Reparametrizations in string field theory

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Reparametrizations in string field theory
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viii, 102 leaves : ; 28 cm.
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Viswanathan, R. Raju, 1963-
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Physics thesis Ph. D
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Thesis:
Thesis (Ph. D.)--University of Florida, 1989.
Bibliography:
Includes bibliographical references (leaves 99-101)
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Typescript.
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Vita.
Statement of Responsibility:
by R. Raju Viswanathan.

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Table of Contents
    Title Page
        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. Super-reparamentrizations
        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 super-reparametrization 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
Full Text










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 room-mate 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. FG05-86-ER40272 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 SUPER-REPARAMETRIZATIONS............................. 39

4.1 The Covariant Formalism .................................... 39

4.2 Linear Representations of the
Super-reparametrization 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
SUPER-REPARAMETRIZATION 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 gauge-covariant 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 anomaly-free algebra in the

process. The analysis is extended to superstrings and representations of the

super-reparametrization 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 ten-dimensional world, as opposed to the four-dimensional 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 non-perturbative

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 gauge-covariant 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 Banks-Peskin20 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 gauge-unfixed equations of motion. The

gauge covariant equations of motion can be obtained from a gauge-fixed set of

equations by the process of successive gauge-unfixing.

Secondly, the role of the reparametrization algebra as a fundamental sym-

metry in bosonic string field theory is studied. The Marshall-Ramond 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 one-dimensional

reparametrizations. The theory is quantized and the anomaly in the algebra

is found. The states in the theory are characterized and a normal-ordered

invariant dynamical operator is constructed, the BRST charge.

Thirdly, the theory is extended to include super-reparametrizations. The

Marshall-Ramond 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 non-anomalous 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 four-parameter 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

four-parameter class of weight zero operators which change the picture num-

ber. The existence of a family of weight-zero operators raises the possibility

that there might exist other BRST-invariant picture-changing 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 gauge-covariant 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 non-zero 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 gauge-covariant equations of motion since one needs to know the off-
shell content of the theory to describe interactions. Since we already know
that the gauge-covariant 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 two-dimensional 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=Kco-bo +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 Banks-Peskin 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 (n-m) 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
two-fold 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 zero-form (0 corresponds to the
physical string field that satisfies the gauge-fixed 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 -- dAnn-1 + aAnn~ n-1


(2.22a)






9

S =-KAn+ + dXn-1 + 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 gauge-covariant 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 non-zero 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 zero-form 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 non-diagonal 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 zero-form 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 +KO-v2 =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 gauge-fixed 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 gauge-fixed
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 + 1-TY = 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 + 0-i = 0 (2.76a)

dw + dA + (K + K)rI + 1-y1
(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
-Krn-1 + I+1 + &9 + Ln-1 = 0 (2.83a)
n nn+ 1 n n-1

-K -1 + i + + + dn_ = 0 (2.83b)

-Knn + K\n + n+ dni = 0 (2.83c)

rin-1 + =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 one-dimensional 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 ( 6rP (a1)P






22
where 6(a1 0-2) 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 Marshall-Ramond 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 (u-oa')
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 space-like 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 Klein-Gordon equation. If the equations of motion are of the form

Ah = 0 (3.22)

where Ah is a 'kinetic' operator, they must be covariant(form-invariant) 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 Va-72^)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 non-covariant 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 one-dimensional 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 x-cos 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

( (3-51)
9 i
,a = (an, + a-n,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 L-n
where the L's are the Virasoro operators

Ln 1 0 an-m am (3.64)
m=-oo
We cannot demand

Mn\[ >= (Ln L-n)\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 normal-ordered 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 anomaly-free algebra

[Mn, Mm] = (n m)Mn+m (n + m)Mn-m. (3.67)

This means that the reparametrization generators are covariant operators even
upon normal-ordering. We note that, by construction, M(o)(as well as its left
and right-moving pieces) is a weight-two operator.






32
The normal-ordered exponentials eik*'L(T) transform covariantly38,39 with
weight k2/2. Similar normal-ordered 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 operator-valued 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 + L-n) (3.70)

Classically, the 0 operator transforms covariantly with weight two, as can be
seen from its commutations with My. The normal-ordered operator, however,
transforms anomalously due to the central charge term in the Virasoro algebra:

[Mn, Dim] = (n m)Dn+m + (n + m)On-m D(n3 n)(6n+m,0 + bn-m,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 ill-defined 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 one-dimensional 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 one-dimensional 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 f3-n) (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 e-in (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 right-moving 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 right-moving 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(Cn-mom) + 2- ) (3.86)
M
The normal-ordered 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 normal-ordered 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)eaOe-aG(,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 minus-one 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 left-moving part of the reparametrization density

including coordinate and ghost contributions. We note that we cannot mix left

and right-moving 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 single-valued 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 half-integer. 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 twenty-six spacetime
dimensions.
The operator K must be an overall normal-ordered expression for it to
make sense. This means that we still have to check the invariance of K after
it has been normal-ordered. 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 right-hand 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 twenty-six 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 l-22 -ab (3.95)

we can see that cL = eL and bL = eCOL are conjugate anti-commuting fields;
these are the usual anti-commuting ghost and anti-ghost respectively of the
bosonic string.






