CHERN-WEIL FORMS ASSOCIATED WITH SUPERCONNECTIONS

Transcription

1 CHERN-WEIL FORMS ASSOCIATED WITH SUPERCONNECTIONS S. PAYCHA Laboratoire de Mathématiques Complexe des Cézeaux Université Blaise Pascal Aubière Cedex F sylvie.paycha@math.univ-bpclermont.fr S. SCOTT Department of Mathematics, King s College London WC2R 2LS sgs@mth.kcl.ac.uk Dedicated to our friend Krzysztof Wojciechowski We define Chern-Weil forms c k IA associated to a superconnection IA using ζ- regularisation methods extended to ΨDO valued forms. We show that they are cohomologous in the de Rham cohomology to tr IA 2k π P involving the projection π P onto the kernel of the elliptic operator P to which the superconnection IA is associated. A transgression formula shows that the corresponding Chern-Weil cohomology classes are independent of the scaling of the superconnection. When P is a differential operator with scalar leading symbol, the k-th Chern-Weil form corresponds to the regularised k-th derivative at t = 0 of the Chern character cht IA and it has a local description c k IA = 1 2p res IA 2k log IA 2 + π P in terms of the Wodzicki residue extended to ΨDO -valued forms Mathematics Subject Classification. Primary 58J20; Secondary 58J40, 47G99. Introduction Given a superconnection IA adapted to a family of self-adjoint elliptic ΨDOs with odd parity parametrised by a manifold B, the ΨDO valued form 1

2 2 e IA2 is trace class so that one can define the associated Chern character ch IA := tr e IA2 1 which defines a characteristic class independent of any scaling t IA t of the super connection. Here, tr is the supertrace associated to the family of ΨDOs, which reduces to the usual trace when the grading is trivial. In the case of a family of Dirac operators the limit lim t 0 tr e IA2 t of the Chern character built from the rescaled Bismut superconnection exists and the local Atiyah-Singer index formula for families provides a formula for it in terms of certain canonical characteristic forms integrated along the fibres see the works of Bismut[1], Bismut and Freed [2] and the book by Berline, Getzler and Vergne [3]. When the fibre of the manifold fibration over B reduces to a point, this family setup reduces to an ordinary finite rank vector bundle situation; IA can be replaced by an ordinary connection on a vector bundle over B with Chern character ch := tr e 2, and the local family index theorem reduces to the usual Atiyah-Singer index formula for a single operator. This can be expressed as a linear combination ch = dimm k=0 1 k c k k! of the associated Chern-Weil forms c k := tr 2k of degree 2k. Conversely, the Chern-Weil forms can be interpreted as the coefficients of a Taylor expansion of the map t ch t := tr e t 2 at t = 0 k t ch t t=0 = 1 k c k. 2 Replacing traces by appropriate regularised traces gives another insight into the family setup close to this finite dimensional description. Given a superconnection IA adapted to a family of self-adjoint elliptic ΨDOs with odd parity parametrised by a manifold B: 1 using regularised traces, one can build Chern-Weil type forms that relate to the Chern character 1 as in 2. 2 If IA is a superconnection associated with a family of differential operators, one can show an a priori locality property for these Chern- Weil forms, without having to compute them explicitly. This can be achieved by expressing them in terms of Wodzicki residues. Let us start with the first of these two issues.

3 3 1. Just as ordinary Chern-Weil forms are traces of the k-th power of the curvature, we define k-th Chern-Weil forms associated with a superconnection IA as weighted traces: c k IA := tr IA2 IA 2k of the k-th power of the curvature IA 2. To do this, we extend weighted traces according to the terminology used in [4] and which were also investigated by Melrose and Nistor [9] and Grubb [5] to families IA, Q of classical ΨDO valued forms setting tr Q IA := ζ IA, Q + π Q[0], z mer z=0 = ζ IA, Q, 0 mer + tr IA π Q[0], 3 including weights which are themselves ΨDO valued forms. Here, π Q[0] is the projection onto the bundle of kernels x B KerQ x of the zero degree part Q [0] of Q we assume that dimkerq x is constant, while ζa, P, z mer is the mixed degree meromorphic differential form studied by Scott in [14] which extends the holomorphic form Tr AP z from a suitable half-plane Rez >> 0. We show that the forms c k IA are closed and that the associated characteristic classes are independent of the scaling of IA. This is equivalent to the fact that the zeta forms ζ IA 2, k := ζ IA 2k, IA 2, 0 mer are exact and independent of scaling this fact was proved in [14]; the closed Chern-Weil forms c k IA differ from these forms by a term trπ IA 2k π involving the projection π on the kernel of the operator IA [0] to which the superconnection IA is adapted. As a consequence of the exactness of ζ IA 2, k we obtain that c k IA is cohomologous to trπ IA 2k π. 2. The second step can be carried out whenever IA is a superconnection associated with a family of differential operators of order p, with scalar leading symbol. In this case, we relate the forms c k IA to the Chern character ch IA via a formula which mimics equation 2: for t > 0, fp t=0 k t ch t IA = 1 k c k IA, 4 where fp denotes the finite part the constant term in the asymptotic expansion as t 0+. With the same assumptions, we show that the following local formula for these weighted Chern forms see equation 19 holds c k IA := 1 2p res IA2k log IA 2 + π P,

