LANDAU-GINZBURG HAMILTONIANS, STRING THEORY AND CRITICAL PHENOMENA

R Howe and P. West Dept. of Mathematics King's College London WC2R 2LS Abstract The identification of fixed points of N=2 super symmetric Landau-Ginzburg models with N=2 super conformai minimal models is reviewed. We also discuss group flows, the role of singularity theory and the Gepner conjecture.

There may be some reasons to believe that The subject of this talk has a bearing on a the resulting space-time theory should possess space number of different areas. It is relevant to our un­ time [1]. This implies that the derstanding of string theory, the theory of critical string should possess N=2 world-sheet supersym­ phenomena and, more surprisingly, to the math­ metry. In this case, we are interested in N—2 su- ematical theory of manifolds. We begin by out­ persymmetric Landau-Ginzburg Lagrangians and lining the connection to the first two subjects. these form the subject of this review. When string theory first emerged in the 70's an obstacle to their being considered as realistic We begin with an elementary account of crit­ models of nature was that they were only consis­ ical phenomena. We include it as an understand­ tent in 26 dimensions for the bosonic string and ing of this subject is basic to the understanding 10 dimensions for the super string. In more re­ of supersymmetric Landau-Ginzburg theory, and cent years it has become apparent that consis­ because it is perhaps not familiar to a number tent strings exist in other dimensions including of theoretical physicists working in this field. In­ four dimensions. We now believe that given any deed, there are a number of incorrect statements two-dimensional conformai system we can form in the literature on N = 2 Landau-Ginzburg mod a superstring theory by coupling it to supergrav- els and we shall comment on some of these later. ity. Such a string is thought to be consistent if A familiar example of a critical point is the there are no worldsheet anomalies, such as the point TC = 647°#,pc = 218a*ra for H20 where conformai anomaly, and, in particular, if it is water and steam coexist. The transition that modular invariant. The former condition is to takes place at this point is a second- order phase be expected as we have a local conformai symme­ transition, i.e. one involving no latent heat. In try while modular invariance ensures that certain general more than two phases can coexist at a space-time anomalies such as gauge and gravi­ critical point; such critical points are known as tational anomalies are absent.* Unfortunately, multi-critical points. there are very, very many such string theories, Critical theory is the study of a system in corresponding to the vast number of different two- the neighbourhood of its critical point. For H20 dimensional conformai systems. There is no known one finds that if P is the density of water and PC method of picking out a preferred string theory. is the density at the critical point then P — CP ~

However, as we shall see, Landau-Ginzburg mod­ (TC — T)P when measured along the water steam

J els provide a framework in which a number of the transition line and PC — P ~ (P — PC) at Tc. The interesting conformai models occur in one system study of critical systems has revealed that they depending upon which fixed point is selected. As possess two remarkable features, non- analyticity such, they may provide a framework for address­ and universality. The behaviour of cer­ ing the above problem of the many theories or tain quantities in the neighbourhood of the crit­ "vacua". ical point is given by the critical indices /?,£,..., which turn out to be non- integral thus implying

* More recently, the consistency of non- critical strings has been the subject of much study.

397 non-analyticity. It was also found that the criti­ Within the field theory context, there are two cal indices (3,6... are the same for systems which known techniques for finding the fixed point and have nothing in common except their dimension­ calculating critical indices in its vicinity: the Wil­ ality and symmetry. This property is referred to son Fisher e-expansion and the Parisi technique. as universality. For example, the critical indices We refer the reader to the reviews of references for the appropriate quantities for the CO2 liquid- [4] and [5]. In the Wilson Fisher ^-expansion, gas transition at 304°1Îl' and for the binary alloy the theory is considered in d dimensions and we Co - Zn at 759°K are the same. perform a perturbation expansion in the coupling and e = d — d where d is the critical dimension The explanation for these features was found c c of the theory, i.e. the dimension in which the in a series of papers culminating in the work of coupling constant is dimensionless and the the­ Wilson [2] who made essential use of the renor- ory classically scale invariant. At the end of the malization group. He supposed that the system calculation we set d to be the actual dimension lies on the critical surface of an infra-red stable of the system. For the three dimensional Ising fixed point. This means that when one looks at model d = 4 and so e = +1. The Parisi tech­ the system at larger and larger scales, the Harnil- c nique relies on the introduction of an appropri­ tonian approaches a scale invariant one. In other ate coupling constant which is well-defined at the words, the system at the fixed point is scale in­ fixed point. The perturbation expansion and the variant and is dominated by long range correla­ renormalisation group calculations can then be tions. Thus, at the critical point, short distance done using the Callan-Symanzik equation. For effects are unimportant, which is the reason for the the anomalous dimension of the universal behaviour. In addition it is possible to field 0 was found by these techniques to be 1.250 give a good description of the theory in terms ± 0.005 leading to a critical index which is cor­ of an effective continuum field theory. Such field rect to a few parts in 104. A similar, but much theories are called Landau-Ginzburg theories. less precise result was found for the two dimen­ A good illustration of this is provided by the sional Ising model [6]. The anomalous dimensions Ising model in three dimensions. The Landau- of various, in general, composite, operators in the Ginzburg Hamiltonian given by field theory are related in a straightforward way to the critical indices of the model. As the former are non-integral, so are the latter and hence the behaviour is non-analytic. We now turn to the discussion of two dimen­ 2 where m (T) oc (T - Tc) + 0((T - Tcf). The sional supersymmetric critical phenomena. Our partition function Z is defined by N=2 supersymmetric Landau-Ginzburg Hamilto­ nian is given by

