Conformal Defects in Neural Network Field Theories

Benjamin Suzzoni; Brandon Robinson; Pietro Capuozzo

arxiv: 2512.07946 · v2 · pith:JZUE573Xnew · submitted 2025-12-08 · ✦ hep-th · cs.LG

Conformal Defects in Neural Network Field Theories

Pietro Capuozzo , Brandon Robinson , Benjamin Suzzoni This is my paper

Pith reviewed 2026-05-21 17:33 UTC · model grok-4.3

classification ✦ hep-th cs.LG

keywords Neural Network Field Theoriesconformal defectsdefect OPEscalar field theoriesconformal symmetrytoy modelscorrelation functions

0 comments

The pith

A formalism enables the construction of conformally invariant defects in neural network field theories.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces a formalism to construct conformally invariant defects inside Neural Network Field Theories. A sympathetic reader would care because defects are key features in many condensed matter and particle physics systems, and this method could make such theories more adaptable using neural networks. The work demonstrates the approach using two toy models based on scalar fields and provides a neural network-based understanding of expansions in their two-point correlation functions that resemble the defect operator product expansion.

Core claim

The central claim is that a formalism exists for building conformally invariant defects in NN-FTs. This is shown explicitly in two toy models of NN scalar field theories. Additionally, the two-point correlation functions in these models can be expanded in a way analogous to the defect OPE, with an interpretation in terms of the neural network.

What carries the argument

The formalism that specifies network architecture and priors to create conformally invariant defects in neural network field theories.

If this is right

Conformally invariant defects can be included in constructions of neural network field theories.
Two specific toy models of scalar neural network field theories successfully incorporate these defects.
Two-point correlation functions admit an expansion similar to the defect operator product expansion.
The expansion receives a direct interpretation based on the neural network structure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Such a construction might allow for numerical studies of defect conformal field theories using machine learning techniques.
It could potentially extend to other types of defects or to theories with different symmetries.
Exploring whether the formalism scales to interacting or higher-dimensional models would test its broader applicability.

Load-bearing premise

Suitable network architectures and prior distributions on the parameters can be chosen to produce a conformally invariant theory that includes well-defined defects whose correlation functions allow an expansion like the defect operator product expansion.

What would settle it

If no choice of network architecture and parameter prior in the toy scalar models produces correlation functions matching those of known conformal defects, the formalism would not hold.

Figures

Figures reproduced from arXiv: 2512.07946 by Benjamin Suzzoni, Brandon Robinson, Pietro Capuozzo.

**Figure 1.** Figure 1: A visual representation of the null cone NC ⊂ R d+1,1 . The Poincar´e section (PS) X+ = 1 and its antipodal locus X+ = −1 are shown in red; they select a unique representative from each projective orbit X ∼ λX within each section. An example of such a null ray is shown in black. As mentioned above, the generators of the conformal symmetries on the PS are induced by the Lorentz symmetry generators of the e… view at source ↗

read the original abstract

Neural Network Field Theories (NN-FTs) represent a novel construction of arbitrary field theories, including those of conformal fields, through the specification of the network architecture and prior distribution for the network parameters. In this work, we present a formalism for the construction of conformally invariant defects in these NN-FTs. We demonstrate this new formalism in two toy models of NN scalar field theories. We develop an NN interpretation of an expansion akin to the defect OPE in two-point correlation functions in these models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper sets up a formalism to add conformal defects to neural network field theories, shows it in two scalar toy models, and gives a network reading of the defect OPE, but the enforcement of symmetry and generality stay thin.

read the letter

The main takeaway is that the authors extend neural network field theories to include conformally invariant defects and demonstrate the idea in two simple scalar cases while linking two-point functions to an OPE-like expansion interpreted through the network parameters. This is new in the NN-FT literature and fills a gap by treating defects as part of the construction rather than an add-on. The toy-model checks are straightforward and show that the correlation functions can be arranged to look like those expected from a defect OPE. That part is useful as a proof of concept for people who want to simulate defects without writing down a full Lagrangian. The approach stays honest about working from network architecture and priors instead of claiming a derivation from first principles. On the downside, the construction appears to rest on selecting architectures and priors that happen to preserve conformal symmetry and produce well-defined defects, yet the paper does not spell out a general recipe or prove that the symmetry holds beyond the examples. The OPE interpretation is presented as an analogy drawn from the network, but without side-by-side comparisons to standard CFT results or checks on higher-point functions it is hard to judge how robust the claim is. The work is aimed at researchers already following NN-FT papers or those who model defects in conformal theories via computational methods. A reader looking for new tools to generate defect correlators would find the formalism and the two examples worth examining. It is worth sending to peer review so referees can press on the symmetry enforcement and see whether the method extends past scalars.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a formalism for constructing conformally invariant defects inside Neural Network Field Theories (NN-FTs) by appropriate choice of network architecture and parameter prior. The construction is illustrated on two toy models of NN scalar field theories, and the authors supply an NN-based reading of an expansion in two-point functions that parallels the defect OPE.

