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Page i Dedication Page ii Acknowledgement Page iii Page iv Table of Contents Page v Page vi Abstract Page vii Page viii Chapter 1. Introduction Page 1 Page 2 Page 3 Page 4 Chapter 2. Free closed string field theory Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page 16 Page 17 Page 18 Page 19 Chapter 3. The bosonic string Page 20 Page 21 Page 22 Page 23 Page 24 Page 25 Page 26 Page 27 Page 28 Page 29 Page 30 Page 31 Page 32 Page 33 Page 34 Page 35 Page 36 Page 37 Page 38 Chapter 4. Superreparamentrizations Page 39 Page 40 Page 41 Page 42 Page 43 Page 44 Page 45 Page 46 Page 47 Page 48 Page 49 Page 50 Page 51 Page 52 Page 53 Page 54 Page 55 Page 56 Page 57 Page 58 Page 59 Chapter 5. Vector and tensor invariants Page 60 Page 61 Page 62 Page 63 Page 64 Page 65 Page 66 Page 67 Page 68 Page 69 Chapter 6. Superbosonization Page 70 Page 71 Page 72 Page 73 Page 74 Page 75 Page 76 Page 77 Page 78 Page 79 Page 80 Page 81 Page 82 Page 83 Page 84 Page 85 Page 86 Appendix A. Reducibility of the superreparametrization representations Page 87 Page 88 Page 89 Page 90 Page 91 Page 92 Appendix B. Explicit construction of invariants Page 93 Page 94 Page 95 Page 96 Page 97 Page 98 References Page 99 Page 100 Page 101 Biographical sketch Page 102 Page 103 Page 104 
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REPARAMETRIZATIONS IN STRING FIELD THEORY By R. RAJU VISWANATHAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1989 To My Parents ACKNOWLEDGEMENTS Many people have assisted me in the completion of this work. First and foremost, I am extremely grateful to my thesis advisor Professor Pierre Ramond from whom I have learnt so much in the past few years. His collaboration, en couragement, patience and timely advice have been invaluable. He has always followed my work closely while at the same time affording me a considerable amount of independence, a method of education that I have found especially beneficial. I wish to thank Professors D. Drake, R. Field, P. Sikivie and C. Thorn for serving on my supervisory committee. I am very grateful to Vincent Rodgers, whose lively discussions and collaboration provided a strong impetus to my research in its initial stages. I wish to thank all the other people from whom I have learnt physics: the faculty members of the physics department, partic ularly those in the particle theory group; the postdoctoral associates in the particle theory group; and especially, all my fellow graduate students from whom I have benefited through many discussions. It is a pleasure to thank Tom McCarty and especially Gary Kleppe. The many discussions that I have had with them have helped me significantly and my collaboration with Gary has been most fruitful. Their friendship has pro vided me with considerable moral support and they have always been around to produce the measure of comic relief necessary to provide a pleasant working environment. I am very grateful to both of them. I am also particularly grateful to Chang Sub Kim, whose friendship and moral support I value greatly. I am especially indebted to my friend and roommate S. Pushpavanam, whom I forgot to acknowledge in my undergraduate project report. I shall make amends by thanking him here for his support and friendship during the past several years. Special thanks also go to P.C. Pratap and my cousin Ravi Viswanath for their encouragement and friendship. I am thankful to all my friends in Gainesville for making my stay here a pleasant one. My gratitude to my mother Kalpagam, my father Ramachandran and my sister Deepa Lakshmi is beyond measure. Their support, care and affection have always been a source of strength for me. So also is my gratitude to my cousin Prema Kumar, with whose family I have spent many a pleasant weekend in Jacksonville. This research was supported in part by the United States Department of Energy under contract No. FG0586ER40272 and by the Institute for Funda mental Theory. TABLE OF CONTENTS page ACKNOWLEDGEMENTS .......................................... iii ABSTRACT ......................................................... v CHAPTERS 1 INTRODUCTION ............................................... 1 2 FREE CLOSED STRING FIELD THEORY..................... 5 3 THE BOSONIC STRING....................................... 20 3.1 Review of the Covariant Formalism .......................... 20 3.2 Representations of theReparametrization Algebra ............ 25 3.3 Quantization and Construction of a Dynamical Invariant................................... 29 4 SUPERREPARAMETRIZATIONS............................. 39 4.1 The Covariant Formalism .................................... 39 4.2 Linear Representations of the Superreparametrization Algebra........................... 41 4.3 Construction of a Dynamical Invariant...................... 54 5 VECTOR AND TENSOR INVARIANTS ...................... 60 5.1 Invariants in the Standard Representation.................... 60 5.2 Algebra of the Bosonic String Tensor Invariants .......................................... 64 5.3 Fermionization of the Superconformal Ghosts..................................... 67 6 SUPERBOSONIZATION....................................... 70 6.1 Construction of the Ghosts .................................. 70 6.2 Construction of Invariants ................................... 79 6.3 Sum m ary.................................................... 85 APPENDICES A REDUCIBILITY OF THE SUPERREPARAMETRIZATION REPRESENTATIONS ...... 87 B EXPLICIT CONSTRUCTION OF INVARIANTS............. 93 REFERENCES ...................................................... 99 BIOGRAPHICAL SKETCH....................................... 102 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy REPARAMETRIZATIONS IN STRING FIELD THEORY By R. Raju Viswanathan May 1989 Chairman: Pierre Ramond Major Department: Physics String theory has recently been recognized as a viable model for the unifi cation of the fundamental forces in nature. Of particular importance is the fact that closed strings contain the graviton as part of their spectrum and could therefore provide us with a consistent quantum theory of gravity. String field theory is a natural arena to examine the dynamics of strings. After the formu lation of a gaugecovariant free closed string field theory, an algebraic approach to string field theory based on reparametrization invariance is discussed. The basic formalism of the algebraic approach is that of the Marshall Ramond formulation of string field theory, where strings are studied as one dimensional spacelike surfaces evolving in time. The formalism is extended to include the bosonized ghost field, yielding an anomalyfree algebra in the process. The analysis is extended to superstrings and representations of the superreparametrization algebra are detailed. Invariant operators are constructed from the coordinates and the ghost fields. It is shown that these operators obey an anomalous algebra. In partic ular, the BRST operator is recovered as the trace of a symmetric spacetime tensor. Alternative representations of the superconformal ghost algebra are vii considered, leading to supersymmetric bosonization formulae. Dynamical in variants besides the BRST operator are shown to exist in the superbosonized theory. CHAPTER 1 INTRODUCTION String theory1',2 is the most recent attempt in physics to unify the fun damental interactions of nature. Unification has long been a central goal of physics. It has been known for the past several decades that the correct descrip tion of microscopic phenomena is in terms of quantum physics. A consistent quantum theory of gravity, however, has eluded physicists. Strings appear to offer new hope in this direction.3'4'5'6'7 A remarkable feature of strings is that they actually predict the dimensionality of spacetime.8'9'10 Superstring theory predicts a tendimensional world, as opposed to the fourdimensional physical universe that we inhabit. If the theory is to be realistic, therefore, it should also predict how and why the extra dimensions only appear at very small length scales. It turns out, however, that one can construct several distinct compact ification schemes for the extra dimensions which yield realistic particle spectra at low energies.11 The low energy predictions depend, among other things, on the topology of the particular compactification scheme that is assumed. Not enough is known about the dynamics of strings to tell us what the preferred compactification of the theory really is. While a fair amount is known about the perturbative aspects12'10'13'14 of strings, it appears that the important issues of low energy physics and the geometry of spacetime require a good deal of knowledge about nonperturbative aspects of strings before they can be successfully dealt with. To resolve these issues requires a more fundamental understanding of strings as building blocks than we have at present. The dynamics of closed strings should presumably determine the geometry of spacetime. String field theory is one of several 2 methods of study that have been proposed towards a better understanding of the dynamics of string theory.15,'16,'17 A proper formulation of string field theory would shed light on issues like compactification if the field equations could be solved. It is therefore crucial to construct a gaugecovariant closed string field theory. It is also important to uncover as much of the algebraic structure of the theory as possible since this could yield significant clues to the dynamics. In this thesis, we shall construct a free closed string field theory as a first step towards understanding the dynamics of closed strings and then examine string field theory from a purely algebraic standpoint. In the algebraic approach, reparametrization invariance is taken to be the fundamental symmetry of the theory. In the process, we shall unearth new invariants in string field theory, apart from recovering the usual BRST formulation.17'18'19'20'21 This dissertation is organized as follows, in essentially chronological order. A formulation of free closed string field theory22 is presented first. The con struction of a free string field theory for closed bosonic strings is detailed using the BanksPeskin20 language of string fields as differential forms. It is shown that it is necessary to introduce an auxiliary field even at the free level in order to construct a lagrangian that yields gaugeunfixed equations of motion. The gauge covariant equations of motion can be obtained from a gaugefixed set of equations by the process of successive gaugeunfixing. Secondly, the role of the reparametrization algebra as a fundamental sym metry in bosonic string field theory is studied. The MarshallRamond for mulation of string field theory is described classically16 and the relevant op erators are identified. Representations of the reparametrization algebra are discussed together with their composition rules. The relevant operators for a general representation are given. The bosonized ghost field is introduced as a 3 connection term in the covariant derivative over the space of onedimensional reparametrizations. The theory is quantized and the anomaly in the algebra is found. The states in the theory are characterized and a normalordered invariant dynamical operator is constructed, the BRST charge. Thirdly, the theory is extended to include superreparametrizations. The MarshallRamond extension to superstrings is described. The algebra of super reparametrizations is derived and its linear representations are given. It is shown that the doublet representation is the only linear representation consist ing completely of covariant fields. Composition rules for products of doublets are given. The ghost doublets are constructed and their structure determines the anomaly in the algebra, which vanishes in the critical dimension, namely ten. The BRST charge is again constructed as an invariant dynamical operator. Next, the superconformal ghosts are fermionized and a set of ghost doublets is catalogued. Invariant vectors are constructed from these and from the co ordinate doublet. New tensor invariants are constructed for the bosonic string and the supersymmetric string. The algebra of these tensors is constructed for the bosonic case and it is found to be anomalous.23 The nonanomalous part of the algebra is projected out by means of a set of matrices. These invariants raise the possibility of a larger symmetry in the theory. In the last chapter, an alternative representation for the superconformal ghosts is constructed with the techniques developed so far. We look for dy namical invariant operators that can be constructed as the integral of heavier components of doublets. A new fourparameter family of solutions is found, and the BRST charge is recovered as a particular combination of the four solu tions. The presence of scalar dynamical invariants besides the BRST operator 4 points to a richer structure underlying the superbosonized theory. The picture changing operator of superstring field theory is also obtained from a general fourparameter class of weight zero operators which change the picture num ber. The existence of a family of weightzero operators raises the possibility that there might exist other BRSTinvariant picturechanging operators in the superbosonized theory besides the usual one. Such operators would have an important role to play in the description of superstring interactions. Finally, the results obtained are summarized. CHAPTER 2 CLOSED STRING FIELD THEORY In this chapter we shall detail the construction of a gaugecovariant free closed string field theory. It has been known for some time now that the open bosonic string has a kinetic operator which is simply the BRST charge of first