Index character associated to the projective Dirac operator

We calculate the equivariant index formula for an infinite dimensional Clifford module canonically associated to any Riemannian manifold. It encompasses the fractional index formula of the projective Dirac operator by Mathai--Melrose--Singer.


Introduction
Mathai, Melrose, and Singer constructed the projective Dirac operator / ∂ pr M for arbitrary Riemannian manifold M as a projective pseudo-differential operator in [MMS06]. Its index ind a / ∂ pr M was defined in terms of the integral kernel I pr / ∂ M of this projective differential operator. They showed the fractional index formula [MMS06, Theorem 2 and Section 9] where the integrand of the right hand side is theÂ-form associated to the Riemannian curvature of M . The equality was proved via local index calculation of the heat kernel associated to I pr / ∂ M . It can be easily seen that the above expression might be a non-integral rational number when M is not spin.
The framework of projective pseudo-differential operators ignited various developments of the study of modules and index theory over the bundles of finite dimensional algebras (or the ones whose fiber is the trace class operator algebra) over manifolds [MMS05,MMS09,AM09,BG10,Gof10].
The projective operator / ∂ pr M is associated to a certain Clifford module of M , but one has to note the difference between (1) and the usual index formula for a Clifford module E over M . Here / ∂ E is the twisted Dirac operator acting on the sections of E and ch(E/S) is the relative Chern character of E. The presence of the relative Chern character ch(E/S) makes the integral in the right hand side to be an integer.
Let π (nat) : SU(N ) V (nat) be the natural representation of SU(N ). The Levi-Civita connection on F SO(n) induces a 1-form ∇ of first order differential operators acting on the space C ∞ (P, V (nat) ). This means, by definition, when X is a vector field over M , one obtains a first order differential operator ∇ X on P .
Since P can be regarded as the bundle of trivializations of the Clifford bundle Cl C (M ), the sections of Cl C (M ) correspond to the PU(N )-invariant functions of P into M N (C). Then the space of functions from P into V (nat) is naturally a module over Cl C (M ). Thus one may define the associated Dirac operator on C ∞ (P, V (nat) ) for any local frame (e i ) n i=1 . The operator / ∂ M can be regarded as a transversely elliptic operator [Ati74] on the SU(N )-manifold P . Hence one obtains a distribution ind SU(N ) (γ, / ∂ M ) on SU(N ) which is invariant under conjugation and satisfies where χ π the character of any irreducible representation π of SU(N ), and p π is the projector onto the π-isotypic component of the action π P ⊗π (nat) on C ∞ (P, V (nat) ), where π P is the action induced by the translation on P .
In [MMS08], it was shown that the SU(N )-equivariant operator / ∂ M descends to the projective Dirac operator / ∂ pr M on the projective spin bundle over the Azumaya bundle Cl C (M ). For such 'descent' one has when φ is a smooth function on SU(N ) which is constantly equal to 1 around e and has small enough support. As mentioned at the end of the introduction of [MMS08], the above formula gives 'the coefficient of the Dirac function' in the distribution ind SU(N ) (γ, / ∂ M ). The aim of this paper is to relate the fractional index formula (1) to the classical index formula (2) of Clifford modules and give a refined formula (Corollary 6) for the contribution of 'higher order derivatives of the Dirac function' in the distribution ind SU(N ) (γ, / ∂ M ). The space L 2 (P, V (nat) ) can be regarded as the space Γ(M, E) of the sections of an infinite dimensional Clifford module E over M = P/ PU(N ). It is defined as the induced vector bundle where we consider the left translation action λ on L 2 (PU(N )) and the trivial action on V (nat) . The precise meaning of the correspondence between L 2 (P, V (nat) ) and E is that we have where the right hand side can be regarded as Γ(M, E). The action ρ ⊗ π (nat) of SU(N ) on L 2 (PU(N )) ⊗ V (nat) commutes with the one λ ⊗ triv of PU(N ). Hence it induces an action on (3). This action corresponds to the action π P ⊗ π (nat) on L 2 (P, V (nat) ).
If one tries to consider an analogue of the equivariant index theorem in the case of E = E, the relative curvature form F E/S of E should be locally given by the action of the curvature of a vector bundle E 0 satisfying E ≃ E 0 ⊗ S. Note that the vector bundle E 0 has infinite rank, hence the fiberwise trace Tr E0 e −F E/S will become infinite. To remedy this, we shall construct an action of SU(N ) on E 0 such that its tensor product with the trivial action on S is the action on E. Then an equivariant choice of curvature on E 0 will allow us to compute the trace of Tr E0 (e −F E/S π(φ)) of the composition of the curvature e −F with the convolution by any auxiliary function φ on SU(N ).
Thus the expression x in M and γ in SU(N ) defines a differential form on M of distributions on SU(N ). Then the analogue of the index formula (2) for the Clifford module E should be

