aa r X i v : . [ m a t h . DG ] J a n An equivariant Kastler-Kalau-Walze typetheorem
Yong Wang
Abstract
In this paper, we prove an equivariant Kastler-Kalau-Walze type theorem forspin manifolds without boundary. For 6 dimensional spin manifolds with bound-ary, we also give an equivariant Kastler-Kalau-Walze type theorem. Then wegeneralize this theorem to the general n dimensional manifold. An equivariantKastler-Kalau-Walze type theorem with torsion is also proved. MSC:
Keywords:
Group action, Bismut Laplacian, noncommutative residue
The noncommutative residue found in [Gu] and [Wo] plays a prominent role innoncommutative geometry. Several years ago, Connes made a challenging observationthat the noncommutative residue of the square of the inverse of the Dirac operator wasproportional to the Einstein-Hilbert action, which we call the Kastler-Kalau-Walzetheorem. In [K], Kastler gave a brute-force proof of this theorem. In [KW], Kalauand Walze proved this theorem in the normal coordinates system simultaneously. In[A], Ackermann gave a note on a new proof of this theorem by means of the heatkernel expansion.On the other hand, Fedosov et al defined a noncommutative residue on Boutet deMonvel’s algebra and proved that it was a unique continuous trace in [FGLS]. In [Wa1]and [Wa2], we generalized some results in [C1] and [U] to the case of manifolds withboundary . In [Wa3], We proved a Kastler-Kalau-Walze type theorem for the Diracoperator and the signature operator for 3 , n dimensional manifold. An equivariant Kastler-Kalau-Walzetype theorem with torsion is also proved. In Section 2, we give an equivariant Kastler-Kalau-Walze type theorem. In Section 3, For 6 dimensional spin manifolds withboundary, we also give an equivariant Kastler-Kalau-Walze type theorem. In Section4, we prove an equivariant Kastler-Kalau-Walze type theorem for a n dimensionalspin manifold with boundary. In Section 5, an equivariant Kastler-Kalau-Walze typetheorem with torsion is given. Let M be a smooth compact Riemannian n -dimensional manifold without boundaryand V be a vector bundle on M . Recall that a differential operator P is of Laplacetype if it has locally the form P = − ( g ij ∂ i ∂ j + A i ∂ i + B ) , (2 . ∂ i is a natural local frame on T M and g i,j = g ( ∂ i , ∂ j ) and ( g ij ) ≤ i,j ≤ m is theinverse matrix associated to the metric matrix ( g i,j ) ≤ i,j ≤ m on M , and A i and B aresmooth sections of End( V ) on M (endomorphism). If P is a Laplace type operatorof the form (2.1), then (see [Gi]) there is a unique connection ∇ on V and an uniqueendomorphism E such that P = − [ g ij ( ∇ ∂ i ∇ ∂ j − ∇ ∇ L∂i ∂ j ) + E ] , (2 . ∇ L denotes the