Zogjomt \] COMITETUL ME STAT PENTRU E NE ROIA NUCLEARA INSTITUTUL. CENTRAL DC FIZICA •UCUftf ŞTI - MAOURILI ROMANIA CENTRAL INSTITUTE OF PHYSICS INSTITUTE FOR PHYSICS AND NUCLEAR ENGINEERING SuchaieAt, P.0.8. 5206, R0MANIA fT-ISI-tO Feb/tua-xy Pour dimensional sigma model coupled to the metric tensor field G.Ghlka and U.VlAlneAcu ABSTRACT: We dlACuAA the ţouK dimensional nonllneax Algma model with an Internal (Tin) In variance coupled to the. metxlc tin A 01 ileld AatlA tying llnAteln equa - tlonA. We dexlve a bound on the coupling constant between the Algma ţleld and the metxlc tenAox, u&lng the theony oţ haxmonlc map*. A special attention IA paid to BlnAtein Apace A and Aome new explicit Aolu - tlonA o&' the model axe conAtxucted. INTRODUCTION In recent years there has been much interest in the finite action solutions of the Euclidean sigma model in two dimensions fl] . This model bears many similarities to a four dimensional non-Abelian gauge theory like scale lnva - r lance, asymptotic freedom, ins canton and rearon solutions , action bounded by a multiple of a topological charge» etc... When trying to generalise this model to a four dl - mensionai one there are difficulties. As long as we use a traditional Lagrangian, the sigma field $ has non-zero dl- mension and the constraint 2($*j =; ^ must introduce a A. dimensional constant "X which destroys the conformai inva - rlance, on the other hand a dimension ess $ implies a higher derivative Lagrangian with many troubles it tha quan­ tum level. A very elegant way to avoid these difficulties was found by de Alfaro, Pubini and Purlan [2] scnstructing a model in which the sigma field is coupled to the metric ten­ sor g v in a generally invariant fashion. Their model, having an internal Cf(iC) in variance, is characterized by the following Lagrangian density where M-,i> «1,2,3,4 , a - 1,2,...,n is "n internal index and the field $ is defined on a four dimensional Riemannian - 2 - manifold N taking values in an Euclidean sphere In Bq. (1,1) K is the coupling constant of the gravitational 2 field with the usual dimensions of a (length) , G ii metric ullr tensor, G « det(G v), R is the scalar curvature of M and F is a coupling constant. For generality a cosmologicei term (^A) was added but we shall deal mainly with the case A -0.Introduc- ing as in paper [2] a new set of quantities g -K~ G ,.X -KA , f *- KF we get _2 in these notations, the metric field has dimensions, of a (length) ^ and f are pure numbers and ţ is dimensionless. In the usual sigma model F is a positive constant and Interpreting K as the usual gravitational constant it follows that f is nonnegatlve. But wishing to keep the model as general as possible we shall not Impose a restriction on the sign of f from the beginning. The authors of [2"] were able to find explicite ly an in- Stanton solution for f " 3 and a naron solution for f - 2 in the 2 cate \ «0. it t rns out that for these values of f the order ot magnitude of the coupling F is not in -the typical range of high energy physics, interpreting X as the usual gravitational constant. But in spite of this problem we consider that the nodal deserves a closer examination. Concerning the cosmological constant we mention that in the last tins many authors reevaluated Its status. One argues [3] that the quantisation of gravity with a cosnologi - - 3 - cai tex» Sa both neoeeeary and feasible and the empirical •aloe of the constant ( very sasll and poasible sero) do not necessarily imply that one should set X - 0 in the bare lim- atein Lagrangian. On the other hand, in the " apacetlme foaai" program [4,5] , the apacetime looks nearly flat when viewed on large length scales, but suffers large fluctuations of the metric and topology on acales on the order of the Planck length. The foam-like structure can be described by introdu­ cing a cosmologicei tem as a Lagrange multiplier for the 4-volume in the path integral approach£6]. In the present paper we shall' use the theory of the harmonic mapa [7} aa a valuable tool to handle the nonlinear sigma model in four dimensions. The role of the harmonic mapa in different fields of particle physics and general relativity was pointed out by Misner [8} . In the last time many papara use the theory of the harmonic mapa in order to investigate the nonlinear sigma model [9] , Grlbov ambiguity in non-Abeli- an gauge theory [loj , general relativity [ll] , etc. He observe that the field $ is in fact a harmonic map from a Riemannlan four