arXiv subjects

Sid Maibach

Publications and source records attributed to Sid Maibach.

4 recordsLinked to original sources

arXiv2026

Universality of the conformal anomaly

We prove a universal property of local real one-dimensional modular functors, which may be thought of as central extensions of the sewing operation on the (infinite-dimensional) Segal moduli spaces. Such modular functors are characterized by one real parameter: their central charge: Our result is an analogue of ``Mumford's theorem,'' that appeared in Segal's monograph [Seg88, Seg04] in the complex case. Algebraically, the modular functors 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 guiding example is the real determinant line bundle encoding the trace anomaly of conformal field theories (CFT) and the restriction function of Schramm-Loewner evolution (SLE) loop measures. Our result thus gives a mathematical explanation for the appearance of the central charge in CFT and SLE, and proves the conjectured uniqueness of Brownian loop measureas the canonical restriction function for SLE.

math-ph↗

arXiv2026

Complex deformations of the circle: Group cohomology and Virasoro uniformization

We approach the question of complexification of the diffeomorphism group of the circle by considering real-analytic maps from the circle into the punctured complex plane with winding number +1. Such complex deformations form an infinite-dimensional manifold with partially defined inversion and composition operations, smooth in the sense of Fr\"olicher structures, and with Lie algebra relations at the identity given by the Witt algebra. With applications to conformal field theory in mind, we compute the second group cohomology group with real coefficients, finding cocycles extending the Bott-Thurston cocycle related to the Gelf'and-Fuks cocycle of the Virasoro algebra, and a natural relative cocycle combining the rotation number and conformal radius of a complex deformation. Complex deformations act naturally on the (infinite-dimensional) Segal moduli spaces of Riemann surfaces with analytically parametrized boundary components. These actions equip said moduli spaces with smooth Fr\"olicher structures. We prove a Virasoro uniformization theorem: the tangent spaces of the Segal moduli spaces are spanned by vector fields induced by the Witt algebra. Finally, we relate the actions of complex deformations to Fenchel-Nielsen coordinates and Schiffer variation on finite-dimensional moduli spaces of hyperbolic surfaces with one marked point on each boundary component.

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arXiv2024

Two-loop Loewner potentials

We study a generalization of the Schramm-Loewner evolution loop measure to pairs of non-intersecting Jordan curves on the Riemann sphere. We also introduce four equivalent definitions for a two-loop Loewner potential: respectively expressing it in terms of normalized Brownian loop measure, zeta-regularized determinants of the Laplacian, an integral formula generalizing universal Liouville action, and Loewner-Kufarev energy of a foliation. Moreover, we prove that the potential is finite if and only if both loops are Weil-Petersson quasicircles, that it is an Onsager-Machlup functional for the two-loop SLE, and a variational formula involving Schwarzian derivatives. Addressing the question of minimization of the two-loop Loewner potential, we find that any such minimizers must be pairs of circles. However, the potential is not bounded, diverging to negative infinity as the circles move away from each other and to positive infinity as the circles merge, thus preventing a definition of two-loop Loewner energy for the prospective large deviations theory for the two-loop SLE. To remedy the divergence, we study a way of generalizing the two-loop Loewner potential by taking into account how conformal field theory (CFT) partition functions depend on the modulus of the annulus between the loops. This generalization is motivated by the correspondence between SLE and CFT, and it also emerges from the geometry of the real determinant line bundle as introduced by Kontsevich and Suhov.

math.CV↗

arXiv2024

From the Conformal Anomaly to the Virasoro Algebra

The conformal anomaly and the Virasoro algebra are fundamental aspects of 2D 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 parametrized 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 Andre 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 K\"ahler 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.

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