quantized string theory. Then the free Lagrangian9,21,'17,'24,'25 for an open string field 4 takes the form < PIIQI(b > where one must define an appropriate inner product. This kinetic operator is a bit unusual in that it carries with it a nonzero quantum number, namely the ghost number. The construction of the BRST charge ensures that it carries a ghost number of one. Since the Lagrangian must not carry any quantum number, the physical string field must have ghost number 1/2, which it does. When one looks at closed strings, the closed string field T has a ghost number of 1 associated with it, since it is this choice that reproduces all of the physical state conditions (L0 + Lo 2)0 = 0 (2.1) (Lo L0) = 0 (2.2) Ln = Ln = 0 (2.3) starting from Q\0 >= 0 (2.4) as an equation of motion, when we look at a state 1\ > that is annihilated by the ghost and antighost annihilation operators. Thus one cannot have a Lagrangian of the above form unless one puts in an appropriate insertion with the correct ghost number. It is necessary to have a Lagrangian which yields gaugecovariant equations of motion since one needs to know the off shell content of the theory to describe interactions. Since we already know that the gaugecovariant equations of motion are of the form19,17 QIT >= 0 (2.5) with the gauge invariance(as a consequence of Q2 = 0) IT > I > +QIA> (2.6) we would like to construct a Lagrangian which yields this unconstrained equa tion. With this motivation, we shall begin our construction by first reviewing the formalism for open strings. The ghost zero mode algebra {co, bo} = 1 (2.7a) cO =0 =0 (2.7b) results in a twodimensional representation18 of states I > and I+ > which satisfy co01 >= I+ > (2.8a) bo\+ >= I > (2.8b) col+ >= bol >= 0 (2.8c) The states I+ > and I > have the inner product relations <1+ >= 1 (2.9a) < I >=< +1+ >=o 0 (2.9b) 7 and have ghost numbers 1/2 and 1/2 respectively. The BRST operator can be expanded in the form Q=Kcobo +d+9. (2.10) The convenience of this form for the BRST operator is that the ghost and antighost zero modes co and b0, respectively, have been separated out. The operators 0 and d are simply the BanksPeskin cohomology generators20 of the Virasoro algebra; they contain, among others, terms trilinear in the ghost modes with the zero modes excluded. The operators appearing in Q then satisfy the algebra20 [K,] =[K,d] = [K,] =0 (2.11) [,] [l, d] = d2 = 2 = 0 (2.12) {d,9} = Kg (2.13) String fields are then viewed as differential forms in this language. A general string field (D has covariant and contravariant indices which simply indicate the number of ghost and antighost oscillators respectively that are associated with it; then (D can be expanded in the form K = (X)Cai..c'b..b6 (2.14) where O(x) stands for a local field and its associated set of coordinate creation operators. Then I)' acting on the vacuum produces the states in the theory. From this definition, it follows that (nm) is the ghost number of the form 1)'. The operators 0 and d act on contravariant and covariant indices respectively to produce forms with one less contravariant index and one more covariant index respectively. The operator I acts to change a contravariant index to a covariant one. 8 Since the vacuum state representation of the ghost zero mode algebra is twofold degenerate, a general string state IA > of ghost number 1/2 can be expanded in the form IA >= (DI > +Sn+1i+ > (2.15) where a summation over n is implied. The zeroform (0 corresponds to the physical string field that satisfies the gaugefixed equation of motion (LO 1)O = KO = 0 (2.16) subject to the 'physical gauge' condition LnO =0 for n > 0 (2.17) or dI = 0. (2.18) The Lagrangian for free open string field theory can be written as =< AIQIA>. (2.19) The corresponding equations of motion QIA >= 0 (2.20) then take the form24,'26 Kn + S 1 + dSn_1 = 0 (2.21a) doln + ,n+l = 0 (2.21b) at the nth level. The gauge invariance at this level which arises as a conse quence of the nilpotency of Q is that these equations are invariant under 6 p dAn +9 1 + IJXn+l 60n  dAnn1 + aAnn~ n1 (2.22a) 9 S =KAn+ + dXn1 + x (2.22b) The gauge parameters An+l and Xn+2 themselves have a further gauge in variance due to the nilpotency of Q; the process continues indefinitely. It is therefore necessary to take an infinite number of levels into account if one wants to completely unfix the gauge. The above infinite set of equations of motion can be compactly summarized as K> + (9 + d)S' = 0 (2.23) (9 + d) + IS1 = 0. (2.24) Taken all together, these equations are then the gaugecovariant equations of motion. We shall demonstrate how these equations can be obtained from the phys ical gauge by the process of gauge unfixing.24'26 Starting from Kg =0 (2.25) and d = 0, (2.26) one makes the gauge change 6I = 9A1 (2.27) which yields the equations Kg1 + KOA1 = 0 (2.28) d4 + d9A1A = 0 (2.29) or(using {fd, 9} = K) Kg + 9KA1 = 0 (2.30a) 10 dc OdA1 + KIA1 = 0 (2.30b) We introduce the Stuckelberg field27,'24,'28 with the variation b = dA1 (2.31) and the subsidiary field S1 with the variation bS1 = KA1 (2.32) to write the above equations (2.30) as KJ + OS1 = 0 (2.33) + 9^ + IS' = 0 (2.34) Using the fact that [0,9 ] = d2 = 02 = 0, we see that these equations are invariant under 1b = dA1 + aA2 ix2 (2.35) S1 = KA1 + ax2 (2.36) The equation of motion of the Stuckelberg field follows from its original varia tion with just A1: KS = KdA1 = dKA1 = dS1 (2.37) = dS1 This relation is of course not invariant under the gauge changes generated by A1 and x2. From the original variation of the Stuckelberg field, we see that it is constrained by dab = 0 (2.38) We can repeat the process starting from the equations (2.37) and (2.38) to obtain (2.39a) 11 di + &D + is2 = 0 (2.39b) where 4(2 is the new Stuckelberg field at this level defined by its variation ID2 dA1 (2.40) and the field S2 is the corresponding subsidiary field defined by its variation 6S? = KA + dx2. (2.41) As before, the equations (2.39) have additional invariances given by S= dA1 + OA2 1x (2.42) 6S = KA2 + dx2 + Ox. (2.43) Now the process stabilizes and at the n th level we get the equations (2.21a) with the gauge invariance (2.21b ); we repeat these here for convenience: K4Dn Kn + Sn+l + dSn1 =0 (2.44) dIn + a tn+1 + =+ 0 (2.45) We will now see that this process of gauge unfixing will be useful for the closed string, where an auxiliary field appears. We shall use the above process to conclude that this auxiliary field contains no propagating degrees of freedom. The BRST charge of the closed string separates into independent left and right moving pieces. It can be written in the form Q = Kc + Kc2 jbl b d2 + d + + a + (2.46) The left and right moving operators are barred and unbarred respectively, and the left and right moving ghost zero modes have a corresponding subscript of 1 or 2 respectively. The operators satisfy [K, ] = [K,d] = [K,9] = 0 (2.47) 12 [0] = [d = d2 = 2 = 0 (2.48) {d,&} = KJ (2.49) and similarly for the right moving operators. The left and right moving opera tors commute or anticommute with one another as they are independent. The ghost zero mode algebra {cl, b} = {c2, b2} = 1 (2.50) c = c2 = bh = j = 0 (2.51) has a standard representation in terms of direct products of open string vacua for the left and right moving sectors, given by the states I  >, I + >, I + > and I + + > in an obvious notation. These states have ghost numbers of 1, 0, 0 and 1 respectively. The nonzero inner products are <++ >=<  I + + >= 1 (2.52) and the action of the zero modes on the vacua is given by cll  > = I+ > c2l  > = I + > ell+ > = ++ > c2l +>= ++> (2.53) b2 ++ > = + > bil +>= > hl +>= > 13 Physical states in the theory are ghost number minus one states. A general state 177 > of ghost number minus one can be expanded in the form 177>= 01\ > +/111 + > +a t+ > +2 t++ > (2.54) As in the case of the open string, the zeroform q contains the physical prop agating degrees of freedom. As stated earlier, it is not possible to construct a diagonal Lagrangian of the form < 71IQI?? > without making a suitable in sertion. It is not easy to find a satisfactory insertion. We shall therefore try to construct a nondiagonal Lagrangian by introducing an additional ghost number zero string field. We define such a ghost number zero string field as IT >= "711 + + > +A + > +1 + > +7rll  > (2.55) The zeroform field 00 in Ir > is the physical string field and its equations of motion in the physical gauge are (L0 + L0 2)0 = (K + K)00 = 0 (2.56) (LO L0)0 = (K K)O = 0 (2.57) The physical gauge conditions read Ln0= =d = 0 (2.58) Ln 0 = d0 = 0. (2.59) The equation (2.57) above is actually just a kinematical constraint equation since it does not contain any time derivatives. The other fields in the expansion of 1t > arise as a consequence of moving out of this gauge. The gauge covariant equations of motion of \r >, which can be obtained by gauge unfixing, can also be obtained by simply acting Q on it. The resulting equations are (d + d + + )or1 + KO Jv2 = 0 (2.60a) 14 (d+d+9+0)p1 +KOv2 =0 (2.60b) (d+d+ + O)V2 K/ 1 Kp1 = 0 (2.60c) (d + d + 0 + 0) + j1 + Iy1 =0 (2.60d) These equations can be obtained from the Lagrangian22'29'30'31 ,C =< r\Q1r> (2.61) which has the correct ghost number of zero. The equation of motion of the field I > (Qir >= 0) reads in component form (d + d + 9 + O)w K7r + 71 = 0 (2.62a) (d + d + 0 + O)A KTr1 + y71 = 0 (2.62b) (d + d + 0 + B)1 Kw + KA = 0 (2.62c) (d+d+ + )7r1 w A = 0 (2.62d) These equations are invariant under the gauge transformations \]q >= QjA > (2.63) and S\r >= QJA > (2.64) Since the equations of motion of the zero forms wA and AO involve kinetic terms, there arises the possibility that Jr > might be a propagating field. However, we note that the kinetic term of w only involves K and that of A only involves K. This suggests that these equations and the kinetic terms arise purely as a consequence of moving out of a set of gaugefixed equations. The removal of the K K constraint on the physical field 0 must correspond to a similar removal of the same constraint on a field in the ghost number zero sector 1r > 15 to which the physical field couples.22 So we shall start from the gaugefixed equations (K K)T 0 (2.65a) dT = dT = 0 (2.65b) The zero form T is the analogue of 00 in the dual space Ir >. The gauge variation of T is 6T = (K + k)a + Op1 + pi. (2.66) This gauge variation results in the gauge transformed equations (K K)T + (K K)[(K + K)a + Op1 + Opil} = 0 (2.67a) dT + d(K + K)a + d9p1 + dOp1 = 0 (2.67b) dT + d(K + K)a + dOp1 + dbp1 = 0 (2.67c) It is understood here that the barred operators only act on barred indices and similarly for the unbarred operators. We shall write p1 and pi as =1 l 1 +A1) (2.68) and p (Q + A1) (2.69) respectively. The role of the fields f1 and A1 will become clear shortly in the equations which follow. The above gauge transformed equations (2.67) can now be written in the form (K K)T + (K + K)T + + 071 + i=0 (2.70a) dT + (K + k)7r1 + 1 + 9T1 + T = 0 (2.70b) 2 Zt2 16 + Kii + T! dT + (K + k)2r 1 + 1TY = 0 (2.70c) 2i 2 where the variations of the various fields are 6T (K+ K)a + a(A1 + Q1) + (Al + Q1) (2.71a) 6f (K k)a + a(Al Q1) O(A1 Q1) (2.71b) 2 2 2 6Y1 = KA1 + k1 (2.71c) ,yi = KA1 + Kf!1 (2.71d) b7rI = da IQ1 (2.71e) 6i = da ji1i (2.71f) 6T1 = !dQi + 2dAl (2.71g) T11 = 1J1 + 2dA1 (2.71h) 2 2 bT1 = ljd(1 + A1) (2.71i) T1 = ld(1 + A1) (2.71j) The zero form T arises as a consequence of unfixing the (K K) constraint on T. At this stage it is convenient to introduce a change of variables for T and T. We define 1 T = (w + A) (2.72) 2 and T 1 (A w). (2.73) Similar definitions also hold for the forms T1, etc. This definition enables us to make contact with the BRST equations (2.62). Then w and A transform as (2.74) 17 6A = Ka + + A+ Ai (2.75) and the equations (2.70) become KA kw + 071 + 0i = 0 (2.76a) dw + dA + (K + K)rI + 1y1 (2.76b) +,Owl +O9\\ + Owl\ + OA\=0 w + djA + (K + K)rI + 17i (2.76c) +Qu 9 + B+ \ + B = 0 a1+o1 + Owl + 1 =0 From the variations defining the fields we get the consistency conditions dw dA + (K k),7i + wl Ai + Ow OA 71 =0 (2.77a) and dw djA + (K k) +&I1 9AI + Ow' OA 1 = 0 (2.77b) The new forms at this level obey the constraint equations dw1=dw\ = w jW=! 0 (2.78a) and similarly for A, as well as wl = dw! (2.78b) dAl = dAI (2.78c) 1 d7Tr jwl = 0 (2.78d) d7ri jw1 = 0 (2.78e) j7rl JL = 0 (2.78f) d7"i 1w, = 0 (2.78g) 18 The equations (2.78) have the further invariances O = 1 + K&i i2 (2.79) 5l = 6kA + K _ [2 (2.80) 61r = 6 (2.81) "1 = bIf2 (2.82) Here the hats over the fields stand for all possible combinations(barred and unbarred) of the covariant and contravariant indices. For instance, f2 stands for F11r, F1, F" and Pil. The operators with hats over them stand for barred and unbarred operators which act wherever possible. The process stabilizes and we obtain the n th level equations22 Krn1 + I+1 + &9 + Ln1 = 0 (2.83a) n nn+ 1 n n1 K 1 + i + + + dn_ = 0 (2.83b) Knn + K\n + n+ dni = 0 (2.83c) rin1 + =rn n n n =0 (2.83d) These equations have a gauge invariance under the gauge variations & =n d,.l + ^+i nP+i1 + K&n (2.84a) 6n = dAnl + An+ I ln+tn1 + K&n (2.84b) b ,nn+i 1 ^n+1 + bfn+2 Khn+li~ n~ d + 9 K + n (2.84c) ^fn 6& n+l !.An+l + .,nn+l n+1 = dann + n+1 + + n (2.84d) We have added and subtracted equations (2.77a) and (2.77b) with equations (2.76b ) and (2.76c ) to obtain equations (2.83a) and (2.83b). The equations 19 (2.83) of course are just the equations of motion (2.62) of the dual field IT >. If one can reverse the above process of gauge unfixing to fix the gauge of the covariant equations (2.62) to just the set of equations (K K)T = 0 (2.85) dT = dT = 0, (2.86) one can show that there are no propagating fields in the zero ghost number sector. However, it is not clear yet if this can be done. We have shown that the gauge covariant equations of motion in the dual sector can be obtained by successively unfixing the gauge in the above set of equations. To summarize, the free Lagrangian for the bosonic closed string involves the coupling of the physical field to a ghost number zero field. The role of the fields in the ghost number zero sector needs to be clarified. Of particular importance is the issue of whether there are propagating fields in the ghost number zero sector, especially propagating fields that are distinct from the propagating modes of the physical field 00. If such fields do exist, they would raise further questions such as the boundedness of the kinetic terms in the lagrangian. Even if there are no such fields at the free level, it is not clear if this state of affairs would continue at the interacting level.