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 $\mathbb{R}_+$-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 $\mathbb{R}_+$-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.