Operator methods in the superstring theory

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Operator methods in the superstring theory
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Thesis (Ph. D.)--University of Florida, 1990.
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Includes bibliographical references (leaves 77-79).
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by Meifang Chu.
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OPERATOR METHODS IN
THE SUPERSTRING THEORY







By

MEIFANG CHU


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


1990




























To My Parents


Digitized by the Internet Archive
in 2010 with funding from
University of Florida, George A. Smathers Libraries with support from Lyrasis and the Sloan Foundation


http://www.archive.org/details/operatormethodsiOOchum












ACKNOWLEDGEMENTS

First of all, I would like to thank Professor Charles B. Thorn, my thesis

advisor, for his guidance, patience and constant encouragement. Secondly, it

is my pleasure to thank the committee members for their supervision on the

dissertation. I am also grateful to the professors and the colleagues in the high

energy theory group for the wonderful time I had in discussing physics with

them. A special thank you is given to Professor C. Hooper and Professor J.

Fry for the assistance they gave me with my visit to Princeton University in

the 1986 fiscal year.

Finally, I wish to thank my family and my close friends in Gainesville for

the sharing and helping through all the highs and lows.

This research was supported in part by the United States Department

of Energy under contracts No. DE-FG-05-86-ER40272 and also in part by the

graduate fellowship of the Institute for Fundamental Theory from the Division

of Sponsored Research at the University of Florida.














TABLE OF CONTENTS


page

ACKNOWLEDGEMENTS ..................................................iii

ABSTRACT ........................................................... v

INTRODUCTION ............................................... .... ...... 1

BRST INVARIANT VERTICES FOR OPEN-CLOSED TRANSITION
OF THE NSR SPINNING STRINGS ...................................5

Open-Closed Transition As An Overlap ..................................5
BRST Invariant Vertices................................................20
Picture Changing And Insertion ........................................ 36

OPERATOR QUANTIZATION OF GREEN-SCHWARZ
HETEROTIC STRINGS .............................................. 44

Covariant Quantization.................................................44
Canonical Quantization ............................................... 51
Quantum Anomaly ......................................................61

CONCLUSION ......................................................... .68

APPENDIX A PATH INTEGRALS FOR OVERLAP VERTICES........71

APPENDIX B USEFUL FORMULAE ................................75

REFERENCES ....................................................... 77

BIOGRAPHICAL SKETCH..................................... ........80












Abtract 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



OPERATOR METHODS IN
THE SUPERSTRING THEORY

By

Meifing Chu

May 1990
Chairman: Charles B. Thorn
Major Department: Physics

Two aspects of the superstring theory are st died in the operator lan-

guage. The interaction between the open string and the closed string is dis-

cussed in the Neveu-Schwarz-Ramond formalism. Transition amplitudes are

given by the overlap vertex of the string wave functions. To ensure the BRST

invariance, a supercurrent-like operator insertion is required. By shifting the

picture of the open string vacuum, this insertion is identified as the picture

changing operator. Its role is to keep both the open string vacuum and the

closed string vacuum in the same picture. This is a consequence of the nonzero

superghost background charge in the RNS formalism.

Also, a canonical quantization of the Green-Schwarz superstring is pur-

sued here. A new method of operator quantization developed by Batalin and

Fradkin is applied to the heterotic string in a semi-light cone gauge. Using a

normal ordering regularization, it is shown that quantum anomalies cancel at

the critical dimension, D = 10.












INTRODUCTION


String theory is considered a promising candidate for unifying all the

fundamental forces of the universe. It describes the propagation and the inter-

actions of a one dimensional object-string. Its low energy limit is equivalent

to an effective field theory of point particles. For example, the low energy limit

of the open string gives the Yang-Mills theory at the tree level 1 and that of

the closed string gives the perturbative gravity of Einstein.2 Among all the

theories for strings, the E8 x Eg heterotic string is the most promising one

to unify the four fundamental interactions. There are two different formalisms

for the first quantized superstring; the Ramond-Neveu-Schwarz (RNS) and the

Green-Schwarz (GS). The RNS formalism is manifestly Lorentz covariant, and

it contains the ten dimensional space time coordinates X' and ten world sheet

spinors ?k. The GS formalism is manifestly supersymmetric and contains the

coordinates XP and the Majorana-Weyl spinor 0 in ten dimensions. It has been

shown that these two formulations are equivalent in the light cone gauge.3 They

are related by the SO(8) triality.

One important content of the string field theory is the description of

string interactions. Similar to the Feynman rules for the particles, string am-

plitudes are described with propagators and vertices. It is natural from the

geometrical point of view that strings can split, join and interchange-a combi-

nation of the first two. If g is the coupling constant for the splitting or joining

of strings, from a crossing symmetry argument,4 the interchange of the strings





2

should be of the order of g2. Including both open strings and closed strings,

there are at least five fundamental interactions,

S= 1 (go 4o o + g2o o o* o0 + g c* 4c c (1.1)

+ go c + g24c o o)

where 4o,c correspond to the open, closed string wave functions. Different

string field theories give different descriptions for the products among the

strings. But on shell, they should all reproduce the Dual Resonance ampli-

tudes. For example, in the light cone parametrization, Mandelstam set up

the path integrals for the scattering amplitudes of the RNS spinning strings.

He gave explicit Neumann functions needed to construct the n-point functions

for the open (or closed) strings. Green and Schwarz constructed the overlap

vertices for the first four types of interactions in the light cone GS formal-

ism. In Witten's open string field theory,5 Cremmer, Schwimmer, Thorn6 and

Gross, Jevicki7 constructed the three-point vertex for the open bosonic strings.

Zwiebach and Sonoda studied the closed string three-point vertex.8'9'10 By

showing that no three-string vertex can generate a BRST invariant off-shell

four-point amplitude, they concluded that covariant closed string field theory

could not be cubic only. As to the interaction between open strings and closed

strings, Shapiro and Thorn11 constructed a covariant transition vertex between

one open string state and one closed string state. However, they discussed the

transition for bosonic strings only.

Among all string vertices being studied,12 the coupling between the open

string and the closed string has received less attention. This is partly because

none of the existing covariant string field theories consistently contains fields

associated with both open and closed strings. So far, there is a self consistent

open string field theory by Witten, but there is no satisfactory covariant string





3

field theory for the closed string alone. The open-closed transition vertex

operator can provide a tool for us to construct a closed string theory from the

existing open string theory. In the first part of the dissertation, I will extend

the open-closed transition of the bosonic string by Shapiro and Thorn" to the

RNS spinning string. These transition vertices will be constructed from the

overlaps of string wave functions. A supercurrent like operator will be inserted

to the vertices in order to maintain the BRST invariance. This insertion will be

later identified as the picture changing operator in order to keep the superghost

vacua for both open and closed strings in the same picture.

Because of the manifest Lorentz covariance, string field theories are usu-

ally studied in the RNS formalism. However, from the point of view that

supersymmetry is very important in explaining finiteness and renormalizabil-

ity, it is more "reasonable" to study the string field theory in the GS formalism.

Unfortunately, the gauge structures of the GS model make the covariant quan-

tization very cumbersome. The second part of this dissertation will focus on

the quantization problems of GS. In particular, a new method proposed by

Batalin and Fradkin will be applied to the operator quantization of the Green-

Schwarz heterotic string in the semi-light cone gauge. This is meant to be

an intermediate step between the light-cone quantization and the covariant

quantization.

The idea of the BF method is to introduce new variables to convert the

second class constraints into first class. One can then construct a nilpotent

BRST charge to generate the constraint algebra. The hamiltonian and other

physical observables can be constructed in the extended phase space by impos-

ing BRST invariance. However, in the BF formalism, quantum corrections to

the observables are ambiguous. They are usually fixed by hand to retain the





4

symmetries of the theory. For the GS heterotic string in the semi-light cone, I

find that there exist unique quantum corrections such that, at D = 10, the nor-

mal ordered BRST charge is nilpotent and the super Poincare algebra closes.

Thus, there is no quantum anomaly in this gauge. This conclusion agrees with

the path integral result by Carlip13 and by Kallosh and Morozov.14

The reason for studying the superstring theory in the operator language

is that the operator method is more efficient and unanibigous in comparison

with the path integral formalism. From the second quantization point of view,

it is desirable to do the first quantization canonically. As to the string interac-

tion, the difference between the two languages can be seen as follows. In the

path integral language, string amplitudes are calculated by integrating over

the string world sheets with vertices inserted at the punctures on the surface.

These vertices correspond to the emissions and the absorptions of the string

states. There are infinitely many string excitation states and thus infinitely

many corresponding vertices. However, in the operator language, it is sufficient

to construct the string propagators and the overlap vertices for the fundamental

couplings in (1.1). All string amplitudes can be obtained from these building

blocks. Therefore, operator methods are chosen to study the superstrings in

this dissertation.











BRST INVARIANT VERTEX FOR OPEN-CLOSED
TRANSITION OF THE RNS SPINNING STRINGS


Open-Closed Transition As An Overlap

The covariant open-closed transition for the bosonic string has been stud-

ied by Shapiro and Thorn.11 They constructed the BRST invariant vertex of

one closed string state transiting into one open string state, and vice versa. I

would like to extend their construction to the RNS spinning string. Although

Green and Schwarz 15 had constructed, in the light cone Green-Schwarz su-

perstring formalism, a supersymmetric transition vertex between the open and

closed strings, their vertex was not covariant. Therefore, it is interesting to

construct the open-closed transition vertex that is both Lorentz covariant and

BRST invariant.

Generally speaking, string interactions can be described by the overlap

among the strings. Here, the overlap means that the string world sheets are

patched smoothly together so that string coordinates are continuous over the

boundaries where the strings are connected. Certain singularities may appear

at the interaction points where the strings join or split. Each string field theory

gives its prescriptions on how strings overlap. Such prescriptions must respect

symmetries like duality and unitarity of superstring and also reproduce the

correct on-shell tree amplitudes. I will use the light cone strip-like world sheet

picture to study the open-closed transition for the RNS spinning strings.

Let < rl be the transition vertex which acts on one open string state

and one closed string state. In order to give reasonable amplit udes, < rF must




6
meet the following requirements. First, string coordinates and physical opera-
tors of the open string should be identified with those of the closed string at
the boundary. The transition vertex should therefore obey the corresponding
overlap conditions. Secondly, unphysical states must decouple from the phys-
ical amplitudes. This means that < FI should be annihilated by the BRST
charge Q. Knowing these properties of the transition vertex, we can divide the
vertex as follows.

< fol =< VBI < VF| < (cb)l < (,/3)|(insertion)

The first four parts are the overlap vertices for the space time coordinates,
the spin variables, the reparametrization ghosts and the superghosts. Shapiro
and Thorn constructed the overlap vertices for the bosonic coordinates and the
reparametrization ghosts (cb). For completeness, I will quote their result here.

Fr Xcr) = q + p + En-O (Ane-2in(r-T) + A -2in(r+))
XY(r, o) = q + pr + En 0 2--A ncos(na)e-nr"
For { (r, a) = q + p + Zn iO 2a~eos()i


B(r, a) = En(Bne-2in(r-) + Bne-2in(r+o)) : (Bn n)
BO(r, a) = 2 E bi cos(n)e-inr
and a similar expansion for the C ghost,

< TI =< VBI < (cb)l
< B I =< 0| exp(W)
00 _\k+1 1 1

00o k+l 1 \ 1 00
k,l=0

+ 4(k + + 1) k I a2k+1 a21+1 E 2k(Ak + Ak) a2k
k,l=0 k=1
00 /k+1 1 1
+_ 2- 2- 1( k ) ( )a2k+l (Al il)
=(2.1)
(2.1)





7
And, the overlap vertex for the reparametrization ghosts,
< (cb)
H( (l)ml+n 1f / I
< exp, I 21c~b- E CA60b
01ep 2(7 + n) n m n -
_l+k+1 1 131
+ 2(1+ k + 1) 2 c k+1l+1 -k c2b
i,k>O m>1
i ln 1 3

+ ^ g ^S 2- V2. '2 fiS (- b2>
+ c o(bi boll) -21 1 cb+
m>l m>1,1>0

2 cm(bm -o- b ; 2k +1- 22n E 1 -k+lb
m> k,n0>
(2.2)
These vertices are constructed from the overlap equations of X/^ and the
bc ghosts. In a similar fashion, we can construct the fermionic part from the
overlap equations of 'P. Of course, one can also construct the overlap vertex
using functional integrals according to Mandelstam.16 The Neumann functions
for each open-closed coupling are given in the appendix. Both methods give the
same transition vertices. We can therefore be very confident about constructing
the transition vertices from the overlaps of strings.
Also, < (fly) will be constructed from the overlaps of the superghosts.
Besides the overlap vertices, there is an insertion operator which was not
present for the pure bosonic string case. In the path integral of the light
cone string amplitudes, Mandelstam inserted the supercurrent at the interac-
tion points to recover Lorentz invariance. Similarly, in our covariant formula-
tion, the supercurrent insertion is needed for BRST invariance. This ensures
that spurious states decouple. There will be ghost contributions also to the
insertion which are determined by imposing BRST invariance of the vertices.
This modified supercurrent insertion can be interpreted as the picture changing
operator to keep all superghost vacua in the same picture.





