APS index theorem for even-dimensional manifolds with non-compact boundary
Maxim Braverman Pengshuai Shi



Department of Mathematics,
Northeastern University,
Boston, MA 02115,
USA

maximbraverman@neu.edu
www.math.neu.edu/ braverman/




Beijing International Center for Mathematical Research (BICMR),
Beijing (Peking) University,
Beijing 100871,
China

pengshuai.shi@gmail.com


[2010]58J28, 58J30, 58J32, 19K56

^††Partially supported by the Simons Foundation collaboration grant #G00005104.


APS index with non-compact boundary]An APS index theorem for even-dimensional manifolds with non-compact boundary






We study the index of  the APS boundary value problem for a strongly Callias-type operator  on a complete Riemannian manifold M. We use this index to define  
the relative η-invariant η(_1,_0) of two strongly Callias-type operators, which are equal outside of a compact set. Even though in our situation the η-invariants of _1 and _0 are not defined, the relative η-invariant behaves as if it were the difference η(_1)-η(_0). We also define the spectral flow of a family of such operators and use it to compute the variation of the relative η-invariant. 



[
    Pengshuai Shi
    August 24, 2017
===================









§ INTRODUCTION



In <cit.> we studied the index of the Atiyah–Patodi–Singer (APS) boundary value problem for a strongly  Callias-type operator on a complete odd-dimensional manifold with non-compact boundary. We used this index to define the relative η-invariant η(_1,_0) of two strongly Callias-type operators  on even-dimensional manifolds, assuming that _0 and _1 coincide outside of a compact set. 

In this paper we discuss an even-dimensional analogue of <cit.>. Many parts of the paper are parallel to the discussion in <cit.>. However, there are two important differences. First, the Atiyah–Singer integrand was, of course, equal to 0 in <cit.>, which simplified many formulas. In particular, the relative η-invariant was an integer. As opposed to it,  in the current paper the Atiyah–Singer integrand plays an important role and the relative η-invariant is a real number. More significantly, the proof of the main result in <cit.> was based on the application of the Callias index theorem, <cit.>. This theorem is not available in our current setting. Consequently, a completely different proof is proposed in Section <ref>.

Our results provide a new tool to study anomalies in quantum field theory. Mathematical description of many anomalies is given by index theorems for boundary value problems, cf. <cit.>, <cit.>. However, most mathematically rigorous descriptions of anomalies in the literature only work on compact manifolds. The results of the current paper allow to extend many of these descriptions to non-compact setting, thus providing a mathematically rigorous description of anomalies in more realistic physical situations. In particular, Bär and Strohmaier, in <cit.>, gave a mathematically rigorous description of chiral anomaly  by considering an APS boundary problem for Dirac operator on a Lorentzian spatially compact manifold. In a recent preprint <cit.> the first author extended the results of <cit.> to spatially non-compact case. The analysis in <cit.> depends heavily on the results of the current paper. Another applications to the anomaly considered in <cit.> will appear in <cit.>. 



We now briefly describe our main results. 





 §.§ Strongly Callias-type operators


A Callias-type operator on a complete Riemannian manifold M is an operator of the form = D+Ψ where D is a Dirac operator and Ψ is a self-adjoint potential which anticommutes with the Clifford multiplication and satisfies certain growth conditions at infinity. In this paper we impose  slightly stronger growth conditions on Ψ and refer to the obtained operator  as a strongly Callias-type operator. Our conditions on the growth of Ψ guarantee that the spectrum of  is discrete.  


The Callias-type index theorem, proven in different forms in  <cit.>, computes the index of a Callias-type operator on a complete odd-dimensional manifold as the index of a certain operator induced by  on a compact hypersurface. Several generalizations and applications of the Callias-type index theorem were obtained recently in <cit.>. 

P. Shi, <cit.>, proved a version of the Callias-type index theorem for the  APS  boundary value problem for Callias-type operators on a complete odd-dimensional manifold with compact boundary. 

Fox and Haskell <cit.> studied Callias-type operators on manifolds with non-compact boundary.  Under rather strong conditions on the geometry of the manifold and the operator  they proved a version of the Atiyah–Patodi–Singer index theorem. 

In <cit.> we studied  the index of the APS boundary value problem on an arbitrary complete odd-dimensional manifold with non-compact boundary. In the current paper, we obtain an even-dimensional analogue of <cit.>. 







 §.§ An almost compact essential support


A manifold C, whose boundary is a disjoint union of two complete manifolds N_0 and N_1, is called essentially cylindrical if outside of a compact set it is isometric to a cylinder [0,ε]× N, where N is a non-compact manifold. It follows that manifolds N_0 and N_1 are isometric outside of a compact set. 

We say that an essentially cylindrical manifold M_1, which contains ,  is an almost compact essential support of  if the restriction of ^* to M∖ M_1 is strictly positive and the restriction of  to the cylinder [0,ε]× N is a product, cf. Definition <ref>. Every strongly Callias-type operator on M which is a product near  has an almost compact essential support. 

Theorem <ref> states that the index of the APS boundary value problem for a strongly Callias-type operator  on a complete manifold M is equal to the index of the APS boundary value problem of the restriction of  to its almost compact essential support M_1. 





 §.§ Index on an essentially cylindrical manifold


Let M be an essentially cylindrical manifold and let  be a strongly Callias-type operator on M, whose restriction to the cylinder [0,ε]× M is a product. Suppose = N_0⊔ N_1 and denote the restrictions of  to N_0 and N_1 by _0 and -_1 respectively (the sign convention means that we think  of  N_0 as the “left boundary” and of N_1 as the “right boundary” of M). Let _B denote the operator  with APS boundary conditions. 

Let () denote the Atiyah–Singer integrand of . This is a differential form on M which depends on the geometry of the manifold and the bundle. Since all structures are product outside of the compact set K, this form vanishes outside of K. Hence, ∫_M() is well-defined and finite. Our main result here is that 

    _B -  ∫_M ()

depends only on the operators _0 and _1 and does  not depend  on the interior of the manifold M and the restriction of  to the interior  of M, cf. Theorem <ref>.  





 §.§ The relative η-invariant

Suppose now that _0 and _1 are self-adjoint strongly Callias-type operators on complete manifolds N_0 and N_1. An almost compact cobordism between _0 and _1 is a pair (M,) where M is an essentially cylindrical manifold with =N_0⊔ N_1 and  is a strongly Callias-type operator on M, whose restriction to the cylindrical part of M is a product and such that the restrictions of  to N_0 and N_1 are equal to _0 and -_1 respectively. We say that _0 and _1 are cobordant if there exists an almost compact cobordism between them. In particular, this implies that  _0 and _1 are equal outside of a compact set. 

Let   be an almost compact cobordism between _0 and _1. Let B_0 and B_1 be the APS boundary conditions for  at N_0 and N_1 respectively. Let _B_0⊕ B_1 denote the index of the APS boundary value problem for .  We define the relative η-invariant by the formula

    η(_1,_0)  =   
    	2 ( _B_0⊕ B_1-∫_M () )
    	  +  _0 +  _1.

It follows from the result of the previous subsection that η(_1,_0) is independent of the choice of an almost compact cobordism.  

If M is a compact manifold, then the Atiayh-Patodi-Singer index theorem <cit.> implies that η(_1,_0)= η(_1)-η(_0). In general, for non-compact manifolds, the individual η-invariants η(_1) and η(_0) might not be defined. However, η(_1,_0)  behaves like it were a difference of two individual η-invariants. In particular, cf. Propositions <ref>–<ref>,  

    η(_1,_0) =  - η(_0,_1), 
    	    η(_2,_0) =  η(_2,_1) +  η(_1,_0).


