Summary. -- In this article I make some comments on the string theory. In ... Consider a relativistic string moving in a d-dimensional curved space-time M d . As.
IL NUOVO CIMENTO
VOL. 110 B, N. 2
Febbraio 1995
Comments on the String Theory. J. A. NIETO (*) (**) Instituto de Fisica de la Universidad de Guanajuato Apartado Postal E-143, C.P. 37500, Ledn, Guanajuato, Mdxico (ricevuto il 27 Settembre 1994; approvato il 26 Ottobre 1994)
Summary. - - In this article I make some comments on the string theory. In particular, I propose to unify the world surface metric g~b($a) and the target space-time metric G,~(x ~) with a unified metric G~(x~($c), ~d) which in some special case may be separable, G~ = ~z-L-ggabG,~. PACS 04.20 - General relativity.
Consider a relativistic string moving in a d-dimensional curved space-time M d . As the string propagates in such a background it sweeps a two-dimensional surface, which (in analogy to the world line of a relativistic point-particle) is called world surface. Let us denote this world surface as B 2 . This world surface swept out by the string in the course of its evolution may be parametrized by two arbitrary parameters ~ a _ (7", (r) (with a = 0, 1), where 7" is a time-like parameter used to describe the evolution of the system and ~ is a space-like parameter describing points along the string. The position of the string with respect to a reference frame attached to the curved space-time may be described by using the coordinates x'(~a), where /~ = 0, 1, ..., d - 1. (The functions x ' ( $ ~) give a map of the world surface B e into the physical space-time Md.) The action associated to the string m a y be written as [1-3] (1)
f S = -1- I d 2 $~-L-~g ab ( ~ c) 3x" 3x~ G~(XtL(~a)) 2 J ~a ~b '
where gab(~ c) is a metric on the world surface, G~,v(x~'($a)) is the metric of the target space-time, and g is the determinant of gab. (*) Supported in part by CONACyT grants 1683-E2909, and F246 E9207, and by CoordinaciSn de InvestigaciSn Cientifica de la UMSNH. (**) Permanent address: Escuela de Ciencias Fisico-Matem~ticas de la Universidad Michoacana de San Nicolas de Hidalgo, Apartado Postal 749, C.P. 58000, Morelia, Michoac~n, M~xico. 225
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J.A. NIETO
Now, I would like to raise the following questions in connection with action (1): If string theory has been proposed as a candidate to unify all fundamental forces, why not to start first by unifying things in the action (1) itself?. For instance, one can notice from (1) that the two metrics gab and Gs~ play different roles. The question is whether these two metrics can be unified in only one metric. In this brief paper I would like to propose the action 1 S = 2
(2)
d ~G~ axs ox~
~a ~ b '
where G ~ ( x ~ ( ~ ) , ~d) is a unified metric with determinant defined by (3)
G -
_
_
d!(2) d/2
~Sl...Sd~Vl".vd,
F. a l a 2 ~ b l b 2
Ga~b~
. . . ,~ClC2 F. d l d 2 ~ S l V l
..
, G ~S d Vd~ d
,
and inverse metric G~ satisfying the relation (4)
~cb
s b
Note that the definition (3) works only for d even. It is not difficult to see that the action (2) reduces to the action (1) in the special case when the unified metric G ~ is separable, (5)
G2(x~(~c),
~d) = Vr-Z_ g g a b ( ~ d ) G ~ , ~ ( X ~ ( ~ ) )
.
Of course, the question now is to see whether a unified metric of the form G ~ is possible. The idea here is to give general arguments in order to justify why a ,,unified, metric of the form G2~ must be possible in physics. The first argument comes by comparing the metric G ~ with a Yang-Mills gauge field A2. One can say that an important step in physics was given when the vector potential A s of the electromagnetism was generalized to the gauge field A~. Symbolically, this transition may be written as (6)
A s ---)A;.
Similarly, one should expect important consequences in physics by making the change (7)
Gs~-.G;b .
In general, the role of the internal index a in A~ may play a very different role than the indices a, b in G~. In the case of the gauge field A~, the index a is related to internal symmetries according to a certain group G, while the index a, b of G~b must be related to geometric aspects of an internal manifold. If the manifold is a group manifold or a coset manifold, the difference between the character of the internal index of A~ and G2~ must stretch to dissapear. Another reason to be physically interested in the unifed metric G ~ is provided by a number of works due to Tapia [4-6] in connection with fourth-rank gravity. The central idea of Tapia's work is to describe the metric properties of space-time not through a quadratic line element, (8)
d8 2 = Gp.vdxt'dx ~ ,
COMMENTS ON THE STRING THEORY
227
but rather through quartic line element, (9)
ds a = G , ~ d x ~ ' d x ~ d x ~ d x ~ .
