Analogues of Eta Invariants for Even Dimensional Manifolds

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ANALOGUES OF ETA INVARIANTS FOR EVEN DIMENSIONAL MANIFOLDS

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of

Philosophy in the Graduate School of the Ohio State University

Zhizhang Xie, BS in Mathematics

Graduate Program in Mathematics

The Ohio State University

2011

Dissertation Committee:

Henri Moscovici, Advisor

James Cogdell

Jeﬀery McNeal c Copyright by

Zhizhang Xie

2011 ABSTRACT

The eta invariant is a secondary geometric invariant, introduced by Atiyah, Patodi and Singer about forty years ago. Ever since, it has been the object of extensive research activity and has found applications in several areas of mathematics and physics. The two results among the rich literature that are most relevant to this thesis are the Atiyah-Patodi-Singer twisted index theorem for trivialized ﬂat bundles and the higher Atiyah-Patodi-Singer index theorem of Getzler and Wu. The former concerns a variational formula for eta invariants on odd dimensional manifolds and its topological interpretation; the latter is a generalization of the Atiyah-Patodi-Singer

L2-index theorem for even dimensional manifolds with boundary in the context of cyclic cohomology and its pairing with K-theory.

In this thesis, we shall prove an analogue for each of these two theorems for the case of manifolds of the opposite dimensional parity. Accordingly, the thesis will consist of two parts.

In the ﬁrst part, we prove an analogue for even dimensional manifolds of the

Atiyah-Patodi-Singer twisted index theorem for trivialized ﬂat bundles over odd dimensional closed manifolds. The corresponding eta invariant is shown to coincide with the one introduced by Dai and Zhang. The fact that this eta invariant is an appropriate analogue of the usual eta invariant for the odd dimensional case is made clear in the second part of the thesis. We conclude the ﬁrst part by establishing an

ii explicit relative pairing index formula for odd dimensional manifolds with boundary, which is the analogue of the pairing formula by Lesch, Moscovici and Pﬂaum.

In the second part we prove an analogue for odd dimensional manifolds with boundary of the higher Atiyah-Patodi-Singer index theorem of Getzler and Wu, where the eta invariant is obtained via the Connes pairing between cyclic cohomology and cyclic homology. The eta invariant defined this way is a natural analogue of that for the odd dimensional case. By comparing our formula with the Toeplitz index formula of Dai-Zhang, we show that, up to an integer, this eta invariant is equivalent to the one defined in the first part of the thesis.

iii To Pusha and Ricu

iv ACKNOWLEDGMENTS

I am greatly indebted to my advisor Henri Moscovici for his continuous support and advice. This thesis grew out of numerous conversations with him. How many times would I have been lost in the world full of greed and pretense, had he not guided me with patience through all these years. With admiration and respect, I thank him for being not only such a great teacher, but also a wonderful friend.

I want to thank Nigel Higson for helpful suggestions. I am grateful to Alexander

Gorokhovsky for a careful reading of the ﬁrst version of the thesis as well as for many helpful comments. I started working on parts of my thesis during my visit at the

Hausdorﬀ Center for Mathematics in Bonn, Germany. And I want to express my thanks to the center for its hospitality and to Matthias Lesch for the invitation, as well as for generously sharing with me his insights into the subject.

I also want to thank James Cogdell and Jeﬀery McNeal for their continuous en- couragement and for agreeing to be on my dissertation committee.

v VITA

2005 ...... B.S. in Mathematics, Zhejiang University, China

2005-Present ...... Graduate Teaching Associate, The Ohio State University, USA

PUBLICATIONS

Zhizhang Xie, Relative index pairing and odd index theorem for even dimensional manifolds, Journal of Functional Analysis, Volume 260, Issue 7, 2011, Pages 2064- 2085

Zhizhang Xie, The odd dimensional analogue of a theorem of Getzler and Wu, Ac- cepted by Journal of Noncommutative Geometry, http://arxiv.org/abs/1011.0721.

FIELDS OF STUDY

Major Field: Mathematics

Specialization: Noncommutative geometry with applications to geometry and topology

vi TABLE OF CONTENTS

Abstract ...... ii

Dedication ...... iii

Acknowledgments ...... v

Vita ...... vi

CHAPTER PAGE

1 Introduction ...... 1

2 Preliminaries ...... 9

2.1 Topological K-theory ...... 9 2.2 K-homology ...... 12 2.3 Cyclic homology and cohomology ...... 15 2.3.1 Cyclic theory of topological algebras ...... 17 2.3.2 Entire cyclic homology and cohomology ...... 19 2.3.3 Chern characters ...... 20 2.3.4 Relation between cyclic theory and de Rham theory . . . . 22

3 Relative index pairing and odd index theorem for even dimensional manifolds ...... 23

3.1 Relative index pairing ...... 23 3.2 Cup product in K-theory ...... 26 3.3 Chern characters and transgression formulas ...... 28 3.4 Odd index theorem on even dimensional manifolds ...... 32 3.5 Equivalence of eta invariants ...... 34 3.6 Relative index pairing for odd dimensional manifolds with boundary 40

4 Higher Atiyah-Patodi-Singer index theorem for odd dimensional manifolds with boundary ...... 46

4.1 Manifolds with boundary ...... 46 4.1.1 Cliﬀord Modules and Dirac Operators ...... 47

vii 4.1.2 b-norm ...... 48 4.1.3 b-trace ...... 49 4.2 JLO Chern Character in b-Calculus ...... 50 4.2.1 JLO Chern Character in b-Calculus ...... 50 4.2.2 Entireness of the b-JLO Chern Character ...... 52 4.3 Odd APS Index Theorem for manifolds with boundary ...... 56 4.4 Spectral Flow ...... 61 4.5 Large Time Limit ...... 68 4.5.1 Technical Lemmas ...... 70 4.5.2 Large Time Limit ...... 74

Bibliography ...... 79

viii CHAPTER 1

INTRODUCTION

This thesis is devoted to the study of the eta invariant by using methods in noncommutative geometry. The eta invariant ﬁrst appeared in a series of papers by Atiyah,

Patodi and Singer as an invariant of the boundary component in their index theorem for manifolds with boundary [APS75a, APS75b, APS76]. Recall that if X is an even dimensional spin manifold with boundary ∂X, then the Atiyah-Singer-index theorem states that Z ∂ ˆ h + η( D) IndAPS(D) = A(X) − , X 2 where D is the Dirac operator on X, IndAPS stands for the index of D under the Atiyah-Patodi-Singer boundary condition, Aˆ(X) is the Aˆ-genus form of X. Denote the Dirac operator on ∂X by ∂D, then h = dim ker ∂D and η(∂D) is the η invariant of

∂D.

The eta invariant is a secondary geometric invariant and its research has turned out to be extremely fruitful during the past forty years. For example, Kreck and

Stolz have used the eta invariant to produce diﬀeomorphism invariants that classify

7-dimensional homogeneous Einstein manifolds with SU(3)×SU(2)×U(1)-symmetry

[KS88]. The eta invariant and its variants have also found applications in mathematical physics. For example, Witten has exploited the close relation of the adiabatic limit of eta invariant to the holonomy of certain determinant line bundles, [Wit85].

We should point out, however, that the eta invariant mentioned above was solely

1 deﬁned for odd dimensional closed manifolds. The main goal of this thesis is to study its analogue for even dimensional closed manifolds and as well as its connections with the index theory of odd dimensional manifolds with boundary.

Accordingly, the thesis consists of two parts. The ﬁrst part concerns a variational formula of the eta invariant. In particular, we will prove an analogue for even dimensional manifolds of the Atiyah-Patodi-Singer twisted index theorem for trivialized

ﬂat bundles over odd dimensional closed manifolds [APS76, Proposition 6.2]. The corresponding eta invariant is shown to coincide with the one introduced by Dai and

Zhang. The fact that this eta invariant is an appropriate analogue of the usual eta invariant for the odd dimensional case is made clear in the second part of the thesis.

We conclude the ﬁrst part by establishing an explicit relative pairing index formula for odd dimensional manifolds with boundary, which is the analogue of the pairing formula by Lesch, Moscovici and Pﬂaum [LMP09, Theorem 7.6]. The second part deals with an analogue for odd dimensional manifolds with boundary of the higher

Atiyah-Patodi-Singer index theorem of Getzler [Get93a] and Wu [Wu93]. More specif- ically, we establish an index theorem on odd dimensional manifolds with boundary where the eta invariant is lifted to a pairing between cyclic cohomology and cyclic homology. This eta invariant is a natural analogue of that for odd dimensional closed manifolds. By comparing our formula with the Toeplitz index formula of Dai-Zhang, we show that, up to an integer, this eta invariant is equivalent to the one deﬁned in the ﬁrst part of the thesis.

To motivate the subject matter of the ﬁrst part, we begin by recalling the APS twisted index theorem for odd dimensional closed manifolds in the following form, cf.

∞ [LMP09, Corollary 7.9]. For (ps)0≤s≤1 ∈ Mk(C (N)), s ∈ [0, 1], a smooth path of projections over N, one has

2 Z 1 Z 1 d ˆ η(psDps)ds = A(N) ∧ Tch•(ps). 0 2 ds N ˆ Here psDps is the Dirac operator twisted by ps, η(psDps) its η-invariant, A(N) the ˆ A-genus form of N and Tch•(ps) is the Chern-Simons transgression form of (ps)0≤s≤1, cf. Section 3.3.

To prove our analogue for even dimensional closed manifolds, we shall replace a path of projections by a path of unitaries. The more interesting issue is what should replace the η-invariant appearing on the left hand side of the above formula. To answer this, let us ﬁrst consider the case where the manifold in question bounds, that is, it is the boundary of some spin manifold. In this case, the η-invariant by Dai and

Zhang [DZ06, Deﬁnition 2.2] is the right candidate, cf. Section 3.6 below. Indeed, suppose the even dimensional manifold Y is the boundary of a spin manifold X and

(Us)0≤s≤1 is the restriction to Y of a smooth path of unitaries over X. Denote the

η-invariant of Dai and Zhang by η(Y,Us) for each s ∈ [0, 1], then

Z 1 Z 1 d ˆ η(Y,Us)ds = A(Y ) ∧ Tch•(Us). (1.0.1) 0 2 ds Y

When Y bounds, it follows from the cobordism invariance of the index of Dirac operators that Ind(D+) = 0, where D+ is the restriction of the Dirac operator over

Y to the even half of the spinor bundle according to its natural Z2-grading. The condition Ind(D+) = 0 is crucial for the deﬁnition of the η-invariant by Dai and

Zhang, however is often not satisﬁed by even dimensional closed spin manifolds in general. To cover the general case, we shall use another approach where we lift the

1 data to S ×Y . The main ingredient of the method of proof is using an explicit formula 1 1 1 0 1 of the cup product K (S ) ⊗ K (Y ) → K (S × Y ), inspired by the Powers-Rieﬀel idempotent construction, cf. [Rie81]. In fact, the formula given for the case when

1 Y = S by Loring in [Lor88] also works for all manifolds in general, cf. Section 3.2

3 below. Our analogue for even dimensional closed spin manifolds of the APS twisted index theorem (Theorem 3.4.1 below) is as follows.

Theorem (I). Let Y be an even dimensional closed spin manifold and (Us)0≤s≤1 ∈

∞ ∞ 1 Uk(C (Y )) a smooth path of unitaries over Y . For s ∈ [0, 1], es ∈ M2k(C (S × Y )) 2πiθ 1 1 is the projection deﬁned as the cup product of Us with the generator e of K (S ). 1 Let DS1×Y be the Dirac operator over S × Y . Then

Z 1 Z 1 d ˆ η(esDS1×Y es)ds = A(Y ) ∧ Tch•(Us). (1.0.2) 0 2 ds Y

The formula for es = eUs is given in Section 3.2. Since the eta invariant is not a homotopy invariant, a priori η(eU DS1×Y eU ) changes as eU varies within the same K-theory class. However, we shall show as a corollary of theorem (I) above that

η(eU DS1×Y eU ) mod Z

is independent of the representative of K-theory class of [eU ]. More importantly, we will show that the η-invariants in the formulas (1.0.1) and (1.0.2) are equal to each other modulo Z (Theorem 3.5.7 below) in the case where Y bounds.

Theorem (II). Suppose Y is the boundary of an odd dimensional spin manifold. If

∞ 2πiθ U ∈ Uk(C (Y )) is a unitary over Y and eU is the cup product of U with [e ] ∈

1 1 K (S ), then one has

η(Y,U) = η(eU DS1×Y eU ) mod Z.

The method of proof is based on a slight generalization of a theorem by Br¨uning and Lesch [BL99, Theorem 3.9], see Proposition 3.5.6 below. In this sense, η(eU DS1×Y eU ) can be thought of as the extension to general even dimensional manifolds of the definition of the η-invariant by Dai and Zhang.

4 The same technique used above also allows us to prove the following analogue

(Theorem 3.6.3 below) for odd dimensional manifolds with boundary of the relative index pairing formula by Lesch, Moscovici and Pﬂaum [LMP09, Theorem 7.6].

Suppose M is an odd dimensional spin manifold with boundary ∂M. By a relative

1 ∞ K-cycle [U, V, us] ∈ K (M, ∂M), we mean U, V ∈ Un(C (M)) are two unitaries over

∞ M with us ∈ Un(C (∂M)), s ∈ [0, 1], a smooth path of unitaries over ∂M such that u0 = U|∂M and u1 = V |∂M . We denote by TU , resp. TV , the Toeplitz operator on M with respect to U, resp. V (see Section 3.5 for details).

1 Theorem (III). Let [U, V, us] be a relative K-cycle in K (M, ∂M). If U and V are constant along the normal direction near the boundary, then

−1 us Ind[D]([U, V, us]) = Ind(TV ) − Ind(TU ) + sf us D[0,1]us; P0

−1 us where sf us D[0,1]us; P0 is the spectral ﬂow of the path of elliptic operators

−1 us (us D[0,1]us; P0 ), s ∈ [0, 1]

us with Atiyah-Patodi-Singer type boundary conditions determined by P0 as in (3.5.4).

This uses Dai and Zhang’s Toeplitz index theorem for odd dimensional manifolds with boundary [DZ06]. We shall give the details in Section 3.6.

It should be mentioned that, although the objects we work with are from classical geometry, the spirit of the proofs is very much inspired by methods from noncommutative geometry, cf. [Con94].

In the second part, we prove an analogue for odd dimensional manifolds with boundary of the higher Atiyah-Patodi-Singer index theorem of Getzler [Get93a] and

Wu [Wu93].

Suppose N is an odd dimensional spin manifold with boundary and carries an

∞ exact b-metric [Mel93], cf. Section 4.1. For g ∈ Uk(C (N)) a unitary over N, let 5 dR Ch•(g) (resp. Ch• (g)) be the Chern character of g in entire cyclic homology of Z C∞(N) (resp. de Rham cohomology of N). In the following, − is the regularized N integral on N with respect to its b-metric (see Section 4.1) and Aˆ(N) is the Aˆ-genus form of N. Let D be the Dirac operator on N and ∂D be its restriction to the boundary ∂N. Denote the higher eta cochain of ∂D by η•(∂D) , introduced by Wu

[Wu93].

Theorem. Let N be an odd dimensional spin manifold with boundary. Endow N with an exact b-metric and let D be its associated Dirac operator. Assume ∂D is invertible.

