Analytic Torsion, the Eta Invariant, and Closed Differential Forms on Spaces of Metrics
by
Phillip Andreae
Department of Mathematics Duke University
Date: Approved:
Mark Stern, Supervisor
Hubert Bray
Lenhard Ng
Leslie Saper
Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2016 Abstract Analytic Torsion, the Eta Invariant, and Closed Differential Forms on Spaces of Metrics
by
Phillip Andreae
Department of Mathematics Duke University
Date: Approved:
Mark Stern, Supervisor
Hubert Bray
Lenhard Ng
Leslie Saper
An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2016 Copyright c 2016 by Phillip Andreae All rights reserved except the rights granted by the Creative Commons Attribution-Noncommercial Licence Abstract
The central idea of this dissertation is to interpret certain invariants constructed from Laplace spectral data on a compact Riemannian manifold as regularized integrals of closed differential forms on the space of Riemannian metrics, or more generally on a space of metrics on a vector bundle. We apply this idea to both the Ray-Singer analytic torsion and the eta invariant, explaining their dependence on the metric used to define them with a Stokes’ theorem argument. We also introduce analytic multi-torsion, a generalization of analytic torsion, in the context of certain manifolds with local product structure; we prove that it is metric independent in a suitable sense.
iv To my mother and father
v Contents
Abstract iv
List of Symbols ix
Acknowledgementsx
1 Introduction1
2 Vector bundles, metrics, and differential forms on spaces of metrics6
2.1 Differential forms...... 6
2.2 Sobolev spaces and algebras of operators...... 9
2.3 Adjoints and traces...... 10
2.4 Flat vector bundles...... 12
3 Analytic torsion of an elliptic complex 15
3.1 Heat kernels and zeta functions...... 15
3.2 Elliptic complexes...... 19
3.3 Analytic torsion...... 22
3.4 A closed form for analytic torsion...... 25
3.4.1 An operator parametrizing the space of metrics...... 25
3.4.2 The closed one-form on the space of metrics...... 27
3.5 Proof of closedness...... 30
3.6 Proof of the metric variation formula...... 35
3.7 Regularization methods...... 37
vi 3.8 Varying the curve...... 42
3.9 Some historical remarks...... 44
4 Multi-torsion on manifolds with local product structure 50
4.1 Motivation: Product manifolds...... 51
4.1.1 Products of zeta functions...... 51
4.1.2 Product manifolds...... 52
4.1.3 Analytic torsion...... 53
4.2 Heat multi-asymptotics and multi-zeta functions...... 54
4.3 The geometry of finite quotients of product manifolds...... 59
4.4 Multi-zeta functions and multi-torsion...... 62
4.5 Proof of heat kernel multi-asymptotics...... 66
4.5.1 Heat kernel on global product...... 67
4.5.2 Heat kernel on the quotient: Selberg principle...... 67
4.5.3 The trace of the heat kernel with an auxiliary operator.... 68
4.5.4 Equivariant heat kernels...... 70
4.5.5 Vanishing of constant terms...... 71
4.6 A closed form for multi-torsion...... 77
4.7 Proof of closedness...... 81
4.8 Proof of metric independence...... 84
5 The eta invariant 88
5.1 The eta invariant...... 88
5.2 A closed form for the eta invariant...... 92
5.3 Proof of closedness...... 93
5.4 Proof of the metric variation formula...... 98
References 102
vii Biography 104
viii List of Symbols
Z The integers.
Zě0 The nonnegative integers.
R The real numbers.
C The complex numbers. Re s The real part of a complex number s.
s The complex conjugate of a complex number s.
DiffpMq The group of diffeomorphisms of a manifold M.
Sk The symmetric group of permutations on k letters.
signpσq The sign of a permutation σ P Sk.
GLpVq The general linear group of invertible linear endomorphisms of a vector space V .
ΩkpM,Aq The space of A-valued k-forms on M, where A is an algebra and M is a space of metrics. (See Section 2.1.)
C8pM,Eq The space of smooth sections of a vector bundle E Ñ M.
L2pM,Eq The space of L2 sections of a vector bundle E Ñ M. (See Section 2.2.)
HapM,Eq The Sobolev space of order a P R of sections of a vector bundle E Ñ M. (See Section 2.2.)
ix Acknowledgements
Thank you to my advisor, Mark Stern, for most of the ideas of this dissertation and for all your support and encouragement. I would also like to thank Mike Lipnowski, for many useful discussions about torsion, and the other members of my dissertation committee: Hugh Bray, Lenny Ng, and Les Saper. The National Science Foundation, the Simons Foundation, and the Duke Univer- sity Graduate School partially funded this research, for which I am very appreciative.
x 1
Introduction
Analytic torsion and the eta invariant are invariants constructed from eigenvalues of Laplace-type operators on differential forms. Their geometric-analytic nature means that their definition requires a choice of Riemannian metric, but in special cases, and/or with a modified interpretation, they are in fact topological invariants, i.e., they are independent of the choice of metric. To elaborate on this point: torsion and eta may be considered invariants in the spirit of the heat equation proof of the Atiyah-Singer index theorem. Unlike the index of an elliptic operator, however, torsion and eta are non-local invariants in the sense that they cannot be expressed as the integral of an integrand that is locally computable in terms of the geometry near a point. This is essentially equivalent to the fact that torsion and eta cannot be computed in terms of the small time asymptotics of the heat kernel; knowledge of the heat trace for all positive times is required. Torsion and eta share the property, however, that their variation with respect to a change in Riemannian metric is a locally computable quantity, expressible in terms of only small time heat asymptotics. In some cases, this local quantity vanishes, giving that the original quantity is independent of the metric. More generally, even if the
1 local quantity does not vanish, this property is enough to construct an associated topological invariant (the Ray-Singer metric or the relative eta invariant) that is independent of the choice of Riemannian metric. The dependence of torsion and eta on the metric used to define them is the subject of this dissertation. Central to our approach is an interpretation involving closed differential forms on the infinite-dimensional manifold consisting of all Riemannian metrics, or more generally the space of all metrics on some vector bundle. We phrase the fact that the variations of torsion and eta are locally computable in terms of a Stokes’ theorem argument on the space of metrics. This perspective suggests the possibility of generalizations involving other closed forms on spaces of metrics. We introduce one such generalization called multi-torsion, which we define on manifolds which are quotients of product manifolds by certain actions of finite groups of diffeomorphisms.
A brief introduction to analytic torsion. Analytic torsion may be thought of as an analogue of an Euler characteristic or index for odd-dimensional manifolds, but as noted above, it differs from an index in its non-local nature. The celebrated Cheeger- M¨ullertheorem ([6], [13], [14], [5]), conjectured by Ray-Singer, states the equality of analytic torsion and Reidemeister torsion, a classical combinatorial invariant that is of historical interest for its use in classifying lens spaces. A deep and surprising con- sequence is that analytic torsion has a connection with torsion in integer cohomology groups; see, for example, the paper of Bergeron-Venkatesh [4]. To be a bit more precise: let M be a compact manifold of dimension n, let g be a Riemannian metric on M, and let F be a flat unitary vector bundle. Using zeta function regularization, one defines the determinants det ∆q of the Laplacians
∆q on F -valued differential q-forms, for q “ 0, . . . , n. Then the associated de Rham
2 analytic torsion T “ T pM, F, gq is defined by the weighted alternating sum
1 n log T pM, F, gq :“ p´1qqq log det ∆ . (1.1) 2 q q“0 ÿ Ray-Singer [16] made the definition (1.1) and proved that if the dimension n is odd
and F is acyclic (i.e., there is no F -cohomology), then T pM, F, gq is independent of the Riemannian metric g, i.e., T is a smooth topological invariant of the pair pM,F q.
Our perspective on analytic torsion. We discuss analytic torsion in the context of
a Z-graded elliptic complex, which generalizes both the de Rham analytic torsion described above and the Dolbeault analytic torsion for complex manifolds, also in- troduced by Ray-Singer [17]. We generalize the metric independence theorem of Ray-Singer to this context and to a generalized class of metrics. For example, in the setting of the previous paragraph, our framework allows us to consider any metric
on the bundle F b Λ‚T ˚M, not just metrics induced as the product of a metric on F and a Riemannian metric on M. More importantly, we provide a new perspective on the metric independence theorem by interpreting analytic torsion as an integral on the space of metrics. To give a rough idea of this perspective, we note that the logarithm of analytic torsion is the regularization of the following divergent integral:
8 n 1 g dt p´1qqq Tr e´t∆q . (1.2) 2 t 0 q“0 ż ÿ (The regularization, involving analytic continuation of a zeta function, is essential,
but we suppress the details for now.) The half-line p0, 8q embeds in the space of
1 metrics by scaling the metric g, i.e., as a curve Cg parametrized by t ÞÑ t g. The
g n q ´t∆q dt one-form q“0p´1q q Tr e t , defined on p0, 8q, is in fact the pullback to Cg of
ωT , a one-formř defined on the entire space of metrics. We prove that ωT is closed. 3 The variation of T pgq with respect to a change in metric g is computable by a Stokes’
theorem argument. The fact that ωT is closed reduces the computation, very roughly, to boundary terms: a term at the limit t Ñ 8 (which vanishes by the decay of the heat kernel) and a term at the limit t Ñ 0. The latter term is locally computable and, furthermore, vanishes under appropriate assumptions. Similar considerations apply to the eta invariant of Atiyah-Patodi-Singer [1], which we interpret as the integral of another closed one-form ωη over the same curve
Cg on the space of metrics, providing a new perspective on the well-known metric independence of the relative eta invariant.
Multi-torsion. The form ωT and the Stokes’ theorem argument computing the vari- ation of T suggests the possibility of new invariants constructed from other closed forms on the space of metrics. We introduce a two-form ωMT defined on the space of product metrics on the product manifold M1 ˆ M2; ωMT is essentially the wedge
M1 M2 product of the torsion one-forms ωT and ωT corresponding to the two factors. We show that ωMT is also defined more generally on the space of metrics inducing a local geometric product structure in a certain sense; we treat in detail the case of the quotient of M1 ˆ M2 by a finite group of diffeomorphisms. In that context, the
´∆ Laplacian ∆ and the heat operator e depend on two parameters t1 and t2, each corresponding to scaling the local product metric in one of the directions. The inte- gral (1.2) over the half-line p0, 8q Ă R is replaced by an integral over the quadrant p0, 8q ˆ p0, 8q Ă R2, which embeds in the space of product metrics. We call the resulting invariant multi-torsion, and we prove that it is independent of the metric in the space of local product metrics.
Outline. We now outline the contents of this dissertation. Chapter 2 fixes some notational conventions and develops some tools related to
4 the notion of spaces of metrics and differential forms thereon. The impatient reader may wish merely to skim Chapter 2 on a first reading.
Chapter 3 introduces analytic torsion T in the context of a Z-graded elliptic complex. We first review some facts about heat kernels and zeta functions that are necessary to the definition of analytic torsion. We then introduce the one-form ωT , explain its relevance to analytic torsion, and prove that it is closed. We prove the metric independence of T (under appropriate conditions) using a Stokes’ theorem argument. We study alternate methods of regularization and show that analytic torsion is invariant under certain deformations of the curve over which ωT is in- tegrated. Finally, in Section 3.9, we discuss briefly the history of Ray-Singer’s de Rham analytic torsion, the Cheeger-M¨uller theorem, and how our results fit in to the picture. Chapter 4 introduces what we call multi-torsion, which generalizes analytic tor- sion to manifolds with a local product structure. To define multi-torsion, we must develop some analytic tools to treat heat kernels depending on two parameters and their associated multi-zeta functions. Then we introduce the closed two-form (more generally, m-form) ωMT and integrate it to produce multi-torsion. Finally, we study the dependence of multi-torsion on the metric and prove a metric independence the- orem. Chapter 5 applies ideas similar to those of Chapter 3 to the eta invariant, which we interpret as the integral of a closed one-form ωη on the space of metrics.
5 2
Vector bundles, metrics, and differential forms on spaces of metrics
This short chapter develops some tools related to the notion of differential forms on the space of metrics on a vector bundle. The final section discusses flat vector bundles.
2.1 Differential forms
Let M be a compact manifold, and let E Ñ M be a real or complex vector bundle. By a metric on E, we shall mean a smoothly varying inner product on the fibers of E; this inner product is euclidean (if E is a real vector bundle) or hermitian (if E is a complex vector bundle). In this chapter, we will assume that E is a real vector bundle, but our results hold in the complex case with minor modifications. Let M denote the set of all metrics on E. M is canonically viewed as an open subset of the Fr´echet space C8pM, Sym2 E˚q, giving M the structure of a Fr´echet manifold. Thus the tangent space to M at a metric h P M is canonically identifiable with C8pSym2 E˚q, and the tangent bundle of M is canonically identifiable with
6 M ˆ C8pSym2 E˚q. We will now introduce the notion of differential forms on M.
Definition 2.1.1. Let A be an associative C-algebra. For k P Zě0, an A-valued k-form on M is a smooth map
ω : M ˆ C8pSym2 E˚qˆk Ñ A
that is multilinear and alternating in the C8pSym2 E˚q entries, i.e., for every h P M,
8 2 ˚ X1,...,Xk P C pSym E q, and σ P Sk, ω satisfies
ωph; Xσp1q,...,Xσpnqq “ signpσqωph; X1,...,Xkq.
The vector space of A-valued k-forms on M will be denoted by ΩkpM,Aq.
ΩkpM,Aq possesses a product structure generalizing the usual wedge product on differential forms. For simplicity of notation, we will omit the traditional wedge symbol “^”, i.e., if κ P ΩkpM,Aq and λ P ΩlpM,Aq, their product will be denoted κλ P Ωk`lpM,Aq.
Remark 2.1.2. Since A need not be a commutative algebra, the product of A-valued forms need not be graded commutative, i.e., if κ P ΩkpM,Aq and λ P ΩlpM,Aq, then κλ is not equal to p´1qklλκ in general. A certain graded commutativity property does hold, however, when we introduce traces; see Lemma 2.3.5 below.
We will denote by δM the exterior derivative. If κ P ΩkpM,Aq, then δMκ P Ωk`1pM,Aq. δM satisfies δM ˝ δM “ 0 and the Leibniz rule: if κ P ΩkpM,Aq and λ P ΩlpM,Aq, then δMpκλq “ pδMκqλ ` p´1qkκpδMλq. If δMω “ 0, then we will say ω is closed. We will be primarily interested in the restriction of forms on M to finite-dimensional submanifolds of M. More precisely:
7 Let U be an open set in Rk (or more generally, a k-manifold) and let φ : U Ñ M be a smooth map. If ω is a C-valued form on M, then we may consider φ˚ω, the pullback of ω by φ, which is a differential form on U in the ordinary sense. We have the usual relation that the exterior derivative commutes with pullback:
δU pφ˚ωq “ φ˚pδMωq, where δU is the de Rham exterior derivative on U. In particular, if ω is closed on M, then φ˚ω is closed on U. Conversely, if φ˚ω is closed on U for every pair pU, φq with φ : U Ñ M, then ω is closed on M. In this sense, to study closed forms on M, it suffices to study their restriction to finite-dimensional submanifolds of M. But we will find the formalism of forms on the infinite-dimensional space M to be useful.
˚ We will abuse notation and write U ω for the integral of φ ω over U. We recall Stokes’ theorem: if U isş a (finite-dimensional) manifold with smooth boundary BU and ω is a form on U, then
δU ω “ ω. żU żBU Of course, Stokes’ theorem also holds on the infinite-dimensional manifold M in the sense that it holds for the pullback of forms to U for every φ : U Ñ M. Stokes’ theorem will be essential to our study of analytic torsion and the eta invariant. A related but distinct remark:
Remark 2.1.3. Rather than considering the space of all metrics on a vector bundle, we sometimes consider a smaller space, generally an (infinite-dimensional Fr´echet) submanifold M1 of the space of all metrics M. When this is the case, implicit in all that we do is that a k-form ω is closed on M1 if and only if
M δ ωph; X1,...,Xk`1q “ 0
for all h P M1 and for all X1,...,Xk`1 tangent to M1. 8 2.2 Sobolev spaces and algebras of operators
Let us choose a volume form vol0 on the base manifold M (or a density if M is not orientable). Then each choice of metric h P M induces an L2-inner product on
smooth sections of E, x¨, ¨yh, defined by
x¨, ¨yh :“ hp¨, ¨qx vol0pxq. (2.1) żM
2 2 (Here hp¨, ¨qx denotes the inner product on the fiber Ex.) The L spaces LhpM,Eq,
2 s and more generally the L -Sobolev spaces HhpM,Eq for s P R, are defined in the usual way. The inner products on these Hilbert spaces depend on the metric h, but any two metrics induce equivalent norms since M is compact, so that it makes sense to refer to the topological vector spaces L2pM,Eq and HspM,Eq, which are independent of the metric.
Remark 2.2.1. There is a redundancy that means a change to the volume form can always be rephrased as a change to the metric for the purposes of the L2 inner
product. More precisely: in (2.1), replacing vol0 by a different volume form ϕ vol0, where ϕ : M Ñ R is a smooth positive function, induces the same L2 inner product
as replacing the metric h by ϕh and keeping vol0 the same. For this reason, we will sometimes decide to fix a volume form vol0, which we will do for the rest of this chapter.
Definition 2.2.2. Let B denote the unital associative algebra of bounded linear maps L2pM,Eq Ñ L2pM,Eq. (The notion of bounded is independent of the metric in M since any two metrics induce equivalent norms.)
Definition 2.2.3. Let Ψ denote the unital associative algebra of pseudodifferential operators C8pM,Eq Ñ C8pM,Eq, and for s P R, let Ψs denote the subset of Ψ consisting of pseudodifferential operators of order (less than or equal to) s.
9 Note that if s ă t, then Ψs Ă Ψt. If s ď 0, then Ψs forms an algebra since it is closed under composition (in general, Ψs ¨Ψt Ă Ψs`t). Thus if s ď 0, then ΩkpM, Ψsq makes sense, with a well-defined product of differential forms.
Remark 2.2.4. Neither B nor Ψ is contained in the other, but if s ď 0, Ψs Ă B. Thus if s ď 0, we may view ΩkpM, Ψsq as a subset of ΩkpM,Bq.
We are primarily interested in B-, Ψ-, and C-valued forms on M.
2.3 Adjoints and traces
We now introduce some useful notions related to adjoints and traces.
Definition 2.3.1. Let h P M be a metric and let φ be an operator either in B or in
Ψ. The h-adjoint of φ, denoted φ˚h , is the operator either in B or in Ψ, respectively, characterized by (for smooth sections a, b of E)
˚h xφa, byh “ xa, φ byh.
We remark that if φ P B, since smooth sections are dense in L2 sections, this de-
fines φ˚h uniquely as an element of B, i.e., as a bounded linear operator on L2pM,Eq.
Definition 2.3.2. If ω P ΩkpM,Aq, where A “ B or A “ Ψ, then the adjoint of ω, denoted ω˚ P ΩkpM,Aq, is defined by
˚ ˚h pω qph; X1,...,Xkqphq :“ pωph; X1,...,Xkqq , where the latter adjoint is as in Definition 2.3.1. We will say that ω is symmetric when ω˚ “ ω.
We will use the notion of a trace-class operator φ P B; for a definition, see, for example, the book of Reed-Simon [18]. The usual definition of the trace depends a priori on a choice of an L2 inner product—and therefore, in our context, on a choice
10 of a metric h P M—but by Lidskii’s theorem, the trace is independent of this choice. Thus it makes sense to denote the trace of a trace-class operator φ P B by Tr φ, with no reference to a choice of metric h P M.
Remark 2.3.3. We will find the following well-known fact useful in later chapters.
Let n be the dimension of the base manifold M. Then if s ă ´n and if φ P Ψs Ă B, then φ is trace-class.
Definition 2.3.4. We will say that ω P ΩkpM,Bq is trace-class when for every
8 2 ˚ h P M and X1,...,Xk P C pM, Sym E q, ωph; X1,...,Xkq P B is trace-class. If
ω is trace-class, Tr ω P ΩkpM, Cq will denote the C-valued k-form on M defined by k pTr ωqph; X1,...,Xkq :“ Tr pωph; X1,...,Xkqq, and Tr ω P Ω pM, Cq will denote the
complex conjugate of Tr ω, defined by pTr ωqph; X1,...,Xkq :“ Tr pωph; X1,...,Xkqq.
We have the following results concerning the adjoint of a product and the inter- action between adjoints and traces:
Lemma 2.3.5. Let κ P ΩkpM,Aq and λ P ΩlpM,Aq, where A “ B or A “ Ψ. Then
pκλq˚ “ p´1qklλ˚κ˚.
κ is trace-class if and only if κ˚ is trace-class; in that case, their traces are complex conjugates of each other:
Tr κ “ Tr κ˚.
If κλ is trace-class, then
Tr κλ “ Tr pκλq˚
“ p´1qkl Tr λ˚κ˚.
If κλ and λκ are both trace-class, then
Tr κλ “ p´1qkl Tr λκ.
11 Proof. The identities follow from the corresponding identities for operators and the alternating property of differential forms.
We have the following useful corollary involving products of 1-forms:
1 Lemma 2.3.6. Let κ1, . . . , κr P Ω pM,Aq, where A “ B or A “ Ψ. Let r “
1 rpr´1q p´1q 2 . Then
˚ ˚ ˚ pκ1 ¨ ¨ ¨ κrq “ r κr ¨ ¨ ¨ κ1 .
If κ1 ¨ ¨ ¨ κr is trace-class, then
˚ Tr κ1 ¨ ¨ ¨ κr “ Tr pκ1 ¨ ¨ ¨ κrq
˚ ˚ “ rTr κr ¨ ¨ ¨ κ1 .
Proof. The lemma follows from Lemma 2.3.5 by induction on r. r is the sign of the following order-reversing permutation on r letters: p1, 2, . . . , rq ÞÑ pr, r ´ 1,..., 1q,
1 which involves 1 ` 2 ` ¨ ¨ ¨ ` pr ´ 1q “ 2 rpr ´ 1q transpositions.
Finally, the operation of taking the trace commutes with the exterior derivative:
Lemma 2.3.7. If κ P ΩkpM,Bq is trace-class, then δMκ is also trace-class, and δMpTr κq “ TrpδMκq.
2.4 Flat vector bundles
In this section we will establish some conventions regarding flat vector bundles.
Let F Ñ M be a vector bundle. A connection ∇F on F Ñ M is said to be flat when its curvature vanishes. The vector bundle F Ñ M is said to be flat when it admits a flat connection. Henceforth, when we refer to a flat vector bundle, we will assume that a fixed flat connection has been chosen. It is well-known that flat vector bundles on M are in one-to-one correspondence
with finite-dimensional representations of the fundamental group π1pMq, i.e., group
12 homomorphisms π1pMq Ñ GLpV q, where V is a finite-dimensional real or complex vector space and GLpV q denotes the group of invertible linear endomorphisms of V . We will now describe this correspondence briefly. ˜ Let M be the universal cover of M, on which π1pMq acts on the left by deck ˜ transformations; let Π : M Ñ M be the covering map. Let ρ : π1pMq Ñ GLpV q be
a representation. Associated to ρ is a flat vector bundle Fρ, the total space of which
˜ ´1 is M ˆ V modulo the equivalence relation px, vq „ pγx, ρpγ qvq for γ P π1pMq. The trivial connection on the trivial bundle M˜ ˆ V Ñ M˜ descends to a flat connection on Fρ Ñ M. Conversely, suppose ∇F is a flat connection on F Ñ M. Then ∇F lifts to a flat connection on the bundle Π˚F Ñ M˜ . Since M˜ is simply connected and the connection is flat, it is not hard to see (via parallel transport) that Π˚F Ñ M˜ is in fact a trivial bundle, and that this construction must be the inverse of a construction of the form described in the previous paragraph.
We will say that a representation ρ : π1pMq Ñ GLpV q on a real euclidean inner product space V is orthogonal when the image of ρ lies in the group of automorphisms that are orthogonal with respect to the inner product on V . Similarly, we will say
that a representation ρ : π1pMq Ñ GLpV q on a complex hermitian inner product space V is unitary when the image of ρ lies in the group of automorphisms that are unitary with respect to the inner product on V . We will also use these terms for flat
vector bundles: we will say that a flat vector bundle F Ñ M is orthogonal or unitary
if the associated representation of π1pMq is orthogonal or unitary, respectively. If F Ñ M is orthogonal or unitary, then the euclidean or hermitian inner product on V descends to a canonical metric on F Ñ M with respect to which the flat connection is metric-compatible.
If F Ñ M is a flat vector bundle, then one may form the F -valued de Rham complex, as we will now describe. For q “ 0,..., dim M, consider the vector bundle
13 F b ΛqT ˚M Ñ M, whose sections are F -valued q-forms. (We will sometimes choose to use the more compact notation ΛqF .) Then the exterior derivative on M and the flat connection on F induce an exterior derivative operator that we will denote by d.
d maps q-forms to pq ` 1q-forms; the identity d ˝ d “ 0 is equivalent to the flatness of the connection on F .
14 3
Analytic torsion of an elliptic complex
3.1 Heat kernels and zeta functions
In this section, we will discuss some well-known general properties of elliptic operators that are necessary to our treatment of analytic torsion. The results on zeta functions are due to Seeley [20]; for an overview of these and many related ideas, see, for example, the book of Gilkey [7]. Let M be a compact orientable manifold of dimension n. Fix once and for all
a volume form vol0 on M. Let E Ñ M be a vector bundle. Let L be an elliptic differential operator on sections of E; we will assume L is nonnegative and formally self-adjoint with respect to an L2 inner product on sections of E induced by a metric
h on E. Let m be the order of L; we will assume m ą 0.
L has nonnegative real eigenvalues 0 ď λ1 ď λ2 ď ¨ ¨ ¨ Ñ 8. The zeta function associated to L is the function of s P C defined for Re s large by
´s ζpsq :“ λj . (3.1) λ ą0 ÿj
ζpsq may be interpreted as the trace of the operator L´s. The sum in (3.1) converges
15 n for Re s ą m and defines a holomorphic function there, and ζpsq admits a unique meromorphic continuation to all of C that is holomorphic at s “ 0. This allows one to define the determinant of L by
d det L :“ exp ´ ζpsq , (3.2) ds ˆ ˇs“0 ˙ ˇ ˇ which one should interpret as a regularized productˇ of the nonzero eigenvalues of L. The zeta function is closely related to the heat operator, as we will now discuss.
For t ą 0, the heat operator e´tL is characterized by, for u P L2pM,Eq, B ` L pe´tLuq “ 0 for t ą 0, and Bt ˆ ˙ lim e´tLu “ u. tÑ0` ` ˘ e´tL is trace-class as an operator L2pM,Eq Ñ L2pM,Eq; furthermore, away from the kernel of L, its trace is what we will call “admissible,” in a sense which we will make precise. To do so, we leave our geometric setting and work in the abstract for a moment.
Definition 3.1.1. A C-valued function hptq, t ą 0, will be said to be admissible when it satisfies both of the following conditions:
(A1) hptq admits an asymptotic expansion for small t in the sense that there exist
real powers p0 ă p1 ă p2 ă ¨ ¨ ¨ Ñ 8 and complex coefficients ap0 , ap1 , ap2 ,...
such that for every real number K, there exists JK and CK such that
JK pj hptq “ apj t ` rK ptq, (3.3) j“0 ÿ
K where the remainder rK ptq satisfies, for 0 ă t ă 1, |rK ptq| ď CK t . In this
pj q case, we will write hptq „ jě0 apj t . (For each q such that t does not appear
in this asymptotic expansion,ř our convention is to set aq “ 0.) 16 (A2) There exist C, ą 0 such that, for t ě 1, |hptq| ď Ce´t.
The utility of this notion of admissibility is that is sufficient to define an associated zeta function via the Mellin transform:
Definition 3.1.2. We define the zeta function associated to hptq by the following, for Re s large: 1 8 dt ζpsq :“ ts hptq . (3.4) Γpsq t ż0 The following result is well-known:
Lemma 3.1.3. Suppose hptq is admissible. Then the integral defining ζpsq in (3.4)
converges for Re s ą ´p0 and defines a holomorphic function there. Furthermore,
ζpsq admits a unique meromorphic extension to all of C; the extension, which we
also denote by ζpsq, is holomorphic at the origin, where its value is ζp0q “ a0, i.e., the coefficient on t0 in the asymptotic expansion of (A1).
Now we return to our geometric setting. We have the following well-known fun- damental result:
Theorem 3.1.4. Let α P C8pEndpEqq. Then Tr αe´tL has an asymptotic expansion in the sense of (A1) taking the form
´tL ´n`2j Tr αe „ a ´n`2j t m . (3.5) m jě0 ÿ (The reader will recall that n is the dimension of M and m is the order of L.)
To obtain the decay as t Ñ 8, as in pA2q, we must remove the kernel of L in the following sense:
2 2 Definition 3.1.5. Let Πker L : L pM,Eq Ñ ker ∆ Ă L pM,Eq denote the orthogonal projection onto ker L. For a trace-class operator A, let
1 Tr A :“ Tr A pI ´ Πker Lq . 17 Then we have:
Theorem 3.1.6. Let α P C8pM, EndpEqq. Then Tr1 αe´tL is an admissible function of t ą 0, i.e., it satisfies conditions (A1) and (A2). Furthermore, the asymptotic expansion of (A1) takes the form
1 ´tL ´n`2j Tr αe „ ´ tr αΠker L ` a ´n`2j t m , (3.6) m j ÿ
i.e., the coefficients are the same as in (3.5), except that the coefficient on t0 differs from that of (3.5) by p´ tr αΠker Lq.
This allows us to make the definition:
Definition 3.1.7. The zeta function associated to the elliptic operator L and α P C8pEndpEqq is defined as in (3.4) with hptq “ Tr1 αe´tL, i.e.,
1 8 dt ζps; L, αq :“ ts Tr1 αe´tL . (3.7) Γpsq t ż0
In the case when α is the identity, we set ζps; Lq :“ ζps; L, Iq.
Remark 3.1.8. To provide motivation, we remark that ζps; L, αq generalizes the zeta function of (3.1) to include the auxiliary operator α. Indeed, the elementary identity
1 8 dt λ´s “ ts e´tλ Γpsq t ż0
shows that ζps; L, αq may be rewritten as, for Re s large,
´s ζps; L, αq “ λ tr αΠλ, λą0 ÿ where the sum is over distinct positive eigenvalues of L and
Lemma 3.1.3 implies immediately:
18 n Theorem 3.1.9. The integral defining ζps; L, αq in (3.7) converges for Re s ą ´ m and defines a holomorphic function there. Furthermore, ζps; L, αq admits a unique
meromorphic extension to all of C; the extension, which we also denote by ζps; L, αq, is holomorphic at the origin, where its value is
ζp0; L, αq “ a0 ´ tr αΠker L, (3.8) i.e., the coefficient on t0 in the asymptotic expansion in (3.6). (If there is no t0 term in (3.5), we set a0 “ 0.)
We record also the useful consequence:
Corollary 3.1.10. If the dimension n is odd and ker L is trivial, then for any α,
ζp0; L, αq “ 0.
Proof. We use (3.8). The asymptotic expansion of Tr αe´tL in (3.5) has terms of the
´n`2j 0 form a ´n`2j t m for j P ě0; if the dimension n is odd, then there is no t term, m Z
i.e., a0 “ 0. Furthermore, if also ker L is trivial, then of course tr αΠker L “ 0.
3.2 Elliptic complexes
We will discuss analytic torsion in the context of a Z-graded elliptic complex, which we will now define.
Suppose that the vector bundle E Ñ M is Z-graded of finite length in the sense r q 0 r that E “ q“0 E for some vector bundles E ,...,E . We assume that there is a differentialÀ operator d : C8pM,Eq Ñ C8pM,Eq such that for each q, d :
8 q 8 q`1 8 q C pM,E q Ñ C pM,E q; the restriction of d to C pM,E q we denote by dq. We assume that d ˝ d “ 0, i.e., we have a chain complex
d d dr 1 0 ÝÑ C8pM,E0q ÝÑ0 C8pM,E1q ÝÑ1 ¨ ¨ ¨ ÝÑ´ C8pM,Erq ÝÑ 0.
19 Let M denote the space of metrics on E such that the subbundles Eq are mutually
orthogonal. As discussed in Section 2.2, for each metric h P M (and having fixed the
2 ˚h volume form), we have an associated L inner product x¨, ¨yh. Let d be the formal
adjoint of d with respect to x¨, ¨yh.
Definition 3.2.1. We will say the complex pE, dq is an elliptic complex when for
some choice of metric h, d ` d˚h is elliptic.
Remark 3.2.2. In fact, if d`d˚h is elliptic for some metric h, then d`d˚h1 is elliptic for every metric h1. Furthermore, this condition is equivalent to an exactness condition on the symbol of d. For details, see Gilkey [7].
Definition 3.2.3. We now define some useful endomorphisms of E. Let Q act on
sections of Eq as multiplication by q. Let p´1qQ act on sections of Eq as multiplication by 1 if q is even and ´1 if q is odd. More generally, if f : Zě0 Ñ C is a function on the nonnegative integers, let fpQq act on sections of Eq as multiplication by fpqq.
Definition 3.2.4. The Laplacian ∆h associated to a metric h P M is ∆h :“pd ` d˚h q2
“dd˚h ` d˚h d.
h ˚h ˚h h q ∆q “ dq´1dq´1 ` dq dq denotes the restriction of ∆ to sections of E . Let m be the order of ∆ as a differential operator. (m is twice the order of d.)
Remark 3.2.5. We use the symbol ∆, which is traditional for Laplacians and is suggestive of the examples below. In general, though, we need not assume that ∆ is a “Laplace-type operator” in the sense of Gilkey [7]. In particular, the order m need not be 2.
h ∆ is elliptic and, with respect to x¨, ¨yh, formally self-adjoint and nonnegative. By elliptic regularity, ∆h has a finite-dimensional kernel, which is the intersection
ker d X ker d˚h and which we may identify with the cohomology, as we now discuss.
20 Definition 3.2.6. For q “ 0, . . . , r, the qth cohomology vector space HqpM,Eq is the quotient ker d HqpM,Eq :“ q . im dq´1
We have the Hodge theorem (for a proof, see, for example, the book of Gilkey [19]):
h q Theorem 3.2.7. For each metric h, the map ker ∆q Ñ H pM,Eq induced by the
h inclusion ker ∆q ãÑ ker dq is an isomorphism of vector spaces.
Definition 3.2.8. The elliptic complex pE, dq will be said to be acyclic when the cohomology is trivial, i.e, when HqpM,Eq “ 0 for q “ 0, . . . , r, or equivalently, when ∆h has trivial kernel.
Examples.
1. The de Rham complex with a flat coefficient bundle. (Recall our discussion of
flat vector bundles in Section 2.4.) Let F Ñ M be a flat vector bundle and let Eq “ F bΛqT ˚M be the bundle of F -valued q-forms. Then the flat connection on F and the usual exterior derivative induce an operator d, giving an elliptic complex. This was the context in which Ray-Singer first introduced analytic torsion [16]. For further comments on this important example, see Section 3.9.
2. The Dolbeault complex with a holomorphic coefficient bundle. This is the ana- logue of the previous example in the complex category. Suppose that M is
a complex manifold and let F Ñ M be a holomorphic vector bundle. Let Ep,q F Λp,qT ˚M be the bundle of F -valued p, q -forms. The boundary “ b C p q operator d in this context is B¯ : C8pM,Ep,qq Ñ C8pM,Ep,q`1q, giving an el- liptic complex for each fixed p. Ray-Singer also introduced analytic torsion in this context [17].
21 Remark 3.2.9. The signature operator and the spinor Dirac operator do not fit into this framework because there is no natural complex, i.e., no natural operator d such that d ˝ d “ 0.
Remark 3.2.10. Mathai-Wu [11] have defined analytic torsion for elliptic complexes that are only Z2-graded, which is more general than our setting. But our interpre- tation of analytic torsion as an integral over a curve in the space of metrics does not make sense in the Z2-graded context. See Remark 3.9.2 for further comments.
3.3 Analytic torsion
We are now prepared to define analytic torsion. Recall the operators Q and p´1qQ from Definition 3.2.3.
Definition 3.3.1. The analytic torsion T pM, E, hq associated to the elliptic complex E and a metric h P M, with associated Laplacian ∆h, is defined by
1 d log T pM, E, hq :“ ζ s; ∆h, p´1qQQ . (3.9) 2 ds ˇs“0 ˇ ` ˘ ˇ Note that by the definition of the zeta-function,ˇ we have that 1 d AC 1 8 log T pM, E, hq :“ ts´1 Tr1p´1qQQe´t∆ dt (3.10) 2 ds Γpsq ˇs“0 ż0 ˇ r ˇ 1 ˇ d “ p´1qqq ζps; ∆ q, (3.11) 2 ds q q“0 ˇs“0 ÿ ˇ ˇ where the superscript AC indicates that analyticˇ continuation is implied.
Remark 3.3.2. We may also phrase the formula for the analytic torsion in terms of determinants. Recall the definition of the determinant of an elliptic operator from (3.2). We see that (3.11) implies
1 r log T pM, E, hq “ ´ p´1qqq log det ∆ , (3.12) 2 q q“0 ÿ 22 i.e, the analytic torsion is a weighted alternating product of determinants.
˚ ˚ There is also a notion of determinant for the operators dqdq and dq dq (which are not elliptic but are nonnegative) that requires projecting away from their infinite-
˚ dimensional kernels. Let ζdq dq psq be defined for Re s large by
8 ˚ 1 s 1 ´td dq dt ζ ˚ psq :“ t Tr e q , dq dq Γpsq t ż0 where, in an abuse of notation, we mean that
˚ 1 ´td dq ´t∆q q ˚ Tr e :“ Tr e Πim dq , (3.13)
˚ ˚ ˚ where Πim dq denotes the orthogonal projection onto the image of dq , on which dq dq is strictly positive. Then we can define the determinant analogously to (3.2):
˚ d log det d d :“ ´ ζ ˚ psq. q q ds dq dq ˇs“0 ˇ ˇ ˚ ˚ ˇ We define ζdqdq psq and det dqdq similarly. Recall the fact that for any t ą 0,
˚ ˚ Tr1 e´t dqdq “ Tr1 e´t dq dq , (3.14)
˚ ˚ which follows because dqdq and dq dq have the same nonzero eigenvalues. We also
1 1 ˚ 1 td d˚ have Tr e´t∆q “ Tr e´tdq dq ` Tr e´ q´1 q´1 , and these facts imply the identities
ζqpsq “ ζ ˚ psq ` ζ ˚ psq dq dq dq´1dq´1
˚ ˚ ζdq dq psq “ ζdqdq psq.
Substituting these into (3.11) shows that the torsion may be written in terms of only dd˚ or only d˚d:
1 r log T pM, E, hq “ p´1qq log det d d˚ (3.15) 2 q q q“0 ÿ 1 r “ ´ p´1qq log det d˚ d . (3.16) 2 q´1 q´1 q“0 ÿ 23 Note the absence of the weighting factor q in (3.15) and (3.16) as compared to (3.12). We remark that the Reidemeister torsion, the classical combinatorial analogue of analytic torsion, has similar expressions in terms of determinants of combinatorial Laplacians. This is explained in the paper of Ray-Singer [16] and seems to have been the motivation for Ray-Singer’s definition of analytic torsion.
We are interested in the dependence of T phq on the metric h P M. The main result of this section is the following generalization of Ray-Singer’s fundamental result, Theorem 2.1 of [16]:
Theorem 3.3.3. Let hpuq be a smooth one-parameter family of metrics in M. Let T phpuqq “ T pM, E, hpuqq be the associated analytic torsion. Then
d log T phpuqq “ a1 puq, (3.17) du 0
1 0 1 Q ´1 dβu ´t∆hpuq where a0puq is the t term in the asymptotic expansion of Tr p´1q βu du e .
For the sake of exposition, we will wait until the next section to define βu, which is a certain endomorphism of E associated to the metric. For now, we merely emphasize the following important corollary:
Corollary 3.3.4. If the dimension n is odd and E is acyclic, then the analytic
torsion T phq is independent of the metric h, for h in the generalized class of metrics M.
Proof. This follows immediately from Theorem 3.3.3 and Corollary 3.1.10.
Remark 3.3.5. Ray-Singer proved Theorem 3.3.3 and Corollary 3.3.4 in the contexts of Examples1 and2 above, in the special case that the metric on E is induced as the product of a metric on the coefficient bundle F and a Riemannian metric on M. We emphasize that our results apply to any metric h in the generalized class M of all metrics on E. For further discussion of this, see Section 3.9.
24 Moreover, our proof of Theorem 3.3.3 involves a novel interpretation of analytic torsion as the integral of a certain closed differential form on a curve in M. This interpretation is the subject of Section 3.4.
Remark 3.3.6. In the case when n is even and/or E is not acyclic, T phq may depend on the metric h, but (3.17) can be used to show that an object called the Ray-Singer metric is a topological invariant, independent of the metric h. The Ray-Singer metric is a certain metric on the determinant line of the cohomology H‚pM,Eq. See [5], for example, for the details.
3.4 A closed form for analytic torsion
We will now introduce ωT , a certain closed one-form on the space of metrics, and we will interpret analytic torsion as the integral of ωT over a certain curve in the space of metrics. In this section and the next, the tools of Chapter2 will be important.
3.4.1 An operator parametrizing the space of metrics
Recall that M denotes the space of metrics on E such that the subbundles Eq are mutually orthogonal. We have fixed a volume form vol0 on M. Now we will fix also
a metric h0 P M that we will consider to be a “basepoint” metric. Having chosen h0, we have the following construction: If h P M is another metric, then there exists a
8 unique operator β “ βh, an automorphism of E, such that for all φ, ψ P C pM,Eq,
hpφ, ψq “ h0pφ, βhψq.
We will view β as a endomorphism-valued function (i.e., a 0-form) on the space of metrics:
β : M Ñ C8pEndpEqq
In the language of Definition 2.1.1, β P Ω0pM, Ψ0q Ă Ω0pM,Bq. We now study some properties of β.
25 Lemma 3.4.1. βh is a pointwise symmetric operator with respect to h0, i.e., on each
fiber Ex, for u, v P Ex we have
h0pu, βhvqx “ h0pβhu, vqx. (3.18)
Proof. We use the fact that h and h0 are euclidean or hermitian inner products on each fiber to compute
h0pu, βhvqx “ hpu, vqx
“ hpv, uqx
“ h0pv, βhuqx
“ h0pβhu, vqx, which proves the claim.
Lemma 3.4.2. If hu is a smooth one-parameter family of metrics with corresponding ´1 9 operators βu :“ βhu , then βu βu is a pointwise symmetric operator with respect to
hu.
Proof. Differentiating the equality (3.18) gives that
´1 9 ´1 9 h0pf, βuβu βugqx “ h0pβuβu βuf, gqx, i.e.,
´1 9 ´1 9 hupf, βu βugqx “ hupβu βuf, gqx.
Recall that for each metric h P M, d˚h (or just d˚ if there is no possibility of
˚0 confusion) denotes the formal adjoint of d with respect to x¨, ¨yh. Denote by d the
formal adjoint of d with respect to x¨, ¨y0. We have the following:
Lemma 3.4.3. d˚h “ β´1d˚0 β.
´1 ˚0 ´1 ˚0 ˚0 Proof. xf, β d βgyh “ xf, ββ d βy0 “ xf, d βgy0 “ xdf, βgy0 “ xdf, gyh. 26 Recall that δM denotes the exterior derivative on differential forms on M.
Definition 3.4.4. Let b denote the following “tautological” one-form on M with values in C8pEnd Eq: b :“ β´1δMβ.
