PQBQEETESE OCR Output

|||i||\\\\||\|l||\\\ ||l1|\\|\l||ll||III| \|\\|I\\\\||QERN LIBRF·|RIES» GENEVH

generalization.

of the mass spectrum. The theory allows a straightforward supersymmetric

and the elementary particle scale which is responsible for the fine-structure

scale which is responsible for the harmonic oscillator spectrum of the string

of the theory is characterized by two completely different scales, the Planck

Virasoro group. After the spontaneous symmetry breaking the mass spectrum

where H is the color subgroup of G and U (1) is the Ca.rtan’s subgroup of the

automatically breaks the Virasoro-Kac-Moody group down to P X H X U(1),

constructing a gauge theory of the Virasoro-Kac-Moody group. The theory

the string excitation modes. We present a. field theory of a closed string by

Poincaré group and G is the grand unification group, which exists among

the Virasoro-Kac·M0ody symmetry associated with P X G, where P is the

principle underlying the string field theory must be the gauge invariance of

We argue that at the pre-confinement level the fundamental dynamical

Abstract

Department of Physics, University of Utah, Salt Lake City, UT 84112, U.S.A.

S. W. Zoh

School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, U.S.A.

Y. M. Cho'

Unified Field Theory of String

/ 5 (,0 5 OKIASSNS—HEP-93/85

object which can also be viewed as a set of infinite number of point particles made of the OCR Output

one means by the string field theory. Intuitively the string is a one-dimensional extended

the 4-dimensional string field theory [2,3]. To answer this question one must first clarify what

is the fundamental principle underlying the string dynamics with which one can construct

One of the important problems in the string theory has been to understand precisely what

spectrum. These are precisely what one can expect from a string field theory.

massive fermions and a finite number of light fermions with spin % and g to the particle

the theory allows a natural supersymmetric generalization, which adds an infinite tower of

multiplet of H, as the pseudo—Goldstone particles of the symmetry breaking. Furthermore

H >< U In addition the theory contains light scalar particles, the dilaton and an adjoint

massless particles made of the graviton as well as the gauge fields of the unbroken subgroup

of an infinite tower of massive spin-two and spin-one particles and a finite number of the

taneous symmetry breaking the mass spectrum of the theory in the bosonic sector consists

of G and U (1) is the Cartan’s subgroup of the Virasoro group. After the inevitable spon

tomatically breaks the symmetry down to P >< H X U (1), where H is the color subgroup

could provide a field-theoretic description of a propagating closed string. The theory au

unification group G, and to present a gauge theory of the Virasoro·Kac·Moody group which

the Virasoro·Kac·Moody symmetry associated with the Poincaré group P and the grand

damental dynamical principle underlying the string theory must be the gauge invariance of

construct the string field theory [2,3]. The purpose of this paper is to argue that the fun

does not know precisely what is the fundamental dynamical principle with which one can

4-dimensional space-time, has not been available till recently. This is because one still

theory of string which could describe the dynamics of a propagating colored string in the

has been studied extensively. In spite of the extensive studies, however, an effective field

theory of everything, the ultimate theory which unifies all the interactions in nature, and

During the last decade the string theory [1,2] has become the leading candidate of the

It has long been recognized that Dif S1, which we call the Virasoro group for simplicity, OCR Output

approach could leads us to a meaningful string field theory.

among the string excitation modes without much dimculty. We believe that only the second

course, is that here we can identify the dynamical principle which governs the interaction

to the quarks and gluons of the strong interaction. The real advantage of this approach, of

modes not as the physical modes, but as the elementary fields of the string which correspond

we adopt the second approach in this paper. This is because we treat the string excitation

the first approach [2,3]. Perhaps this is why all these efforts have been so futile. In contrast

Unfortunately so far almost all the efforts to construct the string field theory have adopted

well as traightforward, because here the underlying dynamical principle is clearly defined.

on to describe their interaction. On the other hand the second approach is economic as

spectrum does not provide a clear indication on what dynamical principle one should rely

first approach is not only impractical but also extremely hazardous, because the hadron

all the hadrons from the beginning, or else start with the quarks and gluons only. The

construct the field theory of the strong interaction. Obviously one could either start with

however, are based on the gross misunderstanding of the problem. To see this let us try to

of the Virasoro-Kac-Moody symmetry in the physical mass spectrum. These objections,

harmonic oscillator spectrum of the string excitation modes. Secondly there appears no trace

physical particle spectrum that one obtains from the string theory is totally different from the

At this point one might object the above argument for the following reasons. First the

the Poincaré group and the grand unification group

field theory must be a gauge theory of the Virasoro-Kac-Moody symmetry associated with

Virasoro-Kac-Moody group, at least in the point limit of the string. More precisely a string

underlying the dynamics of the string excitation modes must be the gauge invariance of the

the interaction among the string excitation modes. We argue that the fundamental principle

principle. In this context one may ask what kind of dynamical principle is responsible for

of the point-like string excitation modes which interact under a well-defined dynamical

string excitation modes. This implies that the string field theory must be a field theory

the corresponding momentum operators. With p,,,,, and :z:"”‘ we can construct two angular OCR Output

Let us denote the string :z:"(0) in terms of its Fourier coefficients :r"”‘, and let p,,,,, be

the string which generates the translation of the string in the 4·dimensional space-time.

To describe the propagation of the string we must introduce the momentum operator of

more symmetry. Consider a propagating closed string described by :c"(0) shown in Fig.2.

To describe the propagation of a string in a curved space-time, however, we need one

fundamental symmetry of the string.

This means that the Kac-Moody group associated with the color S U (3) must be another

that the physics of the string must be independent of how the color is assigned to the string.

form a subclass of the Kac-Moody algebras. Now the quantum chromodynamics requires

which are also called the affine Lie groups are described by the affine Lie algebras which

group of S U (3), which we call the Kac-Moody group for simplicity. In fact the loop groups

a color g to each point 0 of the string. The set of all such mappings forms a group, the loop

S U (3) and 0 is the string coordinate of S1. This mapping shown in Fig.1 naturally assigns

make a mapping g(6) from the string S1 to the color SU (3), where g is an element of the

connects two quarks must be colored. A simple way to put in the colors to the string is to

the strong interaction. But according to the quantum chromodynamics the string which

the color SU As a unified field theory the string theory should be able to describe

Another fundamental symmetry of the string is the Kac-Moody group associated with

is most likely to be formulated as a gauge theory of the Virasoro group [3,4].

dimensional field theoretic description of the string, before an inevitable symmetry breaking,

space-time point with a complete gauge degrees of freedom. This means that any realistic 4

under the reparametrization of the string. Moreover these multiplets should enter each

the infinitely many point-like string excitation modes which must form covariant multiplets

of the string excitation modes. This is because what is left over in this limit is a set of

limit that the string shrinks to a point, this symmetry should become an internal symmetry

been treated as a geometric symmetry of the string world sheet. But notice that in the

is the fundamental symmetry of the string theory. So far, however, this symmetry has

from, in fact completely orthogonal to, the popular approach that one finds in the literature OCR Output

Clearly our approach to the string theory presented in this paper is totally different

gauge theory of the Virasoro-Kac·Moody group.

the following we discuss in detail how to resolve these problems to construct a prototype

to be impossible to construct a sensible field theory with a non-unitary representation. In

particular the second one, pose a really serious problem. So far it has generally been believed

do not form a unitary representation. This creates the unitarity problem. These facts, in

the gauge fields (which should form the adjoint representation) of the Virasoro group simply

sure that the gauge fields form a unitary representation of the gauge group. Unfortunately

creates the ghost problem. Secondly, to guarantee the unitarity of the theory one must make

the Virasoro group does not admit any, positive-definite or not, Gartan-Killing metric. This

make sure that the kinetic energy of the gauge fields becomes positive-definite. However,

need to have a positive·definite Cartan-Killing metric of the underlying symmetry group, to

ghost problem and the unitarity problem [4,7]. First, to construct a gauge theory one

When one tries to construct such a gauge theory one faces two diHicult problems, the

been available till recently. In the following we discuss how to construct such a theory.

so far an explicit construction of a gauge theory of the Virasoro·Kac-Moody group has not

the grand unification group. Although this fact has been recognized by many authors [3,9],

gauge theory of the Virasoro—Kac-Moody symmetry associated with the Poincaré group and

that the unified theory based on the string must be (a supersymmetric generalization of) a

of course, the string theory must also accommodate the electroweak interaction. This means

accommodate the gravitational interaction in the string theory. To unify all the interactions,

fundamental symmetry of the string theory. Indeed it is this symmetry which could naturally

gauge invariance of the Virasoro-Kac·Moody group of the Poincaré group must be another

with the 4-dimensional Poincaré group as we will show in the following. This implies that the

boost and rotation, which together with p,,,,, forms the Virasoro-Kac-Moody algebra associated

internal rotation and the external angular momentum M,,,,,,, which generates the space-time

momentum operators of the string, the internal angular momentum L,. which generates the

construct a genuine gauge theory of the Virasoro—Kac-Moody group associated with the OCR Output

a propagating string in the 4·dimensiona1 space-time in the point limit of the string, and

with the Poincaré group can be identified as the dynamical gauge group which can describe

the Hat space-time. In Section V we show how the Virasoro—Kac-Moody group associated

gauge theory of the Virasoro—Kac-Moody group associated with an arbitrary Lie group G in

gauge theory of the Virasoro group. In Section IV we extend the theory to a prototype

Feigin-Fuks representation of the Virasoro group. In Section III we discuss a prototype

The paper is organized as follows. In Section II we review.the generalized Kaplansky

these features is not clear at this moment.

strings. Unfortunately what kind of dynamical principle, if any, one need to accommodate

take into account the finite size effect of the string and introduce the interaction among the

indeed in the right direction. To obtain the complete string field theory, of course, one must

a string theory. This is a remarkable aspect which shows that our approach to the string is

the gauge theories discussed in this paper have the mass spectrum which is characteristic of

minimum necessary gauge symmetries that is required for a string field theory. Nevertheless

have not taken into account the interaction among the strings. So our theories have only the

degrees of freedom which come from the finite size effect of the string. More importantly we

a point. This is because in our approach we have not yet taken into account the full string

could describe only a single propagating string in the limit that the string is approximated to

provide the ultimate string field theory. Strictly speaking the theories discussed in this paper

have. Of course the theories that we construct in this paper should not be interpreted to

theory which can accommodate all the symmetries that a propagating colored string must

hand in our approach we stay in the 4-dimensional space-time and try to construct a gauge

a 4—dimensional gauge theory which can describe all the known interactions. On the other

approach one starts from a colorless string in a 26-dimensional space-time and try to derive

the two approaches should play a complementary role to each other. In the conventional

new light to the problem of constructing the ultimate theory of everything. This is because

[2,3]. Nevertheless, or precisely because of this, we believe that our approach could shed a

(2.4) OCR Output(L,,,),,°° = [(cx + l)m + ,8 — k]5,,,+,,'°,

given by [4,7]

sentation, and is characterized by two complex parameters oz and B. The representation is

The KFF representation may be understood as a natural extension of the adjoint repre

(2.3)¤,,. - ¢’· = -f,,_,_'·¢·· = —(2m - k)¢’=····.

which acts on an infinite-dimensional vector ¢" as

(L”•)”= fmn= (rn _ n’)6m+nk$k k

From the commutation relation we automatically obtain the adjoint representation L,.

