593 * WHY IS A STRING THEORY ULTRAVIOLET FINITE? Chung-I Tan Physics Department Brown University Providence, Rhode Island 02912, USA Abstract String theories have recently been regarded as candidates for the fundamental theory of nature. In this short review, an elementary exposition is presented on the key ingredients involved for having an ultraviolet finite quantum theory. * This work was supported in part through funds provided by the S. u. Dept. of Energy under Contract No. DE-AC0 2-76 ER03130 .A013 - Task A. 594 I. INTRODUCTION: String theories, developed in the late sixties and early seventies, have enjoyed a great recent comeback, after having remained dormant for more than a ) decetde They have been widely regarded as candidates for the ultimate unified � theory of Nature, chiefly because they are capable of leading to finite theories of quantum gravity. Although they were originally formulated in an S-msctrix framework, they may yet be transformed into some kind of "generalized" quantum field theories. Whatever the outcome, they certainly differ greatly from the conventional quantum field theories (QFT) whkh we have grown acc�.stomed to in recent years. For physicists not working on the subj ect, particularly the experimentalists, the field seems mysterious. In this short review, I will present an elementary exposition on the key ingredients involved for having an ultraviolet finite quantum theory. The ultraviolet behavior of a Feynman integral with loops can always be D characterized by d p.p-q, where D is the number of space-time dimensions and q f is an integer depending on the number of propagators i1wolved. Since one cannot avoid having diagrams with D > q, QFT is meaningless without a certain amount of massaging. Making sense out of nonsensical mathematical expressions has been the preoccupation of field theorists for the past half century . In each case, one always manages to find a "taming operator" such that when it is applied to a Feynman integral, a finite number is obtained. Indeed, the history of QFT can be loosely described as that of taming the ultraviolet divergences. Finding the correct taming operator requires the understanding of quantum physics. First of all, a renormalization procedure muut be adopted. To proceed further, one finds that symmetry plays a crucial role. Starting with the gauge invariance of QED, next the non-abelian gaug" theories, either for QCD or the standard electroweak theory, and more recently supersymmetry and supergravity, it has always been the symmetry which is chiefly responsible for the divergence cancellation. Identifying the relevant symmetry is therefore the key step involved in the construction of a taming operator. 'The question remains whether this scenario can go cm indefinitely within the context of the conventional QFT. The problem becomes particularly acute in regards to gravity--it appears that the usual procedurE! cannot render the theory finite; loop divergence cannot be removed. It c:ould happen that a more clev1er introduction of new symmetries, hitherfore unreC'.ognized, would again lead to a finite quantum gravity within the conventional field theory fram«1work, it is equally plausible that a new avenue of approach should be adopted. Indeed, string theory is the most likely candidate. 595 II. LOOP DIVERGENCE IN A PARTICLE FIELD THEORY : To illustrate the problem, let us consider the simplest situa�ion of a 2 2 -1 -D D bosonic propagator, G(O,x) = J dp (p +m ) exp (ipx) , where dp = (271") d p. Graphically, we normally represent this propagator by a line segment, from the point O to the point x. This symbolic line segment certainly does not correspond to the actual traj ectory of the particle; being a probability amplitude, it can be reexpressed as a sum over all allowed paths from 0 to x. This "world-line" representation becomes particularly useful for our present 2 2 -1 purpose if we consider the "loop" diagram, G(0,0) = fdp (p +m ) , which is alway s ultraviolet divergent . -l Using the identity a = .fds exp (-as) for the integrand and next performing the momentum integration, the propagator can be written as G(0,0) (1/2) D(s) W(s) P(s) f (1) where D(s) is the ordinary integration measure, ds, with the integration -D/2 2 domain O�s<cc , W{s) = (2s) and P(s) = exp (-m s/2) . This can be interpreted as a sum over closed paths according to their path lengths, s. For a fixed length s, the probability amp litude is P(s) and the density of paths is W(s) . The divergence of the original Feynman loop integral has UV