Significance. If the explicit construction succeeds in enforcing conformal invariance while preserving the NN-FT structure, the work would furnish a new, potentially tunable route to defect CFTs. The toy-model demonstrations and the OPE-like expansion constitute concrete, falsifiable steps that could be checked numerically or analytically; these strengths would elevate the paper above purely conceptual proposals.

major comments (2)

[§3] §3 (first toy model): the claim that the chosen architecture and prior produce a conformally invariant defect theory requires an explicit verification that the two-point function satisfies the appropriate conformal Ward identities; the current presentation only states the result without showing the relevant correlation-function calculation or the symmetry-enforcing constraint on the network weights.
[§4] §4 (OPE interpretation): the mapping of the two-point function expansion onto a defect OPE is presented at the level of formal analogy; a concrete check that the extracted coefficients satisfy the expected fusion rules or crossing relations of the underlying CFT is needed to make the interpretation load-bearing rather than suggestive.

minor comments (2)

The abstract and introduction use the phrase 'conformally invariant defects' without a preliminary definition of what conformal invariance means for a defect in the NN-FT setting; a short paragraph recalling the relevant Ward identities would improve readability.
Notation for the network parameters and the prior distribution is introduced piecemeal; a consolidated table or appendix listing all symbols and their ranges would aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and indicate the revisions we will make to strengthen the presentation.

read point-by-point responses

Referee: [§3] §3 (first toy model): the claim that the chosen architecture and prior produce a conformally invariant defect theory requires an explicit verification that the two-point function satisfies the appropriate conformal Ward identities; the current presentation only states the result without showing the relevant correlation-function calculation or the symmetry-enforcing constraint on the network weights.

Authors: We agree that the current presentation would benefit from an explicit verification. In the revised manuscript we will add the calculation of the two-point function in the first toy model and show that it satisfies the relevant conformal Ward identities. We will also detail the constraints on the network weights that follow from the architecture and prior and how these enforce the invariance. revision: yes
Referee: [§4] §4 (OPE interpretation): the mapping of the two-point function expansion onto a defect OPE is presented at the level of formal analogy; a concrete check that the extracted coefficients satisfy the expected fusion rules or crossing relations of the underlying CFT is needed to make the interpretation load-bearing rather than suggestive.

Authors: We acknowledge that the discussion remains at the level of formal analogy in the present version. In the revision we will supply a concrete check for the toy models, verifying that the coefficients obtained from the two-point function expansion obey the expected fusion rules or crossing relations of the underlying defect CFT. revision: yes

Circularity Check

0 steps flagged

No circularity: construction is self-contained

full rationale

The paper introduces a formalism for conformally invariant defects in NN-FTs by specifying network architecture and parameter priors, then demonstrates it in two toy scalar models and interprets a defect OPE-like expansion. No load-bearing step reduces by the paper's own equations or self-citation to its inputs; the central claims rest on explicit construction choices rather than fitted parameters renamed as predictions or uniqueness theorems imported from prior author work. The derivation chain is independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The central construction rests on the background claim that NN-FTs can realize arbitrary field theories, including conformal ones, via architecture and prior choices.

axioms (1)

domain assumption Neural Network Field Theories can represent arbitrary field theories, including conformal ones, by specification of network architecture and parameter prior.
This is the foundational premise stated in the abstract on which the defect construction is built.

pith-pipeline@v0.9.0 · 5601 in / 1324 out tokens · 31099 ms · 2026-05-21T17:33:51.856377+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We present a formalism for the construction of conformally invariant defects in these NN-FTs... develop an NN interpretation of an expansion akin to the defect OPE
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Φ_α(X)=(X·Θ)^α ... binomial expansion ... defect blocks bfbα,s satisfying SO(p+1,1)×SO(q)_N Casimir equations

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Anomalies in Neural Network Field Theory
hep-th 2026-05 unverdicted novelty 7.0

Derives Schwinger-Dyson equations and Ward identities in NN-FT to study anomalies in QFTs via a conserved parameter-space current, yielding a new perspective on symmetries.
Topological Effects in Neural Network Field Theory
hep-th 2026-04 unverdicted novelty 7.0

Neural network field theory extended with discrete topological labels recovers the BKT transition and bosonic string T-duality.
Optimal Architecture and Fundamental Bounds in Neural Network Field Theory
hep-th 2026-04 unverdicted novelty 6.0

α=0 architecture in NNFT minimizes finite-width variance, removes IR corrections, and sets a fundamental SNR bound for correlation functions in scalar field theory.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · cited by 3 Pith papers · 5 internal anchors

[1]

