Locally Anisotropic Gravity and Strings

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annals of physics 256, 3961 (1997)

Locally Anisotropic Gravity and Strings

Sergiu I. Vacaru*Department of Statistical and Nuclear Physics, Institute of Applied Physics,

Academy of Sciences of Moldova, 5 Academy str., Chis inau, 2028, Republic of Moldova

Received August 26, 1996

We shall present an introduction to the theory of gravity on locally anisotropic spacesmodelled as vector bundles provided with compatible nonlinear and distinguished linear con-

nection and metric structures (such spaces are obtained by a nonlinear connection reductionor compactification from higher dimensional spaces to lower dimensional ones and contain asparticular cases various generalizations of KaluzaKlein and Finsler geometry). We shallanalyze the conditions for consistent propagation of closed strings in locally anisotropicbackground spaces. The connection between conformal invariance, the vanishing of the renor-malization group ;-function of the generalized _-model, and field equations of locallyanisotropic gravity will be studied in detail. 1997 Academic Press

1. INTRODUCTION

The relationship between two-dimensional _-models and strings has beenconsidered by several authors [15] in order to discuss the effective low energyfield equations for the massless models of strings. In this paper we shall investigatesome of the problems associated with the theory of locally anisotropic strings beinga natural generalization to locally anisotropic (la) backgrounds (we shall write inbrief la-backgrounds, la-spaces, and la-geometry) of Polyakov's covariant func-

tionalintegral approach to string theory [6]. Our aim is to show that a corre-sponding low-energy string dynamics contains the motion equations for fieldequations on la-spaces.

The first geometric models of la-spaces were formulated by Finsler [7] and werestudied in detail and generalized by Berwald [8] and Cartan [9]. The geometry ofLagrange and Finsler spaces, its extensions and possible applications in physics(locally anisotropic gauge and gravitational theories, statistical physics, andrelativistic optics in locally anisotropic media and other topics) were considered in

a number of works; see references from [10, 11]. It seems likely that classical andquantum physical models admit a straightforward extension of la-spaces with com-patible metric and connections structures. In this case we can define locallyanisotropic spinors [12], formulate the theory of locally anisotropic interactions ofgauge and gravitational fields [13], and consider superspaces with local anisotropy

article no. PH965661

390003-491697 25.00

Copyright 1997 by Academic Press

* E-mail: lisescc.acad.md.


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[14]. Here we remark that modern KaluzaKlein theories can be considered asphysical models on la-spaces with trivial nonlinear connection structure.

There are some arguments for taking into account effects of possible localanisotropy of fundamental interactions. It is well known that a self consistentdescription of radiational and dissipation processes in classical and quantum field

theories requires the addition of higher derivation terms (for instance, in classicalelectrodynamics, radiation is modeled by introducing a corresponding term propor-tional to the third derivation in time of coordinates). Another argument fordeveloping quantum field models on the tangent bundle is the problem of renor-malization of gravitational field interactions in different approaches to quantumgravity. It is quite possible that in order to study the physics of the early universeand propose various scenarios of KaluzaKlein compactification from higherdimensions to the four-dimensional one of the space-time it is more realistic to con-

sider theories with generic local anisotropy caused by fluctuations of quantum high-dimensional space-time ``foam.'' The above mentioned points to the necessity toextend the geometric background of classical and quantum field theories if a carefulanalysis of physical processes with non-negligible beak reaction, quantum andstatistical fluctuations, turbulence, random dislocations, and disclinations in con-tinuous media is proposed.

The bulk of this paper is devoted to a presentation of the essential aspects (whichhas not so far been published in the literature) of locally anisotropic strings and a

discussion of possible generation of la-gravity from string theory.We use MironAnastasiei [10] conventions and basic results on the geometry ofla-spaces and la-gravity.

The plan of the paper is as follows. We begin with some necessary results onla-spaces in Section 2. In Section 3 we study the nonlinear _-model and la-stringpropagation by developing the d-covariant method of la-background field.Section 4 is devoted to problems of regularization and renormalization of thelocally anisotropic _-model and a corresponding analysis of one- and two-loopdiagrams of this model. Questions on duality of la-strings are considered inSection 5, and a summary and conclusions are drawn in Section 6.

2. LOCALLY ANISOTROPIC SPACES

We present a geometric background on vector and tangent bundles (in brief,v-bundles and t-bundles) provided with nonlinear and distinguished connectionsand metric structures [10] which is necessary for our investigations.

Let E=(E, ?, F, Gr, M) be a locally trivial v-bundle, where F=Rm is the typicalvector space, dim=m, and the structural group is taken as Gr=GL(m, R), whereGL(m, R) is the group of linear transforms ofRm. We locally parameterize E bycoordinates u:=(x i, ya), where i, j, k, l, m, ...=0, 1, ..., n&1 and a, b, c, d, ...=1, 2, ..., m.

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Coordinate transforms (xk, ya) (xk$, ya$) on E, considered as a differentiablemanifold, are given by formulas

xk$=xk$(xk), ya$=Ma$a (x) ya, (2.1)

where rank(xk$xk)=n and Ma$a

(x) # Gr. One of the fundamental objects in thegeometry of la-spaces is the nonlinear connection, in brief, the N-connection. Thefirst global definition of an N-connection was given by Barthel [15] (a detailedstudy of N-connection structures in v-bundles and basic references are contained in[10]). Here we introduce the N-connection as a global decomposition of v-bundleE into horizontal (HE) and vertical (VE) subbundles of the tangent bundle (TE):

TE=HEVE. (2.2)

With respect to a N-connection in E one defines a covariant derivation operator

{YA=Yi{

Aa

xi+Nai(x, A)= sa ,

where sa are local linear independent sections ofE, 4=4asa , and Y=Y

isi, is thedecomposition of vector field Y on local basis si on M. Differentiable functionsNai(x, y) are called the coefficients of the N-connection. We have these transforma-tion laws for components Nai under coordinate transforms (2.1):

Na$i$xi$

xi=Ma$a N

ai+

Ma$axi

ya.