8

Before we begin the construction, recall that for the RNS strings, there

are two different sectors in the open string; Ramond sector and Neveu Schwarz

sector, corresponding respectively to space-time fermions and bosons. In closed

strings, there are four different sectors; NS-NS, R-R, NS-R, and R-NS for the

left moving and the right moving modes. The first two sectors describe space-

time bosons and the other two sectors space-time fermions. Therefore, the

couplings between open and closed strings are restricted to three types: (1)

NS open strings couple to NS-NS closed strings; (2) NS open strings couple to

R-R closed strings; (3) Ramond open strings couple to NS-R (or R-NS) closed

strings.

The idea of constructing the overlap vertex < VFI for '# will follow

those for the overlap vertex < VB I for XP in Ref.[11]. Lorentz covariance will

be maintained throughout the construction. Since the overlap of two strings

on a strip is identifying the point a on one string to the point 7r a on the

other string, an operator of conformal weight J acting on the transition vertex

should have the following identities:7


< VoclAJpen( a)- (-)J < VoclAoseda

The phase factor comes from a conformal transformation which takes a of the

closed string to ir a of the open string on a strip.

Let 0o,c denote the right (left) moving fermion field of the open (closed)

string. The standard mode expansions of the i field are,



0,." A= Foo bl/'iror
V r=iR-oo r
fr is a half integer for NS sector
'- e-i2rra I an integer for Ramond sector
2r=-oo Bei2r7





9

Since 4 is a conformal operator of weight 1/2, there is an extra phase i in its

overlap equation.


< VFIl' (7r a) = i< VFI+'c () (2.3)

Let me outline the idea of constructing the transition vertex from the

overlap equations following Ref.[11]. First, one should mode expand the overlap

equations of 0 as follows.


< VFI B~Nr B- NB = 0
\ s>0 /

These equations determine the fermionic vertex directly.


r,s>O

where

Nrs = Nr s Nsr

That is, we should expand these overlap equations as one creation oscillator

acting on the vertex and giving infinite sums of annihilation oscillators.

This particular mode structure is projected out by the so-called moment

functions. They are functions which have the required asymptotic limits of

the open string and the closed string and they isolate only one creation mode.

The guideline of finding these moment functions is to take a chosen polyno-

mial, multiply it with the conformal mapping function to some power that

corresponds to the boundary conditions of the strings. The polynomial will

be determined by the asymptotic limits. It is the conformal mapping function

that controls which interaction prescriptions we use, light-cone, Witten's, etc.

Since the construction of mapping functions varies case by case, I will come

back to this later when explicit calculations are carried out.





10
In general, given a moment function, we can expand it into an infinite

series,
En h e|-n(rFio), as r oo,
h:(, T) =
En hCe-2n(r:ia), as r --oo
The overlap equation Eq. (2.3) is then multiplied by the moment function and

integrated over a at r = 0 where the two strings are connected.

j doa < ri ./O (2.5)
i+(o)h () + i c(a)hI(u)} = 0
Then, one can read off the overlap vertex directly from the overlap equations

according to Eq.(2.4).

At last, to maintain the BRST invariance of the overlap vertex, we should

insert the supercurrent at the interaction point. Note that the conformal map-

ping function from a strip to the upper half plane is /e2r+2i 1, the super-
3
current insertion diverges as 4 near the interaction point.

< rol -< VB < VFIIO =< VBI < VFI(4+O+X)(O,O + e) (2.6)

The decoupling of spurious states will give a good consistency check for the

overlap vertex. Let us now go to each coupling and construct its vertex explic-

itly.


NS Open String Coupled To NS-NS Closed String

For the NS sector, I choose the vacuum of the fermionic strings to be the

one annihilated by all the positive modes of the spin variables.

Bi'O >= 0 for all i and for r >1
r -2

This is purely a choice of convenience. Once the vacuum is chosen, the creation

and annihilation operators are well defined. We can now construct the moment





11

functions h(x) according to the outline that we gave in the beginning of this

chapter. Denote x = eP where p = r + ia.


(a) Single out B3-r + iB-r, r > :


Let h+(x) = P(x-2)(x2 1)/4 -2-1 + o(x). (2.7)
x -- 0

With the power -1/4, the asymptotic limit goes to a negative odd integer power

of x for the NS closed string. P(x-2) is the polynomial that serves to isolate

the creation mode. It can be found by truncating the infinite expansion of the

inverse mapping function. Only the negative modes are kept.

P(-2) = negative modes of [x-2m-1(x2 1)1], as x -+ 0

= ei x(-2m-1 (1 (7)lX21
1=0

Substitute this polynomial into (2.7) and expand h+ in the limits of the open

string and the closed string respectively. Two sets of moment functions are

obtained this way.

as x 0,


h = ei -2m-1 ( )1x21 e-i- ( ()kr2k
1=0 k=0
oo 1 \+1 5 3
= x-2m-1+ (- -q)2+1
1+m+ l f1m
1=0
as x -+ oo,


=i =-2n-l )/x2 k -2k
1=0 k=0

S 7m+1 +13
= e 1+1 K2
1=0 I -n +






where I have used the identities in Appendix B. Choose h- = -i[h+(x)]*, and
substitute them into Eq.(2.5), the following overlap equation is obtained:

S0 I m+1 5 3
< VFI{B-m- i+B-m- + E 1 + (m + )1
2 1=01+M

(1 + i) b21+ } = 0.
1=0 4
(2.8)


(b) Single out B-r iB-r, r > ":


Let h+ = P(x-2)(x2- 1)1/4


z-2m-1 + o(x)


x -- 0


P(x) can be derived using the same trick in (a).

P(^-= e-i-2m-1 x21
1=0
as x -- 0,


h' = e -i -2m-1 -4 \(_lx2
1=0
00 /n+ /m+ 5\
= .-2m-1 -- 1 4( l -
1=0 m+1 m
l= 0


k=O
-41 x21+1
*^^I-f


as x -4 00,


=n
ho = e-i4x-2m-1
1=0


00 1/ n+1 / 5
= e-i! 1
=0 + m --
1=0 '4


)() 1 kz-2k
4)


(7 '
S(-)i21 E
k=0


3(
4)T -21+.
I2


Again, setting h- = i[h+(x)]*, we have
00 l(m+l 5
21=0


+ (1 i)( 1 b2+_ }= 0.
1=04+m-1 m 1


(2.9)


-4 -iB1+ )







(c) Single out b_2m,n, m > 0:
2


Let h+ = P(x)(x2 1)-4


zx2m + o(-)
x -- 00


For the open string, the polynomial can be obtained similarly as in (a) and

(b), but only the positive modes are kept.


in
P(x) = x2m+1 E(-_)
1=0


(1-21


as x -- 00,


in
1 1
S= + (-)
1=0


/\\l


00
-21 -1 I
k=0


5(-/
7


00 1 \l+m
=x2m + 1 + +
I=0


3 \
-)4 X-21-2
in2


m +/
h' = x2m+ 1
l=0


00 1 +m n 5
q1 ]

1=0

After setting h- = -[h+(x)]*, we have


00 1\
x1: k- _) k 2k
k=0


X21+1
km/


S77 21 )
b21+-3


2 o 11+ l -
1 -i=0 Q +(- m (


(43)
in7


(d) Single out b_2m _;, m > 0:
2


Let h+ = P(x2)(x2 1)
X --4 00


as x -- 0,


(1B + iBl+ ))=0.

(2.10)


k1)(
(-^-) x-2k


00 / \l+m /

1+1=+m +
I= 0


< VFl(b-2m-


2mn- + o(-4 ).







2m '+3 /-1-\
Where P(r) = x~'"+ i 1 -21-
1=0


as x -- oo

m1
1=0


/ lx
I
'~ 1 /


00
(_)'.-2'1-
k=0


2m+ )[+11
1=0


as x -- 0 ,


S=2m- 4 (-)1x-21-
1=0
00 1 5
S)'+ 1 A +,,,
1=0 +


With h = -[h+(x)]* we have
VI, 1 )+,m- 5
< -2m- 1 + 1+m m )
1=0
l+i 0 )1+0
2 1= + 7l


Thus, together with (2.8),(2.9),(2.10), these overlap equations determine

the fermionic part of the transition vertex completely. The result is listed in

the following.
< VF=

< Ojexp 1++ 1rn+ Bm+l'+ 1 + 1+iB
1=0
1 + l_1 -+77 15 3
+ 1 b2 [ m+ 1 iBm+ i

I=0 4 m +"m( 2 2 2
i0 + m I m b [ +B


00 1(./ 1+m
+ .1 + 1 + n
1=0


(2.12)


00 /

0k
k=0
( 3
I


14 2
5 3

(-q) iB ,)) =20
)2 2+f
(2.11)


I
(-_ks-2k+2
ki


(-J)(-y_
5 3
M4 4z2-


_)kx2k


S I ) h21+ 2mn+2





15
Next, we insert the supercurrent to the vertex near the interaction point
(a = 0 + e).

< VBFIo =(2e)- V < VB,FI (- )+m (
m,1=0 (2.13)
a2m+l i(Arn Amn)] [b21+1 ((l iB,]
[a 2 2 2
It is straight forward to check that the spurious states decouple from this
vertex, which assures the BRST invariance of the vertex. I also find that
(2.13) agrees with the one from the path integral calculation (see Appendix
A).
Before we go to the other two couplings, let me pause for a moment to
comment about the uniqueness of the fermionic vertex. It is independent of the
moment functions selected, as long as the conformal mapping function stays
the same. For example, let us replace the power -1/4 for the moment functions
in (a) and (c) with 3/4 instead:

h+ = P(x2)(x2 1) z2m+ + o(x )
X -+ 00
h+ = P(x-2(2 1) -2m-1 + o(x)
x -+ 0
These will give us two new sets of overlap equations:
o0 3/ l+m 1 7
+ ()(-117 1)b2
< VF Ib_-2m- + m 1
1=0
S0 l+m 1 7
+ (B11 + iB+ = 0 for 7 > 1
2 l=-4 +-m i 1 ( 2 2_

< VF ((B-rn- + iB-m-, ) + +1 ( (( B + iB+)
1>0
0' 3( (+1+n 1 ) 7
+(1 + i) E l--
-3 1 m-1 2





16
At first glance, these identities may be different from what we obtained earlier.
However, we can replace b_1 by
2

< V i|_ =< VFIl(( 2- ) (B+ + iB,++)
(2.14)

(4 + 41) ) 2
1>o

and the new identities return exactly to the ones we had in Eq.(2.8) and

Eq.(2.10). Therefore, as long as we keep the same conformal mapping factor
in the moment function, any other variation to the moment function should
not affect the final results of the transition vertex.


NS Open String Coupled To R-R Closed String

This coupling involves the Ramond zero modes in the closed string. Its

vacuum has two helicities defined by D0 iDo. To compare the result with
the path integral calculation, we choose the ground state to be I- >, which
satisfies

(Db iDo)l- >= 0.