where (3 = (&T)"1. Although we are dealing with a classical system we can use the techniques of quantum field theory to evaluate the path inte­ where the superfield (j> consists of the component gral. In the tree approximation, we are left with fields (a,XAj/) which are a complex spin 0, a Landau theory [3], which although not correct in complex spin | and an auxiliary field f. The su­ detail, provides at least some of the correct gen­ per Ising model corresponds to taking n = 3; but eral features. for larger n we have n-1 phases coexisting corre­ To follow the strategy of Wilson we must be­ sponding to a multi-critical point. lieve that the above theory possesses a /^-function We were able to show [7], [8], [9], using the with an infra-red stable fixed point even though e -expansion and also the Parisi technique that the theory is not even classically conformally in­ this N—2 supersymmetric model has an infra-red variant. It is easily seen that such a fixed point stable fixed point at which the superconformai is outside the range of usual perturbation theory, theory is described by the nth member of the N=2 for such an expansion must be in a dimensionless supersymmetric minimal series. parameter U — ^, however at the fixed point we have M —> 0 or U —» 00.

398 This result had been previously conjectured The last step in the derivation is to iden­ by Zamalodchikov [10] for N=0 and N=l multi- tify the field theory at a given fixed point with critical models and by Kastor Martinec and Shenker a member of the N=2 supersymmetric minimal [11] for the N=2 case. An alternative, but sub­ series which have c = 3(1 — ^),^ = 3,4,... The sequent, derivation using different methods was primary fields, <^jjm, of these models, are labelled given in reference [12]. by two integers j and m which are related to the We do not have the space in this review to eigenvalues of LQ and the U(l) symmetry gener­ give a full account of these proofs and we refer the ator To by [13] reader to references [7], [8] and [9] for more de­ tails. The essential steps are as follows. We first adapt the method of interest, i.e. the e-expansion or Parisi technique, to the supersymmetric case. We then prove the analogue of the four dimen­ respectively. The "chiral" primary fields are those sional non-renomalization theorem. For the Wilson- with m = ±j. In any conformai field theory, the Fisher e-expansion technique the proof of the non- two point functions are known and in these the­ renormalization theorem is along similar lines to ories those for the chiral primaries are given by

n the standard four-dimensional case, making use < H,T^>Hy-T >~ • O the other hand, in of superspace Feynman diagram techniques. In the Landau- Ginzburg model, the renormalisation the Parisi method the non-renormalization theo­ group summed propagator goes as ~ \zl\i* • Con­ rem is established by using the Callan-Symanzik sequently we identify H = Substituting our equations. The non-renormalization theorem is value for 7* we find H = ~ which agrees with then used to derive a relation between the /? func­ the dimension of the lowest primary field (J>IYI in tion and the anomalous dimension 7 of the field the nth minimal model. allowing us to compute 7 at the fixed point. It was also shown that at the fixed point To give the reader some insight we outline the central charge is c = 3(1 — ^), the anoma­ this last step for the Wilson-Fisher e -expansion lous dimension of the compposite operator ^m is technique. Let us perform the wave function and m/n, the operator product expansion (J)NM ~ coupling constant renormalizations ^n+m j7] an(j finaiiy that the chiral correlators <

