Six-dimensional supergravity : consistency conditions and realizations in string theory

Abstract

We consider the question of which low-energy effective theories with gravity can be realized as string compactifications. In order to make progress on this question, we consider six-dimensional, A = 1 supersymmetric theories with gauge groups, chiral matter and gravity. Stringent constraints imposed by anomaly cancellation make the analysis of large classes of effective theories and string compactifications tractable. We prove that there are only finitely many combinations of non-abelian gauge group and matter that can appear in these theories if the number of tensor multiplets T </= 8. For T >/= 9, we find infinite families of effective theories with anomaly cancellation and positive kinetic terms. We show that anomaly cancellation defines an integral lattice associated with any low-energy theory. F-theory compactified on elliptic Calabi-Yau 3-folds gives a large class of string vacua. For the case of one tensor multiplet, we find an explicit map between the low-energy anomaly data and divisors in the base of an F-theory compactification. In the case of more tensors, a low-energy theory is realized by an F-theory compactification only if the low-energy lattice embeds into the second homology lattice of the base. We find examples of apparently consistent low-energy theories which cannot be realized in F-theory. We construct a subset of 6D, N = 1 vacua by compactifying the type I/heterotic string on a K3 surface where the gauge bundle is assumed to be a sum of U(1) bundles. The gauge group in these vacua is a product of unitary groups and chiral matter. We can construct a lattice from the data describing the low-energy theory, distinct from the lattice determined by the anomalies. A given low-energy theory is realized in this landscape if and only if the corresponding lattice embeds into the K3 lattice [Gamma]3, 19.