(2.1)[e,,,, e,.] = f,,,,,*e;, = (m — n)6,,,+,,"e;,. (k,m,·n.; integers)

Virasoro algebra without the central extension

the generalization of the Kaplansky-Feigin-Fuks representation, we first consider the

one introduced by Kaplansky, and also independently by Feigin and Fuks [10]. To explain

not belong to this representation. The representation that we need is a generalization of the

the Virasoro group, because the gauge fields (which must form an adjoint representation) do

module. Unfortunately this representation is not so useful in constructing a gauge theory of

widely discussed in the literature is the highest~weight representation based on the Verma

some basic facts about the representation of the Virasoro algebra. The representation mo t

Before we construct a prototype gauge theory of the Virasoro group, we need to know

II. GENERALIZED (a,B)·REPRESENTATION

discuss the physical implications as well as unresolved problems of the unified theory.

VII we provide a simple supersymmetric extension of the theory. Finally in Section VIII we

symmetry breaking mechanism and obtain the full mass spectrum of the theory. In Section

Poincaré group and an arbitrary Lie group G. In Section VI we discuss the geometric

or the (a,,B)-representation d. Let ZM be OCR Outputk

For any (cz, B)-representation ¢" we can introduce the complex conjugate representation,

is defined precisely to allow such an invariant scalar product.

which becomes invariant under the Virasoro transformation. In fact the dual representation

¢· w = ¢kU}k, (2.7)

traction, or the scalar product, between dz" and wk

Clearly this defines a representation. With the dual representation we now have the con

(2.6)Cm - wk = (L,,,)k'w; = (um + ,8 — k)wk+,,,.

element of J,) and let

representation, which acts on the dual module J,) which is dual to Wap). Let wk be an

The KFF representation allows us to introduce the (a,,8)'·representation, i.e. , the dual

[7,10].

module has an invariant subspace made of 45* with k 36 0, which has co-dimension one

the (0, 0)-module has a one-dimensional invariant subspace made of ¢°, and ii) the (-1,0)

on = -1, B = n. Otherwise it is irreducible. The reducibility follows from the fact that i)

n is an arbitrary integer. The (1::,,6)-representation is reducible if i) cz = O, B = n, or ii)

Notice that the (oz, ,8)-representation is isomorphic to the (oz, B-{—n)-representation, where

the adjoint representation (2.2) is indeed nothing but the (1,0)-representation.

where 45* is an element of the (oz, B)-module. From the definition it becomes obvious that

(2-5)6m - ¢" = —(L,..)..’°¢” = —[(¤¤ +1)m + /3 — k]¢'°`”',

(oz, B)-module as follows,

The representation acts on an infinite-dimensional vector space l/QM;) which we call the

(ILM, Lnlli= (Lv¤)i(L¤)i$ " (L¤)€k(Lm)!` = (m ` ”·)(L•¤+¤)¥k k

which we call the (cz,B)-representation. Clearly one has

—(L,,.),,t,...,—···— (L,,,),,t;..., (2.13) OCR Output!°""4 !!"""

cn. . ti-"j1 (L,n)‘nt”.·-j+ ·•· + (L'”)j”ti-.q\k··•lk•••l k"‘l

which forms an element of a tensor space Tm. Define the tensor representation L,,, by

t;...j!""l be an infinite-dimensional (p, q)-tensor which has p upper indices and q lower ones

Now we generalize the KFF representation to a tensor representation of (p, q)-type. Let

which becomes a scalar under the Virasoro transformation.

E - w = Tak,

Clearly this defines a representation. With this we have the contraction between 45* and E;.

(2.11)e,,. ·G;, = (L,,,);.`w; = (cfm — B' + k)wk-,,,.

element of VG,) and let

representation, which acts on the dual module V(L_p) which is dual to V(,_p). Let U;. be an

With the (cz, B)-representation we can also introduce the (cz, B) '-representation, i.e., the dual

(2-10)em · (¢")’ = —l(¤' + Um — B` + !¤](¢"+”')°

sentation is that the complex conjugate (¢!‘)‘ of 45* transforms exactly as $2

where ctn is an element of V(,_p). The reason why we call L,. the complex conjugate repre

**’"em -3= —

10 OCR Output

type. This follows from

is an invariant tensor of mixed type, where i is of (a,B) type but j is of (—cz — 1, -5)

is an invariant metric of (—§, 0) type. In fact the (0, 2)-tensor defined by

ggj = 5;+, (2.16)

2) The metric

¢m · 6¢’ = (¤m + B - i)6e+»»’ - [(¤= + Um + B · jl6¢"”` = 0

1) The Kronecker 6;’ is an invariant tensor of an arbitrary (¤,B) type. This is so because

With the above generalization we observe the followings:

type.

see in the following that the most important invariant tensors we use in physics are of this

elements form a tensor representation of mixed type of the Virasoro group. Indeed we will

This shows that one can indeed have a (p,q)·tensor module of mixed type Tm, whose

(2.15)[6,,,, 8,,] · igj = (TTL — 11)€,,,+,, - tgj.

from which it follows

8,,, · {gi = (dm + B — i)£,+,,,j + (dim + B, — j)£;j+,,,,

but j belongs to an (oz', B')-representation. By definition we have

one can define a (0,2)-tensor of mixed type ty, where i belongs to an (a,B)·representation

p+q indices in (2.13) to transform according to different (oz, B)·representations. For example

may proceed further to define a tensor representation of mixed type, by requiring each of the

generalization the ¢" and Wk become a (1,0)-tensor and a. (0,1)-tensor, respectively. One

sentation can easily be generalized to an arbitrary tensor module of (p, q)-type TM. In this

Indeed it is the tensor product of the (a,B)-representations. This shows that KFF repre

From the definition it becomes clear that L,,, forms a representation of the Virasoro group.

11 OCR Output

4) The (1,2)-tensor fijk defined by

cz; - ··· - a,,..1 - n + 1, -51 - ,8; — - ··— 5,,-1) type, respectively.

is an invariant tensor of mixed type, where i,j,···,k are of (cq,;31), (0:;,,8;), ···, (—cq

(2.22)J*f···'· = s,,*+f+···+'·

defined by

ag - — :1:,,-1 — 1, -51 — ,62 - - 5,,-1) type, respectively. Similarly the (·n.,0)-tensor

is an invariant tensor of mixed type, where i,j, ···,k are of (a1,)81),(a;,B;), ···, (-al

d§j...k ·:

B2) type, respectively. In general the (0,n)·tensor defined by

is an invariant tensor of mixed type, where ·i,j, k are of (cx1,B1), (ag, ,32), (—¤z1 -a; -2, -,81

(2.20)jiik = 50‘+i+'=

B;) type, respectively. Similarly, the tensor dl-lk defined by

is an invariant tensor of mixed type, where i,j, k are of (c11,B1), @1;,52), (-0:1 -0:; — 1, -B1

dijk = 6¢+1+» (2-19)

3) The tensor dg), defined by

we = du¢= 61·+1¢= ¢' °l "°

obtain an element of (-:2 — 1, -B)°·module wk by

index-lowering operations. For example given an element of (cz, B)-module ¢", one can easily

type. The importance of the invariant tensors is that they define the index-raising and

is an invariant tensor of mixed type, where i is of (a,H) type but j is of (-cz - 1, -6)

(2.18)J6 = 50i+j

Similarly the (2, 0)-tensor defined by

em *241 = (am + 6 · i)6¢+j+···° + l—(¤ + Um · .8 · jl6¢+1+··° = 0

12 OCR Output

(4;,,,,,5,,,), (a,,,+1,B,,,+1), ···, (¢x,,,+,,,B,,,+,,) type, when the condition al + ··· +c¥m +m— 1 =

becomes an invariant tensor of mixed type, where i,-··,j,k,···,I are of (oq,,B;), ···,

éik...l‘"-j = 5k+...+li+--·+j

6) In general the (m,n)—tensor defined by

(a,,,B,,) type, but lis of (cz; -{-aq + +a,,+n — 1,51 + +,6,.) type.

becomes an invariant tensor of mixed type, where i,j,···,k are of (0:1,51), (:12,52), · - ·,

lJij..·k =

defined by

(:;,,,5,,) type, but I is of (al + oz; + ··· + am,81 + ··· + ,8,,) type. Similarly the (n,1)-tensor

becomes an invariant tensor of mixed type, where i, j,···,k are of (oq,;31), (:12,52), ···,

dijmkl = 5g+j+...+k

1, B1 + B2) type, respectively. In general the (1, n)-tensor defined by

becomes an invariant tensor of mixed type, where i, j, k are of (@,51), (¢z2,B,), (al + ag +

(2.25)if = 6,,**

B2) type, respectively. Similarly the (2,1)-tensor d,,°’ defined by

becomes an invariant tensor of mixed type, where i, j, k are of (aq, ,81), (cg, B2), (aq +a2, B1 +

k kéij= 6j+-i

5) The (1,2)-tensor dif defined by

em ‘ fijk = (m ' i)i¢+mjk + (cm +p - j).fa5+mk ' [(*1 + Um + B ” kl.fijk—m : 0

This follows from

is an invariant tensor of mixed type, where i, j , k are of (1,0), (on, B), (oz, B) type, respectively.