To expand the overlap equations, we use the following moment functions to

single out one creation operator for the closed or open string. Notice that in
the closed string limit, h should go to a negative even power (Ramond sector)

asymptotically.
(a) D-m + iD-m :

h+ = P(x-2)(2 1)- -2m + o(l)
z -4 0
00oo 1/ \l+m 5 3
he = z-21n + l -- + -) -4 21+2
=+m+1 1 m
1=0
00 l+- 3 5

h=o -4 +1 (~m m') (







(b) D-m iD-m :


h+ = P(x-2)(x2 1 -2m+ cost.
x -- 0
00 I -m /3\ 5
h' = x-2m 1 1( ) (+7 2)
+ 1= + m + r m- 1
1=0


(c) g-2m- :
2


S00 / 1 I+t+l / 3\
ho -iW/4 H I+m+1
1=0 -


1 )
2( 2 4


,2m+! 1
2 -4-


x 00
S 4 +4 (j ) x21
h+3 +m m 1
1=0 I


00 I+ 1l+m) 5\/ 32
+ = 32,n+| E -4 ;Z f-4 1-21-
1=0 +

(d) 9-2m- :

h+ = P(x24(x2-li 2 2m+f + o(x-l)
x -+ 00
00 1 l+, / 5 3
/ x \1 -3\
hc = ei/4 E (- x21
-1 l +m\ m
1=0 I
o 2m+-l 0 4 (-4 -21 12
ho = 1 f- + 2
+ Y-+m+l1 11 M
1=0
Substituting these functions into (2.5), one obtains < VFI according to (2.4).
100 1 1+m 5 \ 3
< VFI =< -Iexp( ) (-1) (-) [l + iD,] [Din iDm]

+ 1+22
+ l,r=0
1+1+m77 2 1 by b2m+


1+ i 4 -
+ -I-'1 +
l,m=O1 -I


100 -i l+m _5_( 3
,- )( [D61 + iDt] b2m+2)
=,=o +1- m 2- m
(2.15)


(-_ -21-


I ./ f


( 4[ iDl] b2m+i





18
Again, applying the supercurrent insertion, the BRST invariant vertex
for NS open string coupled to R-R closed string is

< B,FI1o
31
=(2E)-4 < VB,FI ( ,( ) (ml [a,- I i(Am Am)] (2.16)
l,m=O
S1+(l+i
b213 + 2 )(D1 iDj)


Ramond Open String Coupled To NS-R (R-NS) Closed String

This vertex describes the open-closed transition of space-time fermions.
It has rather different features than the vertices we discussed earlier. First,
there are two ways to define the helicity of the vacuum:


either 1(DO2k-1 + iDo2k)[ T>= 1 (d2k-1 id k) T>= 0;
or D+l t >= 1(d + iD)Ifl >= 0.

The path integral result corresponds to the former basis. However, we
find that it is easier to construct the whole vertex with the latter. Second,
because of the asymmetry of the left-moving and right-moving sectors in the
closed string, the moment functions will be generated with a slight modifica-
tion. For the right moving modes, we have

h+ = P(x)(1 x 42l-(v V, +) xk + o(l)
p -: 00

For the left moving modes, we use

h_ = P(x)(1 x2fV(/- T ^/+ ) -- x^ + o(l)
p 00oo

where is the complex conjugate of x. Similar to the other two couplings, these
moment functions are expanded in modes and substituted into the overlap





19
equations. To avoid redundency, I will not list the expansions of the overlap
equations, but list the result of the fermionic vertex in the I T> basis.


< VFI =

Wh =Z

2 2


(21+1 2m C21 2m-1
S4(1 + I + m)


( 212,r 21/+12n2m+
( 21 + 2m + 3
(21 22m -- 21-1l2m-
21+ 2m + 1
(1-1 2m+ -
1 2m+l 2


p$2m
S((2nm- 1 2n

p 2m

Ip2m I
1 2m+1+ )
(lICm -+ (1- lnCm-1-- rn


) [Di + D+ + B .B ]

1+ 1- Di+1

-1 B+ D,,+
1+ m
- DC+2m 1+
)-1 1+1 n-+


-l) (1 + i(-)') d D

1) (- i(-)') dl+ Bm+



+1 m+
S1 i(-)) d din+1


iem+l(d4m+2 Dm+l) + ien(c2 D;)

i2(d2m+l Bm+n) i2+1 (d +l 1 2+)



22
S-1 (D2k-1 iD2k)

AB = (A42k'1 i2A 2)(B2-l T iB2)
k=1


(2.18)


I and m are summed over from 0 to oo. The coefficients (n, en are given in

Appendix B. This vertex agrees with the path integral calculation in Appendix


1-i
4
1+i
4

1-i
4
1-i
4


-*^
^





20
A, as expected. However, if we choose the ground state to be 10 > instead,
the vertex becomes

< V |I


=< liexp(1 (
16


P21i2mn+I P2m~l21+ 1
I-1 ---m+2 DI+ Dm++1
l+m+2--1 'Dr n+l
(P2mrn-121 P21-12 B B
., i---Tm 3 ---- ^+4 B,,,+


+ fP2,nm21 -~ P21-1C2,n+1
4 E ( 21+ 2m + 3


1+' i
+ i2
l#2m+1


P21nm Pl-I (2m+1
S2m I + 1


(1 i(-)) dl+1 Dm+


+ 2 +l)2,+
+ Fn,+2 (2m,+)d,)m+2 Dmn+1


1-i
+8
p#2m


P2nm-1 1 Pl- 12m
2m ni-p


+ ('2,, E +)d,,, + Bm1+
- (1 -i(--)m))) (---) l + 11m2 P+ )


1 -
+ -'+ (-i?'2m+2Dn+l 72rnm+IBSm+
m=0
+ 2 (1 i(-)m) lm+ljdm+1))


(2.19)


where pm and rm are given in Appendix B. For convenience, we shall use
|0 > as the ground state to construct the complete vertex. The insertion
corresponding to this basis is
< rol


00
F (-)( m+1 -i(Am --An)] -
1,771=0
,m=0O
i 1
(D+ (P21 2(21+1)D1+1 + (P21-1

+-i i(-)(-1 2C)d+)
+2".


(2.20)


1-()d+1 Bmi(+1) + Bi


dl+1 dm+1


=< VB,Fp(2E)-~


- 2(21)B1+1




21
BRST Invariant Vertices

In this section, the construction of the open-closed transition vertices

will be completed by including the ghost contributions. The idea is to keep

the BRST invariance of the vertices.


< FIQBRST = 0


(2.21)


To simplify the formula, I will use the following notations:


Lm = da
7 J-W

7r fd7
Lmo =Im
Gro = g ,

bno = bn
Cno = c l

SPr = Or
{7ro = 7r


e"i(4Y(O OAX, + -+^)

a"+ 11)-
2 +



and Lim = -j (m Ln)
and G,.* ( ,, iGr)


1 ~
and b?:, (Bn Bn)
and cn = (CnCn)

and ,. = (Br+ iBr)
and Y (- T irr)
V2_J;


(2.22)


The on-shell conditions, (Lo ao) Istates >= 0, are


1 = { in the NS sector
ao+ = -^ ao0 = 0, Q00 2
0 in the Ramond sector
Also, it is useful to define the following constants

2 for (ijk) =(+ + +),(- +),(- + -),(+ -)
dijk = 1 for (ijk) = (ooo)
0 otherwise


for (ijk) = (- --),(+ +-),(+ +),(- ++)
for (ijk) = (ooo)
otherwise


0





22

With these new notations, the BRST charge can be written as

QBRST =E c n(L- u,,00) + 00, r
-00 -00
m +(m n) : cm -A : djk : Uijk
2-c -c
-00 -00

+ 0(m + r) : ci- rT +r : dijk
-00
(2.23)

To solve Eq.(2.21), let us consider the following ansatz.


< Ir =< VB,FI < ghosts (0 + E 11';, 3 + Zsc/i mj'
r>0,s>O m>1,s>O /
(2.24)

where

< ghosts =< vacuumI exp ( MnjnCm bn + E Yr'is ir )
m>1,n>0 r>0,s>0


It is not difficult to guess the structure of the ansatz. < ghosts is basi-

cally the overlap vertex of bc and #y. The cb ghost overlap vertex was given

in the beginning of this chapter. The coefficients M are listed in Table 2-1.

The superghost overlap vertex (Y) comes from the overlap equations of the

supercurrent. This is because the f superghost has conformal weight 3/2 also.

Therefore, we don't have to derive a separate set of overlap equations for the

superghost vertex.

The new insertion can be understood as an extension of the supercurrent.

The first term I0 is the supercurrent insertion that we constructed in the last

section. The second term 7 b has the same conformal weight 3/2 as the

supercurrent. The c-f terms are included in order to satisfy Eq.(2.21). In

other words, constructing the complete vertex is to find the coefficients M, Y,

W and Z that satisfy Eq.(2.21). Let's substitute the ansatz into Eq.(2.21).









0 /-^ 0




IN






C4
CD C





C,1


o
0 '-c

-tcM


I ^-
1....


0o o 0
-s 2 I



o
11~
*CO
II

Si^
^
*^ C


o -~0
micN


0
mI"







-In
+C
-45,>,


A-


e eo 0 23
I I I

iiII
II II I
.II | I

II





24

The linear terms of C and in (2.21) can be interpreted as a set of Ward

identities for the energy momentum tensor and the supercurrent. Based on the

consistency between the Ward identities and the Virasoro algebras, the terms

of cubic ghosts will cancel among themselves. This argument follows those

in the pure bosonic string case."1 In general, these W\ard identities are of the

following forms :

< VB,FIo(L. + EAI Li K,) = < VBl E rG for m> 1
n>O r>0
(2.25)

< VB,FIIo(Gir ;s G) =< VB,FI(- E W L + T) for r > 0
s>0 p>O
(2.26)

Notice that the right hand sides of these identities are singular at the

interaction points. This is because operators with conformal weights greater

than 1 diverge when acting on the vertex. Moreover, these singularities deter-

mine the ghost insertion (W, Z). To derive these identities, we first construct

the overlap equations of the energy momentum tensor and the supercurrent.


< VBFI(L' + AE. L", k) = 0 m> 1 (2.27)
n>O

< VB,FI(Gi-r ZE Y;ij) -< VB,rFIlIo r > 0 (2.28)
s>O
In the Ramond sector, the zero mode is related to the insertion in the following

way:

< VB,FI( Fo + IpFp) = a < VB,FI (2.29)
p>O
The difference between the Ward identities and the overlap equations is

the I0 insertion. Replacing I0 in the Ward identities with the supercurrent

modes in (2.28) or (2.29), we recover the overlap equations. Thus, the su-

perghosts insertions HI, Z, K and T of the ansatz can be determined from





25

M, Y, CD, w, k in the overlap equations. It should not be too difficult to con-
struct these overlap equations. For the energy momentum tensor, its overlap

equations remain the same as for pure bosonic strings, except the different

constants k due to the extra fermionic contribution. For the supercurrent

conformall weight 3/2), one can use the same set of moment functions for i

(weight 1/2).


NS Open String And NS-NS Closed String

At first, let us discuss the ghost vacua for the NS open string and the

NS-NS closed string. Recall that the reparametrization ghost vacuum is doubly

degenerated. I will choose the up" vacuum for all three sectors.


bn T>= cn T>= cl T>= 0 for n > 1 and for all i (2.30)

This choice will be independent on the sector (NS or Ramond) of the

string and will be used for the other two couplings also. On the other hand, the

superghost vacua are infinitely degenerate and depend on the sector also. We

must first specify the picture of the vacuum for each string. For convenience, I

will choose the superghost vacuum for the NS open string and the one for the

NS-NS closed string to be both in the -1 picture, i.e. they are annihilated by

all the positive modes of the superghosts.

f^ |(-1,-1,-1) >= o 1
for all i and for r> (2.31)
l4 (-1,-1,-1) >=0 2
As mentioned before, the overlap equations for the energy momentum

tensor are the same as in Ref.[ll except the constants k. They can be evaluated

by applying arbitrary string states to the equations and requiring them to hold
exactly in ten dimensions. D = 10. The results are listed in the following.








