta_{m0}. \]

Ambient–ambient check: \(b_{n_1m}b_{n_2m}=\dfrac{\sqrt{n_1!n_2!}}{m!\,2^{r_{1m}+r_{2m}}r_{1m}!r_{2m}!}\rho^{r_{1m}+r_{2m}}\), with \(r_{im}=(n_i-m)/2\). For a nondegenerate scalar family this fixes BOE products—not individual three-point coefficients.

Example 2: a reciprocal nn-dCFT

Same factorized distribution

\(\widehat P(\hat\theta)\widetilde P(\tilde\theta)\), with moments \(\widehat\mu_2,\widetilde\mu_2\).

Defect network

\[ \widehat\varphi_{\widehat\Delta}(X) =(X\bullet\hat\theta)^{-\widehat\Delta} \]

Ambient network

\[ \Phi_\Delta(X)= (X\bullet\hat\theta+X\circ\tilde\theta)^{-\Delta} \]

Positive dimensions

\(\widehat\Delta,\Delta>0\)

positive does not by itself imply unitarity

\[ \Phi_\Delta(X)= \sum_{n=0}^{\infty}\frac{(\Delta)_n}{n!}(-1)^n\, \widetilde\varphi_{-n}(X)\, \widehat\varphi_{\Delta+n}(X) \]

Example 2: One-point functions

spectrum starts at \(\widehat\Delta=\Delta>0\)

defect

\[ \mathbb E[\widehat\varphi_{\widehat\Delta}(X)]=0 \]

ambient

\[ \mathbb E[\Phi_\Delta(X)]=0 \]

With no \(\widehat\Delta=0\) term, there is no coupling to the defect identity.

Example 2:Two-point functions

defect–defect

\[ \begin{aligned} &\mathbb E[ \widehat\varphi_{\widehat\Delta_1}(X_1) \widehat\varphi_{\widehat\Delta_2}(X_2)]\\ &\quad= \frac{\sec(\pi\widehat\Delta_1)} {\Gamma(\widehat\Delta_1)} (\widehat\mu_2X_1\bullet X_2)^{-\widehat\Delta_1} \delta_{\widehat\Delta_1,\widehat\Delta_2}. \end{aligned} \]

For integer \(\widehat\Delta\), \(\sec(\pi\widehat\Delta)=(-1)^{\widehat\Delta}\).

ambient–defect

\[ \begin{aligned} &\mathbb E[ \Phi_{\Delta_1}(X_1) \widehat\varphi_{\widehat\Delta_2}(X_2)] = \frac{ 2^{(\Delta_1-\widehat\Delta_2)/2} \sec(\pi\widehat\Delta_2)} {\Gamma(\Delta_1) \Gamma(\frac{\widehat\Delta_2-\Delta_1}{2}+1)} \\[-1pt] &\qquad\times (\widetilde\mu_2X_1\circ X_1)^{ (\widehat\Delta_2-\Delta_1)/2} (\widehat\mu_2X_1\bullet X_2)^{ -\widehat\Delta_2}. \end{aligned} \]

nonzero for \(\widehat\Delta_2\geq\Delta_1\)

The mixed result has exactly the kinematic form required by defect conformal symmetry.

Example 2: exact scalar one-/two-point data

scalar spectrum and normalization

\[ (\Delta,J)=(\Delta,0),\qquad (\widehat\Delta_k,s)=(\Delta+2k,0), \quad k=0,1,\ldots \]

\[ c_{\hat\Delta}=\frac{\sec(\pi\hat\Delta)}{\Gamma(\hat\Delta)\widehat\mu_2^{\,\hat\Delta}},\qquad \mathcal O_{\hat\Delta}=c_{\hat\Delta}^{-1/2}\varphi_{\hat\Delta} \]

\[ a_\Delta=b_{\Delta\mathbf1}=0, \qquad \langle\widehat{\mathcal O}_{\widehat\Delta}\rangle=0 \]

No coupling of bulk primary to defect identity.

unit-normalized scalar BOE tower

\[ \boxed{\displaystyle b_{\Delta\hat\Delta_k}=\frac1{k!} \left(\frac{\widetilde\mu_2}{2\widehat\mu_2}\right)^k \sqrt{\frac{\Gamma(\Delta+2k)}{\Gamma(\Delta)}}}, \qquad k\geq0 \]

Leading restriction: \(b_0=1\)

The variance ratio controls every higher scalar coupling.

The exact ambient–ambient hypergeometric kernel resums this tower together with the transverse-spin sectors.

Meromorphic and generically nonunitary: \(c_\hat\Delta\) has poles at \(\hat\Delta=(2\ell+1)/2\) and changes sign across them. Outside the bare convergence window these are analytically continued, renormalized data.

Beyond Guassian: Deforming \(P\)

keep every primary architecture · deform only \(P\)

\[ P\propto\exp\!\left[ -\frac{u}{2\widehat\sigma} -\frac{v}{2\widetilde\sigma} +\sum_{a,b}\lambda_{ab}u^av^b \right], \quad u=\widehat\theta^2,\ v=\widetilde\theta^2 \]

\(\widehat\sigma=1.10\)

\(\widetilde\sigma=0.85\)

λ₂₀ = 0

λ₀₂ = 0

λ₁₁ = 0

Only the invariant radial moments \(M_{a,b}=\mathbb E_P[u^av^b]\) change. Spectra and selection rules stay fixed.

Null-projected angular contractions are analytic. The full Gaussian reduction agrees through degree six to \(2.7\times10^{-15}\).

Gaussian reference: the exact coefficients at \(\lambda_{ab}=0\) provide the starting point.

Advance through the Gaussian reference, tangent quartic, normal quartic, cross quartic, and original joint deformation. The parameters on the left and the exact coefficient values on the right are driven by the same state object, so they remain synchronized. Bar heights are the signed coefficients themselves, not ratios and not error bars. Earlier real-vector Monte Carlo projections were off the embedding null cone and have been demoted to sampler diagnostics. This chart uses exact angular contractions and deterministic radial quadrature documented in DEFORMED_PRIOR_STUDY.md. Moving allowed one-/two-point overlaps does not by itself prove interaction; that requires connected higher-point data. Positive coefficients multiplying the leading quartic in the plus-sign convention are non-normalizable unless stabilized by higher negative terms. [Sources] - https://arxiv.org/abs/2512.07946

Chapter 5

What was achieved?

a solvable laboratory · a new toolset for dCFT data

A brief recap

completed here

Extended neural-network conformal field theories to conformal defects.
Built monomial and reciprocal nn-dCFT toy models.
Computed ambient, defect and mixed scalar correlators.
Made explicit the dCFT data of new theories.

next questions

We only considered scalar bosons ⟾ include symmetric traceless tensors? M. Dogra, J. Halverson & J. Naskar · arXiv:2608.15001
We only considered scalar bosons ⟾ fermions?
Can we recover real life CFTs/dCFTs?
Can we compute their anomaly coefficients?C. Ferko, S. Frank, J. Halverson & V. Jejjala · arXiv:2605.12488
Can we guarantee unitarity?
C. Ferko & A. Mutchler · arXiv:2608.21588

Key takeaways

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

Thank you

Questions are welcome — b.suzzoni@benterre.com

arXiv:2512.07946