The N-connection is also characterized by its curvature,

0=12

0aijdxi7dxj

ya

,

where 7 is the antisymmetric tensor product, with coefficients

0aij=Najxi

&Naixj

+NbjNaiyb

&NbiNajyb

,

and by its linearization which is defined as 1a.bi(x, y)=Nai (x, y)y

b. The usuallinear connections |a.b=K

a.bi(x) dx

i in the v-bundle E form a particular class ofN-connections with coefficients parameterized as Nai (x, y)=K

a.bi(x) y

b.If in the v-bundle E a N-connection structure is fixed, we must modify the opera-

tion of partial deviation and introduce a locally adapted (to the N-connection)basis (frame)

$$u:

=\$

$xi=i&N

ai (x, y)

ya

,$

$ya=

ya+ , (2.3)

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instead of the local coordinate basis ua=(x i, ya). The basis dual to $$u:

is written as

$u:=($xi=dxi, $ya=dya+Nai (x, y) dxi). (2.4)

By using bases (2.3) and (2.4) we can introduce the algebra of tensor distinguishedfields (d-fields, d-tensors) on E, C=Cprqs , which is equivalent to the tensor algebraof the v-bundle Ed defined as ?d:HEVETE. An element t #C

prqs , of d-tensor

of type (pqrs), is written in local form as

t=t i1...ipa1...arj1...jqb1...bs

(u)$

$xi1 } } }

$$xirdxj1 } } } dxjr

ya1 } } }

yar$yb1

} } } $ybs

.

In addition to d-tensors we can consider different types of d-objects with groupand coordinate transforms adapted to a global splitting (2.2).

A distinguished linear connection, in brief a d-connection, is defined as a linearconnection D in E conserving as a parallelism the Whitney sum HEVEassociated to a fixed N-connection structure in E. Components 1:.;# of a d-connec-tion D are introduction as

D# \$

$u;+=D($$u#) \$

$u;+=1:.;# \$

$u:+ .

We can define in a standard manner, with respect to locally adapted frame (2.3),the components of torsion T:.;# and curvature R

.:; .#{ of the d-connection D,

T\

$$u# , $$u;+=T:.;# $$u:=D($$u

#) $$u;&D($$u;) $$u#&_

$$u# , $$u;& ,

where

T:.;#=1:.;#&1

:.#;+w

:.;# (2.5)

and

R \$

$u$,

$$u#

,$

$u;+=R .:; .#$$

$u:

=(D($$u $) D($$u#)&D($$u#) D($$u $)&D([$$u$, $$u#]))$

$u;,

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where

R .:; .#$=$1:.;#$u$

&$1:.;$

$u#+1..;#1

:..$&1

.

.;$1:..#+1

:.;.w

.

.#$ . (2.6)

In formulas (2.5) and (2.6) we have used nonholonomy coefficients w#

.:; of locallyadapted frames (2.3) defined so as to satisfy conditions

_$

$u:,

$$u;&=

$$u:

$$u;

&$

$u;$

$u:=w#.:; .

The global decomposition (2.2) induces a corresponding invariant splitting intohorizontal DhX=DhX (h-derivation) and vertical D

vX=DvX (v-derivation) parts of

the operator of covariant derivation D, DX=DhX+D

vX, where hX=X

i($$ui) and

vX=Xa(ya) are, respectively, the horizontal and vertical components of thevector field X=hX+vX on E.

Local coefficients (L i. jk(x, y), La.bk(x, y)) of covariant h-derivation D

h areintroduced as

Dh($$a k) \$

$xj+=L i. jk(x, y)$

$xi, Dh($$a k) \

yb+=La.bk(x, y)

ya

and

Dh($$ak) f=$f

$xk=

fxk

&Nak(x, y)f

ya, (2.7)

where f(x, y) is a scalar function on E.Local coefficients (Ci. jk(x, y), C

a.bk(x, y)) of v-derivation D

v are introduced as

Dv(y c)

\$

$xj

+=C

i. jk(x, y)

$

$xi, Dv(y c)

\

yb+=C

a.bc(x, y),

and

Dv(y c) f=f

yc. (2.8)

By using (2.7), (2.8), (2.5), and (2.6) and straightforward calculations, we obtain(see conventions from [10]) the h- and v-components of torsion,

T:;#=[Ti. jk , T

ija , T

iaj, T

i. ja , T

a.bc],

Ti. jk :=Tijk , T

ija=C

i. ja , T

iaj=&C

ija , T

i. ja=0, T

a.bc :=S

a.bc ,

Ta. ij=$Nai$xj

&$Naj$xi

, Ta.bi=Pa.bi=

Naiyb

&La.bj, Ta. ib=&P

a.bi,



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and of curvature, R .:; .#$=[R. ih . jk , R

.ab . jk , P

. ij.ka , P

.cb .ka , S

. ij.bc , S

.ab .cd],

R . ih . jk=$L i.hj$xh

&$L i.hk$xj

+Lm.hjLimk&L

m.hkL

imj+C

i.haR

a. jk ,

R .ab . jk=$L

a

.bj$xk &$L

a

.bk$xj +L

c.bjLa.ck&Lc.bkLa.cj+Ca.bcRc. jk ,

P . ij.ka=L i. jkyk

&\Ci. jaxk

+L i. lkCl. ja&L

l. jkC

i. la&L

c.akC

i. jc++Ci. jbPb.ka ,

P .cb .ka=Lc.bkya

&\Cc.baxk

+Lc.dkCd.ba&L

d.bkC

c.da&L

d.akC

c.bd++Cc.bdPd.ka ,

S. ij.bc=Ci

. jb

yc &Ci

. jc

yb +Ch. jbCi.hc&Ch. jcCihb ,

S.ab .cd=Ca.bcyd

&Ca.bd

yc+Ce.bcC

a.ed&C

e.bdC

a.ec .

The components of the Ricci d-tensor R:;=R.{: } ;{ with respect to locally adapted

frame (2.4) are as follows: Rij=R.ki. jk , Ria=&

2Pia=&P.ki.ka , Rai=

1Pai=P.ba . ib ,

Rab=S.c

a .bc . We point out that because, in general,

1

Pai{

2

Pai, the Ricci d-tensor isnonsymmetric.We shall also use the auxiliary torsionless d-connection {:.;# (defining the corre-

sponding auxiliary covariant d-derivation denoted as {), constructed in the usualmanner as the second-class Christoffel d-indices, with partial derivations substitutedby d-derivations (2.3) and introduced so as to satisfy the relations

1:.;{={:.;{+T

:.;{ . (2.9)

The Riemann and the Ricci (in this case) d-tensors of the d-connection {:.;# aredenoted respectively as r .:; .#$ and r:; .

