Anomalies in string theory with D-branes
The authors study global anomalies in the world-sheet path integral of Type II superstring theory in the presence of D-branes. A string worldsheet consisting of an oriented surface \(\Sigma\) that is mapped to the spacetime \(Y\) with \(\partial\Sigma\) mapped to the D-brane \(Q\) is considered. There are a field \(\mathbf A\) on the \(Q\) and a Neveu-Schwarz \(\mathbf B\)-field propagating in the bulk of spacetime. In the paper the worldsheet measure is considered which contains two factors : Pfaffian of the worldsheet Dirac operator and the holonomy of a field \(A\). The main result is investigation of the anomaly in the Pfaffian : (a) the Pfaffian is not a function, but rather a section of a line bundle over the space of some parameters; (b) the anomaly is the obstruction to the existence of a global flat section of unit norm. The interpretation of the \(\mathbf A\)-field (for \(\mathbf B=0\) case) as a \(\text{Spin}^c\) connection rather than an ordinary U(1) gauge field is given. If \(\mathbf B\neq 0\) a more complicated interpretation of the \(\mathbf A\)-field is found. In this case D-brane charge takes values in a twisted form of \(K\)-theory.
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