4 4 where the right-side is the residue trace extended these families; this object would not be defined for general families of ΨDOs. This equation generalises formulae obtained by Paycha and Scott in [12] which relate weighted traces of differential operators to Wodzicki residues extended to logarithms see theorem 3 and provides a local expression on the grounds of the locality of the Wodzicki residue. These results are summarised in Theorem 4. The paper is organised as follows 1 ΨDO valued forms 2 Complex powers and logarithms of ΨDO valued forms 3 The Wodzicki residue and the canonical trace extended to ΨDO valued forms 4 Holomorphic families of ΨDO valued forms 5 Weighted traces of differential operator valued forms; locality 6 Chern-Weil forms associated with a superconnection 1. ΨDO valued forms In this section we recall the construction of form valued geometric families of ΨDOs from [14]. Consider a smooth fibration π : M B with closed n- dimensional fibre M b := π 1 b equipped with a Riemannian metric g M/B on the tangent bundle T M/B. Let Λ π = ΛT M/B be the line bundle of vertical densities, restricting on each fibre to the usual bundle of densities Λ Mb along M b. Let E := E + E be a vertical HermitianZZ 2 -graded vector bundle over M and let π E := π E + π E be the graded infinite dimensional Fréchet bundle with fibre C M b, E b Λ Mb 1 2 at b B, where E b is the ZZ 2 -graded vector bundle over M b obtained by restriction of E. By definition, a smooth section ψ of π E over B is a smooth section of E Λ π 1 2 over M, so that ψb C M b, E b Λ Mb 1 2 for each b B. More generally, the de Rham complex of smooth forms on B with values in π E is defined by AB, π E = C M, π T B E Λ π 1 2 with the ZZ 2 -graded tensor product. Let ClE denote the infinitedimensional bundle of algebras with fibre ClE b = Cl M b, E b Λ Mb 1 2. A section Q A B, ClE defines a smooth family of classical ΨDOs with differential form coefficients parametrized by B.

5 5 Such an operator valued form Q is locally described by a vertical symbol qx, y, ξ C U M π U M IR n, π ΛT U B IR N IR N, where π is the fibre product, ξ may be identified with a vertical vector in T b M/B, and U M is a local coordinate neighbourhood of M over which E UM U M IR N is trivialized and IR N inherits the grading of E. With respect to the local trivialisation of π E over U B = πu M one has A U, π E UB AU C M b0, E b0 with M b0 = π 1 b 0 relative to a base point b 0 U B, so that q can be written locally over U B as a finite sum of terms of the form ω k q [k], where ω k A k U B and q [k] C U b0 IR n /{0}, IR N IR N is a symbol in the single manifold sense of form degree zero so that for all multi-indices α, β and each compact subset K U b0 the growth estimate holds α x β y γ ξ q [k]x, y, ξ < C k,α,β,k 1 + ξ q k γ. 5 For clarity we will work only with local symbols which are simple, meaning they have the local form dimb k=0 ω k q [k], with just one term in each form degree, extending by linearity to general sums. The order of a simple symbol is defined to be the dim B + 1-tuple q 0,, q dimb with q k the order of the symbol q [k] ; for simplicity we consider the case where q k is constant on B. In accordance with the splitting of the local symbol into form degree q = q [0] + + q [dim B] the operator Qψx = 1 2π n d vol M/B e ix y ξ qx, y, ξψydξ, M/B IR n for ψ with compact support in U M, splits as Q = Q [0] +Q [1] + +Q [dim B], where Q is a simple family of ΨDOs in so far as locally there is one component Q [k] = ω k Q k A U B, π E UB in each form degree. Q [k] raises form degree by k and Q k b : C M b, E + b C M b, E b is a pseudodifferential operator in the usual sense acting on sections of the bundle E b := E Mb over the fibre M b. The composition of ordinary symbols naturally extends to a composition of families of vertical symbols defined fibrewise by: q q := ω ω q q

6 6 where q q is the ordinary composition of symbols corresponding to the ΨDO algebra multiplication A i B, Ψ ν E A j B, Ψ µ E A i+j B, Ψ ν+µ E. 6 By a standard method the vertical symbol q in x, y-form can be replaced by an equivalent modulo S symbol in x-form. A simple family of vertical symbols q of order q 0,, q dimb is then called classical if for each k {0,, dimb} one has q [k] x, ξ j=0 q [k],jx, ξ with q [k],j x, tξ = t qk j q [k],j x, ξ for t 1, ξ 1. A family of vertical ΨDOs is called classical if each of its local component simple symbols is classical. Definition 1. A smooth family Q A B, ClE of vertical ΨDOs is elliptic if its form degree zero component Q [0] is pointwise with respect to the parameter manifold B elliptic. In this case Q has spectral cut if Q [0] admits a spectral cut. Likewise, it is invertible if Q [0] is invertible. Given that Q admits a spectral cut then it has a well-defined resolvent, which is a sum of simple families of ΨDOs. Setting Q [>0] := Q Q [0] A 1 B, ClE then there is an open sector Γ IC {0} containing the ray L such that on any compact codimension zero submanifold B c of B for large λ Γ one has in A B c, ClE using the idempotence of forms on B In particular Q λ 1 = Q [0] λ k Q [0] λ 1 Q[>0] Q [0] λ 1 k. 7 dimb k=1 Q λ 1 [0] = Q [0] λ Complex powers and logarithms of ΨDO valued forms Here we use the complex powers for ΨDO valued forms introduced in [14] to define and investigate the properties of the logarithm of a simple family of invertible admissible elliptic vertical ΨDOs. Let Q be a smooth family of vertical admissible elliptic invertible ΨDOs, the orders q 0,, q dimb+1 of which fulfill the assumption and q 0 = ordq [0] > 0 q k q 0 k 1. 8

