Well-posedness of the equation for the three-form field in the eleven dimensional supergravity

We analyze the equations for the three-form field - a system of semi-linear gauge-invariant wave equations which arises in the theory of eleven dimensional supergravity. We prove that the Cauchy problem is well-posed globally in time for the fixed-gauge version of the equation for a small compactly supported smooth data. We employ the method of Klainerman vector fields along with a finer analysis of the nonlinearity to establish an integrable decay in the energy estimate.


Introduction
Let K be a compact 7-dimensional Riemannian manifold. Then the product R 3+1 ×K becomes an 11-dimensional Lorentzian manifold. For a 3-form u on R 3+1 × K we use the Hodge star * and the de Rahm differential d of the product to formulate the following Cauchy problem: (1.1b) u(0, ·) = u 0 , u t (0, ·) = u 1 .
In this article, we will prove the following statement.
Theorem 1.1. There exist positive N, ǫ such that the Cauchy problem (1.1) is globally well-posed provided that initial data is localized in the ball of radius 1 in R 3 for every point of K and obeys Moreover, in such a case the solution u satisfies the following estimates |α|≤N ∇ t,x,y Γ α u(t) L 2 (R 3 ×K) ≤ Cǫ(1 + t) 1/12 , |α|≤N −9 ∇ t,x,y Γ α u(t) L 2 (R 3 ×K) ≤ Cǫ, 1 (1 + t) |α|≤N −18 ∇ t,x,y Γ α P 0 u(t) L ∞ (R 3 ×K) where P 0 , P >0 are the spectral projections of the operator ∆ K defined in Section 3, Γ α are compositions of a subset of Klainerman vector fields together with the operator (−∆ K ) 1/2 , which are defined in Section 4, ∇ t,x,y is the gradient in all the derivatives of R 3+1 × K and C is a constant that depends only on N and the geometry of K.
The theorem is true if the Cauchy data is supported in a larger ball but then the constant ǫ has to decrease as a negative power of the size of the support.
The equations have a connection to the theory of supergravity, which we explain in the next section. The mathematical aspects of the supergravity theory have recently drawn the attention in the context of conformal geometry [2], [5], where the space-time was assumed to be a Riemannian manifold. The Lorentzian case was investigated earlier (see [1] and references therein).
Our methods are inspired by the work of Metcalfe, Sogge and Stewart [9] and Metcalfe and Stewart [10] who analyze the quasi-linear wave equation on R 3+1 ×D, where D is a bounded domain in R n with various non-linearities and boundary conditions. Their results do not cover the case in study but we employ some of their ideas in this work.
The article is organized as follows. In section 2 we derive the equation from a gauge-invariant Lagrangian and explain how to fix the gauge. In section 3 we recall some necessary facts from Riemannian geometry. In section 4 we adapt the linear estimates for the wave equation on R 3+1 to the product R 3+1 × K. In section 5 we perform a deeper analysis of the nonlinearity. In section 6, we provide the proof of Theorem 1.1.
We will denote by k a constant which depends only on N and the geometry of K, this constant may change from line to line but for each inequality below there is an apriori computable constant such that the inequality holds. We will also write A B to mean A ≤ kB.

