A collection of theoretical problems in high energy physics


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A collection of theoretical problems in high energy physics
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vi, 85 leaves : ; 29 cm.
McCarty, Thomas, 1957-
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Thesis (Ph. D.)--University of Florida, 1990.
Includes bibliographical references (leaves 82-84).
Statement of Responsibility:
by Thomas McCarty.
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I wish to thank Professor Thomas Curtright, my dissertation advisor, for

many enlightening discussions, and Dr. Ghassan Ghandour for many helpful

and illuminating discussions. I would like to thank Professor Pierre Ramond

for being my surrogate advisor.

I thank the University of Florida particle theory group for useful discussions

during my stay here. I also thank the theory group at the University of Miami,

for their hospitality during my stay there. In particular, I express my gratitude

to Gary Kleppe, Dr. Jun Liu, Dr. Raju Viswanathan and Dr. Ezer Melzer

for many helpful discussions. I express my gratitude to Mr. Marco Monti

for his computer expertise, which was invaluable during the writing of this


I wish to thank my family, especially my brother Sam and his wife Paula,

for their support and encouragement. I also thank all those friends (whom I

do not list, for fear of accidentally omitting someone) who made my time in

graduate school an enjoyable experience. Finally, I thank my wife Julie for her

support and love, which made all this work possible.



ACKNOWLEDGEMENTS ...............

ABSTRACT . . ...

INTRODUCTION ...................

Quantum Backlund Transformations and Conformal Algebras
Angular Realization of SU(1,1) . .
Radial Realization of SU(1,1) . .
Backlund Transformation ...............
Extension of the Radial SU(1,1) . .

. ii

. iv

., 1

. 10
. 10
. 10
. 16

Invariants of the Biicklund Transformation . .

Rigid String in a Curved Ambient Space . .
de-Sitter Static Solution and its Stability . .
Ambient Space Fluctuations to the Hoop Solution .

Classical Solutions of the Canyon Potential Model .

Quantum Instability of the Canyon Hamiltonian .. ..


. 19
. 21
. 25

. 35
. 36
. 39
. 42

. 44
. 46





REFERENCES ...... ... . .82

BIOGRAPHICAL SKETCH ........ ........... .85


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy




August 1990
Chairman: Pierre Ramond
Major Department: Physics

A Baicklund transformation is used to relate two different potential models,

which are both realizations of the algebra SU(1, 1). The Backlund transfor-

mation is used to extend the "angular" realization of the Virasoro algebra to

that of the "radial" realization of the Virasoro algebra. This extension is pos-

sible, only if the Backlund transformation is a similarity transformation. This

example shows how the Bicklund transformation projects out the unphysical

part of the free theories spectrum to coincide with the positive spectrum of the

interacting interacting model.

An exact solution to Liouville theory is found by using a functional Bicklund

transformation. This solution of Liouville theory has the nice property that

it is much more straight forward to compute correlation functions in pertur-

bation theory than by using the previous solutions. This functional solution

is compared to the other known solutions by computing the expectation value

of exp gac to order g6. The correlation functions are found to be exactly the

same at the order of g6 in the coupling constant. However, there is a term of


order g4 that did not appear in the calculation of Braaten, Curtright, Ghan-

dour, and Thorn [1] that is present in our computation. This term is of the

right form to renormalize the mass term that appears in the theory.

The classical stability of a closed rigid string immersed in a curved ambient

space-time is examined in the third chapter of this dissertation. A zero action

solution of the closed rigid string is obtained when the string is immersed into

a two-dimensional de-Sitter space-time. This exact solution is given a radial

perturbation and found to be unstable. Next, the hoop solution of Curtright

et al [2], is perturbed by adding ambient space-time fluctuations via Riemann

normal coordinates and by perturbing its radial coordinate. It is observed that

the space-times of negative curvature are closer to stabilizing the hoop solution

of the rigid string than the spaces of zero and positive curvature.

A renormalization group analysis for a two-dimensional field theory is used

to point out that conclusions on the spectrum of the supermembrane computed

using finite dimensional quantum mechanics in the large N limit are not nec-

essarily valid. This argument arises from analogy with a two dimensional field

theory whose potential has the same functional form as that of the supermem-

brane. It is shown that the Hamiltonian of this field theory is unbounded from

below and is unstable.

Hence, the continuous spectrum determined by the truncation of the su-

permembrane to a finite number of degrees of freedom, i.e. supermembrane

quantum mechanics, is not realistic when the number of degrees of freedom

is allowed to become infinite(i.e. in the limit that the theory becomes a field

theory instead of a quantum mechanical model). This suggests that a phase

transition can occur in the limit that the number of degrees of freedom becomes

infinite. Therefore, any argument of the instability of the supermembrane

based on the spectrum computed using supermembrane quantum mechanics is



The last decade of particle physics has undergone a major revolution in

the way that particle theorists view the unification of forces in nature and the

physics of the Planck scale (1019 GeV). This revolution has occurred because

until recently there was not a consistent realistic theory for quantum grav-

ity: either the theory was not quantum mechanically consistent, or the low

energy particle multiple was inconsistent with the standard model of particle

physics(see ref. 3 for a complete review).

To be able to unify all the forces in nature, requires a proper description of

the quantum geometry of nature. The theory of general relativity, which pro-

vides the classical description of geometry at distances large compared to the

Planck scale, is found to be unrenormalizable [4] when it is quantized. This un-

renormalizability means that quantum corrections to the Einstein-Hilbert ac-

tion have additional ultraviolet structure which were was not originally present

in the action: new terms and coupling constants must be added to the action

that were not originally present in Einstein's action to subtract off these ultravi-

olet divergences. But such terms have other physical effects besides ultraviolet

subtraction. Furthermore, such terms proliferate uncontrollably as higher en-

ergies are probed so the model completely breaks down at very high energy

scales. Supergravity theories are unrenormalizable and are also found to have

an particle spectrum inconsistent with what is observed experimentally.

The revolution that occurred in physics in the last decade was to view dual

resonance models, or strings, as theories of just strong nuclear interactions but

of all the interactions in nature. This revision of thinking has occurred because


the supersymmetric versions of string theories were shown to be free of anoma-

lies [5] for certain choices of the internal symmetry group, therefore strings

reappeared in theoretical physics as the most viable models of unifying all the

forces in nature, including gravity. The reason why strings are thought to be

a theory of gravity is because the closed string spectrum naturally contains

a massless spin-two particle [6] that is identified with the graviton of general

relativity. Strings have been shown to possess excellent ultraviolet behavior

which was the stumbling block of the quantized version of general relativity.

There are only a few different types of string theories [7]. The simplest

model is the bosonic string [8], which is consistent only when the space-time

dimension is twenty-six [9]. The bosonic string is parametrized by XP(a),

which are the space-time coordinates, where p = 0, 1, .,25 labeling the

one time and twenty-five space coordinates of the string, and a is a parameter

labeling the individual points along the string. There are also supersymmetric

strings, where the supersymmetry is either manifest on the worldsheet or on the

space-time [5]. Both versions of these supersymmetric strings have a critical

dimension of ten space-time dimensions. The string theory that is the best

candidate for nature is the heterotic string [10] which combines features of both

the bosonic string and world-sheet supersymmetry of one of the superstring


There are two types of boundary conditions for the bosonic string. These

are open boundary conditions, corresponding to a string with two endpoints

and closed boundary conditions, where XP(0) = XP(2xr) corresponding to a

closed string loop. The heterotic string and most of the other supersymmetric

string theories have only closed string boundary conditions.


We have previously pointed out that strings are only consistent in twenty-

six space-time dimensions for the bosonic string and ten space-time dimensions

for the superstrings and the heterotic string(that is strings without any addi-

tional degrees of freedom). This leads to one of the most important questions

in particle physics today, which is, why are only four of the space-time dimen-

sions presently observed? That is, what does a higher dimensional theory have

to do with what is observed at present particle accelerator energies, and what

happened to the extra dimensions? If we consider the case of the heterotic

string, then the fact that the other six spatial dimensions have curled up has

some extremely important consequences in explaining some aspects of the lower

energy phenomenology [11]. For instance, the topological properties of the six

dimensional space are related to the number of generations, which is the first

time a theory has attempted to predict such an important feature of the stan-

dard model. There still is not a fundamental understanding of the dynamical

reasons which cause the heterotic string to evolve into the four dimensional

universe that is observed today. This problem has led to considerable effort to

construct a string field theory [12] in which it is hoped that nonperturbative

phenomena could lead to a proper understanding of why only four of the di-

mensions of the heterotic string become macroscopic, and the other six spatial

dimensions remain on the order of the Planck size scale, 10-32 centimeters.

There are many other important questions to ask of this theory, such as how

do the masses, coupling constants, families of particles, and the Higgs field

arise from this higher dimensional theory? The other types of questions to ask

of this theory are do black holes exist quantum mechanically, or, put another

way, are singularities allowed? If so, how do they affect the evolution of the


early universe, the evaporation of microscopic black holes [13], and why is the

cosmological constant zero [14]?

One of the topics this dissertation addresses is the classical stability of a

two dimensional version of a string theory that has properties similar to those

of QCD. This theory is called the rigid string, and it is a theory which contains

quartic derivative terms in its action. This theory can be viewed as a type

of compactification from a ten-dimensional superstring in which the string's

worldsheet becomes curved in the process of compactifying.

If strings are so successful, what about other extended objects? For ex-

ample, membranes which instead of sweeping out a worldsheet, like strings

do, sweep out a world volume. That is to say, a membrane is just one higher

dimensional analogue of the string. At this time there is no definite answer

on the quantum consistency of membranes. There are still many outstanding

problems of membranes that have not yet been successfully answered, such as

the determination of the particle spectrum of the theory. Another unanswered

question, is whether bosonic membranes possess a critical dimension like string

theories. These are still outstanding problems because membranes appear to

be interacting theories. It has been shown that a membrane with space-time

supersymmetry [15] could propagate in eleven-dimensional supergravity. This

result suggests that the ground state of the supermembrane is made up of

the states of the eleven-dimensional supergravity supermultiplet. So in some

sense the critical dimension for the supermembrane is known because of the

constraint of supersymmetry but whether this theory is consistent quantum

mechanically is unknown.

Since membranes are an interacting theory, many approximate calculations

[16] have been performed to try to gain some understanding of these theories.


The work of de Wit, Luscher and Nicolai [17], based on a quantum mechanical

calculation of the mass spectrum of supermembranes, showed that the mass

spectrum of this theory is continuous and that there is no mass gap. The final

chapter of this dissertation examines the validity of their quantum mechanics

model. This is done by examining the renormalization group equations of a

two-dimensional model with a potential similar to that of the potential of the

supermembrane quantum mechanics. The main difference between strings and

membranes is that strings possess an extra symmetry that membranes do not

appear to have. This symmetry, known as Weyl symmetry, enables one to

locally gauge away all of the worldsheet metric dependence. The fact that

membranes do not have this symmetry will most likely rule them out as a

possible candidate for a theory of nature.

To properly understand strings requires one to understand two-dimensional

quantum gravity. This is because string perturbation theory is perturbative

with respect to fluctuations of the string worldsheet topology. The quantum

corrections come from the number of handles on the string worldsheet. A

model of two-dimensional gravity known as Liouville theory was constructed

some time ago in the seminal paper by Polyakov [18]. Liouville theory is one

of the first examples of a conformal field theory.

Recently a major breakthrough was made in describing nonperturbative

two dimensional quantum gravity [19]. These models describing non-perturbative

two dimensional gravity are known as multicritical matrix models. These mod-

els were used to sum the entire perturbation expansion of handles on the sphere.

It has been recently shown [20] that Liouville theory is intimately related to

these multicritical matrix models as well as topological field theories. It ap-

pears that there is still much to learn from Liouville theory.


This leads into the first and second chapters of this dissertation where

we use Bicklund transformations to solve an interacting theory in terms of a

free theory. Chapter one examines how this transformation works for simple

potential models and chapter two uses a Backlund transformation to relate

Liouville theory to a two dimensional free field theory.

This dissertation is organized as follows: Following the introduction, the

first chapter examines the use of Backlund transformations to relate two simple

particle models, which are both realizations of the algebra SU(1, 1). Next, the

extension of the radial representation of SU(1, 1) to that of the Virasoro algebra

is made using the Backlund transformation and the Virasoro extension of the

angular realization of SU(1, 1).

The second subject of this dissertation concerns finding a consistent quan-

tum Baicklund transformation for Liouville theory. We develop a perturbation

expansion using the coupling constant of Liouville theory as a small parame-

ter. We compute expectation values for the operator expgaq. We conclude

that the functional Backlund transformation solution of Liouville theory yields

expressions that are almost equivalent to the exact operator results of Braaten,

Curtright, Ghandhour, and Thorn [1], that is the correlation function calcu-

lations agree exactly for the term of order g6. However, a term appears that

is of order g4 that was not present in BCGT's computation. This term in our

functional of solution of Liouville theory has just the right form such that it

can be consistently absorbed by a mass renormalization.