Under rather strong conditions on the manifolds N_0 and N_1 and on the operators _0, _1,  Fox and Haskell <cit.> showed that the  heat kernel of _j (j=0,1) has a nice asymptotic expansion similar to the one  for operators on compact manifolds. Then they were able to define the  individual η-invariants η(_j) (j=0,1). In this situation, as expected, our relative η-invariant is equal to the difference of the individual η-invariants:  η(_1,_0)= η(_1)-η(_0). 

Under much weaker (but still quite strong) assumptions,  Müller, <cit.>, suggested a definition of a relative η-invariant based on analysis of the relative heat kernel <cit.>. This invariant behaves very similar to our η(_1,_0). The precise conditions under which these two invariants are equal are not clear yet. We note that our invariant is defined on a much wider class of manifolds, where the relative heat kernel is not of trace class and can not be used to construct an η-invariant.




 §.§ The spectral flow


Consider a family Å= {^s}_0≤ s≤ 1 of self-adjoint strongly Callias-type operators  on a complete Riemannian manifold. We assume that there is a compact set K⊂ M such that the restriction of ^s to M∖ K is independent of s. Since the spectrum of ^s is discrete for all s, the spectral flow (Å) can be defined in a more or less usual way. By Theorem <ref>, if _0 is a self-adjoint strongly Callias-type operator which is cobordant to ^0 (and hence, to all ^s), then

    η(^1,_0) -  η(^0,_0)
    	 -  ∫_0^1 (d/ds(^s,_0)) ds
    	 =  2 (Å).

The derivative d/ds(^s,_0) can be computed as an integral of the transgression form — a differential form canonically constructed from the symbol of ^s and its derivative with respect to s. Thus (<ref>) expresses the change of the relative η-invariant as a sum of 2(Å) and a local differential geometric expression.






§ BOUNDARY VALUE PROBLEMS FOR CALLIAS-TYPE OPERATORS

In this section we recall some results about boundary value problems for Callias-type  operators on manifolds with non-compact boundary,  <cit.>, keeping in mind applications to even-dimensional case. The operators considered here are slightly more general than those discussed in <cit.>, but all the definitions and most of the properties of the boundary value problems remain the same. 





 §.§ Self-adjoint strongly Callias-type operators


Let M be a complete Riemannian manifold (possibly with boundary) and let E→ M be a Dirac bundle over M, cf. <cit.>. In particular, E is a Hermitian bundle endowed with a Clifford multiplication c:T^*M→(E) and a compatible Hermitian connection ^E.  Let D:C^∞(M,E)→ C^∞(M,E) be the Dirac operator defined by the connection ^E. Let Ψ∈ End(E) be a self-adjoint bundle map (called a Callias potential). Then

    := D+Ψ

is a formally self-adjoint Dirac-type operator on E and

    ^2 = D^2+Ψ^2+[D,Ψ]_+,

where [D,Ψ]_+:=D∘Ψ+Ψ∘ D is the anticommutator of the operators D and Ψ.




We call  a self-adjoint strongly Callias-type operator if


  * [D,Ψ]_+ is a zeroth order differential operator, i.e. a bundle map;

  * for any R>0, there exists a compact subset K_R⊂ M such that

    Ψ^2(x) - |[D,Ψ]_+(x)| ≥ R

for all x∈ M∖ K_R.  In this case, the compact set K_R is called an R-essential support of , or an essential support when we do not need to stress the associated constant.




Condition (i) of Definition <ref> is equivalent to the condition that Ψ anticommutes with the Clifford multiplication: [c(ξ),Ψ]_+ = 0, for all ξ∈ T^*M.






 §.§ Graded self-adjoint strongly Callias-type operators


Suppose now that E=E^+⊕ E^- is a _2-graded Dirac bundle such that the Clifford multiplication c(ξ) is odd and the Clifford connection is even with respect to this grading. Then

    D := (
    
    0     D^- 
    
    D^+     0 
    )

is the _2-graded Dirac operator, where
D^± : C^∞(M,E^±)→ C^∞(M,E^∓)
are formally adjoint to each other. Assume that the Callias potential Ψ has odd grading degree, i.e., 

    Ψ = (
    
    0    Ψ^- 
    Ψ^+     0 
    ),

where Ψ^±∈(E^±,E^∓) are adjoint to each other. Then we have

    = D + Ψ = (
    	
    	0     D^-+Ψ^- 
    
    	D^++Ψ^+     0
    	) =: (
    	
    	0    ^- 
    ^+     0
    	).




Under the same condition as in Definition <ref>,  is called a graded self-adjoint strongly Callias-type operator. In this case, we also call ^+ and ^- strongly Callias-type operators. They are formally adjoint to each other. By an R-essential support (or essential support) of ^± we understand as an R-essential support (or essential support) of . 




When M is an oriented even-dimensional manifold there is a natural grading of E induced by the volume form. We will consider this situation in the next section.




Suppose there is a skew-adjoint isomorphism γ:E^±→ E^∓, γ^*=-γ, which anticommutes with multiplication c(ξ) for all ξ∈ T^*M, satisfies γ^2=-1, and is flat with respect to the connection ^E, i.e. [^E,γ]=0. Then ξ↦γ∘ c(ξ) defines a Clifford multiplication of T^*M on E^+ and the corresponding Dirac operator is D̃^+= γ∘ D^+. Suppose also that γ commutes with Ψ. Then Φ^+=-iγ∘Ψ^+ is a self-adjoint endomorphism of E^+. In this situation, 

    D̃^++iΦ^+ =  γ∘^+:  
    	C^∞(M,E^+) → C^∞(M,E^+)
 
is a strongly Callias-type operator in the sense of <cit.>.







 §.§ Restriction to the boundary


Assume that the Riemannian metric g^M is product near the boundary, that is, there exists a neighborhood U⊂ M of the boundary which is isometric to the cylinder

    Z_r :=  [0,r)×.
 
In the following we identify U with Z_r and denote by t the coordinate along the axis of Z_r. Then the inward unit normal one-form to the boundary is given by τ = dt. 

Furthermore, we assume that the Dirac bundle E is product near the boundary. In other words we assume that the Clifford multiplication c:T^*M→ End(E) and the connection ∇^E have product structure on Z_r, cf. <cit.>.

Let D be a _2-graded Dirac operator. In this situation the restriction of D to Z_r takes the form 

    D  =  c(τ)(_t+Â)=
    	(
    	
    	0     c(τ) 
    
    	c(τ)     0
    	)
    	(
    	_t+A     0 
    
    	0    _t+A^♯),

where

    A: C^∞(,E^+_) →  C^∞(,E^+_)

and 

    A^♯= c(τ)∘ A∘ c(τ):  
    	C^∞(,E^-_) →  C^∞(,E^-_)

are formally self-adjoint operators acting on the restrictions of E^± to the boundary.




It would be more natural to use the notation A^+ and A^- instead of A and A^♯. But since in the future we only deal with the operator A:C^∞(,E^+_)→ C^∞(,E^+_) we remove the superscript “+” to simplify the notation. 




Let =D+Ψ be a graded self-adjoint strongly Callias-type operator. Then the restriction of  to Z_r is given by 

    =  c(τ)(_t+) = (
    	
    	0     c(τ) 
    
    	c(τ)     0
    	)
    	(
    	_t+    0 
    
    	0    _t+^♯),

where

    :=  A-c(τ)Ψ^+: C^∞(,E^+_)
    		 → C^∞(,E^+_).

and ^♯ = A^♯-c(τ)Ψ^-. 
By Remark <ref>, c(τ)Ψ^±∈ End(E^±_) are self-adjoint bundle maps. Therefore  and ^♯ are formally self-adjoint operators. In fact, they are strongly Callias-type operators, cf. Lemma 3.12 of <cit.>.  In particular, they have discrete spectrum. Also,

    ^♯ =   c(τ)∘∘ c(τ).