Preliminaries
Most of the constructions in this section appear in the literature in some way or other. We just recall their definitions in order to fix the notations and conventions. For the sake of simplicity we assume that n = dim(M ) is even and put N = 2 n/2 . Let P be the P U (N ) principal bundle over M induced from the frame bundle F SO(n) by the natural group embedding Let σ be a section of F SO(n) defined on an open set U , andσ = (σ, e) be the section of P defined by σ and the constant mapping M → SU(N ), x → e. Under the identification of (3) any section f of E over U can be expressed using a function ξ : The Clifford module structure on E can be described as One obtains the induced connection form (9) A : TP → su(N ) via (6). Hence it is given by the collection of linear maps T p P → su(N ) which are retractions of the embedding su(N ) → T p P coming from the action map g → p.g from SU(N ) to P , and satisfy the equivariance condition A X = Ad g A g.X for any X ∈ T p P and g ∈ G.
Given a vector field X on M , define the operator A (E,σ) X on the sections of E over U by ). Lemma 1. Let ψ be a function from U to SU(N ). Then one has By Lemma 1, the covariant derivative , which shows that A is a Clifford connection. Finally, we put for any local frame (e i ) n i=1 . For each irreducible representation π of SU(N ), the restriction of / ∂ to the πisotypic component of L 2 (P, V (nat) ) becomes a K-cycle / ∂ π over C(P ) ⋊ SU(N ). Hence we obtain a family of K-cycles (/ ∂ π ) π∈ SU(N ) parametrized by the irreducible representations of SU(N ). Since the π-isotypic component of L 2 (P, V (nat) ) is trivial unless the central part of π agrees with that of the natural representation, our interest is in (/ ∂ π ) π∈ SU(N ) (nat) where SU(N ) (nat) is the collection of the irreducible representation classes whose central character agree with that of the natural representation.
For each π ∈ SU(N ) (nat) , the number ind / ∂ π is finite and agrees with the multiplicity of π in the representation ind ∂ M of SU(N ). Let d π be the dimension of the representation space of π, and χ π be its normalized character χ π (γ) = 1 d π Tr(π(γ)).
Then the sum defines a distribution over SU(N ) invariant under the conjugation.
2.1. Pairing of U(g) and C ∞ (G). When a is an element of the universal enveloping algebra U(su(N )) of SU(N ) can be regarded as a distribution on SU(N ) with support {e} by for φ ∈ C ∞ (SU(N )), where λ(a) is the right invariant differential operator represented by a. The above pairing has another interpretation a, φ = Tr L 2 (SU(N )) (λ(a)λ(φ)) = Tr L 2 (SU(N )) (ρ(a)ρ(φ)).
Let T be a conditional expectation of U(su(N )) onto its center.

KK -element associated to projective Dirac operator
For each character χ on the center of SU(N ), let J χ denote the closure of {f ∈ C(SU(N ), C(P )) | ∀z ∈ Z(SU(N )) : f (gz) = χ(z)f (g)} in C(P ) ⋊ SU(N ). Then the algebra C(P ) ⋊ SU(N ) admits a direct sum decomposition C(P ) ⋊ SU(N ) ≃ ⊕ χ∈ Z(SU(N ) J χ by bilateral ideals. The representation of C(P ) ⋊ SU(N ) on L 2 (P, V (nat) ) factors through the projection onto J (nat) , the factor corresponding to the central character of the natural representation. The bimodule L 2 (P, V (nat) ) is a completion of the following C * -module F over Cl C (M ). The subspace C(P, V (nat) ) of L 2 (P, V (nat) ) admits a Cl C (M )-valued inner product characterized by where we identify p ∈ P x with an algebra isomorphism p * : The completion F of C(P, V (nat) ) with respect to the above inner product admits an action of C(P ) ⋊ SU(N ) as Cl C (M )-compact operators.
Proposition 3. The bimodule F gives a strong Morita equivalence between J (nat) and Cl C (M ).
The operator / ∂ M and the representation of C(P ) ⋊ SU(N ) on L 2 (P, V (nat) ) defines a spectral triple over C(P )⋊SU(N ). Consequently we obtain the associated element α of KK (C(M ) ⋊ SU(N ), C) represented by the phase of / ∂ M as in [BC00], and the map ind / ∂ M : K 0 (C(P ) ⋊ SU(N )) ≃ K 0 (Cl C (M )) → Z. Example 4. As an example of the above construction, consider the case of π = π (nat) . Then the corresponding index ind(/ ∂ π (nat) ) is equal to the signature number of M . Indeed, if one takes the tensor product action ρ ⊗ π ⊗π on C(P ; V (nat) ) ⊗ (V (nat) ) * , its fixed point subspace is identified to the space of the sections of Cl C (M ), and the grading on the former corresponds to the left Clifford action of the volume element. Meanwhile the fixed point subspace is canonically identified to the π (nat) -isotypic component of C(P ; V (nat) ).

