Superstrings from 26 dimensions?

Volume 151B, number 5,6

PHYSICS LETTERS

21 February 1985

SUPERSTRINGS FROM 26 DIMENSIONS? Peter G.O. FREUND 1 Institute for Theoretical Physics, University o f California, Santa Barbara, CA 93106, USA Received 13 November 1984 The finite type I superstdng theories of Green and Schwarz (SO(32) and (?)E a × E8) in ten dimensions are viewed as special dimensional reductions on 16-tori from the non-supersymmetricVeneziano-Nambu-Goto strings in 26 dimensions. Fermions appear as solitons of the two-dimensional string field theory. Various problems of such an approach are pointed out and possible solutions outlined.

Green and Schwarz [1] have discovered that the N = 1 supersymmetric theory of open and closed strings with SO(32) Chan-Paton rules [2] is anomalyfree and very likely finite. The freedom of anomalies, though not the finiteness, persists in the Scherk limit [3] of point-like strings (a' ~ 0). In this limit the theory reduces to a coupled N = 1 supersymmetric SO(32) Yang-Mills plus supergravity system with some additional terms [ 1]. Were one, in this Scherk limit, to change the gauge group G, anomaly cancellation would occur also for G = E 8 × E 8 (which has the same rank and dimension as SO(32)), but for no other compact semi-simple group [3]. The existence of superstrings with E 8 × E 8 symmetry is therefore expected. Alas, no generalization of the Chan-Paton rules to exceptional groups has, so far, been found, and this theory has not yet been constructed. Both the SO(32) and the E 8 X E 8 superstrings (should the latter exist) would allow chiral fermions upon dimensional reduction to four dimensions [4]. They both look phenomenologically promising: one of them could be a "theory of the world", which unifies all interactions (gravity included) and all forms of matter in a consistent manner. Yet, two theories of the world are one theory too many. If one of these two theories had an as yet undiscovered flaw not shared by the other, this, of necessity, would decide the uniqueness problem. Here we contemplate the possibility that both the SO(32) and E 8 × E 8 theories 1 On leave from the Department of Physics and The Enrico Fermi Institute, The University of Chicago,Chicago, IL 60637, USA. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

are equally problem-free, and are but two realizations of one and the same underlying theory. Specifically we propose that this primordial theory involves d > 10 space-time dimensions, and that the SO(32) and E 8 X E 8 superstrings in ten dimensions are but different compactifications of this primordial theory. I shall argue that d = 26 and that the primordial theory is the old non-supersymmetric Veneziano-NambuGoto (VNG) string [5]. Right away this raises a,number of questions: (a) How does the compactification of the VNG string produce the Green-Schwarz groups SO(32) and E 8 X E8? (b) How does one account for the appearance of fermions and supersymmetry in the resulting tendimensional theory? (c) How do the ten-dimensional theories free themselves from the tachyon and from other diseases of the VNG string? (d) How do the Chan-Paton rules in ten dimensions emerge? In an attempt to justify this picture, questions (a)-(d) will be addressed below, though no complete answers will be provided. A string theory is a conformaUy invariant twodimensional o-model. As such, it admits the infinite Virasoro algebra of two-dimensional conformal transformations, and is uniquely characterized by the KacMoody algebra of its "currents". Any two conformally invariant two-dimensional quantum field theories that share the same "current algebra" are equivalent [6]. Consider, on the one hand, the two-dimensional 387

Volume 151B, number 5,6

PHYSICS LETTERS

o-model corresponding to the rank-26 Lie group SO(44) X [:! 4 along with a Wess-Zumino term with quantized coefficient N = 1, at the value of the coupling constant at which the/3-function vanishes and conformal invariance is restored. On the other hand, consider the 26-dimensional VNG string, dimensionally reduced to Minkowski four space-time on a 22torus T22 with all 22 radii equal. Based on results of Frenkel and of Goddard and Olive [7], Witten [8] has shown these two, on the face of it such very different theories, to be equivalent in the above sense. Notice that the rank of the group SO(44) is the same as the dimension of .~22. This torus is the factorization 1:122/A22 of euclidean 22-space I:122 by the 22dimensional hypercubic lattice A22. The crucial feature is, as will be recalled below, that setting the lattice spacing equal to one (unimodular lattice), A22 contains precisely 924 vectors of length squared equal to 2 and that these vectors present the root diagram of the Lie algebra d O (44). In the same way, the (26 - d)-torus T 26-d with all radii equal is R26-d/ A 26--d (A26~/being the (26 - d)-dimensional unimodular hypercubic lattice). Again the vectors of length squared equal to 2 in the A26_ d span the root diagram of dO(52 - 2d), the rank of which is 26 - d , the common dimension of A26_ d and of ~26-d. Now let us specialize to the case where we reduce from 26 down to d = 10 dimensions. The lattice A26_10 = A16 describes the quantization of 16 momentum components when the string theory is compactified on ~16. At every vector k E A16 we can define a Nambu vertex operator U(k,z) [5,7]. Consider now the operators U(ki,z ) i = 1 ..... 480 where k i are the 480 vectors of length squared equal to 2 (k i2 = 2) of the unimodular lattice A16. From these U(ki,z ) by suitable contour integration over the complex variable z, and some further manipulations, we recover [7] 480 operators which, together with the components of the momentum operator in the 16 lattice directions, span the Lie algebra gAl 6 = d O (32). Taking higher moments in the z-integration yields the K a c Moody algebra g'A -" Its representation theory ensures the presence of 4 ~ massless vector particles in the spectrum, i.e., of a gauged SO(32) symmetry. It now appears as if we had arrived at a bosonic string theory in 10 large and 16 compactified dimensions in which a gauged SO(32) symmetry is present. But repeating the just-mentioned construction, this time 388