Levi-civita connection on M . Moreover (with local frames of T ∗ M and V ), ∇ ∂ i = ∂ i + ω i and E are related to g ij , A i and B through ω i = 12 g ij ( A j + g kl Γ jkl Id) , (2 . E = B − g ij ( ∂ i ( ω j ) + ω i ω j − ω k Γ kij ) , (2 . kij are the Christoffel coefficients of ∇ L . Now we let M be a m -dimensional oriented spin manifold with Riemannian met-ric g . We recall that the Dirac operator D is locally given as follows in terms oforthonormal frames e i , ≤ i ≤ n and natural frames ∂ i of T M : one has D = X i,j g ij c ( ∂ i ) ∇ S∂ j = X i c ( e i ) ∇ Se i , (2 . c ( e i ) denotes the Clifford action which satisfies the relation c ( e i ) c ( e j ) + c ( e j ) c ( e i ) = − δ ji , and ∇ S∂ i = ∂ i + σ i , σ i = 14 X j,k (cid:10) ∇ L∂ i e j , e k (cid:11) c ( e j ) c ( e k ) . (2 . ∂ j = g ij ∂ i , σ i = g ij σ j , Γ k = g ij Γ kij . (2 . D = − g ij ∂ i ∂ j − σ j ∂ j + Γ k ∂ k − g ij [ ∂ i ( σ j ) + σ i σ j − Γ kij σ k ] + 14 r, (2 . r is the scalar curvature.Let a compact group G act isometrically on M and preserve the spin structure.This action generates a Killing vector field X . Let L X be the Lie derivation onthe Spinors bundle. The Levi-Civita connection ∇ L lifts a Clifford connection ∇ S .Following Bismut, we define the equivariant Bismut Laplacian. H X = ( D + 14 c ( X )) + L X ; L X = ∇ SX + µ ( X ) , (2 . H X = D + 14 Dc ( X ) + 14 c ( X ) D + ∇ SX + µ ( X ) − | X | . (2 . X = X j ∂ j . So L X = X j ∂ j + X j σ j + µ ( X ) . Then by (10) in [Wa5], we have H X = − g ij ∂ i ∂ j + [ X j − σ j + Γ j + 14 c ( ∂ j ) c ( X ) + 14 c ( X ) c ( ∂ j )] ∂ j + g ij [ − ∂ i ( σ j ) − σ i σ j + Γ kij σ k + 14 c ( ∂ i ) ∂ j ( c ( X )) + 14 c ( ∂ i ) σ j c ( X )+ 14 c ( X ) c ( ∂ i ) σ j ] + 14 r − | X | + 14 X j σ j + µ ( X ) . (2 . D + 14 c ( X )) = − [ g ij ( ∇ ∂ i ∇ ∂ j − ∇ ∇ L∂i ∂ j ) + E ] , (2 . D + 14 c ( X )) + L X = − [ g ij ( e ∇ ∂ i e ∇ ∂ j − e ∇ ∇ L∂i ∂ j ) + e E ] , (2 . e ω i = ω i + 12 g ij X j , e A j = A j + X j , e B = B + 14 X j σ j + µ ( X ) . (2 . e E = E + 14 X j σ j + µ ( X ) − g ij [ ∂ ( 12 g jk X k )+ 12 g il X l ω j + 14 g ik g jl X k X l − g kl X l Γ kij ] . (2 . E = − r + 116 | X | + 12 [ e j ( 14 c ( X )) c ( e j ) − c ( e j ) e j ( 14 c ( X ))] + X i < e i , X > . (2 . e ∇ Y = ∇ Y + 34 g ( X, Y ) . (2 . σ j = 0 , tr σ j = 0 and direct computations, we gettr( r e E ) = dim( S ( T M )) ( − r + 116 | X | + X i < e i , X > − g ij (cid:20) ∂ i ( 12 g jk X k ) + 12 g il X l [ 12 g jα (Γ α + g kl Γ αkl )] (cid:21) − (cid:10) ∂ j , X (cid:11) g ij + 14 g ik g jl X k X l − g kl X l Γ kij (cid:27) + tr( µ ( X )) . (2 . M = m . By [A], we know thatWres( H − m +1 X ) = m − π ) m Γ( m ) tr( r e E ) , (2 . Theorem 1
