Modular invariance and anomaly cancellation formulas in odd dimension
aa r X i v : . [ m a t h . DG ] M a y Modular invariance and anomaly cancellationformulas in odd dimension
Kefeng Liu; Yong Wang
Abstract
By studying modular invariance properties of some characteristic forms, weget some new anomaly cancellation formulas on (4 r −
1) dimensional manifolds.As an application, we derive some results on divisibilities of the index of Toeplitzoperators on (4 r −
1) dimensional spin manifolds and some congruent formulason characteristic number for (4 r −
1) dimensional spin c manifolds. Keywords:
Modular invariance; cancellation formulas in odd dimension; di-visibilities
MSC:
In 1983, the physicists Alvarez-Gaum´e and Witten [AW] discovered the ”mirac-ulous cancellation” formula for gravitational anomaly which reveals a beautiful re-lation between the top components of the Hirzebruch b L -form and b A -form of a 12-dimensional smooth Riemannian manifold. Kefeng Liu [Li1] established higher di-mensional ”miraculous cancellation” formulas for (8 k + 4)-dimensional Riemannianmanifolds by developing modular invariance properties of characteristic forms. Theseformulas could be used to deduce some divisibility results. In [HZ1], [HZ2], for each(8 k + 4)-dimensional smooth Riemannian manifold, a more general cancellation for-mula that involves a complex line bundle was established. This formula was appliedto spin c manifolds, then an analytic Ochanine congruence formula was derived. In[CH], Qingtao Chen and Fei Han obtained more twisted cancellation formulas for 8 k and 8 k + 4 dimensional manifolds and they also applied their cancellation formulasto study divisibilities on spin manifolds and congruences on spin c manifolds.Another important application of modular invariance properties of characteris-tic forms is to prove the rigidity theorem on elliptic genera. For example, see [Li,2,3], [LM], [LMZ1,2]. For odd dimensional manifolds, we proved the similar rigid-ity theorem for elliptic genera under the condition that fixed point submanifolds are1-dimensional in [LW]. In [HY], Han and Yu dropped off our condition and provedmore general odd dimensional rigidity theorem for elliptic genera. In [HY], in orderto prove the rigidity theorem, they constructed some interesting modular forms underthe condition that the 3th de-Rham cohomology of manifolds vanishes.1n parallel, a natural question is whether we can get some interesting cancella-tion formulas in odd dimension by modular forms constructed in [HY]. In this paper,we will give the confirmative answer of this question. That is, by studying modularinvariance properties of some characteristic forms, we get some new anomaly can-cellation formulas on (4 r −
1) dimensional manifolds. As an application, we derivesome results on divisibilities on (4 r −
1) dimensional spin manifolds and congruenceson (4 r −
1) dimensional spin c manifolds. In [CH1], a cancellation formula on 11-dimensional manifolds was derived. To the authors’ best knowledge, our cancellationformulas appear for the first time for general odd dimensional manifolds.This paper is organized as follows: In Section 2, we review some knowledge oncharacteristic forms and modular forms that we are going to use. In Section 3, weprove some odd dimensional cancellation formulas and we apply them to get someresults on divisibilities on the index of Toeplitz operators on spin manifolds. In Sec-tion 4, we prove some odd cancellation formulas involving a complex line bundle.By these formulas, we get some congruent formulas on characteristic number for oddspin c manifolds. The purpose of this section is to review the necessary knowledge on characteristicforms and modular forms that we are going to use.
