Bochner-Kodaira Formulas and the Type IIA Flow
aa r X i v : . [ m a t h . DG ] D ec Bochner-Kodaira Formulas and the Type IIA Flow
Abstract
A new derivation of the flow of metrics in the Type IIA flow is given. It is adapted to theformulation of the flow as a variant of a Laplacian flow, and it uses the projected Levi-Civitaconnection of the metrics themselves instead of their conformal rescalings.
The search for supersymmetric compactifications of string theories has revealed itself tohave deep connections with special geometry. The resulting non-linear partial differentialequations also turned out to be quite rich and interesting in their own right (see e.g.[4, 8, 9, 10, 17, 20]). One invariable feature of particular interest in these equations isthe presence of a cohomological constraint. When no ∂ ¯ ∂ -lemma is available, the mostnatural implementation of these cohomological constraints is by a geometric flow, andthis has resulted in considerable interest in the investigation of such flows in recent years[1, 2, 3, 5, 7, 14, 15, 16, 18, 19].The present paper is mainly concerned with the Type IIA flow, which is a flow in sym-plectic geometry introduced in [6] and motivated by the Type IIA string. More specifically,let ( M, ω ) be a compact 6-dimensional symplectic manifold and ρ A be the Poincar´e dualto a finite combination of Lagrangians. Then the Type IIA flow is the flow of 3-forms ϕ given by ∂ t ϕ = d Λ d ( | ϕ | ⋆ ϕ ) − ρ A (1.1)with an initial data ϕ which is a closed, primitive, and positive 3-form on M . HereΛ is the Hodge contraction operator defined by ω , and ⋆ and | ϕ | are the Hodge staroperator and the norm of ϕ with respect to the metric g ϕ which is compatible with ω and the almost-complex structure J ϕ constructed by Hitchin [13] (see § ϕ , so that itsstationary points are automatically solutions of the system first investigated by Tseng andYau [22]. This system is itself a basic case of the more general equations for supersymmetriccompactifications of the Type IIA string proposed in [12, 21].In [6] it was shown that the Type IIA flow admits at least short-time existence, andcan be continued as long as | ϕ | and the Riemannian curvature of g ϕ remain bounded. Theproof of this last assertion relied heavily on determining the flow of g ϕ . This was one of Work supported in part by the National Science Foundation Grants DMS-1855947 and DMS-1809582. he main results of [6], and it was established using the original formulation (1.1) of theType IIA flow, and the projected Levi-Civita connection ˜ D of a metric ˜ g ϕ conformal to g ϕ (see (2.2 below). A key point was that, with respect to ˜ D , the manifold M has SU(3)holonomy, and the form | Ω | − g ϕ Ω, with Ω = ϕ + i ⋆ ϕ , is covariant constant.The main goal of the present paper is to provide a different derivation of the flow ofthe metrics g ϕ in the Type IIA flow. The new derivation differs from the one in [6] in twoimportant aspects. The first aspect is that it relies on Bochner-Kodaira formulas and adifferent formulation of the Type IIA flow, which is closer in spirit to Bryant’s G flow.From this point of view, it is more easily adaptable to other Laplacian flows. The secondaspect is that it relies instead on the projected Levi-Civita connection D of g ϕ , which isa very natural connection since it coincides with all the unitary Hermitian connectionswith respect to g ϕ on the Gauduchon line. An important additional benefit of this secondderivation is that it provides a check on the formulas obtained in [6], which is non-trivialbecause the calculations in both approaches are particularly long and involved.For simplicity, we focus on the source-free case ρ A = 0. Then we have Theorem 1
Let ( M, ω ) be a -dimensional symplectic manifold, and let t → ϕ ( t ) by theType IIA flow of -forms defined in (1.1) with ρ A = 0 . If g ij = ( g ϕ ) ij is the correspondingflow of metrics, then we have ∂ t g ij = −| ϕ | (cid:8) R ij − ∇ i ∇ j log | ϕ | + 4( N − ) ij − α i α j + α Ji α Jj + 4 α p ( N j pi + N ipj ) (cid:9) (1.2) where ∇ is the Levi-Civita connection of g , R ij is the Ricci curvature, N is the Nijenhuistensor with respect to the almost-complex structure J ϕ , ( N − ) ij = N λpi N pλj , and α is the1-form defined by α = − d log | ϕ | . We begin by providing a brief summary of the setting for the Type IIA flow, which is TypeIIA geometry as introduced in [6].
Let M be an oriented 6-manifold. In [13], Hitchin has shown how to associate to anynon-degenerate 3-form ϕ an almost-complex structure J ϕ . Type IIA geometry arises if,in addition, M is equipped with a fixed symplectic form ω and ϕ is a closed form whichis primitive and positive with respect to ω . The primitive condition means that Λ ϕ = 0,where Λ : A k ( M ) → A k − ( M ) is the standard Hodge contraction operator with respect to ω . It is shown in [6] that ω is then preserved by J ϕ , and the positivity condition meansthat the resulting Hermitian form g ϕ ( X, Y ) = ω ( X, J ϕ Y ) is positive definite and definesa metric. Thus ( J ϕ , g ϕ , ω ) is an almost-K¨ahler manifold. However, the condition in Type2IA geometry that this almost-K¨ahler structure arise from a closed 3-form results in manysubtle properties which are essential for the Type IIA flow.Explicitly, the metric g ϕ is given by( g ϕ ) ij = −| ϕ | − ϕ iab ϕ jkp ω ak ω bp (2.1)where | ϕ | is the norm of the 3-form ϕ with respect to J ϕ , and ω ak is the inverse of thesymplectic form ω , ω ak ω kp = δ ap . The volume form of g ϕ is the same as ω / g ϕ conformally equivalent to g ϕ also plays an important role in Type IIAgeometry, (˜ g ϕ ) ij = | ϕ | ( g ϕ ) ij = − ϕ iab ϕ jkp ω ak ω bp . (2.2)In fact, one of the defining features of Type IIA geometry is that the manifold ( M, J ϕ )have SU(3) holonomy with respect to the projected Levi-Civita connection ˜ D of ˜ g ϕ . Moreprecisely, set ˆ ϕ = ⋆ϕ = J ϕ (2.3)and let Ω be the (3 , ϕ + i ˆ ϕ. (2.4)Then | Ω | − g ϕ Ω is covariantly constant with respect to ˜ D . This was a major reason why thecalculations in [6] were mostly carried out with the connection ˜ D .In the present paper, we shall use instead the unitary connections with respect to g ϕ .Since ω is closed, the Gauduchon line of Hermitian unitary connections with respect to J ϕ collapses to a single connection, which can be viewed as either the Chern connection orthe projected Levi-Civita connection D of g ϕ . Henceforth we drop the subindex ϕ whenthere is no possibility of confusion, and denote g ϕ , ˜ g ϕ , J ϕ simply by g , ˜ g and J . Then theLevi-Civita connection ∇ and the projected Levi-Civita connection D of g are related by D i X m = ∇ i X m − N ipm X p (2.5)where N ipm is the Nijenhuis tensor of J , N kij = 14 ( J ri ∇ r J kj + J kr ∇ j J ri − ( i ↔ j )) . (2.6)In [6], we showed D , Ω = 0 and D , Ω = − α ⊗ Ω (Equation (6.50) in [6]), or equivalently, D m ϕ = 12 ( − α m ϕ − α Jm ˆ ϕ ) , D m ˆ ϕ = 12 ( − α m ˆ ϕ + α Jm ϕ ) . (2.7)Here the 1-form