The following equality holds
Wres( H − m +1 X ) = m − π ) m Γ( m ) Z M " dim( S ( T M )) ( − r + 116 | X | + X i < e i , X > − g ij (cid:20) ∂ i ( 12 g jk X k ) + 12 g il X l [ 12 g jα (Γ α + g kl Γ αkl )] (cid:21) − (cid:10) ∂ j , X (cid:11) g ij + 14 g ik g jl X k X l − g kl X l Γ kij (cid:27) + tr( µ ( X )) (cid:21) . (2 . -dimensional man-ifolds with boundary Let M be a 6-dimensional compact oriented spin manifold with boundary ∂M . Weassume that the metric g M on M has the following form near the boundary, g M = 1 h ( x n ) g ∂M + dx n , (3.1)where g ∂M is the metric on ∂M . Let a compact group G act isometrically on M andpreserve the spin structure. Near the boundary, the group action is not necessarilyproduct action. Let ] Wres denote the noncommutative residue for manifolds withboundary (see [FGLS]). In the following, we want to compute ] Wres[ π + H − X ◦ π + H − X ] . By the definition of the noncommutative residue for manifolds with boundary, we have ] Wres[ π + H − X ◦ π + H − X ] = Z M Z | ξ | =1 trace S ( T M ) [ σ − ( H − X )] σ ( ξ ) dx + Z ∂M Φ , (3 . Z | ξ ′ | =1 Z + ∞−∞ ∞ X j,k =0 X ( − i ) | α | + j + k +1 α !( j + k + 1)! × trace S ( T M ) [ ∂ jx n ∂ αξ ′ ∂ kξ n σ + r ( H − X )( x ′ , , ξ ′ , ξ n ) × ∂ αx ′ ∂ j +1 ξ n ∂ kx n σ l ( H − X )( x ′ , , ξ ′ , ξ n )] dξ n σ ( ξ ′ ) dx ′ , (3 . r − k − | α | + l − j − − , r ≤ − , l ≤ −
2. Interiorterm comes from Theorem 1. We only compute the boundary term. By (2.11), wehave the symbol σ − ( H X ) and σ − ( H X ) of H X By the symbol composition formula,we have 1 = X | α | =0 α ! ∂ αξ ( σ ( H X )) D αx ( σ ( H − X ) , (3 . D αx = ( − i ) | α | ∂ αx . Let σ ( H X ) = p + p + p , σ ( H − X ) = r − + r − + r − + · · · , then we can get r − ( H − X ) = | ξ | − ; p r − + p r − + X j ∂ ξ j D x j r − = 0 . (3 . r − ( H − X ) = r − ( D − ) − √− | ξ | ( X j − < X, ∂ j > ) ξ j , (3 . r − ( D − ) in Lemma 1 in [Wa4].Now we can compute Φ, since the sum is taken over − r − l + k + j + | α | = − , r, l ≤− , then we have the same five cases as in [Wa4]. By r − ( D − ) = r − ( H − X ), we knowthat our cases 1 (i) (ii) (iii) equal cases (i) (ii) (iii) in [Wa4]. So the sum of thesethree cases is zero by [Wa4]. case II) r = − , l = − k = j = | α | = 0 . By (3.2) and an integration by parts and (19) in [Wa4], we getcase II ) = −√− Z | ξ ′ | =1 Z + ∞−∞ trace[ π + ξ n σ − ( H − X ) × ∂ ξ n σ − ( H − X )]( x ) dξ n σ ( ξ ′ ) dx ′ = √− Z | ξ ′ | =1 Z + ∞−∞ trace { ∂ ξ n π + ξ n σ − ( H − X ) × [ σ − ( D − ) −√− | ξ | ( X j − < X, ∂ j > ) ξ j ] } ( x ) dξ n σ ( ξ ′ ) dx ′ . )= case II+ √− Z | ξ ′ | =1 Z + ∞−∞ tr i ξ n − i ) ( −√− | ξ | ( X j − < X, ∂ j > ) ξ j ]( x ) dξ n σ ( ξ ′ ) dx ′ (3 . R | ξ ′ | =1 ξ j σ ( ξ ′ ) = 0 for j < m and the metric on M andthe Cauchy integral formula, we get √− Z | ξ ′ | =1 Z + ∞−∞ i ξ n − i ) ( −√− | ξ | ( X j − < X, ∂ j > ) ξ j ]( x ) dξ n σ ( ξ ′ ) dx ′ −
