SIMIS-APCTP workshop — August 2026
Conformal defects in neural network field theories
symmetry breaking · solvable nn-dCFTs
Outline
Chapter 1
Why NN? Why defect CFTs?
a lot can be found by combining the two
Strong interplay between ML and physics
- Find hidden symmetries in data
- Compute numerical approximations
- Formulate exact conjectures
- Assist theorem proving
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
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.
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
\(\mathcal O_i(\rho x)=\rho^{-\Delta_i}\mathcal O_i(x)\)
\[ \langle\Phi_i(x_1)\Phi_j(x_2)\rangle =\frac{\delta_{ij}}{|x_{12}|^{2\Delta_i}} \]
\(|x_{13}|^{\Delta_i+\Delta_k-\Delta_j}\)
Scalar correlators shown; spinning tensor structures are suppressed.
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
\(q=D-p\)
An ambient operator expands into defect primaries
\[ \begin{aligned} \mathcal O_i(x_\parallel,x_\perp) &=\sum_a b_{ia}|x_\perp|^{\widehat\Delta_a-\Delta_i}\\[-1pt] &\quad\times\mathcal C_{ia}(x_\perp^2\partial_\parallel^2) \widehat{\mathcal O}_a(x_\parallel) \end{aligned} \]
\(\Delta_i,J_i,\lambda_{ijk}\)
\(\widehat\Delta_a,s_a,\widehat\lambda_{abc}\)
\(a_i,b_{ia}\)
\[ \left\{ \begin{aligned} &\Delta_i,J_i,\lambda_{ijk};\, \widehat\Delta_a,s_a,\widehat\lambda_{abc};\\[-2pt] &a_i,b_{ia} \end{aligned} \right\} \]
\[ \begin{aligned} \chi&=-\frac{2X_1\bullet X_2}{\lVert X_1\rVert_\circ\lVert X_2\rVert_\circ},\\ \cos\psi&=\frac{X_1\circ X_2}{\lVert X_1\rVert_\circ\lVert X_2\rVert_\circ} \end{aligned} \]
Chapter 3
Neural-network CFTs
from function approximation to parameter-space field theory
A neural network is a composable map
Neural network:
\(\Phi_{\Theta}:\mathbb R^m\rightarrow\mathbb R\)
Operations: composition, addition and scalar multiplication.
Its functional form is the architecture; its weights and biases are the parameters.
\[ \Phi_{\Theta}(x)=\sum_i \Theta_i \varphi_i(x) \]
The Universal Approximation Theorem (UAT)
A functional integral through the UAT
Network at initialization: weights \(\Theta\) follow a distribution \(P\).
\[ \int\mathcal D\phi\,\mu[\phi]\,\mathcal F[\phi] \]
integrate over field configurations
through \(\Phi_\Theta\)
\[ \lim_{N\to\infty}\int_{\Theta_N} d\Theta\,P_N(\Theta)\, \mathcal F[\Phi_\Theta] \]
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?
\[ Z[J]=\int\mathcal D\phi\,e^{-S[\phi]} e^{\int d^D x\,J(x)\phi(x)} \]
\[ Z[J]=\int d\Theta\,P(\Theta) e^{\int d^D x\,J(x)\Phi_\Theta(x)} \]
\[ \mathbb E_{P(\Theta)} [\Phi_{\Theta,1}\cdots\Phi_{\Theta,n}] \]
inherit the symmetries respected by both \(P(\Theta)\) and \(\Phi_\Theta\).
Key Idea: Symmetries of \(S[\phi]\) are encoded in the probability measure \(P(\Theta)\).