We say that a graded self-adjoint strongly Callias-type operator  is product near the boundary if the Dirac bundle E is  product near the boundary and the restriction of the Callias potential Ψ to Z_r does not depend on t. The operator  (resp. ^♯)  of (<ref>) is called the restriction of ^+ (resp. ^-) to the boundary. 







 §.§ Sobolev spaces on the boundary


Consider a graded  self-adjoint  strongly Callias-type operator :C^∞(M,E)→ C^∞(M,E), cf. (<ref>). The restriction of ^+ to the boundary is a self-adjoint strongly Callias-type operator

    : C^∞(,E_^+) → C^∞(,E_^+).
 
We recall the definition of Sobolev spaces H^s_(,E^+_) of sections over  which depend on the boundary operator , cf. <cit.>.





Set

    C_^∞(,E_^+)	 := { ∈ C^∞(,E_^+): (𝕀+^2)^s/2_L^2(,E_)^2<+∞s∈ }.

For all s∈ we define the Sobolev H_^s-norm on C_^∞(,E_^+) by

    _H_^s(,E_)^2 := (𝕀+^2)^s/2_L^2(,E_)^2.

The Sobolev space H_^s(,E_^+) is defined to be the completion of C_^∞(,E_^+) with respect to this norm.






 §.§ Generalized APS boundary conditions


The eigensections of  belong to H_^s(,E_^+) for all s∈, cf. <cit.>. For I⊂ we denote by 

    H_I^s() ⊂  H_^s(,E_^+)
 
the span of the eigensections of  whose eigenvalues belong to I. 




For any a∈, the subspace

    B = B(a) := H_(-∞,a)^1/2().

is called the the generalized Atiyah–Patodi–Singer boundary conditions for ^+. If a=0, then the space B(0)= H_(-∞,0)^1/2() is called the Atiyah–Patodi–Singer (APS) boundary condition. 

The spaces (a) := H_(-∞,a]^1/2()  and  (0) := H_(-∞,0]^1/2() are called the dual generalized APS boundary conditions and the dual APS boundary conditions respectively.

The space B^=B^(a):=  H_(-∞,-a]^1/2(^♯) is called the adjoint of the generalized APS boundary condition for ^+. One can see that it is a dual generalized APS boundary condition for ^-.




If B is a generalized APS boundary condition for ^+, we denote by ^+_B  the operator ^+ with domain

    ^+_B := {u∈^+_max: u|_∈ B},

where ^+_max denotes the domain of the maximal extension of ^+ (cf. <cit.>). 
We refer to ^+_B  as the generalized APS boundary value problem for ^+. 



Recall that ^- is the formal adjoint of ^+. It is shown in Example 4.9 of <cit.> that the  L^2-adjoint of ^+_B(a)  is given by ^- with the dual APS boundary condition B^(a):

    (^+_B(a))^ =  ^-_B^(a).






Suppose that a graded strongly Callias-type operator (<ref>) is product near . Then the operator ^+_B: ^+_B→ L^2(M,E^-) is Fredholm. In particular, it has finite dimensional kernel and cokernel.


For the case discussed in Remark <ref> this is proven in Theorem 5.4 of <cit.>. Exactly the same proof works in the general case.






 §.§ The index of generalized APS boundary value problems


By (<ref>) the cokernel of ^+_B(a) is isomorphic to the kernel of ^-_B^(a). 




Let ^+ be a strongly Callias-type operator on a complete Riemannian manifold M which is product near the boundary. Let B= H_(-∞,a)^1/2() be a generalized APS boundary condition for ^+ and let B^= H_(-∞,-a]^1/2(^♯) be the adjoint of the generalized APS boundary condition. The integer 

    ^+_B := ^+_B-(^-)_B^ ∈

is called the index of the boundary value problem ^+_B. 



It follows directly from (<ref>) that 

    (^-)_B^ =  -^+_B.








 §.§ More general boundary value conditions

Generalized APS and dual generalized APS boundary conditions are examples of elliptic boundary conditions, <cit.>. In this paper we don't work with general elliptic boundary conditions. However, in Section <ref> we need a slight modification of APS boundary conditions, which we define now. 

 

We say that two closed subspaces X_1, X_2 of a Hilbert space H are finite rank perturbations  of each other if there exists a finite dimensional subspace Y⊂ H such that X_2⊂ X_1⊕ Y and the quotient space (X_1⊕ Y)/X_2 has finite dimension. 

The relative index of X_1 and X_2 is defined by 

    [X_1,X_2]  :=    (X_1⊕ Y)/X_2 -   Y.



One easily sees that the relative index is independent of the choice of Y. We  shall need the following analogue of <cit.>:



Let  be a graded self-adjoint strongly Callias-type operator on M and let B be a generalized APS or dual generalized APS boundary condition for ^+.  If B_1⊂ H^1/2_(,E^+_) is a finite rank perturbation of B, then the operator ^+_B_1 is Fredholm and  

    ^+_B -  ^+_B_1 =  [B,B_1].


The proof of the proposition is a verbatim repetition of the proof of <cit.>.

As an immediate consequence of Proposition <ref> we obtain the following


Let  be the restriction of ^+ to  and let  B_0= H^1/2_(-∞,0)() and _0= H^1/2_(-∞,0]() be the APS and the dual APS boundary conditions respectively. Then 

    ^+__0 =  ^+_B_0 +  .








 §.§ The splitting theorem


Let M be a complete manifold. Let N⊂ M be a hypersurface disjoint from  such that cutting M along N we obtain a manifold M' (connected or not) with  and two copies of N as boundary. So we can write M'=(M∖ N)⊔ N_1⊔ N_2.

Let E=E^+⊕ E^-→ M be a _2-graded Dirac bundle over M and ^±:C^∞(M,E^±)→ C^∞(M,E^∓) be strongly Callias-type operators as in Subsection <ref>. They induce _2-graded Dirac bundle E'=(E')^+⊕(E')^-→ M' and strongly Callias-type operators 

    (')^±:C^∞(M',(E')^±) → C^∞(M',(E')^∓)

on M'. We assume that all structures are product near N_1 and N_2. Let  be the restriction of (')^+ to N_1. Then - is the restriction of (')^+ to N_2 and, thus, the restriction of (')^+ to N_1⊔ N_2 is '=⊕(-). The following Splitting Theorem is an analogue of Theorem 5.11 of <cit.> with the same proof.



Suppose M,^+,M',(')^+ are as above. Let B_0 be a generalized APS boundary condition on . Let B_1=H_(-∞,0)^1/2() and B_2= H_(-∞,0]^1/2(-)= H_[0,∞)^1/2() be the APS and the dual APS boundary conditions for (')^+ along N_1 and N_2, respectively. Then (')^+_B_0⊕ B_1⊕ B_2 is a Fredholm operator and

    ^+_B_0 = (')^+_B_0⊕ B_1⊕ B_2.







 §.§ Reduction of the index to an essentially cylindrical manifold


The study of the index of Callias-type operators on manifolds without boundary can be reduced to a computation on the essential support. For manifolds with boundary we want an analogous subset, but the one which contains the boundary. Such a set is necessarily non-compact, but we want it to be “similar to a compact set". In <cit.>, we introduce the notion of essentially cylindrical manifolds, which replace the role of compact subsets in our study of boundary value problems.