α is defined by α = − d log | ϕ | (2.8)and we used the same notation introduced in [6] for any vector field V and any 1-form W , V Jk = ( J V ) k = J kp V p , W Jk = ( J W ) k = J pk W p . (2.9)In particular, ω ij = g Ji,j , g ij = ω i,Jj , and ω ij = g Ji,j , g ij = ω i,Jj .3 .2 Identities from Type IIA geometry We list here some identities required later. Except for (2.21), they were proved in [6]. ϕ First, the action of J on ϕ is given by ϕ ijk = − ϕ Ji,Jj,k = − ϕ Ji,j,Jk = − ϕ i,Jj,Jk ϕ Ji,j,k = ϕ j,Jj,k = ϕ i,j,Jk . (2.10)Next, bilinears in ϕ with two contractions with ω ij give the metric g ij . But bilinears witha single contraction with either ω ij or g ij simplify as well, ω ij ϕ iab ϕ jcd = | ϕ | ω ac g bd + ω bd g ac − ω bc g ad − ω ad g bc ) g ij ϕ iab ϕ jcd = | ϕ | g ac g bd + ω ca ω bd − ω ad ω cb − g bc g ad ) . (2.11)As a consequence, we also have bilinear identities involving ϕ and ˆ ϕ , for exampleˆ ϕ λkp ϕ iab ω ka ω pb = | ϕ | ω λi . (2.12)This reduces to the previous identity by noting that ˆ ϕ λkp = − ϕ Jλ,kp , so thatˆ ϕ λkp ϕ iab ω ka ω pb = − ϕ Jλ,kp ϕ iab ω ka ω pb = | ϕ | g Jλ,i = | ϕ | ω λi . (2.13) In general, the Nijenhuis tensor satisfies the following identities of a type (0 , N kJi,j = − N Jkij = N ki,Jj , N Ji,j,k = N i,Jj,k = N i,j,Jk . (2.14)Since dω = 0, we also have the Bianchi identity N ijk + N jki + N kij = 0 . (2.15)From this it follows that there are two symmetric tensors quadratic in N , denoted by( N ) ij = N pqi N pqj , ( N − ) ij = N pqi N qpj . (2.16)The relation between the Levi-Civita connection ∇ and the projected Levi-Civita connec-tion D also implies, since D J = 0, ∇ i J kj = − N ij Jk . (2.17)4n Type IIA geometry, we have N − = 2 N − | N | g, | N | = ( N ) λλ = 2( N − ) λλ (2.18)where | N | = N mkp N mkp . We also have the following crucial identity between the Nijenhuistensor and ϕ , N pij ϕ pkl = − N pkl ϕ pij , (2.19)which was proved in [6], Corollary 1. We shall express the desired identities for the curvature tensor of the Levi-Civita connectionin the following convention. The connection ∇ is written as ∇ m V k = ∂ m V k + Γ kmℓ V ℓ , andthe curvature tensor R ij kℓ is defined by[ ∇ i , ∇ j ] V k = R ij kℓ V ℓ . (2.20)The Ricci curvature is then given by R ij = R ipj p .The first curvature identity that we require gives the action of J on Rm , R j,i,Jk,Jℓ = R jikℓ + B ijkℓ B ijkℓ = − D i N jkℓ + 2 D j N ikℓ − N αij N αkℓ , (2.21)This identity can also be expressed as R jipJℓ = R jiJpℓ + 2 D j N iJpℓ − D i N j Jpℓ − N µji N µℓJp . (2.22)To see this, we consider the action of J on a vector field V , R jkpq ( J V ) q = ∇ j ∇ k ( J V ) p − ∇ k ∇ j ( J V ) p = J [ ∇ j , ∇ k ] V p + ( ∇ j ∇ k J − ∇ k ∇ j J ) pλ V λ . (2.23)It follows that R jkpq J qλ = J pq R kjqλ + ∇ j ∇ k J pλ − ∇ k ∇ j J pλ = J pq R kjqλ − ∇ j ( J pµ N kλµ ) + 2 ∇ k ( J pµ N jλµ ) (2.24)or, in more succinct notation, R jkpJλ = R jkJpλ − ∇ j ( N kλJp ) + 2 ∇ k ( N jλJp ) . (2.25)We now convert ∇ derivatives into D derivatives. First lowering indices gives R jik,Jℓ = − R j,i,Jk,ℓ + 2 ∇ j ( N i,ℓ,Jk ) − ∇ i ( N j,ℓ,Jk ) . (2.26)5herefore R j,i,Jk,Jℓ = R jikℓ + 2 J pk ∇ j ( N i,ℓ,Jp ) − J pk ∇ i ( N j,ℓ,Jp ) . (2.27)We write 2 J pk ∇ j ( N i,ℓ,Jp ) = 2 J pk D j ( N i,ℓ,Jp ) − J pk N jiµ ( N µ,ℓ,Jp ) − J pk N jℓµ ( N i,µ,Jp ) − J pk N jpµ ( J nµ N iℓn ) (2.28)Since D J = 0,2 J pk ∇ j ( N i,ℓ,Jp ) = − D j N iℓk + 2 N jiµ N µℓk + 2 N jℓµ N iµk − N j,JkJn N iℓn = 2 D j N ikℓ + 2 N jiµ N µℓk − N jℓµ N ikµ + N jkµ N iℓµ ) (2.29)This last term is symmetric in ( i, j ). Therefore2 J pk ∇ j ( N i,ℓ,Jp ) − ( i ↔ j ) = 2 D j N ikℓ − D i N jkℓ + 2 N jiµ N µℓk − N ijµ N µℓk (2.30)By the Bianchi identity2 J pk ∇ j ( N i,ℓ,Jp ) − ( i ↔ j ) = 2 D j N ikℓ − D i N jkℓ + 2( − N µji − N iµj ) N µℓk − N ij µ N µℓk (2.31)from which the desired identity (2.21) follows.Finally, we shall need the following curvature identity specific to Type IIA geometry(see (6.53) in [6]), R ij = − D s ( N isj + N j si ) − N − ) ij + 12 ∇ i ∇ j log | ϕ | + 12 J pi J qj ∇ p ∇ q log | ϕ | . (2.32) We shall establish Theorem 1 using the formulation of the Type IIA flow as a Laplaciantype flow [6] ∂ t ϕ = − dd † ( | ϕ | ϕ ) + 2 d ( | ϕ | N † · ϕ ) (3.1)where N † : Λ ( M ) → Λ ( M ) is the operator defined by( N † · ϕ ) kj = N µjλ ϕ µkλ − N µkλ ϕ µjλ . (3.2)For our present purposes, it is convenient to rewrite the above expression as ∂ t ϕ = −| ϕ | dd † ϕ − d | ϕ | ∧ d † ϕ + d ( ι ∇| ϕ | ϕ ) + 2 d ( | ϕ | N † · ϕ ) . (3.3)We would like to determine ∂ t g ij explicitly. For this, it is convenient to determine first ∂ t ˜ g ij , since ˜ g ij is a quadratic expression in ϕ , and we have ∂ t ˜ g ij = − (cid:8) ( ∂ t ϕ iab ) ϕ jkp ω ka ω pb + ( i ↔ j ) (cid:9) . (3.4)We shall determine in turn the contribution of each expression in (3.3) to ∂ t ˜ g ij .6 .1 The Bochner-Kodaira formula for the Levi-Civita connection We begin with the contribution of | ϕ | dd † ϕ using a Bochner-Kodaira formula. In general,if M is any compact Riemannian manifold and we express any p -form in components as ϕ = 1 p ! X i , ··· ,i p ϕ i ··· i p dx i ∧ · · · ∧ dx i p = 1 p ! X I ϕ I dx I (3.5)with antisymmetric coefficients ϕ i ··· i p , then the adjoint d † of the de Rham exterior differ-ential with respect to a given metric g ij is given by( d † ϕ ) I ′ = − g ℓm ∇ m ϕ ℓI ′ , (3.6)where ∇ denotes the covariant derivative with respect to the Levi-Civita connection andwe have split the index I into I = ( ℓ, I ′ ), I ′ = ( i , · · · , i p ). It follows that( dd † ϕ ) I = − ( ∇ i ( g ℓm ∇ m ϕ ℓI ′ ) − p X q =2 ( i ↔ i q )) . (3.7)Next, we have ( dϕ ) ℓI = ∇ ℓ ϕ I − p X q =1 ( ℓ ↔ i q ) (3.8)and hence ( d † dϕ ) I = − g ℓm ∇ m ( ∇ ℓ ϕ I − p X q =1 ( ℓ ↔ i q )) . (3.9)Altogether, we obtain the version of the Bochner-Kodaira formula that we need,(( dd † + d † d ) ϕ ) I = − g ℓm ∇ m ∇ ℓ ϕ I + g ℓm p X q =1 [ ∇ m , ∇ i q ] ϕ ··· i q − ℓi q +1 ··· (3.10)In the case of interest, namely 3-forms ϕ with dϕ = 0, we obtain dd † ϕ jkp = − g ℓm ∇ m ∇ ℓ ϕ jkp + g ℓm (cid:8) [ ∇ m , ∇ j ] ϕ kpℓ + [ ∇ m , ∇ k ] ϕ pjℓ + [ ∇ m , ∇ p ] ϕ jkℓ (cid:9) . (3.11) g ℓm ∇ m ∇ ℓ ϕ jkp Recall that the covariant derivatives of ϕ with respect to the projected Levi-Civita con-nection D are given by (2.7). It follows that g ℓm D ℓ D m ϕ = −