* In any case, it is likely that the extra fields involved in the free theory will play a role in building a satisfactory interacting closed string field theory. * The author would like to thank Prof.C.B. Thorn for discussions regarding these issues. CHAPTER 3 THE BOSONIC STRING 3.1 Review of the Covariant Formalism A string can be viewed as a onedimensional object evolving in time. 32,16 As it does so, its shape may change and it may undergo interactions. For definiteness, we shall consider only open strings unless otherwise indicated. The points on the string can be labelled by a continuous parameter which we shall call a. We shall choose this parameter to take the values 0 and 7r at the endpoints of the string; thus, in different Lorentz frames, the string would be viewed as different spacelike surfaces. It is natural to expect that the physics of the theory be independent of the choice of parametrization of the string. We shall use this as our guiding principle throughout, so that the reparametrization group is the fundamental symmetry group.16 The points on the string have definite spacetime coordinates xP(a), where i takes values from 0 to d 1. A natural requirement on the functions xP(0) is that x'(a) = 0 (3.1) at the endpoints. Here and henceforth, a prime will indicate differentiation with respect to a. These functions can therefore be expanded in terms of orthonormal even functions over the interval [0,7r]. The cosines are such a set of functions; therefore we can write 00 / = (r xncosno. (3.2) n=0 21 Now consider making a change in a to a new parametrization 5, such that = a + oef(a) (3.3) We consider only changes in parametrization which leave the endpoints fixed, so that f(O) = f(7r) = 0. (3.4) Under such a change, which is merely a relabelling of points on the string, the spacetime coordinates must not change; we have not moved the string or changed our frame of reference. Therefore, if the xP(a) change to new functions iO(F), we must have )= (3.5) or .TP(a + Ef) = xa((a) (3.6) or ."(a) + 6fx'"(a) = X (3.7) which is correct to order e2. So the functional change in xP(a) is 8fx((a) = .(a) x(a) = efx'(a). (3.8) It is easy to check that the functional changes 8f satisfy the infinite dimensional Lie algebra [6f ]g = 6fg'f'g (3.9) Let us now define a functional derivative operator 6 which obeys [I b(l, X(o2)] b ( 22 where 6(a1 02) is the even delta function over the interval [0, 7r]. The func tional change in x(a) can be represented conveniently in terms of a generator Mf: .fx"(Oa) = iZE[Mf, x((a)] (3.11) where Mf i f(a)x'(). (o) (3.12) The hermitian operators Mf are then the generators of the reparametrization group, and they satisfy the algebra [Mf,Mg] = iMfg,_fg (3.13) In order to construct a string field theory, one now considers functional of x(o), namely objects like A[xa(o)]. Associated with a string xa(a) is such a field functional [a[x(a)]. This functional changes under reparametrizations as 61f[x(Oa)] = ieMf, (3.14) Physically, one expects the field functional to be immune to changes in the parametrization of the string: Mf1[x] = 0 (3.15) for a physical string field. We note that the generators Mf are dimensionless Lorentz scalars. They are independent of the spacetime metric and contain no time derivatives, so that they are purely kinematical objects. Upon quan tization of the coordinates, however, it is not possible to implement this as a requirement on the string field, as we shall see later; it can only be im plemented as a 'weak' condition, i.e., as a statement about matrix elements between physical states. 23 Having imposed reparametrization invariance as a fundamental kinematical constraint on string fields, the MarshallRamond formalism introduces invari ants and covariants of the reparametrization algebra. The physical length of the string can be defined as l = da x'2(T). (3.16) <0 This is clearly an invariant quantity under reparametrizations, since bf V x2() = (fT') (3.17) 2 Ix'12(o') x'2(o) r (f( is a total derivative and f and x'P vanish at the endpoints of the string. Next, under a reparametrization a + F, the delta function (or a'l) changes to (F a=^ oa') (3.18) Since x'(r) changes to da '( ) = x'(a) (3.19) thismeas tat te qantty (uoa') this means that the quantity is an invariant delta functional. We can therefore use = 6x2() as a derivative operator which is a reparametriza tion scalar, so that the operator = da f( ) x,:, ,. _Mf = x =x(a). (3.20) 0 Jo V/x'2() 6X(a) is a formally invariant quantity. Similarly, the object xs" transforms like V/x ''(a) a scalar under reparametrizations. One can write an action for string field theory, just as for point particles, in the form S= f D Tx(a) ) 5, x'(a)) (3.21) fE'. bX^ 24 Here Ei and Ef are the initial and final spacelike surfaces corresponding to the locations of the string and Tx(a) is a suitable functional measure. One can then write down the Feynman path integral with a suitable measure TD[[x] if one wants to calculate amplitudes. A fundamental requirement on the Lagrangian density is that it must be reparametrization invariant. It is natural to expect that, for the bosonic string, the action would yield equations of motion similar to the KleinGordon equation. If the equations of motion are of the form Ah = 0 (3.22) where Ah is a 'kinetic' operator, they must be covariant(forminvariant) under reparametrizations. This means that the commutator of the reparametrization generators Mf with the Ah must itself be another A operator. Further, con sistency demands that the commutator of two A operators be at most a linear combination of M and A operators. One can try to add terms to the dynam rldo, 62 ical operator f0 d7 F 2()ical operator in order to satisfy these closure properties. Further requiring that the covariant d'Alembertian be a Lorentz scalar, it is easy to see that the unique choice for the d'Alembertian is 1g = rdo,[ 62 x,2(0,) 2 Qr 6 x2(a) a12 (3.23) where a' is a constant of dimension (length)2. We will henceforth set a' = 1. The commutation relations satisfied by 0 are [Mf, 0h] = ilfh'f'h (3.24a) [Dh, Og] = iMhg,hg (3.24b.) Since and x"' are reparametrization scalars, the object da 1 62 2(3.25) 0 7T ^ r V wO 25 is an invariant quantity. One can therefore write a formally reparametrization invariant Lagrangian density in the form 1 = j I da D x ( x.) 6 + k( 2 X/2)) [x] (3.26) J0 7r Va72^)9 b x 6x2 where k is a constant. This Lagrangian yields classical equations of motion linear in the generators 0 and M. String fields satisfying the equations Mf = 0 (3.27a) OhI = 0 (3.27b) are particular solutions of the equations of motion. We note that these are free field equations. We will shortly see that these equations only hold in the 'weak' sense once the theory is quantized. 3.2 Representations of the Reparametrization Algebra We have seen that under a reparametrization a + 5 = a + Ef, the string coordinates transform like scalars: P(a) = x1(0). (3.28) This transformation law can be generalized naturally as follows: a quantity A(o) is said to transform covariantly with weight wA if under reparametriza tions it satisfies33'34'35 A(F) = A(a) (d (3.29) In terms of functional changes, this means that(dropping the infinitesimal pa rameter e) 6fA = (fA' + wAf'A) (3.30) 26 The integral of any quantity which transforms with weight one is of course a reparametrization invariant, as we saw for the length of the string. We note that if A(cr) is a covariant field, its derivative is not necessarily covariant: 8fA = (fA' + wAf'A)' (3.31) = (f(A')' + (WA + 1)f'A' + wAf" A) Thus, A'(o) is covariant only if WA = 0. Given two fields A and B, what are the covariant quantities that one can form from these fields? It is clear that if the weights of A and B arewA and wB respectively, the product AB classically transforms covariantly with weight (wA + WB). When the fields are quantized, however, one has to deal with operators, which could lead to ordering problems.It is easy to see that the combination (wAAB' wBA'B) transforms covariantly with weight (WA + wB + 1) since the f" terms in the transformations of A and B cancel. Upon taking more derivatives, one gets terms anomalous in derivatives of f as well as derivatives of the fields, so that it is no longer possible to form covariant combinations. Thus, one has the decomposition rule36 WA 0 WB = (WA + WB) (WA + WB + 1). (3.32) The transformation rule for A(a) can be written conveniently in terms of the generator Mf = i'r du(fA' + wAf'A) (3.33) as 6fA = i[Mf,A]. (3.34) Next we turn to representations in terms of noncovariant fields, or gauge representations.36 Consider an operator 0 which is defined to act on fields 27 of weight w and produce fields of weight w + A. Such an operator has the transformation law 6f0 = fo' + Afo0 [0, f]d w[0, f'I. (3.35) As a particular example of this, consider the operator d P = E aa E2(3.36) which acts on a covariant field A to give a field with weight (wA + A). Then from 6f(PA) =6f(E1A' + E2A) =(6fE)A' + E16f A' + (6f E2)A + E26f A (3.37) = (f(PA)' + (WA + A)f'PA) we can read off the transformations of E1 and E2: 6fE = (fE' + (A 1)f'E1) (3.38) 6fE2 = (fE2 + Af'E2 wf"Ei). (3.39) We note that E1 transforms covariantly, unlike E2. However, the combination E = wEj + (A 1)E2 (3.40) is a new covariant field provided A # 1, so that the representation is reducible in this case. One can form a covariant derivative which raises the weight of a field by one by taking E1 = 1; then d PA = ( + bC)A (3.41) has weight (WA + 1) if C transforms as 6fC= (fC' + f'C) + f" (3.42) =(fC)' + f". 28 Here tb is the weight operator; its value is simply the weight of the field on which it acts. The fact that C transforms inhomogeneously makes it similar to a gauge field or a connection. One can form from the field C the quantities e(a) and 0(a) defined by C(a) = 1 (3.43) w and e(a) = exp(q(a)). (3.44) Here w is a scale factor which is simply the classical weight of the covariant field e(7). The field 0, being the logarithm(at least classically)33 of e, transforms inhomogeneously: 8 = (/' + wf'). (3.45) We shall see later that ordering effects actually change the classical weight of the field e(oa). We note that the covariant derivative of e is zero, which is anal ogous to the statement in Riemannian geometry that the metric is covariantly constant. The 'einbein' field e(a) can therefore be thought of as a metric in the space of onedimensional reparametrizations.36 As we have seen, reparametrizations may be conveniently described in terms of generators involving functional derivatives. Classically, the functional derivative 6e has weight (1 w), since e(a) has weight w. Therefore, in gen 62 62 iewt r e(o) 2 d 2eral, does not transform in the same manner as 6 2 does, i.e., with weight two. As we shall shortly see, we would like to construct a dynamical operator from the field e(a). Since has weight one, it turns out to be more convenient to work with 0(a) rather than e(a). The exponentials e0 classically transform covariantly with weight aw. What polynomial covariants can one form from 0(a)? Since 0(a) transforms 29 inhomogeneously, the answer is actually none. The closest one can get to a covariant quantity is the combination (02 2wo"). This transforms anoma lously: 6f (2 2wO") = _f(02 2w4")' 2f' (42 2wO") + 2w2f". (3.46) This combination then is the analogue of x'2, so that we can use the object 6b2() (2 2w") (3.47) as a dynamical operator(upto a constant) for the field (ao). 3.3 Quantization and Construction of a Dynamical Invariant The functional derivatives 6M( and the coordinates xa(a) can be ex panded in Fourier modes as oo 00 xp(a)= xp + V2 xcos noE (3.48) n=l 6 0 (9 0 6x.() x + V2 E cos noa (3.49) 0 n=1 uXn with [,] 6}= n,m. (3.50) To quantize the string, we now introduce harmonic oscillator modes a, defined by ( (351) 9 i ,a = (an, + an,p) (3.52) for n 7 0. These satisfy [o', aVn] = gpvmbm+nO  (3.53) a The zero mode a0o = iF satisfies [x", ]= ia'. (3.54) The vacuum state is defined by  0 >= 0 (3.55) for all m > 0. The generators Mf = i f fx' .6 (3.56) can be expanded in a Fourier series in terms of sine functions. We note that once we introduce harmonic oscillators, these generators, which are formally metric independent, can be written in terms of the Minkowski metric. This simply corresponds to the fact that we have broken general covariance by in troducing harmonic oscillators in flat space. The Mf's can also be written in the form M j (2^2) (3.57) M f = 4 Ir with X() = x'/(O) + (3.58) and x'/ (O) = x'"(a) i (3.59) The combinations xL(a) and xR(o) are expanded in terms of exponentials: XL(a) = x aa + i eina (3.60) nOO n nO x (0) = x + a'" +i E e n (3.61) nO n 31 They are related by parity: XL(a) = xR(a). It is more convenient to work with exponential functions now that we have split the coordinates in the above manner. We therefore extend the range of a to cover [7r, 7r]. Then the operator M(a) = ix (3.62) ox has Fourier modes Mn = Md= f o ino (ML(0) + MR()) 1 f d na 2 x1 2) (3.63) 2Jf,27r (xR L = Ln Ln where the L's are the Virasoro operators Ln 1 0 anm am (3.64) m=oo We cannot demand Mn\[ >= (Ln Ln)\1 >= 0 (3.65) as a physical state condition since we have already chosen our vacuum to be annihilated by the positive modes am (m > 0). We can at best impose this as a condition on matrix elements of physical states. The normalordered Virasoro operators satisfy the anomalous algebra37 [Ln, Lm] = (n m)Ln+m + d(n3 n)6n+m,O (3.66) The modes Mn of M(a) satisfy the anomalyfree algebra [Mn, Mm] = (n m)Mn+m (n + m)Mnm. (3.67) This means that the reparametrization generators are covariant operators even upon normalordering. We note that, by construction, M(o)(as well as its left and rightmoving pieces) is a weighttwo operator. 