Volume 179, number 1,2 PHYSICS LETTERS B 16 October 1986 THE BRST QUANTIZATION OF THE 0(2) STRING 'v Aleksandar R. BOGOJEVIC and Zvonimlr HLOUSEK 1 Department of Physics, Brown Umverstty, Prowdence, RI 02912, USA Received 31 March 1986, revised manuscript received 28 May 1986 The BRST charge for the N = 2, 0(2) superstring theory is given. The 0(2) algebra is realized a la Kac-Moody, which would seem to cause problems in the BRST formulation, however, the BRST charge is nilpotent due to the supersymmetry of the model. The nilpotency of the BRST charge determines the critical dimension of the model to be D = 2, and the in- tercept of the Regge trajectory becomes zero, making the model tachyon-free. We use the BRST charge to construct a free field theory of the model. 1. Introduction. A covarlant approach to the quan- to construct a nilpotent BRST charge. If this 1s the tization of string theories based on BRST invariance case then the enforcement of the mlpotency of the has been proposed recently [1,2]. An alternative and BRST charge would make the theory fall into the un- equivalent approach is to use exterior differential cal- physical sector (for example the states no longer form culus over the Virasoro algebra [3]. Using the BRST the unitary representations of the Kac-Moody sym- charge it is possible to build a covariant free string metry and consequently, the energy is no longer lagrangian [1,2]. The same ideas work in the case of bounded from below [8]), or to become inconsistent. N = 1 superstrings [4]. It is plausible to assume that Fortunately, this is not true due to the supersymmetry the full interacting covariant string field theory can be of the model. By a different kind of analysis [6,9] it formulated and constructed based on a nonlinear ex- was shown that the critical dimension of the model is tension of the BRST charge [5 ]. D = 2. This enables us to see the subtle way of how the The technique is to build a nilpotent BRST charge, supersymmetry takes care of the anomaly problem for i.e. to first quantize the string theory using the BRST the O (2) group. method. Next, use the BRST charge to construct a Once the nilpotent BRST charge is constructed in BRST lnvarlant kinetic operator for the field theory. the Neveu-Schwarz (NS) and Ramond (R) sectors of The gauge lnvarlant string field theory action is then the theory it becomes possible to construct a covan- given as the BRST invariant measure on the space of ant string field lagrangian. Next one could try to find string functlonals [1 ]. a nonlinear generalization of the BRST charge in order In this paper we work out a BRST charge for the to formulate the interacting theory. The analysis pre- N = 2 supersymmetric, O(2) charged string theory [6]. sented here could be relevant to understanding the The O (2) symmetry is realized as an abelian Kac- structure of compactified strings, and to resolving the Moody symmetry. Based on the result of ref. [7] one problems of the BRST formulation of Kac-Moody al- would expect that the BRST formulation of the mod- gebras. el suffers from a problem that stems from the Kac- Our paper is orgamzed as follows. In section 2 we Moody part, making it appear as if it were not possible briefly review the two-dimensional N = 2 supergravity theory that defines the 0(2) string and give the BRST charge. We show that it is nilpotent provided that the ~" Work supported in part by the US Department of Energy under contract number DE-AC02-76ER03130-19013-T ASK critical dimension of the model is two and the inter- A cept parameter is zero. In section 3 we show that the 1 Corrlna Borden Keen Fellow ghost structure introduced to make the BRST charge 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. 