Although β depends on the choice of basepoint metric h0, a short computation shows that b is well-defined independent of the choice of h0:
Lemma 3.4.5. b does not depend on the choice of basepoint metric h0.
1 1 Proof. Suppose h0 and h0 are two choices of basepoint metric; let β and β correspond
1 to h0 and h0 respectively, i.e., for any metric h P M, we have hpf, gq “ h0pf, βhgq “
1 1 1 1 1 h f, β g . But then we must have h f, β g h f, β 1 β g , which implies β 0p h q 0p h q “ 0p h0 h q h “ 1 8 β 1 β . β 1 may be viewed as a constant map M C End E , which shows h0 h h0 Ñ p q
´1 M 1 ´1 M 1 β δ β β 1 β δ β 1 β “ p h0 hq p h0 hq
1 ´1 ´1 M 1 “ pβhq β 1 βh1 δ βh h0 0
1 ´1 M 1 “ pβhq δ βh, which proves the claim.
Lemma 3.4.2 implies
Lemma 3.4.6. b is symmetric in the sense of Definition 2.3.2.
3.4.2 The closed one-form on the space of metrics
1 Q ´∆ Definition 3.4.7. Let ωT “ Tr p´1q e b.
For the sake of concreteness, we note the following: Consider a curve in M parametrized by hpuq for u in some interval in R. Let βu be the associated operator.
27 Then ωT pulled back to the curve, which we also denote by ωT , is
hpuq dβ ω “ Tr1p´1qQe´∆ β´1 u du T u du
r hpuq dβu “ p´1qq Tr1 e´∆q β´1 du u du q“0 ÿ
We will now explain the significance of ωT . The idea is that the analytic torsion
T may be interpreted as a regularized integral of ωT over a certain curve in the space of metrics M. More precisely:
Let h P M be a metric on E. For each degree p, let hp be the restriction of h to the subbundle Ep.
Remark 3.4.8. Consider the special case in which E “ F b Λ‚T ˚M is the bundle of F -valued forms, where F is a flat unitary bundle. Suppose that h, a metric on
F bΛ‚T ˚M, is induced by a Riemannian metric gTM and the canonical metric on F .
TM 1 TM 1 TM Suppose we scale g by t , i.e., let gt “ t g , and let ht be the induced metric ‚ ˚ p p p on F b Λ T M. Then a short computation shows that ht “ t h . (Here, as in the rest of this chapter, we consider the volume form fixed.)
This motivates the following, which applies to any metric h P M. For t P p0, 8q,
p p p let ht be the metric on E whose restriction to E is t h . This parametrizes a curve
p p Ch in M. The associated operator βht is given by t on E . Thus the one-form
´1 M p b “ β δ β pulled back to Ch is, on E ,
dt b “ p . t
We see that d˚ “ β´1d˚0 β and ∆ are given at time t by
d˚ptq “ td˚0
∆ptq “ t∆h
28 Finally, we have that ωT pulled back to Ch is
h dt ω “ Tr1p´1qQQe´t∆ , T t
which, modulo the regularizing factor ts, is precisely the integrand in the zeta function defining the analytic torsion T . Thus we have proven
n Lemma 3.4.9. For Re s ą m ,
1 ζps; ∆h, p´1qQQq “ tsω , Γpsq T żCh where here ωT denotes the one-form of Definition 3.4.7 pulled back to Ch.
Recall that the analytic torsion T h is defined as 1 d ζ s; ∆h, 1 QQ . p q 2 ds s“0 p´ q Using that 1 “ s ` Ops2q as s Ñ 0 and Lemma 3.4.9, we haveˇ the` heuristic ˘ Γpsq ˇ
AC s 2T phq “ t ωT , ˆżCh ˙ˇs“0 ˇ ˇ where the superscript AC indicates that we mustˇ analytically continue the function in parentheses, which is defined for Re s1, s2 large, to the origin. (This heuristic is only literally true if the aforementioned analytic continuation is holomorphic at the origin.) Formally, setting s “ 0 on the right-hand side leaves ωT ; this is purely Ch formal because the analytic continuation is necessary for theş integral to converge. This suggests the following heuristic interpretation of analytic torsion:
The analytic torsion T phq may be interpreted as the (regularized) integral
of ωT over the curve Ch in the space of metrics M.
In the next section, we will prove that ωT is closed on M. This fact is the essential tool in our proof of the metric variation formula, Theorem 3.3.3. The idea of the proof is simple: If Ch were a closed curve, Stokes’ theorem applied to the closed form ωT 29 would imply that T phq would be independent of h. The problem, of course, is the fact that Ch is not closed (not to mention the fact that regularization is necessary). This means that a boundary term enters the computation. It turns out that this boundary term is just a coefficient in a certain t Ñ 0 heat kernel asymptotic expansion. We give the proof of Theorem 3.3.3 in Section 3.6. First, though, in Section 3.5, we prove that ωT is closed.
3.5 Proof of closedness
This section consists of a proof of:
M Theorem 3.5.1. ωT is closed on M, i.e., δ ωT “ 0.
For each complex number z that is not an eigenvalue of ∆, let Rz denote the resolvent p∆ ´ zq´1.
Remark 3.5.2. The resolvent is a pseudodifferential operator of order ´m, i.e., Rz P
´m Ψ . (Recall that m ą 0 is the order of ∆.) In particular, Rz defines a bounded
2 2 linear map L pM,Eq Ñ L pM,Eq, so in the language of Chapter 2, we view Rz P
0 n N ´Nm Ω pM,Bq. Furthermore, if N is an integer greater than m , then Rz P Ψ is trace-class by Remark 2.3.3.
´∆ We will work not with the heat operator e but with the resolvent Rz for technical reasons (essentially, the resolvent is easier to differentiate). That is, we will work not with ωT but withω ˜T , which we now define.
n Fix once and for all an integer N greater than m . Remark 3.5.2 ensures that Q N p´1q Rz b is trace-class.
1 Definition 3.5.3. Letω ˜T P Ω pM, Cq be the following C-valued one-form on M:
Q N ω˜T :“ Trp´1q Rz b. (3.19)
30 ω˜T depends on a complex number z, but we will usually suppress this dependence.
ωT andω ˜T are related by the Cauchy integral formula: let C be a contour in the
complex plane surrounding the interval rλ1, 8q Ă R Ă C, where λ1 is the first nonzero eigenvalue of ∆. Then we have
1 1 e´∆ pI ´ Π q “ e´z RN dz, ker ∆ 2πi pN ´ 1q! z żC and therefore also
1 1 p´1qQe´∆b pI ´ Π q “ e´z p´1qQRN b dz. ker ∆ 2πi pN ´ 1q! z żC Since the trace and δM commute with the integral, we have 1 1 ω “ e´z ω˜ dz and T 2πi pN ´ 1q! T żC 1 1 δMω “ e´z δMω˜ dz. (3.20) T 2πi pN ´ 1q! T żC ` ˘ Remark 3.5.4. Of course, λ1 depends on the metric, and therefore the choice of C depends on the metric. But this does not affect our computations since on any compact set of metrics K Ă M, λ1 is bounded away from zero. Thus if we restrict our attention to metrics in K, we may choose C independently of the metric for the purposes of the contour integrals above. For similar reasons, our computations are also not affected by the fact that the set of z for which Rz andω ˜T are defined depends on the metric.
We will prove thatω ˜T is closed; by (3.20), this is sufficient to prove that ωT is closed. First, we must prove some intermediate results.
Q N Lemma 3.5.5. We may rewrite ω˜T “ Trp´1q Rz b in either of the following equiv- alent ways: Q N ω˜T “Tr p´1q Rz b (3.21) 1 1 ω˜ “ Trp´1qQRN b ` Tr p´1qQRN b (3.22) T 2 z 2 z 31 Proof. The second equality follows from the first, so it suffices to prove the first. We simply apply Lemma 2.3.5 relating the trace and the adjoint:
Q N Q N ˚ ω˜T “ Trp´1q Rz b “ Tr p´1q Rz b
` N Q ˘ “ Tr bRz p´1q
Q N “ Tr p´1q Rz b,
where the last equality follows from the cyclicity of the trace and the fact that p´1qQ commutes with the resolvent.
For operators or forms a1 and a2, let ra1, a2s “ a1a2 ´ a2a1 and ta1, a2u “
a1a2 ` a2a1 denote their commutator and anticommutator, respectively. We have the following:
Lemma 3.5.6. For a smooth one-parameter family of metrics hu (u P I Ă R) with
˚u ˚hu u hu u hu associated operators βu :“ βhu , d :“ d , ∆ :“ ∆ , and Rz :“ Rz , we have B Bβ d˚u “ d˚u , β´1 u Bu u Bu „ B Bβ ∆u “ d, d˚u , β´1 u Bu u Bu " „ * B B Bβ Ru “ Ru ∆u Ru “ Ru d, d˚u , β´1 u Ru Bu z z Bu z z u Bu z ˆ ˙ " „ *
Proof. The assertions follow from the identity d˚h “ β´1d˚0 β (Lemma 3.4.3) and
B ´1 ´1 Bβ ´1 the standard fact that Bu pβ q “ ´β Bu β (and similarly for the derivative of u u ´1 ´1 Rz “ pz ´ ∆ q ), which follows from differentiating the identity ββ “ I.
This immediately implies:
M ˚u M M Lemma 3.5.7. δ ∆ “ td, rd , bsu and δ Rz “ Rz δ ∆ Rz. ` ˘ 32 We will also need the results:
Lemma 3.5.8. δMb “ ´b b.
2 Proof. Since δM “ 0, we have
` ˘ δMb “ δMpβ´1δMβq
“ δMpβ´1q δMβ
“ ´β´1pδMβqβ´1 δMβ
“ ´b b.
‚ Lemma 3.5.9. Let a1, a2, a3 P Ω pM,Aq, where A “ B or A “ Ψ. Suppose that d
˚ and d commute with each of a1, a2, and a3, and that the forms below are trace-class. Then we have the identity
Q M ˚ Q M Trp´1q a1ba2pδ ∆q a3 “ Trp´1q a1pδ ∆qa2ba3.
˚ Proof. We use the cyclicity of the trace and that d and d commute with a1, a2, and
Q a3 and anticommute with p´1q to compute:
Q ˚ Q ˚ Trp´1q a1ba2pbdd qa3 “ Trp´1q a1pdd bqa2ba3
Q ˚ Q ˚ Trp´1q a1ba2p´d dbqa3 “ Trp´1q a1p´bd dqa2ba3
Q ˚ ˚ Q ˚ ˚ Trp´1q a1ba2p´dbd ` d bdqa3 “ Trp´1q a1pd bd ´ dbd qa2ba3
Summing the three identities above gives the result.
N M N i M N´i`1 Lemma 3.5.10. δ Rz “ Rz δ ∆ Rz . i“1 ÿ ` ˘ Proof. This follows from Lemma 3.5.7.
Now we are prepared to prove the main result of this section.
33 Q N Lemma 3.5.11. The C-valued one-form ω˜T “ Trp´1q Rz b is closed.
Proof. Recall the identity (3.22):
Q N Q N 2˜ωT “ Trp´1q Rz b ` Tr p´1q Rz b.
We will look for cancellation between terms in the derivative of the first term on the right-hand-side and terms in the derivative of the second term on the right-hand side. We have on the one hand
M Q N Q M N Q N M δ Trp´1q Rz b “ Trp´1q pδ Rz qb ` Trp´1q Rz pδ bq
` ˘ N Q i M N´i`1 Q N “ Trp´1q Rz δ ∆ Rz b ´ Trp´1q Rz bb, i“1 ÿ ` ˘ and on the other hand
M Q N Q M N Q N M δ Tr p´1q Rz b “ Trp´1q pδ Rz qb ` Trp´1q Rz pδ bq
` ˘ N Q i M N´i`1 Q N “ Tr p´1q Rz δ ∆ Rz b ´ Tr p´1q Rz bb. i“1 ÿ ` ˘ Note that pbbq˚ “ ´b˚b˚ “ ´bb since b is a symmetric one-form. Now compute
Q N Q N ˚ Trp´1q Rz bb “ Tr pp´1q Rz bbq
N Q “ ´Tr bbRz p´1q
N Q “ ´Tr Rz p´1q bb
Q N “ ´Tr p´1q Rz bb, where we have used the cyclic property of the trace and that p´1qQ commutes with
Rz. It remains to show that the following vanishes:
N Q i M N´i`1 Q i M N´i`1 Trp´1q Rz δ ∆ Rz b ` Tr p´1q Rz δ ∆ Rz b . i“1 ÿ “ ` ˘ ` ˘ ‰ 34 Denote the ith summand by Ai. We claim that for each i, Ai “ 0. To see this, we compute
Q i M N´i`1 Trp´1q Rz pδ ∆q Rz b
Q i M N´i`1 ˚ “ Tr pp´1q Rz pδ ∆q Rz bq
N´i`1 M ˚ i Q “ ´Tr bRz pδ ∆q Rzp´1q
Q i N´i`1 M ˚ “ ´Tr p´1q RzbRz pδ ∆q
Q i M N´i`1 “ ´Tr p´1q Rz pδ ∆q Rz b by Lemma 3.5.9.
This proves that Ai “ 0 and proves the lemma.
This proves that ωT is closed by (3.20), completing the proof of Theorem 3.5.1.
3.6 Proof of the metric variation formula
This section consists of a proof of Theorem 3.3.3, which we now restate for conve- nience.
Theorem 3.6.1. Let hpuq, u P r0, 1s, be a smooth one-parameter family of metrics in M. Let T phpuqq “ T pM, E, hpuqq be the associated analytic torsion. Then
d log T phpuqq “ a1 puq, (3.23) du 0
1 0 1 Q ´1 dβu ´t∆hpuq where a0puq is the t term in the asymptotic expansion of Tr p´1q βu du e .
Proof. The idea of the argument is simple: we apply Stokes’ theorem to the one-form
1 ω. a0puq arises essentially as a boundary term. Of course, we must be careful with the details of the regularization and analytic continuation. Rather than differentiating the analytic torsion, it suffices to compute the differ-
ence in analytic torsions associated to hp0q and hp1q. (As a result, we will use the usual Stokes’ theorem rather than an infinitesimal version.)
35 For A ą ą 0, let Σ “ ΣA, denote the surface with boundary in M parametrized
by pu, tq ÞÑ hpuqt, for u P r0, 1s and t P p, Aq . (We use the subscript t as in Subsection 3.4.2.) The boundary of Σ consists of four curves, described respectively
by u “ 0, u “ 1, t “ , and t “ A. ω pulled back to BΣ, in the coordinates pu, tq, is dt ω “ Tr1p´1qQQe´t∆puq t Bβ ` Tr1p´1qQe´t∆puqβ´1 du. Bu
Since ω is closed, δMptsωq pulled back to Σ is δMptsωq “ sts´1 dt ω
Bβ “ sts´1 Tr1p´1qQe´t∆puqβ´1 dt du. Bu
hpuq Q n Let ζupsq “ ζps; ∆ , p´1q Qq. Let ζu,A,psq be defined for Re s ą m by
s Γpsqζu,A,psq :“ t ω. żChpuq,A, n Stokes’ theorem gives, for Re s ą m , 1 A Bβ Γpsq rζ psq ´ ζ psqs “ sts´1 Tr1p´1qQe´t∆puqβ´1 dt du 0,A, 1,A, Bu ż0 ż 1 Bβ ` ts Tr1p´1qQe´∆puqβ´1 du Bu ż0 1 Bβ ´ ts Tr1p´1qQe´A∆puqβ´1 du Bu ż0 In the limits A Ñ 8 and Ñ 0, the latter two terms tend to zero for Re s large by
n the admissibility conditions (A1) and (A2). Thus we obtain, for Re s ą m , 1 1 8 Bβ ζ psq ´ ζ psq “ sts´1 Tr1p´1qQe´t∆puqβ´1 dt du 0 1 Γpsq Bu ż0 ż0 1 Bβ “ sζ s; ∆puq, p´1qQβ´1 du Bu ż0 ˆ ˙ 36 We have shown that this equality holds for Re s large, but both sides possess unique meromorphic continuations, so in fact the equality holds for all s. Differentiating, we obtain the following expression for the difference in (logarithms of) analytic torsions
associated to the two metrics hp0q and hp1q:
1 d Bβ log T php0qq ´ log T php1qq “ sζ s; ∆puq, p´1qQβ´1 du ds Bu ż0 ˇs“0 ˆ ˆ ˙˙ ˇ ˇ But the derivative in the integrand is justˇ the value of the zeta function at s “ 0:
d Bβ Bβ sζ s; ∆puq, p´1qQβ´1 “ ζ 0; ∆puq, p´1qQβ´1 . ds Bu Bu ˇs“0 ˆ ˆ ˙˙ ˆ ˙ ˇ ˇ Theorem 3.1.9ˇ gives that the latter is given precisely by the coefficient on t0 in the
1 Q ´t∆puq ´1 Bβ asymptotic expansion of Tr p´1q e β Bu . This proves the theorem.
3.7 Regularization methods
Let hptq be an admissible function, satisfying |hptq| ď Ce´t for t ě 1 and hptq “
J pj j“0 apj t ` Optq as t Ñ 0. Recall that the Mellin transform gives the zeta function
řζpsq defined by, for Re s ą ´p0,
1 8 dt ζpsq “ tshptq . (3.24) Γpsq t ż0
Let us assume that the t Ñ 0 asymptotic expansion contains no integer powers of t,
´tL i.e., none of the pj’s is an integer. (This is the case, for example, if hptq “ Tr e for a strictly positive elliptic operator L on an odd-dimensional manifold.) Then
dζ ζp0q “ 0, and ds p0q is the value at s “ 0 of the function zpsq :“ Γpsqζpsq; this value 8 dt may be interpreted as a regularization of the divergent integral 0 hptq t . In this section, we will consider a class of alternative methodsş of regularizing the
aforementioned divergent integral, given by replacing ts in (3.24) by φptqs, where φptq
37 is a more general function. We now discuss the assumptions we will place on φptq, which we do not claim to be sharp.
Assume φptq is positive and smooth for t ą 0. Assume that φptq is Optbq as t Ñ 8 for some b. Assume that there exist a ą 0, p ą 0, and a function uptq such that
φptq “ atpuptq, (3.25)
where uptq is smooth on r0, 1s and up0q “ 1. If φptq satisfies these properties, then we will say that φptq is an “allowable regularizing function”. Let fptq be an admissible function whose t Ñ 0 asymptotic expansion contains
no integer powers of t. Consider the function zφpsq defined by
8 s zφpsq “ φptq fptq dt, (3.26) ż0 whose value at s “ 0 (if it is finite) may be interpreted as a regularization of the
8 hptq divergent integral 0 fptq dt. Note that if we take fptq “ t (which is admissible if
and only if hptq is)ş and φptq “ t, then zφpsq is the function zpsq mentioned in the
first paragraph of this section. Our goal is to study the dependence of zφp0q on the function φptq. First, we prove the following lemma involving Taylor approximations:
Lemma 3.7.1. Suppose uptq is a smooth positive function such that up0q “ 1. Let
K ą 0 be given. Then for s P C, we may write
K´1 s j uptq “ 1 ` s αjptq ` rps, tq, (3.27) j“1 ÿ where each αjptq is a polynomial in t, rps, tq is holomorphic in s, and for each compact set U Ă C, there exists C such that for s P U and 0 ă t ă 1, |rps, tq| ď C|s|tK .
38 Proof. ln uptq is a smooth function with ln up0q “ 0; we will use its Taylor approxi- mation for t near 0, i.e., for any L, we may write as t Ñ 0
L l L`1 ln uptq “ alt ` Opt q. (3.28) l“1 ÿ
Now we expand uptqs “ es ln uptq in powers of s ln uptq:
K´1 uptqs “ 1 ` sjpln uptqqj ` r˜ps, tq (3.29) j“1 ÿ wherer ˜ps, tq is holomorphic in s, and for s in any compact set and 0 ă t ă 1,
K K K K |r˜ps, tq| ď C2|s| | ln uptq| , which implies |r˜ps, tq| ď C3|s| t since ln uptq is Optq as t Ñ 0. (Of course, the constants C2 and C3 depend on the compact set.) (3.28) shows that for each j, pln uptqqj may be written as a polynomial in t, which we define to
K be αjptq, plus a remainder term rjptq that is Opt q as t Ñ 0. The desired remainder term rps, tq is the sum
K´1 j rps, tq “ s rjps, tq ` r˜ps, tq, j“1 ÿ which satisfies the desired estimate, proving the lemma.
Now we return to our study of zφ. We may decompose zφ as follows:
1 8 s s zφpsq “ φptq fptq dt ` φptq fptq dt ż0 ż1
“: zφ,0psq ` zφ,8psq.
zφ,8psq is holomorphic in s, and its value at s “ 0 is
8 zφ,8p0q “ fptq dt, ż1 39 which converges and does not depend on the choice of φptq.
Now we will study zφ,0psq. We substitute the following truncation of the asymp- totic expansion of fptq into the integral defining zφ,0psq:
J pj fptq “ apj t ` RJ ptq, j“0 ÿ where for 0 ă t ă 1, RJ ptq ď Ct. This implies that RJ ptq is integrable on r0, 1s and therefore the integral
1 s φptq RJ ptq dt ż0 converges and is holomorphic at s “ 0, and its value at s “ 0 does not depend on φptq. It remains to study for each j the function of s
1 s pj φptq apj t dt. (3.30) ż0
Recall that we have assumed each pj is not an integer. For clarity of notation, we
will replace pj by a number q that is not an integer and ignore the constant apj , i.e., let us consider the function
1 s q ξφpsq :“ φptq t dt. ż0
Lemma 3.7.1 shows that we may write φptqs as
J s s ps j φptq “ a t 1 ` s αjptq ` rps, tq , ˜ j“1 ¸ ÿ where each αjptq is a polynomial in t and rps, tq is holomorphic in s and |rps, tq| ď
40 C|s|t´q for s in some neighborhood of 0 and 0 ă t ă 1. Thus we have 1 s ps`q ξφpsq “a t dt (3.31) ż0 J 1 s j ps`q `a s t αjptq dt (3.32) j“1 0 ÿ ż 1 `as tps`qrps, tq dt. (3.33) ż0 We have that for any integer k and for Re s sufficiently large, 1 1 as tps`qtk “ as , ps ` q ` k ` 1 ż0 which extends to a meromorphic function that is holomorphic at s “ 0 because q is not an integer. This implies that the meromorphic continuation of each of the
summands in line (3.32) vanishes at s “ 0 because of the factors of sj for j ě 1. The integral in line (3.33) converges for Re s ą ´1 and vanishes at s “ 0 by the estimate |rps, tq| ď C|s|t´q. This leaves the right-hand side of line (3.31), which we can compute explicitly for Re s large, obtaining
1 as , ps ` q ` 1
1 the value of whose meromorphic extension at s “ 0 is q`1 , which clearly does not depend on φ.