21 February 1985

choosing for k i the unit vectors of A16 , one can reconstruct [7] the fermionic oscillators of the NeveuSchwarz and Ramond model as well. Fermions are thus obtained out of bosons, but then a two-dimensional system is involved here, so that, as in the sineGordon and other two-dimensional systems, fermions can arise in an originally bosonic theory [9]. These fermions are solitons and they fall in a supersymmetric pattern (for the Ramond-like excitations, their spinorial character, as far as ten-dimensional spacetime is concerned, is not clear, though). So a kind of dynamical supersymmetry appears. We believe this ten-dimensional supersymmetric theory contains the SO(32) Green-Schwarz superstring. Before continuing on this point, let us now consider the E 8 × E 8 theory. Already in the SO(32) case, we considered the SO(32) o-model with N = 1 WessZumino terms at the zero of its/3-function. To get the proper central extension in the Virasoro algebra, the rank of the group had to be 16. The whole world manifold R 10 X SO(32) is then a 506-dimensional group manifold (Minkowski 10-space R l° is, after all, an abelian group itself) of total rank 26. What in the older literature is referred to as the critical dimension having to be 26, can now be rephrased as the critical rank equalling 26. (This raises the question as to whether 26 or 506 is the "true" dimensionality of space-time in this case.) In that sense, any other candidate group must also have rank 26 which is the case for E 8 X E 8 X R 10. There exists a well-known even unimodular integral lattice F 8 + 1`8, obtained by iterating the 16-dimensional root diagram of E 8 X E 8 [10,11]. I"8 + 1"8 and another lattice 1"16, whose vectors of length squared span the root diagram of d O (32), are the only two 16-dimensional euclidean, even unimodular, integral lattices up to isomorphism [10-12]. The c50 (32) root diagram, as we have seen, also sits in A16 an odd lattice which contains unit vectors. By contrast, there exists no 16-dimensional odd integral euclidean unimodular lattice containing unit vectors such that its length squared two vectors span E 8 X E 8 [13]. The construction of the Neveu-Schwarz-Ramond sector in the E 8 × E 8 case is therefore still open. Maybe dropping unimodularity will help. With the Wess-Zumino parameter set at N = 1 the rank of the group G (other than Minkowski space) had to equal 26 - d, the dimension of the toms T 26-d .

Volume 151B, number 5,6

PHYSICS LETTERS

For Wess-Zumino integers N 5e 1, this is replaced by the condition [14]

r = (26

- d) Nd G + (NS-1)(d

-

26)

with d G the dimension of the group G (for N = 1 this reduces to r = 26 - d, as it should). For d = 10, the case considered here, this allows the further groups G 2 , SU(3), SU(4) ~ SO(6), and SO(8) for N = - 4 8 , N = - 6 , N = - 6 4 , and N = +8, respectively. (Similar considerations for d = 4 have been made earlier by Friedan [15], see also ref. [4] .) We now return to the problem of recovering the SO(32) superstring theory from the compactified VNG string and to the questions (b)--(d) posed in the introduction. First of all, the VNG string has a tachyon [5] which means that one is expanding around an unstable vacuum, which, as pointed out by Gross and Witten [16], is not even a classical solution. The tachyon cannot possibly appear in the superstring, on account of tendimensional supersymmetry. But in a superstring theory, there are two kinds of supersymmetry, the twodimensional conformal supersymmetry on the string's world sheet, and the Poincar6 supersymmetry of the ten-dimensional host space. In the original NeveuSchwarz-Ramond formulation the latter supersymmetry was not enforced and tachyons were present. They were then eliminated via a projection onto an even "G-parity" sector [17]. Coming from 26 dimensions, whatever supersymmetry is achieved via the Frenkel-Goddard-Olive construction is also of a world-sheet type, and a further projection is called for onto a ten-dimensional Poincar6 supersymmetric sector. Second, the original 26-dimensional theory is not finite or even consistent [16]. Toroidal compactifications can be contemplated toward final dimensions other than 10. But only in the critical ten-dimensional case is there a chance of obtaining a non-anomalous super-Virasoro algebra. Maybe this drives the inconsistent [16] 26-dimensional theory down to ten dimensions. The original 26-dimensional string has no fermions and no self-dual antisymmetric tensor fields. Then it cannot have gauge or gravitational anomalies. This may be the reason why the reduced superstring theories are anomaly-free themselves.

