Conformal structures with explicit ambient metrics and conformal G_2 holonomy PDF
Preview Conformal structures with explicit ambient metrics and conformal G_2 holonomy
CONFORMAL STRUCTURES WITH EXPLICIT AMBIENT G 7 2 METRICS AND CONFORMAL HOLONOMY 0 0 2 PAWEŠNUROWSKI n a 2 5 J Abstra (3t,.2)Givenageneri -plane(cid:28)eld[ogn]a -dimensionalmanifoldwe on- siderits -signature onformalmetri asde(cid:28)nedin[7℄. Every onformal 1 [g] lass obtained in this way has very spe ial onformal holonomy: it must 3 G2 be ontained in the split-real-form of the ex eptional group . In this note 2 5 [g] weshowthatforspe ial -plane(cid:28)eldson -manifoldsthe onformal lasses ] G have the Fe(cid:27)erman-Graham ambient metri s whi h, ontrary to the general Fe(cid:27)erman-Graham metri s given as a formal power series [2℄, an be written D inanexpli itform. Wepropose tostudytherelationsbetweenthe onformal G2 [g] G2 . -holonomyofmetri s andthepossiblepseudo-Riemannian -holonomy h ofthe orresponding ambientmetri s. t a m [ (3,2) 1. The -signature onformal metri s 2 v Consider an equation 1 z′ =F(x,y,y′,y′′,z) with Fy′′y′′ =0, 9 (1.1) 6 8 y =y(x) z =z(x) x fortworealfun tions , ofonerealvariable . Tosimplifynotation 1 p=y q =y ′ ′′ 0 introdu e new symbols and . Equation (1.1) is totally en oded in the 7 system of three 1-forms: 0 ω1 = dz F(x,y,p,q,z)dx / − h ω2 = dy pdx at (1.2) ω3 = dp−qdx, m − J (x,y,p,q,z) : living on a 5-dimensional manifold parametrized by . In parti ular, v γ(t)=(x(t),y(t),p(t),q(t),z(t)) J every solution to (1.1) is a urve ⊂ on whi h all i ω1,ω2,ω3 X the forms identi ally vanish. r Weintrodu eanequivalen erelationbetweenequations(1.1)whi hidenti(cid:28)esthe a equations having the same set of solutions. This leads to the following de(cid:28)nition: z = F(x,y,y ,y ,z) z¯ = F¯(x¯,y¯,y¯,y¯ ,z¯) ′ ′ ′′ ′ ′ ′′ De(cid:28)nition 1.1. TJwo equJ¯ations and(x,y,p = y ,q = y ,z), ′ ′′ de(cid:28)ned on spa es and parametrized, respe tively, by (x¯,y¯,p¯ = y¯,q¯ = y¯ ,z¯) ′ ′′ and φ:J, arJe¯said to be (lo ally) equivalent, i(cid:27) there exists a (lo al) di(cid:27)eomorphism → transforming the orresponding forms ω1 =dz F(x,y,p,q,z)dx ω¯1 =dz¯ F¯(x¯,y¯,p¯,q¯,z¯)dx¯ − − ω2 =dy pdx ω¯2 =dy¯ p¯dx¯ − and − ω3 =dp qdx ω¯3 =dp¯ q¯dx¯ − − Date:February 2,2008. Thiswork wassupported inpart bythePolishMinisterstwoNauki iInformatyza jigrant nr: 1P03B07529andtheUSInstitute forMathemati sand ItsAppli ationsinMinneapolis. 1 2 PAWEŠNUROWSKI via: φ (ω¯1) = αω1+βω2+γω3 ∗ φ (ω¯2) = δω1+ǫω2+λω3 α,β,γ,δ,ǫ,λ,κ,ν J ∗ , with fun tions on su h that φ (ω¯3) = κω1+µω2+νω3 ∗ α β γ det δ ǫ λ =0. 