12 ( ∇ µ α µ ) ϕ (3.12)7nd [ D m , D ℓ ] ϕ = 12 ( − D m α ℓ + D ℓ α m ) ϕ + 12 ( − D m α Jℓ + D ℓ α Jm ) ˆ ϕ = − N mℓj α j ϕ + 12 N ℓmj α j ϕ + 12 ( − D m α Jℓ + D ℓ α Jm ) ˆ ϕ. (3.13)Now the difference between ∇ and D on vectors is given by (2.5). On 3-forms, it is givenby ∇ ℓ ϕ jkp = D ℓ ϕ jkp − ϕ λkp N ℓjλ − ϕ jλp N ℓkλ − ϕ jkλ N ℓpλ = D ℓ ϕ jkp − E ℓ ; jkp , (3.14)where E ℓ ; jkp = ϕ λkp N ℓjλ + ϕ jλp N ℓkλ + ϕ jkλ N ℓpλ . (3.15)Similarly, we write ∇ m D ℓ ϕ jkp = D m D ℓ ϕ jkp − D µ ϕ jkp N mℓµ − D ℓ ϕ µkp N mj µ − D ℓ ϕ jµp N mkµ − D ℓ ϕ jkµ N mpµ := D m D ℓ ϕ jkp − E m ; ℓjkp , (3.16)and hence g mℓ ∇ m ∇ ℓ ϕ jkp = g mℓ D m D ℓ ϕ jkp − g mℓ E m ; ℓjkp − g mℓ ∇ m E ℓ ; jkp . (3.17)We begin by computing the contributions of g mℓ ∇ m E ℓ ; jkp , g mℓ ∇ m E ℓ ; jkp = ( g ℓm ∇ m ϕ λkp ) N ℓjλ + ( g ℓm ∇ m ϕ jλp ) N ℓkλ + ( g ℓm ∇ m ϕ jkλ ) N ℓpλ + ϕ λkp g ℓm ∇ m N ℓjλ + ϕ jλp g ℓm ∇ m N ℓkλ + ϕ jkλ g ℓm ∇ m N ℓpλ = ( g ℓm D m ϕ λkp ) N ℓjλ + ( g ℓm D m ϕ jλp ) N ℓkλ + ( g ℓm D m ϕ jkλ ) N ℓpλ − g ℓm ( E m ; λkp N ℓjλ + E m ; jλp N ℓkλ + E m ; jkλ N ℓpλ )+ ϕ λkp g ℓm ∇ m N ℓjλ + ϕ jλp g ℓm ∇ m N ℓkλ + ϕ jkλ g ℓm ∇ m N ℓpλ . (3.18) E ℓ ; jkp Consider the contributions of the second row on the right hand side of the last equation.Paired with ϕ iab ω ka ω pb , it gives g ℓm E m ; λkp N ℓj λ ϕ iab ω ka ω pb = g ℓm ( ϕ µkp N mλµ + ϕ λµp N mkµ + ϕ λkµ N mpµ ) N ℓjλ ϕ iab ω ka ω pb = (I + II + III) · ϕ iab ω ka ω pb (3.19)with (I) · ϕ iab ω ka ω pb = g ℓm ϕ µkp N mλµ N ℓj λ ϕ iab ω ka ω pb (II) · ϕ iab ω ka ω pb = g ℓm ϕ λµp N mkµ N ℓj λ ϕ iab ω ka ω pb (III) · ϕ iab ω ka ω pb = g ℓm ϕ λkµ N mpµ N ℓj λ ϕ iab ω ka ω pb . (3.20)8ext, we have(I) · ϕ iab ω ka ω pb = −| ϕ | g ℓm g µi N mλµ N ℓj λ = −| ϕ | N ℓλi N ℓjλ = | ϕ | ( N ) ij (3.21)and, using (2.11), we compute(II) · ϕ iab ω ka ω pb = | ϕ | g ℓm ( ω λi g µa + ω µa g λi − ω λa g µi − ω µi g λa ) N mkµ N ℓj λ ω ka = | ϕ | g ℓm ( ω λi J kµ − δ kµ g λi + δ kλ g µi − ω µi J kλ ) N mkµ N ℓj λ = | ϕ | ω λi N mkJk N mj λ − N mkk N mjλ + N mki N mj k − ω µi N mkµ N mj Jk ) . Now N mkk = 0, and by the Nijenhuis tensor identities, N mkJk = N m,Jkk = N Jm,kk = 0 . (3.22)Furthermore, we have by definition N mki N mjk = − ( N ) ij , while − ω µi N mkµ N mj Jk = g µν J νi N mkµ N mjJk = N mk Ji N mj Jk = N m,Jk,i N mj Jk = − N mki N mj k = ( N ) ij (3.23)and hence (II) · ϕ iab ω ka ω pb = 0 . (3.24)Since (III) can be obtained from (II) by the simultaneous interchange a ↔ b and k ↔ p ,we also have (III) · ϕ iab ω ka ω pb = 0 . (3.25)We consider next the expression g ℓm E m ; jλp N ℓkλ ϕ iab ω ka ω pb = g ℓm ( ϕ µλp N mj µ + ϕ jµp N mλµ + ϕ jλµ N mpµ ) N ℓkλ ϕ iab ω ka ω pb = (IV + V + VI) ϕ iab ω ka ω pb . (3.26)The contributions of the term (IV) worked out to be 0,(IV) · ϕ iab ω ka ω pb = | ϕ | ω µi g λa + ω λa g µi − ω µa g λi − ω λi g µa ) N mj µ N mkλ ω ka = | ϕ | ω µi J kλ − δ kλ g µi + δ kµ g λi − ω λi J kµ ) N mj µ N mkλ . (3.27)The first two terms on the right hand side vanish individually, since ω µi J kλ N mj µ N mkλ = ω µi N mj µ N mJλλ = 0 δ kλ g µi N mj µ N mkλ = N mj µ N mkk = 0 . (3.28)9f the remaining two terms, we have obviously δ kµ g λi N mj µ N mkλ = N mj k N mki = − N mkj N mki = − ( N ) ij , (3.29)while − ω λi N mj Jk N mkλ = g λν J νi N mj Jk N mkλ = J νi N mj Jk N mkν = N mj Jk N mk,Ji = N mj Jk N mJk,i = − N mj k N mki = ( N ) ij (3.30)so they cancel each other out and we obtain, as claimed,(IV) · ϕ iab ω ka ω pb = 0 . (3.31)The next group of terms is given by(V) · ϕ iab ω ka ω pb = ϕ jµp N mλµ N mkλ ϕ iab ω ka ω pb = − ϕ jµp g µν ( N ) νk ϕ iab ω ka ω pb = − | ϕ | ω ji g µa + ω µa g ji − ω ja g µi − ω µi g ja ) ω ka ( N ) νk g µν . (3.32)The first term on the right produces 0, since it can be computed as ω ji ω kν ( N ) νk . Thisterm vanishes due to the anti-symmetrization of k and ν . We are left with(V) · ϕ iab ω ka ω pb = − | ϕ | − δ kµ g ji g µν + δ kj g µi g µν − ω µi J kj g µν )( N ) νk = − | ϕ | −| N | g ij + ( N ) ij + ( N ) Ji,Jj ) (3.33)Since we have( N ) Ji,Jj = N mkJi N mk,Jj = − N m,Jki N m,Jk,j = N mki N mkj = ( N ) ij (3.34)we are left with (V) · ϕ iab ω ka ω pb = | ϕ | | N | g ij − | ϕ | N ) ij . (3.35)Finally, we observe that(VI) · ϕ iab ω ka ω pb = g ℓm ϕ jλµ N mpµ N ℓkλ ϕ iab ω ka ω pb = 0 . (3.36)We can readily see this in a complex frame. Since ϕ ∈ Λ , ⊕ Λ , , the only components of ϕ jλµ which are not 0 must have both barred or both unbarred indices. But the contractionwith g ℓm implies that the indices ℓ and m must be mixed. But then for N mpµ N ℓkλ notto be 0, the indices λ and µ must be mixed too, contradicting the requirement that theymust be both barred or both unbarred. This establishes our claim.10e still have one more contribution from the second row of (3.18), given by g ℓm E m ; jkλ N ℓpλ ϕ iab ω ka ω pb (3.37)but which can be recognized as coinciding with the term that we just computed g ℓm E m ; jλp N ℓkλ ϕ iab ω ka ω pb = | ϕ | | N | g ij − | ϕ | N ) ij (3.38)upon the renaming of indices a ↔ b , p ↔ k .It is convenient to summarize the formula which we have obtained as a lemma: Lemma 1
We have g ℓm ( E m ; λkp N ℓjλ + E m ; jλp N ℓkλ + E m ; jkλ N ℓpλ ) ϕ iab ω ka ω pb = | ϕ | | N | g ij . (3.39) E m ; ℓjkp The term E m ; ℓjkp involves D µ ϕ jkp , D ℓ ϕ µkp , D ℓ ϕ jµp , and D ℓ ϕ jkµ . We use (2.7) to evaluatethe contribution of each term in turn, D µ ϕ jkp N mℓµ ϕ iab ω ka ω pb = −
12 ( α µ ϕ jkp + α Jµ ˆ ϕ jkp ) N mℓµ ϕ iab ω ka ω pb = 12 | ϕ | ( α µ g ji − α Jµ ω ji ) N mℓµ (3.40)Upon symmetrization in i and j , and contracting with g ℓm , we obtain g ℓm D µ ϕ jkp N mℓµ ϕ iab ω ka ω pb + ( i ↔ j ) = | ϕ | g ij α µ g mℓ N mℓµ = 0 (3.41)where we have used the fact that N is of type (0 ,
2) to write g mℓ N mℓµ = g Jm,Jℓ N mℓµ = g mℓ N Jm,Jℓµ = − g mℓ N mℓµ and therefore g mℓ N mℓµ = 0 . (3.42)Next, we consider the term D ℓ ϕ µkp N mj µ ϕ iab ω ka ω pb = −
12 ( α ℓ ϕ µkp + α Jℓ ˆ ϕ µkp ) N mj µ ϕ iab ω ka ω pb (3.43)= 12 | ϕ | ( α ℓ g µi − α Jℓ ω µi ) N mj µ = 12 | ϕ | ( α ℓ N mji + α Jℓ N mj,Ji ) . The first term on the right symmetrizes to 0. So does the second, using the fact that N isa type (0 , N mj,Ji = N Jm,j,i which is antisymmetric in the last two indices.11e consider now D ℓ ϕ jµp N mkµ ϕ iab ω ka ω pb = −
12 ( α ℓ ϕ jµp + α Jℓ ˆ ϕ jµp ) N mkµ ϕ iab ω ka ω pb . (3.44)We work out separately the contributions of the two terms ϕ jµp and ˆ ϕ jµp on the right handside. First, we have α ℓ ϕ jµp N mkµ ϕ iab ω ka ω pb = | ϕ | α ℓ ( ω ji g µa + ω µa g ji − ω ja g µi − ω µi g ja ) N mkµ ω ka = | ϕ | α ℓ ( ω ji J kµ − δ kµ g ji + δ kj g µi − ω µi J kj ) N mkµ . (3.45)We claim that, upon symmetrization in i and j , the net result is 0. This is obviously trueof the term ω ji , while N mkk = 0 and N mji also symmetrizes to 0. The fourth term can berewritten as ω µi J kj N mkµ = − g µν J νi N m,Jj µ = − J ν i N m,Jj,ν = N m,Jj,Ji (3.46)which symmetrizes to 0. We come to the contribution of the term involving ˆ ϕ ,ˆ ϕ jµp ϕ iab ω ka ω pb = − ϕ Jj,µp ϕ iab ω ka ω pb = − | ϕ | ω Jj,i g µa + ω µa g Jj,i − ω Jj,a g µi − ω µi g Jj,a ) ω ka = − | ϕ | − g ij g µa + ω µa ω ji + g aj g µi − ω µi ω ja ) ω ka = − | ϕ | − g ij J kµ − δ kµ ω ji + J kj g µi + δ kj ω µi ) . (3.47)Dropping the term ω ji since it symmetrizes to 0, we arrive atˆ ϕ jµp ϕ iab ω ka ω pb N mkµ + ( i ↔ j ) = − | ϕ | − g ij N m,Jµµ + J kj N mki + N mj µ ω µi ) + ( i ↔ j )= − | ϕ | N m,Jj,i − N mj,Ji ) + ( i ↔ j )= − | ϕ | N Jm,j,i − N Jm,j,i ) + ( i ↔ j ) = 0 . (3.48)The last term D ℓ ϕ jkµ makes an identical contribution as D ℓ ϕ jµp , upon renaming the sum-mation indices a ↔ b , k ↔ p . Thus its contribution is also 0. In summary, we haveestablished Lemma 2
