Holomorphic supergravity in ten dimensions and anomaly cancellation

A bstract We formulate a ten-dimensional version of Kodaira-Spencer gravity on a Calabi-Yau five-fold that reproduces the classical Maurer-Cartan equation governing supersymmetric heterotic moduli. Quantising this theory’s quadratic fluctuations, we show that its one-loop partition function simplifies to products of holomorphic Ray-Singer torsions and exhibits an anomaly that factorises as in SO(32) and E 8 × E 8 supergravity. Based on this, we conjecture that this theory is the SU(5)-twisted version of ten-dimensional N = 1 supergravity coupled to Yang-Mills and show that it is related to the type I Kodaira-Spencer theory of Costello-Li via a non-local field redefinition. The counter-terms needed to cancel the anomaly and retain gauge invariance for the one-loop effective theory reconstruct the differential of a recently discovered double-extension complex. This complex has non-tensorial extension classes and its first cohomology counts the infinitesimal moduli of heterotic compactifications modulo $$ \mathcal{O}\left({\alpha}^{\prime 2}\right) $$ O α ′ 2 corrections.