Eleven Dimension Supergravity Equations on Edge Manifolds Xuwen Zhu MIT Xuwen Zhu (MIT) 11-Dim Supergravity 1/29 Supergravity Equations The 11 dimensional bosonic supergravity equations on M = B7 S4 × are defined for a metric g and a 4-form F γ γ γ R = 1 (F F 1 2 3 1 F F γ1γ2γ3γ4 g ) αβ 12 αγ1γ2γ3 β − 12 γ1γ2γ3γ4 αβ d F = 1 F F ∗ − 2 ∧ dF = 0 Result overview All the smooth solutions in edge form with a given conformal infinity near the Freund-Rubin solution are parametrized by a 3-form and two functions on the bounding 6-sphere of B7. Xuwen Zhu (MIT) 11-Dim Supergravity 2/29 Background: physics From the representation of super Lie algebra [Kac, 1978] Eleven is the maximal dimension in order to be physical [Nahm, 1975] Existence shown in [Cremmer-Scherk, 1977] Lower dimension cases can be obtained through dimension reductions that yield more smaller pieces [Nieuwenhuizen, 1985] Recent development in relation with AdS/CFT correspondence [Witten, 1997] [Blau-Figueroa-O’Farrill-Papadopoulos, 2002] Generalization of Einstein equations on a manifold Ric(g) = (n 1)Rg − Change of signature: Lorentzian to Riemannian by complexifying the time direction: t i t → ∗ Xuwen Zhu (MIT) 11-Dim Supergravity 3/29 Derivation of Supergravity Equations Euler-Lagrange equations from the following Lagrangian 1 1 L(g, A)= RdVg ( F F + A F F). ZM − 2 ZM ∧ ∗ ZM 3 ∧ ∧ Three terms: classical Einstein-Hilbert action, Yang-Mills type, Maxwell type A is a 3-form such that F is the field strength F = dA, need not to be globally defined Xuwen Zhu (MIT) 11-Dim Supergravity 4/29 Supergravity Equations The 11 dimensional bosonic supergravity equations on M = B7 S4 × are defined for a metric g and a 4-form F 1 1 R = (F F γ1γ2γ3 F F γ1γ2γ3γ4 g ) αβ 12 αγ1γ2γ3 β − 12 γ1γ2γ3γ4 αβ 1 d F = F F ∗ −2 ∧ dF = 0 Remark If F = 0, then any Einstein vacuum solutions would solve the equations. Xuwen Zhu (MIT) 11-Dim Supergravity 5/29 A family of product solutions X 7: Einstein manifold with negative scalar curvature α< 0. K 4: Einstein manifold with positive scalar curvature β > 0. Consider M = X K , then × X 6αgAB 0 Rαβ = K 0 3βgab Let F = c VolK . The contraction part is then 2 X c 2gAB 0 (F F)αβ = − K ◦ 12 0 4gab Therefore any set (c, α, β) satisfying c2/6 = 6α, c2/3 = 3β − corresponds to a solution. Xuwen Zhu (MIT) 11-Dim Supergravity 6/29 Freund-Rubin solution The parameters c2/6 = 6α, c2/3 = 3β − give a family of solutions (X 7 K 4, c Vol ) with × K 7 4 Ric(X )= 6αgX , Ric(K )= 3βgK . In particular, when c = 6 there is the following product solution Definition (Freund-Rubin solution) The Freund-Rubin Solution is defined on B7 S4 as × 1 (g0, V0) = (gH7 gS4 , 6 VolS4 ) × 4 Xuwen Zhu (MIT) 11-Dim Supergravity 7/29 Review: Poincare-Einstein´ metric on the n-ball Poincare-Einstein´ metric: Einstein metric with a conformal infinity. Definition (Conformal infinity) For gˆ a metric on ∂M, we say a metric g on M has a conformal infinity [gˆ] if for a boundary defining function x, x 2g extends to M¯ and is conformal to gˆ on ∂M. Existence of Poincare-Einstein´ metric near the standard hyperbolic metric on Bn+1: Theorem [Graham-Lee, 1991] Let hˆ be the standard metric on Sn. For any smooth Riemannian metric gˆ on Sn which is sufficiently close to hˆ in C2,α norm if n > 4 or C3,α norm if n = 3, for some 0 <α< 1, there exists a smooth metric g on the interior of M, with conformal infinity [gˆ] and satisfies Ric(g)= ng. − Xuwen Zhu (MIT) 11-Dim Supergravity 8/29 Review of the Poincare-Einstein´ Proof Proof. Use DeTurck term φ(t, g) to break the gauge Compute the linearization of the operator Q = Ric φ(t, g)+ n − Using indicial roots computation and Cheng-Yau Maximum Principle [Cheng-Yau, 1980] to show that dQ is an isomorphism between weighted Sobolev spaces Then use a perturbation argument to show that the nonlinear operator Q is a bijection between the boundary parameter and the Einstein metric. Xuwen Zhu (MIT) 11-Dim Supergravity 9/29 Edge manifold Edge operator theory [Mazzeo, 1991] is introduced on a compact manifold M with boundary, where the boundary has a fibration structure: F / ∂M π B Edge vector fields are regular in the interior and parallel to fibers on the boundary. In local coordinates, Lie algebra e is spanned V by x∂x , x∂ I , ∂ j . { y z } By duality, edge bundles are given by tensor products of dx/x, { dy I /x, dzj } The edge metric is of the form dx 2 dxdy I dy I dy J dxdzj dy I dzj g = a +a +a +a +a +a dzi dzj . 