# Equivariant Spectral Flow and Equivariant $\eta$-invariants on Manifolds
with Boundary

###### Abstract.

In this article, we study several closely related invariants associated to
Dirac operators on odd-dimensional manifolds with boundary with an action of
the compact group $H$ of isometries. In particular, the equality between
equivariant winding numbers, equivariant spectral flow, and equivariant Maslov
indices is established. We also study equivariant $\eta$-invariants which play
a fundamental role in the equivariant analog of Getzler’s spectral flow
formula. As a consequence, we establish a relation between equivariant
$\eta$-invariants and equivariant Maslov triple indices in the splitting of
manifolds.

###### Key words and phrases:

Equivariant, spectral flow, $\eta$-invariant, Maslov index, Maslov triple
index, winding number, $\zeta$-determinant, splitting of $\eta$-invariant

###### 2020 Mathematics Subject Classification:

58J28, 58J30, 58J32.

Johnny Lim1 222Corresponding author., Hang Wang2

1 School of Mathematical Sciences, Universiti Sains Malaysia, Malaysia

2Research Center for Operator Algebras, East China Normal University, China

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Equivariant spectral flow
  4. 4 Equivariant Maslov index
  5. 5 Equivariant winding number
  6. 6 Equivariant Maslov triple index
  7. 7 Equivariant $\zeta$-determinants
  8. 8 Equivariant Scott-Wojciechowski Theorem
  9. 9 Equivariant $\eta$-invariants

##  1\. Introduction

The notion of spectral flow goes way back to the seminal paper of Atiyah-
Patodi-Singer [4]. Informally, it measures the net change of spectral
differences of self-adjoint elliptic operators across the real line
${\mathbb{R}}.$ In 96’, Phillips [22] provides an equivalent description of
spectral flow in terms of grid partitions. We generalise this to the
equivariant case (Definition 3.2) where it takes value in trace when evaluated
over a grid partition. By adopting this approach, the complication of counting
the crossings is simplified by merely considering the intersection of a
spectral path with the vertical “wall” of each partitioned grid. Under this
setting, equivariant spectral flow is well-defined (Proposition 3.4) and is an
isomorphism (Theorem 3.5) between the $H$-equivariant fundamental group of the
non-trivial component of the algebra of self-adjoint equivariant Fredholm
operators and the representation ring $R(H).$

On the other hand, Getzler [13] shows that the spectral flow of a path can be
expressed in terms of Atiyah-Patodi-Singer’s $\eta$-invariants [1, 2], which
comprises the evaluation at the endpoints and the variation of
$\eta$-invariants along the path. Kirk and Lesch [15] provide another proof of
this (in terms of reduced $\eta$-invariants) via the path-lifting method to
the universal cover. In this paper, we generalise this well-known formula to
the equivariant case in Theorem 9.5. Moreover, for our future purposes, we
also establish the gluing formula (Theorem 9.10) of equivariant
$\eta$-invariants along the boundary splitting via a closed hypersurface.
Readers can refer to [8, 20] for expositions on gluing formulae for spectral
flow and $\eta$-invariants.

Another spectral invariants closely related to $\eta$-invariants are the
$\zeta$-function $\zeta(D^{2},s)$ of the square of Dirac operators and its
variation $\dot{\zeta}(D^{2},s)$ evaluated at $s=0,$ with
$\dot{\zeta}(D^{2},0)$ being the most sensitive one in the sense that its
value changes upon a perturbation by a smoothing operator. See [28]. We
discuss an equivariant generalisation of both of these invariants explicitly
in Section 7, as well as defining the equivariant $\zeta$-determinant of $D$
(Definition 7.11). Inspired by Kirk-Lesch, we extend our discussion of
$\zeta$-determinants to the equivariant Scott-Wojciechowski theorem
(Conjecture 8.3) – we merely state this as a conjecture, instead of a theorem,
as it is not our intention to re-prove [24] in full details in the equivariant
settings. Despite this, we reasonably justify why our settings are compatible
and extend to theirs.

In addition, we also show an identification (Theorem 9.7) between the
equivariant spectral flow of a path of Dirac operators and the Maslov index of
a pair of possibly infinite-dimensional Lagrangian $H$-spaces associated to
the Calderón projection and boundary conditions. As an intermediate step, we
give a description of the winding number (Definition 5.6) of equivariant
unitary paths associated to a Lagrangian pair, which corresponds to a pair of
boundary conditions. These can be viewed as slight generalisations of [15],
which is itself an infinite version of the classical Maslov index, e.g. in
[21, 9].

As a summary, a large portion of this paper is dedicated in discussing the
correspondence

$\begin{subarray}{c}\text{\normalsize equivariant}\\\ \text{\normalsize
spectral
flow}\end{subarray}\quad\longleftrightarrow\quad\begin{subarray}{c}\text{\normalsize
equivariant}\\\ \text{\normalsize Maslov
index}\end{subarray}\quad\longleftrightarrow\quad\begin{subarray}{c}\text{\normalsize
equivariant}\\\ \text{ \normalsize winding number}\end{subarray}$

where the corresponding ingredients are respectively

$\begin{subarray}{c}\text{\normalsize an equivariant path $D_{P}(t)$}\\\
\text{\normalsize connecting $P(t)$ and
${\mathcal{P}}_{M}(t)$}\end{subarray}\;\longleftrightarrow\;\begin{subarray}{c}\text{\normalsize
a pair of $H$-Lagrangians}\\\
\text{\normalsize$({\mathcal{L}}_{P}(t),{\mathcal{L}}_{{\mathcal{P}}_{M}}(t))$}\end{subarray}\;\longleftrightarrow\;\begin{subarray}{c}\text{\normalsize
an equivariant path of unitary}\\\ \text{ \normalsize operators
$T^{*}(t)K(t).$}\end{subarray}$

In other words, they provide an unified overview of “spectral intersection
counting” via different perspectives. With these established, we can construct
a boundary correction term for Toeplitz operators on manifolds with boundary
with compact group actions in the sequel [16]. In particular, the correction
term comprises an equivariant $\eta$-invariant and an equivariant spectral
flow of a path of perturbed Dirac-type operators. This new invariant serves as
an equivariant analog of the Dai-Zhang even-type $\eta$-invariant in [11].

The organisation of this paper is as follows. Section 2 serves as station
where the setups, assumptions, and notations are given and explained. In
Section 3, we study the equivariant spectral flow of self-adjoint equivariant
Fredholm operators via grid partitions. In Section 4, we study the equivariant
Maslov index of pairs of $h$-Lagrangians. In Section 5, we discuss the
equivariant winding number of some associated unitary operators and its
relation with equivariant Maslov indices. Combining the notions in Sections 4
and 5, we study an equivariant analog of Maslov triple indices which can be
expressed as an algebraic combination of equivariant Maslov indices associated
to three pairs of $H$-Lagrangians. Sections 7 and 8 are dedicated to the study
of equivariant $\zeta$-determinants of self-adjoint elliptic equivariant
Fredholm operators and its fundamental role in the equivariant analog of
Scott-Wojciechowski’s theorem. Lastly, in Section 9 we discuss the equivariant
$\eta$-invariants of Dirac operators on manifolds with boundary and its
relation with the equivariant Maslov triple index associated to a triple of
boundary projections.

Acknowledgement. The year 2020 is truly a challenging year that has impacted
our lives profoundly. The first author wishes to express his gratitude to
Universiti Sains Malaysia for keeping his job throughout the pandemic. Part of
the work was done while the first author visited the Research Center for
Operator Algebras at East China Normal University back in October 2019. He is
grateful to the center for financial support. The second author acknowledges
support from grants NSFC-11801178 and Shanghai Rising-Star Program
19QA1403200.

##  2\. Preliminaries

Let $M$ be a smooth compact odd-dimensional ${\text{spin}^{c}}$ manifold with
boundary $\partial M.$ Assume that an identification between a neighborhood of
$\partial M$ and $(-\epsilon,0]\times\partial M$ is fixed. Let $E$ be a
complex vector bundle over $M,$ equipped with a Hermitian metric and a unitary
connection $\nabla^{E}$. For $S^{c}(TM)$ the spinc spinor bundle over $M$,
consider the tensor product bundle $S^{c}(TM)\otimes E$ equipped with a
compatible connection. Assume that the spinc structure is fixed on $M.$ For
simplicity, we denote the space of $L^{2}$-sections $L^{2}(S^{c}\otimes E)$ by
$L^{2}(E)$ with the underlying bundle $S^{c}$ implicitly implied. Let
$D^{E}:L^{2}(E)\to L^{2}(E)$ be the twisted Dirac operator on $M$ which takes
the form

(2.1) $D^{E}=\gamma\Big{(}\frac{\partial}{\partial t}+A\Big{)}$

near the cylinder $(-\epsilon,0]\times\partial M$ equipped with the product
metric. Here, $\gamma:E_{|_{\partial M}}\to E_{|_{\partial M}}$ is a bundle
isomorphism such that

(2.2) $\gamma^{2}=-{\mathrm{Id}},\;\;\;\gamma^{*}=-\gamma$

and $A:L^{2}(E_{|_{\partial M}})\to L^{2}(E_{|_{\partial M}})$ is the twisted
Dirac operator on $\partial M.$ In particular, since $\partial M$ is closed,
$A$ is essentially self-adjoint and has point spectrum. Moreover, in view of
(2.1) and (2.2), $A$ anti-commutes with $\gamma,$

(2.3) $\gamma A=-A\gamma.$

As a consequence, the space $L^{2}(E|_{\partial M})$ splits into the direct
sum of

${\mathcal{E}}_{i}:=L^{2}(E_{i})=\ker(\gamma-i)\quad\text{ and
}\quad{\mathcal{E}}_{-i}:=L^{2}(E_{-i})=\ker(\gamma+i)$

whose grading is given by $\gamma.$

Let $\\{\lambda,\psi_{\lambda}\\}$ be the spectral decomposition of $A.$ Let
$P^{+}:L^{2}(E_{|_{\partial M}})\to L^{2}(E_{|_{\partial M}})$ be the Atiyah-
Patodi-Singer (APS) spectral projection defined by

$P^{+}(\sum_{\lambda}a_{\lambda}\psi_{\lambda})=\sum_{\lambda>0}a_{\lambda}\psi_{\lambda}.$

Let $G$ be a compact group of isometries acting on $M$ preserving the fixed
${\text{spin}^{c}}$ structure. For $h\in G,$ we consider $H=\overline{\langle
h\rangle}$ to be the closed subgroup of $G$ generated by $h.$

######  Assumption 1.

Let $h\in H.$ Assume that $h$ commutes with the Dirac operator $D.$

In view of (2.1), it follows that $h$ also commutes with the self-adjoint
operator $A,$ the Clifford multiplication $\gamma,$ and the corresponding
spectral projections $P$ associated to $A$ as in Definition 2.1 below.

######  Definition 2.1.

Let $h\in H.$ The $h$-equivariant self-adjoint Fredholm Grassmannian of $A,$
denoted by ${\mathrm{Gr}}_{h}(A),$ is the set of all $h$-equivariant
projections $P:L^{2}(E_{|_{\partial M}})\to L^{2}(E_{|_{\partial M}})$ such
that

  1. (1)

$P$ is a zeroth order pseudodifferential projection;

  2. (2)

$P^{2}=P$ and $P^{*}=P,$ i.e. $P$ is an orthogonal projection;

  3. (3)

$\gamma P\gamma^{*}=1-P,$ i.e. $P$ is Lagrangian;

  4. (4)

$(P,P^{+})$ is a Fredholm pair, i.e. $PP^{+}:\text{im}(P^{+})\to\text{im}(P)$
is a Fredholm operator.

This is an immediate equivariant adaptation of [15, Definition 2.1]. Note that
the classical APS projection $P^{+}\notin{\mathrm{Gr}}_{h}(A)$ if and only if
$A$ is not invertible. In fact, $A$ is in general not invertible in (2.1).
Now, we modify $P^{+}$ by a finite rank projection $P_{{\mathcal{L}}},$ i.e.

(2.4) $P^{\partial}({\mathcal{L}}):=P^{+}+P_{{\mathcal{L}}}$

where ${\mathcal{L}}\subset\ker(A)$ is a Lagrangian subspace such that
$\gamma({\mathcal{L}})={\mathcal{L}}^{\perp}\cap\ker(A).$ Since $h$ commutes
with $P_{{\mathcal{L}}}$, its corresponding Lagrangian ${\mathcal{L}}$ is an
$h$-space for $h\in H.$ Then, one can verify that
$P^{\partial}({\mathcal{L}})\in{\mathrm{Gr}}_{h}(A),$ which shows that the set
${\mathrm{Gr}}_{h}(A)$ is indeed non-empty. The projection
$P^{\partial}({\mathcal{L}})$ is a priori dependent on the choice of a
Lagrangian subspace ${\mathcal{L}}.$ Henceforth, we shall assume a choice of
${\mathcal{L}}$ is fixed and write $P^{\partial}=P^{\partial}({\mathcal{L}}).$

Consider the subset

(2.5)
${\mathrm{Gr}}^{\infty}_{h}(A)=\\{P\in{\mathrm{Gr}}_{h}(A)\;|\;P-P^{+}\in\Psi^{-\infty}\\}$

where $\Psi^{-\infty}$ stands for the algebra of smoothing operators. Recall
that there is a canonical projection in ${\mathrm{Gr}}(A)$ that depends only
on $M$ and $D,$ which is commonly known as the Calderón projection
${\mathcal{P}}_{M}$ on $M.$ It is defined as the orthogonal projection onto
the Cauchy-data space

(2.6) ${\mathcal{L}}_{M}:=r\\{\ker D:H_{1/2}(E)\to H_{-1/2}(E)\\}\subset
L^{2}(E|_{\partial M})$

where $r$ denotes the restriction to the boundary. See [7]. By assumption, $h$
commutes with ${\mathcal{P}}_{M}.$ This $h$-equivariant Calderón projection is
an element of ${\mathrm{Gr}}_{h}(A).$ In fact, an equivariant refinement
obtained from [23] implies that
${\mathcal{P}}_{M}\in{\mathrm{Gr}}^{\infty}_{h}(A).$

Let $P\in{\mathrm{Gr}}_{h}(A)$ be a self-adjoint spectral projection with
respect to the boundary (tangential) operator. Then, the operator $D$ equipped
with $P$ whose domain restricts to

$\text{Dom}(D_{P})=\\{s\in L^{2}(E)\;|\;s\in H^{1}(E),P(s_{|_{\partial
M}})=0\\},$

is self-adjoint and Fredholm.

As explained in [15, §2], every orthogonal projection
$P\in{\mathrm{Gr}}_{h}(A)$ takes the explicit form

(2.7) $P=\frac{1}{2}\begin{pmatrix}{\mathrm{Id}}&T^{*}\\\
T&{\mathrm{Id}}\end{pmatrix}$

for some odd unitary operator $T:{\mathcal{E}}_{i}\to{\mathcal{E}}_{-i}.$ On
the other hand, given such an odd unitary operator $T,$ the matrix in $T$ on
the right side of (2.7) satisfies all four conditions in Definition 2.1. We
shall use the notation $T$ and $\Phi(P)$ interchangeably. The following lemma
is a characterisation of $h.$

######  Lemma 2.2.

Let $h\in H.$ Let $P\in{\mathrm{Gr}}_{h}(A).$ By Assumption 1, the isometry
$h$ can be characterized by the form

(2.8) $h=\begin{pmatrix}a&0\\\ 0&\Phi(P)a\Phi(P)^{*}\end{pmatrix}.$

###### Proof.

As $h$ acts as an isometry on $M,$ which lifts to act on
${\mathbb{Z}}_{2}$-graded Hilbert spaces $L^{2}(E)=L^{2}_{+}(E)\oplus
L^{2}_{-}(E)$ and $L^{2}(E_{|_{\partial M}})=L^{2}_{+}(E_{|_{\partial
M}})\oplus L^{2}_{-}(E_{|_{\partial
M}})={\mathcal{E}}_{i}\oplus{\mathcal{E}}_{-i},$ whose gradings are given by
$\gamma,$ we write $h$ in the form

$h=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}.$

Let $T=\Phi(P).$ Note that

$hP=\frac{1}{2}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\begin{pmatrix}{\mathrm{Id}}&T^{*}\\\
T&{\mathrm{Id}}\end{pmatrix}=\begin{pmatrix}a+bT&b+aT^{*}\\\
c+dT&d+cT^{*}\end{pmatrix}$

and

$Ph=\frac{1}{2}\begin{pmatrix}{\mathrm{Id}}&T^{*}\\\
T&{\mathrm{Id}}\end{pmatrix}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}=\begin{pmatrix}a+T^{*}c&b+T^{*}d\\\ c+Ta&d+Tb\end{pmatrix}$

By the commutative relation $Ph=hP,$ we obtain

$h=\begin{pmatrix}a&T^{*}cT^{*}\\\ TbT&TaT^{*}\end{pmatrix}.$

On the other hand, by Assumption 1, $h$ must also commute with
$\gamma={\mathrm{Diag}}(i,-i)$ which satisfies (2.2). Then, we can verify by a
direct computation that $h$ commutes with $\gamma$ if and only if $TbT=0$ and
$T^{*}cT^{*}=0.$ Thus, the isometry $h$ is a diagonal matrix

$h=\begin{pmatrix}a&0\\\
0&TaT^{*}\end{pmatrix}\in{\mathcal{U}}({\mathcal{E}}_{i})\oplus{\mathcal{U}}({\mathcal{E}}_{-i}).$

∎

######  Remark 2.3.

In general, the explicit form of the isometry $h$ does not depend on a
particular projection. Hence, shall no confusion occur we will denote
$h={\mathrm{Diag}}(a,WaW^{*})$ for some
$W\in{\mathcal{U}}({\mathcal{E}}_{i},{\mathcal{E}}_{-i}).$

Let $(P,Q)$ be a Fredholm pair, i.e.
$PQ:\operatorname{im}(Q)\to\operatorname{im}(P)$ is a Fredholm operator. By
Assumption 1, the pair is compatible with the $h$-action, i.e. $(PQ)h=h(PQ)$
for all $h\in H.$ In view of (2.7), the product of the corresponding matrices
is well-defined. In this case, for $i=1,2,3,$ let $T_{i}$ be the corresponding
odd unitary operators in defining $P,Q$ and $h$ respectively, then
$T^{*}_{i}T_{j}$ must commute with $a$ for all $i,j=1,2,3$ with $i\neq j$ and
$T_{3}^{*}(T_{1}T^{*}_{2})T_{3}$ must also commute with $a.$ The explicit
formula for the general matrix multiplication $P_{\gamma}\cdots
P_{\beta}P_{\alpha}$ is significantly more convoluted to state so we leave
this to interested readers.

######  Lemma 2.4.

Let $h={\mathrm{Diag}}(a,WaW^{*})\in H.$ Let $P,Q\in{\mathrm{Gr}}_{h}(A)$ be
given by (2.7), i.e.

