THE UNIVERSAL GERBE and LOCAL FAMILY INDEX THEORY 1. Introduction Bundle Gerbes on Differentiable Manifolds Are Introduced by Mu

THE UNIVERSAL GERBE AND LOCAL FAMILY INDEX THEORY

ALAN L. CAREY, BAI-LING WANG

Abstract. The goal of this paper is to apply the universal gerbe developed in [CMi1] and [CMi2] and the local family index theorems to give a uniﬁed viewpoint on the known examples of geometrically interesting gerbes, including the determinant bundle gerbes in [CMMi1], the index gerbe in [L] for a family of Dirac operators on odd dimensional closed manifolds. We also discuss the associated gerbes for a family of Dirac operators on odd dimensional manifolds with boundary, and for a pair of Melrose-Piazza’s Cl(1)-spectral sections for a family of Dirac operators on even dimensional closed manifolds with vanishing index in K- theory. The common feature of these bundle gerbes is that there exists a canonical bundle gerbe connection whose curving is given by the degree 2 part of the even eta-form (up to an locally deﬁned exact form) arising from the local family index theorem.

1. Introduction Bundle gerbes on differentiable manifolds are introduced by Murray in [Mur] as differential geometric objects corresponding to Brylinski’s description in terms of sheaves of groupoids for Giraud’s gerbes in [Bry]. Associated with a smooth submersion π : Y M, let Y [2] the fiber product of π with the → obvious groupoid structure. A bundle gerbe over M, as defined in [Mur], consists of a sujective G submersion π : Y M, and a principal U(1)-bundle over Y [2] = Y Y together with a → G ×π groupoid multiplication on , which is compatible with the natural groupoid multiplication on G Y [2]. We represent a bundle gerbe over M as the following diagram G (1) G

π1 / Y [2] / Y π2

M with the bundle gerbe product given by an isomorphism (2) m : π∗ π∗ π∗ 1 G⊗ 3 G⊗ → 2 G [3] as principal U(1)-bundles over Y = Y π Y π Y , and πi, i=1,2, 3, are three natural projection [3] [2] × × from Y to Y by omitting the entry in i position for πi. The tautological bundle gerbes in terms of path ﬁbrations and the lifting bundle gerbes associated to a central extension of groups are given as examples in [Mur]. The bundle gerbe in its stable isomorphism class admits a canonical isomorphism class given by its local systems: the local bundle gerbe (Cf. [MS]). When M is covered by a good cover U such that all their ﬁnite intersections are contractible and the submersion π admits a { α} 1 2 ALAN L. CAREY, BAI-LING WANG

local section sα over Uα, then the local bundle gerbe can be described by a family of local U(1)-bundles over U = U U , the pull-back of via (s ,s ): {Gαβ} { αβ α ∩ β} G α β (3) αβ αβ G F π 1 // αβ Uαβ / α Uα π2 F F M and the bundle gerbe product (2) is given by an isomorphism (4) φ : = , αβγ Gαβ ⊗ Gβγ ∼ Gαγ over Uαβγ, such that φαβγ is associative as a groupoid multiplication. A related local picture of gerbes has been introduced by Hitchen [Hit]. A Cˇech cocycle f can be obtained from the isomorphism (4) by choosing a section s { αβγ} αβ of , i.e., Gαβ φ (s s )= f s αβγ αβ ⊗ βγ αβγ · αγ for a U(1)-valued function f over U . The equivalence class of f doesn’t depend on αβγ αβγ { αβγ} the choices of local section s , and represents the Dixmier-Douady class of the bundle gerbe { αβ} in 2 3 Z HCˇech(M,U(1)) ∼= H (M, ), where U(1) is the sheaf of continuous U(1)-valued functions over M. The geometry of bundle gerbes, bundle gerbe connections and their curvings are studied in [Mur]. On the corresponding local bundle gerbe (3), a bundle gerbe connection is a family of U(1)-connections on which is compatible with the isomorphism (4), i.e., {∇αβ} {Gαβ} A = s /s satisﬁes αβ ∇αβ αβ αβ −1 (5) Aαβ + Aβγ + Aγα = fαβγ dfαβγ.

A curving then is a locally defined 2-form Bα, the so-called B-field, such that, (6) B B = dA , α − β αβ over U U . We remark that the B-field B is unique, up to locally defined exact 2- α ∩ β { α}α forms. For the B-field B + dC , then we need to modify the gerbe connection by adding { α α}α C C to A without any effect on the compatibility condition (5). This local description { α − β} { αβ} of connection and curving for the canonical local bundle gerbe defines an element

[(fαβγ, Aαβ ,Bα)] in the degree three Deligne cohomology group H3 (M, 3). The curvature of B-field is given Del D by dB , note that { α} dBα = dBβ, on Uαβ is a globally defined 3-form which represents the image of the Dixmier-Douady class in H3(M, R) i up to a factor of , just like the case of line bundles with connections. In this paper, we 2π often suppress this factor and identify the bundle gerbe curvature with the differential form representing the image of the Dixmier-Douady class in H3(M, R). THE UNIVERSAL GERBE AND LOCAL FAMILY INDEX THEORY 3

To be precise, a B-field of a bundle gerbe over M should be a Hopkins-Singer differential G cocycle as defined in [HS] whose equivalence class is the corresponding degree three Deligne cohomology class. Recall that the following commutative diagram / H3 (M, 3) / H3(M, Z) Del D

3 / 3 ΩZ(M) H (M, R)

3 where ΩZ(M) is the space of closed 3-form on M with integral periods. In this sense, we often say that the curvature of the B-field represents the image of the Dixmier-Douady class of the corresponding bundle gerbe in H3(M, R), even loosely speaking, the curvature of the B-field represents the Dixmier-Douady class of the corresponding bundle gerbe. We remark that there is a subtlety here relating to the gauge transformations of the B-field. To illustrate this subtlety, given a globally defined closed 2-form ω (not necessarily of integer periods), with respect to a cover U of M we can write ω = dθ as an exact 2-form by { α} |Uα α Poincare Lemma. Then on the one hand, one can see that (1, 0,ω ) |Uα represents a trivial degree three Deligne class in H3 (M, 3) if and only if the closed 2-form Del D ω has integer periods. This claim follows from the exact sequence