(2-23)"fa= (Le); = l(¤ + 1)i+ B · kl6;+j

13 OCR Output

with the invariant tensor d;j = 5;+,**,

and (-cz — 1, —B)—representation 45:, one can define the scalar product by contracting them

by contracting the indices of the same types. For example, given an (cx, B)-representation 45f

and the vector products, among different tensors. Clearly the scalar products can be defined

The existence of the invariant tensors allows us to define the products, the scalar products

mp) ,,,,,,_,_,,0, + iuzllaibecomes an invariant tensor of mixed type, where i, j, - ··, k are of (a,H) type but I is of

(2.32)e.‘*··'* = .4s.*+f+···+'·

j are of (:.1,5) type but k is of (2oz — 1,26) type. Similarly the (11., 1)-tensor élumk defined

This means that the structure constants form an invariant tensor of mixed type, where ·i and

= —f;j (2.31)

@1/“ = (J · 1)*%+1

in the n indices i, j, ···, k. In particular when n = 2, one has

,·nB) type. Notice that by construction the é,,___,,‘ tensor is totally antisymmetricnm _ 1

is an invariant tensor of mixed type, where i, j, ···, Ic are of (a,,6) type but I is of (na

1 k k’ --- k···*

(2.30)A = aa I J J J1 · *2 _ _ _ *n—1

1 e P --- ¢···*

with

éijmkl = A6j+j+...+g

7) The (1,11.)-tensor é,j___,,' defined by

(1,,,+] +€Xm+,, alldpl+‘··+Bm=Bm+1+··•+Bm+n is 88»tlSl'lCd.

14

(2-36) OCR Output" '(¢>1 * ¢¤)= Jr/¢l¢i = ¢{"¢'§""»

45:, one can define the J-product (¢1 =•= 45,)* by

By the same reasoning, given an (0:1, B1)-representation 45,; and an (042, ,82)-representation

us to define the gauge coupling of the Virasoro group.

From the physical point of view, the f-product plays a very important role because it allows

cm - (¢>< w)» = [(¢»··¢) >< wh + [¢>< (em ·~)]¤· = (m — k)(¢>< ~)¤·+·»·

em · (E >< w)» = [(¢····€) >< wh + [E >< (em - ~)]» = (Mn + B — k)(£>< w)¤·+»· (2-35)

em · (E >< (cm ·¢)]" = —[(¤+1)m+6 — k](E >

15 OCR Output

wedge product. Given n (a,B)·representations eff, 45,;, ···, ¢:, we can define the é-product

Now the existence of the é-tensor allows us to define the é-product which we also call the

symmetric.

to obtain an element of an (na + n — 1,·nB)'—module. Obviously the d-product is totally

= ¢kwk, ·· - wk', = wg, wh ··· w;,-k,_ -;,, -...-k”_, , (2.40)ihmk”

(w");. = (w=•=w=•=····•=w),,

allows us to define the n·th power (w"),. of an arbitrary (oz, B)°·representation wk by

to obtain an element of (al +a2 +···+ ol,. +n- 1, B1 +,8, +- ··+B,,)‘-module. The d-product

(2.39)w; = dl"""°w1;w;j ···w,,;,,

wx by

(cxg,B;)’-representation wu, ···, (a,,,B,,)'-representation wnk, one can define the d-product

to obtain an element of the (na, 115)-module. Similarly, given an (al, 51)*-representation wu,

».,l,...¤.,.4 (2-33)= ¢2"‘>"‘¢"’¢"” = ¢'°‘¢"’···¢"“'°"'°°""""""

(¢”)" = (¢=•·¢*--·=•·¢)"

or in general the n-th power (¢")" by

(¢2)k = * ¢)k = ¢m¢l¢—m,

allows us to define the square (¢°)* of an arbitrary (on, B)-representation ¢"

to obtain an element of the (al + az + ··· + an, B1 + ,8; + ··· + B,)-module. The d-product

(2-37)' '¢= ¢2;j...;.¢§¢$ ···¢Z,

can define the d-product ¢l by

representation 45,;, an (ag, B;)-representation 45,;, ···, and an (:1,,,5,,)-representation 45:, one

to obtain an element of the (al + az, B1 + ,82)-module. In general, given an @21,,6;)

16

(2-46) OCR Output¢··- · (¢"1)" = [(¤· Um + B + kl(¢'1)"'”‘

we {ind that (¢'1)" should form a (—a, -5)-representation,

= O, (2.45)

cm _ |;(¢-1)n¢k—n:| = _ m)6·0k-m

One can easily show that the inverse representation so defined is unique From

(¢—·1 * ¢)k = (¢-1)m¢= 50k—m ]:

element 50* of the (0,0)-representation,

of an (oz, B)·representation 45*, by requiring the d~product of 45* and (¢’1 )" to be the invariant

Virasoro transformation. Based on this fact one can define the inverse representation (45*)*

has a one-dimensional invariant subspace made of qi" = 50*, which is invariant under the

Now we introduce the inverse representation. Remember that the (0,0)-module lQ0_0)

totally antisymmetric.

to obtain an element of the (na + ,nB)'-module. Obviously the é-product is

= (wl A wg A · - · A w,,); (2.43)

wl = éawiawzj ···¢~w»ijmk

Similarly given n (ex, 5)*-representations wu, wu, ···, w,,;,, we can define the é-product by

(2.42)= —(¢0 A ¢1)"

¢" = (¢·1 A ¢¤)'° = (k - 2m)¢¥‘¢5''”‘

we have

which must form an element of an (na — 22,-1-l,n;3)-module. In particular when n = 2(}

(2.41)= (¢»1 A qs, A - - · A ¢,,)"

¢= é,,...,.¢;¢; — - - ¢: = A&+j+...+t‘¢;¢a - - ·¢:' ’

17 OCR Output

of the Virasoro group.

representation. Notice that the adjoint representation does not form a unitary representation

is nothing but the condition that one need to define a. positive-definite norm of an (a, B)

which is invariant under the Virasoro group. This shows that the unitarity condition (2.49)

||¢|I= (¢)¢» (2-51)° '“'"

for any unitary (oz,B)-representation 45* as

The existence of the Hermitian scalar product allows us to define the positive-definite norm

Lfn : L-,,,. (2.50)

Notice that the unitarity condition can be written as

For this reason we call an (a,B)·representation unitary if the condition (2.49) is satisfied.

B - B' = 0. (2.49)

or + of + 1 = 0

if and only if

This means that the Hermitian product (2.48) becomes invariant under the Virasoro group

em · ((x|¢)) = —[(¤ + ¤' + 1)m + B - 6'l(x)¢'“'""”

From the fact that must form an element of V(,,,_p), we have

(x|¢) = (x)¢ (2-48)"‘"

(X|¢) defined by

and X" be two elements of an (cx, B)-module, and consider the following Hermitian product

Finally we introduce the concept of the unitarity for the (o¤,,8)-representation. Let 45*

Notice that the inverse (az'] );, forms a (—a — 2, —,8)°-representation.

(ul-1 * (4.))], = (w_1)mwk-m = 6kO

wk by

subspace made of wk = 6;,°, we can define the inverse (w'1);, of a given (cr, B)°-representation

Similarly from the fact that the (—1,0)'-module Vijm) has a one-domensional invariant

18

2) The covariant derivatives transform covariantly under the gauge transformation,

(3.4) OCR Output6F,,,," = i(£ x F,,,,)

formation,

1) The field strength FW" transforms as an adjoint representation under the gauge trans

definitions we observe the followings:

the conventional one in that the coupling strength is characterized by ie, not e. With the

Notice that our definition of the field strength and the covariant derivative is different from

(3.3)D,,w;, = 6,,w;, — ie(A,, X w);, = 6,,w;, — ie(czm + B — k)A,,”‘w;,+,,,.

v..¢= 6..¢+ ww., >= 6.,¢+ eeua + um + n — k1A.:"¢* * '··*"’·’·

and the covariant derivative D,, by

(3.2)6,,A.,— 6.,A,,+ ie(2m — k)A,,”‘A,,""”‘,·· " "

1·*,_,= 0,,,4,- 0,,4,+ ie(A,, X ,4,)*’= * *

where e is the coupling constant. Next, we define the field strength FW" by

6`wk = —i(§>< w);. = —i(am + B — k)f'”w;,+,,,,

(3.1)6¢'° = i(§>< 45)* = i[(a + 1)m + B - k]§”‘¢'°

. . - $#4,.= —;[6.,€+ ¤¤(A»>

19 OCR Output

i‘EL·‘·’F,.JF,.J — i·— [— P + E. 4 2 4 A 4A= —

·*'£·isr'·F..JF.J - §< D.·w)» (3-5)

6(D»¢”) = i(£>< Dm)

20

B“= A“0 O

W=w+¢

rc = w-1

So with

(3.12) OCR Output< (¢°)" >= w‘60".

be clear from (2.5). At the vacuum we have

The vacuum breaks the Virasoro group down to the Cartan’s subgroup U (1), which must

(3.11)'° -k *= U6g= (/$60

of 45* has the minimum value with

expectation value. To obtain the desired symmetry breaking notice that the Higgs potential

try breaking, because it becomes singular if ¢" does not develop a non-vanishing vacuum

notice that the Lagrangian becomes physically meaningful only with a spontaneous symme

Clearly the Lagrangian describes a genuine gauge theory of the Virasoro group. But

kinetic term for the gauge fields.

group does not have any Cartan-Killing metric which one can use to define a gauge invariant

The introduction of the dimensional parameter is absolutely necessary, because the Virasoro

An important aspect of the Lagrangian (3.10) is that it contains a dimensional parameter rc.

the scalar products of two (-15, 0)·representations obtained with the invariant metric 5;+j

invariant, which must be clear from (2.33). The other terms are invariant because they are

(-3,0)-representation due to (2.38). So the first term in the Lagrangian is explicitly gauge

check the gauge invariance notice that F,.} forms a (1, 0)-representation, but (456)* forms a

sor d;,;, = 5;+,+;,° and the invariant metric di, = 6}+}, of the (-15,0)-representation. To

transformation (3.1), which is made possible due to the existence of the invariant ten

the Lagrangian is explicitly real. Furthermore it is manifestly invariant under the gauge

where we have introduced a scale parameter rc to keep mi" dimensionless. Notice that

21 OCR Output

In our approach we show explicitly how to construct a gauge theory of the Virasoro group

stance one can not construct a gauge theory of the Virasoro group with any known method.

eral. Indeed it becomes unitary only if the condition (2.49) is satisfied. Under this circum

the (oz, B)-representation the Virasoro group does not form a unitary representation in gen

but also does not admit any bi-invariant Cartan-Killing metric. To make the matters worse,

available till recently. The reason is obvious. The Virasoro group is not only non-compact,

At this point one may wonder why a gauge theory of the Virasoro group has not been

terized by the string tension in the real world.

has a. dimensional parameter oc, which obviously is related to the hadronic scale w charac

group U (1) of the Virasoro group. We emphasize that just like the string theory the theory

which guarantees the charge conjugation invariance of the theory under the unbroken sub

doubly degenerate. The degeneracy follows from the Hermiticity of the Lagrangian (3.10),

Only A,,° remains massless, and all the massive modes except the Higgs field cp become

(3.16)m = x/ip.

leaves a Higgs field cp whose mass m becomes

after the spontaneous symmetry breaking. Besides, the spontaneous symmetry breaking

mk = w|k| = |k|, (k;integer) (3.15)EE-(/E2 2 »\into the massive vector field B,," which has the following mass

where G,,,' is the field strength of Bf. From this we find that the gauge field .4,,* transforms

+highe·r-order terms, (3.14)

é(6u‘P)(6u‘P) — #’v¤’ + 5

““”L = _iGy—kG“Vk _ _g_€2w2(g)2B—kBk

the Lagrangian (3.10) can be written as

(3.33)* 23..= AJ + $(,;-)3.¢'· (3 7é 3).