We have proven that the value of ξφpsq at s “ 0 does not depend on φ, which
proves that the same is true for every term in zφ,0psq of the form (3.30). To summarize, we have proven the following:
Proposition 3.7.2. If fptq is an admissible function with no integer powers of t in its t Ñ 0 asymptotic expansion, then the value at s “ 0 of the meromorphic extension of 8 φptqsfptq dt ż0 41 is independent of the choice of allowable regularizing function φptq.
Applying this to analytic torsion, we obtain:
Corollary 3.7.3. If the dimension of M is odd and E is an acyclic elliptic complex, then the associated analytic torsion is given by
AC s 2 ln T pM, E, hq “ φptq ωT , ˆżCh ˙ˇs“0 ˇ ˇ ˇ where φptq is any allowable regularizing function, viewed as a function Ch Ñ R.
3.8 Varying the curve
The interpretation of the analytic torsion T phq as a regularized integral of ωT over the curve Ch raises the question of whether there are other interesting curves in the space of metrics over which to integrate ωT . In this section, we will show that, assuming odd dimension and acylicity, this regularized integral is constant on a certain family of curves that includes Ch. This should not be surprising, as Stokes’ theorem applied to the closed form ωT gives that the integral is invariant under compactly supported perturbations of Ch. We will now show that the same is true for more general perturbations of Ch.
Fix a metric h P M. Recall that the curve Ch is parametrized by t ÞÑ ht, where
q q q q on E , we scale the metric according to ht “ t h . This motivates the following generalization in which we scale the metric on Eq by a more general function of t.
Let f “ fpt, qq : p0, 8q ˆ t0, . . . , ru Ñ p0, 8q be a function that is smooth in t. We
q will take the operator β to be multiplication by fpt, qq on E ; that is, let hf,t be the metric on E such that its restriction to Eq is
q q hf,t :“ fpt, qqh .
42 q In this notation, ht is hf,t if we take f to be fpt, qq “ t . Let Ch,f be the curve in M
parametrized by t P p0, 8q ÞÑ hf,t. For ease of notation, for 1 ď q ď r, let ρpt, qq be the ratio
fpt, qq ρpt, qq :“ . (3.34) fpt, q ´ 1q
We will use the notation ρ9pt, qq and f9pt, qq for derivatives with respect to t. Let us impose the assumptions for each q that ρ9pt, qq ą 0 (so that ρpt, qq is an invert-
´1 ible function of t with smooth inverse ρ p¨, qq) and that limtÑ0` ρpt, qq “ 0 and
limtÑ8 ρpt, qq “ 8.
q We compute the following operators pulled back to Ch,f , acting on sections of E :
f9pt, qq b “ dt fpt, qq
d˚ptq “ ρpt, qqd˚
∆ “ ρpt, qqdd˚ ` ρpt, q ` 1qd˚d, where d˚ “ d˚h is associated to the original metric h. We obtain that the one-form
ωT pulled back to Ch,f is
r 9 q fpt, qq 1 ρ t,q d d˚ 1 ρ t,q 1 d˚d ω “ p´1q Tr e´ p q q´1 q´1 ` Tr e´ p ` q q q dt, T f t, q q“0 p q ÿ ´ ¯ where we are using the notation of (3.13). Recall from (3.14) that for any c ą 0,
˚ ˚ Tr1 e´c dqdq “ Tr1 e´c dq dq ,
Thus we may rewrite ωT as
r 9 9 q fpt, qq fpt, q ´ 1q 1 ρ t,q d d˚ ω “ p´1q ´ Tr e´ p q q´1 q´1 dt. T f t, q f t, q 1 q“0 ˜ p q p ´ q¸ ÿ 43 ´1 We will make a change of variables. Let τq “ ρpt, qq, i.e., t “ ρ pτq, qq. A short
1 computation, using dt “ ρ1pt,qq dτq and the definition (3.34) of ρpt, qq, shows that under this change of variables, we have the equality of one-forms
f9pt, qq f9pt, q ´ 1q dτ ´ dt “ q . ˜fpt, qq fpt, q ´ 1q¸ τq
This implies that ωT transforms under this change of variables to
r q 1 τ d d˚ dτq ω “ p´1q Tr e´ q q´1 q´1 . (3.35) T τ q“0 q ÿ
Proposition 3.7.2 shows that, assuming acyclicity and odd dimension, the method of regularization is irrelevant. So we may choose any method to regularize the integral
1 of 2 ωT over Ch,f ;(3.35) gives that we obtain
r q ˚ p´1q ln detpdq´1dq´1q. q“0 ÿ
But recall the identity (3.15), which shows that this is precisely the usual analytic torsion.
3.9 Some historical remarks
This section contains some remarks both on our results and on the history of ana- lytic torsion, including discussion of generalized metrics, Morse variations, and the Cheeger-M¨ullertheorem.
Let us now specialize to the case when the elliptic complex pE, dq is the de Rham complex associated to a flat vector bundle F Ñ M, which was introduced briefly in Example1 on page 21. Recall that this means that E “ F b Λ‚T ˚M, and d is induced by the flat connection on F and the usual exterior derivative on ordinary
44 differential forms. In this section, we will give a mostly non-technical, brief (and far from exhaustive) account of the history of the study of analytic torsion in this context, and we will relate some of our results to this history. We will discuss spaces of metrics, the Cheeger-M¨ullertheorem, and in particular Bismut-Zhang’s proof of
the Cheeger-M¨ullertheorem, which uses a special case of our closed form ωT and a metric variation involving a Morse function. In this context, the space of metrics M that we have considered throughout this
chapter is the space of all metrics on F b Λ‚ such that the F b Λq’s are mutually orthogonal for q “ 0, . . . , n. We will now introduce a distinguished subset of M.
Definition 3.9.1. Let MRiem be the set of metrics on E that are induced as the tensor product of some metric hF on F and some Riemannian metric on M, which induces a metric on the exterior bundle Λ‚T ˚M.
MRiem is strictly contained in M. To explain this rather loosely, for a metric
in MRiem, the induced inner products on q-forms for q “ 0, . . . , n are related in the sense that they are all induced by a single Riemannian metric. But for a generic metric in M, the inner products on q-forms need not be related in any way; we are allowed to pick metrics on Eq for each q independently. Ray-Singer defined the analytic torsion T in the case when F is orthogonal (so that F possesses a canonical metric) and proved that T is independent of the Rie- mannian metric, assuming acyclicity and odd dimension; their results trivially extend to unitary F . Motivated by the formal similarity between T and the Reidemeister torsion τ and by computations of T and τ on lens spaces by Ray [15], Ray-Singer conjectured the equality of T and τ in the orthogonal case. This conjecture, now known as the Cheeger-M¨ullertheorem, was proved independently by Cheeger [6] and M¨uller[13] using different techniques. Later, M¨uller[14] generalized the theorem to unimodular bundles using an extension of Cheeger’s techniques. To our knowledge,
45 this paper of M¨ullerwas the first to observe that analytic torsion T phq may be defined for any metric h P MRiem (i.e., for any metric on F and for any Riemannian metric). Bismut-Zhang [5] proved a further generalization of the Cheeger-M¨ullertheorem; we will comment a bit further on their proof later in this section. Now we will make a a few comments about our results. We emphasize that the closedness of ωT (Theorem 3.5.1), the variation of T with respect to the metric (Theorem 3.3.3), and the metric independence theorem (Corollary 3.3.4) apply on the entire space of metrics M and not just on MRiem. In that sense, the latter two results generalize the metric independence theorem of Ray-Singer [16] and the metric anomaly formula of Bismut-Zhang [5], in which they considered only metrics in the distinguished subset MRiem.
Remark 3.9.2. Mathai-Wu [11] have introduced analytic torsion for Z2-graded elliptic complexes. This is more general than our Z-graded context, since a Z-graded complex
can always be “rolled up” into a Z2-graded complex. In particular, Mathai-Wu’s twisted de Rham torsion ([9], [10]) and twisted Dolbeault torsion [11], involving a
closed flux form H of odd degree, do not fit into our Z-graded framework unless the degree of H is one. Mathai-Wu consider an appropriate analogue of our space of all metrics M, and they prove metric independence of the torsion on this space under appropriate assumptions.
An examination of our definition of the form ωT and the proof of its closedness
shows that they extend without change to the Z2-graded setting. But crucially, our
interpretation of analytic torsion as a regularized integral of ωT over a curve in the
space of metrics does not make sense in the Z2-graded setting, since the curve Ch relies in an essential way on the Z-grading. For this reason, we have chosen not to
discuss Z2-graded complexes in detail.
We have not yet found an interesting application of our generalized metric inde-
46 pendence theorem, but we note that the generalization from orthogonal and unitary bundles to arbitrary metrics on F is essential to Bismut-Zhang’s proof of the Cheeger- M¨ullertheorem [5]. Their proof involves a “conformal” perturbation of the metric on F involving a Morse function. We will now discuss their techniques a bit further
and relate them to our closed form ωT .
Following Bismut-Zhang, let f : M Ñ R be a smooth function. Let gF be a metric on the coefficient bundle F . For T P R, consider the metric gF,T on F defined by
gF,T :“ e´2T f gF .
Note that we obtain the original metric gF if we take T “ 0, i.e., gF “ gF,0. Let gTM be a Riemannian metric on M. Then gTM and gF,T induce a metric on F b Λ‚T ˚M; we will denote this metric by hT .
Remark 3.9.3. The metric hT is “conformal on F ” in the sense that it involves multiplication of gF by a positive function, which scales the metric on F -valued forms by the same factor in all degrees. It is not conformal in the usual sense because it does not involve a conformal change (or any change) to the Riemannian metric, which would scale the metric on F -valued forms differently in each degree.
The operator β “ βT associated to hT is βT “ e´2T f , i.e., multiplication by the function e´2T f . We compute the following operators and forms pulled back to the curve parametrized by T ÞÑ hT : b “ ´2f dT
d˚,T “ e2T f d˚,0e´2T f
∆T “ de2T f d˚,0e´2T f ` e2T f d˚,0e´2T f d
Q ´∆T ωT “ ´2 Trp´1q fe dT, where f and e˘2T f denote the operators defined by multiplication by f and e˘2T f , respectively.
47 Remark 3.9.4. As Bismut-Zhang discuss, there is an equivalent alternative approach. Rather than considering the metric hT , one could instead fix the metric and vary the
operator d as follows: let d˜T :“ e´T f deT f . Then pd˜T q˚ :“ eT f d˚e´T f is the adjoint of dT with respect to the L2 inner product induced by the fixed metric gF . The
associated Laplacian is ∆˜ T :“ d˜T pd˜T q˚ ` pd˜T q˚d˜T . (These operators were introduced
T by Witten, who related the asymptotics of e´∆˜ as T Ñ 8 to Morse theory [23].) It is easy to see that ∆˜ T and ∆T are conjugate (∆˜ T “ e´T f ∆T eT f ), which implies that they have the same eigenvalues. Thus these two approaches are equivalent for our purposes; we will choose to phrase what follows in terms of ∆T .
1 We can also scale the Riemannian metric in the usual way, i.e., by a factor t ‚ ˚ 1 TM F,T for t ą 0. Let hpt, T q be the metric on F b Λ T M induced by t g and g “ e´2T f gF ; the map pt, T q ÞÑ hpt, T q, for t P p0, 8q and T P R, parametrizes a surface Σ “ ΣpgTM , gF , fq in the space of metrics MRiem Ă M. Recall that the scaling
1 TM ˚ t ÞÑ t g scales d and ∆ by a factor of t; we obtain that the torsion one-form ωT pulled back to Σ, which we will denote by α (following Bismut-Zhang), is
T T α “ Trp´1qQQe´t∆ dt ´ 2 Trp´1qQfe´t∆ dT.
Bismut-Zhang prove that α is closed on the surface Σ, although they do not make our more general observation that α is the pullback to Σ of a closed form defined on all of M. The closedness of α is essential to the argument of Bismut-Zhang that computes the difference between the analytic torsion T and the Reidemeister torsion τ. We will now give a very rough outline of some of the ideas of their argument. Let f be a Morse function. By the metric independence theorem for analytic torsion, the computation is not affected by choosing a special Riemannian metric gTM that satisfies (among other things) that the associated gradient vector field of f satisfies the Smale transversality conditions. This gives a Thom-Smale complex
whose cohomology may be identified with H‚pM,F q; the so-called Milnor metric,
48 the details of which we will omit, is computed from this Thom-Smale complex and is known to equal the Reidemeister metric in the unimodular case. Rather than directly comparing T and τ (or more precisely, the Ray-Singer metric and the Reidemeister metric), Bismut-Zhang compare the Ray-Singer metric and the Milnor metric. Their basic approach is similar to our approach to proving Theorem 3.3.3 (cf. page 35): they integrate α over the curve Γ that is the boundary of the
rectangle R “ R,A,T0 , where R is the subset of Σ on which ď t ď A and 0 ď T ď T0. Crucially, by Stokes’ theorem, this integral vanishes. By taking the limits Ñ 0,
T0 Ñ 8, and A Ñ 8 and carefully keeping track of the rates of divergence of the integrals along each of the four components of Γ, Bismut-Zhang compute the difference between the Ray-Singer metric and the Milnor metric. They prove that this difference vanishes in the unimodular case, thus generalizing the Cheeger-M¨uller theorem.
49 4
Multi-torsion on manifolds with local product structure
The interpretation of analytic torsion as the integral of a closed one-form naturally raises the question of the existence of new invariants constructed from other closed differential forms on the space of metrics, possibly of degree greater than one. In this
chapter, we introduce one such closed m-form ωMT on the space of metrics inducing a local geometric product structure on certain manifolds. ωMT is inspired by the m-fold product of ωT in the following sense: on the space of product metrics on a
M1 Mm Mj global product manifold M1 ˆ ¨ ¨ ¨ ˆ Mm, ωMT “ ωT ¨ ¨ ¨ ωT , where for each j, ωT denotes the torsion one-form associated to Mj. ωMT also makes sense on manifolds with only a local product structure; we treat in detail the case when m “ 2 and
the base manifold is a quotient of M1 ˆ M2 by a finite group of diffeomorphisms, in which case we define a quantity that we call multi-torsion.
Multi-torsion is essentially a regularized integral of the two-form ωMT over a two- dimensional submanifold of the space of metrics. The regularization involves taking a second derivative of a certain multi-zeta function, a function of two complex variables which may be of independent interest. Under appropriate assumptions, we prove the
50 independence of multi-torsion in the class of local product metrics using a Stokes’ theorem argument.
4.1 Motivation: Product manifolds
Our goal in this chapter will be to define multi-torsion on certain manifolds with a local product structure. In this section, we begin with some motivating observations in the case of a global product.
4.1.1 Products of zeta functions
We start with some simple observations. Let ζ1 and ζ2 be two zeta functions, each
related to an admissible function hj by the Mellin transform; i.e., for Re s large and for j “ 1, 2, we have: 1 8 dt ζ psq “ tsh ptq . j Γpsq j t ż0
Define ζps1, s2q as the following product:
ζps1, s2q :“ ζ1ps1qζ2ps2q.
Observe that B2ζ dζ dζ “ 1 2 . (4.1) Bs1Bs2 ds1 ds2 ˇps1,s2q“p0,0q ˇs1“0 ˇs2“0 ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ We may interpret ζps1, s2q as a “multi-Mellin transform,” i.e., as an integral over the quadrant p0, 8q ˆ p0, 8q Ă R2 by Fubini’s theorem:
ζps1, s2q “ ζ1psqζ2psq
1 8 8 “ ts1´1ts2´1hpt , t q dt dt (4.2) Γps qΓps q 1 2 1 2 1 2 1 2 ż0 ż0
where hpt1, t2q :“ h1pt1qh2pt2q. This raises the question:
51 Question 4.1.1. For which functions hpt1, t2q does the integral (4.2) define a mero- morphic function ζps1, s2q, and in which interesting geometric situations do such hpt1, t2q arise?
In this chapter, we will provide a partial answer to this question. In particular, we will observe that the product case suggests a condition on what we will call the
“multi-asymptotics” of hpt1, t2q that is sufficient to define a meromorphic function
ζps1, s2q; we discuss this condition in Section 4.2.
4.1.2 Product manifolds
We now describe a simple geometric setting in which the zeta functions of the previous section might arise. Consider two compact manifolds M1 and M2 of dimensions n1 and n2, respectively. For j “ 1, 2, denote by πj the projection M1 ˆ M2 Ñ Mj. Let
E1 Ñ M1 and E2 Ñ M2 be vector bundles endowed with Laplace-type operators ∆1 and ∆2, respectively. Assume that each ∆j has trivial kernel. Consider the product
˚ ˚ vector bundle E :“ π1 E1 b π2 E2 Ñ M1 ˆ M2. ∆1 and ∆2 induce commuting sub- elliptic operators, which we will also denote by ∆1 and ∆2, on sections of E. Define
∆pt1, t2q, an elliptic operator depending on two positive parameters t1 and t2, by
∆pt1, t2q :“ t1∆1 ` t2∆2 (4.3)
´tj ∆j To match the notation of Section 4.1.1, for j “ 1, 2, let hjptjq :“ Tr e , and let
´∆pt1,t2q ζjpsjq be the associated zeta function. Let hpt1, t2q :“ Tr e . Then
hpt1, t2q “ h1pt1qh2pt2q. (4.4)
The “multi-zeta function” ζps1, s2q :“ ζ1ps1qζ2ps2q may be interpreted as follows:
1 8 8 ζps , s q “ ts1´1ts2´1 hpt , t q dt dt . (4.5) 1 2 Γps qΓps q 1 2 1 2 1 2 1 2 ż0 ż0
52 4.1.3 Analytic torsion
We will now make a simple observation about analytic torsion in this context of
product manifolds. Suppose that F1 Ñ M1 and F2 Ñ M2 are flat, acyclic, unitary vector bundles. To match the notation of the previous subsection, for j “ 1, 2, let
‚ ˚ Ej “ Fj b Λ T Mj. A choice of Riemannian metrics on M1 and M2 induces Laplace operators ∆1 and ∆2 on sections of E1 and E2, respectively. In this context, the analytic torsions Tj :“ T pMj,Fjq are defined.
˚ ˚ We may consider the product vector bundle F “ π1 F1 b π2 F2 Ñ M1 ˆ M2, which
‚ ˚ ˚ ˚ is flat. Let E “ F bΛ T pM1ˆM2q; then E is canonically isomorphic to π1 E1bπ2 E2.
The grading (by degree of forms) on each Ej induces a bigrading on E in the following
q q q sense: E Eq1,q2 . where Eq1,q2 : π˚E 1 π˚E 2 and E j : F Λqj T ˚M “ q1,q2 “ 1 1 b 2 2 j “ j b j denotes theÀ bundle of Fj-valued qj-forms on Mj. E also possesses a grading by total degree of forms: let Eq F ΛqT ˚ M M Eq1,q2 . As usual, let 1 Q “ ˆ p 1 ˆ 2q – q1`q2“q p´ q
q q Qj be multiplication by p´1q on E . For j “ 1, 2, letÀ Qj and p´1q be multiplication
qj q1,q2 Q Q1 Q2 by qj and p´1q , respectively, on E . Note that p´1q “ p´1q p´1q . We have
Q1 ´t1∆1 Q2 ´t2∆2 Q ´∆pt1,t2q pTrp´1q Q1e qpTrp´1q Q2e q “ Trp´1q Q1Q2e .
Let ζT ps1, s2q be defined for Re s1 and Re s2 large by 8 8 1 dt1 dt2 ζ ps , s q “ ts1 ts2 Trp´1qQQ Q e´∆pt1,t2q . T 1 2 Γps qΓps q 1 2 1 2 t t 1 2 ż0 ż0 1 2 Then the observation (4.1) shows that
B2 ζT ps1, s2q “ plog T1qplog T2q, (4.6) Bs1Bs2 ˇps1,s2q“p0,0q ˇ ˇ the product of the logarithmsˇ of the respective analytic torsions of the factors F1 and
F2. If n1 and n2 are both odd, then this expression is independent of the metrics on
F1 and F2. (Recall that we assumed both F1 and F2 to be acyclic.) 53 The above considerations motivate a generalization to geometric settings more general than product manifolds. We will consider one such setting in which a multi- zeta function is defined: a quotient of a product manifold by a finite group. The
essential feature is a local product structure inducing a Laplacian ∆pt1, t2q depending on two parameters, as in (4.3). In the next section, we will discuss the relevant
´∆pt1,t2q properties of hpt1, t2q “ Tr e .
4.2 Heat multi-asymptotics and multi-zeta functions
We retain the notation of the previous section, except that we will use α to index the
two factors to avoid confusion in the notation. For α “ 1, 2, we have an admissible
function hα, each admitting an asymptotic expansion for small tα.