now been translated into the divergence of the world-line sum, (2) , at s=O. That is , the divergence in this sum comes from the small loop UV contribution, as is expected intuitively . More interestingly, the zero-point (vacuum) energy of a point particle can also be expressed in 2 terms of the propagator: V(m ) = - Tr ln G, which is again a divergent 2 one-loop integral . In the world-line representation, V(m ) is again given by ( 1) if one substitutes the measure by D(s) -ds/s. Again, the = UV divergence can be associated with the divergent contribution from small loops in the world-line representation. Although these examples are of no imme diate physical consequence, other divergences must be regulated in order to provide a meaningful quantum field theory. Using the vacuum energy as a typical examp le of a potentially divergent loop integral , we shall see next that in a string theory, due to the presence of a new symmetry, ultraviolet divergences are removed automatically. 596 III. MODULAR INVARIANCE AND ULTRAVIOLET-FINITENESS : Whereas a particle corresponds to a point, a string can be parametrized by a line segment, (a circle in the case of a closed string) , a closed string evolves, it traces out a cylindrical surface in the As space-time continuum. Therefore, the vacuum energy for a closed string, in constrast to a closed path in the csse of a particle, is charac terized by s two dimensional torus. Instead of a world-line sum, a ''world-sheet" sum over sll distinct tori can be used to represent its vacuum energy. Is the ultraviolet divergence present doe to "short" tori? One must first learn how to label a torus. Given a parallelogram, a torus can be constructed by identifying the opposite sides. Since a parallelogram can be specified by two independent basis vectors w ' w in a l 2 plane, any function f over the torus can in turn be thought of as a periodic function on a two-dimensional Euclidean space such that f(r) f(r+n w +n w ) with n , n arbitrary integers . Furthermore, since a 1 1 2 2 1 2 two-vector can equally well be represented by a complex number, we will treat r, w and w as complex numbers in what follows . It is slso clear 1 2 that only the length ratio and the relative phase between w and w are 1 2 relevant, which can be characterized by the ratio i: = w /w , with the 2 1 convention Im(T)>O. We therefore tentatively arrive at a conclusion that distinct tori can be specified by points in the upper-half complex plane. The vacuum energy for a free closed string can next be obtained, e.g. , 2 by summing V(m ) over its particle spectrum. In particular, we can identify the parameter s, up to a factor, with Im(-r ). Writing T as T +iT , l 2 2) one arrives at an exp ression similar to (1) V(string) -(1/2) D(torus) W(T ) P(T) (2) .{ 2 where P(T) can be expressed in terms of known func tions. The integration measure for the torus can be chosen as dT dT /T • What remains to be 1 2 2 specified is the domain of integration. The association of tori with points of the upper-half complex plane is not one-to-one. For instance, it is not difficult to convince oneself that changing the pair of vectors (w ,w ) to (w ,w +w ) yields the same torus. 1 2 1 2 1 Therefore, there is a redundancy , described by the transformation T: T+T+l. Provided that the integrand in (2) also remains in'•ariant under T, we can immediately restrict the integration domain to the strip T >0 and -1/2� T 2 l �1/2. In this parametrization for the world-sheet sum, T is analogous to 2 the parameter s of (1) , so that it characterizes the "length" of a torus. 597 l A similar analysis shows that we also have s symmetry S: T+-T- , which can be understood as changing the pair (w ,w ) into (w -W ,w ). This 1 2 1 2 2 allows us to further restrict I T l>l. Together with our earlier restriction, we arrive at the fundamental domain of integration for (2l: T >l, -l/2�T I l l �1/2, 0�T <o:i. In particular, the "zero-length" torus region has now been 2 excluded and the vacuum energy is now ultraviolet finite ! The totality of transformations generated by S and T forms the "modular group". It is a symmetry group of any function defined over distinct tori. At a classical level, a string theory obviously possesses this property . However, this symmetry may not be preserved at a quantum level, a phenomenon we have grown accustomed to, generically referred to as an "anomaly". One of the most far-reaching findings of the string studies is the fact that this symmetry can be preserved at certain critical space-time dimension, e.g.