Physical Motivation.
The supergravity theory is a model of classical physics, which describes the low-energy, classical limit of the superstring theory. The model describes the interaction of the field of gravity with other fields. In one of the simplest setups, one considers an 11-dimensional Lorentzian manifold as a space-time, with gravity field g and a field, whose strength is described by a closed differential 4-form F . The Lagrangian is prescribed only locally by restricting the attention to an open, topologically trivial subset U of the space-time. Then one solves the equation for a potential of F on U: Then the Lagrangian is where R, dv and * are the (scalar) Ricci curvature, the volume form and the Hodge * corresponding to g, respectively. The reader should consult [14] and textbooks for physical aspects of this theory.
2.2. The Lagrangian and the equation. We will simplify our setup to consider the product manifold R 3+1 × K as the fixed space-time, where K is a 7-dimensional compact Riemannian manifold without a boundary. The metric on the product space will be the product of the Minkowski metric and the metric on K. We will also assume that the field strength F is not only closed but also exact, namely there exists a global 3-form u, such that du = F. Then u will be the dynamical variable, for which we define a classical field theory Lagrangian The formal Euler-Lagrange equations are We will take the Hodge-dual on both sides of the equation and use the notation δ = * d * to arrive to the following equation Since our space time is a product manifold then most of the operators which act on it can be decomposed in a natural way as operators acting on either on R 3+1 or K. We will denote by subscript the operators acting on R 3+1 and by subscript ⊥ the operators acting on K. For instance, we will have The tensor notation should be understood in terms of operations on differential forms Ω(R 3+1 ) and Ω(K). The equation (2.4) we will colloquially write 2.3. Hodge star and form Laplacian. Let us recall a few simple facts regarding the Hodge star operator, which is an operator that takes differential n-forms to differential (11 − n)-forms. Let x i , i = 0..3 be the coordinates on R 3+1 and x i , i = 4..10 be a coordinate patch on K at a point where the metric tensor is the identity and its derivative vanish, i.e. normal coordinates. Then the Hodge dual * for an n- .10 is defined as follows: Thus the * operator exchanges the components of the forms, multiplying those containing the time x 0 coordinate by −1. Next, we define δ which takes n-forms to (n − 1)-forms by Lastly we define We have the following facts about * and δ where g is the Lorentzian metric on R 3+1 × K and dvol is the volume form. • The operator * is an isometry and in particular d * u = 0 if and only if δu = 0. • δ is the Lorentzian adjoint of d in the sense that • In normal coordinates and with notational conventions of relativity we have .. , where f (R) is a linear, zeroth-order tensor that depends on the Riemann curvature tensor.
. The first identity is the consequence of the fact that d 2 = 0. This fact also implies δ 2 = 0 which leads to the second identity above.

Gauge fixing.
There is an obvious gauge freedom in equation (2.3) -if u is a solution of the equation then for any two-form w, the three-form u + dw is a solution as well. Therefore, we will fix the gauge by requiring that u satisfies which is equivalent to δu = 0. This choice of gauge is similar to the Lorenz gauge of the Maxwell equations, where for a one-form a one requires d * a = 0. Since we work with 3-forms on a product manifold, the gauge is structurally more complicated. We will give a proof that equation (2.7) is a valid gauge choice in the end of this section but first we rewrite the equation (2.9) using the gauge. We defined the Laplace(d'Alembert)-Beltrami operator on forms as R 3+1 ×K u = −δd − dδ 1 . On a product manifold, the operator decomposes into R 3+1 ×K = R 3+1 − ∆ K ,where ∆ K is the Laplace-Beltrami operator on Ω(K) (the space of differential forms). Thus we can rewrite the main equation (2.3) as . Therefore, the equation (2.3) with the gauge choice (2.7) becomes (2.9) R 3+1 u − ∆ K u = * (du ∧ du). 1 We make a consistent effort to have the operator ∆ K be negative on Riemannian manifolds. With this convention Let us now address the validity of the gauge choice. Moreover, we can chooseÃ such that Proof. Let A be as above. Denote δA = e. The two-form e is the error we wish to eliminate by finding a two-form b such that δ(A + db) = 0. We solve the equation We have from equation (2.10) Thus δb solves the homogeneous wave equation. We will prove that δb = 0 by choosing suitable Cauchy data for b at the hypersurface t = 0. Our goal is to make the Cauchy data for δb be zero. The Cauchy data that we prescribe for b in the normal coordinates are as follows: We check the Cauchy data for δb at t = 0.
This is because the Cauchy data for b 0α is zero and the function b βα = 0 for α = 0 at t = 0 and so are the spatial derivatives. To see that the time derivative of δb at t = 0 is zero, we employ the equation for b Observe that (∆ R 3 + ∆ K )b 0α = 0 since the function at t = 0 is zero and we take spatial derivatives and zero order term which are linear in b to compute the Laplacian. The second and the third terms cancel because since A 00α = 0 by antisymmetry then by equation (2.12c). Therefore (δb) α | t=0 = 0 and thus δb obeys a homogeneous wave equation with zero Cauchy data, which makes it identically zero. Therefore Observe that the support of the Cauchy data for b is contained in the support for A and thus the statement on the support follows from finite speed of propagation.
We now wish to prove that the gauge condition δu = 0 persists for the equation (2.9). For that we need to discuss the initial conditions. Since the equation is of second order, the natural Cauchy data is u| t=0 and ∂ ∂t u| t=0 . If we express (2.3) through the field strength F = du we have The natural initial condition for this first order equation is F | t=0 but we first need to observe that there is a certain compatibility condition in (2.13), which is not of the evolution form . For that we recall the notion of the interior product of a form by a vector field. Let α be an n-form and X be a vector field, then the interior product of α by X, denoted by α⌋ X is an n − 1 form defined by α⌋ X (X 1 , X 2 , .., X n−1 ) = α(X, X 1 , X 2 , ..X n−1 ).
We will be interested in interior products by ∂ ∂t = ∂ ∂x 0 . Such a construction can be simply described as freezing the first index of the form α to be the zero (i.e. time) index. Thus in coordinates With this notation we prove the following observation Proof. We will give the proof in normal coordinates. We have Only the first term contains the time derivative but F 00a 1 a 2 = 0 due to antisymmetry of F .
Thus applying ⌋ ∂ ∂t to (2.13) and restricting it to time t = 0 we see that both sides of the equality depend only on F | t=0 so they are functions of a gauge invariant part of the Cauchy data and express a compatibility condition, which must hold in both gauge-invariant and gauge-fixed versions of the equations. We thus make the following definition.
We now can prove that the gauge condition δu = 0 persists in the equation for the compatible Cauchy data.
Proof. We apply R 3+1 ×K to δu to see that We check the Cauchy data: δu| t=0 = 0 by assumption. The term ∂ ∂t δu| t=0 vanishes because of the equation and the compatibility condition. We prove that in normal coordinates. We have If a, b = 0 then the last two terms above are spatial derivatives of δu which is zero when we compute at t = 0. Therefore, for a, b = 0 where we applied the compatibility condition to conclude the last equality. Next we assume without loss of generality that a = 0, b = 0 then since This vanishes because it is a sum of spatial derivatives of components of δu which vanish at t = 0.
Corollary 2.5. Let u solve the equation