α pα hαptαq „ a α ptαq j , (4.7) pj jě0 ÿ
Each hα also satisfies the estimate, for tα ě 1,
´αtα |hαptαq| ď e . (4.8)
“Multiplication” of these conditions implies that the product hpt1, t2q “ h1pt1qh2pt2q has certain “multi-asymptotic” properties, which we now discuss. Since there are now two variables, there are now four cases to consider. We will
label the four conditions on hpt1, t2q as (MA1)-(MA4):
(MA1) (Both t1 and t2 small.) For every real number K, we may write
1 2 JK JK 1 2 pj pj hpt1, t2q “ pt1q 1 pt2q 2 ap1 ,p2 j1 j2 j “0 j “0 ÿ1 ÿ2 1 2 IK IK p1 1 p2 2 11 i2 ` pt1q bp1 pt2q ` pt2q bp2 pt1q i1 i2 i “0 i “0 ÿ1 ÿ2
` rK pt1, t2q,
54 K K 1 K where for 0 ă t1, t2 ă 1, |rK pt1, t2q| ď CK t1 t2 , |bp1 pt2q| ď CK,i1 pt2q , and i1 2 K |bp2 pt1q| ď CK,i2 pt1q . i2
(MA2) (t1 small, t2 large.) For every real number K we may write
1 IK p1 1 1 i1 hpt1, t2q “ pt1q cp1 pt2q ` sK pt1, t2q i1 i “0 ÿ1
1 ´2t2 1 where for 0 ă t1 ă 1 and t2 ě 1, |cp1 pt2q| ď e and |sK pt1, t2q| ď i1
1 K ´2t2 CK pt1q e .
(MA3) (t1 small, t2 large.) (This is (MA2) with the roles of t1 and t2 reversed.) For every real number K we may write
2 IK p2 2 2 i2 hpt1, t2q “ pt2q cp2 pt1q ` sK pt1, t2q i2 i “0 ÿ2
2 ´1t1 2 where for 0 ă t2 ă 1 and t1 ě 1, |cp2 pt1q| ď e and |sK pt1, t2q| ď i2
2 K ´1t1 CK pt2q e .
(MA4) (Both t1 and t2 large.) If t1, t2 ě 1,
´1t1 ´2t2 |hpt1, t2q| ď Ce e
Definition 4.2.1. We will say that a smooth C-valued function hpt1, t2q is “multi- admissible” when it satisfies conditions (MA1)-(MA4) above.
The utility of this notion of multi-admissible is that it is sufficient to define an associated multi-zeta function.
Definition 4.2.2. We define the multi-zeta function associated to hpt1, t2q by the
following, for Re s1 and Re s2 large:
1 8 8 dt dt ζps , s q :“ ts1 ts2 hpt , t q 1 2 . (4.9) 1 2 Γps qΓps q 1 2 1 2 t t 1 2 ż0 ż0 1 2 55 Lemma 4.2.3. Suppose hpt1, t2q is multi-admissible. Then the integral defining
1 2 ζps1, s2q in (4.9) converges for Re s1 ą ´p0 and Re s2 ą ´p0 and defines a holo- morphic function there. ζps1, s2q admits a unique meromorphic extension to all of
2 C ; the extension, which we also denote by ζps1, s2q, is holomorphic at the origin, where its value is ζp0, 0q “ a0,0. Furthermore, we have the vanishing results:
0 Bζ 1. If no pt1q terms appear in (MA1) or in (MA2), then p0, 0q “ 0. Bs2
0 Bζ 2. If no pt2q terms appear in (MA1) or in (MA3), then p0, 0q “ 0. Bs1
Proof. Let fps1, s2q be the integral in (4.9):
8 8 dt dt fps , s q :“ ts1 ts2 hpt , t q 1 2 . 1 2 1 2 1 2 t t ż0 ż0 1 2
Then 1 ζαps1, s2q “ fps1, s2q. (4.10) Γps1qΓps2q
The gamma function Γ is meromorphic and nonvanishing, so 1 is holomorphic Γps1qΓps2q
1 everywhere. Since Γpzq “ z ` Op1q as z Ñ 0,
1 3 “ s1s2 ` Op|ps1, s2q| q as ps1, s2q Ñ p0, 0q. (4.11) Γps1qΓps2q
Thus it suffices to show that fps1, s2q admits a meromorphic extension and to study its behavior near p0, 0q. We write the domain of the integral in (4.9) as the union of the sets
1. where 0 ă t1, t2 ă 1,
2. where 0 ă t1 ă 1 and t2 ě 1,
3. where 0 ă t2 ă 1 and t1 ě 1, and
56 4. where t1, t2 ě 1.
The respective conditions (MA1)-(MA4) ensure that the integral over each of these
nj four sets converges absolutely if Re sj ą 2 for j “ 1, 2. Computing the integrals give that for any real number K we have
J1 J2 K K 1 fps1, s2q “ 1 2 ap1 ,p2 ps ´ p qps ´ p q j1 j2 j “0 j “0 1 j1 2 j2 ÿ1 ÿ2 1 2 IK IK 1 ˜1 1 ˜2 ` 1 bp1 ps2q ` 2 bp2 ps1q s ´ p i1 s ´ p i2 i “0 1 i1 i “0 2 i2 ÿ1 ÿ2
` r˜K ps1, s2q
1 IK 1 1 1 ` 1 c˜p1 ps2q ` s˜K ps1, s2q s ´ p i1 i “0 1 i1 ÿ1 2 IK 1 2 2 ` 2 c˜p2 ps2q ` s˜K ps1, s2q s ´ p i2 i “0 2 i2 ÿ2 ˜ ` hps1, s2q
where the new functions of s1, s2 (marked by tildes), each of which is holomorphic
57 for Re s1, Re s2 ą ´K, are defined in terms of the functions in (MA1)-(MA4) by 1 dt2 ˜1 s2 bp1 ps2q :“ pt2q bp1 pt2q i1 i1 t ż0 2 1 dt1 ˜2 s1 bp2 ps1q :“ pt1q bp2 pt1q i2 i2 t ż0 1 1 1 dt1 dt2 r˜ ps , s q :“ pt qs1 pt qs2 r pt , t q K 1 2 1 2 K 1 2 t t ż0 ż0 1 2 8 1 s2 dt2 c˜p1 ps2q :“ t2 cp1 pt2q i1 i1 t ż1 2 8 1 dt1 dt2 s˜1 ps , s q :“ pt qs1 pt qs2 s1 pt , t q K 1 2 1 2 K 1 2 t t ż1 ż0 1 2 8 2 s1 dt1 c˜p2 ps1q :“ t1 cp2 pt1q i2 i2 t ż1 1 1 8 dt1 dt2 s˜2 ps , s q :“ pt qs1 pt qs2 s2 pt , t q K 1 2 1 2 K 1 2 t t ż0 ż1 1 2 1 8 dt1 dt2 h˜ps , s q :“ pt qs1 pt qs2 hpt , t q . 1 2 1 2 1 2 t t ż0 ż1 1 2
This uniquely defines the meromorphic extension of f to the set where Re s1, Re s2 ą ´K. Since K is arbitrary, this proves that f has a unique meromorphic extension
2 to all of C . We see that as ps1, s2q Ñ p0, 0q,
1 ˜1 1 1 ˜2 2 1 fps1, s2q “ a0,0 ` b0p0q ` c˜0p0q ` b0p0q ` c˜0p0q ` O p1q . s1s2 s1 s2 ´ ¯ ´ ¯ Using (4.10) and (4.11), this shows
ζp0, 0q “ a0,0
Bζ ˜1 1 p0, 0q “ b0p0q ` c˜0p0q Bs2
Bζ ˜2 2 p0, 0q “ b0p0q ` c˜0p0q. Bs1
The final vanishing assertions of the theorem follow.
58 Remark 4.2.4. There is a natural generalization to functions hpt1, . . . , tmq of m vari- ables, for which one may attempt to define a multi-zeta function ζps1, . . . , smq via an Mellin-type transform involving an integral over p0, 8qm Ă Rm. One should then replace the four conditions (MA1)-(MA4) with 2m conditions, corresponding to whether each of the m variables t1, . . . , tm is greater than or less than one. We will not pursue this further.
Remark 4.2.5. The multi-zeta functions we consider here bear some formal resem- blance to the zeta functions of Shintani [22], but we do not know if there is any deeper connection.
4.3 The geometry of finite quotients of product manifolds
We will now generalize the setting of Subsection 4.1.3, in which we considered a
˚ ˚ product of acyclic, unitary, flat bundles F “ π1 F1 b π2 F2 Ñ M1 ˆ M2. We will now consider quotients by certain actions of finite groups.
Definition 4.3.1. Let V Ñ N be a vector bundle. We will say φ is a V -diffeomorphism when φ : N Ñ N is a diffeomorphism that lifts to a diffeomorphism (also called φ) φ : V Ñ V on the total space that acts as a linear isomorphism on fibers, i.e., for
each fiber Vx, φx :“ φ|Vx : Vx Ñ Vφpxq is a linear isomorphism. If V is endowed with a metric and φ|Vx : Vx Ñ Vφpxq is in addition an orthogonal map with respect to that metric, then we will say φ is a V -isometry.
Let DiffpV Ñ Nq denote the group of V -diffeomorphisms, and let IsompV Ñ Nq denote the group of V -isometries.
A V -diffeomorphism φ induces a pullback operator φ˚ : C8pN,V q Ñ C8pN,V q
˚ ´1 defined by φ pσqpxq “ pφxq σpφpxqq. This extends also to a pullback operator φ˚ : C8pN, Λ‚V q Ñ C8pN, Λ‚V q.
59 Let Γ be a finite subgroup of IsompF1 Ñ M1q ˆ IsompF2 Ñ M2q, which we view as a subgroup of IsompF Ñ M1 ˆ M2q. For j “ 1, 2, let Γj denote the projection of
Γ onto IsompFj Ñ Mjq. We remark that each Γj is a subgroup of IsompFj Ñ Mjq, but Γ need not be the product group Γ1 ˆ Γ2.
We will assume Γ, viewed as a group of diffeomorphisms of M1 ˆ M2, acts prop- erly discontinuously in the sense that for allx ˜ P M1 ˆ M2, there exists an open neighborhood U Ă M1 ˆ M2 containingx ˜ such that γpUq X U “H for all γ P Γz Id. We will now discuss the possibility of fixed points. The properly discontinuous assumption implies that for γ “ pγ1, γ2q P ΓztIu, if γ1 P DiffpM1q has a fixed point, then γ2 P DiffpM2q must not have a fixed point (and vice versa). But we will allow exactly zero or one of γ1 and γ2 to have fixed points. Recall the notion of a nondegenerate fixed point:
Definition 4.3.2. Let N be a manifold and let φ : N Ñ N be a diffeomorphism. A
fixed point x P N of φ is said to be nondegenerate when I ´ dφx : TxN Ñ TxN is an isomorphism.
Theorem 4.4.8 involves the assumption that for j “ 1, 2, every γj P ΓjztIu has nondegenerate fixed points when viewed as a diffeomorphism of Mj, but we need not assume this in general.
Let MΓ be the quotient MΓ :“ pM1 ˆM2q{Γ. The properly discontinuous assump- tion implies that MΓ is a smooth manifold of dimension n :“ n1 ` n2. The isometric action of Γ on F induces a quotient vector bundle FΓ Ñ MΓ with an induced metric.
Furthermore, for every x P MΓ, there is an open neighborhood U Ă MΓ containing ˜ ˜ ˜ ˜ x such that U is diffeomorphic to a product U1 ˆ U2 for some U1 and U2, open sets in M1 and M2, respectively; the restriction of FΓ to U is isometric to the product of
˚ ˚ the restrictions π F1| ˜ b π F2| ˜ . U˜1 U1 U˜2 U2
This local product structure gives a bigrading of FΓ-valued forms, as in Subsection
60 ‚ ˚ 4.1.3. We will mimic the notation of that subsection: let EΓ :“ FΓ bΛ T MΓ Ñ MΓ.
q ,q q ,q Then E E 1 2 , where E 1 2 is locally isometric to π˚F Λq1 T ˚M Γ “ q1,q2 Γ Γ p 1 1 b 1q b
˚ q2 ˚ Q1 Q2 Q pπ2 F2 b Λ TÀM2q. Then we have the operators Q1, Q2, p´1q , p´1q , and p´1q , defined as in Subsection 4.1.3.
Let d be the exterior derivative on sections of EΓ. The local product structure ensures that there is a global decomposition of d: d “ d1 `d2, where d1 maps pq1, q2q- forms to pq1 ` 1, q2q-forms, and d2 maps pq1, q2q-forms to pq1, q2 ` 1q-forms. d1 and d2 commute. We would also like a global decomposition of d˚ and ∆. To do so, we need to consider metrics that induce a local geometric product structure on MΓ. We will now discuss the relevant space of metrics.
Let MΓj denote the set of Γj-invariant Riemannian metrics on Mj. A choice of metrics h1 P MΓ1 and h2 P MΓ1 induces a product Riemannian metric h1 ˆ h2 on M1 ˆ M2. Let Mprod denote the set of metrics arising in this way; Mprod is
identifiable with MΓ1 ˆ MΓ2 .
Consider a metric h “ h1ˆh2 P Mprod. h is Γ-invariant, and therefore descends to a well-defined metric, which we will also call h, on the quotient MΓ. Furthermore, h induces a local geometric product structure in the following sense: the identification ˜ ˜ ˜ ˜ U – U1 ˆ U2 from above is a Riemannian isometry, where U1 ˆ U2 Ă M1 ˆ M2 is given the product metric induced by h1 and h2. In addition to the decomposition
˚ ˚ ˚ d “ d1 ` d2, under these hypotheses we have that d “ d1 ` d2 and
∆ “ ∆1 ` ∆2, where (4.12)
˚ ˚ ∆j “ djdj ` dj dj for j “ 1, 2.
61 Our assumptions apply the vanishing of the following (anti)commutators:
td1, d2u “0 (4.13)
˚ td1, d2 u “0 (4.14)
˚ td2, d1 u “0 (4.15)
rdi, ∆js “0 (4.16)
˚ rdi , ∆js “0 (4.17)
r∆1, ∆2s “0 (4.18) where (4.16) and (4.17) hold for i, j “ 1, 2.
Remark 4.3.3. The conditions outlined above are essentially what is required to define the closed form ωMT , which we will introduce in Section 4.6.
4.4 Multi-zeta functions and multi-torsion
The decomposition in (4.12) of the Laplacian on FΓ-valued forms on MΓ suggests the consideration of the following generalized Laplacian:
Definition 4.4.1. For t1, t2 ą 0, let ∆pt1, t2q :“ t1∆1 ` t2∆2.
∆pt1, t2q is elliptic and nonnegative (in fact, strictly positive by our acyclicity assumption), so its associated heat operator e´∆pt1,t2q is defined. In Section 4.6, we will interpret ∆pt1, t2q as the Laplacian associated to a scaled metric. Recall the definition of multi-admissible from Definition 4.2.1. We will show that the heat trace Tr e´∆pt1,t2q is multi-admissible. More generally, to study the dependence of the multi-torsion on the metric, we will also need to consider certain
8 ‚ auxiliary operators of the following form: For j “ 1, 2, let αj P C pMj, End Λ Fjq, and suppose that αj commutes with the action of Γj. Then α1 and α2 induce an
8 ‚ operator α “ α1 b α2 P C pM1 ˆ M2, Λ F q that commutes with the action of Γ, and
8 ‚ α descends to an operator αΓ P C pMΓ, End Λ FΓq. 62 We have the following fundamental result:
´∆pt1,t2q Theorem 4.4.2. Let αΓ be as above. Then Tr αΓe is multi-admissible.
We will prove this theorem in Section 4.5, where we will also elaborate on the
´∆pt1,t2q details of the multi-asymptotics of Tr αΓe . This allows us to define the associated zeta function using (4.9), i.e.:
8 ‚ n1 Definition 4.4.3. Let αΓ P C pMΓ, End Λ FΓq be as above. For Re s1 ą 2 and
n2 Re s2 ą 2 , define ζps1, s2; ∆pt1, t2q, αΓq by the following:
ζps1, s2; ∆pt1, t2q, αΓq
8 8 1 dt1 dt2 :“ ts1 ts2 Tr α e´∆pt1,t2q . (4.19) Γps qΓps q 1 2 Γ t t 1 2 ż0 ż0 1 2
We immediately obtain:
Corollary 4.4.4. Suppose both factors are acyclic. Then the integral defining
2 ζps1, s2; ∆pt1, t2q, αq in (4.19) converges on the set of points ps1, s2q P C such that
n1 n2 Re s1 ą 2 and Re s2 ą 2 and defines a holomorphic function there. Furthermore, 2 ζps1, s2; ∆pt1, t2q, αq admits a unique meromorphic extension to all of C ; the exten- sion is holomorphic at the origin.
Proof. This follows from Theorem 4.4.2 and Lemma 4.2.3.
Remark 4.4.5. Recall that if ker ∆ is nontrivial, the Definition 3.1.7 of the usual zeta function via the Mellin transform requires the use of Tr1 e´t∆ (as opposed to Tr e´t∆) to ensure decay as t Ñ 8. In defining the multi-zeta function, we have assumed acyclicity in both factors, which means that we need not worry about projecting away from ker ∆pt1, t2q. Moreover, this “double acyclicity” assumption seems quite essential without extensive modification, as it is essential to the decay of Tr e´∆pt1,t2q in the limits t1 Ñ 8 (for fixed t2) and t2 Ñ 8 (for fixed t1). (This decay is encoded
63 in the multi-admissible conditions (MA2)-(MA4), and it follows essentially from the
triviality of the kernels of both ∆1 and ∆2.) This decay, in turn, is essential to the
integrability of Tr e´∆pt1,t2q in the integral in (4.19).
Definition 4.4.6. For ph1, h2q P Mprod, we define multi-torsion MT “ MT pMΓ,FΓ, h1, h2q by
2 B Q MT pMΓ,FΓ, h1, h2q “ ζps1, s2; ∆pt1, t2q, p´1q Q1Q2q Bs1Bs2 ˇps1,s2q“p0,0q ˇ ˇ ˇ If we wish to emphasize only the dependence of the multi-torsion on the metrics
h1 and h2, we will denote it by MT ph1, h2q.
Q We see that the multi-zeta function ζps1, s2; ∆pt1, t2q, p´1q Q1Q2q appearing in
n1 n2 Definition 4.4.6 may be written as (for Re s1 ą 2 and Re s2 ą 2 )
Q ζps1, s2; ∆pt1, t2q, p´1q Q1Q2q
1 8 8 dt dt s1 s2 Q ´∆pt1,t2q 1 2 “ t t Trp´1q Q1Q2e Γps qΓps q 1 2 t t (4.20) 1 2 ż0 ż0 1 2 8 8 1 s s q q t ∆q1 t ∆q2 dt1 dt2 “ t 1 t 2 p´1q 1` 2 q q Tr e´p 1 1 ` 2 2 q , Γ s Γ s 1 2 1 2 t t p 1q p 2q 0 0 q ,q 1 2 ż ż ÿ1 2
q1 q2 where t1∆1 ` t2∆2 denotes the restriction of t1∆1 ` t2∆2 to pq1, q2q-forms.
(4.6) shows that in the case when Γ is the trivial group, i.e., when MΓ is simply
the product M1 ˆM2, MT is the product of the logarithms of the analytic torsions of
the factors. In that case, if n1 and n2 are both odd, then MT is trivially independent of the metric by Corollary 3.3.4. We will now study the dependence of MT on the metric in the general case. First we have the following vanishing result:
Proposition 4.4.7. For j “ 1, 2, if nj is even and Γj acts by orientation-preserving
diffeomorphisms, then MT ph1, h2q “ 0 for any ph1, h2q P Mprod.
64 Proof. Without loss of generality, we may assume j “ 1 for the sake of clarity. The
‚ ˚ Hodge star on M1 induces an operator ˚1 on sections of F b Λ T pM1 ˆ M2q. Since
Γ1 acts by orientation-preserving isometries, ˚1 descends to a well-defined invertible
operator on the quotient, which we also denote by ˚1. ˚1 maps pq1, q2q-forms to
pn1 ´ q1, q2q-forms. Thus we have the commutation relations
˚1Q1 “ pn1I ´ Q1q˚1
˚1Q2 “ Q2˚1
Q n1 Q ˚1p´1q “ p´1q p´1q ˚1
˚1∆pt1, t2q “ ∆pt1, t2q˚1
´∆pt1,t2q The last implies that ˚1 commutes also with e . We compute the following, using the cyclicity of the trace:
Q ´∆pt1,t2q ´1 Q ´∆pt1,t2q Trp´1q Q1Q2e “ Tr ˚1 p´1q Q1Q2e ˚1
n1 Q ´∆pt1,t2q “ p´1q Trp´1q pn1I ´ Q1qQ2e .
In the case when n1 is even, this implies
n1 Trp´1qQQ Q e´∆pt1,t2q “ Trp´1qQQ e´∆pt1,t2q, 1 2 2 2
which we claim vanishes. This vanishing shows that the zeta function of (4.20) vanishes identically, which proves the proposition.
Q ´∆pt1,t2q To prove the claim that Trp´1q Q2e “ 0, we will use what is essentially a standard argument that can be used to prove the McKean-Singer formula [12] relating the index to the heat equation. We start by differentiating with respect to
B t1. Since ∆pt1, t2q “ ∆1, which commutes with ∆ and therefore with the resolvent Bt1 p∆ ´ zq´1, we obtain
B Q ´∆pt1,t2q Q ´∆pt1,t2q Trp´1q Q2e “ ´ Trp´1q Q2e ∆1. Bt1
Q ´∆pt1,t2q ˚ Q ´∆pt1,t2q ˚ “ ´ Trp´1q Q2e d1d1 ´ Trp´1q Q2e d1 d1.
65 But this is zero for the following reason: the cyclicity of the trace and the fact that
´∆pt1,t2q Q d1 commutes with Q2 and e and anticommutes with p´1q imply that
Q ´∆pt1,t2q ˚ Q ´∆pt1,t2q ˚ Trp´1q Q2e d1d1 “ ´ Trp´1q Q2e d1 d1.