7 7 Under these assumptions one obtains an operator norm estimate in AB as λ in Γ Q λi 1 l M/B = O λ 1 where l M/B : A B, ClE AB is the vertical Sobolev endomorphism norm associated to the vertical metric. Lemma 1. Let Q be an admissible elliptic invertible ΨDO valued form on B with spectral cut. Then Q z = i 2π λ z C,r Q λi 1 dλ defines a family of ΨDOs in A B, ClE which is a finite sum of simple ΨDO families with holomorphic orders αz = q 0 z + α0 where q 0 = ordq [0] and the constant term α0 is determined by the q i and the form degree. In particular, z Q z [0] = Q [0]. Here λ z is the branch of λ z defined by λ z = λ z e iz Argλ, 2π Argλ < and r being a sufficiently small positive number, C,r is a contour defined by C,r = C 1,,r C 2,,r C 3,,r with C 1,,r = {λ = λ e i + > λ r}, C 2,,r = {λ = re iφ φ 2π} and C 3,,r = {λ = λ e i 2π r λ < + }. When Q [0] has non negative leading symbol we can choose = π 2 in which case this complex power is the Mellin transform of the corresponding heatoperator form: Q z = 1 Γz 0 t z 1 e tq dt. Remark 1. When Q is not invertible, we can apply the Lemma to Q+π Q[0] which is invertible. Here π Q[0] denotes the projection onto the kernel of Q [0]. Proof. Let us first check the last formula, which is very straightforward.

8 8 As for ordinary pseudodifferential operators, we write Q z = i 2π Q λi 1 dλ = i 2π = 1 Γz = 1 Γz λ z C R, C R, Γz dt t z 1 0 i 2π t z 1 e t λ dt Q λi 1 dλ dt t z 1 e t Q. C R, e t λ Q λi 1 dλ To prove the first part of the lemma, we follow [14]. Let us first assume 8. Following Seeley s analysis, one can define the complex power Q z = Q λi 1 dλ for Re z > 0 in the usual way provided Q i 2π C R, λ z satisfies assumption 8. Since Q λi 1 = Q [0] [0] λi 1 it follows that Q z z = Q [0] [0]. The order of Q z is derived from the [d] expression of Q λi 1, which from 7 can be rewritten as [d] Q λ 1 [d] p 1+ +p k =d = Q [0] λ dimb k 9 k=1 Q[0] λ 1 W[p1] Q[0] λ 1... W[pk ] Q[0] λ 1 where W := Q [>0] so that W [pj] = Q Q [>0] [pj], a simple family of ΨDOs of pure form degree p j {1,..., dim B}. Let α j = ordw [pj]. This contributes to Q z a simple family of ΨDOs order q [d] 0 z q 0 k + α α k. It follows that the order of each simple component of the complex power is holomorphic with derivative at 0 independent of k. Complex powers can be extended to operators which do not satisfy assumption 8. By 9 dimb λ m Q λ 1 [d] = m0 λ k=0 p p k =d m 0+ +m k =m Q[0] λ 1 W[p1] m1 λ Q[0] λ 1... W[pk ] m k λ Q [0] λ Since mi λ Q [0] λi 1 is of order qm i + 1, taking m sufficiently large, we can ensure that m λ Q [0] λi 1 l M/B = O λ 1

9 9 without assumption 8. In this way, using integration by parts we may define Q z 1 i = λ m z λ m Q λi 1 dλ. 11 z 1 z m 2π C R, The order computation is essentially unchanged. Let Q be a smooth family of vertical admissible elliptic invertible ΨDOs. The logarithm is also built as in the single operator case by defining log Q := z z=0 Q z. As in the single operator case the logarithm is not quite a family of classical ΨDOs, but the non-logarithmic component is located only in the form degree zero component: Lemma 2. Let Q A B, ClE be a family of differential form valued vertical classical invertible elliptic ΨDOs with spectral cut. Then log Q [0] = log Q [0]. If Q moreover satisfies assumption 8, then Remark 2. log Q log Q [0] A B, ClE. Since the order of the components of Q z [d] are of the form qz + α0, from Lemma 1, one expects log Q [d] to contain log ξ terms, which would contradict the statement of Lemma 2. A closer look shows that the terms of the form ξ qz when d > 0 come with a factor of z and therefore do not yield any log ξ term when differentiated with respect to z at z = 0. A priori ξ is b B-dependent, this reflecting the fact that the decomposition depends on the choice of metric on the fibre M b. that arise in Q z [d] Proof. We drop the index to simplify notations. First, we show that log Q [0] = log Q [0]. For some integer m chosen large enough, we have for Rez > 0: Q z 1 1 = log λ λ m z λ m Q λi 1 dλ z 1 z m 2iπ C

10 10 and hence log Q = 1 m 1 1 m! 2iπ C log λ λ m m λ Q λi 1 dλ. From 9 it follows that setting d = k = 0 yields: log Q [0] = 1 log λ λ m Q [0] λi m 1 dλ = log Q [0]. 2iπ C It follows that log Q = log Q [0] + log Q[>0]. To see that log Q log Q [0] A B, ClE, recall that log Q = z Q z z=0 and that for each z the operator Q z is represented with respect to local trivializations by a local polyhomogeneous symbol of mixed differentialform degree q z b, x, ξ = i λ z q[λ]b, x, ξ wb, x, ξ q[λ]b, x, ξ k dλ 12 2π C 0 where is the vertical form-valued symbol product, C 0 is a finite closed key-hole contour enclosing the spectrum of the leading vertical symbol of Q λ 1 [0] = P λ 1, and where Opq[λ] Q λ 1, Opw W := Q P. The symbol q[λ] w q[λ] k in 12 is polyhomogeneous, with an asymptotic expansion into terms of decreasing homogeneity, each of mixed form degree, in the usual way. In general this is a complicated expression. Nevertheless, the log-type of log Q can be inferred just from the leading symbol top homogeneity. This is a consequence of the following simple lemma. Lemma 3. q z z=0 is the identity vertical symbol I defined by I [0] = I, 0, 0,... and I [p] = o := 0, 0, 0,..., p > 0, where I [p] indicates the component of form degree p, and the sequence on the right-side are the homogeneous terms. Proof. This is immediate from 12 since q[λ] w q[λ] k is Oλ 2 for k > 0 general ΨDO s. When k = 0 then the integrand is Oλ 1, and we have the usual situation of form degree zero operators. Let q z wx, ξ denote the mixed-form degree term in the asymptotic expansion of Q z of homogeneity w, and let q z w max x, ξ be the leading symbol with maximum homogeneity. Then q z w max x, ξ = i 2π C 0 λ z gx, ξ, λ dλ, 13