In the third chapter, we construct a zero action solution and study the

stability of a closed rigid string immersed in a two dimensional de-Sitter space-

time. Next, we perturb the hoop solution of Curtright et al by using Riemann

normal coordinates in the ambient space. This is done to try to understand


how the sign of the local ambient curvature affects the classical stability of the

string. It is found that a space of negative curvature is closer to stabilizing the

rigid string.

The fourth and final chapter concerns the construction of static soliton

solutions to the cross canyon potential model of a two-dimensional field the-

ory. Next, a renormalization group calculation is performed to show that the

Hamiltonian of the crossed canyon potential is unbounded from below. Which

in turn suggests that the model undergoes a phase transition


Backlund transformations provide maps between various fields, for example

S-+ 4, and therefore relate functionals of one field to those of another, Q[4] =

q[(]. If Q is a conserved charge then the Backlund transformation is just a re-

expression of the symmetry of the 4 system in terms of those of the 4 system

and vice versa. This has been well established for classical field theories and a

more detailed explanation can be found in [21].

We will investigate some of the properties that quantum Bicklund trans-

formations possess. We do this in the context of simple potential models for a

single point particle. We consider how the Backlund transformation maps the

angular realization of SU(1, 1), and its extension to the Virasoro algebra, ap-

propriate for a particle moving freely on a circle onto the radial representation

of SU(1, 1), and determine its extension to the Virasoro algebra, appropriate

for a particle moving radially in an r2 + r-2 potential. The angular and radial

realizations of SU(1, 1), while quite different in their appearance, are directly

related via Backlund transformation, which interchanges the the two realiza-

tions. This is not very surprising since it is well known that all single particle

models with identical spectra, have canonical transformations relating one sys-

tem to to the other. That is, there are unitary transformations which relates

the the one basis to the other, where we have assumed that the energy spectra

of the theories in question are not degenerate.

For the system that we will study in this work, there is a simple canonical

transformation relating the 0-realization to that of the r-realization. This

canonical transformation is then just the Backlund transformation. Indeed,



there are many parallels with the Liouville field theory, which will be the

subject of chapter two. The major parallel is that one can view the Biicklund

transformation as mapping of the Hamiltonian of an interacting theory for a

particle moving in a r-dependent potential to that of a free Hamiltonian with

the particle moving on the circle. This can be made more precise because there

exists a Biicklund transformation which maps the interacting two dimensional

Liouville field theory onto a noninteracting two dimensional pseudoscalar field

theory. There will be a thorough explanation of this in the next chapter of this


Angular Realization of SU(1,1)

The angular realization of SU(1, 1) and its extension to the Virasoro alge-

bra for a free quantum particle moving on a circle is given by

In = exp(inO)(i&g n/2 + + irn). (1)

This realization contains two parameters, 3 and y. The In's realize the

centerless algebra

[In, Imr = (n m)ln+m, (2)

for any choice of the parameters # and -. If / and 7 are real parameters, the

In's satisfy the hermiticity conditions

S= -n, (3)

when acting on any arbitrary functions of 0. Note that 10 is the Hamiltonian

for this angular realization of SU(1, 1).

The quadratic Casimir invariant for the SU(1, 1) subalgebra generated by

141 and 10 is given by

C[O] = 1+14_1 lo(l0 + 1) = 2 + 1/4. (4)

Note that the Casimir operator is independent of 3. Also, if 7 is real, as re-

quired by the hermiticity properties in (3), then C[O] > 1/4. The eigenvalues

of 10 p are 0, 1, 2, 0oo which is the Principal series of SU(1, 1)s irre-

ducible representation. It will be shown that / is the groundstate energy of

the radial Hamiltonian.

Radial Realization of SU(1,1)

Now we consider the second-order realization of SU(1, 1) [22] in terms of

the radial variable r given by the following:

LO = -(r)2 a/r2 + r2/16, (5)

L = _()2 a/r2 r2/16 F 1/2(r + 1/2), (6)

where L0 is the Hamiltonian of the radial variable system. The eigenvalues

of L0 form the Discrete series of SU(1, 1)s irreducible representation. The

eigenvalues for L0 0 are 0, 1, 2, oo. Where / is groundstate energy of the

radial Hamiltonian. This realization of SU(1, 1) contains just one parameter

a. Now the Casimir operator for the radial realization takes the form

C[r] = L+1L_1 Lo(Lo + 1) = (3 + 4a)/16. (7)

Comparing C[r] to C[O] we note that C[r] < 1/4 when a < 1/4.

Blcklund Transformation

The generator of the transformation between the 0- and r- theories is given


r2 i(A + 1)
F(r, 0) = 30 i/2A ln(r) + cot(0/2) + 2 ln(sin(0/2)). (8)

This form of the generating function comes from the well known classical results

for the simple harmonic oscillator and from [23].

This generating function can be written as a similarity transformation in
the following manner,

S-1LOS = 10, (9)

where the S is the similarity transformation. The function S can be expressed
in terms of the r-and 0-eigenfunctions as

S= dqrN(Lo)l)Lo(77o10, (10)

where dq = drdO and N(H) is an energy dependent function which causes (10)

to be a similarity transformation rather than an unitary transformation. The

functions appearing in ieta)Lo and (11to are the eigenfunctions which satisfy

the following equations

(Lo En) r)Lo =0,
(Io En))10to =0,
where En are the energy eigenvalues of the operators LO and l0.

Now (9) can be expressed as

J dNN(H)LoI1)Lo0 (I1 = / dN(H)h7)Lo(l 1olo. (11)

It is found that the Hamiltonians of the two different systems are related

LO exp(iF(r, 8)) = exp(iF(r, 9))10. (12)

Then the result is

(a,.F)2 i(r)F a/r2 + r2/16 = -9eF + 3. (13)

In fact, (10) was used to determine F up to possible linear terms in 0. From
this we can solve for A in terms of a, and obtain (2 A)A = 4a, which yields

A= 1 + 1- 4. (14)

For the interacting r-theory, note that a > 1/4 corresponds to an attractive
radial potential which is strong enough that a particle would "fall into the
origin" [24]. This simple physical constraint therefore requires that a < 1/4
which was previously pointed out to be equivalent to C[r] < 1/4 Therefore, for
a < 1/4 there is a possible conflict between the physics of the r- and the 8-
theories. The hermiticity properties of (3) imposed on the results of (4) exclude
the allowed physical range for the Casimir operator of the r-theory. However,
there is a way around this dilemma, because the transformation between the
two different theories can be a similarity transformation instead of a unitary
transformation. This will be commented on in greater detail when we discuss
the extension of the radial theory to the Virasoro algebra.
Now using the result (11) in (10) yields

Lo exp iF(r, 8) = lo exp iF(r, 0) = 1/16 csc 2(0/2)[r -2i(A+l) sin 0] exp iF(r, 0).


However, (11) does not determine the parameter /. This can be solved for

by relating r- and 9- wavefunctions to each other. The Biicklund transforma-

tion can be thought of as a "nonlinear" transformation whose kernel is just

the transformation exp iF(r, 0). Then with this in mind, we can relate the two

different eigenfunctions to each other via

PE(r) = tE J dOexpiF(r, O)E(O), (16)
where r7E is an energy-dependent normalization, which is needed to avoid the

problems mentioned about the inconsistency of the Casimir operators. This

transformation can be inverted. Assuming that P and y are real; we obtain

OEE(9) = r77E drexp(-iF(r, ))%E(r). (17)
It is a straightforward exercise to obtain the radial eigenfunctions and the

eigenfunctions of the angular Hamiltonian. For the radial eigenfunctions we use

the method of series and the boundary conditions that I(r) -+ rx/2 as r -+ 0,

and that T(r) -+ exp(-r2/8) as r -+ oo. Then by demanding orthogonality of

the eigenfunctions we can determine the proper measure for this vector space

and normalize the eigenfunctions. Similarly it is simple to find the 0b(0) by

asking that they be periodic functions of 0. To solve for /P we use (14) and

that the groundstate eigenfunctions be

g0(r) = 2-(A+1)/2(r-1/2(A + 1)/2))rA/2 exp(-r2/8),



On(0) = 1/(2r) exp(inO), n Z+. (19)

Inserting these expressions and changing variables via z = exp(-i9) yields

S= (A + 1)/4 + N,N E Z+. (20)

For simplicity we choose N = 0. This comes about by asking that there
are no branch cuts about z = 0 in (14), when we perform the contour integral
on z, and that the contour integral does not enclose the branch cut at z = 1.
This calculation appears in Appendix A.
In general the ''n(r) can be simply computed from

n(r) = iE(2i)A+)/2rA/2/(27) f dz exp(r(1 + z) )z-(n+), (21)
8(1 z)

where again z = exp(-iO). Therefore, to obtain the excited states of the radial
eigenfunctions, one merely uses the residue theorem. We point out that if

the value of n in appearing (17) was allowed to be negative, then (18) would
be zero for the contour that we have chosen. So this is a natural restriction
of the spectrum of states appearing in the 0-realization of SU(1, 1) in order
that it maps onto the radial realization. Now we consider the effects that the

other SU(1, 1) generators have when acting on the Biicklund transformation.
Equating these generators acting on the Baicklund transformation, we obtain

L exp iF(r, 0) = l1 exp iF(r, 0), (22)

which is valid if and only if

y = i(A 1)/4 = +iv- 4a. (23)

Note, that the subscripts on the LHS of (19) become F on the RHS. This

is necessary to be consistent with the SU(1, 1) commutators and (10). Thus we

have expressed /, 7, and A in terms of a, and only one free parameter appears

in the expressions for the conformal charges and the generating function. Also,

note that

/ = 1/2 i. (24)

Collecting together the results for the various realizations for the Casimir

operator, we have

C[O] = C[r] = 72 + 1/4 = (13 + 1), (25)

or this can be expressed as

C[9] = C[r] = (4a + 3)/16 = (3 A)(1 + A)/16. (26)

Now returning to the problem previously mentioned about the incompati-

bility of the Casimir operators in the r- and O-realizations, the Bicklund trans-

formation exp(iF(r, 8)) completely sidesteps the issue by choosing a purely

imaginary value for -, which is completely inconsistent with the hermiticity

conditions stated in (3). However, the transformation preserves all of the

commutation relations and other operator identities. This is only possible if

the transformation is not unitary when acting on arbitrary states, but only a


similarity transformation. The r +-+ 0 mapping differs from a unitary trans-

formation because of the presence of L0-dependent factors. We have previ-

ously pointed out such ambiguities in exp iF. Although the final form of the

9-dependent Virasoro generators does not obey the general hermiticity condi-

tions given in (3), there is not a problem with unitarity in the model since

the space of 0-dependent functions has been restricted. In particular, only the

n > 0 are allowed for the single valued functions in (17), or else the condition

on the spectrum resulting from (18), and that was explicitly shown in (19) to


The situation that occurs here is very similar to that which occurs in two

dimensional conformal field theory when the "Coulomb gas" is mapped onto

the the conformal series with central charge less than one [25].

Extension of the Radial SU(1,1)

Next, consider the action of the full Virasoro algebra on the generating

functional. This is fairly easy to do for potential models, at least formally,

because the full algebra can be realized in terms of rational functions of the

SU(1, 1) generators [26]. Also, by constructing L2 using difference equations,

it can be seen by induction that the result [26] is (25) below. Thus for n > 0

Ln = (L + ) + ) (L1), (27)
L(Lo + P + n)

L-n = (Lo-n)r(Lo + 1 ) (- (28)

Due to the fact that the Casimir operator has the form C = (/? + 1), we

can interchange /3 (1 /) in these expressions. This will be very useful


in the following manipulations, along with the fact that the different r-and 0-
dependent generators of the SU(1, 1) realizations commute among themselves.
Acting with Ln as given in (25) on the transformational functional yields

r(Lo + 3)
Ln exp iF = (Lo + n) (L + ) (L1)n exp iF. (29)
r(LO + 0 + n)
Now we use (19) to rewrite this as

r(L/ + 1 )
Ln exp iF = (LO + n3) (Lo + (11)n exp iF. (30)
r(Lo + P + n)
Then we use the fact that the different generators of the realizations com-

mute and pull the l-n's through the various functions of L0 to obtain

S r(L0 + 3)
Ln exp iF = (1-1)n(Lo + pn) exp iF. (31)
r(LO + P + n)
Again we replace LO with l0 acting on the generating functional to yield

)r(lo + 10 + n)
Ln exp iF = (1-n)n(l0 + np) exp iF. (32)
r(lo + P + n)
Making use of the fact that (l_1)mf(lo) = f(10 m)1m1, (29) becomes

F(10 n + /)
Ln exp iF = (lo n + npo) (-_l)n exp iF. (33)
r(lo + P)
That is, again for n > 0,

Ln exp iF = l-n exp iF, (34)

n(o r(/00 + )) (- (35)
Ln = (10 n + n+) (1_1)n. (35)
r(lo + p)

Tracing through the same steps yields

L-n exp iF = In exp iF, (36)

In = (l + n ng) + -n (1). (37)
r(io + + 1 ()

Now it is a simple exercise to show that the expression for In and l-n reduce

to (2) upon substitution of the explicit forms for 1o and l into (33) and (34)


The results of (32) and (34) are a nice feature of the r <-+ 0 transformation.

