Higher-Order Corrections to Non-Compact Calabi-Yau Manifolds in String Theory

At the leading order, the low-energy effective field equations in string theory admit solutions of the form of products of Minkowski spacetime and a Ricci-flat Calabi-Yau space. The equations of motion receive corrections at higher orders in α′, which imply that the Ricci-flat Calabi-Yau space is modified. In an appropriate choice of scheme, the Calabi-Yau space remains Kähler, but is no longer Ricci-flat. We discuss the nature of these corrections at order α′, and consider the deformations of all the known cohomogeneity one non-compact Kähler metrics in six and eight dimensions. We do this by deriving the first-order equations associated with the modified Killing-spinor conditions, and we thereby obtain the modified supersymmetric solutions. We also give a detailed discussion of the boundary terms for the Euler complex in six and eight dimensions, and apply the results to all the cohomogeneity one examples. 1 Research supported in part by DOE grant DE-FG03-95ER40917 2 Research supported in part by the EC under TMR contract HPRN-CT-2000-00131 and by PPARC under SPG grant PPA/G/S/1998/00613.