11 11 where gx, ξ, λ = q m [λ]x, ξ w ν x, ξ q m [λ]x, ξ... w ν x, ξ q m [λ]x, ξ is the ordinary form-valued matrix product i.e. not a symbol product of leading order symbols b m [λ]x, ξ of P λi P of order m > 0, and w ν the leading ΨDO -order symbol of W which has maximum homogeneity ν. Each q m has the quasi-homogeneity property for t > 0 and so q m [t m λ]x, tξ = t m q m [λ]x, ξ gx, tξ, t m λ = t mk+1+kν gx, ξ, λ. Hence, making the change of variable λ = t m µ in 13 we have q z w max x, tξ = t mz mk+kν f z w max x, ξ. 14 It follows that q z has an expansion into terms of mixed form degree q z x, ξ j 0 q z mz mk+kν jx, ξ. From Lemma 3 we therefore have q z mz mk+kν jx, ξ z=0 = δ j,0 I 15 since from the lemma terms of positive form degree do not contribute. The final conclusion for log q = z q z z=0 now follows in the usual way by differentiating q z mz mk+kν jx, ξ = ξ mz mk+kν j q z mz mk+kν j x, ξ ξ with respect to z, then evaluating at z = 0 and using 15 to get log q j 0 log q j with log qx, ξ = m log ξ δ j,0 I + ξ kν m j z z=0 q z x, ξ ξ Note, since z z=0 q z x, ξ/ ξ has order zero, that provided ordp = m ordw which we assumed the second term is of order no larger than zero.. Remark 3. A formal argument based on the Campbell-Hausdorff formula provides some intuition why the second part of the lemma holds. We show

12 12 here how the Campbell-Hausdorff formula for ordinary ΨDOs obtained by Okikiolu [10] formally extended to families of vertical ΨDOs, yields that log Q [0] is a classical vertical ΨDO. Indeed, the splitting Q = Q [0] +Q [>0] yields log Q = log [ ] Q [0] + Q [>0] = log Q [0] I + Q 1 [0] Q [>0] log Q [0] + logi + Q 1 0 Q [>0] + k=2 C k log Q [0], logi + Q 1 [0] Q [>0] where C k M, N stands for a linear combination of Lie polynomials of degree k in M and N given by: C k M, N = j=1 c j α l +β l >0, P l j=1 αj+βj+1=k ad M α1 ad N α1 ad M α1 ad N α1 N for some coefficients c j IR and where ad MM := [M, M ]. Under assumption 8, the operator Q 1 [0] Q [>0] has a vanishing form degree zero part and negative orders β [1],, β [dimb+1]. The logarithm therefore coincides with the logarithm on bounded operators and yields an asymptotic expansion: logi + Q 1 [0] Q 1 k 1 k [>0] Q 1 k [0] [>0] Q, k=1 which shows that logi + Q 1 [0] Q [>0] is a classical vertical ΨDO. It follows that each of the C k M, N s is also a classical vertical ΨDO. Indeed, these are built up from iterated brackets of an ordinary corresponding to the 0-form degree part logarithmic ΨDO log Q [0] and the classical vertical ΨDO logi + Q 1 [0] Q [>0]. The ordinary symbol analysis shows that such brackets are classical and hence that log Q log Q [0] A B, ClE as claimed in the lemma. This argument therefore provides a heuristic but maybe more intuitive explanation of the lemma. 3. The Wodzicki residue and the canonical trace extended to geometric families Both the Wodzicki residue and the cut-off integral defined on ordinary classical symbols extend to smooth families of vertical differential form valued classical symbols.

13 13 Let Q AB, ΨE be a simple family of ΨDOs of non-integer order, so that in each form degree ordq [k] R\Z. Then working in local coordinates U M on M where Q is represented by a smooth family of symbols σ Q x, ξ, we find that the local matrix valued forms a σ Q x, ξ d ξ Tx M patch together to determine a global section of the bundle π T B EndE Λ π over M; this is proved by an obvious fibrewise version of the usual existence proof of the Kontsevich Vishik canonical trace. Taking the fibrewise trace we consequently have an element TR x Q := tr x σ Q x, ξ d ξ C M, π T B Λ π Tx M which can then be integrated over the fibres to define the canonical trace for families of non-integer order ΨDOs, a differential form on the parameter manifold B, by TRQ := TR x Q AB. M/B In the case when each component of Q has order less than n, then TRQ = Tr Q, the usual fibrewise trace. In a similar way, for a simple family Q AB, ΨE of ΨDOs of any real or complex order one has a residue trace density b res x Q := tr x σ Q x, ξ n d Sξ C M, π T B Λ π Sx M where σ Q x, ξ n = dim B k=0 σ Qx, ξ n [k] is the homogeneous part of the local symbol of homogeneity n. This can then be integrated over the fibres to define the residue trace for families of arbitrary order ΨDOs, once more defining a differential form on the parameter manifold B, by resq := res x Q dx AB. M/B Notice that if all the components of Q have non-integer ΨDO order then resq vanishes; as in the case of a single operator, the functionals TR and res are roughly complementary. a d ξ := 1 2π n d ξ where dξ is the ordinary Lebesgue measure on T x M IRn b In the following formula d S ξ := 1 2π n d S ξ where d S ξ is the canonical volume measure on the cotangent unit sphere S xm.