6 κ µ ν Itfollowsthatequation(1.1) onsideredmoduloequivalen erelationofDe(cid:28)nition (3,2) [g ] F 1.1 uniquely de(cid:28)nes a onformal lass of -signature metri s on the spa e J (x,y,p,q,z) . In oordinates this lass may be des ribed as follows. Let D =∂ +p∂ +q∂ +F∂ x y p z J be a total di(cid:27)erential asso iated with equation (1.1) on . Then a representative g [g ] F F of the onformal lass may be written as g =[ DF2F2 +6DF DF F2 6DF F F2 F qq qq q qqq qq − qqq p qq − 3DDF F3 +9DF F3 9F F3 + qq qq qp qq − pp qq 9DF F F3 18F F F3 +3DF F4 qz q qq − pz q qq z qq − 6DF F2F +6F F2F 8DF DF F F + q qq qqp p qq qqp− q qq qq qqq 8DF F F F +3DDF F2F 3DF F2F qq p qq qqq q qq qqq − p qq qqq − 3DF F F2F +4(DF )2F2 8DF F F2 z q qq qqq q qqq − q p qqq − 3(DF )2F F +4F2F2 +6DF F F F q qq qqqq p qqq q p qq qqqq − 3F2F F 6DF F F2F +6F F F2F p qq qqqq − q q qq qqz p q qq qqz − 3DF F3F +12F F3F +3F2F F q qq qz p qq qz qq qqq y − 6DF F F2F +4DF F3F +6F F2F F + qqq q qq z qq qq z q qq qqp z 8DF F F F F 4DF F2F F (1.3) qq q qq qqq z − q qq qqq z − 9F F3F +F F2F F 8DF F F2 F + qp qq z p qq qqq z − q q qqq z 8F F F2 F +6DF F F F F 6F F F F F + p q qqq z q q qq qqqq z − p q qq qqqq z 18F3F +6F2F2F F +3F F3F F qq qy q qq qqz z q qq qz z − 2F4F2+F F2F F2+4F2F2 F2 qq z q qq qqq z q qqq z − 3F2F F F2 9F2F3F ] (ω˜1)2+ q qq qqqq z − q qq zz [ 6DF F2 6F2F 8DF F F + qqq qq − qq qqp− qq qq qqq 8DF F2 8F F2 6DF F F + q qqq − p qqq − q qq qqqq 6F F F 6F F2F +6F3F + p qq qqqq − q qq qqz qq qz 2F2F F 8F F2 F +6F F F F ] ω˜1ω˜2+ qq qqq z − q qqq z q qq qqqq z [ 10DF F3 10DF F2F +10F F2F qq qq − q qq qqq p qq qqq − 10F4F +10F F2F F ] ω˜1ω˜3+ qq z q qq qqq z G2 CONFORMAL STRUCTURES WITH EXPLICIT AMBIENT METRICS AND CONFORMAL HOLONOMY3 30F4 ω˜1ω˜4+[ 30DF F3 30F F3 30F F3F ] ω˜1ω˜5+ qq q qq − p qq − q qq z [ 4F2 3F F ] (ω˜2)2 10F2F ω˜2ω˜3+30F3 ω˜2ω˜5 20F4 (ω˜3)2 qqq − qq qqqq − qq qqq qq − qq 1 where ω˜1 = dy pdx − ω˜2 = dz Fdx F (dp qdx) q − − − ω˜3 = dp qdx (1.4) − ω˜4 = dq ω˜5 = dx. Itfollowsfromthe onstru tiondes ribedinRef. [7℄thatwhentheequation(1.1) φ g F undergoes a di(cid:27)eomorphism of De(cid:28)nition 1.1, the above metri transforms onformally. [g ] (3,2) F The onformal lass of metri s is veryspe ialamongall the -signature onformalmetri sindimension5: theCartannormal onformal onne tionforthis so(4,3) lass, instead of having values in full Lie algebra, has values in its ertain 14-dimensionalsubalgebra. Thissubalgebraturnsouttobeisomorphi tothesplit g so(4,3) 2 real form of the ex eptional Lie algebra ⊂ . Thus, onformal metri s [g ] F provide an abundan e of examples of metri s with an ex eptional onformal holonomy. This holonomy is always a subgroup of the non ompa t form of the G F 2 ex eptional Lie group . We strongly believe that randomly hosen fun tion , F = 0 [g ] qq F su h that 6 , give rise to onformal metri s with onformal holonomy G 2 equal to . [g ] F It is interesting to study the onformal lasses from the point of view of F the Fe(cid:27)erman-Graham ambient metri onstru tion [2℄. Sin e for ea h de(cid:28)ning [g ] F equation(1.1)wehavea