$P=\frac{1}{2}\begin{pmatrix}{\mathrm{Id}}&T^{*}\\\
T&{\mathrm{Id}}\end{pmatrix},\quad
Q=\frac{1}{2}\begin{pmatrix}{\mathrm{Id}}&S^{*}\\\
S&{\mathrm{Id}}\end{pmatrix}$

for some odd unitary operators $T,S,W:{\mathcal{E}}_{i}\to{\mathcal{E}}_{-i}.$
Then,

  1. (1)

$(P,Q)$ is a Fredholm pair if and only if
$-1\notin{\text{spec}}_{ess}((a-{\mathrm{Id}})+T^{*}S)),$

  2. (2)

$(P,Q)$ is an invertible pair (i.e.
$PQ:\operatorname{im}(Q)\to\operatorname{im}(P)$ is invertible) if and only if
$-1\notin{\text{spec}}((a-{\mathrm{Id}})+aT^{*}S),$

  3. (3)

$\ker(P)\cap\operatorname{im}(Q)\cong\ker(a({\mathrm{Id}}+T^{*}S)),$

  4. (4)

$h(P-Q)\in\Psi^{-\infty}$ if and only if
$({\mathrm{Id}}-T^{*}S)a\in\Psi^{-\infty}.$

###### Proof.

We follow an argument of [15]. Let $L^{2}(E|_{\partial
M})={\mathcal{E}}_{i}\oplus{\mathcal{E}}_{-i}.$ Then, any element in
$L^{2}(E|_{\partial M})$ can be written as $(x,Sy)^{T}$ for
$x,y\in{\mathcal{E}}_{i}$ and for some
$S:{\mathcal{E}}_{i}\to{\mathcal{E}}_{-i}.$ One verifies that
$\operatorname{im}(Q)$ is the set
$\\{(x,Sx)^{T}\;|\;x\in{\mathcal{E}}_{i}\\}.$ Then, the composition $h(PQ)$
takes the form

(2.9) $hPQ=\frac{1}{2}\begin{pmatrix}a({\mathrm{Id}}+T^{*}S)\\\
Ta({\mathrm{Id}}+T^{*}S)\end{pmatrix}$

where we use the fact that both $T^{*}W$ and its adjoint $W^{*}T$ commute with
$a.$ Hence, $(P,Q)$ is a Fredholm pair if and only if
$a({\mathrm{Id}}+T^{*}S)={\mathrm{Id}}+[(a-{\mathrm{Id}})+aT^{*}S]$ is
Fredholm, which is exactly when
$-1\notin{\text{spec}}_{ess}((a-{\mathrm{Id}})+aT^{*}S).$ Similarly, $(P,Q)$
is an invertible pair if and only if the condition
$-1\notin{\text{spec}}((a-{\mathrm{Id}})+aT^{*}S)$ holds. For (3), in view of
(2.9) the intersection $\ker(P)\cap\operatorname{im}(Q)$ coincides with
$\ker(a({\mathrm{Id}}+T^{*}S)).$ For the last part, one verifies that

(2.10) $h(P-Q)=\frac{1}{2}\begin{pmatrix}0&a(T^{*}-S^{*})\\\
(T-S)a&0\end{pmatrix}.$

Since $T,S$ and $a$ are unitary, the operator $(T-S)a$ is smoothing if and
only if $({\mathrm{Id}}-T^{*}S)a$ is smoothing. ∎

Let ${\mathcal{H}}$ be a complex separable Hilbert space with a fixed basis.
Let $H$ be the compact group of isometries on $M.$ Let

(2.11) ${\mathcal{H}}^{\prime}=L^{2}(H,{\mathcal{H}})$

be the space of square integrable functions on $H$ with values in
${\mathcal{H}}.$ These $L^{2}$ functions are integrated with respect to an
$H$-invariant Haar measure. By the Peter-Weyl theorem, the $H$-module
${\mathcal{H}}^{\prime}$ can be expressed as an isomorphism

(2.12) $L^{2}(H,{\mathcal{H}}^{\prime})\cong\sum_{i}(\dim
V_{i})V_{i}\otimes{\mathcal{H}}$

where the sum runs through irreducible complex representations $V_{i}$ of $H$
for all $i.$ Let
${\mathcal{B}}({\mathcal{H}}^{\prime}),{\mathcal{F}}({\mathcal{H}}^{\prime}),GL({\mathcal{H}}^{\prime})$
and ${\mathcal{U}}({\mathcal{H}}^{\prime})$ be the Banach algebra of bounded
operators, the subalgebra of Fredholm operators of the form $I+{\mathcal{K}},$
the general linear group and the unitary group of ${\mathcal{H}}^{\prime}$
respectively. The natural $H$-action on ${\mathcal{H}}^{\prime}$ induces one
on ${\mathcal{B}}({\mathcal{H}}^{\prime}),$ which is given by the commutative
diagram

${{\mathcal{H}}^{\prime}}$${{\mathcal{H}}^{\prime}}$${{\mathcal{H}}^{\prime}}$${{\mathcal{H}}^{\prime},}$$\scriptstyle{h}$$\scriptstyle{T}$$\scriptstyle{T^{h}}$$\scriptstyle{h}$

i.e.

(2.13) $T^{h}(f)=hT(h^{-1}f)$

for $f\in L^{2}(h,{\mathcal{H}}^{\prime})$ and
$T,T^{h}\in{\mathcal{B}}({\mathcal{H}}^{\prime}).$ Such an $H$-action respects
the algebra structure and is continuous with respect to the norm topology of
${\mathcal{B}}({\mathcal{H}}^{\prime}).$ The induced actions on the spaces
${\mathcal{F}}({\mathcal{H}}^{\prime}),GL({\mathcal{H}}^{\prime})$ and
${\mathcal{U}}({\mathcal{H}}^{\prime})$ are defined similarly.

We end this section by recalling a definition of equivariant fundamental
groups in [17].

######  Definition 2.5.

Let $X$ be an $H$-equivariant CW-complex. Let $x_{0}\in X$ be a basepoint. The
equivariant fundamental group $\pi_{1}(X,x_{0})_{H}$ is defined to be the set
of $h$-equivariant homotopy classes of paths from $x_{0}$ to $hx_{0}$ for
$h\in H.$ In particular, if $\varphi(h)=[x_{0}\mapsto hx_{0}]$ and
$\phi(g)=[x_{0}\mapsto gx_{0}]$ are two homotopy classes of paths translated
under group elements, then the composition law is given by
$\varphi(h)\circ\phi(g):=(\varphi h_{*}\phi)(hg).$

##  3\. Equivariant spectral flow

Let $\widehat{{\mathcal{F}}}({\mathcal{H}}^{\prime})$ be the subalgebra of
${\mathcal{F}}({\mathcal{H}}^{\prime})$ of all self-adjoint equivariant
Fredholm operators. By [5], the space
$\widehat{{\mathcal{F}}}({\mathcal{H}}^{\prime})$ consists of three connected
components:

(3.1)
$\widehat{{\mathcal{F}}}({\mathcal{H}}^{\prime})=\widehat{{\mathcal{F}}}_{+}({\mathcal{H}}^{\prime})\oplus\widehat{{\mathcal{F}}}_{-}({\mathcal{H}}^{\prime})\oplus\widehat{{\mathcal{F}}}_{*}({\mathcal{H}}^{\prime}).$

Here, $\widehat{{\mathcal{F}}}_{+}({\mathcal{H}}^{\prime})$ (resp.
$\widehat{{\mathcal{F}}}_{-}({\mathcal{H}}^{\prime})$) consists of those
operators whose essential spectrum lie entirely in ${\mathbb{R}}^{+}$ (resp.
${\mathbb{R}}^{-}$); whereas
$\widehat{{\mathcal{F}}}_{*}({\mathcal{H}}^{\prime})$ consists of those
operators whose essential spectrum lie simultaneously in both
${\mathbb{R}}^{+}$ and ${\mathbb{R}}^{-}.$

In fact, for an element
$T\in\widehat{{\mathcal{F}}}_{+}({\mathcal{H}}^{\prime})$ and for all $0\leq
s\leq 1,$ the linear homotopy

$T(s)=(1-s)T+s{\mathrm{Id}}$

contracts $T$ to the identity operator ${\mathrm{Id}}.$ Similarly, the
homotopy $T(s)=(1-s)T+s(-{\mathrm{Id}})$ contracts
$T\in\widehat{{\mathcal{F}}}_{-}({\mathcal{H}}^{\prime})$ to $-{\mathrm{Id}}.$
This concludes the following statement.

######  Lemma 3.1.

Both of the spaces $\widehat{{\mathcal{F}}}_{\pm}({\mathcal{H}}^{\prime})$ are
contractible.

Let
$\pi:{\mathcal{B}}({\mathcal{H}}^{\prime})\to{\mathcal{Q}}({\mathcal{H}}^{\prime})={\mathcal{B}}({\mathcal{H}}^{\prime})/{\mathcal{K}}({\mathcal{H}}^{\prime})$
be the canonical projection into the Calkin algebra of
${\mathcal{H}}^{\prime}.$ Let ${\mathcal{J}}$ be the group of invertible
elements of ${\mathcal{Q}}({\mathcal{H}}^{\prime}).$ By definition, we have
${\mathcal{F}}=\pi^{-1}({\mathcal{J}}).$ Let
$\widehat{{\mathcal{J}}}\subset{\mathcal{J}}$ be the subspace that contains
all self-adjoint invertible elements of
${\mathcal{Q}}({\mathcal{H}}^{\prime}).$ Then, by [7, Lemma 16.3] the
projection $\pi$ restricts to a homotopy equivalence
$\widehat{{\mathcal{F}}}_{*}({\mathcal{H}}^{\prime})\xrightarrow{\sim}\widehat{{\mathcal{J}}}_{*}({\mathcal{H}}^{\prime})$
where $\widehat{{\mathcal{J}}}_{*}({\mathcal{H}}^{\prime})$ is the non-trivial
connected component of $\widehat{{\mathcal{J}}}.$ More precisely,

$\widehat{{\mathcal{J}}}_{*}({\mathcal{H}}^{\prime})=\\{j\in{\mathcal{Q}}({\mathcal{H}}^{\prime})|j^{*}=j,\text{spec}(j)\cap{\mathbb{R}}^{+}\neq\emptyset\textit{
and }\text{spec}(j)\cap{\mathbb{R}}^{-}\neq\emptyset\\}$

where the pre-image of each $j\in\widehat{{\mathcal{J}}}_{*}$ takes the form
$\frac{1}{2}(T+T^{*})\in\widehat{{\mathcal{F}}}_{*}$ for which
$\pi(T)=\pi(T^{*})=j.$ The induced $H$-action on $\widehat{{\mathcal{J}}}_{*}$
is given by that of restricted on $\widehat{{\mathcal{F}}}_{*}$ and by the
equivariance of projections.

Moreover, as with the case where invertible elements can be retracted into
unitary elements, the standard $C^{*}$-algebra retraction

(3.2)
$j_{s}=j\Big{(}(1-s){\mathrm{Id}}+\frac{s}{\sqrt{j^{*}j}}\Big{)},\;\;\;\text{for
}s\in[0,1],$

defines the unitary groups

$\displaystyle\widehat{J}({\mathcal{H}}^{\prime})=\\{j\in{\mathcal{Q}}({\mathcal{H}}^{\prime})\;|\;j^{*}=j,j^{2}={\mathrm{Id}},j\neq\pm{\mathrm{Id}}\\},$
$\displaystyle\widehat{J}_{*}({\mathcal{H}}^{\prime})=\\{j\in\widehat{J}({\mathcal{H}}^{\prime})\;|\;{\text{spec}}(j)=\\{-1,+1\\}\\}.$

By the unitary retraction (3.2), we conclude that
$\widehat{{\mathcal{F}}}_{*}({\mathcal{H}}^{\prime})$ is homotopy equivalent
to $\widehat{J}_{*}({\mathcal{H}}^{\prime}).$ See [7, Theorem 16.4].

For $h\in H,$ let $P^{+}$ be the $h$-equivariant positive spectral projections
of $D$ onto the Lagrangian subspace
${\mathcal{L}}_{P^{+}}({\mathcal{H}}^{\prime}).$ Let $P^{-}$ be the
complementary projection of $P^{+}$ onto the Lagrangian subspace
${\mathcal{L}}_{P^{-}}({\mathcal{H}}^{\prime}).$ Then, $P^{+}-P^{-}$ defines
an involution in $\widehat{J}_{*}({\mathcal{H}}^{\prime}).$ Choose and fix a
finite dimensional projection $P_{k}$ of $P^{-}$ onto an $h$-subspace $V_{k}$
of dimension $k$ of Ran($P^{-}$). Then, in view of
$\widehat{{\mathcal{F}}}_{*}({\mathcal{H}}^{\prime})\simeq\widehat{J}_{*}({\mathcal{H}}^{\prime}),$
every equivariant Fredholm operator
$T\in\widehat{{\mathcal{F}}}_{*}({\mathcal{H}}^{\prime})$ corresponds to some
involution up to finite rank projection $P^{+}-P^{-}+P_{k}.$ Then, any element
of $\pi_{1}(\widehat{{\mathcal{F}}}_{*}({\mathcal{H}}^{\prime}))_{H}$ is a
family $\\{B_{k}(t):=P^{+}-P^{-}+2tP_{k}\\}_{t\in S^{1}}.$ For simplicity, we
shall denote a general element of
$\pi_{1}(\widehat{{\mathcal{F}}}_{*}({\mathcal{H}}^{\prime}))_{H}$ by $B_{t},$
to mean a family of self-adjoint equivariant Fredholm operators labelled by
$t\in S^{1}.$

######  Assumption 2.

By choosing a good partition of a grid, we mean to choose a partition
$0<t_{0}<\cdots<t_{n}=1$ of the ‘$x$-axis’ the interval $[0,1],$ and a
partition of the ‘$y$-axis’, usually a set of real number
$\\{a_{1},a_{2},\ldots,a_{n}\\}$ which is the image of some continuous
function, such that the function is non-singular at the four corner vertices.
This can always be achieved by some $\epsilon$-perturbations. A refinement of
a chosen partition refers to the $1/2^{i}$-division of the partition into
smaller grids such that it sums to the original grid.

######  Definition 3.2.

Let $h\in H.$ By Assumption 2, fix a good partition of a grid where the
$x$-axis is the interval $[0,1]$ and the $y$-axis is a set
$\\{a_{j}\\}^{n}_{j=1}$ of the image of spectral projections. For each
$j=1,2,\cdots,n,$ let $P^{+}_{j}$ be the positive spectral projection of
$B_{t}\in\pi_{1}(\widehat{{\mathcal{F}}}_{*}({\mathcal{H}}^{\prime}))_{H}$
onto the $y$-axis over the partitioned grid with the subinterval
$[t_{j-1},t_{j}]$ of $[0,1].$ Then, we define the $h$-equivariant spectral
flow of $B_{t}$ by

$\displaystyle{\mathrm{sf}}_{h}:\pi_{1}$
$\displaystyle(\widehat{{\mathcal{F}}}_{*}({\mathcal{H}}^{\prime}))_{H}\longrightarrow{\mathbb{C}}$
(3.3) $\displaystyle{\mathrm{sf}}_{h}(B_{t})=\sum^{\infty}_{j=0}$
$\displaystyle\Big{[}{\mathrm{Tr}}\big{(}h|_{{\mathbb{E}}_{j}(t_{j})}\big{)}-{\mathrm{Tr}}\big{(}h|_{{\mathbb{E}}_{j}(t_{j-1})}\big{)}\Big{]}$

where ${\mathbb{E}}_{j}(t_{i})$ denotes the corresponding
$P^{+}_{j}(t_{i})$-eigenspaces for $i=j-1,j.$

######  Proposition 3.3.

For $h\in H,$ the $h$-equivariant spectral flow is a homotopy invariant. That
is, for a homotopy
$H(s,t):[0,1]\to\pi_{1}(\widehat{{\mathcal{F}}}_{*}({\mathcal{H}}^{\prime}))_{H}$
between $B(t)$ and $B^{\prime}(t)$ such that $B(0)=B^{\prime}(0)$ and
$B(1)=B^{\prime}(1),$ we have

${\mathrm{sf}}_{h}(B(t))={\mathrm{sf}}_{h}(B^{\prime}(t)).$

###### Proof.

The proof follows verbatim from [22, Proposition 3]. In particular, the main
strategy is to break a homotopy $H:I\times
I\to\widehat{{\mathcal{F}}}_{*}({\mathcal{H}}^{\prime})$ into the sum of
‘shorter’ homotopies in each grid. This is achievable due to compactness and
we can cover $I\times I$ by a finite set of preimages
$\\{H^{-1}(N_{i})\\}_{i}$ where each $N_{i}$ is a neighbourhood of $B(t)$
which covers some grid $J_{i}\times J_{j}.$ ∎

Proposition 3.3 allows us to consider continuous deformations of spectrum with
fixed points. In particular, given a continuous path which crosses the
$x$-axis over a region in a partitioned grid, as in Figure 1, we can consider
a homotopy of paths with fixed endpoints, as in Figure 2. By homotopy
invariance, the equivariant spectral flows of the green line and the blue line
are equal, i.e. the net ‘counting’ is the same as that of in Figure 3. The
local equivariant spectral flow is thus given by
${\mathrm{Tr}}(h|_{{\mathbb{E}}(t_{j})})-{\mathrm{Tr}}(h|_{{\mathbb{E}}(t_{j-1})}).$

Figure 1.
Figure 2.

Figure 3.

The elegant method of Phillips [22] simplifies the notion of spectral flow
introduced by Atiyah-Patodi-Singer [3] which measures the net change of
crossings across zero counted with multiplicities. In particular, Phillips’s
counting method no longer depends on the crossings over the $x$-axis because
via homotopy we only need to consider the crossings on the positive ‘quadrant’
with respect to the $y$-axis. In the equivariant case, this corresponds to
taking the trace of the induced linear map on the ${\mathbb{E}}(t_{j-1})$ and
${\mathbb{E}}(t_{j})$ respectively.

In general, there are much more complicated paths of spectrum. However, by
Phillips’s method, over an appropriate partition, we may reduce this to only
three simple types of path crossings and they span all other possible
combinations. The first type has already been described above. The second type
are shown in Figure 4-6. They depict the net effect of a crossing from the
negative to the positive ‘quadrant’. The local equivariant spectral flow is
${\mathrm{Tr}}(h|_{{\mathbb{E}}(t_{j})}).$ Similarly, Figure 7-9 depicts the
net effect of a crossing from the positive to the negative ‘quadrant’, for
which the local equivariant spectral flow is
$-{\mathrm{Tr}}(h|_{{\mathbb{E}}(t_{j-1})}).$

Figure 4.
Figure 5.

Figure 6.

Figure 7.
Figure 8.

Figure 9.

Figure 10.
Figure 11.

Figure 12.

The third type is the combination of crossed paths of the first and the second
types, as shown in Figure 10-12. In the positive ‘quadrant’ of Figure 11, we
construct a new homotopy partly from the green line and the light blue line
respectively and further deform smoothly to the orange line. We perform
similar procedure for the bottom ‘quadrant’. The net effect is exactly as in
Figure 12 where the local equivariant spectral flow is
${\mathrm{Tr}}(h|_{{\mathbb{E}}(t_{j})})-{\mathrm{Tr}}(h|_{{\mathbb{E}}(t_{j-1})}).$

######  Proposition 3.4.

For $h\in H,$ the equivariant spectral flow ${\mathrm{sf}}_{h}$ is well-
defined, i.e. it is independent of the refinement of the grid partition and it
only depends on the continuous function
$B(t):[0,1]\to\widehat{{\mathcal{F}}}_{*}({\mathcal{H}}^{\prime}).$

###### Proof.

Without loss of generality, we choose a good grid partition such that there
are only one to two eigenvalues inside the grid over $[t_{j-1},t_{j}].$ Let
$t_{d}$ be any element in $[t_{j-1},t_{j}]$ (its corresponding line of
spectrum is the vertical orange line, Figure 13). Let
$P^{+}_{d}(t_{d})\Big{|}^{a}_{0}$ be the intersection (green dot, Figure 13)
between the line $t=t_{d}$ and a path of spectrum. Then,

$\displaystyle{\mathrm{sf}}_{h}(B_{t})_{\begin{subarray}{c}t_{j-1}\leq t\leq
t_{j}\\\ 0\leq P^{+}\leq a\end{subarray}}$
$\displaystyle=\Big{(}{\mathrm{Tr}}\big{(}h|_{{\mathbb{E}}_{j}(t_{j})}\big{)}-{\mathrm{Tr}}\big{(}h|_{{\mathbb{E}}_{j}(t_{d})}\big{)}\Big{)}+\Big{(}{\mathrm{Tr}}\big{(}h|_{{\mathbb{E}}_{d}(t_{d})}\big{)}-{\mathrm{Tr}}\big{(}h|_{{\mathbb{E}}_{d}(t_{j-1})}\big{)}\Big{)}$
$\displaystyle={\mathrm{Tr}}\big{(}h|_{{\mathbb{E}}_{j}(t_{j})}\big{)}-{\mathrm{Tr}}\big{(}h|_{{\mathbb{E}}_{j}(t_{j-1})}\big{)}$

where the second equality follows from the fact that $P^{+}$ is canonical,
that is, $P^{+}_{j}=P^{+}_{d}.$ This shows that ${\mathrm{sf}}_{h}$ is
independent of vertical cut, see Figure 13. On the other hand, let the orange
line be any horizontal cut as shown in Figure 14 which splits the grid $[0,a]$
into $[0,a^{\prime}]$ and $[a^{\prime},a]$. Then, it is evident from
definition that

${\mathrm{sf}}_{h}(B_{t})_{\begin{subarray}{c}t_{j-1}\leq t\leq t_{j}\\\ 0\leq
P^{+}\leq
a\end{subarray}}={\mathrm{sf}}_{h}(B_{t})_{\begin{subarray}{c}t_{j-1}\leq
t\leq t_{j}\\\ 0\leq P^{+}\leq
a^{\prime}\end{subarray}}+{\mathrm{sf}}_{h}(B_{t})_{\begin{subarray}{c}t_{j-1}\leq
t\leq t_{j}\\\ a^{\prime}\leq P^{+}\leq a\end{subarray}}.$

Thus, it is independent of horizontal cut. The general case follows
immediately by summing over all grids.

Figure 13. Vertical cut.
Figure 14. Horizontal cut.

∎

Let
${\mathrm{sf}}_{H}:\pi_{1}(\widehat{{\mathcal{F}}}_{*}({\mathcal{H}}^{\prime}))_{H}\longrightarrow
R(H)$ be the equivariant spectral flow such that the evaluation at every $h\in
H$ is given by (3.2). Here, $R(H)$ denotes the representation ring of $H,$
regarded as the ring of characters of $H.$

######  Theorem 3.5.

The spectral flow
${\mathrm{sf}}_{H}:\pi_{1}(\widehat{{\mathcal{F}}}_{*}({\mathcal{H}}^{\prime}))_{H}\longrightarrow
R(H)$ is an isomorphism.

