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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 (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. \ ^e(^>\\/j JL M Applying Eg. (2.7) to Eq. (2.5) and observing that *7 Ţ i.1'. )\ i3 non-.negative? we have 4 1. \itf )V^x £ 0 if V[A^Vo,which means that the map CP is totally jeodo^ic (fe<~ Appendix A), then in Eq. (2.8) w* nave equality. We iot'= als that for constant energy density, Eq. (2.3) an e --»>pressed simply as (2.J) QUf) * 0 we shall apply Eqs, (2.8) and (2.9) in order to obtain some information about the couoling constant f. In our case the f eld Ct- takes values in an Euclidean sphere N • sn and - 8 - using Eq.(2.1) we shall first evaluate Q U«f) for X-0: (2.10) where *= JrV iţi1- RrV We used explicitely that on a sphere (following the notations fro» Eisenhart i_14j : w.ti- ihe se^ l)/al ur /at^re KQ » l. Actually, Eq. (2.10) was duced !'• sJlghMy more general conditions, i.e. for a mani- toi'i N *'i*-ii constant sectional curvature . Using Eg.s P.8) and (2.10) we get (2.11) (i+ţ} ^ Iţfv^ In order to obtain a bound on f, we shall use a gener^U inequality for a nfdlmensional Riemann manifold ţ_15 J x 2 (2.12) m»j\ >R 9 The «quality holds in the above aquation if and only if v. is an Einstein api (2.i:i) p _ K a Fro* Bqs. (2.11) and (2.12) we obtain for • • 4 (2.14) -ţ ^ assuming that the Ricci tenaor R v ^o (In fact if IL^^O the aodel is trivial). We remark that the bound (2.14) was deduced for any dimensions of the final sphere Sn . On the otb«*r hand this bound is reached for an Einstsin manifold M am* a totally geodesic nap from that manifold to a aphere. That ia exactly the lnstanton solution found in paper T 21 • Now, if the cosmologicei constant is taken Into ac - count for Q(i«f) we obtain a more involved expression t <2'15' -iT.JixSx'RliMVwVn-O] f 2 L 4 x Of course, we can use condition (2.8), but a sere transparent result Is obtained using Eq. (2.9) valid for constant energy density. We get (for f > -1) (2.16) 1'4.1+ 2A- and the equality holda also for totally geodesic maps and taking M as Einstein space. For an Einstein space, R < C - 10 - ia a «officiant condition to ba conpact [«, li] • dacraaaaa «ban ona adda a coauoloalcal tarn for a apnea with R <0. now It la worth diacaaaina about the aolvtlona of &]•. (2.1), (2.2) involving Blnatnln «paean. In ganaral an hamonlc nap dona not xnlata thn natrlca of the nanlfolda. A particular claaa of hnrannlc napa la rapraaantnd by tha nlnlaal inunmiona (ana Aypandla A) for which C2-17) Vv--V^"^v^ and ton anargy danalty la € (•) - 2. Slaea tha nodal ( with >\o ) la invariant andar tha ruacallaa (2.3), Eq. ( 2.17) can ba ra\aaad mqoirlna only a proportionality batwnan laft and right hand aidaa and conaaqnuntly Bq. (2.1) will dnfian i ran Elnateln aanlfold (2.13). Por a onapart Binatain nanlfold thnra ara bound» on tha conatant of proportionality batwnan ^ y and 4 \i] . OOing thaaa boanda It la poaalbla to dă daca a ICMVX bound (a non-poaltlva valon) for tha coupling conatant f la tnma of tha Salar, algnatura charactarl«tica and thn volcaa of tha nanlfold. Thora la a wnllknown conatructlon of adnlnal ianar - alon of a aanlfold ll Into a aphara. Svppoaa that $ mall* saa a alnlMl lauaraion In a aphnra & SB~ of radina h. la rC*. Than $ aatlaflaa [l7, 18 3 ^v (2.19) . <\ ~ ~ Q - _£. a such that Eq.(2.18) becoaes (2.20) ^ Ş _ - 4 <£ z From Eq. (2.1) with A = 0 we get so that (2.22) -f , - L. which expresses the coupling constant f in terms of the scalar curvature R and an eigenvalue M- of the Laplace - Beltrami operator on the manifold M. The dimension of the sphere Sn~ is related to the multiplicity (n) of that eigen value li. . As it was expected, f is given in an invariant way, any rescaling of the metric being cancelled in the above ratio. We mention that as a rule title construction can be applied to any Einstein compact, homogeneous manifold H since it admits a minimal immersion into some Euclidec sphere , but in what follows we shall restrict ourselves to simplo manifolds as spheres, tori ,complex projective spaces. -. 