22 OCR Output

where m, n, k are the indices of the Virasoro group and a, b, c are the indices of the associ

lT¤m,T¢»»l = fma».°'°Tex = f¤b°T¤m+m

[L·»»T¤··] = fm ¤»°"Ta = -aT.-.+.. (4-1)

[L,,,, L,.] = f,,,,,"Lk = (m — n)L,,,+,,

and the Kac-Moody group whose algebra is given by

The Virasoro-Kac-Moody group is the semi-direct product group of the Virasoro group

modifies the harmonic oscillator mass spectrum of the string.

scales, the scale which determines the harmonic oscillator spectrum and the scale which

theory after the symmetry breaking is characterized by two completely independent mass

massless gauge fields of the unbroken subgroup. Remarkably the mass spectrum of the

interpreted as the pseudo-Goldstone fields of the symmetry breaking, in addition to the

tower of massive spin-one fields and a finite number of the light scalar fields which can be

subgroup of the Virasoro group. The spontaneous symmetry breaking generates an infinite

symmetry down to H >< U (1), where H is a subgroup of the group G and U (1) is the Cartan’s

G [6,8]. The theory we present in this section automatically breaks the Virasoro-Kac-Moody

theory of the Virasoro-Kac-Moody symmetry associated with an arbitrary non-abelian group

colored string. So we extend the gauge theory of the Virasoro group to construct a gauge

Moody group for simplicity, of the color SU (3) should be a fundamental symmetry of a

The quantum electrodynamics requires that the loop group, or what we call the Kac

IV. GAUGE THEORY OF VIRASORO-KAC-MOODY GROUP

of the Virasoro group, which is all that we need to guarantee the unitarity of the theory.

breaking all the physical fields become explicitly unitary under the unbroken U (1) subgroup

The mechanism for the restoration of the unitarity is simple and clear. After the symmetry

of the Lagrangian (3.10) becomes explicitly positi·ve·definite. This must be clear from (3.14).

the gauge fields of the Virasoro group do not form a unitary representation, the Hamiltonian

in the absence of a positive-definite bi-invariant metric. Notice that, in spite of the fact that

23 OCR Output

(l)A,Alah; = A:-k= ¢c—k_

which we choose to be H ermitian,

As for the Kac-Moody group let A,,°" be the gauge potential and ¢°" be a scalar multiplet

an (oz, B)-representation.

which confirms that FW" and D,,¢" transform covariantly as an adjoint representation and

(4-4)6e(D»¢") = i[(¤ + Um + B — kl£”‘(D»¢"“”`)»

5€F,,,.* = i(2m — k){”‘F,,,,'°`”`

we have

representation of the Virasoro group. Remember that under the gauge transformation (3.1)

group first, and let Auk be the Hermitian gauge potential and ¢" be a Hermitian (0:,B)

theory of the Virasoro group and the Kac-Moody group. So we consider the Virasoro

Any gauge theory of the Virasoro-Kac-Moody group must be simultaneously a gauge

where (fm, ¢x“”‘) are the infinitesimal parameters of the Virasoro-Kac-Moody group.

¤mn°,6&¢ck = u¤m(T)bCk¢bn = fbca¤m¢bk-m

°'°`”'6£¢°k = ·i{”‘(L,,,),,’°¢d' = i[(cz +1)m + B — k]f”‘¢

representation, we obtain the following product representation 45

matrix which acts only on the indices of the Virasoro group. When T, becomes an adjoint

of the associated group which acts only on the indices of G and Tm is an infinite-dimensional

Notice that T,,,, consists of two matrices T, and T,,,, where T, is an arbitrary representation

(4-2)(T¤¤•)B¤“ = (T¤)Ba(Tm)••k = (T¤)B°6m+••

(L,,,),,’° = [(a + l)m + ,8 — k]5,,,+,,

group T,,,, given by

trary (oz, B)-representation of the Virasoro group L,,, and a representation of the Kac-Moody

Virasoro-Kac-Moody group allows what we call the product representation made of an arbi

ated group G. Notice that here again we exclude the central extension for simplicity. The

24 OCR Output

further require

Virasoro transformation (3.1) and the Kac-Moody transformation (4.6) simultaneously, and

To construct a gauge theory of the Virasoro-Kac-Moody group we now consider the

as we will see in the following.

gauge transformation. The non-linear realization has a very important physical implication

under the gauge transformation (4.6). Obviously this is due to the non-linear nature of the

group. In particular this shows that it is D,,¢°", but not D,,¢°", which transforms covariantly

The result shows that F,,,,°" and D,,¢°" form an adjoint representation of the Kac-Moody

6a(D»¢°") = f·¤»°¤°”‘(D.·¢°’°`”‘) (4-9)

#”65FVCk = f°bcaamFvbk—m

mation (4.6) we have

The justification of the above definition follows from the fact that under the gauge transfor

25 OCR Output

6aF¤“yck = i5ma¤nFpvk-m + f°bca¤mF1“ybk—m•

6£F',_,'* = i(m - k)§”‘F,,,‘°""”‘

manifest in the definition. From (4.13) we find

Notice again that the semi-direct product structure of the Virasoro-Kac-Moody group is

+gf°bcA“¤mAVbk-m •

@,,4;* - 0,.4,;* 4 e4(m - 1¤)(.4,,···A_,=’····· - .4,···A;’·····)

(4-13)*·m*·+1... 4./1.4.1

ek ek ekmFwd = aufiv_ ayjiu+ fm °"A“Jiy¤n + fam ”dAp¤mAvn)

Moody group we define the field strength of A“°" by

ture of the Virasoro-Kac-Moody group. To obtain the field strength of the Virasoro-Ka.c

to A,,", the gauge transformation (4.12) explicitly incorporates the semi-direct product struc

This must be clear because, up to the normalization factor ·i% which can easily be absorbed

tantly it tells that (A,,",/1;*) forms the gauge potential of the Virasoro-Kac-Moody group.

The result shows that ,4,f" forms a (0, 0)-representation of the Virasoro group. More impor

(4.12)s,,,.4;'· = -§a,,a=’· 4 i§ma‘”"A,,"“"‘ 4 ;,,,¤¤·*··A,_°’·····.

5€A,f* = i(m — k)§”‘A,f"'”‘

and obtain from (3.1), (4.6), and (4.10)

(4.11)Ay = .4,:* - §.4,,···¢=*····,

potential A,,‘* by

group. To introduce the gauge potential of the Virasoro—Kac-Moody group we define a new

Notice that the last equation tells that ¢°" forms a (-1, 0)-representation under the Virasoro

6‘€¢ek = _,ik£m¢ck—m.

‘ In 6,4,= -T(a,,g)4s·"· 4 4(m - k); A,, (4.10)-7Il =’·1 * "·='·

5,;A,," = 0, 6,;¢" = 0

26 OCR Output

L = L1 ·i· Lg

consider the following Lagrangian [6]

Kac-Moody group. Let 45* be a Hermitian (-1, 0)-representation of the Virasoro group, and

With the preliminaries we can easily write down a prototype gauge theory of the Virasoro

representation as well as the Kac-Moody group as an adjoint representation.

This indeed shows that F“,,°" transforms covariantly under the Virasoro group as a (0,0)

,__,,,,(4.17)&&i¤'*=’= = f,,,=a····1·*='·····.

6£F‘W°" = i(m - k)§"‘F,_,°"""

and find

,,,, (4.16)= F‘,°" + %F,,,,”‘¢°'°`”‘

I- f-”I HI - -”I F...r= Fw + -[A.. — A. .

6eF»·»°" = i(m — k)€”`F»·°"`”` + E[(6..£”')(1l¢°"‘”')·-(6»£”‘)(D»¢°'°`”‘)]

From (4.7) and (4.10) we have

the Virasoro-Kac-Moody group is a semi-direct product of two simple groups.

of the Virasoro·Kac-Moody group allows two coupling constants e and g. This is because

Kac-Moody group (again up to the normalization factor i§). Notice that the gauge theory

group, because it shows that (Fwk, 1?’,,,,°") forms an adjoint representation of the Virasoro

The result tells that (FW", F`,,,,°*) indeed forms the field strength of the Virasoro-Kac-Moody

27

Fw F.»· ;(D»¢ )(D»¢)1 ~ c_ - 1 - c_ - k°'° "°"

C = ·7F»·· Fw · ;(D»¢ )(D»¢ )1 -k k 1 -1: k

Now, expanding the Lagrangian around the vacuum, we obtain

(4.20) OCR Output¢°'° = ;¢°6`¤’° + cpd.