That is, the Backlund transformation naturally provides an extension of the

second-order radial realization of SU(1, 1) to include the full Virasoro algebra.

This follows immediately from the usual extension of the angular realization.


This chapter considers the problem of using Backlund transformations to

construct a functional solution to Liouville theory. As we pointed out in the

previous chapter, Backlund transformations provide a re-expression of the sym-

metries of one dynamical system to those of a very different-appearing dynam-

ical system. This comes about because the interacting system is an integrable

theory. The Baicklund transformation relating Liouville field theory to that of
a free pseudoscalar field theory in two dimensions is given by

Oab (2m/g) exp(g$) sinh(gi) = Qr = II

4a90 + (2m/g) exp(gO) cosh(gO) = 9r = rT. (38)

The integrability conditions for this pair of equations give rise to the Liou-

ville and free field wave equations

(ar2 9a2) + (4m2)/g exp 2gO = 0,

(ar2 9a2), = 0. (39)

The above equations (36) are just canonical transformations. A generating

functional at a fixed r is given by

r p (40)
F[o, 0] = da[0o' 2m/g2exp gosinhgo] (40)


where 8' = a7, Canonical moment are determined from F[?Pq] in the stan-

dard manner through taking functional derivatives at a fixed r phone a

6F 6F
n = (41)

Then it is simple to evaluate these derivatives and obtain the original pair of

equations appearing (36).

For the classical field theory, this is essentially all that needs to be done.

It is now possible to solve for exp go, and then II and thus obtain expressions

for functionals of the Liouville field 0 in terms of those of the free field 4,

G[II, ] = g[7r, ], at any fixed r. For the quantum theory, this is just the

starting point.

One way to quantize the theory is to use operator methods to convert the

classical expressions for g[7r, '] into well defined free field operator expressions

in such a way that the locality and conformal transformation properties that

are expected to hold for the original G[II, 0] as equal time expressions involving

canonical interacting fields are maintained. This is a very complicated proce-

dure, but it was carried out [1]. The physical properties of the expressions

obtained using this procedure are not very transparent.

Another equivalent way to proceed is to use the Schrodinger equal-time

formalism [23]. This is in some sense a natural approach to the problem con-

sidering the functional derivative relations in (38). To carry out this approach

requires further knowledge about the quantum interpretation of the classi-

cal generating functional for the Biicklund transformation, and the quantum

corrections to it. The first step towards understanding this method is to expo-

nentiate the classical generating functional, up to factors of h.

Now, the exponential of F[0, 4] acts on tensor products of two different

functional spaces: one for 0, the other for 4. Then a natural way to view

exp iF[zC, 4] is as a transformation from one space to the other. The next
question to ask is, what are the invariants of this functional transformation?

Invariants of the Backlund Transformation
Now we compute the invariants of the functional Backlund transformation;
these will turn out to be the components of the stress-energy tensor. The stress-
energy tensors for the Liouville theory and the free theory are given respectively
by the following

62 2 2
000 = 1/2(--- + 2 24 + (4m2/g2)exp 2g),
01o =(-i6+ia6 ),(4

62 12
000= 1/2(- 2 + 2ia 6),
b 64 (43)
001=(--i tH +0").

Now we show that the components of the stress energy tensor yield the
same expression when acting on the Backlund functional. First we point out
that the Bicklund transformation can be integrated by parts to obtain

27r da( -' 2m/g2 exp gQ sinh g) = 2 dur(?b' + 2m/g2 exp g sinh gi).

We assume that the functional derivatives are taken at the same point,

o00 exp(iF) = 1/2[2im exp(gO) sinh(gV) + 0'2 + 4,2 2("
4m/gt' exp(g4) sinh(gO) + 4m2/g2 cosh2(2go)],
where we have used

exp(iF) = -2im exp(gq) sinh(go) (0' 2m/g exp(g)) sinh(g))2, (46)


6 exp(iF) = i(O' 2m/g exp(g)) sinh(go)) (47)

to arrive at (43), and the fact that cosh2 gi = 1 + sinh2 go. Similarly we

functionally act on exp iF with respect to 800 and obtain

800 exp(iF) = 1/2[2im exp(g) sinh(go) + ,'2 + /,2 + 4m2/g2 exp 2g cosh2 gi

-24" 4m' exp g 0 sinh go],
so the two expressions are equal, i.e.,

000 exp iF = 800 exp iF. (49)

Also, it is simple to see that 001 exp iF = 801 exp iF where we have in-

cluded the the anomaly term in the stress energy tensors. So the equivalence of

(43) and (45) means that the energy density is an invariant of the transforma-

tion functional and so are the other components of the stress-energy tensors.

As in the case of the simple potential models in the previous chapter, the
invariance of the energy densities allows the construction of energy eigenfunc-

tionals for the Liouville theory in terms of those for the free field theory, which

are just essentially Schr6dinger wave functional transforms. Then this is ex-

pressed as

E[] = JD exp(iF[4, )D'WE[-]. (50)


Using equation (45) acting on (46) and functionally integrating by parts,
yields Eo00oE[]- = Eq[O] if and only if o00o[4'] = ET [i0],.where the integration
is a functional integral over all field configurations at a fixed time, and is not a
path integral. Again, equation (47) can be thought of as a "nonlinear" Fourier

Now we need to solve for the free wavefunctional for the free field theory.

First, we must solve equations (36) at a fixed time and with periodic boundary

conditions. This yields the mode expansions for 0(a) and O(a), which are
respectively given by

O(a) = q + i/lV n-l(an exp(-ine) + bn exp(ina)),

(a) = Q + i/V' E n-1(An exp(-ino) + Bn exp(ino)). (51)
where, for convenience, the nonzero mode part of the Liouville field will be de-
noted as 4 = i/V EnO n-l(an exp(-ina) + bn exp(ina)) and the nonzero

mode part of the free field as i = i/ V EnO n-1(An exp(-ina)+Bn exp(ina)).

Where the expressions an,bn,An and Bn are Fourier coefficients and not to be
confused with creation and annihilation operators.

Now, solving for '[], using 900 with the surface term discarded

62 2
1/2(- 2+] = 0. (52)

This has a solution of the form

%(E = 0) = exp -1/2 0 (0') a(). (53)
)o V aa

To verify that this is a solution merely take two functional derivatives of
(50) where we assume the functional derivatives are at different points, so that
the the infinite zero point energy can be explicitly seen,

Sp) (a) I(E = 0) = Io, l(a)laplC(p)W(E = 0) |1al(a p)W(E = 0),

where aO,.6(o-a) = 1/(2r) EnO Inl exp in(a-p). When a = p this expression
diverges quadratically. This is just the usual zero-point energy associated with
each non-zero mode, and this must be removed to have a well defined 000.
Then to show that (51) is a solution requires that

12r da = j0 da'. (55)

This can be done by first showing that

da (-9 = 2 = d2a 1 al,, (56)

and then integrating the first expression in (91) by parts using periodic bound-
ary conditions. Now the expression obtained from functionally differentiating
T twice becomes in the limit p --+ a

1/2 da-o, 2 2(E = 0) = dar'i'. (57)
0 6 )
Now, inserting (54) into (50), since this expression is zero, (51) is a solution
to (50).
Now the wavefunctional at nonzero energy but zero momentum can be
achieved by multiplying (51) by exp(ipQ). Then this wavefunctional has energy

27r daoo'p = p2/4rp, (58)

where we have used the fact that 1/27r(6 + -). Then the energy of the
functional field Ep = p2/47r = g2k2/47r where p = gk and g is the coupling
constant of the Liouville action.

Now we express the functional Bficklund transformation in a more con-
venient form, so that the evaluation of (47) in a perturbation expansion in
the coupling constant g is well defined when the coupling constant is a small
parameter. With this in mind, it is very useful to make the following identifi-

exp go sinh go =2expgq sinhgQ + exp g(q + Q)(exp g(i + 4) 1)
exp g(q Q)(exp g(O 0) 1).
Using (94) in (86) with the wavefunctional at nonzero energy yields

I(D) = DQ exp i(gkQ 4rm/g2 exp gq sinh gQ) x
S"0 .2r (60)
I D exp / do[-i(q 1/218lj4] + Wi]
where we have factorized the measure with respect to the zero mode and the
nonzero modes. Now (57) can be expanded in terms of the function W1 where
W1 is given by

Wi = -mi/g2[expg(q + Q)F+(pi) expg(q Q)F.(pl)], (61)

where the functions F+ and F_ are defined by

F+(pi) = dpl(n7expg((PI) + (iPl))- 1)
f2i (62)
F+(PI) = 1 dpl( 7expg((pj) -(pl)) 1).
Note that we have inserted in the above expressions a constant r7. This
constant will be chosen such that 7 = exp -g2(1/(27r). The importance of this
will be expanded upon shortly.
Expanding the exponential in (58) in terms of W1 yields

,(0) = dQ exp i(gkQ 47rm/g2 exp gq sinh gQ x
/ 2l (63)
Sdi 1/n!Wl exp -1/2 / d n=0 0
The following identity [27] is useful in relating the zero mode part of (60)
to the modified Bessel functions Kv(x):

dt exp[ix sinhit(k in)] = 2 expr/2(im + k)K,(x),n E Z, v = n + ik,
where x = 47rm/g2 exp gq for our purposes. Note that for the lowest order of
perturbation theory n = 0, in (61), yields

(0)(k) = 2/gexp(7rk/2)Kik(x), (65)

which is the zero mode solution found by D'Hoker and Jackiw [28]. With the
above established, it is now a simple matter to expand I in terms of W1, use

(62), complete the square of the various nonzero mode expressions involving
, and then compute the Gaussian integrals over the ) nonzero modes. As an
example of this last statement, consider the following expression:

d\101 2i'b + 2/3g$S(a p), (66)

where S(p) = ]n# exp inp, that is, the zero mode of the delta function is
removed. Changing variables in (63) to

lb = i + g(p), (67)

then when (64) is inserted back into (63) (and note that we have dropped the
index on the derivative but it is with respect to a) yields

-1 11i + $ I4 #292 2(a p)l-1'S(ao p) 2ifg 6(a p) =
I"! (68)
5la,1 2i,'V + 203g~(a p).

We point out that the expression #2Sa-(_p)ll-l_-p) = P2/rT Z 1 1/n =

/321/7r is just the usual "normal-ordering" divergence that appears in opera-
tor expressions. The normal ordering divergence coefficient /, will be equal to
one and hence this is why we have chosen r7 appropriately to cancel out this
divergence. This choice of q will cancel all of the terms g2(1/(27r) that arise
from the integration of the k nonzero modes to every order in perturbation
theory. We will see later on in the calculation that there will be no "normal
ordering terms coming from the integration over the 0 nonzero modes.
The perturbation expansion for D is given by the following expression:

S(k) = (0)(k) + 4(1)(k) + (2)(k) + 3(k)..., (69)

and the various functions that appear in these terms are defined by

An = 2g-1-2nmnexp(kcr/2),


G = D exp -1/2 I daal I: (70)

where the various expressions in (66) are defined by

i(0)(k)= 2g-1 exp(k7r/2)Kik(x)G, (71)

41(k) = -2AG expgqg[Kl+ik(x)F+(Pl)
K-l+ik(x)F-(p)], 2)

.(2)(k)) = A2/2G exp 2gq[K2+ikF+(pl)F+(p2) + l-2+ikF-(pI )F-(P2)
+ Kik(F+(P1)F-(p2) + F+(P2)F-(p))].

.(3)(k) = -A3/6G exp 3gq[K3k+ikF+(pl)F+(P2)F+(p3)+

F+(pl)F+()F-(P2) +P2F+(P3)F(pl))+
K_ 1+ik( F+(pl )F- 2)F-(P3)+

F+(p2)F-(Pl)F-() 3) + F+(P3)F-(pI)F-(p2)+
K-3+ikF-(p )F-(p2)F-(p3))].
The complex conjugate of 4*(k')) is easily obtained from (66) by changing
k -* -k' and the term -it'k changes sign which will cause the projection
operators p which are defined in the appendix on Liouville theory to change
to p:.
The explicit form of (69),(70). and (71) appear in the app,,,ndli; on Liouville
theory after integration over the v rlonzrr modes hs brn p(xrforned. Note
that we have kept terms to cubic ordr in exp gq and that the subscript n

denotes the different shifted arguments of the Gaussian integrals, which is an
overall infinite multiplicative factor which will be divided out of the correlation
Now we make use of the above expressions of the wavefunctional and its
complex conjugate to compute the expectation value of exp ago(O) to order
g6 in the coupling constant. We do this to make contact with the Liouville
quantum mechanics calculation of Braaten ,Curtright, Ghandour, and Thorn
[1]. Again, we complete the square of the massive modes, where the massive
modes are the 4 fields, and then integrate over all possible field configurations
at a given time, which yields

(,P(k')lexp agO(0)|l(k)) = ({(0)(k') exp ag (O) (0)(k))+

(<(1)(kc'lexp ag O(0)lD(1)(k))+

((1)(k')|exp agO(0)|$(O)(k))+

((0) (k'lexp agO(O) l(X)(k))+

(p(2)(k')[exp ag(0)| (O)( k))+
(0(0)(k')jexp ag(O) I(2)(k))+ (75)

,D(3) (k') exp agc(0) 1(O)(k))+
((O)(k' exp ,agq(0)1(3)(k))+

(4(1)(k'lexp ag(O)) (2)(k))+

(D(2)(k')lexp agO(0)JO(1)(k))+
terms of order exp g(a + 4)q.
Where the brackets contain the integration over both the zero mode and the
nonzero modes of the field 0. This functional form is chosen for the expectation
value for two reasons, the first reason being is that the vertex operators of string
theory are of this form and the second reason being, is that monomials in the


field 0(0) can be calculated by differentiating the correlation function with

respect to a, n times and then setting a = 0 to obtain the desired power of

the expectation value of 0.