onformal lassofmetri s indimension(cid:28)ve,thensin e (cid:28)ve is odd, Fe(cid:27)erman-Graham guarantees [2℄ that there is a unique formal power (4,3) [g ] F series of a Ri i-(cid:29)at metri of signature orresponding to . Moreover, F g F sin e given the metri is expli itely determined by formula (1.3), we see that F g F starting with real analyti , the metri is real analyti . Thus, every analyti F g F of (1.1) leads to analyti and then, in turn, via Fe(cid:27)erman-Graham, leads g˜ (4,3) F to a unique real analyti ambient metri of signature . Sin e both the g˜ F Levi-Civita onne tion for and the Cartan normal onformal onne tion forthe g F orresponding5-dimensionalmetri havevaluesin (possiblysubalgebrasof)the so(4,3) sameLiealgebra , it isinterestingto askabout the relationsbetweenthem. We dis uss these relations on examples. 2. The strategy for onstru ting expli it examples of ambient metri s We start with the Fe(cid:27)erman-Graham result [2℄ adapted to the 5-dimensional [g ] F situation of onformal metri s . g [g ] J F F Let be a repreJsentaRtive oRf the onformal lass (0de<(cid:28)nte,du)on Rby (1R.3). + + Consider a manifold × × . Introdu e oordinates on × in g 1 F Note that formula for di(cid:27)ers from the one given in Ref. [7℄ by tilde signs over the all g F omegas. In Ref. [7℄,when opying the al ulated metri , by mistake, we forgot to put these g F tildesignsovertheomegas. Hen e,inRef. [7℄,formulafor istrue,providedthatoneputsthe tildesignsover theomegasand supplementsitbythede(cid:28)nitions(1.4)ofthetildedomegas. 4 PAWEŠNUROWSKI J R R π : J R R J + + × × . We have a naJturalJproRje tionR × × → , whi h enables + us to pullba k forms from to × × . Ommiting the pulba k sign in the π (g ) ∗ F expressionslike we de(cid:28)ne a formal power series ∞ gˇ = 2dtdu+t2g utα+u2β+u3t 1γ+ ukt2 kµ . (2.1) F − F − − X − k k=4 α,β,γ,µ k =4,5,6,.... α,β,γ,µ k k HereJ J R, R gˇ,arepullba ksofsymmetri bJilinRearfoRrms + F + from to × × . Thus isaformalbilinear form on × × . Thisformal (4,3) u = 0 bilinear form has signature in some neighbourhood of . The following theorem is due to Fe(cid:27)erman and Graham [2℄. gˇ F Theorem 2.1. Among all the bilinear forms whi h, via (2.1), are asso iated g g˜ F F with metri of (1.3) there is pre isely one, say , satisfying the Ri i (cid:29)atness Ric(g˜ )=0. F ondition g α,β,γ,µ g˜ F k F Given ,allthebilinearforms in aretotallydetermined. Another issueisto al ulatethemexpli itely. Forexample,itisquitedi(cid:30) ultto(cid:28)ndthegen- µ k eral formulas for the higher order forms . Nevertherless the expli it expressions α,β,γ for the forms are known [4, 5℄. We write them below in the form obtained α β γ α = α dxidxj ij ij ij ij by C R Graham. We de(cid:28)ne the oe(cid:30) ients , and by , β = β dxidxj γ = γ dxidxj (xi) = (x,y,p,q,z) J ij ij , , where are