Q ´∆pt1,t2q We have shown that for each t2 ą 0, Trp´1q Q2e is constant in t1. But Theo-
Q ´∆pt1,t2q Q ´∆pt1,t2q rem 4.4.2 gives that Trp´1q Q2e Ñ 0 as t1 Ñ 8, so in fact Trp´1q Q2e vanishes for all t1, t2 ą 0.
Thus the interesting case (at least under the orientation-preserving assumption) is when both dimensions n1 and n2 are odd. The metric independence of analytic torsion, Corollary 3.3.4, generalizes to the following, which is the main result of this chapter.
Theorem 4.4.8. For j “ 1, 2, if nj is odd and if for every γj P Γj, γj is either orientation-preserving or has nondegenerate fixed points as a diffeomorphism of Mj, then MT ph1, h2q is independent of the metric hj.
We will give the proof in Section 4.8. The proof involves a Stokes’ theorem argument and relies crucially on two things:
• Our interpretation of MT as an integral of a certain closed two-form ωMT over a surface in the space of metrics, which we discuss in Section 4.6; and
• The results of Subsection 4.5.5, in which we explain the relevance of the hy- potheses of Theorem 4.4.8 to the vanishing of constant terms in certain multi- asymptotic expansions. These constant terms arise essentially as boundary terms in the Stokes’ theorem argument.
4.5 Proof of heat kernel multi-asymptotics
In this section we will study the multi-asymptotics of the heat kernel on MΓ. Our primary goal will be to prove Theorem 4.4.2. In the last part of this section (Sub-
66 section 4.5.5), we will also make some observations about constant terms in certain multi-asymptotic expansions that are essential to the metric independence theorem, Theorem 4.4.8.
A note on notation: We will use letters such as x and y to denote points in MΓ.
Tildes will denote points in M1 ˆ M2, e.g., if x, y P M, thenx, ˜ y˜ P M1 ˆ M2 will denote points lying over x, y, respectively. Subscripts will denote projections onto
the factors M1 and M2, i.e., ifx ˜ P M1 ˆ M2, thenx ˜1 “ π1px˜q andx ˜2 “ π2px˜q.
4.5.1 Heat kernel on global product
Mj First we must understand the heat kernel on M1ˆM2. For j “ 1, 2, let k pt;x ˜j, y˜jq P
‚ ‚ HomppΛ Fjqy˜j , pΛ Fjqx˜j q denote the heat kernel on Fj-valued forms on Mj. Recall
˚ ˚ that F denotes the flat product bundle π1 F1 b π2 F2 Ñ M1 ˆ M2. The heat kernel
M1ˆM2 k pt1, t2;x, ˜ y˜q on F -valued forms is the tensor product of the factor heat kernels in the following sense:
M1ˆM2 M1 M2 k pt1, t2;x, ˜ y˜q “ k pt1;x ˜1, y˜1q b k pt2;x ˜2, y˜2q. (4.21)
‚ ‚ Implicit in the notation is the canonical identification of fibers pΛ F qz˜ – pΛ F1qz˜1 b
‚ pΛ F2qz˜2 .
4.5.2 Heat kernel on the quotient: Selberg principle
Suppose π : X˜ Ñ X is a finite-sheeted Riemannian cover, i.e., a finite-sheeted smooth covering map that is a local isometry. Suppose also that W˜ Ñ X˜ and W Ñ X are unitary flat bundles, and that π extends to a smooth covering map of the total
spaces W˜ Ñ W that is a linear isometry on fibers. For x P X andx ˜ P π´1pxq, let
˚ ‚ ˜ ‚ πx˜ : pΛ W qx Ñ pΛ W qx˜ be the induced identification of fibers. Let G be the group of deck transformations of the cover; G is a finite group that acts by W˜ -isometries.
˚ ‚ ˜ ‚ ˜ For each g P G, let gx˜ : pΛ W qgpx˜q Ñ pΛ W qx˜ be the induced identification of fibers. 67 Then we have the following relationship between the heat kernels on N and N˜, which is a simple version of the Selberg principle [21]:
N ˚ ´1 ˚ N˜ ˚ k pt; x, yq “ pπx˜q ˝ gx˜ ˝ k pt; gpx˜q, y˜q ˝ πy˜, gPG ÿ wherex ˜ P π´1pxq andy ˜ P π´1pyq are any lifts of x, y P X. As a consequence, we have the following in our setting:
MΓ ˚ ´1 ˚ M1ˆM2 ˚ k pt1, t2; x, yq “ pπx˜q ˝ γx˜ ˝ k pt1, t2; γpx˜q, y˜q ˝ πy˜, (4.22) γPΓ ÿ wherex ˜ “ px˜1, x˜2q andy ˜ “ py˜1, y˜2q are any lifts of x and y. Using (4.21) and (4.22), we obtain the following, which we record as a proposition:
Proposition 4.5.1.
MΓ k pt1, t2; x, yq
˚ ´1 ˚ M1 ˚ M2 ˚ “ pπx˜q ˝ γ1 k pt1; γ1px˜1q, y˜1q b γ2 k pt2; γ2px˜2q, y˜2q ˝ πy˜ (4.23) γ“pγ1,γ2qPΓ ÿ ` ˘ 4.5.3 The trace of the heat kernel with an auxiliary operator
We now study the heat kernel on the diagonal. Proposition 4.5.1 implies that at a point px, xq on the diagonal of MΓ ˆ MΓ, we have
MΓ k pt1, t2; x, xq
˚ ´1 ˚ M1 ˚ M2 ˚ “ pπx˜q ˝ γ1 k pt1; γ1px˜1q, x˜1q b γ2 k pt2; γ2px˜2q, x˜2q ˝ πx˜ γ“pγ1,γ2qPΓ ÿ ` ˘
68 Let α “ α1 b α2 and αΓ be as in the hypotheses of Theorem 4.4.2. Then the integral
´∆pt1,t2q kernel of αΓe has the value at px, xq
MΓ pαΓqxk pt1, t2; x, xq
˚ ´1 ˚ M1 ˚ M2 ˚ “ pαΓqx pπx˜q ˝ γ1 k pt1; γ1px˜1q, x˜1q b γ2 k pt2; γ2px˜2q, x˜2q ˝ πx˜ γ“pγ1,γ2qPΓ ÿ ` ˘ π˚ ´1 α γ˚kM1 t ; γ x˜ , x˜ “ p x˜q ˝ p 1qx˜1 1 p 1 1p 1q 1qb γ“pγ1,γ2qPΓ ÿ ` α γ˚kM2 t ; γ x˜ , x˜ π˚ p 2qx˜2 2 p 2 2p 2q 2q ˝ x˜ ˘ ˚ ´1 ˚ ´1 since pαΓqx pπx˜q “ pπx˜q αx˜ and α “ α1 b α2. We now take the trace on the fiber at x. By the cyclicity of the trace and the multiplicativity of the trace of a tensor product, we obtain
MΓ trpαΓqxk pt1, t2; x, xq
tr α γ˚kM1 t ; γ x˜ , x˜ tr α γ˚kM2 t ; γ x˜ , x˜ . “ p 1qx˜1 1 p 1 1p 1q 1q p 2qx˜2 2 p 2 2p 2q 2q γ“pγ1,γ2qPΓ ÿ ` ˘ ` ˘
To prove Theorem 4.4.2, it suffices to prove that for each γ,
hγpt1, t2q :“
tr α γ˚kM1 t ; γ x˜ , x˜ tr α γ˚kM2 t ; γ x˜ , x˜ vol x p 1qx˜1 1 p 1 1p 1q 1q p 2qx˜2 2 p 2 2p 2q 2q MΓ p q żMΓ ` ˘ ` ˘ is a multi-admissible function of t1, t2. Let D Ă M1 ˆ M2 be a fundamental domain for the cover M1 ˆ M2 Ñ MΓ. Then we have
hγpt1, t2q “
tr α γ˚kM1 t ; γ x˜ , x˜ tr α γ˚kM2 t ; γ x˜ , x˜ vol x . p 1qx˜1 1 p 1 1p 1q 1q p 2qx˜2 2 p 2 2p 2q 2q MΓ p q żD ` ˘ ` ˘ But by the Γ-invariance of all the relevant operators, this is the same (up to the size of Γ, which is the number of sheets in the cover) as integrating over all of M1 ˆ M2,
69 which we can decompose as a product: 1 ˚ M1 hγpt1, t2q “ tr pα1q γ k pt1; γ1px˜1q, x˜1q |Γ| x˜1 1 żM1ˆM2 ` ˘ tr α γ˚kM2 t ; γ x˜ , x˜ vol x˜ p 2qx˜2 2 p 2 2p 2q 2q M1ˆM2 p q 1 ` ˘ ˚ M1 “ tr pα1q γ k pt1; γ1px˜1q, x˜1q volM px˜1q |Γ| x˜1 1 1 ˆżM1 ˙ ` ˘ tr α γ˚kM2 t ; γ x˜ , x˜ vol x˜ p 2qx˜2 2 p 2 2p 2q 2q M2 p 2q ˆżM2 ˙ ` ˘ All that remains is to prove that for j “ 1, 2, the following is an admissible function of tj:
tr pα q γ˚kM1 pt ; γ px˜ q, x˜ q vol px˜ q. (4.24) j x˜j j 1 j j j Mj j Mj ż ´ ¯
˚ ´tj ∆j Note that this is precisely the trace of the operator αjγj e , viewed as an operator
2 ‚ 2 ‚ L pMj, Λ Fjq Ñ L pMj, Λ Fjq.
4.5.4 Equivariant heat kernels
In this subsection, we will consider expressions of the form (4.24). We will now introduce new notation to emphasize the generality in which the following results hold. Consider a compact Riemannian manifold N of dimension l and a flat unitary vector bundle V Ñ N. Let φ be an isometry of N that extends to a V -isometry; we need not assume that φ has nondegenerate fixed points. Let Λ‚V :“ V b Λ‚T ˚N denote the bundle of V -valued differential forms. φ induces a map φ˚ on sections of
Λ‚V . Let ∆ be the Laplacian on sections of Λ‚V . Let σ P C8pEndpΛ‚V qq. Recall the notion of an admissible function of t ą 0 from Definition 3.1.1. The following fundamental fact is well-known:
Lemma 4.5.2. Under the assumptions from above, Tr1 σ φ˚ e´t∆ is admissible.
70 For a proof, see Gilkey [7]. We remark that we have stated the lemma in terms of Tr1 σ φ˚ e´t∆ to ensure decay as t Ñ 8, but the essential content of the lemma is that Tr σ φ˚ e´t∆ admits a t Ñ 0 asymptotic expansion. In the case that we are interested in, our acyclicity assumption implies that we need not worry about the kernel of ∆, i.e., Tr1 σ φ˚ e´t∆ “ Tr σ φ˚ e´t∆. Lemma 4.5.2 shows that the terms of the form (4.24) are all admissible. This proves that for each γ P Γ, hγpt1, t2q is multi-admissible. Summing over the group elements γ P Γ completes the proof of Theorem 4.4.2, which we restate here:
8 ‚ Theorem 4.5.3. For j “ 1, 2, let αj P C pMj, End Λ Fjq, and suppose that αj commutes with the action of Γj. Then α1 and α2 induce an operator α “ α1 b
8 α2 P C pM,F1q that commutes with the action of Γ, and α descends to an operator
8 ‚ ´∆pt1,t2q αΓ P C pMΓ, End Λ FΓq. Then Tr αΓe is multi-admissible.
´∆pt1,t2q The reader will recall that the multi-admissibility of Tr αΓe was essential to our definitions of the multi-zeta function and multi-analytic torsion.
4.5.5 Vanishing of constant terms
´∆pt1,t2q Having proven the multi-admissibility of Tr αΓe , we now study the constant terms in its multi-asymptotic expansion. The results of this subsection will be es- sential in proving the metric independence result of Theorem 4.4.8, whose proof we complete in Section 4.8.
Theorem 4.4.8 involves the assumptions that the dimension nj of Mj is odd and that each γj P Γj, viewed as a diffeomorphism of Mj, either is orientation- preserving or has nondegenerate fixed points. We will now explain the relevance of these assumptions. We retain the notation of the previous subsection. First, we discuss the orientation- preserving assumption and what it has to do with dimension.
71 Lemma 4.5.4. Suppose φ is an isometry of the Riemannian manifold N. Let X0 be a submanifold of N that is a connected component of the fixed point set of φ. Let
c be the codimension of X0 in N. Then the parity of c depends on whether φ is orientation-preserving, i.e.:
• φ is orientation-preserving if and only if c is even, and equivalently,
• φ is orientation-reversing if and only if c is odd.
Lemma 4.5.5. Suppose φ is orientation-preserving and l is odd. Then there is no t0 term in the asymptotic expansion of Tr σ φ˚ e´t∆.
We omit the proof, but the result follows essentially from an examination of the approximate heat kernel of (4.26) below, using Lemma 4.5.4. For a proof, see the proof of Proposition 2 in [8]. We will now study the case when φ has nondegenerate fixed points, implying the codimension of each fixed point in N is l, the dimension of N. Lemma 4.5.4 shows that in that case, l is even if and only if φ is orientation-preserving. If φ has nondegenerate fixed points, the φ-heat trace Tr σ φ˚ e´t∆ is in fact bounded as t Ñ 0. We will need the following result computing the constant term in that case:
Lemma 4.5.6. Suppose that φ has nondegenerate fixed points. Let Fixpφq denote the finite set of fixed points in N. Let α P C8pN, End Λ‚V q. Then as t Ñ 0,
˚ tr α φ 1 ˚ ´t∆ x0 x0 Tr αφ e “ ` O t 2 . | detpI ´ dφx q| x PFixpφq 0 0 ÿ ´ ¯
˚ ‚ ‚ Proof. Let φx : pΛ V qφpxq Ñ pΛ V qx be the linear map between fibers such that
˚ ˚ ˚ ´t∆ pφ pσqq pxq “ φx pσpφpxqqq. The operator α φ e has an integral kernel whose value at px, xq on the diagonal of M ˆ M is
˚ ‚ αx φx kpt; φpxq, xq P EndppΛ V qxq, (4.25) 72 Recall that we may approximate the heat kernel k near the diagonal by an approxi- mate heat kernel pK , where pK has the form
dist2px, yq K pK pt; x, yq “ p4πtq´l{2 exp ´ tju px, yq, (4.26) 4t j j“0 ˆ ˙ ÿ
where u0px, xq “ I. For any a, by choosing K sufficiently large, we may replace k by pK in (4.25) at the expense of an error term whose trace is of order ta, which we may ignore for the purposes of proving the lemma. We have ˚ K tr αx φx p pt; φpxq, xq
dist2pφpxq, xq K “ p4πtq´l{2 exp ´ tj tr α φ˚ u pφpxq, xq. 4t x x j j“0 ˆ ˙ ÿ We wish to integrate the above expression times the volume form over M. The
Gaussian factor ensures that for any ą 0, the points x such that distpφpxq, xq ą do not contribute to the asymptotic expansion. Thus it suffices to integrate over an arbitrarily small neighborhood of each fixed point of φ.
Let x0 P M be a fixed point of φ. Let T “ dφx0 . Choose normal coordinates
1 l x “ px , . . . , x q centered at x0. In these coordinates, we have dist2pφpxq, xq “ |T x ´ x|2 ` rpxq, where rpxq is O |x|3
˚ ˚ ` ˘ αx φx u0pφpxq, xq “ αx0 φx0 ` Op|x|q
volpxq “ p1 ` Op|x|qq dx where dx “ dx1 ¨ ¨ ¨ dxl. Thus ˚ K tr αx φx p pt; φpxq, xq volpxq (4.27) |T x ´ x|2 ` rpxq “ p4πtq´l{2 exp ´ tr α φ˚ ` Op|x|q ` Optq dx. 4t x0 x0 ˆ ˙ ` ˘ 2 2 Since detpI ´ T q ‰ 0, there exists δ1 ą 0 such that |T x ´ x| ě δ1|x| . Using this and the fact that rpxq is Op|x|3q, there exists ą 0 and δ ą 0 such that for |x| ď ,
|T x ´ x|2 ` rpxq ě δ|x|2. (4.28)
73 For this , we will integrate (4.27) over the set on which |x| ă and show that we obtain ˚ tr αx0 φx 1 0 ` O t 2 , | detpI ´ T q| ´ ¯ which suffices to prove the lemma. A standard computation shows the key fact that
2 ˚ |T x ´ x| tr αx0 φx 1 ´l{2 ˚ 0 2 p4πtq exp ´ tr αx0 φx0 dx “ ` O t . |x|ă 4t | detpI ´ T q| ż ˆ ˙ ´ ¯
1 1 (For a proof, see [19]. The O t 2 is not sharp.) The change of variables x “ t 2 y ´ ¯ ´δ|x|2 1 1 ´l{2 a b 2 a`b shows that the L norm of p4πtq exp 4t x t is O t , which implies that ´ ¯ ´ ¯ we may ignore the terms of the form Op|x|q and Op|t|q in (4.27). Thus it suffices to
1 show that the following is O t 2 : ´ ¯ |T x ´ x|2 rpxq p4πtq´l{2 exp ´ exp ´ ´ 1 dx (4.29) 4t 4t ż|x|ă ˆ ˙ ˆ ˆ ˙ ˙
We write the set on which |x| ă as the union of the sets A and B, defined by:
n 1 A :“ tx P R : t 3 ď |x| ă u
n 1 B :“ tx P R : |x| ă t 3 u.
1 (We may assume t 3 ă .) For x P A, we use the estimate (4.28), which gives
|T x ´ x|2 rpxq p4πtq´l{2 exp ´ exp ´ ´ 1 dx 4t 4t żA ˆ ˙ ˇ ˆ ˙ ˇ ˇ ˇ ´δ|x|2 ˇ ˇ ď 2 p4πtq´l{2 exp dx,ˇ ˇ 1 4t ż|x|ět 3 ˆ ˙
δ 1 1 ´ 3 2 which is O exp ´ 8 t , which is in particular O t . ´ ´ ¯¯ ´ ¯ 74 1 |x|3 rpxq 3 3 Let x P B, i.e., |x| ă t . Then t ă 1; thus since rpxq is Op|x| q, t ă C. ˇ ˇ rpxq |rpxq| |x|3 ˇ ˇ Taylor’s theorem implies exp ´ 4t ´ 1 ď C1 4t ď C2 t , so ˇ ˇ ˇ ´ ¯ ˇ ˇ |T x ´ ˇx|2 rpxq p4πtq´ˇl{2 exp ´ ˇ exp ´ ´ 1 dx 4t 4t żB ˆ ˙ ˇ ˆ ˙ ˇ ˇ ˇ ´δ |x|2 |xˇ|3 ˇ ďC p4πtq´l{2 exp 1 ˇ dx, ˇ 2 4t t żB ˆ ˙
1 which is O t 2 . This proves the lemma. ´ ¯ We have the following vanishing result in the case in which the operator α of Lemma 4.5.6 takes a certain special form:
Corollary 4.5.7. Suppose that φ has nondegenerate fixed points. Let gpuq be a smooth one-parameter family of φ-invariant Riemannian metrics on N.
Q ´1 d˚ ˚ ´t∆gpuq 1{2 Then Trp´1q ˚ du φ e is Opt q as t Ñ 0.
Proof. We will show that for each fixed point x0,
d˚ trp´1qQ ˚´1 x0 φ˚ “ 0, (4.30) x0 du x0
which suffices to prove the claim by Lemma 4.5.6.
We simply conjugate the left-hand side of (4.30) by ˚x0 , using the facts that
Q l Q d˚ d˚ ˚ ˚ ˚p´1q “ p´1q p´1q ˚, ˚ du “ ´ du ˚ (since ˚ squares to a constant), and ˚φ “ or.φ ˚ where or. “ 1 if φ is orientation-preserving and or. “ ´1 if φ is orientation-reversing. But since φ has nondegenerate fixed points, φ is orientation-preserving if and only
l if l is even, which means or. “ p´1q . The cyclicity of the trace and the facts above show that d˚ d˚ trp´1qQ ˚´1 x0 φ˚ “ tr ˚ p´1qQ ˚´1 x0 φ˚ ˚´1 x0 du x0 x0 x0 du x0 x0 d˚ “ ´ trp´1qQ ˚´1 x0 φ˚ , x0 du x0
75 which proves (4.30) and completes the proof of the corollary.
The following proposition concerns constant terms in multi-asymptotic expan- sions. It is essential to Theorem 4.4.8, whose proof we will give in Section 4.8.
Proposition 4.5.8. Suppose the dimension n1 is odd and for every γ1 P Γ1, γ1 is either orientation-preserving or has nondegenerate fixed points as a diffeomorphism
d˚1 of M1. Suppose α1 is of the special form α1 “ ˚1 du for a smooth one-parameter 0 family of metrics h1puq on M1. Then there are no pt1q terms in the multi-asymptotic expansions of (MA1)-(MA3) for the multi-admissible function
Q ´∆pt1,t2q Trp´1q αΓe .
A similar statement holds with the roles of 1 and 2 reversed.
Proof. By the discussion in Subsections 4.5.3 and 4.5.4, it suffices to prove the claim
Q1 ˚ ´t1∆1 0 that for each γ1 P Γ1, Trp´1q α1γ1 e has no pt1q term in its small t1 asymptotic expansion. There are two cases:
First, if γ1 is orientation-preserving, then the claim follows from Lemma 4.5.5
since n1 is odd.
Second, if γ1 has nondegenerate fixed points, then the claim follows from Corollary 4.5.7.
We obtain the following corollary:
Corollary 4.5.9. Retain the assumptions of Proposition 4.5.8, including the special form of α1. Then
B Q ζ s1, s2; ´∆pt1, t2, uq, p´1q αΓ “ 0. Bs2 ˇps1,s2q“p0,0q ˇ ` ˘ ˇ A similar statementˇ holds with the roles of 1 and 2 reversed.
Proof. This follows from Proposition 4.5.8 and Lemma 4.2.3.
76 4.6 A closed form for multi-torsion
In this section we will introduce the closed form ωMT and interpret multi-torsion as
the integral of ωMT over a surface in the space of metrics.