Let M be a Riemannian manifold. Let ∇ T M be theassociated Levi-Civita connection on
T M and R T M = ( ∇ T M ) be the curvature of ∇ T M . Let b A ( T M, ∇ T M ) and b L ( T M, ∇ T M ) be the Hirzebruch characteristic formsdefined respectively by (cf. [Zh]) b A ( T M, ∇ T M ) = det √− π R T M sinh( √− π R T M ) , b L ( T M, ∇ T M ) = det √− π R T M tanh( √− π R T M ) . (2 . F , F ′ be two Hermitian vector bundles over M carrying Hermitian connection ∇ F , ∇ F ′ respectively. Let R F = ( ∇ F ) (resp. R F ′ = ( ∇ F ′ ) ) be the curvature of ∇ F (resp. ∇ F ′ ). If we set the formal difference G = F − F ′ , then G carries aninduced Hermitian connection ∇ G in an obvious sense. We define the associatedChern character form asch( G, ∇ G ) = tr " exp( √− π R F ) − tr " exp( √− π R F ′ ) . (2 . t , let ∧ t ( F ) = C | M + tF + t ∧ ( F ) + · · · , S t ( F ) = C | M + tF + t S ( F ) + · · · F , which live in K ( M )[[ t ]] . The following relations between these operations hold, S t ( F ) = 1 ∧ − t ( F ) , ∧ t ( F − F ′ ) = ∧ t ( F ) ∧ t ( F ′ ) . (2 . { ω i } , { ω ′ j } are formal Chern roots for Hermitian vector bundles F, F ′ respectively, then ch( ∧ t ( F )) = Y i (1 + e ω i t ) . (2 . S t ( F )) = 1 Q i (1 − e ω i t ) , ch( ∧ t ( F − F ′ )) = Q i (1 + e ω i t ) Q j (1 + e ω ′ j t ) . (2 . W is a real Euclidean vector bundle over M carrying a Euclidean connection ∇ W , then its complexification W C = W ⊗ C is a complex vector bundle over M car-rying a canonical induced Hermitian metric from that of W , as well as a Hermitianconnection ∇ W C induced from ∇ W . If F is a vector bundle (complex or real) over M , set e F = F − dim F in K ( M ) or KO ( M ). We first recall the four Jacobi theta functions are defined as follows( cf. [Ch]): θ ( v, τ ) = 2 q sin( πv ) ∞ Y j =1 [(1 − q j )(1 − e π √− v q j )(1 − e − π √− v q j )] , (2 . θ ( v, τ ) = 2 q cos( πv ) ∞ Y j =1 [(1 − q j )(1 + e π √− v q j )(1 + e − π √− v q j )] , (2 . θ ( v, τ ) = ∞ Y j =1 [(1 − q j )(1 − e π √− v q j − )(1 − e − π √− v q j − )] , (2 . θ ( v, τ ) = ∞ Y j =1 [(1 − q j )(1 + e π √− v q j − )(1 + e − π √− v q j − )] , (2 . q = e π √− τ with τ ∈ H , the upper half complex plane. Let θ ′ (0 , τ ) = ∂θ ( v, τ ) ∂v | v =0 . (2 . θ ′ (0 , τ ) = πθ (0 , τ ) θ (0 , τ ) θ (0 , τ ) . (2 . SL ( Z ) = ( a bc d ! | a, b, c, d ∈ Z , ad − bc = 1 ) the modular group. Let S = −
11 0 ! , T = ! be the two generators of SL ( Z ). They act on3 by Sτ = − τ , T τ = τ + 1. One has the following transformation laws of thetafunctions under the actions of S and T (cf. [Ch]): θ ( v, τ + 1) = e π √− θ ( v, τ ) , θ ( v, − τ ) = 1 √− (cid:18) τ √− (cid:19) e π √− τv θ ( τ v, τ ); (2 . θ ( v, τ + 1) = e π √− θ ( v, τ ) , θ ( v, − τ ) = (cid:18) τ √− (cid:19) e π √− τv θ ( τ v, τ ); (2 . θ ( v, τ + 1) = θ ( v, τ ) , θ ( v, − τ ) = (cid:18) τ √− (cid:19) e π √− τv θ ( τ v, τ ); (2 . θ ( v, τ + 1) = θ ( v, τ ) , θ ( v, − τ ) = (cid:18) τ √− (cid:19) e π √− τv θ ( τ v, τ ) , (2 . θ ′ ( v, τ + 1) = e π √− θ ′ ( v, τ ) , θ ′ (0 , − τ ) = 1 √− (cid:18) τ √− (cid:19) τ θ ′ (0 , τ ) . (2 . Definition 2.1