oordinates on . α β γ ij ij ij Then Graham's expressionsfor , and are [4℄: α =2 , ij ij P β = B + k , ij − ij Pi Pjk 3γ =B k 2W Bkl+4 B k 4 kB +4 klC (2.2) ij ij;k − kijl Pk(i j) − Pk ij P (ij)k;l − 2CklC +C klC +2 k C l 2W k ml, i ljk i jkl P k;l (ij) − kijlP mP where = 1(R 1Rg ), Pij 3 ij − 8 Fij g =g dxidxj F Fij is the S houten tensor for the metri , W =R 2( g g ) ijkl ijkl i[k Fl]j j[k Fl]i − P −P is its Weyl tensor, C = ijk ij;k ik;j P −P is the Cotton tensor, and B =C k klW ij ijk; −P kijl is the Ba h tensor. F Of ourse all the above quantities an be expli itely al ulated on e , and in g F turn the metri , is hosen. F =F(x,y,p,q,z) Intherestofthepaperwewill hoseparti ularfun tions ,and α,β,γ we will al ulate the orresponding forms for them. We will give examples F γ of 's for whi h the bilinear form is identi ally vanishing, γ 0. (2.3) ≡ F Given su h 's we will onsider g¯ = 2dtdu+t2g utα+u2β. F F − − G2 CONFORMAL STRUCTURES WITH EXPLICIT AMBIENT METRICS AND CONFORMAL HOLONOMY5 g¯ g˜ F F Note that oin ides with the ambient metri up to the terms quadrati in t,u g¯ F the ambient oordinates . If by han e the bilinear form satis(cid:28)es the Ri i (cid:29)atness ondition Ric(g¯ ) 0, F ≡ g˜ F then by the uniqueness of the ambient metri stated in Theorem 2.1, it will g˜ F oi ide with the ambient metri : g¯ g˜ . F F ≡ g¯ F The uniqueness result of Theorem 2.1, together with the Ri i (cid:29)atness of , is γ powerfullenoughtoguaranteethatnotonlythe oe(cid:30) ient intheambientmetri g˜ µ k =4,5,6,...., F k identi ally vanishes, but that all the oe(cid:30) ients , vanish too! g˜ g F F Thus the strategy of (cid:28)nding expli it ambient metri s for is as follows: F = F(x,y,p,q,z) g F • (cid:28)nd for whi h the orresponding metri has identi- γ ally vanishing form of (2.2); g¯ F F • al ulate the approximate ambient metri for su h ; Ric(g¯ ) g¯ F F • he k if the Ri i tensor of is identi ally vanishing; F g¯ F • if you have with the aboveproperties then the approximatemetri is g˜ g F F the ambient metri for . 3. Conformally Einstein example g F As the (cid:28)rst example, following Ref. [7℄, we al ulate and its approximate g¯ F ambient metri for a very simple equation: z′ =F(y′′), with Fy′′y′′ =0. 6 [g ] F 2 It was shown in Ref. [7℄ that the onformal lass may be represented by 15(F )10/3g = ′′ F − 30(F )4 [ dqdy pdqdx ] + [ 4F(3)2 3F F(4) ] dz2+ ′′ ′′ − − 2 [ 5(F′′)2F(3) 4F′F(3)2+3F′F′′F(4) ] dpdz+ − − 2 [15(F )3+5q(F )2F(3) 4FF(3)2+4qF F(3)2+3FF F(4) ′′ ′′ ′ ′′ − − 3qF F F(4) ] dxdz+ ′ ′′ [ 20(F )4+10F (F )2F(3)+4(F )2F(3)2 3(F )2F F(4) ] dp2+ ′′ ′ ′′ ′ ′ ′′ (3.1) − − 2 [ 15F (F )3+20q(F )4+5F(F )2F(3) 10qF (F )2F(3)+ ′ ′′ ′′ ′′ ′ ′′ − − 4FF′F(3)2 4q(F′)2F(3)2 3FF′F′′F(4)+3q(F′)2F′′F(4) ] dpdx+ − − [ 30F(F )3+30qF (F )3 20q2(F )4 ′′ ′ ′′ ′′ − − − 10qF(F )2F(3)+10q2F (F )2F(3)+4F2F(3)2 ′′ ′ ′′ − 8qFF F(3)2+4q2(F )2F(3)2 3F2F F(4)+ ′ ′ ′′ − 6qFF F F(4) 3q2(F )2F F(4) ] dx2. ′ ′′ ′ ′′ − gˆ = e2Υ(q)g F F As noted in Ref. [7℄ this metri is onformal to a Ri i (cid:29)at metri Υ=Υ(q) with a onformal s ale satisfying se ond order ODE: 90F 2(Υ Υ2) 60F F(3)Υ +3F F(4) 4F(3)2 =0. ′′ ′′ ′ ′′ ′ ′′ − − − 2 −15(TFh′e′)1m0/e3tri presented here di(cid:27)ers from this of [7℄ by a onvenient onformal fa tor equal to . 