∞ ∂ For g ∈ Uk(C (N)) a unitary over N, if k[ D, g]k < λ where λ the lowest nonzero eigenvalue of |∂D|, then

Z −1 ˆ dR • ∂ ∂ sf(D, g Dg) = − A(N) ∧ Ch• (g) + η ( D), Ch•( g) . (1.0.3) N

−1 −1 Here sf(D, g Dg) is the spectral flow of the path Du = (1 − u)D + ug Dg with u ∈ [0, 1] (see Section 4.4). In order for sf(D, g−1Dg) to be well-defined, the infimum of the essential spectrum inf specess(Du) of Du has to be greater than zero for each u. The latter condition is fulfilled if and only if the restriction Du to the boundary ∂N is invertible for each u. Thus the almost flatness condition k[∂D, g]k < λ ensures that sf(D, g−1Dg) is well-defined.

b • We consider the b-analogue Ch (Dt) of the odd Chern character by Jaﬀe-Lesniewski- Osterwalder [JLO88], cf. Section 4.2. The theorem is proved by interpolating between

b • the limit of Ch (Dt) as t → ∞ and its limit as t → 0, where Dt = tD. In fact, the limit at t = ∞ does not exist in general. However, when evaluated at Ch•(g) with g

b • satisfying the almost ﬂat condition above, the limit of Ch (Dt) as t → ∞ does exist and gives the spectral ﬂow sf(D, g−1Dg). To prove this, i.e. the equality

b • −1 lim h Ch (Dt), Ch•(g)i = sf(D, g Dg), (1.0.4) t→∞

6 we ﬁrst show (see Proposition 4.4.6) that

1 ε Z 2 2 −1 b ˙ −ε Du sf(D, g Dg) = lim √ Tr(Due )du. (1.0.5) ε→∞ π 0

This is a generalization to the b-calculus setting of Getzler’s spectral ﬂow formula for closed manifolds, cf. [Get93b, Corollary 2.7]. Once we show Eq. (1.0.5), the proof of

Eq. (1.0.4) reduces to

1 t Z 2 2 b ˙ −t Du b • lim √ Tr(Due )du = lim h Ch (Dt), Ch•(g)i. (1.0.6) t→∞ π 0 t→∞

In turn, to verify this, we consider a multiparameter version of the Chern character

b Ch( A) (see [Get93b], also Section 4.5, for the precise deﬁnition). Each side of Eq. (1.0.6) corresponds to one term in the formula obtained by applying Stokes theorem

b to Ch(t A) for each ﬁxed t. We then show the vanishing of the rest of the terms as t → ∞, hence prove the validity of Eq. (1.0.6), cf. Section 4.5. The rest of the proof follows along the lines of Getzler’s even counterpart, cf. [Get93a].

b b • Due to the fact that Tr is not a trace, Ch (Dt) is not a closed cochain. The integral of its boundary from 0 to ∞ gives the odd eta cochain η•(∂D) on the right hand side of (1.0.3). As a corollary of the main theorem, by comparing Eq. (1.0.3) with

Dai-Zhang’s Toeplitz index formula for odd dimensional manifolds with boundary

[DZ06], we obtain

• ∂ ∂ ∂ η ( D), Ch•( g) = η(∂N, g) mod Z where η(∂N, ∂g) is the eta invariant of Dai-Zhang.

A brief outline of the thesis is as follows. Chapter 2 consists of some standard preliminaries of noncommutative geometry. Chapter 3 and Chapter 4 are the main part of the thesis.

For chapter 3, we ﬁrst recall some results about index pairings for manifolds with boundary in Section 3.1. Then Section 3.2 is devoted to the explicit formula of the

7 cup product in K-theory mentioned earlier. This allows us to carry out explicit calculations for Chern characters in Section 3.3. With these preparations, we prove an analogue for even dimensional manifolds of the APS twisted index theorem in

Section 3.4. In Section 3.5, we show the equality of the two a priori diﬀerent eta invariants. In Section 3.6, we prove the odd-dimensional counterpart of the relative index pairing formula by Lesch, Moscovici and Pﬂaum [LMP09, Theorem 7.6].

For chapter 4, we recall some facts from b-calculus in 4.1. We deﬁne a b-analogue of the JLO Chern character and prove its entireness in Section 4.2. Then we state our main result (Theorem 4.3.1 below) in Section 4.3. We prove the main step of the proof in Section 4.4 and 4.5.

8 CHAPTER 2

PRELIMINARIES

In this chapter, we recall some of the standard facts from cyclic (co)homology and index theory. Most of the material is taken from [Con94, Lod92] and [Ati67, APS73,

Bla86, BHS07].

2.1 Topological K-theory

K-theory was ﬁrst introduced by Grothendieck in algebraic geometry. In the late

1950’s, Atiyah and Hirzebruch developed a K-theory based on vectors bundles over compact Hausdorﬀ topological spaces. This Atiyah-Hirzebruch K-theory is now called the topological K-theory, which has been an essential tool in the proof of the Atiyah-

Singer index theorem.

Topological K-theory is an extraordinary cohomology theory. For X compact

Hausdorff space, K0(X) is defined to be the Grothendieck group of the abelian semigroup V(X), the set of isomorphism classes of vector bundles over X. Each continuous map f : X → Y induces a natural homomorphism f ∗ : K0(Y ) → K0(X). In particular, let ϕ : X → pt be the unique map from X to the set of one point, then ∗ ∼ 0 0 one has ϕ : Z = K (pt) → K (X). The reduced K-group of X is defined to be 0 ∗ 0 −n 0 n K (X)/ϕ (Z), denoted by Ke (X). Now for each n ∈ N, K (X) := Ke (S X) where

9 SnX is the n-th suspension of X. The famous Bott periodicity theorem states that

Ke −n−2(X) ∼= Ke −n(X).

Hence if we put Kn = K−n, then the topological K-theory becomes an extraordinary

(Z/2Z-graded) cohomology theory, by applying Bott periodicity. In fact, there is a natural isomorphism induced by the Chern character map:

0 =∼ ev Ch : K (X) ⊗ C −→ H (X; C)

ev where H (X; C) is the direct sum of all even-dimensional singular cohomology groups. 1 =∼ odd Similarly, Ch : K (X) ⊗ C −→ H (X; C). To extend topological K-theory from compact Hausdorﬀ space to noncommutative topology, we recall a famous theorem of Gelfand, which asserts that the category of compact Hausdorﬀ topological spaces is (contravariantly) equivalent to the category of commutative C∗-algebras, and the equivalence functor is called Gelfand-tranform.

In particular, every commutative C∗-algebras is isomorphic to C(X) the algebra of continuous functions over a compact Hausdorﬀ space X. Moreover, by Swan-

Serre theorem, there is one-to-one correspondence between vector bundles over X and projective modules of C(X). This allows us to translate everything from the category of compact Hausdorﬀ spaces to the category of commutative C∗-algebras.

In particular, we obtain the topological K-theory of commutative C∗-algebras. In fact, commutativity is unnecessary for the deﬁnition of K-theory and the theory can be verbatim extended to all (noncommutative and commutative ) C∗-algebras.

To fix some notation, we briefly recall some basic definitions of the topological

K-theory of C∗-algebras. Let A be a (unital) C∗-algebra, that is, a ∗-Banach algebra

∗ 2 such that ka ak = kak for all a ∈ A. We deﬁne ∼h to be the equivalence relation on [ the set of idempotents in M∞(A) = Mn(A) such that e ∼h f if and only if there n is a norm-continuous path of idempotents et in M∞(A) such that e0 = e and e1 = f. 10 Let V(A) be the set of equivalence classes of idempotents in M∞(A), which becomes an abelian semigroup under direct sum. We deﬁne K0(A) to be the Grothendieck group of V(A).

For each C∗-algebra A, we deﬁne its suspension algebra as

SA = {f : → A | f is continuous and lim kf(x)k = 0} R x→±∞ and K1(A) := K0(SA). Alternatively, we denote

GLn(A) = {x ∈ Mn(A) | x is invertible }

and its identity component by GLn(A)0. Notice that the embedding GLn(A) →

GLn+1(A) by x 7→ (x, 1) maps GLn(A)0 to GLn+1(A)0. We deﬁne

K (A) := GL (A)/GL (A) = lim GL (A)/GL (A) 1 ∞ ∞ 0 −→ n n 0 where GL (A) = lim GL (A) and GL (A) = lim GL (A) . One veriﬁes that this al- ∞ −→ n ∞ 0 −→ n 0 ternative deﬁnition of K1(A) is naturally isomorphic to K0(SA), cf. [Bla86, Theorem 8.2.2].

Like every other homological/cohomological theory, there are natural operations on the topological K-theory of C∗-algebras. For C∗-algebras A and B, we deﬁne the natural cup product map

∪ : K0(A) ⊗ K0(B) → K0(A ⊗ B)

2 2 by ([e], [f]) 7→ [e ⊗ f] for e = e ∈ M∞(A) and f = f ∈ M∞(B). In fact, by ∼ the natural isomorphism K1(A) = K0(SA), we can extend the cup product to all

K-groups ∪ : Ki(A) ⊗ Kj(B) → Ki+j(A ⊗ B). See also the discussion in Section 3.2. Topological K-theory can actually be applied to larger classes of topological algebras. In fact, in order to incorporate smooth structures, we shall consider the class of local C∗-algebras. A local C∗-algebra A is a dense subalgebra of a C∗-algebra A¯ 11 such that it is closed under holomorphic functional calculus, i.e. for every x ∈ A and f a holomorphic function (f(0) = 0 if A is non-unital) on a neighborhood of the spectrum of x (in A¯), f(x) ∈ A. We also require that all matrix algebras over

A have the same property. For example, C∞(M) the algebra of smooth functions over M is a local C∗-algebra, as a dense subalgebra of C(M), where M is a smooth closed manifold. The same results mentioned above apply to local C∗-algebras. In ∼ ¯ fact, we have Ki(A) = Ki(A). So we see that, at the K-theory level, there is no real distinction between a local C∗-algebra and its completion. However, the diﬀerence becomes obvious when one studies more subtle geometric invariants of the algebra.

Among these invariants are cyclic homology and cohomology which will be discussed in Section 2.3.

2.2 K-homology

K-homology is the dual theory of topological K-theory. It can be deﬁned formally by the abstract machinery of algebraic topology. Following Atiyah’s suggestions [Ati70], concrete analytic realizations of K-homology were provided by Brown, Douglas and

Fillmore [BDF77] and Kasparov [Kas75]. A geometric deﬁnition of K-homology was also introduced by Baum and Douglas [BD82] by using manifolds, bordisms and so on. In an eﬀort to make the main idea as plain as possible, we shall follow closely

Atiyah’s paper [Ati70]. In particular, most of the material in this section is straightly taken from [Ati70].

It is well-known that a topological space not only has cohomology groups but also homology groups. In fact, one way to deﬁne homology groups (as the dual of cohomology groups) is by putting

N−m Hm(X; Z) = H (DN X; Z)

12 N+1 where DN X is a Spanier-Whitehead N-dual of X, i.e. we embed X in R and take N+1 DN X to be a deformation retract of the complement R − X. Hence, abstractly we can deﬁne K-homology as the dual of K-theory by putting

0 K0(X) = K (D2N X).

0 There is then a natural pairing between K0(X) and K (X), however a pairing defined this way is rather formal. To give a more analytic definition of K0(X), we shall first consider (abstract) elliptic operators on a topological space.

Let X be a compact Hausdorﬀ topological space and C(X) the algebra of continuous functions on X. An elliptic operator P on X consists of the following data:

(1) H is a C(X) Hilbert module, i.e. H is a Hilbert space and there is a norm

continuous action of C(X) on H,

(2) P : H → H is a bounded linear operator and P is Fredholm,

(3) [P, f] := P ◦ f − f ◦ P is a compact operator for any f ∈ C(X).

Here is a motivational example. Let X be a closed smooth manifold. For any pseudo-diﬀerential operator P of order zero (acting on sections of a vector bundle

V ), one has that: for any continuous function f, the commutator [P, f] is a compact operator. Moreover, P is a bounded linear operator on L2(X,V ) and P is Fredholm if

P is elliptic. Therefore, an elliptic pseudo-diﬀerential operator gives rise an abstract elliptic operator in the sense of the deﬁnition above.

Denote by Ell(X) the set of all (abstract) elliptic operators on X. Then there is a natural map

ψ : Ell(X) → K0(X).

In fact, we shall give the construction of a map

0 Ell(X) → HomZ(K (X), Z). 13 This nearly gives the map ψ, because there is a homomorphism

0 K0(X) → HomZ(K (X), Z) which becomes an isomorphism after tensoring with the rationals Q. Thus the construction will certainly give a map

Ell(X) → K0(X) ⊗Z Q, although the more general map ψ is not that much harder, minor a few technical details.

By Swan-Serre theorem, given V a vector bundle on X, there is an idempotent

2 eV = eV ∈ Mr(C(X)) such that V is isomorphic to the bundle of ranges of eV . Fix P an abstract elliptic operator on X, it naturally induces an operator

r r P ⊗ 1 : H ⊗ C → H ⊗ C

Now for each vector bundle V on X, we deﬁne

r r Q := P ⊗ eV = (1 ⊗ eV )(P ⊗ 1)(1 ⊗ eV ): H ⊗ eV C → H ⊗ eV C .

Then it is not hard to verify that Q satisfies the all the conditions of an elliptic operator. Hence Q ∈ Ell(X). Although our construction of Q depends of the choice of eV , the index of Q depends only on V and not on the choice of eV , and so for each fixed P , the assignment V 7→ Ind(Q) defines a map

0 K (X) → Z.

0 Therefore, we have constructed a natural map Ell(X) → HomZ(K (X), Z).

With some eﬀorts, one can verify that in fact the map ψ : Ell(X) → K0(X) is surjective. Thus a natural question which arises is how to describe explicitly the kernel of the map ψ. In other words, we need to describe explicitly the equivalence relations which must be imposed on Ell(X) to produce an isomorphism. We refer to

[Kas75] for a detailed account on this, see also [Bla86]. 14 2.3 Cyclic homology and cohomology

The appropriate generalization of de Rham cohomology/homology to noncommutative geometry was found by Connes in 1980’s [Con85]. For an algebra A, we put

⊗n Cn(A) = A ⊗ (A/C)

with its elements denoted by a0 ⊗ a1 ⊗ · · · ⊗ an or (a0, a1, ··· , an), and we deﬁne

n X i b(a0, ··· , an) = (−1) (a0, ··· , aiai+1, ··· , an), i=0

n X ni B(a0, ··· , an) = (−1) (1, ai, ··· , an, a0, ··· , ai−1). i=0 It is routine to verify that b2 = B2 = bB + Bb = 0. The nth cyclic homology of A,

th denoted by HCn(A), is deﬁned to be the n homology of the following bicomplex:

......

b b b C (A) o C (A) o C (A) 2 B 1 B 0 b b C (A) o C (A) 1 B 0 b C0(A)

A variant of the cyclic homology is the periodic cyclic homology. The latter is a

Z/2Z-graded. The even (resp. odd) periodic cyclic homology of A, denoted HP+(A)

(resp. HP−(A)), is deﬁned to be the even (resp. odd) homology of the following

15 bicomplex: ......

b b b ··· o C (A) o C (A) o C (A) B 2 B 1 B 0 b b ··· o C (A) o C (A) B 1 B 0 b ··· o C (A) B 0 Cyclic cohomology (resp. periodic cyclic cohomology) is the dual theory of cyclic homology (resp. periodic cyclic homology). If we put

n C (A) = Hom (Cn(A), C) , then nth cyclic cohomology of A, denoted by HCn(A), is deﬁned to be the nth cohomology of the total complex of the following bicomplex:

. . . .O .O .O

b b b C2(A) B / C1(A) B / C0(A) O O b b C1(A) B / C0(A) O b

C0(A) where b and B are the dual maps of those deﬁned above. And the even (resp. odd) period cyclic cohomology of A, denoted by HP +(A) (resp. HP −(A)), is deﬁned to

16 be the even (resp. odd) cohomology of the total complex of the following bicomplex:

. . . .O .O .O

b b b ··· B / C2(A) B / C1(A) B / C0(A) O O b b ··· B / C1(A) B / C0(A) O b B ··· / C0(A)

2.3.1 Cyclic theory of topological algebras

In this subsection, let us recall some basic facts of cyclic homology/cohomology of topological algebras.

Let A be locally convex topological algebra, for which the product map m :

A × A → A is separately continuous. We are mainly concerned with complete topological algebras, so we shall assume all the topological algebras are locally convex and complete throughout the discussion below. Some of the material is taken from

[BGJ95].

For two topological algebras A1 and A2, their inductive tensor product A1⊗A2 is defined to be the completion of A1 ⊗ A2 with respect to the finest compatible tensor product topology, àla Grothendieck [Gro55, page 89]. Let us recall some standard properties of the inductive tensor product:

(1) If B is another topological algebra, then the space of continuous linear maps from

A1⊗A2 to B is isomorphic to the space of separately continuous bilinear forms

from A1 × A2 to B. In particular, the topological dual of A⊗A2 is isomorphic to

the space of separately continuous bilinear forms on A1 × A2.

17 (2) For a topological algebra A, its product map m : A × A → A induces naturally

a continuous map m : A⊗A → A.

(3) If both A1 and A2 are Fr´echet, then A1⊗A2 is equal to the projective tensor

product A1⊗Ab 2.

(4) For a closed smooth manifold M (resp. N), we endow C∞(M) (resp. C∞(N))

with the standard Fr´echet topology. Then

∞ ∞ ∞ ∞ ∼ ∞ C (M)⊗C (N) = C (M)⊗bC (N) = C (M × N).

To deﬁne the cyclic homology of a topological algebra A, we just need to re-

⊗n place the algebraic tensor product A ⊗ (A/C) by the inductive tensor product ⊗n A⊗(A/C) . Similarly, for cyclic cohomology, we use

n ⊗n C (A) = Homtop(A⊗(A/C) , C)

⊗n which is the set of all continuous linear maps from A⊗(A/C) to C. It is easy to see that b and B are continuous with respect to the topology of the inductive tensor product.

As an example, let us consider A = C∞(M), where M is a closed smooth manifold.

A is a locally convex Fr´echet topological algebra under the standard Fr´echet topology.