14 14 As in the case of a single operator, res : AB, ΨE AB defines a trace, vanishing on graded brackets [Q 1, Q 2 ] of families ΨDOs, while TR vanishes provided [Q 1, Q 2 ] has non-integer order components. Let us now extend the Wodzicki residue to forms on B with values in ΨDOs of logarithmic type. Let A AB, ΨE be a family of differential operators. Since by Lemma 2 the operator valued form log Q log Q [0] is classical and since res x A log Q [0] dx defines a global top degree form on M, as A is a family of differential operators, by the results of [12], so does res x A log Q = res x Alog Q log Q [0] + res x A log Q [0]. Hence, in this case, we may define the differential form by resa log Q = res x A log Q dx AB. M/B The fact that we restrict to differential operators A ensures the independence of this extended residue on the choice of the metric on the fibre M b since a change of metric brings in a vertical multiplication operator, which combined with the vertical differential operator A modifies the expression by another differential operator, for which the Wodzicki residue will vanish. We comment that this is possibly taking place in a ZZ 2 -graded context, where the residue density is the super residue density and so forth. 4. Holomorphic families of ΨDO valued forms We call a family Q z = dimb k=0 Q z [k] A B, ClE parametrised by z W IC holomorphic if in each local trivialization of π E over a neighbourhood U B of b, Q z = ω UB k Q k,z for some ω k AU B we have [k] that z Q k,z Cl M b, E b is holomorphic family of ΨDOs parametrised by W in the usual single operator sense following Kontsevich and Vishik [7], Lesch [8], see also [12]. In particular, the corresponding symbols q z then define a holomorphic family of symbols in the usual sense. Definition 2. We call a holomorphic regularisation procedure a map R which to any A A B, ClE associates a holomorphic family A z AB, ClE such that A 0 = A and with order αz such that α [k] 0 0 or any k {0,, dimb + 1}. Similarly, one defines holomorphic regularisation procedures on the level of symbols in such a way that a regularisation procedure R : A A z induces one for the corresponding symbols.

15 15 Let us illustrate these definitions with two examples. Example 1. 1 For any holomorphic map H such that H0 = 1, the map R H : q Hz ξ z q defines a holomorphic regularisation procedure on local classical vertical symbols. For a certain choice of H it gives back dimensional regularisation. 2 Given a family Q A B, ClE of differential form valued vertical classical invertible elliptic ΨDOs with spectral cut such that Q moreover satisfies assumption 8, the map R Q : A A Q z is a holomorphic regularisation procedure on vertical classical ΨDOs called ζ-regularisation. Theorem For any family z q z := dimb+1 k=0 q z [k] b, x, ξ of classical symbols locally parametrised by b B and holomorphic on an open subset W IC with order z αz = α [0] z,, α [dimb+1] z such that z α [k] b z does not vanish for any k, then the functions T x M q z b [k] b, x, ξdξ are meromorphic with simple poles in α[k] b 1 ZZ W. The pole of the map z Tx M q z b [k] b, x, ξdξ at a point z 0 in α [k] b 1 ZZ W is expressed in terms of a Wodzicki residue: 1 Res z=z0 q z [k] b, x, ξdξ = α[k] b q z0 res z0 [k] b. 16 T x M b 2. As a consequence, given a holomorphic family z Q z := dimb+1 k=0 Q z [k] at point b on W IC of differential form valued vertical classical ΨDOs with holomorphic order z αb = α [0] bz,, α [dimb] bz such that z α [k] b z does not vanish for any k {0,, dimb}, the map z TR Q z b [k] is meromorphic with simple poles in α [k] b 1 ZZ W. The pole of TR Q z b [k] at a point z 0 in α [k] b 1 ZZ is expressed in terms of the Wodzicki residue of Q z0 [k] at point b B: Res z=z0 TR Q z b [k] 1 = α[k] b z0 res Q z0 b [k]. 17

16 16 Proof. The similar result for ordinary classical ΨDOs [7] applied to each q z b [k] and each Q z b [k] yields the result. On the grounds of this theorem, the holomorphic regularisation R Q : A A Q z on differential form valued vertical classical ΨDOs gives rise to meromorphic maps z ζ A, Q, z := TR A Q z with simple poles so that it makes sense to extract the finite part at z = 0 denoted by ζa, Q, 0 mer. As a consequence of the above theorem, we have as in the case of ordinary ΨDOs, the following formula relating the Wodzicki residue with the complex residue at z = 0 resa = q 0 Res z=0 TR A Q z = q 0 Res z=0 TR A Q z [0] since A Q z has order α [k] z = q 0 z + α [k] 0 with q 0 the order of Q [0]. Indeed, notice this formula is independent of the choice of Q apart from q 0 = ordq [0]. Definition 3. Let Q A B, ClE be a family of differential form valued vertical classical invertible elliptic ΨDOs with spectral cut such that Q moreover satisfies assumption 8. Provided the dimension of the kernel kerqb [0] is independent of b, the map b Π Qb[0] built from the orthogonal projection onto this kernel is smooth and for any A A B, ClE, tr Q A [k] TR := lim z 0 A Q + π Q[0] z := ζa, Q, 0 mer [k] + tr A [k] π Q[0], 1 [k] z Res z=0tr A Q + π Q[0] z defines a differential form tr Q A on B called the Q-weighted trace of A. Let us compare these weighted traces to the finite part of heat-operator regularised traces. When Q [0] has non negative leading symbol the operator A e ɛq is traceclass for positive ɛ and we can write these formulae are similar to the ones used by Higson [6] to derive the local formula for the Chern character in a [k]