2 2 3 3 3 0 Ω (M)/ΩZ(M) H (M, ) H (M, Z) 0. −→ −→ D −→ −→ In this sense, the gauge transformation of the B-field is given by line bundles with connection on M. On the other hand, (1,θ θ ,ω )=(1,θ θ , dθ ) α − β |Uα α − β α always represents a trivial degree three Deligne class in H3 (M, Z), as (1,θ θ , dθ ) is a Del α − β α coboundary element. In this paper, a curving on a bundle gerbe is always defined up to a locally defined exact 2-form in this latter sense so that the corresponding B-field is well-defined up to a degree three Deligne coboundary term of form (1,θ θ , dθ ), here dθ may not be a globally α − β α α defined exact 2-form. We hope that this remark will clarify any confusion when we talk about curvings on a bundle gerbe, such as degree 2 part of the even eta-form up to an exact 2-form. Earlier applications to various aspects of quantum field theory are extensively exploited in [BoMa][BCMMS] [CMu1][CMMi1][CMMi2] [CMi1][CMi2] [EM][GR] [Mic] which cover a wide range of applications. For example they provide intrinsic geometric meaning to string structures on manifolds (a problem originating from string theory), give a topological sense to local Hamiltonian or commutator anomalies (also referred to as Schwinger terms) for the gauge group action on fermionic Fock spaces, describe the geometry of Wess-Zumino-Witten actions and finally give a framework for the study of twisted K-theory for D-brane charges. A significant application comes from the construction of the universal gerbe in [CMi1][CMi2] where Carey and Mickelsson analysed the obstruction to obtaining a second quantization for a smooth family of Dirac operators on an odd dimensional spin manifold. Explicit computations of the Dixmier-Douady class for the universal gerbe were also done in [CMi1][CMi2]. Using techniques from local family index theory, in [L], Lott gives a construction of the higher degree analogs of the determinant line bundle for a family of Dirac operators on odd-dimensional 4 ALAN L. CAREY, BAI-LING WANG closed spin manifolds, which realized Carey and Mickelsson’s universal gerbe in [CMi1][CMi2] in this particular case. The connective structures and gerbe holonomy are discussed by Bunke in [Bu1]. The construction in Lott’s paper [L] is rather technical and doesn’t work for a family of Dirac operators on odd-dimensional manifolds with boundary. In the present paper, we will apply the universal gerbe developed in [CMi1] [CMi2] to give a unified viewpoint on the known examples of geometrically interesting gerbes and also construct gerbes for a family of Dirac operators on odd dimensional manifolds with boundary and for a pair of Melrose-Piazza Cl(1)-spectral sections for a family of Dirac operators on even dimensional closed manifolds with vanishing index in K-theory. These latter gerbes exhaust all bundle gerbes corresponding to the Dixmier-Douady classes in the image of the Chern character map on the K1-group of the underlying manifold. In all these examples of gerbes, there exists a canonical bundle gerbe connection whose curving, up to an exact 2-form, is given by the degree 2 part of the even eta-form arising from the local family index theorem. The framework for our study of the geometry of bundle gerbes in this paper is provided by the four famous local family index theorems developed by Bismut[Bis2], Bismut-Freed[BF], Bismut-Cheeger[BC2] and Melrose-Piazza [MP1][MP2], which are the corner-stones of local family index theory. For the convenience of readers and to set up various notations, we give a brief review of local family index theory in Section 2, and state these four local family index theorems as Theorems 2.2, 2.4, 2.6 and 2.7. In Section 3, we recall the construction of the determinant bundle gerbe in [CMMi1] and study its geometry in Theorem 3.1. In Section 4, we give a complete proof of the existence theorem (Theorem 4.1) of the canonical bundle gerbe associated to any element in K1-group using the universal gerbe developed in [CMi1][CMi2]. In Section 5, we apply the universal gerbe and the canonical bundle gerbe constructed in section 4 to discuss the index gerbe associated to a family of Dirac operators on odd dimensional manifold with or without boundary. This gives a simple construction of the gerbe, which was discussed in [L] and [Bu1], for a family of Dirac operators on an odd dimensional closed manifold. Two new examples of bundle gerbes are obtained, one for a family of Dirac operators on odd dimensional manifolds with boundary, and the other for a pair of Melrose-Piazza Cl(1)-spectral sections for a family of Dirac operators on even dimensional closed manifolds with vanishing index in K-theory. Some relations among these bundle gerbes are also briefly included at the end of this section. Our descriptions of the universal gerbe and its applications demonstrate on the one hand that these powerful local family index theorems provide simple and clear perspectives to study those bundle gerbes associated to families of Dirac operators, and on the other hand that the bundle gerbe theory also motivates the further understanding of local family index theory for manifolds with corners. For a single manifold with corners, Fredholm perturbations of the Dirac type operators and their index are throughly studied by Loya and Melrose in [LM]. A different approach to these latter questions is contained in [Bu2]. We intend the present paper as a different and much more accessible method for tackling these issues. Acknowledgement ALC acknowledges advice from P. Piazza on spectral sections and also thanks V. Mathai for conversations on his joint work with R. Melrose related to the theme THE UNIVERSAL GERBE AND LOCAL FAMILY INDEX THEORY 5 analysed in this paper. BLW wishes to thank Xiaonan Ma for many invaluable comments and his explanations of eta forms.

2. Review of local family index theory In this section, we briefly review local index theory and eta forms. Local family index theorems refine the cohomological family index theorem of Atiyah-Singer, while eta forms transgress the local family index theorem at the level of differential forms. The main reference for this section is Bismut’s survey paper [Bis1] and the references therein. Let π : X B be a smooth fibration over a closed smooth manifold B, whose fibers are → diffeomorphic to a compact, oriented, spin manifold M. Let(V,hV , V ) be an Hermitian vector ∇ bundle over X equipped with a unitary connection. Let gX/B be a metric on the relative tangent bundle T (X/B) (the vertical tangent subbundle of TX), which is fiberwise a product metric near the boundary if the boundary of M is non- empty. In the latter case, ∂X B is also a fibration with compact fiber ∂M. Let S be the → X/B corresponding spinor bundle associated the spin structure on (T (X/B),gX/B). Let T H X be a smooth vector subbundle of TX, (a horizontal vector subbundle of TX), such that (7) TX = T H X T (X/B), ⊕ and such that if ∂M is non-empty, T H X T (∂X). Then T H (∂X) = T H X is a hori- |∂X ⊂ |∂X zontal subbundle of T (∂X). Denote by P v the projection of TX onto the vertical tangent bundle under the decomposition (7). As shown in Theorem 1.9 of [Bis2], (T H X,gX/B) determines a canonical Euclidean connection X/B as follows. Choose a metric gX on TX such that T H X is orthogonal to ∇ T (X/B) and such that gX/B is the restriction of gX to T (X/B). Then X/B = P v X ∇ ◦ ∇ where X is the Levi-Civita connection on (TX,gX ). ∇ Write x/2 Aˆ(x)= , sinh(x/2) Ch(x)= exp(x). Then the corresponding characteristic classes can be represented by closed differential forms on X as i( X/B )2 Aˆ(T (X/B), X/B)= det1/2 Aˆ( ∇ ) , ∇ 2π i( V )2 Ch(V, V )= T r Ch( ∇ ) . ∇ 2π ∞ For any b B, let /Db be the Dirac operator acting on C (Xb, (SX/B V ) X ), where ∈ ⊗ | b X = π−1(b) is the fiber of π over b. Then /D is a family of elliptic operators over B b { b}b∈B acting on an infinite dimensional vector bundle H where H = C∞(X , (S V ) ). b b X/B ⊗ |Xb 2 The Hermitian metrics on the spinor bundle SX/B and V induce a natural L Hermitian metric on H, as we can identify smooth sections of H with smooth sections of S V ) on X/B ⊗ X and integrate along the fiber using the Riemannian volume form. 6 ALAN L. CAREY, BAI-LING WANG

Denote by ξ˜ the horizontal lift of a vector field ξ TB. Define ∈ T (ξ , ξ )= P v([ξ˜ , ξ˜ ]). 1 2 − 1 2 Let f be a local frame of TB, and let f i be the dual frame of T ∗B. Set { i} { } 1 c(T )= f if j c(T (f ,f )), 2 i j where c( ) is the Clifford action of T (X/B) on S . Then c(T ) Λ(T ∗B) ˆ End(H), the · X/B ∈ ⊗ graded tensor product when H is Z2-graded in the case of even fibers. Define a connection H on H by ∇ S ⊗V H s = X/B s, ∇ξ ∇ξ˜ where SX/B ⊗V is the induced connection on S V from X/B and V , ξ TB, and s is ∇ X/B ⊗ ∇ ∇ ∈ a smooth section of S V on X. X/B ⊗ Let dvolM be the Riemannian volume form along the fibers M, the Lie derivative operator

Lξ˜ acting on tensors along the ﬁber M deﬁnes a tensor

Lξ˜(dvolM ) divM (ξ)= . dvolM 1 Then H,u = H + div (ξ) defines a unitary connection on H (cf. [Bis2]). The curvature of ∇ξ ∇ξ 2 M H,u takes values in first order differential operators acting along the fibers of π. ∇ξ For convenience in the expression of local index formulae, let φ be the endomorphism of Λ(T ∗B) such that φ(ω)=(2πi)−degω/2ω.

