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interaction along a chain;

and ~=[a(~)]%=A.~= 0.3. which describes the interaction between

the chains. Here, ~i is the species of the atom localized in the i-th chain for configuration ~. For comparison we give the histogram of the density of states of a one-dimensional chain of 4,000 atoms calculated using the method of negative eigenvalues. Figure 2 illustrates the rapid convergence of the density of states with increasing number of chains and the vanishing of the singularities characteristic of the one-dimensional case. Thus, for t/T = 0.3 the density of states hardly changes already for M ~ 5, With decreasing value of the parameter t convergence occurs even more rapidly, as the calculations showed. As regards efficiency, the method proposed in this paper for analyzing the single#article spectra of stochastic quasione-dimensional systems is not inferior to the method of negative eigenvalues, but at the same time it enables one to solve a much larger group of problems by virtue of the information contained in the averaged Green's functions. LITERATURE CITED
i.

2 3 4 5 6 7
8

G. E. T. Goncalves de Silva and Belita Koiller, Solid State Commun., 40, 215 (1981). Mark O. Robins and Belita Koiller, Phys. Rev. B, 2/7, 7703 (1983). M. Hwang, R. Podloucky, A. Gonis, and A. Freeman, Phys. Rev. B, 33, 765 (1986). Youyan Liu and K. A. Chao, Phys. Rev. B, 3-3, I010 (1986)o I. D. Mikhailov and L. V. Zhuravskii, Teor. Eksp. Khim., No. 3, 322 (1987). I. D. Mikhailov, Fiz. Met. Metalloved., 32, 1141 (1971). J. Hubbard, Phys. Rev. B, i__9, 1828 (1979). P. Dean, Proc. R. Soc. London, Ser. A, 25__~4,507 (1960).

ULTRAVIOLET FINITENESS OF NONLINEAR TWO-DIMENSIONAL SIGMA MODELS ON AFFINE-METRIC MANIFOLDS V. V. Be!okurov and V. E. Tarasov The two-loop counterterms of a nonlinear two-dimensional boson sigma model whose target space is an arbitrary affine-metric manifold are calculated, Examples are given of nonflat manifolds that lead to ultraviolet-finite sigma models. Study of nonlinear two-dimensional sigma models has recently become particularly interesting in connection with the development of string theory. The target manifolds of ultraviolet-finite sigma models determine the spaces of the compactified additional dimensions [1,2]. The condition of ultraviolet finiteness determines the equations of motion of the string modes [3-5]. The action of the bosonic sigma model has the form

I(~) = ~ d2x G~,(~) ~
where the integration is over a two-dimensional

~"~J,
The fields ~(x) take

(1)

Minkowski space.

values on some manifold M. metric-consistent connection of the sigma model leads to words, finiteness holds only

It is generally assumed that M is a Riemannian manifold with [6,7]. In this case the condition of ultraviolet finiteness the requirement of vanishing of the Riemann tensor. In other for flat manifolds M.

In this paper we consider as space M an arbitrary affine-metria manifold for which consistency of the connection with the metric is not assumed. In this case the connection has the form

r~,j = {L~} + D%

(2)

Scientific-Research Institute of Nuclear Physics at the Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika~ Vol. 78, No. 3, ppo 471-474~ March, 1989. Original article submitted February 2, 1988.

334

0040-5779/89/7803-0334512.50 9 1989 Plenum Publishing Corporation


where

{k

i]} ------~-G ~p (0~G~ ~- 0~G~p-- 0pG~) is the Christoffel t

symbol,

D~i~=-~/~G~ (K~v~+K~vi-Ki~)+ 2Q (o>~ +Q~
is the connection defect, K~=V~G~ is the nonmetricity tensor, Q~--F [m is the torsion tensor. For the Riemannian manifolds usually considered [6,7], the relations K~=0, Q~=0 hold. The counterterms in the Sigma model can be conveniently calculated by the background field method. In this method, the action is expanded in powers of the quantum field
h __ k

(3 )

dt
d~%~

t=o

where xi(t) is the geodesic determined by the equation

dP .