17 17 non commutative geometric setup: tr A e ɛq = ɛ n du tr n 0 A e u0ɛq [0] Q [>0] e un 1ɛQ [0] Q [>0] e unɛq [0] n = ck ɛ k +2n 1 k + n 1! tr A Q k1 [>0] Qkn [>0] e ɛq [0], n 0 k 0 where ck is defined by induction for any multiindex k = k 1,, k n by ck 1 = 1 and ck 1,, k n = ck 1,, k n 1 k 1 + k n k 1 + k n 1 + n 1. k n! For an operator B, the operator B i is also defined by induction; B 0 := B and for any non negative integer i, B i+1 := [Q [0], B i ] so that B i = ad i Q [0] B. The sum over n is finite for each fixed form degree d whereas the sum over k = k 1,, k n IN n is a priori infinite. However, if Q [0] is assumed to have scalar leading symbol, then B k has order b + kq 0 1 where b is the order of B and q 0 the order of Q [0] and A Q k1 [>0] Qkn [>0] has order [d] a + nq d + k q 0 1 where q d is the order of Q [d]. It follows that for each fixed multiindex k, there are coefficients α jk, j k 0 and β k such that ɛ k +2n 1 tr A Q k1 [>0] Qkn [>0] e ɛq [0] ɛ 0 j k =0 ɛ 0 j k =0 α jk ɛ q 0 k +q 0 2n 1+j k a+nq d + k q 0 1 dimm b q 0 α jk ɛ q 0 2n 1+j k a nq d + k dimm b q 0 + β k log ɛ + β k log ɛ so that the fractional powers of ɛ increase with k ; in the ɛ 0 limit, they will not contribute for large enough k. Extracting a finite part when ɛ 0, we can therefore define for any non negative integer d: fp ɛ=0 tr A e ɛq [d] = fp ɛ=0 ck ɛ k +2n 1 k + n 1! tr A Q k1 [>0] Qkn [>0] e ɛq [0]. [d] n 0 k 0

18 18 Since for ordinary ΨDOs A, Q we have a folklore result, the proof of which can be found e.g. in a survey by Paycha [11] fp ɛ=0 trae ɛq = tr Q A + γ resa where γ is the Euler constant, weighted traces coincide with heat-kernel regularised traces for operators with vanishing residue, so this holds in particular for differential operators. Applying this to each operator A Q k1 [>0] Qkn [>0] we get that provided [d] Q [d] and A [d] are differential operators for any non negative integer d, then: fp ɛ=0 tr A e ɛq = tr Q A Weighted traces of differential operator valued forms; locality A connection on E ΛM b 1 2 induces a connection Hom on Cl E which locally reads Hom = d+[θ, ] if reads = d+θ. Applying Theorem 1 to the holomorphic family Q z = A [, Q + π Q[0] z ] where A AB, ClE yields: Theorem 2. Let Q A B, ClE be a differential form on B with values in vertical classical elliptic ΨDOs with spectral cut and with kernel Ker Qb [0] independent of b. Let A AB, ClE. Given a connection on E ΛM b 1 2 then we have the equality of forms: d tr Q A = tr Q [, A] + 1a+1 q 0 res A [, log Q + π Q[0] ] where q 0 is the order of Q [0] and where a is the degree of A as a form. Proof. For simplicity we assume Q is invertible, but the proof extends to the non invertible case replacing Q by Q + π Q[0] in the complex powers. The proof goes as in [4] where Q was a ΨDO valued 0-form; indeed we have d tr Q A tr Q [, A] = fp z=0 dtraq z TR[, A] Q z = 1 a fp z=0 TR A [, Q z ] A [, Q z = 1 a Res z=0 TR ] z = 1a+1 res A [, log q Q] 0

19 19 where we have applied Theorem 1 to the holomorphic family Q z = A [, Q z ] to get the last identity using the fact that the degree k part of Q z has order q 0 z + cst. Applying Theorem 1 to the holomorphic family Q z = A Q + π Q[0] z where A AB, ClE is such that A [i] is a differential operator for any non negative integer i leads to the a description of the weighted trace of a differential operator valued differential form in terms of a Wodzicki residue. In order to make these notes self-contained, we include the full proof for ΨDO valued forms although it mimics the proof derived in [12] in the case of ordinary ΨDOs. As in [12] we use the following preliminary lemma. Lemma 4. Let A AB, ClE be a family of vertical ΨDOs such that A [i] b is a differential operator on M b at any point b B and for any non negative integer i. Then, for any x M b, for any positive real number α z ξ α z σ A x, ξ d ξ Tx M b is meromorphic with simple poles and if fp z=0 z = 0 we have: denotes its finite part at 0 = Tx M b σ A b, x, ξ d ξ = fp z=0 ξ α z σ A b, x, ξ d ξ. Tx M b Proof. The fact that x T x M ξ αz σ b A b, x, ξ d ξ defines a meromorphic function with simple poles follows from Theorem 1 applied to σ z b, x, ξ = ξ α z σ A b, x, ξ of order αz = α z + α where α is the order of A. let us fix a non negative integer i. The symbol of the differential operator A [i] reads σ A[i] b, x, ξ = orda [i] k=0 σ k b, x, ξ where for any multiindex k = k 1,, k dimmb, σ k b, x, ξ = ab, xξ k is positively homogeneous. Hence, its cut-off integral on the cotangent space at x M b reads here Bb,x 0, R