1 3D
< VB,FI(+- L+ L,,, L) 32'(m -1)) = 0

VBFI( + 2(1 + m) 1 m 1
1>0

( -'1 2 Lol+1, =0
2 1 2-21-1 1 -1 2


< VB,FI-2m L L 2V(Lrm L+) (m + 1)) = 0

< VB,FI L2m-1 + 2(1 + + m) ( 2.32


+ 2V2 i 2 + 1 21( = 0
1>0-

The overlap equations for the supercurrent (weight 3/2) are similar to
those for (weight 1/2), except for an opposite sign between the open string
and the closed string.

< VB,FIJ0(r a) 2 Ti < VB,FIJ()

We can mode expand the equations with the same moment functions for V).
However, the supercurrent has a, higher weight than 0. It is more singular when
acting on the overlap vertex. This appears at the interaction point with an
insertion IO. Such singular contribution (,' ) can be determined by requiring
overlap equations to hold exactly when acting on arbitrary string states.
1 \I+m / 5 3
< VBFG+m 1 =< VBF (- GI ) 1
2+ 1+l +m 1 / +m
mY ++ 1+ gin 2 (2.33)
+ i8 -4f2 9+'21++ (-) n
11 ( 1) 3m) 2 f \2 n





27
5 3
< V G- =< VF) (-1)r +




< VBF, =2m-=< VBFl(-( +1+ (n 2

r (2 1-) __( )I(flm-
(2.35)
1 5+m 3 5 3



< VB,Fg-2m-1 =< V IB,F ) +?) 921+
S1+(1+ i) m 1 5 3







S24 G-+ (_)m -
f+l-m I m I+ m
(2.35)





(2.36)

These identities tell us the coefficients Y, w, and a's, see Table 2-2. To

derive the Ward identities, we insert I0 next to the vertex in the above overlap

equations and take all the creation operators to the left. The coefficients for

the superghost insertion in the ansatz can be determined.


< r = < VB,F, < (cb)l < (70)1(lo + E irji" + ZE zcP-i i)

Wrn = E( f -ws 9yik 2iu) 2k +nikj
S r n-s r,n-s s8 r+n
s>O
r-1
+2 r-M s,n+ r+iio. 6n o(21 -.


ZW = M ( 3n)w L idklj- + (S + il

n>O s=1

+ 3 + -_ 1I, + -LyG1.
-3mom 1 o-
wk+m(r + )dijk .',owl b 2m+,r
+ ( +s6 3 1o i 3m 1
+ (, + ) -WO a
2 4 ms+ 2 m-
(2.37)











-*, l o-. CD +)
31v '3
+





0 ,I-I

+ +
+ I I + I E
II "T I II








-- -f- I *
II II







1 I> II0




+ s-. o
II ---I II

II I II










o+
II I I I









A I 4 C -
II II I I



^ ^^ ^






and the constants
rnf i ., -l ikl 1 io -io"
K = ki + Z(s + -)Wso sdikI I ,,0o + --"o
s=l
m k3m 1
+ &i312nW o (2.38)
2 ,7-2
T = w sr-sUij- + (Ulio + uti+),.(r2 -) fk + 02kr+
2s=

NS Open String And R-R Closed String

Due to the presence of the Ramond zero modes, the superghost vacuum
has two different helicities J >. For convenience, let us choose the superghost
vacuum to be

{f|- >=7- >= 0 for n, > 1
-1 and POLI- >= 71-+- >=
O{I- >= 7- >= 0 for r 0 >

Also, the on-shell condition for the Ranmond sector differs from the NS sector,
we must re-evaluate the constants k in the overlaps of the L's.

< VB,F\Li L L L) 1D (3nm + 1) = 0
W2 2 0-) -
im _1 3
< VB,F (Lm + E 2( + m 1) L-
1>0
i (_)t+m 3
+ 2W 2m-n 21 ) 1 m-1 +1) 0

< VB,F\ L2m LO 2V2(L L) t(3m 1) = 0
VB,FI( L-2rn-1 21l-
(1+)I+7+ 1 1 3 =o

+l+m 21+ 1
+ 2v\2-i 27) -LT) =
12m l- + 1 Am1 /"
1>0
To mode expand the supercurrent overlap equations, we can use the same
moment functions for o's. They are








1>0
+ 1+1+ 13 5
(-1) (-4)921+ )

1 8 1_)l+m +1 3 5
1 0\+7m+l 3 5\
< VB,FF-m =< VBFI( 1 +1 1+) +


+E 1 ( ) 921+1 mm


< VB,F IF =< VB,FI(E-) (- )F++ + +i (-)~ I)921+
1>1 1>0
1+i
+ 1 + 10 +
2\/-2



!>3o
1lC+m /-3\ /-5
2 1>0+ m




S_ 1+I+m (3 I)5
< VB,FrlI-2 =< VB,F\ (+ -4 1 21m

13 5 3

(2.40)
These equations give Y, w, and CD listed in Table 2-3. After I0 is inserted
next to the overlap vertex in (2.40), W, Z, K, T in the superghost insertion
of the transition vertex can all be determined.










Ml-' q iu),T I'1- CClp I
+ IO _. I +




I -,I Z+ IIm
+ +













'i IV










+
II+ II




















+-
AI II
a o0 00











II I II t 3










+1 I
I i I I








AA A6
I II








E E 9S













aII II 3D '3 ^



*


0



bo
V
<4.)



0
u
x








k-
a
8


m





hl-r'e

II









orn = i kr kn-sklj + li kil 2 kn-s Irsu k
s>0
+ 6,0u++J + 2f u+lu + 21u+il" l 2ukijk+r 2u+kjyik
-- rl n+r rn
oo
Zmr s= in-s,rd" ( + z w>- -n.And'(r -
s>0 n=0
m d+'il j w+mdkil(r + 3m) r d+lj Mi
2+ mr +2r+ 2 mr
+ 3 ,+w d+i+ 6,. m d+i-
And

Km = k + (p + I)k -p diki + 7nd+l +i+
p0 "22? 2" V2

aTr' = 2u+iik +2 w+k -nu+il + +i r2 + iowoD(r2 1
22 4
n=1
r, s are positive half integers for the open string and positive integers for the

closed string. And, a = .+i

Ramond Open String And NS-R (R-NS) Closed String

Because of the asymmetry between the left moving sector and the right

moving sector of the closed string, the superghost vacuum is chosen as follows.

Bn|I >= rnl >=0 for n> 1
1
BrlQ >= NolR >= 0 for r >

PnIn >= n7 2>=0 for n> 1

(0lo + iBo)l0 >= ( 1o iro)l >= 0

Constants k in the overlap equations of the energy momentum tensor are re-

evaluated also in this basis.


VB,FI(L+ L+ (L L 3) (3m 2)) = 0
-,n 032V2-







< VB,FI(L-m


2(1+ m) (-
1>0


( 3
M -1
on1lLi


(-a (-)+m /-1
+ 2m 21- 1 i
I>0


(31)
m 1 21+1i=


< VBF L12m LO- 2(L ( L+) D(3m + 2)) =0
< VB,Fi("-, 111 0~L~- +) 16


)1+ (_)l+ +m 1
- 2(1+1+ m) 1

+ 2/i -)'+
2>0 + 1 21
__O


)l (n)Ln+l
2 L2 =+1

S1 m I 3


In order to expand the overlap equations for the supercurrent, we need
to modify the moment functions by subtracting the polynomial at x = 1.
The complication is due to the singularities at the interaction point and the
asymmetry in the closed string.


h(x) = [P(x) P(1)](x2 1)(-v x 1 +x)


SXm + o(1).
X --+ 0,00


Expanding h in the open and closed string limits, we obtain the following
overlap equations for the supercurrent.


((2pP2mn-1 C2mP2p-1 G
8(p + m + 1) P


m > 1
i (2mP2p (2p+lP2m-1
4 2p 2m+3 p+l


(pP2m-1 (2mPp-l (1 2i(-)P)fp+ + 721m+1 / O
2m p 2 (2


- (2m E2m+1n+ )f2m+ + --- (2r+1 + 7'2r-1 2772r)I)
4 4,F r=0


< VB,FIF-m-1
=< V ((I ( 2m+lP2p (2p+lP2nm ,
< F8(p + m + 2) Fp+


1+i
16


E
p=02m+l


(2m+lPp-1 (1,P2,m (1- i())f + 2jn+2
2m + 1 p 2 2


S1 m+1
-(2m+2 e2m+l)f2m+2 2, -
r=0


< VB,FIGM-1
2


=< VB,F\
1-i
16


(p

p 2m


- iF)


m > 0
i (2pP2,n (2m+1P2p-1
4 2p+ 2m +3 2)


- iF)


f l2r+llo
r=0


< VB,FI(L_2m-I








1-i
( 2 ) T ( iVi p CpP2m C2m+lPp-1
=< VBF (1 i(-)P) p+2m 2n+1Pp3 fp+i
i + i C(2p+lP2m C2n+1P2p
+ 1+i 911 p+1
+4 2p 2m-
pmn
1- i 2pP22m 2m+1P2p-1 2i r+f0
+ 4 2p 21-1 2G+ + 2mn+2(, iF0)



1l+i
++ (E2m+2- f^mn+l+)Fn+l + + i (2C2m+l (2m (2m+2) 10


( 2) < V I ()pP2m-1 (2mPp- +1
Va vFl8 (- -)P) p + 2m + 2 +
1 + i C2p+lP2m-1 C2mP2p F
4 2p+ 1 2m
1- Ci(2pP271-1 C2mP2p-1 1 --i fo
+ 2p 2n + 2 2,+1 + FLo
p~m
1-i +i
(e2n E + 4 1+ (2(2m (2m-1 C2m+l) Io

for m > 0,
along with the following three identities:
1-i *"
< VB,FI (G_ + -E[72q+2Gq+_ + i'l2q+1Fq + "--Sq7qfq+l + 2 ) = 0
q>O
< VB,FI(f-l+ + + Sq+i lIfq 2112q+lFq + 272G+lG 1o1) = 0

< VBF l + iFo + S[ q 1SqIq+f + i?72q+2Fq+1 + 12q+lGq+]
q>o
+ 10) =

where Sq (1 i(-1)).
(2.41)
The coefficients Y, w, and w for the overlaps of the superghosts can be read


off from these equations. They are.listed in Table 2-4.
























0 0




w o I




S .
0

Q -

i 0






*0
o 5
+ S









E0
S o0
O


E








II *
Sf


E E -
+< + I +


I I
>LII 11"
S ,S | I
's _1 i- ^ I w" I "
I I 's> *w
II II II II

+ +

+ p 1M -
S O E
ffo 1,


I I





I I
O -
'3



I I
I -rrl
'3


+

-



c,. +
II II II
+ I o
+ |4
^ir E


r

c'1-
I I I



I





3G
Similar to the other two couplings, the coefficients of the superghost
insertions can be written in terms of the coefficients in Table 2-4.

o.d = --" 5 + gl'---Sn
s>0
6i o( 6,o + 2w + 1r) i+(- i'no + iwi.4 + 2iY )
4 r
+ 2w +r 6ioAf + 2iAl+j Si+
mk ,ir "" i 3n 3m 3Zm
aZr = ( + )i -sr j- -n r ) (r + )W
s>O n=O
3m 1 3 1.3 1 w+( 'o o
+ -3m(Wm + -i )wr 32( 2
2 2i 2 ,01-20 2)"06r)
+ ioY + i+o mY+ + ,, + 'irM.i t
(2.42)

and

Km = ki + E(s + )WM s + ( + ) + o io + mi
n12 + M-O+ + + 4

aTr = biOk + 2i6bi+k+ + A.w + 2 k
s>0
(2.43)
where m,n stand for integers, and r,s are intergers for the Ramond sector (

i = + ) and half integers for the NS sector ( i = ). Since the coefficients
in Table 2-4 determine the superghost vertex and (2.42), (2.43) determine

the superghost insertions, we have completed the construction of the BRST

invariant vertex for the Ramond open string coupled to NS-R closed string.