Now, we shall analyze the compatibility conditions of the N- and d-connectionsand metric structures on the v-bundle E. A metric field on E, G(u)=G:;(u) du

: du;,is associated to a map G(X, Y) :TuE_TuER, parameterized by a nondegeneratesymmetric matrix ( Gij

Gaj

G iaGab

), where

G ij=G \

xi,

xj+ ,G ia=G \

xi,

ya+ , andG ab=G \

ya ,

yb+ .We choose a concordance between N-connection and G-metric structures by

imposing conditions G($$xi, ya)=0, or, equivalently, Nai(x, u)=G ib(x, y)Gba(x, y), where G ba(x, y) are found to be components of the matrix G :;, which isthe inverse to G :; . In this case the metric G on E is defined by two independent

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d-tensors, gij(x, y) of type (20

00) and hab(x, y) of type (

00

02), and with respect to

locally adapted basis (2.3) can be written as

G(u)=G:;(u) $u: $u;=gij(x, y) dx

idxj+hab(x, y) $y

a$yb. (2.10)

The D-connection 1:.;#

is compatible with the d-metric structure G(u) on E if oneholds equalities

D:G;#=0.

Having defined the d-metric (2.10) in E we can introduce the scalar curvature of thed-connection R0=G:;R:;=R+S, where R=gijRij and S=habSab . The scalarcurvature of the auxiliary torsionless d-connection {:;# from (2.9) can be written ina similar manner, r^=G:;r:; .

Both the d-metric (2.10) and the N-connection on E define the so-called canonicd-connection [10], %1i. jk=(%Li. jk , %L

a.bi, %C

i. jk , %C

a.bc), with local components

expressed as

%L i. jk=12

gip \$gpj$xk

+$gpk$xj

&$gjk$xk+ ,

%La.bi=Naiyb

+12

hac \$hbc$x i

&Ndiyb

hdc&Ndiyc

hdb+ ,

%Ci. jb=12

gikgjkyb

, %Ca.bc=12

had\hdcyb

+hdbyc

&hbcyd+ .

Putting the components of the connection %1:.;# instead of the components of thearbitrary d-connection 1:.;# into formulas (2.5) and (2.6), we can calculate respec-tively the adapted coefficients of the canonic torsion %T:.;# , curvature %R

.:; .#$ , and

Ricci d-tensor %R:; .Finally, in this section, we note that modern approaches to generalized Lagrange

and Finsler geometry [10] are constructed in a similar manner by consideringN-connection structures in the tangent bundle (TM, {, M), Mn=(M, gij(x, y)), andby using metric coefficients gij(u)=hij(u) from (2.10) (parametization g ij=12 (

2Lyiyj), where L=L(u) is a Lagrangian on M, holds for Lagrange geometry[16, 10]; if L=42(u), where 4(u) is the so-called Finsler metric [711], we cangenerate models of Finsler geometry on the tangent bundle). We are not concernedin this work with problems connected with gravitational and matter field theorieson Finsler spaces and theirs extensions (see [1014]), but we emphasize that theconcept of la-space is a general one, including the just mentioned ones as well, fortrivial N-connections and different models of KaluzaKlein theories. We candevelop scenarios of compatification from higher dimensions to a four-dimensionalone, for instance, by postulating different types of spontaneous symmetry breakingof local multidimensional gravitational symmetry, or, in a more consequentialmanner in the framework of a la-space, by using N-connections for modeling



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dynamical variants of ``splitting'' space-time dimensions and for analyses of physicalconsequences for possible relic effects of locally anisotropic fluctuations.

3. LA-STRINGS AND SIGMA-MODELS

In this section we present a generalization of some necessary results on the non-linear _-model and string propagation to the case of la-backgrounds. Calculationson both types of locally anisotropic and isotropic spaces are rather similar if weaccept the Miron and Anastasiei [10] geometric formalism. We emphasize that onla-backgrounds we have to take into account the distinguished character, byN-connection, of geometric objects.

3.1. Action of Nonlinear _-Model and Torsion of La-SpaceLet a map of a two-dimensional (2d), for simplicity, flat space M2 into a la-space

! define a _-model field u+(z)=(xi(z), ya(z)), where z=[zA, A=0, 1] are two-dimensional complex coordinates on M2. The moving of a bosonic string in la-space is governed by the nonlinear _-model action (see, for instance, [1, 13, 4] fordetails on locally isotropic spaces):

I=

1

*2 |d2

z_

1

2 -# #AB

Au+

(z) Bu&

(z) G+&(u)

+n~3

=ABAu+ Bu

&b+&(u)+*2

4?-# R (2 )8(u)& , (3.1)

where *2 and n~ are interaction constants, 8(u) is the dilation field, R(2 ) is the cur-vature of the 2d world sheet provided with metric #AB, #=det(#AB), and A=z

A,and tensor =AB and d-tensor b+& are antisymmetric.

From the viewpoint of string theory we can interpret b:; as the vacuum expecta-tion of the antisymmetric, in our case locally anisotropic, d-tensor gauge field B:;#(see considerations for locally isotropic models in [17, 18] and the WessZuminoWitten model [19, 20], which lead to the conclusion [21] that n~ takes only integervalues and that in the perturbative quantum field theory the effective quantumaction depends only on B ... and does not depend on b ...).

In order to obtain some motions of the la-strings compatible with the N-connec-tion, we consider the relations between the d-tensor b:; , the strength B:;#=$[:b;#] ,

and the torsion T:

.;# :$:b;#=T:;# , (3.2)

with the integrability conditions

0aijab;#=$[iTj] ;# , (3.3)

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where 0aij are the coefficients of the N-connection curvature. In this case we canexpress B:;#=T[:;#] . Conditions (3.2) and (3.3) define a simplified model of la-strings when the _-model antisymmetric strength is induced from the la-backgroundtorsion. More general constructions are possible by using normal coordinatesadapted to both N-connection and torsion structures on la-background space. For

simplicity, we omit such considerations in this work.Choosing the complex (conformal) coordinates z=@0+i@1, z=@0&i@1, where @A,A=0, 1, are real coordinates, on the world sheet we can represent the two-dimen-sional metric in the conformally flat form:

ds2=e2. dz dz, (3.4)

where #zz=12 e

2. and #zz=#zz=0.Let us consider an la-field U(u), u # !, taking values in G as the Lie algebra of

a compact and semisimple Lie group,

U(u)=exp[i.(u)], .(u)=.: (u) q: ,

where q: are generators of the Lie algebra with antisymmetric structural constantsf:; # satisfying conditions

[q: , q; ]=2if:;#

q#

, tr(q: q; )=2$

:; .