A modular form over Γ, a subgroup of SL ( Z ), is a holomorphicfunction f ( τ ) on H such that f ( gτ ) := f (cid:18) aτ + bcτ + d (cid:19) = χ ( g )( cτ + d ) k f ( τ ) , ∀ g = a bc d ! ∈ Γ , (2 . χ : Γ → C ⋆ is a character of Γ. k is called the weight of f .Let Γ (2) = ( a bc d ! ∈ SL ( Z ) | c ≡ ) , Γ (2) = ( a bc d ! ∈ SL ( Z ) | b ≡ ) , be the two modular subgroups of SL ( Z ). It is known that the generators of Γ (2)are T, ST ST , the generators of Γ (2) are ST S, T ST S (cf.[Ch]).If Γ is a modular subgroup, let M R (Γ) denote the ring of modular forms over Γwith real Fourier coefficients. Writing θ j = θ j (0 , τ ) , ≤ j ≤ , we introduce fourexplicit modular forms (cf. [Li1]), δ ( τ ) = 18 ( θ + θ ) , ε ( τ ) = 116 θ θ ,δ ( τ ) = −
18 ( θ + θ ) , ε ( τ ) = 116 θ θ . (2 . q : δ ( τ ) = 14 + 6 q + 6 q · · · , ε ( τ ) = 116 − q + 7 q + · · · ,δ ( τ ) = − − q − q + · · · , ε ( τ ) = q + 8 q + · · · , (2 . · · · ” terms are the higher degree terms, all of which have integral coeffi-cients. They also satisfy the transformation laws, δ ( − τ ) = τ δ ( τ ) , ε ( − τ ) = τ ε ( τ ) . (2 . Lemma 2.2 ([Li1]) δ ( τ ) (resp. ε ( τ ) ) is a modular form of weight (resp. ) over Γ (2) , δ ( τ ) (resp. ε ( τ ) ) is a modular form of weight (resp. ) over Γ (2) andmoreover M R (Γ (2)) = R [ δ ( τ ) , ε ( τ )] . Let M be a (4 r −
1) dimensional Riemannian manifold. SetΘ ( T C M ) = ∞ O n =1 S q n ( g T C M ) ⊗ ∞ O m =1 ∧ q m ( g T C M ) , Θ ( T C M ) = ∞ O n =1 S q n ( g T C M ) ⊗ ∞ O m =1 ∧ − q m − ( g T C M )Θ ( T C M ) = ∞ O n =1 S q n ( g T C M ) ⊗ ∞ O m =1 ∧ q m − ( g T C M ) . (3 . g from M to the generallinear group GL ( N, C ) with N a positive integer (see [Zh]). Let d denote a trivialconnection on C N | M . We will denote by c n ( M, [ g ]) the cohomology class associatedto the closed n -form c n ( C N | M , g, d ) = (cid:18) π √− (cid:19) ( n +1)2 Tr[( g − dg ) n ] . (3 . C N | M , g, d ) associated to g and d by definition isch( C N | M , g, d ) = ∞ X n =1 n !(2 n + 1)! c n +1 ( C N | M , g, d ) . (3 . ∇ u on the trivial bundle C N | M defined by ∇ u = (1 − u ) d + ug − · d · g, u ∈ [0 , . (3 . d ch( C N | M , g, d ) = ch( C N | M , d ) − ch( C N | M , g − · d · g ) . (3 . g : M → SO ( N ) and we assume that N is even and large enough. Let E denote the trivial real vector bundle of rank N over M . We equip E with thecanonical trivial metric and trivial connection d . Set ∇ u = d + ug − dg, u ∈ [0 , . R u be the curvature of ∇ u , then R u = ( u − u )( g − dg ) . (3 . E and g extends to a unitary automorphismof E C . The connection ∇ u extends to a Hermitian connection on E C with curvaturestill given by (3.6). Let △ ( E ) be the spinor bundle of E , which is a trivial Hermitianbundle of rank 2 N . We assume that g has a lift to the Spin group Spin( N ): g △ : M → Spin( N ) . So g △ can be viewed as an automorphism of △ ( E ) preserving theHermitian metric. We lift d on E to be a trivial Hermitian connection d △ on △ ( E ),then ∇ △ u = (1 − u ) d △ + u ( g △ ) − · d △ · g △ , u ∈ [0 ,