2.1. Local family index theorem for closed manifolds. Case 1: Even dimensional fibers We assume that the fibers of π : X B are even dimensional closed manifolds. Then the → spinor bundle S = S+ S− is Z -graded, hence, H = H+ H− with X/B X/B ⊕ X/B 2 ⊕ H± = C∞(X , (S± V ) ) b X/B ⊗ |Xb and /D is a family of self-adjoint, elliptic odd operators over B interchanging { b}b∈B C∞(X , (S+ V ) ) and C∞(X , (S− V ) ). Let /D be the restriction of /D to b X/B ⊗ |Xb b X/B ⊗ |Xb ±,b b C∞(X , (S± V ) ), so b X/B ⊗ |Xb 0 /D−,b /Db = /D+,b 0 ! Then the family /D defines a K0-element { b}b∈B Ind( /D ) K0(B). + ∈ The Atiyah-Singer family index formula is given by (8) Ch Ind( /D ) = π [Aˆ(T (X/B))Ch(V )] Heven(B, R) + ∗ ∈ where π∗ the push-forward map given by the integration along the fibers on the level of differential forms. In order to get explicit differential forms, we need to introduce a Bismut superconnection for Z graded vector bundles. Let E = E E be a Z graded complex vector space, let 2 + ⊕ − 2 THE UNIVERSAL GERBE AND LOCAL FAMILY INDEX THEORY 7

τ = 1 on E deﬁning the Z -grading. The supertrace for a trace class operator A End(E) ± ± 2 ∈ is deﬁned to be

T rs(A)= T r(τA). Note that the supertrace vanishes on any supercommutator. For a Hermitian Z vector bundle E with a unitary connection E preserving the grading, 2 ∇ then the super Chern character for (E, E ) is deﬁned to be ∇ i( E)2 Ch (E, E )= T r Ch( ∇ ) . s ∇ s 2π

The Bismut superconnection on the inﬁnite dimensional Z2-graded bundle H, adapted to /D, is deﬁned to be

H,u c(T ) (9) At = + √t /D , ∇ − 4√t acting on Ω∗(B) ˆ C∞(B, H). Note that the ﬁberwise ellipticity of A implies that the heat ⊗ t operator exp( A2) − t is ﬁberwise trace class in the supertrace sense.

Definition 2.1. Define α = φT r exp( A2) , t s − t (10) 1 ∂At 2 βt = φT rs exp( A ) . √2πi ∂t − t The following local family index theorem of ([Bis2] and [BGV]) refines the Atiyah-Singer family index formulae (8), ∂α Theorem 2.2. The forms α and β are real and satisfy t = dβ . The form α is closed t t ∂t − t t and its cohomology class [αt] is constant and [α ]= Ch(Ind( /D )) Heven(B, R). t + ∈ Moreover αt and βt have the following asymptotic behaviours. (a) As t 0, → α = π Aˆ(T (X/B), X/B)Ch(V, V ) + (t), t ∗ ∇ ∇ O β = (1). t O (b) Assume that Ker /D is a smooth subbundle of H with Ker /D,u given by the orthogonal ∇ projection of H,u on Ker /D. As t , ∇ → ∞ Ker /D,u − 1 αt = Chs(Ker /D, )+ (t 2 ), − 3 ∇ O β = (t 2 ). t O Define the odd eta form on B to be ∞ ηõdd = βtdt. Z0 Then this odd eta form ηõdd satisfies the transgression formula (11) dη˜ = π Aˆ(T (X/B), X/B)Ch(V, V ) Ch (Ker /D, Ker /D,u). odd ∗ ∇ ∇ − s ∇ 8 ALAN L. CAREY, BAI-LING WANG

Case 2: odd dimensional fibers When M is an odd dimensional closed manifold, the family /D is a family of self-adjoint { b}b∈B operators on H. Then the Atiyah-Singer index theory tells us that Ind( /D) K1(B) ∈ with its odd Chern character given by (12) Ch(Ind( /D)) = π [Aˆ(T (X/B))Ch(V )] Hodd(B, R). ∗ ∈ To define the Bismut superconnection in this setting, we need to introduce an odd variable such that σ2 = 1. We may do this by tensoring on a complex vector space on which the Clifford element σ acts as well as a Z -grading. Then for any complex vector bundle E on B, E C[σ] 2 ⊗ and End(E) C[σ] are Z -graded in the obvious way. Define the σ-trace to be the functional ⊗ 2 Λ(T ∗B) ˆ End(E) C[σ] Λ(T ∗B), ⊗ ⊗ → such that if ω Λ(T ∗B), and A End (E), then ∈ ∈ T rσ(ωA)=0 T rσ(ωAσ)= ωT r(A). Define the Bismut superconnection for the odd dimensional case to be

H,u c(T )σ (13) At = + √t /Dσ . ∇ − 4√t Definition 2.3. Define 1 σ 2 α = (2i) 2 φT r exp( A ) , t − t (14) 1 ∂A β = φT rσ t exp( A2) . t √π ∂t − t Then we have the following local family index theorem for the odd case ([BF]). ∂α Theorem 2.4. The forms α and β are real and satisfy t = dβ . The form α is closed, t t ∂t − t t its cohomology class [αt] is constant and [α ]= Ch(Ind( /D)) Hodd(B, R). t ∈ Moreover αt and βt have the following asymptotic behaviours. (a) As t 0, → α = π Aˆ(T (X/B), X/B)Ch(V, V ) + (t), t ∗ ∇ ∇ O β = (1). t O (b) Assume that Ker /D is a smooth subbundle of H. As t , → ∞ − 1 αt = (t 2 ), O − 3 β = (t 2 ). t O Define the even eta form on B to be ∞ η˜even = βtdt. Z0 Then this even eta form η˜even satisfies the following transgression formula (15) dη˜ = π Aˆ(T (X/B), X/B)Ch(V, V ) . even ∗ ∇ ∇ THE UNIVERSAL GERBE AND LOCAL FAMILY INDEX THEORY 9

2.2. Local family index theorem for manifolds with boundary. Case 1: Even dimensional ﬁbers. Assume that M is an even dimensional manifold with boundary ∂M. The family of boundary Dirac operators /D∂M for the odd dimensional ﬁbration ∂X B, if Ker /D is a smooth { b }b∈B → bundle over B, assigns an even eta-formη ˜even on B satisfying the transgression formula (15)

dη˜ = Aˆ(T (X/B), X/B)Ch(V, V ). even ∇ ∇ Z∂M With the assumption of invertibility of the boundary Dirac operators, Bismut and Cheeger proved the local family index theory in [BC2].