A well-known theorem of Connes states that

n ∼ dR dR HC (A) = ker dn ⊕ Hn−2(M) ⊕ Hn−4(M) ⊕ · · ·

where dn : Dn → Dn−1 is the usual dual diﬀerential map on Dn the space of n-

dR dimensional currents on M and Hi is the usual i-th de Rham homology group. Moreover, + ∼ dR − ∼ dR HP (A) = Hev (M) and HP (A) = Hodd(M)

18 dR dR where Hev (M) (resp. Hodd(M)) is the direct sum of the even (resp. odd) dimensional de Rham homology groups. Hence we see that the cyclic cohomology groups of

C∞(M) recovers the usual de Rham homology groups of M. On the other hand, we have that the cyclic cohomology (or homology) groups of the C∗-algebra C(M) are almost trivial. In fact, HC2n(A) = HC0(A) and HC2n+1(A) = 0 for all nuclear C∗- algebras. This is quite contrary to K-theory, as we pointed out earlier that C∞(M) ∗ ∞ ∼ viewed as a dense subalgebra of C(M) is a local C -algebra and Ki(C (M)) =

Ki(C(M)).

2.3.2 Entire cyclic homology and cohomology

In order to treat infinite-dimensional noncommutative spaces, Connes has introduced the entire cyclic homology/cohomology [Con88]. In order to keep the notation simple, let us assume A is a Banach algebra for now. We call an even chain element c0 + Y c2 + ··· + c2k + · · · ∈ C+(A) = C2k(A) entire if kc0 + c2 + · · · kλ is finite for some k λ > 0, where λ2k ka0 + a2 + · · · kλ = sup ka2kk. k Γ(k) Entire odd chain elements are defined similarly. The space of even (resp. odd) entire

ω ω chains will be denoted by C+(A) (resp. C−(A)). One checks that b and B are

ω ω continuous maps from C±(A) to C∓(A), hence

ω ω b + B : C±(A) → C∓(A) is a well deﬁned chain complex. The resulted homology is called the entire cyclic

ω homology of A, denoted H±(A).

± Similarly, the entire cyclic cohomology of A, denoted Hω , is deﬁned to be the

± homology of the cochain complex (Cω (A), b + B) where

± ω Cω (A) := Homtop(C±(A), C). 19 2.3.3 Chern characters

For an algebra A, let Mr(A) be the algebra of r × r-matrices with entries in A. Then there is a natural trace map tr : Cn(Mr(A)) → Cn(A) by X tr(α0, ··· , αn) = ((α0)i0i1 , (α1)i1i2 , ··· , (αn)ini0 ).

0≤i0,··· ,in≤r

In particular, for mi ∈ Mr(C) and ai ∈ A,

tr(m0a0 ⊗ m1a1 ⊗ · · · ⊗ mnan) = tr(m0m1 ··· mn)a0 ⊗ a1 ⊗ · · · ⊗ an.

It is easy to verify that b ◦ tr = b ◦ tr and B ◦ tr = tr ◦ B, therefore the trace map induces a homomorphism

tr : C±(Mr(A)) → C±(A).

2 Let e ∈ Mr(A) be an idempotent, i.e. e = e. We deﬁne the Chern character of e to be ∞ X (2k)! 1 Ch (e) := tr(e) + (−1)k tr e − , e, ··· , e (2.3.1) • k! 2 k=1 2k

For g ∈ GLN (A), we deﬁne its Chern character to be ∞ X k −1 −1 Ch•(g) := (−1) k! tr g , g, ··· , g , g 2k+1 (2.3.2) k=0 We have

(b + B)Ch•(e) = 0 and (b + B)Ch•(g) = 0.

In other words, Ch•(e) (resp. Ch•(g)) is closed and deﬁnes a periodic cyclic homology class in ∈ HP+(A) (resp. HP−(A)). In the case of a Banach algebra A, we actually

ω ω have Ch•(e) ∈ C+(A) (resp. Ch•(g) ∈ C−(A)) deﬁne a homology class in the even (resp. odd) entire cyclic homology of A.

Moreover, let h : [0, 1] → GLr(A) a smooth path of invertible elements. Then we have the following transgression formula: d Ch•(ht) = (b + B)Chf •(h, t) dt 20 where the secondary Chern character Chf •(h, t) of h is deﬁned as

−1 ˙ Chf •(h, t) = Tr(ht ht) ∞ k X k+1 X −1 ⊗(j+1) −1 ˙ −1 ⊗(k−j) + (−1) k! Tr ht ⊗ ht) ⊗ ht ht ⊗ (ht ⊗ ht) . k=1 j=0 If we denote Z 1 Tch•(h) = Chf •(h, t)dt, 0 then

Ch•(h1) − Ch•(h0) = (b + B)Tch•(h).

There is a similar transgression formula for the even Chern character.

When A = C∞(M), we also have the usual Chern characters in the de Rham

• k cohomology HdR(M) of M. Let Ω (M) be the space of k-dimensional diﬀerential forms on M. Consider the following de Rham chain complex:

Ω0(M) −→d Ω1(M) −→·d · · or more generally,

0 d⊗1 1 d⊗1 Ω (M) ⊗ Mn(C) −−→ Ω (M) ⊗ Mn(C) −−→· · ·

k k The natural trace map tr : Ω (M) ⊗ Mn(C) → Ω (M) by ω ⊗ a 7→ tr(a)ω for k ω ∈ Ω (M) and a ∈ Mn(C), also deﬁnes a chain complex homomorphism from • • Ω (M) ⊗ Mn(C) to Ω (M), that is, tr ◦ (d ⊗ 1) = d ◦ tr. If no confusion is likely to occur, we shall also write d for d ⊗ 1.

Now we deﬁne

∞ X 1 ChdR(e) := tr(e) + (−1)k tr e(de)2k . (2.3.3) • k! k=1

2 ∞ ∞ for each idempotent e = e ∈ Mn(C (M)) = C (M) ⊗ Mn(C), and ∞ X k! ChdR(g) := tr (g−1dg)2k+1 . (2.3.4) • (2k + 1)! k=0 21 ∞ dR for each invertible g ∈ GLn(C (M)). It is straightforward to verify that Ch• (e) dR (resp. Ch• (g) ) is closed and gives a even (resp. odd) dimensional cohomology class

• in HdR(M).

2.3.4 Relation between cyclic theory and de Rham theory

∞ ⊗n Let A = C (M) and Cn(A) = A⊗(A/C) . Consider the following two diagrams:

b b b C0(A) / C1(A) / C2(A) / ···

ϕ0 ϕ1 ϕ2 Ω0(A) 0 / Ω1(A) 0 / Ω2(A) 0 / ···

B B B C0(A) / C1(A) / C2(A) / ···

ϕ0 ϕ1 ϕ2 Ω0(A) d / Ω1(A) d / Ω2(A) d / ··· where 1 ϕ (a ⊗ a ⊗ · · · ⊗ a ) = a da ··· da n 0 1 n n! 0 1 n

It is easy to see that bϕn = 0 and ϕnB = dϕn−1. Therefore, ϕ induces a homo-

+ ∞ ev − ∞ od morphism from HP (C (M)) to HdR(M) (resp. from HP (C (M)) to HdR(M) ). Moreover, a straightforward calculation shows that

dR dR ϕ(Ch•(e)) = Ch• (e) and ϕ(Ch•(g)) = Ch• (g).

22 CHAPTER 3

RELATIVE INDEX PAIRING AND ODD INDEX

THEOREM FOR EVEN DIMENSIONAL MANIFOLDS

3.1 Relative index pairing

Let M be a compact smooth manifold with boundary ∂M 6= ∅. Following [BDT89,

Sec. 2], consider an elliptic ﬁrst order diﬀerential operator

∞ ∞ D : Cc (M \ ∂M, E) → Cc (M \ ∂M, E)

∞ where Cc (M \ ∂M, E) is the space of compactly supported smooth sections of the Hermitian vector bundle E. Such an operator has a number of extensions to become

2 a closed unbounded operator on H = L (M \ ∂M, E), e.g. Dmin and Dmax the minimum extension and the maximum extension respectively. Consider De a closed extension of D such that

Dmin ⊂ De ⊂ Dmax, (3.1.1) that is, D(Dmin) ⊂ D(De) ⊂ D(Dmax). Let   0 D∗  e  B =   De 0 and   0 T ∗ 2 −1/2   Fe = B(B + 1) =   T 0

23 ∗ −1/2 ∗ ∗ ∗ −1/2 with T = De(De De + 1) and T = De (DeDe + 1) . Denote by C0(M \ ∂M) the space of continuous functions vanishing at inﬁnity. Then the ∗-representation of

C0(M \ ∂M) on H ⊕ H given by scalar multiplication, together with Fe, deﬁnes an element in KK(C0(M \ ∂M), C), see [BDT89] for the precise construction. Such a K-homology class turns out to be independent of the choice of a closed extension of

D [BDT89, Proposition 2.1 ], and will be denoted [D]. Similarly for each formally symmetric elliptic operator, one constructs a cycle in KK(C0(M \∂M), Cl1) [BDT89,

Section 2], where Cl1 is the Cliﬀord algebra with one generator. For each [D] ∈

KK(C0(M \ ∂M), Cl•), one has the index pairing map

• Ind[D] : K (M \ ∂M) → Z.

An element in K0(M\∂M) is represented by a triple (E, F, α) with E, F vector bundles over M\∂M and α : E → F a bundle homomorphism whose restriction near infinity is an isomorphism, cf. [Ati67]. Moreover, we can choose connections over the bundles E and F such that the forms Ch•(E) and Ch•(F ) coincide near infinity. Under this assumption, one can write down an explicit formula for the index pairing map: Z Ind[D]([E, F, α]) = ωD ∧ [Ch•(E) − Ch•(F )] . M even Here ωD ∈ HdR (M\∂M) is the dual of the Chern character of the K-homology class [D], as explained in the introduction of Chap. I in [Con85]. In the case where M is ˆ a spin manifold and D the Dirac operator over M, one has ωD = A(M). Similarly, in the odd case, an element in K1(M \ ∂M) consists of two unitaries U and V over M \ ∂M and a homotopy h between U and V near infinity. Moreover, we can assume that U and V are identical near infinity and the homotopy h becomes

24 the identity map near inﬁnity, cf. e.g.[HR00, Prop. 4.3.14], in which case the index pairing map has the following cohomological expression 1:

Z Ind[D]([V, U, h]) = − ωD ∧ [Ch•(V ) − Ch•(U)] . M

Note that the boundary data are conspicuously absent in the above formulas.

Indeed, by deﬁnition, K•(M\∂M) is essentially the (reduced) K-group of the one point compactiﬁcation of M\∂M. The information from the boundary is therefore completely eliminated from the picture. In order to recover that, we shall turn to the relative K theory of the pair (M, ∂M), denoted K•(M, ∂M), cf. [LMP09]. A relative

0 ∞ K-cycle [p, q, hs] ∈ K (M, ∂M) is a triple where p, q ∈ Mn(C (M)) are two projec-

∞ tions over M and hs ∈ Mn(C (∂M)), s ∈ [0, 1], is a path of projections over ∂M such

1 that h0 = p|∂M and h1 = q|∂M . Similarly, a relative K-cycle [U, V, us] ∈ K (M, ∂M) is

∞ ∞ a triple where U, V ∈ Un(C (M)) are two unitaries over M with us ∈ Un(C (∂M)), s ∈ [0, 1], a smooth path of unitaries over ∂M such that u0 = U|∂M and u1 = V |∂M . First notice that K•(M, ∂M) ∼= K•(M \∂M). Hence the above index pairing induces

• a map Ind[D] : K (M, ∂M) → Z. The issue now is to ﬁnd an explicit formula which incorporates geometric information of the boundary. For even dimensional manifolds with boundary, this is done by Lesch, Moscovici and Pﬂaum[LMP09, Theorem 7.6].

We shall give an analogous formula for odd dimensional manifolds with boundary in the Section 3.6.

1We adopt the negative sign here in order to be consistent with our sign convention throughout the thesis.

25 3.2 Cup product in K-theory

Let A and B be local Fr´echet algebras. The cup product between K1(A) and K1(B) is deﬁned by

∼ × : K1(B) ⊗ K1(A) = K0(SB) ⊗ K0(SA) → K0(SB ⊗ SA) = K0(B ⊗ A) where SA (resp. SB) is the suspension of A (resp. B), the isomorphism is the Bott isomorphism and

K0(SB) ⊗ K0(SA) → K0(SB ⊗ SA) is given by

[p] × [q] = [p ⊗ q]. (3.2.1)

∞ 1 In the case where B = C (S ), we shall give an explicit formula of this cup 2πiθ ∞ 1 ∼ product. Since e is a generator of K1(C (S )) = Z, it suﬃces to give this formula 2πiθ for [e ] × [U] with U ∈ Uk(A).

Lemma 3.2.1 (cf. [Lor88]). With the above notation,

2πiθ [e ] × [U] = [eU ]   f g + hU   ∞ 1 where eU =   ∈ M2k(C (S ) ⊗ A) is a projection with f, g and h hU ∗ + g 1 − f 1 nonnegative functions on S satisfying the following conditions

1. 0 ≤ f ≤ 1,

2. f(0) = f(1) = 1 and f(1/2) = 0,

2 1/2 2 1/2 3. g = χ[0,1/2](f − f ) and h = χ[1/2,1](f − f ) .

Proof. It is not diﬃcult to see that

∞ 1 ∞ 1 × : K1(C (S )) ⊗ K1(A) → K0(C (S ) ⊗ A) 26 is the same as the standard isomorphism [WO93, Section 7.2]

∞ 1 ΘA : K1(A) → K0(SA) ⊂ K0(C (S ) ⊗ A)

∞ 1 after identifying K1(C (S )) with Z. The inverse of this map is constructed as follows, cf. [HR00, Proposition 4.8.2][WO93, Section 7.2]. The group K0(SA) is generated by formal differences of normalized loops of projections over A. Such a loop is a projection-valued maps p : [0, 1] → Mn(A) with p(0) = p(1) ∈ Mn(C). For each ∗ loop, there is a path of unitaries u : [0, 1] → Un(A) such that p(t) = u(t)p(1)u(t)   1 0   and u(0) = 1n.Without loss of generality, we can assume p(0) = p(1) =  . 0 0   v 0   This implies that u(1) is of the form  . Then one checks that [p] 7→ [v] is a 0 w well-defined inverse to ΘA. To see that our formula agrees with the usual definition, it suffices to show that   1 0 Θ−1(e ) = U. First notice that e (0) = e (1) =   and e (θ) is a projection A U U U   U 0 0 1 over A for each θ ∈ S = R/Z, hence eU is a normalized loop of projections. Now consider the following path of unitaries over A,   1/2 f1(θ) + f2(θ)U (1 − f) (θ) U(θ) =    1/2 ∗ (1 − f) (θ) −f1(θ) − f2(θ)U   1 0 1/2 1/2   where f1 = χ[0,1/2]f and f2 = χ[1/2,1]f . In particular, U(0) =   and 0 1   U 0   U(1) =  . By a direct calculation, one verifies 0 −U ∗   1 0   ∗ eU (θ) = U(θ)   U(θ) 0 0

27 from which the lemma follows.

We will also make use of the following lemma in Section 3.3, cf. [Lor88, Lemma

2.2].

1 Lemma 3.2.2. For f, g and h nonnegative functions on S = R/Z satisfying the following conditions

1. 0 ≤ f ≤ 1,

2. f(0) = f(1) = 1 and f(1/2) = 0,

2 1/2 2 1/2 3. g = χ[0,1/2](f − f ) and h = χ[1/2,1](f − f ) , we have Z 1 (k − 1)!(k − 1)! (2 − 4f)h0h2k−1 + 4f 0h2k dθ = 0 (2k − 1)! Proof. Notice that

Z 1 Z 1 Z 1 k!k! f 0h2kdθ = (f(θ) − f 2(θ))kdf(θ) = (x − x2)kdx = , 0 1/2 0 (2k + 1)! and integration by parts gives

Z 1 2 Z 1 (2 − 4f)h0h2k−1dθ = f 0h2kdθ. 0 k 0

3.3 Chern characters and transgression formulas

Throughout this section, although we deal with commutative algebras, we shall use the similar formalism for the Chern character in K-theory as in cyclic homology

[Con85, Chap. II], [Lod92, Chap. VIII]. Let M be a compact smooth manifold with or without boundary. 28 ∞ 2 ∗ Recall that for p ∈ Mn(C (M)) such that p = p and p = p, we deﬁne ∞ X 1 Ch (p) := tr(p) + (−1)k tr p(dp)2k ∈ Heven(M). • k! dR k=1 ∞ For U ∈ Un(C (M)), we put ∞ X k! Ch (U) := tr (U −1dU)2k+1 ∈ Hodd(M). • (2k + 1)! dR k=0 In fact, to follow topologists convention and for the purpose of notational simplicity, we shall use the following properly rescaled version of the Chern charaters. ∞ X 1 1 Ch (p) := tr(p) + (−1)k tr p(dp)2k ∈ Heven(M), (3.3.1) • (2πi)k k! dR k=1 ∞ X 1 k! Ch (U) := tr (U −1dU)2k+1 ∈ Hodd(M). (3.3.2) • (2πi)k+1 (2k + 1)! dR k=0 ∞ For each U ∈ Un(C (M)), let eU be the projection as in Lemma 3.2.1. If no confusion is likely to arise, we also write e instead of eU .

