$\alpha'$ corrections to self-dual gravitational instantons

TL;DR

This paper investigates the effects of $oldsymbol{rac{ ext{alpha}'}{ ext{corrections}}}$ on self-dual gravitational instantons, showing that the metric remains unchanged while dilaton and axion fields are corrected, with explicit results for specific instantons.

Contribution

It provides the first explicit analysis of $oldsymbol{rac{ ext{alpha}'}{ ext{corrections}}}$ to self-dual gravitational instantons within the Cano--Ruipérez framework, including boundary terms and corrected solutions.

Findings

The metric of self-dual spaces does not receive $oldsymbol{rac{ ext{alpha}'}{ ext{corrections}}}$.

Dilaton and axion fields are corrected due to couplings to topological densities.

The $oldsymbol{rac{ ext{alpha}'}{ ext{corrections}}}$ do not alter the Euclidean action of the Eguchi--Hanson instanton at first order.

Abstract

We study the $\alpha'$ corrections to self-dual gravitational instantons in the context of the four-dimensional Cano--Ruip\'erez action, which can be obtained by the compactification of the Bergshoeff--de Roo heterotic string effective action on $\mathbb{T}^{6}$ followed by a truncation and a field redefinition. We show that the metric of spaces of self-dual curvature does not receive any corrections, but their (initially trivial) dilaton and axion fields do, owing to their couplings to Gauss--Bonnet and Pontrjagin densities. We find the generic form of the corrections of the dilaton and axion fields for the Gibbons--Hawking multi-instanton solutions and their explicit form for the particular cases of the Euclidean Taub--NUT and Eguchi--Hanson spaces. We construct the boundary terms required to define a well-posed Dirichlet variational principle in the Euclidean Cano--Ruip\'erez theory, including the contributions associated with the Gauss--Bonnet and Pontrjagin terms. The boundary terms are normalized for asymptotically-locally-Euclidean solutions, and we evaluate with them the Euclidean action of the $\alpha'$-corrected Eguchi--Hanson instanton showing that the total action receives no corrections to first order in $\alpha'$. We also show that, at zeroth order in $\alpha'$, one can construct Euclidean solutions similar to the string theory D-instanton with non-trivial dilaton and axion on the background of a self-dual purely gravitational instanton which remains unmodified. We also compute the $\alpha'$ corrections to these solutions.