Trace and chiral anomalies in string and ordinary field theory from Feynman diagrams for nonlinear sigma models

We write general one-loop anomalies of string field theory as path integrals on a torus for the corresponding nonlinear sigma model. This extends the work of Alvarez-Gaum\'e and Witten from quantum mechanics to two dimensions. Higher world-volume loops contribute in general to nontopological anomalies and a formalism to compute these is developed. We claim that (i) for general anomalies one should not use the propagator widely used in string theory but rather the one obtained by generalization from quantum mechanics, but (ii) for chiral anomalies both propagators give the same result. As a check of this claim in a simpler model we compute trace anomalies in quantum mechanics. The propagator with a center-of-mass zero mode indeed does not give the correct result for the trace anomaly while the propagator for fluctuations $q^i (\tau)$ satisfying $q^i (\tau = -1) = q^i (\tau = 0) = 0$ yields in $d=2$ and $d=4$ dimensions the correct results from two- and three-loop graphs. We then return to heterotic string theory and calculate the contributions to the anomaly from the different spin structures for $d=2$. We obtain agreement with the work of Pilch, Schellekens and Warner and that of Li in the sector with spacetime fermions. In the other sectors, where no explicit computations have been performed in the past and for which one needs higher loops, we find a genuine divergence, whose interpretation is unclear to us. We discuss whether or not this leads to a new anomaly.