32 The normalordered exponentials eik*'L(T) transform covariantly38,39 with weight k2/2. Similar normalordered exponentials with the coordinates xp(a) are not covariant since they are afflicted with ordering anomalies. Polynomials of order greater than two in xi(a) and its derivatives are not covariant since they contain operatorvalued anomalies under reparametrizations. We now turn to the dynamical operator 3= 2if h(a) ( &2(a) _Xo) (3.68) This can be rewritten in terms of xI and x' as 1h h(o) L 4'R(O)) (3.69) The density O(T) has Fourier modes [n = Dein = (Ln + Ln) (3.70) Classically, the 0 operator transforms covariantly with weight two, as can be seen from its commutations with My. The normalordered operator, however, transforms anomalously due to the central charge term in the Virasoro algebra: [Mn, Dim] = (n m)Dn+m + (n + m)Onm D(n3 n)(6n+m,0 + bnm,O). 6 (3.71) So we cannot use the 13 operator as a covariant equation of motion, unless we can somehow get rid of the anomaly. Also, we cannot yet construct a measure of suitable weight for use in the construction of an invariant operator(as in (3.26)); objects like v' ) are now illdefined since we have quantized the theory. As a possible solution to both of these problems, we introduce an extra 'ein bein' field variable40'41'33 e(or) (the same one which appeared in the covariant 33 derivative in the space of onedimensional reparametrizations) and quantize it. The motivation for introducing the einbein field comes from the analogy with the case of the point particle. The free point particle action (with i = ) m In dr V"2 (3.72) can be replaced by42,'43,'44 Sdrf{[l/e(r)]2 + m2e()} (3.73) where e(r) is an einbein field which transforms as a total derivative under reparametrizations in r. So the einbein field serves in this case to eliminate the need for square roots, and at the same time provides an action for the massless point particle. The einbein field in our case is also introduced with the view of eliminating square roots in the action; it acts as a 'metric' in the space of onedimensional reparametrizations. The price we pay is just that the string field now also depends on the extra field we have introduced. It turns out to be more convenient to work with the field O1=lne(a) rather than with e itself.33 Now the string functional 4 also depends on O(or), in addition to the coordinates. 19 This field has the inhomogeneous transformation law mentioned in the previous section: b = (f/ + wf'). (3.74) We could work directly with the covariant field e(a), but then we would run into problems when we tried to construct a dynamical operator, since the functional derivative would then have a weight different from one. The reparametrization generator for this field takes the form35 MO= if ,tY/ + Wf_ (3.75) f 10 7r ~ j We can expand 0(a) in modes, just like the coordinates: 00 0(u) = 0 + V2_ Y ncos no, (3.76a) n=l 6 a9 = a + V2 cos no (3.76b) n=l We can quantize q just as we did the coordinates by introducing harmonic oscillators: On= (fOn f3n) (3.76c) nv2 0n = ( + l) (3.76) 0(u) eL(o) + OR(o) (3.76e) 2 OL(7) =0  foa + i E neino (3.76f) n n0O R(a) = 00 + flo + i ein (3.76g) n n$O CL(O) = CR(a) (3.76h) The modes On satisfy [fn, fmr] = r7n'n+m,O (3.77a) [q0, #0] = (3.77b) Here the parameter q takes the values 1;7 = 1 means that 0(a0) has ghost like excitations. The vacuum state is defined by Oni0 >= 0 (3.78) for all n > 0. The left and rightmoving pieces of 0 can be written as 0L ='+ i3769 L 01 +i?760(3.79) 35 R 0 (3.80) Correspondingly, the dynamical operator is I=7i'rd, 2 (o'2 2w')) (3.81) as mentioned in the previous section. We note that 77 = 1 corresponds to negative kinetic energy for the field 0(o). The factor of q here is necessary for separability of the left and the rightmoving pieces of the M and 0 operators. We can write Mf =ML+ MR q da( 2 w(of/ _/ (3.82) 2 1o7r (f 2 L R and O ff 7 do (f2 + f( + 1) (3.83) 2 7r 2 Ls +R or, in terms of Fourier modes, M = Lo Lon (3.84) On = (L + Lo). (3.85) Here the Virasoro operators for the field 4 are n= 2 Z(Cnmom) + 2 ) (3.86) M The normalordered Ln obey the algebra [L", LoJ =(n m)L+m + A(12w2n3,q + 3 n)bn+m,o (3.87) 36 The linear term in the anomaly can be absorbed by a shift in L0; the cubic term in the anomaly of the algebra of the total Virasoro generators (LO + Lx) vanishes for d + 12w2t + 1 = 0. (3.88) Clearly, ij must be minus one to yield sensible values of D(since w is real). The only normalordered covariants one can form from 0 are the normal ordered exponentials ea1L(U) (and similarly for CR(a)) defined by38,'39 ea1OL(a) := exp(ai E LeinU)eaOeaG(,o+ia/2)exp(ai E e" i). (3.89) n<0 n>0 The quantity ea1L(U) transforms covariantly with the weight a(w arf/2). We need a weight minusone object as an integration measure Mo(a) in order to construct a dynamical invariant operator of the form (as in (3.26)) K = 7 (O)ML(Oa) (3.90) where ML(a) is the total leftmoving part of the reparametrization density including coordinate and ghost contributions. We note that we cannot mix left and rightmoving modes here since it would lead to equations of motion that are inconsistent with the definition of the vacuum state. Since the exponential eakI is the only possible covariant that could provide us with a suitable measure, we must have a(w a?/2) = 1. (3.91) This gives us a 1 W = a (3.92) 2 a Requiring the exponential eaOL to be singlevalued as a changes from Tr to 7r, we see that a must be an integer since the eigenvalues of /0 increase in steps of 37 i. We note that /o, being the 'momentum' of the 0 field, is a reparametrization invariant. The states of the theory are thus labelled by their eigenvalues under /30, in addition to the values of the spacetime momentum. These eigenvalues label the ghost numbers of the states. Since d must be a positive integer, we see from (3.88) that w must be a halfinteger. So we can only have a = +1 or a = 2; we have w = 3/2 for a = 1 or 2 and w = 3/2 for a = +1 or +2. For either of these possibilities, the theory predicts twentysix spacetime dimensions. The operator K must be an overall normalordered expression for it to make sense. This means that we still have to check the invariance of K after it has been normalordered. Let us set Q = J : eaeLML(O): (3.93) f rK_27r and check its invariance. We find35 n(n + 1) do, in d a2 aw 2 aO [LnQ] = jir dO,n(i.i  naw)e (394) 2 7r do 2 3 3 We see that the righthand side is a total derivative if and only if a2 = 1 and aw 3/2. We make the choice a = 1 corresponding to w = 3/2. So we get a unique invariant scalar operator in twentysix dimensions. This is of course the usual BRST charge, and it is not hard to check that it is nilpotent. The field 0(a) is then the bosonized ghost field. From the operator product eaOL(rl)ebOL('2) eaL(01)+bL(a2) (2isin l22 ab (3.95) we can see that cL = eL and bL = eCOL are conjugate anticommuting fields; these are the usual anticommuting ghost and antighost respectively of the bosonic string. 38 As mentioned earlier, the states are labelled by the eigenvalues of i00, which are halfinteger; this is simply the ghostnumber operator. The free field theory action is given by S =< IQI > (3.96) and it yields the usual equation of motion QI >= 0. (3.97) As a consequence of the nilpotency of Q, this has the wellknown gauge invari ance I >* J > +QIA > which eliminates states of negative norm.45 In the next chapter we shall generalize the theory to include fermions. CHAPTER 4 SUPERREPARAMETRIZATIONS 4.1 The Covariant Formalism The MarshallRamond extension to include superreparametrizations in the formalism is based on the introduction of two anticommuting quantities, the generalized Dirac gamma matrices Fq (a) (i = 1,2). These hermitian operators obey the anticommutation rules16 {r'(o,), rI!(o)} = 26ijg'6(a a'). (4.1) As for the coordinates, the delta function here is defined over the interval [0,7r]. Since the delta function is a weightone object, these fermions are weightone half objects under reparametrizations. Two sets of matrices are necessary for the construction of a dynamical operator, as will be seen later. We note that these matrices can be replaced by the equivalent set Fr(a), (, defined by r"(o) = rP(a) + SF() (4.2) ( 5 6r,(a)) 43 rP = i r(4) Since these are weightonehalf quantities, the reparametrization generators are given by Mf = i (fUr' + f'ir). 6 (4.4) or equivalently by M = f ()rr. (4.5) 39 40 This generator acts on string functionals which are spacetime spinors. In par ticular, the wave functional of the string is such a spinor. In order to obtain Diraclike firstorder equations of motion for the string field, the Lagrangian density must also contain a firstorder operator. Since Fr(a) is a weightonehalf quantity, the objects (x'2)1/4Fr(o) trans form as reparametrization scalars. Using this fact, one can build a Lagrangian density of the form j (X(12)1/4 (if. + r2 ) (4.6) where the i is included for hermiticity. This Lagrangian density is reparametrization invariant by construction. Used in an action of the form S =< TIL1\ > (4.7) it yields equations of motion of the form r daf(f) (0 iF1 + F2 x' I >= 0. (4.8) The operators 1 0 d ) (irl + r2.x') (4.9) satisfy the classical algebra [Mf, Pg] = iPfg,_f/g/2 (4.10a) {JPf ,Pg} = 40fg (4.10b) [f r'g] = iQfg'/2f'g (4.10c) Here Mf stands for the total reparametrization generators including the coor dinates and the F's. The operator Qf is defined by interchanging F1 and IF2 in Pf: Qf = f f(0) (iF2 + + F i) (4.11) 41 The Of operator now includes contributions from the F's: 7 do, /82 1 = I f () x() + (2(r'() + r ) (4.12) The algebra is completed by noting that the following commutations hold: {Qf, Qg} = 4Ofg (4.13a) {.f, Qg} = 4Mfg (4.13b) [Qf7, g] = ifgI/2f'g (4.13c) [Qf,Mg] = iQfg,'/2_fg (4.13d) The above equations of motion are then covariant, at least clasically. Upon quantization, however, an anomaly arises in (4.13a) and in (4.10b ), which means that the corresponding equations of motion are no longer covariant. The anomaly needs to be cancelled, and we can do this as for the bosonic string by simply adding extra fields. Before doing this, however, one needs to take a closer look at the superreparametrization algebra and its representations, to which we now turn. The results obtained in the rest of this chapter are based on the author's work in ref.36. 4.2 Linear Representations of the Superreparametrization Algebra For superstring field theory we seek a kinematical supersymmetry36 trans formation f which is the "square root" of the reparametrization 65f in the sense that [f(6l), g(2)]F(o) = = 2fg(612)F(a) (4.14) 42 for any field F(oa), where the 's are anticommuting parameters. The com mutation relations of the 's with the reparametrizations 5 can be determined from the Jacobi identity [[ ],tbh] + [bh,,] + [[,6hIf] = 0. (4.15) We first note that the commutator of a reparametrization 6h with a super reparametrization of must be bilinear in f, h and their derivatives; this is clear from (4.15) and (4.14). Furthermore, derivatives of order higher than one are excluded due to the presence of the first term in (4.15) (since this identity should hold for arbitrary functions). The commutator must therefore have the form [Wh(), f()01 = a'hf'+,6h'f(6) (4.16) Using this relation and (4.14) in (4.15), we find that a = 1 and/3 = , i.e. [WOOf(0] = Ohf,hf/2(E0. (4.17) Henceforth, the parameters e and 6 will not be indicated explicitly unless clarity warrants it. Given fields transforming in a specified manner under reparametrizations we can deduce their possible transformation properties under superreparametrizations. First consider the case of a field a(a), either commuting or anticommuting, transforming covariantly under reparametrizations with weight Wa, for which we postulate the transformation law fa = fb (4.18) where b is a field of opposite type (commuting or anticommuting) from a. (4.17) tells us that fSgb = ({f6g ,gf,_gff/2)a (4.19) 43 and upon evaluating the right hand side of (4.19) we find 6gb= gb' (wa+ )g'b, (4.20) i.e. b transforms covariantly with weight wb = Wa + . Assuming that a and b form a closed multiple involving no other fields(we shall show later that adding extra fields does not generate new irreducible representations), the most general form for the transformation of b under a superreparametrization is 1b An a (4.21) n where the An's are functions of f and its derivatives. Using (4.14) with g = f, we find 9fffb = 6ffb (4.22) so that Anfb = ffb' + 2wbff'b. (4.23) n Since b and its derivatives are all independent, we can equate coefficients on either side to solve for the An's. We find that the only nonzero An's are AO = 2waf' and A1 = f, i.e. fb = (fa' + 2waf'a). (4.24) We have discovered one type of multiplet6 on which the superreparametrization algebra is represented. The representation can be written in matrix form: (a) = f(f +2Waf' Of) (a), (4.25) f )b Ga / +a 2W whereas the transformation bf is written as 6b(a) (fd+ f Wa 0 )(a) (. a (f wa