1] (3 . ∇ u on E to △ ( E ). Let Q j ( E ), j = 1 , , Q ( E ) = △ ( E ) ⊗ ∞ O n =1 ∧ q n ( g E C ) ,Q ( E ) = ∞ O n =1 ∧ − q n − ( g E C ); Q ( E ) = ∞ O n =1 ∧ q n − ( g E C ) . (3 . g on E have a lift g Q j ( E ) on Q j ( E ) and ∇ u have a lift ∇ Q j ( E ) u on Q j ( E ). Following[HY], we defined ch( Q j ( E ) , g Q j ( E ) , d, τ ) for j = 1 , , Q j ( E ) , ∇ Q j ( E )0 , τ ) − ch( Q j ( E ) , ∇ Q j ( E )1 , τ ) = d ch( Q j ( E ) , g Q j ( E ) , d, τ ) , (3 . Q ( E ) , g Q ( E ) , d, τ ) = − N/ π Z Tr " g − dg θ ′ ( R u / (4 π ) , τ ) θ ( R u / (4 π ) , τ ) du, (3 . j = 2 , Q j ( E ) , g Q j ( E ) , d, τ ) = − π Z Tr " g − dg θ ′ j ( R u / (4 π ) , τ ) θ j ( R u / (4 π ) , τ ) du. (3 . c ( E C , g, d ) = 0, then for any integer l ≥ j = 1 , ,
3, ch( Q j ( E ) , g Q j ( E ) , d, τ ) (4 l − are modular forms of weight 2 l over Γ (2),Γ (2) and Γ θ respectively. Let (see [HY, Def. 2.3])Φ L ( ∇ T M , g, d, τ ) = b L ( T M, ∇ T M )ch(Θ ( T M ) , ∇ Θ ( T M ) , τ )ch( Q ( E ) , g Q ( E ) , d, τ );(3 . W ( ∇ T M , g, d, τ ) = b A ( T M, ∇ T M )ch(Θ ( T M ) , ∇ Θ ( T M ) , τ )ch( Q ( E ) , g Q ( E ) , d, τ );(3 . ′ W ( ∇ T M , g, d, τ ) = b A ( T M, ∇ T M )ch(Θ ( T M ) , ∇ Θ ( T M ) , τ )ch( Q ( E ) , g Q ( E ) , d, τ ) . (3 . c ( E C , g, d ) = 0, thenfor any integer l ≥ j = 1 , ,
3, Φ L ( ∇ T M , g, d, τ ) (4 l − , Φ W ( ∇ T M , g, d, τ ) (4 l − ′ W ( ∇ T M , g, d, τ ) (4 l − are modular forms of weight 2 l over Γ (2), Γ (2) and Γ θ respectively. We have D Φ L ( ∇ T M , g, d, τ ) , [ M ] E = − Ind( T ⊗ △ ( T M ) ⊗ Θ ( T M ) ⊗ ( Q ( E ) , g Q ( E ) )); D Φ W ( ∇ T M , g, d, τ ) , [ M ] E = − Ind( T ⊗ Θ ( T M ) ⊗ ( Q ( E ) , g Q ( E ) )); D Φ ′ W ( ∇ T M , g, d, τ ) , [ M ] E = − Ind( T ⊗ Θ ( T M ) ⊗ ( Q ( E ) , g Q ( E ) )) , (3 . T ⊗ · · · ) denotes the index of the Toeplitz operator. Let {± π √− x j } for1 ≤ j ≤ r − T M ⊗ C . Similar to the computations in [Li1],we haveΦ L ( ∇ T M , g, d, τ ) = 2 r − r − Y j =1 x j θ ′ (0 , τ ) θ ( x j , τ ) θ ( x j , τ ) θ (0 , τ ) ch( Q ( E ) , g Q ( E ) , d, τ ); (3 . W ( ∇ T M , g, d, τ ) = r − Y j =1 x j θ ′ (0 , τ ) θ ( x j , τ ) θ ( x j , τ ) θ (0 , τ ) ch( Q ( E ) , g Q ( E ) , d, τ ); (3 . ′ W ( ∇ T M , g, d, τ ) = r − Y j =1 x j θ ′ (0 , τ ) θ ( x j , τ ) θ ( x j , τ ) θ (0 , τ ) ch( Q ( E ) , g Q ( E ) , d, τ ) . (3 . ( T C M ) ⊗ Q ( E ) and Θ ( T C M ) ⊗ Q ( E ) admit formal Fourier expansionin q as Θ ( T C M ) ⊗ Q ( E ) = A ( T C M, E ) + A ( T C M, E ) q + · · · , Θ ( T C M ) ⊗ Q ( E ) = B ( T C M, E ) + B ( T C M, E ) q + · · · , (3 . A j and B j are elements in the semi-group formally generated by Hermi-tian vector bundles over M . Moreover, they carry canonically induced Hermitianconnections. If B j ( T C M, E ) = B j, ( T C M ) ⊗ B j, ( E ) , we let f ch( B j ( T C M, E )) = ch( B j, ( T C M ))ch( B j, ( E ) , g B j, ( E ) , d ) . If ω is a differential form over M , we denote ω (4 r − its top degree component. Ourmain results include the following theorem. Theorem 3.1 If c ( E, g, d ) = 0 , then n b L ( T M, ∇ T M )ch( △ ( E ) , g △ ( E ) , d ) o (4 r − = 2 r − N [ r ] X l =1 − l h l , (3 . where each h l , ≤ l ≤ [ r ] , is a canonical integral linear combination of n b A ( T M, ∇ T M ) f ch( B j ( T C M, E )) o (4 r − , ≤ j ≤ l and h , h are given by (3.25) and (3.30). Proof.