Lemma 3.3.1. ∞ X 1 k Ch (e ) = − 4f 0h2k + (2 − 4f)h0h2k−1 dθ ∧ tr(U −1 · dU)2k−1 • U (2πi)k k! k=1 Proof. Notice that     f 0 g0 + h0U 0 h dU     de =   dθ +   , h0U ∗ + g0 −f 0 h dU ∗ 0 which implies    2k f g + hU 0 h dU 2k      tr(e(de) ) = tr      (3.3.3) hU ∗ + g 1 − f h dU ∗ 0    j−1 j=2k X f g + hU 0 h dU + tr      ∗   ∗  j=1 hU + g 1 − f h dU 0    (2k−j) f 0 g0 + h0U 0 h dU        dθ    . (3.3.4) g0 + h0U ∗ −f 0 h dU ∗ 0 

29 Since most of the matrices appearing in the above summation only have oﬀ diagonal entries, a straightforward calculation gives the following equalities.

(3.3.3) = h2ktr (U −1 · dU)2k

(3.3.4) = −(−1)kk (2 − 4f)h0h2k−1 + 4f 0h2k dθ ∧ tr(U −1 · dU)2k−1.

On the other hand,

tr((U −1 · dU)2k) = −tr (U −1 · dU)(U −1 · dU)2k−1 from which it follows that (3.3.3) vanishes. This ﬁnishes the proof.

As a consequence of lemma (3.2.2) and lemma (3.3.1), one has the following

1 corollary. From now on, integration along the ﬁber S will be denoted by π∗.

Corollary 3.3.2. π∗Ch•(eU ) = −Ch•(U).

∞ Consider a smooth path of unitaries Us ∈ Un(C (M)) with s ∈ [0, 1], or equiva-

∞ lently U ∈ Un(C ([0, 1] × M)). The secondary Chern character Chf •(Us) is given by the formula

∞ X 1 k! −1 ˙ −1 2k Chf •(Us) := tr U Us(U dUs) . (2πi)k+1 (2k)! s s k=0

Then Ch•(U) can be decomposed as

Ch•(U) = Ch•(Us) + ds ∧ Chf •(Us)

where Ch(Us) (see (3.3.2) above) and Chf •(Us) do not contain ds. Applying de Rham diﬀerential to both sides gives us the following transgression formula

∂ Ch•(Us) = dChf •(Us). ∂s

30 ∞ Similarly, if es ∈ Mm(C (M)) is a smooth path of projections, or equivalently a

∞ projection e ∈ Mm(C ([0, 1] × M)), then

Ch•(e) = Ch•(es) + ds ∧ Chf •(es). with

∞ X k+1 1 1 2k+1 Chf •(es) := (−1) tr (2es − 1)e ˙s(des) . (2πi)k+1 k! k=0

Applying Corollary (3.3.2) to Ch•(U) and Ch•(eU), one has

π∗Ch•(eU) = −Ch•(U), which implies

ds ∧ π∗Chf •(es) = ds ∧ Chf •(Us).

Denote the Chern-Simons transgression forms by

Z 1 Tch•(es)0≤s≤1 := ds ∧ Chf •(es), 0 Z 1 Tch• (Us)0≤s≤1 := ds ∧ Chf •(Us). 0 We summarize the results of this section in the following proposition.

∞ ∞ Proposition 3.3.3. Consider U ∈ Un(C (M)) and Us ∈ Un(C (M)) for s ∈ [0, 1].

2πiθ 1 1 Let e, resp. es, be the cup product of U, resp. Us, with e a generator of K (S ) as in Lemma 3.2.1. Then

π∗Ch•(e) = −Ch•(U) and

π∗Tch•(es)0≤s≤1 = Tch•(Us)0≤s≤1.

31 3.4 Odd index theorem on even dimensional manifolds

In this section, we shall prove our analogue for even dimensional closed manifolds of the APS twisted index theorem.

Let us ﬁrst recall the APS twisted index theorem and ﬁx some notation. Let N be a closed odd dimensional spin manifold and /D its Dirac operator. If p is a projection

∞ in Mn(C (N)), then p induces a Hermitian vector bundle, denoted Ep, over N. With the Grassmannian connection on Ep, let p( /D⊗In)p be the twisted Dirac operator with coeﬃcients in Ep. For notational simplicity, we also write p /Dp instead of p( /D ⊗ In)p.

∞ Then by [LMP09, Corollary 7.9], for ps ∈ Mk(C (N)) a smooth path of projections over N, one has

Z ˆ ξ(p1 /Dp1) − ξ(p0 /Dp0) = A(N) ∧ Tch•(ps) + sf(ps /Dps)0≤s≤1 (3.4.1) N where η(p /Dp ) + dim ker(p /Dp ) ξ(p /Dp ) = i i i i i i 2 the reduced eta invariant of pi /Dpi. Here sf(ps /Dps)0≤s≤1 is the spectral ﬂow of

(ps /Dps)0≤s≤1. Notice that the vector bundle on which ps /Dps acts may vary as s moves along [0, 1]. To make sense of the deﬁnition of such a spectral ﬂow, we introduce a

∞ ∗ ∗ path of unitaries us ∈ Un(C (N)) over N with usp0us = ps so that p0us /Dusp0 acts on

∗ the same vector bundle Ep0 . sf(ps /Dps)0≤s≤1 is then deﬁned to be sf(p0us /Dusp0)0≤s≤1

∗ the spectral ﬂow of the family (p0us /Dusp0)0≤s≤1. Now by [KL04, Lemma 3.4], formula (3.4.1) is equivalent to

Z 1 Z 1 d ˆ η(ps /Dps)ds = A(N) ∧ Tch•(ps). (3.4.2) 0 2 ds N

Theorem 3.4.1. Let Y be a closed even dimensional spin manifold and (Us)0≤s≤1 ∈

∞ ∞ Uk(C (Y )) a smooth path of unitaries over Y . For s ∈ [0, 1], let es ∈ M2k(C (Y ))

32 2πiθ 1 1 be the projection deﬁned as the cup product of Us with the generator e of K (S ). 1 Let DS1×Y be the Dirac operator over S × Y . Then

Z 1 Z 1 d ˆ η(esDS1×Y es)ds = A(Y ) ∧ Tch•(Us). (3.4.3) 0 2 ds Y

1 Proof. Applying formula (3.4.1) to S × Y , one has Z ˆ 1 ξ(e1DS1×Y e1) − ξ(e0DS1×Y e0) = A(S × Y ) ∧ Tch•(es) + sf(esDS1×Y es). S1×Y ˆ 1 ∗ ˆ 1 ∗ ˆ ˆ 1 1 1 Notice that A(S ×M) = π1A(S )∧π2A(M) and A(S ) = 1, where π1 : S ×M → S , 1 1 1 resp. π2 : S ×M → M, is the projection from S ×M to S , resp. M. By Proposition Z ˆ 3.3.3, the integral on the right side is equal to A(Y ) ∧ Tch•(Us). Now the formula Y Z 1 Z 1 d ˆ η(esDS1×Y es)ds = A(Y ) ∧ Tch•(Us) 0 2 ds Y follows from the equality [KL04, Lemma 3.4]

Z 1 1 d ξ(e1DS1×Y e1) − ξ(e0DS1×Y e0) = sf(esDS1×Y es) + η(esDS1×Y es)ds. 0 2 ds

Remark 3.4.2. Mod Z, the reduced η-invariant ξ(esDS1×Y es) is equal to the reduced

η-invariant ξ(Y,Us) deﬁned by Dai and Zhang, cf. [DZ06, Deﬁnition 2.2], at least when Y bounds. See Theorem 3.5.7 below.

Notice that in the deﬁnition of eU , we do have a choice to make for the function f.

Since the eta invariant is not a homotopy invariant, a priori η(eU DS1×Y eU ) changes as eU varies within the same K-theory class. To emphasis the dependence on f, let us write   f g + hU   eU,f =   . hU ∗ + g 1 − f

Let f[0] and f[1] be two functions satisfying 33 (1) 0 ≤ f[i] ≤ 1,

(2) f[i](0) = f[i](1) = 1 and f[i](1/2) = 0.

Let f[s] = sf[0] + (1 − s)f[1] with s ∈ [0, 1], then f[s] satisﬁes the above two conditions for all s ∈ [0, 1]. We denote   f[s] g[s] + h[s]U e˜s =   .  ∗  h[s]U + g[s] 1 − f[s]

If we replace es in Theorem 3.4.1 bye ˜s, then the same proof shows that

Z 1 Z 1 d ˆ η(˜esDS1×Y e˜s)ds = A(Y ) ∧ Tch•(U) = 0. (3.4.4) 0 2 ds Y where the second equality follows from the fact that Tch•(U) = 0. Therefore, we have

Z 1 1 d ξ(˜e1DS1×Y e˜1) − ξ(˜e0DS1×Y e˜0) = η(˜esDS1×Y e˜s)ds + sf(˜esDS1×Y e˜s)0≤s≤1 0 2 ds

= sf(˜esDS1×Y e˜s)0≤s≤1

In particular,

ξ(˜e1DS1×Y e˜1) = ξ(˜e0DS1×Y e˜0) mod Z.

3.5 Equivalence of eta invariants

Throughout this section, we assume M is an odd dimensional spin manifold with boundary ∂M. Denote by SM the spinor bundle over M. Let D be the Dirac operator over M, then near the boundary

d D = c(d/dx) + D∂ dx where D∂ is the Dirac operator over ∂M and c(d/dx) is the Cliﬀord multiplication

n by the normal vector d/dx. Then D ⊗ In is the Dirac operator acting on SM ⊗ C , 34 n when we use the trivial connection on the bundle M × C over M. If no confusion is likely to arise, we shall write D instead of D ⊗ In. Now a subspace L of ker D∂ is Lagrangian if c(d/dx)L = L⊥ ∩ker D∂. In our case, since ∂M bounds M, the existence of such a Lagrangian subspace follows from the

2 n cobordism invariance of the index of Dirac operators. Let L>0(SM ⊗ C |∂M ) be the positive eigenspace of D∂, i.e. the L2-closure of the direct sum of eigenspaces with positive eigenvalues of D∂. Then the projection

∂ P := P∂M (L) = P∂M + PL

imposes an APS type boundary condition for D, where P∂M , resp. PL, is the orthog-

2 n 2 n 2 n onal projection L (SM ⊗ C |∂M ) → L>0(SM ⊗ C |∂M ), resp. L (SM ⊗ C |∂M ) → L.

Let us denote the corresponding self-adjoint elliptic operator by DP ∂ .

2 n ∂ Let L≥0(SM ⊗ C ; P ) be the nonnegative eigenspace of DP ∂ and PP ∂ the orthogonal projection

2 n 2 n ∂ PP ∂ : L (SM ⊗ C ) → L≥0(SM ⊗ C ; P ).

∞ ∂ −1 More generally, for each unitary U ∈ Un(C (M)) over M, the projection UP U imposes an APS type boundary condition for D and we shall denote the corresponding elliptic self-adjoint operator by DUP ∂ U −1 . Similarly, let PUP ∂ U −1 be the orthogonal projection

2 n 2 n ∂ −1 PUP ∂ U −1 : L (SM ⊗ C ) → L≥0(SM ⊗ C ; UP U )

2 n ∂ −1 where L≥0(SM ⊗ C ; UP U ) is the nonnegative eigenspace of DUP ∂ U −1 . With the above notation, we deﬁne the Toeplitz operator on M with respect to

U as follows, cf.[DZ06, Deﬁnition 2.1].

Deﬁnition 3.5.1.

TU := PUP ∂ U −1 ◦ U ◦ PP ∂ .

35 Dai and Zhang’s index theorem for Toeplitz operators on odd dimensional manifolds with boundary [DZ06, Theorem 2.3] states that Z ˆ ∂ −1 ∂ Ind(TU ) = − A(M) ∧ Ch•(U) − ξ(∂M, U) + τµ(UP U ,P , PM ) (3.5.1) M where PM is the Calder´onprojection associated to the Dirac operator D on M (cf.

∂ −1 ∂ [BBW93]) and τµ(UP U ,P , PM ) is the Maslov triple index [KL04, Deﬁnition 6.8]. The reduced η-invariant ξ(∂M, U) will be deﬁned after the remarks.

Remark 3.5.2. Notice that the integral in (3.5.1) differs from Dai and Zhang’s by a constant coefficient (2πi)−(dim M+1)/2. This is due to the fact that our definition of 1 characteristic classes follows topologists’ convention, i.e. factors such as ( )k/2 are 2πi already included.

∂ −1 ∂ Remark 3.5.3. The Maslov triple index τµ(UP U ,P , PM ) is an integer. For

∞ ∞ unitaries U, V ∈ Un(C (M)), if there a path of unitaries us ∈ Un(C (∂M)) with s ∈ [0, 1] such that u0 = U|∂M and u1 = V |∂M , one has

∂ −1 ∂ ∂ −1 ∂ τµ(UP U ,P , PM ) = τµ(VP V ,P , PM ), cf.[KL04, Lemma 6.10].

To deﬁne ξ(∂M, U), let us ﬁrst consider D[0,1] the Dirac operator over [0, 1] × ∂M.

If no confusion is likely to arise, we shall write U for both U|∂M and the trivial lift of

U|∂M from ∂M to [0, 1] × ∂M. Let

ψ,U −1 D[0,1] := D[0,1] + (1 − ψ)U [D[0,1],U] (3.5.2) over [0, 1]×∂M, where ψ is a cut-oﬀ function on [0, 1] with ψ ≡ 1 near {0} and ψ ≡ 0 near {1}. With APS type boundary conditions determined by P ∂ on {0} × ∂M and

−1 ∂ ψ,U Id − U P U on {1} × ∂M, D[0,1] becomes a self-adjoint elliptic operator, denoted ψ,U U D[0,1]; P0 . See proposition 3.5.6 for an explanation of the choice of notation. 36 Similarly,

ψ,U −1 D[0,1](t) := D[0,1] + (1 − tψ)U D[0,1],U . (3.5.3)

ψ,U U ψ,U U Denote by D[0,1](t); P0 the elliptic operator D[0,1](t) with boundary condition P0 . ψ,U ψ,U Note that D[0,1](1) = D[0,1].

Deﬁnition 3.5.4. ([DZ06, Deﬁnition 2.2])

ψ,U U ψ,U U η(∂M, U) := ξ(D[0,1]; P0 ) − sf D[0,1](t); P0 0≤t≤1 where ψ,U U ψ,U U dim ker(D ; P0 ) + η(D ; P0 ) ξ(Dψ,U ; P U ) = [0,1] [0,1] . [0,1] 0 2

Remark 3.5.5. η(∂M, U) is independent of the cut-oﬀ function ψ [DZ06, Proposition

5.1].

In order to show the equality η(∂M, U) = ξ(eU DS1×∂M eU ) mod Z, we need to ψ,U d ∂ relate the operator e D 1 e to D , where D 1 = c(d/dθ) + D is U S ×∂M U [0,1] S ×∂M dθ 1 2πiθ 1 1 the Dirac operator over S × ∂M and eU is the cup product of U with e ∈ K (S ). Recall that       f g + hU D 1 0 f g + hU    S ×∂M    eU DS1×∂M eU =  ∗     ∗  hU + g 1 − f 0 DS1×∂M hU + g 1 − f       1 0 D 1 0 1 0   ∗  S ×∂M    ∗ = U   U   U   U 0 0 0 DS1×∂M 0 0 where   1/2 1/2 1/2 f1 + f2 U (1 − f) U =    1/2 1/2 1/2 ∗ (1 − f) −f1 − f2 U with f1 = χ[0,1/2]f and f2 = χ[1/2,1]f. Then viewed as an operator over [0, 1] × ∂M,

∗ −1 U (eU DS1×∂M eU )U = D[0,1] + f2U D[0,1],U 37 with the boundary condition

n β(0, x) = Uβ(1, x), for ∀x ∈ ∂M and β ∈ Γ([0, 1] × ∂M; S ⊗ C ).

∂ 2 n 2 n Let H := L ({0} × ∂M; S ⊗ C ) ⊕ L ({1} × ∂M; S ⊗ C ), then the above boundary condition can be written as   1 1 −U   ∂   β = 0, for ∀ β ∈ H . 2 −U −1 1

From now on, let us assume ψ = 1 − f2. In particular, one has

∗ ψ,U U (eU DS1×∂M eU )U = D[0,1].

