Local gauge and Lorentz invariance of the heterotic string theory*

The Hull-Witten proof of the local gauge and Lorentz invariance of the Qmodel describing the propagation of the heterotic string in arbitrary background field is extended to higher orders in cy’. The modification of the transformation laws of the antisymmetric tensor field under these symmetries is discussed. Finally we point out the existence of an anomaly in the naive N = f supersymmetry transformation, and show that it is cancelled by the same counterterms which restore local Lorentz and gauge invariance of the a-model. Submitted to Physics Letters B * Work supported by the Department of Energy, contract DE AC03 76SF00515. It has recently been conjectured[l-41 that the classical equations of motion of the massless fields in string theories may be interpreted as fixed point equations of the appropriate a-model. As a result we expect the symmetries of the classical equations of motion to be reflected in the a-model, and vice-versa. For example, in the closed bosonic string theory, and the type II superstring theory, the general coordinate and local Lorentz invariance of the classical equations of motion are consequences of the invariance of the corresponding non-linear a-models under reparametrization of the internal manifold[5]. Th e existence of such symmetries, although expected, is hard to prove otherwise. Usually the effective action involving the massless fields is constructed from the scattering amplitude involving the massless fields, and the rule for constructing these scattering amplitudes do not exhibit any general coordinate or local Lorentz invariance. The heterotic string theory[6] is expected to have general coordinate, local Lorentz and local gauge invariance. One would expect that these symmetries should be manifest in the o-model which describes the propagation of the string in arbitrary background field. Indeed, these symmetries are present in the classical a-model action. However, a close look at the model tells us that these symmetries are anomalous. In fact, the presence of these anomalies was shown to be responsible for the appearance of the Chern-Simons terms in the classical equations of motion[l,2], which apparently destroys the local gauge and Lorentz invariance. Ultimately, however, we must recover local gauge and Lorentz invariance of the equations of motion. As was shown by Hull and Witten[7], to one loop order the anomalous variation of the effective a-model action under these symmetries may be cancelled by redefining the transformation laws of the antisymmetric tensor field. The purpose of this paper is to extend their proof to higher orders in the a-model perturbation theory, and to derive the exact transformation laws of the antisymmetric tensor field under local Lorentz and gauge transformations. The light-cone gauge action for the a-model describing the propagation of