An essentially cylindrical manifold  C is a complete Riemannian manifold whose boundary is a union of two disjoint manifolds, ∂ C=N_0⊔ N_1, such that 


  * there exist a compact set K⊂ C, an open Riemannian manifold N, and an isometry C∖ K≃ [0,δ]× N;


  * under the above isometry N_0∖ K ={0}× N and N_1∖ K ={δ}× N. 

See Figure 1 for an example of essentially cylindrical manifolds.




[height=5cm]Fig1.pdf
(98,40)N_0
(98,8)N_1
(47,23)K


Figure 1. An essentially cylindrical manifold  C







Let ^+ be a strongly Callias-type operator on M. An almost compact essential support of  ^+ is a smooth submanifold M_1⊂ M  with smooth boundary, which contains  and such that there exist a (compact) essential support K⊂ M and ε∈ (0,r) such that 

    M_1∖ K  =  (∖ K)× [0,ε] 
    	⊂ Z_r.



An almost compact essential support is a special type of essentially cylindrical manifolds. It is shown in <cit.> that for any strongly Callias-type operator which is product near  there exists an  almost compact essential support. Figure 2 illustrates the idea of building an almost compact essential support.



[height=8cm]Fig2.pdf
(-5,47)
(34,47)N_1
(15,36)K
(15,-3)M_1
(50,-4)M




Figure 2. The idea to obtain an almost compact essential support M_1







 §.§ The index on an almost compact essential support

Suppose M_1⊂ M is an almost compact essential support of ^+. Set N_0:= and let N_1⊂ M be such that _1=N_0⊔ N_1 as in Definition <ref>. The restriction of  to a neighborhood of N_1 need not be product. Since in this paper we only consider boundary value problems for operators which are product near the boundary, we first deform  to a product form. It is shown in <cit.> that there exists a perturbation of all the structures such that 


  * the new structures are product near N_1. In particular, the corresponding Callias-type operator  ' is product near N_1.

  * ' has a compact essential support, which is contained in M_1.

  * The difference -' vanishes near  and outside of a compact subset of M_1. In this situation we say that ' (or (')^±) is  a compact perturbation of  (or ^±). 


Let M_1⊂ M be an almost compact essential support of ^+. Let (')^+ be a compact perturbation of ^+ which is product near the boundary (cf. <cit.>). Let  be the restriction of ^+ to . It is also the restriction of (')^+. We denote by -_1 the restriction of (')^+ to N_1. Theorem 6.10 of <cit.> claims that the index of an elliptic boundary value problem on M can be reduced to an index on an almost compact essential support:




Suppose M_1⊂ M is an almost compact essential support of ^+ and let _1=⊔ N_1.  Let (')^+ be a compact perturbation of ^+ which is product near N_1 and  such that there is a compact essential support for (')^+ which is contained in M_1. Let B_0 be a generalized APS boundary condition for ^+. View (')^+ as an operator on M_1 and let 

    B_1  =  H_(-∞,0)^1/2(-_1)  =  
    	H_(0,∞)^1/2(_1)
 
be the APS boundary condition for (')^+ at N_1. Then

    ^+_B_0 =   (')^+_B_0⊕ B_1.









§ THE INDEX OF OPERATORS ON ESSENTIALLY CYLINDRICAL MANIFOLDS


In this section we discuss the index of strongly Callias-type operators on even-dimensional essentially cylindrical manifolds. It is parallel to <cit.>, where the odd-dimensional case was considered. 

From now on we assume that M is an oriented even-dimensional essentially cylindrical manifold whose boundary = N_0⊔ N_1 is a disjoint union of two non-compact manifolds N_0 and N_1. Let E be a Dirac bundle over M. As pointed out in Remark <ref>, there is a natural _2-grading E=E^+⊕ E^- on E. Let ^+: C^∞(M,E^+)→ C^∞(M,E^-) be a strongly Callias-type operator as in Definition <ref> (these data might or might not come as a restriction of another operator to its almost compact essential support. In particular, we don't assume that the restriction of ^+ to N_1 is invertible). Let _0 and -_1 be the restrictions of ^+ to N_0 and N_1 respectively. 

We first recall some definitions from <cit.>.




 §.§ Compatible data


Let M be an essentially cylindrical manifold and let =N_0⊔ N_1. As usual, we identify a tubular neighborhood of  with the product 

    Z_r :=  (  N_0×[0,r) ) ⊔ (  N_1×[0,r) )
    	 ⊂ M.





We say that another essentially cylindrical manifold M' is compatible with  M if there is a fixed isometry between Z_r and a neighborhood Z_r'⊂ M' of the boundary of M'. 


Note that if M and M' are compatible then their boundaries are isometric. 


Let M and M' be compatible essentially cylindrical manifolds and let Z_r and Z_r' be as above. Let E→ M be a _2-graded Dirac bundle over M and let  ^+: C^∞(M,E^+)→ C^∞(M,E^-)  be a strongly Callias-type operator whose restriction to Z_r is product and such that M is an almost compact essential support of ^+.  This  means that there is a compact set K⊂ M such that M∖ K= [0,ε]× N and the restriction of ^+ to M∖ K is product (i.e.  is given by (<ref>)). Let 
 E'→ M' be a _2-graded Dirac bundle over M' and let (')^+: C^∞(M',(E')^+)→ C^∞(M',(E')^-) be a strongly Callias-type operator, whose restriction to  Z_r' is product and such that  M' is an almost compact essential support of (')^+. 




In the situation discussed above we say that ^+ and (')^+ are compatible if there is an isomorphism E|_Z_r≃ E'|_Z_r' of graded Dirac bundles which identifies the restriction of ^+ to Z_r with the restriction of (')^+ to Z_r'.  



Let _0 and -_1 be the restrictions of ^+ to N_0 and N_1 respectively (the sign convention means that we think  of  N_0 as the “left boundary” and of N_1 as the “right boundary” of M). 
Let B_0=H^1/2_(-∞,0)(_0) and B_1=H^1/2_(-∞,0)(-_1)= H^1/2_(0,∞)(_1) be the APS boundary conditions for ^+ at N_0 and N_1 respectively. Since ^+ and (')^+ are equal near the boundary, B_0 and B_1 are also APS boundary conditions for (')^+.

We denote by α_ AS(^+) the Atiyah–Singer integrand of ^+. It can be written as

    α_ AS(^+) :=  (2π i)^- M Â(M)  ch(E/S)

where Â(M)  and ch(E/S) are  the differential forms representing the Â-genus of M and the relative Chern character of E, cf.  <cit.>. Note that α_ AS(^+) depends only on the metric on M and the Clifford multiplication on E and thus is independent of the potential Ψ.


Since outside of a compact set K, M and E are product, the interior multiplication by / t annihilates  . Hence, the top degree component of  vanishes on M∖ K. We conclude that the integral  ∫_Mα_ AS(^+) is well-defined and finite. Similarly, ∫_M'α_ AS((')^+) is well-defined and finite.




Suppose ^+ is a strongly Callias-type operator on an oriented even-dimensional essentially cylindrical manifold M such that M is an almost compact essential support of  ^+. Suppose that the operator  (')^+ is compatible with ^+. Let =N_0⊔ N_1 and let B_0= H^1/2_(-∞,0)(_0) and B_1= H^1/2_(-∞,0)(-_1)= H^1/2_(0,∞)(_1) be the APS boundary conditions for ^+ (and, hence, for (')^+) at  N_0 and N_1 respectively. Then 

    ^+_B_0⊕ B_1 - ∫_Mα_ AS(^+) 
    	 =   (')^+_B_0⊕ B_1 - ∫_M'α_ AS((')^+).