Embedding space gives a linear action
Chapter 4
Neural-network defect CFTs
break the symmetry, choose an invariant distribution, pick an architecture
Ingredients for nn-dCFTs
\[ SO(1,D+1)\rightarrow SO(1,p+1)\times SO(q) \]
with \(q=D-p\)
\[ P(g\theta)=P(\theta),\qquad g\in G_{\rm defect} \]
ambient \(\Phi_\Delta(X)\)
defect \(\widehat\varphi_{\widehat\Delta,s}(X)\)
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
\[ \Phi_n(X)= \sum_{d=0}^{n}\binom nd \underbrace{(X\circ\tilde\theta)^{n-d}}_{\text{normal degree }n-d} \underbrace{(X\bullet\hat\theta)^d}_{\text{tangent degree }d} \]
Each term is a smaller network with a definite scaling dimensions.
For \(n=4\): exactly five network factors ⟾ no infinite defect tower.
Example 1: One-point functions
\[ \mathbb E[\widehat\varphi_{\widehat n}(X)] \overset{\mathrm{P.S.}}{=}0,\qquad \widehat n>0 \]
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.
Example 1: Two-point functions
\[ \mathbb E[\widehat\varphi_{\widehat n_1}(X_1) \widehat\varphi_{\widehat n_2}(X_2)] =\delta_{\widehat n_1,\widehat n_2} \Gamma(\widehat n_1+1) (\widehat\mu_2X_1\bullet X_2)^{\widehat n_1} \]
\[ \begin{aligned} &\mathbb E[\widehat\varphi_{\widehat n_1}(X_1)\Phi_{n_2}(X_2)] =\frac{\Gamma(n_2+1)} {2^{(n_2-\widehat n_1)/2} \Gamma(\frac{n_2-\widehat n_1}{2}+1)} (\widehat\mu_2X_1\bullet X_2)^{\widehat n_1}\\ &\hspace{2.3cm}\times [(\widetilde\mu_2-\widehat\mu_2)X_2\circ X_2]^{ (n_2-\widehat n_1)/2} \end{aligned} \]
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
Example 1: exact scalar one-/two-point data
\[ \rho=\frac{\widetilde\mu_2-\widehat\mu_2}{\widehat\mu_2},\qquad \mathcal O_n=\frac{\Phi_n}{\sqrt{n!\,\widehat\mu_2^{\,n}}},\qquad \widehat{\mathcal O}_m=\frac{\widehat\varphi_m}{\sqrt{m!\,\widehat\mu_2^{\,m}}}. \]
\[ (\Delta_n,J_n)=(-n,0),\qquad (\widehat\Delta_m,s_m)=(-m,0) \]
For a fixed \(\mathcal O_n\), only
\[ m=n,n-2,n-4,\ldots\geq0, \qquad r_{nm}\equiv\frac{n-m}{2}\in\mathbb N \]
\[ \boxed{\displaystyle a_n=b_{n0}=\delta_{n\,\mathrm{even}} \frac{\sqrt{n!}}{2^{n/2}(n/2)!}\,\rho^{n/2}} \]
\[ \boxed{\displaystyle b_{nm}=\delta_{r_{nm}\in\mathbb N} \frac{\sqrt{n!/m!}}{2^{r_{nm}}r_{nm}!}\,\rho^{r_{nm}}} \]
\[ \langle\mathcal O_n(X)\widehat{\mathcal O}_m(Y)\rangle =b_{nm}(X\circ X)^{r_{nm}}(X\bullet Y)^m, \qquad \langle\widehat{\mathcal O}_m\rangle=\delta_{m0}. \]
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
\[ \mathbb E[\widehat\varphi_{\widehat\Delta}(X)]=0 \]
\[ \mathbb E[\Phi_\Delta(X)]=0 \]
With no \(\widehat\Delta=0\) term, there is no coupling to the defect identity.
Example 2:Two-point functions
\[ \begin{aligned} &\mathbb E[ \widehat\varphi_{\widehat\Delta_1}(X_1) \widehat\varphi_{\widehat\Delta_2}(X_2)]\\ &\quad= \frac{\sec(\pi\widehat\Delta_1)} {\Gamma(\widehat\Delta_1)} (\widehat\mu_2X_1\bullet X_2)^{-\widehat\Delta_1} \delta_{\widehat\Delta_1,\widehat\Delta_2}. \end{aligned} \]
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.