Similarly to the computations in [Li1, P.35] and by (2.26) in [HY] and thecondition c ( E, g, d ) = 0, we haveΦ W ( ∇ T M , g, d, − τ ) (4 r − = τ r r − N Φ L ( ∇ T M , g, d, τ ) (4 r − . (3 . W ( ∇ T M , g, d, τ ) (4 r − is a modular form of weight 2 r over Γ (2). By Lemma 2.2,we haveΦ W ( ∇ T M , g, d, τ ) (4 r − = h (8 δ ) r + h (8 δ ) r − ε + · · · + h [ r ] (8 δ ) r − r ] ε [ r ]2 , (3 . h l , ≤ l ≤ [ r ] , is a real multiple of the volume form at x . By (2.20)(3.21) and (3.22), we getΦ L ( ∇ T M , g, d, τ ) (4 r − = 2 r − N (cid:20) h (8 δ ) r + h (8 δ ) r − ε + · · · + h [ r ] (8 δ ) r − r ] ε [ r ]1 (cid:21) . (3 . q j , j ≥ h = 0 and each h l ≤ l ≤ [ r ], can be expressed through a canonicalintegral linear combination of n b A ( T M, ∇ T M ) f ch( B j ( T C M, E )) o (4 r − , ≤ j ≤ l . Direct computations shows thatΘ ( T C M ) ⊗ Q ( E ) = 1 − ( g T C M + g E C ) q +( g T C M + ∧ g T C M + ∧ g E C + g T C M ⊗ g E C ) q + · · · . (3 . q of (3.22), we get h = ( − r − n b A ( T M, ∇ T M ) f ch( B ( T C M, E )) o (4 r − = ( − r − n b A ( T M, ∇ T M )ch(
E, g, d ) o (4 r − . (3 . q of (3.22), we get h = ( − r − n b A ( T M, ∇ T M ) f ch( B ( T C M, E )) o (4 r − + [ − − r ( r − h . (3 . f ch( B ( T C M, E )) = ch( ∧ g E C , g, d ) + ch( g T C M )ch( E, g, d ); (3 . ∧ g E C = S ( C N ) + ∧ E C − E C ⊗ C N ; (3 . ∧ g E C , g, d ) = ch( ∧ E C , g, d ) − N ch( E C , g, d ) . (3 . h = ( − r − h b A ( T M, ∇ T M )ch( T C M )ch( E, g, d ) + b A ( T M, ∇ T M )ch( ∧ E, g, d ) i (4 r − +[( − r − (7 − N − r ) − r −
2) + 8( − r ] h b A ( T M, ∇ T M )ch(
E, g, d ) i (4 r − . (3 . ✷ Corollary 3.2
Let M be a (4 r − -dimensional spin manifold and c ( E, g, d ) = 0 ,then
Ind( T ⊗ △ T M ⊗ ( △ ( E ) , g △ ( E ) )) = − r − N [ r ] X l =1 − l h l , (3 . where each h l , ≤ l ≤ [ r ] , is a canonical integral linear combination of Ind( T ⊗ ( B j ( T C M, E ))) . Corollary 3.3
Let M be a (4 r − -dimensional spin manifold and c ( E, g, d ) = 0 .If r is even, then Ind( T ⊗ △ T M ⊗ ( △ ( E ) , g △ ( E ) )) ≡ N − ) . If r is odd, then Ind( T ⊗ △ T M ⊗ ( △ ( E ) , g △ ( E ) )) ≡ N +2 ) . Corollary 3.4
Let M be a -dimensional manifold and c ( E, g, d ) = 0 . Then wehave n b L ( T M, ∇ T M )ch( △ ( E ) , g △ ( E ) , d ) o (11) = 2 N +2 n b A ( T M, ∇ T M )ch(
E, g, d ) o (11) . (3 . Corollary 3.5
Let M be a -dimensional manifold and c ( E, g, d ) = 0 . Then wehave n b L ( T M, ∇ T M )ch( △ ( E ) , g △ ( E ) , d ) o (15) = 2 N − h − (113 + N ) b A ( T M, ∇ T M )ch(