f 0 fb )f')[b) (4.26) b  dd, rWa +lr 44 The representation is the same regardless of the Grassmann character of a or b. For this type of multiple, we will refer to the component a transforming according to (4.18) as the light component, and to b which transforms according to (4.24) as the heavy component. An important difference between the two components is that if the integral of the heavy component is reparametrization invariant (i.e. if it has weight one), then it is also superreparametrization invariant, as is evident from the transformation law (4.24). The integral over the light component is never superreparametrization invariant. An example of this type of representation is provided by the string coor dinates x1. These transform according to (4.18) into the generalized Dirac matrices Fr: f = frP (4.27) f F = f x' (4.28) Because xP has weight zero, the multiple (F/, x'I') also transforms as (4.25), with FIP as the light component. This multiple is of more direct use in string field theory because it is translationally invariant. Given two doublets (a, b) and (c, d), it will be useful to know all of the dif ferent covariant superreparametrization representations which can be built out of products of these fields and their derivatives. One can form eight quantities which transform covariantly: weight w: A1 = ac 1 weight w + : A2 = ad andA3 = bc 2 weight w + 1: A4 = bd and A5 = wca'c Waac' 3 2 (4.29) weight w + : A6 = (wc + I )a'd Wad' and (4.29) 2 6 w +2a a A7 = wcb'c (Wa + I)bc' weight w + 2: A8 = (wc + )bd (Wa + )bd' 2 z2 45 In these equations, w Wa + wc. Among these quantities, three combina tions may be identified as doublets: (Ai,A2 + A3), with weight (w,w + ) (4.30a) 1 (waA2 TF wcA3, A5 + wA4), with weight (w + w + 1) (4.30b) (2A5 + A4, 2A7 + 2A6), with weight (w + 1, w + ) (4.30c) The upper (lower) sign of the 's in these equations is to be read in the case where a is the commuting (anticommuting) member of its multiple. The heavy component of both (4.30b ) and (4.30c ) reduce to total derivatives in the cases in which their weight is one, so they yield only trivial invariants. The remaining two quantities in (4.29) are members of a multiple containing noncovariant quantities. We have thus demonstrated the decomposition36 2w 0 2v = 2v+w 2v+w+x (e 2v+w+l D (non covariant) (4.31) 2 We will use the symbols Oa, Ob, Oc to denote the three ways of combining two doublets to obtain a third given in (4.30); i.e. (a, b) Oa (c, d) (ac, ad + be) , (a, b) Ob (c, d) (Waad wcbc, waac' F WcaC + wbd) , (a, b) Gc (c, d) (2(wca'c Waac') + bd, 2wcb'c F (2wa + 1)bc' + (2wc + 1)a'd 2waad') . Note that for the Gb and Gc coupling schemes, the heavy component can only yield trivial invariants. The fact that the only covariant representation of the superreparametrization algebra found in the direct product of two doublets is again a doublet sug gests that no other covariant irreducible representations exist. We shall now 46 prove that the doublet representation given by (4.18) and (4.24) is the only irreducible linear representation of the superreparametrization algebra whose basis elements are a finite number of fields which transform covariantly under reparametrizations. We will show that given a set of covariant fields which transform into each other under superreparametrizations, the representation can be reduced into a series of doublets.36 We will use the notation aw,i to denote the ith field of weight w in the collection, where i = 1 to Nw for each value of w. Consider the fields a, i, where w0 is the lowest weight in the set. Since the superreparametrizations increase the weight by , these fields must transform into weight wo + fields. We can choose the basis for these fields so that ofawo,i = faw+j, i = 1 to Nwo (4.32) Applying a second superreparametrization operator, the covariance of awo,i requires f= (fawo,i + 2wof'awo,i) i = 1 to Nwo (4.33) i.e. the combinations (awo,i, awo+ i.) form Nwo independent doublets. We now show that with an appropriate choice of basis, the elements of these doublets do not appear elsewhere in the representation. First consider the other elements, i.e. Nwo < i < Nwoi+. The most general possible transformation law satisfying (4.17) for these elements is N.0 wo+f = E Aij(f a',j + 2wof'awoj) fawo+l,i (4.34) j=l 47 with an appropriate choice of basis for the weight wo + 1 elements. By changing the basis for the weight wo + elements we can obtain elements which do not transform into the weight w0 elements. Redefining N.0 awo+, + awo+,i E aijawo+ j,' (4.35) j=l we obtain awo+,i = fawo+l,i (4.36) We now show that the elements in the doublets (awo, awo+) do not appear elsewhere in the algebra. (Here the subscripts i are left as implicitly under stood). Let aw be the first (i.e. lowest weight) element whose transformation law involves one of these elements. Then there are two cases to be considered: 1) w w0 = n is an integer, and the transformation of aw involves awo+. aw could possibly have the transformation law dmf dnma + aw EAm dam dnm X (4.37) m where Ai are coefficients and Xf is some quantity which does not involve the elements in the doublet. Xf is found to transform to ;2/ 2"' Am dmf dnm SfXf = f aw + 2wffaw +EAm dm dAnm (fawo + 2wof'awo). (4.38) m The transformation of Xf involves awo; since we assumed that no field of lower weight than w has this property, Xf must have higher weight; the only possibility is Xf = faw+. (4.39) 48 Since Xf has no derivatives of f, the only possible Ai's which could be nonzero are those which are multiplied by f, which in this case is only A0. Then we find faw+1 =Ao (fawo+i)'+ 2wof'(faw+)) (4.40) + f(ofaw)' + 2wf'ofaw It is easy to see by substituting from (4.37) that this cannot be satisfied unless AO=O. 2) w w0 is half integral (w = w0 + n 1), and the transformation of aw involves awo. The details of this case are similar to case 1. We find n dmf dnm oaw Am "n aw + faw+ m=l (4.41) and fj dm f dnm + f ffaw+ = E dmf dnm (faw0+) + f2aw + 2wff'aw m (4.42) requiring Am = 0 except for m = 0 and m = n; then a+1 =AOn (f2 awo 2woff'awo) + dan (fa0 2wof'aw +fd(AOan(faw+) + Ann awo+1 + 2wf' (Ao ( o(faw+)+ An oa+) (4.43) which again cannot be satisfied unless A0 and An are zero. We have shown that the lowest weight fields are parts of doublets which decouple from all other fields under superreparametrizations. One may apply the same procedure to what remains, again and again until the whole repre sentation is reduced to doublets. So any arbitrary representation in terms of covariant quantities may be reduced to doublets. 49 We have demonstrated above that all representations with covariant com ponents are doublets with weight (w, w + ). There exist other types of representations with components that transform like gauge fields, i.e. non covariantly.36 Such representations as well as the covariant derivatives can be constructed, using techniques introduced in the bosonic case. Let T be a 2 x 2 matrix of operators acting on a doublet (a, b) of weight Wa, and producing a doublet (A, B) of weight WA: (A) =(F, F12)(a) (4.44 B)\F21 F22,)\bJ Application of the doublet transformation laws then yields the following transformation equations for the matrix elements of : ffF11 = d F12f j: 2waF12a fF21 F22 = fdFd___2 d fF12 df F2f d o, T dc a (4 .4 5 ) SF12 =+ F11ffF22 f 1F21 =(F22f fF) f df 2waF2d 2wAFd jo, dodf df The upper signs in these equations are to be read when F12 and F21 are anticommuting operators, and the lower signs when F11 and F22 are the anti commuting operators. Since application of another such matrix to the doublet (A, B) must yield yet another doublet (C, D), the matrix product of two F matrices must satisfy the same transformation law (4.45). As FT is allowed to contain derivatives, the ordering of products in these equations is important even classically. These equations have many solutions, depending on the number of derivatives present in the F's. In the following we discuss several simple cases; the general case is presented in Appendix A. 50 First, assume that F contains no derivatives. We immediately deduce that F12 = F11 T F22 = 0. (F11,F21) transform as a doublet, yielding a way of composing two doublets (a, b) and (c, d) to make another doublet which we recognize as the a coupling scheme. We can rewrite this in matrix form as ( ac O0 (a a0 (4.46) (bcad ac =(b a d c As before, if Wa+Wc = , the integral of the heavy component of the compound doublet is an invariant. One can also use Grassmann notation with 0 = a identified as the nilpotent component. We now investigate representations built from F operators containing at most first derivatives: d T" = G + *. (4.47) We consider only the case in which the offdiagonal components of F are anticommuting. From (4.45) we find H12 = 0 and Hll = H22 (4.48) and fHll = f(H21 + G12) 1 fg21 = f(G22 Gl H'l) + 2(A + 1)f'H21 2 fGil = fG21 2waf'G12 (4.49) .f G22 = fG12 fG21 f'(2wAG12 + H21) fG12 = f(Gll G22) +f'Hll ffG21 = fGl + 2f(waG22 WAG11) + 2waf"Hll. Under reparametrizations, some of these fields transform covariantly and some anomalously. It is easy to see that H1l has weight A 1, G12 and H21 51 have weight A 1; all of these fields are covariant. However, fG = D(A)Gll + waf"H ,fG22= DA)G22 + (wa + )f"Hn (4.50) bfG21 = + G21 + waf"H21 where D^ _(fd+w$). The two fields H1i and 121 +G12 transform into one another as a doublet. The other four fields transform into these fields, so it would seem that the representation is irreducible. However, except for rather special values of the parameters, it is possible to find linear combinations of the fields and their derivatives whose transformation laws decouple into doublets. Specifically, (2A 1)G12 + H21 and 2(A 1)(Gl G22) dH11 (4.51) form a doublet. Clearly, for A = 1, this doublet is not independent of the first doublet. The combinations (1 2wA)G1 + 2wa(G22 H') and 2waH'l+(12A)G21 (4.52) form a third new doublet, except for A = 1. Thus, except for these two values of A, this sixfield representation can be reduced into three doublets. However, for A = 1, if it is also true that Wa = 0, then this sextet representation can still be reduced into three doublets, given by (Hll,/H21 +G12), (G11, G21) and (H21,G22GllHil). (4.53) However, whenever Wa : 0, we know that the transformation of G21 under reparametrizations has a noncovariant term proportional to 121, which has zero weight when A = '. Thus it is impossible to cancel this anomalous term by adding a derivative of H21 to G21. On the other hand, components of 52 the doublet transform covariantly under reparametrizations, which leads us to conclude that it is not possible to split the sextet into doublets in this case. Rather, the sextet splits into a doublet and a quartet. Its members are given by (G21,G11,G22,G12) = (G21 2waG12,(2wa + 1)Gll 2waG22, 2waG11 (2wa 1)G22 H', H21), with the transformation laws #fG21 = fG'l f'[(2wa + 1)G11 2waG22] fGil = fG21 2waf'Gi2  _(4.54) fG22 = fG21 fG12 2wafG12 fG12 = f(Gl G22). Under reparametrizations, G11 and ?22 have weight , ?12 has weight zero, and G21 has weight one. All of these fields transform covariantly except for G21, which transforms as f = D G21 + Waf"G12. (4.55) This quartet representation of the superreparametrization algebra is irre ducible. Finally, when A = 1, with Wa 5 0, we obtain the quartet with slightly different transformation laws, namely (021, Gu, 022,0 12) = (G21 2waG'l2, Gll 2wa(Gli G22), G22 H'I1, 2waGi2 H21), The 'hatted' fields have the transformations fG2l = dG 1 + f'(2waG22 il1) fGnll = fG21 2waf'G12 .fG22 = fG^2 f'G12(4.56) G612 = f(11 022). 53 These two representations can be understood as special cases of the generic quartet obtained by setting Hll = H21 + G12 = 0 in the sextet transformation laws. In general, for representations with more derivatives, it is not possible to completely reduce the representation into dou blets, as we shall see in Appendix A. We conclude this section with the building of the covariant derivative which is the direct generalization of the one we have constructed in the bosonic case. Our starting point will be the quartet with A = 1, and with the offdiagonal elements behaving as fermions, because the derivative operator appears only below the diagonal. As this involves some changes of signs from the above, we repeat the transformation laws of the quartet: ffGl = fG21 2wf'G12 SG22 = fG21 2wf'G12 fG'2 (4.57) ffG12 = f(Gl CG22) SfG21 = (fGil)' + 2wf'(G22 Gil). All components except G21 transform covariantly, with weights (, 1,0,1), respectively. Let us define the new constructs A 2G il (Gl  G22); A _= G2n Gn ;n1~ X = ;7, C = G12 D = In G12, wG12 wG12 lnG12, in terms of which the transformation laws read (using (2 = X2 = 0) fD = f(, Of C = fD', (4.58) OfX = fA 2f' + fx(, (4.58) Of A = (fx)' 2f'C fxD' fAC. 54 These are nonlinear, but A transforms exactly like the bosonic connection. It is interesting to note that the point D = constant, ( = 0 is stable under superreparametrizations, leaving us with the anomalous doublet 9fX = fA 2f', (4.59) f A = (fx)'. (4.59) Since A transforms as a total derivative, one can then identify A with the derivative of the bosonic 0 field. It is not possible to build an anomaly free rep resentation of the reparametrization algebra with a suitable integration mea sure by just using this doublet.36 We shall see later that it is necessary to use two such doublets for this purpose. 