Example 2: exact scalar one-/two-point data
\[ (\Delta,J)=(\Delta,0),\qquad (\widehat\Delta_k,s)=(\Delta+2k,0), \quad k=0,1,\ldots \]
\[ c_{\hat\Delta}=\frac{\sec(\pi\hat\Delta)}{\Gamma(\hat\Delta)\widehat\mu_2^{\,\hat\Delta}},\qquad \mathcal O_{\hat\Delta}=c_{\hat\Delta}^{-1/2}\varphi_{\hat\Delta} \]
\[ a_\Delta=b_{\Delta\mathbf1}=0, \qquad \langle\widehat{\mathcal O}_{\widehat\Delta}\rangle=0 \]
No coupling of bulk primary to defect identity.
\[ \boxed{\displaystyle b_{\Delta\hat\Delta_k}=\frac1{k!} \left(\frac{\widetilde\mu_2}{2\widehat\mu_2}\right)^k \sqrt{\frac{\Gamma(\Delta+2k)}{\Gamma(\Delta)}}}, \qquad k\geq0 \]
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.
Beyond Guassian: Deforming \(P\)
\[ P\propto\exp\!\left[ -\frac{u}{2\widehat\sigma} -\frac{v}{2\widetilde\sigma} +\sum_{a,b}\lambda_{ab}u^av^b \right], \quad u=\widehat\theta^2,\ v=\widetilde\theta^2 \]
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 \(2.7\times10^{-15}\).
Chapter 5
What was achieved?
a solvable laboratory · a new toolset for dCFT data
A brief recap
- Extended neural-network conformal fields to flat defects.
- Built monomial and reciprocal nn-dCFT toy models.
- Computed ambient, defect and mixed scalar correlators.
- Made explicit the dCFT data of new theories.
- We only considered scalar bosons ⟾ include symmetric traceless tensors?
- We only considered scalar bosons ⟾ fermions?
- Can we recover real life CFTs/dCFTs?
- Can we compute their anomaly coefficients?
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
Special-boundary Ising is not the only target
Tricritical Ising \((+0)\)
One shared scalar head with \(\widehat\Delta=3/5\), plus two identity skips.
\[ \frac{B_{\epsilon\widehat\epsilon'}} {B_{\sigma\widehat\epsilon'}} =-2^{3/4} \]
Exact and interacting: a sharp test that cannot be passed by fitting each mixed coefficient independently.
\(O(2)\) normal boundary
An odd scalar tilt head and an even scalar displacement head, both protected:
\[ \widehat\Delta_t=2,\quad b_{\phi t}=.5272(2), \qquad \lambda_{ttt}=0 \]
The residual \(\mathbb Z_2\) makes the connected three-point test exact.
3d Ising magnetic line
Scalar defect-changing operators now provide a one-dimensional crossing problem:
\[ \Delta_c=.818(28),\quad C_1=1.25(12),\quad C_2=1.01(26) \]
A clean scalar domain-wall test: no transverse-spin displacement is inserted.
The optimizer finds a structural no-fit
\[ \widehat\Delta=\Delta+2k,\qquad k=0,1,\ldots, \qquad \mathbf1\notin\mathrm{BOE} \]
\(\widehat\Delta_\sigma-\Delta_\phi=-0.1650\)
below tower\(\widehat\Delta_\epsilon-\Delta_\epsilon=-0.1306\)
below tower\(a_\epsilon\neq0\) requires the defect identity
channel absent
Add scalar representations—not symmetry breaking
\[ \Phi_\Delta^{\rm ext} =\Phi_\Delta^{\rm core} +a_\Delta r_\perp^{-\Delta}\mathbf1 +\sum_A\gamma_{\Delta A}\, \mathcal B_{\Delta,\widehat\Delta_A} [\widehat{\mathcal O}_A] \]
\[ \mathcal B_{\Delta,\widehat\Delta}[\widehat{\mathcal O}] =r_\perp^{\widehat\Delta-\Delta} {}_0F_1\!\left( ;\widehat\Delta+1-\frac p2; -\frac{r_\perp^2\widehat\nabla^2}{4} \right)\widehat{\mathcal O} \]
Identity: deterministic skip. New primaries: centered heads at one fixed \(\widehat\Delta_A\), with the full descendant completion.