We point out that the terms: (('1)(k'lexp ago(0)l}(1)), ((1l)(k')lexp agc(0) 4(0)(k)),

(D(O)(ck'exp agO(0)I(1l)(k)), (,(1)(k'Jexp ago(0) 1(2)(k)), and (D(2)(k')[exp ago(0) [(l))

appearing in (72) are zero. It is shown in the appendix by considering generic

nonzero mode matrix elements of each of the above matrix elements are zero.

Then it is explained that the same mechanism that caused the particular ma-

trix nonzero mode matrix to vanish, also cause the rest of the nonzero mode

matrix elements to be zero. These terms vanish because there is always an an-

gular variable pn, which is not suitably correlated with the rest of the angular

variables. When the exponentials are expanded and integrated term by term,

these uncorrelated pn integrate to zero. Then the parts of the angular integrals

that are nonzero when integrated always occur in pairs with a relative minus

sign and cancel against each other. This lack of correlation occurs between the

angular variables occurs because the in the wavefunctionals are integrated

independently of each other. Then it appears that the only matrix elements

that will be nonzero are of the form ((O0)(k')lexp agQ(0)1.(n)(k)) and its com-

plex conjugate. We have explicitly shown this to order exp g(a + 3)q in our


Listing the nonzero matrix elements appearing in (72):

(O(0)(k') exp agtb(0) lO)(k)) =Mo f dq exp gaqKik'(z)Kik(x)

((0O)(k')lexp agq(O)')(2)(k)) =Mo/2 dq exp g(a + 2)qK-ik,(x)[K2+ik(x)M+2(1, 2)
+ iK-2+ik(x)M-2(1, 2) + Kik(x)(M+-(1, 2)
+ M-+(1, 2))]
(1(2)(k')exp ag(0)I(0)(k)) =Mo/2 /dqexpg(a + 2)qKik(x)[K2_ik M_2(1, 2)

+ Ki-2-ik,(x)M+2(1, 2) + K-ik,(x)(M-+(1, 2)
+ M+-(1, 2))]
(,(O)(k'lexp ag(O)l)(3)(k)) = Mo/6 dqexp g(a + 3)qK-ik'(x)[

K3+ik(x)M3(1, 2, 3)
+ Kl+ik(x)M++-(1,2,3) + -1+ik(x)M-+ (1,2,3)
+ I-3+ikM-3(1,2,3)]
((O)(fk'lexpago(0) )(k)) = Mo/6 dqexpg(a + 3)qK-ik(x)[

+ Klik_(x)M--+(1,2,3) + K--ik,(x)M+--(1,2,3)

+ C-3-ik'M3(1, 2, 3)],
where the limits of the q integral are from -oo to oo and the Mi that appear
in (73) are defined by

MO =4g-2 exp(k + k')r/2( Dp exp -1/2 dojalf)2x

(J D exp -1/2 do~I^9k)

Mn =Momng-1-2n
2 2
Mi2(1,2) =(2r)2[(9 )2(2 + 2(g)3C3] (77)
2 2
M+:F(1, 2) =(2)2 [-( 2 + a2 )C31

M3 =2g6(3
M++-(1,2,3) =M--+(1,2,3) = -6g6g3.
The form and the evaluation of these ) nonzero modes can be found in
detail in the appendix on Liouville theory.
We can simplify the higher order modified Bessel functions appearing in
(73) by making use of the identity [27]

-1 Tv+ = 2vx-lT, (78)

where Tv = exp ivirKv(x). This enables us to express all of the Kn+ik in terms
of Kik(x),Kl+ik(x) and K_-+ik(x). The same procedure can be carried out
for Kn-ik'. We give one example for IK3+ik(x),

1K3+ik(x) =Kl+ik(x) + 2(2 + ik)m-1 exp -gqKik(x)
2 (79)
+ 4(2 + ik)(1 + ik)(-m-1)2 exp -2gqK1+ik(x).
Now we integrate over the zero mode q and use the identity [29]

j dzz- K(az)K,(bz) =2-(2+A)A-1(r(c)-1r((c + P + V)r (c + V V)

(C- + + V) ((c-I-v)
x r( )r( ),
2 2

where c = 1 A.
To make use of (77) we change variables, exp gq = z With this change
of variable, the limits of (73) change to to that of (77) and for our purposes
a = 47rm/g2
Now we substitute (73) into (72) and make use of the reduction formula of
(75) and as an example we list the following matrix element,

(4,()(k')JexpagOB(O)) (2)(k)) =Mo( g)3Zo[C3/(2a(a + 1)

(2 a2)(a4 + 2a2(k12 + k2) + (k2 k2)2]

+ ((2 + 4C3)(a + 1/a(k'2 k2) 2k2)],
where the function

Zo =1/(2g3r(a))(2-- )(1/2(a + i(k + k'))r(1/2(a i(k + k'))x
r(1/2(a + i(k k'))r(1/2(a i(k k')), (82)

Z0 = (4(O)(k')lexp crg(0))D(O)(k)).
We have listed (79) since every matrix element has this factor in common
and it is just the matrix element of the ground state.
Now adding up all of the matrix elements we obtain the final result of the
correlation function which is

(M(k')lexp agf(O))[(k)) =MOZo[l + 2(9-)2(2(a (k2 + k'2)
+ ( )(3(-a4 + 4/3a3 + 8/3a 2a(2 a)(k2 + k'2)
(k'2 k2)2 + terms of order g8].
The terms of order g6 correspond exactly with the computation of Braaten,
Curtright, Ghandour, and Thorn who computed this correlation function using


Liouville quantum mechanics perturbed by the nonzero modes. The terms of

order g4 can be handled in the following manner. The term k'2 + k2 can be

absorbed into the normalization of the wavefunctional and the a dependent

term gives a finite renormalization to the mass term that appears in prefactor

of (49). This term then changes to

2 2
2Irm 2r(m + Am)

where Am = -g4/48.

Note that any other power of a or terms with k or k' dependence multiply-

ing the term of order g4 could not have renormalized the mass term or been

swept up in the wavefunctional normalization.

The next step in the computation is to verify that |D(k)) obeys the Virasoro


L=o(k)) =0, (84)


f2 r
L = da exp ino[0o o 01] (85)

000 and O01 are defined in (42). Verifying that (84) is satisfied has not

been shown yet but the calculation is in progress. This calculation is crucial

to understand how this mass renormalization of order g4 is consistent with

conformal invariance.


Polyakov [30] and others [31] have pointed out the possibility of adding

an extrinsic curvature term to the action of a relativistic string. These actions

with an extrinsic curvature term can be viewed as the string's worldsheet being

embedded into a higher dimensional spacetime. These rigid strings appear to

arise quite naturally from Green-Schwarz actions in which the fermions have

been functionally integrated out of the action [32]; so this model is in some

sense a compactification which preserves many features of the original string

theory. This structure is what led Polyakov to consider the rigid string as a

model for QCD. This paper investigates the effects that an extrinsic curvature

term coming from a curved ambient space have on the classical stability of a

rigid string. The rigid string action has the novel feature that classical string

configurations can be investigated whose Euler number is different from zero,

depending on the ambient space's curvature.

Braaten and Zachos [33] have shown that the static solution of the rigid

string, that is, the hoop solution found by Curtright et al [2], is unstable

against small radial perturbations. Perturbations about the other coordinates

were found to be stable. This hoop solution, which assumed a flat ambient

space was shown not to have a stable classical ground state; that is, the vac-

uum energy could be made arbitrarily negative. This implied that the radius

of the hoop could be made arbitrarily large. Hence Braaten and Zachos con-

cluded that the rigid string should be regarded as only an effective theory and

that other terms would have to be added to the action in order to guarantee the

existence of a classical ground state. The present work looks for nonflat spaces



that can stabilize the rigid string against small fluctuations, in the same man-

ner that anti-de-Sitter Space was used to stabilize an extended supergravity

theory with gauged SO(N) internal symmetries [34]. That is this supergravity

theory had a Hamiltonian that was unbounded from below, until an anti-de-

Sitter background was added to the theory. The addition of the anti-de-Sitter

background stabilized the supergravity theory with gauged SO(N) internal


Rigid String in a Curved Ambient Space

The derivation of the rigidity term is briefly reviewed [35] in the presence of

a curved ambient metric. A complete derivation of the extrinsic curvature term

and its generalization appears in the appendix on the geometry of immersions.

The induced metric on the string worldsheet is assumed to be the usual one of

string theory, except that an ambient metric is used to contract the spacetime


gab = X'XG (86)

where Gy, is the ambient space metric. We will briefly comment on the validity

of this equation at the end of this section. When one considers a single normal

to the string worldsheet, one obtains the following relations

G ,XP?,V = 0 and GpvIr y = 1. (87)

Note that only the simplest case of just one normal is being considered

here, though it is simple enough to generalize to more than one normal.

Capv is the Christoffel connection of the ambient space, Da is the covari-

ant derivative on the string worldsheet, and b is the worldsheet Christoffel


connection. Two worldsheet covariant derivatives acting on an ambient space

vector field yield

DaDbX = aXP rc X, (88)


rab = g(cG X X + CaXax). (89)

Now the second fundamental form can be expressed in terms of (85) and

the ambient space metric as

Kab = GuvDaDbXXrl' + CavXaX r"l. (90)

The second fundamental form can be covariantly differentiated and anti-

symmetrized on the last two indices; then, using Ricci's identity, the projection

of the Riemann tensor onto the string's worldsheet, and the condition for the

integrability of the second fundamental form one obtains

Rabcd = Kab Kd IacKbd + R 6 X ,aX XdX (91)
A,8-yb a c7 b di

DcKab DbKac = RA XaXXX (92)

where the superscript A denotes ambient space Riemann tensor in order to

avoid confusing it with the worldsheet Riemann tensor.

The world sheet Ricci curvature can be related to the Gauss-Bonnet theo-

rem to obtain the following expression:

2 2n h= d2z/[(gabKab)2 gab cdacK
J (93)
-+R A oQ' x37 ab v-X ,6 cd,
+Ra- X,.a bY .,,c.,, J'
where n is the number of handles, h is the number of boundaries, and since

only closed strings are being considered, h will be set to zero. Since (93) is a

topological invariant, one of the terms can be removed from (93). Then any

two terms from (93) can be added to the minimal surface area term of the

usual bosonic string, yielding a highly nontrivial interacting theory. In this

work the term gabg cdKacdKbd is removed. Then the action for the rigid string

in a curved ambient space becomes

I = T dz g [1 R((gabKab)2- A X X -tXX6 d9ab ] cd, (94)

where R2 = S/T, T is the string tension, and S is the rigidity parameter.

Note that this action is written as a minimal surface action plus the rigidity

terms. This is because the induced metric which is the variation to the above

equation, treating the worldsheet metric as an independent variable, yields

equations that are no longer just algebraic equations but are nonlinear partial

differential equations of the worldsheet metric, the ambient metric and the

tangent vectors. This is true also for the case when the ambient space is flat.

The equation for the induced metric is computed in the appendix A. So we are

assuming without proof that the induced metric is gab = GpXX and it is

quite possible that there could be additional terms with higher order derivatives

appearing in the induced metric. This is still an unsolved problem.

The equations of motion of the rigid string in a curved ambient space

are given succinctly by the following expression (the full expression for them

appears in the appendix of the geometry of immersions):

6I/6X 8m6I/6X,m + Dm8nl/6X'n = 0. (95)

de-Sitter Static Solution and its Stability
The classical motions of a closed string are considered in a two dimen-
sional de-Sitter space. de Sitter space is the maximally symmetric solution
to Einstein's equation with scalar curvature R = 2a2. The three dimensional
hyperboloid is coordinatized by r7ijYiyj = a2 where 77ij = (-, +, +) embedded
into R3. Now upon making the following transformations, the ambient met-
ric can be diagonalized YO = asinh(XO/a) Y1 = acosh(X0/a)cos X1 and
Y2 = acosh(X0/a)sinX1. The range of X0 and X1 is respectively given by
is -oo < X0 < oo and 0 < X1 < 27r. Then the d.S. metric is

d2s = -(dX0)2 + a2 cosh2(XO/a)(dX')2. (96)

Next the lab gauge is chosen where r = X0 and a = X1. Then the induced
metric on the worldsheet is given by

1 0 (97)
ab=( 0 a2 cosh2(/a) (97)

The second fundamental form for this induced metric can be written as

Kab = (-9CGa pC tXXX + C.,pXX b). (98)

This can be simplified to

'tab = 7T(-CUtvr + Cpvr)X{aXv + 7OrCO V)X:
Ka b = 777 3 b I ) b~y


Then from (13) and the symmetry on the first two indices of the ambient
space connection, it is simple to see that Kab = 0.

