Volume 192, number 3,4 PHYSICS LETTERS B 2 July 1987 TOMONAGA-DIRAC-SCHWINGER FORMULATION OF THE FERMIONIC STRING Roberto ZUCCHINI l New York University, New York, NY 10003, USA Received 5 March 1987 We present the Tomonaga-Dirac-Schwinger formulation of the fermionic string. The constraint algebra of the classical theory reproduces upon suitable quantization the superconformal algebra of the BRST formalism. The supergeometrodynamicdegrees of freedom are related to the ghost and superghost variables in a simple manner. I. Introduction. In recent time it has become clear that the BRST formalism is of fundamental impor- tance for a gauge invariant and (super)conformal anomaly free formulation of string theory. In such an approach ghost variables are needed. These either are used for the computation ofa functionaljacobian [ 1 ] or are simply added as an enlargement of the phase space of the theory [2,3] or enter as extra compo- nents of opposite statistics of the various fields of the theory [4,5]. Recently, Das and Rubin [6,7] have proposed a Tomonaga-Dirac-Schwinger [ 8 ] formulation of the bosonic string theory. The BRST formalism is built in their approach and the ghost variables are inter- preted as quantized world-sheet parameters of the orthonormal gauge. Their symmetric treatment of the world-sheet parameters could also provide a mani- festly dual bosonic string field theory. The purpose of this communication is that of gen- eralizing the results of refs. [6,7] to the fermionic string. Our discussion applies equally to both the first- and the second-quantized version of the theory. 2. Description of the formalism. The action of the fermionic string is [9] tl S=jd2¢ (1) Supported in part by NSF Grant No. PHY-8413569. C[7 It ~ Ct b -- ~:ab~O el X [ rlCaeckeJOkX~'0lx~ + ie~k(u~'p ~ Ok ~ --ieckeJzkPdpc~'(O,X~-- ~izz~u)] • (1 cont'd) Here x u is the string spacetime coordinate, ~u is its fermion counterpart, em a is the two-dimensional graviton and Zm, is the two-dimensional gravitino. The associated energy-momentum tensor and super- current are, respectively, Y,~m=OLZlOe,,,a, fl~'~=-OLZlOZmo,. (2) The Euler-Lagrange equations of em a and X,~, con- strain ~-'-a m and jm, to be zero [9]. However, only four of the resulting equations are independent: J-~° = 0, J°" =0. (3) The reparametrization and super-Weyl invariance of the action S allow the orthonormal gauge fixing In this paper we use the followingnotation. Early latin letters denote tangent space indices. Middle latin letters denote coor- dinate indices. Early greek letters denote spinor indices. Mid- dle greek lettersdenote spacetimeindices.The valuesof tangent space indices are always underlined. We have qoo= -q~, = - 1, qo~= qm =0; Coo =e~ =0, co, = -e~o = 1. For any tangent space vector- v ~ we set v -÷=2 °~(v-° + v~),v±=2-~/2(vo + v~ ). The p matrices are p9 =ia 2, pt = a~ . For a spinor 0n, we set 0'~ = ~a0p where et2 = _~21 =0, eI 1=e22 = 0. We alsoset 0+ = 0L2, 0±=0 ~,2. For two spinors q, ~ we set ~/(=r/~p9~. Finally, O,,=OIO~'", O~=OIO0% On= -O/dOn. 346 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)