In particular, ^+_B_0⊕ B_1-∫_Mα_ AS(^+) depends only on the restrictions _0 and -_1 of ^+ to the boundary.


The rest of the section is devoted  to the  proof of this theorem.




In <cit.> the odd dimensional version of Theorem <ref> was considered. Of course, in this case  vanishes identically and Theorem 7.5 of <cit.> states that the indexes of compatible operators are equal. The proof in <cit.> is based on an application of a Callias-type index theorem and can not be adjusted to our current situation. Consequently, a completely different proof is proposed below.




 §.§ Gluing the data together


We follow <cit.> to glue  M with M' and ^+ with (')^+.

Let -M' denote  the  manifold M' with the opposite orientation. We identify a neighborhood of the boundary of -M' with the product 

    -Z_r' :=  (  N_0×(-r,0] ) ⊔ (  N_1×(-r,0] )

and consider the union 

    M̃ :=  M∪_N_0⊔ N_1 (-M').

Then Z_(-r,r):= Z_r∪ (-Z_r') is a subset of M̃ identified with the product 

    (  N_0×(-r,r) ) ⊔ (  N_1×(-r,r) ).

We note that M̃ is a complete Riemannian manifold without boundary. 
 

Let E_=E^+_⊕ E^-_ denote the restriction of E=E^+⊕ E^- to . The product structure on E|_Z_r gives a grading-respecting isomorphism ψ:E|_Z_r→ [0,r)× E_. Recall that we identified Z_r with Z_r' and fixed an isomorphism between the restrictions of E to Z_r and E' to Z_r'. By a slight abuse of notation we use this isomorphism to view ψ also as an isomorphism E'|_Z'_r→ [0,r)× E_.


Let Ẽ→M̃ be the vector bundle over M̃ obtained by gluing E and E' using the isomorphism c(τ):E|_→ E'|_'. This means that we fix isomorphisms 

    ϕ: Ẽ|_M→ E,   ϕ': Ẽ|_M'→ E',

so that 

    ψ∘ϕ∘ψ^-1    =  𝕀: [0,r)× E_ → [0,r)× E_, 
    ψ∘ϕ'∘ψ^-1    =  
    		1× c(τ): [0,r)× E_ → [0,r)× E_.

Note that the grading of E is preserved while the grading of E' is reversed in this gluing process. Therefore Ẽ=Ẽ^+⊕Ẽ^- is a _2-graded bundle.

We denote by c':T^*M'→(E') the Clifford multiplication on E' and set c”(ξ):= -c'(ξ). Then Ẽ is a Dirac bundle over M̃ with the Clifford multiplication

    c̃(ξ) : =     c(ξ),     ξ∈ T^*M;
       c”(ξ)=-c'(ξ),     ξ∈ T^*M'.

One readily checks that (<ref>) defines a smooth odd-graded Clifford multiplication on Ẽ. Let D̃:C^∞(M̃,Ẽ)→ C^∞(M̃,Ẽ) be the _2-graded Dirac operator. Then the isomorphism ϕ of (<ref>) identifies the restriction of D̃^± with D^±, the isomorphism ϕ' identifies the restriction of D̃^± with -(D')^∓, and isomorphism ψ∘ϕ'∘ψ^-1 identifies the restriction of D̃^± to -Z_r' with

    D̃^±|_Z_r' =  - c'(τ)∘ (D')^±_Z'_r∘ c'(τ)^-1.
 

Let (Ψ')^± denote the Callias potentials of (')^±, so that (')^±=(D')^±+(Ψ')^±. Consider the bundle maps Ψ̃^±∈ (Ẽ^±,Ẽ^∓) whose restrictions to M are equal to Ψ^± and whose restrictions to M' are equal to -(Ψ')^∓. The two pieces fit well on Z_(-r,r) by Remark <ref>. To sum up the constructions presented in this subsection, we have



The operators ^± :=  D̃^± +Ψ̃^± are strongly Callias-type operators on M̃, formally adjoint to each other, whose restrictions to M are equal to ^± and whose restrictions to M' are equal to -(D')^∓-(Ψ')^∓ = -(')^∓.


The operator ^+ is a strongly Callias-type operator on a complete Riemannian manifold without boundary. Hence, <cit.>, it is Fredholm. We again denote by α_ AS(^+) the Atiyah–Singer integrand of ^+. It is explained in the paragraph before Theorem <ref> that the integral ∫_M̃α_ AS(^+) is well defined.



^+=∫_M̃α_ AS(^+). 




Since M̃ is a union of two essentially cylindrical manifolds, there exists a compact essential support K̃⊂M̃ of  such that M̃∖K̃ is of the form S^1× N. We can choose  K̃ to be large enough so that the restriction of  to a neighborhood W of  M̃∖K̃≃ S^1× N is a product of an operator on N and an operator on S^1. Then the restriction of  to this neighborhood vanishes. We can also assume that K̃ has a smooth boundary Σ=S^1× L. 

Let ^+ be a compact perturbation of ^+ (cf. Subsection <ref>) in W which is product both near Σ and on W and whose essential support is contained in K̃. Then 

    ^+  =  ^+.

We cut  along Σ and apply the Splitting Theorem <ref>[Since Σ is compact we can also use the splitting theorem  for compact hypersurfaces,   <cit.>.] to get

    ^+
    	 = ^+_K̃ + ^+_∖K̃,

where ^+_K̃ stands for the index of the restriction of ^+ to K̃ with APS boundary condition, and ^+_∖K̃ stands for the index of the restriction of ^+ to ∖K̃ with the dual APS boundary condition.

Since ^+ has an empty essential support in ∖K̃, by the vanishing theorem <cit.>, the second summand in the right hand side of (<ref>) vanishes. The first summand in the right hand side of (<ref>) is given by the Atiyah–Patodi–Singer index theorem <cit.> (Note that  Σ is outside of an essential support of ^+ and, hence, the restriction of ^+ to Σ is invertible. Hence, the kernel of the restriction of ^+ to Σ is trivial) 

    ^+_K̃ = ∫_K̃α_ AS(^+) - 1/2η(0),

where η(0) is the η-invariant of the restriction of ^+ to Σ. 

As (^+)= (^+)≡ 0 on W and ^+≡^+ elsewhere, we have  

    ∫_K̃α_ AS(^+)  =  ∫_K̃α_ AS(^+)  =  ∫_α_ AS(^+).

To finish the proof of the lemma it suffices now to show η(0)=0.

Let ω be the inward (with respect to K̃) unit normal one-form along Σ. Recall that Σ=S^1× L. We denote the coordinate along S^1 by θ. Suppose that {ω,dθ,e_1,⋯,e_m} forms a local orthonormal frame of T^* on Σ. Then the restriction of ^+=D̂^++Ψ̂^+ to Σ can be written as

    ^+_Σ = -∑_i=1^mc̃(ω)c̃(e_i)∇_e_i^Ẽ-c̃(ω)c̃(dθ)∂_θ-c̃(ω)Ψ̂^+

which maps C^∞(Σ,Ẽ^+|_Σ) to itself. We define a unitary isomorphism Θ on the space C^∞(Σ,Ẽ|_Σ) given by

    Θ u(θ,y) := -c̃(ω)c̃(dθ)u(-θ,y).

One can check that Θ anticommutes with ^+_Σ. As a result, the spectrum of ^+_Σ is symmetric about 0. Therefore η(0)=0 and lemma is proved.