Review of Hodge theory
The objects of our study are 3-forms on R 3+1 ×K. The basic example of such a form would be u ∧ u ⊥ where u ⊥ is a k-form (for k = 0, 1, 2, 3) and u is a 3 − k form on R 3+1 . The action of the Hodge-Laplacian of K is clearly ∆ K (u ∧u ⊥ ) = u ∧(∆ K u ⊥ ) and it extends through density on all the forms on R 3+1 × K. Moreover, if we use the eigenvectors of ∆ K , ∆ K e λ = −λ 2 e λ , we can further decompose any form on R 3+1 × K as a series u(x, y) = λ u λ (x)e λ (y), where x is a variable on R 3+1 and y is the variable on K. Thus, we envision the equation being the system of equations on differential forms on R 3+1 which are indexed by λ, in which case the equation will become where B's are bilinear differential operators. Thus we see that u λ for λ = 0 evolve under a non-linear wave equation, while u λ for λ = 0 evolve under a non-linear Klein-Gordon equation. This analysis follows the ideas of Metcalfe, Sogge and Stewart [9] and Metcalfe and Stewart [10], who analyze the wave equation on R n+1 ×D, where D is a bounded domain in R m with various boundary conditions. Their analysis splits the function to eigenfunctions of the Laplacian on D with appropriate boundary conditions.
In this section, we recall some properties of the eigenvectors of ∆ K which we need for the proof. The material is taken from textbooks, [4, section 2.1] and [12, section 5.8]. For the rest of this section we will deal only with forms on K. We will continue to employ the subscript ⊥ to maintain consistency. We begin with the following facts. (1) The operator ∆ K is a differential operator acting on the space of forms The operator ∆ K has a self-adjoint nonpositive-definite extension to the space of L 2 -valued forms on K.
These are spectral projections on the zero-,non-zero subspace of the spectrum of −∆ K , respectively.
3.1. Hodge Theory. The range of P 0 , i.e all the forms ω that satisfy ∆ K ω = 0 are called the harmonic forms. We have the following simple fact.
The full version of this claim can be found in [4, Proposition 2.1.5]. It is the basis of Hodge theory in algebraic topology. We will not require any of it in this paper but we will quote the following theorem for the sake of beauty. Theorem 3.3. Every non-empty de-Rham cohomology class of K contains precisely one harmonic form.
See [4, Theorem 2.2.1] for the proof. Thus existence and properties of harmonic forms are connected to the topological properties of the manifold. For instance the sphere S 7 will have only two harmonic forms -the constant 0-form and the volume 7-form. The torus T 7 will have 7 n linearly independent harmonic n-forms. Observe that both of these statements are independent of the choice of the Riemannian metric.
3.2. Elliptic regularity for ∆ K . We recall some basic regularity results for the form Laplacian. We have the following estimates See [12,Proposition 8.1] for proof. This inequality has the following immediate corollaries: Corollary 3.6. for every N, there are constants A N such that for every See the discussion leading to [12,Equation (8.20)] for the proofs. The practical conclusion that we will draw from these two corollaries is that when measuring smoothness of the solution in K variables, we can ignore the question completely for u = P 0 u and use the (−∆ K ) 1/2 operator for u = P >0 u.