12 - f. SIMPLE EXPLICIT SOLUTIONS In this section we shall present a few sets of solu tions of Eqs. (2.1), (2.2) for the field <$> and the corres ponding metric field <\ . , beginning with those of finite action. The first known example is the instanton solution of de Alfaro et al [2^ for a sigma model with an internal 5) invariance. The manifold M is chosen to be S and the map <£ is the identity between two S spheres. One can parametrize 4 in 4 the S sphere using the stereographlc projection on $\ . h 2* X (3.1) i +X = 1#2,3,4 and the field <$> being the identity map (3 2.) ^* - \ ) « " 1,2, ...5 . We can choose the same metric on both S spheres r and the identity map realizes a minimal Immersion. Due to re- scaling freedom (2.3) we can put on the initial S sphere a metric proportional to the final one (*.J) 4 . : cC — -. OfcV V - (7c?rr and a simple calculation gives f • 3. This is in agreement with our observation from previous section concerning the bound (2.14). Indeed the identity map (3.2) is a totally - 13 - geodesic map and, with the metric (3.3), the S sphere is an Einstein manifold. If a cosmological term is taken into account then from (2.16) one gets ,3.4, 1 , j + 52L where R is the scalar curvature of the initial S sphere.Of course the action is finite and the degree of the map is one (the topological characteristic of this instanton solution). Having in mind that the eigenvalues of the Laplacian on G4 are (3.5) p^fctfc**) , *- >1 , fe € '£ with the multipllcites (3.6) m l*+2V. (ili-5) we can consider minimal immersions of the sphere S in S1™*" . Tha model will have an internal CMm.) invariance and the coupling f will be given by (2.22) (for A - 0). (3.7) -P. K% being a positive rational number smaller than 3. In what follows we shall study other finite action solutions trying to construct new minimal immersions between four dimensional manifolds M and various Sn~ spheres. - 14 A class of solutions will be of the form M = M, x M_ where M. is a sphere or a torus and CO ; M —??v ^ IK will be minimal isometric immersions, x'aen the field <£> will be constructed as a napping & ; M ^\ -^C/'**1 £_ flO"1**'1* (which corresponds to an internal U(^t f^ +2) in variance ) setting [l8] (3.8) ^(M^C^*^»***^^) where XL and tf are points in Mj^ and Mj respectively and ^ is a real constant. Since we shall restrict ourselves to final spheres of not too high dimensions we shall choose in our constructions the lowest possible p. - dimensions. 2 2 For example we can take M - S x S , cf beinq the identity maps. Denoting by ( (3.9) ^L(*iţpr,S)= («rtfrftts* ţCt*frAix«C4>6jO»fc***** Ainft From Eq. (2.1) we have (3.10) * 4 2 2 where C1 and C2 are two constants multiplying the standard metrics of the initial S2 spheres, "sing Eq. (2.2) we get - 15 - ^V = c***= - 1/iL • ^.ct-cf" so that (3.11) î-2-i.xV In this example we can take A -0 ( f » 2) since -> 2 r M - S" x S is an Einstein space and the field ţ» realises, up to a multiplicative constant, a minimal immersion. The next example, is M - S x S and £ : M-**S choosing in Eq. (3.8) o> : S -* S as the identity map and S as a polynomial map z —+zn. In a similar manner as above we get \- Crtre- - i*u*> Hhu^-0- 7> 4 A ^(aW^-csv*) •-2-Ut1 * In . ve equations the constant C •is related to the con- wants and ci which can multiply the standard metrics on S and i ' respectively as follows 3 1 xi' this example M « 8 x S is not an linstein manifold so t iat A can not be put equal to sero. In fact we use a AfO 16 - jjst to compensate tha vanishing of that part of the Riccl tenaor which