Here G ~ (x ~) is a completely symmetrized fourth-rank tensor. Tapia used the metric G,~z to construct a new theory of gravity, which reduces to general relativity in the vacuum case. Further, he was able to show that a conformal field theory possesses an infinite-dimensional symmetry group only if the dimension of space-time is equal to the rank of the metric. So, a conformal field theory with infinite symmetries may be constructed in four-dimensions with a fourth-rank metric G,~z rather than with a second-rank metric G,~ as in string theory. Finally, Tapia argued that in the low-energy regime of the new gravitational theory the fourth-rank metric G ~ must lead to separable spaces: (10)
G , ~ = g(,~ g~z),
where the brackets in (10) mean completely symmetrized. In this case the line element (9) factorizes and one is back to the usual Riemannian case. What is important is that Tapia's work provides enough motivation to be physically interested in a fourth-rank metric G,~z. Now, what I would like to argue is that the unified metric G ~ may be closely related to the metric G ~ proposed by Tapia. Let us assume that the metric G ~ corresponds to (d + 2) dimensional space-time. So, the indices ~, ~ run from 0 to d + + 2 - 1. In this context one can try to write G ~ in the block form: (11)
G;,~(x;,)=(G~ G~abI k G ab~
G abcd] '
with G ~ ( x ~', x a ) , and a, b = d, d + 1. So the quantity G~,vab(X ~', x a) is very similar to the quantity G ~ (x '(~ a), ~ a) associated here to string theory. The only differences are that in G , ~ ( x ' , xa), x ~is not necessarily function of the coordinates x ~ , which may be indentified with the parameters ~ of G ~ ( x ~ ( ~ ) , ~ ) and also that the indices a, b appear as low-case instead of upper-case indices. Further, the form (11) which breaks the metric G ~ in a block form reminds us the procedure used in Kaluza-Klein theories. From this point of view one can think of (11) as some kind of Kaluza-Klein procedure in which one is making a transition of the form (12)
Md+2-o M d • B 2 ,
with the difference that B 2 is here a non-compact space. Finally, I would like to comment on the possibility to have a metric G ~ in (2) not necessarily symmetric in the indices t~, v and a, b. If this is the case, there must be a special case in which the metric G ~ is separable in the form (13)
G a~ = ~
gab G,~ + ~abA,~,
where A,v is antisymmetric gauge field. Notice that the first term is very similar to the expression (10) corresponding to separable spaces of G ~ according to (11).
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j . A . NIETO
Using (13) the action (2) becomes (14)
S
1 2
d2~
gab ~x~ ~x~ G~(x) + s ab ~x~ Y ~ ~E.b ~ ~ bA~(x)
This action is usually referred to as a non-linear sigma-model. By adding to the action (14) the term (15)
S' = I d25~L-g(~)R(~) r
one obtains the more general sigma-model. The action (14) is renormalizable and Weyl-invariant, while the term (15) is renormalizable but not Weyl-invariant [7-10]. The arbitrary functions G,~(x), B,y(x) and r correspond to the three massless modes of a closed string: graviton, antisymmetric tensor and dilaton. A nice picture arising by quantum-mechanical considerations of the actions (14) and (15) is that the general non-linear sigma-model corresponds to a string propagating in a condensate of its massless modes. What this means is that although classically the functions G,~(x), B,~ (x) and r are arbitrary, quantum-mechanically they must be a solution of the field equations (16)
R~ - 1H~H~ 4
(17) (18)
-
2(d
26) 3=
+ 2V~V~r = 0, = 0, 1
+ 4(Vr 2 - 4V~r - R + ---H 2 = 0, 2
which are obtained by the calculation of the fl-functions to one-loop order [7-10]. In these equations H,~ = V[,A,~I is the antisymmetric tensor field strength. From this perspective one may conjecture that similar quantum-mechanical techniques must be applied to the action (2), leading to the interpretation that the action (2) may be understood as string propagating in a condensate of the unified metric G~. So, although the metric G2~ may be arbitrary at the classical level, at the quantum level it must satisfy some still unknow field equations. Thus, the expressions (5) and (13) may be understood as two possible solutions of such field equations.
I wish to thank J. Socorro and G. Moreno for helpful discussions. Furthermore, I would like to thank Dr. 0bregSn and the IFUG for its hospitality.
REFERENCES [1] BRINK L., DI VECCHIAP. and HOWE P., Phys. Lett. B, 65 (1976) 471. [2] POLYAKOVA. M., Phys. Lett. B, 103 (1981) 207. [3] For a review of string theory, see GREEN M., SCHWARZJ. and WITTEN E., Superstring Theory, Vol. I and II (Cambridge University Press) 1987; KAKU M., Introduction to Superstrings (Springer-Verlag, New York, N.Y.) 1990.
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[4] T~L~ V., Fourth-rank gravity, a progress report, ICTP preprint (1992). [5] ThP~ V., Motion in fourth-rank gravity, ICTP preprint (1992). [6] TAP~h V., Beyond two dimensions, communication of the Joint Institute for Nuclear Research E2-91-316, Dubna (1991). [7] FRADKIN E. S. and TSYETLINA. A., Nucl. Phys. B, 261 (1985) 1. [8] CALLAN C. G., FRIEDAN D., MARTINIC E. and PERRY M. J., Nucl. Phys. B, 262 (1985) 593. [9] SEN A., Phys. Rev. Lett., 55 (1985) 1846. [10] CALLAS C. G., Topics in the string theory, in Proceedings of the Winter School on High Energy Physics, Superstrings and Grand Unification, P u ~ India) 1988 edited by T. PRADHAS (World Scientific) 1988.
