aalto1 untyped-item.component.html
From the conformal anomaly to the Virasoro algebra
Loading...
Access rights
openAccess
CC BY
CC BY
Creative Commons license
Except where otherwised noted, this item's license is described as openAccess
publishedVersion
URL
Journal Title
Journal ISSN
Volume Title
A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
This publication is imported from Aalto University research portal.
View publication in the Research portal (opens in new window)
View/Open full text file from the Research portal (opens in new window)
View publication in the Research portal (opens in new window)
View/Open full text file from the Research portal (opens in new window)
Unless otherwise stated, all rights belong to the author. You may download, display and print this publication for Your own personal use. Commercial use is prohibited.
Authors
Date
Major/Subject
Mcode
Degree programme
Language
en
Pages
45
Series
Proceedings of the London Mathematical Society, Volume 130, issue 4, pp. 1-45
Abstract
The conformal anomaly and the Virasoro algebra are fundamental aspects of two-dimensional conformal field theory and conformally covariant models in planar random geometry. In this article, we explicitly derive the Virasoro algebra from an axiomatization of the conformal anomaly in terms of real determinant lines, one-dimensional vector spaces associated to Riemann surfaces with analytically parameterized boundary components. Here, analytical orientation-preserving diffeomorphisms and deformations of the circle naturally act on the boundary components. We introduce a sewing operation on the real determinant lines over the semigroup of annuli, which then induces central extensions of the diffeomorphism group, as well as of the complex deformations. Our main theorem shows that on the one hand, the cocycle associated to the central extension of diffeomorphisms is trivial, while on the other hand, the Lie algebra cocycle associated to the central extension of complex deformations is nontrivial, yielding the imaginary part of the Gel'fand–Fuks cocycle. We thus answer a question, partly negatively and partly affirmatively, discussed by André Henriques and Dylan Thurston in 2011. The proof uses concrete computations, which we aim to be accessible to a wide audience. We also show an explicit relation to loop Loewner energy, anticipating the real determinant lines to be pertinent to locally conformally covariant (Malliavin–Kontsevich–Suhov) measures on curves and loops, as well as to Kähler geometry and geometric quantization of moduli spaces of Riemann surfaces. Inherently, the conformal anomaly and real determinant line bundles are expected to be universal, following a classification of modular functors.
Description
Publisher Copyright: © 2025 The Author(s). Proceedings of the London Mathematical Society is copyright © London Mathematical Society.
Keywords
Other note
Citation
Maibach, S & Peltola, E 2025, 'From the conformal anomaly to the Virasoro algebra', Proceedings of the London Mathematical Society, vol. 130, no. 4, e70040, pp. 1-45. https://doi.org/10.1112/plms.70040