α′ corrections to self-dual gravitational instantons

A bstract We study the α ′ corrections to self-dual gravitational instantons in the context of the four-dimensional Cano-Ruipérez action, which can be obtained by the compactification of the Bergshoeff-de Roo heterotic string effective action on T⁶ 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-Ruipérez 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 α ′ -corrected Eguchi-Hanson instanton showing that the total action receives no corrections to first order in α ′ . We also show that, at zeroth order in α ′ , 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 α ′ corrections to these solutions.