The action of the WessZuminoWitten-type la-model should be written as

I(U)=1

4*2 |d2z tr(AU

AU&1)+n~1[U], (3.5)

where 1[U] is the standard topologically invariant functional [21]. For pertur-bative calculations in the framework of the model (3.1) it is enough to know thatas a matter of principle we can present the action of our theory as (3.5) and use

d-curvature r .:; .#$ for the torsionless d-connection {:.;# , see (2.9), and strength B:;# ,respectively expressed as

R:;#$= f:;{f#

${

V::V;;V

##V

$$ and B:;#='f:

;{

V::V;;V

{{ ,

where a new interaction constant '#n~*22? is used and V:: is a locally adaptedvielbein associated to the metric (2.20):

G:;=

V

::V

;;$

:;

and

G

:;

V

::V

;;

=$

:;

. (3.6)

For simplicity, we shall omit underlining of indices if this will not give rise toambiguities.

Finally, in this subsection, we remark that for '=1 we obtain a conformallyinvariant two-dimensional quantum field theory (being similar to those developedin [22, 19]).



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3.2. The D-Covariant Method of La-Background Field and _-Models

Suggesting the compensation of all anomalies we can fix the gauge for the two-dimensional metric when action (3.1) is written as

I[u]=

1

2*2 |d2

z{

G:;'AB

+

2

3 b:;=AB

= Au

:

Bu;

, (3.1a)

where 'AB and =AB are, respectively, the constant two-dimensional metric andantisymmetric tensors. The covariant method of the background field (as generalreferences see [2326]) can be extended for la-spaces. Let consider a curve in !parameterized as \:(z, s), s # [0, 1], satisfying autoparallel equations

d2\:(z, s)

ds2 +1:.;#[\]

d\;

ds

d\#

ds =

d2\:(z, s)

ds2 +{:.;#[\]

d\;

ds

d\#

ds =0,

with boundary conditions \(z, s=0)=u(z) and \(z, s=1)=u(z)+v(z). For sim-plicity, hereafter we shall consider the d-connection 1:;# to be defined by d-metric(2.10) and N-connection structures, i.e., 1:;#=%1

:;# . The tangent d-vector `

:=(dds) \:, where `:| s=0 =`

:(0 ) is chosen as the quantum d-field. Then the expansion

of the action I[u+v(`)] (see (3.1a)), as a power series on `, I[u+v(`)]=k=0 Ik ,where Ik=(1k!)(d

kdsk) I[\(s)]| s=0 defines d-covariant, depending on the back-

ground d-field interaction vortexes of locally anisotropic _-model.In order to compute Ik it is useful to consider relations

dds

A\:=A`

:,dds

G:;=`{ ${G:; , AG:;=A\

{ ${G:; ,

to introduce auxiliary operators

({ A`):=({A`):&G:{T[:;#][\] =ABB\#`;,

({A`):=[$:; A+{

:.;# A\

#] `;, (3.7)

{(s) !*=`: {:!*=

dds

!*+{*.;#[\(s)] `;!#,

having properties {(s) `:=0, {(s) A\:=({A`)

:, {2(s) A\:=r .:; .#$`

;`# A\$, and

to use the curvature d-tensor of d-connection (3.7),

r;:#$=r;:#$&{#T[:;$]+{$T[:;#]&T[{:#] G{*T[*$;]+T[{:$] G

{*T[*#;] .

Values Ik can be computed in a manner similar to that in [18, 21, 26], but inour case by using corresponding d-connections and d-objects. Here we present thefirst four terms in explicit form:

48 SERGIU I. VACARU


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I1=1

2*2 |d2z2G:;({ A`)

: Au;,

I2=1

2*2 |d2z[({ A`)

2+ r;:#$`;`#('AB&=AB) Au

: Bu;],

I3=1

2*2 |d2z {

43

(r;:#$&G{=T[=:;] T[{#$]) Au

:({ A`$) `;`#

+43

{:T[$;#] Au;=AB({ B

#) `:`$+23

T[:;#]({ A`:) =AB({`;) `#

+13

({*r;:#$+4G{=T[=*:] {;T[#${]) Au

: Au$`#`*

+13

({:{;T[{#$]+2G*=G.,T[:*.] T[=;$] T[,{#]

+2r .*: .;#T[:{$]) Au#=AB(B

{) `:`;= ,

I4=1

4*2 |d2z{\

12

{:r#;${&G*=T[=;#] {:T[*${]+ Au;({ A`{) `:`#`$

+13

r;:#$({ A`:)({ A`$) `;`#

+\1

12{:{;r$#{*+

13

r .}$ .{#r;}:*&12

({:{;T[#{=]) G=?T[?$*]

&12

r .}: .;#T[}{=] G=?T[?$*]+

16

r .}: .;=T[}#{] G=?T[=$*]+ Au# Au*`:`;`$`{

+_1

12{:{;{#T[*${]+

12

{:(G}=T[=*$]) r;}#{

+12

({:T[?;}]) G?=G}&T[=#$] T[&*{]&

13

G}=T[=*$] {:r;}${&_Au

$=ABBu{`*`:`;`#+

1

2

[{:{;T[{#$]+T[}${] r.}: .;#+r

.}: .;$T[}#{]]

_Au#=AB({ B

$) `:`;`{+12

{:T[$;#]({ A`B) =AB({ B

#) `:`$= . (3.8)

Now we construct the d-covariant la-background functional (we use methods, inour case correspondingly adapted to the N-connection structure, developed in



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[18, 26, 27]). The standard quantization technique is based on the functionalintegral

Z[J]=exp(iW[J])=|d[u] exp[i(J+uJ)], (3.9)

with source J:

(we use condensed denotations and consider that computations aremade in the Euclidean space). The generation functional 1 of one-particleirreducible (1PI) Green functions is defined as 1[u]=W[J(u)]&u } J[u], whereu=2W$J is the mean field. For explicit perturbative calculations it is useful toconnect the source only with the covariant quantum d-field ` and to use instead of(3.9) the new functional

exp(iW[u, J])=|[d`] exp[i(I[u+v(`)]+J} `)]. (3.10)

It is clear that Feynman diagrams obtained from this functional are d-covariant.Defining the mean d-field (u)=2W$J(u) and introducing the auxiliary d-field

`$=`& , we obtain from (3.9) a double expansion on both classical and quantumla-backgrounds:

exp(i1[u, ])=|[d`$] exp {i\I[u+v(`$+ )]&`$21

$ += . (3.11)

The manner of fixing the measure in the functional (3.10) (and as a consequencein (3.11)) is obvious: [d`]=6u -|G(u)| 6n+m&1:=0 d`:(u). Using vielbein fields (3.6),we can rewrite the measure in this form: [d`]=6u6

n+m&1:=0 d`

: (u). The structure

of renormalization of_-models of type (3.10) (or (3.11)) is analyzed, for instance, in[19, 26, 27]. For la-spaces we must take into account the N-connection structure.