Theorem 2.5. Assume that Ker /D∂M = 0 for any b B. With the Atiyah-Patodi- b ∈ Singer boundary condition, the family of Dirac operators /DM has a well-deﬁned index { +,≥} Ind( /DM ) K0(B), whose Chern character is given by +,≥ ∈

Ch Ind( /DM ) = Aˆ(T (X/B), X/B)Ch(V, V ) η˜ Heven(B, R). +,≥ ∇ ∇ − even ∈ ZM Recall that the Atiyah-Patodi-Singer boundary condition is given by the family of orthogonal projection operators P on the direct sum of the eigenspaces of /D∂M associated to the non- { ≥,b} b negative spectrum for any b B. The invertibility condition ensures that P≥,b is a smooth ∈ M { } 2 family of boundary conditions. The Dirac operator /D+,≥;b acts on the space of L1 sections of (S+ V ) with boundary component belonging to KerP . Then /DM is a family of X/B ⊗ |Xb ≥,b { +,≥} Fredholm operators. For the general situation, Melrose-Piazza introduced the notion of a spectral section for the family of boundary Dirac operators /D∂M in [MP1]. A spectral section is a smooth family { b }b∈B of self-adjoint pseudo-diﬀerential projections

P : C∞(∂X , (S V ) ) C∞(∂X , (S V ) ), b b ∂M/B ⊗ |∂Xb → b ∂M/B ⊗ |∂Xb ∂M such that there is R> 0 for which /Db s = λs implies that

Pbs = s λ>R, P s =0 λ< R. b − The spectral section exists if and only if Ind( /D∂M )=0 K1(B). In our case, Ind( /D∂M )=0 ∈ follows from the cobordism invariance of the Atiyah-Singer index theory. For a spectral section P , Melrose-Piazza also constructed a similar Bismut superconnection (13) where /D∂M is replaced by an interpolator between /D∂M for t << 1 and a suitable pertur- ∂M bation /D + AP (with AP depending on P ) for t >> 1. The family AP,b b∈B is a family of ∂M { } P self-adjoint smooth operators such that /Db + AP,b is invertible. Then an even eta formη ˜even can be deﬁned as in Theorem 2.4 , modulo exact forms, depending only on /D∂M and { b }b∈B the spectral section P . They established the local family index theorem for even dimensional manifolds with nonempty boundary ([MP1]) generalizing the Bismut-Cheeger Theorem 2.5.

Theorem 2.6. Let π : X B be a smooth fibration with fiber diffeomorphic to an even → dimensional spin manifold M with boundary and the geometric data given as in the beginning of this section. With the boundary condition given by a spectral section P for /D∂M , the { b }b∈B 10 ALAN L. CAREY, BAI-LING WANG family of Dirac operators /DM has a well-defined index Ind( /DM ) K0(B), with the { +,P ;b}b∈B +,P ∈ Chern character form given by

Ch Ind( /DM ) = Aˆ(T (X/B), X/B)Ch(V, V ) η˜P Heven(B, R). +,P ∇ ∇ − even ∈ ZM Case 2: odd dimensional fibers Now we assume that the fibers of the fibration π : X B are odd dimensional manifolds → with non-empty boundary. The family of boundary Dirac operator /D∂M consists of self- { b }b∈B adjoint, elliptic, Z2-graded odd differential operators acting on (16) H = C∞(X , (S+ V ) ) C∞(X , (S− V ) ). b X/B ⊗ |Xb ⊕ b X/B ⊗ |Xb The odd dimensional Dirac operator /DM acting on C∞(X , (S V ) ) is neither Fred- b b X/B ⊗ |Xb holm nor self-adjoint without suitable boundary conditions. The Fredholm boundary conditions as developed in [Sco0] form the restricted smooth Grassmannian with respect to the chirality polarization (16). To get self-adjointness, we need to impose a Cl(1)-condition (where Cl(1) denote the Clifford algebra Cl(R)):

c(du)PW + PW c(du)= c(du), with u being the inward normal coordinate near the boundary ∂M, and PW is the orthogonal projection corresponding to the Fredholm boundary condition W in the restricted smooth Grassmannian. In order to get a continuous family of self-adjoint Fredholm extensions from /DM , { b }b∈B Melrose and Piazza introduced a Cl(1)-spectral section P and constructed an odd eta form η˜P which, modulo exact forms, depends only on /D∂M and the Cl(1)-spectral section odd { b }b∈B ([MP2]).

Theorem 2.7. ([MP2]) Let π : X B be a smooth fibration with fiber diffeomorphic to an odd → dimensional spin manifold M with boundary and the geometric data given as in the beginning of this section. With the boundary condition given by a Cl(1)-spectral section P for /D∂M , { b }b∈B the family of Dirac operator /DM has a well-defined index { P ;b}b∈B Ind( /DM ) K1(B), P ∈ with the Chern character form given by

(17) Ch Ind( /DM ) = Aˆ(T (X/B), X/B)Ch(V, V ) η˜P Hodd(B, R). P ∇ ∇ − odd ∈ ZM 2.3. Determinant line bundle. Let π : X B be a smooth fibration over a smooth manifold → B, whose fibers are diffeomorphic to a compact, oriented, even dimensional spin manifold M. Let (V,hV , V ) be a Hermitian vector bundle over X equipped with a unitary connection. ∇ Let gX/B be a metric on the relative tangent bundle T (X/B) (the vertical tangent subbundle of TX), which is fiberwisely a product metric near the boundary SX/B be the corresponding spinor bundle associated the spin structure on (T (X/B),gX/B). ∂M With a choice of spectral section P for the family of boundary Dirac operators /Db b∈B, M { } we have a family of Fredholm operators /D+,P ;b b∈B over B. There is a honest determinant M { } line bundle, denoted by Det( /D+,P ), over B given by Det( /DM ) =Λtop(Ker /DM )∗ Λtop(Coker /DM ). +,P b +,P ;b ⊗ +,P ;b THE UNIVERSAL GERBE AND LOCAL FAMILY INDEX THEORY 11

Using zeta determinant regularization, as in [Qu][BF][Woj][SW], a Hermitian metric and a M unitary connection /D+,P can be constructed on Det( /DM ) such that ∇ +,P M (18) c (Det( /DM ), /D+,P )= π Aˆ(T (X/B), X/B)Ch(V, V ) η˜P , 1 +,P ∇ ∗ ∇ ∇ − even (2) P Note that the eta formη ˜even is defined modulo exact forms, the above equality (18) holds modulo exact 2-forms. For a smooth fibration X over a smooth manifold B, with closed even-dimensional fibers partitioned into two codimensional 0 submanifolds M = M M along a codimension 1 0 ∪ 1 submanifold ∂M = ∂M . Assume that the metric gX/B is of product type near the collar 0 − 1 neighborhood of the separating submanifold. Let P be a spectral section for for /D∂M0 , { b }b∈B then I P is a spectral section for /D∂M1 . Scott showed ([Sco]) that the determinant − { b }b∈B line bundles for these three families of /DM , /DM0 and /DM1 satisfy the { +;b}b∈B { +,P ;b}b∈B { +,I−P ;b}b∈B following gluing formulae as Hermitian line bundles: (19) Det( /DM ) = Det( /DM0 ) Det( /DM1 ), + ∼ +,P ⊗ +,I−P M moreover the splitting formulae for the curvature of /D+ implies that ∇ M M0 M1 M /D M0 /D M1 /D (20) c (Det( /D ), + )= c (Det( /D ), +,P )+ c (Det( /D ), +,I−P ). 1 + ∇ 1 +,P ∇ 1 +,I−P ∇ See ([MaP][Pi1] [Pi2]) for another explicit expressions of the corresponding Quillen metrics M M0 M1 and their unitary connections on Det( /D+ ),Det( /D+,P ) and Det( /D+,I−P ), which are called the b-Quillen metrics and the b-Bismut-Freed connections.