###### Proof.

Inspired by [7], this is equivalent to show that the following diagram

${\centering\pi_{1}(\widehat{{\mathcal{F}}}_{*}({\mathcal{H}}^{\prime}))_{H}\@add@centering}$${K^{1}_{H}(S^{1})}$${R(H)}$$\scriptstyle{{\mathrm{Ind}}_{a}}$$\scriptstyle{{\mathrm{sf}}_{H}}$$\scriptstyle{ch_{H}}$

commutes. Here, the equivariant $K^{1}$-theory of $S^{1}$ is isomorphic to the
homotopy classes of equivariant maps $[S^{1},U(\infty)]_{H},$ which is
$\pi_{1}(U(\infty))_{H}$ by Bott’s Periodicity. Hence,
$ch_{H}:K^{1}_{H}(S^{1})\to\pi_{1}(U(\infty))_{H}\cong R(H)$ is an
isomorphism. On the other hand, the top horizontal map is given by

$\displaystyle{\mathrm{Ind}}_{a}:\pi_{1}(\widehat{{\mathcal{F}}}_{*}({\mathcal{H}}^{\prime}))_{H}$
$\displaystyle\longrightarrow K^{1}_{H}(S^{1})$ $\displaystyle B_{k}(t)$
$\displaystyle\longmapsto[{\mathcal{W}}_{t}]$

where ${\mathcal{W}}_{t}=-\exp(i\pi B_{k}(t))$ is a unitary map which equals
$\exp(2\pi it)$ on $V_{k}$ and ${\mathrm{Id}}$ on $V_{k}^{\perp}.$ Here,
$V_{k}$ is the image of the infinite ‘negative half’ projection
$P_{-}({\mathcal{H}}^{\prime})$ with $P_{+}+P_{-}={\mathrm{Id}}.$ In
particular, the exponential map $B_{k}(t)\mapsto\exp(i\pi
B_{k}(t)+{\mathrm{Id}})$ can be shown to be a homotopy equivalence between
$\widehat{F}_{\ast}$ and $U(\infty),$ see [22, 5, 7]. Thus, their first
equivariant fundamental groups are isomorphic given by the induced map
${\mathrm{Ind}}_{a}.$ Moreover, by a direct computation

(3.4) $\displaystyle ch_{H}({\mathrm{Ind}}_{a}(B_{k}(t)))$
$\displaystyle=\frac{1}{2\pi
i}\int_{0}^{1}{\mathrm{Tr}}\Big{(}h{\mathcal{W}}^{-1}(t)\frac{d{\mathcal{W}}(t)}{dt}\Big{)}dt=\frac{1}{2\pi
i}\int_{0}^{1}{\mathrm{Tr}}(h|_{V_{k}})e^{-2\pi it}(2\pi i)e^{2\pi it}dt$
$\displaystyle={\mathrm{Tr}}(h|_{V_{k}})\int_{0}^{1}dt={\mathrm{sf}}_{h}(B_{k}(t))$

for all $h\in H.$ In particular, one deduces that the family ${B_{1}(t)}$ is
mapped to a generator ${\mathrm{Tr}}(h|_{V_{1}})$ in $R(H)$ via
${\mathrm{sf}}_{h}.$ Equation (3.4) implies that
${\mathrm{sf}}_{h}=ch_{H}\circ{\mathrm{Ind}}_{a}.$ Since both $ch_{H}$ and
${\mathrm{Ind}}_{a}$ are both isomorphisms, so is ${\mathrm{sf}}_{H}.$ ∎

######  Proposition 3.6.

For $h\in H,$ ${\mathrm{sf}}_{h}$ satisfies the following properties.

  1. (1)

For $t\in[0,1],$ let $B_{k}(t)$ and $B^{\prime}_{k}(t)$ be in
$\pi_{1}(\widehat{{\mathcal{F}}}_{*}({\mathcal{H}}^{\prime}))_{H}$ such that
$B_{k}(1)=B^{\prime}_{k}(0).$ Denote the concatenation by
$B_{k}(t)*B^{\prime}_{k}(t).$ Then,

${\mathrm{sf}}_{h}(B_{k}(t)*B^{\prime}_{k}(t))={\mathrm{sf}}_{h}(B_{k}(t))+{\mathrm{sf}}_{h}(B^{\prime}_{k}(t)).$

  2. (2)

The map ${\mathrm{sf}}_{h}$ is natural under any $h$-equivariant isomorphism
on $\widehat{{\mathcal{F}}}_{*}({\mathcal{H}}^{\prime}).$

  3. (3)

If $h$ acts trivially, then the $h$-equivariant spectral flow coincides with
the classical spectral flow of Phillips, i.e.
${\mathrm{sf}}_{h}={\mathrm{sf}}.$ Furthermore, if $h$ acts trivially and
$D(t)$ is a path of self-adjoint invertible Fredholm operator, then
${\mathrm{sf}}_{h}(D(t))=0.$

##  4\. Equivariant Maslov index

In this section, we discuss an equivariant analog of Maslov indices of
Lagrangian paths ala Kirk-Lesch [15]. Let ${\mathcal{H}}^{\prime}$ be the
model $H$-module Hilbert space as above and let
$\gamma:{\mathcal{H}}^{\prime}\to{\mathcal{H}}^{\prime}$ be an unitary map
satisfying (2.2). If we consider a symplectic form on ${\mathcal{H}}^{\prime}$
defined by $\omega(x,y)=\langle x,\gamma y\rangle,$ then the triple
$({\mathcal{H}}^{\prime},\langle,\rangle,\gamma)$ forms a Hermitian symplectic
Hilbert space.

######  Definition 4.1.

Given a Hermitian symplectic Hilbert space
$({\mathcal{H}}^{\prime},\langle,\rangle,\gamma),$ define the set of
Lagrangian subspaces of ${\mathcal{H}}^{\prime}$ by

${\mathfrak{L}}({\mathcal{H}}^{\prime})=\\{{\mathcal{L}}\subset{\mathcal{H}}^{\prime}\;|\;\gamma({\mathcal{L}})={\mathcal{L}}^{\perp}\cap{\mathcal{H}}^{\prime}\\}.$

######  Remark 4.2.

The image of any orthogonal projection $P$ on ${\mathcal{H}}^{\prime}$
satisfying the condition $\gamma P\gamma^{*}={\mathrm{Id}}-P$ is a Lagrangian
subspace. Hence, ${\mathcal{L}}({\mathcal{H}}^{\prime})$ is non-empty.
Moreover, ${\mathrm{Gr}}_{h}({\mathcal{H}}^{\prime})$ can be identified with
${\mathcal{L}}({\mathcal{H}}^{\prime})$ by

(4.1) $P\mapsto\operatorname{im}(P),\quad\quad
P_{\mathcal{L}}\mapsfrom{\mathcal{L}}.$

The $h$-action on each ${\mathcal{L}}$ is induced by the restriction of
${\mathcal{H}}^{\prime}.$ This means that each Lagrangian ${\mathcal{L}}$ is
naturally an $h$-subspace.

######  Definition 4.3.

Let
${\mathfrak{L}}^{2}({\mathcal{H}}^{\prime})=\\{({\mathcal{L}}_{1},{\mathcal{L}}_{2})\;|\;{\mathcal{L}}_{1},{\mathcal{L}}_{2}\in{\mathfrak{L}}({\mathcal{H}}^{\prime})\\}$
be the set of pairs of Lagrangian subspaces of ${\mathcal{H}}^{\prime}.$ Then,
the subset of Fredholm pairs and invertible pairs are respectively given by

(4.2) $\displaystyle{\mathfrak{L}}^{2}_{F}({\mathcal{H}}^{\prime})$
$\displaystyle=\\{({\mathcal{L}}_{1},{\mathcal{L}}_{2})\in{\mathfrak{L}}^{2}({\mathcal{H}}^{\prime})\;|\;({\mathcal{L}}_{1},{\mathcal{L}}_{2})\text{
is a Fredholm pair}\\}$ (4.3)
$\displaystyle{\mathfrak{L}}^{2}_{*}({\mathcal{H}}^{\prime})$
$\displaystyle=\\{({\mathcal{L}}_{1},{\mathcal{L}}_{2})\in{\mathfrak{L}}^{2}({\mathcal{H}}^{\prime})\;|\;({\mathcal{L}}_{1},{\mathcal{L}}_{2})\text{
is an invertible pair}\\}.$

Let ${\mathcal{L}}_{M}$ be the canonical Lagrangian subspace associated to the
canonical Calderon projection ${\mathcal{P}}_{M}$ of $M.$ Then, write
${\mathfrak{L}}^{0}_{F}({\mathcal{H}}^{\prime})$ (resp.
${\mathfrak{L}}^{0}_{*}({\mathcal{H}}^{\prime})$ ) to be the Lagrangian subset
with respect to ${\mathcal{L}}_{M},$ i.e. all Lagrangian pairs are taking the
form $({\mathcal{L}},{\mathcal{L}}_{M})$ such that
$({\mathcal{L}},{\mathcal{L}}_{M})$ is a Fredholm (resp. invertible) pair.
Then, a Maslov cycle ${\mathcal{M}}_{{\mathcal{L}}_{M}}$ at
${\mathcal{L}}_{M}$ is defined by

(4.4)
${\mathcal{M}}_{{\mathcal{L}}_{M}}={\mathfrak{L}}^{0}_{F}({\mathcal{H}}^{\prime})\backslash{\mathfrak{L}}^{0}_{*}({\mathcal{H}}^{\prime}).$

######  Remark 4.4.

Recall that $({\mathcal{L}}_{1},{\mathcal{L}}_{2})$ is a Fredholm pair for
${\mathcal{L}}_{i}\in{\mathfrak{L}}({\mathcal{H}}^{\prime})$ if
${\mathcal{L}}_{1}\cap{\mathcal{L}}_{2}$ is a finite dimensional $h$-subspace
of ${\mathcal{H}}^{\prime}$ and ${\mathcal{L}}_{1}+{\mathcal{L}}_{2}$ is a
closed $h$-space with finite codimension. In particular, the Lagrangian pair
$({\mathcal{L}}_{1},{\mathcal{L}}_{2})$ is invertible if
${\mathcal{L}}_{1}\cap{\mathcal{L}}_{2}=\\{0\\}$ and
${\mathcal{L}}_{1}+{\mathcal{L}}_{2}={\mathcal{H}}^{\prime}.$ Hence,
${\mathcal{M}}_{{\mathcal{L}}_{M}}$ is non-trivial as it does not include all
Lagrangian subspaces that intersect the Cauchy-data space ${\mathcal{L}}_{M}$
transversally.

######  Definition 4.5.

For $h\in H.$ Denote by

(4.5)
${\mathrm{Gr}}_{h}({\mathcal{H}}^{\prime})=\\{P\in{\mathcal{B}}({\mathcal{H}}^{\prime})|P^{2}=P=P^{*},\gamma
P\gamma^{*}={\mathrm{Id}}-P\\}$

the Lagrangian grassmannian of equivariant orthogonal projections on
${\mathcal{H}}^{\prime}.$ Then, we define

(4.6)
$\displaystyle{\mathrm{Gr}}^{2}_{F}({\mathcal{H}}^{\prime}):={\mathrm{Gr}}^{2}_{F,h}({\mathcal{H}}^{\prime})$
$\displaystyle=\\{(P,Q)\;|\;P,Q\in{\mathrm{Gr}}_{h}({\mathcal{H}}^{\prime}),(P,Q)\text{
is a Fredholm pair}\\}$ (4.7)
$\displaystyle{\mathrm{Gr}}^{2}_{*}({\mathcal{H}}^{\prime}):={\mathrm{Gr}}^{2}_{*,h}({\mathcal{H}}^{\prime})$
$\displaystyle=\\{(P,Q)\;|\;P,Q\in{\mathrm{Gr}}_{h}({\mathcal{H}}^{\prime}),(P,Q)\text{
is an invertible pair}\\}.$

From (4.1), an induced associated map between
${\mathrm{Gr}}^{2}_{F}({\mathcal{H}}^{\prime})$ and
${\mathfrak{L}}^{2}_{F}({\mathcal{H}}^{\prime})$ is given by

(4.8) $(P,Q)\mapsto(\ker
P,\operatorname{im}Q),\quad\quad({\mathrm{Id}}-P_{{\mathcal{L}}_{1}},P_{{\mathcal{L}}_{2}})\mapsfrom({\mathcal{L}}_{1},{\mathcal{L}}_{2}).$

An immediate adaptation of Lemma 2.4 to
${\mathrm{Gr}}_{h}({\mathcal{H}}^{\prime})$ is as follows.

######  Lemma 4.6.

Let $h={\mathrm{Diag}}(a,WaW^{*})\in H.$ Let
$P,Q\in{\mathrm{Gr}}_{h}({\mathcal{H}}^{\prime}).$ Then, $(P,Q)$ is an
equivariant Fredholm (resp. invertible) pair if and only if
$-1\notin{\text{spec}}_{ess}((a-{\mathrm{Id}})+aT^{*}S)$ (resp.
$-1\notin{\text{spec}}((a-{\mathrm{Id}})+aT^{*}S)).$ There is an isomorphism
between the $h$-space $\ker(P)\cap\operatorname{im}(Q)$ and the space
$\ker(a({\mathrm{Id}}+T^{*}S)).$

######  Definition 4.7.

For $h\in H,$ let
${\mathcal{L}}(t)=({\mathcal{L}}_{1}(t),{\mathcal{L}}_{2}(t)):[0,1]\to{\mathfrak{L}}^{2}_{F}({\mathcal{H}}^{\prime})$
be a continuous path of $h$-equivariant Fredholm Lagrangian pairs. Choose a
good grid partition. Then, the $h$-equivariant Maslov index

${\mathrm{Mas}}_{h}:\pi_{1}({\mathfrak{L}}^{2}_{F}({\mathcal{H}}^{\prime}))_{H}\to{\mathbb{C}}$

of the path ${\mathcal{L}}(t)$ is defined by

(4.9)
${\mathrm{Mas}}_{h}({\mathcal{L}}_{1}(t),{\mathcal{L}}_{2}(t)):=\sum^{n}_{j=1}\Big{[}{\mathrm{Tr}}\big{(}h|_{{\mathcal{L}}_{1}(t_{j})\cap{\mathcal{L}}_{2}(t_{j})}\big{)}-{\mathrm{Tr}}\big{(}h|_{{\mathcal{L}}_{1}(t_{j-1})\cap{\mathcal{L}}_{2}(t_{j-1})}\big{)}\Big{]}.$

######  Lemma 4.8.

For $h\in H,$ the equivariant Maslov index ${\mathrm{Mas}}_{h}$ is well-
defined.

###### Proof.

Suppose a good grid partition is chosen. Consider the grid over
$[t_{j-1},t_{j}].$ Without loss of generality, we may assume that the paths
${\mathcal{L}}_{1}(t)$ and ${\mathcal{L}}_{2}(t)$ do not intersect
transversally at the four vertices of the grid. Then, for any
$t_{d}\in[t_{j-1},t_{j}],$ by a similar argument as in the proof of
Proposition 3.4, we have

$\displaystyle{\mathrm{Mas}}_{h}({\mathcal{L}}_{1}(t),{\mathcal{L}}_{2}(t))_{\begin{subarray}{c}t_{j-1}\leq
t\leq t_{j}\end{subarray}}$
$\displaystyle=\Big{(}{\mathrm{Tr}}\big{(}h|_{{\mathcal{L}}_{1}(t_{j})\cap{\mathcal{L}}_{2}(t_{j})}\big{)}-{\mathrm{Tr}}\big{(}h|_{{\mathcal{L}}_{1}(t_{d})\cap{\mathcal{L}}_{2}(t_{d})}\big{)}\Big{)}$
$\displaystyle+\Big{(}{\mathrm{Tr}}\big{(}h|_{{\mathcal{L}}_{1}(t_{d})\cap{\mathcal{L}}_{2}(t_{d})}\big{)}-{\mathrm{Tr}}\big{(}h|_{{\mathcal{L}}_{1}(t_{j-1})\cap{\mathcal{L}}_{2}(t_{j-1})}\big{)}\Big{)}$
$\displaystyle={\mathrm{Tr}}\big{(}h|_{{\mathcal{L}}_{1}(t_{j})\cap{\mathcal{L}}_{2}(t_{j})}\big{)}-{\mathrm{Tr}}\big{(}h|_{{\mathcal{L}}_{1}(t_{j-1})\cap{\mathcal{L}}_{2}(t_{j-1})}\big{)}.$

Here, we include the possibility of paths whose ends do not meet the initial
of the path in the next grid. In the case where a grid partition is fixed and
the paths intersect transversally at some $t_{d}\in[t_{j-1},t_{j}],$ then we
compute ${\mathrm{Mas}}_{h}$ at $t_{d}\pm\epsilon$ for some $\epsilon>0$ such
that
${\mathcal{L}}_{1}(t_{d}\pm\epsilon)\cap{\mathcal{L}}_{2}(t_{d}\pm\epsilon)\neq
0.$ ∎

We state the following properties of ${\mathrm{Mas}}_{h}$ without proof.

######  Proposition 4.9.

For $h\in H,$ ${\mathrm{Mas}}_{h}$ satisfies the following properties.

  1. (1)

(Reversal) Let
$(\widetilde{{\mathcal{L}}_{1}}(t),\widetilde{{\mathcal{L}}_{2}}(t))$ be the
Lagrangian paths $({\mathcal{L}}_{1}(t),{\mathcal{L}}_{2}(t))$ traversing in
the opposite direction. Then,

${\mathrm{Mas}}_{h}(\widetilde{{\mathcal{L}}_{1}}(t),\widetilde{{\mathcal{L}}_{2}}(t))=-{\mathrm{Mas}}_{h}({\mathcal{L}}_{1}(t),{\mathcal{L}}_{2}(t)).$

  2. (2)

(Homotopy invariance) Let
$({\mathcal{L}}_{1}(t,s),{\mathcal{L}}_{2}(t,s)):[0,1]\times[0,1]\to{\mathfrak{L}}^{2}_{F}({\mathcal{H}}^{\prime})$
be a continuous path with
${\mathcal{L}}_{1}(t,0)={\mathcal{L}}_{1}(t),{\mathcal{L}}_{1}(t,1)={\mathcal{L}}^{\prime}_{1}(t)$
and
${\mathcal{L}}_{2}(t,0)={\mathcal{L}}_{2}(t),{\mathcal{L}}_{2}(t,1)={\mathcal{L}}^{\prime}_{2}(t).$
Then,

${\mathrm{Mas}}_{h}({\mathcal{L}}_{1}(t),{\mathcal{L}}_{2}(t))={\mathrm{Mas}}_{h}({\mathcal{L}}^{\prime}_{1}(t),{\mathcal{L}}^{\prime}_{2}(t)).$

  3. (3)

(Path additivity) Let $({\mathcal{L}}_{1}(t),{\mathcal{L}}_{2}(t))$ and
$({\mathcal{L}}^{\prime}_{1}(t),{\mathcal{L}}^{\prime}_{2}(t))$ be two
Lagrangian paths in
$\pi_{1}({\mathfrak{L}}^{2}_{F}({\mathcal{H}}^{\prime}))_{H}$ such that
${\mathcal{L}}_{i}(1)={\mathcal{L}}^{\prime}_{i}(0)$ for $i=1,2.$ Then,
${\mathrm{Mas}}_{h}$ is additive with respect to the concatenation of paths,
i.e.

$\displaystyle{\mathrm{Mas}}_{h}(({\mathcal{L}}_{1}(t)$
$\displaystyle*{\mathcal{L}}^{\prime}_{1}(t),{\mathcal{L}}_{2}(t)*{\mathcal{L}}^{\prime}_{2}(t)))$
$\displaystyle={\mathrm{Mas}}_{h}(({\mathcal{L}}_{1}(t),{\mathcal{L}}_{2}(t))+{\mathrm{Mas}}_{h}({\mathcal{L}}^{\prime}_{1}(t),{\mathcal{L}}^{\prime}_{2}(t))).$

  4. (4)

The map ${\mathrm{Mas}}_{h}$ is natural under any $h$-equivariant isomorphism
between ${\mathcal{H}}^{\prime}.$

  5. (5)

If $h$ acts trivially, then the $h$-equivariant Maslov index coincides with
the classical Maslov index, i.e. ${\mathrm{Mas}}_{h}={\mathrm{Mas}}.$ If,
furthermore, ${\mathcal{L}}_{1}(t)\cap{\mathcal{L}}_{2}(t)$ is of constant
dimension for all $t,$ then
${\mathrm{Mas}}_{h}({\mathcal{L}}_{1}(t),{\mathcal{L}}_{2}(t))=0.$

##  5\. Equivariant winding number

Let ${\mathcal{H}}^{\prime}$ be the $H$-module Hilbert space defined by
(2.11). Let ${\mathcal{U}}({\mathcal{H}}^{\prime})$ be the unitary group of
${\mathcal{H}}^{\prime}$. Let ${\mathcal{U}}_{F}({\mathcal{H}}^{\prime})$
(resp. ${\mathcal{U}}_{*}({\mathcal{H}}^{\prime})$) be the subset that
consists of the unitary operator $T\in{\mathcal{U}}({\mathcal{H}}^{\prime})$
satisfying $-1\notin{\text{spec}}_{ess}(T)$ (resp.
$-1\notin{\text{spec}}(T)$). They are not groups since for any unitary
operators $T,S$ in ${\mathcal{U}}_{F}({\mathcal{H}}^{\prime})$ or in
${\mathcal{U}}_{*}({\mathcal{H}}^{\prime}),$ their product $TS$ does not
necessarily lie in the set. On the other hand, let
${\mathcal{U}}_{\mathcal{K}}({\mathcal{H}}^{\prime})$ (resp.
${\mathcal{U}}_{\mathrm{Tr}}({\mathcal{H}}^{\prime})$) be the subgroup that
consists of the unitary operator $T\in{\mathcal{U}}({\mathcal{H}}^{\prime})$
such that $T-{\mathrm{Id}}\in{\mathcal{K}}({\mathcal{H}}^{\prime})$ (resp.
$T-{\mathrm{Id}}$ is trace class).