4. REGULARIZATION AND ;-FUNCTIONS OF LA-_-MODEL

The aim of this section is to study the problem of regularization and quantumambiguities in ;-functions of the renormalization group and to present the resultson one- and two-loop calculi for the la-_-model (LAS-model).

4.1. Regularization and Renormalization Group ;-Functions

Because our _-model is a two-dimensional and massless locally anisotropictheory we have to consider both types of infrared and ultraviolet regularizations

(in brief, IR- and UV-regularization). In order to regularize IR-divergences anddistinguish them from UV-divergences, we can use a standard mass term in theaction (3.1) of the LAS-model

I(m)=&m~2

2*2 |d2z G:;u

:u;.

50 SERGIU I. VACARU


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For regularization of UF-divergences it is convenient to use the dimensionalregularization. For instance, the regularized propagator of quantum d-fields ` lookslike

(`

:(u1

)`

;(u2

))

=$

:;

G(u1

&u2

)=i*

2

$

:;

|

dqp

(2?)q

exp[&ip(u1&u2)]

(p2&m~2+i0),

where q=2&2=.The d-covariant dimensional regularization of UF-divergences is complicated

because of the existence of the antisymmetric symbol =AB. One introduces [28, 26]the general prescription

=LN'NM=MR=(=) 'LR and =MN=RS=|(=)['MS'NR&'MR'NS],

where 'MN is the q-dimensional Minkowski metric, and (=) and |(=) are arbitraryd-functions satisfying conditions (0)=|(0)=1 and depending on the type ofrenormalization.

We use the standard dimensional regularization, with dimensionless scalar d-fieldu:(z), when expressions for unrenormalized G (ur):; and B

(k, l):; have a d-tensor

character, i.e., they are polynoms on d-tensors of curvature and torsion and theird-covariant derivations (for simplicity, in this subsection we consider *2=1; in

general one-loop 1PI-diagrams must be proportional to (*2

)l&1

).RG ;-functions are defined by relations (for simplicity we shall omit index R forrenormalized values)

+d

d+G:;=;

G(:;) (G, B), +

dd+

B[:;]=;B[:;] (G, B), ;:;=;

G(:;)+;

B[:;] .

By using the scaling property of the one-loop counterterm under global conformal

transforms

GG:;4(l&1)G (k, l):; , B

(k, l):; 4

(l&1 )B(k, l):; ,

we obtain

;G(:;)=& :(1, l)

l=1

lG (1, l)(:;) , ;B[:;]=& :

l=1

lB(1, l)[:;]

in the leading order on = (compare with the usual perturbative calculus from [30]).The d-covariant one-loop counterterm is taken as

2I(l)= 12 |d2z T(l):; ('AB&=AB) Au: Bu;,



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where

T(l):;= :l

k=1

1(2=k)

T(k, l):; (G, B). (4.1)

For instance, in the three-loop approximation we have;:;=T

(1, 1):; +2T

(1, 2):; +3T

(1, 3):; . (4.2)

In the next subsection we shall also consider constraints on the structure of;-functions connected with conditions of integrability (caused by conformalinvariance of the two-dimensional world-sheet).

4.2. One-Loop Divergences and RG-Equations of the LAS-Model

We generalize the one-loop results [31] to the case of la-backgrounds. If inlocally isotropic models one considers a one-loop diagram, for the LAS-model thedistinguished by N-connection character of the la-interactions leads to the necessityto consider four one-loop diagrams (see Fig. 1). To these diagrams one correspondscounterterms:

I(c)1 =I(c, x2)1 +I

(c, y2)1 +I

(c, xy)1 +I

(c, yx)1

=&1

2 I1 |d2

z r ij('AB

&=AB

) Axi

Bxj

& 12 I1 |d2z rab('AB&=AB) Aya Byb

& 12 I1 |d2z r ia('AB&=AB) AxiBya

& 12 I1 |d

2

z rai('

AB&=

AB) Ay

a Bx

i,

where I1 is the standard integral

I1=G(0)

*2=i|

dqp(2?)2

1p2&m~2

=1(=)

4?q2(m~2)==

14?=

&1?

ln m~+finite counterterms.

There are one-loops on the base and fiber spaces or describing quantum interac-tions between fiber and base components of d-fields. If the la-background d-connec-tion is of distinguished LeviCivita type we obtain only two one-loop diagrams (onthe base and in the fiber) because in this case the Ricci d-tensor is symmetric. Itis clear that this four-multiplying (doubling for the LeviCivita d-connection) ofthe number of one-loop diagrams is caused by the ``indirect'' interactions withthe N-connection field. Hereafter, for simplicity, we shall use a compactified(nondistinguished on x- and y-components) form of writing out diagrams and

52 SERGIU I. VACARU


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Fig. 1. The Feynman diagrams for the one-loop ;-functions of the LAS-model.

corresponding formulas and emphasize that really all expressions containingcomponents of d-torsion generate four irreducible types of diagrams (with respectiveinteraction constants) and that all expressions containing components of d-curvaturegive rise in a similar manner to six irreducible types of diagrams. We shall take into con-sideration these details in the subsection where we shall write the two-loop effectiveaction.

Subtracting in a trivial manner I1 , I1+subtractions=14?=, we can write theone-loop ;-function in the form:

; (1 ):; = 12?r:;= 12?

(r:;&G:{T[{#,]T[;#,]+G

{+ {+T[:;{]).

We also note that the mass term in the action generates the mass one-loopcounterterm

2I(m)1 =m~2

2I1 |d2z {

13

r:;u:u;&u:{

:.;#G

;#= .

The last two formulas can be used for a study of effective charges as in [21]where some solutions of RG-equations are analyzed. We shall not consider in thispaper such methods connected with the theory of differential equations.