6 PAWEŠNUROWSKI F =F(q) [g ] F Thus,sin eforea h the onformal lass ontainsaRi i(cid:29)at metri , G 2 its onformalholonomymust be a propersubgroup of the non ompa t form of . Aninterestingfeatureofthis onformal lassisthatit isveryspe ialamongallthe g F onformal lassesasso iatedwithequation(1.1). Notonlyhas veryspe ial on- formalholonomy,makingitverysimilartotheLorentzian4-dimensionalBrinkman metri s;moreover,sin eitsWeyltensorhasessentiallyonlyonenonvanishing om- ponent(see Ref. [7℄ fordetails) it isnot weaklygeneri (see Ref. [3℄ forde(cid:28)nition). [g ] N F This makes analogousto the Lorentzian type metri s in 4-dimensions,su h as for example, Fe(cid:27)erman metri s. g F Having of (3.1) we used the symboli omputer al ulation program Math- γ emati a to al ulate its asso iated form of (2.2). We he ked that this form identi ally vanishes. We further used Mathemati a to al ulate the orresponding g¯ F approximate ambient metri . On doing that we obsereved that, surprisingly, β the bilinear form is also identi ally vanishing. The expli it formula for the ap- proximate ambient metri is given below: g¯ =t2g 2 dtdu 2tuF 4/3Pdq2, F F ′′ (3.2) − − with 4F(3)2 3F F(4) P = − ′′ , 90(F )10/3 ′′ g g¯ J R R F F + and given by (3.1). The metri is de(cid:28)ned lo ally on × × with (x,y,p,q,z,t,u) (4,3) oordiantes . It obviously has signature . We also he ked, Ric(g¯ ) 0 F againusingMathemati a,that ≡ . Thus,weful(cid:28)ledthestrategyoutlined g¯ F inSe tion2. Thisenablesusto on ludethat of(3.2) oin ideswiththeambient g˜ g F F metri for . To give expressionsfor the Cartan normal onformal onne tion g g˜ =g¯ F F F for andtheLevi-Civita onne tionfor we(cid:28)rstintrodu eanonholonomi (θ1,θ2,θ3,θ4,θ5) J oframe on given by θ1 =dy pdx − θ2 =dz Fdx F (dp qdx) ′ − − − θ3 = 2 (F )1/3(dp qdx) −√3 ′′ − 30(F )10/3θ4 = 3F F F(4) 4F F(3)2 10(F )2F(3) dp qdx + ′′ ′ ′′ ′ ′′ (cid:0) − − (cid:1)(cid:0) − (cid:1) 4F(3)2 3F′′F(4) dz Fdx +30(F′′)3dx (cid:0) − (cid:1)(cid:0) − (cid:1) θ5 = (F′′)2/3dq. − g F In this oframe the metri is simply: g =2θ1θ5 2θ2θ4+(θ3)2. F − By means of the anoni al proje tion π(x,y,p,q,z,t,u)=(x,y,p,q,z) (θ1,θ2,θ3,θ4,θ5) t(θh1e, θo2f,rθa3m,θe4,θ5) J R Ran be pulba ked to (cid:28)ve linearly independent forms + on × × . They an be suplemented by θ0 =dt and θ6 =du (θ0,θ1,θ2,θ3,θ4,θ5,θ6) J R R + to form a oframe on the ambient spa e × × . G2 CONFORMAL STRUCTURES WITH EXPLICIT AMBIENT METRICS AND CONFORMAL HOLONOMY7 J The Cartan normal onformal onne tion, when written on in the oframe (θ1,θ2,θ3,θ4,θ5) reads: 0 0 0 0 0 Pθ5 0 − θ1 0 Qθ2+ 9 Pθ3 1 θ4 1 θ3 0 Pθ5 2√3 √3 −2√3 − θ2 0 0 1 θ5 0 1 θ3 0 √3 −2√3 ω =θ3 0 2√3Pθ5 0 1 θ5 1 θ4 0 . G2 − √3 −√3 θ4 0 0 2√3Pθ5 0 Qθ2+ 9 Pθ3 0 − 2√3 θ5 0 0 0 0 0 0 0 θ5 θ4 θ3 θ2 θ1 0 − − Here: 40F(3)3 45F F(3)F(4)+9F 2F(5) Q= − ′′ ′′ . 