######  Lemma 5.1.

For $h\in H.$ Let ${\mathrm{Gr}}^{2}_{F}({\mathcal{H}}^{\prime})$ and
${\mathrm{Gr}}^{2}_{*}({\mathcal{H}}^{\prime})$ be as in Definition 4.5. Then,
there are diffeomorphisms

(5.1)
${\mathcal{U}}_{F}({\mathcal{E}}_{i})\cong{\mathrm{Gr}}^{2}_{F}({\mathcal{H}}^{\prime}),\quad\quad{\mathcal{U}}_{*}({\mathcal{E}}_{i})\cong{\mathrm{Gr}}^{2}_{*}({\mathcal{H}}^{\prime}).$

###### Proof.

Each Fredholm (resp. invertible) pair $(P,Q)$ is mapped to $T^{*}S,$ where
$T=\Phi(P)$ and $S=\Phi(Q)$ in view of (2.7). In particular,
$T^{*}S\in{\mathcal{U}}_{F}({\mathcal{E}}_{i})$ (resp.
$T^{*}S\in{\mathcal{U}}_{*}({\mathcal{E}}_{i})$). Conversely, for any unitary
operators $T:{\mathcal{E}}_{i}\to{\mathcal{E}}_{-i},$ we have
$P=\frac{1}{2}\begin{pmatrix}{\mathrm{Id}}&T^{*}\\\
T&{\mathrm{Id}}\end{pmatrix}\in{\mathrm{Gr}}_{h}({\mathcal{H}}^{\prime}).$
Similarly, for an unitary $S:{\mathcal{E}}_{i}\to{\mathcal{E}}_{-1}$ we get
$Q=\frac{1}{2}\begin{pmatrix}{\mathrm{Id}}&S^{*}\\\
S&{\mathrm{Id}}\end{pmatrix}\in{\mathrm{Gr}}_{h}({\mathcal{H}}^{\prime}).$ One
verifies that such a $(P,Q)$ is a Fredholm (resp. invertible) pair. ∎

######  Lemma 5.2.

${\mathcal{U}}_{\mathcal{K}}({\mathcal{H}}^{\prime})$ is weakly $H$-homotopy
equivalent to ${\mathcal{U}}_{F}({\mathcal{H}}^{\prime}).$

###### Proof.

This is an equivariant adaptation of [15, Lemma 6.1]. Let
$\pi:{\mathcal{B}}({\mathcal{H}}^{\prime})\to{\mathcal{Q}}({\mathcal{H}}^{\prime})$
be the quotient map as in Section 3. Then,
$\pi:{\mathcal{U}}_{F}({\mathcal{H}}^{\prime})\to{\mathcal{U}}_{*}{\mathcal{Q}}({\mathcal{H}}^{\prime})$
is a $H$-Hilbert bundle with fibres
${\mathcal{U}}_{\mathcal{K}}({\mathcal{H}}^{\prime}).$ The action of $H$ acts
accordingly by the rule $(gh)(u)=u(g^{-1}h)$ for $g,h\in H$ and
$u\in{\mathcal{U}}_{\mathcal{K}},{\mathcal{U}}_{F}$ and
${\mathcal{U}}_{*}{\mathcal{Q}}.$ By [18, Theorem 3.5],
${\mathcal{U}}_{*}{\mathcal{Q}}({\mathcal{H}}^{\prime})\simeq
GL({\mathcal{Q}}({\mathcal{H}}^{\prime}))$ is $H$-contractible. ∎

Since ${\mathcal{U}}_{\mathrm{Tr}}({\mathcal{H}}^{\prime})$ is known to be
homotopy equivalent to ${\mathcal{U}}_{\mathcal{K}}({\mathcal{H}}^{\prime}),$
the following holds.

######  Corollary 5.3.

$\pi_{1}({\mathcal{U}}_{\mathrm{Tr}}({\mathcal{H}}^{\prime}))_{H}\cong\pi_{1}({\mathcal{U}}_{\mathcal{K}}({\mathcal{H}}^{\prime}))_{H}\cong\pi_{1}({\mathcal{U}}_{F}({\mathcal{H}}^{\prime}))_{H}.$

Without loss of generality, consider an equivariant path of unitary operators
$f:[0,1]\to{\mathcal{U}}_{F}$ whose endpoints are in ${\mathcal{U}}_{*},$ i.e.
the path $f(t)$ is an element of
$\pi_{1}({\mathcal{U}}_{F}({\mathcal{H}}^{\prime}),{\mathcal{U}}_{*}({\mathcal{H}}^{\prime}))_{H},$
or equivalently, of $\pi_{1}({\mathcal{U}}_{F}({\mathcal{H}}^{\prime}))_{H}$
by the $H$-contractibility of ${\mathcal{U}}_{*}.$ If the endpoints of $f(t)$
are not in ${\mathcal{U}}_{*}({\mathcal{H}}^{\prime}),$ then we may perturb
the endpoints $f(k)$ by $e^{-i\epsilon}$ for some $\epsilon>0$ such that
$-1\notin{\text{spec}}(f(k)e^{-i\epsilon})$ for $k=0,1.$ This means that for
$t\in[0,1],$ by a perturbation of the endpoints, a general equivariant path
$f(t)$ can be concatenated to another path
$p(t)\in{\mathcal{U}}_{\ast}({\mathcal{H}}^{\prime})$ such that $p(0)=f(1)$
and $p(1)=f(0),$ forming a loop $f(t)\ast p(t).$ This concatenation is
independent of the choice of $p(t).$ This suggests that the consideration of
an equivariant loop can be relaxed a little to an equivariant path of unitary
operators, with perturbed endpoints if necessary.

As inspired by [19, Lecture 10], we give the following definition.

######  Definition 5.4.

For $h\in H,$ define the equivariant Fredholm determinant by

${\mathrm{det}}_{h}:\pi_{1}({\mathcal{U}}_{\mathrm{Tr}}({\mathcal{H}}^{\prime}))_{H}\to{\mathbb{C}}$
(5.2)
${\mathrm{det}}_{h}(f(t)):=\exp\Big{(}\int^{1}_{0}{\mathrm{Tr}}_{h}(f(t)^{-1}\frac{d}{dt}f(t))dt\Big{)}$

where ${\mathrm{Tr}}_{h}(f^{-1}df):={\mathrm{Tr}}(hf^{-1}df).$

######  Lemma 5.5.

For $h\in H,$ the equivariant Fredholm determinant ${\mathrm{det}}_{h}$ is
multiplicative, i.e.
${\mathrm{det}}_{h}(f_{t}g_{t})={\mathrm{det}}_{h}(f_{t}){\mathrm{det}}_{h}(g_{t}).$

###### Proof.

Let
$f_{t}=f(t),g_{t}=g(t)\in\pi_{1}({\mathcal{U}}_{\mathrm{Tr}}({\mathcal{H}}^{\prime}))_{H}.$
By definition, we have
${\mathrm{Tr}}_{h}((f_{t}g_{t})^{-1}d(f_{t}g_{t}))={\mathrm{Tr}}(h(f_{t}g_{t})^{-1}d(f_{t}g_{t})).$
Then, by the conjugation invariance of trace, we have

$\displaystyle{\mathrm{Tr}}(h(f_{t}g_{t})^{-1}d(f_{t}g_{t}))$
$\displaystyle={\mathrm{Tr}}(hg^{-1}_{t}f^{-1}_{t}[(df_{t})g_{t}+f_{t}(dg_{t})])$
$\displaystyle={\mathrm{Tr}}(g^{-1}_{t}(hf^{-1}_{t}(df_{t}))g_{t}+hg^{-1}_{t}(dg_{t}))$
$\displaystyle={\mathrm{Tr}}(hf^{-1}_{t}(df_{t}))+{\mathrm{Tr}}(hg^{-1}_{t}(dg_{t})).$

By taking integration and exponential, the lemma follows. ∎

######  Definition 5.6.

For $h\in H.$ Assume that a good grid partition is chosen. Define the
equivariant winding number by

$w_{h}(f(t)):\pi_{1}({\mathcal{U}}_{F}({\mathcal{H}}^{\prime}))_{H}\to{\mathbb{C}}$
(5.3) $w_{h}(f(t))=\sum^{n}_{\begin{subarray}{c}j=1\\\
0\leq\theta_{i}<\epsilon_{i}\end{subarray}}\left[{\mathrm{Tr}}\left(h|_{\ker\left(f(t_{j})-e^{i(\pi+\theta_{j})}\right)}\right)-{\mathrm{Tr}}\left(h|_{\ker\left(f(t_{j-1})-e^{i(\pi+\theta_{j-1})}\right)}\right)\right].$

In analogy to [15, Proposition 5.7], we have an equivariant analog of some
properties of winding number.

######  Proposition 5.7.

For $h\in H,$ $w_{h}$ satisfies the following properties.

  1. (a)

(Reversal) Let $\widetilde{f}(t)$ be the path $f(t)$ traversing in the
opposite direction. Then,

$w_{h}(\widetilde{f}(t))=-w_{h}(f(t)).$

  2. (b)

(Homotopy invariance) Let
$f(t),f^{\prime}(t)\in\pi_{1}({\mathcal{U}}_{F}({\mathcal{H}}^{\prime}))_{H}$
be such that $f(0)=f^{\prime}(0).$ Then,

(5.4) $w_{h}(f(t))=w_{h}(f^{\prime}(t)).$

  3. (c)

(Path additivity) Let $f(t)=f_{1}(t)\ast f_{2}(t)$ be the concatenation of two
paths. Then,

(5.5) $w_{h}(f(t))=w_{h}(f_{1}(t))+w_{h}(f_{2}(t)).$

  4. (d)

If $f\equiv f(t)$ for all $t,$ then,

(5.6) $w_{h}(f)=0.$

  5. (e)

Let $f:[0,1]\to{\mathcal{U}}_{{\mathrm{Tr}}}({\mathcal{H}}^{\prime})$ is a
$C^{1}$-curve. Then,

(5.7) $w_{h}(f(t))=\frac{1}{2\pi
i}\Big{(}\int^{1}_{0}{\mathrm{Tr}}\log(hf(t))dt-{\mathrm{Tr}}\log(hf(1))+{\mathrm{Tr}}\log(hf(0))\Big{)}.$

######  Theorem 5.8.

Let $h={\mathrm{Diag}}(a,WaW^{*})\in H.$ Let
$({\mathcal{L}}_{1}(t),{\mathcal{L}}_{2}(t))\in\pi_{1}({\mathfrak{L}}^{2}_{F}({\mathcal{H}}^{\prime}))_{H}.$
Let $T(t)$ and $S(t)$ be the associated unitary operators of
${\mathcal{L}}_{1}(t)$ and ${\mathcal{L}}_{2}(t)$ respectively. Then,

(5.8)
${\mathrm{Mas}}_{h}({\mathcal{L}}_{1}(t),{\mathcal{L}}_{2}(t))=w_{h}(T^{*}(t)S(t))=w(aT^{*}(t)S(t))$

where $w$ is the usual winding number.

###### Proof.

Let
$(P(t),Q(t))\in\pi_{1}({\mathrm{Gr}}^{2}_{F}({\mathcal{H}}^{\prime}))_{H}.$ By
(4.8), we have the map
$({\mathcal{L}}_{1}(t),{\mathcal{L}}_{2}(t))\mapsto({\mathrm{Id}}-P_{{\mathcal{L}}_{1}(t)},P_{{\mathcal{L}}_{2}(t)}).$
Then, the claim follows from Definition 4.7, Definition 5.6 and Lemma 4.6 that
there is an isomorphism between the $h$-space
$\ker(P(t))\cap\operatorname{im}(Q(t))$ and
$\ker(a({\mathrm{Id}}+T^{*}(t)S(t)))$ for all $t.$ ∎

Let $U\in{\mathcal{U}}_{F}({\mathcal{H}}^{\prime}).$ Let
${\mathcal{H}}_{0}:=\ker(U+{\mathrm{Id}})$ and
${\mathcal{H}}_{1}:={\mathcal{H}}_{0}^{\perp}.$ Decompose
${\mathcal{H}}^{\prime}$ into the sum

(5.9) ${\mathcal{H}}^{\prime}={\mathcal{H}}_{0}\oplus{\mathcal{H}}_{1}.$

Then, the unitary operator $U$ splits accordingly into
$-{\mathrm{Id}}_{{\mathcal{H}}_{0}}\oplus\widetilde{U}$ where
$-1\notin\text{spec}(\widetilde{U}).$ For $t\in[0,1],$ construct a
contractible path by

$f(t)=-{\mathrm{Id}}_{{\mathcal{H}}_{0}}\oplus\exp(t\log\widetilde{U}),$

cf.[15, Lemma 6.1(2)]. Let $h={\mathrm{Diag}}(a,WaW^{*})$ where
$a\in{\mathcal{U}}({\mathcal{E}}_{i}).$ Consider an $h$-equivariant path
$f_{a}(t)$ whose $h$-action is given by

(5.10) $f_{a}(t)=a\odot
f(t):=-a\>{\mathrm{Id}}_{{\mathcal{H}}_{0}}\oplus\exp(t\log(a\widetilde{U})),$

where $a$ is independent of $t\in[0,1].$ Similarly, by splitting
$V\in{\mathcal{U}}_{\mathcal{K}}({\mathcal{H}}^{\prime})$ into
$-{\mathrm{Id}}_{{\mathcal{H}}_{0}}\oplus\widetilde{V}$ with respect to (5.9),
we consider

(5.11)
$g_{a}(t)=-a\>{\mathrm{Id}}_{{\mathcal{H}}_{0}}\oplus\exp(t\log(a\widetilde{V})).$

Then, by the ideal property of the group of compact operators
${\mathcal{K}}={\mathcal{K}}({\mathcal{H}}^{\prime}),$ the multiplication

(5.12)
$q_{a}(t):=a\odot(f(t)g(t))=-a{\mathrm{Id}}_{{\mathcal{H}}_{0}}\oplus\exp(t\log(a\widetilde{U}\widetilde{V}))$

is well-defined. This also works for
$U\in{\mathcal{U}}_{\mathcal{K}}({\mathcal{H}}^{\prime}),V\in{\mathcal{U}}_{F}({\mathcal{H}}^{\prime})$
or $U,V\in{\mathcal{U}}_{\mathcal{K}}({\mathcal{H}}^{\prime}).$

The following is a slight generalisation of double index in [15, Definition
6.2].

######  Definition 5.9.

Let $h={\mathrm{Diag}}(a,WaW^{*})\in H.$ Let
$U\in{\mathcal{U}}_{F}({\mathcal{H}}^{\prime})$ and
$V\in{\mathcal{U}}_{\mathcal{K}}({\mathcal{H}}^{\prime}).$ Let
$f(t)\in\pi_{1}({\mathcal{U}}_{\mathcal{F}}({\mathcal{H}}^{\prime}))_{H}$ and
$g(t),q(t)\in\pi_{1}({\mathcal{U}}_{\mathcal{K}}({\mathcal{H}}^{\prime}))_{H}.$
Define the $h$-equivariant double index of $U$ and $V$ by

$\displaystyle\tau_{h}$
$\displaystyle:{\mathcal{U}}_{\mathcal{F}}({\mathcal{H}}^{\prime})\times{\mathcal{U}}_{\mathcal{K}}({\mathcal{H}}^{\prime})\longrightarrow{\mathbb{C}}$
(5.13) $\displaystyle\tau_{h}(U,V)$
$\displaystyle:=w_{h}(f(t))+w_{h}(g(t))-w_{h}(q(t)).$

where $w_{h}(f(t)):=w(f_{a}(t))$ and the same for $g(t)$ and $q(t)$
respectively.

######  Proposition 5.10.

  1. (a)

If either one of the unitary operators is the identity ${\mathrm{Id}},$ then

$\tau_{h}(U,V)=0.$

  2. (b)

If $V=U^{-1},$ then

$\tau_{h}(U,U^{-1})=w_{h}(f(t))+w_{h}(f^{-1}(t)).$

###### Proof.

For (a), if both $U$ and $V$ are ${\mathrm{Id}},$ then

$\tau_{h}({\mathrm{Id}},{\mathrm{Id}})=2w_{h}({\mathrm{Id}})-w_{h}({\mathrm{Id}})=w_{h}({\mathrm{Id}})=w(a)=0$

where the last equality follows from Proposition 5.7(d) as $a$ is independent
of $t.$ If $U={\mathrm{Id}}$ but $V\neq{\mathrm{Id}},$ then
$q_{a}(t)=g_{a}(t)$ and $\tau_{h}({\mathrm{Id}},V)=w(g_{a}(t))-w(g_{a}(t))=0.$
The same applies when $V={\mathrm{Id}}$ but $U\neq{\mathrm{Id}}.$ For part
(b), let $V=U^{-1}.$ Choose222Once we fix both ends $g_{a}(0)={\mathrm{Id}}$
and $g_{a}(1)=U^{-1},$ this is possible to obtain by homotopy invariance.
$g_{a}(t)=f^{-1}_{a}(t)$ for all $t.$ Then,

$\displaystyle\tau_{h}(U,U^{-1})$
$\displaystyle=w_{h}(f(t))+w_{h}(f^{-1}(t))-w_{h}(f(t)f^{-1}(t))$
$\displaystyle=w_{h}(f(t))+w_{h}(f^{-1}(t))-w_{h}({\mathrm{Id}})$
$\displaystyle=w_{h}(f(t))+w_{h}(f^{-1}(t)).$

∎

######  Remark 5.11.

  1. (1)

When the unitary operator $a$ is the identity operator, then the above
definition and proposition reduce to the non-equivariant version of [15].

  2. (2)

All paths here are in the general position. Hence, it is valid to consider the
unitary operator $U\in{\mathcal{U}}_{F}({\mathcal{H}}^{\prime})$ as an
endpoint of $f_{a}(t)$ due to the perturbation term $e^{\pm i\theta}$ in
(5.3). This would ensure that $f_{a}(t)\pm e^{i\theta}\cdot{\mathrm{Id}}$ is
compact.

######  Corollary 5.12.

Let $h={\mathrm{Diag}}(a,WaW^{*})\in H.$ For $t\in[0,1],$ let
$({\mathcal{L}}_{0}(t),{\mathcal{L}}_{1}(t))$ be the pair of equivariant
Lagrangian paths which correspond to the equivariant path of unitary operators
$T^{*}(t)S(t).$ Let $U=T^{*}(1)S(1).$ Then,

(5.14)
${\mathrm{Mas}}_{h}({\mathcal{L}}_{0}(t),{\mathcal{L}}_{1}(t))+{\mathrm{Mas}}_{h}({\mathcal{L}}_{1}(t),{\mathcal{L}}_{0}(t))=\tau_{h}(aU,aU^{-1}).$

###### Proof.

By Theorem 5.8, we have
${\mathrm{Mas}}_{h}({\mathcal{L}}_{1}(t),{\mathcal{L}}_{0}(t))=w_{h}(S^{*}(t)T(t)).$
Since both $T(t)$ and $S(t)$ are paths of unitary operators, we have
$S^{*}(t)T(t)=(T^{*}(t)S(t))^{-1}.$ By Proposition 5.10(b), we obtain

$\displaystyle{\mathrm{Mas}}_{h}({\mathcal{L}}_{0}(t)$
$\displaystyle,{\mathcal{L}}_{1}(t))+{\mathrm{Mas}}_{h}({\mathcal{L}}_{1}(t),{\mathcal{L}}_{0}(t))$
$\displaystyle=w_{h}(T^{*}(t)S(t))+w_{h}((T^{*}(t)S(t))^{-1})=\tau_{h}(U,U^{-1}).$

∎

This tells us that, in general, the equivariant Maslov index of a Lagrangian
pair is non-commutative. For a non-trivial pair, this is commutative when the
inverse path $g(t)$ is also the reversal of $f(t)=T^{*}(t)S(t),$ i.e. for all
$t\in[0,1],$

$(T^{*}(t)S(t))^{-1}=T(1-t)S(1-t).$

Explicitly, the reversal is given by

$\overline{g}(t)=-{\mathrm{Id}}_{{\mathcal{H}}_{0}}\oplus\exp((1-t)\log\widetilde{U})=-{\mathrm{Id}}_{{\mathcal{H}}_{0}}\oplus\widetilde{U}\exp(-t\log\widetilde{U})$

and it is evidently to see that $q(t)$ reduces to the identity, which leads to
the vanishing of $\tau_{h}(U,U^{-1})$ for $U=T^{*}(1)S(1).$ Next, since any
two unitary operators can be connected to the identity operator by (5.10), we
can therefore consider general $h$-equivariant paths whose endpoints are not
necessarily the identity operators. This allows us to define a double index
relative to the identity.