4.3. Two-Loop ;-Functions for the LAS-Model

In order to obtain two-loops of the LAS-model we add to the list (3.8) theexpansion

2I(c)1| 2=&12 I1 |d2z [r^:;('AB&=AB)({ A`:)({ B ;)+({{ r:;+ r:#G#$T[$;{])

_('AB&=AB) `{({ A`:) Bu

;+({{ r:;&T[:{#] r#.;)('

AB&=AB) Au:({ B`

;) `{

+('AB&=AB)( 12 {#{{ r:;+12 r=;r

.=# .{:+

12 r:=r

.=# .{;+T[:{=] r

=$T[$#;]

+G+&T[&;#] {{ r:+&G+&T[:#&] {{ r+;) Au

: Bu;`{`#]



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and the d-covariant part of the expansion for the one-loop mass counterterm

2I(m)1| 2 =(m~)2

2I1 |d2z

13

r:;`:`;.

The nondistinguished diagrams defining two-loop divergences are illustrated inFig. 2. We present the explicit form of corresponding counterterms computed byusing methods, in our case adapted to la-backgrounds, developed in [26, 28].

For a counterterm of the diagram (:) we obtain

(:)=&12 *2(Ii)

2 |d2z [( 14 2r$.& 112 {$ {.r^+ 12 r$:r:.& 16 r: .;.$ ..r:;

+ 12 r: .;#.$ r:.;#+

12 G${T

[{:;] 2T[.:;]+12 G.{r

:$T[:;#] T

[{;#]

& 16 G;{T[$:;] T[.#{] r:#+G#{T[$:{] {(:{;) T[.;#]+ 34 G}{r: .;#.$ T[;#}] T[:.{]

& 14 r}:;#T[$;#] T[}:.]) Au

$ Au.

+ 14 [{; 2T[$.;]&3r

#;:...$ {:T[;#.]&3T[:;$] {

#r;:..#.

+ 14 r:# {:T[#$.]+

16 T[$.:] {

:r&4G#{T[{;$] T[:}.] {;(G#=T

[:}=])

+2G${G;= {:(G

:&T[&;#]) T[#}{]T[=}.]] =

ABAu$ Bu

.].

In order to compute the counterterm for diagram (;) we use integrals:

limuv

i(A`(u) A`(v)) =i|

dqp(2?)q

p2

p2&m~2=m~2I1

(containing only a IR-divergence) and

J#i|d2p

(2?)21

(p2&m~2)2=&

1(2?)2 |d

2kE1

(k2E+m~2)2

(being convergent). In result we can express

(;)= 16 *2(I21+2m~ I1 J) |d2z r;:#$r;#('AB&=AB) Au: Bu;.

In our further considerations we shall use the identities (we can verify them bystraightforward calculations)

r[# .$](: .;) =&{(:(G;) {T[{#$]), r[;:#$]=2G

}{T[{[:;] T[#$] }] .

In the last expression we have three types of antisymmetrizations on indices, [{:;],[#$}], and [:;#$],

{ $T[:;#] { .T[:;#]= 916 ( r[;:#] $& r$[:;#])( r

[;:#]............& r

. [:;#]............)

& 94 r[:;#$] r[:;#..........]+

94 r

:;#......[$ r[.] :;#]+

94 r

: .;#. [$ r[.] :;#] . (4.3)

54 SERGIU I. VACARU


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Fig. 2. Two-loops diagrams for the LAS-model.

The momentum integral for the first of the diagrams (#),

|dqpdqp$(2?)2q

pApB(p2&m~2)([k+q]2&m~2)([p+q]2&m~2)

,

diverges for a vanishing exterior momenta k+ . The explicit calculus of the corre-sponding counterterm results in

#1=&2*2

3q I21 |d

2z [(r:(;#) $+G.{T[{:(;] T[#)$.])

_(r; .#$.+ &G+{G}=T[{;}]T[#$=]) Au: Au

+

+({(;T[$) :#]) {;(G+{T

[{#$]) =LN'NM=MR Lu

: Ru+

&2(r:(;#) $+G.{T[:{(;] T[#) $.]) {

;(G+=T[=$#]) =MR Mu

: Ru+]. (4.4)



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The counterterm of the sum of next two (#)-diagrams is chosen to be the la-exten-sion of that introduced in [28, 26]:

#2+#3=&*2|(=)

10&7q18q

I21 |d2z [{ AT[:;#] { AT[:;#]

+6G{=T[$:{] T[=;#] { .T[:;#]('AB&=AB) Au

$ Bu.]. (4.5)

In a similar manner we can compute the rest of the counterterms:

$= 12 *2(I21+m

2I1 J) |d2z r:(;#) $ r;#('AB&=AB) Au: Bu$,

==

1

4 *2

I2

1

|d2

z('AB

&=AB

)[2r$.+r:

$ r:.+r:

. r$:

&2(G:{T[$;:] T[.#{] r;#&G.{T

[{:;] {: r$;+G${T[{:;] {: r;.] Au

$ Bu.,

@= 16 *2m~2I1 J|d2z r;:#$r;#('AB&=AB) Au: Bu$,

'= 14 *2|(=)(I21+2m~

2I1 J) |d2z r;.:#$T[;.{]T[#.{]('AB&=AB) Au: Bu$.

By using relations (4.3) we can represent terms (4.4) and (4.5) in the canonicalform (4.1) from which we find the contribution in the ;$.-function (4.2):

#1 : &2

3(2?)2r$(:;) # r

#(:;).........&

(|1&1)(2?)2 {

43

r[#(:;) $]r[#(:;)..........]+r[:;#$] r

[:;#..........]= ,

#2+#3 :(4|1&5)

9(2?)2

{{ $T[:;#] { .T

[:;#]+6G{=T[$:{]T[=;#] { .T[:;#]

&(|1&1)

(2?)2r[:;#$] r

[:;#..........]= ,

' :|1

(2?)2r:.$;.T[:{=] T

[;{=]. (4.6)

Finally, in this subsection we remark that the two-loop ;-function cannot be

written only in terms of curvature r:;#$ and d-derivation { : (similarly to the locallyisotropic case [28, 26]).

4.4. Low-Energy Effective Action for La-Strings

The conditions of vanishing of;-functions describe the propagation of a string inthe background of la-fields G:; and b:; . (In this section we chose the canonic

56 SERGIU I. VACARU


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d-connection %1:.;# on E.) The ;-functions are proportional to d-field equationsobtained from the on-shell string effective action

Ieff=|du -|#| Leff(#, b). (4.7)

The adapted to N-connection variations of (4.6) with respect to G+& and b+& can bewritten as

2Ieff$G:;

=W:;+12

G:;(Leff+complete derivation) and2Leff$b:;

=0.

