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 $$ \mathcal{O}\left({\alpha}^{\prime 2}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msup> <mml:mi>α</mml:mi> <mml:mrow> <mml:mo>′</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mfenced> </mml:math> corrections.