We have g mℓ E m ; ℓjkp ϕ iab ω ka ω pb + ( i ← j ) = 0 . (3.49)12 .2.3 Completion of the calculations for ∇ ℓ E ℓ ; jkp The terms from ∇ ℓ E ℓ ; jkp in (3.18) whose contributions we have not worked out as yet arethe following ( g ℓm D m ϕ λkp ) N ℓjλ + ( g ℓm D m ϕ jλp ) N ℓkλ + ( g ℓm D m ϕ jkλ ) N ℓpλ + ϕ λkp g ℓm ∇ m N ℓjλ + ϕ jλp g ℓm ∇ m N ℓkλ + ϕ jkλ g ℓm ∇ m N ℓpλ . = ϕ λkp ( − α ℓ N ℓjλ + ∇ ℓ N ℓjλ ) −
12 ˆ ϕ λkp α Jm g ℓm N ℓjλ + ϕ jλp ( − α ℓ N ℓkλ + ∇ ℓ N ℓkλ ) −
12 ˆ ϕ jλp α Jm g ℓm N ℓkλ + ϕ jkλ ( − α ℓ N ℓpλ + ∇ ℓ N ℓpλ ) −
12 ˆ ϕ jkλ α Jm g ℓm N ℓpλ = VII + ˆVII + VIII + ˆVIII + IX + ˆIX . (3.50)Again, we evaluate each contribution in turn. We have(VII) · ϕ iab ω ka ω pb = −| ϕ | g λi ( − α ℓ N ℓjλ + ∇ ℓ N ℓj λ ) = | ϕ | ( 12 α ℓ N ℓji − ∇ ℓ N ℓji ) = 0 (3.51)upon symmetrization in i ↔ j . Next,( ˆVII) · ϕ iab ω ka ω pb = − α Jm ϕ Jλ,k,p N mj λ ϕ iab ω ka ω pb = 12 | ϕ | α Jm ω λi N mj λ = − | ϕ | α Jm N mj,Ji = − | ϕ | α Jm g mℓ N ℓ,j,Ji = − | ϕ | α Jm g mℓ N Jℓ,j,i (3.52)which produces 0 upon symmetrization in j and i . Next,(VIII) · ϕ iab ω ka ω pb = ϕ jλp ( − α ℓ N ℓkλ + ∇ ℓ N ℓkλ ) ϕ iab ω ka ω pb = | ϕ | ω ji g λa + ω λa g ji − ω ja g λi − ω λi g ja ) ω ka ( − α ℓ N ℓkλ + ∇ ℓ N ℓkλ )= | ϕ | ω ji J kλ − δ kλ g ji + δ kj g λi − ω λi J kj )( − α ℓ N ℓkλ + ∇ ℓ N ℓkλ )= | ϕ | (cid:8) g ji ( 12 α ℓ N ℓλλ − ∇ ℓ N ℓλλ ) − α ℓ N ℓji + ∇ ℓ N ℓji − α ℓ N ℓ,Jj,Ji − ω λi J kj ∇ ℓ N ℓkλ (cid:9) (3.53)Note that, the first two terms are zero because N ℓλλ = 0; the next three terms also adds upto 0 upon symmetrization in i and j . Indeed, the last term is also zero upon symmetrizationin i and j because it is anti-symmetric about i and j as ω λi J kj ∇ ℓ N ℓkλ = ω λi g kp J pj ∇ ℓ N ℓkλ = ω λi ω pj ∇ ℓ ( N ℓkλ g kp ) = ω λi ω pj ∇ ℓ N ℓpλ = − ω λj ω pi ∇ ℓ N ℓpλ p ↔ λ and using the antisymmetry of N .The next term to be considered is( ˆVIII) · ϕ iab ω ka ω pb = − α Jm ˆ ϕ jλp g ℓm N ℓkλ ϕ iab ω ka ω pb = 12 α Jm ϕ Jj,λ,p g ℓm N ℓkλ ϕ iab ω ka ω pb = | ϕ | ω Jj,i g λa + ω λa g Jj,i − ω Jj,a g λi − ω λi g Jj,a ) ω ka α Jm N mkλ = | ϕ | − g ij J kλ − δ kλ ω ji + J kj g λi + ω λi δ kj ) α Jm N mkλ = | ϕ | − g ij α Jm N mJλλ − α Jm N mλλ + α Jm N mJj,i − α Jm N mj,Ji )Using the fact that N is a tensor of type (0 , Lemma 3
We have ( g ℓm ∇ m ∇ ℓ ϕ jkp ) ϕ iab ω ka ω pb + ( i ↔ j ) = | ϕ | (cid:8) ∇ µ α µ + | N | (cid:9) g ij . Turning next to the curvature contributions, we write g ℓm [ ∇ m , ∇ j ] ϕ kpℓ = − g ℓm ( R mj λk ϕ λpℓ + R mj λp ϕ kλℓ + R mj λℓ ϕ kpλ )= − R ℓjλk ϕ λpℓ − R ℓj λp ϕ kλℓ + R jλ ϕ kpλ = − R ℓjλk ϕ λpℓ + R ℓjλp ϕ λkℓ + R j λ ϕ kpλ (3.54)We consider for the moment only the contribution of the last term. R j λ ϕ kpλ ϕ iab ω ka ω pb = −| ϕ | R j λ g λi = −| ϕ | R ji . (3.55)The next curvature contribution is similar g ℓm [ ∇ m , ∇ p ] ϕ jkℓ = − g ℓm ( R mpλj ϕ λkℓ + R mpλk ϕ jλℓ + R mpλℓ ϕ jkλ )= − R ℓpλj ϕ λkℓ + R ℓpλk ϕ λjℓ + R pλ ϕ jkλ (3.56)and the corresponding last term gives R pλ ϕ jkλ ϕ iab ω ka ω pb = | ϕ | R pλ ( ω ji g λb + ω λb g ji − ω jb g λi − ω λi g jb ) ω pb = | ϕ | R pλ ( ω ji J pλ − δ pλ g ji + δ pj g λi − ω λi J pj )= | ϕ | − R g ji + R ji + R Jj,Ji ) (3.57)14here we have dropped the term proportional to ω ji since it symmetrizes to 0. Theremaining terms gives an identical contribution. Indeed, g ℓm [ ∇ m , ∇ k ] ϕ pjℓ = − g ℓm ( R mkλp ϕ λjℓ + R mkλj ϕ pλℓ + R mkλℓ ϕ pjλ )= − R ℓkλp ϕ λjℓ + R ℓkλj ϕ λpℓ + R kλ ϕ pjλ . (3.58)Considering for the moment only the contribution of the last term, we can write R kλ ϕ pjλ ϕ iab ω ka ω pb = R kλ ( ω pb ϕ pjλ ϕ bia ) ω ka = | ϕ | R kλ ( ω ji g λa + ω λa g ji − ω ja g λi − ω λi g ja ) ω ka = | ϕ | R kλ ( ω ji J kλ − δ kλ g ji + δ kj g λi − ω λi J kj )= | ϕ | − R g ji + R ji + R Jj,Ji ) (3.59)where we have dropped the antisymmetric term ω ji just as before. Assembling all theterms, we have proved the following lemma Lemma 4
We have the following formula g ℓm ([ ∇ m , ∇ j ] ϕ kpℓ + [ ∇ m , ∇ k ] ϕ pjℓ + [ ∇ m , ∇ p ] ϕ jkℓ ) ϕ iab ω ka ω pb + ( i ↔ j )= − | ϕ | R ji − | ϕ | R g ij + | ϕ | R ij + | ϕ | R Jj,Ji + F (3.60) where the term F is given by F = (cid:8) ( R ℓj λp − R ℓpλj ) ϕ λkℓ + ( − R ℓj λk + R ℓkλj ) ϕ λpℓ + ( R ℓpλk − R ℓkλp ) ϕ λjℓ (cid:9) ϕ iab ω ka ω pb +( i ↔ j ) (3.61) F We begin with( R ℓj λp − R ℓpλj ) ϕ λkℓ ϕ iab ω ka ω pb = | ϕ | R ℓj λp − R ℓpλj )( ω λi g ℓb + ω ℓb g λi − ω λb g ℓi − ω ℓi g λb ) ω pb = | ϕ | R ℓj λp − R ℓpλj )( ω λi J pℓ − δ pℓ g λi + δ pλ g ℓi − ω ℓi J pλ )= | ϕ | − R Jpj,Ji,p + R ℓJℓJi,j ) − | ϕ | − R ji + R ℓℓij )+ | ϕ | R ij λλ − R iλλj ) + | ϕ | R Ji,jλJλ − R Ji,Jλλj ) (3.62)This reduces to( R ℓj λp − R ℓpλj ) ϕ λkℓ ϕ iab ω ka ω pb = | ϕ | − R Jpj,Ji,p + R ℓJℓ,Ji,j + R Ji,jλJλ − R Ji,Jλλj ) + | ϕ | R ij . R ℓjλp − R ℓpλj ) ϕ λkℓ ϕ iab ω ka ω pb = | ϕ | − R Ji,Jλλj + R Ji,jλJλ ) + | ϕ | R ij . We work out the next term, which after relabeling is( − R ℓj λk + R ℓkλj ) ϕ λpℓ ϕ iab ω ka ω pb = ( R ℓjλk − R ℓkλj ) ϕ λℓp ϕ iab ω ka ω pb = ( R ℓjλp − R ℓpλj ) ϕ λℓk ϕ iba ω pb ω ka , (3.63)and is therefore identical to the previous term,( − R ℓjλk + R ℓkλj ) ϕ λpℓ ϕ iab ω ka ω pb = | ϕ | − R Ji,Jλλj + R Ji,jλJλ ) + | ϕ | R ij . (3.64)We work out the final term. We start with( R ℓpλk − R ℓkλp ) ϕ λjℓ ϕ iab ω ka ω pb = − R pkλℓ ϕ λjℓ ϕ iab ω ka ω pb (3.65)by the Bianchi identity R ℓpλk + R pkλℓ + R kℓλp = 0. Applying the identity (2.21) gives − R pkλℓ ϕ λjℓ ϕ iab ω ka ω pb = ( − R pkJλ,Jℓ + B kpλℓ ) ϕ λjℓ ϕ iab ω ka ω pb = − R pkλℓ ϕ Jλ,j,Jℓ ϕ iab ω ka ω pb + B kpλℓ ϕ λjℓ ϕ iab ω ka ω pb = R pkλℓ ϕ λ,j,ℓ ϕ iab ω ka ω pb + B kpλℓ ϕ λjℓ ϕ iab ω ka ω pb (3.66)Therefore − R pkλℓ ϕ λjℓ ϕ iab ω ka ω pb = 12 B kpλℓ ϕ λjℓ ϕ iab ω ka ω pb (3.67)and hence ( R ℓpλk − R ℓkλp ) ϕ λjℓ ϕ iab ω ka ω pb = 12 B kpλℓ ϕ λjℓ ϕ iab ω ka ω pb (3.68)By definition of B , B kpλℓ = − D k N pλℓ + 2 D p N kλℓ − N αkp N αλℓ . (3.69)Therefore( R ℓpλk − R ℓkλp ) ϕ λjℓ ϕ iab ω ka ω pb = − D k N pλℓ ϕ λjℓ ϕ iab ω ka ω pb + D p N kλℓ ϕ λjℓ ϕ iab ω ka ω pb − N αkp N αλℓ ϕ λjℓ ϕ iab ω ka ω pb (3.70)= − D k N pλℓ ϕ λjℓ ϕ iab ω ka ω pb − N αkp N αλℓ ϕ λjℓ ϕ iab ω ka ω pb . We start with the last term. By the Bianchi identity N ijk + N kij + N jki = 0, N αkp N αλℓ ϕ λjℓ ϕ iab ω ka ω pb = − N αkp [ N ℓαλ ϕ λjℓ + N λℓα ϕ λjℓ ] ϕ iab ω ka ω pb (3.71)16ecall the identity (2.19) for switching indices on contractions of N and ϕ . Therefore N αkp N αλℓ ϕ λjℓ ϕ iab ω ka ω pb = N αkp [ N ℓλj ϕ ℓαλ + N λjℓ ϕ λℓα ] ϕ iab ω ka ω pb = − N αkp N ℓλj ϕ αℓλ ϕ iab ω ka ω pb . (3.72)Applying the identity (2.19) again, N αkp N αλℓ ϕ λjℓ ϕ iab ω ka ω pb = 2 N αℓλ N ℓλj ϕ αkp ϕ iab ω ka ω pb . (3.73)We can now apply the bilinear identities (2.11), so that N αkp N αλℓ ϕ λjℓ ϕ iab ω ka ω pb = − | ϕ | N αℓλ N ℓλj g αi = − | ϕ | N iℓλ N ℓλj . (3.74)Next, we need to handle the D N terms in (3.70). By the Bianchi identity N ijk + N kij + N jki = 0, we have − D k N pλℓ ϕ λjℓ ϕ iab ω ka ω pb = 2 D k N ℓpλ ϕ λjℓ ϕ iab ω ka ω pb + 2 D k N λℓp ϕ λjℓ ϕ iab ω ka ω pb (3.75)This is − D k N pλℓ ϕ λjℓ ϕ iab ω ka ω pb = − D k N ℓλp ϕ ℓλj ϕ iab ω ka ω pb . (3.76)To apply the bilinear identities (2.11), we will need to switch some indices. Lemma 5 D k N pij ϕ pλl = − D k N pλl ϕ pij + N pλ,Jl α Jk ϕ pij . (3.77) Proof:
Differentiating identity (2.19) gives D k N pij ϕ pλl + N pij D k ϕ pλl = − D k N pλl ϕ pij − N pλl D k ϕ pij . (3.78)Using the formula (2.7), we obtain D k N pij ϕ pλl = − D k N pλl ϕ pij + 12 N pij α k ϕ pλl + 12 N pij α Jk ˆ ϕ pλl + 12 N pλl α k ϕ pij + 12 N pλl α Jk ˆ ϕ pij . (3.79)Using (2.19) and ˆ ϕ pλl = − ϕ pJλ,l = − ϕ Jp,λl , we simplify this to D k N pij ϕ pλl = − D k N pλl ϕ pij − N pij α Jk ϕ p,Jλ,l − N pλl α Jk ϕ Jp,ij . (3.80)Using (2.19) again and N Jpλl = − N pλ,Jl , we obtain the desired identity. Q.E.D.Applying now this lemma to (3.76), we find − D k N pλℓ ϕ λjℓ ϕ iab ω ka ω pb = (4 D k N ℓλj − N ℓλJj α Jk ) ϕ ℓλp ϕ iab ω ka ω pb (3.81)17e can now use the bilinear identities − D k N pλℓ ϕ λjℓ ϕ iab ω ka ω pb = | ϕ | ( D k N ℓλj − N ℓλJj α Jk )( ω ℓi g λa − ω λi g ℓa − ω ℓa g λi + ω λa g ℓi ) ω ka = | ϕ | ( D k N ℓλj − N ℓλJj α Jk )( ω ℓi J kλ − ω λi J kℓ + δ kℓ g λi − δ kλ g ℓi )= | ϕ | ( − D k N JiJkj + N JiJkJj α Jk ) + | ϕ | ( D k N JkJi,j − N JkJi,Jj α Jk )+ | ϕ | ( D k N kij − N ki,Jj α Jk ) + | ϕ | ( − D k N ikj + N ikJj α Jk ) . This simplifies to − D k N pλℓ ϕ λjℓ ϕ iab ω ka ω pb = | ϕ | ( − D k N ikj + 2 D k N kij + 2 N ikj α k − N kij α k ) . (3.82)Substituting (3.74) and (3.82) into (3.70),( R ℓpλk − R ℓkλp ) ϕ λjℓ ϕ iab ω ka ω pb = | ϕ | ( − D k N ikj + 2 D k N kij + 2 N ikj α k − N kij α k ) + 2 | ϕ | N iℓλ N ℓλj (3.83)By the Bianchi identity,2 | ϕ | N iℓλ N ℓλj = 2 | ϕ | ( − N λiℓ − N ℓλi ) N ℓλj = 2 | ϕ | N λℓi N ℓλj − | ϕ | N ℓλi N ℓλj (3.84)and hence( R ℓpλk − R ℓkλp ) ϕ λjℓ ϕ iab ω ka ω pb = | ϕ | ( − D k N ikj + 2 D k N kij + 2 N ikj α k − N kij α k )+2 | ϕ | ( N − ) ij − | ϕ | ( N ) ij The result is F = | ϕ | (cid:26) ( − R Ji,Jλλj − R Jj,Jλλi ) + ( R Ji,jλJλ + R Jj,iλJλ ) + 2 R ij − D k N ikj + D k N j ki ) + 2( N ikj + N j ki ) α k + 4( N − ) ij − N ) ij (cid:27) (3.85) Lemma 6
We have the following formula g ℓm ([ ∇ m , ∇ j ] ϕ kpℓ + [ ∇ m , ∇ k ] ϕ pjℓ + [ ∇ m , ∇ p ] ϕ jkℓ ) ϕ iab ω ka ω pb + ( i ↔ j )= | ϕ | (cid:26) − R ji − Rg ij + R ij + R Jj,Ji − ( R Ji,Jλλj + R Jj,Jλλi ) + ( R i,JjλJλ + R j,JiλJλ ) + 2 R ij − D k N ikj + D k N jki ) + 2( N ikj + N jki ) α k + 4( N − ) ij − N ) ij (cid:27) (3.86)18 .2.6 Contributions of the curvature terms, continued We now simplify Lemma 6 by applying identities for the action of J on the Riemanncurvature tensor. We start with the terms − R Ji,Jλλj − R Jj,Jλλi (3.87)which can be manipulated using the relation (2.21) into − R Ji,Jλλj − R Jj,Jλλi = − R j λJλ,Ji − R iλJλ,Jj = − R j λλi − R iλλj − B λjλi − B λiλj = 2 R ij − B λjλi − B λiλj . (3.88)Next, we have the terms R i,JjλJλ + R j,JiλJλ . (3.89)By the Bianchi identity, R i,JjλJλ + ( i ↔ j ) = − R j,JλλJi − R Jλ,Jiλj + ( i ↔ j )= − R jλJλJi − R j λJi,Jλ + ( i ↔ j )= g λµ R j,λ,Jµ,Ji − R j λJi,Jλ + ( i ↔ j ) (3.90)Using the relation (2.21), R i,JjλJλ + ( i ↔ j ) = g λµ R j,λ,µ,i − R jλi,λ + g λµ B λ,j,µ,i − B λjiλ + ( i ↔ j )= − R ij − R ij + { B λjλi − B λjiλ + B λiλj − B λijλ } (3.91)Therefore R Ji,jλJλ + R Jj,iλJλ = − R ij + { B λjλi − B λjiλ + B λiλj − B λijλ } . (3.92)The next term in Lemma 6 that we consider is R Jj,Ji . This term becomes R Jj,Ji = g λµ R λ,Jj,µ,Ji = − g λµ R λ,Jj,Jµ,i − g λµ B Jj,λ,Jµ,i = − g λµ R i,Jµ,Jj,λ − g λµ B Jj,λ,Jµ,i = g λµ R i,Jµ,j,Jλ + g λµ B Jµ,i,j,Jλ − g λµ B Jj,λ,Jµ,i (3.93)and thus R Jj,Ji = R ij + B λijλ − B Jj λJλ,i . (3.94)Substituting (3.88), (3.92) and (3.94) into Lemma 6, we obtain g ℓm ([ ∇ m , ∇ j ] ϕ kpℓ + [ ∇ m , ∇ k ] ϕ pjℓ + [ ∇ m , ∇ p ] ϕ jkℓ ) ϕ iab ω ka ω pb + ( i ↔ j )= −| ϕ | R g ij − D k N ikj + D k N jki ) + 2( N ikj + N jki ) α k + 4( N − ) ij − N ) ij − B λjiλ − B JjλJλ,i (3.95)19sing the definition of B , − B λjiλ − B Jj λJλ,i = − [ − D λ N jiλ + 2 D j N λiλ − N αλj N αiλ ] − [ − D Jj N λJλ,i + 2 D λ N Jj,Jλ,i − N αJjλ N α,Jλ,i ]= − N ) ij (3.96)where we use the symmetries of N to get the last equality. Therefore Lemma 7
We have the following formula g ℓm ([ ∇ m , ∇ j ] ϕ kpℓ + [ ∇ m , ∇ k ] ϕ pjℓ + [ ∇ m , ∇ p ] ϕ jkℓ ) ϕ iab ω ka ω pb + ( i ↔ j ) (3.97)= | ϕ | (cid:8) − Rg ij + 2 D k N ijk + 2 D k N jik + 2( N ikj + N j ki ) α k + 4( N − ) ij − N ) ij (cid:9) By (3.11), we have( −| ϕ | dd † ϕ ) jkp ϕ iab ω ka ω pb = ( | ϕ | g ℓm ∇ m ∇ ℓ ϕ jkp ) ϕ iab ω ka ω pb − | ϕ | ( g ℓm (cid:8) [ ∇ m , ∇ j ] ϕ kpℓ +[ ∇ m , ∇ k ] ϕ pjℓ + [ ∇ m , ∇ p ] ϕ jkℓ (cid:9) ) ϕ iab ω ka ω pb By Lemma 3 and Lemma 7, we obtain( −| ϕ | dd † ϕ ) jkp ϕ iab ω ka ω pb + ( i ↔ j ) = | ϕ | (cid:8) ∇ µ α µ + | N | (cid:9) g ij + | ϕ | (cid:8) Rg ij + 2( − D k N ij k − D k N jik ) − N ikj + N j ki ) α k − N − ) ij + 8( N ) ij (cid:9) Altogether,
Lemma 8
We have the following formula ( −| ϕ | dd † ϕ ) jkp ϕ iab ω ka ω pb + ( i ↔ j ) (3.98)= | ϕ | (cid:26) Rg ij − D k N ijk + D k N jik ) + ( ∇ µ α µ + | N | ) g ij − N ikj + N jki ) α k − N − ) ij + 8( N ) ij (cid:27) . Returning to (3.3), we study the contributions of the second term − d | ϕ | ∧ d † ϕ . We let α = − d log | ϕ | as before, and write − d | ϕ | = | ϕ | α, ( d † ϕ ) kp = − g µβ ∇ β ϕ µkp . (3.99)20ince( − d | ϕ | ∧ d † ϕ ) jkp = ( − d | ϕ | ) j ( d † ϕ ) kp + ( − d | ϕ | ) p ( d † ϕ ) jk + ( − d | ϕ | ) k ( d † ϕ ) pj (3.100)we have( − d | ϕ | ∧ d † ϕ ) jkp = | ϕ | (cid:0) − α j g µβ ∇ β ϕ µkp − α p g µβ ∇ β ϕ µjk − α k g µβ ∇ β ϕ µpj (cid:1) (3.101)Using previous notation, ∇ β ϕ µkp = D β ϕ µkp − E β ; µkp . (3.102)By the formula (2.7), we conclude ∇ β ϕ µkp = − α β ϕ µkp + 12 α Jβ ϕ Jµ,kp − E β ; µkp . (3.103)Therefore( − d | ϕ | ∧ d † ϕ ) jkp = | ϕ | (cid:18) α j g µβ α β ϕ µkp − α j g µβ α Jβ ϕ Jµ,kp + α j g µβ E β ; µkp + 12 α p g µβ α β ϕ µjk − α p g µβ α Jβ ϕ Jµ,jk + α p g µβ E β ; µjk + 12 α k g µβ α β ϕ µpj − α k g µβ α Jβ ϕ Jµ,pj + α k g µβ E β ; µpj (cid:19) (3.104)which simplifies to( − d | ϕ | ∧ d † ϕ ) jkp = | ϕ | (cid:0) α j g µβ E β ; µkp + α p g µβ E β ; µjk + α k g µβ E β ; µpj (cid:1) := (I) + (II) + (III) . (3.105)We now work out the bilinears.(I) · ϕ iab ω ka ω pb = | ϕ | α j g µβ ( ϕ λkp N βµλ + ϕ µλp N βkλ + ϕ µkλ N βpλ ) ϕ iab ω ka ω pb (3.106)Since N µµλ = 0 and we can relabel p ↔ k and a ↔ b ,(I) · ϕ iab ω ka ω pb = 2 | ϕ | α j g µβ ( ϕ µλp N βkλ ) ϕ iab ω ka ω pb . (3.107)By the bilinear identities(I) · ϕ iab ω ka ω pb = | ϕ | α j g µβ N βkλ ( g µi ω λa − g λi ω µa − g µa ω λi + g λa ω µi ) ω ka = | ϕ | α j g µβ N βkλ ( − g µi δ kλ + g λi δ kµ − J kµ ω λi + J kλ ω µi )= | ϕ | α j N JkkJi − α j N Ji,Jλλ ) = 0 (3.108)21sing the type (0 ,