Linear Estimates
In this subsection, we would like to obtain decay estimates for the linear inhomogeneous equation. We will leverage this decay by employing the following subset of Klainerman vector fields: We augmentΓ with the operator (−∆ K ) 1/2 We will index the set Γ by i = 1..11 and for a multi-index I = (I 1 , I 2 , .., I |I| ) ∈ {1, .., 11} |I| we define the composition We will introduce some notation to simplify the presentation. We will denote for an integer N, an abstract vector valued function: Accordingly we will interpret the following notations We will also have a similar notation for the gradients All those norm will be taken at a certain time t, which we will drop from the notation when there is no ambiguity. We will fix coordinate patches on K, with the appropriate partition of unity. That will turn our objects into vector valued functions on R 3+1 × R 7 , so that we will apply the vector fieldsΓ simply by applying them on every component of u.

4.1.
Linear estimates in R 3+1 . We recall the following estimates in R 3+1 which we seek to generalize to the product case R 3+1 × K.
This proposition is proven in [10, Proposition 3.1]. Although [10] proves it with zero Cauchy data, the estimate with non-zero Cauchy data is proven in the same manner.

4.2.
Linear estimates in R 3+1 × K. We turn to obtaining estimates for the equation Since the spectral projections P A commute with this equation, we will split the equation into two equations with the spectral projections applied to initial data as well. By elliptic regularity and the estimate for the wave equation, Proposition 4.1, we have the following estimate.
Proof. We wish to apply Proposition 4.1 with B = 1. For that we need to switch to a new coordinate τ = t + 2 then the proposition applies with one reservation: the vector fields in τ, x, y coordinates are different from the vector fields in t, x, y coordinates but they are expressible in terms of sums of the old ones since ∂ ∂τ = ∂ ∂t and Thus, the Proposition 4.1 applies with possibly a different constant and (t, x, y) vector fields to show that for every y ∈ K Apply elliptic regularity (Corollary 3.5) to dominate (P 0 F )(s, ·, y) by its L 2 (K) norm.

then
(1 + t) 3/2 |P >0 u(t, x, y)| ∇Γ (9) P >0 u(0, x, y) 2 The proof of the theorem follows almost verbatim the proof of [3, Proposition 7.3.5] with the exception of the following modification of [3, Lemma 7.3.4] Lemma 4.5. Let K be a compact manifold. Let u ∈ Ω(K) solve the equation Proof of Lemma 4.5. We combine the energy estimate for the equation with the Sobolev embedding for a 7-dimensional manifold: We relegate the rest of the proof of the Theorem 4.4 to the appendix. We combine the two decay estimates above with the possibility to apply Γ α which are symmetries of the equation in the following statement.
with initial data concentrated in the ball of radius 1 for every y ∈ K and suppF (·, ·, y) ⊆ {(t, x) : |t − |x|| ≤ 1} for every y ∈ K then the following estimate holds:

Energy estimates.
We combine the energy estimates for the solution of R 3+1 ×K u = F with the fact that the operators Γ are symmetries of the equation and use the notation introduced above.
Proposition 4.7. Let u be the solution of R 3+1 ×K u = F then for any M ≥ 0 we have for any M ≥ 0.