Thus the static action for the rigid string reduces to

S = -aT d2z cosh(r/a) 1 2Ra-2] (100)

This de-Sitter geometry can only be a solution of the rigid string if the

coupling constants in the theory are chosen such that 1 = 2R2a-2, that is, if

there is zero action. Note that in the lab gauge the second fundamental form

had to vanish and the ambient space had to have constant positive curvature;
otherwise the ambient curvature term could not have cancelled against the

minimal surface term in (90).

Now fluctuations about this de-Sitter background are considered. This is

done by letting the radial variable a -- a+ y, where y(r, o) and then expanding

the action about its lowest energy state of T = 0. This is because all of the

terms in the action grow as exp(r/a). Also transverse coordinates zP are added,

and these directions are assumed to be flat in the ambient space. Then the

perturbed worldsheet metric becomes

(/-1 + y2 + izy' yy' + iz' z
gab Y' + Pz (a + y)2 cosh2(r/(a y)) (101)

where i = ,rz and z' = Oaz. Now the fluctuations to the ambient curvature,

the second fundamental form, and the square root of the determinant, are

expanded out and the quadratic parts of the fluctuations are kept. Then the

resulting action is

I = -Sa d2z[R2 2a-2 + y/a(R2 2a2) 2y2/a4

+1/2(R' + 2/a2)(y'y'/a2 +2 + z z,/a2 (102)
-(y"/a2 )2 (z-"/a2 )( ,))(/a2
Now the above action is Fourier transformed by

x(r, c) = 1/27 f dw exp i(wr + no)Xn(o),
n (103)
Xn(w) = 1/2r J dr exp(-iWr) f du exp(-ino)x,
so that the quadratic part of the action becomes

I, = -Sa f dw (lynI2w + n2/a2)2
+1/2(R,2 + 2/a2)(w2 n2/2) 2/a4] (104)
+lz12[(2 2/,2)2 + 1/2(R 2 + 2/a2)(w2 n2/2)].
Now the conditions for stability can be determined by requiring that the
eigenfrequencies be real. Solving for the eigenfrequency of the radial and trans-
verse fluctuations, the following expressions are obtained:

W2 = n2/a2 1/4(R0-2 + 2/a2) /-2/a4 + 1/4(R2 + 2/a2)2, (105)

2zl = n n2/, 2 2/a2 1/2(Ro2 + 2a-2). (106)

The radial fluctuation (n = 0) eigenfrequencies are complex for all values
of the parameters in the theory. The transverse (n = 0) eigenfrequencies have
one zero mode and one complex mode that is also unstable. These instabilities
just reflect the fact that the extrinsic curvature terms come into the action


with the wrong sign. The de-Sitter space acts as a source of negative energy

density for the action.

Ambient Space Fluctuations to the Hoop Solution

In this section, we consider perturbations of the hoop solution of Curtright

et al [2] about its Minkowski background by using Riemann normal coordinates.

The ambient space metric in Riemann normal coordinates is given by

G, = rlv, 1/3Rf X X 1/6DfR XXXXK +.... (107)

This hoop solution also uses the lab gauge for the time, X0 = 7, and

X1 = a. Then the unit radius vector of the hoop is given by i = (cos aC +

sin ca) The worldsheet metric is given by the following expression when it

is perturbed by the ambient curvature fluctuations and a radial perturbation,

where a -* a + y in just the same manner as it was done in the de-Sitter case

except that a = 1:

(Goo + Grry2 Grry'
ab = Grr y' Grr((1+ y) + ') (108)

The only nonflat fluctuations are assumed for G00, where G00 is given by

Goo = -1 1/3R0101(1 y)2 + .... (109)

The radial part of the ambient metric is assumed constant, Grr = 1. Only

radial fluctuations are considered for the hoop solution, since the perturbations

around the other coordinates were found to be stable by Braaten et al [33].

The quadratic part of this action is

Iq = -SA d2z[-(y Ay")2 + M + Ny2 + Ly'y], (110)

where the coefficients A, M, N, and L are respectively given by

A = 1 1/3R, M = 1/2(1 RO2) R(5/2 RO2)+ 1/2R2, (111)

N = 1 + R(4 5/3RO2) + R2(1/9RO2 + 37/6) + order (R3), (112)

L = (1/2RO-2 3/2A2 + 6A(1 R) + 1/2R2). (113)

The quadratic part of the action is Fourier transformed, and the expression
for w2 is given by

=1/4(1 R2 R(5 1/3RO-2) + 2 )

1/2 /[1 RU2 R(5 1/3RO2) + R2]2 + 4(1 + R(4 5/3Ro2) + order (R2)
For w2 to be real requires that T/S = R 2 < 1 and that the scalar curva-
ture R should be negative. The dominant terms in the square root are

65/4 1/2RO2 + R(27/2 12/5R22) + o(R2). (115)

To make wo real requires that the terms inside the square root must be
zero. This cancellation forces R < 0. Since this is a weak background field, R
can not be large without violating the assumption of a weak gravitational field.
Therefore R can not be made large enough to cause the expression under the
radical to vanish. Hence the rigid string perturbed about a Minkowski ambient
space is still unstable.


It has been argued recently that the mass spectrum of the quantized super-

membrane is continuous from zero, and that smooth membranes are not stable

against collapsing into fibrous configurations [17]. Thus, quantum supermem-

branes are to be regarded as flawed extensions of quantum superstrings.

There are two logically independent features for the argument for the col-

lapse of the supermembrane into string-like configurations. The first feature

requires the implicit assumption that curvature terms do not appear in the

effective action of the supermembranes. The second feature requires the as-

sumption that the mass spectrum of a quantum field theory may be obtained

by first truncating the theory to a finite number of degrees of freedom, calcu-

lating the spectrum from the truncated theory, and then letting the number of

degrees of freedom become infinite.

Now, consider the first assumption, that the effective action does not con-

tain curvature terms. This assumption makes plausible the heuristic argument

[17] that there is an energy barrier to prevent the collapse of any smooth mem-

brane configuration into a network of filaments with infinite extrinsic curvature,

but zero area. This heuristic argument is probably correct with the given initial

assumption. However, without proof that there are no such curvature terms

appearing in the effective action, the argument is logically incomplete and does

not establish that such a collapse can occur. Rather, it only focuses attention

on the possibility that curvature terms could exist in the membrane effective



Such curvature terms have been considered [36]. In marked contrast to

the assumption in de Wit et al [17], however, Curtright [36] has argued that

curvature terms could indeed exist and maintain the local supersymmetry on

the world-volume swept out by the evolving membrane. The possible curvature

terms were explicitly written down and shown to be locally supersymmetric

to the lowest nontrivial order. Nevertheless, since no one has performed the

requisite calculation to see whether such curvature terms actually arise in the

membrane effective action, this issue is still open. Therefore, this first feature

of the argument against quantum supermembranes is obviously not conclusive.

The second feature of the argument against supermembranes is more quan-

titative. The supermembrane is first truncated to a finite number of degrees of

freedom, essentially a SU(N) invariant quantum mechanics model [37]. Then

it follows from an analysis of the resulting supersymmetric SU(N) model, that

the spectrum is continuous; that is, the spectrum is essentially the same as

it is for the classical limit of the theory. Since it holds for any finite N, it

is claimed that the spectrum remains continuous in the limit of an infinite

number of degrees of freedom, i.e., N --+ oo. However, it is well known that

phase transitions [38] may occur to obviate such naive conclusions about the

properties of SU(oo) quantum mechanical models.

In the remainder of this chapter, we will not discuss any further the issue

of curvature terms appearing in the effective action of membranes. Rather, we

will explore the issue of finite versus infinite degrees of freedom in the quantum

field theory arguments. We will examine some features of the finite versus

infinite situation for a simple field theory model whose dynamical properties

are very similar to those of the membrane theory. In the context of this model,

we will argue that the quantum properties in the case of an infinite number


of degrees of freedom are indeed quite different from those of the truncated

version with a finite number of degrees of freedom. We present evidence that

phase transitions in such models are to be expected and lead to a quantum

system whose properties are very remote from those of the classical limit.

In this work we do not determine the properties of the quantum phase of

supermembranes. This is still beyond reach. Nevertheless, we believe that our

arguments indicate the possible directions for further study and suggest that

supermembranes are still interesting and viable models.

This chapter is organized in the following manner: In section one, we

write down the quantum field theory model similar to the quantum mechanical

supermembrane model and construct classical soliton solutions. In section two,

we present two arguments to show that the field theory is unbounded from

below when N -+ oo.

Classical Solutions of the Canyon Potential Model

Classically, the model we consider is given by the Lagrangian density

L = 1/2[(X)2 + (aY)2] g2X2Y2 (116)

and whose Hamiltonian density is

H = 1/2[(9X)2 + (OY)2] + g2X2y2. (117)

At this point we make contact with the membrane and the supermembrane

Hamiltonian densities when written in the light cone gauge. Their respective

Hamiltonians are given by

2m = = =[ Jd2a(Pa + V(X), (118)


where V(X) = 1/2(eabaaXVPbXp)2 and

27-sm = Mm = da(pAaA + 1/4fABEfCEDXa Xb X (119)
b ABa (119)
i/2fABCXax A 7a .

In (119) the indices A, B, are the SU(N) labels and the fABC are the

SU(N) structure constants. Note the similar structure of the potential for
all three models listed above and that M2 and M2m are the mass squared

operators with the center of mass momentum excluded for the membrane and

supermembrane respectively.
Now we return to the canyon potential field theory model and examine the
classical motions of this system. The equations of motion are given by

OX = 2g2XY2,
DY = 2g2yX2,
where we have chosen the metric to be goo = -1 = -gaa- We point out

the following relation from the equations of motion: if one multiplies through
the X equation of motion by X and the Y equation of motion by Y, then one
obtains the identity that XOX = YOY. The physical meaning of this will be

explained shortly.

The above Lagrangian density can be recast into polar coordinates and this


= 1/2[(OR)2 + R2(9)2] g2R4/8[1 cos 40]. (121)

Note that for fixed R the similarity of the above equation to the Sine-

Gordon equation [39], where it is well known that soliton solutions exist.

It is possible to construct a static soliton solution from (120) assuming

that R is fixed. A solution to the 0 equation of motion can be obtained, but


it does not satisfy the R equation of motion. Both of the equations have the

same functional form, but their arguments differ by a factor of V/2. Which

is unfortunate because the equations described by the Sine-Gordon solution

exhibited the behaviour we desired. The desired behaviour, would be when

one of the fields, say X(a = oo) = 1 and Y(a = oo) = 0 and then this system

evolves to X(a = -oo) = 0 and Y(a = -oo) = 1. Physically, all this means

is that the solitons can switch from one trough of a canyon to another trough

of a canyon perpendicular to the one it started out in. This type of behaviour

would show classically the soliton would not be restricted to moving along the

same trough, which the analogy with the supermembrane would mean that,

the supermembrane would not be constrained into evolving into long stringlike

configurations. We were not able to construct a classical solution with this


We were able to construct one exact solution to (116) and it is a zero action

solution. This solution found by setting = 0 is given by

X = cosh(exp(-aa)) + b),
Y = exp(--aa).
where a = gV/2 and b is arbitrary constant to be determined. We must choose

the parameter b to be real for the solution to be stable. If the range of a is

chosen to be 0 < a < oo, then the most physical choice of the parameter b, is

to choose b = -1. Then the behaviour of the solution is as follows, X(0) = 1

and Y(0) = 1, then the solitons evolve to the configuration, X(oo) = 1.543 and

Y(oo) = 0. The main thing to notice is that the X field does not go to zero,

so that the soliton does not switch into a trough perpendicular to the trough

it started out in.


The second solution that we discuss is when X = Y to obtain a solution

for (114) we must add a mass term, then the solution is:

X = = (au) sn(au), (123)
(4V o (au)
where 0l(au) and 0o(au) are the Jacobi theta functions [40]. The argument of

(123) is u = 2Ka/pi, where K is the quarter period of the expression

,12 9
K = d (124)
J0 V/1 k2 sin2(8)
and k is the modulus.

The mass term that must be added for the above function to be a solution

is m2 = -(k2+ 1)a2 and the coupling constant g2 = k2a2. The function sn(au)

behaves exactly like the usual sine function, that is it has a period of 4K and

oscillates between 1. This solution corresponds to the particles being trapped
between the maxima of the canyon potential.

Quantum Instability of the Canyon Hamiltonian

To define the quantum version of the model, it must be renormalized.

Since this is a two-dimensional spacetime, the model is super-renormalizable.

Alternatively, for the quantum theory to be well defined, the Hamiltonian of

the system must be free of ultraviolet divergences. This can be achieved by

normal ordering the operators at some mass, say ip. 7I could be normal ordered

at a different mass scale. Thus we have

'H = 1/2N,[(DX)2 + (0Y)2 + 2g2X2Y2], (125)

In addition to removing the zero point energies of the fields, X and Y,

which corresponds to normal ordering the kinetic terms, it is also necessary


to insure ultraviolet finiteness to normal order the interaction term X2y2.

