u model approach to the heterotic string theory

Relation between the equations of motion for the massless fields in the heterotic string theory, and the conformal invariance of the o model describing the propagation of the heterotic string in arbitrary background massless fields is discussed. It is emphasized that this Q model contains complete information about the string theory. Finally we discuss the extension of the Hull-Witten proof of local gauge and Lorentz invariance of the a-model to higher order in cr’, and the modification of the transformation laws of the antisymmetric tensor field under these symmetries. Presence of anomaly in the naive N = k supersymmetry transformation is also pointed out in this context. Talk presented at the Unified String Theories, Santa Barbara, California, July 29 August 16, 1985 * Work supported by the Department of Energy, contract DE AC03 76SF00515. I shall begin my talk by discussing the relation between the fixed point equations of the Q model and the equations of motion for the various massless fields in the heterotic string theory. 11-31 I shall work in the light cone gauge and consider a background where the graviton field g;i(z), the antisymmetric tensor field Bii (5) , and the gauge field A?( 5 ac ) q uire vacuum expectation value (vev) only in the eight transverse directions and are independent of the longitudinal coordinates z” and zg. The dilaton field CJ~ is taken to be independent of all space-time coordinates, in which case it may be absorbed in various fields and coupling constants 4 and never appear explicitly in our analysis.* The action for the first quantized heterotic string 5) in such a background is given by, I)-3),6) where (Y’ is the string tension, X’ are the eight bosonic fields, Xi are the eight lefthanded Majorana-Weyl spinors and $P are the 32 right-handed Majorana-Weyl spinors respectively. We are working in the Neveu-Schwarz-Ramond representation, so that the X’9.s transform in the vector representation of SO(8), whereas the $‘js transform in the 32 representation of SO(32) or (16,1)+(1,16) representation of the SO(16) @ SO(16) subgroup of &3 @ Eg. Also here,