E, g, d )+ b A ( T M, ∇ T M )ch( T C M )ch( E, g, d ) + b A ( T M, ∇ T M )ch( ∧ E, g, d ) i (15) . (3 . ( T M ) = 2 + 2( T C M − r + 1) q + 2[( − r + 3) T C M + T C M ⊗ T C M + (4 r − r − q + · · · . (3 . Q ( E ) , g Q ( E ) , d ) = ch( △ ( E ) ⊗ ( E C q + ((2 − r ) E C + ∧ E C ) q + · · · ) , g, d ) . (3 . ( T M ) ⊗ Q ( E ) = 2 △ ( E ) ⊗ E C q + h △ ( E ) ⊗ ((2 − r ) E C + ∧ E C )+2( T C M − r + 1) ⊗ △ ( E ) ⊗ E C ] q + O ( q ) . (3 . ≤ l ≤ [ r ](8 δ ) r − l ε l = 2 r − l h r − l ) q + (288 r − rl +2048 l + 512 l − r ) q + O ( q ) i . (3 . q in (3.23), we get Theorem 3.6 If c ( E, g, d ) = 0 , then n b L ( T M, ∇ T M )ch( △ ( E ) ⊗ E C , g, d ) − r b L ( T M, ∇ T M )ch( △ ( E ) , g △ ( E ) , d ) o (4 r − = − r +4+ N [ r ] X l =1 − l lh l . (3 . Corollary 3.7
Let M be a (4 r − -dimensional spin manifold and c ( E, g, d ) = 0 .If r is even, then Ind( T ⊗△ ( T M ) ⊗ ( △ ( E ) ⊗ E C , g )) − r Ind( T ⊗△ ( T M ) ⊗ ( △ ( E ) , g )) ≡ N +4 ) . (3 . If r is odd, then Ind( T ⊗△ ( T M ) ⊗ ( △ ( E ) ⊗ E C , g )) − r Ind( T ⊗△ ( T M ) ⊗ ( △ ( E ) , g )) ≡ N +7 ) . (3 . q in (3.23), we get Theorem 3.8 If c ( E, g, d ) = 0 , then n b L ( T M, ∇ T M ) h (19 − r )ch( △ ( E ) ⊗ E C , g, d ) − r − r )ch( △ ( E ) , g, d )+ch( T C M )ch( △ ( E ) ⊗ E C , g, d ) + ch( △ ( E ) ⊗ ∧ E C , g, d ) io (4 r − = 2 r +9+ N [ r ] X l =1 − l l h l . (3 . orollary 3.9 Let M be a (4 r − -dimensional spin manifold and c ( E, g, d ) = 0 .Write A for the index of the Toeplitz operator determined by the left hand of (3.38).If r is even, then A ≡ N +9 ) . If r is odd, then A ≡ N +12 ) . Corollary 3.10 If dim M = 11 and c ( E, g, d ) = 0 , then n b L ( T M, ∇ T M )ch( △ ( E ) ⊗ E C , g, d ) − b L ( T M, ∇ T M )ch( △ ( E ) , g, d )+2 N b A ( T M, ∇ T M )ch(
E, g, d ) o (11) = 0 . (3 . Corollary 3.11 If dim M = 11 and c ( E, g, d ) = 0 , then n b L ( T M, ∇ T M ) [ − △ ( E ) ⊗ E C , g, d ) − △ ( E ) , g, d )+ch( T C M )ch( △ ( E ) ⊗ E C , g, d ) + ch( △ ( E ) ⊗ ∧ E C , g, d ) io (11) = 2 N [ b A ( T M, ∇ T M )ch(
E, g, d )] (11) . (3 . Let M be a 4 r − ξ be a rank two realoriented Euclidean vector bundle over M carrying with a Euclidean connection ∇ ξ .SetΘ ( T C M, ξ C ) = ∞ O n =1 S q n ( g T C M ) ⊗ ∞ O m =1 ∧ q m ( g T C M − f ξ C ) ⊗ ∞ O r =1 ∧ q r − ( f ξ C ) ⊗ ∞ O s =1 ∧ − q s − ( f ξ C ) , Θ ( T C M, ξ C ) = ∞ O n =1 S q n ( g T C M ) ⊗ ∞ O m =1 ∧ − q m − ( g T C M − f ξ C ) ⊗ ∞ O r =1 ∧ q r − ( f ξ C ) ⊗ ∞ O s =1 ∧ q s ( f ξ C ) , Θ ( T C M, ξ C ) = ∞ O n =1 S q n ( g T C M ) ⊗ ∞ O m =1 ∧ q m − ( g T C M − f ξ C ) ⊗ ∞ O r =1 ∧ q r ( f ξ C ) ⊗ ∞ O s =1 ∧ − q s − ( f ξ C ) . (4 . c = e ( ξ, ∇ ξ ) be the Euler form of ξ canonically associated to ∇ ξ . SetΦ L ( ∇ T M , ∇ ξ , g, d, τ ) = b L ( T M, ∇ T M )cosh ( c ) ch(Θ ( T C M, ξ C ) , ∇ Θ ( T C M,ξ C ) ) · ch( Q ( E ) , g Q ( E ) , d, τ ) , Φ W ( ∇ T M , ∇ ξ , d, g, τ ) = b