Holomorphic supergravity in ten dimensions and anomaly cancellation

Abstract

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 $E_8\times E_8$ 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 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 order $α'^2$ corrections.

Figures (2)

• Figure 1: The double complex encoding the BV description of the complex $(\Omega^{0,\bullet}(Q), \bar{D})$ which computes the infinitesimal moduli space of the Hull--Strominger system (plus spurious degrees of freedom) on a complex five-manifold. The original complex is promoted to a double complex to account for gauge symmetries.
• Figure 2: The double complex obtained by starting from Figure \ref{['fig:10d-Q-complex-BRST']} and adding a further diagonal. The ghost number of each space shown on the left. This BV complex is the ten-dimensional version of the complex of that appeared in Ashmore:2023vji and, crucially, admits a symplectic pairing which can be used to define an action.