Picture Changing And Insertion

From the construction in the last section, it may seem that the transition

vertices have very complicated insertion operators. However, with a more

careful study, we can find the similarity between the insertion and the well
known picture changing operator. In order to have a better understanding of

this insertion operator, I decided to reconstruct the transition vertex for the NS





37
open string and NS-NS closed string in a different superghost vacuum. Notice

that, in the earlier construction for the transition vertices, the string states

(open or closed) are chosen to be in the -1 picture.17,18 Suppose we choose

the superghost vacuum for the open string in 0-picture and keep those of the

closed string in -1 picture. The total ghost number of the vertex will then be

-2, which corresponds to the correct background charge for the superghosts.

This indicates that (0, -1, -1) is a more natural picture to construct the open-

closed transition vertex and no insertion will be necessary.

Let's denote the vacuum state with the picture numbers I(Po, P+, P-) >

for the open string, and for the left/right moving sectors of the closed string

respectively. For example, the vacuum IQ0 > in (0, -1, -1) > satisfies

3
ysojo >= 0 for s >
-2

1
O*O >= 0 for s2 -2

Since, only the superghost components will be affected by the choice of the

pictures for the ghost vacuum, we will rederive the superghost part of the

transition vertex. What we did earlier for the overlap of these superghosts in

the coupling between the NS open string and the NS-NS closed string remains

correct except that the mode expansions require some rearrangement. The

reason is that, in this new picture, g-1/2 is an annihilation operator. We can

substitute I0 in the overlap equations by g-1/2 according to (2.35). After some

shuffling, a new set of coefficients Y's are obtained in this picture. They are

listed in Table 2-5.

i = ,r > 1; = o,r > 3
1
s j = ,s > j =o,s > -






38












0 I 0+ +







cI I
0 I









n CD |D 0 (

C + 0
8 11 c .t -1 0
C --- --------











Z c
U2 + 0






II -
S I aI




-- -















o t
A 11 1 II II
& W C O

'
.s^ w- ld





39
In this picture, the insertion 10 does not come into the overlap equations

at all. This implies that the overlap vertex for the superghosts in this picture

is simply

< (y3P),(O,-1,-1) = < Do exp( ir,,i)
3=4

The complete transition vertex is the tensor product of the four overlap vertices

< r, (0, -1, -1) =< VB,FI < (bc)I < (7y),(0,-1,-1)1 (2.44)

It is easy to prove that (2.44) is a BRST invariant operator. When the

BRST charge acts on this transition vertex, it simply reproduces the overlap

equations of the energy momentum tensor and the supercurrent. Now the issue

becomes clear to us: The natural overlap vertex tends to mix the superghost

vacua of the open and closed string in different pictures. In order to keep all

the superghost vacua in the same picture, a. picture changing operator X is

inserted to the vertex at the interaction point. In the following, I will argue

that this conjecture is indeed correct by showing the equivalence of the two

transition vertices for the coupling of one NS open string state and one NS-NS

closed string state.

Let us insert the picture changing operator1" directly to the vertex in

(2.44). Since the superghosts in the vertex are unbosonized, it is easier to first

unbosonize those in the picture changing operator.

X(z) = b(#)[PO + 2by] c&#b(3) + 6'(#3)zb (2.45)

Notice that one should normal order this operator before inserting it to the

vertex. The creation modes in X act on the overlap vertex and bring down the

sums of annihilation modes as follows.








< r,(o, -1, -1)I(o,o)

S< B,F, (-, -1,-1)l exp(E A' ,,^c' E r i)
m< rs>1




4( b- -1) (- -b 1 e2,m+
m>O

+ ()P+ ()4q~ [-i2v/b-p bO2p+1 ] eij7+q+l + 702q+11]
p,q>O
(2.46)

By checking the transitions of a few low lying states, I find that this

lengthy vertex produces the same amplitudes as those from the transition ver-
tex < r, (-1, -1, -1)| in (2.37). A similar check can be done in the bosonized
language also. I adopt the notations used by Thorn [ 20]:

7 = e07 and P = e-01a

where { (z) = E nz-" satisfy {(n,7m} = 6n,-m
I (z) = ZT,nz-

and 4(z) = cnz-n satisfy [,i, Om] = mbn,-m

Also eO = V : eZEn-- : z-0-

For example, the first few low lying states are
7_-I(-1) >= 77-V (-1) >
2
7-3 (-1) >= (r-2 n-1-l0-1)V (-1) >
2

2 21 -


9_A 1(-1) >= (26-2 + 2-1-)V- (-1) >

-1 >= (36-3 + 2(-2 -1 + -1(2 1 -2))V-1(-1) >
2






The picture changing operator is more frequently used in the bosonized form.

X(z) = eOP P- c Oz e2((26b 0, i + 9zb t + 2b r- Oz 9) (2.47)

Inserting it to the new vertex (2.44) at the interaction point, we have
< r, (0, -1,-1) x(0,0)

=< n,0 < VB,Flexp( M',nnc + +Z A,~,nnl
m>l1 m>1
n>0 n>0

Smn> 2f k2k+1 k
m,nl> kL>1


- (-),P+'
p,q2>O


(-2q) ic-p+i + 2V2co2p+l]


+

[i-,q+1 2V o2q+1])
MnCn + Y mnin +- z ,nmn
b- 5*' ~ AT~q~q


m>l
n=>O
- i2 ) VI


m,n>l


+ < oI < VB,Flexp(A Mncm + iiij E Nij n;j
m>l m">1 m,n>l
n>O n>0
1 V2
+ T( 2k+1 -+ V ) Vri) x 8V
k>1


([-i ()P


S[i 1(-)q q
q>1


( b 7- (-) b2p+l
prO


1 T_1
-- -) q (--) 2q+1
-,E "q>Oq


+[V2 1] [x[2 ( ) 2q + 1 2q+
q>1 q>O

+ 2[i 1(-)'( -(-)q 2( 7 2q+.4
qP1 q >0

x (-)"'( -d + 02+11])
(2.48)





42

Note that the overlap vertex for the superghosts has been bosonized. It

is not difficult to see that the scalar fields O's overlap the same way as the

bosonic coordinates X's, see Eq.(2.1), except the linear terms corresponding

to the background charge in the particular picture. The rl, C ghosts (weight

1 and 0) overlap similarly to the b, c ghosts except an opposite sign for the

open-closed crossing terms, i.e.
M~0 = Mo00 MCC = MCC Mc = Mc MOC ~ -MC
mn mn mn mn mn mn mn mn

BRST invariance is manifest here because the picture changing operator

is BRST invariant by itself. Once again, this vertex in (2.48) gives the same

transition amplitudes as those of the vertex in (2.37). Although only the first

few terms of the insertion were checked explicitly with a few low lying states,

the results should hold for all higher terms in the insertion because both vertices

are BRST invariant. Therefore, the extended supercurrent insertion in (2.37)

can be identified as the picture changing operator acting on the vertex in the

(0, -1, -1) picture.

< F, (0, -1, -1)IX(0,0)

= < VB,F(-1, -1, -1)1 < ghost I(Il + W wi-y + Z c f8y) (2.49)

= < r, (-1, -1, )1

From the above examination on the transition between a NS open string

state and a NS-NS closed string state, I believe that the same interpretation

of the insertion as the picture changing operator holds in the other two types

of open-closed transitions.

To summarize, the covariant open-closed transition vertices for the NSR

spinning strings were constructed based on the overlap of string wave func-

tions and the extended supercurrent ansatz inserted at the interaction point.





43

Superghost contributions are determined by the BRST invariance of the tran-

sition vertices. By changing the picture of the transition vertex for the NS

open string coupled to the NS-NS closed string, the role of the extended super-

current insertion is understood as a picture changing operator. One should be

able to identify the insertion with the picture changing operator in the other

two types of open-closed transition in a similar fashion.











OPERATOR QUANTIZATION OF THE
GREEN-SCHWTARZ HETEROTIC STRINGS


Covariant Quantization

In 1983, Green and Schwarz proposed a covariant string action in which

the space-time supersymmetry was built manifestly. This theory is described

by the bosonic coordinates XI' and the Majorana Weyl space-time spinors 0

in ten dimensions. Both of them are scalars on the world sheet. It has a

local fermionic symmetry (K symmetry) which implements a global spacetime

supersymmetry in the theory. For the Type II strings, there is an extra bosonic

symmetry (A symmetry). In the light cone gauge, this theory has been proved

to be equivalent to the usual light-cone superstring in the RNS formalism.3

Because of the manifest space-time supersymmetry, it is desirable to construct

a string field theory in the Green-Schwarz formalism.

Many people21,22,13,23,24,25 have been trying to quantize the theory co-

variantly since then. Yet, very little success has been made. The difficulties

for achieving a canonical quantization of this theory come from the fact that

the fermionic constraint is a mixture of first class constraints and second class

constraints. Moreover, these constraints are linearly dependent on one another

(so-called reducible constraints). Hori and Kamimura were able to separate
these constraints covariantly, but the separated constraints remain reducible.

So far, only the Super Harmonic Space technique26,27 was able to overcome

these problems. The drawback is that at least 112 new harmonic variables must





45

be introduced into the theory. The huge number of these auxiliary variables

(pure gauge) makes this method less practical and less attractive.

Comparatively, the Lagrangian quantization method seems to be easier.

The mixture of the first class and second class constraints is not a problem at

all. However, the linear dependence of the constraints requires the introduction

of an infinite tower of ghosts for ghosts.28 Batalin and Vilkoviski (BV) formu-

lated a systematic way to quantize theories like this.29 If lucky, one may be

able to handle these infinite ghosts effectively. Most attempts of quantizing the

Green-Schwarz superstrings have been following this direction. I will review

the recent works on the BV quantization of the Green-Schwarz superstring and

discuss the frustration of these attempts. In the next section, I will pursue the

canonical quantization of GS superstring. For simplicity and consistency, I will

discuss only the heterotic strings.

The Green-Schwarz heterotic string action can be written covariantly in

ten dimensions as21


S= d2a (illa + ie2 alxerqp+ a4- + (3.1)

II = -aXP iOFk aO and e01 = -e10 = 1. 01 are the left moving world
sheet Majorana fermions, I = 1,2,..., 32. They are Lorentz singlets with some

internal quantum numbers such as SO(32) or Eg x E8.

Like the other string theories, this covariant action has a Weyl symmetry

and a local reparametrization symmetry. Its right moving sector has a local

fermionic symmetry, called K symmetry, which is closely related to the global

space time supersymmetry of the theory. In this article, we consider only the

right moving sector. The transformation laws under these gauge symmetries

are given as follows:








Reparametriza t ion

6XI = eC0AX

60 = e-0_0

= +

Sga = 8-7 -0 fg97f _97eg

K supersymmetry

`6X = i0FL60

60 = 2ill PrK+

`5 =0

6gap = -16 g=(Pi P =0)

where P0 1 (g2 Ea^/fg) = e7
2 e=F
Global super Poincare

6XL, = aPlX + b iOF"P7

60 = apvO + rl



6gf = 0

From the degree counting, the Majorana-Weyl spinor 0 in the ten dimen-

sional heterotic string has sixteen components. The gauge degree of freedom

for the K supersymmetry allows us to eliminate eight of them. As pointed

out by Kallosh [28], the parameter t+ itself is another sixteen component

Majorana-Weyl spinor. Obviously, twice too many gauge parameters have

been introduced. This means that not all the components in K+ are linearly

independent. In fact, this can be seen when we set K+ to be









=+ = P-t_++

4++ will parametrize another symmetry of the action, because II2 = 0 is one of

the equations of motion. The same argument applies to tc++ and K+++, etc.

Correspondingly, one must introduce infinite Fadeev-Popov ghosts c+, c++,

etc.28 Before applying the BV quantization, let us fix the world sheet Lorentz

parameter and the Weyl parameter such that the metric can be represented by

the vielbeins e.

According to BV, the quantized action is the solution of the master equa-

tion.
6,.1S 6iS _,.S 61S
(S, S) = --- = (3.2)
,- i Z Zi -

In general, this solution is written as an expansion of the antifields 0*:


S = SO+E Sick+ SJ. .+Snonmin, Snonmin -= E *ra (3.3)
i ij a
These antifields are defined as the functional derivatives of the gauge fermion

Q with respect to the minimal sectors 4 in the theory, i.e. the coordinates,

moment, and the Fadeev-Popov ghosts. are the rest of the field contents

including the antighosts and the extra ghosts. The gauge fermion is chosen so

that O*s correspond to the gauge fixing conditions in the nonminimal part of

the action.