20 20 is the ball of radius R centered at 0 in Tx M b : σ A[i] b, x, ξ d ξ = fp R σ A[i] b, x, ξ d ξ B b,x 0,R T x M b Similarly, = = = = = orda [i] k=0 orda [i] k=0 = fp R R k+n k + n a k b, x fp R B b,x 0,R ξ k d ξ R a k b, x fp R r k+n 1 dr S x M b fp z=0 σ A[i] b, x, ξ ξ z d ξ Tx M b orda [i] k=0 orda [i] k=0 orda k=0 a k b, x fp z=0 T x M b ξ z ξ k d ξ 0 ξ k d ξ = 0. R a k b, x fp z=0 fp R r k+n z 1 dr 0 S x M b a k b, x fp z=0 fpr R k+n z ξ k d ξ = 0. Sx M b S x M b ξ k d ξ ξ k dξ The fact that the finite part vanishes in the line before last follows from the fact that fp R R k+n z vanishes for Rez sufficiently small, as the finite part of a meromorphic extension of a function which vanishes on some half plane. We are now ready to prove the main result of this section: Theorem 3. Let Q A B, ClE be a differential form on B with values in vertical classical elliptic ΨDOs with spectral cut such that Q moreover satisfies assumption 8 and has kernel Ker Qb [0] with constant dimension. Let A AB, ClE such that A [i] is a differential operator for any non negative integer i then we have the equality of forms: tr Q A = 1 q 0 res A log Q + π Q[0]

21 21 where q 0 is the order of Q [0] and π Q[0] Ker Q [0]. the orthogonal projection onto the Proof. Here again, we prove the result for invertible Q; the proof then extends to the non invertible case replacing Q by Q + π Q[0] in the complex powers. Since Q z [0] = Q z [0] has order q 0 z with q 0 the order of Q [0], dropping the subscript to simplify notations, we write for any b B tr Q Ab := fp z=0 dx M b = fp z=0 dx M b + fp z=0 dx M b = fp z=0 dx M b T x M b T x M b T x M b T x M b tr x σa Q zb, x, ξ d ξ tr x σa Q zb, x, ξ ξ q0z σ A b, x, ξ + d ξ ξ q0 z tr x σ A b, x, ξ d ξ tr x σa Q zb, x, ξ ξ q0 z σ A b, x, ξ by Lemma 4 = Res z=0 dx d ξ M tr x σa Q zb, x, ξ ξ q0 z tr x σ A b, x, ξ. b z T x M b Applying Theorem 1 to σ z b, x, ξ := trxσ A Q z b,x,ξ ξ q 0 z tr xσ A b,x,ξ z yields for any d {1,, dimb} then tr Q 1 Ab [k] = Res z=0 dx d ξ tr x σ z b, x, ξ [k] = M b Tx M b α[k] b 0 [ [ trx σ A Q zb, x, ξ ξ q0 z ] ] tr x σ A b, x, ξ dx d ξ M b Sx M z b z=0 [k] [ = 1 dx d ξ d [ trx σ q 0 M b Sx M dz A Q zb, x, ξ ξ q0z tr x σ A Q zb, x, ξ ] ] z=0 b since α [k] z = ordσ z [k] = q 0 z + α [k] 0 [ and since trx σ A Q z ξ q0 z ] tr x σ A z=0 = 0 = 1 d ξ [tr x σ log Q A b, x, ξ q 0 log ξ tr x σ A b, x, ξ] q [k] 0 M b S x M b = 1 q 0 [res A log Q b] [k]. [k]

22 22 6. Chern-Weil forms associated with a superconnection Definition 4. A super connection introduced by Quillen [16], see also [1], [3] on π E adapted to a smooth family of formally self-adjoint elliptic ΨDOs P A 0 B, Cl q E with odd parity is a classical ΨDO IA on A B, π E of odd parity with respect to the ZZ 2 -grading such that: and IAω σ = dω σ + 1 ω ω IAσ ω AB, σ A B, π E IA [0] := P where as before, IA = dimb i=0 IA [i] and IA [i] : A B, π E A +i B, π E. The curvature of a super connection IA is given by IA 2 A B, ClE. Notice that IA 2 [0] = P 2 so that IA 2 is elliptic with spectral cut π.we know from the previous paragraphs that provided Ker IA 2 b [0] = Ker P b is independent of b: ζ IA 2k, IA 2 + π P, z := TR IA 2k IA 2 + π P z π P will denote the orthogonal projection onto the kernel of P, is a ΨDO valued form in A B, ClE so that we can define its finite part: tr IA2 IA 2k := ζ IA 2k, IA 2 + π P, 0 mer. Theorem 4. Let IA be a super connection on π E adapted to a smooth family of formally self-adjoint elliptic ΨDOs P A 0 B, Cl p E of odd parity which satisfies assumption 8. Let us further assume that the kernel Ker IA 2 b [0] = Ker P b is independent of b. Then for any non negative integer k, 1 the associated Chern forms c k IA := tr IA2 IA 2k are closed forms on B which are cohomologous in de Rham cohomology to tr IA 2k π P. 2 The corresponding Chern-Weil classes are independent of the scaling of IA with fixed kernel and we have the following transgression formula t c k IA t = dτ k IA t where τ k IA t = k tr IA2 t İAt IA 2k 1 t 1 p res İAt IA 2 t + π P k 1