+

F%

d%' d~~

dt

dt

=

O.

Note that the symmetric part of the connection (2) occurs in the equation for the geodesic in the considered case. The covariant expansion of the action (i) in powers of the quantum field differs from the corresponding expansion for Riemannian manifolds [7,8] by the presence of the additional terms

I m ----- d2x{ G~ V~ ~V~~ + 2Go;~Wu~0"r ' 4- (~

4-~/~Go:~;O~tO~O~cP:},

i(~)
8 8

[~/~ Gv~;

~o~p ~,~4- ~/~ ~;
$ 8

4- '/~ a~; ~; ~; ,~,]~"V

~0.,~ ~ -t-

8

s

.

We have here introduced the notation
$

~,=~,+2Q~a~.
8

, ~,+2Q ~~IQ ~ ,,I,,,

~--20[~IF ~ +2F , ~[~I ,, ~I~, ~I~ r ~_

A~;:=V ~A,=V~A~+Q~A~, V~A~=O~A~-F~A~,

In each order, the obtained counterterms have the structure of Eq. (i) and reduce to a renormalization of the metric tensor Gij [6]. The divergentsingle-loop counterterm is

T(1,1).__ t ~3--~ where 2~=~=~==dbG ~ In calculating

( ,~(ij) 4---~- G~j;a;a--2Gi[a;blGj[a;b]) 8 I we use dimensional regularization,

(5)

the counterterms,

2 + n = 2 -- 2e and introduce an auxiliary mass term to eliminate the infrared divergences. The divergent two-loop counterterm (including the expansion of the expression (5) to terms quadratic in proportional to e -2 and e -I. The term proportional to means of pole equations [7]. Therefore, we write down of a simple pole: contribution obtained from the the quantum field) contains terms e -2 can be obtained from (5) by the answer only for the coefficient

T(,, m~j=

~

335


*/~ G~[c;hi; .,7~i(~[b)~] -- 2Cat; ~; b,~i(ab)i + a~/a G~[p; qlGp(b; a)j41(a[b)q] -- a/.~ Gj[~,; b]6~; (r

ca)~ @

$ 1/2 ~i(a5)i (8Gp[q; ajGp[q; b] -- Gpq, aGpq; b ~

4Gaiv;qlGb[p;

ql) "J-

4/3 ~e(ab)dGi[a; e]Cj[b; d] -~ 8/9 Gild; a]Gj[d, ,5] (Ac(ab)c + 3/4 ,y~.(ab)) -'~/~ G~[< ~ (3~(~1~; t~) + 4~(.~)., ~) -- ~/~ G~; ~(~)~; ~+

Gca; b; iGab; c; ~ -- a/~ Gi(a; b); i (Gcc; (a; b) + 4G~; ~; ~) -- G~; (o; ~)Gar o; ~ -k 8G~[v; q]Gp(~; a)G~[q; (a]; ~) + 2G~[o; ~];~G~; qGq[p, a] -47 1/4 Gi~; (p; q) (SGa[b; p]Gaib; r -- Gab; pGab; q

~Gp[a; b]Gq[a., b]) @ 2God; (a; b)Gi[a; c]Gj[b; d] "Jr"1/3 Gi[p; a]Gj[p; b] X

8Gila; b]Gj[c; d]G~[c; d]; b + X~/3G~ia; ~]Gii~;e]G(blp; pi; e) + /3 Gi[a; b]Gj[a; c]Gb[p; q]Ge[p; a] -- 5/~ Gi[a; b]Gj[c; d] (Gae: pGbd; p --~ (6) Repeated subscripts denote summation with the tensor 89 ab , for example,

It is readily shown that on the transition to a Riemannian manifold Eqs. (5) and (6) lead to the well-known expressions [6,7]. We give examples of nonflat manifolds for which the single-loop and two-loo p counter ~ terms (5) and (6) vanish. In particular, this occurs if we have fulfillment of the conditions
0
or

(7)

(s)
In these cases the Riemann tensor tively,

R~mln=Ol{~mn}--3n{~ml)~y{kpl}{Pmn}--{~p~}{PmZ ~

is~ respec-

Just as finiteness of the sigma model with Wess-Zumino term on parallelizable manifolds was proved [9], we should be able to show that in each order of perturbation theory ultraviolet finiteness of the sigma model on the affine-metric manifold is ensured by the condition
S i , , ,

special

cases

of which

are

provided

by the

conditions

(7)

and (8).