corraaponda to tha aphere C . Another example ia M - T2 x S2 where T2 la the flat torua S1 x S1 and $ ia -tha map $ : T2 x S2-* S6 2 3 m realized aa above with i- ***$> - CAN*- 2\X _ C*iV- with the obvious relation t i- (3.13) ^ i - | X C Many other examples can be constructed aa above , in volving final apherea of higher dimensiona. For example, we can minimally immerse Ş x S x S x S into S which givae of course, a vanishing Ricci tensor and a f i* 0 can be acco - modated chooaing an appropiata X j* 0. A more interacting mi nimal immeraion Into S ia the Veronese mapping; \T} \ «CP -> 8 . The dimension 7 of'the final fohere is the lowest possible 2 n dimension for minimal immersion of CP intn S due to tha fact that tha multiplicity of the first eigenvalue of the Laplacian 2 on CP ia 8 . The caLculatlon of f can be dona directly or - 17 - applying Sq. (2.22) getting f = 2. If we add a cosmologi ca i term we have o 2 wh^re C is a constant which can multiply the standard 2 me*:r. on CP In all above example» the metric on the initial four Hrens.onal manifold M was obtained (up to a possible re - s -^inq) as the pull-back of the Euclidean metric of the f:.r.al sphere S "" . If M >is an Einstein manifold we have s which will be obviously satisfied by any four dimensional Einstein pace, of course for A * 0 any vacuum solutions is a 1 lower» Finally, we shall discuss a solution of the meron type for which th* action is not finite. Such a solution can be realized by applying the manifold M on a three dimension» 3 4 al rphere S C S . The simplest procedure is to start with a conformally flat space M ' - 18 - 2 «-(**) (3.16) a ^'v*0~ I d ^|V -1,2,3,4 and to make a Rienannian harmonic subversion zp M --*• S^ (3.17) xu —» *T-- followed by the identity map I : S3-* S3. Then the field will be defined as the composition of these two maps 4* * 4'1 We must remark that this composition of maps will satisfy formally Eq. (2.2) but it is not a proDer harmonic map since the energy of the map is not finite. In order to determine the function T(jx.'i) from Fa. (3,16) we shall use Eq. (2.1). For^this purpose it is neces sary to evaluate *:he Riccl tensor for a con formally flat space [l4] : (3.18) r r l T x where T' - A.Tlx'1 y". ^ !l . Prom Eq. (2.1) we get 1 2T (3.19) jl£l 4x- 4 ST' 4 *_ X* c " -- - i- The solution of this system of equations will contain two arbitrary positive constants a, b 1/2. 3.21) «U-i t U-4A3 We note tha we mu t have f ^2 In order to obtain a r 1 metric For f * 2 we re over the meron sol tlon f om f 3 wi':h A* 0, a and b d n ,.:..:«. 4. CONCLUSIONS This oaper can be considered aa an llluatratlon of t'e utility of the harmonic map*» in the study of some non - lfn»ar flelJ equations which appear» In physical theories.The harmonic maps turn out to be of real utility to understand beittt soue of the nonlinearitiaa that occur in Einstein ec ations for general relativity. We were able to find some restrictions on the coupling between the metric tenaor and the algma field. .Mso we exprei- aed this coupling for a claaa of aolutiona Involving Einstein spaces in terns of geometrical quantltiea. Since these results do not depend on the dinanaion of the ephere in which the field tai.ea valuea, we were tempted to conalder higher C (n) (r. £. 