Constraints on the topology of axionic wormhole solutions

Abstract

The authors examine the elements entering into the Giddings and Strominger (1988) construction of axion induced topology change. Euclidean instantons with non-vanishing axionic charge must have manifolds with non-trivial third cohomology. Furthermore, boundary conditions on the axion field and extrinsic curvature of Euclidean instantons mediating topology change restrict the topology of any compact boundary 3-manifold to one that admits positive scalar curvature. These compact 3-manifold are S3, S3/ Gamma where Gamma is a finite group, S2*S1 or a connected sum of these manifolds. For the case of zero axion charge this class is enlarged to include the non-orientable manifolds whose double covers are manifolds in the previous class and the corresponding connected sums and if the axiom field vanishes identically on the boundary, one of the ten flat 3-manifolds. These allowed 3-manifolds form a very small subset of all compact 3-manifolds. Therefore, in general only a limited class of topology changes can be mediated by axionic Euclidean instantons. Finally, explicit axionic Euclidean instanton solutions of Giddings and Strominger type with topology Sigma *R and conformal metric ansatz where Sigma is a compact 3-manifold in this limited class are constructed. They find that the conformal metric ansatz implies the Sigma must be a homogeneous space. This further restricts the class of manifolds to be geometrically Sigma =S3/ Gamma where Gamma is a finite group. They construct solutions for these 3-manifolds and discuss their asymptotic behaviour.