######  Definition 5.13.

Let $h\in H.$ For $t\in[0,1],$ let $f(t)$ and $g(t)$ be any $h$-equivariant
continuous paths in ${\mathcal{U}}_{F}({\mathcal{H}}^{\prime})$ and
${\mathcal{U}}_{\mathcal{K}}({\mathcal{H}}^{\prime})$ respectively. Define the
relative $h$-double index of $f(t)$ and $g(t)$ by

(5.15) $\displaystyle\tau_{h}(f(t),g(t))$
$\displaystyle:=\tau_{h}(f(1),g(1))-\tau_{h}(f(0),g(0))$
$\displaystyle=w_{h}(f(t))+w_{h}(g(t))-w_{h}(q(t)).$

To see that this is well-defined, we consider the following skeleton path
diagram.

Let $f$ and $g$ be any $h$-equivariant paths as in Definition 5.13. Let
$\overline{\phi}$ and $\overline{\varphi}$ be the reversal of $\phi$ and
$\varphi$ respectively. Then, $f=\chi_{1}\ast\overline{\phi}$ and
$g=\chi_{2}\ast\overline{\varphi}.$ By Proposition 5.10(c), we have

(5.16) $\displaystyle w_{h}(f)=w_{h}(\chi_{1})+w_{h}(\overline{\phi}),\quad
w_{h}(g)=w_{h}(\chi_{2})+w_{h}(\overline{\varphi}),$ (5.17) $\displaystyle
w_{h}(fg)=w_{h}((\chi_{1}\ast\overline{\phi})(\chi_{2}\ast\overline{\varphi}))=w_{h}(\chi_{1}\chi_{2})+w_{h}(\overline{\phi}\overline{\varphi}).$

The last equality can be seen as follows. Construct a path explicitly by

(5.18)
$(\chi_{1}\ast\overline{\phi})(\chi_{2}\ast\overline{\varphi})=\begin{cases}-{\mathrm{Id}}\oplus\exp((1-2t)\log(\widetilde{\overline{\phi}}(0)\widetilde{\overline{\varphi}}(0))),\quad\quad
0\leq t\leq\frac{1}{2},\\\
-{\mathrm{Id}}\oplus\exp((2t-1)\log(\widetilde{\chi}_{1}(1)\widetilde{\chi}_{2}(1))),\quad\frac{1}{2}\leq
t\leq 1,\end{cases}$

where the four operators split into the form
$r=-{\mathrm{Id}}_{{\mathcal{H}}_{0}}\oplus\widetilde{r}$ such that
$-1\notin\text{spec}(\widetilde{r}).$ Then, we observe that the concatenation
of the paths (5.18) coincide with
$(\chi_{1}\chi_{2})\ast(\overline{\phi}\overline{\varphi}).$ By Proposition
5.10(c), we obtain (5.17).

Then, by (5.16),(5.17), Proposition 5.10(a) and (5.15), we obtain

$\displaystyle\tau_{h}(f(1),g(1))$ $\displaystyle=w_{h}(f)+w_{h}(g)-w_{h}(fg)$
$\displaystyle=w_{h}(\chi_{1})+w_{h}(\overline{\phi})+w_{h}(\chi_{2})+w_{h}(\overline{\varphi})+w_{h}(\chi_{1}\chi_{2})+w_{h}(\overline{\phi}\overline{\varphi})$
$\displaystyle=\tau_{h}(\chi_{1}(1),\chi_{2}(1))-\tau_{h}(\phi(1),\varphi(1)).$

This justifies the term “relative” for the double index where $f$ and $g$ are
general paths with possibly non-trivial endpoints. Now, we conclude some
properties of the $h$-double index in the following, which are essentially an
extension from that of the $h$-winding number $w_{h}$. They are also a slight
generalisation of [15, Proposition 6.3].

######  Proposition 5.14.

For $h\in H,$ $\tau_{h}$ satisfies the following properties.

  1. (1)

(Reversal) Let $\widetilde{f}(t)$ and $\widetilde{g}(t)$ be the respective
paths $f(t)$ and $g(t)$ with both traversing in the opposite direction. Then,

$\tau_{h}(\widetilde{f}(t),\widetilde{g}(t))=-\tau_{h}(f(t),g(t)).$

  2. (2)

(Homotopy invariance) For $i=0,1,$ let $f(t),f^{\prime}(t)$ be two paths in
${\mathcal{U}}_{F}({\mathcal{H}}^{\prime})$ such that $f(i)=f^{\prime}(i)$ and
$g(t),g^{\prime}(t)$ be two paths in
${\mathcal{U}}_{\mathcal{K}}({\mathcal{H}}^{\prime})$ such that
$g(i)=g^{\prime}(i).$ Then,

$\tau_{h}(f(t),g(t))=\tau_{h}(f^{\prime}(t),g^{\prime}(t)).$

  3. (3)

(Path additivity) Consider

$f(t)=\begin{cases}f_{1}(2t),\quad\quad\quad 0\leq t\leq\frac{1}{2},\\\
f_{2}(2t-1),\;\quad\frac{1}{2}\leq t\leq 1.\end{cases}$

Let $f(t)=f_{1}(t)\ast f_{2}(t)$ and $g(t)=g_{1}(t)\ast g_{2}(t)$ be the
concatenation of two paths in ${\mathcal{U}}_{F}({\mathcal{H}}^{\prime})$ and
${\mathcal{U}}_{\mathcal{K}}({\mathcal{H}}^{\prime})$ respectively. Then,

$\tau_{h}(f(t),g(t))=\tau_{h}(f_{1}(t),g_{1}(t))+\tau_{h}(f_{2}(t),g_{2}(t)).$

  4. (4)

Let $t\in[0,1].$ Let $f\equiv f(t)$ and $g\equiv g(t)$ be two constant paths
in ${\mathcal{U}}_{F}({\mathcal{H}}^{\prime})$ and
${\mathcal{U}}_{\mathcal{K}}({\mathcal{H}}^{\prime})$ respectively. Then,

$\tau_{h}(f,g)=0.$

##  6\. Equivariant Maslov triple index

Inspired by [15], we also consider an equivariant version of Maslov triple
indices. For simplicity, we write
$P_{t}:=P(t),T_{t}:=T(t),{\mathcal{L}}_{i}:={\mathcal{L}}_{i}(t),$ and etc in
the following.

######  Definition 6.1.

Let $h\in H.$ Let $P_{t},Q_{t},N_{t}$ be continuous paths in
${\mathrm{Gr}}_{h}({\mathcal{H}}^{\prime})$ such that every pair
$(P_{t},Q_{t}),(Q_{t},N_{t})$ and $(P_{t},N_{t})$ are $h$-Fredholm and at
least one of the differences $P_{t}-Q_{t},Q_{t}-N_{t}$ and $P_{t}-N_{t}$ is
compact. Let $T_{t},S_{t}$ and $R_{t}$ be the associated unitary operators of
$P_{t},Q_{t}$ and $N_{t}$ respectively. Then, define the $h$-Maslov triple
index $\tau^{h}_{\mu}$ of $(P_{t},Q_{t},N_{t})$ by

(6.1)
$\tau^{h}_{\mu}(P_{t},Q_{t},N_{t}):=\tau_{h}(T^{*}_{t}S_{t},S^{*}_{t}R_{t}).$

This is well-defined relative to two endpoints as a consequence of the well-
definedness of $\tau_{h}$ and thus inherits the properties in Proposition
5.14. Moreover, equivariant Maslov triple indices can be expressed as some
mixed terms of equivariant Maslov double indices:

(6.2) $\displaystyle\tau^{h}_{\mu}$
$\displaystyle(P_{1},Q_{1},N_{1})-\tau^{h}_{\mu}(P_{0},Q_{0},N_{0})$
$\displaystyle=\tau_{h}(T^{*}_{1}S_{1},S^{*}_{1}R_{1})-\tau_{h}(T^{*}_{0}S_{0},S^{*}_{0}R_{0})$
$\displaystyle=w_{h}(T^{*}_{t}S_{t})+w_{h}(S^{*}_{t}R_{t})-w_{h}(T^{*}_{t}R_{t})$
$\displaystyle={\mathrm{Mas}}_{h}({\mathcal{L}}_{0},{\mathcal{L}}_{1})+{\mathrm{Mas}}_{h}({\mathcal{L}}_{1},{\mathcal{L}}_{2})-{\mathrm{Mas}}_{h}({\mathcal{L}}_{0},{\mathcal{L}}_{2}).$

######  Proposition 6.2.

For $h\in H$, $\tau^{h}_{\mu}$ satisfies the following properties.

(6.3) $\displaystyle\tau^{h}_{\mu}(Q_{t},P_{t},N_{t})$
$\displaystyle=-\tau^{h}_{\mu}(P_{t},Q_{t},N_{t})+\tau_{h}(T^{*}_{1}S_{1},(T^{*}_{1}S_{1})^{-1})$
(6.4) $\displaystyle\tau^{h}_{\mu}(P_{t},N_{t},Q_{t})$
$\displaystyle=-\tau^{h}_{\mu}(P_{t},Q_{t},N_{t})+\tau_{h}(S^{*}_{1}R_{1},(S^{*}_{1}R_{1})^{-1})$
(6.5) $\displaystyle\tau^{h}_{\mu}(N_{t},Q_{t},P_{t})$
$\displaystyle=-\tau^{h}_{\mu}(P_{t},Q_{t},N_{t})+\tau_{h}(T^{*}_{1}S_{1},(T^{*}_{1}S_{1})^{-1})$
$\displaystyle\quad\qquad+\tau_{h}(S^{*}_{1}R_{1},(S^{*}_{1}R_{1})^{-1})-\tau_{h}(T^{*}_{1}R_{1},(T^{*}_{1}R_{1})^{-1}).$

###### Proof.

For (6.3), we obtain by a direct computation

$\displaystyle\tau^{h}_{\mu}(P_{t},Q_{t},N_{t})+\tau^{h}_{\mu}(Q_{t},P_{t},N_{t})$
$\displaystyle=\tau_{h}(T^{*}_{t}S_{t},S^{*}_{t}R_{t})+\tau_{h}(S^{*}_{t}T_{t},T^{*}_{t}R_{t})$
$\displaystyle=w_{h}(T^{*}_{t}S_{t})+w_{h}(S^{*}_{t}R_{t})-w_{h}(T^{*}_{t}R_{t})$
$\displaystyle\quad\quad+w_{h}(S^{*}_{t}T_{t})+w_{h}(T^{*}_{t}R_{t})-w_{h}(S^{*}_{t}R_{t})$
$\displaystyle=\tau_{h}(T^{*}_{1}S_{1},(T^{*}_{1}S_{1})^{-1}).$

Equalities (6.4) and (6.5) can be obtained similarly, so we leave them to
readers. ∎

######  Corollary 6.3.

Let $h\in H.$

  1. (a)

If $P_{t}=Q_{t}$ for all $t,$ then

$\tau^{h}_{\mu}(P_{t},P_{t},N_{t})=0.$

  2. (b)

If $Q_{t}=N_{t}$ for all $t,$ then

$\tau^{h}_{\mu}(P_{t},Q_{t},Q_{t})=0.$

  3. (c)

If $P_{t}=N_{t}$ for all $t,$ then

$\tau^{h}_{\mu}(P_{t},Q_{t},P_{t})=\tau_{h}(T^{*}_{1}S_{1},(T^{*}_{1}S_{1})^{-1}).$

  4. (d)

If $P_{t}=Q_{t}=N_{t}$ for all $t,$ then

$\tau^{h}_{\mu}(P_{t},P_{t},P_{t})=0.$

##  7\. Equivariant $\zeta$-determinants

######  Definition 7.1.

Let $h\in H.$ Define the $h$-equivariant $\zeta$-function of a self-adjoint
elliptic equivariant differential operator $D=D_{P}$ of degree $d>0$ by

(7.1)
$\zeta_{h}(D,s):={\mathrm{Tr}}(hD^{-s})=\sum_{\lambda\in{\text{spec}}(D)\backslash\\{0\\}}{\mathrm{Tr}}(h^{*}_{\lambda})\lambda^{-s}$

where $h^{*}_{\lambda}$ is the induced linear map on the $\lambda$-eigenspace.

######  Definition 7.2.

Let $\\{\lambda,\phi_{\lambda}\\}$ be a spectral decomposition of $D=D_{P}.$
Let

$K(t,x,y)=\sum_{\lambda}e^{-\lambda
t}\phi_{\lambda}(x)\otimes\bar{\phi}_{\lambda}(y)$

be the smooth kernels of the heat operator $e^{-tD}.$ Let $h\in H$ for which
the induced map $h^{*}$ acts with respect to the second component $(x,hy),$
then

(7.2) ${\mathrm{Tr}}(h^{*}K(t,x,hy))=\sum_{\lambda}e^{-\lambda
t}\phi_{\lambda}(x)\otimes\bar{\phi}_{\lambda}(hy).$

Define the corresponding $h$-kernel by

(7.3) $K(h,t,x,y):=\int_{M}{\mathrm{Tr}}(h^{*}K(t,x,hy))d\text{vol}(y)$

and the $h$-trace by

(7.4)
${\mathrm{Tr}}(he^{-tD}):=\int_{M}{\mathrm{Tr}}(h^{*}K(t,x,hx))d\text{vol}(x)=\sum_{k}{\mathrm{Tr}}(h^{*}_{\lambda_{k}})e^{-\lambda_{k}t}.$

######  Remark 7.3.

It was shown in [14] and [6, §6.3] that the heat kernel $K(h,t)$ has an
asymptotic expansion as $t\to 0.$ For $x$ not in the fixed point set of $h,$
the heat kernel $K(h,t,x,x)$ decays rapidly as $t\to 0.$ By an argument of
partition of unity, the trace ${\mathrm{Tr}}(he^{-tD})$ has an asymptotic
expansion as $t\to 0$ of the form

(7.5) ${\mathrm{Tr}}(he^{-tD})\sim
t^{-n/d}(a_{0}(D)+a_{1}(D)t^{1/d}+\cdots+a_{N}(D)t^{N/d}+O(t^{(N+1)/d}))$

where $d$ is the order of the $h$-equivariant elliptic differential operator
$D$ and each $a_{i}(D)$ is the integral of some function on an $n$-dimensional
manifold $M^{n}.$

######  Lemma 7.4.

For $h\in H,$ the $h$-equivariant $\zeta$-function of $D$ is the Mellin
transform of ${\mathrm{Tr}}(he^{-tD}).$

###### Proof.

For
$\lambda_{p_{i}}\leq\lambda_{p_{i+1}}\leq\cdots<0<\lambda_{1}\leq\lambda_{2}\leq\cdots,$
the trace ${\mathrm{Tr}}(he^{-tD})$ is given by the sum
$\sum^{\infty}_{k=p_{i}}{\mathrm{Tr}}(h^{*}_{\lambda_{k}})e^{-\lambda_{k}t}.$
Set $\tau=\lambda t.$ Then, we compute

$\displaystyle\int^{\infty}_{0}t^{s-1}{\mathrm{Tr}}(h^{*}_{\lambda})e^{-\lambda
t}dt=\int^{\infty}_{0}\frac{\tau^{s-1}}{\lambda^{s-1}}{\mathrm{Tr}}(h^{*}_{\lambda})e^{-\tau}\lambda^{-1}d\tau=\frac{{\mathrm{Tr}}(h^{*}_{\lambda})}{\lambda^{s}}\int^{\infty}_{0}\tau^{s-1}e^{-\tau}d\tau={\mathrm{Tr}}(h^{*}_{\lambda})\lambda^{-s}\Gamma(s).$

Thus,

$\displaystyle\zeta_{h}(D,s)$
$\displaystyle=\sum_{k}{\mathrm{Tr}}(h^{*}_{\lambda_{k}})\lambda^{-s}_{k}=\frac{1}{\Gamma(s)}\sum_{k}\int^{\infty}_{0}t^{s-1}{\mathrm{Tr}}(h^{*}_{\lambda_{k}})e^{-\lambda_{k}t}dt$
$\displaystyle=\frac{1}{\Gamma(s)}\int^{\infty}_{0}t^{s-1}\sum_{k}\Big{(}{\mathrm{Tr}}(h^{*}_{\lambda_{k}})e^{-\lambda_{k}t}\Big{)}dt=\frac{1}{\Gamma(s)}\int^{\infty}_{0}t^{s-1}{\mathrm{Tr}}(he^{-tD})dt.$

∎

In contrast to $\eta$-functions, the $\zeta$-function is often defined for
unbounded operators such as the square of Dirac operators whose discrete
spectrum is positive. In this case, the appropriate heat trace is the
renormalised heat trace

${\mathrm{Tr}}(he^{-tD|_{P^{+}}})=\sum^{\infty}_{k=1}{\mathrm{Tr}}(h^{*}_{\lambda_{k}})e^{-\lambda_{k}t}=\sum^{\infty}_{k=p_{i}}{\mathrm{Tr}}(h^{*}_{\lambda_{k}})e^{-\lambda_{k}t}-\sum^{0}_{k=p_{i}}{\mathrm{Tr}}(h^{*}_{\lambda_{k}})e^{-\lambda_{k}t}.$

######  Corollary 7.5.

For $h\in H,$ the $h$-equivariant $\zeta$-function of the operator
$D|_{P^{+}}$ is given by

(7.6)
$\zeta_{h}(D,s)=\sum^{\infty}_{k=1}{\mathrm{Tr}}(h^{*}_{\lambda_{k}})\lambda_{k}^{-s}=\frac{1}{\Gamma(s)}\int^{\infty}_{0}t^{s-1}{\mathrm{Tr}}(he^{-tD|_{P^{+}}})dt.$

######  Remark 7.6.

In this case, the renormalised heat trace also has an asymptotic expansion as
$t\to 0$ which is in general different than that of (7.5). In particular, the
corresponding coefficient $\tilde{a}_{i}(D^{2})$ takes the form

$\tilde{a}_{i}(D)=a_{i}(D)-\frac{1}{r!}\sum^{0}_{k=n_{p}}(-{\mathrm{Tr}}(h^{*}_{\lambda_{k}})\lambda_{k})^{r}$

when $r=(i-n)/d\in\\{0,1,2,\ldots\\}.$ The existence and an explicit
description of the expansion were shown by Donnelly [12, Theorem 4.1].
Moreover, by a slight generalisation of Gilkey’s result [14], the residue of
$\Gamma(s)\zeta_{h}(D,s)$ at $s=(n-i)/d$ is

(7.7) $\text{Res}_{s=(n-i)/d}(\Gamma(s)\zeta_{h}(D,s))=\tilde{a}_{i}(D).$

At $s=0,$ the gamma function is known to have a Laurent expansion

(7.8) $\Gamma(s)=\frac{1}{s}+\gamma+p(s)$

where $\gamma$ is the Euler constant and $p(s)$ is the part of the expansion
with positive powers of $s.$ As it has no pole at $s=0,$ the function
$\zeta_{h}(D,s)$ is regular at $s=0.$ The constant term is

(7.9)
$\zeta_{h}(D):=\zeta_{h}(D,0)=\text{Res}_{s=0}(\Gamma(s)\zeta_{h}(D,s))=\tilde{a}_{n}(D)$

for $n=\dim M.$ We shall call $\zeta_{h}(D)$ the $h$-equivariant
$\zeta$-invariant of $D.$

######  Proposition 7.7.

Let $P\in{\mathrm{Gr}}_{h}^{\infty}(A).$ Let
$P^{\partial}\in{\mathrm{Gr}}_{h}^{\infty}(A)$ be the perturbed APS projection
as given by (2.4). Then, there exists a constant $c_{1},c_{2}>0$ such that the
estimate

(7.10)
$|{\mathrm{Tr}}(he^{-tD^{2}_{P}})-{\mathrm{Tr}}(he^{-tD^{2}_{P^{\partial}}})|\leq
c_{1}\sqrt{t}{\mathrm{Tr}}|{\mathcal{K}}|e^{c_{2}t||{\mathcal{K}}||}.$

###### Proof.

This is a slight generalisation of [27, Corollary 2.2, Theorem 3.2]. Assume
that a fixed $h\in H$ commutes with ${\mathcal{K}}_{2}=g^{-1}[A,g]$ where $g$
is a unitary operator on $L^{2}(M,S)$ acts as the identity on the interior
$M\backslash([0,1]\times\partial M)$ and as a path of a certain unitaries in
${\mathrm{Gr}}_{\infty}(A)$ on the cylinder $[0,1]\times\partial M,$ c.f. [27,
Lemma 1.2,(1.10)]. Note that $h$ is independent of $t\in[0,1]$ and the
commutation does not affect each factor of the local expression of heat
kernels, we observe that the method of Duhamel’s principle still works, in
which as a consequence of [27, Proposition 1.1,2.1,3.2] we have an upper
estimate

(7.11)
$|{\mathrm{Tr}}(he^{-tD^{2}_{P}})-{\mathrm{Tr}}(he^{-tD^{2}_{P^{\partial}}})|\leq
c\cdot{\mathrm{Tr}}|h^{*}|\sqrt{t}{\mathrm{Tr}}|{\mathcal{K}}|e^{c_{0}t|h|||{\mathcal{K}}||}$

where ${\mathrm{Tr}}(h^{*})=\sum_{k}{\mathrm{Tr}}(h^{*}_{\lambda_{k}})$ is the
trace of the induced linear map. ∎

It follows that we have an equivariant adaptation of the classically known
fact regarding the independence of the choice of boundary projections for
$\zeta$-invariants of $D^{2},$ see [27, Proposition 0.5].