2) and trace-free property of N . Next,(II + III) · ϕ iab ω ka ω pb = 2 | ϕ | α p g µβ ( ϕ λjk N βµλ + ϕ µλk N βj λ + ϕ µjλ N βkλ ) ϕ iab ω ka ω pb = 2 | ϕ | α p (0 + ϕ µλk N µj λ + ϕ µjλ N µkλ ) ϕ iab ω ka ω pb (3.109)The first term is2 | ϕ | α p ( ϕ µλk N µjλ ) ϕ iab ω ka ω pb = − | ϕ | α p N µjλ ( ϕ µλk ϕ iba ω ka ) ω pb = − | ϕ | α p N µjλ ( g µi ω λb − g λi ω µb − g µb ω λi + g λb ω µi ) ω pb = − | ϕ | α p N µjλ ( − g µi δ pλ + g λi δ pµ − J pµ ω λi + J pλ ω µi )= − | ϕ | − α p N ij p + α p N pji + α p N Jpj,Ji − α p N Ji,jJp )= | ϕ | ( α p N ij p − α p N pji ) (3.110)For the second term, we use the identity (2.19) to obtain2 | ϕ | α p ( ϕ µjλ N µkλ ) ϕ iab ω ka ω pb = 2 | ϕ | α p ( − ϕ µkλ N µj λ ) ϕ iab ω ka ω pb = − | ϕ | α p N µj λ ϕ µλk ϕ iba ω ka ω pb . (3.111)This term is identical to the one above. Therefore(II + III) · ϕ iab ω ka ω pb = 2 | ϕ | ( α p N ij p − α p N pji ) (3.112)Altogether, ( − d | ϕ | ∧ d † ϕ ) jkp ϕ iab ω ka ω pb = 2 | ϕ | ( α p N ijp − α p N pji ) . (3.113)Therefore( − d | ϕ | ∧ d † ϕ ) jkp ϕ iab ω ka ω pb +( i ↔ j ) = 2 | ϕ | ( α p N ij p − α p N pji + α p N jip − α p N pij ) . (3.114)By the Bianchi identity N pij + N j pi + N ijp = 0, and hence( − d | ϕ | ∧ d † ϕ ) jkp ϕ iab ω ka ω pb + ( i ↔ j )= 2 | ϕ | ( α p N ij p − α p ( − N ipj − N jip ) + α p N jip − α p ( − N j pi − N ij p ))= 2 | ϕ | α p ( N ij p + N ipj + N jip + N jip + N j pi + N ij p ) (3.115)Thus we have established the following lemma: Lemma 9 ( − d | ϕ | ∧ d † ϕ ) jkp ϕ iab ω ka ω pb + ( i ↔ j ) = − | ϕ | α p ( N j pi + N ipj ) . (3.116)22 .3.2 Interior product Returning to (3.3), we study the contributions of the third term d ( ι ∇| ϕ | ϕ ). We can write( ι ∇| ϕ | ϕ ) kp = g µν ( ∂ ν | ϕ | ) ϕ µkp = −| ϕ | g µν α ν ϕ µkp (3.117)since α i = − ∂ i log | ϕ | , or ∂ j | ϕ | = −| ϕ | α j . Next, d ( ι ∇| ϕ | ϕ ) jkp = ∇ j ( ι ∇| ϕ | ϕ ) kp + ∇ p ( ι ∇| ϕ | ϕ ) jk + ∇ k ( ι ∇| ϕ | ϕ ) pj . (3.118)We start with ∇ j ( ι ∇| ϕ | ϕ ) kp = | ϕ | α j α µ ϕ µkp − | ϕ | ∇ j α µ ϕ µkp − | ϕ | α µ ∇ j ϕ µkp (3.119)Since ∇ j ϕ µkp = − α j ϕ µkp + 12 α Jj ϕ Jµ,k,p − E j ; µkp , (3.120)we have ∇ j ( ι ∇| ϕ | ϕ ) kp = 32 | ϕ | α j α µ ϕ µkp −| ϕ | ∇ j α µ ϕ µkp − | ϕ | α µ α Jj ϕ Jµ,k,p + | ϕ | α µ E j ; µkp . (3.121)We now work out the bilinears.( 32 | ϕ | α j α µ ϕ µkp ) ϕ iab ω ka ω pb = − | ϕ | α j α µ g µi = − | ϕ | α i α j , (3.122)( −| ϕ | ∇ j α µ ϕ µkp ) ϕ iab ω ka ω pb = | ϕ | ∇ j α i , (3.123)( − | ϕ | α µ α Jj ϕ Jµ,k,p ) ϕ iab ω ka ω pb = 12 | ϕ | α µ α Jj g Jµ,i = − | ϕ | α Ji α Jj . (3.124)Therefore( ∇ j ( ι ∇| ϕ | ϕ ) kp ) ϕ iab ω ka ω pb = − | ϕ | α i α j + | ϕ | ∇ j α i − | ϕ | α Ji α Jj + | ϕ | α µ E j ; µkp ϕ iab ω ka ω pb . (3.125)Next, we work out the two next contributions of this term with the indices ( jkp ) cyclicallypermuted. After forming bilinears, these two extra terms are identical.( ∇ p ( ι ∇| ϕ | ϕ ) jk ) ϕ iab ω ka ω pb + ( ∇ k ( ι ∇| ϕ | ϕ ) pj ) ϕ iab ω ka ω pb = 2( ∇ p ( ι ∇| ϕ | ϕ ) jk ) ϕ iab ω ka ω pb (3.126)As before, we have ∇ p ( ι ∇| ϕ | ϕ ) jk = 32 | ϕ | α p α µ ϕ µjk −| ϕ | ∇ p α µ ϕ µjk − | ϕ | α µ α Jp ϕ Jµ,j,k + | ϕ | α µ E p ; µjk (3.127)23orming bilinears,( 32 | ϕ | α p α µ ϕ µjk ) ϕ iab ω ka ω pb = − | ϕ | α p α µ ( ω µi g jb − ω ji g µb − ω µb g ji + ω jb g µi ) ω pb = 38 | ϕ | α p α µ ( − ω µi J pj + ω ji J pµ − δ pµ g ji + δ pj g µi )= 38 | ϕ | ( α Jj α Ji + α Jµ α µ ω ij − α µ α µ g ij + α j α i ) , (3.128)and( −| ϕ | ∇ p α µ ϕ µjk ) ϕ iab ω ka ω pb = 14 | ϕ | ∇ p α µ ( ω µi g jb − ω ji g µb − ω µb g ji + ω jb g µi ) ω pb = 14 | ϕ | ∇ p α µ ( ω µi J pj − ω ji J pµ + δ pµ g ji − δ pj g µi )= | ϕ | − J nj ∇ n α q J qi − J pµ ∇ p α µ ω ji + ∇ µ α µ g ij − ∇ j α i ) , and( − | ϕ | α µ α Jp ϕ Jµ,j,k ) ϕ iab ω ka ω pb = 18 | ϕ | α µ α Jp ( ω Jµ,i g jb − ω ji g Jµ,b − ω Jµ,b g ji + ω jb g Jµ,i ) ω pb = 18 | ϕ | α µ α Jp ( − g µi J pj + ω ji δ pµ + J pµ g ji − δ pj g Jµ,i )= | ϕ | α i α j + α p α Jp ω ji − α µ α µ g ij + α Ji α Jj ) . (3.129)Altogether,( ∇ p ( ι ∇| ϕ | ϕ ) jk ) ϕ iab ω ka ω pb = | ϕ | (cid:18) α Jj α Ji + 3 α Jµ α µ ω ij − α µ α µ g ij + 3 α j α i − ∇ Jj α Ji − ∇ Jµ α µ ω ji + 2 ∇ µ α µ g ij − ∇ j α i + α i α j + α p α Jp ω ji − α µ α µ g ij + α Ji α Jj (cid:19) + | ϕ | α µ E p ; µjk ϕ iab ω ka ω pb (3.130)It follows that2( ∇ p ( ι ∇| ϕ | ϕ ) jk ) ϕ iab ω ka ω pb = | ϕ | (cid:18) α Jj α Ji + 2 α Jµ α µ ω ij − α µ α µ g ij + 4 α j α i − J nj ∇ n α q J qi − J pµ ∇ p α µ ω ji + 2 ∇ µ α µ g ij − ∇ j α i (cid:19) +2 | ϕ | α µ E p ; µjk ϕ iab ω ka ω pb (3.131)We can now combine all of our calculations. By (3.118), (3.125), (3.131),24 emma 10 ( dι ∇| ϕ | ϕ ) jkp ϕ iab ω ka ω pb + ( i ↔ j )= | ϕ | (cid:8)
12 ( ∇ j α i + ∇ i α j ) − α i α j + α Ji α Jj − α µ α µ g ij −
12 ( J pj J qi ∇ p α q + J pi J qj ∇ p α q )+ ∇ µ α µ g ij (cid:9) + | ϕ | α µ E j ; µkp ϕ iab ω ka ω pb + | ϕ | α µ E i ; µkp ϕ jab ω ka ω pb +2 | ϕ | α µ E p ; µjk ϕ iab ω ka ω pb + 2 | ϕ | α µ E p ; µik ϕ jab ω ka ω pb . (3.132)It remains to evaluate the E terms. | ϕ | α µ E j ; µkp ϕ iab ω ka ω pb + 2 | ϕ | α µ E p ; µjk ϕ iab ω ka ω pb + ( i ↔ j ) (3.133)We start with | ϕ | α µ E j ; µkp ϕ iab ω ka ω pb = | ϕ | α µ ( ϕ λkp N jµλ + ϕ µλp N jkλ + ϕ µkλ N jpλ ) ϕ iab ω ka ω pb (3.134)which by symmetry is | ϕ | α µ E j ; µkp ϕ iab ω ka ω pb = | ϕ | α µ ( ϕ λkp N jµλ ) ϕ iab ω ka ω pb + 2 | ϕ | α µ ( ϕ µλp N jkλ ) ϕ iab ω ka ω pb (3.135)The first term is | ϕ | α µ ( ϕ λkp N jµλ ) ϕ iab ω ka ω pb = −| ϕ | α µ N jµλ g λi = −| ϕ | α µ N jµi . (3.136)The second term is2 | ϕ | α µ ( ϕ µλp N jkλ ) ϕ iab ω ka ω pb = | ϕ | α µ N jkλ ( g µi ω λa − g λi ω µa − g µa ω λi + g λa ω µi ) ω ka = | ϕ | α µ N jkλ ( − g µi δ kλ + g λi δ kµ − J kµ ω λi + J kµ ω µi )= | ϕ | α µ N jkλ ( − g µi δ kλ + g λi δ kµ − J kµ ω λi + J kλ ω µi )= | ϕ | − α i N jλλ + α µ N jµi + α µ N j,Jµ,Ji − α Ji N j,Jλλ )= | ϕ | α µ N jµi − α µ N jµi + 0) = 0 . (3.137)Therefore | ϕ | α µ E j ; µkp ϕ iab ω ka ω pb = −| ϕ | α µ N jµi = | ϕ | α µ N jiµ . (3.138)Next, we consider2 | ϕ | α µ E p ; µjk ϕ iab ω ka ω pb = 2 | ϕ | α µ ( ϕ λjk N pµλ + ϕ µλk N pjλ + ϕ µjλ N pkλ ) ϕ iab ω ka ω pb := (˜I + ˜II + ˜III) · ϕ iab ω ka ω pb . (3.139)25e start with(˜I) · ϕ iab ω ka ω pb = − | ϕ | α µ N pµλ ( ϕ λkj ϕ iab ω ka ) ω pb = − | ϕ | α µ N pµλ ( ω λi g jb − ω ji g λb − ω λb g ji + ω jb g λi ) ω pb = − | ϕ | α µ N pµλ ( ω λi J pj − ω ji J pλ + δ pλ g ji − δ pj g λi )= | ϕ | α µ N Jj,µ,Ji + ω ji α µ N Jλ,µλ − g ji α µ N λµλ + α µ N jµi )= | ϕ | − α µ N jµi + 0 − α µ N jµi ) = 0 . (3.140)Similarly, we can also compute ( ˜II) · ϕ iab ω ka ω pb = 0 (3.141)The third term is ( ˜III) · ϕ iab ω ka ω pb = 2 | ϕ | α µ ( ϕ µjλ N pkλ ) ϕ iab ω ka ω pb (3.142)It can be rearranged using the symmetry p ↔ k , a ↔ b ( ˜III) · ϕ iab ω ka ω pb = 2 | ϕ | α µ ϕ µjλ ( N pkλ − N kpλ )2 ϕ iab ω ka ω pb = −| ϕ | α µ ϕ λµj N λpk ϕ iab ω ka ω pb (3.143)and then using the Bianchi identity. By the identity N λpk ϕ λµj = − N λµj ϕ λpk ,( ˜III) · ϕ iab ω ka ω pb = −| ϕ | α µ N λµj ϕ λpk ϕ iba ω ka ω pb . (3.144)We can now use the bilinear identity.( ˜III) · ϕ iab ω ka ω pb = | ϕ | α µ N λµj g λi = | ϕ | α µ N iµj = −| ϕ | α µ N ijµ . (3.145)Substituting our results into (3.139), we obtain2 | ϕ | α µ E p ; µjk ϕ iab ω ka ω pb = −| ϕ | α µ N ijµ . (3.146)Combining the above equation with (3.138), | ϕ | α µ E j ; µkp ϕ iab ω ka ω pb + 2 | ϕ | α µ E p ; µjk ϕ iab ω ka ω pb + ( i ↔ j )= | ϕ | α µ N jiµ − | ϕ | α µ N ijµ + ( i ↔ j ) = 0 . (3.147)Therefore the E terms do not contribute, and we are left with: Lemma 11 ( dι ∇| ϕ | ϕ ) jkp ϕ iab ω ka ω pb + ( i ↔ j ) = | ϕ | (cid:8)