5) invarlance modela. We found that euch modela jupport complicacud topoiogiea for tha Euclidean four dimensional apace M. 20 - S ch kind ot apacaa appeara a)sc in othar related problems in phyal* like gravitational inatantons, spacetlve foam, odel*, etc . The fact hat we und that ther arff; in pr ciple other aolutl » th f < 3 nana that the model is flexib ao that there 1 a hope *o ob • other aolu - t'.ona with th coupling F \n tie ty ca r of high energy physlca. Proa this point of view the model see deserve attentlo Of course the net used in this paper can be extended to other nodala for wh ch the field g> takes va<- lues in a net complicated apace [20J . Apptndix A J tltUlHTS 0. HARMONIC MAPS Let M N be a ooth nan folds of dimensions a, n equiped wi Rieman*ian me -ics g, h reapectively. Let (X1 - x (v • r M da ta smoth local coordinates on . N. In " as ordinate we have ...» AA» 3 and the usual Christoffel symbols of M and N will be denoted by M P^ , Npft reap.ctivaly. Given a smooth map (A. 2) ftt\ f» = 3x* 1^ 7*- -i^^ (A. 3) ^ where \7 Is the covarlant differentiation on Mi» If the raetrlce g, h on H, N are related by CP h • g the map a> Is a Riemannlan immersion. A map with vanishing second fundamental for» is said to be totally geodesic. The energy of the nap CP la defined by the formula (a generalization of the classical Integral of Dlrlchlet ) (A.4) E( * ^ t ((*> Vj" The map cP Is called harmonic If It Is an extremal of the energy Integral E(«P). The corresponding Euler - Lagrange equations are v[tf - f (v (A where Ale the Laplace-Beltraml operator and 7(if) is called the tension field of CP . if cP 1* an harmonic map and an Iso metric Immersion then CP is called minimal Immersion. - 22 - The minimal immersions axa the extremals of the volume functional (A.7) V(«?^* ^^(^i)!^^ M It la uaaful to consider the composition of two mapa U> : M-*M' and ty : M'—>N and to calculate the tenaion from Eq. (A.6) (A. 8) -c(f^)r JLf .x(<^) + Twe ^H'C'*1?,^) or explicitely in coordinatea : We are intereatad to study maps between a manifold K and a sphere S which is isomatrically immersed in Rn . In that case ^ is the Immersion f : Sn"X^j^n and assuming that vţ>: M-»S is harmonic wa obtain from Eqs. (A.6), (A.8) and (A.9) i - 23 - (A.10) T T 3 1 Moreover it is possible to prore that cP t M Sn~l Is harmonic if and only if Bq. (A.7) holds. REFERENCES [lj See for example : D.J.Gross : Nuci. Phya B132, U39 (1978) P.Tataru - Nihai : "On the two dimensional (ft 3) (J*-nonlinear model"» preprint Sternberg Univ. (1978) V. de Alfaro, S.Fubini and G.F-rlan:"Non-linear sigma model and classical solutions"t preprint ICTP-Trieste 78/69 (1978). H. Eichenherr : Nucl. Phys. Bl»6 , 215 (1978) H.J. Borchers ana W.D.Garber : "Analiticity of solutions of the lT(N) nonlinear 0* -model", preprint Univ. Gottingen (July >979) H.J.Borchers and W.D.Garber : "Local theory of solutions for the (7 <2k*l) 0"-model", prepr. Univ. Gottingen (October 1979) D.Haison: "Some facts about classical nonlinear [O f. de Aliaro, S.Fabiai and 6. Parian.- N.C. SO A. S23 (1*7$) f»3 S.H.Christ«aMn and H.J.Daff: "Quantizing gravity with a coeaological constant", preprint Univ.California, Santa Barbara CTP-79-01 COctobar 1979). i%] J.A. Mheeler, ia "Relativity croups and topology", ad. B.S. and CM. Da Witt, Gordan and Breach, Mew York (196*). [Si S.V.Havking s Mucl. Phys. Bit» , 3H9 (19/8) [6] 6.H.eibbons and S.v.Havking : Cons. Hath. Phya. 6Ş, 291 (1979) [73 For a recent review see : J.Eells and L.Leaaire : Bull. London Hath. Soc. 10,1 (1968) [83 C.W.Misner : Phys. Rev. 18, -ill, fl97g). [93 H.Forger : "Instantons in nonlinear •" -nodels, gauge theories and general relativity ", preprint Freie «Jhiv. Berlin July 1978). V.D.Garoer, S.N.M. Ruijenaars and E.Seller : Ann. of Phya. 119 , 305 (1979) P. Tataru-Mihai : Muovo Ciaento 51A, 169 (1979) F.Gursey, M.A.Jafarizadeh and H.C.Tze : Phys. Lett 98B , 282 (1979). TlOl 6.6hika and M.Visinescu : "Vac un degeneracies in terns of hamonic naps", preprint fentrc* Institute of Physics, Bucharest FT-16» (1979) and Phys. Rev (in press) [ill A.Eri» : J.Hath.Phys. 18, 820 (1977). Y.Nutku and M.Halil : Phys. Rev. Lett. 39, 1379 (1977) [12} S.Bochnar : Trans.of the Aner. Math. Soc. «7* 1W (1979) E.Gava, R.Jengo, C.Omero and R.Percacci : MJCI. Phys. B151 , >»57 (1979). CENTRAL INSTITUTE OF PHYSICS Documentation Office Bucharest, FOB. 5206 ROMANIA