######  Lemma 7.8.

For $h\in H,$ the value of the $h$-equivariant $\zeta$-function at $s=0$ is
constant on ${\mathrm{Gr}}_{h}^{\infty}(A),$ i.e. for any
$P,Q\in{\mathrm{Gr}}_{h}^{\infty}(A),$ we have

(7.12) $\zeta_{h}(D^{2}_{P})=\zeta_{h}(D^{2}_{Q}).$

###### Proof.

For $P,Q,P^{\partial}\in{\mathrm{Gr}}_{h}^{\infty}(A),$

$\displaystyle\zeta_{h}(D^{2}_{P})-\zeta_{h}(D^{2}_{P^{\partial}})$
$\displaystyle=\lim_{s\to
0}\frac{1}{\Gamma(s)}\int^{\infty}_{0}t^{s-1}{\mathrm{Tr}}(he^{-tD^{2}_{P}}-he^{-tD^{2}_{P^{\partial}}})dt$
$\displaystyle=\lim_{s\to
0}s\int^{\infty}_{0}t^{s-1}{\mathrm{Tr}}(he^{-tD^{2}_{P}}-he^{-tD^{2}_{P^{\partial}}})dt\quad\quad\text{from
}\eqref{eq:gammafn1}$

By the estimate (7.11), we have

$\displaystyle|\zeta_{h}(D^{2}_{P})-\zeta_{h}(D^{2}_{P^{\partial}})|$
$\displaystyle\leq\lim_{s\to
0}s\int^{\infty}_{0}t^{s-1}|{\mathrm{Tr}}(he^{-tD^{2}_{P}}-he^{-tD^{2}_{P^{\partial}}})|dt$
$\displaystyle\leq c\cdot\lim_{s\to 0}s\int^{\infty}_{0}t^{s-\frac{1}{2}}dt=0$

where the second inequality follows from Proposition 7.7. Lastly, the claim
follows
$|\zeta_{h}(D^{2}_{P})-\zeta_{h}(D^{2}_{Q})|\leq|\zeta_{h}(D^{2}_{P})-\zeta_{h}(D^{2}_{P^{\partial}})|+|\zeta_{h}(D^{2}_{P^{\partial}})-\zeta_{h}(D^{2}_{Q})|\leq
0.$ ∎

######  Proposition 7.9.

Let $h\in H.$ Let $D^{2}$ be the square of a self-adjoint $h$-equivariant
Dirac operator $D.$ Then,

(7.13)
$\dot{\zeta}_{h}(D^{2})=\int^{\infty}_{0}\frac{1}{t}{\mathrm{Tr}}(he^{-tD^{2}})dt-\frac{\tilde{a}_{n}(D^{2})}{s}\Big{|}_{s=0}-\tilde{a}_{n}(D^{2})\cdot\gamma$

###### Proof.

We adopt an approach of Singer in [25]. Let
$p(s)=\int^{\infty}_{0}t^{s-1}{\mathrm{Tr}}(he^{-tD^{2}})dt=\frac{\tilde{a}_{n}(D^{2})}{s}+b+f(s)$
where $b$ is some constant and $f(s)$ consists of all the terms with positive
powers of $s.$ Then, (7.6) can be written as

$\zeta_{h}(D^{2},s)=\frac{p(s)}{\frac{1}{s}+\gamma+p(s)}=\frac{sp(s)}{1+s\gamma+p_{1}(s)}$

where we use the expansion (7.8). Its derivative at $s=0$ can be computed as

$\displaystyle\dot{\zeta}_{h}(D^{2})=\dot{\zeta}_{h}(D^{2},0)$
$\displaystyle=\frac{d}{ds}\Big{|}_{s=0}\zeta_{h}(D^{2},s)=\frac{d}{ds}\Big{|}_{s=0}(sp(s))(1+s\gamma+p_{1}(s))^{-1}$
$\displaystyle=\frac{s\dot{p}(s)+p(s)}{1+s\gamma+p_{1}(s)}\Big{|}_{s=0}-\frac{(sp(s))(s\gamma+s\dot{p}_{1}(s))}{(1+s\gamma+p_{1}(s))^{2}}\Big{|}_{s=0}$
$\displaystyle=\int^{\infty}_{0}\frac{1}{t}{\mathrm{Tr}}(he^{-tD^{2}})dt-\frac{\tilde{a}_{n}(D^{2})}{s}\Big{|}_{s=0}-\tilde{a}_{n}(D^{2})\cdot\gamma$

where $p(0)=\int^{\infty}_{0}\frac{1}{t}{\mathrm{Tr}}(he^{-tD^{2}})dt$ and
$\zeta_{h}(D^{2})=\tilde{a}_{n}(D^{2}).$ ∎

######  Remark 7.10.

The proof of Proposition 7.9 extends without change to the operator
$D|_{P^{+}}$ which is exactly the derivative of $\zeta_{h}(D|_{P^{+}})$ of
(7.6) with respect to the second equality. In this case, another description
of $\dot{\zeta}_{h}(D|_{P^{+}})$ can be obtained from the first equality of
(7.6). More precisely,

(7.14) $\displaystyle\dot{\zeta}_{h}(D|_{P^{+}})$
$\displaystyle=\frac{d}{ds}\Big{|}_{s=0}\zeta_{h}(D|_{P^{+}},s)=\sum^{\infty}_{k=1}{\mathrm{Tr}}(h^{*}_{\lambda_{k}})\frac{d}{ds}\Big{|}_{s=0}(\lambda_{k})^{-s}$
$\displaystyle=-\sum^{\infty}_{k=1}{\mathrm{Tr}}(h^{*}_{\lambda_{k}})\log(\lambda_{k})=-\sum^{\infty}_{k=1}\log\Big{(}\lambda_{k}^{{\mathrm{Tr}}(h^{*}_{\lambda_{k}})}\Big{)}$
$\displaystyle=-\log\Bigg{(}\prod^{\infty}_{k=1}\lambda^{{\mathrm{Tr}}(h^{*}_{\lambda_{k}})}_{k}\Bigg{)}.$

######  Definition 7.11.

Let $h\in H.$ Define the $h$-equivariant regularised $\zeta$-determinant of
$D=D_{P}$ by

(7.15) ${\mathrm{det}}^{h}_{\zeta}(D|_{P^{+}})=e^{-\dot{\zeta}^{h}_{D}(0)}.$

In view of Remark 7.10 and Definition 7.11, it was observed by Singer [25]
that it is possible to extend to the case of self-adjoint elliptic operators
whose symbols are not necessarily positive definite. In particular, such an
operator can have positive $\\{\lambda_{k}\\}$ or negative $\\{-\mu_{k}\\}$
eigenvalues. Then, the corresponding $\zeta^{h}$-function can be written as

(7.16) $\displaystyle\zeta^{h}_{D}(s)=\sum_{k}$
$\displaystyle{\mathrm{Tr}}(h^{*}_{\lambda_{k}})\lambda^{-s}_{k}+\sum_{k}(-1)^{-s}{\mathrm{Tr}}(h^{*}_{\mu_{k}})\mu^{-s}_{k}$
$\displaystyle=\sum_{k}$
$\displaystyle\Big{(}\frac{{\mathrm{Tr}}(h^{*}_{\lambda_{k}})\lambda^{-s}_{k}-{\mathrm{Tr}}(h^{*}_{\mu_{k}})\mu^{-s}_{k}}{2}+\frac{{\mathrm{Tr}}(h^{*}_{\lambda_{k}})\lambda^{-s}_{k}+{\mathrm{Tr}}(h^{*}_{\mu_{k}})\mu^{-s}_{k}}{2}\Big{)}$
$\displaystyle+(-1)^{-s}\sum_{k}\Big{(}\frac{{\mathrm{Tr}}(h^{*}_{\lambda_{k}})\lambda^{-s}_{k}+{\mathrm{Tr}}(h^{*}_{\mu_{k}})\mu^{-s}_{k}}{2}-\frac{{\mathrm{Tr}}(h^{*}_{\lambda_{k}})\lambda^{-s}_{k}-{\mathrm{Tr}}(h^{*}_{\mu_{k}})\mu^{-s}_{k}}{2}\Big{)}.$

Let
$\eta^{h}_{D}(s)=\sum_{k}{\mathrm{Tr}}(h^{*}_{\lambda_{k}})\lambda^{-s}_{k}-{\mathrm{Tr}}(h^{*}_{\mu_{k}})\mu^{-s}_{k}$
be the equivariant $\eta$-function of $D.$ (Compare with Definition 9.1). Let

$\displaystyle\zeta^{h}_{D^{2}}(s/2)$
$\displaystyle=\sum_{k}{\mathrm{Tr}}(h^{*}_{\lambda_{k}})(\lambda^{2})^{-\frac{s}{2}}_{k}+{\mathrm{Tr}}(h^{*}_{\mu_{k}})(\mu^{2})^{-\frac{s}{2}}_{k}$
$\displaystyle=\sum_{k}{\mathrm{Tr}}(h^{*}_{\lambda_{k}})|\lambda_{k}|^{-s}+{\mathrm{Tr}}(h^{*}_{\mu_{k}})|\mu_{k}|^{-s}$
$\displaystyle=\sum_{k}{\mathrm{Tr}}(h^{*}_{\lambda_{k}})\lambda^{-s}_{k}+{\mathrm{Tr}}(h^{*}_{\mu_{k}})\mu^{-s}_{k}$

be the equivariant $\zeta$-function of $D^{2}$ of the variable $s/2.$ (Compare
with Definition 7.1). Then, (7.16) can be rewritten as

(7.17)
$\zeta^{h}_{D}(s)=\frac{\eta^{h}_{D}(s)+\zeta^{h}_{D^{2}}(s/2)}{2}+(-1)^{-s}\Bigg{(}\frac{\zeta^{h}_{D^{2}}(s/2)-\eta^{h}_{D}(s)}{2}\Bigg{)}.$

Choose $(-1)^{-s}=e^{-i\pi s}.$ Then, the derivative $\dot{\zeta}^{h}_{D}(0)$
of (7.17) is

$\displaystyle\dot{\zeta}^{h}_{D}(0)$
$\displaystyle=\frac{\dot{\eta}^{h}_{D}(0)}{2}+\frac{\dot{\zeta}^{h}_{D^{2}}(0)}{4}-\frac{i\pi(-1)^{-s}}{2}\big{(}\zeta^{h}_{D^{2}}(s/2)-\eta^{h}_{D}(s)\big{)}|_{s=0}$
$\displaystyle\quad+(-1)^{-s}\Big{(}\frac{\dot{\zeta}^{h}_{D^{2}}(0)}{4}-\frac{\dot{\eta}^{h}_{D}(0)}{2}\Big{)}|_{s=0}$
$\displaystyle=\frac{\dot{\zeta}^{h}_{D^{2}}(0)}{2}-\frac{i\pi}{2}(\zeta^{h}_{D^{2}}(0)-\eta^{h}_{D}(0)).$

######  Corollary 7.12.

Let $h\in H.$ The $h$-equivariant $\zeta$-determinant of $D$ is given by

(7.18)
${\mathrm{det}}^{h}_{\zeta}(D):=\exp\Big{(}\frac{i\pi}{2}(\zeta_{h}(D^{2})-\eta_{h}(D))-\frac{1}{2}\dot{\zeta}_{h}(D^{2})\Big{)}.$

##  8\. Equivariant Scott-Wojciechowski Theorem

Next, we conjecture an equivariant adaptation of the Scott-Wojciechowski
Theorem, which is roughly speaking, allow us to express the ratio of the
$h$-equivariant $\zeta$-determinants of the Dirac operators (associated with
two boundary projecitons) as the Fredholm determinant of the respective
unitaries associated to the projections. The original version was first
introduced by Scott and Wojciechowski [24]. Using this, Kirk and Lesch proved
a remarkable relation between the difference of the $\eta$-invariants of Dirac
operators (associated with two different boundary projections) and the
Fredholm determinant of the corresponding unitary operators, see [15, §4].

To begin with, we propose a suitable modification to the settings in [24]. The
main reason for such a modification is due to the convention that we have used
in our definitions above such as the equivariant winding number $w_{h}$ and
the equivariant double index $\tau_{h}$. For instance, in our case the unitary
operator under the $h$ action is taken to be $aT^{*}S$ acting from
${\mathcal{E}}_{i}$ to ${\mathcal{E}}_{i};$ whilst in the case of [24] the
unitary operator is acting from ${\mathcal{E}}_{-i}$ to ${\mathcal{E}}_{-i}.$
If this was the case, then it would imply the unitary operators are acted on
by $WaW^{*},$ which creates unwanted complication. In the following, we do not
intent to prove the equivariant version of the Scott-Wojciechowski Theorem in
details, but only to lay out some suggestions of our modification to their
settings. Therefore, we will leave out most of the original details out for
the interested readers.

Let ${\mathcal{H}}^{\prime}=L^{2}(H,{\mathcal{H}})$ be our model Hilbert space
as defined by (2.11). Let ${\mathcal{F}}({\mathcal{H}}^{\prime})$ be the space
of equivariant Fredholm operators on ${\mathcal{H}}^{\prime}.$ Let
${\mathcal{F}}_{0}({\mathcal{H}}^{\prime})$ be the connected component of
index zero. Fix $B\in{\mathcal{F}}_{0}({\mathcal{H}}^{\prime}).$ Set

${\mathcal{F}}_{B}={\mathcal{F}}_{B}({\mathcal{H}}^{\prime})=\\{S\in{\mathcal{F}}({\mathcal{H}}^{\prime})\;|\;S-B\text{
is trace-class}\\}.$

Fix a trace-class operator $\mathscr{B}$ such that $S=B+\mathscr{B}$ is
invertible. Let $h\in H,$ define the determinant line bundle of $B$ by

$\text{Det}(B)={\mathcal{F}}_{B}\times_{\sim_{h}}{\mathbb{C}}$

where the equivalence relation $\sim_{h}$ incorporated with the $h$-action is
given by

$(R,z)_{h}:=h\cdot(R,z)=(S(S^{-1}R),z)_{h}=(aS(S^{-1}R),z)_{1_{H}}\sim(S,z\cdot{\mathrm{det}}_{F}(aS^{-1}R))_{1_{H}}$

for $h\in H,R,S\in{\mathcal{F}}_{B}$ and $z\in{\mathbb{C}}.$ Here, $\cdot$
denotes the $h$-action; $1_{H}$ denotes the identity element in $H$; the third
equality is the action by $h={\mathrm{Diag}}(a,WaW^{*})$ on $S(S^{-1}R)$ and
the last equivalence follows from the multiplicative property of Fredholm
determinant ${\mathrm{det}}_{F}.$ Let $[R,z]_{h}$ be the equivalence class of
$(R,z)_{h}.$ Define the determinant element by

${\mathrm{det}}(B):=[B,1]_{1_{H}},$

and observe that ${\mathrm{det}}(B)\neq 0$ if and only if $B$ is invertible.
Here, the last entry denotes the identity element $1=e^{2\pi
i}\in{\mathbb{C}}.$

Recall from (2.5) that ${\mathrm{Gr}}_{h}^{\infty}(A)$ is the space of all
equivariant boundary projections which differ from the APS projection $P^{+}$
by a smoothing operator. We identify the Grassmannian
${\mathrm{Gr}}_{h}^{\infty}(A)$ with the group
${\mathcal{U}}_{\infty}({\mathcal{E}}_{i})$ of unitary operators that differ
from ${\mathrm{Id}}_{{\mathcal{E}}_{i}}$ by a smoothing operator as follows.
Let ${\mathcal{P}}_{M}\in{\mathrm{Gr}}_{h}^{\infty}(A)$ be the canonical
equivariant Calderón projection on $M$. Let $K:C^{\infty}(\partial X,E_{i})\to
C^{\infty}(\partial X,E_{-i})$ be the associated unitary operator such that
the Cauchy-data space ${\mathcal{L}}_{M}$ (cf. (2.6)) coincides with
$\text{graph}(K).$ Let $P\in{\mathrm{Gr}}_{h}^{\infty}(A)$ be any projection.
Let $T:{\mathcal{E}}_{i}\to{\mathcal{E}}_{-i}$ be the associated unitary
operator such that

$\text{Ran}(P)=\text{graph}(T^{-1}).$

In particular, the map $P\mapsto T^{-1}K$ defines an isomorphism
${\mathrm{Gr}}_{h}^{\infty}(A)\cong{\mathcal{U}}_{\infty}({\mathcal{E}}_{i})$
where
${\mathcal{U}}_{\infty}({\mathcal{E}}_{i})=\\{V\in{\mathcal{U}}_{F}({\mathcal{E}}_{i})\;|\;V-{\mathrm{Id}}\text{
is smoothing}\\}.$ Note that the commutativity of $h$ and
$P\in{\mathrm{Gr}}_{h}^{\infty}(A)$ imply
$a\in{\mathcal{U}}_{\infty}({\mathcal{E}}_{i})$ and
$WaW^{*}\in{\mathcal{U}}_{\infty}({\mathcal{E}}_{-i}).$

Let

$V(P)=\begin{pmatrix}K^{-1}T&\textbf{0}\\\
\textbf{0}&{\mathrm{Id}}_{{\mathcal{E}}_{-i}}\end{pmatrix},\quad\text{ and so
}V^{-1}(P)=\begin{pmatrix}T^{-1}K&\textbf{0}\\\
\textbf{0}&{\mathrm{Id}}_{{\mathcal{E}}_{-i}}\end{pmatrix}.$

Then, one can check that in terms of the homogeneous structure of the
Grassmannian

$P=V^{-1}(P){\mathcal{P}}_{m}V(P)$

where $P$ and ${\mathcal{P}}_{M}$ are expressed in terms of (2.7) with the
unitaries $T$ and $K$ respectively. Consider the composition
${\mathcal{S}}(P)=P{\mathcal{P}}_{M}:L_{M}\to\text{Ran}(P).$ Then, [24, Lemma
1.1] reads as follows.

