Volume 233, number 1,2 PHYSICS LETTERS B 21 December 1989 ON THE CONFORMAL ANOMALY OF THE GREEN-SCHWARZ HETEROTIC STRING IN CURVED N= 1, D= 10 SUPERSPACE * Parthasarath! MAJUMDAR 1 Robert N OERTER Department of Physws and Astronomy, Umversttyof Maryland at CollegePark, CollegePark, MD 20742-411I, USA and Anton E VAN DE VEN Department of Phystcs, New York Umverstty, New York, NY 10003, USA Received 14 February 1989, revised manuscript received 3 October 1989 It is shown that the a-model describing a Green-Schwarz heterotic stnng propagating m a curved N= 1, D= 10 superspace possesses a non-vanishing one loop world sheet conformal anomaly even when the background superfields satisfy their classical equations of motion The anomaly is shown to disappear for a flat background superspace and for non-trivial bosomc background fields We conjecture that the existence of this anomaly ~mphes that it must cancel against an anomaly m the fermiomc x-sym- metry (which must therefore exist) to preserve quantum consistency of the theory I. Introduction It is now an oft-quoted adage, amply supported by detailed analyses of bosomc and spinning strings propagating m arbitrary background fields, that the requtrement of d= 2 conformal lnvartance on the a- model forces the background fields to satisfy their classical equations of motton [ 1 ]. Corrections to these classical equations of motion (1 e. higher orders in a' ) have also been computed for these string the- ones [2]. Furthermore, heurlsUc arguments which clarify the relattonshlp between the vamshmg of the d=2 anomahes and the on-shell restriction on the background fields have been advanced recently [2,3 ] One would therefore expect s~mllar conclusions to hold for manifestly spaceUme supersymmetrtc for- maUons (i e the Green-Schwarz (GS) formula- tton), of the heterotlc string, propagating in a curved superspace In fact, the requtrement of d= 2 confor- * Research supported in part by NSF grant #PHY87-46846, PHY85-17475 and PHY86-05207 l On leave from Saha Institute of Nuclear Physics, Calcutta 700 009, India mal and x-mvarlance for GS nonhnear a-models in D= 10 superspace has already been shown to put the background superfields on their mass shell at the tree level of the a-model [4 ]. Further, the ultraviolet be- havior of this class of a-models has been investigated in some detail, for on-shell D= 10 backgrounds, the model ts found to be ultravtolet finite modulo a su- per&ffeomorphlsm, upto the two-loop level [ 5 ]. For a D= 4 supergravlty background including a dilaton multlplet, the requirement of ultraviolet fimteness (modulo a super&ffeomorphtsm) at the one-loop level forces the background fields to satisfy a one-pa- rameter famdy of equations which includes their ~'classlcal equations of motion [ 6 ] Thus, in either sit- uation, one finds that ultravtolet finiteness is consls- f6nt with the field equations, but (in D = 4) they are rot necessarily imphed This is as expected, we should eally calculate the d= 2 conformal (Weyl) anomaly and show that requmng it to vanish lmphes the back- ground super field equations It is the purpose of this letter to point out that the naive expectation ~s not borne out by actual calculations, Le., there exists a non-vamshmg one-loop d= 2 conformal anomaly for on-shell D= l0 background superfields (recall ref. 0370-2693/89/$03 50©ElsevlerSciencePubhshersBV (North-Holland) 123