(4.6), and (4.10). To obtain the mass spectrum after the symmetry breaking we let

p° and U (1) is the Cartan’s subgroup of the Virasoro group. This must be clear from (3.1),

U (1), where H is the subgroup of G which forms the little group of the adjoint multiplet

Notice that the vacuum breaks the Virasoro-Kac-Moody group spontaneously down to H ><

= #60. = w6¤ (4-19)**°'·*

Lagrangian

To discuss the physical content of the theory we choose the following vacuum for the

precludes any polynomial potential for the scalar fields from the Lagrangian.

does not contain any Higgs-type potential for the scalar fields. In fact the gauge invariance

miticity of the field strengths and the covariant derivatives. Remarkably the Lagrangian

forms a scalar product under the gauge transformation. The reality follows from the Her

it is explicitly real. The gauge invariance follows from the fact each term in the Lagrangian

mation of the Virasoro-Kac-Moody group described by (3.1), (4.6), and (4.10). Furthermore

defined by (2.19). Notice that the Lagrangian is explicitly invariant under the gauge transfor

where dw, = 5;+j+;,° is one of the invariant tensors (of mixed type) of the Virasoro group

4 p. 21 ¢—i—`° ci' c` 1 —i—` °' d ' c' ·(·) ’F,.., F,.} ’·—(%) ’(D»¢ )(D»¢’)»

L_ ·1 —·;dw=(j) Fw FW · ;d

28 OCR Output

the pseudo-Goldstone fields of the symmetry breaking.

guarantee the masslessness of the scalar fields. This implies that the scalar fields become

is Hat. Notice, however, that after the symmetry breaking there is no symmetry which can

fact that the vacuum of the theory is degenerate, because the potential of the scalar fields

Nambu-Goldstone fields [11] of the symmetry breaking. The interpretation follows from the

Furthermore the scalar fields

29

"51 = —iF»»"‘F»»" - i(D.·

30 OCR Output

an infinitesimal reparametrization of the string given by 56 = one has

equivalently by its Fourier components ::"”‘, where 0 is the string coordinate of S1. Under

in the 4-dimensional space-time described in Fig.2. We can denote the string by :¢:"(9), or

associated with the Poincaré group. To explain this consider a propagating closed string

the string in the 4-dimensional space·time, which is described by the Kac-Moody symmetry

because in the string theory the gravitation must be generated by the Poincaré invariance of

But in the string theory this is not a proper way to introduce the gravitation. This is

one can easily make the theory generally and Virasoro-Kac-Moody invariant.

transformation. So with the minimal coupling of the gravitation to the Lagrangian (4.18)

(0,0)-representation, which forms a one-dimensional invariant subspace under the Virasoro

because the space-time metric gw, can always be viewed as the zeroth component of a

ance without compromising the Virasoro-Kac-Moody invariance of the theory. This is

and make it generally invariant. Notice that one can always achieve the general invari

troduce the gravitation is to add the Einstein-Hilbert Lagrangian to the Lagrangian (4.18)

So far we have not introduced the gravitation in the theory. The simplest way to in

V. THEORY OF CLOSED STRING

G as the grand unification group and H as the electromagnetic and color subgroup of G.

So physically one could regard L; as a grand unified theory based on the string, by identifying

[5]. Here again the spontaneous breaking of the Kac-Moody group down to H is inevitable.

own right, which is completely different from the one proposed by Cadavid and Finkelstein

realized. The gauge theory of the Kac-Moody group described by L2 is very interesting in its

independently describes a gauge theory of the Kac-Moody group which is again non-linearly

group and the U(1) subgroup of the Virasoro group. However notice that the Lagrangian L;

So in this case the Virasoro-Kac-Moody group is explicitly broken down to the Kac-Moody

·=·’·=* =·*·*cz = —}F,..F,..—§,.¢>.

31 OCR Output

= —nM“yyn+”[LM, Mm]

= -np,,,,,+,,[L»·»p»·»] (5.4)

= (m — n)L,,,+,,[LM, Ln]

m»»».z>»»]

= -—i6`,,"6,,,"{ppm ==""]

Lorentz tensor made of the d-product, of z"”‘ and p,,,,,. From the deiinition we have

Notice that Lk is a. Lorentz scalar made of the f-product, but M,,,,k is an anti-symmetric

= i(¢»”‘1¤»»+···—¤¤»’”1>..»·+»·)·

= idkm”(xumPv¤ " $vmP#¤)

Mawk = i(zu * Pvlk ' i($v * Pu)k (53)

= _1mzP.-puk+m

Lk = i($u X Pu)k = ifkm”$“mPu¤

external rotation, that we can construct with pp;. and x"". They are given by

two angular momentum operators of the string, Lk for an internal rotation and MW; for an

the translation of the string in the 4-dimensional space-time. Now we show that there are

which should form a (0, 0)‘—representa.tion of the Virasoro group. Obviously pp;. describes

ppg = = -1:6,*,

operator pu;. corresponding to z"k is given by

This shows that z"* form a (0,0)-representation of the Virasoro group. The momentum

61** = i(m - k)f"‘z"""”

Compoucntwisc this can bc written as

(5-1)= ·£(9)

5m" = z“’(0) — z"(0) 2 z“’(0’) — x"(0) — [:z:"(0') — z"(9)]

32 OCR Output

so far.

Kaluza-Klein theory may be more closely related to the string theory than we have thought

string field theory. This is a remarkable observation which implies that the 5-dimensional

Klein theory [13,14], with a trivial generalization, could be interpreted as a model for a

group and the grand unification group. This means that the original 5-dimensional Kaluza

prototype gauge theory of the Virasoro-Kac-Moody symmetry associated with the Poincaré

4-dimensional space-time. Indeed after the dimensional reduction the theory becomes a

dimensional Einstein·Dirac-Yang·Mills theory and to make the dimensional reduction to the

group. Now we show that a natural way to construct such a theory is to start from the 5

Virasoro-Kac-Moody symmet1·y associated with the Poincaré group and the grand unihcation

propagating colored string in the Ldimensional space-time, we need a gauge theory of the

generate the gravitational interaction in the string theory. This implies that, to describe a

group, at least in the point limit of the string. So it must be this symmetry which should

space-time is generated by the Virasoro-Kac-Moody symmetry associated with the Poincaré

The above discussion makes it clear that the propagation of a string in the Ldimensional

generates the space-time propagation of the string.

coordinate and momentum generates the internal propagation, while the d-product of them

generators of the Virasoro group. It is really remarkable that the f-product of the string

without ever changing its space-time position. This rotation, of course, is generated by the

of the internal space S1 of the string. This is because the string could rotate in space-time

interpret them as the internal angular momentum generators which generate the rotation

time. Notice, however, that L,,, form the generators of the Virasoro group. But here we can

generators of the string which generate the boost and rotation in the 4-dimensional space

group. This means that M,,,,,,, is indeed the space-time (i.e., the external) angular momentum

Clearly p,,,,, and M,,,,,,, describe the Kac-Moody symmetry associated with the Poincaré

lM»¤·m» Mcdnl = _’Ip¤M»¤m+¤ + ’h»BMv¤m+¤· "7·»¤M»¤m+¤ + ’l·»¤MuBm+¤·

[v»···» Mm] = m.¤p¤·»+,. — nnsz¤¤».+..

33

A: = A: — ercA,,¢° (5.9) OCR Output

In the block·diagonal basis, however, the gauge potentials become (Au", ¢“) where

= D,,¢°— {80A,f = D,,¢°.

F»s° = @»¢° + 9f·»°A»°¢°· {6vAf (5-8)

F,,,° = 8,,/1,,° - 8,,A,,° + g f,;,°A,,°A,,

field strengths of the potentials are given by

Let the gauge potentials of G in the coordinate basis (6,, ® 65) be (Au", ¢“). Clearly the

of S1.

Notice that here we have identified :1:5 with x0, where 6 is the dimensionless string coordinate

7 O 0 ¢°

we have

FW = GMA., — 8,,/1,, — e(A,,6gA., — A,,8,A,,),

(5.6)[5,,,5.,] = —ex.F,,,,65, [6),,65] = e(6gA,,)65

6,, = 6,, — ercA,,85

characterizes the size of the string. But in the block-diagonal basis (6,, ® 65) where

where e is the coupling constant of the Virasoro group and ac is a. scale parameter which

crc¢°A,,'YMN = (5.5)

·y + e°1c°¢°A A, ercA ¢° uv u u

general metric on M5 can be written as [14,15]

string S1 and the 4-dimensional space-time M4 In a coordinate basis (6,, EB B5) the most

5-dimensional manifold M5 which itself forms a fibre bundle B(M4,S*) made of a closed

dle P(M5,G), whose fibre is the grand unification group G and the base manifold is a

Let us start from the Einstein-Dirac-Yang-Mills theory based on the principal fibre bun

34

6e(D»¢°) = ·(@o£)D»·¢° - £6a(D»¢°)·

0€F,_,° = -56,F,; (5.14) OCR Output

0£F,,., = (6,§)F,,,, - {0,F,,,,

Restricting the general coordinate transformation to the Virasoro transformation we obtain

6¢° = —£°‘@a¢°· %(@e£°)}l-.5 - ¤(@e£°‘)Aa¢°— (6¤£)¢°—£@¤¢°M: = —

35

6a¢ek = + f°bca¤.m¢bk-vn,

(5.18) OCR Outputsa,-A,;='· = -§a,,¤=* + 5-;ma·=···A,,'··"· + j,,,·=¤·*··A_,°'·····

6;),,,* = 0, 6,;A,," = 0, 5;¢" = 0

Componentwise this can be written as

6,;¢° = ——i6,a° + f,(,°a°¢°.

(5.17)»s5-A; = -§é,.¤ + ;.,,=5·A,}·

6;*7,,,, = O, 6`,;A,, = 0, 6,;¢ = O

Under the infinitesimal gauge transformation of G generated by 6Z(:v, 0), we have

Virasoro group, respectively.

which shows that Fwk, Fwd, D,,¢°'° forms a (1,0), (0,0), (-1,0) representation of the

6e(D»¢°") = -ik€”‘(D»¢°"`”‘).

(5.16)6`€F,,,,°" = i(m — k)§”‘F,,,,°"’”‘5€F,,," = i(2m — k)£”‘F,,,,"`”‘

have

exactly the gauge potential and the field strength of the Virasoro group. From (5.14) we

of the Virasoro group, respectively. More importantly it shows that A,} and F,,,,become"The result shows that ·y,,,,", .#l,f", 48*, 45** forms a (0, 0), (0, 0), (-1,0), (-1, 0) representation

s£¢¤'· = -5kg···¢='····.

=*=····.s€A;’· = e(m - 1¤)g···A,_

(5.15)6£¢" = —ik§”‘¢'°“”‘

, 6eA»= ··:@»£+ =(2m — k)£"‘A»_,,, "1 '“ '°

6€'Yuvk = i(m· " k)Em°`Yuvk-m

Upon the Fourier decomposition (5.13) can be written as [7,8]

36 OCR Output

group.

which again confirms that Fw" and F,,,,°" form the field strengths of the Virasoro-Kac-Moody

(5.21)5&F“_,='· = i§m¤·*··F,,_,’···*· + j_,=¤.···1¤,__,'·'·····,

from (5.19) we have

Notice that this is precisely the field strength that we have introduced in (4.13). Indeed

,,_,= F} - e»1·*···4»¤'··*··.