4.3 Construction of a Dynamical Invariant As mentioned earlier, the string coordinates transform into the generalized gamma matrices under a superreparametrization. We can separate F(oa) and 6(a) into left and rightmoving parts as we did for the coordinates: p(a) + i (4.60) S=p (o) ((4) (4.61) These of course transform into the left and rightmoving parts of the coordi nates: .fF = fx'L (4.62) f R = fx' (4.63) They satisfy the commutations ro{ (al), rIF(a2)} = igl6(ali a2) (4.64) 55 {I( Ol), r (o2)} = ig l a2) (4.65) {r (0),r(2)} = 0. (4.66) Here the delta function on the right hand side is defined over the interval [7r,)r]. We note that (xp,F ) and (F',x'j) are both doublets. The latter is more useful since it is translationally invariant. The generator of super reparametrizations for these fields is then d o/L n. X'L (4.67) and similarly = 1 frR 'xR. (4.68) We note that with our normalization for the gamma matrices, /AIL(a) is hermitian. The operators A.f satisfy the classical algebra {Mf, g} = 2iMfg (4.69a) and similarly {.M ,.ML} = 2iMg (4.69b) Also, we have [MfA = A/fg'fIg/2 (4.69c) Here the operator ML now includes contributions from the F's (as given in (4.4)): M 2=7/ f () x2+ rt rLF ) (4.70) While (4.69a) and (4.69c) are fine at the quantum level, (4.69b ) picks up an anomaly upon quantization. For a general representation, it is easy to show that the anomaly in the algebra of ML with ML takes the form f g ML .ML I I [Mf/g = + 2Lr(Af"g + Bfg), (4.71) f 9 M 9,_fg + 27r 56 where A and B are constants which depend on the representation. The anomaly Cf,g in the anticommutation relation {A^jL,ALL} = 2iMfg + Cfg (4.72) can be related to A and B through the Jacobi identity. Specifically, the identity (the superscript L has been suppressed) [I{f, Alg,},Mh] + {[Ag, Mh],..f} {[Mh,kMf],AMg} = 0 (4.73) tells us that Cfg must be C2, =/ (4Afg" + Bfg). (4.74) It can be seen from the commutations that classically, (A L (oa), 2M(cr)) forms a covariant weight (3/2,2) doublet, but the covariance is spoiled due to quantum ordering effects. How can we form an invariant dynamical operator which yields consistent equations of motion? We want invariance under reparametrizations as well as superreparametrizations, so we would like to construct the dynamical invariant as the integral of the heavier component of a weight (1/2,1) doublet.36 Starting from the above (AIL, 2ML) doublet, if we could restore covariance, we could multiply it by a (1, 1/2) doublet to get a (1/2,1) doublet. We recall that the field cL = eOL that we had before was a weight 1 field. So we define its partner 7YL to be a weight onehalf field: ffCL = fTL = i{04f, CL} (4.75) fTYL = (fc' 2fc) (4.76) Henceforth we shall, for convenience, drop the sub(super)script 'L'; it will be understood that all fields(unless otherwise mentioned) have this sub(super)script. 57 Here the generator for this superreparametrization can be written in the form 9h = i f (fc '2f'c)) (477) Now b = e~ is conjugate to c, so is simply the field b. Similarly, the field 4is the field conjugate to 7, which we shall call P. Since 7 has weight 1/2, f must have weight 3/2. Also, b has weight two. Therefore (fl, b) is a (3/2,2) doublet pair: ff3 = fb (4.78) 9fb = (f3' + 3f'3) (4.79) and [(,(')] = (o a') (4.80) The fields f3 and 7 have the mode expansions 7(=) /E 7ne (4.81) 0(0) =i E Onin" (4.82) The modes 7n and /fn are hermitian and antihermitian respectively and satisfy [On, tm] = 6m+n (4.83) Then .Agh can be written in the form 4gh = dij ff(yb + 3c#/ + 2c#') (4.84) f 27r The corresponding reparametrization generator for the ghosts can of course be obtained by anticommuting two A4 operators; apart from a cnumber anomaly, {Mgh, h}= 2iMgh (4.85) where M gh do fp (cb' I~ _t, O/ Mf = f'(c'r + 2c'b + 2 + 2 ) (4.86) The total anomaly in the algebra of A4" = (A4', + Alh) is proportional to 3 c= d 2(6w2 6wb + 1) + 2(6w2 6w8 + 1) (4.87) and so cancels in ten spacetime dimensions. Now we would like to construct an invariant dynamical Lorentz scalar oper ator for use in our Lagrangian. We want to construct it, as mentioned earlier, as the integral of the heavy component of a (1/2,1) doublet. It turns out, however, that the heavier component of the product (c,7 ) a (Ato, 2Mtot) = (cAtot, 2cMtot 7 Jtot) (4.88) is not covariant upon normalordering, despite the fact that the total M and A4 operators are now anomalyfree. This arises because of additional ordering ambiguities in the product (4.88). It turns out that the correct prescription is to include only half the naive ghost contribution to (A, 2M). Then the invariant dynamical operator we have is39,'36 Q = ir L ( ,r() 2cMz,r(o)) + 1 (Y7 gh(a) 2cMgh(O')) (4.89) This hermitian operator is invariant upon overall normalordering, and is sim ply the nilpotent BRST charge of superstring theory. Again, nilpotency here turns out to be a property of the invariant; we do not require it at the outset, but end up with it anyway. The free action constructed from this invariant has the simple form (4.90) 59 We note that this action is secondorder in time derivatives, unlike the usual action for a fermion. The supersymmetry of the theory mixes first and second order operators, and it is therefore necessary to include them both in the construction of the dynamical invariant. However, it can be shown that the gaugefixed form of this action is indeed firstorder. In the next chapter we shall compile a list of invariants and 'fermionize' the superghosts. CHAPTER 5 VECTOR AND TENSOR INVARIANTS 5.1 Invariants in the standard representation One may ask what other invariants it is possible to construct in the bosonic and supersymmetric theories. In the case of the bosonic string, the following objects36 are invariants: (1) The momentum vector J Z rJ 7r 6du A 6  (2) The ghost number NG = _i r r da 6 ^ 0 0O 7r6(o) (3) The Lorentz generators ,.V j "I r do, ( 6 \ O JO ir P x x )v (4) The symmetric spacetime tensor Q V ~7r :r [ L L   L + 30L) : The BRST operator is obtained by taking the trace of the above symmetric tensor: Q = gVQV. (5.1) 61 This invariant tensor depends on the spacetime geometry. Its most interesting property is that its spacetime trace is the BRST operator. We shall shortly look at the algebra satisfied by this tensor. We note that like the BRST charge, this tensor is a ghost number one object. The algebra of this tensor generates another symmetric tensor, as we shall see. The results of this chapter are based on the work of ref.[23,36]. We note that although the dilatation operator D = f f0 x7*r: has the right weight to be a classical invariant, it transforms anomalously due to ordering effects. Thus the largest spacetime symmetry seems to be that of the Poincare group. We remark that there does not exist an invariant 26vector which serves as the string position in spacetime. This is not too surprising since the theory is not (spacetime) conformally invariant. On the other hand, by specializing the Poincare generators to the relevant spacelike surfaces, we can define a physical position for the string in 25 (at equal time) or 24 (light cone) space dimensions. One can now look for a bigger list of invariants in the supersymmetric theory. It is possible to construct invariants in the supersymmetric theory by combining the various doublets present with one another according to (4.30). The fundamental doublets present in the theory are (x,)/ with weight (0, 1), (F,x')P with weight (1 1) 2 (5.2) (e', ,) with weight (1,k), and (I,e0) with weight (3 ,2). As before, we leave it understood that all fields represent left movers only, and that all exponentials of fields are implicitly normal ordered. In taking 62 products of such exponentials, the normal ordering must be carefully taken into account. Using the BakerHausdorf identity eAeB = eA+Be[A,B]/2 which is true for any operators A and B which commute with their commutator, we find ea(ffl) :: eb2) :=: ea^aiic)+b(02) [2i sin Orl2+ ] 0?ab (53) \ 2 (5.3) where e is a small positive number needed for convergence, and 77 is the sign of the commutator of the modes of the field (see eqn. (3.76)). From this we see that if 77 = 1, {e(Ua),e(U2)} = (l 2) (5.4) First consider looking for invariants made up of the product of two of these doublets. We have seen that such an invariant must be constructed from the form (4.30a). To use (F, x')u we would need to combine it with a doublet of weight (0, ); the only such thing here is (x, F)u, and this combination produces a trivial invariant. The other two doublets (eS, 7) and (3, e0) have the right weight to be combined and yield an invariant. The invariant so constructed has the form Jfdo, (C, 10 Oa (,8, e0) = ['7 [7(r ~+ 55 I d(OO ) (5.5) which we recognize as the ghost number (the righthandside above is under stood to be normal ordered). Next we may look for further invariants by taking products of three doublets.36 These may be constructed by taking any two of the above doublets together according to any of the three product rules (4.30), then combining the result 63 of this with another doublet according to (4.30a) (the other two would yield trivial invariants) in such a way as to achieve a final result with weight (", 1). Note that such triple products do not in general satisfy associativity. With the four doublets present, there axe 192 possible combinations, 12 of them with the proper weight to be invariants. These fall into three categories. These we now list: (1) Products which give zero upon integration. These are: {(X,P) Ob (x,(r)} ga (x, r) {(x, r) o (r, x')} a (x,r) and {(x,r) a (x,r)} a (r,x') (2) Certain products involving only the ghost fields. These are: {(e", ) c (3, eO)} Oa (e, 7) {(eO, ) c (eO, Y)} a (, e') These two products are identical when evaluated. They reproduce the part of the BRST charge (4.89) involving ghosts only, which we will denote by Qghost Qghost = if j d : [ +2e + 2y/3'e + 37,(eO)' 11 r(5.6) e0(0'2 + 3" + )/3' + 303y')]: However, this quantity by itself is not invariant after overall normal ordering. (3) Certain products involving both ghost and coordinate fields, which yield a second rank tensor. These are: {(r, x'), Gc (eo, 7)1 Oa (x, r) 64 {(x, r)oc (e, Y)} a (r,x')V {(rf, X') Ob (e,)} 0a (r,X')r {(X, r)p gc (r, x')'} a (e, 7) These four expressions are identical up to total derivatives, and so lead to the same invariant; the result is a secondrank tensor Q0 = fJ d : [O(xlxl' + r'pr1 ) yi'x" + ( v)1 :. (5.7) 0r 71" It is the supersymmetric generalization of the bosonic invariant metriclike tensor we have previously discussed. The diagonal elements of this tensor transform anomalously; however, in d = 10 it is possible to form the anomaly free combination QP = Q&V + 1 pQghost (5.8) As in the bosonic case, taking the trace of this tensor operator reproduces the BRST charge:36 Q = QVg,1, (5.9) 5.2 Algebra of the bosonic string tensor invariants The bosonic string tensor invariants Qv satisfy an interesting algebra. We recall that this tensor was given by the expression Q!' j : [x 6L L + 30L) : 65 This symmetric spacetime tensor is an anticommuting operator of ghost num ber one. Its algebra with the BRST charge Q is23 {Q, QV} = 2iBRv (5.10) where the symmetric tensor BR' is defined by RI"' = I : II(~n g' x): 27r 26 (5.11) = fr da / I V, 9 x() : .f : e27r (xx 26 ' The tensor BP' has ghost number two. It is easy to check that B"' is indeed invariant under reparametrizations. Further, since the BRST charge is the trace of Q', the nilpotency of Q ensures that BJ' is traceless; the statement that BA' is traceless is equivalent to the statement that Q is nilpotent. We note that BIv commutes with QP, and therefore also with the BRST charge. Also, [B",',BPO] = 0. (5.12) The algebra of the components of QI" with themselves is more complicated. One finds {Q", QP'} = i(gVPBP' + gIPBva + gIaBL'P + gVRBt'P) + I.[(g "BP' + gP7BRV) (g9cgVP + gIPVgV g 1 gP)C (5.13) where C is the object da = f drae20L XI XIL 5.4 c=jL cc'x x = if xL 2L L (5.14) C r 2. 27'clX L r 2 This object is invariant, but it is not normalordered. In fact, it has operator anomalies upon normalordering. As a means of projecting out the anomalous 66 part of the algebra, we can introduce a set of 26 x 26 matrices oai such that the projections23 Q = am"Q. = Tr(alQ) (5.15) obey a nonanomalous algebra, where the trace is taken over spacetime indices. Then it can be easily checked that the QI's obey the nonanomalous algebra {QI, QJ} = 2i{aI, aJ},vBPV + 3(BTr(aJ) + BjTr(a1)) (5.16) 13 provided that the a's satisfy Tr(a=a) Tr(a')Tr(aJ). (5.17) 26 We note that the spacetime metric itself satisfies this equation. By taking out the trace part of these matrices, this condition becomes equivalent to the requirement TraI = Tr(alaj) = 0. (5.18) The number p of such independent matrices in d dimensions can be determined from the obvious relation P d(d+ 1) 1 P(P+ 1) (5.19) 2 2 which yields p = d 1. So there are exactly 25 matrices in every such set in 26 dimensions. Of course, there is an infinite number of such sets that one can construct. It is easier to work with a matrices with one covariant and one contravariant index, since then the trace is the usual sum of diagonal elements and the matrix multiplication is easy to do. Then the a's have the standard form a (TrA A) (5.20) 67 where a is a 25vector and A is a symmetric 2525 matrix such that they satisfy the constraints Tr(AIAJ) + TrAITrAJ = 2a aj (5.21) Nilpotency of any one of the a's is equivalent to demanding that A2 = aaT (5.22a) and Aa = (TrA)a. (5.22b) A solution of these equations is in terms of the null vector ap which has the components (TrA, a). Then we have the simple relation Ov = apav (5.23) for the components of the nilpotent matrix. It is easy to see that there can exist at most one nilpotent in the set of the aIs. Such a nilpotent would of course correspond to a nilpotent Q"'projection Q1. It would be interesting to look at the cohomology of this nilpotent. 