23 23 for any smooth one parameter family IA t of superconnections associated with P of order p. 3 If P has scalar leading symbol and if IAb is a differential operator at each point b B then the Chern-Weil classes relate to the Chern character by fp t=0 t k tr e t IA2 = 1 k c k IA. 4 If IAb is a differential operator at each point b B then the associated Chern forms have a local description in terms of the Wodzicki residue: c k IA = 1 2p res IA 2k log IA 2 + π P. 19 Moreover, τ k is also local and we have: τ k IA t = k 2p res İAt IA 2 t + π P k 1 log IA 2 t + π P 1 p res İAt IA 2 t + π P k 1. Proof. Ad 1 Theorem 2 applied to A = IA 2k and Q = IA 2 with q 0 = 2p yields the closedness. Indeed, using the fact that = IA commutes with integer powers of IA and with e IA2, we have: d tr IA2 IA 2k = fp ɛ 0 d tr IA 2k e ɛ IA2 = fp ɛ 0 tr[ IA, IA 2k ] e ɛ IA2 + fp ɛ 0 tr IA 2k [ IA, e ɛ IA2 ] = 0, since d tr = tr IA. Furthermore, from [14] we know that ζ IA 2, k := ζ IA 2k, IA 2, z mer z=0 is exact, so that tr IA2 IA 2k = ζ IA 2k, IA 2, z mer z=0 + tr IA 2k π P is cohomologous to tr IA 2k π P. Ad 2 Applying Theorem 2 to = t, A = IA 2k t, Q := IA 2 t with IA t a smooth family of superconnections parametrised by IR associated with a

24 24 family P t with constant kernel and corresponding projection π P we get t c k IA t = tr IA2 t t IA 2k t 1 2p res IA 2k t t log IA 2 t + π P = = 1 p k i=1 k 1 p i=1 1 0 = k tr IA2 t tr IA2 t 1 0 tr IA2 t IA 2i 1 t [ IA t, İA t] IA 2k i t res IA 2k t IA 2 t + π P 1 λ [ IA t, İA t] IA 2 t + π P λ dλ res [ IA 2i 1 t [ IA t, İA t] IA 2k i t IA 2 t + π P k IA 2 t + π P 1 λ [ IA t, İA t] IA 2 t + π P λ dλ ] IA t, İA t IA 2k 1 t 1 p res [ IA t, İA t IA 2 t + π P k IA 2 t + π P 1] = k d tr IA2 t İAt IA 2k 1 t 1 p d res İAt IA 2 t + π P k 1 where we have used the fact that t IA 2 t = [ IA t, İA t] as well as the cyclicity of the Wodzicki residue combined with the fact that it vanishes on finite rank operators which we used to replace IA 2 t by IA 2 t + π P in the third equality. t IA2 Ad 3 First of all, since t e = t 0 ds e t s IA2 IA 2 e s IA2 see e.g. t IA2 formula 2.6 in [3] and since the exponential e commutes with any power of IA we have: k t cht IA = k t tre t IA2 = 1 k tr IA 2k e t IA2. Since weighted traces coincide with the ordinary trace on trace-class operators and since the operator valued form e is trace-class for positive t IA2 t as a consequence of the ellipticity of the self-adjoint operator P and we have fp t=0 t k tr e t IA2 = 1 k fp t=0 tr IA 2k t e IA2 = 1 k tr IA2 IA 2k = 1 k c k IA. Here we have used the fact that the leading symbol of P is scalar to make sense of the heat-kernel regularised trace fp t=0 tr IA 2k e t IA2 and formula 18 to identify the weighted trace with the heat-kernel regularised trace since, by assumption, all the operators involved are differential operators.

25 25 Ad 4 Applying Theorem 3 to A = IA 2k, Q := IA 2 then yields the local formula for Chern forms announced in the last part of the theorem. In that case, τ k is also local and we have τ k IA t = k 2p res İAt IA 2k 1 t log IA 2 t + π P 1 p res İAt IA 2 t + π P k 1 = k 2p res İAt IA 2 t + π p k 1 log IA 2 t + π P 1 p res İAt IA 2 t + π P k 1 using here again, the fact that the Wodzicki residue vanishes on finite rank operators in order to replace IA 2 t by IA 2 t + π P. References 1. J.-M. Bismut, The Atiyah-Singer index theorem for families of Dirac operators: two equation proofs, Invent. Math. 83, J.-M. Bismut, D. Freed, The analysis of elliptic families I and II, Comm. Math. Phys. 106, and Comm. Math. Phys. 107, N. Berline, E. Getzler, M. Vergne, Heat kernels and Dirac operators. Grundlehren der Mathematischen Wissenschaften 298, Springer Verlag, Berlin A. Cardona, C. Ducourtioux, J.-P. Magnot, S. Paycha, Weighted traces on algebras of pseudo-differential operators and the geometry of loop groups, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 n.4, G. Grubb, A resolvent approach to traces and zeta Laurent expansions, AMS Contemp. Math. Proc. 366, See also arxiv: math.ap/ N. Higson, The local index formula in noncommutative geometry, Trieste Lecture Notes 2004; The residue index theorem of Connes and Moscovici, Clay Mathematics Proceedings M. Kontsevich, S. Vishik, Geometry of determinants of elliptic operators, Func. Anal. on the Eve of the XXI century, Vol I, Progress in Mathematics 131, ; Determinants of elliptic pseudodifferential operators, Max Planck Preprint M. Lesch, On the non commutative residue for pseudo-differential operators with log-polyhomogeneous symbols, Ann. Global Anal. Geom. 17, R. Melrose, V. Nistor, Homology of pseudo-differential operators I. Manifolds with boundary, arxiv: funct-an/ K. Okikiolu, The multiplicative anomaly for determinants of elliptic operators, Duke Math. J. 79, ; K. Okikiolu, The Campbell-Hausdorff

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