(5.20)¤'·"·*···+,5,,,, ,,,_A;A,]

ck ck,Z _ 8VA[|+ GHApnJ§`V¤n + fdfll nckAp¤mAv”)

group are given by Fwand I7W, where Fw‘k is defined byk `°"symmetry associated with G. The corresponding field strengths of the Virasoro-Kac-Moody

be absorbed to Au", Au" and /1,,** form the gauge potentials of the Virasoro-Kac-Moody

space-time. Secondly (5.18) tells us that, up to the normalization factor ig which can easily

the non-linear realization is nothing but the gauge transformation in the higher-dimensional

non·linear realization discussed in the last section. Here we see clearly that the origin of

that we have discussed in (4.6). This provides the natural geometric interpretation to the

Moody transformation. Notice that, with nc = 5%, this is precisely the non·linear realization

nevertheless admits the unique covariant derivative which transforms linearly under the Kac

transformation is non-linearly realized. The scalar rnultiplet 49* transforms non—linearly, but

with G. But there are two points to be noticed here. First (5.18) tells us that the gauge

Obviously the gauge transformation corresponds to the Kac-Moody symmetry associated

6.; = r..:¤··"

37

V5\I1 = (E6, — EQEMN - g¢°T,)\Il, (5.26) OCR Output-- 1 1 · MN

. Vu`? = (au ” $9:4 EMR ' 9Au T¤)\I’1 mv

and gauge covariant derivative given by

where R5 is the 5-dimensional curvature of the connection (IL, and {7M is the generallyM"-

(5.25)+g[(=T¤yMV,,,=1») - (¤iwMvM¤1»)*] - moninn

1 - 1 ‘¤ = _? {MRS + z"””‘*'°°"`MP°"M=“

Einstein-Cartan-Dirac·Yang-Mills Lagrangian

choose 1;),;);, = diag (—,+,+,+,+). With this we consider the following 5-dimensional

where 7,; is the 4-dimensional ·y-matrices, 15 = l?¥e"""" 7,1,7,7, = ——i·y6·yi··y;·y§, and we

{‘m»1n} = -2*1mv, (5-24)

’m = ('Y;2¤i'Y5)

dimensional ·y-matrix yy we let

AM° = (A,,°,¢“) and to the connection GL= (Q,,,$2) in M5. As for the 5EN M&`MN

fermions \I! of the grand unification group G which couples minimally to the gauge fields

Notice that ef becomes the vierbein of 7,,,,. Next consider an arbitrary representation of

(5.23)WM =. 0 l e"P 0 ¢

Clearly we have 1MN = eM"eNL. But in the block-diagonal basis (5.6) we have

(5.22)WM =. *A l e" °'°¢ "2 0 ¢

Lorentz indices. In the coordinate basis (6,, EB 65) the fiinfbein is given by

gauge potential of the Lorentz group in M5, where M and N represent the 5-dimensional

eMM of the metric (5.5) and an arbitrary linear connection QLMN which plays the role of the

Now we include the fermions to the theory. To do this we first introduce the fiinfbein

38

6,;\I( = a°T,\I¤. (5.31) OCR Output

0,,-,e,,¤· = 0, s&n,,= 0, sio= 0m’i”°

group. Under the gauge transformation of G we have

Notice that the last equation shows that *1** forms a (0,0)-representation under the Virasoro

5\Ir" = -§“”‘3,,\I¤'°’"‘ + i(m — k)§”'¤I¤"“”‘.

6Qb?1Vk = _£¤m6¤QIlZ}Vk-m __ imfamnuygk-m _ ik£mQAZNk—m

'°"’”‘N°°'’”'w°6e,,= —(é,,f)c,,,`”‘—§“”‘8,,,e,,""‘”‘ + i(m — k)§"‘e,,

which can be written, after Fourier decomposition, as

6¤I# = —£"‘6,,¤Ir — {B6)?

(5.29)som = -g·=a,,,o””° - (a,g·)sz,,*”’ - (8,§)Qm° -ga,o””°

m..= —

39

('YLEMN + EMNWL) = ‘Y[z.‘7M*m] = €r.M1vpq (5.37) OCR OutputEP Q.

where eLMNpQ is the 5-dimensional e·tensor with eaim = 1 and we have used the identity

M ¤M”P°-{-%[(\i}'yVM\I’)—(\i’1MVM\I’)l] - m0\l*\I! - }€,,MN,q0·i»2w }, (5.00)

1 1 C c Cs = —?¢ {@·(Rs + CMLMCNLN - CMLNCNLM) + ;·‘YMP‘YNQFM1v FPQ

reduce it to the following Lagrangian

basis. Inserting this to the Lagrangian (5.25) and removing the total divergences we can

commutation coefficients of the basis, which does not vanish when we use the block-diagonal

nection FLwhich is different from the torsional covariant derivative VM, and fMN'“ is theMN

Notice that here VM is the torsion—free covariant derivative defined by the Christoffel con

S5 = VMCN_ VNCM+ CMLCN_ CMLCN_ fMNCL,MN MLN NLM LMNMN

scalar curvature of the contortion given by

where R5 is the 5-dimensional scalar curvature of the metric ·yMN, and S5 is the 5-dimensional

R5 = R5 + S5

5-dimensional contortion tensor. With the decomposition (5.33) we have

where PLis the 5-dimensional Christoffel connection of the metric ·‘yMN and GLis theENM]?

(5.33)nL’°”° = 1*,,"”° + 0,,*-***

we decompose the connection into

dent variable. To express the Lagrangian in terms of the dynamically independent variables

Notice that in the Lagrangian (5.25) the connection QLMN is not treated as an indepen

5.;·’~I!" = a°”T.\I!"'”‘.

s&e,,·¤= = 0, 6..0,,**** = 0, 0,,-simk = 0

Componentwise this can be written as

40 OCR Output

+12wG(‘Y’¤,»‘I’)° + g"G(‘i"f·”*`I’)2 >·- - - ·¤ .W-- ¢F‘I’¤""‘I’ +§y¢"¢°[(‘I’T¤‘I’)·

42 OCR Output

161rG 4rc- _k 1 Lu) = ·bL{R'° + -5(¢`3)'°`”"”""'9‘”“9°°“[(Dv9»¤)"(Dv9·»¤)" · (Dv9»··)‘°(De9¤¤)"]

dt

grangian L5 down to M4, we obtain

After the dimensional reduction of the 5·dimensionaI Einstein-Cartan-Dirac·Yang-Mills La

Now we are ready to write down the gauge theory of the Virasoro-Kac-Moody group.

the Virasoro group.

This shows that _q,,,," and 1b" forms a (-1,0)-representation and a (i,0)-representation of

6·¢k = _£¤m6¤1/)k—m + %cm£¤mA¤n¢k-m-n + + 1)m _ k]€f!I,¢’k·7D·

, (5.46)—-ik§”'g,,,"“”‘

,Pp “69*1,]: = _(飤)mg¤vk—m _ (évfcqmgak-vn ___ iemE¤mA¤ngvk—m—n _ Eamaaglwk-m

or componentwise,

51/¤ = —£“6..=/¤ + §¢(@a£°‘)A¤¢ + }(8¤E)¢—£@o¢,

(aeflypv ··€609»·· (5-45)

6y.»» = -(6..£°)a¤»— (@»£°)y»¤—£°@¤sn~ - ¢(@¤£°)A¤y»»

the metric gw, not 7,,,,. Notice that after the conformal transformation (5.43) one has

Notice that here Vu represents the generally and gauge covariant derivative determined by

DMP = (6c + 7%‘)'P·1 6 ?

V C Vnib = lan ' Zpuaaé " eAu60 + T(60Au) ' 9Au Tcl¢'1 * - ap

DM = [Gp · ==A»@e —¢(@vA»)]¢

Dew = (6a — %)y»~6 fDugab = lan ”¢A»8v —€(89An)l9¤B

OCR OutputFava = %9ap(Du9vB + Dvgub ” Dbguvl

43 OCR Output

not, in fact can not, contain any Higgs-type potential for the scalar multiplets 49and ¢* °"

string field theory which describes the grand unification. Again the Lagrangian (5.47) docs

we have discussed in the last section. So Cm should form another essential part of of the

absence of the fermion source, precisely the gauge theory of the Kac-Moody symmetry which

of the Kac-Moody symmetry associated with G. Indeed in this limit LU) describes, in the

in the limit where the 5-dimensional metric (5.5) becomes flat LO) describes a gauge theory

Virasoro-Kac-Moody symmetry associated with the grand unification group. Furthermore

essential part of the string field theory. Now, notice that LU) describes a gauge theory of the

size is that any propagating string must have this symmetry, and that LO) should form an

been pointed out in a fundamental paper by Dolan and Duff [17]. But here what we empha

theory should possess the Virasoro-Kac-Moody symmetry of the Poincaré group has first

nothing but the Kaluza—Klein theory [7,14]. The fact that the 5-dimensional Kaluza—Klein

Virasoro-Kac-Moody symmetry associated with the Poincaré group. In fact Cm describes

ant under the Virasoro-Kac-Moody group. Notice that Lu) describes the gauge theory of the

of the Lagrangian is obvious. Indeed each term in the Lagrangian is manifestly gauge invari

mation group and the Lorentz group) and the grand unification group. The gauge invariance

group associated with the Poincaré group (more precisely the general coordinate transfor

Clearly the Lagrangian (5.47) describes the gauge theory of the Virasoro-Kac-Moody

Again, without the torsion, all the quartic fermionic couplings in Lp) should disappear.

_ - +12rG(¢¤'¢)'(¢¤..»¢)+ ;*G(¢‘r"1s¢)" '(¢vns¢)- _ - 3 .. “’" '

-Qk-l—m—nh&mDhn1Z’Uwl _ eNk-m—nFm_$Upv1))néw)p(0V&)(¢)&¢#v(/

im-%k-m-n0(¢)¢m75¢n + %g(¢—%)k—l—m—n¢d[(1/;mTc¢n) _ (1I,mTc¢n)f]

7II D — fn fl §[—*1 - iw ¥>w — (Dm ¢ 1Z _ - ° - · 0 —• -'HI·f\ -* ''* ‘· '*(2) = _(\/§)—k[ ;‘_¢k—m—n—p-qgu.vmgaBnIE¤“¤cpj;°vvp¤q + g(¢—2)k—l-m—ngpvl(‘Dp¢)¤n(D“¢)m

99»¤»¤:9»»+7~¢(¢)1 ·7I\•Yl·· D 3 —·•”I-fl TTI fl °°°'°‘° "'“”"°‘° F'F" + (¢ 2)* I ""'(D¢) (D¢)

44 OCR Output

To discuss the geometric symmetry breaking notice that L5 has the unique vacuum

spectrum of the string.

the “hadronic” scale which is responsible for the fine structure of the harmonic oscillator

mass scales, the Planck scale which is responsible for the harmonic oscillator spectrum and

unification group. So the mass spectrum of the theory is characterized by two independent

which is responsible for the spontaneous breaking of the Kac-Moody symmetry of the grand

another mass scale which comes from the vacuum expectation value of the scalar multiplet

spectrum of the Virasoro group in the Lagrangian (5.47). On the other hand the theory has

gravitation. This means that the Planck scale sets the scale of the harmonic oscillator

details, that the Lagrangian (5.47) includes the Virasoro·Kac·Moody generalization of the

Notice that the difference between the Lagrangian (5.47) and (4.18) is, aside from the

pseudo-Goldstone fields.

tower of massive spin-two, spin-one, and spin-half fields as well as massless gauge fields and

grangian (5.47) we show explicity how the geometric symmetry breaking creates an infinite

spin-two particles as well as the massive spin·one particles. Starting from the effective La

markably the symmetry breaking that we discuss in this section creates a set of massive

broken by the underlying geometry of the theory, not by any ad hoc Higgs potential. Re

mechanism [19]. In this geometric symmetry breaking the gauge symmetry is automatically

discuss in this section is of another type, which we call the geometric symmetry breaking

gauge fields to the massive vector fields. The symmetry breaking mechanism which we

anism [18] the symmetry is broken by the Higgs potential, which transforms the massless

spontaneous symmetry breaking mechanism in gauge theories. In this Higgs·Kibble mech

One of the most important developments in theoretical physics is the discovery of the

VI. GEOMETRIC SYMMETRY BREAKING

the Lagrangian.