38
As mentioned earlier, the states are labelled by the eigenvalues of -i00,
which are half-integer; this is simply the ghost-number operator. The free field

theory action is given by
S =< |IQI| > (3.96)

and it yields the usual equation of motion

Q|I >= 0. (3.97)

As a consequence of the nilpotency of Q, this has the well-known 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
SUPER-REPARAMETRIZATIONS

4.1 The Covariant Formalism



The Marshall-Ramond extension to include super-reparametrizations in the
formalism is based on the introduction of two anti-commuting quantities, the
generalized Dirac gamma matrices Fq (a) (i = 1,2). These hermitian operators
obey the anti-commutation 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 weight-one object, these fermions are weight-one-
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 weight-one-half 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
Dirac-like first-order equations of motion for the string field, the Lagrangian
density must also contain a first-order operator.
Since Fr(a) is a weight-one-half 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'/2-f'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] = i-fgI/2-f'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 super-reparametrization 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 Super-reparametrization 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 super-reparametrizations.
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 super-reparametrization 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
An-fb = 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 non-zero 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 super-reparametrization
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 super-reparametrization
invariant, as is evident from the transformation law (4.24). The integral over

the light component is never super-reparametrization 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 super-reparametrization 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
non-covariant 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 super-reparametrization
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 super-reparametrization 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 super-reparametrizations, 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 super-reparametrizations

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 super-reparametrization 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 dn-ma +-
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 dn-m
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 dn-m
oaw Am -"n aw + faw+
m=l


(4.41)


and


fj dm f dn-m + 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 =AO--n (-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 super-reparametrizations. 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 =+ F11f-fF22

f 1F21 =(F22f fF) f df 2waF2d 2wAFd
j-o, 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 off-diagonal 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^ _-(f-d+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+(1-2A)G21 (4.52)

form a third new doublet, except for A = 1. Thus, except for these two values
of A, this six-field 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,G22-Gll-Hil). (4.53)

However, whenever Wa : 0, we know that the transformation of G21 under
reparametrizations has a non-covariant 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 super-reparametrization 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 off-diagonal
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 2-G 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 non-linear, but A transforms exactly like the bosonic connection.
It is interesting to note that the point D = constant, ( = 0 is stable under
super-reparametrizations, 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 super-reparametrization. We can separate F(oa) and

6(a) into left and right-moving 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 right-moving 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
super-reparametrizations, 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 one-half 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 super-reparametrization can be written in the
form
9h = i -f (fc '2f'c)-) (4-77)

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 anti-hermitian 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 c-number 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 normal-ordering, despite the fact that the total M and
A4 operators are now anomaly-free. 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 normal-ordering, 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 second-order 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

gauge-fixed form of this action is indeed first-order. 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 space-time 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 space-time 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 space-time symmetry seems to be that of the

Poincare group. We remark that there does not exist an invariant 26-vector

which serves as the string position in space-time. This is not too surprising

since the theory is not (space-time) conformally invariant. On the other hand,

by specializing the Poincare generators to the relevant space-like 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,e-0) 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 Baker-Hausdorf 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 Orl-2+ ] -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, e-0) have the right
weight to be combined and yield an invariant. The invariant so constructed
has the form
Jfdo, (C, 10 Oa (,8, e-0) = ['7 [7(r ~+ 55
I d-(-O-O ) (5.5)

which we recognize as the ghost number (the right-hand-side 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, e-O)} 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 second-rank 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 metric-like
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 normal-ordered. In fact, it has operator
anomalies upon normal-ordering. 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 non-anomalous algebra, where the trace is taken over spacetime indices.
Then it can be easily checked that the QI's obey the non-anomalous 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 25-vector 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 well-known34 that the superconformal ghosts /3 and 7-y 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 7-y to transform covariantly with the right weights, 17 and
Must have weights 1 and 0 respectively, and ex and e-X must have weights
-2 and respectively. This is true, since the normal-ordered 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 non-linear representation of the super-reparametrization
algebra. However, various combinations of these fields belong to doublet
representations.36 These are as follows:

[eO, r/ex] with weight (-1, -)

[i'e-X, e-] with weight (3 2)

[, e-ex] with weight (0,1

-i(e(e-X)' + 2(e)'e-X, 77 (i'e-2Xe20)'] with weight (, 1)

[ex, re-oe 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:
(, e-ex) a (F, x')A = [6F, x'1 e-eXF]p (5.29)

and

[- i(e0(e-x)' + l(e)'ex),77 (ie-2xe2)'] a(x, ) =

[-iXe((e-)' + 1(el)'X), x(7 (i6'e-2Xe2)') 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 e--xFP) (5.31)
27r

and

Y= d o (x"t( (i6'e-2Xe2)') iF('(e-X)' + 1(e)'e-x)) (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 normal-ordering.
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 super-reparametrization algebra whose components transform covariantly.