3. Determinant bundle gerbe In [CMMi1], Carey-Mickelson-Murray applied the theory of bundle gerbes to study the bundle of fermionic Fock spaces parametrized by vector potentials on odd dimensional manifolds and explained the appearance of the so-called Schwinger terms for the gauge group action. We briefly review this construction here. Let M be a closed, oriented, spin manifold with a Riemannian metric gM . Let S be the spinor bundle over M and let (V,hV ) be a Hermitian vector bundle. Denote by the space of unitary connections on V and by G the based gauge transformation A group, that is, those gauge transformations fixing the fiber over some fixed point in M. With proper regularity on connections, the quotient space /G is a smooth Frechet manifold. A Carey-Mickelson-Murray constructed a non-trivial bundle gerbe on /G, and gave a formula A for its Dixmier-Douady class. We summarise their construction in terms of the local family index theorem. Let λ R, denote ∈ U = A λ / spec( /D ) , λ { ∈ A| ∈ A } where the Dirac operator /D acts on C∞(M,S V ). Let H be the space of square integrable A ⊗ sections of S V . For any A U , there is a uniform polarization: ⊗ ∈ λ H = H (A, λ) H (A, λ) + ⊕ − given by the spectral decomposition of H with respect to /D λ into the positive and negative A − eigenspaces. Denote by PH±(A,λ) the orthogonal projection onto H±(A, λ). Fix λ R and a reference connection A such that the Dirac operator /D λ is 0 ∈ 0 ∈ A A0 − 0 invertible, denote by PH±(A0,λ0) the orthogonal projection onto H±(A0, λ0). 12 ALAN L. CAREY, BAI-LING WANG

Over Uλ, there exists a complex line bundle Detλ, which is essentially the determinant line bundle for the even dimensional Dirac operators on [0, 1] M coupled to some path of × connections in with respect to the generalized Atiyah-Singer-Patodi boundary condition A ([APS]). The path of vector potentials A(t) can chosen to be any smooth path connecting A0 and A U , for simplicity, we take ∈ λ (21) A = A(t)= f(t)A + (1 f(t))A − 0 for t [0, 1] where f is zero on [0, 1/4], equal to one on [3/4, 1] and interpolates smoothly ∈ between these values on [1/4, 3/4]. The generalized Atiyah-Singer-Patodi boundary condition is determined by the orthogonal spectrum projection P = P P , λ H+(A0,λ0) ⊕ H−(A,λ) that is, the even dimensional Dirac operator on [0, 1] M acts on the spinor ﬁelds whose × boundary components belong to H (A , λ ) at 0 M and H (A, λ) at 1 M. − 0 0 { }× + { }× Note that Pλ can be thought as a spectral section in the sense of Melrose-Piazza [MP1] for the family of Dirac operators /D /D , { A0 ⊕ A}A∈Uλ on 0 M 1 M. Then the eta formη ˜Pλ associated with /D /D and the { }× ⊔{ }× even { A0 ⊕ A}A∈Uλ spectral section Pλ is an even diﬀerential form on Uλ, which is unique up to an exact form. Pλ Denote by /DA the even dimensional Dirac operator /DA with respect to the APS-type boundary condition Pλ. Then the even dimensional Dirac operator

Pλ Pλ 0 /DA,− /DA = Pλ /DA,+ 0 ! is a Fredholm operator, whose determinant line is given by

top Pλ ∗ top Pλ Det (A)=Λ (Ker /DA ) Λ (Coker /DA ). λ ,+ ⊗ ,+ Pλ The family of Fredholm operators /DA , parametrized by A U deﬁnes a determinant { } ∈ λ line bundle over Uλ, which is given by

Detλ = Detλ(A). A[∈Uλ The determinant line bundle Detλ can be equipped with a Quillen metric and a Bismut-Freed unitary connection whose curvature can be calculated by the local family index theorem. Over U ′ = U U ′ , there exists a complex line bundle Det ′ such that λλ λ ∩ λ λλ ∗ Det ′ = Det Det ′ . λλ λ ⊗ λ These local line bundles Det ′ over , form a bundle gerbe as established in [CMMi1]. { λλ } A In the following, we will apply the local family index theorem to study the geometry of this determinant bundle gerbe. To apply the local family index theorem, we should restrict ourselves to a smooth finite dimensional submanifold of . For convenience however, we will formally A work on the infinite dimensional manifold directly. A Consider the trivial fibration [0, 1] M over with fiber [0, 1] M an even dimensional × × A A × manifold with boundary. Over [0, 1] M , there is a Hermitian vector bundle V which is × × A the pull-back bundle of V . THE UNIVERSAL GERBE AND LOCAL FAMILY INDEX THEORY 13

There is a universal unitary connection on V, also denoted by A, whose vector potential at (t, x, A) is given by

(22) A(t, x)= A(t)(x), where A(t) is given by (21). Denote by Ch(V, A) the Chern character of (V, A). Now we can state the following theorem regarding the geometry of the bundle gerbe over /G constructed in [CMMi1]. A Theorem 3.1. The local line bundles Det ′ descend to local line bundle over /G, which { λλ } A in turn deﬁnes a local bundle gerbe over /G, the corresponding Dixmier-Douady class is give A by Aˆ(T M, T M )Ch(V, A) . ∇ (3) ZM Pλ Moreover, the induced unitary connection and the even eta form η˜even (up to an exact 2-form) descend to a bundle gerbe connection and curving on the local bundle gerbe over / A G

Proof. Equip Detλ with the Quillen metric and its unitary connection, then its ﬁrst Chern class is represented by the degree 2 part of the following diﬀerential form:

Aˆ(T M, T M )Ch(V, A) η˜Pλ . ∇ − even Z[0,1]×M In this formula T M is the Levi-Civita connection on (TM,gM ), Aˆ(T M, T M ) represents the ∇ ∇ ˆ Pλ A-genus of M, andη ˜even is the even eta form on Uλ associated to the family of boundary Pλ Dirac operators and the spectral section Pλ. Note thatη ˜even (modulo exact forms) is uniquely determined, see Theorem 2.6.