dimensional manifold M into a (n-1) dimensional sphere (for an internal (/ (n) lnvarlance). On the other h**d, from the equations of motion we find that the metric on M is such that the Ricci tensor is related to the "pull-back" of the metric of the final sphere S .These observations will permit us to find some constraints on the parameter f if the space M is compact as we shall mainly assume in our paper. - 4 - Of course the nee of the compact spaces is a techni­ cal restriction which allows as to find solutions with finite action. This doss not swan necessarily that tarn actual space- tine is cospact and one can interpret it as a normalization prescription like introducing a box in ordinary quanta» meche- nics. On the other hand it is known that starting with a non- coapact manifold N It Is difficult to obtain finite action so­ lutions. Por example Garber et al [9] proved that the d-di - menslonal T -model ( <$: f\ —• S^ ) with d>2 has no re­ gular finite solution. We must mention also that starting with a compact manifold it is essential to search solutions satisfy- w ing the constraint (1.2). This can be put in connection with the fact that compact, orientable Riemannlan manifolds without boundary do not support non trivial harmonic functions £12J Hence this kind of manifolds is not interesting for an Eucli­ dean relat1vistic theory with energy-momentum tensor construc­ ted from a free scalar field without mass. In Section 2 we derive the general constraints imposed by the harmonic property of the map <§ and by the Einstein equations for the metric. We are able to derive a bound on the coupling f which might allow a hope to obtain a solution with acceptable value for F , at least for a vanishing cos­ mologicei term. Special attention is paid to Einstein mani - folds which form an important class of possible solutions if the metric tensor of II is proportional to the pull-back of the metric of the final sphere. Por this class of solutions we express the coupling f in a geometrical way» namely as the ratio of the scalar curvature of M and an eigenvalue "f - 5 - tha Laplace - Beltrami oparattor on thia manifold. In Section 3 wa construct eeveral aolutiona with finita action. They involve for * various topologia* lika S4, s2 x S2, 2 CP which appaar alao in tha study of tha gravitational lnatan- tona . Binatain aquations on conpact «anifold have solutions with multiple topologia* which can play an inportant rola in tha path integral in tha spaca of metric* [5, 6] . Finally wa discuss the meren type solution (infinita action) assuming that tha aanlfold N la conformally flat. In tha last section we aake son» abort convent* concer­ ning the aodel and it* possible generalisations. A collection of useful formulae about the harmonic maps is presented in the Appendix. I. GENERAL PROPERTIES In this «ection w* «hall derive soma general properties of tha solution* of the model described in the ItUKoduvUon assuming that the four dimensional manifold M is compact without boundary. Tha loler - Lagrange equations corresponding to the Lagranglan (1.3) are where the left hand side of Eq.(2.2) is the usual Laplace - ! - 6 Beltrami operator defined on K. As it was observed in paper [2] for A - 0 the field Bqs. (2.1) and (2.2) are invariant under the simple resettling. and consequently any solution of the metric field g is defined in this case op to a multiplicative constant. It is important to observe that any finite action solution of Bq. (2.2) is a harmonic map betMeen the manifold K with the metric g ^v. anad aman Euclideaniirr 11 mmn spherinneiei JS* isome trically immersed in R n. In fact Bq. (2.2) is just Bq. (A.tO) in which the field $ is the composition of a map <f tM^s""1 and of th« Riemannlan immersion ^ s Sn~ -•R,". In our case the energy density of the harmonic map ^ is (see Eqs. (A.5) (A.10)). (2.4) Taking into account this observation» we can use the general properties of the harmonic maps to obtain some features of th solutions of the model Sells and Sampson ţ.13^ , using the general method ini­ tiated by Bonhner [l2*] and the Ricci identities, were able to obtain *he following equation satisfied by the energy den­ sity of an harmonic map u> ; M + H 7 - 2 (2.5) ACW- W(Acf)) + QU^ With i> (r where R'^^d is the Rlemann - Christoffel curvature tensor on N (with the metric h_. ) and R ^, is the Ricci tensor on M (with the metric g ). Assuming that the Rleir.annian manifold M ti? compact without boundary from Stokes' theorem we have 4.