 §.§ Proof of Theorem <ref>


Recall that we denote by B_0 and B_1 the APS boundary conditions for ^+=^+|_M at N_0 and N_1 respectively. Let (”)^+ denote the restriction of ^+ to -M'=M̃∖ M. Let _0 and _1 be the dual APS boundary conditions for (”)^+ at N_0 and N_1 respectively. By the Splitting Theorem <ref>,

    ^+  =  ^+_B_0⊕ B_1 +   (”)^+__0⊕_1.

By Lemma <ref>, we obtain 

    ^+_B_0⊕ B_1 +   (”)^+__0⊕_1 = ∫_Mα_ AS(^+) + ∫_M'α_ AS((”)^+),

which means

    ^+_B_0⊕ B_1 -  ∫_Mα_ AS(^+)
    	 = 
    	- (”)^+__0⊕_1 + ∫_M'α_ AS((”)^+).


By Lemma <ref>, (”)^+= -(')^-. Thus _0⊕_1 is the adjoint of the APS boundary condition for (-')^+ (cf. Definition <ref>). Therefore,

    (”)^+__0⊕_1 =   (-')^-__0⊕_1 =  - (-')^+_B_0⊕ B_1 =  
    	-  (')^+_B_0⊕ B_1,

where we used (<ref>) in the middle equality. Also by the construction of local index density,

    α_ AS((”)^+) = α_ AS((-')^-)
    	 = α_ AS((')^-)
    	 =  -α_ AS((')^+).

Combining these equalities with (<ref>) we obtain (<ref>). □







§ THE RELATIVE Η-INVARIANT


In the previous section we proved that  on an essentially cylindrical manifold M the difference _B_0⊕ B_1- ∫_M ()  depends only on the restriction of  to the boundary, i.e., on the  operators _0 and -_1. In this section we use this fact to define the relative η-invariant η(_1,_0) and show that it has properties similar to the difference of η-invariants η(_1)-η(_0) of operators on compact manifolds. For special cases, <cit.>, when the index can be computed using heat kernel asymptotics, we show that  η(_1,_0) is indeed equal to the difference of the η-invariants of _1 and _0. In the next section we discuss the connection between the relative η-invariant and the spectral flow.

In the case when _0, _1 are operators on even-dimensional manifolds, an analogous construction  was proposed in <cit.>. Even though the definition of the relative η-invariant  for operators on odd-dimensional manifolds proposed in this section is slightly more involved than the definition in <cit.>, we show that most of the properties of η(_1,_0) remain the same. 




 §.§ Almost compact cobordisms

Let N_0 and N_1 be two complete odd-dimensional Riemannian manifolds and let _0 and _1 be self-adjoint strongly Callias-type operators on N_0 and N_1 respectively, cf. Definition <ref>. 




An almost compact cobordism between _0 and _1 is a pair (M,), where M is an essentially cylindrical manifold with =N_0⊔ N_1 and  is a graded self-adjoint strongly Callias-type operator on M such that 


  * M is an almost compact essential support of ;

  *  is product near ;

  * The restriction of ^+ to N_0 is equal to _0 and the restriction of ^+ to N_1 is equal to -_1.

If there exists an almost compact cobordism between _0 and _1 we say that operator _0 is cobordant to operator _1.


 

An almost compact cobordism is an equivalence relation on the set of self-adjoint strongly Callias-type operators, i.e.,


  *  If _0 is cobordant to _1 then _1 is cobordant to _0. 


  *  Let _0,_1 and _2 be self-adjoint strongly Callias-type operators on odd-dimensional complete Riemannian manifolds N_0, N_1 and N_2 respectively. 
Suppose _0 is cobordant to _1 and _1 is cobordant to _2. Then _0 is cobordant to _2.





The proof is a verbatim repetition of the proof of Lemmas 8.2 and 8.3 of <cit.>. 





Suppose _0 and _1 are cobordant self-adjoint strongly Callias-type operators and let (M,) be an almost compact cobordism between them. Let B_0= H_(-∞,0)^1/2(_0) and B_1= H_(-∞,0)^1/2(-_1) be the APS boundary conditions for ^+. The relative η-invariant is defined as 

    η(_1,_0)  =   2 ( ^+_B_0⊕ B_1
    	- ∫_M (^+) )
    	  +  _0 +  _1.


Theorem <ref>  implies that  η(_1,_0) is independent of the choice of the cobordism (M,). 



Sometimes it is  convenient to use the dual APS boundary conditions _0= H_(-∞,0]^1/2(_0) and _2= H_(-∞,0]^1/2(-_1) instead of B_0 and B_1. It follows from Corollary <ref> that the relative η-invariant can be written as

    η(_1,_0)  =   
    		2 ( ^+__0⊕_1- ∫_M (^+) )
    	  -  _0 -  _1.







 §.§ The case when the heat kernel has an asymptotic expansion

In <cit.>, Fox and Haskell studied the index of a boundary value problem on manifolds of bounded geometry. They showed that under certain conditions (satisfied for natural operators on manifolds with conical or cylindrical ends) on M and ,  the heat kernel e^-t(_B)^*_B is of trace class and its trace has an asymptotic expansion similar to the one on compact manifolds. In this case the η-function, defined by a usual formula 

    η(s;)  :=  ∑_λ∈ spec() sign(λ) |λ|^s, 
    		    Re s ≪ 0,
 
is an analytic function of s, which has a meromorphic continuation to the whole complex plane and is regular at 0. So one can define the η-invariant of  by η()= η(0;). 




Suppose now that  is an operator on an essentially cylindrical manifold M which satisfies the conditions of  <cit.>. We also assume that  is product near =N_0⊔ N_1 and that M is an almost compact essential support of . Let _0 and -_1 be the restrictions of ^+ to N_0 and N_1 respectively. Let η(_j) (j=0,1) be the η-invariant of _j.   Then 

    η(_1,_0)  =  η(_1) -  η(_0).





An analogue of this proposition for the case when M is odd is proven in <cit.>. This proof extends to the case when M is even without any changes. 







 §.§ Basic properties of the relative η-invariant



Proposition <ref> shows that under certain conditions the η-invariants of _0 and _1 are defined and η(_1,_0) is their difference. We now show that in general case, when η(_0) and η(_1) do not necessarily exist,  η(_1,_0) behaves like it were a difference of an invariant of N_1 and an invariant of N_0. 



	
Suppose _0 and _1 are cobordant self-adjoint strongly Callias-type operators. Then 

    η(_0,_1) =  - η(_1,_0).





Let -M denote the manifold M with the opposite orientation and let M̃:= M∪_(-M) denote the double of M. Let  be an almost compact cobordism between _0 and _1. Using the construction of  Section <ref> (with '=) we obtain a graded self-adjoint strongly Callias-type operator  on M̃ whose restriction to M is isometric to . Let ” denote the restriction of  to -M=M̃∖M. Then the restriction of (”)^+ to N_1 is equal to _1 and the restriction of (”)^+ to N_0 is equal to -_0.

Let   

    _0     =  H^1/2_[0,∞)(_0) =   H^1/2_(-∞,0](-_0),
    _1     =   H^1/2_[0,∞)(-_1)  =  H^1/2_(-∞,0](_1)
 
be the dual APS boundary conditions for (”)^+.  By (<ref>), 

    (”)^+__0⊕_1 - ∫_M'α_ AS((”)^+)
    			 = 			-^+_B_0⊕ B_1 +  ∫_Mα_ AS(^+).


Since ” is an almost compact cobordism between _1 and _0 we conclude  from (<ref>) that

    η(_0,_1) =  
    	2 (  (”)^+__0⊕_1- 
    	  	∫_M'α_ AS((”)^+) )
    	 -  _0 -  _1.

Combining  (<ref>) and (<ref>) we obtain (<ref>).