The induced connection on Detλλ′ implies that its ﬁrst Chern class is given by

η˜Pλ η˜Pλ′ . even − even (2)

Pλ Note that the eta formη ˜even is unique up to an exact form (in a way analogous to the B-ﬁeld), P Pλ′ 2 hence, η˜ λ η˜even is a well-deﬁned element in H (M, R). From the transgression formula even − (2) for the eta forms and the Stokes formula, we know that over U ′ , λλ

(23) d(˜ηPλ ) = d(˜ηPλ′ ) = Aˆ(T M, T M )Ch(V, A) . even (2) even (2) ∇ (3) ZM Here use the fact that the contribution from 0 M vanishes as a diﬀerential form on , as { }× A we ﬁxed a connection A over 0 M. We also use the same notation (V, A) to denote the 0 { }× Hermitian vector bundle over M and the universal unitary connection A. × A As the gauge group acts on and Det covariantly, by quotienting out , we obtain the A λ G bundle gerbe over /G described in the theorem. A

Remark 3.2. The Dixmier-Douady class in Theorem 3.1 is often non-trivial. For example note that when dimM < 4, there is no contribution from the Aˆ-genus and in particular, for dimM =1 or 3, using the Chern-Simons forms, non-triviality was proved by explicit calculation in [CMMi1]. 14 ALAN L. CAREY, BAI-LING WANG

The above construction can be generalized to the ﬁbration case as in the local family index theorem for odd dimensional manifolds with boundary as in section 2. When the ﬁbers are closed odd dimensional spin manifolds, this leads to the index gerbe as discussed by Lott ([L]). In the next section, we discuss the universal bundle gerbe as in [CMi1][CMi2], which provides a unifying viewpoint for those bundle gerbes constructed from various determinant line bundles.

4. The universal bundle gerbe Let H be an inﬁnite dimensional separable complex Hilbert space. Let a.s be the space of F∗ all self-adjoint Fredholm operators on H with positive and negative essential spectrum. With the norm topology on a.s, Atiyah-Singer [AS1] showed that a.s is a representing space for F∗ F∗ the K1-group, that is, for any closed manifold B,

(24) K1(B) = [B, a.s] ∼ F∗ the homotopy classes of continuous maps from B to a.s. F∗ As a.s is homotopy equivalent to (1), the group of unitary automorphisms g : H H F∗ U → such that g 1 is trace class, we obtain − (25) K1(B) = [B, (1)]. ∼ U In [CMi1], a universal bundle gerbe was constructed on (1) with the Dixmier-Douady class U given by the basic 3-form 1 (26) T r(g−1dg)3, 24π2 the generator of H3( (1), Z). U For any compact Lie group, this basic 3-form gives the so-called basic gerbe. We prefer however to call it the universal bundle gerbe in the sense that many examples of bundle gerbes on smooth manifolds are obtained by pulling back this universal bundle gerbe via certain smooth maps from B to (1). U Note that the odd Chern character of K1(B) = [B, (1)] is given by U n! (27) Ch([g]) = ( 1)n T r (g−1dg)2n+1 , − (2πi)n+1(2n + 1)! n≥0 X for a smooth map g : B (1) representing a K1-element [g] in K1(B). This odd Chern → U character formula was proved in [Getz]. The following theorem was established in [CMi1] as the ﬁrst obstruction to obtaining a second quantization for a smooth family of Dirac operators (parametrized by B) on an odd dimensional spin manifold. By a second quantization, we mean an irreducible representation of the canonical anticommutative relations (CAR) algebra, the complex Cliﬀord algebra Cl(H H¯ ), which is ⊕ compatible with the action of the quantized Dirac operator. Such a representation is given by the Fock space associated to a polarization on H. For a bundle of Hilbert space over B, a continuous polarization always exists locally. This leads to a bundle gerbe over B.

Theorem 4.1. For any K1-element [g] K1(B). There exists a canonical isomorphism class ∈ of bundle gerbe whose Dixmier-Douady class is given by the degree 3 part of the odd Chern character Ch([g]). THE UNIVERSAL GERBE AND LOCAL FAMILY INDEX THEORY 15

Proof. The canonical bundle gerbe we are after is essentially the pull-back bundle gerbe from the universal bundle gerbe over (1) under a smooth map g : B (1) representing [g]. U →U Consider the polarized Hilbert space = L2(S1, H) with the polarization ǫ given by the H Hardy decomposition. = L2(S1, H)= . H H+ ⊕ H− That is we take the polarisation given by splitting into positive and negative Fourier modes. Then the smooth based loop group Ω (1) acts naturally on . From [PS], we see that U H (28) Ω (1) ( ,ǫ). U ⊂Ures H Here ( ,ǫ) is the restricted unitary group of H with respect to the polarization ǫ, those Ures H g U(H) such that the off-diagonal block of g is Hilbert-Schmidt. It was shown in [CMi1] that ∈ the inclusion Ω (1) ( ,ǫ) is a homotopy equivalence. U ⊂Ures H We know that the holonomy map from the space of connections on a trivial (1)-bundle over U S1 provides a model for the universal Ω (1)-bundle. Hence, (1) is a classifying space for Ω (1). U U U From K1(B) = [B, (1)], we conclude that elements in K1(B) are in one-to-one correspon- ∼ U dence with isomorphism classes of principal ( ,ǫ)-bundles over B. Ures H Associated to the basic three form (26) on (1), is the universal gerbe realized by the central U extension (29) 1 U(1) ˆ ( ,ǫ) 1. → → Ures →Ures H → Note that the fundamental representation of ˆ acts on the Fock space (see [PS]) Ures = Λ( ) Λ( ¯ ) FH H+ ⊗ H− This gives rise to a homomorphism ( ,ǫ) PU( ) which induces a PU( ) principal Ures H → FH FH bundle from the ( ,ǫ) principal bundle associated to the K1-element [g]. Ures H Denote by the resulting PU( ) principal bundle corresponding to a smooth map Pg FH g : B (1) representing [g]. The corresponding lifting bundle gerbe (as defined in [Mur]) is → U the canonical bundle gerbe over B for which we are looking. Recall that the lifting bundle gerbe associated a principal PU( )-bundle can be described FH locally as follows. Take a local trivialization of with respect to a good covering of B = U . Pg α α Assume over U , the trivialization is given by a local section α S s : U , α α →Pg|Uα such that the transition function over U = U U is given by a smooth function αβ α ∩ β γ : U PU( ). αβ αβ → FH We can define a local Hermitian line bundle over U by the pull-back the complex line Lαβ αβ bundle associated to (29) via γ . These local Hermitian line bundles define a bundle αβ {Lαβ} gerbe over B. From the odd Chern character formula (27), we know that 1 Ch([g]) = T r(g−1dg)3, (3) 24π2 which implies that the Dixmier-Douady class of is exactly the degree 3 part of Pg Ch([g]) H3(B, Z). ∈ 16 ALAN L. CAREY, BAI-LING WANG

For an Hermitian vector bundle (V,hV ) over an oriented, closed, spin manifold M with a Riemannian metric denote by the space of unitary connections on V and by the based A G gauge transformation group as in section 3. In [CMi2], an explicit smooth mapg ˜ : /G (1) was constructed: firstly assign to any A → U A a unitary operator in (1) by ∈ A U /D A g(A)= exp(iπ A ), → − /D + χ( /D ) | A| | A| where χ is any positive smooth exponentially decay function on [0, ) with χ(0) = 1. As g(A) ∞ is not gauge invariant, g(Au)= u−1g(A)u, for any u G, it doesn’t define a smooth map from /G to (1). In order to get a gauge invari- ∈ A U ant map, a global section for the associated U(H)-bundle U(H) is needed. This exists due A×G to the contractibility of U(H). This section is given by a smooth map r : U(H) such that A → r(Au)= u−1r(A). Then the required mapg ˜ : /G (1) is given byg ˜(A)= r(A)−1g(A)r(A). A →U Let B be a smooth submanifold of /G. The restriction ofg ˜ to B defines an element A in K1(B), which is exactly the family of Dirac operators over B associated to the universal connection A on V (see section 3 for the definition). From the local family index theorem, we know that 1 Aˆ(T M, T M)Ch(V, A) = T r(˜g−1dg˜)3 ∇ (3) 24π2 ZM in H3(B, R). One can verify that the canonical bundle gerbe constructed in the proof of The- orem 4.1 agrees with the determinant bundle gerbe constructed in section 3. In section 2 of [CMMi1], both of these constructions were presented. Explicit computations of the corresponding Dixmier-Douady class were given in [CMMi1], see [CMi2] for more examples.