Analysis of Nonlinearity
In this section, we will treat the bilinear form (u 1 , u 2 ) → * (du 1 ∧du 2 ). Recall from Subsection 2.3 the * operator exchanges the components of the forms, multiplying those containing the time x 0 coordinate by −1. The * operator loses the simple form when the metric on K is no longer the identity, but because of tensoriality, it will be multiplied by a function depending only on x 4 , ..x 10 , which due to compactness will be bounded above and below. Therefore, when we take L 2 (R 3 × K)norm at a certain time, we will consider * v to be L 2 equivalent to v. Furthermore, we will be interested in the action of Γ operators on * (du ∧ du). Clearly, the operators which act on R 3+1 componentwise will commute with * . The equation (2.6) shows that the Laplacian ∆ K commutes with * simply because ∆ K = 10 i=4 ∂ 2 ∂x 2 i at that point and the relation is tensorial. Thus any function of ∆ K will commute with * and we have (−∆ K ) 1/2 * v = * (−∆ K ) 1/2 v. From this discussion we conclude that for any multi-index α, time t, with constants which depend only on the manifold K.

5.1.
The splitting of the nonlinearity. Recall that the operator d splits into d = d + d ⊥ . Also any form u on R 3+1 × K can be written as u = P 0 u + P >0 u. Therefore, we can write where we used that d ⊥ P 0 = 0, which is the content of Claim 3.2. With this we proved the following splitting of the nonlinearity: Observe that all the operators in Γ besides (−∆ K ) 1/2 are vector fields and thus obey Leibniz's rule. So assume first that in the composite operator Γ α there are no (−∆ K ) 1/2 operators. We treat the Hodge dual * as a constant coefficient operator, which only permutes between different components. Next, we employ the Jacobi identity to write I as where C α ′ α ′′ are constants. Observe that Γ i commutes with ∂ ∂x i , for i ≥ 4 and for i, j ≤ 4 we have Thus we commute ∂ ∂x i with Γ's to get for some constants C β ′ β ′′ ij . In the expression above only one of the |β ′ |, |β ′′ | can be larger then M/2. We split the sum accordingly We then obtain the required estimate by applying the L 2 norm to |I| and using the appropriate Hölder inequalities.
In case Γ α contains k (−∆ K ) 1/2 operators, we note that (−∆ K ) 1/2 commutes with all the other Γ operators as the rest of Γ operate on R 3+1 only. By elliptic regularity Now H k (K) norm obeys a Jacobi's "inequality", which is a primitive form of the Kato-Ponce estimates, see [6],[13, Chapter II, Prop. 1.1] and thus the rest of the proof proceeds in a similar fashion. 5.3. Null form. Continuing with the notation of Claim 5.1, we need the observation that B(P 0 u 1 , P 0 u 2 ) = * (d P 0 u 1 ∧d P 0 u 2 ) is a null form and the estimates that follow from it.
We will give two proofs of the proposition.
Fourier. It is enough to compute B(ω 1 , ω 2 ) for ω i = A i e ik i x for two parallel null-vectors, with A i being constant. If B(ω 1 , ω 2 ) vanishes in such a case then B is a null-form. But When two vectors in the wedge product are parallel -the wedge product vanishes. Thus B(ω 1 , ω 2 ) = 0.