Based on the transformation laws of the various gauge symmetries listed

above, Gates et al. solved the master equation for the Green-Schwarz heterotic

string.30 The quantized action has a finite number of terms with the antifields

only up to quadratic.


S = SO + S1 + S2 + Snonmin


(3.4)








S1 = x*Pa + (A[+P ])[+P- + aAab + 2-_.lrcilP

+ (et* P_*P+ (*i+)(A+(- 4clA+0)

+ [e-* P P- ac-

e(P P+ PI*P- + ( C+ *nc"c, e+*e-)] A-

+ c [pcn+1Pp + +ncnA-. (Aacn)a]
0
+ s4 (3.5)

S2 = Ax* A[+P* 2_(circl) cFcrn+)]
0
+ (e+* P*P+ +) [(X* A[+P )(circi)

+ 4*(Elc2) + c cncn+2]
0
0x n
Snonmin = :IFF + c n m*- ,
1n=l m=l
where C and i are the reparametrization ghosts. co = Cn(+) are the
Faddeev-Popov ghosts and cm for n > m > 1 are the extra ghosts and antighost
at the nth level. c* is defined as 6* oFX* 2P -A_]oP. And, Sf are the
four ghost coupling terms. Its details are not crucial and will not affect the
following analysis. I will not quote them here.
The real task now is to find the right gauge conditions such that the
gauge fixed action can be linearized, or manageable. From the experience on
the superparticles,31 the authors in references [24], [30], and [ 32] tried the
derivative gauge, a0+ = 0. At first glance, they seemed to over gauge fix.
But actually, eight of these conditions correspond to one of the equations of
motion, IIf~la'0+ = 0. Therefore, this gauge choice is consistent within the





49
BV formulation. There are many ways to choose the gauge conditions for the

ghosts. For example, Lindstr6m et al chose the gauge fermion to be
00 n
S= e+e + n- m+1~ 1 m (3.6)
n=0 m=0
The pyramid structure of the ghosts is30










\ \ \






Bergshoeff and Kallosh arranged the gauge fermion slightly differently.33
00 00
S= -e- + E On"'n+(9m,n + +m+l,n+l) (3.7)
n=0 n=m
where the pyramid of the ghosts is


00,0

\

01,0 p0,0

\ \

02,0 ^10 81,1

\ \ \


03,0 2,0 02,1 #1,1





50

Once the gauge fermion is chosen, the antifields can be determined.




Subsituting them into Eq.(3.5) for the solution of the master equation, we

obtain the complete gauge fixed action. Not too surprisingly, the action has

complicated interaction terms and kinetic terms.

At first, these three groups claimed that after some field redefinitions the

gauge fixed action becomes quadratic and the theory is essentially free!


Sg = (P+P -- -Xo+X + E S++Oa(
So n ) + (3.8)

n=1 m=l
Moreover, the conformal anomalies of the string variables and the ghosts cancel

among themselves with simple regulnrization with


fn = lim Efun
n n
oo n
10 26 16(-)"+12[6(2m n)(2m n 1) + 1] = 0.
n=0 m=0
However, more careful study shows that BRST invariance is lost after the

field redefinitions. Fisch and Henneaux34 pointed out that these field redefi-

nitions are noninvertible mappings. They found infinite residual symmetries

in the shifted action. Various modifications on the gauge fermions and mod-

ifications on the field redefinitions have been investigated.33 So far, there is

no satisfactory way of removing the constraints on the new fields in the "lin-

earized" action, neither of realizing BRST symmetry in the quantized theory.

Therefore, it is an open question whether the covariantly quantized action in

the derivative gauge corresponds to a free theory.





51

In summary, the frustration of Lagrangian quantization of the Green-

Schwarz superstrings is that the right gauge fermion which would lead to a

linearized (free) action has not yet been found ,or a more sophisticated but

unknown treatment is necessary. It seems that in order to study string field

theory from this, one must introduce more variables and more symmetries

to overcome the difficulties that we face currently. In other words, direct

modifications in the original Green-Schwarz action such as Siegel's23 seems

unavoidable. For example, LindstrSm et al have tried to enlarge the phase space

to replace all second class constraints by "virtual" first class constraints. So

far, they have succeeded in the covariant quantization of the superparticle.35 It

is unclear whether their generalization will work also for the superstrings. This

is out of the scope of this dissertation and I shall not say anymore about this.

Let us now move on to the next section and discuss the canonical quantization.


Canonical Quantization

As I explained earlier, the covariant canonical quantization of the Green-

Schwarz superstring is much more difficult than the Lagrangian quantization.

Unfortunately, even with the Lagrangian formalism, its covariant quantization

is still not very satisfactory. On the other hand, the path integral quantization

results are sometimes ambiguous due to the dependence on the regularization

of the measure. From the second quantization point of view, it is preferable

to have the canonical first quantization. In the following, I will introduce

the quantization method of Batalin and Fradkin to canonically quantize the

Green-Schwarz heterotic string in the semi-light cone gauge.

In the path integral quantization by Carlip13 and later by Kallosh and

Morozov,14 a partially covariant gauge was considered; namely, the conformal





52

gauge for the metric and the light cone gauge for the space-time spinors.

gaQ = r0o0 F+0 = 0 semi-light cone gauge (3.9)

Kallosh and Morozov's results indicated that the gauge fixed action is free of
the conformal anomaly. They even went further and discussed how the one,
two, three-point functions vanished by counting the zero modes of the spinors
in the action. It is therefore interesting to understand the machinery behind
the anomaly cancellation in the operator language, i.e. Q2 = 0. Hopefully, the
canonical quantization in this semi-light cone gauge will teach us something

useful for the canonical quantization in a covariant gauge.
In the operator formulation, the Hamiltonian for the Green-Schwarz het-
erotic string is
H = d2a(PtX + C Lo) (3.10)

L0 is the Lagrangian in (3.1). The dot (the prime) denotes the time (spatial)

derivative. From the symmetry transformations listed in the last section, the
corresponding constraints can be found with the Noether method. After some

rearrangements such that there are no secondary constraints,22 the constraints
for the reparametrization and the K symmetry are

L(a) =(P X')2 2(0'
2 (3.11)
Fa(a) =Ca + i(oF")a(P X' + i6ro'),
Since we are interested in the right-moving sector where supersymme-

try is implemented, let's examine the constraint algebra under the Poisson
brackets:
{ L(), L(a')p B = -2,L(a)6(a a') 4L(a)o)6(a a')

{L(a),Fa(a')} PB = -2Fa(a)O,(a a') (3.12)

{Fa(o), F(a')}pB = 2i(cr()ab(P X' + 2iBr8ao)6(a a').





53
It is clear that the fermionic constraint F is a mixture of the first class con-

straint and the second class constraint. Only eight of them are the first class

constraints which generate the K symmetry. The advantage of choosing the

semi-light cone gauge is that we can separate these constraints easily by the

following projections:

-f r= FF:, + + r = 1i ll = 1, :l = 0. (3.13)


After we impose the gauge conditions, the 1+ projection corresponds to the

conjugate momentum of F+0 and can be solved explicitly. The 1- projection

corresponds to the original second class constraint in the theory.

FI : (l+ = iBrF(P X')k

Fl- : G = (l- iM0O- = 0 where 3(a) = (P X')
(3.14)

We are now left with a theory whose first class constraint L and second class

constraint G are irreducible.

It is traditional to use Dirac brackets for the canonical quantization in

the presence of the second class constraints. In general, these Dirac brackets

can be very complicated and ordering ambiguities can cause serious problems.

Consequently, it is very difficult to implement BRST quantization with Dirac

brackets. Batalin and Fradkin proposed a new method to overcome these prob-

lems. The idea is very simple. By introducing new variables, the second class

constraints can be converted into first class. These effective first class con-

straints then generate virtual symmetries in the extended phase space, which

allows us to eliminate these new variables through gauge fixing and recover

the original second class constraints. These constraint algebra can be gener-

ated by the nilpotency condition of the BRST charge. There are two different

formulations for this new canonical quantization.36,37 In the following, I will





54
apply these two different formulations separately to the quantization of the
Green-Schwarz heterotic strings.



Old Formulation
Define the equal-time supercommutator for operators A and B:

[A, B] = AB (-)e(A)(B)BA. (3.15)

where e(A) denotes the Grassmann parity of the operator A. Under these
supercommutators, the conjugate relations among the string variables and their
moment are preserved.

[XP(a),P"(a')] = ihgP'(a a').
(3.16)
[Ca(), 0b(O')] = itlbb6(U 0').
For example, in the semi-light cone gauge, the constraint algebra reduces to

[L(), L(a')] = -29aLS(a a') 4L&a6(a a')

[L(a), Ga(a')] = -2Ga(a)a,6(a a')
(3.17)
[Ga(a), Gb(O)] = WabM6(a a')

Wab -2i(Cr)ab, M(U) (p x')+
This constraint algebra can be generated from one equation, namely, the nilpo-
tency condition of the BRST charge QB.


[QB, QB = 0 (3.18)

Let us introduce the reparametrization ghosts B, C for L and the spinor
ghosts 7, fP for G.
(C) = e(B) = (L) + 1 = 1
[C(a), B(')] = b(a ') ghl(C) = -gh(B) = 1 (3.19)
gh2(C) = gh2(B) = 0




55
( e(7) = ()= e(G) 1 = 0
[ra(a), b(a')] = 6 56(a a') gh2-) = -gh2(6f) = 1 (3.20)
ghl = ghl(#) =0
To convert G into a first class constraint, we introduce new variables VO
which have the same Grassmann parity as the constraint itself. And they obey
the following commutation relation:
[ca(a), b(ar)] = _-abb(a a')
(3.21)
e(I) = 1 and ow = I
Then a linear term in 4 is added to the naive BRST charge for G to make it
nilpotent.
[, Q2] = 0

Q2 = da-a(Ga + abM' ) (3.22)
It is easy to see that the integrand is now an effective first class constraint
which anticommutes with itself. In general, the solution for Q2 will be a series
expansion in 7 and j, such that

e(Q2) = 1 and ghl(Q2) = 0, gh2(Q2) = 1.


To construct the complete BRST charge, we add in addition to Q2 a
nilpotent charge Q1, which generates the Virasoro algebra for the first class
constraint and at the same time anticommutes with Q2.

QB = Q1 + Q2, [Q,Q11 = [Q1, Q21 = 0

Since Q2 is linear in D, we can write Q2 = 02 + a>" Equivalently, Qi obeys

[Q1, Q2] = [Q1, a]

If we take the commutator of this equation with the conjugate of .a,
i.e. (Sa, fb] = 6ab, this condition can be viewed as a first order differential
equation with respect to 4. It has the following solution







+-4
Q1 =QleD 'D

with Aia =[, [t2, Qa]]

and [Q(, Pa] = [ l, ia] =0

where Qi is independent of <. On the other hand,


[Ql,0Ql]=0 Q l0e eDDi =0

Batalin and Fradkin suggested that (Q can be solved with a series ex-
pansion in powers of ghost s as follows.


Q1 = J(CL+ VICCB + UICy +...) (3.23)

(Q1) = 1 and ghl(Qi) = 1, gh2(Q1) = 0
However, with the above expansion, there exists no solution! Modification such
as allowing second class constraints at the first order (linear in C) is necessary
for a possible solution to exist. This situation happens whenever the original
first class constraints do not form a closed algebra with the new effective first
class constraints. Adding a term quadratic in the second class constraints to
L, the nilpotent BRST charge can be solved right away.


QB =Q1 + Q2
/ da(C(L + M-lbGaaGGb) + 2CaCB Ca() + CyfM-^aM

+ a(G. 4bAfb 1
+ 7a(G + .abM ))
(3.24)

It turns out that the linear term is the same as writing the L constraints
in the starred variables which are used to substitute the Dirac brackets. In
other words, one can define the starred variables as








q* q {q, Gi}A'Gj, Ai = {G, Gj}

then {q,p}D = {q*,P*}

we find L(X*, P*, 0*, *) = L + MAIGa, GbC^ab

It is easy to check that L* now forms a closed algebra with the new effective

first class constraint. In other words, the BRST charge generates the constraint

algebra of two first class constraints, L* and the extended G constraint.