5. Index gerbe as induced from the universal bundle gerbe Let π : X B be a smooth fibration over a closed smooth manifold B, whose fibers are → diffeomorphic to a compact, oriented, odd dimensional spin manifold M. Let (V,hV , V )be a ∇ Hermitian vector bundle over X equipped with a unitary connection. X/B Let g be a metric on the relative tangent bundle T (X/B) and let SX/B be the spinor bundle associated to (T (X/B),gX/B). Let T H X be a horizontal vector subbundle of TX. Then (T H X,gX/B) determines a connection X/B on T (X/B) as in section 2. ∇ 5.1. Family of Dirac operators on odd dimensional closed manifolds. When the fibers of π are closed, the family of Dirac operators /D defines a K1-element { b}b∈B Ind( /D) K1(B), ∈ with Ch(Ind( /D)) = π Aˆ(T (X/B), X/B)Ch(V, V ) Hodd(B, R) as given by the local ∗ ∇ ∇ ∈ family index theorem (Theorem 2.4). Then Theorem 4.1 provides a canonical bundle gerbe M over B whose Dixmier-Douady G class is given by π Aˆ(T (X/B), X/B)Ch(V, V ) . ∗ ∇ ∇ (3) One can repeat the construction in section 3 to get the gerbe connection and curving on this bundle gerbe. This is similar to the construction of Lott in [L]. THE UNIVERSAL GERBE AND LOCAL FAMILY INDEX THEORY 17

Now we can prove the following theorem, which was obtained in [L] by Lott using diﬀerent construction.

Theorem 5.1. Let π : X B be a smooth fibration with fibers closed odd dimensional spin → manifolds M with the geometric data given as in the beginning of this section. Then the associated family of Dirac operators defines a canonical bundle gerbe M over B equipped with a G Hermitian metric and a unitary gerbe connection whose curving (up to an exact form) is given by the locally defined eta form such that its bundle gerbe curvature is given by

Aˆ(T (X/B), X/B)Ch(V, V ) . ∇ ∇ (3) ZM Proof. Cover B by U (for λ R) such that λ ∈ U = b B /D λ is invertible . λ { ∈ | b − }

Over Uλ, the bundle of Hilbert spaces of the square integrable sections along the ﬁbers has a uniform polarization: H = H+(λ) H−(λ), b b ⊕ b the spectral decomposition with respect to /DM λ into the positive and negative eigenspaces. P,b − Denote by P the smooth family of orthogonal projections P onto H+(λ) along the λ { λ,b}b∈Uλ b above uniform polarization over Uλ. Note that the restriction of Ind( /D) on Uλ is trivial by the result of Melrose-Piazza in [MP1], as /D over Uλ admits a spectral section. So over Uλ, its Chern character form is exact:

(30) π Aˆ(T (X/B), X/B)Ch(V, V ) = dη˜Pλ ∗ ∇ ∇ even whereη ˜Pλ , unique up to an exact form, is the even eta form on U , associated to /DM even λ { b }b∈Uλ and the spectral section Pλ. ′ For λ λ , over U ′ = U U ′ , consider the fibration [0, 1] X B with even dimensional ≥ λλ λ ∩ λ × → fibers, the boundary fibration has two components 0 X and 1 X. The spectral section { }× { }× can be chosen to be

(31) P ′ = P (I P ′ ). λλ λ ⊕ − λ Pλλ′ Associated with this spectral section Pλλ′ , there exists an even eta formη ˜even on Uλλ′ , modulo exact forms, depending only on Pλλ′ and the family of boundary Dirac operators. From the deﬁnition of the eta form, we have

η˜Pλλ′ =η ˜Pλ η˜Pλ′ , even even − even which should be understood modulo exact even forms on Uαβ. Then the family of Dirac operators with boundary condition given by the spectral section

Pλλ′ , denoted by [0,1]×M /D b∈U ′ , { Pλλ′ ,b } λλ is a smooth family of Fredholm operators, which has a well-deﬁned index Ind( /D[0,1]×M ) in Pλλ′ 0 K (Uλλ′ ). 18 ALAN L. CAREY, BAI-LING WANG

The corresponding determinant line bundle over Uλλ′ , denoted by Detλλ′ , is a Hermitian line bundle equipped with the Quillen metric and the Bismut-Freed unitary connection. Its ﬁrst Chern class is given by the local family index formula (18):

X/B V P ′ (32) π Aˆ(T ([0, 1] X/U ′ ), )Ch(V, ) η˜ λλ . ∗ × λλ ∇ ∇ − even (2) Note that in this situation, the contribution from the chara cteristic class X/B V π Aˆ(T ([0, 1] X/U ′ ), )Ch(V, ) ∗ × λλ ∇ ∇ vanishes, as it has no component in [0, 1]-direction. Hence, the first Chern class of Detλλ′ is represented by the η˜Pλλ′ = η˜Pλ′ η˜Pλ . even (2) even − even (2) We understand that these eta forms are well-defined only modulo exact 2-forms, so the above equation should be viewed at the cohomological level. This implies that the even eta form, up to an exact 2-form, Pλ η˜even )(2), which is only defined over Uλ, is the curving (up to an exact 2-form) for the Bismut-Freed connection on Detλλ′ . Hence, the gerbe curvature is given by

d(˜ηPλ ) = d(˜ηPλ′ ) = Aˆ(T (X/B), X/B)Ch(V, V ) . even (2) even (2) ∇ ∇ (3) ZM

5.2. Family of Dirac operators on odd dimensional manifolds with boundary. Now we assume that the fibers of the fibration π : X B are odd dimensional manifolds with → non-empty boundary. A Cl(1)-spectral section P for the family of boundary Dirac operators /D∂M provides { b }b∈B a well-defined index for the family of self-adjoint Fredholm operators /DM : { P } Ind( /DM ) K1(B). P ∈ Then Theorem 4.1 provides a canonical bundle gerbe M over B whose Dixmier-Douady GP class is given by the local family index theorem 2.7: (33) π (Aˆ(T (X/B), X/B)Ch(V, V )) η˜P . ∗ ∇ ∇ − odd (3) P Here we should emphasize thatη õdd is an odd eta form associated to a perturbation of ∂M ∂M /Db b∈B by a family of self-adjoint smooth operators AP,b b∈B such that /Db + AP,b { } { }P is invertible as in [MP2]. Note that previously, we always defineη õdd up to an exact form. It would be interesting to understand the bundle gerbe connection and its curving such that the bundle gerbe curvature is given by (33). Note that the argument for the previous case can’t be applied here, as now [0, 1] M is a manifold with corner near ∂M, the local family index × theorem for such a manifold is not understood, and the even eta form for a family of Dirac operators on odd dimensional manifolds with boundary which transgresses the odd local index form (33) has not been found. For a single manifold with corners, Fredholm perturbations of Dirac operators and their index formulae have been developed by Loya-Melrose [LM]. For a family of Dirac operators on a manifold with corners, even in the case of corners up to codimension two, their Fredholm perturbations and the corresponding local family index theorems are still open. THE UNIVERSAL GERBE AND LOCAL FAMILY INDEX THEORY 19