Now consider   cos2 tP ∂ + sin2 t(I − P ∂) − cos t sin tU P U =   (3.5.4) t   − cos t sin tU −1 cos2 t(Id − U −1P ∂U) + sin2 tU −1P ∂U for 0 ≤ t ≤ π/4(cf.[KL04, Equation 5.13], [BL99, Section 3]). This is a path of projections in B(H∂) such that   P ∂ 0 P U =   0   0 Id − U −1P ∂U and   1 1 −U P U =   . π/4   2 −U −1 1

ψ,U U For each t ∈ [0, π/4], the Dirac operator D[0,1], with the boundary condition Pt , is a ψ,U U self-adjoint elliptic operator, denoted by (D[0,1]; Pt ). With the above notation, we have the following slight generalization of a theorem by Br¨uningand Lesch [BL99, Theorem 3.9] .

Proposition 3.5.6. d η(Dψ,U ; P U ) = 0. dt [0,1] t 38 Proof. Following [BL99, Section 3], we deﬁne     0 U 0 U     τ :=   =   , U −1 0 U ∗ 0   c(d/dθ) 0   γe :=   , 0 −c(d/dθ)   D∂ 0   Ae :=   . 0 −U −1D∂U

ψ,U where Ae is determined by D[0,1] near the boundary, by noticing that d Dψ,U = c(d/dθ) + D∂ [0,1] dθ near {0} × ∂M and d Dψ,U = c(d/dθ) + U −1D∂U [0,1] dθ n near {1} × ∂M. Since c(d/dθ) U = U c(d/dθ) ∈ End(S ⊗ C ), it follows that

2 ∗ τAe + Aτe = 0 = τγe + γτ,e τ = 1, τ = τ .

Moreover, one veriﬁes by calculation (cf. [BL99, Eqs. (3.11) to (3.13)])

U U γPe t = (I − Pt )γe;

U 2 [Pt , Ae ] = 0;

U U U Pt APe t = cos(2t)|Ae|Pt .

Then by [BL99, Theorem 3.9], it suﬃces to ﬁnd a unitary µ : H∂ → H∂ such that

2 µ = −I, µτ + τµ = µγe + γµe = µAe + Aµe = 0.

Let   0 U   µ :=   . −U −1 0 This ﬁnishes the proof. 39 Now the equality η(∂M, U) = ξ(eU DS1×∂M eU ) mod Z follows as a corollary. To be slightly more precise, we have the following result.

Theorem 3.5.7.

ψ,U U ψ,U U η(∂M, U) = ξ(eU DS1×∂M eU ) − sf(D[0,1]; Pt ) − sf(D[0,1](t); P0 )0≤t≤1.

In particular,

η(∂M, U) = ξ(eU DS1×∂M eU ) mod Z.

Proof. By [KL04, Lemma 3.4],

ψ,U U ξ(eU DS1×∂M eU ) − ξ(D[0,1]; P0 ) Z π/4 ψ,U U d 1 ψ,U U = sf(D[0,1]; Pt )0≤t≤π/4 + η(D[0,1]; Pt )dt. 0 dt 2 The formula now follows from the deﬁnition of η(∂M, U) and the proposition above.

3.6 Relative index pairing for odd dimensional manifolds

with boundary

In this section, we shall use the Toeplitz index theorem for odd dimensional manifolds with boundary by Dai and Zhang to prove our analogue of the index pairing formula by Lesch, Moscovici and Pﬂaum [LMP09, Theorem 7.6].

First let us recall the even case. Let X be an even dimensional spin manifold with boundary ∂X. We assume its Riemannian metric has product structure near the boundary. The associated Dirac operator takes of the following form     − d D − + D∂X    dx  DX =   =  d  D+ + D dx ∂X near the boundary, where D∂X is the Dirac operator over ∂X. 40 Deﬁnition 3.6.1. Let P = χ (D ) and D+ be the elliptic operator D+ with ≥0 [0,∞) ∂X P≥0 the APS boundary condition P , cf.[APS75a]. Then Ind (D+) := Ind(D+ ). ≥0 AP S P≥0

0 Recall that a relative K-cycle in K (X, ∂X) is a triple [p, q, hs] such that p, q ∈

∞ ∞ Mn(C (X)) are two projections over X and hs ∈ Mn(C (∂X)), s ∈ [0, 1], is a path of projections over ∂X such that h0 = p|∂X and h1 = q|∂X . If p and q are constant along the normal direction near the boundary, then the relative index pairing by

Lesch, Moscovici and Pﬂaum [LMP09, Theorem 7.6] states that

+ + Ind[DX ]([p, q, hs]) = IndAP S(qD q) − IndAP S(pD p) + sf(hsD∂X hs)0≤s≤1.

Now let M be an odd dimensional spin manifold with boundary ∂M. We assume its Riemannian metric has product structure near the boundary. The Dirac operator ∼ D over M naturally induces an element in KK(C0(M \ ∂M), c1) = K1(M, ∂M) cf. [BDT89, Section 2], from which one has the relative index pairing map

1 Ind[D] : K (M, ∂M) → Z. (3.6.1)

As an intermediate step, let us ﬁrst show a pairing formula by using the lifted data

1 on S × M. The method of proof is similar to the one used in proving Theorem 3.4.1. 1 Denote the Dirac operator over S × M by Db and its restriction to the half-spinor bundles by Db +.

1 Lemma 3.6.2. For a relative K-cycle [U, V, us] ∈ K (M, ∂M), that is, U, V ∈

∞ ∞ Un(C (M)) are two unitaries over M with us ∈ Un(C (∂M)), s ∈ [0, 1], a smooth path of unitaries over ∂M such that u0 = U|∂M and u1 = V |∂M . If U and V are constant along the normal direction near the boundary, then

Ind[D]([U, V, us])

+ + 1 = IndAP S(eV Db eV ) − IndAP S(eU Db eU ) + sf(eus DS ×∂M eus )0≤s≤1

41 1 Proof. A relative K-cycle [U, V, us] ∈ K (M, ∂M) naturally induces a relative K-cycle

0 [eU , eV , eus ] ∈ K (M, ∂M). By [LMP09, Theorem 7.6],

+ + 1 [U, V, us] 7→ IndAP S(eV Db eV ) − IndAP S(eU Db eU ) + sf(eus DS ×∂M eus )0≤s≤1 (3.6.2)

1 is a well-deﬁned map from K (M, ∂M) to Z. We need to show that it does agree with the relative index pairing induced by that of K1(M\∂M). As before (cf. Section

3.1 above), we can assume U|[0,)×∂M = V |[0,)×∂M and us = U|∂M = V |∂M , for all s ∈ [0, 1]. It suﬃces to prove the lemma for representatives of relative K-cycles of this special type. Notice that such a representative also deﬁnes an element in K1(M\∂M) by its restriction to M\∂M and recall from Section 3.1 that the index map (3.6.1) has the following explicit formula:

Z ˆ Ind[D]([V, U, us]) = − A(M) ∧ [Ch•(V ) − Ch•(U)] . M

Now by the APS index theorem for manifolds with boundary,

Z + ˆ 1 IndAP S(eU Db eU ) = A(S × M) ∧ Ch•(eU ) − ξ(eU DS1×∂M eU ) (3.6.3) S1×M Z ˆ = − A(M) ∧ Ch•(U) − ξ(eU DS1×∂M eU ). (3.6.4) M where the second equality follows from Proposition 3.3.3. There is a similar equation where we replace U by V . It follows that the image of a representative of the special type as above, under the map (3.6.2), is equal to

Z ˆ − A(M) ∧ [Ch•(V ) − Ch•(U)] . M

This agrees with the relative index map (3.6.1).

Using this lemma and another two lemmas below, we shall now prove our main result in this section.

42 1 Theorem 3.6.3. For a relative K-cycle [U, V, us] ∈ K (M, ∂M), that is, U, V ∈

−1 us Ind[D]([U, V, us]) = Ind(TV ) − Ind(TU ) + sf us D[0,1]us; P0

−1 us −1 us where sf us D[0,1]us; P0 is the spectral ﬂow of the path of elliptic operators (us D[0,1]us; P0 ),

us s ∈ [0, 1], with APS type boundary conditions P0 as in (3.5.4).

Proof. By formula (3.5.1), we have

Ind(TV ) − Ind(TU ) Z ˆ ∂ −1 ∂ = − A(M) ∧ Ch•(V ) − ξ(∂M, V ) + τµ(VP V ,P , PM ) M Z ˆ ∂ −1 ∂ + A(M) ∧ Ch•(U) + ξ(∂M, U) − τµ(UP U ,P , PM ) M Z ˆ = − A(M) ∧ [Ch•(V ) − Ch•(U)] + ξ(∂M, U) − ξ(∂M, V ) M

∂ −1 ∂ ∂ −1 ∂ since τµ(UP U ,P , PM ) = τµ(VP V ,P , PM ) by [KL04, Lemma 6.10]. Notice that

−1 us ξ(∂M, U) − ξ(∂M, V ) + sf us D[0,1]us; P0 0≤s≤1

ψ,U U ψ,U U = ξ(eU DS1×∂M eU ) − sf(D[0,1]; Pt ) − sf D[0,1](t); P0

ψ,V V ψ,V V − ξ(eV DS1×∂M eV ) + sf(D[0,1]; Pt ) + sf D[0,1](t); P0

−1 us + sf us D[0,1]us; P0 0≤s≤1 which is equal to

1 1 1 ξ(eU DS ×∂M eU ) − ξ(eV DS ×∂M eV ) + sf(eus DS ×∂M eus )0≤s≤1

43 by the lemmas below. Hence

−1 us Ind(TV ) − Ind(TU ) + sf us D[0,1]us; P0 0≤s≤1 Z ˆ = − A(M) ∧ [Ch•(V ) − Ch•(U)] M

1 1 1 − ξ(eV DS ×∂M eV ) + ξ(eU DS ×∂M eU ) + sf(eus DS ×∂M eus )0≤s≤1

+ + 1 = IndAP S(eV Db eV ) − IndAP S(eU Db eU ) + sf(eus DS ×∂M eus )0≤s≤1 which is equal to Ind[D]([U, V, us]) by Lemma 3.6.2.

Lemma 3.6.4.

ψ,us us sf D[0,1] ; P0 0≤s≤1

ψ,U U ψ,V V 1 = sf(D[0,1]; Pt ) − sf(D[0,1]; Pt ) + sf(eus DS ×∂M eus )0≤s≤1

Proof. Consider the (t, s)-parametrized family of operators

ψ,us us D[0,1] ; Pt (0≤t≤π/4 ; 0≤s≤1)

us where Pt is deﬁned as in Eq. (3.5.4). Note that     ∂ P 0 1 1 −us us   us   P0 = and Pπ/4 = .  −1 ∂  2  −1  0 Id − us P us −us 1

Hence

ψ,us us 1 eus DS ×∂M eus = (D[0,1] ; Pπ/4).

Consider the following diagram Dψ,V ; P V [0,1] t ψ,V V o ψ,V V D[0,1]; P0 D[0,1]; Pπ/4 O O ψ,us us ψ,us us D[0,1] ; P0 D[0,1] ; Pπ/4

ψ,U U / ψ,U U D[0,1]; P0 D[0,1]; Pπ/4 ψ,U U D[0,1]; Pt 44 where the arrows stand for smooth paths connecting the corresponding vertices. Now the lemma follows from the homotopy invariance of the spectral ﬂow.

Now let

ψ,us −1 D[0,1] (t) := D[0,1] + (1 − tψ)us D[0,1], us , then the same argument above proves the following lemma.

Lemma 3.6.5.

−1 us sf us D[0,1]us; P0 0≤s≤1 ψ,U U ψ,V V ψ,us us = sf D[0,1](t); P0 − sf D[0,1](t); P0 + sf D[0,1] ; P0 0≤s≤1

Proof. Consider the (s, t)-parametrized family of operators

ψ,us us D[0,1] (t),P0 0≤t,s≤1 cf. the following diagram Dψ,V (t); P V [0,1] 0 ψ,V V o ψ,V V D[0,1](0); P0 D[0,1](1); P0 O O ψ,us us ψ,us us D[0,1] ,P0 D[0,1] ; P0

ψ,U U / ψ,U U D[0,1](0); P0 D[0,1](1); P0 ψ,V U D[0,1](t); P0

ψ,us ψ,us ψ,us −1 Notice that D[0,1] (1) = D[0,1] and D[0,1] (0) = us D[0,1]us. The lemma follows by the homotopy invariance of the spectral ﬂow.

45 CHAPTER 4

HIGHER ATIYAH-PATODI-SINGER INDEX THEOREM

FOR ODD DIMENSIONAL MANIFOLDS WITH

BOUNDARY

4.1 Manifolds with boundary

We denote by cq(C) the complex Cliﬀord algebra with odd generators ci, 1 ≤ i ≤ q and relations

cicj + cjci = −2δij.

∗ This is a Z2-graded ∗-algebra with ci = −ci. Let M be an odd dimensional spin manifold with boundary. We ﬁx a Riemannian metric, say w, and a spin structure on M. Furthermore, we assume the Rieman- nian metric is of product type near the boundary, that is, on [0, ε)x × ∂M a collar neighborhood of ∂M, it takes the form

w = (dx)2 + h where h is the Riemannian metric on ∂M. Denote by Mc the manifold obtained by attaching an inﬁnite cylinder (−∞, 0] × ∂M to M along ∂M:

Mc = (−∞, 0] × ∂M ∪∂M M

The Riemannian metric M extends naturally to a Riemannian metric on Mc, still denoted by w. 46 Notice that (M,c w) is isometric to a standard b-manifold, that is, a manifold with boundary carrying a b-metric. To see this, one performs the change of variable

x x 7→ r = e on the cylindrical end. This replaces (−∞, 0]x×∂M by a compact cylinder

[0, 1]r ×∂M. Moreover, the metric w induces a metric on N = [0, 1]×∂M ∪∂M M under the coordinate change. In particular, the induced metric restricted on [0, 1]r × ∂M takes the form dr ( )2 + h r which is an exact b-metric on N, cf. [Mel93] and [Loy05]. Unless otherwise speciﬁed, all b-metrics in this thesis are assumed to be exact.

4.1.1 Cliﬀord Modules and Dirac Operators

Consider N = [0, 1]r × ∂M ∪∂M M with an exact b-metric. The set of b-vector ﬁelds, that is, vector ﬁelds on N tangential to ∂N, is closed under Lie bracket. By Swan-

Serre Theorem, such vector fields are smooth sections of a vector bundle bTN over N, called the b-tangent bundle of N, cf. [Mel93, Lemma 2.5]. We denote the b-cotangent bundle by bT ∗N and the set of b-differential forms with coefficients in V by

bΩk(N, V) = Γ(N, V ⊗ Λk(bT ∗N)). where V is a vector bundle over N. By a Clifford module over N of degree q, we mean a Z2-graded Hermitian vector bundle E over N with commuting graded ∗-actions of b ∗ the Clifford algebra cq(C) and the Clifford bundle c( T N), cf. [Get93a]. The spinor bundle S of N naturally induces a Clifford module of degree 1 as   0 c(ω) b ∗ 1|1   follows. Each ω ∈ Γ(N, c( T N)) acts on S⊗C by   and the generator c(ω) 0   0 1   1|1 + − e1 of c1(C) acts by  , where C = C ⊕ C is Z2-graded. −1 0

47 4.1.2 b-norm

∞ In this subsection, we introduce a b-norm on Cexp(Mc). We shall use this b-norm to

∞ prove the entireness of the b-JLO Chern character in section 4.2. Here Cexp(Mc) is the space of smooth functions on Mc which expands exponentially on the inﬁnite cylinder

∞ (−∞, 0]x × ∂M, cf.[Loy05]. A smooth function f ∈ C (Mc) expands exponentially on (−∞, 0]x × ∂M if ∞ X kx f(x, y) ∼ e fk(y) k=0 ∞ for (x, y) ∈ (−∞, 0]x × ∂M, where fk(y) ∈ C (∂M) for each k. More precisely, we have N−1 X kx Nx f(x, y) − e fk(y) = e RN (x, y) k=0 where all derivatives of RN (x, y) in x and y are bounded.

∞ ∞ Remark 4.1.1. Notice that Cexp(Mc) becomes exactly C (N) if one performs the change of variable x → ex on the cylindrical end.

Restricted onto (−∞, 0]x × ∂M,

x a = ac + e a∞

∞ ∞ for a ∈ Cexp(Mc), with ac, a∞ ∈ C (Mc) and ac constant with respect to x. We deﬁne

∞ a norm on Cexp(Mc) by

b kak := kak1 + 2ka∞k1

1 1 where kak1 is the usual C -norm of a and ka∞k1 is the C -norm of a∞.