5.3 Fermionization of the Superconformal Ghosts It is wellknown34 that the superconformal ghosts /3 and 7y can be rewritten in terms of quantities X, rq, and , as follows: /3 = i'e (5.24) 7 = 7e (5.25) 68 The commutation relation between the conjugate fields P3 and 7Y can then be reproduced if we choose the fermionic fields t and r7 to be conjugate, and if X is a field whose modes Xn satisfy the commutations [Xn, Xm] = +n8n,m. (5.26) This field transforms anomalously with an inhomogeneous term: 6X = YX' f'). (5.27) In order for /3 and 7y to transform covariantly with the right weights, 17 and Must have weights 1 and 0 respectively, and ex and eX must have weights 2 and respectively. This is true, since the normalordered exponential eaX transforms covariantly with weight a(a + 2)/2. Of course, the anomaly in the supersymmetry algebra still cancels for d = 10. Upon investigating the super symmetry transformations of these fields, we find that the fields themselves (q, X, 6 and 77) form a nonlinear representation of the superreparametrization algebra. However, various combinations of these fields belong to doublet representations.36 These are as follows: [eO, r/ex] with weight (1, ) [i'eX, e] with weight (3 2) [, eex] with weight (0,1 i(e(eX)' + 2(e)'eX, 77 (i'e2Xe20)'] with weight (, 1) [ex, reoe 2x] with weight (3,1). 2 (5.28) Given these doublets (the first two pairs are conjugate to one another), one can of course compose them to form further doublets. We note the curious fact 69 that one can obtain vector invariants from the following two ways of composing doublets: (, eex) a (F, x')A = [6F, x'1 eeXF]p (5.29) and [ i(e0(ex)' + l(e)'ex),77 (ie2xe2)'] a(x, ) = [iXe((e)' + 1(el)'X), x(7 (i6'e2Xe2)') ir(e _eX)' + ( ) (5.30) In both cases, the heavier components are vectors with weight one and therefore yield invariants when integrated over a; these are X = j (,XP exFP) (5.31) 27r and Y= d o (x"t( (i6'e2Xe2)') iF('(eX)' + 1(e)'ex)) (5.32) T 27r 2" We note that YP transforms like a coordinate under translations; thus an invariant coordinate can be defined in the supersymmetric theory, unlike in the bosonic theory. One can presumably form more of these vector invariants by taking products of more doublets; however, except for the above two, all of these seem to be either total derivatives (and hence trivial), or have anomalies upon overall normalordering. In the next chapter, we shall arrive at a supersymmetric bosonization scheme that uses only fields that form a linear representation of the super reparametrization algebra. CHAPTER 6 SUPERBOSONIZATION 6.1 Construction of the ghosts We have seen that the doublet is the only irreducible representation of the superreparametrization algebra whose components transform covariantly. However, a field (o) transforming according to (3.45) may be used, along with a weight1/2 anticommuting field which we will call s(o), to provide another representation: ff = fs 9fS = (f0' + 2wf') (6.1) Note that this inhomogeneous representation coincides with the usual doublet in the case when w = 0; we shall refer to it as the anomalous doublet. As men tioned earlier, it can be easily checked that a single anomalous doublet does not allow for the construction of an invariant dynamical operator, even though it provides for an anomalyfree representation of the superreparametrization algebra. This is because a single such doublet simply does not have a sufficient number of degrees of freedom; we have seen that the standard representation as well as the previously introduced fermionization required two bosonic and two fermionic degrees of freedom. It is therefore natural to consider the pos sibility that two such anomalous doublets46'47'48 might provide a satisfactory representationclearly they would naively possess the correct number of de grees of freedom. The results of this chapter are based on the author's work in ref.48. 71 For the above multiple (0, s), the generators have the form S+ _i /Tf' ] (6.2a) , f + (fs f+0' +f2w')l (6.2b) f27r TO$ 6s As before, this type of multiple may be separated into left and right moving pieces, which are defined by the relations OR + OL (6.3a) 2 = L'R ) (6.3b) S = sL (6.3c) 2 = (SR L) (6.3d) where 77 = 1. All left movers commute with right movers over the interval [0, 7r]. The generators Mf and .Mf spilt into pieces containing only one type of mover: M = Mf + Mf (6.4a) A.f = + .A (6.4b) The generators for the standard doublet can be similarly split into left and right movers only if its weights are (0,1/2). It is a remarkable fact that the string coordinates xP and their superpartners form a multiple with precisely these weights. For the rest of this chapter we shall only deal with leftmoving fields and it is understood that similar remarks hold for right movers. We saw earlier that the ghosts of bosonic string theory may be described either in terms of the anticommuting variables b(a) and c(a), or the commuting variable 0(a). The relations between these quantities were as follows: b =: e( : (6.5a) c =: e : (6.5b) The exponentials of 0 satisfy the product relation(since 77 = 1 for the field ) e ao(a) e bo(2) : (2i sin ( a2) ab ea.)+b0(C2) (6.6) Using (6.6) we may invert (6.5) as a) = i c(a)b(a) (6.7) It is not obvious that the Fock space created by the modes On of q(a) is isomorphic to that created by the modes of the fermionic ghosts b(a) and c(oa). There is a wellknown proof38'49 of this equivalence, using Jacobi's triple prod uct identity to relate the partition functions. Here we give another argument. In either the fermionic or bosonized ghost representation, the full Fock space may be generated by acting with the Virasoro operators on a certain subspace which is referred to as the highest weight states. These are defined to be those states which are annihilated by Ln for all n > 0. Acting with the other L's (those with n < 0) reproduces the full Fock space. The space of highest weight states is labelled by the eigenvalues of normalordered operators which commute with all the L's, i.e. which are reparametrization invariant. Using only the fermionic ghosts, the only such operator is the ghost number, defined as NG = : b(a)c(a): (6.8a) With the bosonized ghosts, the only such operator is the zero mode pd =(6.8b) 73 These two quantities have the same eigenvalue spectrum, and in fact (6.7) shows that they are actually identical except for a factor of i. Thus the space of highest weight states is the same in both representations. As we have seen, the ghosts in the supersymmetric theory could be bosonized according to /3 = iW'eX (6.9a) 7 = 7/ex (6.9b) This bosonization of superghosts does not have supersymmetry in the new variables ), X, (, and 17, as mentioned earlier; their transformation laws under supersymmetry are nonlinear. For instance, the field X' transforms as follows: #fX' = i (f(i' 'ex + rex) + 2if''eOX) (6.10) Thus the superreparametrization invariance of the theory is no longer as sim ply implemented, and this can be inconvenient for some applications. An alternate bosonization of the conformal and superconformal ghosts has been introduced46'47'50 which does not sacrifice the superfield structure of the ghosts. This bosonization is as follows: b = ue (6.11a) 2 c = uea' (6.11b) 7 : (9/ au)ea : (6.11c) 1 / 1 = (6.11d) 2a In (6.11), (V, u) and (s, ii) are supermultiplets transforming according to (6.1). Under reparametrizations, 9o and i3 transform like bosonized fields, i.e. 74 with an inhomogeneous term as in (3.45). The two multiplets are defined to be conjugate to each other in the sense that [4=0 2id (O7l G2) (6.12a) {u(71), ii(72)} = 2i6(al  a2) (6.12b) [=(l), (2)] =0 (6.12c) [(al), (u2)] = 0 (6.12d) {U(M ), u(72)} =0 (6.12e) {U(OI), i(72)} = 0 (6.12f) At this stage, we have changed our conventions a little; the commutations of the conjugate fields b, c and #, y now have factors of i and i respectively due to the commutations (6.12). Because all modes of p commute among themselves, the exponentials in (6.11) have their classical weight, namely wa, where wV is the coefficient of the inhomogenous term in (3.45). Since the weights of u and ii must be because of (6.1), we must have wa = _. Also, we must have wo = a in order to maintain covariance of y. Then all of the ghosts transform with the appropriate weights. It is easy to partially invert (6.11) to obtain S= 2a/c (6.13a) ' = 2a : (7,/ + bc): (6.13b) We recall from the previous chapter that this latter expression is just propor tional to the superghost number. 75 We will now derive (6.11) using the representation theory of the super reparametrization algebra. Consider a pair of selfconjugate doublets (w, s) and (,, t), with the transformation laws48 fw = fs (6.14a) fs = (fw' + 2wf') (6.14b) fo = ft (6.14c) ft = (fo' + 2vf') (6.14d) where w and v are cnumbers. As before, we consider leftmovers only. The fields w, f, s and t have the respective Fourier expansions O(U) = Wo Op + i : Lo ina (6.15a) n O n 0( = 00 ape + i E nina (6.15b) nOO n s(a) = sne1in (6.15c) t(o) = Ztn en (6.15d) where the modes satisfy [Wm,Wn] = mSm,n (6.16a) [Pwwml = i6mO (6.16b) [Im, bn] = m6m,n (6.16c) [PP, Om] ibm,O (6.16d) {Sn,Sm} = i6m+n (6.16e) {tn,tm} = im+n (6.16f) 76 The choice of signs in the commutations above is necessary in order to re produce the ghost algebra. The generators for the reparametrizations and the superreparametrizations can of course be easily written down. With the choice of signs we have made, the total anomaly in the algebra of Af with A.g is proportional to B = d + 2 w2 + v2 (6.17) 8 We now investigate the question of what quantities may be formed with these fields which will transform as the ghosts. The basic covariant doublets are ( aw aseaw) and (ebe, btebe). for any constants a and b. As usual, these doublets may be combined with the rules (4.30) to yield additional covariant doublets. There is no ordering problem at this stage. The results are / ea b ab(6.18a) ((as + bt)eaw eb(. ab [(w + a/2)t (v b/2)s] ea eb O a(w + a2)eaw(eb)' b(v b/2)(eaw) eb ) (6.18b) + [a(w + a/2) + b(v b/2)] absteaw eb 2b(v b/2)(eaw)IebO 2a(w + a/2)eaw(ebO)1 + absteaebV 2ab(v b/2)seaw(ebO)' a(2a(w + a/2) + l)(seaw)'eb j (6.18c) +(2b(v b/2) + 1)bt(ea))eb 2ab(w + a/2)eaw(teb)' The lighter components of these doublets transform with weights a(w + a/2) + b(v b/2) plus 0, 1/2, and 1, respectively. Since c is anticommuting, 77 and the lighter component of its multiple, its multiple must be (6.18b ) for some a, b. This has the right weight if a(w + a/2) + b(v b/2) = 3/2 (6.19) Then we must have {C(7l),c(u2)} =0, which will be true only if a2 = b2 (6.20a) and (w + a/2)2 = (v b/2)2. (6.20b) Thus we see a(w + a/2) = b(v b/2) = 3/4. Then (6.17) is satisfied if the number of spacetime dimensions d is ten. We will choose a = b. Then (removing overall multiplicative constants) c = (t s)eawea (6.21a) S= (0' J + 2ast)ea, ea (6.21b) We obtain the conjugate doublet (/f, b) from (4.30a ) by taking the opposite value for the constants in (4.30a): / = e aweap (6.21c) b = a(s + t)eea (6.21d) To make the connection with (6.11), we define the combinations O = w + 0 (6.22a) (6.22b) 78 u = s + t (6.22c) u= s t (6.22d) Substituting (6.22) into (6.21), we recover (6.11). We now turn to the question of whether the spectrum of states is equiv alent in the superbosonized representation.48 Actually the question is easier to answer here than in the bosonic theory, since here both before and af ter bosonization, the theory posesses two fermionic variables and two bosonic variables. However, the superbosonized fields p and <3 both have invariant zero modes. We need to determine the spectrum of eigenvalues for these op erators. The superghosts /3 and 7 satisfy boundary conditions of the form 3(a + 27r) = +3(a) ( in the NeveuSchwarz sector, + in the Ramond sector). From (6.l1d ) we see that the eigenvalue spectrum of py which satisfy these conditions is pp = i22714 (NS sector) or in (R sector), where n is any integer. Since b and c must be single valued, the modes of u and iu will be halfintegral in the NS sector and integral in the R sector. Finally, (6.11c ) does not put any constraint at this stage on the eigenvalues of the zero mode pC. However, (6.13b) shows that 2p is equal to the superghost number, so its eigenval ues are halfintegral in the NS sector and integral in the R sector. The other invariant zero mode pp generates a set of eigenstates that are not present in the standard representation. Furthermore, in the superbosonized representation, the zero mode Virasoro generator L0 has the form L gh =~ 1 2 E ( Un nnnunn + Sn'n + n+cn) + pppoP (6.23a) n>0 By comparison, with the usual ghosts, LO = E n(bncn + cnbn + 0nTn 7nin) (6.23b) n>0 79 Comparing these two expressions, we see that while (6.23b ) is bounded from below, (6.23a), because of the term pp is not bounded from either direction. Clearly, the space of states is different in the two representations. Some sort of truncation of the spectrum is therefore necessary if we want to have equivalent state spaces. By restricting our attention to those states in the theory which satisfy py = 2a2pp, we get in L0 a term proportional to N2, which makes L0 = SG, bounded from below48 and agrees with the superghost number dependence of (6.23b ). 