Another remark about this BRST charge QB is that, in this old formu-

lation of Batalin and Fradkin, the rank of the charge can generally be higher.

Fortunately, in this semi-light cone gauge, it turns out to be first rank only

(cubic in the ghosts). This is favourable for the second quantization. Also,
analogous to the bosonic superghosts for the covariant NSR superstrings, we

have bosonic superghosts -y, # for the fermionic constraints G. Like the con-

straints, these ghosts are space time spinors and world sheet scalars. Their

contributions to the conformal anomaly will be discussed in the last section.

The BRST charge not only generates the constraint algebra for the the-

ory, its kernel also defines the physical space. As a conserved charge, QB

commutes with the Hamiltonian. From the consistency of the constraints, all

physical observables should be BRST invariant operators. For example, the

Super Poincare generators should be extended in the enlarged phase space so

that they commute with QB. It is important to check the closure of this algebra

after the extension and after gauge fixing.

To preserve the gauge fixing conditions, the naive Lorentz transformation

laws must be compensated by the transformations from the reparametrization

and the K symmetry. The corresponding generators can be found with the

Noether method.









Jo+- = (dP+X- P-X+ C(0)

JO+= Jd(P+X' P'X+) (3.25)
(3.25)
Joi= d(P-Xi pi- + i(P X')kBF-okio)

jJ = I da(Pj' PjX' + Ca'iG)
QOa = f d(a + i(r-)a.M 2i(P X')k(Irl+).)
In the enlarged phase space, one should modify these generators in the
same way as the unitarizing Hamiltonian discussed in Ref.[ 36]. They will be
extended to series expansions in powers of the ghosts with zero total ghost num-
ber. Adjusting the expansions with second class constraints, one can impose
BRST invariance of these generators. The results are

P11 =-Po
J+- = Jo+-_
J+i = +i
J-i = Ji + jd( M- (P X')i(apya + 0)
+ M-( X)kCGki9 M-2(P Xt)kGaCkiG)
Ji = J i + da(M- Gaijc + twaiJ)

Qa = Qoa -f daM-'(P X')k(GTr+r)a
If the semi-light cone gauge fixing is consistent, then the super Poincare
algebra should close up to a BRST transformation. At the classical level, these
extended generators indeed give the standard super Poincare algebra.








[pP, pv] =O.

[PI, JVP] = gPVpp g7p1pv

[P", Qa] =0
(3.26)
[JPV, JPl = gVPJ'Ua + gPjJVP gV JPP gP PJV + [Q, S"fpl]

[Jpv.Qa] = -QbOa + [Q, Sa']

[Qa, Qb] = -2i(CP-)abP + [Q, Sab]
Notice that the S's are determined up to BRST only. The nonzero ones are
listed as follows.

S-i+j = da (c' jfl.- + GwiJ'M-1)

Sijk = Jda(M + A M^G,)aklaij

S-i-j = daM-3B(( + iMOi-)&ai'(( + iMiT-) (3.27)

i = 1 /doBM 2( + iMI-)b(,r+r)ba

Sab= 2i(CP+)ab daBM-1.

At the quantum level, this closure usually provides an important check
for the critical dimension after the normal ordering is taken into account. A
discussion of the quantum anomaly will be given in the last section.

New Formulation
Because the BRST charge constructed from the old formulation is in
general of higher rank, its practical use will be severely reduced. That is
why Batalin, Fradkin and Fradkina37 derived another formulation in which
the BRST charge will, by construction, be first rank only. The punch line is
to extend the second class constraints G to a series expansion in powers of
the new variables ( such that the new constraints G commute(anticommute)




60
among themselves. Similarly, one can extend the original first class constraint
L also in a series expansion of 4, such that L commutes with 6. Consequently,
in the extended phase space, these "tilded" first class constraints lead to a first
rank BRST charge. For example, for the Green-Schwarz heterotic strings in
the semi-light cone gauge, we can extend the second class constraint Ga to

Ga(a) = Ga + wabbM2 such that [da, Gb] = 0


The extended reparametrization constraint is
~ a b
L(a) = L 2Gaa-(4 a A) wabo

so that [L, Ga] = 0.

For example, Virasoro algebra is now extended to

[L(7), L(u')] = -20aL6(o a') 4La,6(a a')
(3.28)
4cabGaa(A-f- 1a(Gb6(U ao)))

Now that all constraints are first class, the BRST charge is simply

Q= Q1 + Q2
= / d (CL + 2COCB + 4COCab8UG, baMl- + aGa) (3.29)

Physical observables, such as the Hamiltonian or super Poincare gener-
ators, will be extended in a similar fashion. They should commute (anticom-
mute) with the BRST charge. Or equivalently, they should satisfy

[A, Ga] = 0 and [A, Qi] = Ga
This can be done by expanding in a power series of
This can be done by expanding A in a power series of #:


A = A+ Aa+ AabaIn + ...




61
For example, we can extend the super Poincare algebra generators in the en-
larged phase space as follows.


P = JdaPP

J+- = doaP+X- P-X+ I GaaM- + C Gabod.aflM 1')

J+i= da (P+Xi iX+)

J- = d (P-Xi P'X- + i(P X')F- aOki

21
+ i(P X')kM-Or-r1 r -(P XI)kM-l -waki
+ (M-4 + 0)rirk8a-(c3)M-l(P X')L

+ 2Co ,c fpi rkrFM-2(P ,')k)

Ji = Jdu(p'Xj P'Xi i0 + tj G+ M-i

awoi' + M-1,2(CM)Gai )

Qa = du(C+ i6r-M 2i(P Xi)krkl+ 2i M-(P X')k4b(CFT)ba
+ 4iM-'(P X')kO1(CJb)cbd(crF)da)


Quantum Anomaly
In this last section, I will examine the quantum consistency of the semi-
light cone quantization discussed in the previous section. Due to the operator
product singularities, an ordering prescription for the operators must be given
to regularize the infinities. It is important to check whether the classical BRST
charge remains nilpotent and whether the global symmetry, such as super
Poincare, are respected or not. The nilpotency of the BRST charge implies
that the quantized theory is free of the conformal anomaly. The closure of
the super Poincare algebra assures that the noncovariant gauge fixing does not
destroy Lorentz invariance and space-time supersymmetry of the theory.




62
For a free theory, one can use the normal ordering for regularization.
Unfortunately, in this semi-light cone gauge, 0 does not have the usual kinetic
term and it couples to oX+. Without solving the energy spectrum explicitly,
which is extremely difficult because of the interaction, we can expand BX+
around a constant momentum, as in the true light cone, and treat the action
perturbatively. Therefore, string coordinates and their conjugate moment can
be mode expanded according to the periodic boundary conditions for the closed
string. More specifically, we are choosing a particular basis for the Fock space.
In this basis, the vacuum is annihilated by all the positive modes and by the
zero modes of the moment C, b, and /3.
Taking into account of this normal ordering, the quantization that was
discussed in the earlier section must be re-examined. The failure of retaining
the BRST nilpotency and/or closing the super Poincard algebra will indicate
a quantum anomaly in the quantized theory. Let us now examine the results
obtained in the old formulation. A straight forward calculation shows that the
normal ordered BRST charge does not anticommute with itself. Denote : A :
as the normal ordering for an operator A.

[: Q:,: Q:] =(i)2 CpCp ( 11)p3 +(- + 5)p
p>0

+ (ih)2 daCC 6-iM-18M) + M-^18)
(3.30)

Naively, one thinks that Q is anomalous in ten dimensions, and there
exist also M dependent terms that violate Lorentz covariance explicitly. This
is not so surprising if we compare the result with the Liouville action derived
from the heat kernel calculation.14
exp(4 ) ((- 13 + 4 x 2)I1ln pl2 + 4(21n p + In ju 2)01n lu12)

u = _X+




63
There exist u-terms which violate Lorentz covariance explicitly. If the argument
in Ref.[36] is correct, such terms can be removed by shifting the X- in the
action. We can add o(h) corrections into the BRST charge to remove the
M dependent anomaly in (3.30). From the power counting, there are four
independent terms that produce the types of anomaly that appear in (3.30):

Q= Q + dC (aM18 M + b(MA- M) + dM-1 OaM +ti

where a, b, d, t are free parameters. Therefore, the anomalies that violate
Lorentz covariance will disappear, provided a + b = -1

[: :,: :] = 2 Cp ( 11 12a)p3 + ( + 5 3t)
6n V 2 2
p>0
(3.31)
The consequence of this shifting is that the nilpotency of : Q : does not

determine the critical dimension. There is an extra free parameter a appearing
in the cubic term. An interesting observation is that in Ref.[ 38] Kraemmer

and Rebhan pointed out a corresponding ambiguity in the path integral result.

Let us rewrite the Liouville action as

d 2z (( 13)1alnp 2 + 4.26(1l1n p12 + In u 12 n p)

+ 4-2(1- )lnI u 2aInp + 41n lul1280Inu2)

The second term is a total derivative! This allows us to put an arbitrary

parameter in front of the total derivative. Therefore, the critical dimension can

not be determined from the heat kernel method. We have seen this ambiguity
from the operator quantization method where the free parameter a appears in
the shift of the BRST charge in (3.31).

To determine the critical dimension, let us examine the closure of the
super Poincare algebra. Similar to the BRST charge, one expects that its





64
generators will have quantum corrections also. These corrections can be found
by imposing BRST invariance to the generators. With the shifted Q, it is not
difficult to find that all the super Poincard generators commute with Q except
:J-k:

[: J-k: Q :] = (i1 2) daC (aM.-OM + ( + )(M-~M)2

+ (d + 2)M-laM + t)(P- X')k
(3.32)

If J-k were BRST invariant, the nilpotency of the BRST charge in (3.30) will
imply that the critical dimension is 22 instead of 10. But the Green-Schwarz
heterotic string is constructed in ten dimensions, it is more likely that J-k is
not a BRST invariant operator at D = 10. Indeed, one can add the following
corrections to J-k and shift the anomalies in (3.32).

j-k = j-k + h/ d (-(P X'i)kM.- + u(P X)kM-1).

such that the commutator of j-k and the BRST charge Q becomes
[: j- :,: Q :]

=ih2 J daC- 6 ((a + )M-'M + (b + z)(M-1M)2 (3.33)

+ (d + 2u + 2i)-1. + t)(P X)k
Note that the coefficients z, u in front of the quantum corrections are
arbitrary at this moment, since the coefficients for the quantum corrections
of the BRST charge, a, b, d, t are free parameters also. Thus, the quantum
corrections to Q and J's are still ambiguous. In the following, we will see
how they can be determined uniquely from the closure of the super Poincard
algebra. Because some of these generators have cubic and quartic operators,
one must be careful with the ordering of their commutators. More specifically,
I will regularize the supercommutators as follows.








[: AB :,: CD:] =: AB :: CD: -_()(A+EB)(CC+'D) : CD:: AB :

The quantum ordering effects appearing in the algebra (3.26) are the
commutators involving j-k and the supersymmetry charge Qa.
j-k I: j-l:]

= ih Jda(L + M-GoaG Msa(yAM 7 ) + ht)M-3YaklY

+ .( ) i)2 daM-2 ((P -. X,)ka(P X')1 (P X,)l (P X')k

+ h2 da M-30^ ak + lM-aOakIaO
7r2
+ 2' a 2
M 2 2
+ [(-4 z)M-5(OaM )2 ( 8 ) OMa + (-i + 2unx)M-40aM](aki&
+ [(-- )M-3(a9M)2 +(2 + Z)-2a2M +(- + r)M-2aM] "OwOklO
4 8 2
+ [( + z)M-4(aM)2 + ( 3 )M-3^ + ( 2u7r)M~-3aM]C(kIO
2 42 2
Y = ~+ i0r-M.
(3.34)
The operators in the integrands are all normal ordered. The commutator of
: J-k : with the supersymmtry charge Qa gives
[: j-k. : ]: O :

= -ih Qbabk

+ de(f z 5 Mz(4 M) +( 5)M^8)M
+ dax(- 4(aoM)2 + (z + )M-2M1
5i (3.35)
+ (u7r )MA-39MM) c(r+ k)ba

+ 8 da ((-W )M- (aQM) + (- + z)M-^M
r + (iu + M-216 4
+ (iur + )M-2 M)b(r-r+rP)b





66
To close the algebra, these anomalies must be BRST trivial at the critical
dimension, in which case the integrals can be written as the commutators of
Q with S-k-I and S^k given in (3.27).

