Anomalies in string theory with $D$-branes

Abstract

Introduction.This paper is devoted to studying global anomalies in the worldsheet path integral of Type II superstring theory in the presence of D-branes.We will not consider orientifolds (or Type I superstrings) and so our string worldsheets will be oriented Riemann surfaces, mapped to a spacetime manifold Y, which is endowed with a spin structure since the model contains fermions.The first case to consider is that of closed worldsheets, without boundary.In this case, global anomalies cancel, and the worldsheet measure is globally well-defined, given that Y is spin.This is the content of Corollary 4.7 of the present paper; for Y = M 10 one has (Theorem 4.8) the further result, related to conformal invariance, that the global anomaly vanishes separately for left-and right-moving degrees of freedom.By contrast, for heterotic strings, global anomaly cancellation gives a restriction that is not evident from the point of view of the low energy effective field theory [W2,F2].But we will find that Type II global worldsheet anomalies give some novel results when D-branes are present.Thus, we assume the existence of an oriented submanifold Q of spacetime on which strings can end.We then consider a string worldsheet consisting of an oriented surface X that is mapped to Y with <9£ mapped to Q. On Q, there is a field A that is conventionally regarded as a U(l) gauge field, though as we will explain this is not the correct interpretation globally.Another important part of the story is the Neveu-Schwarz i?-field, which propagates in the bulk of spacetime.Assume for simplicity that B = 0.In this case, the worldsheet measure contains two interesting factors which we write as follows: *