90F 5 ′′ (θ0,θ1,θ2,θ3,θ4,θ5,θ6) Nowweuse oframe to writedownthe Levi-Civita onne - g˜ F tion for . We have g˜ =g θiθj, F ij i,j =0,1,2,...6 g ij with the indi es range: , and the matrix given by 0 0 0 0 0 0 1 − 0 0 0 0 0 t2 0 0 0 0 0 t2 0 0 − g = 0 0 0 t2 0 0 0 . ij 0 0 t2 0 0 0 0 − 0 t2 0 0 0 2tuP 0 − 1 0 0 0 0 0 0 − g˜ J R R F + The Levi-Civita onne tion for on × × , when written in the oframe (θ0,θ1,θ2,θ3,θ4,θ5,θ6) reads: 0 0 0 0 0 tPθ5 0 − 1tθ1+ tu2Pθ5 1tθ0 Qθ2+ 2√93Pθ3 √13θ4 −2√13θ3 tu2Pθ0− 3utQθ5− 1tPθ6 −1tPθ5 1θ2 0 1θ0 1 θ5 0 1 θ3 0 t t √3 −2√3 ω = 1θ3 0 2√3Pθ5 1θ0 1 θ5 1 θ4 0 . LC t − t √3 −√3 1θ4 0 0 2√3Pθ5 1θ0 Qθ2+ 9 Pθ3 0 t − t 2√3 1θ5 0 0 0 0 1θ0 0 t t 0 tθ5 tθ4 tθ3 tθ2 tθ1 uPθ5 0 − − − 8 PAWEŠNUROWSKI Σ = (x,y,p,q,z,t,u): u = 0, t = 1 θ0 0 θ6 Note that on { } we trivially have ≡ ≡ . ω Σ ω ω Thus,restri tingtωheformulafor LC to ,wesπeethat G2 ≡ω LC|Σ. O(cid:27)thissetthe two onne tions: LC and the pullba ked-by- - onne tion G2, di(cid:27)er signi(cid:28) antly. ω π (ω ) To see this it is enough to observe that ontrary to LC, the onne tion ∗ G2 π (ω ) has torsion. Indeed writing the (cid:28)rst Cartan stru ture equations for the ∗ G2 in (θ0,θ1,θ2,θ3,θ4,θ5,θ6) the oframe we (cid:28)nd that the torsion is: 0 θ0 θ1 Pθ5 θ6 − ∧ − ∧ θ0 θ2 − ∧ dθi+π∗(ωG2)ij ∧θj = −θ0∧θ3 . θ0 θ4 − ∧ θ0 θ5 − ∧ 0 Σ The vanishing of this torsion on the initial hypersurfa e on(cid:28)rms our earlier ω ω statemant that the two onne tions G2 and LC oin ide there. dω +ω ω LC LC LC It is interesting to note that the urvature ∧ does not depend t u ∂ ∂ t u on , and isanihilated by and . Thus it anbe onsideredto be a2-formon Σ dω +ω ω . Assu hitispre iselyequaltothe urvature G2 G2∧ G2 ofthe onne tion ω G2: 0 0 0 0 0 0 0 0 0 A 0 0 0 0 5 0 0 0 0 0 0 0 dωG2 +ωG2 ∧ωG2 =dωLC +ωLC ∧ωLC =0 0 0 0 0 0 0θ2∧θ5, 0 0 0 0 0 A5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 where 224F(3)4+336F F(3)2F(4) 51F 2F(4)2 80F 2F(3)F(5)+10F 3F(6) A = − ′′ − ′′ − ′′ ′′ . 5 100F 20/3 ′′ 4. Non- onformally Einstein example [g ] F To get quite di(cid:27)erent example of we onsider equation (1.1) in the form: z =y 2+a y6+a y5+a y4+a y3+a y2+a y +a +bz, ′ ′′ 6 ′ 5 ′ 4 ′ 3 ′ 2 ′ 1 ′ 0 a ,i = 0,1,...,6, b i where and are real onstants. This equation has the de(cid:28)ning fun tion F =q2+a p6+a p5+a p4+a p3+a p2+a p+a +bz 6 5 4 3 2 1 0 [g ] F and, via (1.3), leads to a onformal lass represented by a metri 15(2) 2/3g =[9a +2b2+27a p+54a p2+90a p3+135a p4]dy2+ − F 2 3 4 5 6 [15a +2(b2 