The form ωMT makes sense in a somewhat more general setting than that in which we defined multi-torsion. What is essential is that there is a sufficiently nice decomposition of d, d˚, ∆, and δM. More precisely, we make the following assump- tions: Let M be a compact manifold of dimension n. (In the setting of Section 4.3,
Q M “ MΓ and n “ n1 ` n2.) Let pE, dq be an elliptic complex on M, and let p´1q be the grading operator as usual. We assume that d admits a decomposition
d “ d1 ` ¨ ¨ ¨ ` dm,
Q such that for all i, j, di and dj commute and di anticommutes with p´1q . This implies that for any metric on E, we have a decomposition of the associated d˚:
˚ ˚ ˚ d “ d1 ` ¨ ¨ ¨ ` dm.
We assume that our space of allowable metrics M satisfies the following: for any
˚ ˚ ‹ metric in M, the associated di ’s satisfy that for all i, j, di and dj commute and that
‹ if i ‰ j, di and dj commute. Then we also have the decomposition of the Laplacian
m ˚ ˚ ˚ ∆ “ k“1 ∆k, where ∆k :“ dkdk ` dkdk. dk and dk commute with ∆ and therefore ´1 withř functions of ∆, in particular the resolvent R “ Rz “ pz ´ ∆q . Furthermore, we require that the M-exterior derivative δM decomposes as
M M M δ “ δ1 ` ¨ ¨ ¨ δm
M ˚ such that δi dj “ 0 for i ‰ j.
Fix a basepoint metric h0 P M and a volume form vol0; let x¨, ¨y0 be the induced L2-inner product. As in Subsection 3.4.1, let β P Ω0pM,Bq be the operator defined
77 ´1 M by, for h P M, x¨, ¨yh “ x¨, βh¨y0. The 1-form b “ β δ β now decomposes as
b “ b1 ` ¨ ¨ ¨ ` bm,
´1 M M ˚ M ˚ ˚ where bj “ β δj β. We obtain that δ dj “ δj dj “ rdj , bjs. We assume that bj
˚ commutes with di and di for i ‰ j, and that each bj is symmetric in the sense of Definition 2.3.2. (Thus we assume that a generalization of Lemma 3.4.6 holds. We need not assume this in the geometric setting of Section 4.3, since in that setting this fact holds automatically for the same reason that Lemma 3.4.6 holds.)
We will now introduce ωMT andω ˜MT , which are m-forms that generalize the one-forms ωT andω ˜T , respectively, of Chapter3.
´1 Recall that Rz denotes the resolvent p∆ ´ zq . Fix an integer N such that porder ∆qN ą n, where n is the dimension of the base manifold M. (order ∆ denotes the order of ∆ as a differential operator; in Chapter3, we referred to this integer as m, but in this chapter, we reserve m for the degree of the forms ωMT andω ˜MT .) Then
N ´porder ∆qN by Remark 2.3.3, Rz P Ψ is trace-class, which ensures the operator-valued forms introduced below are trace-class. (See also Remark 3.5.2.) We must now introduce some rather cumbersome combinatorics and associated notation. Let CN,m denote the finite set of compositions (i.e., ordered partitions) of the positive integer N into m positive integer summands; i.e, CN,m is the set of multiindices I “ pi1, . . . , imq such that each ij is a strictly positive integer and
N “ i1 `¨ ¨ ¨`im. We will view Sm´1 as the permutation group of the set t2, . . . , mu.
For σ P Sm´1, I “ pi1, . . . , imq P CN,m, and z a complex number not in the spectrum of ∆, let Rz,σ,I be defined by
i1 i2 im Rz,σ,I :“ Rz b1 Rz bσp2q ¨ ¨ ¨ Rz bσpmq.
Let Tσ,I and T σ,I be defined by Q Tσ,I :“ sign σ Trp´1q Rz,σ,I ,
Q T σ,I :“ sign σ Tr p´1q Rz,σ,I .
78 (Tσ,I and T σ,I depend on z, but we suppress this. Note that Tσ,I is not necessarily
equal to the complex conjugate of T σ,I .)
Our m-formω ˜MT is defined by
1 1 ω˜MT :“ Tσ,I ` T σ,I , |CN,m|pm ´ 1q! 2 σPSm´1,IPCN,m ÿ ` ˘
where |CN,m| denotes the number of elements in the finite set CN,m, so that the
normalizing constant |CN,m|pm´1q! is the number of summands in the sum. (pm´1q!
is the number of elements in Sm´1; we have not bothered to compute |CN,m|.) Again, we suppress the dependence on z. We have symmetrized in a sense over all orderings of the factors 2, . . . , m; we could have also chosen to symmetrize over all orderings of all the factors 1, . . . , m, which would have been equivalent by the graded cyclicity of the trace.
ωMT is defined in terms ofω ˜MT via the following contour integral:
1 1 ω :“ e´z ω˜ dz, (4.31) MT pN ´ 1q! 2πi MT żC where C is a contour in C enclosing r0, 8q. We remark that there is not an obvious simpler formula for ωMT involving the heat operator (analogous to Definition 3.4.7) because b1, . . . , bm do not commute with the resolvent Rz. But in the most important special case, the contour integral in (4.31) is easy to compute, giving a heat operator; see (4.32) below. The main result of this section is
Theorem 4.6.1. Under the assumptions above, ω˜MT and ωMT are closed on M.
The next section will be dedicated to proving thatω ˜MT is closed, which suffices to prove Theorem 4.6.1. First, though, we remark on the significance of ωMT in the context of multi-torsion:
79 We note that the geometric setting of Section 4.3 satisfies the assumptions above,
with m “ 2 and with M “ Mprod. In that setting, we have a bigrading of E:
E Eq1,q2 . Let h h h be a metric on E. Let hq1,q2 be the restriction of h “ q1,q2 “ 1 ˆ 2
q1,q2 h h h to E À and let ∆ “ ∆1 `∆2 be the associated Laplacian. For t1, t2 ą 0, let ht1,t2 be
q1,q2 q1 q2 q1,q2 q1 q2 q1,q2 the metric whose restriction to E is t1 t2 h , so that βpt1, t2q “ t1 t2 on E .
Then we see that for j “ 1, 2, we have, on the surface Σh in Mprod parametrized by
pt1, t2q ÞÑ ht1,t2 ,
dtj bj “ qj tj
˚ ˚ dj ptjq “ tjd phq
h h ∆pt1, t2q “ t1∆1 ` t2∆2
Pulled back to Σh, every summand in the sum in (4.6) is the same; i.e., we have for every σ, I that
1 Q h h ´N dt1 dt2 Tσ,I ` T σ,I “ Trp´1q Q1Q2pt1∆1 ` t2∆2 ´ zq , 2 t1 t2 ` ˘ which follows from the graded cyclicity of the trace, the adjoint-trace relation of
Lemma 2.3.5, and the fact that Q1 and Q2 commute with the resolvent Rz. Thus
the 2-formω ˜MT pulled back to Σh is
Q h h ´N dt1 dt2 ω˜MT “ Trp´1q Q1Q2 pt1∆1 ` t2∆2 ´ zq . t1 t2
Performing the contour integral gives in turn that ωMT pulled back to Σh is
h h dt dt Q ´pt1∆ `t2∆ q 1 2 ωMT “ Trp´1q Q1Q2 e 1 2 . (4.32) t1 t2
Thus we have proven the following generalization of Lemma 3.4.9:
Lemma 4.6.2. For Re s1, Re s2 large,
1 ζps , s ; ∆hpt , t q, p´1qQQ Q q “ ts1 ts2 ω . 1 2 1 2 1 2 Γps qΓps q 1 2 MT 1 2 żΣh 80 1 2 Using that Γpsq “ s ` Ops q as s Ñ 0 and Lemma 4.6.2, and recalling Definition
4.4.6 of the multi-torsion MT ph1, h2q, we have shown that
AC s1 s2 MT ph1, h2q “ t1 t2 ωMT , Σ ˆż h ˙ˇps1,s2q“p0,0q ˇ ˇ where the superscript AC indicates that we must analyticallyˇ continue the function in parentheses to the origin. Formally, setting s1 “ s2 “ 0 on the right-hand side leaves
ωMT ; this is purely formal because the regularization is necessary for the integral Σh şto converge. This suggests the following heuristic interpretation of multi-torsion, generalizing our interpretation of analytic torsion:
The multi-torsion MT phq may be interpreted as a regularized integral of
ωMT over the surface Σh in the space of metrics Mprod.
4.7 Proof of closedness
This section is dedicated to the proof of Theorem 4.6.1, which we restate for conve- nience:
Theorem 4.7.1. Under the assumptions above, ω˜MT and ωMT are closed on M.
M Without loss of generality, it suffices to prove that δ1 ω˜MT “ 0, which we will do. Recall that M ˚ ˚ ˚ ˚ δ1 ∆ “ d1d1 b1 ´ d1b1d1 ` d1 b1d1 ´ b1d1 d1
M ˚ ˚ ˚ ˚ ˚ pδ1 ∆q “ b1d1d1 ´ d1b1d1 ` d1 b1d1 ´ d1 d1b1
We will find the following lemma, which generalizes Lemma 3.5.9, useful:
Lemma 4.7.2. Consider operator-valued forms A, B, and C. Assume that d1 and
˚ d1 commute with each of A, B, and C, and that the forms below are trace-class. Then we have the identity
Q M ˚ Q M Trp´1q Ab1Bpδ1 ∆q C “ Trp´1q Apδ1 ∆qBb1C. 81 ˚ Proof. We use the cyclicity of the trace and that d1 and d1 commute with A, B, and C and anticommute with p´1qQ to compute:
Q ˚ Q ˚ Trp´1q Ab1Bpb1d1d1 qC “ Trp´1q Apd1d1 b1qBb1C
Q ˚ Q ˚ Trp´1q Ab1Bp´d1 d1b1qC “ Trp´1q Ap´b1d1 d1qBb1C
Q ˚ ˚ Q ˚ ˚ Trp´1q Ab1Bp´d1b1d1 ` d1 b1d1qC “ Trp´1q Apd1 b1d1 ´ d1b1d1 qBb1C
Summing the three identities above gives the result.
We will now compute the derivatives of Tσ,I and T σ,I in the 1-direction:
M Q i1 i2 im δ1 pTσ,I q “ ´ sign σ Trp´1q Rz b1b1Rz bσp2q ...Rz bσpmq
i1 Q α M i1´α`1 i2 im ` sign σ Trp´1q Rz pδ1 ∆qRz b1Rz bσp2q ¨ ¨ ¨ Rz bσpmq α“1 ÿ m ij
` Tσ,I;j,β j“2 β“1 ÿ ÿ where
j´1 Q i1 β M ij ´β`1 im Tσ,I;j,β :“ p´1q sign σ Trp´1q Rz b1 ¨ ¨ ¨ bσpk´1qRz pδ1 ∆qRz bσpjq ¨ ¨ ¨ Rz bσpmq.
Similarly,
M Q i1 i2 im δ1 pT σ,I q “ ´ sign σTr p´1q Rz b1b1Rz bσp2q ...Rz bσpmq
i1 Q α M i1´α`1 i2 im ` sign σTr p´1q Rz pδ1 ∆qRz b1Rz bσp2q ¨ ¨ ¨ Rz bσpmq α“1 ÿ m ij
` T σ,I;j,β j“2 β“1 ÿ ÿ where
j´1 Q i1 β M ij ´β`1 im T σ,I;j,β :“ p´1q sign σTr p´1q Rz b1 ¨ ¨ ¨ bσpk´1qRz pδ1 ∆qRz bσpjq ¨ ¨ ¨ Rz bσpmq.
M We will now show δ1 ω˜MT “ 0 via several lemmas. 82 Lemma 4.7.3.
Q i1 i2 im sign σ Trp´1q Rz b1b1Rz bσp2q ¨ ¨ ¨ Rz bσpmq σPS ,IPC m´ÿ1 N,m
Q i1 i2 im ` sign σTr p´1q Rz b1b1Rz bσp2q ¨ ¨ ¨ Rz bσpmq “ 0. σPS ,IPC m´ÿ1 N,m Proof.
Q i1 i2 im sign σ Trp´1q Rz b1b1Rz bσp2q ¨ ¨ ¨ Rz bσpmq
Q i1 i2 im ˚ “ sign σTr p´1q Rz b1b1Rz bσp2q ¨ ¨ ¨ Rz bσpmq
1 i i ` 2 mpm`1q im 2 ˘ 1 Q “ sign σp´1q Tr bσpmqRz ¨ ¨ ¨ bσp2qRz b1b1Rz p´1q
1 i i 2 mpm`1q Q 2 1 im “ sign σp´1q Tr p´1q Rz b1b1Rz bσpmqRz ¨ ¨ ¨ bσp2q.
Let r be the permutation reversing the order of p2, . . . , mq. Note that sign r “
1 pm´2qpm´1q 1` 1 mpm`1q 1` 1 mpm`1q p´1q 2 “ p´1q 2 , so that signpr ˝ σq “ p´1q 2 sign σ. Sum-
ming over σ P Sm´1 and I P CN,m, we obtain the lemma.
Lemma 4.7.4.
i1 Q α M i1´α`1 i2 im sign σ Trp´1q Rz pδ1 ∆qRz b1Rz bσp2q ¨ ¨ ¨ Rz bσpmq σPS ,IPC α“1 m´ÿ1 N,m ÿ
i1 Q α M i1´α`1 i2 im ` sign σTr p´1q Rz pδ1 ∆qRz b1Rz bσp2q ¨ ¨ ¨ Rz bσpmq “ 0 σPS ,IPC α“1 m´ÿ1 N,m ÿ Proof. We compute:
Q α M i1´α`1 i2 im sign σ Trp´1q Rz pδ1 ∆qRz b1Rz bσp2q ¨ ¨ ¨ Rz bσpmq
Q α M i1´α`1 i2 im ˚ “ sign σTr p´1q Rz pδ1 ∆qRz b1Rz bσp2q ¨ ¨ ¨ Rz bσpmq
1 i i α 1 ` 2 mpm`1q im 2 1´ ` M ˘ ˚ α Q “ sign σp´1q Tr bσpmqRz ¨ ¨ ¨ bσp2qRz b1Rz pδ1 ∆q Rz p´1q
1 i i α 1 2 mpm`1q im 2 M 1´ ` α Q “ sign σp´1q Tr bσpmqRz ¨ ¨ ¨ bσp2qRz pδ1 ∆qRz b1Rz p´1q
1 i i α 1 2 mpm`1q Q 2 M 1´ ` α im “ sign σp´1q Tr p´1q Rz pδ1 ∆qRz b1Rz bσpmqRz ¨ ¨ ¨ bσp2q, which proves the lemma by an argument similar to the previous lemma.
83 Lemma 4.7.5. For each j (2 ď j ď m),
ij
Tσ,I;j,β ` T σ,I;j,β “ 0.
σPSm´1,IPCN,m β“1 ÿ ÿ ` ˘
Proof. We compute
j´1 Tσ,I;j,β “ p´1q sign σ
Q i1 β M ij ´β`1 im Trp´1q Rz b1 ¨ ¨ ¨ bσpj´1qRz pδ1 ∆qRz bσpjq ¨ ¨ ¨ Rz bσpmq
“ p´1qj´1 sign σ
Q i1 β M ij ´β`1 im ˚ Tr p´1q Rz b1 ¨ ¨ ¨ bσpj´1qRz pδ1 ∆qRz bσpjq ¨ ¨ ¨ Rz bσpmq
j´1 ` 1 mpm`1q ˘ “ p´1q sign σp´1q 2
im ij ´β`1 M ˚ β i1 Q Tr bσpmqRz ¨ ¨ ¨ bσpjqRz pδ1 ∆q Rz bσpj´1q ¨ ¨ ¨ b1Rz p´1q
j´1 1 mpm`1q “ p´1q sign σp´1q 2
im ij ´β`1 β i2 M i1 Q Tr bσpmqRz ¨ ¨ ¨ bσpjqRz b1Rz bσpj´1q ¨ ¨ ¨ b2Rz pδ1 ∆qRz p´1q
j´1 1 mpm`1q`jpm´j`1q “ p´1q sign σp´1q 2
Q ij ´β`1 β i2 M i1 im Tr p´1q Rz b1Rz bσpj´1q ¨ ¨ ¨ b2Rz pδ1 ∆qRz bσpmqRz ¨ ¨ ¨ bσpjq.
Summing over σ P Sm´1, I P CN,m, and β P t1, . . . , iju proves the lemma.
M The previous three lemmas prove that δ1 ω˜MT “ 0. The same argument shows
M that δk ω˜MT “ 0 for all k. Thus we have proven Theorem 4.6.1.
4.8 Proof of metric independence
We will now prove Theorem 4.4.8 concerning the metric independence of multi-torsion under appropriate assumptions. We restate the theorem for convenience:
84 Theorem 4.8.1. For j “ 1, 2, if nj is odd and if for every γj P Γj, γj is either
orientation-preserving or has nondegenerate fixed points as a diffeomorphism of Mj,
then MT ph1, h2q is independent of the metric hj.
Proof. We will mimic the proof of Theorem 3.3.3 in Section 3.6. Again, the essential idea is to apply Stokes’ theorem. The hypotheses ensure that the boundary terms vanish.
Without loss of generality, we assume j “ 1. So we suppose that n1 is odd.
We fix a metric h2 on M2 and consider a curve h1puq, u P r0, 1s, of metrics on
M1. For 1, 2,A1,A2 ą 0, let B “ B1,2,A1,A2 be the cube in M parametrized by
pt1, t2, uq ÞÑ hpuqt1,t2 , for 1 ď t1 ď A1, 2 ď t2 ď A2, 0 ď u ď 1. ωMT pulled back to
BB, in the coordinates pt1, t2, uq, is Q Q Q ´∆pt1,t2,uq 1 2 ωMT “ Trp´1q e dt1dt2 t1 t2 β Q Q ´∆pt1,t2,uq ´1 B 1 2 ` Trp´1q e β1 dudt2. Bu t2
s1 s2 (Note that there is no dudt1 component.) δpt1 t2 ωMT q is
β Q s1 s2 s1´1 s2 Q ´∆pt1,t2,uq ´1 B 1 2 δpt1 t2 ωMT q “ s1t1 t2 Trp´1q e β1 dudt1dt2 Bu t2
Assume that Re s1, Re s2 are both large. We apply Stokes’ theorem on the cube B to obtain
M s1 s2 s1 s2 δ1 pt1 t2 ωMT q “ t1 t2 ωMT , (4.33) ¡B ijBB
The integral over B is
M s1 s2 δ1 pt1 t2 ωMT q (4.34) ¡B
1 A2 A1 Bβ1 Q2 “ s ts1´1ts2 Trp´1qQe´∆pt1,t2,uqβ´1 dt dt du. (4.35) 1 1 2 1 Bu t 1 2 ż0 ż2 ż1 2 85 The boundary BB is a cube with six faces, described by u “ 0, u “ 1, t1 “ 1, t1 “ A1, t2 “ 2, and t2 “ A2. Since ωMT vanishes on the latter two faces, the integral over BB consists of four terms:
A2 A1 Q1 Q2 ts1 ts2 Trp´1qQe´∆pt1,t2,1q dt dt pu “ 1q (4.36) 1 2 t t 1 2 ż2 ż1 1 2
A2 A1 Q1 Q2 ´ ts1 ts2 Trp´1qQe´∆pt1,t2,0q dt dt pu “ 0q (4.37) 1 2 t t 1 2 ż2 ż1 1 2
1 A2 Bβ1 Q2 ` s1 ts2 Trp´1qQe´∆p1,t2,uqβ´1 dt du pt “ q (4.38) 1 2 1 Bu t 2 1 1 ż0 ż2 2
1 A2 Bβ1 Q2 ` As1 ts2 Trp´1qQe´∆pA1,t2,uqβ´1 dt du pt “ A q (4.39) 1 2 1 Bu t 2 1 1 ż0 ż2 2
` For fixed 2 and A2, we will now study the limits 1 Ñ 0 and A1 Ñ 8. By the heat
` kernel estimates, the term (4.38) tends to 0 as 1 Ñ 0 , and the term (4.39) tends
to 0 as A1 Ñ 8. Thus from (4.33) we obtain
1 A2 8 Bβ1 Q2 s ts1´1ts2 Trp´1qQe´∆pt1,t2,uqβ´1 dudt dt 1 1 2 1 Bu t 2 1 ż0 ż2 ż0 2
A2 8 Q1 Q2 “ ts1 ts2 Trp´1qQe´∆pt1,t2,1q dt dt 1 2 t t 1 2 ż2 ż0 1 2
A2 8 Q1 Q2 ´ ts1 ts2 Trp´1qQe´∆pt1,t2,0q dt dt . 1 2 t t 1 2 ż2 ż0 1 2
` We may now take the limits 2 Ñ 0 and A2 Ñ 8 to obtain
1 8 8 Bβ1 Q2 s ts1´1ts2 Trp´1qQe´∆pt1,t2,uqβ´1 dudt dt 1 1 2 1 Bu t 2 1 ż0 ż0 ż0 2 8 8 Q1 Q2 “ ts1 ts2 Trp´1qQe´∆pt1,t2,1q dt dt 1 2 t t 1 2 ż0 ż0 1 2 8 8 Q1 Q2 ´ ts1 ts2 Trp´1qQe´∆pt1,t2,0q dt dt . 1 2 t t 1 2 ż0 ż0 1 2
86 Multiplying both sides by 1 gives that Γps1qΓps2q
1 Bβ s ζps , s ; ´∆pt , t , uq, p´1qQβ´1 1 Q qdu 1 1 2 1 2 1 Bu 2 ż0 Q Q “ ζps1, s2; ∆pt1, t2, 1q, p´1q Q1Q2q ´ ζps1, s2; ∆pt1, t2, 1q, p´1q Q1Q2q.