69 (North-Holland Physics Publishing Division) Volume 179, number 1,2 PHYSICS LETTERS B 16 October 1986 nflpotent also makes the partition function equal to 2 one in each sector of the model. This means that there S~ = ~ (8 q ~jpl _ e~jOv~2) ~vTu~l = 0, (2.3b) is only one physical state in each sector. In the NS sec- I=1 tor it is a ground state scalar and m the R sector it is a 2 ground state fermion. Therefore we must conclude T= ~i ~ etT~i~ tj = 0. (2.3c) that the free 0(2) string theory is trivial. We also give i,]=l the BRST mvariant kinetic operators and construct a The constraints (2.3a), (2.3b) describe conformal in- gauge-fixed free string action and list its invariances. variance and N = 2 supersymmetry respectively and (2.3c) is a consequence of the O (2) symmetry of the 2. The BRST charge of the 0(2) string. Let us first model. give a brief review of the first quantized formulation The constraints form a closed algebra, the symme- of the O (2) string. The actmn of a two-dimensional, try algebra of the model. This means they are first N = 2, 0 (2) symmetric supergravlty (gravlton, two class. Quantlzation can be implemented by construct- Majorana gravitinos, vector particle) interacting with mga Fock space reahzation of this algebra. In the D complex (0,1/2) matter multiplets (complex scalar, 0(2) model this can be accomplished by using two two Majorana spinors, complex auxiliary field) that de- sets ofbosonlc and two sets of fermionic ladder opera- fines the model is given by [10] tors: i,a " s = fd2 * + ½i Tu I)v* Otn , Ot~ b] = rt~tl ~n+m,otTab , L;bi'a r ~I,b~ _ ~z]~ ab ,u s j--u Vr+s,0q , where I,]= 1,2;a,b = 0,1 .... ,D- 1;n,m @ Z in the + (aucb + ~) ~Tv'yuXv - ½FF*] , (2.1) NS sector, and r, s C Z + ½ in the R sector. The vacu- um is defined by where D u = Ou + ½w#75, Xu = (1/x/2)(X 1 + ix2), ~b = ¢1 + icb2, ~ = (1/X/~)(~I + i~2), and other conven- al0>=b 'al0>= 0 (n,r>0). tions are as m ref. [9]. As a consequence of ordering ambiguities the al- The action (1) is invariant under coordinate, gebra of constraints acquires central charges, and be- Lorentz, N = 2 supersymmetry, Weyl, vector gauge, comes chiral gauge and super-conformal transformations [9,10]. As usual, we choose the gauge: eu _ 6a, Xu = O, [Ln, Lm] = (n - m)Ln+m + ¼D(n 3 - n) ~n+m,0, A u = 0, in order to linearize the action (2.1). This gauge chome is possible due to the symmetries cited. (G~, G~) = 28t]Lr+s + 2Ie*J(r - s) rr+ s The lineanzed version of the action thus becomes + ¼D(4r 2 - 1) 8r+s, 0 , S =fdZ~(~au.au** + ½,~¢gu'V- ½FF*). (2.2) [Ln, G[] = (In -r) G.+r,i [Tn, Tml=~Dn~n+m,O, Together with (2.2) one needs a set of constraints and appropriate boundary conditions. They follow from [Ln, Tm]=-mTn+m, [Tn, Gtr] =~,ex' i],,t~n+r. (2.4) (2.1). The constraints, written in terms of real fields, The L's generate conformal symmetry, G's supersym- are given by metry and the T's generate the 0(2) symmetry. Note 2 that the algebra (2.4) is the N = 2 super-Vlrasoro al- 1 i p t gebra, with an abehan Kac-Moody subalgebra. Physi- O# v = [~t~l~#rb i _ sg#v~p ¢ ~ dp cal states are defined by (L n - flfn,0)lphys) = Tn Iphys) = Gr~ Iphys) = 0 + ¼iv~i(Tua v + 7vOu) xifi] = 0, (2.3a) (n, r t> 0). (2.5) 70 Volume 179, number 1,2 PHYSICS LETTERS B 16 October 1986 Given the algebra (2.4) one can use a standard pre- QBRST = Q1 + Q2, (2.6) scription [11 ] for forming BRST invariant hamilto- nmns of constrained systems. with Let H 0, Ca (a = 1 ..... k) be the origmal hamiltoman and the first class constraints of some system satisfy- Q1 = ~(Ln - flSn,0) r/-n + ~ TnX-n mg the Poisson bracket relations n n -- C 2 {~ba ' ~bb)P.B. _ Uab d~c, {Ho, ~ba}P.B. = v,,4~bb • +~ ii Gr~-r , Here we allow q~a to be bosonic or fermionic, which r i=1 means that the Poisson brackets are defined with the appropriate Lorentz grading ,1 (0 for bosons, 1 for 1 ~ (n - m) :~n+mr/-mr/-n: Qz=~n,m fermions). (For our purposes it is enough to assume that the structure constants ucab and va are ordinary c-numbers.) Then there exist classes of quantum - ~ m :y`n+m~._mrl_n : hamiltonians H invariant under fermiomc (BRST) n)m transformation satisfying nilpotency: H = H 0 + C b ObaCa + [QBRST, ~21, + (½. ,) -i , - :~n+r~-rr/-n: n,r QBRST = dPa wt-,a -. "~l ~.--)t" "~?a. Uab C ~c Cb C a 1~ -i I - i i [H, QBRST] = 0, 4 n,r eij :~n+r~-rX-n: -- ~ :r/r+s~-r~-s : r,s where C a and Ca are ghost and antighost fields, respec- tively (corresponding to q~a) satisfying + (r- s) d / :X,+sg-,g-s : , [ca(x), Cb(Y)}l xo=y o = 6ab8 (x -- y) . r)s Physical states are given by a single condition: where/3 is the undetermined parameter present due to normal ordering ambiguitxes in L 0.