The gauge invariance simply forbids any polynomial potential for the scalar multiplets from

45

(@°— %)< G x U (1), where U (1) is

< A,,°* >= 0, = 0. (6.2)

*< *7,,,,k >= 1],,,,60, < A,} >= O, = 5;;

Componentwise the vacuum is described by

< A,,“>= 0, = 0. (6.1)

=n,,.,, =0, =1

46 OCR Output

written as

Notice that, to the lowest order, the general coordinate transformation (5.13) can be

independently can not represent the J, = :l:2, :l:1,0 states of pw"

Virasoro group by the vacuum (6.2). Again it must be clear from (6.7) that hw", Au", ep"

describe the massive spin-two fields generated by the spontaneous symmetry breaking of the

which we need for the self-consistency of the theory. This shows that pw" defined by (6.7)

that the last two equations of (6.8) can be interpreted as the constraint equation for pw",

This is precisely the Fierz-Pauli equation for a massive spin-two field pw" [20]. Notice

(6—8)(B2 - %)¤.»" = 0, 6·p»¤" = 0, ess" = 0- (¤ at 0)

Now without any gauge condition we find that for n gé 0 (6.4) can be written as [7]

$ (6.7)... pw" = h“y” + i;(8,,A,," + 6,,A,,") - ;;6,,8.,

47

(@»A»° - @»A»°)(@»A»° - @»A»°) —§(6»

48 OCR Output

around the vacuum (6.2) we obtain

becomes the invariant subgroup of the spontaneous symmetry breaking. Expanding Lim

Notice that the vacuum (6.2) breaks the Kac-Moody symmetry of Bm down to G, which

Now, we discuss the spontaneous symmetry breaking of Cm in the Lagrangian (5.47).

itself, automatically generates the desired symmetry breaking [19].

breaking mechanism the geometry, more precisely the scalar curvature of the unified space

often contains a large number of arbitrary coupling constants. In this geometric symmetry

is that it breaks the symmetry without the introduction of an ad hoc Higgs potential which

absorbing these Nambu-Goldstone fields. The virture of the geometric symmetry breaking

group for n 36 0. Thus the gauge fields of the Kac-Moody group h,,,,” become massive

scalar fields cp" remain the Nambu-Goldstone fields of the spontaneously broken Virasoro

Nambu-Goldstone fields of the spontaneously broken Kac-Moody group for n 36 0, while the

Kac-Moody group. So the gauge fields of the Virasoro group A,,” themselves become the

4-dimensional general coordinate transformation and the U (1) subgroup of the Virasoro

Virasoro group, which must be obvious from (6.9). This symmetry is broken down to the

Kac-Moody generalization of the 4~dimensiona.l general coordinate transformation and the

that the gauge symmetry of the unified theory is not just the Virasoro group, but the

tower of massive spin-two particles in the 5-dimensional Kaluza-Klein unification. Notice

So far we have shown how the geometric symmetry breaking in LU) creates an infinite

is to show how the quantum correction can indeed create a small mass to the dilaton.

symmetry breaking, should become a pseudo-Goldstone field. An interesting problem here

the dilaton to remain massless. So the dilaton, which is nothing but the Higgs field of the

dilaton field

49 OCR Output

[8]

vanishing expectation value. So the Lagrangian (5.47) in general has the following vacuum

dimensional reduction, however, there is nothing which can forbid ¢>“" to acquire a non

Certainly the vacuum (6.2) is unique from the 5-dimensional point of view. After the

the unbroken subgroup U(1) of the Virasoro group.

Lagrangian (5.47), which guarantees the charge conjugation invariance of the theory under

massive modes are doubly degenerate. The degeneracy follows from the Hermiticity of the

fields of the unbroken subgroup U (1) x G, the dilaton, and the scalar multiplet ¢"°. All the

by the Planck scale. Furthermore it contains the massless fields made of graviton, the gauge

massive spin-two, spin-one, and spin-half fields whose mass scale is completely determined

vacuum (6.2) the mass spectrum of the Lagrangian (5.47) consists of an infinite tower of

scalar multiplet ¢“° which forms an adjoint representation of G. This shows that with the

tower of the massive spin-half fields ·¢", the massless gauge fields C,,°° of G, and the massless

S0 Bm contains an infinite tower of the massive spin-one fields C,,“” (·n. 36 O), an infinite

(6.17)mq} + (%)’1/2'”1/v".

the fermion mass term can be written as

(6.16)”——» ei°""‘¢”, tan 0,, = gmc,

Notice that after the chiral rotation for the fermions 1b"

(6.15)C,,°” = A,,°" + if8,,¢°” (n gé 0).

npUdo = Alo

where H ,,,,°" is the field strength of C,,°" given by

+higher-order interactions,

(6-14)él$"”*r“(@..¢)" — (@»¤ll)`”*1"¤/¤”l ·—

50 OCR Output

responsible for the harmonic oscillator spectrum of the string and the elementary particle

fields is characterized by two completely diferent mass scales, the Planck scale rc which is

as the Planck scale. So with the vacuum (6.18) the mass spectrum of the fermions and vector

Notice that the mass matrix of the vector fields is identical to (4.23), as far as we identify u

Here the superscript u (in am "‘ cz ¤£(2) = "YHW Huv" §M¤imbnCu Cub'. " K(8u‘P 0)(6#'P Ol1 c- 1 kCk

we have

¢u (6.19)°’° = °50" + = n,,,,60, < A,/>= 0, < 45* >= 50"‘

51 OCR Output

We also choose the 5-dimensional 7-matrix given by (5.24). But it is more convenient to

BM = (Bm X)

‘I’M° = (wif. ‘I’°) (7-1)

M(QHQQ)QMN = IFINMIV

by (5.22) and let

to the internal symmetry S O(2). In the coordinate basis (6,,® {6,), we express the fninfbein

nection OM, one complex gravitino \I!M1 +i\I1M°, and a gauge field BM which correspondsMNThe 5-dimensional supergravity multiplet consists of the fiinfbein eMM , the linear con

viewed as an effective field theory of superstring.

the simplest supersymmetric gauge theories of the Virasoro-Kac—Moody group which can be

group [24]. Clearly the theory that we obtain from the 5-dimensional supergravity is one of

Virasoro-Kac-Moody group which is associated with the 4-dimensional N = 2 super-Poincaré

the dimensional reduction, the 5-dimensional supergravity reduces to a gauge theory of the

symmetric generalization of the 5-dimensional Einstein’s theory [23] and show that, after

In this section we consider the 5-dimensional supergravity which is the N = 2 super

VII. SUPERSYMMETRIC GENERALIZATION

become the pseud0·Goldstone fields of the symmetry breaking.

which can guarantee the scalar fields to remain massless. So the scalar fields ¢p"° is likely to

the scalar fields ¢°‘°. Notice, however, that after the symmetry breaking there is nothing

follows from the fact that the vacuum (6.18) is degenerate, because there is no potential for

be interpreted as the Nambu-Goldstone fields of the symmetry breaking. The interpretation

ken subgroup H massless. But the theory also has the massless scalar fields ¢p°‘° which can

Obviously the spontaneous symmetry breaking leaves the gauge fields C,,°‘° of the unbro

of the spin-two particles remains unchanged and is characterized by the Planck scale.

scale ;4“ which is responsible for the fine-structure of the mass spectrum. The mass spectrum

52

6\I¤° = —§°8,\I*°— (8g§°‘)\I!,,° + %;¢"(8,§°)·y,,¤I¤°— (6q§)\I'°— f6g\I!°` 1

L 6‘I’»° = —(@»E°‘)‘I’¤°—€°'6¤‘I’»° + g;¢`1(@»£°)1»‘I’»°— :(@»€)‘I’°·£@v‘I’»°I 1

6¢ = —£°‘@¤¢ -·¢(@e£°‘)A¤¢· (@aE)¢··£6e¢ (7-5) OCR Output

- L _ Q ,, _ 5A,. = —(6,,f°')A, —§°‘6,,A,, -— -5545 °(8gf )e,,"e,,,; - i6,,f —§5gA,,]..

6-gun = "(éu£¤)e¤p " fuacreup " £86€uB

Under the general coordinate transformation (5.12) we have

does not couple to the fermion field in the 5-dimensional supergravity [23].

the field strength of the abelian gauge field BM. Here we emphasize that the gauge field BM

where R5 is the 5-dimensional curvature expressed by the connection OL , and GMN isMw

(7.4)+\I!L°·‘yM\I!N°\l!P\I!Q+ 2i\1!LEMNVp\IiQ°]°° "

+sqM'w”°\i»M·—1»N°—i»,·=1»Q° + éM”*°°[§§BLc;M,,cpQ + §¢=i»L=qM¤1»,,=G,,,

1 - 3 3 . — C C C = ·L/·Rs + WMPWNQGMNGPQ ··;‘¤'YMP'YNQGMN‘I’p ‘I’q'?{T€mEEThe Lagrangian for the 5-dimensional supergravity is given by

B, = B, — e»cA,,X.

u ",W° = ‘I!°— e»:A,\I!° (7.3)

uQMN = _ e~A"QMN

where

BM : (Bw X)•

(7-2)‘7’M° = (‘7’,.°. ‘I’°)

M(“’)GM]? = 01VI1VQ1VfN

fiinfbein has the block-diagonal form (5.23), and the gravitinos and the gauge field are given

work in the block-diagonal basis (6,, EB 726Q) defined in (5.6). In the block-diagonal basis the

53

6,95 = i'é°‘I!°

6zAu = '$"¢_1Ec(‘I’uc " e’°Au‘I’c) + $¢-2-€c'Y#`I’c

5,e,/2 = iE:'°·y’2(\I¥,,°— ercA,,\I!°)

Under the supersymmetry transformation generated by c°(:z:, 0) we have

(7.7) OCR OutputQB; = i(m - k){”‘B”"'”‘

6£\P“°’° = i(m — k)§"`\I!“°k`”'

block-diagonal basis forms a (0,0) and (0,0) representation of the Virasoro group

‘* "a definite representation of the Virasoro group. Notice, however, that \i!#and Buin the

(-1, 0) representation of the Virasoro group. But \I!,,°’° and Bf do not appear to constitute

The result confirms that efk forms a (0,0)—representation but 45*, X", and \I!°" forms a

6xk = ___£¤m8¤X!c—m _ il£_£GfRB¤k·7D _ ikfmxh-m.