Lemma 4.1.2. bk · k is a well-deﬁned multiplicative norm.

x Proof. Note that (a + b)∞ = a∞ + b∞ and (ab)∞ = acb∞ + a∞bc + e a∞b∞. So it is clear that

bkλak = |λ| · bkak

bka + bk ≤ bkak + bkbk

48 To prove the norm is multiplicative, we ﬁrst notice that kack ≤ kak , kdack ≤ kdak and

x x x d(e a∞b∞) = (e dx)a∞b∞ + e d(a∞b∞).

Thus we have

x 2kacb∞ + a∞bc + e a∞b∞k1

x x = 2kacb∞ + a∞bc + e a∞b∞k + 2kd(acb∞ + a∞bc + e a∞b∞)k

+ 2kak · kdb∞k + 2kda∞k · kbk

≤ 2kak1 · kb∞k1 + 2ka∞k1 · kbk1 + 4ka∞k1 · kb∞k1

By applying the inequality kabk1 ≤ (kak + kdak)(kbk + kdbk), we obtain

bkabk ≤ bkak · bkbk.

4.1.3 b-trace

∞ For f ∈ Cexp(Mc), we have

x f = fc + e f∞ on the cylindrical end (−∞, 0] × ∂M, where fc is constant with respect to x .

∞ Deﬁnition 4.1.3. The regularized integral of f ∈ Cexp(Mc) with respect to the b- metric is deﬁned to be

Z Z Z x − f dvol := f|M dvol + e f∞ dvol. Mc M (−∞,0]×∂M

b −∞ For A ∈ Ψ (M,c V), let KA be its Schwartz kernel and KA|∆ the restriction of KA to the diagonal ∆ ⊂ Mc × Mc. Then the ﬁberwise trace of KA|∆, denoted

49 ∞ by tr(KA|∆), is a function in Cexp(Mc), cf. [Loy05]. We deﬁne the b-trace of A ∈ bΨ−∞(M,c V) to be Z b Tr(A) := − tr(KA|∆) dvol. Mc

When V is Z2-graded, then we deﬁne the b-supertrace of A by Z b Str(A) = − str(KA|∆) dvol, Mc where str is the ﬁberwise supertrace on EndZ2 (V).

4.2 JLO Chern Character in b-Calculus

In this section, we shall deﬁne the b-JLO Chern character and prove its entireness.

1|1 Let Mc be as before and S be the spinor bundle over Mc. Then S1 = S ⊗ C is a

Cliﬀord module over Mc of degree 1, where the generator e1 of c1(C) acts on S1 by   0 1    . Let D be the Dirac operator on Mc and denote −1 0   0 D D   b 1 =   ∈ Ψ (Mc; S1). D 0

Notice that D is odd and self-adjoint, and (graded) commutes with the action of c1(C).

4.2.1 JLO Chern Character in b-Calculus

b m For A ∈ Ψ (Mc; S1), we deﬁne 1 bStr (A) := √ bStr(e A). (1) 2 π 1

More generally, for A ∈ bΨm(Mc; V) with V a Clifford module of degree q, we define 1 bStr (A) := bStr(e ··· e A), (q) (4π)q/2 1 q where e1, ··· , eq are generators of cq(C). 50 Definition 4.2.1. The b-JLO Chern character of D is defined as

b n b Ch (D)(a0, ··· , an) = ha0, [D, a1], ··· , [D, an]i

Z 2 2 2 b −σ0D −σ1D −σnD = Str(1) a0e [D, a1]e ··· [D, an]e dσ ∆n where [−, −] stands for the graded commutator.

A straightforward calculation gives the following lemma.

Lemma 4.2.2.

b 2k+1 Ch (D)(a0, ··· , a2k+1)

1 Z 2 2 2 b −σ0D −σ1D −σ2k+1D = √ Tr(a0e [D, a1]e ··· [D, a2k+1]e )dσ π ∆2k+1

Recall in the case of closed manifolds, the JLO odd Chern character is deﬁned to be

2k+1 Ch (D)(a0, ··· , a2k+1)

1 Z 2 2 2 −σ0D −σ1D −σ2k+1D = √ Tr(a0e [D, a1]e ··· [D, a2k+1]e )dσ. π ∆2k+1

Hence bCh•(D) is a natural generalization of the JLO odd Chern character to the b-calculus setting.

∞ Lemma 4.2.3. (cf. [Get93b, Lemma 4.4 ]) For g ∈ Ur(Cexp(Mc)), we have

∞ b • X b • −1 h Ch (D), k! Str(p, ··· , p)2k+1i = h Ch (D), Ch•(g ) − Ch•(g)i, k=0   0 g−1 where p =   ∈ C∞ (M) ⊗ End( r|r) with r|r = ( r)+ ⊕ ( r)− being -   exp c C C C C Z2 g 0 graded.

51 Proof. Notice that   0 [D, g−1] D   [ , p] =   [D, g] 0 and

[D, p]e−σD2 [D, p]e−τD2   −[D, g−1]e−σD2 [D, g]e−τD2 0 =   .  2 2  0 −[D, g]e−σD [D, g−1]e−τD

It follows that

bhp, [D, p], ··· , [D, p]i

= (−1)(k+1) bhg−1, [D, g], ··· , [D, g−1], [D, g]i

− (−1)(k+1) bhg, [D, g−1], ··· , [D, g], [D, g−1]i

= (−1)k bhg, [D, g−1], ··· , [D, g], [D, g−1]i

− (−1)k bhg−1, [D, g], ··· , [D, g−1], [D, g]i

Hence follows the lemma.

4.2.2 Entireness of the b-JLO Chern Character

For A ∈ bΨ−∞(M,c V), we denote Z Z M b end Tr (A) := tr(KA|∆) and Tr (A) := − tr(KA|∆). M (−∞,0]×∂M

When V is a Cliﬀord module of degree 1, we deﬁne

Z Z M b end Str(1) (A) := str(1)(KA|∆) and Str(1) (A) := − str(1)(KA|∆). M (−∞,0]×∂M

52 end b end When A|(−∞,0]×∂M is of trace class, we also write Str(1) (A) instead of Str(1) (A). Now

b let us give an upper bound in terms of kaik for

Z 2 2 2 b −σ0D −σ1D −σnD Str(1) a0e [D, a1]e ··· [D, an]e dσ (4.2.1) ∆n

Z 2 2 2 M −σ0D −σ1D −σnD = Str(1) a0e [D, a1]e ··· [D, an]e dσ (4.2.2) ∆n

Z 2 2 2 b end −σ0D −σ1D −σnD + Str(1) a0e [D, a1]e ··· [D, an]e dσ (4.2.3) ∆n

For the ﬁrst summand (4.2.2), by standard diﬀerential calculus on compact manifolds, one has

Z 2 2 2 M −σ0D −σnD M −D b b b Str(1) a0e ··· [D, an]e dσ ≤ Tr (e ) ka0k ka1k · · · kank ∆n cf. [GS89, Lemma 2.1].

For the second summand (4.2.3), ﬁrst notice that on (−∞, 0] × ∂M,

x [D, a] = c(dac) + e [c(a∞dx) + c(da∞)]

= C + exB

where C = c(dac) and B = [c(a∞dx) + c(da∞)] with c(−) denoting the Cliﬀord multiplication. Similarly, we write

x x [D, ai] = Ci + e Bi and a0 = C0 + e B0,

b where Ci is constant along the normal direction x. Notice that kBik ≤ kaik and

b kCik ≤ kaik. The term (4.2.3) can be written as a sum of terms of the following two types:

Z 2 2 b end −σ0D −σnD (I) Str(1) C0e ··· Cne dσ, ∆n

Z 2 2 2 2 b end −σ0D −σiD x −σi+1D −σnD (II) Str(1) C0e ··· e e Bi e ··· Cne dσ . ∆n

53   0 D Let us denote the Dirac operator × ∂M by D and write D =  R. By R R R   DR 0 −σD2 −σD2 [LMP09, Proposition 3.1], (e R − e )|(−∞,0]×∂M is of trace class and there is a constant K0 such that

2 2 −σD −σD Tr(e R − e )|(−∞,0]×∂M ≤ K0 for all 0 ≤ σ ≤ 1 . (4.2.4)

−σD2 −σD2 Type I. Since ke R k ≤ 1 and ke k ≤ 1, one has

2 2 b end −σ0D −σnD Str(1) C0e ··· Cne |(−∞,0]×∂M n Y 2 2 b end −σ0D −σnD ≤ (n + 1)K0 kCik + Str(1) C0e R ··· Cne R |(−∞,0]×∂M i=0 n Y = (n + 1)K0 kCik i=0 where the last equality follows from the fact

2 2 b end −σ0D −σnD Str(1) C0e R ··· Cne R |(−∞,0]×∂M = 0 by the deﬁnition of the b-trace.

Type II. Due to the presence of the factor ex,

2 2 2 2 −σ0D −σiD x −σi+1D −σnD C0e ··· e e Bi e ··· Cne is of trace class.

Without loss of generality, it suﬃces to give an upper bound for

2 2 2 end x −σ0D −σ1D −σnD Str(1) e B0e A1e ··· Ane where Ai = Bi or Ci as deﬁned above for 1 ≤ i ≤ n. First by the inequality (4.2.4),

2 2 2 end x −σ0D −σ1D −σnD Str(1) e B0e A1e ··· Ane n Y ≤ (n + 1)K0kB0k kAik i=1

2 2 2 end x −σ0D −σ1D −σnD + Str(1) e B0e R A1e R ··· Ane R |(−∞,0]×∂M

54 Now we can rewrite

2 2 2 x −σ0D −σ1D −σnD e B0e R A1e R ··· Ane R

2 2 2 x −σ0D −β1x β1x −σ1D −β2x β2x −βnx βnx −σnD = (e B0e R e )(e A1e R e )e ··· e (e Ane R )

with 1 > β1 > β2 > ··· > βn > 0. By [LMP09, Proposition 3.7], there is a constant K0 such that

2 − dim M+1 σ βix −σiD −βi+1x 0 −σi 2 i ke Aie R e k −1 ≤ K kAik(βi − βi+1) σ σi i for all i. Notice that

− dim M+1 σ − dim M+1 σ ln(σ) σ 2 = e 2 is bounded on [0, 1]. If we take βi = (n + 1 − i)/(n + 1), then by H¨olderinequality one has

2 2 2 end x −σ0D −σ1D −σnD Str(1) e B0e R A1e R ··· Ane R |(−∞,0]×∂M n n+1 Y ≤ K1 (n + 1)kB0k kAik i=1 for some ﬁxed constant K1. Applying the estimates above, we have the following proposition.

Proposition 4.2.4. bCh•(D) is an entire cyclic cochain.

Proof.

b n | Ch (D)(a0, ··· , an)|

b = ha0, [D, a1], ··· , [D, an]i

Z 2 2 2 b −σ0D −σ1D −σnD = Str(1) a0e [D, a1]e ··· [D, an]e dσ ∆n 2n(n + 1) (Kn + 2K ) ≤ 1 0 bka kbka k · · · bka k (4.2.5) n! 0 1 n

b • ω It follows that Ch (D) deﬁnes a continuous linear functional on C−(A), i.e. an entire

− cyclic cochain in Cω (A) 55 4.3 Odd APS Index Theorem for manifolds with boundary

In this section, we shall state and prove the main theorem of the second of the thesis.

Let Mc be an odd dimensional spin b-manifold with a b-metric as before and D its associated Dirac operator. Recall that the spinor bundle S of Mc naturally induces

1|1 a Cliﬀord module of degree 1, denoted by S ⊗ C , where the generator e1 of c1(C)   0 1 1|1   1|1 + − acts on S ⊗ C by  , where C = C ⊕ C is Z2-graded. We put −1 0

  0 D D   D D =   and t = t . D 0

We deﬁne

b n b Ch (D, t)(a0, a1, ··· , an) := hha0, [Dt, a1], ··· , [Dt, an]ii

∞ for ai ∈ Cexp(Mc). Here

Z 2 2 b b −σ0(dDt+D ) −σn(dDt+D ) hhA0,A1, ··· ,Anii := Str(1)(A0e t ··· Ane t )dσ, ∆n with ∞ 2 Z 2 2 2 −s(dDt+D ) X k −σ0sD −σ1sD −σksD e t := (−s) e t dDte t ··· dDte t dσ. k k=0 ∆ Notice that ( cf. [Get93a, Lemma 2.5])

n b b X b hhA0,A1, ··· ,Anii = hA0, ··· ,Ani − hA0, ··· ,Ai, dtD,Ai+1, ··· ,Ani i=0 Therefore

b k b Ch (D, t)(a0, ··· , an) = ha0, [Dt, a1], ··· , [Dt, an]i n X i b − (−1) dt ha0, [Dt, a1], ··· , [Dt, ai], D, [Dt, ai+1], ··· , [Dt, an]i. (4.3.1) i=0

56 Recall that i Z ∞ dI(Q, λ) bTr[Q, K] = ∂Tr I(K, λ) dλ 2π −∞ dλ −∞ if either Q or K is in Ψb (M,c V), where I(Q, λ) (resp. I(K, λ)) is the indicial family of Q (resp. K), cf. [Loy05, Theorem 2.5]. For the Dirac operator D above, we have

I(D, λ) = ∂D + iλc(ν) where ν = dx is the normal cotangent vector and c(ν) is the Cliﬀord multiplication of ν, cf. [Get93b, Proposition 5.4]. In the following identities ([Get93a, Lemma 6.3]), we assume that the indicial family I(Ai, λ) of Ai is independent of λ and commutes with c1(C) and c(v). Let us denote the degree of Ai with respect to the Z2-grading by |Ai|.

1. If we denote εi = (|A0| + ··· + |Ai−1|)(|Ai| + ··· + |An|), then

b εi b hhA0,A1, ··· ,Anii = (−1) hhAi, ··· ,An,A0, ··· ,Ai−1ii,

2. n X εi b b (−1) hh1,Ai, ··· ,An,A0, ··· ,Ai−1ii = hhA0,A1, ··· ,Anii, i=0 3.

b 2 hhA0, ··· ,Ai−1, [dDt + Dt ,Ai],Ai+1, ··· ,Anii

b b = hhA0, ··· ,Ai−1Ai, ··· ,Anii − hhA0, ··· ,AiAi+1, ··· ,Anii

4. We denote δi = |A0| + ··· + |Ai−1|, then

b ∂ ∂ ∂ d hhA0,A1, ··· ,Anii − hh A0, A1, ··· , Anii n X δi b = (−1) hhA0,A1, ··· ,Ai−1, [d + Dt,Ai],Ai+1, ··· ,Anii i=0

57 Here we deﬁne

Z ∂ ∂ 2 ∂ ∂ 2 ∂ ∂ ∂ ∂ b ∂ −σ0(d Dt+ D ) ∂ −σn(d Dt+ D ) hh A0, A1, ··· , Anii = Str(2) A0e t ··· Ane t dσ, ∆n where     1 0 1 0 c(ν) b b     Str(2)(B) = Str (e1e2B) with e1 =   , e2 =   4π −1 0 c(ν) 0 and   0 ∂D ∂D ∂D ∂D   t = t with =  . ∂D 0

∞ Now if we let A0 = a0 and Ai = [Dt, ai] for 1 ≤ i ≤ n, where ai ∈ Cexp(Mc), then it is easy to see that

[d + Dt,A0] = [d + Dt, a0] = [Dt, a0] and

2 [d + Dt,Ai] = [d + Dt, [Dt, ai]] = [dDt + Dt , ai] with 1 ≤ i ≤ n .

Therefore, by applying the identities above to A0 = a0 and Ai = [Dt, ai], we see that ([Get93a, Theorem 6.2])

(d − b − B)bCh•(D, t) = Ch•(∂D, t). (4.3.2) where

n ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ Ch ( D, t)( a0, a1, ··· , an) = hh a0, [ Dt, a1], ··· , [ Dt, an]ii.

∗ b • Let α ∈ Ω (0, ∞) be the diﬀerential form α = h Ch (D, t), Ch•(g)i, then

• ∂ ∂ dα = hCh ( D, t), Ch•( g)i.

By the fundamental theorem of calculus,

Z t2 • ∂ ∂ α(t2) − α(t1) = h Ch ( D, t), Ch•( g)i. t1 58 Therefore, if both of the following limits exist, then

• ∂ ∂ lim α(t) − lim α(t) = − η ( D), Ch•( g) . t→∞ t→0 Z ∞ Here η•(∂D) = Ch•(∂D, t), which is the higher eta cochain of Wu [Wu93], although 0 our normalization factor is diﬀerent from that of Wu. More explicitly,

2k+1 ∂ η ( D)(a0, a1, ··· , a2k+1)

k 1 X Z ∞ = (−1)i dt ∂ha , [∂D , a ], ··· , [∂D , a ], ∂D, [∂D , a ], ··· , [∂D , a ]i 2π 0 t 1 t i t i+1 t 2k+1 t i=0 0 where

Z ∂ 2 ∂ 2 ∂ −σ0 D −σn D hA0, ··· ,Anit = Str(A0e t ··· Ane t )dσ. ∆n Let us also write η•(∂D) = η•(∂D). This higher eta cochain η•(∂D) has a ﬁnite radius of convergence, cf. [Wu93, Proposition 1.5]. In order for

• ∂ ∂ hη ( D), Ch•( g)i to converge, we shall make the following assumptions throughout the rest of the thesis:

1. ∂D is invertible and the lowest eigenvalue of |∂D| is λ;

2. k[∂D, ∂g]k = kd ∂gk < λ.

