Universality of the conformal anomaly
Abstract
The sewing operation is the fundamental algebraic structure of the (infinite-dimensional) Segal moduli spaces as the conformal cobordism category. We prove a universal property of local ℝ+-extensions of the sewing operation, characterizing them by one real parameter: their central charge. Our result is an analogue of ``Mumford's theorem,'' that appeared in Segal's monograph [Seg88, Seg04], classifying complex extensions, also known as one-dimensional modular functors. Algebraically, the real extensions are characterized by real-valued cocycles on pairs of surfaces under sewing. We identify the disk-disk cocycle as the universal Liouville action, also known as loop Loewner energy. The real determinant line bundle encodes the trace anomaly of euclidean two-dimensional conformal field theories (CFT) and the restriction function of Schramm-Loewner evolution (SLE) loop measures. It serves as the guiding example in our setup and proof strategy, providing the universal ℝ+-extension. Our results thus give a mathematical explanation for the appearance of the central charge in CFT and SLE, and prove the conjectured uniqueness of Brownian loop measures as the canonical restriction function for SLE.
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BibTeX
@misc{maibach2026universality,
title = {{Universality of the conformal anomaly}},
author = {Sid Maibach and Eveliina Peltola},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.23201},
eprint = {2608.23201},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.23201},
}
RIS
TY - UNPB
TI - Universality of the conformal anomaly
AU - Sid Maibach
AU - Eveliina Peltola
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2608.23201
UR - https://arxiv.org/abs/2608.23201
AB - The sewing operation is the fundamental algebraic structure of the (infinite-dimensional) Segal moduli spaces as the conformal cobordism category. We prove a universal property of local ℝ_+-extensions of the sewing operation, characterizing them by one real parameter: their central charge. Our result is an analogue of ``Mumford's theorem,'' that appeared in Segal's monograph [Seg88, Seg04], classifying complex extensions, also known as one-dimensional modular functors. Algebraically, the real extensions are characterized by real-valued cocycles on pairs of surfaces under sewing. We identify the disk-disk cocycle as the universal Liouville action, also known as loop Loewner energy. The real determinant line bundle encodes the trace anomaly of euclidean two-dimensional conformal field theories (CFT) and the restriction function of Schramm-Loewner evolution (SLE) loop measures. It serves as the guiding example in our setup and proof strategy, providing the universal ℝ_+-extension. Our results thus give a mathematical explanation for the appearance of the central charge in CFT and SLE, and prove the conjectured uniqueness of Brownian loop measures as the canonical restriction function for SLE.
ER -