12 ( ∇ j α i + ∇ i α j ) − α i α j + α Ji α Jj − α µ α µ g ij −
12 ( J pj J qi ∇ p α q + J pi J qj ∇ p α q ) + ∇ µ α µ g ij (cid:9) .4 N † term: d ( | ϕ | N † · ϕ ) Recall from the definition of the operator N † that ( N † ϕ ) kj = 2 N µjλ ϕ µkλ , and thus d ( | ϕ | N † · ϕ ) jkp = ∇ j ( | ϕ | ( N † · ϕ ) kp ) + ∇ p ( | ϕ | ( N † · ϕ ) jk ) + ∇ k ( | ϕ | ( N † · ϕ ) pj ):= I + II + III . (3.148) We start with the first term ∇ j ( | ϕ | ( N † · ϕ ) kp ) = − | ϕ | α j N µpλ ϕ µkλ + 2 | ϕ | ∇ j ( N µpλ ϕ µkλ ) (3.149)= − | ϕ | α j N µpλ ϕ µkλ + 2 | ϕ | ∇ j N µpλ ϕ µkλ + 2 | ϕ | N µpλ ∇ j ϕ µkλ We now work out the bilinears term by term − | ϕ | α j N µkλ ϕ µpλ ϕ iab ω ka ω pb = 12 | ϕ | α j N µkλ ( ω µi g λa + ω λa g µi − ω µa g λi − ω λi g µa ) ω ka = 12 | ϕ | α j N µkλ ( ω µi J kλ − δ kλ g µi + δ kµ g λi − ω λi J kµ )= 12 | ϕ | α j ( − N Ji,kJk − N ikk + N kki + N Jkk,Ji ) = 0 . The first two terms are zero due to anti-symmetry of N in the second and third indices. Thethird and fourth terms are also zero since g ml N mlj = 0 and N Jkk,Ji = − N kJk,Ji = N kki .Next, we work with the second group of terms in (3.149):2 | ϕ | ∇ j N µpλ ϕ µkλ ϕ iab ω ka ω pb = 12 | ϕ | ∇ j N µpλ ( ω µi g λb + ω λb g µi − ω µb g λi − ω λi g µb ) ω pb = 12 | ϕ | ∇ j N µpλ ( ω µi J pλ − δ pλ g µi + δ pµ g λi − ω λi J pµ )= 12 | ϕ | ( ∇ j N µpλ ω µi J pλ − ∇ j N ipp + ∇ j N ppi − ∇ j N µpλ ω λi J pµ )= 12 | ϕ | ( ∇ j N µpλ ω µi J pλ − ∇ j N µpλ ω λi J pµ ) (3.150)The last two terms require extra work since J may not be covariantly constant under ∇ . ω µi ∇ j N µpλ J pλ = ω µi ( ∇ j ( N µpλ J pλ ) − N µpλ ∇ j J pλ ) = ω µi ( ∇ j N µpJp + 2 N µpλ N jλJp )= 2 ω µi N µpλ N jλJp = − N Ji,pλ N jλJp = − N i,Jpλ N jλJp = 2 N ipλ N jλp . Similarly, we can compute −∇ j N µpλ ω λi J pµ = 2 N µJp,i N jµJp = − N µpi N jµp . (3.151)27ltogether, we have2 | ϕ | ∇ j N µpλ ϕ µkλ ϕ iab ω ka ω pb = | ϕ | ( N ipλ N jλp − N λpi N jλp ) . (3.152)Next, we consider the last group of terms in (3.149).2 | ϕ | N µpλ ∇ j ϕ µkλ ϕ iab ω ka ω pb (3.153)Since ∇ j ϕ µkλ = − α j ϕ µkλ + 12 α Jj ϕ Jµ,kλ − E j ; µkλ , (3.154)then2 | ϕ | N µpλ ∇ j ϕ µkλ ϕ iab ω ka ω pb = 2 | ϕ | N µpλ ( − α j ϕ µkλ + 12 α Jj ϕ Jµ,kλ − E j ; µkλ ) ϕ iab ω ka ω pb . We work out the bilinears term by term2 | ϕ | N µpλ ( − α j ϕ µkλ ) ϕ iab ω ka ω pb = −| ϕ | α j N µpλ ϕ µkλ ϕ iab ω ka ω pb = − | ϕ | α j N µpλ ( ω µi g λb + ω λb g µi − ω µb g λi − ω λi g µb ) ω pb = − | ϕ | α j N µpλ ( ω µi J pλ − δ pλ g µi + δ pµ g λi − ω λi J pµ )= − | ϕ | α j ( − N Ji,pJp − N ipp + N ppi + N pp,Ji ) = 0 . | ϕ | N µpλ ( 12 α Jj ϕ Jµ,kλ ) ϕ iab ω ka ω pb = | ϕ | α Jj N µpλ ϕ Jµ,kλ ϕ iab ω ka ω pb = − | ϕ | α Jj N µpλ ( ω Jµ,i g λb + ω λb g Jµ,i − ω Jµ,b g λi − ω λi g Jµ,b ) ω pb = − | ϕ | α Jj N µpλ ( − g µi J pλ − δ pλ ω µi + J pµ g λi + δ pµ ω λi )= − | ϕ | α Jj ( − N ipJp + N Ji,pp + N Jppi − N pp,Ji ) = 0 . | ϕ | N µpλ ( − E j ; µkλ ) ϕ iab ω ka ω pb = − | ϕ | N µpλ E j ; µkλ ϕ iab ω ka ω pb = − | ϕ | N µpλ ( N jµℓ ϕ ℓkλ + N jkℓ ϕ µℓλ + N jλℓ ϕ µkℓ ) ϕ iab ω ka ω pb (3.155)The first term in the above last line is easy to handle2 | ϕ | N µpλ N jµℓ ϕ ℓkλ ϕ iab ω ka ω pb = 12 | ϕ | N µpλ N jµℓ ( ω ℓi g λb + ω λb g ℓi − ω ℓb g λi − ω λi g ℓb ) ω pb = 12 | ϕ | N µpλ N jµℓ ( ω ℓi J pλ − δ pλ g ℓi + δ pℓ g λi − ω λi J pℓ )= 12 | ϕ | ( − N µJλλ N jµ,Ji − N µpp N jµi + N µpi N jµp + N µJℓ,Ji N jµℓ = 12 | ϕ | ( N µpi N jµp − N µpi N jµp ) = 0 . (3.156)28he third term can also be handled in the similar way2 | ϕ | N µpλ N jλℓ ϕ µkℓ ϕ iab ω ka ω pb = 12 | ϕ | N µpλ N jλℓ ( ω µi g ℓb + ω ℓb g µi − ω µb g ℓi − ω ℓi g µb ) ω pb = 12 | ϕ | N µpλ N jλℓ ( ω µi J pℓ − δ pℓ g µi + δ pµ g ℓi − ω ℓi J pµ )= 12 | ϕ | ( − N Ji,pλ N jλJp − N ipλ N jλp + N ppλ N jλi + N Jppλ N jλ,Ji )= 12 | ϕ | ( N ipλ N jλp − N ipλ N jλp ) = 0 . (3.157)For the second term in (3.155), we will use the Bianchi identity and switch the indices asbefore, N pij ϕ pkl = − N pkl ϕ pij , obtaining N µpλ N jkℓ ϕ µℓλ = − N µpλ ( N ℓjk + N kℓj ) ϕ µℓλ = N µpλ N ℓjk ϕ ℓµλ − N kℓj N µpλ ϕ µℓλ = − N µpλ N ℓµλ ϕ ℓjk + N kℓj N µℓλ ϕ µpλ (3.158)Therefore, − | ϕ | N µpλ N jkℓ ϕ µℓλ ϕ iab ω ka ω pb = 2 | ϕ | ( N µpλ N ℓµλ ϕ ℓjk − N kℓj N µℓλ ϕ µpλ ) ϕ iab ω ka ω pb = − | ϕ | N µpλ N ℓµλ ( ω ℓi g jb + ω jb g ℓi − ω ℓb g ji − ω ji g ℓb ) ω pb + 12 | ϕ | N kℓj N µℓλ ( ω µi g λa + ω λa ω µi − ω µa g λi − ω λi g µa ) ω ka = − | ϕ | N µpλ N ℓµλ ( ω ℓi J pj − δ pj g ℓi + δ pℓ g ji − ω ji J pℓ )+ 12 | ϕ | N kℓj N µℓλ ( ω µi J kλ − δ kλ g µi + δ kµ g λi − ω λi J kµ )= − | ϕ | ( − N µJj λ N Ji,µλ − N µj λ N iµλ + N µpλ N pµλ g ji )+ 12 | ϕ | ( − N Jλℓj N Ji,ℓλ − N λℓj N iℓλ + N µℓj N µℓi + N Jµℓj N µℓ,Ji )The right hand side can be readily simplified as follows, − | ϕ | ( − N µj λ N iµλ + N µpλ N pµλ g ji ) + 12 | ϕ | ( − N λℓj N iℓλ + N µℓj N µℓi − N Jµℓj N Jµℓi )= − | ϕ | ( − N µj λ N iµλ + N µpλ N pµλ g ji + 2 N λℓj N iℓλ − N µℓj N µℓi )= − | ϕ | (2 N µλj N iµλ − N λℓj N iλℓ + ( N − ) λλ g ji − N µℓj N µℓi )= −| ϕ | ( 12 ( N − ) λλ g ij − N µℓj N µℓi ) . (3.159)Putting the above computations into (3.153), we obtain2 | ϕ | N µpλ ∇ j ϕ µkλ ϕ iab ω ka ω pb = −| ϕ | ( 12 ( N − ) λλ g ij − N µℓj N µℓi ) (3.160)29herefore, we obtain the first term (I) in (3.148):(I) · ϕ iab ω ka ω pb = | ϕ | ( N ipλ N jλp − N λpi N jλp ) − | ϕ | ( 12 ( N − ) λλ g ij − N pλj N pλi )= − | ϕ | ( N − ) λλ g ij + | ϕ | ( N ipλ N jλp − N λpi N jλp + N pλj N pλi )Using the Bianchi identity, we readily find N ipλ N jλp − N λpi N jλp + N pλj N pλi = − N λj p N pλi (3.161)Therefore, (I) · ϕ iab ω ka ω pb = − | ϕ | ( N − ) λλ g ij + | ϕ | N λpj N pλi . (3.162) Next we work out the contributions of (II) in (3.148). The contributions from (III) willturn out to be similar.12 II = 12 ∇ p ( | ϕ | ( N † · ϕ ) jk ) = − ∇ p ( | ϕ | ( N † · ϕ ) kj )= | ϕ | α p N µj λ ϕ µkλ − | ϕ | ∇ p N µjλ ϕ µkλ − | ϕ | N µj λ ∇ p ϕ µkλ (3.163)Again, we will work out the bilinears term by term. | ϕ | α p N µjλ ϕ µkλ ϕ iab ω ka ω pb = 14 | ϕ | α p N µj λ ( ω µi g λb + ω λb g µi − ω µb g λi − ω λi g µb ) ω pb = 14 | ϕ | α p N µj λ ( ω µi J pλ − δ pλ g µi + δ pµ g λi − ω λi J pµ )= 14 | ϕ | α p ( − N Ji,jJp − N ijp + N pji + N Jpj,Ji )= 14 | ϕ | α p ( − N ij p − N ij p + N pji + N pji )= 12 | ϕ | α p ( − N ij p + N pji ) (3.164)Next, we deal with the second term in (3.163) −| ϕ | ∇ p N µjλ ϕ µkλ ϕ iab ω ka ω pb = − | ϕ | ∇ p N µjλ ( ω µi g λb + ω λb g µi − ω µb g λi − ω λi g µb ) ω pb = − | ϕ | ∇ p N µjλ ( ω µi J pλ − δ pλ g µi + δ pµ g λi − ω λi J pµ )= 14 | ϕ | ( ∇ p N ij p − ∇ p N pji ) − | ϕ | ( ω µi ∇ p N µjλ J pλ − ω λi ∇ p N µjλ J pµ ) (3.165)30or the second group of terms in (3.165) , we need to take care of ∇ J , ω µi ∇ p N µjλ J pλ − ω λi ∇ p N µj λ J pµ (3.166)= ω µi ∇ p N µjλ J pλ − ω µi ∇ p N λj µ J pλ = ω µi ∇ p ( N µj λ + N λµj ) J pλ = − ω µi ∇ p N