Superspace Analysis Of_lqcaj _ Lorentz and Gauge Anomalies in the Heterotic String Theory *

Local Lorentz and gauge anomalies in the sigma model describing the propagation of the heterotic string in an arbitrary background field is discussed in the superspace formulation. An expression for these anomalies is written down in terms of the superfields, and is shown to cancel against an anomalous variation of the antisymmetric tensor field. World sheet supersymmetry is manifest throughout this analysis. Submitted to Physics Letters B * Work supported by the Department of Energy, contract DE A CO 3 76s FOO 5 15. In a previous paper[l] we analyzed local gauge and Lorentz anomalies in the a-model describing the propagation of the heterotic string in an arbitrary background field. The analysis was done in the component field language, and it was shown that the gauge and Lorentz invariance of the sigma model+ could be restored by adding suitable counterterms to the LagrangiaK Although these counterterms are not supersymmetric by themselves, it was shown that to one loop order the full effective action remains supersymmetric. I In this paper we shall repeat the analysis using superfield formulation[Z]. Although superfield formulation has been used extensively to study anomalies in supersymmetric gauge theories[3], t o our knowledge it has not been directly applied to study a-model anomalies[4]. In this formulation, we always maintain explicit supersymmetry. As we shall show, there is a supersymmetric extension of the local gauge or Lorentz anomaly in the sigma model. This term may be expressed in terms of superfields, and hence may be regarded as gauge (or Lorentz) variation of some manifestly supersymmetric non-local effective action. This action differs from the standard expression for the anomalous action, written in terms of the component fields, by a local counterterm, which is precisely the counterterm which was added by hand in Ref.[l] in order to restore local gauge and Lorentz invariance, as well as supersymmetry invariance. We shall also show that this anomalous variation of the effective action may be cancelled completely by redefining the transformation laws of the antisymmetric tensor field under local gauge and Lorentz transformations[5]. The dynamical fields in the first quantized heterotic string in the light cone gauge are the eight bosonic coordinates Xi, eight left-handed MajoranaWeyl fermions Xi, and thirty two right handed Majorana-Weyl fermions $J”. Let us introduce thirty two auxiliary scalar fields P, an anticommuting parameter 0 _ ,t Here by local gauge and Lorentz transformations we mean certain transformations on the parameters of the a-model, as given in Eq.(5). I nvariance under these transformations implies that the theory remains unchanged if we change the parameters of the v-model as in Eq.(5).