Instead, we will apply the bundle gerbe theory to find this even eta form which transgresses the odd local index form (33) and discuss its implications for local family index theory for a family of Fredholm operators on a manifold with corners of a particular type, [0, 1] M. × For the fibration π : X B with odd dimensional manifolds with non-empty boundary as → fiber, the Melrose-Piazza Cl(1)-spectral section P for the family of boundary Dirac operators /D∂M gives rise to the family of self-adjoint Fredholm operators /DM with discrete { b }b∈B { P,b}b∈B spectrum. Cover B by U with λ R such that λ ∈ U = b B /DM λ is invertible . λ { ∈ | P,b − } M Note that, over Uλ, the family of self-adjoint Fredholm operators /DP,b has trivial index in 1 { } K (Uλ). This follows from the fact that, over Uλ, we have a uniform polarization of Hilbert space of the square integrable sections along the fibers with boundary condition given by the Cl(1)-spectral section P , H = H+ (λ) H− (λ), P,b P,b ⊕ P,b the spectral decomposition with respect to /DM λ into the positive and negative eigenspaces. P,b − Denote by P the smooth family of orthogonal projections P onto H+ (λ) along the λ { λ,b}b∈Uλ P,b above uniform polarization over Uλ. P,λ Hence, there exists an even form ηeven on Uλ, unique up to an exact form, satisfying the following transgression formula over Uλ: (34) dηP,λ = π (Aˆ(T (X/B), X/B)Ch(V, V )) η˜P . even ∗ ∇ ∇ − odd We remark that these ηP,λ should be even eta forms a la Melrose-Piazza’s construction { } in ([MP2]) associated with the family of self-adjoint Fredholm operators /DM and the { P,b}b∈Uλ spectral section Pλ over Uλ. From the abstract bundle gerbe theory in [Mur], we know that this locally defined even P,λ form ηeven (2) on Uλ is the curving, up to an exact form on Uλ, for the bundle gerbe induced from the universal gerbe as in Theorem 4.1 equipped with a gerbe connection, such that gerbe curvature is given by

d(ηP,λ ) = Aˆ(T (X/B), X/B)Ch(V, V ) η˜P . even (2) ∇ ∇ − odd (3) ZM Consider the fibration [0, 1] X B with even dimensional fibers. Now the fibers are × → manifolds with corners [0, 1] M, with two codimension two corners given by × 0 ∂M, 1 ∂M, { }× { }× and three codimension one faces 0 M, 1 M and [0, 1] ∂M. { }× { }× × ′ For λ λ , over U ′ , associated with the codimension one boundary fibration corresponding ≥ λλ to components 0 X and 1 X, we can choose a spectral section for the codimension one { }× { }× boundary Dirac operators /D{0}×M /D{1}×M to be P ⊕ P (35) P ′ = P (I P ′ ). λλ λ ⊕ − λ The corresponding family of Dirac operators [0,1]×M,+ /D b∈U ′ , { P,Pλλ′;b } λλ 20 ALAN L. CAREY, BAI-LING WANG

over Uλλ′ can be written as a family of operators of the form

∂ M 2 + 2 − (36) + /D : L ([0, 1] X , (S V ) ; P, P ′ ) L ([0, 1] X , (S V ) ) ∂t P,b 1 × b ⊗ |Xb λλ → × b ⊗ |Xb 2 + 2 where L ([0, 1] X , (S V ) ; P ′ ) denotes the L -integrable sections with boundary con- 1 × b ⊗ |Xb λλ 1 ditions given by P along ∂(X ) and P ′ along 0 X 1 X . b λλ { }× b ⊔{ }× b Note that /DM is a family of self-adjoint pseudodiﬀerential Fredholm operators with { P,b}b∈B ∂ discrete spectrum. From the explicit time evolution equation associated with + /DM , we ∂t P,b [0,1]×M,+ can see that /D b∈U ′ is a family of Fredholm operators, which induce a determinant { P,Pλλ′;b } λλ line bundle from the universal determinant line bundle over the space of Fredholm operators, P denoted by Detλλ′ . It would be interesting to know whether the methods of this paper can be extended to answer the following: P (a) Is there a Quillen-type metric and its unitary Bismut-Freed-type connection on Detλλ′ ? (b) Is the curvature of this unitary Bismut-Freed-type connection given by a yet to be deﬁned local family index theorem for manifolds with corners [0, 1] M such that the × degree 3 part implies the transgression formula (34)? Both these questions are also very important from the index theory viewpoint (see for example [LM] [Bu2]).

Remark 5.2. From Proposition 4 in [MP2], we know that if P1 and P2 are two Cl(1)-spectral sections for /D∂M , then the Atiyah-Singer suspension operation [AS1] defines a difference { b }b∈B element [P P ] K1(B), 2 − 1 ∈ such that Ind( /DM ) Ind( /DM ) = [P P ] K1(B), P1 − P2 2 − 1 ∈ whose Chern character is given by η˜P2 η˜P1 . It was also shown in Proposition 12 in [MP2] odd − odd that for a fixed P , as P ranges over all Cl(1)-spectral sections, [P P ] exhausts K1(B). 1 2 2 − 1 Apply Theorem 4.1 again to [P P ], we obtain a canonical bundle gerbe associated 2 − 1 GP2P1 to [P P ] K1(B) such that its Dixmier-Douady class is given by 2 − 1 ∈ η˜P2 η˜P1 . odd − odd (3) Moreover, we have the following isomorphism relating the bundle gerbes associated to families of Dirac operators on odd-dimensional manifolds with boundary equipped with two different Cl(1)-spectral sections for the family of boundary Dirac operators M = M . GP1 ∼ GP2 ⊗ GP2P1 Take a smooth fibration X over B, with closed odd-dimensional fibers partitioned into two codimensional 0 submanifolds M = M M along a codimension 1 submanifold ∂M = ∂M . 0 ∪ 1 0 − 1 Assume that the metric gX/B is of product type near the collar neighborhood of the separating submanifold. Let P be a Cl(1)-spectral section for /D∂M0 , then I P is a Cl(1)-spectral { b }b∈B − section for /D∂M1 . Then we have the following splitting formula for the canonical bundle { b }b∈B gerbe M obtained in Theorem 5.1: G M = M0 M1 . G ∼ GP ⊗ GI−P THE UNIVERSAL GERBE AND LOCAL FAMILY INDEX THEORY 21

Compare this with the corresponding splitting formula (19) for the determinant line bundle for the splitting of a smooth ﬁbration with closed even-dimensional ﬁbers.

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A.L. Carey, Mathematical Sciences Institute, Australian National University, Canberra ACT, Australia [email protected]

B.L. Wang, Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057, Zürich, Switzerland [email protected]