Connections on infinite dimensional equivariant bundles
Now we give a more precise description of (4). Let A be the curvature 1-form of (9). The associated connection 2-form in A 2 (P, su(N )) basic is given by for vector fields X and Y on P .
Let σ be a section of P → M defined on an open set U of M . When X and Y be vector fields on M , consider the operator on the sections of E over U as in (10). This is independent of the choice of σ, and the operators Ω Then F X,Y commutes with the Clifford action, and that of ρ ⊗ π (nat) (SU(N )). Similarly we obtain 2k-forms of endomorphisms of E by

4.2.
Computation of index character. On one hand, the first part in the right hand side of (14) can be thought as the action of the relative curvature form F On the other hand, the second part can be thought as the action of the spin curvature i,j R LC X,Y e i , e j c i c j . Hence the quantity (12) can be translated to (15) Tr L 2 (PU(N );V (nat) ) (λ(e −Ω )ρ(φ)) = N Tr L 2 (SU(N )) χ (nat) (λ(e −Ω )ρ(φ)).
Theorem 1. Let π be an irreducible representation with central character χ (nat) . The distribution ind SU(N ) (γ, / ∂ M ) satisfies Proof. Let E π be the π-isotypic component of E. Since the action of Cl C (M ) on E commutes with the action of SU(N ), the vector bundle E π is a Clifford submodule of E. We show that the relative Chern character ch(E π /S) of the Clifford module V π is represented by d π Tr(e π(Ω) ). Then the assertion will follow from the index formula (2) for Clifford modules. Let p π denote the projector SU(N ) χ π (g)g in the convolution algebra L 1 (SU(N )).
Then E π is the image of π P ⊗ π (nat) (p π ). When σ is a section of P , the sections Γ(dom σ; E π ) of E π over the domain of σ can be identified to the space of the functions for x ∈ dom σ. By the SU(N )-invariance of ∇ on E, the associated curvature form of E π is given by F ρ(p π ).
Remark 5. Consider the case of π = π (nat) as in Example 4. Thus the left hand side of (17) is equal to the signature number of M . In the right hand side, the term Tr V (nat) (π (nat) (Ω j )) is equal to the j-th component of the relative Chern character ch(∧ * T * M/S). Hence we recover the signature formula We obtain the following formula whose conceptual meaning is (5).
Corollary 6. The distribution ind SU(N ) (γ, / ∂ M ) can be written as where φ is any test function in C ∞ (SU(N )).
Proof. Since both sides are invariant under conjugation for φ, we may assume that φ is invariant under conjugation. By continuity and linearity, we may assume that φ is a character of some irreducible representation of SU(N ). Then the assertion follows from Lemma 2 and Theorem 1. Now, we can recover the fractional index formula (1) as a particular case of Corollary 6. Proof. For each j, the operator T ((Ω (σ) ) j ) is represented by a SU(N )-biinvariant differential operator of degree 2j. Suppose that φ agrees with the constant function 1 on a neighborhood of e. Then one has Tr(ρ(φ)) = φ(e) = 1, Tr(ρ(φ)F j ) = T ((Ω (σ) ) j ), φ = 0 (j > 0).
Hence one has Tr(γe −F ), φ = 1 in this case. Consequently one obtains which proves the assertion.
Remark 8. Since / ∂ M is formulated as a transversely elliptic operator on the SU(N )manifold P , the Kirillov type formulation of the equivariant index formula for such operators by Berline-Vergne [BV96a,BV96b] might be also employed to prove the above result. Our presentation is rather based on the Atiyah-Segal-Singer type formulation of the equivariant index theorem.