21 February 1985

Now to the Chan-Paton rules: These rules were inspired by the two-component duality exhibited by hadronic reactions [18]. In string theory they occupy a rather peculiar place. One produces a detailed dynamics for a string devoid of internal attributes and then appends these Chan-Paton rules as an afterthought, not at the lagrangian level, but when calculating S-matrix elements. What is the dynamics behind these rules? One can require the internal attributes to be carried at the ends of the string. In the SO(32) case, the ends carry the fundamental representation 32, so that the open strings carry the representations contained in the product 32 X 32. The rules vouch for the consistency of this picture. A priori, one could have expected strings in higher representations of SO(32) as well. But the rules assure the consistency of having intermediate one-partiele states in all channels only in those representations contained in the product 32 X 32. For E 8 X E 8 such a truncation does not seem possible, as suggested also by the o-model picture. This raises the general question as to what algebraic combination laws for the internal attributes of strings are compatible with duality. At the simplest level, is there a setA o f E 8 X E 8 representations including ( I , 1) + (1,248) + (248,1) which is "closed" under crossing, in that particle poles in representations from the set A in the s-channel involve, by duality, poles again only in representations from the set A in the t- and u-channels for all combinations of external particles. One may also ask whether there are string models, which, while producing the same a' ~ 0 Scherk limit as the SO(32) superstring, involve for finite a' not just the representations contained in the product 32 X 32 but also higher ones. In other words, are the Chan-Paton rules the unique prescription even in the SO(32) case? A general theory of internal attribute algebraic rules is needed (with usual Chan-Paton rules only O(n) and Sp(n) groups are allowed; E 8 X E 8 is ruled out [19]). In conclusion then, the rank 16 of the groups SO(32) and E 8 X E 8, along with Witten's a-model considerations, suggest that one derive these superstrings from a 26-dimensional string theory. The VNG string is an obvious candidate, and the more obvious difficulties (lack of fermions, supersymmetry .... ) can be solved taking advantage of the remarkable properties of two-dimensional field theories (fermionic solitons, Kac-Moody algebras .... ). Ultimately, only 389

Volume 151B, number 5,6

PHYSICS LETTERS

a detailed derivation of the ten-dimensional superstring from the 26-dimensional VNG string can vindicate this admittedly quite unorthodox point of view. Still, one cannot but find it ironical that a theory originally meant to describe the hadronic S-matrix and then abandoned, should now, whether in tenor 26-dimensional guise, make a comeback as a field theory " o f the world". I wish to thank Professor J.R. Schrieffer and Professor R. Sugar for their kind hospitality at the Institute for Theoretical Physics and Professor D. Friedan, Professor J. Schwarz and Professor E. Witten for their patient criticism and comments. This material was based upon research supported in part by the National Science Foundation under Grant Nos. PHY8301221 and PHY77-27084, supplemented by funds from the National Aeronautics and Space Administration.

References [1] M.B. Green and J. Schwarz, Phys. Lett. 149B (1984) 117; 151B (1985) 21. [2] J.E. Paton and H.M. Chan, Nucl. Phys. B10 (1969) 516; T. Matsuoka, K. Ninomiya and S. Sawada, Prog. Theor. Phys. 42 (1969) 56;

390

21 February 1985

H. Harari, Phys. Rev. Lett. 22 (1969) 562; J. Rosner, Phys. Rev. Lett. 22 (1969) 689. [3] J. Scherk, Nucl. Phys. B31 (1971) 222. [4] E. Witten, Nucl. Phys. B, to be published; S. Randjibar Daemi, A Salam and J. Strathdee, Phys. Lett. 132B (1983) 56. [5] See, e.g., C. Rebbi, Phsys. Rep. 12C (1974) 1. [6] E. Witten, Commun. Math. Phys. 92 (1984) 455. [7] I.B. Frenkel, preprint; P. Goddard and D. Olive, preprint. [8] E. Witten, unpublished; see also P.G.O. Freund, P. Oh and J.T. Wheeler, Nucl. Phys. B, to be published. [9] S. Coleman, Phys. Rev. D l l (1975) 1088; S. Mandelstam, Phys. Rev. D11 (1975) 3026. [10] E. Witt, Abh. Math. Sem. Univ. Hamburg 14 (1941) 323. [11] J. Milnor, Proc. Natl. Acad. Sci. USA 51 (1964) 542. [12] J.-P. Serre, A course in arithmetic (Springer, Berlin, 1973) pp. 55,110. [13] M. Kneser, Arch. Math. 8 (1957) 241. [14] A. Rocha-Caridi, private communication to D. Friedan. [15] D. Friedan, private communication. [16] D. Gross and E. Witten, private communication. [17] J. Schwarz, Phys. Rep. 8C (1973) 269. [18] R. Dolen, D. Horn and C. Schmid, Phys. Rev. Lett. 19 (1967) 402; P.G.O. Freund, Phys. Rev. Lett. 20 (1968) 235; H. Harari, Phys. Rev. Lett. 20 (1968) 1395. [19] N. Marcus and A. Sagnotti, Phys. Lett. l19B (1982) 97.