Note that (<ref>) implies that 

    η(,) =  0

for every self-adjoint strongly Callias-type operator . 



Let _0,_1 and _2 be self-adjoint strongly Callias-type operators which are cobordant to each other. Then 

    η(_2,_0) =  η(_2,_1) +  η(_1,_0).



Let M_1 and M_2 be essentially  cylindrical manifolds such that _1= N_0⊔ N_1 and _2= N_1⊔ N_2. Let _1 be an operator on M_1 which is an almost compact cobordism between _0 and _1. Let _2 be an operator on M_2 which is an almost compact cobordism between _1 and _2. Then the operator _3 on M_1∪_N_1 M_2 whose restriction to M_j (j=1,2) is equal to _j is an almost compact cobordism between _0 and _2. 

Let B_0 and B_1 be the APS boundary conditions for _1^+ at N_0 and N_1 respectively. Then _1= H^1/2_[0,∞)(_1) is equal to the dual APS boundary condition for _2^+. Let B_2 be the APS boundary condition for _2^+ at N_2. From Corollary <ref> we obtain 

    η(_2,_1)  =   
    		2 (  (_2^+)__1⊕ B_2- ∫_M (_2^+) )
    	  -  _1 +  _2.

By the Splitting Theorem <ref>

    (_3^+)_B_0⊕ B_2 =   (_1^+)_B_0⊕ B_1 +   (_2^+)__1⊕ B_2.

Clearly, 

    ∫_M_1∪ M_2(_3^+) =  ∫_M_1 (_1^+) +  ∫_M_2 (_2^+).

Combining (<ref>), (<ref>), and (<ref>) we obtain (<ref>). 







§ THE SPECTRAL FLOW


Suppose Å:= {^s}_0≤ s≤ 1 is a smooth family of self-adjoint elliptic operators on a closed  manifold N.  Let (^s)∈/ denote the mod  reduction of the η-invariant η(^s).  Atiyah, Patodi, and Singer, <cit.>, showed that s↦(^s) is a smooth function whose derivative d/ds(^s) is given by an explicit local formula. 
Further, Atiyah, Patodi and Singer, <cit.>, introduced a notion of spectral flow (Å) and showed that it  computes the net number of integer jumps of η(^s), i.e., 

    2 (Å) =  η(^1) -  η(^0) -  ∫_0^1 (d/ds(^s) )  ds.

In this section we consider a family of self-adjoint strongly Callias-type operators Å= {^s}_0≤ s≤ 1 on a complete  Riemannian manifold. Assuming that the restriction of ^s to a complement of a compact set K⊂ N is independent of  s, we show that for any operator _0 cobordant to ^0  the  mod   reduction (^s,_0) of the relative η-invariant depends smoothly on s and 

    2 (Å) =  η(^1,_0)- η(^0,_0) -  ∫_0^1 (d/ds(^s,_0) )  ds.


 





 §.§ A family of boundary operators

Let E_N→ N be a Dirac bundle over a complete odd-dimensional Riemannian manifold N. We denote the Clifford multiplication of T^*N on E_N by c_N:T^*N→(E_N). Let Å= {^s}_0≤ s≤ 1 be a family of self-adjoint strongly Callias-type operators ^s:C^∞(N,E_N) → C^∞(N,E_N).




The family Å= {^s}_0≤ s≤ 1 is called almost constant if there exists a compact set K⊂ N such that the restriction of ^s to N∖ K is independent of s. 
 


Consider the cylinder M:=[0,1]× N  and denote by t the coordinate along [0,1]. Set 

    E^+= E^- :=  [0,1]× E_N.

Then E=E^+⊕ E^-→ M is naturally a _2-graded Dirac bundle over M with 

    c(dt)  :=  [      0 -𝕀_E_N;  𝕀_E_N      0 ]

and 

    c(ξ) : =  [      0 c_N(ξ); c_N(ξ)      0 ],     for  ξ∈ T^*N.




The family Å= {^s}_0≤ s≤ 1 is called smooth if  

    : =  c(dt) ( _t+
    		[  ^t   0;   0 -^t ] ): 
    	C^∞(M,E)→ C^∞(M,E)

is a smooth differential operator on M. 



Fix a smooth non-decreasing function κ:[0,1]→ [0,1] such that κ(t)=0 for t≤1/3 and κ(t)=1 for t≥2/3 and consider the operator 

    : =  c(dt) ( _t+
    		[  ^κ(t)      0;      0 -^κ(t) ] ): C^∞(M,E)→ C^∞(M,E).

Then  is product near . If Å is a smooth almost constant family of self-adjoint strongly Callias-type operators then (<ref>) is a strongly Callias-type operator for which M is an almost compact essential support. Hence it is a non-compact cobordism (cf. Definition <ref>) between ^0 and ^1. 




 §.§ The spectral section


If Å= {^s}_0≤ s≤ 1 is a smooth almost constant family of self-adjoint strongly Callias-type operators then it satisfies the conditions of the Kato Selection Theorem <cit.>, <cit.>. Thus there is a family of eigenvalues λ_j(s) (j∈)  which depend continuously on s. We order the eigenvalues so that λ_j(0)≤λ_j+1(0) for all j∈ and λ_j(0)≤ 0 for j≤ 0 while λ_j(0)>0 for j>0. 

Atiyah, Patodi and Singer <cit.> defined the spectral flow (Å) for a family of operators satisfying the conditions of the Kato Selection Theorem  (<cit.>, <cit.>) as an integer that counts the net number of eigenvalues that change sign when s changes from 0 to 1. 
Several other equivalent definitions of the spectral flow based on different assumptions on the family Å exist in the literature. For our purposes the most convenient is the  Dai and Zhang's definition <cit.> which is based on the notion of spectral section introduced by Melrose and Piazza <cit.>.




A spectral section for Å is a continuous family = {P^s}_0≤ s≤ 1 of self-adjoint projections such that there exists a constant R>0 such that for all 0≤ s≤ 1, if ^su= λ u then

    P^su  =   
    		0,      if  λ< -R;
     
    		u,      if  λ> R.



If Å satisfies the conditions of the Kato Selection Theorem, then the arguments of the proof of  <cit.> show that Å admits a spectral section. 








 §.§ The spectral flow

Let = {P^s} be a spectral section for Å. Set B^s:=  P^s. Let B_0^s:= H^1/2_(-∞,0)(^s) denote the APS boundary condition defined by the boundary operator ^s.   Since the spectrum of ^s is discrete, it follows immediately from the definition of the spectral section that for every s∈[0,1] the space B^s is a finite rank perturbation of B_0^s, cf. Section <ref>. Recall that  the relative index [B^s,B_0^s] was defined in Definition <ref>. Following Dai and Zhang <cit.> (see also <cit.>) we give the following definition.




Let Å= {^s}_0≤ s≤ 1 be a smooth almost constant  family of self-adjoint strongly Callias-type operators which admits a spectral section = {P^s}_0≤ s≤ 1. Assume that the operators ^0 and ^1 are invertible. Let B^s:=  P^s and B_0^s:= H^1/2_(-∞,0)(^s). The spectral flow (Å) of the family Å is defined by the formula

    (Å) :=  [B^1,B_0^1]  -  [B^0,B^0_0].


By Theorem 1.4 of <cit.> the spectral flow is independent of the choice of the spectral section  and computes the net number of eigenvalues that change sign when s changes from 0 to 1.






Let -Å denote the family {-^s}_0≤ s≤ 1. Then 

    (-Å)  =  -(Å).



The lemma is an immediate consequence of Lemma 5.7 of <cit.>.