The theory given by 7i is now well defined, at least to any given order in

perturbation theory. It is certainly a simple infinite degree of freedom version

of the quantum mechanics [41] defined by

H = 1/2[(X)2 + (Y)2 + 2g2X2y2]. (126)

where X = OrX. The preceding facts are presumably well known. However,

it is less well known that the above theory has an energy spectrum somewhat

remote from that suggested by the finite degree of freedom model. In fact, the

energy density is unbounded from below [42].

This may be shown by choosing an appropriate trial state and computing

(H). For example, choose the state to be "coherent" so that (X) # 0 is

constant over a large region, but (Y) = 0. In addition, choose the state to be

"superfluidic" [43] for the Y field but not for the X field, so that (X)2 = (X2)

but that (y2) < 0. The latter is possible because the normal ordered operator

Np(Y2) is not positive definite. Now take the limit when (X) -- oo. Then in
that limit, (7-) -+ -oo.

To go into a little more detail, let 10, p) be the vacuum annihilated by a Y

field lowering operator of mass p. Then take an expectation value of N,(Y2)

in a different vacuum, say 10, v). Then we obtain the well known result

(0, INp(Y2)10, v) = (47r)-1 ln(2/u2)(0, v0I, i), (127)


N,(Y2) = N_(Y2) + (47)-1 ln(2/v2).



Hence if we choose v2 > 12, then we can arrange for (7{) -- -oo as

previously described.


The first two sections of this dissertation used Bicklund transformations to

study the interacting theories in terms of free theories. For the simple potential

models in chapter one we saw that the transformation had to be a similarity

transformation and not a unitary transformation. This feature was consistent

only because the Baicklund transformation naturally restricted the spectrum

of the free theory, which normally would cause a lack of unitarity. However

the Biicklund transformation simply projected out the negative energy angu-

lar solutions preserving the unitarity of the radial realization of the Virasoro


In chapter two, we find that the functional Biicklund transformation is a

solution to Liouville theory when we simply expand the nonzero modes and

integrate term by term. We find that normal ordering divergences appear in the

same manner that they do in the usual operator formulation. These divergences

were cured by multiplying the nonzero mode interaction term by a factor of

exp and this "normal ordered" the functional Backlund transformation

to all orders of perturbation theory. This is because the only "normal ordering"

divergences arose from the integration over the ) nonzero modes and not from

the 0 nonzero modes. We next computed the correlation function exp gao and

showed that the order g6 part was exactly the same as that of the correlation

function computed by BCGT using Liouville quantum mechanics perturbed

by the nonzero modes. Our calculation had a g4 contribution with a linear

dependence on a, we showed that this could be consistently absorbed into a

finite renormalization of the mass parameter appearing in the theory. To obtain


deeper insight into what this renormalization of the mass parameter means,

we have to see if the wavefunctional we have found obeys the Virasoro algebra

and compute the central charge of the Virasoro algebra. Once this calculation

is carried out and properly understood there are many interesting questions

to ask of this exact solution to Liouville theory. First, how does the Bicklund

transformation project out the nonconformal parts of the free field functional.

How is this solution related to the nonperturbative two dimensional quantum

gravity results at genus one?

The third chapter of this dissertation, examined closed rigid strings im-

mersed in a space of positive constant curvature and a Minkowski space with

small ambient space curvature fluctuations were found to be unstable. When

the two-dimensional de-Sitter solution was perturbed with flat transverse co-

ordinates, these perturbations were found to unstable in marked contrast to

results of of the hoop calculation by Braaten and Zachos. This means that a

space of constant curvature only further destabilizes the rigid string. This has

a simple physical explanation in this theory because the constant curvature

acts like a source of negative energy density; thus, it drives the theory away

from being stable.

The ambient space curvature fluctuations about the hoop solution sup-

ports the previous comments, and suggests that an ambient space of negative

curvature could classically stabilize the rigid string. This is reasonable if one

considers the rigid string in the path integral formulation, and observes that the

extrinsic curvature terms have an overall positive sign in the action. Integrating

over the quadratic parts of the integral yields an integral that is divergent. We

suggest that for a certain range of the rigidity and the string tension, a space

of constant curvature might permit the rigid string to be classically stable.


A topic for future work would be to try to construct a solution to the

rigid string in three dimensional anti-de-Sitter space. However, solving (A29)

appears to be a highly nontrivial calculation. If a classical solution could be

found it would be interesting to first see if it could be stable classically, and

then see if it were stable with respect to quantum corrections. This is because

higher derivative theories are notorious for being difficult to quantize.

It would be interesting to compute the first loop corrections in a curved am-

bient space whose metric is Euclidean, to study how the ambient space curva-

ture affects the asymptotically free behavior of rigid string found by Polyakov.

The last topic of this work concerns, a two dimensional field theory whose

potential is quite similar to that of the supermembrane quantum mechanical

model. We showed that by a renormalization group calculation that the Hamil-

tonian was unbounded from below, and that a system with a finite number of

degrees of freedom can act rather differently when the number of degrees is

allowed to become infinite, that is, the system can undergo a phase transition.

Therefore the two different phases of the theory can have radically different

behavior. So we make the point that the conclusion of De Wit et al is not

conclusive, and that supermembranes are not ruled out as being quantum me-

chanically consistent theories.


Now we show that there are no branch cuts for the Backlund transforma-

tion appearing in chapter one of this dissertation. By making the following

transformation z = exp(-i9), the expression for the radial wavefunction be-

,0(r) = iNz(-2i)(A+1)/2 dz/z(1 z)-(A+1)/2rA/2 exp -(r2(1+ z)/8(l z)).

There is a simple pole at z = 0 and a branch cut at z = 1; however if the
contour is distorted so that only the pole at z = 0 contributes to the integral.
That is, if the contour only passes around the pole at z = 0 and avoids the
point z = 1 the expression for the radial eigenfunctions are well defined.

Then the radial wavefunction is given by

b0(r) = 27r(-2i)(A+)/2NzrA/2 exp -(r2/8). (A2)

Now acting with Lo on the first expression, the integral representation of
the radial wavefunction is given by

LOo(r) = iNz [ dzz-[Eo(l + z)/(l z) r2z/4(z )-2]
Jc (A3)
(1 z)-(A+1)/2rA/2 exp(-r2(1 + z)/8(1 z))

where N' is equal to N(-2i)(A+1)/2

Now the above expression for L0 is replaced by E0 + 10. In (12) we used
the expression of LO acting on exp -iF*(r, 0) instead of acting on exp iF(r, 0).



With this in mind LO is replaced by EO + l0 in the integral representation.

Where E0 = (A + 1)/4 and 10 = izdz. Then equation (A3) becomes

LoV'o(r) = iN' dzz-l(izz + E0)(1 z)-(A+)/2 A/2 exp -r2(1+ z)/8(1- z),


which integrated by parts, it is simple to see that the first term when acting

on exp iF(r, z) has no poles or branch cuts in the contour that we have chosen,

and that the second term acting on exp iF(r, z) has a simple pole at the origin

and has the proper energy eigenvalue of E0.

The next topic is to determine the normalization of both the radial wave-

functions and the proportionality constant of the integral representation Nz.

To determine the normalizations of the radial wavefunctions one must first

find the proper measure. This is found by constructing o0(r) and 01(r) and

demanding that they are orthogonal to each other when they are integrated

over from 0 to oo. The expressions for the groundstate and the first excited

state two wavefunctions are given by

Co(r) =norA/2 exp -r2/8

01(r) =nl(E0 r2/8)rA/2 exp -r2/8. (A5)

02(r) =n2[E0(2E0 + 2) (2E0 + 1/2)r2 + r4/16]rA/2 exp -r2/8.
These were found from solving the differential equation for the radial vari-

able to construct the ground state eigenfunction and using L_1 to generate the

next two higher energy wavefunctions.

When this is carried out the proper measure is 1. The normalization for

0 and i1 are given respectively by

no =2-A/2(r((A + 1)/2)-1/2
n1 =2-(A+1)/2(F((A + 1)/2)(Eo + 1))-1/2(A6)
Now using equation (A2) and the above value for n0 is given by we can deter-
mine the coefficient appearing in (A4)

Nz = 1/(27)(-i)-(A+1)/22-(3A+2)/4(r(A + 1)/2)-1/2. (A7)

Now the nth level radial wavefunction can be computed from the following

On(r) = iNz dzz-(n+l)(1 z)-(A+1)/2r/2 exp -r2(1 + z)/8(1 z). (A8)

This expression can be easily computed by doing a contour integral, and
we obtain the following result:

On(r) = 2rN'r/2z[(1 z)-(A+)/2exp -r2(1 + z)/8(1 z)]=0/n! (A9)

Evaluating this expression gives the exact same form of the first three
wavefunctions as given in (A5) .


In this appendix we will go through some of the details needed to calculate
the Liouville wavefunctional to order exp 3gq and how to compute the nonzero
mode matrix elements for both the 4 and < fields. We start with the expressions

(71), (72), (73) and (74) and then integrate out the 4 nonzero modes. The
generic integration will involve expressions of the form

Pn = -1/21014 + i$'" + g(61 62 6 n) (B1)

now we change variables to

n = +( -8 l 91a^+g1-1[S1 2* -* n]. (B2)

Then (Bl) becomes

Pn =,n lal9n g(p1 p- n)
g2(i 6n ,)1-1(-i Sn). (

p+ =(1 + i; )
P+ =(1 -- ),
S=(1 (B4)
4,n =4(pn)

Sn =8(o pn).
For concreteness we choose the following example of the linear terms in 4

are of the form 41 22. Then (B3) becomes

P2 = \Q\4 9(pS1 + P62)
92(S~ \2) --61(S1 S2).
Where it is understood that the operators p+ and p- only act on the various
delta functions and not upon the fields 0(ar). Before calculating (B5), we list
some helpful identities that will be used to evaluate the rest of the nonzero
mode matrix elements:

6n = 1/(27r) E (exp +im(cr Pn) + exp -in(a Pn),
P6n = 1/7r E expFi(a Pn),
[o|-ln = 1/(27r) E 1/m(exp i(a pm) + exp -i(a rhon). (B6)
Using the results of (B6) in (B5) and integrating over a yields

j daP2 = 0 da(laO g(p1 + Pi2))
oo (B7)
+ g2 1/n(cos n(pl P2) + -92/7r1.
Now we use the above results to carry out the integration for the various
nonzero modes matrix modes of the field i occurring in (68), (69), (70) and
(71) with the result:

( = NoKi-k()exp -1/2 2 d- ld8, (B8)

l() = -Nlexp -1/2 jf daO[i0O I dpl[Kl+k(x)(exp g(/ da(p+1(a)) 1)

K-l+ik(x)((exp g(J p-6Si(a)) 1)],

2(2) =1/2N2exp-1/2 f27 d4a|9\ 4 J dpldp2[
(expg 2/x 1/n cosn(pl p2) exp g J da(p+ 1(c) + p+(a2q('))
exp g J d(p+l {(a)) exp g J du(p+p2(a)) + 1)K2+ik+

(exp g2/7r 1/n cosn(pl P2) expg J da(p-S1j(c') + P-2'(a))
expg J dc(p-81(o)) expg J da(p-_2(a)) + l)-_2+ik

+ (exp -g 2/7r 1/n cos n(pl P2) exp g J da(p+r ((ao) + P- 2'(7))
+ (exp -g2/r 1 1/n cos n(pl p2) exp g f do(p+8b2 t() + p- 51())
exp g J do(p+1 (oa)) exp g f do(p+624(o))

expg J da(p- S (u)) exp g J d(p- 2((o))

+ 1)Kik(x)],

#(3) = 1/6N3exp -1/2 2f dojai/ /dP dP2dp3[
00 00
+ (exp g2/ir 1/n cos n(p p2)exp g2/r 1/n cos n(pi P2)
n=l n=l
exp g2/ I 1 (/n cos n(p2 P3)
expg J da(p+Sij(u) + p+A2 () + p+364(U)
exp g2/7 1/n cosn(pl P2) expg dco(p+iS4 (cr) + p+ 2q(o))
n= 1
-expg2 /, 7 1/n cosn(p1 P2) expg do7(p4il (or) +p+A3q$(a))

00 /
exp g2/ 1/n cos n(P2 P3) exp g dc (p+ 2(cr) + p+ 3(o))
+ exp g(J dp+61(oa) l)KJ3+ik
00 00
(exp g2/r 1/n cos n(pl P2)exp g2/ir 13 1/n cos n(pl P2)
n=l n=l
exp g2/7r E 1/n cos n(p2 P3)

exp g J d7(p- S1 ) + P-62 (() + P- 3())
exp g2/r y 1/n cos n(pl P2) exp g do(p-16(a) + p-A2(a)
exp g2/r 1/n cos n(pi P2) exp g dor(p-^i5(cr) + p-83(o)
exp g2/7r l/n cos n(P2 P3) exp g do(p-2q(o) + p3A(o)
+ exp g J do(p-1(1 (a)) + exp g J do(p- 6 (cr))