3a )p2 3a p3+9a p4+30a p5+60a p6 0 2 3 4 5 6 − − − 20bpq+5q2+15bz]dx2+ (4.1) [15a +4(3a b2)p 9a p2 48a p3 105a p4 180a p5+ 1 2 3 4 5 6 − − − − − 3 A5 Weusetheletter todenote thenonvanishing omponentofthe urvature tobeina or- dan ewith[7℄and Cartan's paper [1℄. Notehowever that inorder toavoid ollisionofnotations A5 a5 betweenthepresentandthenextse tions weuse apital insteadof ofpaper [7℄. G2 CONFORMAL STRUCTURES WITH EXPLICIT AMBIENT METRICS AND CONFORMAL HOLONOMY9 20bq]dxdy+20dp2 − 10(bp+q)dpdx+10bdpdy 30dqdy 15dxdz+30pdqdx. − − This metri is not onformal to an Einstein metri . The qui kest way to he k C W g ijk ijkl F this is the al ulation of the Cotton, , and the Weyl, , tensors for . On e these tensors are al ulated, it is easy to observe that they do not admit a Ki C +KlW = 0 ijk lijk ve tor (cid:28)eld su h that . As a onsequen e the metri is not a onformal C-spa e metri . This proves our statement sin e every onformally Einstein metri is ne essarily a onformal C-spa e metri (see e.g. Ref. [3℄). g F Re all that of (4.1), as a member of the family of metri s (1.3), de(cid:28)nes a [g ] H F onformal lass with onformal holonomy redu ed tothenon ompa tgroup G 2 or to one of its subgroups. But sin e the metri (4.1) is not onformal to an H = G 2 Einstein metri , we do not have an immediate reason to on lude that 6 . H =G 2 We onje ture that here and try to prove it in a subsequent paper [6℄. g˜ g F F It is remarkable that the ambient metri for of (4.1) assumes a very ompa t form: g˜ =t2g 2 dtdu F F − − 2 tu [ 1 ( 2a +4b2+3a p+6a p2 20a p3 120a p4)dx2 20 − 2 3 4 − 5 − 6 − 9 (a 10a p2 40a p3)dxdy 9 (a +5a p+15a p2)dy2 ] + 20 3− 5 − 6 − 10 4 5 6 u2 [ 3 (a 10a p+60a p2)dx2+ 9 (a 12a p)dxdy+ 81 a dy2 ]. 20(2)2/3 4− 5 6 4(2)2/3 5− 6 4(2)2/3 6 Thisis he kedby applyingourstrategydes ribed in Se tion 2to the metri (4.1). As in the previous example, using Mathemati a, we al ulated the bilinear form γ γ 0 g¯ F for (4.1). It turned out to be equal to zero, ≡ . Then we al ulated , g¯ F and he ked that it is Ri i (cid:29)at. Thus we on luded that oin ides with the g˜ g˜ g¯ F F F ambient metri for . The above given formula for is therefore just , whi h we al ulated using (2.2). g F We (cid:28)nd this example as a sort of mira le. Apriori there is no reason for to have the ambient metri trun ated at the se ond order in terms of the ambient t u parameters and . We are intrigued by this fa t. Now, following the general pro edure outlined in [7℄, we introdu e a spe ial g F oframe for given by: θ1 =dy pdx − θ2 =dz Fdx 2q(dp qdx) − − − θ3 = 24/3(dp qdx) − √3 − θ4 =2 1/3dx − 15(2)1/3θ5 =(9a +2b2+27a p+54a p2+90a p3+135a p4)(dy pdx)+ 2 3 4 5 6 − 10b(dp qdx) 30dq+ − − 15(a +2a p+3a p2+4a p3+5a p4+6a p5+2bq)dx. 1 2 3 4 5 6 g F In this oframe the metri is: g =2θ1θ5 2θ2θ4+(θ3)2. F − As in the previous se tion, we