The invariance of action (4.6) with respect to N-adapted diffeomorphisms givesrise to the identity

{;W:;&T[:;#]

2Ieff$b;#

=&12

{:(Leff+complete derivation)

(in the locally isotropic limit we obtain the well-known results from [32, 33]). Thispoints to the possibility of writing out the integrability conditions as

{;;(:;)&G:{T[{;#];[;#]=&

12 {:Leff. (4.8)

For the one-loop ;-function, ;

(1 )

:; =(12?) r:; , we find from the last equations

{;; (1 )($;)&G${T[{;#];(1 )[;#]=

14?

{$ \R0+13

T[:;#] T[:;#]+ .

We can take into account two-loop ;-functions by fixing an explicit form of

|(=)=1+2|1 =+4|2 =2+ } } } ,

when

wHVB(=)=1

(1&=)2, |HVB1 =1, |

HVB2 =

34

(the t'HooftVeltmanBos prescription [34]). Putting values (4.6) into (4.8) wehave this two-loop approximation for la-field equations

{;; (2 )($;)

&G${T[{;#]; (2 )

[;#]=

1

2(2?)2{$

_&

1

8r:;#$r

:;#$+1

4r:;#$G{=T

[:;{]T[=#$]

+14

G;=G:}T[:{_]T[={_] T

[}+&]T[;+&]

&1

12G;=G#*T[:;{] T

[:#.]T[=.}] T[*{}]& ,



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which can be obtained from effective action

Iefft|$n+mu -|#| _&r^+13

T[:;#] T[:;#]&

:$4 \

12

r:;#$r:;#$&G{=r:;#$T

[:;{]T[=#$]

&G;=G:"T[:#}]T[=#}] T

["_`]T[;_`]

+13

G;=G$}T[:;#] T[:$.]T[."=] T

[#"}]+& . (4.9)

The action (4.9) (for 2?:$=1 and in the locally isotropic limit) is in good concor-dance with the similar ones on usual closed strings [35, 26].

We note that the existence of an effective action is assured by the Zamolodchikovc-theorem [36] which was generalized [37] for the case of the bosonic nonlinear_-model with dilation connection. In a similar manner we can prove that suchresults hold good for la-backgrounds.

5. DUALITY OF LA-_-MODELS

The quantum theory of la-strings can be naturally considered by using theformalism of functional integrals on ``hypersurfaces'' (see Polyakov's works [38]).In this section we shall study the duality of la-string theories.

Two theories are dual if their nonequivalent second-order actions can begenerated by the same first order action. The action principle assures the equiv-alence of the classical dual theories. But, in general, the duality transforms affect thequantum conformal properties [40]. In this subsection we shall prove this for the

la-_-model (3.1) when the metric # and the torsion potential b on the la-back-ground ! do not depend on the coordinate u0. If such conditions are satisfied wecan write for (3.1) the first-order action

I=1

4?:$ |d2z { :

n+m&1

:, ;=1

[-|#| #AB(G00 VAVB+2G0:VA(Bu:)+G:;(Au:)(Bu;))

+=AB(b0:VB(Au:)+b:;(Au

:)(Bu;))]

+=ABu0(AVB)+:$ -|#| R(2 )8(u)= , (5.1)

where string interaction constants from (3.1) and (5.1) are related as *2=2?:$.This action will generate an action of type (3.1) if we shall exclude the Lagrange

multiplier u0. The dual to the (5.1) action can be constructed by substituting

58 SERGIU I. VACARU


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VA expressed from the motion equations for fields VA (also obtained from action(5.1)),

I=1

4?:$ |d2z [-|#| #ABG :;(A u:)(Bu;)+=ABb:;(A u:)(Bu;)

+:$ -|#| R(2)8(u)],

where the new metric and the torsion potential are introduced respectively as

G 00=1

G00, G 0:=

b0:G00

, G :;=G:;&G0:G0;&b0:b0;

G00

and

b0:=&b:0=G0:

G00, b:;=b:;+G

0:b0;&b0:G0;G00

(in the formulas for the new metric and torsion potential indices : and ; takevalues 1, 2, ..., n+m&1).

If the model (3.1) satisfies the conditions of one-loop conformal invariance (seedetails for locally isotropic backgrounds in [3]), one holds these la-field equations

1:$

n+m&253 +_4(

{8)2

&4 {2

8&r&13 T[:;#] T

[:;#]

&=0,r(:;)+2 {(:{;) 8=0, r[:;]+2T[:;#] {#8=0. (5.2)By straightforward calculations we can show that the dual theory has the sameconformal properties and satisfies the conditions (5.4) if the dual transform iscompleted by the shift of the dilation field 8 =8& 12 log G00 .

The system of la-field equations (5.2), obtained as a low-energy limit of thela-string theory, is similar to Einstein equations written on a la-space [10]. Wenote that the explicit form of locally anisotropic energy-momentum source in (5.2)is defined from well defined principles and symmetries of string interactions and thisform is not postulated, as in usual locally isotropic field models, from some generalconsiderations in order to satisfy the necessary conservation laws on la-space whoseformulation is very sophisticated because of the nonexistence of a global and evena local group of symmetries of this type of space. Here we also remark that theLAS-model with dilation field interactions does not generate in the low-energy limitthe EinsteinCartan la-theory because the first system of equations from (5.2)represents some constraints (being a consequence of the two-dimensional symmetryof the model) on torsion and scalar curvature which cannot be interpreted as somealgebraic relations between the locally anisotropic spin-matter source and torsion.As a matter of principle we can generalize our constructions by introducing interac-tions with gauge la-fields and considering a variant of the locally anisotropic chiral_-model [41].



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6. SUMMARY AND CONCLUSIONS

Let us try to summarize our results, discuss their possible implications, and makethe basic conclusions. First, we emphasize that following the Miron and Anastesieiapproach [10] to the geometry of la-spaces, the possibility and manner of formula-

tion of classical and quantum field theories on such spaces become evident. Herewe also note that in la-theories we have an additional geometric structure, theN-connection. From a physical point of view it can be interpreted, for instance,as a fundamental field managing the dynamics of splitting high-dimensional space-time into the four-dimensional and compactified ones. We can also consider theN-connection as a generalized type of gauge field which reflects some specifics ofla-field interactions and the possible intrinsic structure of la-spaces.