132 Ω X m Vol ∂M (3 . is the canonical volume of the sphere S . case III ) r = − , l = − , k = j = | α | = 0case III ) = − i Z | ξ ′ | =1 Z + ∞−∞ trace[ π + ξ n σ − ( H − X ) × ∂ ξ n σ − ( H − X )]( x ) dξ n σ ( ξ ′ ) dx ′ . (3 . A = √− | ξ | ( X j − < X, ∂ j > ) ξ j . Socase III ) = case
III ) + √− Z | ξ ′ | =1 Z + ∞−∞ trace[ π + ξ n A∂ ξ n σ − ( H − X )]( x ) dξ n σ ( ξ ′ ) dx ′ . (3 . III ) = case
III ) + 132 Ω X m Vol ∂M (3 . II ) + case III ) = 0, so by (3.6) (3.7) and (3.10), we getΦ = 0 So we get the following theorem
Theorem 2
For dimensional spin manifolds with boundary, the following equalityholds ] Wres[ π + H − X ◦ π + H − X ] = 14 π Z M "( − r + 116 | X | + X i < e i , X > − g ij (cid:20) ∂ i ( 12 g jk X k ) + 12 g il X l [ 12 g jα (Γ α + g kl Γ αkl )] (cid:21) − (cid:10) ∂ j , X (cid:11) g ij + 14 g ik g jl X k X l − g kl X l Γ kij (cid:27) + tr( µ ( X )) (cid:21) . (3 . Remark.
We may extend this theorem to the higher dimensional case.In order to get the nonzero boundary term, we use a function f on M to perturb ] Wres[ π + H − X ◦ π + H − X ]. We find only the term case I ii) change and other terms doesnot change. We get Theorem 3
For dimensional spin manifolds with boundary, the following equalityholds ] Wres[ π + f H − X ◦ π + H − X ] = 14 π Z M f "( − r + 116 | X | + X i < e i , X > − g ij (cid:20) ∂ i ( 12 g jk X k ) + 12 g il X l [ 12 g jα (Γ α + g kl Γ αkl )] (cid:21) (cid:10) ∂ j , X (cid:11) g ij + 14 g ik g jl X k X l − g kl X l Γ kij (cid:27) + tr( µ ( X )) (cid:21) − π Ω Z ∂ M ∂ x n f | x n =0 d Vol ∂ M . (3 . Let M be a n = n + 2-dimensional compact oriented spin manifold with boundary ∂M and n is an even integer. We will compute ] Wres[ π + H − X ◦ π + H − n +1 X ]. By thedefinition of ] Wres (see [FGLS]), we only compute the termΦ ′ = Z | ξ ′ | =1 Z + ∞−∞ ∞ X j,k =0 X ( − i ) | α | + j + k +1 α !( j + k + 1)! × trace S ( T M ) [ ∂ jx n ∂ αξ ′ ∂ kξ n σ + r ( H − X )( x ′ , , ξ ′ , ξ n ) × ∂ αx ′ ∂ j +1 ξ n ∂ kx n σ l ( H − n +1 )( x ′ , , ξ ′ , ξ n )] dξ n σ ( ξ ′ ) dx ′ , (4 . r − k − | α | + l − j − − n, r ≤ − , l ≤ − n . Similarto Section 3 and [WW], we divide Φ ′ into five terms and the first three terms havethe same expressions with the three term in [WW].case II : r = − , l = 1 − n, k = j = | α | = 0case II ) = −√− Z | ξ ′ | =1 Z + ∞−∞ trace[ ∂ ξ n π + ξ n σ − ( H − X ) × σ − n ( H − n +1 )]( x ) dξ n σ ( ξ ′ ) dx ′ (4 , σ − n ( H − n +1 X ) = n − σ − n +22 σ − ( H − X ) − √− n − X k =0 ∂ ξ µ σ − n + k +2 ∂ x µ σ − ( σ − ) k . (4 . σ − n ( H − n +1 X ) = σ − n ( D − n +2 ) − √− | ξ | − ( X j − < X, ∂ j > ) ξ j n − σ − n +2 , (4 . II = case II + √− Z | ξ ′ | =1 Z + ∞−∞ trace (cid:20) ξ n − i ) ×| ξ | − ( X j − < X, ∂ j > ) ξ j n − | ξ | − n +4 (cid:21) σ ( ξ ′ ) dx ′ := case II + A, (4 . R | ξ ′ | =1 ξ j σ ( ξ ′ ) = 0 for j < m and the metric on M and the Cauchy integral formula,we get A = (2 − n )2 n − ( n + 1)! X n Ω( S n ) " ξ n ( ξ n + i ) n ( n +1) | ξ n = i . (4 . caseIII, r = − , l = 2 − n, k = j = | α | = 0case III ) = − i Z | ξ ′ | =1 Z + ∞−∞ trace[ π + ξ n σ − ( H − X ) × ∂ ξ n σ − n ( H − n X ))]( x ) dξ n σ ( ξ ′ ) dx ′ . (4 . III = case
III − √− Z | ξ ′ | =1 Z + ∞−∞ trace[ π + ξ n ( −√− | ξ | − ( X j − < X, ∂ j > ) ξ j ) × ∂ ξ n σ − n ( H − n X )]( x ) dξ n σ ( ξ ′ ) dx ′ := case III + B (4 . B = n − n + 1)! 2 n +1 X n Ω( S n ) " ξ n ( ξ n + i ) n ( n +1) | ξ n = i . (4 . ′ = Φ + A + B = Φ + 3 n − n + 1)! 2 n − X n Ω( S n ) " ξ n ( ξ n + i ) n ( n +1) | ξ n = i . (4 . Theorem 4
The following equality holds ] Wres[ π + H − X ◦ π + H − n +1 X ] = n − π ) n Γ( n ) 2 n Z M "( − r + 116 | X | + X i < e i , X > − g ij (cid:20) ∂ i ( 12 g jk X k ) + 12 g il X l [ 12 g jα (Γ α + g kl Γ αkl )] (cid:21) − (cid:10) ∂ j , X (cid:11) g ij + 14 g ik g jl X k X l − g kl X l Γ kij (cid:27) + tr( µ ( X )) (cid:21) − n − n + 1 πi ( n + 2)! 2 n − A Ω( S n ) Z ∂ M K dvol ∂M + 3 n − n + 1)! 2 n − Ω( S n ) " ξ n ( ξ n + i ) n ( n +1) | ξ n = i Z ∂ M X n dvol ∂M , (4 . here K is an extrinsic curvature and A is a constant (see (3.42) in [WW]). Let T be a real three form on M . Let D T = D + T where T denotes the three forminduces the Clifford action. Then D T is self adjoint operator and D T = △ + 14 r + 32 dT − || T || , (5 . H TX = ( D T + c ( X )) + L X . So H TX = H X + 32 dT − || T || + 14 T c ( X ) + 14 c ( X ) T, (5 . E H TX = E H X + 32 dT − || T || + 14 T c ( X ) + 14 c ( X ) T, (5 . Theorem 5
The following equality holds
Wres(( H TX ) − m +1 ) = m − π ) m Γ( m ) 2 m Z M "( − r + 116 | X | + X i < e i , X > − g ij (cid:20) ∂ i ( 12 g jk X k ) + 12 g il X l [ 12 g jα (Γ α + g kl Γ αkl )] (cid:21) − (cid:10) ∂ j , X (cid:11) g ij + 14 g ik g jl X k X l − g kl X l Γ kij (cid:27) + tr( µ ( X ) + 32 dT − || T || + 12 T c ( X )) (cid:21) . (5 . Theorem 6
Let dT = 0 and i X T = 0 and X is small, we have lim t → str[exp( − tH TX )( x, x )] d vol M = (2 π √− − n b A ( F − Tg ( X )) , (5 . F − Tg ( X ) = R − T + µ ( X ) and µ ( X ) is the moment map.9 cknowledgement: The work of the author was supported by NSFC. 11271062and NCET-13-0721.
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