+ exp g do(p-63b(a)) 1)K-3+ik(x)
00 00
+ (exp g2/7r /ncosn(pl p2)exp g2/7r 1 /ncosn(pl P2)
n=1 n=1
exp g2/r E 1/n cos n(p2 P3)
exp g J da(p+61qi(o) + P+62 ()) + P-S63(o))
exp -g2/r 1/n cos n(pl p2) expg I dc(p+1 (Sa) + p+3a(~))

exp -g2/rr 1/n cos n(pl p3) exp g J da(p+ S1(o)) + p-S3(o-))

exp -g2/7r 1/n cosn(P2 P3) expg da(p+ 2 (O) P- 3
+ exp g d(p+l q(a)) + exp g J da(p+$2^(a))

+ exp g J du(p_-3(o)) 1 + perm(p3 P2) + perm(pl P3))Kl+ik
00 00
+ (exp g2/ir E 1/n cos n(pl p2)exp -g2 /x 1/n cos n(pl P3)
n=l n=l
exp -g2/7r E 1/n cos n(p2 P3)
exp g J da(p- i1(o) + p- 2 b(o) + p+63(o))
exp g2/7 1/n cos(n(pl p2) exp g d(p-51( ) + p-2 a))
exp -g2/r 1/n cosn(pl P3) exp g do(p-i5^5(a) + p4+3 S(3))
exp -g2/ir E 1/n cos n(P2 P3) exp g f d(pS2(cr) + p+A6q())
+ exp g do(p-61 ()) + exp g(j da(pP-22(u))

+ expg Jdcri(p+63^o() + 1

+ perm(pl -+ P3) + perm(p2 +- P3))K-l+ik].
Where the prefactors appearing in the various wavefunctionals were defined by
Nn = An exp ngq and the An were defined in chapter two.
Now we carry out the evaluation of the nonzero mode matrix elements oc-
curring in (72). The first matrix element we consider is (I(0)(k')jexp agq(0)l4(O)(k))
and it is of the form

f D exp -0 do(q jq ga4o), (B12)

Where each wavefunctional contributes a factor of exp -1/2 f r" dao^O al to the
evaluation of (72). A change of variables is made to complete the square by
shifting ( -- 4

S= $ ga/21a-1'o, (B13)

now this is inserted into the function appearing in the exponential of (B12)

and the result is:

ilal g2a2/46aIDo1 = ~jlal gab06. (B14)

The left hand side of (B14) is put into (B12) and evaluating the a integral


g2 2/4/ do ol-lo = g2 2(1/4. (B15)

Again the usual "normal ordering" divergence shows up that appears in the
operator formulation of exp gaf(0). This infinite term will be divided out of

the correlation function. Using the results of (B15) and (B14) (B12) becomes

SDe exp + dojoa5|expg a2(1/(47r). (B16)

From (B16) there will be an infinite term from integrating this Gaussian

integral over all the infinite number of nonzero modes. This term will be

canceled by the normalization of the wavefunctional. These terms will show

up in every matrix element that we compute and we will not bother with them

from here on.

This procedure of shifting the field q will be repeated over and over again

when evaluating the nonzero mode integration of (72). Therefore, we will

simply write down the original expression appearing in the various matrix

elements and what it becomes after shifting variables to complete the square.

The next matrix elements we consider are of order exp g(a + 1)q. These are
the terms are given by (4(1)(k') exp ag((0) \(0)(k)) and (.(0)(k') exp ago(O)j'(1)(k))
and their nonzero mode matrix elements are of the form

I dp1(exp ,r doa[j(](exp -g(pi6a8) 1). (B17)

So we make the following change of variables

+ = g/211-1(p S + aSo). (B18)

Then (B17) becomes

= d exp[/4 dpa(p21 + ao0)1K1-(pSi + &eo)] 1). (B19)
0 0
There is an overall factor of f Dq exp fd 19810 multiplying (B19). Now
to compute the expression appearing in the exponential of (B18), we use the
identities in (B6) and the fact that fS2 da exp i(n m)a = 6n,m to obtain

02 7r co00
Sda[(pi6 + ao)ll-1 (p + a o)] = 4/7r 1/nexp inpl + ga2L/ .
Now the expression appearing in (B20) is inserted into (B18) and then
expanded in a power series in g2, the result is

I= dpl([1 + b 1/mexpinp + ..+
00 00
bn/mn! .. 1/(ml mn) exp fi(ml + + mn)pl] 1),
m=l mn=l

and b = g2/(27r). Only the first term is nonzero because all integrals
have either all positive or negative frequencies in the exponential, therefore
they integrate to zero. All of the matrix elements of order exp(a + 1)gq are
of this form and they are zero. This is simple to understand because the
term appearing in the exponent for these nonzero matrix elements is either a
positive or negative projection operator and only the positive frequencies or
the negative frequencies are left after the a integration has been done. Where
the frequencies are the positive integers.
The next order in the calculation of the matrix elements of (72) involves
terms of the type exp g(a+l)q. The terms to evaluate are (D(l)(k') exp agO(0)O)4()(k))

, ((O)(k')lexp ago(0))fQ(2)(k)) and and the previous term's complex conjugate.
Considering the first term in the above list, its nonzero mode structure is of
the form,

I1, = JJ dp/dp2 Deexp J ddaPO9l[exp dog(p-b 1 + aSo)4(a)) 1]

[expJ dg(pTS2 + -o) ()) 1].
Where it is to be understood that the integrals over the variables a, Pn are from
0 to 27r. The case that could possibly be nonzero is when the above product
involves a p+ and a p_. This follows from what we observed in the evaluation
of the matrix elements of order expg(a + 1)q strings of only p+ or p- project
out all the opposite frequency states and then there power series expansion is
observed to trivially integrate term by term to zero. Considering a term with
a p+ and p- and examining only the exponential term with both p+ and p-
present because terms with a single p are equal to one after being evaluated,

II+- = 41al g(p+Al + P-S2 + O0)q. (B23)

We change variables as before and obtain

II+- = i101 g2/4(p+b1 + p- 2 + -ao)9\-1(P+S +P-62 + a6o) (B24)

Now the II+,_ is integrated over a and the result is

dall, = b 1/n(2 expin(p 2) + (exp inp + exp in2). (B25)
We have suppressed the "normal ordering" term coming from the function
exp gaf(0). (B25) is inserted back into the exponential expanded and inte-
grated over pl and P2. The integrated expansion of (B25) will contain only
positive frequencies of pl and negative frequencies involving P2 and therefore
they all integrate to zero except for the leading term in the expansion. When
(B22) has been integrated it is zero because only the 1 survive the angular
integration and they cancel against each other. Similarly, the term with p+
and p_ switched only flips the sign on the frequency multiplying the variables

pl and P2 and it also vanishes.
Next we evaluate terms of the type (4(O)(k')(expag (O)[^(2)(k)). There
will be two different types of terms will occur that have not been encountered
before. These terms are of the form

12 = J dpdpd2 J D exp J da(4,9\j g(p5i + P62 + Ao0))

x exp b 2/n cos n(p P2),


I,q= = J dpdP2 f exp Jda(19i\i| g(pil +PT2 + Ao))
oo (B26)
x exp -b E 2/n cos n(pi P2).
As usual we complete the square on both terms which yields:

2 = f dpidP2expa2,
I = / dpdp2 expb2,
where the functions a2 and bi2 are defined respectively by

a2 = 2/4 do[(pSil + PA?2 + a0o)l&lj(pSl + P:2 + a6o)]
+ g2/ E n-1 cos n(p2 P1),

b2 g=92/4 da[(Pi1 + PFT2 + Ao60)(I-11(pt + PFS2 + 60)]
g2/7 n-1 cos n(p2 Pl).
Now evaluating the a integrals in appearing in (B28)and (B29), then (B28)

M2 =g2/4 da[(pilt + PA2 b+ o60)IO1-(Pi + P62 + a6)]
o oo (B30)
M2 =b 1/n(2(exp b E a/n(exp minp + exp inP2).
n=l n=1
Note that only positive or negative frequencies appear in the exponential
terms multiplied by a. Similarly, computing (B29) yields

M2 =-g2/4 do[(p1 + PT62 + abo)IPal -'1( + P62 + (30)]
oo 0 (B31)
M2 =b E o/n(exp inpi + exp finP2).
Finally we have the expressions for .2 and I- ready to be integrated

over P1 and P2,

2 = dpldp2 exp b 1/n(cos n(pi P2) + a[exp inpl + exp inp2]),

I+ = dpIdp2 exp b 1/n(sin n(pl P2) + a[exp inpl + exp FinP2]).

To facilitate the evaluation of these integrals, it is simpler to expand out

all the exponentials independently. With this in mind the expansion of (B32)


2 = dpidP2[
00 00
(1 + b /nexpin(p -P2) + b2/2 Y l/nmexpi(n + m)(p + p2) + ) x
n=1 n,m=l
00 00
(1 + l1/n exp -in(p1 P2) + b2/2 l1/nm exp -i(n + m)(pl P2) + ) x
n=l n,m=l
00 00
(1 + ab 1/n exp finpl + 2b2/2 1/nmexp i(n + m)pl +..)x
n=l n,m=l

(1 + ab 1/n exp finp2 + +a2b2/2 1/nm exp i(n + m)p2 + ).
n=l n,m=l

Now we perform the angular integration and list only the nonzero parts of
(B34) to order g6 which is of order b3, the result is

oo oo
12 = (2r)2(1 + b2 E 1/nm6n,m + b2 1/nml(61,n+m). (B35)
n,m=l n,m,l=l

These sums are evaluated by using the Riemann Zeta functions, which are
defined by E'1 1/(nP) = (p and Em=l 1/(nm[n + m]) = 2(3. Now (B35)

can be expressed as

I2 = (27r)2(1 + b2 + 2b3C3 + terms of order b4. (B36)

Notice that there is no a dependence in this expression because the frequen-
cies were either all positive or negative in the exponentials that were multiplied
by a and that the sums are over the positive integers. So all of these integrate
to zero. Now we quote the results for Il: using the same techniques that were
used to evaluate 442,

I=F = (27r)2(1 b22 + a2b3(3 + terms of order b4 (B37)

Now making contact with chapter two where we have defined the various

nonzero mode matrix elements,

Md =0,

M2 =(I2(1,2) I(1) 1(2) (27)2),

M2 =(27r)2((g2/(27r))22 + 2(g2/(27r))3(3), (B38)

M:F =(I:(1, 2) 1(1) I:F(2) (2r)2),

M =(2x)2 (-(g2/(2))2(2 + a2 ~/(27))3(3).


Now we move on to evaluate nonzero mode matrix elements of order exp g(a+

3)q. The terms that we consider first are of the type (0(1)(k')\exp agO(0O)l1(2)(k))

and its complex conjugate. We show that one of the particular matrix element

is zero that the same mechanism will apply to the other matrix elements of this

form and that they are also zero. Now we consider the following expression

M_1,2(1,2,3) = dpldp2dp3 J D exp- da(^1015 a6-~)

[exp J dagp-S3 1][exp J drg(p+ 1 + P+6S2)
oo (B39)
x exp b 2/n cos n(pi P2)
exp dagp+165 exp f dagp+862 + 1].

So we multiply out all of these terms, complete the square and perform the

a integral and the result is

M-1,2(1,2,3) = /dpiddp[dps[expb 1/n(2expin(p1 P3) + 2expin(p2 P3))x

exp ab E 1/n[exp inP1 + exp inp2 + exp -inp3]
x exp b 2/n cos n(pI P2)

exp b 1/n(a[exp inp1 + exp inp2] x

exp b > 2/n cos n(pI P2).

Note that the other six terms had been previously evaluated in showing that

the matrix element (4(1)(k')Jexp agO(0) I(1)(k)) vanished. We expand the ex-

ponentials appearing in (B40) and make the observation that all the exponen-

tials in the first term of (B40) have negative frequencies multiplying p3. All of


these expressions will integrate to zero and the only term that will contribute to

the first term in (B40) is the cosine term and similarly the a dependent terms

in the second exponential have positive frequencies and will integrate to zero.

The only term that contributes to the second exponential is the cosine term

again but these two exponential terms have a relative minus sign between them

and cancel each other out. The other terms of this order are also of this form,

so that there is always an uncorrelated pn which will not contribute once the pn

integrals have been performed. The parts of the exponentials that are nonzero

always come in with a relative minus sign between them and cancel. There-

fore matrix elements of the form (I(1)(k')lexpcagO()Jl(1)(k)) are identically

zero. This lack of correlation occurs because the two different wavefunctionals

Sintegrals were evaluated independently of each other.