jλµ J pλ = − ω µi ( ∇ p ( N j λµ J pλ ) − N j λµ ∇ p J pλ ) = ω µi ∇ p N Jjpµ − ω µi N j λµ N pλJp = ω µi ∇ p N Jjpµ = ∇ p N Jj pµ J ℓµ g ℓi = ∇ p ( N Jjpµ J ℓµ g ℓi ) − N Jjpµ ∇ p J ℓµ g ℓi = ∇ p N j pi − N jpµ N pµi (3.167)Putting this back into the calculation, −| ϕ | ∇ p N µjλ ϕ µkλ ϕ iab ω ka ω pb = 14 | ϕ | ( ∇ p N ij p − ∇ p N pji − ∇ p N j pi ) + 12 | ϕ | N j pµ N pµi . = 12 | ϕ | ( ∇ p N ij p − ∇ p N pji ) + 12 | ϕ | N j pµ N pµi (3.168)where we used the Bianchi identity − N j pi = N ij p + N pij to obtain the last equality above.Now, we deal with the ∇ N terms using the projected Levi-Civita connection ∇ p N ij p − ∇ p N pji = D p N ij p − N αj p N piα − N iαp N pjα − N ijα N ppα (3.169) − D p N pji + N αji N ppα + N pαi N pj α + N pjα N piα = D p N ij p − D p N pji − ( N αjp − N pjα ) N piα − ( N iαp − N pαi ) N pjα since N ppα = 0. Next, apply the Bianchi identity of N to the last two terms, and get ∇ p N ij p − ∇ p N pji = D p N ijp − D p N pji + N j pα N piα + N αpi N pjα . (3.170)So, we have −| ϕ | ∇ p N µjλ ϕ µkλ ϕ iab ω ka ω pb = 12 | ϕ | ( D p N ijp − D p N pji ) (3.171)+ 12 | ϕ | ( N j pα N piα + N αpi N pjα + N jpα N pαi )= 12 | ϕ | ( D p N ijp − D p N pji ) + 12 | ϕ | N αpi N pjα Next, we deal with the last term in (3.163). Since ∇ p ϕ µkλ = − α p ϕ µkλ + α Jp ϕ Jµ,kλ − E p ; µkλ , we have −| ϕ | N µj λ ∇ p ϕ µkλ = | ϕ | N µjλ ( 12 α p ϕ µkλ − α Jp ϕ Jµ,kλ + E p ; µkλ ) . (3.172)We work out the bilinears term by term.12 | ϕ | α p N µj λ ϕ µkλ ϕ iab ω ka ω pb = 18 | ϕ | α p N µjλ ( ω µi J pλ − δ pλ g µi + δ pµ g λi − ω λi J pµ )= 18 | ϕ | α p ( − N Ji,jJp − N ijp + N pji + N Jpj,Ji )= 14 | ϕ | α p ( − N ij p + N pji ) (3.173)31 | ϕ | α Jp N µjλ ϕ Jµ,kλ ϕ iab ω ka ω pb = − | ϕ | α Jp N µjλ ( − g µi J pλ − δ pλ ω µi + J pµ g λi + δ pµ ω λi )= − | ϕ | α Jp ( − N ij Jp + N Ji,jp + N Jpji − N pj,Ji )= − | ϕ | α Jp ( − N ijJp + 2 N Jpji ) = 14 | ϕ | α p N jpi (3.174)by the Bianchi identity satisfied by N . The terms E lead to | ϕ | N µjλ E p ; µkλ ϕ iab ω ka ω pb = | ϕ | N µj λ ( N pµℓ ϕ ℓkλ + N pkℓ ϕ µℓλ + N pλℓ ϕ µkℓ ) ϕ iab ω ka ω pb We compute the three terms | ϕ | N µjλ N pµℓ ϕ ℓkλ ϕ iab ω ka ω pb = 12 | ϕ | N µj λ N pµℓ ( ω ℓi g λb + ω λb g ℓi − ω ℓb g λi − ω λi g ℓb ) ω pb = 12 | ϕ | N µj λ N pµℓ ( ω ℓi J pλ − δ pλ g ℓi + δ pℓ g λi − ω λi J pℓ )= 12 | ϕ | ( − N µjJp N pµ,Ji − N µj p N pµi + N µji N pµp + N µj,Ji N pµJp )= 12 | ϕ | ( N µj p N pµi − N µjp N pµi ) = 0 (3.175) | ϕ | N µjλ N pλℓ ϕ µkℓ ϕ iab ω ka ω pb = 12 | ϕ | N µj λ N pλℓ ( ω µi g ℓb + ω ℓb g µi − ω µb g ℓi − ω ℓi g µb ) ω pb = 12 | ϕ | N µj λ N pλℓ ( ω µi J pℓ − δ pℓ g µi + δ pµ g ℓi − ω ℓi J pµ )= 12 | ϕ | ( − N Ji,j λ N pλJp − N ij λ N pλp + N pj λ N pλi + N Jpj λ N pλ,Ji )= 12 | ϕ | ( N pj λ N pλi − N Jpjλ N pλ,Ji ) = 0 (3.176)The second term in (3.175) is more complicated, we first note that, by interchanging indices k ↔ p and a ↔ b , N µj λ N pkℓ ϕ µℓλ ϕ iab ω ka ω pb = − N µjλ N kpℓ ϕ µℓλ ϕ iab ω ka ω pb (3.177)It follows that N µjλ N pkℓ ϕ µℓλ ϕ iab ω ka ω pb = 12 N µjλ ( N pkℓ − N kpℓ ) ϕ µℓλ ϕ iab ω ka ω pb = 12 N µjλ ( N pkℓ + N kℓp ) ϕ µℓλ ϕ iab ω ka ω pb = − N µj λ N ℓpk ϕ µℓλ ϕ iab ω ka ω pb (3.178)32here we use Bianchi identity to get the last equality. Now, we can ready to use theidentity N pkℓ ϕ pij = − N pij ϕ pkℓ to handle the second term in (3.175) | ϕ | N µj λ N pkℓ ϕ µℓλ ϕ iab ω ka ω pb = − | ϕ | N µj λ N ℓpk ϕ µℓλ ϕ iab ω ka ω pb = 12 | ϕ | N µjλ N ℓµλ ϕ ℓkp ϕ iab ω ka ω pb = − | ϕ | N µj λ N ℓµλ g ℓi = 12 | ϕ | N µλj N iµλ . (3.179)So, | ϕ | N µjλ E p ; µkλ ϕ iab ω ka ω pb = 12 | ϕ | N µλj N iµλ . (3.180)Putting the above calculation together, we obtain −| ϕ | N µj λ ∇ p ϕ µkλ ϕ iab ω ka ω pb = 14 | ϕ | α p ( − N ij p + N pji ) + 14 | ϕ | α p N j pi + 12 | ϕ | N µλj N iµλ = 12 | ϕ | N µλj N iµλ + 12 | ϕ | α p N jpi (3.181)using Bianchi identity N pji + N ipj + N jip = 0.Back to (3.163), using (3.164) (3.171) and (3.181), we complete the calculation for (II):(II) · ϕ iab ω ka ω pb = | ϕ | α p ( − N ij p + N pji ) + | ϕ | ( D p N ij p − D p N pji ) (3.182)+ | ϕ | N λpi N pjλ + | ϕ | N µλj N iµλ + | ϕ | α p N j pi = | ϕ | D p N jip + 2 | ϕ | α p N jpi − | ϕ | N pλi N pλj Note that N pji = 0 up to the symmetrization for ( i ↔ j ). So, terms involving N pij vanishup to the symmetrization for ( i ↔ j ). For the two quadratic terms about N , we useBianchi identity to obtain the last line. Thus(II) · ϕ iab ω ka ω pb = | ϕ | (cid:8) D p N jip + 2 α p N j pi − N pλi N pλj (cid:9) (3.183) Next, we consider (III) in (3.148). We simply observe that by switching the indices k ↔ p and a ↔ b and exploiting the antisymmetry of ( N † ϕ ) kj in j and k , we may write(III) · ϕ iab ω ka ω pb = ∇ k ( | ϕ | ( N † · ϕ ) pj ) ϕ iab ω ka ω pb = −∇ p ( | ϕ | ( N † · ϕ ) kj ) ϕ iab ω ka ω pb = ∇ p ( | ϕ | ( N † · ϕ ) jk ) ϕ iab ω ka ω pb = (II) · ϕ iab ω ka ω pb . (3.184)We can now put (I), (II) and (III) all together,( d ( | ϕ | N † · ϕ )) jkp · ϕ iab ω ka ω pb + ( i ↔ j ) = −| ϕ | ( N − ) λλ g ij + | ϕ | (cid:8) N λpj N pλi + ( i ↔ j ) (cid:9) +2 | ϕ | (cid:8) D p N jip + 2 α p N j pi − N pλi N pλj + ( i ↔ j ) (cid:9) emma 12 In conclusion, we have d ( | ϕ | N † · ϕ ) jkp · ϕ iab ω ka ω pb + ( i ↔ j ) (3.185)= | ϕ | (cid:8) D p N jip + D p N ijp ) + 4 α p ( N jpi + N ipj ) − ( N − ) λλ g ij − N ) ij + 2( N − ) ij (cid:9) g ij Assembling all the terms in (3.3) and putting them in (3.4), we obtain the flow of ˜ g ij , ∂ t ˜ g ij = − (cid:26) ( −| ϕ | dd † ϕ ) jkp ϕ iab ω ka ω pb − ( d | ϕ | ∧ d † ϕ ) jkp ϕ iab ω ka ω pb (3.186)+( dι ∇| ϕ | ϕ ) jkp ϕ iab ω ka ω pb + (2 d ( | ϕ | N † · ϕ )) jkp ϕ iab ω ka ω pb + ( i ↔ j ) (cid:27) By (3.98), (3.116), Lemmas 11 and 12, and the identity (2.18), ∂ t ˜ g ij = −| ϕ | (cid:26) D k N ij k + D k N jik ) + Rg ij + 2 ∇ µ α µ g ij + 12 ( ∇ j α i + ∇ i α j ) (3.187) −
12 ( J pj J qi ∇ p α q + J pi J qj ∇ p α q ) − α i α j + α Ji α Jj − α µ α µ g ij + 4 α p ( N jpi + N ipj ) (cid:27) Recall that ˜ g ij = | ϕ | g ij . Therefore ∂ t log det˜ g = | ϕ | − g ij ∂ t ˜ g ij = | ϕ | (cid:8) − ∇ µ α µ − R + 12 | α | (cid:9) . (3.188)Since det˜ g = | ϕ | det g and ∂ t det g = 0 as the volume form of g equals to ω /
3! and ω isfixed, we have ∂ t log | ϕ | = 16 ∂ t log det ˜ g (3.189)Then, we conclude ∂ t log | ϕ | = | ϕ | (cid:8) − ∇ µ α µ − R + 2 | α | (cid:9) (3.190)The flow of g ij = | ϕ | − ˜ g ij is ∂ t g ij = | ϕ | − { ∂ t ˜ g ij − ( ∂ t log | ϕ | ) g ij } . (3.191)Substituting the equations derived above, ∂ t g ij = −| ϕ | (cid:26) D p N ij p + D p N jip ) − ∇ i ∇ j log | ϕ | + J pi J qj ∇ p ∇ q log | ϕ | − α i α j + α Ji α Jj + 4 α p ( N j pi + N ipj ) (cid:27) (3.192)using α i = − ∂ i log | ϕ | . The Ricci curvature of g ij is given by (2.32). Substituting thisinto (3.192), we obtain the flow of g ij as stated in Theorem 1. Q.E.D.34 eferences [1] L. Bedulli and L. Vezzoni, On the stability of the anomaly flow , arXiv:2005.0670, to appear in Math.Res. Lett.[2] R. Bryant,
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