Let us denote by

∞ dR X 1 k! −1 2k+1 ∗ Ch (g) := tr (g dg) ∈ Ω (Mc) • (2πi)k+1 (2k + 1)! k=0 the Chern character of g in de Rham cohomology of Mc. Then we have the following main theorem of the second part of the thesis.

59 Theorem 4.3.1. Let Mc be an odd dimensional spin b-manifold with a b-metric and

∂ ∞ D its associated Dirac operator. Assume D is invertible. For g ∈ Uk(C (Mc)) a unitary over Mc, if kd ∂gk < λ where λ is the lowest nonzero eigenvalue of |∂D|, then Z −1 ˆ dR • ∂ ∂ sf(D, g Dg) = − A(Mc) ∧ Ch• (g) − η ( D), Ch•( g) . Mc

b∇2/4πi 1/2 where Aˆ(Mc) = det with b∇ the Levi-Civita b-connection associ- sinh b∇2/4πi ated to the b-metric on Mc.

Proof. We need to identify the limits of α(t) for t = ∞ and t = 0. In the case of closed manifolds, the local formula for the limit of α(t) as t → 0 follows from Getzler’s asymptotic calculus, cf. [BF90, CM90, Get83]. A direct calculation in the b-calculus setting is carried out in [LMP09, Section 5 & 6]. In particular, we have

b • lim α(t) = limh Ch (D), Ch•(g)i t→0 t→0 Z ˆ dR = − A(Mc) ∧ Ch• (g) Mc Now the theorem follows immediately once we have

lim α(t) = sf(D, g−1Dg), t→∞ which we will prove in Proposition 4.5.8 below.

Corollary 4.3.2. With the same notation as above,

• ∂ ∂ ∂ η ( D), Ch•( g) = −η¯(∂M,c g) mod Z

Proof. Hereη ¯(∂M,c ∂g) is the eta invariant of Dai and Zhang [DZ06]. Without loss of generality, we can assume the unitary g is constant along the the normal direction of the cylindrical end. In this case, we have Z Z ˆ dR ˆ dR − A(Mc) ∧ Ch• (g) = A(M) ∧ Ch• (g) Mc M 60 by deﬁnition of the regularized integral. Now comparing the above theorem with the Toeplitz index theorem on odd dimensional manifolds by Dai and Zhang [DZ06,

Theorem 2.3], we have

• ∂ ∂ ∂ η ( D), Ch•( g) = −η¯(∂M,c g) mod Z.

This equality provides more evidence for the naturality of the Dai-Zhang eta invariant for even dimensional closed manifolds.

4.4 Spectral Flow

In this section, we proceed to explain the notion of spectral ﬂow and prove an analogue of Getzler’s formula for spectral ﬂow (cf. [Get93b, Corollary 2.7]) in the b-calculus setting.

Following Booss-Bavnbek, Lesch and Phillips [BBLP05], we deﬁne the notion of spectral ﬂow as follows.

Definition 4.4.1. Let Tu : H → H for u ∈ [0, 1] be a continuous path of (possibly unbounded) self-adjoint Fredholm operators, then its spectral flow, denoted by sf(Tu)0≤u≤1, is defined by

sf(Tu)0≤u≤1 = wind (κ(Tu)0≤u≤1)

−1 where κ(T ) = (T − i)(T + i) is the Cayley transform of T and wind (κ(Tu)0≤u≤1) is the winding number of the path κ(Tu)0≤u≤1 (see also [KL04, Section 6]). We also write sf(T0,T1) for the spectral ﬂow if it is clear what the path is from the context.

Actually, in this part of the thesis where we are concerned with smooth paths of self-adjoint Fredholm operators, we use the following equivalent working definition 61 of the spectral flow (cf. [BBLP05, Section 2.2]). Let Tu : H → H for u ∈ [0, 1] be a smooth path of (possibly unbounded) self-adjoint Fredholm operators. For a fixed u0 ∈ [0, 1], there exists (a, b) ⊂ [0, 1] such that

1. u0 ∈ (a, b) (unless u0 = 0 or 1, in which case u0 = a = 0 if u0 = 0 and u0 = b = 1

if u0 = 1);

2. dim ker(Tu) ≤ dim ker(Tu0 ) for all u ∈ (a, b).

By shrinking the neighborhood (a, b) if necessary, we can assume that the essential spectrum of |Tu| for u ∈ (a, b) is bounded below uniformly by λ0 and the spectrum of Tu in (−λ0, λ0) consists of discrete eigenvalues. We can further assume Tu has the same number of eigenvalues (counted with multiplicities) in (−λ0, λ0), for all u ∈ (a, b). By perturbation theory of linear operators (cf. [Kat95, II.6, V.4.3, VII.3

]), there are smooth functions βk on (a, b) such that {βk(u)}k gives a complete set of eigenvalue of Tu in (−λ0, λ0). Let nb (resp. na) be the number of nonnegative eigenvalues of Tb (resp. Ta ) in (−λ0, λ). Then we deﬁne the spectral ﬂow of (Tu)a≤u≤b to be

sf(Tu)a≤u≤b := (nb − na). (4.4.1)

We call an interval (a, b) as above together with u0 ∈ (a, b) a pointed gap interval. It is easy to see that the formula (4.4.1) is additive with respect to disjoint pointed gap intervals. Let us cover [0, 1] by ﬁnitely many intervals, say [ai, bi]0≤i≤n such that each (ai, bi) is a pointed gap interval, with bi = ai+1, u0 = a0 = 0, un = bn = 1 and uj ∈ (aj, bj) for 1 ≤ j ≤ n − 1. Then we deﬁne

n X sf(Tu)0≤u≤1 := sf(Tu)aj ≤u≤bj . (4.4.2) j=0

By additivity of formula (4.4.1), we see that sf(Tu)0≤u≤1 deﬁned as such is independent of the choice of pointed gap intervals. 62 Let Mc be an odd dimensional spin b-manifold with a b-metric as before and D

−1 its associated Dirac operator. Let Du = (1 − u)D + ug Dg. Since by assump-

∂ ∂ ∂ tion k[ D, g]k < λ, we see that Du is invertible for all u ∈ [0, 1]. It follows that inf specess(|Du|) > 0 for all u ∈ [0, 1]. Thus {Du}0≤u≤1 is an analytic family of self-adjoint Fredholm operators.

Following from the discussion above, we see that for each ﬁxed u0 ∈ [0, 1], there exists (a, b) ⊂ [0, 1] and λ0 > 0 such that the spectrum of Du in (−λ0, λ0) consists of discrete eigenvalues for all u ∈ (a, b). Moreover, we can assume Du has the same number of eigenvalues in (−λ0, λ0), for all u ∈ (a, b). We put

Au := DuPu, (4.4.3)

Bu := Du(I − Pu) + Pu, (4.4.4)

Cu := Du(I − Pu), (4.4.5)

where Pu is the spectral projection of Du on (−λ0, λ0). Let βk be the smooth functions on (a, b) such that {βk(u)}k gives the complete set of eigenvalues (counted with multiplicities) of Au. Since {Du}0≤u≤1 is an analytic family of operators, βk is an analytic function of u ∈ (a, b). It follows that for each k, βk either has only ﬁnitely many isolated zeroes or is itself constantly zero. Hence by shrinking (a, b) as much as needed, we can assume βk either is a constant zero function or has only one zero in (a, b). In the latter case, by shrinking (a, b) again if necessary, we can assume the isolated zeros can only happen at u0. Moreover, for each u ∈ (a, b), there is a set of orthonormal eigenvectors {φk(u)}1≤k≤m such that Auφk(u) = βk(u)φk(u) and the vector-valued function φk is analytic with respect to u for each 1 ≤ k ≤ m. Following Getzler [Get93b], we deﬁne the truncated eta invariant of D to be

Z ∞ Z ∞ 1 b −sD2 −1/2 2 b −t2D2 ηε(D) := √ Tr(De )s ds = √ Tr(De )dt. π ε π ε

63 and the reduced (truncated) eta invariant of D to be

η (D) + dim ker(D) ξ (D) = ε . ε 2

The following lemma is a natural extension of [Get93b, Proposition 2.5] to the b- calculus setting.

Lemma 4.4.2.

dηε(Bu) 2ε 2 2 = −√ bTr(B˙ e−ε Bu ) + E (u) du π u ε where Eε(u) is deﬁned by

∞ 1 2 Z Z h 2 2 2 2 i 2 b −st Bu 2 ˙ −(1−s)t Bu Eε(u) = −√ t Tr e Bu , Bue dsdt π ε 0 ∞ 1 2 Z Z h 2 2 2 2 i 2 b −st Bu ˙ −(1−s)t Bu − √ t Tr e Bu , BuBue dsdt. π ε 0

Proof. Using Duhamel’s principle, we have

∞ d 2 Z 2 2 b ˙ −t Bu ηε(Bu) = √ Tr(Bue )dt du π ε ∞ 1 2 Z Z 2 2 2 2 b −st Bu 2 ˙ ˙ −(1−s)t Bu − √ Tr Bue t (BuBu + BuBu)e dsdt π ε 0 ∞ ∞ 2 Z 2 2 4 Z 2 2 b ˙ −t Bu b 2 ˙ 2 −t Bu = √ Tr(Bue )dt − √ Tr(t BuBue )dt + Eε(u), π ε π ε

Integration by parts shows that

∞ ∞ Z 2 2 1 Z d 2 2 b 2 ˙ 2 −t Bu b ˙ −t Bu Tr(t BuBue )dt = − t Tr(Bue )dt ε 2 ε dt ∞ 1 2 2 t=∞ 1 Z 2 2 b ˙ −t Bu b ˙ −t Bu = − Tr(Bue )t + Tr(Bue )dt 2 t= 2 ε

2 2 b ˙ −t Bu Since Bu is invertible, Tr(tBue ) goes to 0 as t → ∞. It follows that

d 2ε 2 2 η (B ) = −√ bTr(B˙ e−ε Bu ) + E (u). du ε u π u ε

64 Corollary 4.4.3. For u ∈ (a, b),

dηε(Cu) 2ε 2 2 = −√ bTr(C˙ e−ε Cu ) + E (u) du π u ε

Proof. By deﬁnition, we have

Z ∞ −t2 ηε(Cu) = ηε(Bu) − K e dt, ε d d where K = rank(P ) is independent of u ∈ (a, b). Thus η (C ) = η (B ). Notice u du ε u du ε u that

2 2 b ˙ −ε Bu Tr(Bue )

2 2 2 2 2 2 2 2 b ˙ −ε Cu b ˙ −ε Pu b ˙ −ε Pu b ˙ −ε Cu = Tr(Cue ) + Tr(Pue ) + Tr(Cue ) + Tr(Pue ).

2 2 2 2 2 2 2 2 b ˙ −ε Cu ˙ −ε Pu ˙ −ε Pu ˙ −ε Cu = Tr(Cue ) + Tr(Pue ) + Tr(Cue ) + Tr(Pue ).

2 2 2 2 2 2 ˙ −ε Pu ˙ −ε Pu ˙ −ε Cu since Pue , Cue and Pue are all trace class operators. In fact, since Pu is a projection and the rank of Pu remains constant for each u ∈ (a, b), using Duhamel’s formula, we have

2 2 1 2 2 −1 d 2 2 Tr(P˙ e−ε Pu ) = Tr (P˙ P + P P˙ )e−ε Pu = Tr(e−ε Pu ) = 0. u 2 u u u u 2ε2 du

By the very deﬁnition of Cu (see Formula (4.4.5) above), we have PuCu = CuPu = 0.

2 2 −ε Pu In particular, Tr(Cue ) ≡ 0. Therefore,

d 2 2 2 2 2 2 0 = Tr(C e−ε Pu ) = Tr(C˙ e−ε Pu ) + Tr C e−ε Pu (P P˙ + P˙ P ) du u u u u u u u 2 2 ˙ −ε Pu = Tr(Cue )

2 2 ˙ −ε Cu Similarly, Tr(Pue ) = 0. We conclude that

2 2 2 2 b ˙ −ε Bu b ˙ −ε Cu Tr(Bue ) = Tr(Cue ).

Hence follows the corollary.

65 Lemma 4.4.4. For τ ∈ (a, b) and τ 6= u0, we have

d 2ε 2 2 ˙ −ε Au ηε(Au) = − √ Tr(Aue ) . du u=τ π u=τ Proof. Notice that Z ∞ X 2 −t2β2(u) η (A ) = √ β (u)e k dt ε u π k k ε Z ∞ X 2 2 2 = √ hA e−t Au φ (u), φ (u)idt. π u k k k ε

If βk(τ) 6= 0, then the same argument from Lemma 4.4.2 shows that

∞ d Z 2 2 2 2 −t Au ˙ −ε Au hAue φk(u), φk(u)idt = − εhAue φk(u), φk(u)i . du ε u=τ u=τ

If βk(τ) = 0, then βk ≡ 0 on (a, b) by our choice of the interval (a, b). In particular,

Auφk(u) = 0 for all u ∈ (a, b). Then d hA φ (u), φ (u)i = hA˙ φ (u), φ (u)i + hA φ˙ (u), φ (u)i + hA φ (u), φ˙ (u)i du u k k u k k u k k u k k ˙ = hAuφk(u), φk(u)i.

˙ It follows that hAuφk(u), φk(u)i = 0. Hence

2 2 ˙ −ε Au ˙ hAue φk(u), φk(u)i = hAuφk(u), φk(u)i = 0 for all u ∈ (a, b). This ﬁnishes the proof.

Corollary 4.4.5. For u ∈ (a, b) and u 6= u0,

d 2ε 2 2 η (D )du = −√ bTr(D˙ e−ε Du ) + E (u). du ε u π u ε

Proof. Notice that

ηε(Du) = ηε(Au) + ηε(Cu) and

2 2 2 2 b ˙ −ε Cu b ˙ −ε Au Tr(Aue ) = Tr(Cue ) = 0.

The corollary follows from the above lemmas. 66 + If we denote by Q the cardinality of the set {βk | βk(u0) = 0 and βk(a) > 0} and

− Q be the cardinality of the set {βk | βk(u0) = 0 and βk(a) < 0}, then

+ − dim ker Du0 = dim ker Du + Q + Q for u ∈ (a, u0). (4.4.6)

Since ∞ 2 Z 2 2 lim √ λe−t λ dt = ±1, ± λ→0 π ε it follows that

+ − lim ηε(Du) = ηε(Du ) + Q − Q . (4.4.7) − 0 u→u0 − Recall that by deﬁnition sf(Da,Du0 ) = Q and η (D ) + dim ker(D ) ξ (D ) = ε u0 u0 . ε u0 2 Therefore, the diﬀerence of equation (4.4.6) and equation (4.4.7) gives

sf(Da,Du ) = ξε(Du ) − lim ξε(Du). 0 0 − u→u0

Similarly, sf(Du ,Db) = lim ξε(Du) − ξε(Du ). Thus we have 0 + 0 u→u0

sf(Da,Db) = lim ξε(Du) − lim ξε(Du). + − u→u0 u→u0 With the above results combined, we have the following proposition.

Proposition 4.4.6.

1 ε Z 2 2 −1 b ˙ −ε Du sf(D, g Dg) = lim √ Tr(Due )du ε→∞ π 0

Proof. Let us cover [0, 1] by ﬁnitely many pointed gap intervals [ai, bi], 0 ≤ i ≤ n, with ui ∈ [ai, bi] such that bi = ai+1 with u0 = a0 = 0, un = bn = 1 and uj ∈ (aj, bj) for 1 ≤ j ≤ n − 1. Then X sf(D0,D1) = ξε(D1) − ξε(D0) + lim ξε(Du) − lim ξε(Du) u→u+ u→u− i i i 1 Z 1 d = ξε(D1) − ξε(D0) − ηε(Du)du 2 0 du 1 1 ε Z 2 2 1 Z b ˙ −ε Du = ξε(D1) − ξε(D0) + √ Tr(Due )du − Eε(u)du. π 0 2 0 67 Z 1 −1 Notice that ξε(g Dg) = ξε(D) and Eε(u)du vanishes when ε → ∞, hence follows 0 the proposition.

4.5 Large Time Limit

In this section, we prove the equality

−1 b • sf(D, g Dg) = lim h Ch (tD), Ch•(g)i. t→∞ This is the last step remaining to prove Theorem 4.3.1. We follow rather closely

Getzler’s proof for closed manifolds [Get93b].