00 x 2 0I x 2 IJ x 2 0j x Ij x ij Xuwen Zhu (MIT) 11-Dim Supergravity 10 / 29 Edge Sobolev Space Definition (Edge Sobolev space) s On an edge manifold M, the edge Sobolev space He (M) is defined as s 2 k 2 H (M)= u L (M) V u L (M), Ve e(M), 0 k s e { ∈ | e ∈ ∈V ≤ ≤ } Another related Lie algebra is the b-vector field [Melrose, 1992], Vb which is tangent to the boundary, and spanned locally by x∂x , ∂y , ∂z . { } Definition (Weighted hybrid Sobolev space) x δHs,k (M)= x δu Hs(M) V i u Hs(M), V , 0 i k e,b { ∈ e | b ∈ e b ∈Vb ≤ ≤ } Xuwen Zhu (MIT) 11-Dim Supergravity 11 / 29 Semi-Fredholm property of elliptic edge operator j α β Elliptic edge operator L = j+|α|+|β|≤m aj,α,β(x∂x ) (x∂y ) (∂z ) is defined for the principal edgeP symbol j α β aj,α,βξ1ξ2 ξ3 j+|αX|+|β|=m Example: Laplacian on H2 S1 × Theorem [Mazzeo, 1991] If an elliptic edge operator L Diffm(M) has two properties: ∈ e (a) Constant indicial roots over the boundary; t (b) Its normal operator L0 and adjoint L0 has the unique continuation property, δ k then L is essentially injective (resp. surjective) on x He (M) for a weight parameter δ 0 (resp. δ 0) with δ / Λ= Re θ + 1/2 : θ spec L , ≫ ≪ ∈ { ∈ b } and in either case has closed range. Xuwen Zhu (MIT) 11-Dim Supergravity 12 / 29 Main theorem Define three bundles over S6 = ∂B7: ∞ 6 3 ∗ 6 V1 := v1 C (S ; T S ) : S6 v1 = iv1 . { ∈ ^ ∗ } V := v ξ : v C∞(S6; R),ξ E cl (S4) 2 { 2 ⊗ 16 2 ∈ 16 ∈ 16 } V := v ξ : v C∞(S6; R),ξ E cl (S4) 3 { 3 ⊗ 40 3 ∈ 40 ∈ 40 } Theorem For k 0, δ (0, 1), s 2, fixing a conformal infinity [hˆ] that is close ≫ ∈ ≥6 to [gˆ0] at the boundary S , then given any smooth section 3 k v = (v1, v2, v3) of the bundle V with a sufficiently small H norm, ⊕i=1 i there is a unique (g, F) ∞(M; Sym2(eT ∗M) e 4 T ∗M) such that ∈C ⊕ (a) (g g , F V ) x −δHs,k (M; W ) and has a leadingV expansion − 0 − 0 ∈ e,b given by v; (b) (g, F) satisfy the supergravity equation S(g, F)= 0 with g having the conformal infinity [hˆ]. Xuwen Zhu (MIT) 11-Dim Supergravity 13 / 29 Gauge term ∗ −1 We apply the DeTurck type term: φ(g,t) = δt (tg) δt Gt g to get the full nonlinear system 4 4 Q : S2(T ∗M) (M) S2(T ∗M) (M) ⊕ → ⊕ ^cl ^cl g Ric(g) φ F F − (g,t) − ◦ F 7→ d (d F + 1 F F) ∗ ∗ 2 ∧ Then we get the following gauge elimination: Proposition (Gauge elimination lemma) If (k,H) satisfies the linearized equation dQg(k, H)= 0, then we can ˜ ˜ find a 1-form v and k = k + Lv ♯ g such that dSg(k, H)= 0. Xuwen Zhu (MIT) 11-Dim Supergravity 14 / 29 Linearization of the gauged operator The operator Q : W W has the following linearization at the point → (g0, V0): 4 4 2 e ∗ e 2 e ∗ e dQg0,V0 : Γ(Sym ( T M) (M)) Γ(Sym ( T M) (M)) ⊕ ^ → ⊕ ^ k ∆k + LOT H 7→ d (d H + 6V H + 6d k1 1 + 3d(7σ 4τ) V ) ∗ ∗ S ∧ ∗H , − ∧ H where the lower order term matrix LOT is as follows: kIJ 6TrS(k)tIJ − − 6k1,1 3 S H1,3 +TrH (k)tIJ + 2 S H 0 4 tIJ − ∗ LOT = ∗ ( , ) 4kij + 8TrS(k)tij 6k1,1 3 S H1,3 − ∗ H0 4t − ∗S , ij Xuwen Zhu (MIT) 11-Dim Supergravity 15 / 29 Splitting with the product structure From the form equation 7 6d 7k +3d (Tr 7 (k) Tr 4 (k)) V +d H +d H = 0 H ∗ (1,1) S H − S ∧ S ∗ (0,4) H ∗ (1,3) d H + d H + 6d 7 k = 0 S ∗ (1,3) H ∗ (2,2) S ∗ (1,1) d H + d H = 0 S ∗ (2,2) H ∗ (3,1) d H + d H + W H = 0 S ∗ (3,1) H ∗ (4,0) ∧ (4,0) From the metric equation: 1 1 sk + k + 6k 3 H = 0 2△ Ij 2△H Ij Ij − ∗S (1,3) 1 ( s + )k k 6Tr (k)t + Tr (k)t + 2H t = 0 2 △ △H IJ − IJ − S IJ H IJ (0,4) IJ 1 ( + )k + 4k + 8Tr (k)t H t = 0 2 △S △H ij ij S ij − (0,4) ij Xuwen Zhu (MIT) 11-Dim Supergravity 16 / 29 Inidicial operator: introduction Definition (Indicial operator) Let L : Γ(E1) Γ(E2) be an edge operator between two vector → bundles over M.