M-theory compactifications, G₂-manifolds and anomalies

This diploma thesis has three major objectives. Firstly, we give an elementary introduction to M-theory compactifications, which are obtained from an analysis of its low-energy effective theory, eleven-dimensional supergravity. In particular, we show how the requirement of N=1 supersymmetry in four dimensions leads to compactifications on G_2-manifolds. We also examine the Freund-Rubin solution as well as the M2- and M5-brane. Secondly, we review the construction of realistic theories in four dimensions from compactifications on G_2-manifolds. It turns out that this can only be achieved if the manifolds are allowed to carry singularities of various kinds. Thirdly, we are interested in the concept of anomalies in the framework of M-theory. We present some basic material on anomalies and examine three cases where anomalies play a prominent role in M-theory. We review M-theory on R^10 x S^1/Z_2 where anomalies are a major ingredient leading to the duality between M-theory and the E_8 x E_8 heterotic string. A detailed calculation of the tangent and normal bundle anomaly in the case of the M5-brane is also included. It is known that in this case the normal bundle anomaly can only be cancelled if the topological term of eleven-dimensional supergravity is modified in a suitable way. Finally, we present a new mechanism to cancel anomalies which are present if M-theory is compactified on G_2-manifolds carrying singularities of codimension seven. In order to establish local anomaly cancellation we once again have to modify the topological term of eleven-dimensional supergravity as well as the Green-Schwarz term.