Volume 179, number 4 PHYSICS LETTERS B 30 October 1986 BRST COHOMOLOGY IN SUPERSTRING THEORIES Nobuyosht OHTA t Theory Group, Department of Phy~w~. Umversttv of Tevas. Austin. TX 78712. USA Recmved 7 July 1986 Followmg the work of Freeman and Olive. it Is shown that the operators count,ng the number of non-transverse modes of the superstrmgs m lhe Neveu-Schwarz-Ramond formahsm are expressed as the antlcommutators of the BRST charges Q with other operators Also the cohomology of Q and the transverse state projection operators are discussed m terms of Q A useful formulation of the covarlant quantlzatlon of string theories 1s provided by the BRST formalism [ 1-4] In this quatlzatlon, physical states are, m general, identified with the cohomology class of the BRST charge Q [5], and this condition is sufficient to ehmlnate the negative-norm states. In order to get further mstght into string theories and to make a connection with the usual covariant quantizatlon [ 6 ], it is therefore necessary to examme the cohomology class of Q. Recently Freeman and Olive [7] have analyzed the space H~ of the cohomology classes for the open bosonlc string m the critical dtmenslon. By expressing the operator countmg the number of nontransverse modes as the anttcommutator of Q with another operator, they were able to clarify HQ and make a connection between the transverse state projectton operator and Q. This also leads to a simplification of the calculation of planar one- loop graphs [ 7 ]. In this paper, we generalize their work to the open superstrings tn the Neveu-Schwarz-Ramond formalism and discuss the cohomology classes of the BRST charges QNs and QR, using our previous results on the BRST quantlzatton of these models [ 3 ]. The application of the present results to the computation of the one-loop amplitudes will be discussed elsewhere [ 8 ]. Let us start with the reparametrlzation-, Weyl- and supersymmetry-lnvarlant lagrangtan [ 9 ]. 5go = e( - ½e~ 'e~O,x~'Ojxu - ½i~'y' 0, ~ + ½1~, 7'7JO~O,x~ - 1 ~q~gtj7,7 j ~, ) , (l) where x ~' and gO'are the bosontc and fermionlc coordinates, and e~' and ~,, are the zweibem and Rarlta -Schwln- ger fields, respectively. In the following, we restrict our attention to the case in which e~' can be taken to be the Minkowskl metric and ~u, can be put to zero. After fixmg the gauge and introducing the Faddeev-Popov ghosts C', C~, and C for the reparametrlzatlon, local Lorentz and local supersymmetry transformations, and antighosts (~, C'~, C'4 and (',, the lagrangian takes the form [3] Zf= -- ½O'x"O,x~,-- ½t(/.~"7'O,O,,+l(_" , (0o C° -OIC') +i('3(Oo Cl -Co,) - t('4(O,C ° --Col ) + (', (O,+ ½7,S)C, (2) where S is a scalar field in the supergravIty multiplet. For the open superstrmgs, the mode expansions ofbosonlc variables are given by 1( ) ,,~o ~exp(lnz) cos na , (3) ,/Tr Permanent address Instnute of Physics. College of General Education, Osaka Umverslty, Toyonaka 560. Japan 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) 347