6.2 Construction of Invariants Let us now consider the question of what invariant operators exist in the superbosonized theory, in particular, the BRST charge18'19, which is normally constructed as a product of doublets. Neglecting ordering effects for the mo ment, we may combine (c,7) with the doublet (A.L(a),2ML(a)) using the rule (4.30a) to form the covariant doublet (Pl((), QcI(a)). Then we find Pcl = (3a(t s)r. x' tswl + 3 P i 2 2 + 3(s t) [(a2/2 + 3/4)s' (a2/2 3/4)t'] )eawea (6.24) and Q = 3a(t s)(x'2 + rrF' w 2 + 2ww" + 012 2v" s's + t't)eaw a +[3(L' /') + 3a2st][r x' sw' + 2ws' + tO' 2vt'eaw a. 2 (6.25) Here w and v are the weight parameters of w and f respectively, given (from (6.20) and (6.19)) by a 3 w ( + 4) '2 4a' and a 3 2 4a Unfortunately, ordering effects spoil the covariance of these quantities. We can remedy this problem by adding terms to Q in order to make it both nilpotent and truly superreparametrization invariant. To find all possible invariant op erators we will simply write down all possible covariant quantities and try to assemble an operator doublet (P(o), Q(cr)) for which the integral of the heavier weight component Q = f dcrQ(ar) is invariant.48 It is simplest to first deter mine the form of the lighter weight component P(a) and transform it to get Q. We write down all possible terms Pi which are weight up to anomalies. The correct P will be some linear combination of these. We then require that P transform as the lighter component of a doublet of the standard form (4.25), and that the anomalies cancel. This is accomplished by demanding that the f" and fill terms in 6fP and the f' and f" terms in f P add to zero. These restrictions select out a four parameter set of solutions. This calculation is described in Appendix B. The result is Q = AQA + BQB + CQc + DQD (6.26) where A,B, C and D are any constants and QA = (2a2 + 9)Qi + 3aQ2 + 3aQ3 2aQ4 d (6.27a) 4a(9 + a2)Q5 + 4a3Q6 2a(27 + 2a2)Q7 + 4a3Q8 54aQ9 QB =d(2a2 + 9)Qi 3aQ2 3aQ3 + 2aQ4 + 4a3Q5 + 4a(9 a2)Q6 + 4a3Q7 + 2a(27 2a2)Q8 54aQo10 (6.27b) 18a 2 QC= (9 + 2a2)Q1 3(2a2 9)Q2 3(27 + 2a2 )Q + 2(27 + 2a2)Q4 + 4a2(2a2 9)Q5 4a2(27 + 2a2)Q6 (6.27c) + 4a2(27 + 2a2)Q7 4a2(27 + 2a2)Q8 + 54aQll QD = 18a(2a2 9)Qi + 3(27 + 2a2)Q2 + 3(9 + 2a2)Q3 2a)a 2(2a2 27)Q4 4a2(2a2 27)Q5 + 4a2(9 + 2a2)Q6 (6.27d) 4a2(2a2 27)Q7 + 4a2(27 + 2a2)Q8 + 54aQ12 where the Qi's are defined as follows: QI =((t s)(x'2 + F'r) + (01 w' + 2ast)x' )eawea (6.28a) Q2 = a[s" + 2as'' + 2s(aw" + a2w/2)]e awea (6.28b) Q3 = a[t" + 2at'O' + 2t(a4" + a2e2)]eaea (6.28c) Q4 = a2[8'0 + a(s + t)w't' + t,']e]aea (6.28d) Q5 = [atw'2 + ass' t + asw oeaea (6.28e) Q6 = [aso12 astt' atw'O']eaeaO (6.28f) Q7 = [tw" ass' t + s 101]eaweao (6.28g) Q8 = [t'w + s" ast't]eaweaV (6.28h) Q9 = [sw" + s'w + ass' t]eeaea (6.28i) QlO = [tO" + t'' astt']eaw a (6.28j) Q1 = [s" + asJ' + atwjeawea0 (6.28k) Q12 = [t" + at+ + aset/IeaweaO (6.281) 82 We now need to investigate whether any of the invariant Q operators ob tained from the above Q(a)'s are nilpotent. Consider a general Q of the form Q 1 (2cMx'r(OT) 7_YX'rF(0)) + Qgh (6.29) Using the (anti)commutation relations of the A4 and and the M operators, Q2 = 0 is equivalent to the conditions48 {Qgh,C(O)} = i (17(a)2 2c(a)c'(a)) (6.30a) [Qgh,()] = i(c'(a)7(a) 2c(aT)7'(a)) (6.30b) Q2h = /J (4 )/( ) + c"'(a)c(a)) (6.30c) The expressions on the right hand side of (6.30a )can be evaluated in terms of the new ghosts; for instance, the first one is j( 12 2cc') = i: (12 a 'uu 2Hi' ap + 2au2u) e2ap :, (6.31) 2 \2 We find that (6.26) satisfies these conditions for A = 1/3 (6.32) B = 1 (6.33) (9 + 14a2) (6.34) D = (2a2 + 3) (6.35) 8a3 (6.3) so that this combination (up to an overall constant) is indeed a nilpotent operator. We can of course also derive the expression for the BRST charge in terms of the new ghosts by substituting for the ghosts in the old expression for Q and redoing the normalordering. In terms of the old ghosts, we have Qgh = 2 J : y(7b 2/3'c 3/0c') c(4c'b 3/y/) : (6.36a) 83 The terms in Q expressed in terms of the new ghosts are as follows: '2 2b = ( + 2au'(' 2a2iuu' auO")eap (6.37) 2 7 10'c= + + ('p + iv")a (6.38) 2 2"2 cc 2 2 au (6.39) 71c'= (ua"  + upi' 2aiip")eao (6.40) 2u a = (uip" + auip'' aii'uui)ea (6.41) All expressions on both sides of this equation are understood to be normal ordered. For completeness, we give the final form for the integrand of the nilpotent operator Q: Q(a) =2iea9MJ'r (' aiu )eaA./Jlr (u12 + 2au'(' 2a2iuu' au3" 1u y (6.42) 2 222 3 ~, ,3 ,1 221 1 a + 2au '' + u'ilu 2a Vu 2 + 2a2 ii2 + 4aii'cJ)ea We note that this differs somewhat from the expression given in ref.46. As another application of our methods, we consider the construction of the picturechanging operator.34'17'48 This operator has weight zero and is constructed as the anticommutator of the BRST charge with the field (O) (see (6.9)). The bosonized field X can be written in terms of the new fields as (this can be seen from the operator product y/) X = ap'  (6.43) 2a 2 so that the relation eX = ,ga/2a (6.44) 84 holds. Also, the fields ' and ,7 can be written as = ue/2a (6.45) S= (2au' + u')e0/2a (6.46) Since the picture changer has a term of the form ex x u ea/2ap x (6.47) we can use our method to write down a general weight zero operator with this term in it. We note that it is essential for the picture changing operator to transform without any f' or f" terms under 1, since all amplitudes calcu lated with it must of course be invariant under superreparametrizations. We again find a four parameter family of operators, this time of weight zero. The independent solutions are (with the constant a = 1 for convenience) 11 1 PA=P1 + P5 P6 + P7 P8 10 2 2 (6.48a) + P9 + P1O 2 2 7 7 7 21p PB=P2+P P6 + P7 21 P8 11 4 4 8 8 (6.48b) +11P 21P1 1 P + 8 4 P 81= P 45P6 + P7 135P Pc=P3P 5P6P64 P8 1 71 4 38 8 (6.48c) PD =P4 + 9P5 11P6 + 5 P7 5 P8 2 8 4 (6.48d) P9 33 P10 + P11 3 (3 + the P12 2r4 8 8 Here the Pi's are given by Pl = x F(t se e32 (6.49a) 85 P2 = (ew/2)te3/2 (6.49b) P3 = e/2(e3/2)" (6.49c) P4 = (ew/2)'(e3O/2)' (6.49d) P5 = st(e/2)'e3/2 (6.49e) P6 = stew/2(e3O/2)' (6.49f) P7 = Stew12e3/2 (6.49g) P8 = stl'ew/2e3/2 (6.49h) P9 SS=: : ew/2e3/2 (6.49i) P10o =: t': e/2e/2 (6.49j) Pl = wi"ewl/2e3b/2 (6.49k) P12 = O"ew/2e3O/2 (6.491) Any combination of PA, PB, PC and PD is of course a weight zero opera tor. By comparison of coefficients, the usual BRSTinvariant picturechanging operator corresponds to the combination 9228 10568 p, 48p X = 1PA 9 6 B + 16 C D (6.50) We do not yet know if other combinations of these four operators exist which are also BRSTinvariant. It would be of potential interest to find these, if they do exist, since they would be of particular use in the construction of interactions for superstring field theories. 6.3 Summary We have shown that a superbosonized representation of the superconformal ghosts in terms of two doublets can be obtained using our algebraic techniques. 86 The space of states was shown to be larger than the usual one and a correct subspace was identified by means of a suitable constraint. A search for dy namical invariants produced a hitherto unknown fourparameter class of such objects. These new objects, being dynamical invariants, are worthy of fur ther investigation. The picturechanging operator of superstring field theory was identified as one member of a fourparameter class of weight zero opera tors which change the picture number. The existence of these operators offers interesting possibilities for building interacting superstring field theories. APPENDIX A REDUCIBILITY OF THE SUPERREPARAMETRIZATION REPRESENTATIONS We shall start with the master equations (4.45) for the transformation of a doublet (a, b) into (A, B). We shall assume, as before, that the bosonic or fermionic character of the light component is left unchanged36 by the transfor mation matrix. We recapitulate the master equations here for convenience: d (fFll)a = F12(f + 2waf')a fF21a (ff F12)b = Ffb fF22b (A.1) (f fF21)a = F22(fd + 2waf')a f(Flla)' 2wAf'Fl1la (ffF22)b = F21(fb) f(F12b)' 2wAf'Fl2b We expand each of the F's in a finite series of derivative operators: F = Gn dn dan z dcr" Let the highest order derivative operator appearing in the expansion of F21 have order N. Then it is easy to see that we have two families of representations. In the first case, the highest derivative operator in the expansions of F11 and F22 has order N, and that in F12 has order N 1; in this case we have the constraint that GN = GN "11 = 22 Thus the representations in this case consist of (4N + 2) independent fields (the G's). In the second case, the highest derivative operator in the expansions of Fll, F22 and F12 has order N 1; in this case we have the constraint that =N GN1. (A.2) 21 8712 87 88 The representations in this case consist of 4N independent fields. Note that this case can be obtained from the previous one by setting GN = GN = 0 and 11 22 imposing the constraint (A.2). In either case, we can obtain the equations for the supersymmetry trans formations of the G's by equating the coefficients of derivatives of a and b in the equations (A.1). We get the following equations: N / N fG rl = n Gr2f(nr+l) ( n ) 2wa n n 12f(nr+l) (n) fGr 1G = 1: G^J 12 r ) 2wal: G1 r^ f G21 n=r1 v / n=r N N NN ffG12 =E Gn f(nr ) r2  n=r N / \N \ ifGnr N >1 ~n jf(nr+l) (n N n 2war f(nr+l) (n) rf G2 1 Z: G22 [r + 2wa 1:G22J rI n=r 1 / n=r Grl' fGrl1 2wAf'GI N f = G Glfn, (n) f(Gr2' + G21) 2wAf'Gl2 n=r (A.3) If we define a generalized covariant derivative operator of order N as N N ndn 0 = : Af dun ^4 dan n=0 and demand that it act on a covariant field F of weight wF to produce a new covariant field of weight (wF + A), we can read off the transformations of the An from (3.35). They are NI\ f A r= DAr)Ar+ [ +wFQ1] f(ml)Am. (A.4) m=r+l v It can be checked that the above transformations on the G's indeed satisfy f f= 6/' 89 so that they indeed form a representation of the supersymmetry algebra. Under a reparametrization 6f, G'I and G' transform with weight (A r), Gr2 with weight (A r 2), and Goi with weight (A r + 1), apart from anomaly terms which have the same form as in (A.4). Specifically, N frl n r1+ z r(r)m + wa(m)]f(mr+l)Gm 11 = D(A )G + 1 \+ + Wa (Ml f( 11 m=r+l = (Ar) + m [ ) + (1 + Wam f(mr+l)Gm bf G12 = D 2G + = 2+ (Wa + 1)(n)] f(m 1r+l) 2 m=r+l I G 5f (Ar!) mr 6f G^' =D22+ G rr1 +( Wa ()] f( 1 22 m=r+l 2 \r]J (A.5) We shall now consider the reducibility of these representations. As we have shown earlier, the only irreducible representations in terms of covariant fields are doublets. Our modus operandi shall hence consist of starting from the lowest weight field in the representation (which necessarily is the member of a covariant doublet) and working our way up the weight 'ladder', trying to form a covariant doublet at each stage. The existence of a new doublet at each level implies that the fields in all the previous levels decouple completely from those at this level and at all further levels. To illustrate this procedure, let us first look at the (4N + 2)field representation. The lowest weight field in this multiple is GNl(= G2), which has weight (A N). This transforms as SfGN f(Gl1 N +G). 11^ (12 + 21) We look now for a different linear combination of GN1 and GN which trans forms as the light component of a doublet(i.e., the transformation does not involve f). We find that 1(GN1 +aGN) = f [(1 a)(GN1 G22Nl) aGN] +(N+aN2aA)fG. (A.6) Requiring that the f' term vanish, we get a = N/(2A N) (A.7) So the combinations (2A N)GN1 + NGN and 2(A N)(GN1 GN1) NGN form a new doublet, provided that a~l, i.e.,A 0 N. If A = N, this doublet is the same as before; therefore, there is no reducibility at this level. In this case, as we shall see in a moment, the next level separates out, leaving us with an irreducible quartet at this level. Continuing this process, let us consider f (Gl1 +G 1+ ') = f [(1 + )(G1N2 + GN1) + (/ + 9')GN1' + yGN/] (A.8) f' [(N 1 + 2wa + 2WA + 7)GaN1 + (N + 9)GN] This combination transforms without the f' term if we choose S2wa + N 1 and N(N 1 + 2wa) (A.9) = N 2wA 2w N N (A9) Hence the combinations (N 2wA)Gf1 + (N 1 + 2wa)(GN1 NGj<) and (2N 2A 1)(GN2+ GN1) + (N + 2w) ((1 N)G 1' NGN') form a new doublet, as long as A # Ni. If A = N 1, there is no reducibility at this level; so far only the first doublet has decoupled completely. The decomposition of the sextet presented in the third chapter follows this same pattern. Going a step further up the ladder of weights, we can now look at the transformation of a different linear combination of the fields GN2 GN, 12 '21 GN1' and GN', namely, 12 2 (G1N2 + ,G2N1 + ,GN'11 pGN. ( A.10) 12 + 21 N12i (A.1O) This time, however, in addition to the fl term, there is an f" term; both of these terms must vanish if we want a reduction into doublets. This yields four conditions for the three parameters ji, v and p, which are in general consistent only if the relation (2N 1 2A)(A + 2Wa) (N 1 + 2wa) = 0 (A.11) is satisfied. We note that this relation has A = N 1 as a possible solution; thus, in this case, the representation is reducible at this level. This is in complete contrast to the previous two stages of reduction, where the doublets would decouple except for special values of the weights. If A = N 1, three doublets have by now completely decoupled. For higher levels, the number of constraints increases faster than the number of coefficients in the combinations of fields. Then reducibility breaks down in general, leaving us with larger and larger irreducible(noncovariant) multiplets. 92 Next we consider the case of the 4Nfield representations. Now the lowest weight field in the multiple is GN1, with weight (A N + ) and GN = GN = 0. We have fG1 = f(GN G1). (A.12) As before, we consider the transformation of a different linear combination of the fields on the right hand side: f (GNil+ +GNl) f ((1 + )(GN2 G+ + ) aG1 (A.13)' (A.13) f'(N 1 + 2wa aN + 2awA)G'1 The f' term vanishes if we choose N 1+ 2wa a 2wA N Then the combinations (2wA N)G N1 + (N 1 + 2wa)GN1 and (2wA + 2wa 1)(G1N2 + GN1) + (N 1 + 2wa)GfN1' form a new doublet if a $ 1, i.e., if Wa + wA # 1. Let us move on to the next level and look at the f' term in (Gi/N+ # NV 1 + 1 rNl'1 1(+2 + 21 + 12 ) which is f' ((N 1 + y)GfN1 ,3(N 1 + 2wa)GN1 GfN1). This is zero if and only if /3 = Y = 0 and N = 1, in which case this level does not even exist. As we go up to higher levels, we find as before that there are too many constraints on too few parameters, so that in general, only the first doublet decouples completely from these representations. 