-eke···¤1»='····

gqlck = _E¤m8¤\I,ck-m _ im£¤m\I,¤¤k—m _ g_%£¤m(¢-1,Y¤\I,c)k—m(@»£)‘I’`”‘ + i(m ·· k)£”"I’» °’“`”`—;”°°"

CWI ·Hl CWI **773 CWI •· C *7D 6‘I’»°'° = ·(@»E )‘I’¤°" ·£ @¤‘I’»°'° - {fe (¢ 1‘n·‘I’» )’°1

—8,,fk + i(2m - k)f”‘A,,"

Q6Apk = _(5“£¤)mA¤l:-m _ £¤m6¤Apk-m _ i%£¤m(¢-2eu§e¤p)k-m

pk-mm ¤mNcsea= "(5u€)c¤_m " €¤m8¤cupk_m + °i(m "' k)6mcu

After the Fourier decomposition (7.5) can be written as

6x = —6°‘@¤x - §(

57 OCR Output

we obtained in the last section. To simplify ACI] we put

indices are contracted with the flat metric 7;,,,,. Notice that L; in (7.19) is identical to Em

¢p° is the dilaton, GW" is the field strength of Bu", and now all the 4-dimensional Lorentz

where p,,,,” is the spin-two field deined by (6.7) and (6.11), A,,° is the U (1) gauge potential,

+4·i¢,,°'”0'“’ 8,,1/v°" + 4i¢°'”0""8,,1l:,,'

+4i1c-n,7;40,vp6V1l)pm _ 4§¢“c—nUpv¢v¤sZ“[]

··fI fl **7I fl Bu ’= ·G,... 6,... + ;(D»X) (Dpxl3 · - 3 ~ ~{E

(7-19)(6uAv0 ‘ 6vAu0)(8uAv0 _ 6vAu0) ‘‘§(6u‘P0)(8u‘P0)

-.. .. -.. .. ZF(P¤¤ P¤¤· Pas Pan l1¤’

+2(8uPw_”)(6vP¤¤”) " 2(8#Pu¤_”)(‘9vPv¤n)l

Ez 2 -{ §[(@»p¤¤`”)(@»p¤¤”) — (@»p¤¤`”)(6»P¤¤”)

becomes

the Virasoro group. Then, to the lowest order with e°rc° = 161rG, the effective Lagrangian

down to the direct product of the N = 2 super-Poincaré group and the U (1) subgroup of

The vacuum (7.18) breaks the supersymmetric Virasoro-Kac-Moody group spontaneously

(¤!*,.°") = (¢°") = (BJ`) = (X") = 0 (7-18)

(¤»'°") = 6./°6¤”» (Ap"} = 0» (¢”) = 6c"

to the lowest order around the vacuum

To investigate the physical content of the 4·dimensiona.l eEective action we expand Cc

with the N = 2 super·Poincaré group.

Notice that Leg describes a gauge theory of the Virasoro-Kac·M0ody symmetry associated

+higher-order interaction terms.

OCR OutputOCR OutputOCR Output+4i(¢—1)k-l—·m-n,l/;“dUmnn(VV1l])m + 4i(¢—1)k—l—m—n1ZdU;wm(V“¢u)m

58

(7-22) OCR Output+3s¤°(

59 OCR Output

confinement theory based on the elementary excitations of the string, not the one that deals

We emphasize again that the string field theory we discussed in this paper is a pre

moment.

size effect of the string and the string-string interaction into the theory is not clear at this

string is approximated to a point. Exactly how one can successfully accommodate the finite

paper can describe a free field theory of a self-interacting single string in the limit that the

does not take into account the string·string interaction. So the theory described in this

of freedom which come from the finite size effect of the string. More importantly the theory

theory. This is because the theory has not properly taken into account the full string degrees

Admittedly the theory described in this paper may not describe a full·fledged string field

harmonic oscillator spectrum which is precisely what one can expect from a string theory.

cation group. More importantly the unified theory has a remarkable mass spectrum, the

Virasoro-Kac-Moody symmetry of the 4-dimensional Poincaré group and the grand unifi

imum necessary symmetries required to describe the colored string, the (supersymmetric)

point limit of the string. From the physical point of view of the unified theory has the min

dynamics of a. propagating colored string in the 4—dimensional space-time, at least in the

persymmetric gauge theory of the Virasoro-Kac-Moody group, which could describe the

string excitation modes. Furthermore we have shown how to construct a prototype su

sociated with the Poincaré group and the grand unification group among the elementary

string field theory must be the gauge invariance of the Virasoro-Kac-Moody symmetry as

In this paper we have shown that the fundamental dynamical principle underlying the

VIII. DISCUSSION

universe and play the role of the dark matter in the present universe [22,25].

the gravitation [22]. Furthermore in cosmology it could generate the inflation in the early

perhaps more important than the axion. It could create a fifth force which could modify

This implies that in this type of unified theories the dilaton could play an important role,

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that it describes a theory which is not based on the unitary representations of the underlying

1. From the field theoretic point of view a most remarkable characteristics of the theory is

We conclude with the following remarks:

the moment not clear and remains an open question.

Whether the central extension is absolutely necessary at the quantum level, however, is at

extension. Similarly the renormalizability of the theory could require the central extension.

For example, the theory could develop an anomaly which could force us to make the central

quantum level, one might need the central extension of the Virasoro-Kac-Moody symmetry.

the necessary gauge symmetry of the string, at least at the classical level. But at the

crucial role [2,3]. As we have shown the central extension is not required for us to describe

extension. In the popular string theory, on the other hand, the central extension plays a

In this paper we have considered the Vira.soro—Kac-Moody symmetry without a central

acceptable unified field theory of string.

be renormalizable or even finite. With this observation one could regard the theory as an

extra interactions neccessary to describe the full string theory, the theory could be made to

to describe the full string theory which is believed to be finite. This implies that, with the

theory. As we have emphasized, however, with a proper generalization the theory is likely

conventional requirements of the renormalizability, which is a serious drawback of the unified

Moody group is the renormalizability. Clearly the Lagrangian (5.47) does not satisfy the

One of the objections one might raise against the gauge theory of the Virasoro-Kac

very interesting question whose answer we do not know at the moment.

way that one can keep the full symmetry unbroken in this type of theory. This is indeed a

depend on which type of the vacuum one chooses. So one might ask whether there is any

this is one of the most important aspect of the theory, because the physics will crucially

been unable to find a. sensible vacuum which can keep the full symmetry intact. Obviously

leaves the full Virasoro—Kac-Moody symmetry unbroken. This is because so far we have

observe that the theory allows only the Higgs phase, but not the symmetric phase which

with all the physical states explicitely. From this point of view it is very interesting to

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elementary particle scale which provides the fine·structure to the mass spectrum.

effective potential [26]. According to this scenario, the quantum correction can generate the

the vacuum (6.18) through the quantum correction which induces a Coleman-Weinberg type

the renormalizability of the theory, one may start from the vacuum (6.2) but end up with

scale through the higher—order quantum correction. Indeed it is quite possible that, assuming

provides us with the possibility that the hadronic scale could be generated from the Planck

structure to the harmonic oscillator spectrum of the theory. Furthermore, the unified theory

the scale of the harmonic oscillator spectrum and the hadronic scale which provides the fine

theory is characterized by two completely different mass scales, the Planck scale which sets

3. From the physical point of view it is remarkable that the mass spectrum of the unified

of the theory.

non-linear realization and the geometric symmetry breaking is due to the geometric origin

gauge fields have a singular kinetic energy without the symmetry breaking. Obviously the

without a spontaneous symmetry breaking the theory does not make any sense, because the

does not admit a vacuum which remains invariant under the gauge transformation. Besides

potential from the Lagrangian, the symmetry breaking must take place. The theory simply

[19]. In spite of the fact that the non-linear gauge invariance precludes any Higgs-type

Furthermore the non-linear realization makes the spontaneous symmetry breaking geometric

totally different from the standard Callan-Coleman-Wess-Zumino non-linear realization [12].

gauge theory. Notice that the non·linear realization described by (4.6) is of a novel type,

2. Another remarkable characteristics of the theory is that it is a non-linearly realized

this paper we have shown how to circumvent this sacred no-go theorem in field theory.

a sensible field theory with a non-unitary representation is considered to be impossible. In

under the unbroken subgroup. This guarantees the unitary of the theory. So far constructing

This is because after the symmetry breaking all the physical fields become explicitly unitary

the Hamiltonian of the theory becomes explicitly positive-definite as we have demonstrated.

field in Section III) form a unitary representation under the Virasoro group. Nevertheless

symmetry group. Indeed none of the fields which appear in the Lagrangians (except the scalar

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be more closely related to each other than we have thought so far.

Nevertheless our analysis implies that the string theory and the Kaluza-Klein theory may

symmetry need not neccessarily be described by the Lagrangians discussed in this paper.

so that it is still possible that the string theory which has the desired Virasoro-Kac-Moody

Moody symmetry in the string theory could manifest itself in a completely different way,

with the Poincaré group and the grand unification group. Of course the Virasoro-Kac

theory which has the gauge invariance of the Virasoro-Kac-Moody symmetry associated

been forced to end up with this type of Lagrangians in an effort to construct a string field

however, that our original intention was not to discuss the Kaluza-Klein theory. We have

from the (supersymmetric) 5-dimensional Einstein-Dirac-Yang-Mills theory. We emphasize,

the 5·dimensional Kaluza-Klein theory. They are obtained by the dimensional reduction

Obviously the string theory Lagrangians discussed in the last two sections are based on

an effective field theory of a closed superstring.

grand unification group. With the supersymmetric generalization the theory could become

theory to obtain a gauge theory of a supersymmetric Kac-Moody group associated with

does not include the grand unification group. But obviously one can easily generalize the

5. The supersymmetric generalization of the unified theory discussed in the last section

theoretical prediction of the mass of the dilaton so far.

however, one must know the mass of the dilaton. Unfortunately there has been no reliable

the present expe