######  Lemma 8.1.

$V(P)$ differs from ${\mathcal{S}}(P)$ by a smoothing operator, i.e.
$V(P)-{\mathcal{S}}(P)\in\Psi^{-\infty}.$

###### Proof.

The proof here is modified from that of [24, Lemma 1.1] by using our new
setting. The operator $V(P)$ acts by its inverse from
${\mathcal{L}}_{M}=\text{graph}(K)$ to $\text{Ran}(P)=\text{graph}(T^{-1})$

$\begin{pmatrix}x\\\
Kx\end{pmatrix}\longrightarrow\begin{pmatrix}T^{-1}K&\textbf{0}\\\
\textbf{0}&{\mathrm{Id}}_{{\mathcal{E}}_{-i}}\end{pmatrix}\begin{pmatrix}x\\\
Kx\end{pmatrix}=\begin{pmatrix}T^{-1}(Kx)\\\
Kx\end{pmatrix}=\begin{pmatrix}T^{-1}y\\\ y\end{pmatrix}.$

By a direct computation, the composition ${\mathcal{S}}(P)$ can be written as

${\mathcal{S}}(P)=P{\mathcal{P}}_{M}=\frac{1}{4}\begin{pmatrix}{\mathrm{Id}}+T^{-1}K&T^{-1}+K^{-1}\\\
T+K&{\mathrm{Id}}+TK^{-1}\end{pmatrix}.$

Then,

$\displaystyle{\mathcal{S}}(P)\begin{pmatrix}x\\\ Kx\end{pmatrix}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}({\mathrm{Id}}+T^{-1}K)x\\\
({\mathrm{Id}}+TK^{-1})Kx\end{pmatrix}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}({\mathrm{Id}}+K^{-1}T)(T^{-1}y)\\\
({\mathrm{Id}}+TK^{-1})y\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\frac{{\mathrm{Id}}+K^{-1}T}{2}&\textbf{0}\\\
\textbf{0}&\frac{{\mathrm{Id}}+TK^{-1}}{2}\end{pmatrix}\begin{pmatrix}T^{-1}y\\\
y\end{pmatrix}.$

This shows that with respect to $\text{Ran}(P)$ the composition
${\mathcal{S}}(P)$ is given by
${\mathrm{Diag}}(\frac{{\mathrm{Id}}+K^{-1}T}{2},\frac{{\mathrm{Id}}+TK^{-1}}{2}).$
The rest of the claim follows from the fact that $K^{-1}T$ (resp. $TK^{-1}$)
differs from ${\mathrm{Id}}_{{\mathcal{E}}_{i}}$ (resp.
${\mathrm{Id}}_{{\mathcal{E}}_{-i}}$) by a smoothing operator. ∎

As a consequence, $\det(V(P))=[V(P),1]_{1_{H}}$ defines an element of the
determinant line bundle $\text{Det}_{h}({\mathcal{S}}(P)).$ For $h\in H,$ let
$B:{\mathcal{L}}_{M}\to\text{Ran}(P)$ be an invertible $h$-equivariant
Fredholm operator such that $B-{\mathcal{S}}(P)$ is a trace class operator.
Then,

$\displaystyle{\mathrm{det}}(B)$
$\displaystyle=[B,1]_{1_{H}}=[V(P)(V^{-1}(P)B),1]_{1_{H}}=[V(P),1\cdot{\mathrm{det}}_{F}(V^{-1}(P)B)]_{1_{H}}$
$\displaystyle={\mathrm{det}}_{F}(V^{-1}(P)B)\cdot[V(P),1]_{1_{H}}={\mathrm{det}}_{F}(V^{-1}(P)B)\cdot{\mathrm{det}}(V(P)).$

######  Definition 8.2.

For $h\in H$, define the $h$-canonical determinant of $D_{P}$ by

(8.1)
${\mathrm{det}}^{h}_{\mathcal{C}}(D_{P}):={\mathrm{det}}^{h}_{\mathcal{C}}{\mathcal{S}}(P):={\mathrm{det}}_{F}(hV^{-1}(P){\mathcal{S}}(P)).$

It is straightforward to verify that

$hV^{-1}(P){\mathcal{S}}(P)={\mathrm{Diag}}\Big{(}\frac{a({\mathrm{Id}}+T^{-1}K)}{2},\frac{WaW^{*}({\mathrm{Id}}+TK^{-1})}{2}\Big{)}.$

The determinant on the right side of (8.1) is the Fredholm determinant of the
diagonal matrix acting on the graph of $K.$ Hence,

${\mathrm{det}}^{h}_{\mathcal{C}}(D_{P})={\mathrm{det}}_{F}\Big{(}\frac{a({\mathrm{Id}}+T^{-1}K)}{2}\Big{)}.$

(Compare with [24, Lemma 1.3]). To see the well-definedness on the right side,
we rewrite

$\displaystyle\frac{a({\mathrm{Id}}+T^{-1}K)}{2}$
$\displaystyle=\frac{{\mathrm{Id}}+aT^{-1}K-({\mathrm{Id}}-a)}{2}$
$\displaystyle={\mathrm{Id}}+\frac{[aT^{-1}K-({\mathrm{Id}}-a)]-{\mathrm{Id}}}{2}.$

Since ${\mathrm{Id}}-a$ is smoothing for
$a\in{\mathcal{U}}_{\infty}({\mathcal{E}}_{i})$, so is
$[aT^{-1}K-({\mathrm{Id}}-a)]-{\mathrm{Id}}.$ Hence,
$[aT^{-1}K-({\mathrm{Id}}-a)]-{\mathrm{Id}}$ is of trace class and
$\frac{1}{2}a({\mathrm{Id}}+T^{-1}K)$ is of determinant class. This argument
also works if we replace $K$ by any other projection
$S\in{\mathrm{Gr}}^{\infty}_{h}(A).$

An equivariant version of the Scott-Wojciechowski Theorem reads as follows.

######  Conjecture 8.3.

For $h\in H,$ the following equality holds over
${\mathrm{Gr}}_{h}^{\infty}(A)$

(8.2)
${\mathrm{det}}^{h}_{\zeta}(D_{P})={\mathrm{det}}^{h}_{\zeta}(D_{{\mathcal{P}}_{M}})\cdot{\mathrm{det}}_{F}\Big{(}\frac{a({\mathrm{Id}}+T^{-1}K)}{2}\Big{)}.$

We state this as a conjecture instead of a theorem. The basis of believing
that the conjecture is true are (i) the equivariant modification above is
compatible with that of Scott and Wojciechowski, so the proof is only expected
to subject to minor changes; (ii) the formula of equivariant
$\zeta$-determinants given in Corollary 7.12 turns out to have the same form
as the one they are working with.

By using our settings above, an equivariant version of [15, Theorem 4.2] is
given as follows.

######  Theorem 8.4.

Let $h\in H.$ Let $P,Q\in{\mathrm{Gr}}_{h}^{\infty}(A)$ and $T=T(P),S=S(Q)$ be
the respective associated unitary operators. Then,

(8.3) $\exp(2\pi
i(\tilde{\eta}_{h}(D_{P})-\tilde{\eta}_{h}(D_{Q})))={\mathrm{det}}_{F}(aT^{*}S).$

###### Proof.

We adopt an elegant method introduced by Kirk and Lesch. First, assume a
special case where $P={\mathcal{P}}_{M}$ and such that the composition
${\mathcal{P}}_{M}Q$ is invertible, i.e. $({\mathcal{P}}_{M},Q)$ is an
$h$-equivariant invertible pair. Then, the corresponding boundary problems
$D_{{\mathcal{P}}_{M}}$ and $D_{Q}$ are both invertible. Let
$K=K({\mathcal{P}}_{M})$ is the associated unitary operator. By Lemma 7.8, we
know that the equivariant $\zeta$-invariant is independent of the projections,
i.e. $\zeta_{h}(D^{2}_{{\mathcal{P}}_{M}})=\zeta_{h}(D^{2}_{Q}).$ By
Conjecture 8.3 and Corollary 7.12, we have

$\displaystyle{\mathrm{det}}_{F}\Big{(}\frac{a({\mathrm{Id}}+K^{*}S}{2}\Big{)}$
$\displaystyle=\frac{{\mathrm{det}}^{h}_{\zeta}(D_{Q})}{{\mathrm{det}}^{h}_{\zeta}(D_{{\mathcal{P}}_{M}})}=\frac{\exp\Big{(}\frac{i\pi}{2}(\zeta_{h}(D^{2}_{Q})-\eta_{h}(D_{Q}))-\frac{1}{2}\dot{\zeta}_{h}(D^{2}_{Q})\Big{)}}{\exp\Big{(}\frac{i\pi}{2}(\zeta_{h}(D^{2}_{{\mathcal{P}}_{M}})-\eta_{h}(D_{{\mathcal{P}}_{M}}))-\frac{1}{2}\dot{\zeta}_{h}(D^{2}_{{\mathcal{P}}_{M}})\Big{)}}$
$\displaystyle=\exp\Big{(}\frac{i\pi}{2}(\eta_{h}(D_{{\mathcal{P}}_{M}})-\eta_{h}(D_{Q}))+\frac{1}{2}(\dot{\zeta}_{h}(D^{2}_{{\mathcal{P}}_{M}})-\dot{\zeta}_{h}(D^{2}_{Q}))\Big{)}.$

By taking the modulus on both sides, we see that

$\Big{|}{\mathrm{det}}_{F}\Big{(}\frac{a({\mathrm{Id}}+K^{*}S)}{2}\Big{)}\Big{|}=\exp\Big{(}\frac{1}{2}(\dot{\zeta}_{h}(D^{2}_{{\mathcal{P}}_{M}})-\dot{\zeta}_{h}(D^{2}_{Q}))\Big{)}$

and thus

(8.4)
${\mathrm{det}}_{F}\Big{(}\frac{a({\mathrm{Id}}+K^{*}S)}{2}\Big{)}\Big{/}\Big{|}{\mathrm{det}}_{F}\Big{(}\frac{a({\mathrm{Id}}+K^{*}S)}{2}\Big{)}\Big{|}=\exp\Big{(}\frac{i\pi}{2}(\eta_{h}(D_{{\mathcal{P}}_{M}})-\eta_{h}(D_{Q})\Big{)}.$

On the other hand, by Lemma 5.5, we have

${\mathrm{det}}_{F}\Big{(}a\Big{(}\frac{{\mathrm{Id}}+K^{*}S}{2}\Big{)}\Big{)}={\mathrm{det}}_{F}(a){\mathrm{det}}_{F}\Big{(}\frac{{\mathrm{Id}}+K^{*}S}{2}\Big{)}.$

The right side is well-defined because $a={\mathrm{Id}}+(a-{\mathrm{Id}})$ for
$a\in{\mathcal{U}}_{\infty}({\mathcal{H}}^{\prime}),$ so $a-{\mathrm{Id}}$ is
smoothing and is of trace class. Hence, $a$ is of determinant class.
Similarly,

$\frac{{\mathrm{Id}}+K^{*}S}{2}={\mathrm{Id}}+\frac{K^{*}S-{\mathrm{Id}}}{2}.$

Since $K^{*}S-{\mathrm{Id}}$ is smoothing and is of trace class,
$\frac{1}{2}(K^{*}S-{\mathrm{Id}})$ is of determinant class. Then,

$\displaystyle{\mathrm{det}}_{F}\Big{(}a\Big{(}\frac{{\mathrm{Id}}+K^{*}S}{2}\Big{)}\Big{)}$
$\displaystyle={\mathrm{det}}_{F}(a){\mathrm{det}}_{F}(K^{*}S){\mathrm{det}}_{F}\Big{(}\frac{{\mathrm{Id}}+S^{*}K}{2}\Big{)}$
$\displaystyle={\mathrm{det}}_{F}(aK^{*}S){\mathrm{det}}_{F}\Big{(}\frac{{\mathrm{Id}}+S^{*}K}{2}\Big{)}.$

Note that $\frac{1+S^{*}K}{2}$ is in general not unitary and hence its
Fredholm determinant is a complex number whose modulus is a non-zero real
number. By taking modulo, we obtain

(8.5)
${\mathrm{det}}_{F}\Big{(}a\Big{(}\frac{{\mathrm{Id}}+K^{*}S}{2}\Big{)}\Big{)}\Big{/}\Big{|}{\mathrm{det}}_{F}\Big{(}a\Big{(}\frac{{\mathrm{Id}}+K^{*}S}{2}\Big{)}\Big{)}\Big{|}={\mathrm{det}}_{F}(aK^{*}S).$

From (8.4) and (8.5), we have

$\exp\Big{(}\frac{i\pi}{2}(\eta_{h}(D_{{\mathcal{P}}_{M}})-\eta_{h}(D_{Q})\Big{)}={\mathrm{det}}_{F}(aK^{*}S).$

Let $P,Q\in{\mathrm{Gr}}_{h}^{\infty}(A)$ be arbitrary. Then,

$\displaystyle\exp\Big{(}\frac{i\pi}{2}(\eta_{h}(D_{{\mathcal{P}}})-\eta_{h}(D_{Q})\Big{)}$
$\displaystyle=\exp\Big{(}\frac{i\pi}{2}(\eta_{h}(D_{{\mathcal{P}}})-\eta_{h}(D_{{\mathcal{P}}_{M}})\Big{)}\exp\Big{(}\frac{i\pi}{2}(\eta_{h}(D_{{\mathcal{P}}_{M}})-\eta_{h}(D_{Q})\Big{)}$
$\displaystyle={\mathrm{det}}_{F}(aT^{*}K){\mathrm{det}}_{F}(aK^{*}S)$
$\displaystyle={\mathrm{det}}_{F}(aT^{*}S)$

where the third line follows from the multiplicative property of equivariant
Fredholm determinants. ∎

##  9\. Equivariant $\eta$-invariants

######  Definition 9.1.

For $h\in H,$ the $h$-equivariant $\eta$-function of a self-adjoint elliptic
equivariant differential operator $D$ is defined by

(9.1) $\eta_{h}(D,s):=\sum_{\lambda\neq
0,\lambda\in{\text{spec}}(D)\backslash\\{0\\}}{\mathrm{Tr}}(h^{*}_{\lambda})\frac{\text{sgn}(\lambda)}{|\lambda|^{s}}$

where $h^{*}$ is the induced linear map on the $\lambda$-eigenspace. The
equivariant $\eta$-invariant $\eta_{h}(D)$ is defined to be $\eta_{h}(D,0).$

######  Lemma 9.2.

For $h\in H,$ $\eta_{h}(D,s)$ is the Mellin transform of
${\mathrm{Tr}}(hDe^{-tD^{2}}).$

###### Proof.

Recall that the $\Gamma$-function with the variable $\frac{1}{2}(s+1)$ is
given by

(9.2)
$\Gamma\Big{(}\frac{s+1}{2}\Big{)}=\int^{\infty}_{0}t^{\frac{s-1}{2}}e^{-t}dt.$

Set $\tau=\lambda^{2}t.$ Then,

$\displaystyle\int^{\infty}_{0}t^{\frac{s-1}{2}}{\mathrm{Tr}}(h^{*}_{\lambda})\lambda
e^{-\lambda^{2}t}dt$
$\displaystyle=\int^{\infty}_{0}\Big{(}\frac{\tau}{\lambda^{2}}\Big{)}^{\frac{s-1}{2}}{\mathrm{Tr}}(h^{*}_{\lambda})\frac{\lambda}{\lambda^{2}}e^{-\tau}d\tau=\lambda^{-1}\int^{\infty}_{0}\frac{|\lambda|}{|\lambda|^{s}}{\mathrm{Tr}}(h^{*}_{\lambda})\tau^{\frac{s-1}{2}}e^{-\tau}d\tau$
$\displaystyle=\Big{(}\int^{\infty}_{0}\frac{|\lambda|}{|\lambda|^{s}}{\mathrm{Tr}}(h^{*}_{\lambda})\tau^{\frac{s-1}{2}}e^{-\tau}d\tau\Big{)}\frac{|\lambda|}{\lambda}|\lambda|^{-s}$
$\displaystyle=\Gamma\Big{(}\frac{s+1}{2}\Big{)}{\mathrm{Tr}}(h^{*}_{\lambda})\text{sgn}(\lambda)|\lambda|^{-s}.$

Thus,

$\displaystyle\eta_{h}(D,s)$
$\displaystyle=\frac{1}{\Gamma\big{(}\frac{s+1}{2}\big{)}}\sum_{k}\int^{\infty}_{0}t^{\frac{s-1}{2}}{\mathrm{Tr}}(h^{*}_{\lambda_{k}})\lambda
e^{-\lambda^{2}_{k}t}dt$
$\displaystyle=\frac{1}{\Gamma\big{(}\frac{s+1}{2}\big{)}}\int^{\infty}_{0}t^{\frac{s-1}{2}}\sum_{k}\Big{(}{\mathrm{Tr}}(h^{*}_{\lambda_{k}})\lambda
e^{-\lambda^{2}_{k}t}\Big{)}dt$
$\displaystyle=\frac{1}{\Gamma\big{(}\frac{s+1}{2}\big{)}}\int^{\infty}_{0}t^{\frac{s-1}{2}}{\mathrm{Tr}}(hDe^{-tD^{2}})dt.$

∎

######  Definition 9.3.

Let $h\in H.$ Let $\epsilon>0.$ Define the truncated $\eta$-function of $D$ by

(9.3)
$\eta_{h,\geq\epsilon}(D,s)=\frac{1}{\Gamma\big{(}\frac{s+1}{2}\big{)}}\int^{\infty}_{\epsilon}t^{\frac{s-1}{2}}{\mathrm{Tr}}(hDe^{-tD^{2}})dt.$

######  Definition 9.4.

Let $h\in H.$ Define the reduced equivariant $\eta$-invariant of $D$ by

(9.4) $\tilde{\eta}_{h}(D)=\frac{\eta_{h}(D)+{\mathrm{Tr}}(h|_{\ker
D})}{2}\quad\in{\mathbb{C}}.$

We now give an equivariant version of Getzler’s spectral flow formula in terms
of reduced $\eta$-invariants. See also [15, Lemma 3.3].

######  Theorem 9.5.

Let $h\in H.$ For $t\in[0,1],$ let $P_{t}$ be a smooth path in
${\mathrm{Gr}}(A_{t})$ and let $D_{P}(t):=D_{P(t)}(t)$ be a smooth path of
self-adjoint equivariant Dirac operators. Then,

(9.5)
${\mathrm{sf}}_{h}\\{D_{P}(t)\\}_{t\in[0,1]}=\tilde{\eta}_{h}(D_{P}(1))-\tilde{\eta}_{h}(D_{P}(0))-\frac{1}{2}\int\frac{d}{dt}\eta_{h}(D_{P}(t)).$

###### Proof.

Note that our definition of equivariant spectral flow are based on the
approach of Phillips [22], which is another approach to that of Atiyah in [3].
This coincides with the context that Getzler is working on, c.f. [13,
Definition 2.3]. Hence, the equivariant notion in Definition 3.2 extends to
Getzler’s setting. In the following we only highlight the relevant changes to
Getzler’s setting without reciting all details. For $h\in H,$ consider the
associated equivariant one-form

$\alpha^{h}_{\epsilon}(X)=\sqrt{\frac{\epsilon}{\pi}}{\mathrm{Tr}}(hXe^{-\epsilon
D^{2}})$

for some self-adjoint bounded linear operator $X$ on
${\mathcal{H}}^{\prime}=L^{2}(H,{\mathcal{H}}).$ Then, $\alpha^{h}_{\epsilon}$
has locally bounded first derivatives and is closed
($d\alpha^{h}_{\epsilon}=0)$. Moreover, $d\eta_{h,\epsilon}(D_{P})$ can be
derived to be $2\alpha^{h}_{\epsilon}$ on
$\Phi_{0}=\\{D\>|\>\dim\ker(D)=0\\}.$ Then, the integral $\int
d\eta_{h,\epsilon}$ is invariant under isotopies of the smooth paths $D(t).$
In particular, there is an isotopy from $D_{P}(t)$ to a path transversal to
$\Phi_{1}=\\{D\>|\>\dim\ker(D)=1\\},$ so we may assume that $D_{P}(t)$ is
transversal to $\Phi_{1}.$ Let $\\{t_{i}\\}^{k}_{i=1}$ be the ordered sequence
of points in the open interval $(0,1)$ for which $D_{P}(t)$ intersects
$\Phi_{1}.$ By [13, Proposition 2.5], $(D_{P}(t))^{*}\eta_{h,\epsilon}$ is a
smooth function in $t\in[0,1]$ except at the points $t_{i}.$ With the
truncated $\eta-$function

$\eta_{h,\epsilon}(\lambda)=\frac{1}{\sqrt{\pi}}\int^{\infty}_{\epsilon}t^{-\frac{1}{2}}{\mathrm{Tr}}(h\lambda
e^{t\lambda^{2}})dt$

that changes signs across the $x-$axis, we have

(9.6) $\displaystyle{\mathrm{sf}}_{h}\\{D_{p}(t)\\}_{t\in[0,1]}$
$\displaystyle=\frac{1}{2}\sum^{k}_{i=1}\Big{(}\lim_{t\to
t^{+}_{i}}\eta_{h,\epsilon}(D_{P}(t))-\lim_{t\to
t^{-}_{i}}\eta_{h,\epsilon}(D_{P}(t))\Big{)}$
$\displaystyle=\frac{1}{2}\Big{(}\eta_{h,\epsilon}(D_{P}(1))-\eta_{h,\epsilon}(D_{P}(0))-\int_{D_{P}(t)}d\eta_{h,\epsilon}(D_{P}(t))\Big{)}$
$\displaystyle=\frac{1}{2}\Big{(}\eta_{h,\epsilon}(D_{P}(1))-\eta_{h,\epsilon}(D_{P}(0))-\int^{1}_{0}\frac{d}{dt}\eta_{h,\epsilon}(D_{P}(t))dt\Big{)}$

where the second line follows from the Fundamental Theorem of Calculus. From
Definition 9.1, for some $\epsilon>0,$ we can rewrite the equivariant
$\eta$-function $\eta_{h}(D_{P}(t),s)$ as

(9.7) $\displaystyle\eta_{h}(D_{P}(t))$
$\displaystyle=\sum_{\begin{subarray}{c}\lambda\in{\text{spec}}(D_{p}(t))\\\
0<|\lambda|<\epsilon\end{subarray}}{\mathrm{Tr}}(h^{*}_{\lambda})\frac{\text{sgn}(\lambda)}{|\lambda|^{s}}+\sum_{\begin{subarray}{c}\lambda\in{\text{spec}}(D_{p}(t))\\\
\epsilon\leq|\lambda|\end{subarray}}{\mathrm{Tr}}(h^{*}_{\lambda})\frac{\text{sgn}(\lambda)}{|\lambda|^{s}}$
$\displaystyle=\eta_{h,<\epsilon}(D_{P}(t),s)+\eta_{h,\geq\epsilon}(D_{P}(t),s)$

for $\Re(s)$ sufficiently large. Here, $\eta_{h,\geq\epsilon}$ is the
truncated $\eta$-invariant from (9.3). Note that over $(-\epsilon,\epsilon)$
the $\eta$-invariant takes the form

(9.8)
$\eta_{h,<\epsilon}(D_{P}(t))=\sum_{\begin{subarray}{c}\lambda\in{\text{spec}}(D_{p}(t))\\\
0<|\lambda|<\epsilon\end{subarray}}{\mathrm{Tr}}(h^{*}_{\lambda})\text{sgn}(\lambda)=\sum_{\begin{subarray}{c}\lambda\in{\text{spec}}(D_{p}(t))\\\
-\epsilon<t<\epsilon\end{subarray}}\pm{\mathrm{Tr}}(h|_{\ker(P^{+}(t))})$

where $P^{+}(t)$ denotes the positive spectral projection over
$(-\epsilon,\epsilon),$ see (3.2) in Definition 3.2. Then, by the additivity
of spectral flow, we obtain