According to modern views, the theories of fundamental field interactions should

be a low energy limit of the string theory. One of the basic results of this work isthe proof of the fact that in the framework of la-string theory is contained a moregeneral, locally anisotropic, gravitational physics. To do this we have developed alocally anisotropic nonlinear sigma model and studied its general properties. Wehave shown that the condition of selfconsistent propagation of a string on ala-background imposes corresponding constraints on the N-connection curvature,la-space torsion, and antisymmetric d-tensor. Our extension of background fieldmethod for la-spaces has a distinguished by N-connection character and the main

advantage of this formalism is doubtlessly its universality for all types of locallyisotropic or anisotropic spaces.Finally, we note that from the viewpoint of string fundamental ideas only, some

primary changes in the established material have been introduced. But ourapproach is not a simple straightforward repetition of standard material in thecontext of some sophisticated geometries. Our main purposes were to show thatthe la-gravity (more generally, the la-field theory) is also naturally contained inthe framework of low-energy string dynamics and to propose a corresponding

geometric and computational technique necessary for further self-consistentinvestigations, for instance, of anisotropic multidimensional models.

REFERENCES

1. C. Lovelace, Phys. Lett. B35 (1984), 75.2. S. Fradkin and A. A. Tseytlin, Phys. Lett. B158 (1985), 316; Nucl. Phys. B261 (1985), 1.3. C. G. Callan, D. Friedan, E. J. Martinec, and M. J. Perry,

Nucl.

Phys.

B262 (1985), 593.

4. A. Sen, Phys. Rev. Lett. 55 (1986), 1846.5. S. P. de Alwis, Phys. Rev. D 34 (1986), 3760.6. A. M. Polyakov, Phys. Lett. B103 (1981), 107.7. P. Finsler, ``Uber und Fla chen in allgemeinen Raumen,'' dissertation, University of Go tingen, 1918;

Birkhauser, Basel, 1951.8. L. Berwalld, Math. Z. 25 (1926), 40.9. E. Cartan, ``Les Espaces de Finsler,'' Hermann, Paris, 1934.

60 SERGIU I. VACARU


23/23

10. R. Miron and M. Anastasiei, ``The Geometry of Lagrange Spaces: Theory and Applications,''Kluwer, Dordrecht, 1994; R. Miron and M. Anastasiei, ``Vector Bundles. Lagrange Spaces. Applica-tions to Relativity,'' Editura Academiei Roma$ne, Bucharest, 1987 [in Romanian].

11. H. Rund, ``The Differential Geometry of Finsler Spaces,'' Springer-Verlag, Berlin, 1959; G. S.Asanov, ``Finsler Geometry, Relativity and Gauge Theories,'' Reidel, Dordrecht, 1985; R. Miron andT. Kawaguchi, Int. J. Theor. Phys. 30 (1991), 1521; A. A. Vlasov, ``Statistical Distribution Func-

tions,'' Nauka, Moscow, 1966 [in Russian]; G. Yu. Bogoslovskyi, ``The Theory of LocallyAnisotropic Space-Time,'' Moscow University Press, Moscow, 1982 [in Russian]; A. Bejancu,``Finsler Geometry and Applications,'' Horwood, Chichester, England, 1990.

12. S. Vacaru, J. Math. Phys. 37 (1996), 508; e-print: gr-qc9604015.13. S. Vacaru and Yu. Goncharenko, Int. J. Theor. Phys. 34 (1995), 1955; e-print: gr-qc9604013.14. S. Vacaru, ``Nonlinear Connections in Superbundles and Locally Anisotropic Supergravity,'' in

preparation; e-print: gr-qc9604016.15. W. Barthel, J. Reine. Angew. Math. 212 (1963), 120.16. J. Kern, Arch. Math. 25 (1974), 438.17. A. A. Tseytlin, Int. J. Mod. Phys. A 4 (1989), 1257.

18. C. M. Hull, ``Lectures on Non-linear Sigma Models and Strings,'' DAMPT, Cambridge, England,1987.

19. E. Witten, Commun. Math. Phys. 92 (1984), 455.20. J. Wess and B. Zumino, Phys. Lett. 37 (1971), 95.21. E. Braaten, T. L. Curtright, and C. K. Zachos, Nucl. Phys. B260 (1985), 630.22. A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Nucl. Phys. B241 (1984), 333.23. S. V. Ketov, ``Introduction into Quantum Theory of Strings and Superstrings,'' Nauka, Novosibirsk,

1990 [in Russian].24. B. S. DeWitt, ``Dynamical Theory of Groups and Fields,'' Gordon and Breach, London, 1965.25. L. Alvarez-Gaume, D. Z. Freedman, and S. Mukhi, Ann. Phys. 134 (1981), 85.

26. S. V. Ketov, ``Nonlinear Sigma-Models in Quantum Field Theory and String Theory,'' Nauka,Novosibirsk, 1992 [in Russian].

27. R. S. Howe, G. Papadopulos, and K. S. Stelle, Nucl. Phys. B296 (1988), 26.28. S. V. Ketov, Nucl. Phys. 294 (1987), 813.29. B. E. Fridling and E. A. M. Van de Ven, Nucl. Phys. B268 (1986), 719.30. G. t'Hooft, Nucl. Phys. B61 (1973), 455.31. T. L. Curtright and C. K. Zachos, Phys. Rev. Lett. 53 (1984), 1799.32. D. Zanon, Phys. Lett. B191 (1987), 363.33. C. G. Callan, I. R. Klebanov, and M. J. Perry, Nucl. Phys. B278 (1986), 78.34. G. t'Hooft and M. Veltman,

Nucl.

Phys.

B44 (1971), 189; M. Bos,

Ann.

Phys. (

USA) 181 (1988),

177.35. D. J. Gross and J. H. Sloan, Nucl. Phys. B291 (1987), 41.36. A. B. Zamolodchikov, Pis'ma J. T. P. 43 (1986), 565.37. A. A. Tseytlin, Phys. Lett. B194 (1987), 63.38. A. M. Polyakov, Phys. Lett. B 103 (1981), 207211; ``Gauge Fields and Strings,'' Contemporary

Concepts in Physics, Harwood Academic, Chur, Switzerland, 1987.39. J. H. Schwarz, ``Superstrings, The First 15 Years of Superstring Theory: Reprints 6 Commentary,''

World Scientific, Singapore, 1985.40. T. H. Busher, Phys. Lett. B201 (1988), 466.

41. I. Jack, Nucl. Phys. B234 (1984), 365.


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