(ZI)...(ZN) > calculated in the Landau-Ginzburg model and the conformai field theory correspond­

th where \i is the renormalization scale,

3(1 - 2qi) where qL = ~ is the UL{1) charge of . Using the definitions /? = fij-g |So>a»7 = In references [14] and [16] no calculations start­ |/x^/nZ |fl0jAj we readily find that ing from a Landau-Ginzburg Hamiltonian were performed. However, it was assumed that the perturbation series relevant to criticality satifies At the fixed point = 0 and so, to all orders a strong version of the non-renormalisation the­ orem, namely that there are no quantum correc­ of perturbation theory, tions to the effective action of the form of subsu- perspace integrals of local functionals. This as­ sumption, repeated in other papers, is in fact not true as was shown by an explicit calculation in Hence, in this case, unlike in previous examples [8]. A more detailed account of this will be given of the e-expansion, we find an exact result for the elsewhere. anomalous dimension. We can generalise the discussion of Landau- Ginzburg Hamiltonians given in this review to the case of many superfields

399 However, it is not clear that these two such disparate subjects are related in the sense that every potential which is in the singularity the­ ory classification list necessarily corresponds to a fixed point and so to critical behaviour in the physical sense. Indeed, one finds that there exist where examples of potentials in the singularity listing that are not quasi-homogeneous and so according to the previous discussion cannot correspond to fixed points. To examine the fixed points and correspond­ For an N=2 supersymmetric with one super- ing critical behaviour of such a Hamiltonian one field, however, one can show [9] that all fixed must use the Parisi method since the e-expansion points are of the form (/)n and this does coincide relies on the theory possessing a dimension for with the list of such potentials given in singularity which all the coupling are dimensionless. Carry­ theory. In this case one can also carry out explic­ ing out the analogue of the previous calculations itly the flow from one fixed for the many field case [9] we find that the /? - point to another [9]. functions are given by We end this review with two unsolved prob­ lems. Zamalodchikov has conjectured [10] a rela­ tion between the N=0 and N=l minimal series and fixed points of the corresponding Landau- Ginzburg models. The techniques described here which established this conjecture for the N=2 case where 7^ is the matrix of anomalous dimen­ were also used for the N=0 case, but the low or­ sions. The reader may recover the results for der results of the e-expansion were far away from the one field case given earlier. At a fixed point the minimal series results [19]. It is not possible

an one /3»i...tn = 0 d obtains a relation between to conclude one way or the other from these low the anomalous dimensions 7? and the couplings. order results, but these authors know of no con­ In fact, 7? is a positive definite symmetric matrix crete argument that supports the Zamolodchikov which may be diagonalized by a field redefinition. conjecture for N=0 and 1. This relation then has a simple interpretation in Finally, we will discuss to what extent the in­ terms of W, namely at the fixed point, W satisfies teresting conjecture of Gepner [20] concerning the relation between strings propagating on Calabi- Yau manifolds and strings constructed from ten­ sor products of N=2 super conformai field theories This means W is a quasi-homogeneous func­ is true. tion of degree 1, a result conjectured in reference We now know [7,8,9] that N=2 superconfor- [14]. mal theories are related to fixed points in Landau- There exists a considerable literature on the Ginzburg Hamiltonians. For example, 5 tensored relation between critical phenomena and singu­ copies of c = I N=2 minimal model corresponds larity theory [17]. It was realised that at the crit­ to the potential Yli^i- ^ was observed empiri­ ical point the effective potential of the Landau- cally in reference [11] that the potential that oc­ Ginzburg model is massless (i.e. Q^Q^J U=O= 0) curs in the Landau-Ginzburg model is the same and the equation of motion at zero momentum as the embedding polynomial of the correspond­ implies 6VQ!? \=o= 0. In singularity theory [18] ing Calabi-Yau manifold in CP71. For the exam­ a potential is called degenerate and critical if the ple above, the Calabi Yau manifold is defined by above two criteria are met. All such potentials setting the embedding polynomial £\ z\ — 0 in (of low modality) have been classified up to field complex 5-space. Thus one could prove the Gep­ redefinitions. This relation between singularity ner conjecture if one could show that the Landau- theory and critical phenomena was also reiter­ Ginzburg Hamiltonian at the fixed point was such ated more recently [14,15] within the context of that it constrained the potential to vanish and Landau-Ginzburg models. the kinetic term had the appropriate metric, al­ though it is not entirely clear what metric this should be. In this context we note that the ma-