Chalabi, C

A. Chalabi, C. P. Herzog, K. Ray, B. Robinson, J. Sisti and A. Stergiou,Boundaries in free higher derivative conformal field theories,JHEP04(2023) 098, [2211.14335]

work page arXiv 2023
[2]

C. P. Herzog and Y. Zhou,An interacting, higher derivative, boundary conformal field theory,JHEP12(2024) 133, [2409.11072]

work page arXiv 2024
[3]

Halverson, A

J. Halverson, A. Maiti and K. Stoner,Neural Networks and Quantum Field Theory,Mach. Learn. Sci. Tech.2(2021) 035002, [2008.08601]

work page arXiv 2021
[4]

Building Quantum Field Theories Out of Neurons,

J. Halverson,Building Quantum Field Theories Out of Neurons,2112.04527

work page arXiv
[5]

Neural network field theories: non-Gaussianity, actions, and locality,

M. Demirtas, J. Halverson, A. Maiti, M. D. Schwartz and K. Stoner,Neural network field theories: non-Gaussianity, actions, and locality,Mach. Learn. Sci. Tech.5(2024) 015002, [2307.03223]

work page arXiv 2024
[6]

Maiti, K

A. Maiti, K. Stoner and J. Halverson,Symmetry-via-Duality: Invariant Neural Network Densities from Parameter-Space Correlators,2106.00694

work page arXiv
[7]

Conformal Fields from Neural Networks,

J. Halverson, J. Naskar and J. Tian,Conformal Fields from Neural Networks,2409.12222

work page arXiv
[8]

Boundary and Defect CFT: Open Problems and Applications

N. Andrei et al.,Boundary and Defect CFT: Open Problems and Applications,J. Phys. A53 (2020) 453002, [1810.05697]

work page internal anchor Pith review Pith/arXiv arXiv 2020
[9]

Defects in conformal field theory

M. Bill` o, V. Gon¸ calves, E. Lauria and M. Meineri,Defects in conformal field theory,JHEP 04(2016) 091, [1601.02883]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[10]

Radial coordinates for defect CFTs

E. Lauria, M. Meineri and E. Trevisani,Radial coordinates for defect CFTs,JHEP11(2018) 148, [1712.07668]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[11]

Fefferman, S

C. Fefferman, S. Mitter and H. Narayanan,Testing the manifold hypothesis,Journal of the American Mathematical Society29(2016) 983–1049

work page 2016
[12]

A. S. Wightman and L. Garding,Fields as operator-valued distributions in relativistic quantum theory,Arkiv Fys.Vol: 28(12, 1964) . – 31 –

work page 1964
[13]

Osterwalder and R

K. Osterwalder and R. Schrader,Axioms for Euclidean Green’s functions,Commun. Math. Phys.31(1973) 83–112

work page 1973
[14]

P. A. M. Dirac,Wave equations in conformal space,Annals Math.37(1936) 429–442

work page 1936
[15]

F. A. Dolan and H. Osborn,Conformal four point functions and the operator product expansion,Nucl. Phys. B599(2001) 459–496, [hep-th/0011040]

work page internal anchor Pith review Pith/arXiv arXiv 2001
[16]

Conformal constraints on defects

A. Gadde,Conformal constraints on defects,JHEP01(2020) 038, [1602.06354]

work page internal anchor Pith review Pith/arXiv arXiv 2020
[17]

Nishioka, Y

T. Nishioka, Y. Okuyama and S. Shimamori,Method of images in defect conformal field theories,Phys. Rev. D106(2022) L081701, [2205.05370]

work page arXiv 2022
[18]

Lauria, P

E. Lauria, P. Liendo, B. C. Van Rees and X. Zhao,Line and surface defects for the free scalar field,JHEP01(2021) 060, [2005.02413]

work page arXiv 2021
[19]

Bianchi, A

L. Bianchi, A. Chalabi, V. Proch´ azka, B. Robinson and J. Sisti,Monodromy defects in free field theories,JHEP08(2021) 013, [2104.01220]

work page arXiv 2021
[20]

Frank, J

S. Frank, J. Halverson, A. Maiti and F. Ruehle,Fermions and Supersymmetry in Neural Network Field Theories,2511.16741

work page arXiv
[21]

Chalabi, C

A. Chalabi, C. P. Herzog, A. O’Bannon, B. Robinson and J. Sisti,Weyl anomalies of four dimensional conformal boundaries and defects,JHEP02(2022) 166, [2111.14713]

work page arXiv 2022
[22]

Robinson,Virasoro Symmetry in Neural Network Field Theories,2512.24420

B. Robinson,Virasoro Symmetry in Neural Network Field Theories,2512.24420. – 32 –

work page arXiv

[1] [1]

Chalabi, C

A. Chalabi, C. P. Herzog, K. Ray, B. Robinson, J. Sisti and A. Stergiou,Boundaries in free higher derivative conformal field theories,JHEP04(2023) 098, [2211.14335]

work page arXiv 2023

[2] [2]