Now we consider terms of the form (4(0)(k'lexp agO(0),,(3)(k)) now that

we have established how the various nonzero mode matrix elements arise and

are evaluated, we will now just write the expression and quote the results:

M3 = ddpldp2dp3[exp b 1/n(a exp inp + exp inp2 + exp inp3)

exp b : 2/n(cos n(pl P2) + cos n(p1 P3) + cos n(p2 P3))
(27r)(I2(1, 3) + -2(2,3) + I12(1,2))

+ (27r)2(+ (1) + (2) + +(3)) (27,3]

M3 =2g6(3,

M++ (1,2, 3)= /dpldp2dp3[exp ab 1/n(expinpl + exp inP2 + exp -inp3)

x expb b 2/n(cosn(p1 rho2) + sinn(p1 P3) + sinn(p2 P3))
(27r)(I+- (1, 3) + +-(2,3) + 1++(1,2))

+ (27r)2(I+(1) + I+(2) + 1-(3) (27r)3

+ perms(pl p3) + perm(p2 P3)]

M++-(1, 2, 3) = 6g6'3,

M-+(1, 2,3) = dpldp2dp3[exp ab E 1/n(exp -inpl + exp -inP2 + exp inp3)

x exp b 2/n(cosn(pl rho2) + sinn(pl P3) + sin n(2 P3))

(27r)(I_+(1, 3) + -+(2, 3) + --(1, 2))

+ (27r)2(I_(1) + -(2) + +(3) (27r)3
+ perms(pl -+ p3) + perm(p2 +- P3)]

M_-+(1,2,3)= 6g 63.
Where we have used the expressions for the various Is which have been defined

earlier in the appendix.


In general one can consider a map from a n dimensional manifold into an

N dimensional manifold 0 : Mn -- MN. If 0 is C' and locally a one to one

mapping then 0-1 : 0(8) -+ Mn for any open neighborhood 0 of a point in

Mn. Then 0 is an immersion. However if 4 is a one to one mapping for all

of the manifold Mn then 0 is an embedding. Immersions allow for the case of

intersections while embeddings do not. For the case that we are considering Mn

is the immersed or the embedded submanifold and MN is the ambient or the

embedding manifold. N- n is the codimension of Mn in MN. The coordinates

of Mn are given by ya ,a = 1, -, n and the coordinates on MN are given by
X/, P = 1, -, N. We assume that the ambient space metric on MN is G1v so

that the invariant line element on MN is given by dS2 = GpvddX dXv. This

causes an induced metric on Mn. Then the immersed manifold's invariant line

element is given by

ds2 = GydXudX'" = GXXbdyadyy (Cl)

Where X = -5 and where the manifold MN is restricted to the image

points of Mn. Then the expression for the induced metric gab(y) is given by

d2s = gabdyadyb (C2)

gab = G Xal X (C3)

This expression is bilinear in the tangent vectors, where it is understood

that XP(y) is restricted to the submanifold by the map I : fn" --_, N. The



XP/ is a set of N scalar functions on M" and Xa are vectors on the submanifold

M". Since the codimension of the submanifold Mn in the ambient space MN

is N n, then there will be N n vectors which are orthogonal to the XI,

normalized to unity, are given by the following expressions:

GpXfi9Y = 0

Guvz = 67 j (C4)

i,j = 1,--, N- n.
The functions Guv and 71i are tensor functions in the ambient space but

are scalars functions when restricted to the submanifold Mn. Therefore G~,

and 77 are invariant under the reparametrizations of Mn.

Now the Gauss-Weingarten and the Gauss-Codazzi-Mainardi equations will

be derived. Next make the identification that Xa = DaX" where Da is the

covariant derivative of the submanifold Mn. Then covariantly differentiate the

induced metric gab using Da to obtain the following

Dcgab = agbc abgcd cbd
rabc = 1/2(aagbc + abac Ocgab).
Where tabc is the affine connection on Mn, gab = G yXX, and rabc is given


rabc = 1/2(Oa(GT X 'X) + Ob(G7aXjcX,) aOc(GaXpXX))
5c b c ax a (cb)
rabc = 1/2((Gyp,a + G-tp Ga,)XaXX + 2G Xf) (C6)
Fabc = CaXyaXX X, + GapX Xc,

where we have used the fact that

DaGpv = X -a Gpv (C7)

and Ca#7 is the affine connection on the ambient space and Xa = abX,
X,ab bX
that is, the comma just denotes ordinary differentiation.
Now covariantly differentiating gab yields

Dagbc = DaGvXbX"c + GLv[(DaXb)X + X (DaXv]. (C8)

Now forming the combination Dbgac + Dcgba Dagbc = 0 and using (C7)
yields the following

Gv,Xa DbX + CaX'X X" = 0 (C9)

and this can be rewritten as

GvX, (DbX c + C XXfc) = 0. (C10)

The expression inside the parenthesis is a vector that is orthogonal to the
subspace M", therefore a vector perpendicular to the subspace M" without
altering (C10). This new term that we add to (C10) is denoted by the following
expression K]b. Then K'b is defined by

DaX + X X" C = K' = K
Kib = Gpv7zi(DaX1 + Ct XX ,ab)
where the second line of the above equation has been projected out. It is
simple to see that Kab = Kba by observing the symmetry in F'b from (A6)
and since Capy is symmetric in a and P. These expressions are known as the

Gauss formulas and Kab is called the second fundamental form. Then there
are N n such of these symmetric second rank tensors on the submanifold .1In

when the codimension is N n.

Now covariantly differentiating the other two equations that define the
subspace Mn in (C4) to obtain the following

Da(GvX -u) = 0
GvDaXT~Y = -G/tv,17a (C~,v + Cvpj)X aXbi7i .
Now (C10) is used to plug into the left hand side of (C12) which yields

Gy i({Kab XaX Cp) = -GOX ,a (C/P, + C,,^P)X, X,aXb
ia ,b a / X/1 L ,-#bz
Kab = G,,vX ,la CvzXV"aXri .
Where we have used the identity G/tv,7 = Cyyv, + Cvy,7 in the previous expres-
sion. Now covariantly differentiating the expression Gpvii5fr' = bij yields the
following result

G,aX e j + Gtv(r,7 e, + 7) = 0 (C14)

Again using the same identity that was used in (C13) the above expression
can be rewritten as

(Cp,, + CVpX7ar7ij + Gpv(77aj1 + 7 ja = 0. (C15)

Which yields two equations of the same form which is given by

77(G,,,a + CypX, ) = 0 (C16)

but whose normals are pointing in different directions. This equation can be
rewritten in the following form

Gvrl u"(?lia + C/ XA 9fl) = 0. (C17)

Now a vector orthogonal to Gyvqri can be added to (C17) without changing
the result of (C16). This new term will be denoted by AX'b and then (C16)
changes to

77a + C X ",1' = AX. (C18)

Now (C18) is inserted into (C13) to express Ab in terms of the second
fundamental form Kiab. Then the result is AC = -gbKab. Then (C18)

o = -gbKiabXI C(,XCC. (C19)

Now we want to find the conditions for integrability of DaDbXP, to do this
we make use of the Ricci identity

[Da, Db]X, = Xc gef Rfabc. (C20)

Where Rfabc is the Riemann tensor on the submanifold Mn which is con-
structed out of the gab We use the result previously derived in (C11) and
express DbXc in terms of the second fundamental form and the ambient space
connection multiplied by the appropriate tangent vectors. Then we covariantly
differentiate with with respect to Da and obtain the following result

D,,DbDXP = Da(K c?" CapX X, )

D,,DbDcs = (DaKIc), + ab + [(DaX )Xc + X,(DaXc)]CJ

+ XO'XDaCp.
Now ,we m~ik use of (C19) along with (C11) to obtain the following ex-

KD a(bef1) X" + C 7
DaDbD ( = (Da. Ke'(ef Kia fX, + Cf )

+(a + KP X)Cp XX c 0C p (C22)

XXXbCC + X X. b cC
,a- ,C ,b 0 Ap a CM Apy
Now antisymmetrizing on the a and b indices on the submanifold yields

[Da, Db]Xp = (DaKib Dbiac)7ii + X gef (IacKJe KibcK e)6ij
+a/37 a,b ,C*

Where RA is the Riemann tensor in the ambient space MN, that is
formed from Guv. Where

S= GpA(Ca,/ C- ,a + CC CjC p). (C24)

where the same equation was used on the right hand side of (C20) but is
constructed out of the subspace connections cab. Now we equate (C20) with
(C23) which yields
(gac l hcj abb+jgefXu lli
X gefRfcba = (bKc Kh i ab+igefX + (Kc;a K,;b)
+RAf Xca XX
c37 ,c,b ,a"
Where the semicolon denotes covariant differentiation on any tensor it acts on.
Now the tangent and orthogonal components of (C23) can be projected out
respectively by multiplying by GiXd to yield

Rdcba =(KbKic K cd'Ib)bij + Xcfl ,b ,a (C26)

and by multiplying by 77' to obtain

(Kca Kc,) Rap X"flX (C27)
bc;a -- I ac;b )ij -- Y c ,b 1Q7.

For the ambient space-times that we are considering it is simple to see
that the second fundamental forms are integrable, that is (C27) is zero. Since
de-Sitter space is a space of constant curvature, its Riemann tensor can be
written as

RIva'3 = a0(GuaGv, GpGva) (C28)

When (C28) is inserted into (C27) it is zero because of the first expression
in (C4).
Next, we show explicitly what the equations of motion are for the rigid
string immersed in an ambient gravitational field. Where we have assumed
that gb = GX ,X X. We vary the action which is given by

I = T d2zfg [l R2((gabKab)2 RA XaxflX X X6gabgcd)] (C28)
ay6 ,ay ,e ,b 9.

with respect to 6XI to obtain the following equation.

6l/SXA 8m6I/6Xm + an9n6I/6Xn =0.

S= (-g)1/22C/, (R2 GK r bv ab ab R X ab cd

(-g)12 abgab (GTA b GI (gcdXc[2C(GaX, bXd)+

CL+XC,XX Xd + Gap,AXabX + C ,AX, Xd + C A)
cb Id Id 0 ,-f ,d a
G- pvK"Xb (g ,c[2CsA(GpXabX + C#XaX XXd)+

Gap,AX bX) + Capy,AXX XYX ] + C A XOX) + 4GpvKPbc"C

(-g)1/2gab cdA vXaX 4RA p X vX -
--gs)'/2s~bs~ g g k l lx~a' ,b ,d .vaca- ,ca.,c,b",d' OA

- am [((-g)12 oGa)(XoGuXX XG,,X X)

+ 1 mG(X GyXXv X GX 0X 1))

[(-g)-(R2 Gr'P K.vbg ab RA QX X" VX
0a p. iB-,a ,c ,b d
2GKv ab abgcd -1(GaX bX + Cy XX Xd)X

2GpyvK babgab gd g-1(GapXbXd + Ca3yXaXX ,dX)Xc
P _a abgl + Afl Xacd)
4G,, Kal i ab 9abb X-1 + RA x'3XYXVXO ab cd9

[Xciab + X,"Kab]g- 1G vG, (c XbX, + dm a)(GX, C X,aX,d)

- G gab ab amCpX" + 6mCQX,)(GO-. + G+OKb)-
RA 9 ab cdcmcnxpXv + .rnx 7 ,u Y '- -'v'f YrnA' v -l l/ t Y^ Y
a A ,c ,bX,d + c "A ,a ,d A ,a ,C ,d A d ,a ,c,
R X11 X11 aXfXX 6cmbXX, + +gdm WXX+gX
-A ..Xa-X, X gd c Gax-, bmX-gab A(cmX +a a )]
+,m9,(-g)1/2G,,bbmg g ab abtl cdGpX Xzlt) g gcdG cpXIX iK .
+ a ((br 9 GAXX + (b gcd G#'^x )K)
The last calculation of this appendix is the variation of the action with
respect to the subspace metric gab, to show the highly nontrivial equation
that must be solved to compute the induced metric in terms of the tangent
vectors XPa, the ambient space metric Gy, and the other derivatives of these
fields. The variation of the action treating gab as an independent field yields
the following expressions

6I/69ef = 0, (C30)

1/2(R 2 gab Kab2 R X X X abcd) 2ge gbf KababK ab
0cfl7 a c 3 b dg ab
S2gcegdfgab(G,3XaYbX + CapX aX XdX)Xcg Kab- G/v-
I f- ,a ,b d ab
RA xaX X t X,6 (gae bf cd + gabgcedf = 0.
76 a c ,b(C31)


It appears finding an exact solution to this general equation is not very

likely. It would be very important to see whether or not terms with quartic

derivatives appeared in the induced metric.


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Thomas McCarty was born at Ramey Air Force Base, Puerto Rico, on

January 6, 1957. After graduating from South Dade High School, he attended

the University of Florida and attained an undergraduate degree in nuclear

engineering in 1980. He started in the Physics Department at the University of

Florida in August of 1981. He was attracted to the Physics Department at the

University of Florida following the formation there of a particle theory.group

in 1980. He started to do research under the tutelage of Assistant Professor

Thomas L. Curtright in 1984. His research, as that of his advisor, has been

wide and varied. His current work is on understanding higher spin massless

fields and their conformal properties in 2 and 3 space-time dimensions.

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.

Pierr amond
Professor of Physics, Chairman

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.

Richard D. Field
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.

es N. Fry
sociate 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.

Richard Woodard
Assistant 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.

Ulrich H. Kurzweg
Professor of Aerospace Engineering,
Mechanics, and Engineering Science
This dissertation 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.

August 1990 a~tA_ _-<-
Dean, Graduate school

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