400 jority of the considerations of the proof of refer­ [7] P.S. Howe and P.C. West, Phys. Lett. B223 ences [7] [8] [9] apply to the chiral operators and (1989) 377. we did not identify the non-chiral primary fields [8] P.S. Howe and P.C. West, Phys. Lett. B227 with Landau-Ginzburg operators. Although we (1989) 397. took a canonical kinetic term our considerations [9] P.S. Howe and P.C. West, Phys. lett. B244 were rather insensitive to this assumption. This (1990) 270. would not be the case for the non-chiral operators [10] A. Zamolodchikov, Sov. J. Nucl. Phys. B241 whose identification would presumably fix the ki­ (1984) 333. netic term, and hence establish in what sense the [11] D. Kastor, E. Martinec and S. Shenker, Nucl. Gepner conjecture is true. A clear demonstra­ Phys.B316 (1985) 590. tion of this point would provide an interesting [12] S. Cecotti, L. Girardello and A. Pasquinucci, connection between quantum field theory (criti­ Nucl. Phys. B328 (1989) 701. cal phenomena) and the geometry of Calabi Yau [13] M. Ademollo et al, Phys. Lett. 62B (1976) manifolds. In [21] Green, Vafa and Warner as­ 105; Nucl. Phys. Bill (1976) 77; V.A. Fa- sumed the connection between Landau-Ginzburg teev and A.B. Zamolodchikov, Sov. Phys. models and minimal N=2 super conformai models JETP 2 (1985) 215; P. Di Vecchia, J.L. Pe­ and purported to show the relation of Landau- tersen and H.B. Zheng, Phys. Lett. 162B Ginzburg models to Calabi-Yau manifolds. An (1985) 327; P. Di Vecchia, J.L. Petersen and essential step in this "proof was setting the ki­ M. Yu, Phys. Let. 172B (1986) 211; P. netic term to zero at the fixed point. In this case, Di Vecchia, J.L. Petersen, M. Yu and H.B. however, we have a system for which interactions Zheng at different spatial points are not correlated. In [14] C. Vafa and N. Warner, Phys. Lett. B218 the language of the lattice, the spins at different (1989) 51. lattice sites are not correlated. The system is ef­ [15] E. Martinec, Phys. Lett. B217 (1989) 431. fect frozen and there is no renormaJization group [16] W. Lerche, C.Vafa and N. Warner, Nucl. Phys flow from this Hamiltonian. The most important B324 (1989) 427. characteristic of a system at its critical point is [17] see, e.g. T. Poston and I. Stewart, Catas­ that it is a system which is dominated by long trophe theory and its applications, Pitman range correlations. Thus the Hamiltonian with 1978; M. Vendrik, Physica A99A (1979) 103 zero kinetic term has nothing to do with the fixed and 513; K. Okada, Solid State 22 points corresponding to the N=2 minimal series (1987) 254 and 392. nor is it in the same universality class. [18] V.I. Arnold, Singularity theory, Cambridge Univ. Press, 1981; V.I. Arnold, S.M. Gusein- Zade and A.N. Varchenko, Singularities of differentiable maps, vols. I and II, Birkhauser 1985. References [19] P.S. Howe and P.C. West, Phys. Lett. B223 (1989) 371 [1] T. Banks, L. Dixon, D. Friedan and E. Mar- [20] D. Gepner, Nucl. Phys. B296 (1987) 757; tinec, Nucl. Phys. B299 (1988) 613. Phys. Lett. B199 (1987) 380. [2] K.Wilson and and J. G. Kogut, Physics Re­ [21] B. Greene, C. Vafa and N. Warner, Nucl. ports 12C (1974) 75; K. Wilson, Reviews Phys. B324 (1989) 321. of Modern Physics 55 (1983) 583, and refer­ ences therein. See also, e.g. E. Ma, "Modern Theory Critical Phenomena" Benjamin. [3] L.D. Landau, Phys.Z Sov. Jet. 11 (1937) 545, 26. [4] K.G. Wilson and M.E. Fisher, Phys. Rev. Lett. 28 (1972) 240; K. Wilson, Phys. Rev. Lett. 28 (1972) 548. [5] G. Parisi, Journal of 23 (1980) 49. [6] J.C. Le Guillou and J. Zinn-Justin, Phys. Rev. Lett. 39 (1977) 95.

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