A ( T M, ∇ T M )cosh( c ( T C M, ξ C ) , ∇ Θ ( T C M,ξ C ) )11 ch( Q ( E ) , g Q ( E ) , d, τ ) , Φ ′ W ( ∇ T M , ∇ ξ , d, g, τ ) = b A ( T M, ∇ T M )cosh( c ( T C M, ξ C ) , ∇ Θ ( T C M,ξ C ) ) · ch( Q ( E ) , g Q ( E ) , d, τ ) . (4 . {± π √− x j | ≤ j ≤ r − } and {± π √− u } be the Chern roots of T C M and ξ C respectively and c = 2 π √− u. Through direct computations, we get (cf. [HZ2])Φ L ( ∇ T M , ∇ ξ , g, d, τ ) = 2 r − r − Y j =1 x j θ ′ (0 , τ ) θ ( x j , τ ) θ ( x j , τ ) θ (0 , τ ) θ (0 , τ ) θ ( u, τ ) θ ( u, τ ) θ (0 , τ ) θ ( u, τ ) θ (0 , τ ) · ch( Q ( E ) , g Q ( E ) , d, τ ); (4 . W ( ∇ T M , ∇ ξ , τ ) = r − Y j =1 x j θ ′ (0 , τ ) θ ( x j , τ ) θ ( x j , τ ) θ (0 , τ ) θ (0 , τ ) θ ( u, τ ) θ ( u, τ ) θ (0 , τ ) θ ( u, τ ) θ (0 , τ ) · ch( Q ( E ) , g Q ( E ) , d, τ ); (4 . ′ W ( ∇ T M , ∇ ξ , τ ) = r − Y j =1 x j θ ′ (0 , τ ) θ ( x j , τ ) θ ( x j , τ ) θ (0 , τ ) θ (0 , τ ) θ ( u, τ ) θ ( u, τ ) θ (0 , τ ) θ ( u, τ ) θ (0 , τ ) · ch( Q ( E ) , g Q ( E ) , d, τ ) . (4 . W ( ∇ T M , ∇ ξ , g, d, − τ ) (4 r − = τ r r − N Φ L ( ∇ T M , ∇ ξ , g, d, τ ) (4 r − . (4 . Proposition 4.1 If c ( E C , g, d ) = 0 , then for any integer l ≥ and j = 1 , , , Φ L ( ∇ T M , ∇ ξ , g, d, τ ) (4 l − , Φ W ( ∇ T M , ∇ ξ , g, d, τ ) (4 l − and Φ ′ W ( ∇ T M , ∇ ξ , g, d, τ ) (4 l − are modular forms of weight l over Γ (2) , Γ (2) and Γ θ respectively. We know that (3.22) and (3.23) hold in the twisted case. We define B j ( T C M, ξ C , E )similarly to B j ( T C M, E ). Similarly to Theorem 3.1, we have
Theorem 4.2 If c ( E, g, d ) = 0 , then ( b L ( T M, ∇ T M )cosh ( c ) ch( △ ( E ) , g △ ( E ) , d ) ) (4 r − = 2 r − N [ r ] X l =1 − l e h l , (4 . here each e h l , ≤ l ≤ [ r ] , is a canonical integral linear combination of (cid:26) b A ( T M, ∇ T M )cosh( c f ch( B j ( T C M, ξ C , E )) (cid:27) (4 r − , ≤ j ≤ l . By the definition, we haveΘ ( T M, ξ C ) ⊗ Q ( E )= 1 + (3 f ξ C − g T C M − g E C ) q + h − f ξ C ⊗ ( g T C M + g E C )+( g T C M + ∧ g T C M + ∧ g E C + g T C M ⊗ g E C )+(3 f ξ C ⊗ f ξ C + 2 S ( f ξ C ) + ∧ ( f ξ C ) + f ξ C ) i q + · · · . (4 . q and q in (3.22), we get f h = ( − r − (cid:26) b A ( T M, ∇ T M )cosh c E, g, d ) (cid:27) (4 r − . (4 . f h = ( − r − (cid:20) b A ( T M, ∇ T M )ch( T C M − ξ C )ch( E, g, d )cosh c b A ( T M, ∇ T M )ch( ∧ E, g, d )cosh c (cid:21) (4 r − +[( − r − (7 − N − r ) − r −
2) + 8( − r ] · (cid:20) b A ( T M, ∇ T M )ch(
E, g, d )cosh c (cid:21) (4 r − . (4 . Corollary 4.3
Let M be a (4 r − -dimensional spin c manifold and c ( E, g, d ) = 0 .If r is even, then − N * b L ( T M, ∇ T M )cosh ( c ) ch( △ ( E ) , g △ ( E ) , d ) , [ M ] + ≡ e h r (mod 64 Z ) . (4 . If r is odd, then − − N * b L ( T M, ∇ T M )cosh ( c ) ch( △ ( E ) , g △ ( E ) , d ) , [ M ] + ≡ e h r − (mod 64 Z ) . (4 . Corollary 4.4