However, a field (o) transforming according to (3.45) may be used, along with

a weight-1/2 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 anomaly-free representation of the super-reparametrization

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

representation-clearly 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 left-moving
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 well-known 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 normal-ordered 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'e-X (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' 'e-x + rex-) + 2if''eO-X) (6.10)

Thus the super-reparametrization 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 self-conjugate 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 c-numbers. As before, we consider left-movers 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
super-reparametrizations 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 -awe-ap (6.21c)

b = a(s + t)e-e-a (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 Neveu-Schwarz 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 half-integral
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 -2-p is equal to the superghost number, so its eigenval-

ues are half-integral 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 n-nnunn + S-n'n + -n+cn) + pppoP (6.23a)
n>0
By comparison, with the usual ghosts,

LO = E n(b-ncn + c-nbn + 0-nTn 7-nin) (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 super-reparametrization 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 re-doing the normal-ordering. 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 picture-changing operator.34'17'48 This operator has weight zero and is
constructed as the anti-commutator 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 super-reparametrizations. 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=P3-P 5--P6-P64 ---P-8
1 71 4 38 8 (6.48c)


PD =P4 + 9P5 -1--1P6 + 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 = (e-w/2)te3/2 (6.49b)

P3 = e-/2(e3/2)" (6.49c)

P4 = (e-w/2)'(e3O/2)' (6.49d)

P5 = st(e-/2)'e3/2 (6.49e)

P6 = -ste-w/2(e3O/2)' (6.49f)

P7 = Ste-w12e3/2 (6.49g)

P8 = stl'e-w/2e3/2 (6.49h)

P9 SS=: : e-w/2e3/2 (6.49i)

P10o =: t': e-/2e/2 (6.49j)

Pl = wi"e-wl/2e3b/2 (6.49k)

P12 = O"e-w/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 BRST-invariant picture-changing
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 BRST-invariant. 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 four-parameter class of such

objects. These new objects, being dynamical invariants, are worthy of fur-

ther investigation. The picture-changing operator of superstring field theory

was identified as one member of a four-parameter 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
SUPER-REPARAMETRIZATION 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 -GN-1. (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(n-r+l) ( n )- 2wa n n 12f(n-r+l) (n) fGr
1G = 1: G^J 12 r ) 2wal: G1 r^ f G21
n=r-1 v / n=r
N
N NN
ffG12 =E Gn f(n-r ) r2 -
n=r
N / \N \
ifGnr N >1 ~n jf(n-r+l) (n N n 2war f(n-r+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= DA-r)Ar+ [ +wFQ-1] f(m-l)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(m-r+l)Gm
11 = D(A )G + 1 \+- + Wa (Ml f( 11
m=r+l
= (A-r-) + m [ ) + (1 + Wam f(m-r+l)Gm


bf G12- = D -2-G +
= 2+ (Wa + 1)(n)] f(m 1r+l) 2
m=r+l I G

5f (A-r!) m-r
6f G^' =D22+ G rr-1 +( 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(Gl-1 N +G).
11^ --(12 + 21)-

We look now for a different linear combination of GN-1 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)(GN-1 G22N-l) aGN] +(N+aN-2aA)fG.
(A.6)
Requiring that the f' term vanish, we get

a = N/(2A N) (A.7)

So the combinations
(2A N)GN-1 + NGN

and
2(A N)(GN1- GN-1)- 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 (Gl-1 +G -1+ ') =
f [(1 + )(G1N-2 + GN-1) + (/ + 9')GN-1' + yGN/] (A.8)

f' [(N 1 + 2wa + 2WA + 7)GaN-1 + (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)Gf-1 + (N 1 + 2wa)(GN-1 NGj<)








and

(2N 2A 1)(GN-2+ GN-1) + (N + 2w) ((1 N)G 1' NGN')


form a new doublet, as long as A # N-i. 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 GN-2 GN-,
12 '21
GN-1' and GN', namely,
12 2
(G1N-2 + ,G2N-1 + ,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(non-covariant) multiplets.






92
Next we consider the case of the 4N-field representations. Now the lowest
weight field in the multiple is GN-1, with weight (A- N + ) and GN =
GN = 0. We have
fG-1 = f(GN G-1). (A.12)

As before, we consider the transformation of a different linear combination of
the fields on the right hand side:
f (GN-il+ +GN-l) 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 N-1 + (N 1 + 2wa)GN-1

and
(2wA + 2wa 1)(G1N-2 + GN-1) + (N 1 + 2wa)GfN-1'

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 rN-l'1
1(+2 + 21 + 12 )

which is

f' ((N 1 + y)GfN-1 ,3(N 1 + 2wa)GN-1 GfN-1).

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.