Rigorous Construction of Conformal Field Theories

Rigorous Construction of Conformal Field Theories

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.

A conformal field theory has correlation functions transforming covariantly under conformal maps. A rigorous path integral must define those correlations or measures rather than only formal integral symbols. Bootstrap consistency constrains spectra and operator-product coefficients through compatibility, including crossing relations.

Catalogue wording retained for traceability.

How can rigorous examples of conformal field theories be constructed, including rigorous path integrals and justification of the bootstrap hypothesis?

Scope and source qualification.

This programme contains several construction and justification questions. The source does not specify one target dimension, spectrum or universal existence claim.

Short English statement

Construct rigorous conformal field theories and justify their path-integral and bootstrap descriptions.

Research attempt

Model, date and investigated scope.

OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Derived normalized finite Gaussian vertex correlations and their neutral-charge scaling, with an explicit limit/bootstrapping gap.

Sources and literature check.

Read the full supplied definition, catalogue wording, source qualifications and listed references for 30005927. No fresh status or novelty search was performed. The argument below proves its auxiliary statements directly and does not import an unverified external theorem.

A cutoff Gaussian construction test.

Let $X=(X_1,\ldots,X_n)$ be a real centered Gaussian vector with a positive semidefinite covariance matrix $C$. Define normalized imaginary vertex factors $V_j=\exp(i\alpha_jX_j+\alpha_j^2 C_{jj}/2)$, for real $\alpha_j$. Completing the square in the Gaussian characteristic function gives \[ \mathbb E\prod_jV_j =\exp\left(-\sum_{i<j}\alpha_i\alpha_j C_{ij}\right). \] Singular covariance matrices follow by adding $\epsilon I$ and taking the explicit formula's limit. This defines every finite-cutoff correlation without a formal path integral.

A covariance and scaling check.

If a proposed two-dimensional cutoff has off-diagonal limit $C_{ij}=-\log|x_i-x_j|$ up to controlled background terms, its displayed factor becomes $\prod_{i<j}|x_i-x_j|^{\alpha_i\alpha_j}$. For neutral charges $\sum_i\alpha_i=0$, dilation by $\lambda>0$ gives the factor \[ \lambda^{\sum_{i<j}\alpha_i\alpha_j} =\lambda^{-\frac12\sum_i\alpha_i^2}. \] The algebra thus passes a scaling test with individual dimensions $\alpha_i^2/2$. It is also invariant under simultaneous permutations of the insertions and their charges.

Where bootstrap is still missing.

The off-diagonal logarithmic formula is a conditional covariance ansatz here, not a proved measure construction on every domain. Permutation symmetry of finitely many correlations does not establish convergence of operator-product expansions, completeness of intermediate states or crossing in a conformal-block decomposition. The first gap is a compatible ultraviolet and zero-mode limit producing the required positive Hilbert-space or probabilistic structure and convergent expansions. The broad source construction and bootstrap programme is not settled by this elementary Gaussian consistency calculation.

Verification and adversarial audit.

Checked covariance positivity assumptions, Gaussian normalization sign, charge-neutral scaling exponent and permutation symmetry versus actual bootstrap completeness. Model self-audit only; no independent human mathematical verification. Applied contribution prompt A before drafting and B afterwards. Checked covariance positivity assumptions, Gaussian normalization sign, charge-neutral scaling exponent and permutation symmetry versus actual bootstrap completeness. First remaining gap: Construct compatible cutoff limits and prove convergent operator-product/conformal-block decompositions for a specified source CFT. The inherited source status is preserved; original-target completion remains zero.

Coordinator checked the normalized Gaussian vertex expectation and neutral-charge dilation exponent. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.

Python 3 and local file tools used to read definitions and serialize manuscripts. Calculations are symbolic reasoning; no numerical experiment or theorem prover. Contribution prompt A was followed by prompt B. GPT-6 Astra with ultra reasoning drafted the research and performed its stated self-audit; any additional coordinator contribution is identified in the verification and audit. A GPT-6-based Codex coordinator assembled the manuscript and checked repository integration; its exact variant and reasoning configuration were not exposed. No human mathematical contribution, independent human proof check or formal proof certification is supplied. Complete derivations and the consulted source references are included here; private reasoning transcripts are not published.

Final status and first remaining gap.

UNRESOLVED. Construct compatible cutoff limits and prove convergent operator-product/conformal-block decompositions for a specified source CFT.

Completion Estimate

COMPLETION: 0% This is an elementary auxiliary calculation or an explicit test of a restricted strategy. No new portion of the full source target is established.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath, record 30005927 (OWR-14298373-005). Catalogue curation under CC BY 4.0; definitions and English exposition added locally. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Statistical Physics and Random Surfaces $2024$, Oberwolfach Reports, DOI 10.4171/owr/2024/25, pp. 1387--1444. Original report and mathematical context. Incorporated report material remains under the source CC BY-SA 4.0 terms unless otherwise noted. https://doi.org/10.4171/owr/2024/25.