Let M be a -dimensional manifold and c ( E, g, d ) = 0 . Then wehave ( b L ( T M, ∇ T M )cosh ( c ) ch( △ ( E ) , g △ ( E ) , d ) ) (11) = 2 N +2 (cid:26) b A ( T M, ∇ T M )ch(
E, g, d )cosh c (cid:27) (11) . (4 . orollary 4.5 Let M be a -dimensional manifold and c ( E, g, d ) = 0 . Then wehave ( b L ( T M, ∇ T M )cosh ( c ) ch( △ ( E ) , g △ ( E ) , d ) ) (15) = 2 N − (cid:20) − (113 + N ) b A ( T M, ∇ T M )cosh c E, g, d )+ b A ( T M, ∇ T M )ch( T C M − ξ C )ch( E, g, d )cosh c b A ( T M, ∇ T M )ch( ∧ E, g, d )cosh c (cid:21) (15) . (4 . f ch(Θ ( T M ) ⊗ Q ( E )) = 2ch( △ ( E ) ⊗ E C , g, d ) q + O ( q ) . (4 . q in (3.23), we get Theorem 4.6 If c ( E, g, d ) = 0 , then ( b L ( T M, ∇ T M )cosh ( c ) ch( △ ( E ) ⊗ E C , g, d ) − r b L ( T M, ∇ T M )cosh ( c ) ch( △ ( E ) , g △ ( E ) , d ) ) (4 r − = − r +4+ N [ r ] X l =1 − l l e h l . (4 . Corollary 4.7
Let M be a (4 r − -dimensional spin c manifold and c ( E, g, d ) = 0 .If r is even, then − − N * b L ( T M, ∇ T M )cosh ( c ) ch( △ ( E ) ⊗ E C , g, d ) − r b L ( T M, ∇ T M )cosh ( c ) ch( △ ( E ) , g △ ( E ) , d ) , [ M ] + ≡ − r e h r (mod 64 Z ) . (4 . If r is odd, then − − N * b L ( T M, ∇ T M )cosh ( c ) ch( △ ( E ) ⊗ E C , g, d ) − r b L ( T M, ∇ T M )cosh ( c ) ch( △ ( E ) , g △ ( E ) , d ) , [ M ] + ≡ − r − e h r − (mod 64 Z ) . (4 . orollary 4.8 If dim M = 11 and c ( E, g, d ) = 0 , then ( b L ( T M, ∇ T M )cosh ( c ) ch( △ ( E ) ⊗ E C , g, d ) − b L ( T M, ∇ T M )cosh ( c ) ch( △ ( E ) , g, d )+2 N b A ( T M, ∇ T M )ch(
E, g, d )cosh c (cid:27) (11) = 0 . (4 . Remark.
1. By the Dai-Zhang family index theorem associated to the Toeplitzoperator in [DZ], similarly to [HL], we can easily extend our anomaly cancellationformulas to the family case.2. Similarly to [Li1], [HLZ] and [LW1], we also extend our anomaly cancellationformulas to these cases.
Acknowledgement
The work of the first author was supported by NSF. The workof the second author was supported by NSFC. 11271062 and NCET-13-0721. Theauthors are indebted to Dr. F. Han for helpful comments.
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NankaiTracks in Mathematics Vol. 4, World Scientific, Singapore, 2001.Center of Mathematical Sciences, Zhejiang University Hangzhou Zhejiang 310027,China and Department of Mathematics, University of California at Los Angeles, LosAngeles CA 90095-1555, USAEmail: [email protected]; [email protected]
School of Mathematics and Statistics, Northeast Normal University, ChangchunJilin, 130024, ChinaE-mail: [email protected]@nenu.edu.cn