C. P. Herzog and Y. Zhou,An interacting, higher derivative, boundary conformal field theory,JHEP12(2024) 133, [2409.11072]

work page arXiv 2024

[3] [3]

Halverson, A

J. Halverson, A. Maiti and K. Stoner,Neural Networks and Quantum Field Theory,Mach. Learn. Sci. Tech.2(2021) 035002, [2008.08601]

work page arXiv 2021

[4] [4]

Building Quantum Field Theories Out of Neurons,

J. Halverson,Building Quantum Field Theories Out of Neurons,2112.04527

work page arXiv

[5] [5]

Neural network field theories: non-Gaussianity, actions, and locality,

M. Demirtas, J. Halverson, A. Maiti, M. D. Schwartz and K. Stoner,Neural network field theories: non-Gaussianity, actions, and locality,Mach. Learn. Sci. Tech.5(2024) 015002, [2307.03223]

work page arXiv 2024

[6] [6]

Maiti, K

A. Maiti, K. Stoner and J. Halverson,Symmetry-via-Duality: Invariant Neural Network Densities from Parameter-Space Correlators,2106.00694

work page arXiv

[7] [7]

Conformal Fields from Neural Networks,

J. Halverson, J. Naskar and J. Tian,Conformal Fields from Neural Networks,2409.12222

work page arXiv

[8] [8]

Boundary and Defect CFT: Open Problems and Applications

N. Andrei et al.,Boundary and Defect CFT: Open Problems and Applications,J. Phys. A53 (2020) 453002, [1810.05697]

work page internal anchor Pith review Pith/arXiv arXiv 2020

[9] [9]

Defects in conformal field theory

M. Bill` o, V. Gon¸ calves, E. Lauria and M. Meineri,Defects in conformal field theory,JHEP 04(2016) 091, [1601.02883]

work page internal anchor Pith review Pith/arXiv arXiv 2016

[10] [10]

Radial coordinates for defect CFTs

E. Lauria, M. Meineri and E. Trevisani,Radial coordinates for defect CFTs,JHEP11(2018) 148, [1712.07668]

work page internal anchor Pith review Pith/arXiv arXiv 2018

[11] [11]

Fefferman, S

C. Fefferman, S. Mitter and H. Narayanan,Testing the manifold hypothesis,Journal of the American Mathematical Society29(2016) 983–1049

work page 2016

[12] [12]

A. S. Wightman and L. Garding,Fields as operator-valued distributions in relativistic quantum theory,Arkiv Fys.Vol: 28(12, 1964) . – 31 –

work page 1964

[13] [13]

Osterwalder and R

K. Osterwalder and R. Schrader,Axioms for Euclidean Green’s functions,Commun. Math. Phys.31(1973) 83–112

work page 1973

[14] [14]

P. A. M. Dirac,Wave equations in conformal space,Annals Math.37(1936) 429–442

work page 1936

[15] [15]

F. A. Dolan and H. Osborn,Conformal four point functions and the operator product expansion,Nucl. Phys. B599(2001) 459–496, [hep-th/0011040]

work page internal anchor Pith review Pith/arXiv arXiv 2001

[16] [16]

Conformal constraints on defects

A. Gadde,Conformal constraints on defects,JHEP01(2020) 038, [1602.06354]

work page internal anchor Pith review Pith/arXiv arXiv 2020

[17] [17]

Nishioka, Y

T. Nishioka, Y. Okuyama and S. Shimamori,Method of images in defect conformal field theories,Phys. Rev. D106(2022) L081701, [2205.05370]

work page arXiv 2022

[18] [18]

Lauria, P

E. Lauria, P. Liendo, B. C. Van Rees and X. Zhao,Line and surface defects for the free scalar field,JHEP01(2021) 060, [2005.02413]

work page arXiv 2021

[19] [19]

Bianchi, A

L. Bianchi, A. Chalabi, V. Proch´ azka, B. Robinson and J. Sisti,Monodromy defects in free field theories,JHEP08(2021) 013, [2104.01220]

work page arXiv 2021

[20] [20]

Frank, J

S. Frank, J. Halverson, A. Maiti and F. Ruehle,Fermions and Supersymmetry in Neural Network Field Theories,2511.16741

work page arXiv

[21] [21]

Chalabi, C

A. Chalabi, C. P. Herzog, A. O’Bannon, B. Robinson and J. Sisti,Weyl anomalies of four dimensional conformal boundaries and defects,JHEP02(2022) 166, [2111.14713]

work page arXiv 2022

[22] [22]

Robinson,Virasoro Symmetry in Neural Network Field Theories,2512.24420

B. Robinson,Virasoro Symmetry in Neural Network Field Theories,2512.24420. – 32 –

work page arXiv