LITERATURE CITED
1.

2. 3.
4.

P. Candelas, G. Horowitz, A. Strominger, and E. Witten, Nuclo Phys. B, 258, 46 (1985)~ D. Nemeschansky and S. Yankielowicz, Phys. Rev. Lett., 54, 620 (1985)~ C. G. Callan, D. Friedan, E. J. Martinec, and M. J. Perry, Nucl. Phys. B, 262, 593

(1985).
5. 6. 7. E. A. D. L. S. Fradkin and A. A. Tseytlin, Phys. Lett. B, 158, 316 (1985). Sen, Phys. Rev. D, 32, 2102 (1985). H. Friedan, Ann. Phys. (N.Y.), $63, 318 (1985). Alvarez-Gaume, D. Z. Freedman, and S. Mukhi, Ann. Phys. (N.Y.), !34, 85 (!981)o

336


8. 9.

S. Mukhi, Nuc. Phys. B, 264, 640 (1986). S. Mukhi, Phys. Lett. B, 16__/2,345 (1985).

STRING OPERATOR FORMALISM AND FUNCTIONAL INTEGRAL IN THE HOLOMORPHIC REPRESENTATION A. S. Losev, A. Yu. Morozov, A. A. Roslyi, and S. L. Shatashvili The connection between a functional integral over open Riemann surfaces [i] and the operator formalism on closed Riemann surfaces [2] is discussed. The states in the operator formalism are a holomorphic representation of the functional integral. i. Several recent papers [1,2] have been devoted to the calculation of functional integrals I l over Riemann surfaces Z with boundary r. The integrals I E are important in string theory, since they satisfy the so-called sewing algebra: if the surface Z is obtained from the surfaces E l and 7. by identifying some components of the boundary of the 2 surface Y l with the same number of components of the boundary of the surface 72, then

Iz
where y is plication, a surface grals over

=

lz,yl~,,

the common part of the boundary of the surfaces E l and Y2, and ~ is some multiwhich will be described below. Using this "sewing," we can construct 17. for of arbitrarily high genus from simple blocks, for example, from functional inte"trousers" (spheres with three deleted disks). 9 f~ is a

In the coordinate approach [i] for scalars with action

functional of the values of the field on the boundary and is equal to Iz (~)r)= (det0 Az)-'l' exp (--S(@h(r where Ch(~r) is a harmonic function on E, equal to Cr on the boundary, and det 0 A~. is the determinant on functions on E with zero-value boundary conditions. In this approach, "sewing" along the common boundary y means integration over the fields on ~: ( 1)

The operator approach [2] considers quantization near boundaries, which leads to a certain Heisenberg algebra. In this approach, 17. is regarded as a state in the representation space of this algebra that can be obtained from the vacuum by a certain Bogolyubov transformation, the "sewing" being specified by the scalar product in the Hilbert space corresponding to the common boundary. In this paper we show that for scalars the operator formalism is the holomorphic representation for the functional integral. 2. For simplicity 9 we consider the case of a connected boundary. to the case of a disconnected boundary is trivial. The generalization

It follows from (i) that S(~ F) determines IZ(r F) up to a constant factor (det 0 AZ)-89 Therefore, we first calculate S(~F), and we then find det 0 A Z from the sewing algebra. Let z be a coordinate near F such that Izl = 1 on the boundary and Izl > 1 for other points of Z near the boundary. The field #r on the boundary can be specified by means of coefficients #n in the following Fourier series: Cr=~0+Z
9 .+0

~"
Ylnl

e+"% r

Institute of Theoretical and Experimental Physics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 78, No. 3, pp. 475-479, March, 1989. Original article submitted October 17, 1988. 0040-5779/89/7803-0337512.50 9 1989 Plenum Publishing Corporation 337