Recall that we have   0 D D   b 1 =   ∈ Ψ (Mc; S1), D 0   0 ∂D ∂D   b 1 =   ∈ Ψ (∂M; S1) ∂D 0 and   0 g−1 p =   ∈ C∞ (M) ⊗ End( r|r)   exp c C g 0

r|r r + r − with C = (C ) ⊕ (C ) being Z2-graded. Let us put

b 1 r|r Du = (1 − u)D − upDp ∈ Ψ (Mc; S1 ⊗s C )

r|r r|r for u ∈ [0, 1], where S1 ⊗s C is the super-tensor product of S1 and C . We see immediately that   (1 − u)D + ug−1Dg 0 D   u =   0 (1 − u)D + ugDg−1   D + ug−1[D, g] 0   =   . 0 D + ug[D, g−1]

68 ∂ ∂ We denote Du,s = Du + sp (resp. Du,s = Du + sp), where (u, s) ∈ [0, 1] × (−∞, 0].

b ∂ ∂ Consider the superconnections A = d + Du,s and A = d + Du,s, where d is the standard de Rham diﬀerential on the parameter space [0, 1] × (−∞, 0]. We have

b 2 2 2 ˙ A = Du + s[Du, p] + s + duDu + dsp,

∂ 2 ∂ 2 ∂ ∂ 2 ∂ ˙ ∂ A = Du + s[ Du, p] + s + du Du + ds p.

Recall that the indicial family I(D, λ) of D is

I(D, λ) = ∂D + iλc(ν) where ν = dx is the normal cotangent vector and c(ν) is the Cliﬀord multiplication of ν, cf. [Get93b, Proposition 5.4]. Therefore, we have

∂ I(Du, λ) = Du + iλc(ν)

∂ ∂ I(Du,s, λ) = Du + iλc(ν) + s p,

∂ ˙ ∂ I(dDu,s, λ) = du Du + ds p,

2 ∂ 2 2 ∂ ∂ 2 I(Du,s, λ) = Du + λ + s[ Du, p] + s .

b Consider the Chern character of A, deﬁned by

b 2 b b − A Ch( A) := Str(1)(e ).

Denote Γu the contour {u} × [0, ∞) and γs the contour [0, 1] × {s}. By Stoke’s theorem, we have

Z Z Z Z Z Ch( b ) − Ch( b ) + Ch( b ) − lim Ch( b ) = dCh( b ). A A A s→∞ A A Γ1 Γ0 γ0 γs [0,1]×[0,∞) (4.5.1)

69 4.5.1 Technical Lemmas

In this section, let us prove several technical lemmas. Notice that by deﬁnition, we have

1 Z 2 2 2 2 b b −σ(Du+s[Du,p]+s ) ˙ −(1−σ)(Du+s[Du,p]+s ) Ch( A) = −du Str(1) e Due dσ 0 1 Z 2 2 2 2 b −σ(Du+s[Du,p]+s ) −(1−σ)(Du+s[Du,p]+s ) − ds Str(1) e p e dσ 0

Lemma 4.5.1.

1 Z Z 2 b b ˙ −Du Ch( A) = − Str(1)(Due )du γ0 0

Proof. Since

2 2 b −σDu ˙ −(1−σ)Du Str(1)[e , Due ]

∞ −σ(∂D2+λ2) ! −1 Z d(e ) ∂ 2 2 ∂˙ −(1−σ)( D +λ ) = Str(1) Due dλ = 0, 2πi −∞ dλ it follows that

1 1 Z Z Z 2 2 b b −σDu ˙ −(1−σ)Du Ch( A) = − du Str(1)(e Due )dσ Γ0 0 0 1 Z 2 b ˙ −Du = − Str(1)(Due )du 0

Lemma 4.5.2. Z lim Ch( b ) = 0 s→∞ A γs Proof. First notice that a similar argument as that in Lemma 4.5.1 shows that

1 Z 2 2 2 2 b −σ(Du+s[Du,p]+s ) −(1−σ)(Du+s[Du,p]+s ) Str(1) e p e dσ 0

2 2 b −Du−s[Du,p]−s = Str(1)(p e ).

70 Using Duhamel’s principle, we have

2 2 b −Du−s[Du,p]−s Str(1)(p e ) ∞ 2 Z 2 2 2 X −s n b −σ0Du −σ1Du −σnDu = e (−s) Str(1) p e [Du, p]e ··· [Du, p]e dσ. n n=0 ∆ The estimates in Section 4.2.2 show that

Z 2 2 2 b −σ0Du −σ1Du −σnDu Str(1) pe [Du, p]e ··· [Du, p]e dσ ∆n Kn + 2K ≤ 2n(n + 1) 1 0 bkpkn+2. n! for some constants K0 and K1. In fact K0 and K1 can be chosen independent of u, since there is constant C such that

2 2 −σD −σDu Tr(e R − e )|(−∞,0]×∂M ≤ C for all σ, u ∈ [0, 1] (cf. [LMP09, Proposition 3.1]). Hence

2 2 2 b b −Du−s[Du,p]−s 0 −s +2K kpks Str(1)(p e ) ≤ K e Z 0 b −s2/2 for some constants K and K . Therefore Ch( A) = O(e ) as s → ∞, hence γs follows the lemma.

Lemma 4.5.3.

∞ Z Z 1 X Ch( b ) = − Ch( b ) = hbCh•(D), k!Str(p, ··· , p) i A A 2 2k+1 Γ0 Γ1 k=0

71 b 2 2 2 Proof. When u = 0, we have A = D +s[D, p]+s +dsp. Using Duhamel’s principle, we see that

b Ch( A)|Γ0 ∞ 2 Z 2 2 X 2k+1 −s b −σ0D −σ1D = (−s) e Str(1) e [D, p]e ··· n n=0 ∆

2 2 2 e−σiD (−dsp)e−σi+1D ··· [D, p]e−σnD dσ

∞ 2k+1 X 2 X = s2k+1e−s ds bh1, [D, p], ··· , [D, p], p, [D, p], ··· , [D, p]i k=0 i=0 ∞ X 2k+1 −s2 b = s e ds hp, [D, p], ··· , [D, p]i2k+1. (4.5.2) k=0 where we have used the fact (cf.[Get93a, Lemma 6.3 (2)])

2k+1 X b b h1, [D, p], ··· , [D, p], p, [D, p], ··· , [D, p]i = hp, [D, p], ··· , [D, p]i2k+1. i=0 It follows that

∞ Z 1 X Ch( b ) = hbCh•(D), k!Str(p, ··· , p) i. A 2 2k+1 Γ0 k=0

b 2 2 2 When u = 1, we have A = D1 + s[D1, p] + s + dsp. Since D1 = −pDp, we see that

−D2 −D2 [D1, p] = −[D, p] and e 1 = pe p .

Furthermore, we notice that p[D, p]p = [D, p]. Combining these with a calculation similar to that in Equation (4.5.2) with D replaced by D1, we see that

∞ Z 1 X Ch( b ) = − hbCh•(D ), k!Str(p, ··· , p) i A 2 1 2k+1 Γ1 k=0

Lemma 4.5.4.

b 2 ∂ 2 b b − A − A dCh( A) = − Str[Du,s, e ] = Str(2)(e ) 72 b − b 2 Proof. Since [ A, e A ] = 0,

b 2 b 2 b 2 b − A b − A b − A d Str(1)(e ) = Str(1)[d, e ] = − Str(1)[Du,s, e ].

It follows that

∞ 1 Z dI(D , λ) b 2 b u,s − A dCh( A) = Str(1) I(e , λ) dλ 2πi −∞ dλ ∞ 1 Z b 2 − A = Str(1) ic(ν)I(e , λ) dλ 2πi −∞ ∞ 1 Z b 2 − A = √ Str(2) I(e , λ) dλ π −∞ ∞ 1 Z 2 ∂ 2 −λ − A = √ e dλ Str(2)(e ) π −∞

∂ 2 − A = Str(2)(e )

∂ 2 − A By Duhamel’s principle, the 2-form components in Str(2)(e ) can be expanded as

∞ X X k−2 −s2 ∂ ∂ ∂ (−s) e h1, [ Du, p], ··· , [ Du, p], p , [ Du, p], ··· , |{z} k=2 1≤i

∂ ∂ ˙ ∂ ∂ [ Du, p], Du , [ Du, p], ··· , [ Du, p]iu duds |{z} j−th ∞ X X k−2 −s2 ∂ ∂ ∂ ˙ ∂ − (−s) e h1, [ Du, p], ··· , [ Du, p], Du , [ Du, p], ··· , |{z} k=2 1≤i

∂ ∂ ∂ [ Du, p], p , [ Du, p], ··· , [ Du, p]iu duds (4.5.3) |{z} j−th where

Z ∂ 2 ∂ 2 ∂ 2 b −σ0 Du −σ1 Du −σn Du hA0, ··· ,Aniu = Str(2) A0 e A1e ··· Ane dσ. ∆n Recall that (cf.[GS89, Lemma 2.2])

n X (|A0|+···+|Ai−1|)(|Ai|+···+|An|) hA0, ··· ,Aniu = (−1) h1,Ai, ··· ,An,A0, ··· ,Ai−1iu i=0 73 ∂ ∂ ˙ ∂ Since Du, Du and p are of odd degree and [ Du, p] is of even degree, one has

(4.5.3)

∞ k−1 X X k−2 −s2 ∂ ∂ ∂ ˙ ∂ ∂ = (−s) e hp, [ Du, p], ··· , [ Du, p], Du , [ Du, p], ··· , [ Du, p]iu duds. |{z} k=2 i=1 i−th Let us deﬁne

n ∂ ∂ ∂ Chf ( Du,V )(a0, ··· , an) = ι(V )ha0, [ Du, a1], ··· , [ Du, an]iu (4.5.4) where

X |V |(|A0|+···+|Ai|) ι(V )hA0, ··· ,Aniu := (−1) hA0, ··· ,Ai,V,Ai+1, ··· ,Aniu 0≤i≤n Then the calculation above shows that ∞ ∂ 2 2 k − A X k −s ∂ ∂ ˙ Str(2)(e ) = − (−s) e hChf ( Du, Du), Str(p, ··· , p)kiu duds. k=0 We summarize this in the following lemma.

Lemma 4.5.5.

1 ∞ Z ∂ 2 1 Z • − A ∂ ∂ ˙ X Str(2)(e ) = h Chf ( Du, Du)du, k! Str(p, ··· , p)2k+1i 2 [0,1]×[0,∞) 0 k=0

Proof. Notice that Str(p, ··· , p)k = 0 for k even, and ∞ Z 2 k! s2k+1e−s ds = . 0 2

4.5.2 Large Time Limit

∂ ∞ ∂ ∂ Recall that D is invertible and g ∈ Uk(C (N)) is a unitary such that k[ D, g]k < λ with λ the lowest nonzero eigenvalue of |∂D|. In the following, we also write g for ∂g if there is no confusion. Notice that

k[∂D, g−1]k = k − g−1[∂D, g]g−1k ≤ k[∂D, g]k, and similarly k[∂D, g]k ≤ k[∂D, g−1]k. Hence k[∂D, g−1]k = k[∂D, g]k. 74 b Lemma 4.5.6. Let At = d + tDu,s. Then Z b lim dCh( At) = 0. t→∞ [0,1]×[0,∞)

Proof. Notice that

|∂D + ug−1[∂D, g]| ≥ |∂D| − ukg−1[∂D, g]k ≥ λ − uk[∂D, g]k (4.5.5)

|∂D + ug[∂D, g−1]| ≥ λ − uk[∂D, g]k (4.5.6)

When u = 1, ∂D + ug−1[∂D, g] = g−1 ∂Dg. The lowest eigenvalue of |g−1 ∂Dg| is also

λ, since g is a unitary. Therefore, a similar argument as above shows that

|∂D + ug−1[∂D, g]| ≥ λ − (1 − u)k[∂D, g]k, (4.5.7)

|∂D + ug[∂D, g−1]| ≥ λ − (1 − u)k[∂D, g]k. (4.5.8)

∂ Thus | Du| is bounded below by

∂ ∂ λu := max{λ − uk[ D, g]k, λ − (1 − u)k[ D, g]k}.

Then there exists a constant C such that

2∂ 2 2 2 Tr(e−t Du ) ≤ Ce−t λu

2 ∂ 2 for all t ≥ 1, where we may take C = sup eλu Tr(e− Du ), cf. [GS89, Theorem C]. One u∈[0,1] ∂ ∂ also notices that k[ Du, p]k = (1 − 2u)k[ D, p]k < λu. Therefore we have

2k+2 ∂ ∂ ∂ ˙ ∂ ∂ t hp, [ Du, p], ··· , [ Du, p], Du, [ Du, p], ··· , [ Du, p]iu,2k+1

1 2∂ 2 ≤ t2k+2Tr(e−t Du )kpk · k[∂D , p]k2k+1k∂D˙ k (2k)! u u

C 2 2 ≤ t2k+2e−t λu k[∂D , p]k2k+1kp[∂D, p]k. (2k)! u

75 Hence

Z Z ∂ 2 b − A dCh( At) = Str(2)(e t ) [0,1]×[0,∞) [0,1]×[0,∞) 1 ∞ Z • ∂ ∂ ˙ X = h Chf (t Du, t Du)du, k! Str(p, ··· , p)2k+1i 0 k=0 Z 1 ∞ X C 2 2 ≤ t · kp[∂D, p]k e−t λu (k[∂D , p]k · t)2k+1du k! u 0 k=1 1 Z ∂ 2 2 2 ≤ t2C0 e(k[ Du,p]k −λu)t du 0

where the last term goes to 0 when t → ∞. This ﬁnishes the proof.

∞ ∂ ∂ Lemma 4.5.7. If g ∈ Uk(C (N)) is a unitary such that k[ D, g]k < λ with λ the lowest nonzero eigenvalue of |∂D|, then

b • b • −1 lim h Ch (tD), Ch•(g)i = − lim h Ch (tD), Ch•(g )i t→∞ t→∞

Proof. Recall the equation (4.3.2):

(d − b − B)bCh•(D, t) = Ch•(∂D, t).

−1 Since Ch•(g) + Ch•(g ) − 2Ch•(1) = (b + B)Tch•(h) where h is a smooth path of     1 0 g 0     unitaries connecting   and   (cf. Section 2.3.3), we have 0 1 0 g−1

b • −1 lim h Ch (tD), Ch•(g) + Ch•(g )i t→∞

b • −1 = lim h Ch (tD), Ch•(g) + Ch•(g ) − 2Ch•(1)i t→∞

b • = lim h(b + B) Ch (tD), Tch•(h)i t→∞

• ∂ = − lim hCh (t D), Tch•(h)i t→∞

76 Now a similar argument (without the parameter u involved) as that in Lemma 4.5.6 above shows that

• ∂ lim hCh (t D), Tch•(h)i = 0. t→∞

Hence follows the lemma.

Proposition 4.5.8.

−1 b • sf(D, g Dg) = lim h Ch (tD), Ch•(g)i. t→∞

Proof. Notice that   D + ug−1[D, g] 0 D   u =   0 D + ug[D, g−1]   0 D + ug−1[D, g]     D + ug−1[D, g] 0    =   .  0 D + ug[D, g−1]     D + ug[D, g−1] 0

Hence

1 1 Z 2 1 Z −1 2 b ˙ −Du b −1 −(D+ug [D,g]) Str(1)(Due )du = √ Tr(g [D, g]e )du 0 π 0 1 1 Z −1 2 − √ bTr(g[D, g−1]e−(D+ug[D,g ]) )du. π 0

It follows from Theorem 4.4.6 that

1 Z 2 2 b ˙ −t Du lim Str(1)(tDue )du t→∞ 0 = sf(D, g−1Dg) − sf(D, gDg−1) = 2 sf(D, g−1gD) since sf(D, gDg−1) = −sf(D, g−1Dg).

77 Now by applying Lemma 4.5.2, 4.5.3 and 4.5.1 to equation (4.5.1), we have

∞ 1 Z 2 2 b • X b ˙ −t Du − h Ch (tD), k!Str(p, ··· , p)2k+1i − Str(1)(tDue )du k=0 0 Z b = dCh( At) [0,1]×[0,∞)

Therefore,

1 Z 2 2 b ˙ −t Du lim Str(1)(tDue )du t→∞ 0 Z b • −1 b = lim h Ch (tD), Ch•(g) − Ch•(g )i − lim dCh( At). t→∞ t→∞ [0,1]×[0,∞)

It follows from Lemma 4.5.6 that

−1 b • −1 2 sf(D, g Dg) = lim h Ch (tD), Ch•(g) − Ch•(g )i. t→∞

Now applying Lemma 4.5.7, we have

−1 b • sf(D, g Dg) = lim h Ch (tD), Ch•(g)i. t→∞

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