[: -k-I :,: :]

= iA d (L+ M-1GoaG MAa(#fM-1))M-3YeakY

+-- da1 _M^ k r akl
J 1 2
+ 3-2M 1ak02- 3M -3oM(akHCO
2 2
5+ ar)-42o.M + (i +dr)M-4OaM](aktCO
+ [(-5 + br)M-5(~aM)2 + (+ + ar)M-4OM + (i + dr)M4M](a1k

+ [(3 + M-3 M2 1+ at)-2M + (d)M-2aM]OwaklO
+ [(5 b)M-4(aOM)2 ( + M- M ( + d)kl
2 2
(3.36)
and

[: Q :,: sa :
= 2e 9 (- + )M-4(OA)2 + (13 + M2)M-3n M



+ -- d [(- + i )MA3(aM)2 + (15 + i)M-2


+(-- + i)M-2y. A -r+rk

+ t (ih2) : Sa-k:
(3.37)
Comparing the nilpotency condition (3.30) with (3.34) and (3.35), I find
that all quantum anomalies cancel and the algebra will close, provided that

z= u =0
1 2.
+-+ b=t=0, a = d=--ti
27 7T


4-+ D=10





67
The anticommutator of the supercharge with itself does not have a quan-
tum anomaly, but the commutator of Sab and the BRST charge has an anomaly
that vanishes only at D = 10.

: Qa :,: Qb :] = 2h(cr")abP, 2h(Cr+)ab I dM-'(L + M-1^G&oG)

[: Q :,: Sb :] = -2h(Cr+)ab daM-(L + M-1GOc G)

+ 2i(Cr+)ab2 dr(1 + ar)M-2^M
2r 2
Therefore, our conclusion is that the critical dimension of the Green-
Schwarz heterotic string is ten! We have uniquely determined not only the
quantum corrections to the BRST charge Q but also those to the super Poincard
generators which close the algebra up to a BRST transformation. This com-
pletes the realization of the conformal anomaly cancellation in the operator


language.












CONCLUSION


The open-closed transition for the RNS spinning string and the first

quantization of the Green-Schwarz heterotic string have been studied using

the operator methods. For the open-closed transition, three types of couplings

between the NS (R) open string and the R-R, NS-NS (R-NS) closed string

were discussed in detail. The transition vertices were constructed based on

the overlap equations of the string variables, the energy momentum tensor and

the supercurrent. Lorentz covariance is manifest in the construction. Ghost

contributions and the insertions are determined by requiring the vertices to

be BRST invariant. By shifting the picture of the superghost vacuum for the

open string, we understand that the insertion operator not only ensures BRST

invariance but also plays the role of a picture changing operator in order to

keep all superghost vacua in the same pictures.

In the operator quantization of the Green-Schwarz heterotic string, in-

stead of using the Dirac brackets, I applied the new method of Batalin and

Fradkin to derive the BRST charge in the semi-light cone gauge. It was found

that the old BF method failed to solve a general gauge theory whose original

first class constraints and the new effective first class constraints did not form a

closed algebra. In this dissertation, a slight modification was suggested to the

ansatz which solved the generating equations of the constraint algebra. After

this work had been completed, Batalin and Fradkin generalized their formula-

tion by modifying the generating equations rather than modifying the ansatz.





69

The results of these two modifications give the same BRST charge discussed

in this dissertation.

The classical super Poincare generators are constructed in the extended

phase space. The extensions are made according to BRST invariance of these

generators. To study the quantum ordering effects, I chose a particular normal

ordering prescription and re-examined the nilpotency of the BRST charge and

the super Poincare algebra for the quantization performed in the old formu-

lation. With the proper quantum corrections to the BRST charge and the

Lorentz boost J-k, the super Poincare algebra closes up to a BRST trivial

transformation at the critical dimension, D = 10, and conformal anomalies

also cancel at the critical dimension.

I have also applied the new formulation by Batalin, Fradkin and Fradkina

to study the semi-light cone gauge. In this new operator formulation, the

BRST charge is manifestly first rank (cubic in the ghosts) and no modification

is needed as in the BF formulation. I did not discuss the quantum anomaly for

the quantization in BFF method. Presumably, the anomalies will also cancel

once the proper quantum correction terms to the BRST charge and the super

Poincare generators are found.

The result on the operator quantization in the semi-light cone gauge is

very encouraging for the Green-Schwarz formalism. One can further study the

BRST cohomology of this theory and possibly construct a noncovariant string

field theory out of it. But one should remember that the ultimate goal is to

formulate a covariant superstring field theory. The success in the semi-light

cone quantization is only one small step towards the ultimate goal. So far,

there has been no success in achieving a reasonable covariant quantization the

Green-Schwarz superstring. In the path integral quantization, the covariantly





70

quantized action gives highly nonlinear interactions. In the operator quantiza-

tion, one hopes to be able to apply BF or BFF method in a covariant gauge.

But, these methods only work for theories with irreducible first class and sec-

ond class constraints. One must generalize the operator quantization method

by BF or BFF to theories with reducible gauge constraints. Otherwise, one

should look for alternatives such as enlarging the phase space and the sym-

metries in order to overcome these difficulties. More study is necessary before

one can rule out the possibility of achieving a covariant quantization of the

Green-Schwarz superstring.










APPENDIX A. PATH INTEGRAL
FOR OVERLAP VERTICES




According to Mandelstam's prescription of light cone string interaction

in the RNS formalism, the transition amplitudes between one open string and
one closed string can be written as16
T
f dtH
Ixc,c >=< xc, o1 e _To IXO, o >
< o o -SoD C( 0)) b>(= = XV e-So 6 (X(,0) X,(,O)) (0) T c,(,0))
Xcopc
= J vxD"(-So(X,,)+f{F'X+G+++G }- bk -k k- Bk )

with a = 7 a.
(A.1)
where So is the classical action for the fermionic string in ten dimensions.
T v
So(X, ) = dr d((VX)2 + i+(r iO9)+ + 0-(r + ijo)-)
-Tc 0
The sources F and G specify the external boundary states. For example,
the source term G for the spin variables is

G(a, r) = +C(o, 0)6(r + Tc) + o(7r a, 0)6(r To).

However, this transition amplitude in (A.1) is not Lorentz invariant.
Mandelstam suggested that in order to maintain the Lorentz invariance of

the amplitudes in the light cone gauge, a supercurrent must be inserted at the

interaction point.










To evaluate this integral, one can use the Neumann's function technique.
We need to construct the Neumann's functions for the following equations.

o+g+,b = 27r+b62(p- p')
a-K_,b = 27rb6 62(p p')

The right handed modes are related to the left handed through complex
conjugation, therefore we also have


K-,b(p, p') = K+b,(p*, ')

Thus, the amplitude becomes

J DXD exp (so(X, ) J1/GalKa,bt",Gb' + surface terms) (insertion)

The surface terms will vanish provided the Neumann's functions satisfy the
proper boundary conditions. In general, these Neumann's functions are very

complicated for finite times, Tc, T. However, the leading contributions come
from the limit when Tc, T go to oo. One can Fourier expand these functions

and evaluate the amplitude. Thus, a transition vertex can be extracted in
terms of oscillators.

A.1 NS open string and NS-NS closed string.

The boundary condition for this type of transition is that the Neumann's
functions have to be anti-periodic for both the open string region and the closed
string region. Such a function is already known in the upper half plane,

1
z z





73

Since the K's have the same conformal weight as the spin variables, the correct
Neumann's function in the p plane is simply

(1 (p)- p'
K--(p, p') =
z z \-z ,')

After doing the Fourier expansion, the evaluation of the amplitude is

straight forward. We found that the result agrees with the one we did in the

overlap method.


A.2 NS open string and R-R closed string.

In order to obtain periodicity in the closed string boundary and antiperi-

odicity in the open string boundary, we modify the Neumann's function in the

p plane to be

K1 z -i z'+i 2 p 2 p' 2
K__(p,p')= __ \ )a
z z' z' i z + i) lz z


A.3 Ramond open string and R-NS closed string.

This boundary condition is a little more complicated than the previous

ones. It requires antiperiodicity in the open string and opposite periodicities

for the right moving modes and left moving modes in the closed string. This

can be done with the following function:

1 z 2 8 2 8 2
K--(p, p') z z'

For the NS-R closed string, all we have to do is to exchange the left moving part

with the right moving part. Due to the zero modes in the Ramond sector, the

Neumann's functions also carry the helicities (upperindices), and the relations

among them are





74

Kab++ = ab-- = 0

Kab+-(z, z) = -Kab-+ (, Z)
After substituting the Neumann's function, the path integral in (A.1)

gives the same overlap vertex as from the overlap calculation in the basis whose
vacuum is chosen to be I T>.












APPENDIX B. USEFUL FORMULAE


B.1 Combinatorial Relations


E
1=0


1=0
'a) (,-a


mo
1 a =a
^k-1 1
1= 0


k -b 1) ( a + b

b ,i= a+b-1a
-1 k-1b

a (-a-1 (am
a+k k+rnm\


B.2 Identities Of The (,q, p's ...etc.

The definitions of the coefficients are
00
(nn- (1 + x)-4(1 X)-4
n=O

xn ( 1 + i
?77n 4l(l+ (- x)-4
n=O
Pn -- n + (n+2 7n Cn C n
m
m+l ~ 2T2r r2r+l) c2m
r=0
They obey the following recursion relations,


7p C p-
2p


p (_)k (2)
L,=0
1=0


(B.5)


En =- 7n r7n-1


m
= (2r 72'r-1)
r=0


for p > 1


(B.6)


(B.7)


for p = 2k
for p odd


(B.1)


(B.2)


(B.3)


(B.4)


-1)
mn


(-a1
1 rm


m

1=0


a a-1
1 k-m-


i-2






Sk _-)k for p = 2k + 1
S2(-3)()k for p= 2k + 2
1=0 { fop= 2k+2

(-) 1 772m-21 = (2m C2m-1
1=0

(-) 1 72m+I-21 = (2m+I1 C2m
1=0
Note that (n, 7n, ,n vanish for negative index n.


B.3 Gamma matrices.


(B.8)


(B.9)

(B.10)


Following the notation by Peskin [ 39], r's are the 32 x32 gamma matrices
in ten dimensions.


ro=f -') r9=(0 -) rk = ( y
(i 0 -i 00 -

where y7 are the 16x16 gamma matrices for SO(8). That is,

{y,7,8} = 2kI

And C is the complex conjugate matrix such that

C- = CT = -C and C,,C-1 = -rT


r


(B.11)


(B.12)


I- /


(B.13)


i)












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BIOGRAPHICAL SKETCH


Meifang Chu was born in Hsin-Chu, Taiwan, as the youngest daughter

of Chin-Lian Chu and Hsei-Quei-Chu Chu. After her graduation from a local

high school there, she went to Taipei Municipal Teachers' College and later

became a teacher in the East Gate Elementary School in Taipei. She taught

there for two years before she continued to pursue her Ph.D. degree in physics

at the University of Florida. She has wanted to be a theorist in particle physics

since high school. She expects to continue research in the high energy theory

after her degree is completed.












I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.



Charles B. Thorn, Chairman
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.



Pierre Rmond, Cochairman
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.



Pierre Sikivie
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.



ItichardField
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope
and quality, as a dissertation for the degree of Doctor of Philosophy.



Paul Robinson
Assistant Professor of Mathematics












This thesis was submitted to the Graduate Faculty of the Department of
Physics in the College of Liberal Arts and Sciences and to the Graduate School
and was accepted as partial fulfillment of the requirements for the degree of
Doctor of Philosophy.



May 1990
Dean, Graduate School
























































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UNIVERSITY OF FLORIDA

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