Universality of the conformal anomaly

Abstract

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.