Coefficients. Let
Then Observe that we needed to assume that only dx p and dx s are co-vectors on R 3+1 ; all the other indices could have belonged to either K or R 3+1 . In order to see that the sign in front of the second term is negative, we count the transpositions needed to transform one needs four to bring dx p to the front and then another three to bring dx s behind dx i dx j dx k therefore there are 7 transpositions in total and the sign is minus, As the proof of Proposition 5.3 shows the form B(ω 1 , ω 2 ) = * (d ω 1 ∧ d ω 2 ) is the sum of the forms Q ij applied to different components. We have the following estimate Remark. The proof is taken from [8, Lemma 1.1]. We reproduce it here to stress that we have the estimate involving only vector fields Γ and not the full set of Klainerman vector fields, which includes the radial scaling field x 0 For i = 0, we have We wish to prove a variant of the basic estimate specialized to the null form.
Proof. The bilinear form B is a linear combination of the forms Q ij . The forms Q ij are preserved under applications of the Γ operators (with the exception of (−∆ K ) 1/2 ) because of the following formula which appears in [8,Lemma 1.2] and which can be obtained by direct calculation where m αβ are coefficients of the Minkowski metric. Thus we apply Lemma 5.4 and after grouping the multi-indices with order less then M 2 and applying the Hölder estimates we get the result like in Proposition 5.2 . For (−∆ K ) 1/2 we apply Kato-Ponce estimates.
6. Proof of Theorem 1.1 To prove the theorem we seek to establish the following apriori bounds for the solutions of (1.1): where δ is smaller then 1 12 . Any of the global in time estimates above implies uniqueness, existence and well-posedness for the semilinear wave equations by employing the local theory which is explained in [11,Chapter 2] or [3, section 6.2]. We will prove the estimates by bootstrapping. Namely, we will prove that (6.1) imply i.e. the right-hand side can be made ǫ 2 instead of ǫ. Thus our goal is to establish the following statement. Proposition 6.1. Let N be an integer that satisfies Then there exists ǫ > 0 small enough such that, for every T > 0, if a solution u to the Cauchy problem (1.1) with initial data that satisfies such that u 0 (·, y), u 1 (·, y) are localized in a ball of radius 1 for every y ∈ K and if the inequalities (6.1a),(6.1b), (6.1c) hold for every 0 < t ≤ T then the inequalities (6.2a),(6.2b), (6.2c) hold for every 0 < t ≤ T , where δ is a positive exponent which depends on ǫ and is smaller then 1 12 Remark. There are two considerations that affect the smallness of ǫ. One is that we will see that we can replace ǫ in the right-hand side of (6.1) by ǫ 4 + kǫ 2 , where k is will an apriori computable constant that depends only on N and the geometry of the manifold K. Thus we will need to decrease ǫ to achieve the inequality Another consideration is that the exponent δ depends linearly on ǫ, δ = Cǫ with C depending on N and the geometry of K. We have to decrease ǫ so that δ = Cǫ ≤ 1 12 . We prove the implication (6.2a) in the next lemma, while the implications (6.2b),(6.2c) are proved in Lemma 6.5. Proof. We use the energy estimate for the equation Γ (N ) u = Γ (N ) ( * du∧ du).
We now use the Proposition 5.2 to estimate the nonlinearity. We have Since N 2 ≤ N − 18 we employ the assumption (6.1c) in equation (6.5) to conclude After applying Gronwall inequality we conclude which is the required inequality.
We require the following interpolated intermediate result.

Acknowledgements
The author would like to thank his doctoral adviser Daniel Tataru for suggesting the problem and several key ideas in the proof, Jason Metcalfe for discussing the technical aspects of [9] and [10], Robin Graham for exposing the author to questions of supergravity and explaining the results of conformal geometry, Jon Aytac for pointing to the work of Choquet-Bruhat [1] and Tobias Schottdorf for helping improve the exposition.
Appendix A. Proof of the theorem 4.4 We will denote τ = t + 2. We will again employ operators Γ in τ variable, which are different from the operators in t variable but can be expressed as a linear combination with coefficients independent of u. See proof of Proposition 4.3 for details regarding this substitution. We will use the following statement Introduce the hyperbolic polar coordinates (τ, x) = ρω, ρ = (τ 2 − |x| 2 ) 1 2 , ω ∈ S 2 . Then the equation becomes where ∆ H is the Laplacian on the hyperbolic space. We have Thus v = ρ 3 2 u obeys We decompose the right-hand-side dyadically in time. Let χ be a smooth function, supported on [ 1 2 , 2] s.t.