We have shown that the equality holds for Re s1 and Re s2 large, but both sides pos- sess unique meromorphic continuations to all of C2, so in fact the equality must hold everywhere for the meromorphic continuations. In particular, using the Definition 4.4.6 of multi-torsion, we have the following equality of derivatives at the origin:
2 1 B Q ´1 Bβ1 s1 ζps1, s2; ´∆pt1, t2, uq, p´1q β1 Q2qdu Bs1Bs2 Bu ˇps1,s2q“p0,0q ż0 ˇ ˇ “ MT pˇ1q ´ MT p0q.
The left-hand side is equal to
1 B Q ´1 Bβ1 ζps1, s2; ´∆pt1, t2, uq, p´1q β1 Q2qdu. Bs2 Bu ż0 ˇps1,s2q“p0,0q ˇ ˇ ˇ By Corollary 4.5.9, the integrand vanishes if n1 is odd. This proves MT p1q´MT p0q “ 0 and proves the theorem.
87 5
The eta invariant
In this chapter we study the eta invariant, a regularized signature of a certain square root of the Laplacian on manifolds of dimension congruent to 3 modulo 4. We show that the eta invariant allows for an interpretation similar to that for analytic torsion, as the integral of a closed one-form on the space of Riemannian metrics. We phrase the well-known fact that the relative eta invariant is metric independent in terms of essentially the same Stokes’ theorem argument that computes the dependence of analytic torsion on the metric. We consider only the eta invariant as defined by Atiyah-Patodi-Singer in the first of the series of papers [1], [2], [3], in which they studied the index theorem for manifolds with boundary.
5.1 The eta invariant
All the results of this section are due to Atiyah-Patodi-Singer [1].
Let M be a compact oriented manifold of dimension n “ 4k ´ 1. Let F Ñ M be a flat unitary vector bundle with flat connection ∇F and metric hF . Let Λ‚F “ F b Λ‚T ˚M be the bundle of F -valued forms.
88 Let MTM denote the space of Riemannian metrics on M i.e., metrics on the
tangent bundle TM Ñ M. A Riemannian metric g P MTM induces a metric, which in an abuse of notation we will also call g, on the exterior bundle Λ‚T ˚M. We will consider only metrics on Λ‚F that are induced as tensor products of the fixed metric hF and a Riemannian metric g. As usual, F ’s flat connection and the exterior derivative on M induce an operator
d on sections of Λ‚F . A metric g P MTM determines operators d˚ and ∆ and
(having fixed an orientation of M) a Hodge star operator ˚ “ ˚g, which extends to an endomorphism of Λ‚F .
Let p´1qQ be as usual. Let σ “ p´1qp on forms of degree 2p or 2p ´ 1, i.e., on q-forms,
1, q ” 0 pmod 4q ´1, q ” 1 pmod 4q σ “ $ ’´1, q ” 2 pmod 4q &’ 1, q ” 3 pmod 4q ’ %’ Let B be the first order operator
B :“ ˚dp´1qQ ´ d˚ σ. ` ˘ B is a self-adjoint operator and B2 “ ∆. Therefore B has real eigenvalues, and for each eigenvalue λ of B, λ2 is an eigenvalue of ∆; conversely, every eigenvalue of ∆ arises in this way.
Recall the definition (3.7) of the zeta function ζps; L, P q. The eta function ηps; B,P q is defined by
s ` 1 ηps; B,P q :“ ζ ; B2,PB . (5.1) 2 ˆ ˙
In the special case when P is the identity, we set
ηps; Bq :“ ηps; B,Iq. (5.2)
89 The eta invariant η “ ηpM, F, gq is defined by ηpM, F, gq :“ ηp0; Bpgqq, (5.3) where here our notation emphasizes the dependence of B on g. We must explain why ηps; Bq is finite at s “ 0. First, though, we make a couple of observations:
Remark 5.1.1. Note that by the definition of the zeta function, we have for Re s large that 8 1 s`1 ´tB2 dt ηps; B,P q “ t 2 Tr P Be . (5.4) Γ s`1 t 2 ż0 ` ˘ Since B annihilates harmonic forms, we need not project away from the harmonic forms, i.e., “Tr1” need not appear in (5.4).
Remark 5.1.2. The identity for Re s large 8 ´s 1 s`1 ´tλ2 dt signpλqλ “ t 2 λe Γ s`1 t 2 ż0 ` ˘ shows that for Re s large ηps; Bq “ signpλqλ´s, λ‰0 ÿ where the sum is over the nonzero eigenvalues of B. This shows that ηp0; Bq may be interpreted as a regularized signature of B, i.e., a regularized difference between the number of positive eigenvalues of B and the number of negative eigenvalues of B.
s`1 2 Lemma 5.1.3. ζ 2 ; ∆,P has a simple pole at s “ 0 whose residue is 1 a´1{2, Γp 2 q
` ˘ ´1{2 ´t∆ where a´1{2 is the coefficient on t in the t Ñ 0 asymptotic expansion of Tr P e .
Thus a priori, ηps; Bq has a simple pole at s “ 0 whose residue is the coefficient
´ 1 ´tB2 on t 2 in the t Ñ 0 asymptotic expansion of Tr Be . But in fact, by a theorem of Atiyah-Patodi-Singer [1], that coefficient vanishes, and furthermore, Tr Be´tB2 is
1 1 2 O t , so that ηpsq is holomorphic for s ą ´ 2 . This ensures that the definition ´ ¯ (5.3) of the eta invariant makes sense.
90 Remark 5.1.4. In fact, for our purposes, we do not need to assume the aforementioned theorem of Atiyah-Patodi-Singer; rather, we could simply take the definition of the eta invariant to be the finite part of ηps; Bq at s “ 0 (i.e., ignore the potential pole).
The eta invariant is not a topological invariant, but we do have the following result of Atiyah-Patodi-Singer [1], a proof of which we will give in Section 5.4 using the closed form introduced in Section 5.2:
Theorem 5.1.5. For a smooth one-parameter family of Riemannian metrics gpuq, the derivative of the eta invariant is d 2 ηpM, F, gpuqq “ a pgpuqq, (5.5) du 1 1{2 Γ 2 ` ˘ ´1{2 where a´1{2pgpuqq denotes the t coefficient in the t Ñ 0 asymptotic expansion of
´1 d˚ ´t∆gpuq Tr ˚ d, ˚ du e . “ ‰ In particular, the variation of the eta invariant is a locally computable quantity. This implies the following, which states that a relative version of the eta invariant, also known as the rho invariant, is topological:
Corollary 5.1.6. Let ηpM, Cr, gq denote the eta invariant associated to the trivial flat rank r bundle with total space M ˆ Cr. Let F be a rank r unitary flat bundle.
Then the relative eta invariant ηrelpM, F, gq defined by
r ηrelpM, F, gq :“ ηpM, F, gq ´ ηpM, C , gq (5.6) is independent of the metric g P MTM .
Proof. The coefficient a1{2 is the integral over M of an integrand fpxq that depends only on the local geometry near x P M. The bundle F is locally isometric to the
r trivial bundle M ˆ C of the same rank, so their coefficients a1{2 agree. This fact d and Theorem 5.1.5 imply that du ηrelpM, F, gpuqq “ 0, proving the claim. 91 5.2 A closed form for the eta invariant
In this section we will interpret the eta invariant as an integral on the space of Riemannian metrics MTM . Henceforth, δ will denote the exterior derivative on MTM .
First, we will rephrase the formula for ηps; Bq in a more convenient way. Note that ˚d maps q-forms to p4k ´ 2 ´ qq-forms, implying that the only contribution to Tr ˚dp´1qQe´t∆ is from p2k ´ 1q-forms; similarly, d˚ maps q-forms to p4k ´ qq-forms, implying that the only contribution to Tr d ˚ e´t∆ is from p2kq-forms. Furthermore, conjugation by ˚ (recall that ˚´1 “ ˚ since the dimension n is odd) gives the equality of traces Tr ˚dp´1qQe´t∆2k´1 “ ´ Tr ˚de´t∆2k´1
“ ´ Tr ˚ ˚ de´t∆2k´1 ˚
“ ´ Tr d ˚ e´t∆2k
“ ´ Tr d ˚ e´t∆.
This implies 1 ´ Tr Be´t∆ “ Tr ˚de´t∆ 2
“ Tr d ˚ e´t∆ so that we may rewrite (5.4) (with P “ I) as the following identity, for Re s large: 8 ´2 s`1 ´t∆ dt ηps; Bq “ t 2 Tr ˚de , (5.7) Γ s`1 t 2 ż0
s`1 ` ˘ i.e., ηps; Bq “ ´2ζ 2 ; ∆, ˚d . Let b “ ˚´1δ˚,` a “tautological”˘ one-form that is a special case of the form b of Chapter 3.
TM Definition 5.2.1. Let ωη be the following one-form on M : Tr ˚rd, bse´∆
92 In the next section, we will prove that ωη is closed.
TM We will now explain the relevance of ωη to the eta invariant. Let g P M . Let
TM 1 Cg be the curve in M parametrized by, for 0 ă t ă 8, t ÞÑ t g. We compute that,
after pulling back to Cg, we have
Q´ n ˚ptq “ t 2 ˚g
n dt b “ Q ´ I 2 t ´ ¯ dt rd, bs “ ´d t ∆ptq “ t∆g, implying
´ 1 ´t∆g ωη “ ´t 2 Tr ˚gd e 2k´1 dt
Using (5.4), we have
2 s ηpsq “ t 2 ω . Γ s`1 η 2 żCh ` ˘ Thus we have the interpretation:
2 The eta invariant ηpgq may be interpreted as Γp1{2q times the (regularized)
TM integral of ωη over the curve Cg in the space of Riemannian metrics M .
In fact, the theorem of Atiyah-Patodi-Singer [1] shows that the regularization is not necessary, i.e., the eta invariant is literally equal to the convergent integral
2 ωη. Γp1{2q Ch ş 5.3 Proof of closedness
This section is devoted to a proof of
TM Theorem 5.3.1. ωη is closed on M . 93 As with the form associated to analytic torsion, to prove this theorem, we will
´1 work not directly with the heat operator but with the resolvent Rz “ p∆ ´ zq . Fix
an integer N so that 2N ´ 1 ą n, which ensures thatω ˜η, defined by
N ω˜η “ Tr ˚rd, bsRz ,
is trace-class. By the Cauchy integral formula, ωη andω ˜η are related by the identity
´∆ ωη “ Tr ˚rd, bse
1 1 “ e´z ω˜ dz, 2πi pN ´ 1q! η żC
so to prove ωη is closed, it suffices to proveω ˜η is closed, i.e., δω˜η “ 0, which we will now do. Recall some facts:
δ˚ “ ˚b
δb “ ´bb
˚ δRz “ Rztd, rd , bsuRz
δrd, bs “ ´rd, bbs
t˚, bu “ 0
p˚dq˚ “ ˚d (on p2k ´ 1q-forms)
The trace identities from Chapter 2 will of course be useful. Because taking the adjoint introduces a complex conjugate, we will also need the following analogue of Lemma 3.5.5:
N Lemma 5.3.2. We may rewrite ω˜η “ Tr ˚rd, bsRz in either of the following equiva- lent ways:
N ω˜η “Tr ˚ rd, bsRz (5.8) 1 1 ω˜ “ Tr ˚rd, bsRN ` Tr ˚ rd, bsRN (5.9) η 2 z 2 z 94 Proof. The second equality follows from the first, so it suffices to prove the first. Note
that since ˚ anticommutes with b, ˚rd, bs may also be written as the anticommutator t˚d, bu, which is symmetric because both ˚d and b are symmetric (in the relevant degree). We now apply Lemma 2.3.5 relating the trace and the adjoint:
N N ˚ ω˜η “ Tr Tr ˚rd, bsRz “ Tr ˚rd, bsRz
` N ˘ “ Tr Rz ˚ rd, bs
N “ Tr ˚ rd, bsRz ,
where the last equality follows from the cyclicity of the trace.
Using the expression (5.9) forω ˜η, we compute
N N i ˚ j 2δω˜η “ Tr ˚brd, bsRz ´ Tr ˚rd, bbsRz ` Tr ˚rd, bsRztd, rd , bsuRz (5.10) i`j“N`1 ÿ N N i ˚ j ` Tr ˚ brd, bsRz ´ Tr ˚ rd, bbsRz ` Tr ˚ rd, bsRztd, rd , bsuRz (5.11) i`j“N`1 ÿ We expand the first term on the right-hand side of (5.10) and show cancellation with the corresponding term of line (5.11):
N N Tr ˚bdbRz “ ´ Tr b ˚ dbRz
N ˚ “ ´Tr pb ˚ dbRz q
N “ Tr Rz b ˚ db
N “ Tr b ˚ dbRz
N “ ´Tr ˚ bdbRz .
N N ´ Tr ˚bbdRz “ ´ Tr bb ˚ dRz
N ˚ “ ´Tr pbb ˚ dRz q
N “ Tr Rz bb ˚ d
N “ Tr bb ˚ dRz .
95 The second term on the right-hand side of line (5.10) vanishes:
N N ´ Tr ˚dbbRz ` Tr ˚bbdRz
N N “ ´ Tr bbRz ˚ d ` Tr ˚bbdRz
N N “ ´ Tr bb ˚ dRz ` Tr ˚bbdRz
N N “ ´ Tr ˚bbdRz ` Tr ˚bbdRz
“ 0.
The same argument shows that the second term on line (5.11) also vanishes. Now we have left to show that
i ˚ j i ˚ j Tr ˚rd, bsRztd, rd , bsuRz ` Tr ˚ rd, bsRztd, rd , bsuRz “ 0. i`j“N`1 ÿ ` ˘
In fact, we’ll show that the pi, jq-summand is zero for every i, j. Expanding the three (anti-)commutators in the “Tr” term of pi, jq-summand gives eight terms, three of which are trivially zero since d2 “ pd˚q2 “ 0:
i ˚ j ´ Tr ˚dbRzdbd Rz “ 0
i ˚ j ´ Tr ˚bdRzdrd , bsRz “ 0
The same argument shows that the corresponding three terms terms of the “Tr ” term vanish. Also, we have
i ˚ j i ˚ j ˚ Tr ˚dbRzdd bRz “ Tr p˚dbRzdd bRzq
j ˚ i “ ´Tr Rzbdd Rzb ˚ d
i ˚ j “ ´Tr ˚ dbRzdd bRz
“ 0.
96 We’re left with
i ˚ j i ˚ j Trt˚d, buRzrd , bsdRz “ Tr ˚dbRzd bdRz
i ˚ j ´ Tr ˚dbRzbd dRz
i ˚ j ` Tr b ˚ dRzd bdRz
i ˚ j ´ Tr b ˚ dRzbd dRz,
plus the corresponding terms from the “Tr ” term. The four terms on the right-hand
side we denote by Aij,Bij,Cij,Dij, respectively; the corresponding terms from the
“Tr ” term we denote by Aij, Bij, Cij, Dij, respectively. (Note that our notation is
not meant to suggest that Aij is the complex conjugate of Aij.) We compute
i ˚ j Bij “ ´ Tr ˚dbRzbd dRz
i ˚ j ˚ “ ´Tr p˚dbRzbd dRzq
j ˚ i “ Tr Rzd dbRzb ˚ d
j ˚ i “ ´Tr b ˚ dRzd dbRz
j ˚ i “ ´Tr bRzd d ˚ dbRz
i ˚ j “ Tr ˚ dbRzbd dRz
“ ´Bij where we’ve used that ˚dd˚ “ d˚d˚.
i ˚ j Dij “ ´ Tr b ˚ dRzbd dRz
i ˚ j ˚ “ ´Tr pb ˚ dRzbd dRzq
j ˚ i “ Tr Rzd dbRz ˚ db
i j ˚ “ Tr bRz ˚ dbRzd d
i ˚ j “ Tr b ˚ dRzbd dRz
“ ´Dij.
97 i ˚ j Aij “ Tr ˚dbRzd bdRz
i ˚ j ˚ “ Tr p˚dbRzd bdRzq
j ˚ i “ ´Tr Rzd bdRzb ˚ d
j ˚ i “ Tr b ˚ dRzd bdRz
“ Cji.
On the other hand,
j ˚ i Aij “ ´ Tr Rzd bdRzb ˚ d
j i ˚ “ ´ Tr RzbdRzb ˚ dd
j i ˚ “ ´ Tr RzbdRzbd d˚
j i ˚ “ ´ Tr ˚RzbdRzbd d
i ˚ j “ Tr b ˚ dRzbd dRz
“ ´Dij
The same arguments show Aij “ Cji “ ´Dij. Since Dij `Dij vanishes, so do Aij `Aij and Cji ` Cji. This concludes the proof of Theorem 5.3.1.
5.4 Proof of the metric variation formula
In this section, we will prove Theorem 5.1.5, which we restate for convenience:
Theorem 5.4.1. For a smooth one-parameter family of Riemannian metrics gpuq, the derivative of the eta invariant is d 2 ηpM, F, gpuqq “ a pgpuqq, (5.12) du 1 ´1{2 Γ 2 ` ˘ ´1{2 where a´1{2pgpuqq denotes the t coefficient in the t Ñ 0 asymptotic expansion of
´1 d˚ ´t∆gpuq Tr ˚ d, ˚ du e . “ ‰ 98 Proof. Suppose gpuq is a smooth family of metrics for u P r0, 1s. Our goal will be to prove
1 1 ηpM, F, gp1qq ´ ηpM, F, gp0qq “ a pgpuqq du, (5.13) 2Γ 1 1{2 2 ż0 ` ˘ which suffices to prove (5.12).
TM For A ą ą 0, let Σ “ ΣA, denote the surface with boundary in M
parametrized by pu, tq ÞÑ gpuqt, for u P r0, 1s and t P r, As . (We use the sub- script t as in Subsection 3.4.2.) The boundary of Σ consists of four curves, described
by u “ 0, u “ 1, t “ , and t “ A. We compute the following operators pulled back to Σ:
Q´ n ˚pu, tq “ t 2 ˚gpuq
b “ ˚´1δ˚
n dt d “ Q ´ I ` ˚´1 ˚ du 2 t gpuq du gpuq ´ ¯ dt d rd, bs “ ´d ` d, ˚´1 ˚ du t gpuq du gpuq „
This implies that ωη pulled back to BΣ is
1 ´t∆g dt ω “ ´ t 2 Tr ˚ d e 2k´1 η gpuq t
1 ´1 d ´t∆g ` t 2 Tr ˚ d, ˚ ˚ e 2k´1 du gpuq gpuq du gpuq „
M s Since ωη is closed, δ t 2 ωη pulled back to Σ is
M s `s s ´1˘ δ t 2 ω “ t 2 dt ω η 2 ` ˘ s s ´ 1 ´1 d ´t∆g “ t 2 t 2 Tr ˚ d, ˚ ˚ e 2k´1 dtdu. 2 gpuq gpuq du gpuq „
(Recall that only forms of degree 2k ´ 1 contribute to the trace.)
99 Assume Re s is large. We apply Stokes’ theorem to obtain
M s s δ t 2 ωη “ t 2 ωη. ij żBΣ Σ ` ˘
The integral over Σ is
M s δ t 2 ωη ij Σ ` ˘ 1 A s s ´ 1 ´1 d ´t∆g “ t 2 t 2 Tr ˚ d, ˚ ˚ e 2k´1 dtdu. 2 gpuq gpuq du gpuq ż0 ż „
The integral over BΣ consists of four terms:
s t 2 ωη żBΣ A s 1 ´t∆gp1q dt “ t 2 p´qt 2 Tr ˚ d e 2k´1 gp1q t ż A s 1 ´t∆gp0q dt ´ t 2 p´qt 2 Tr ˚ d e 2k´1 gp0q t ż 1 s 1 ´1 d ´∆g ` 2 2 Tr ˚ d, ˚ ˚ e 2k´1 du (5.14) gpuq gpuq du gpuq ż0 „ 1 s 1 ´1 d ´A∆g ´ A 2 A 2 Tr ˚ d, ˚ ˚ e 2k´1 du. (5.15) gpuq gpuq du gpuq ż0 „
In the limits Ñ 0 and A Ñ 8, the terms (5.14) and (5.15) vanish by the heat kernel estimates. Thus we obtain
1 8 s s ´ 1 ´1 d ´t∆g t 2 t 2 Tr ˚ d, ˚ ˚ e 2k´1 dtdu 2 gpuq gpuq du gpuq ż0 ż0 „ 8 s 1 ´t∆gp1q dt “ t 2 p´qt 2 Tr ˚ d e 2k´1 gp1q t ż0 8 s 1 ´t∆gp0q dt ´ t 2 p´qt 2 Tr ˚ d e 2k´1 . gp0q t ż0 100 s`1 Dividing by Γ 2 gives the equality
1 s s`` 1˘ d ζ ; ∆gpuq, ˚ d, ˚´1 ˚ du “ ηps; Bgp1qq ´ ηps; Bgp1qq. 2 2 gpuq gpuq du gpuq ż0 ˆ „ ˙
This equality holds for all s by the uniqueness of the analytic continuations.
We now evaluate at s “ 0. The right-hand side gives the difference in eta invari- ants ηpM, F, gp1qq ´ ηpM, F, gp0qq. The left-hand side gives, because of the factor of s, the residue at s “ 0 of the function
1 1 d ζ s; Bgpuq, ˚ d, ˚´1 ˚ du. (5.16) 2 gpuq gpuq du gpuq ż0 ˆ „ ˙
1 By Lemma 5.1.3, the residue of the integrand at s “ 0 is precisely Γp1{2q times the
1 d gpuq ´ 2 ´1 ˚ ´t∆ t coefficient in the small t asymptotic expansion of Tr ˚ d, ˚ du e . This proves (5.13). “ ‰
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103 Biography
Phillip Andreae was born December 9, 1988, in Knoxville, Tennessee. In 2006, he graduated from Bearden High School. In 2010, he earned a B.S. with highest honors in mathematics and physics from Emory University. In 2016, he earned a Ph.D. in mathematics from Duke University. Beginning in August 2016, Phillip will be an assistant professor of mathematics at Meredith College in Raleigh, North Carolina.
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