Every term has homogeneity \(-\Delta\) and is scalar under \(SO(1,p+1)\times SO(q)\). Ising \(\mathbb Z_2\) keeps \(\widehat\sigma\) and \(\widehat\epsilon\) in separate sectors.
A symmetry-preserving prior family fixes connected moments
\[ \begin{aligned} P(o,e)\propto\exp\!\Big[ &-\tfrac12(o^2+e^2) -g_4(o^2+e^2)^2\\ &-g_6(o^6+e^6) +\kappa_{21}o^2e+\kappa_{03}e^3-he \Big] \end{aligned} \]
\(o\mapsto-o\), \(e\mapsto e\); solve \(h\) from \(\langle e\rangle=0\), then normalize both heads to unit two-point norm.
Every amplitude multiplies a complete scalar primary intertwiner: spectra, covariance and \(SO(1,p+1)\times SO(q)\) stay intact.
Exact tricritical Ising \((+0)\)
\(\kappa_3=-.226886\) gives \(\lambda_{\psi\psi\psi}=-.5445423896\), with \(\langle\psi\rangle=0\).
3d Ising · special boundary
\((\kappa_{21},\kappa_{03})=(.575915,.358899)\) fits \((\lambda_{\sigma\sigma\epsilon},\lambda_{\epsilon\epsilon\epsilon})=(.86,1.02)\).
3d Ising · ordinary boundary
Scanning \(0\leq\kappa_{21}\leq4\) alone peaks at \(1.251<1.44\). Sharing \(\kappa_{03}=.358899\) and fitting \(\kappa_{21}=.812545\) reaches \(\lambda_{\sigma\sigma D}=1.44\).
Held-out amplitude diagnostics
The same fitted prior fixes the standardized amplitude cumulants \(\kappa_{\sigma^4}\), \(\kappa_{e^4}\) and \(\kappa_{\sigma^2e^2}\); full position-dependent crossing is tested separately.
Exact curve, compact held-out test
\[ \widehat\Delta_\psi=\frac35,\qquad \frac{B_\epsilon^\psi}{B_\sigma^\psi}=-2^{3/4}, \qquad \lambda_{\psi\psi\psi}=-.5445423896 \]
Fix the leading \(\psi\)-exchange singularity, then use only two global crossing-symmetric shape coefficients:
\[ G_c(z)=\lambda^2 \big[z^{-3/5}+(1-z)^{-3/5}\big] +c_0+c_1z(1-z). \]
Connected tests reveal what is—and is not—solved
Tricritical Ising · exact
Exact BOE ratio + 3pt; compact 4pt ansatz has holdout RMSE \(4.81\times10^{-3}\).
\(O(2)\) normal · constrained
\(\lambda_{ttt}=0\) exactly; integrated \(t^4\) needs its contact completion.
Magnetic line · held-out agreement
Domain-wall crossing predicts \(C_2=1.031\) vs \(1.01(26)\): \(0.08\) times its quoted error.
Ordinary Ising · checks pass
\(\lambda_{\sigma\sigma D}=1.44\); Ward holdout differs by \(0.23\) times its quoted \(b\)-error; scalar crossing RMS \(1.98\times10^{-3}\).
Special Ising · checks pass
Two connected 3pt targets fit; scalar crossing RMS \(4.04\times10^{-4}\).