(9.9)
$\eta_{h,<\epsilon}(D_{P}(1))-\eta_{h,<\epsilon}(D_{P}(0))={\mathrm{Tr}}(h|_{\ker(D_{P}(0))})-{\mathrm{Tr}}(h|_{\ker(D_{P}(1))}).$

From the first two terms on the right side of (9.6), (9.8), and (9.9), we
deduce that

(9.10)
$\displaystyle\eta_{h,\geq\epsilon}(D_{P}(1))-\eta_{h,\geq\epsilon}(D_{P}(0))$
$\displaystyle=\big{[}\eta_{h}(D_{P}(1))-\eta_{h}(D_{P}(0))\big{]}-\big{[}\eta_{h,<\epsilon}(D_{P}(1))-\eta_{h,<\epsilon}(D_{P}(0))\big{]}$
$\displaystyle=\big{[}\eta_{h}(D_{P}(1))+{\mathrm{Tr}}(h|_{\ker(D_{P}(1))})\big{]}-\big{[}\eta_{h}(D_{P}(0))+{\mathrm{Tr}}(h|_{\ker(D_{P}(0))})\big{]}.$

Together with the fact that $\eta_{h,>\epsilon}(D_{P}(t))$ is smooth and
$\eta_{h,>\epsilon}(D_{P}(t))\equiv\eta_{h}(D_{P}(t))\mod R(H),$ we obtain
from (9.6) and (9.10) that

$\displaystyle{\mathrm{sf}}_{h}\\{D_{p}(t)\\}_{t\in[0,1]}$
$\displaystyle=\frac{1}{2}\Big{(}\eta_{h,\epsilon}(D_{P}(1))-\eta_{h,\epsilon}(D_{P}(0))-\int
d\eta_{h,\epsilon}(D_{P}(t))\Big{)}$
$\displaystyle=\frac{1}{2}\big{[}\eta_{h}(D_{P}(1))+{\mathrm{Tr}}(h|_{\ker(D_{P}(1))})\big{]}-\frac{1}{2}\big{[}\eta_{h}(D_{P}(0))+\big{[}{\mathrm{Tr}}(h|_{\ker(D_{P}(0))})\big{]}$
$\displaystyle\quad\quad-\frac{1}{2}\int d\eta_{h}(D_{P}(t))$
$\displaystyle=\tilde{\eta}_{h}(D_{P}(1))-\tilde{\eta}_{h}(D_{P}(0))-\frac{1}{2}\int
d\eta_{h}(D_{P}(t)).$

This completes the proof. ∎

The next lemma is, strictly speaking, a special case of [15, Theorem 4.4] in
which the unitary operator associated to a boundary projection takes an
explicit form that uses error functions. Let

$\varphi(x)=\frac{2}{\sqrt{\pi}}\int^{x}_{0}e^{-t^{2}}dt$

be the error function of $x.$ Note that $\varphi\in C^{1}({\mathbb{R}})$ is a
special case of normalizing functions satisfying

$\varphi^{-1}(0)=\\{0\\},\quad\lim_{x\to-\infty}\varphi(x)=-1,\quad\lim_{x\to\infty}\varphi(x)=1,$
$\varphi^{\prime}\in C_{0}({\mathbb{R}})\text{ with
}\varphi^{\prime}(0)>0\text{ and }\varphi^{\prime}\geq 0.$

See [26]. For any $P\in{\mathrm{Gr}}_{h}(A),$ consider the associated unitary
operator $T=\exp(i\pi\varphi(D_{P})).$ From (2.7), there is an involution
$\sigma=2P-{\mathrm{Id}}$ for any given $P,$ where $T$ sits at the bottom left
block and $T^{*}$ sits at the upper right block of $\sigma,$ satisfying the
anti-commutativity property $\sigma\gamma=-\gamma\sigma.$

######  Lemma 9.6.

Let $h\in H.$ For $t\in[0,1],$ let $D_{P}(t)$ be a smooth path of equivariant
Dirac operators with invertible endpoints $D_{P}(0)$ and $D_{P}(1).$ Then,

(9.11) $\tilde{\eta}_{h}(D_{P}(1))-\tilde{\eta}_{h}(D_{P}(0))=\frac{1}{2\pi
i}{\mathrm{Tr}}_{h}\log(T^{*}K)$

where $T=\exp(i\pi\varphi(D_{P}(1)))$ and $K=\exp(i\pi\varphi(D_{P}(0)))$ for
$\varphi(D_{P}(i))$ denote the error function evaluated at $D_{P}(i)$ with
$i=0,1.$

###### Proof.

Define $V_{t}=-\exp(i\pi\varphi(D_{P}(t))).$ Then, for $t\in[0,1],$

$V_{t}^{-1}=-\exp(-i\pi\varphi(D_{P}(t))),\quad\frac{dV_{t}}{dt}=-i\pi\exp(i\pi\varphi(D_{P}(t)))\big{(}\frac{d}{dt}D_{P}(t)\big{)}\varphi^{\prime}(D_{P}(t)).$

Hence,

(9.12) ${\mathrm{Tr}}_{h}\left(V_{t}^{-1}\frac{dV_{t}}{dt}\right)=\frac{2\pi
i}{\sqrt{\pi}}{\mathrm{Tr}}\left(h\dot{D}_{P}(t)e^{-D^{2}_{P}(t)}\right).$

On the other hand, from the proof of Theorem 9.5, which follows from [13],
that with $\epsilon=1$ we obtain

(9.13)
$\frac{d}{dt}\eta_{h}(D_{p}(t))=\frac{2}{\sqrt{\pi}}{\mathrm{Tr}}(h\dot{D}_{P}(t)e^{-D^{2}_{P}(t)}).$

Hence, from (9.12) and (9.13), we have

(9.14) $\frac{1}{2}\frac{d}{dt}\eta_{h}(D_{p}(t))=\frac{1}{2\pi
i}{\mathrm{Tr}}_{h}\left(V_{t}^{-1}\frac{dV_{t}}{dt}\right).$

Consider the unitary path
$U_{t}=\exp(i\pi[t\varphi(D_{P}(0))+(1-t)D_{P}(1)]).$ By a direct computation,
we obtain

(9.15)
${\mathrm{Tr}}_{h}\left(U_{t}^{-1}\frac{dU_{t}}{dt}\right)={\mathrm{Tr}}_{h}\log\exp(i\pi(\varphi(D_{P}(0))-\varphi(D_{P}(1))))$

where the right side is independent of $t.$ Let
$\tilde{D}_{P}(t):=tD_{P}(0)+(1-t)D_{P}(1)$ be the linear path of equivariant
Dirac paths associated to $U_{t}.$ One can check that $\tilde{D}^{2}_{P}(t)$
equals $D^{2}_{P}(1)$ when $t=0$ and $D^{2}_{P}(0)$ when $t=1.$ By applying
(9.14) to $U_{t}$ and from (9.15), we have

$\displaystyle\frac{1}{2}[\eta_{h}(D_{P}(1))-\eta_{h}(D_{P}(0))]$
$\displaystyle=\frac{1}{2}\int^{1}_{0}d\eta_{h}(\tilde{D}_{P}(t))=\frac{1}{2\pi
i}\int^{1}_{0}{\mathrm{Tr}}_{h}\left(U_{t}^{-1}\frac{dU_{t}}{dt}\right)dt$
$\displaystyle=\frac{1}{2\pi
i}\int^{1}_{0}{\mathrm{Tr}}_{h}\log\exp(i\pi(\varphi(D_{P}(0))-\varphi(D_{P}(1))))dt$
$\displaystyle=\frac{1}{2\pi i}{\mathrm{Tr}}_{h}\log(T^{*}K).$

By the invertibility assumption of $D_{P}(i)$ for $i=0,1,$ there is no
contribution from the $h$-trace on the trivial kernels. Hence, we obtain

$\displaystyle\tilde{\eta}_{h}(1)-\tilde{\eta}_{h}(0)=\frac{1}{2}[\eta_{h}($
$\displaystyle D_{P}(1))+{\mathrm{Tr}}(h|_{\ker D_{P}(1)})]$
$\displaystyle-\frac{1}{2}[\eta_{h}(D_{P}(0))+{\mathrm{Tr}}(h|_{\ker
D_{P}(0)})]=\frac{1}{2\pi i}{\mathrm{Tr}}_{h}\log(T^{*}K).$

∎

######  Theorem 9.7.

Let $h\in H.$ For $t\in[0,1],$ let $D_{P}(t)$ be a smooth path of equivariant
Dirac operators with endpoints $D_{P}(0)=D_{P}$ and $D_{P}(1)=D_{Q}.$ Let
$({\mathcal{L}}(t),{\mathcal{L}}_{M}(t))$ be a smooth path in
${\mathfrak{L}}_{F}({\mathcal{H}}^{\prime})$ whose endpoints
${\mathcal{L}}(0)$ (resp. ${\mathcal{L}}(1)$) correspond to the orthogonal
projections $P$ (resp. $Q$). Then,

(9.16)
${\mathrm{sf}}_{h}\\{D_{P}(t)\\}={\mathrm{Mas}}_{h}({\mathcal{L}}(t),{\mathcal{L}}_{M}(t)).$

###### Proof.

First, let $P(t)\in{\mathrm{Gr}}_{h}^{\infty}(A(t)).$ Since the path
$D_{{\mathcal{P}}_{M}}(t):=D_{{\mathcal{P}}_{M}(t)}(t)$ is invertible for all
$t,$ its equivariant spectral flow is
${\mathrm{sf}}_{h}\\{D_{{\mathcal{P}}_{M}}(t)\\}_{t\in[0,1]}=0.$ Let

$T(t)=\exp(i\pi\varphi(D_{P}(t))),\quad
K(t)=\exp(i\pi\varphi(D_{{\mathcal{P}}_{M}}(t)))$

where $\varphi$ denotes the error function. By Theorem 9.5, Lemma 9.6, (5.7),
and Theorem 5.8, we have

$\displaystyle{\mathrm{sf}}_{h}\\{D_{P}(t)\\}$
$\displaystyle={\mathrm{sf}}_{h}\\{D_{P}(t)\\}-{\mathrm{sf}}_{h}\\{D_{{\mathcal{P}}_{M}}(t)\\}$
$\displaystyle=\left[\tilde{\eta}_{h}(D_{P}(1))-\tilde{\eta}_{h}(D_{{\mathcal{P}}_{M}}(1))\right]-\left[\tilde{\eta}_{h}(D_{P}(0))-\tilde{\eta}_{h}(D_{{\mathcal{P}}_{M}}(0))\right]$
$\displaystyle\quad\quad-\frac{1}{2}\int^{1}_{0}\frac{d}{dt}\left(\tilde{\eta}_{h}(D_{P}(t))-\tilde{\eta}_{h}(D_{{\mathcal{P}}_{M}}(t))\right)dt$
$\displaystyle=\left[\frac{1}{2\pi
i}{\mathrm{Tr}}_{h}\log(T^{*}(1)K(1))\right]-\left[\frac{1}{2\pi
i}{\mathrm{Tr}}_{h}\log(T^{*}(0)K(0))\right]$
$\displaystyle\quad\quad-\frac{1}{2\pi
i}\int^{1}_{0}\frac{d}{dt}\left({\mathrm{Tr}}_{h}\log(T^{*}(t)K(t))\right)dt$
$\displaystyle=w_{h}(T^{*}(t)K(t))={\mathrm{Mas}}_{h}({\mathcal{L}}(t),{\mathcal{L}}_{M}(t)).$

Once this is established, the generalization to arbitrary
$P(t)\in{\mathrm{Gr}}_{h}(A(t))$ can be done by using a similar trick (with
appropriate modification) in the proof of [15, Theorem 7.5]. ∎

Let $M^{\pm}$ be the compact spin manifold with boundary obtained by cutting
$M$ into two disjoint parts along the boundary $Y=\partial M.$ Assume an
identification between the boundary of the split manifolds $M^{\pm}$ and their
collar neighborhoods $\partial M^{+}\times(-\epsilon,0]$ and $\partial
M^{-}\times[0,\epsilon)$ respectively. Near the collar neighborhood, the Dirac
operator takes the product form

$D=\begin{pmatrix}\gamma&0\\\
0&-\gamma\end{pmatrix}\left(\frac{d}{dx}+\begin{pmatrix}A&0\\\
0&-A\end{pmatrix}\right).$

Since $h$ commutes with $\gamma$ and $A,$ it commutes with $D$ in this
splitting. Hence, it extends to act on the components of $L^{2}(E|_{\partial
M^{\pm}})=L^{2}(E|_{Y})\oplus L^{2}(E|_{Y})$ respectively. The following is
regarding the splitting of equivariant $\eta$-invariants with respect to
$M^{\pm}.$ The proof can be obtained almost verbatim by the arguments of [15],
which require a special angle-valued homotopy $P(\theta,P)$ of projections.
Its spectral flow is not computable in general, but it leads to a complete
splitting formula of $\eta$-invariants. Kirk and Lesch also established the
equality
${\mathrm{sf}}\\{D^{M^{\pm}}_{P(\theta,P)}\\}={\mathrm{sf}}\\{D^{M^{\pm}}_{P_{t}\oplus({\mathrm{Id}}-P_{t})}\\}$
with respect to the splitting. By adapting the equivariance assumption by
elements of $H$ above to their method, we obtain the following.

######  Theorem 9.8.

Let $h\in H.$ Let $M=M^{+}\cup_{Y}M^{-}.$ Let
$P_{t}\in{\mathrm{Gr}}_{h}(A_{t})$ be an equivariant path connecting
${\mathcal{P}}_{M^{+}}$ at $t=0$ and $P$ at $t=1.$ Similarly, let
${\mathrm{Id}}-P_{t}$ be another path connecting ${\mathcal{P}}_{M^{-}}$ and
${\mathrm{Id}}-P.$ Let $D^{M^{+}}_{P}$ and $D^{M^{-}}_{{\mathrm{Id}}-P}$ be
the Dirac operators equipped with $P$ and ${\mathrm{Id}}-P_{t}$ respectively.
Then,

(9.17)
$\tilde{\eta}_{h}(D^{M})=\tilde{\eta}_{h}(D^{M^{+}}_{P})+\tilde{\eta}_{h}(D^{M^{-}}_{{\mathrm{Id}}-P})-{\mathrm{sf}}_{h}\\{D^{M^{\pm}}_{P_{t}\oplus{\mathrm{Id}}-P_{t}}\\}.$

######  Remark 9.9.

It is noteworthy to point out that there was an attempt to give an alternate
proof to this splitting formula of equivariant $\eta$-invariants by using a
trick of Dai-Zhang [10] by cutting a manifold into three parts. It relies on
the behaviour of the invariants under a homotopy between two certain operators
whose difference is a bounded operator. Unfortunately, $\eta$-invariants are
not homotopy invariants in general. In special cases where this is true,
another simplified proof of Theorem 9.8 can be obtained by using merely the
properties of spectral flow, which does not require the angle-valued homotopy
above.

The opposite symplectic structures between the two collar neighborhoods induce
a flip of signs, e.g. $\gamma$ and $A$ over $\partial
M^{+}\times(-\epsilon,0]$ becomes $-\gamma$ and $-A$ over $\partial
M^{-}\times[0,\epsilon).$ Then, one can show that for compatibility reasons
(for $P$ to satisfy the Lagrangian property and lies in
$\in{\mathrm{Gr}}_{h}(-A)$), the induced relation between the association map
is given by $T_{-\gamma}(P)=-T_{\gamma}({\mathrm{Id}}-P)^{*}.$ As a
consequence, we have

(9.18)
${\mathrm{Mas}}^{\gamma}_{h}({\mathcal{L}}_{1}(t),{\mathcal{L}}_{2}(t))={\mathrm{Mas}}^{-\gamma}_{h}({\mathcal{L}}_{2}(t),{\mathcal{L}}_{1}(t)).$

######  Theorem 9.10.

With the above settings, for $h\in H,$ the following equality holds

$\tilde{\eta}_{h}(D^{M})=\tilde{\eta}_{h}(D_{P}^{M^{+}})+\tilde{\eta}_{h}(D_{{\mathrm{Id}}-P}^{M^{-}})+\tau^{h}_{\mu}({\mathrm{Id}}-{\mathcal{P}}_{M^{-}},P,{\mathcal{P}}_{M^{+}}).$

###### Proof.

Let $P\in{\mathrm{Gr}}_{h}(A).$ Since
$({\mathrm{Id}}-{\mathcal{P}}_{M^{-}})-{\mathcal{P}}_{M^{+}}$ is of trace
class, it is compact and its equivariant Maslov triple index
$\tau^{h}_{\mu}({\mathrm{Id}}-{\mathcal{P}}_{M^{-}},P,{\mathcal{P}}_{M^{+}})$
is well-defined for $P.$ For $t\in[0,1],$ consider a smooth equivariant path
$P(t)$ in ${\mathrm{Gr}}_{h}(A(t))$ connecting ${\mathcal{P}}_{M^{+}}$ and
$P.$ Let $\mathring{{\mathcal{L}}}$ be the Lagrangian subspace corresponding
to the orthogonal projection ${\mathrm{Id}}-P_{{\mathcal{L}}}.$

Note that $(\mathring{{\mathcal{L}}}_{M^{-}}(t),{\mathcal{L}}_{M^{+}}(t))$ is
constant for all $t,$ its equivariant Maslov index vanishes, i.e.
${\mathrm{Mas}}^{\gamma}_{h}(\mathring{{\mathcal{L}}}_{M^{-}}(t),{\mathcal{L}}_{M^{+}}(t))=0.$
Then, by (9.18), we have

(9.19)
$\displaystyle{\mathrm{Mas}}^{-\gamma}_{h}(\mathring{{\mathcal{L}}}(t),{\mathcal{L}}_{M^{-}}(t))+{\mathrm{Mas}}^{\gamma}_{h}({\mathcal{L}}(t),{\mathcal{L}}_{M^{+}}(t))$
$\displaystyle={\mathrm{Mas}}^{\gamma}_{h}(\mathring{{\mathcal{L}}}_{M^{-}}(t),{\mathcal{L}}(t))+{\mathrm{Mas}}^{\gamma}_{h}({\mathcal{L}}(t),{\mathcal{L}}_{M^{+}}(t))-{\mathrm{Mas}}^{\gamma}_{h}(\mathring{{\mathcal{L}}}_{M^{-}}(t),{\mathcal{L}}_{M^{+}}(t)).$

This is an algebraic combination of three double Maslov indices, where the
first, the second and the third paths of equivariant Lagrangians are
$\mathring{{\mathcal{L}}}_{M^{-}}(t),{\mathcal{L}}(t)$ and
${\mathcal{L}}_{M^{+}}(t)$ respectively. By (6.2), we have

(9.20)
$\displaystyle{\mathrm{Mas}}^{\gamma}_{h}(\mathring{{\mathcal{L}}}_{M^{-}}(t),{\mathcal{L}}(t))+{\mathrm{Mas}}^{\gamma}_{h}({\mathcal{L}}(t),{\mathcal{L}}_{M^{+}}(t))-{\mathrm{Mas}}^{\gamma}_{h}(\mathring{{\mathcal{L}}}_{M^{-}}(t),{\mathcal{L}}_{M^{+}}(t))$
$\displaystyle=\tau^{h}_{\mu}({\mathrm{Id}}-{\mathcal{P}}_{M^{-}},P,{\mathcal{P}}_{M^{+}})-\tau^{h}_{\mu}({\mathrm{Id}}-{\mathcal{P}}_{M^{-}},{\mathcal{P}}_{M^{+}},{\mathcal{P}}_{M^{+}})=\tau^{h}_{\mu}({\mathrm{Id}}-{\mathcal{P}}_{M^{-}},P,{\mathcal{P}}_{M^{+}}).$

Finally, we obtain

$\displaystyle\tilde{\eta}_{h}(D^{M})$
$\displaystyle=\tilde{\eta}_{h}(D_{P}^{M^{+}})+\tilde{\eta}_{h}(D_{1-P}^{M^{-}})+{\mathrm{sf}}_{h}\\{D^{M}_{P}(t)\\}+{\mathrm{sf}}_{h}\\{D^{M^{-}}_{{\mathrm{Id}}-P}(t)\\}$
$\displaystyle=\tilde{\eta}_{h}(D_{P}^{M^{+}})+\tilde{\eta}_{h}(D_{1-P}^{M^{-}})+{\mathrm{Mas}}^{\gamma}_{h}({\mathcal{L}}(t),{\mathcal{L}}_{M^{+}}(t))+{\mathrm{Mas}}^{-\gamma}_{h}(\mathring{{\mathcal{L}}}(t),{\mathcal{L}}_{M^{-}}(t))$
$\displaystyle=\tilde{\eta}_{h}(D_{P}^{M^{+}})+\tilde{\eta}_{h}(D_{1-P}^{M^{-}})+\tau^{h}_{\mu}({\mathrm{Id}}-{\mathcal{P}}_{M^{-}},P,{\mathcal{P}}_{M^{+}})$

where the second equality follows from Theorem 9.7 and the third equality
follows from (9.19) and (9.20). ∎

######  Remark 9.11.

A consequence of Theorem 9.10 will be illustrated in forming the boundary
correction terms of the equivariant Toeplitz index theorem on odd-dimensional
manifolds with boundary in [16].

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