use the anoni al proje tion π(x,y,p,q,z,t,u)=(x,y,p,q,z) 10 PAWEŠNUROWSKI (θ1,θ2,θ3,θ4,θ5) (θ1,θ2,θ3,θ4,θ5) topJullbRa kthRe oframe to(cid:28)velinearlyindependentforms + on × × , whi h are further suplemented by θ0 =dt and θ6 =du (θ0,θ1,θ2,θ3,θ4,θ5,θ6) J R R + to form a oframe b = 0 on thJe ambiJent sRpa e R× × . + It turns out that if the oframes on and × × de(cid:28)ned in this way are suitable to analyze the relations between the Cartan normal onformal ω [g ] ω g˜ b = 0 onne tion G2 for F and the Levi-Civita onne tion LC for F. If 6 ω (θ1,θ2,θ3,θ4,θ5) ω the one tion G2 in the oframe and the onne tion LC in the (θ0,θ1,θ2,θ3,θ4,θ5,θ6) t = 1 u = 0 oframe do not oin ide on , . We will not analyze this ase here. Restri ting to the b=0 ase we (cid:28)nd the following: ω (θ1,θ2,θ3,θ4,θ5) • the onne tions G2 in the oframe and the onne tion ω (θ0,θ1,θ2,θ3,θ4,θ5,θ6) t=1 u=0 LC in the oframe oin ide on , . π (ω ) (θ0,θ1,θ2,θ3,θ4,θ5,θ6) • the torsionof ∗ G2 in the oframe is nonvanish- t=1 u=0 ing o(cid:27) the set , dω +ω ω LC LC LC • unliketheexampleofthepreviousse tionthe urvature ∧ t u sigini(cid:28) antly depends on and . t = 1 u = 0 dω +ω ω • even on , , the urvature G2 G2 ∧ G2 and the restri tion dω +ω ω LC LC LC of ∧ do not oin ide. 5. A knowledgements IamverygratefultoTPBranson,MEastwoodandWMillerJr,theorganizers of the 2006 IMA Summer Program (cid:16)Symmetries and Overdetermined Systems of Partial Di(cid:27)erential Equations(cid:17), for invitating me to Minneapolis to parti pate in this very fruitful event. The topi overed by this note is inspired by the talk of C R Graham whi h I heard in Minneapolis during the program. In parti ular, I am very obliged to C R Graham for sending me the formulas (2.2), whi h I used to prepare the examples in luded in this note. Referen es [1℄ Cartan E, (cid:16)LessystemesdePfa(cid:27)a inq variables et les equations aux derivees partielles du se onde ordre(cid:17) Ann. S . Norm. Sup.27109-192 (1910) [2℄ Fe(cid:27)erman C, Graham C R, (cid:16)Conformal invariants(cid:17), in Elie Cartan et mathematiques d'aujourd'hui,Asterisque,horsserie(So iete MathematiquedeFran e, Paris)95-116 (1985) [3℄ Gover A R, Nurowski P, "Obstru tions to onformally Einstein metri s in n dimensions" Journ. Geom. Phys.56450-484 (2006) [4℄ GrahamCR,private ommuni ations,unpublished [5℄ deHaroS,SkenderisK,SolodukhinSN,(cid:16)Holographi re onstru tionofspa etimeandrenor- malization in the AdS/CFT orresponden e(cid:17), Comm. Math. Phys. 217 (2001), 594(cid:21)622, hep-th/0002230 [6℄ LeistnerTh,NurowskiP,inpreparation [7℄ NurowskiP,"Di(cid:27)erentialequationsand onformalstru tures" Journ.Geom.Phys.5519-49 (2005) Instytut Fizyki Teorety znej, Uniwersytet Warszawski, ul. Ho»a 69, Warszawa, Poland E-mail address: nurowskifuw.edu.pl
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