 §.§ Deformation of the relative η-invariant


Let Å= {^s}_0≤ s≤ 1 be a smooth almost constant family of self-adjoint strongly Callias-type operators on a complete odd-dimensional Riemannian manifold N_1. Let _0 be another self-adjoint strongly Callias-type operator, which is cobordant to ^0. In Section <ref> we showed that  ^0 is cobordant to ^s for all s∈ [0,1]. Hence, by Lemma <ref>.(ii), _0 is cobordant to ^1. In this situation we say the _0 is cobordant to the family Å. 




The following theorem is the main result of this  section.



Suppose 
Å  =  {^s: C^∞(N_1,E_1)→ C^∞(N_1,E_1)}_0≤ s≤ 1
is a smooth almost constant family of self-adjoint strongly Callias-type operators on a complete odd-dimensional Riemannian manifold N_1. Assume that ^0 and ^1 are invertible. Let _0:C^∞(N_0,E_0)→ C^∞(N_0,E_0) be an invertible self-adjoint strongly Callias-type operator on a complete Riemannian manifold N_0 which is cobordant to the family Å. Then the   mod  reduction (^s,_0)∈/  of the relative η-invariant depends smoothly on s∈[0,1] and 

    η(^1,_0) -  η(^0,_0)
    	 -  ∫_0^1 (d/ds(^s,_0)) ds
    	 =  2 (Å).



The proof of this theorem occupies Sections <ref>–<ref>.








 §.§ A family of almost compact cobordisms


Let M be an essentially cylindrical manifold whose boundary is the disjoint union of N_0 and N_1. First, we construct a smooth family ^r (0≤ r≤1) of graded self-adjoint strongly Callias-type operators on the manifold

    M' :=  M∪_N_1([0,1]× N_1 ),

such that for each r∈ [0,1] the pair (M',^r) is an almost compact cobordism between _0 and ^r. 

 



Let :C^∞(M,E)→ C^∞(M,E) be an almost compact cobordism between _0 and ^0. Let E_0 and E_1 denote the restrictions of E to N_0 and N_1 respectively. 

Let M' be given by (<ref>) and let E'→ M' be the bundle over M' whose restriction to M is equal to E and whose restriction to the cylinder [0,1]× N_1 is equal to [0,1]× E_1. 

We fix a smooth function ρ:[0,1]× [0,1]→ [0,1] such that for each r∈ [0,1]


  * the function s↦ρ(r,s) is non-decreasing.

  * ρ(r,s)=0 for s≤1/3 and ρ(r,s)= r for s≥2/3.

Consider the family of strongly Callias-type operators ^r:C^∞(M',E')→ C^∞(M',E') whose restriction to M is equal to  and whose restriction to [0,1]× N_1 is given by 

    ^r : =  c(dt) ( _t+
    		[  ^ρ(r,t)        0;        0 -^ρ(r,t) ] ).

Then ^r is an almost compact cobordism between _0 and ^r. In particular, the restriction of ^r to N_1 is equal to -^r. 

Recall that we denote by -Å the family {-^s}_0≤ s≤ 1.
Let = {P^s} be a spectral section for -Å. Then for each r∈ [0,1] the space  B^r:=  P^r is a finite rank perturbation of the APS boundary condition  for ^r at {1}× N_1. Let B_0:= H^1/2_(-∞,0)(_0) be the APS boundary condition for ^r at N_0. Then, by Proposition <ref>, the operator  ^r_B_0⊕ B^r is Fredholm. Recall that the domain ^r_B_0⊕ B^r consists of sections u whose restriction to '=N_0⊔ N_1 lies in B_0⊕ B^r. 




^r_B_0⊕ B^r= ^1_B_0⊕ B^1 for all r∈ [0,1]. 


For r_0,r∈ [0,1], let π_r_0r: B^r_0→ B^r
denote the orthogonal projection. Then for every r_0∈[0,1] there exists ε>0 such that if |r-r_0|<ε then π_r_0r is an isomorphism. As in the proofs of <cit.> and <cit.>, it induces an isomorphism

    Π_r_0r : ^r_0_B_0⊕ B^r_0 → ^r_B_0⊕ B^r.

Hence

    (^r_B_0⊕ B^r∘Π_r_0r) 
    		 =  ^r_B_0⊕ B^r.

Since for |r-r_0|<ε

    ^r_B_0⊕ B^r∘Π_r_0r: ^r_0_B_0⊕ B^r_0 → L^2(M',E')

is a continuous family of bounded operators, ^r_B_0⊕ B^r∘Π_r_0r is independent of r. The lemma follows now from (<ref>).






 §.§ Variation of the reduced relative η-invariant
	

By Definition <ref>, the mod  reduction of the relative η-invariant is given by 

    (^r,_0) :=  -2 ∫_M' (^r).
 
It follows that (^r,_0) depends smoothly on r and 

    d/dr(^r,_0) =  -2 ∫_M' d/dr(^r).

A more explicit local expression for the right hand side of this equation is given in Section <ref>. For the moment we just note that (<ref>) implies that

    ∫_0^1 (d/ds(^s,_0)) ds  =  
    	-2 ∫_M' ((^1)- (^0) ).

 




 §.§ Proof of Theorem <ref>

 Since the operators _0, ^0, and ^1 are invertible, we have 

    η(^j,_0)= 2 ( ^j_B_0⊕ B^j_0  -  ∫_M' (^j)
    	 ),      j=0,1.

Thus, using (<ref>), we obtain

    η(^1,_0)- η(^0,_0) -  ∫_0^1 (d/ds(^s,_0) )  ds 
     	 =  
     	2 ( ^1_B_0⊕ B^1_0 -  ^0_B_0⊕ B^0_0 ).


Recall that, by Proposition <ref>,

    ^r_B_0⊕ B^r  =  ^r_B_0⊕ B^r_0 +  [B^r,B^r_0],

where B^r=  P^r and B^r_0= H^1/2_(-∞,0)(^r) are defined in Subsection <ref>.
Hence, from (<ref>) we obtain

    1/2( η(^1,_0)- η(^0,_0) -  ∫_0^1 (d/ds(^s,_0) )  ds  )
    	
     =  ( ^1_B_0⊕ B^1- [B^1,B^1_0] )  - ( ^0_B_0⊕ B^0- [B^0,B^0_0] )
    	
    Lemma <ref>=	-[B^1,B^1_0] +  [B^0,B^0_0]  =  -(-Å) 
    	 Lemma <ref>=  (Å).

□





 §.§ A local formula for variation of the reduced relative η-invariant

It is well known that there exists a family of  differential forms β_r (0≤ r≤ 1), called the transgression form such that 

    dβ_r =  d/dr(^r).

The transgression form depends on the symbol of ^r and its derivatives with respect to r. For geometric Dirac operators one can write very explicit formulas for β_r. For example, if ^r is the signature operator (so that ^r is the odd signature operator) corresponding to a family ^r of flat connections on E,  then β_r= L(M)∧d/dr^r, where L(M) is the  L-genus of M, cf, for example, <cit.>. For general Dirac-type operators, a formula for β_r is more complicated, cf. <cit.>.

We note that since the family ^r is constant outside of the compact set K, the form β_r vanishes outside of K. Hence, ∫_'β_r is well defined and finite. Thus we obtain from (<ref>) that 

    d/dr(^r,_0)  =  -2∫_M' dβ_r =  
    	-2∫_' β_r  = 	2 (∫_{1}× N_1β_r  -  ∫_N_0 β_r).

Hence, (<ref>) expresses η(^1,_0)- η(^0,_0) as a sum of 2(Å) and a local differential geometric expression 2∫_'(∫_0^1β_r) dr.
 






amsplain




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