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% Author Package file for use with AMS-LaTeX 1.2
\controldates{29-MAR-2001,29-MAR-2001,29-MAR-2001,29-MAR-2001}
 
\documentclass{era-l}
\issueinfo{7}{03}{}{2001}
\dateposted{April 2, 2001}
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\copyrightinfo{2001}{American Mathematical Society}
%\def{\copyrightyear}{2001}
%\copyrightinfo{2001}{American Mathematical Society}

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\theoremstyle{remark}
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\hyphenation{dif-fer-en-tial Dol-beault Fred-holm geo-metry
geo-metrically Grass-mannian Grass-mannians mani-fold mani-folds
para-meterization pseudo-dif-fer-en-tial }



\begin{document}

\title[Zeta Determinants of EBVPs]{Relative
zeta determinants and the geometry of the determinant line bundle}

\author{Simon Scott}
\address{Department of Mathematics, King's College, London WC2R 2LS, U.K.}
\email{sscott@mth.kcl.ac.uk}

\commby{Michael Taylor}

\date{December  15, 1999, and, in revised form, September 15, 2000}

\subjclass[2000]{Primary 58G20, 58G26, 11S45; Secondary 81T50}

\begin{abstract} The spectral $\zeta$-function regularized geometry
of the determinant line bundle for a family of first-order
elliptic operators over a closed manifold encodes a subtle
relation between the local family's index theorem and fundamental
non-local spectral invariants. A great deal of
interest has been directed towards a generalization of this theory
to families of elliptic boundary value problems. We give here
precise formulas for the relative zeta metric and curvature in
terms of Fredholm determinants and traces of operators over the
boundary. This has consequences for anomalies over manifolds with
boundary.
\end{abstract}

\maketitle


\section{Introduction}

In this paper we study the zeta-function regularized geometry of
the determinant line bundle for a family of first-order elliptic
boundary value problems (EBVPs). By an EBVP we mean an elliptic
differential operator over a compact manifold with boundary,
endowed with a global (injectively elliptic) boundary condition.
In \cite{Sc99} it was shown that the determinant line bundle for
such a family has a natural Hermitian metric defined by the
Fredholm determinant of a canonically associated operator over the
boundary. The resulting canonical metric has an analytical status
which is essentially opposite to that of the usual zeta function
metric \cite{BiFr86,Qu85}. That the two metrics, and their
associated connections, are nevertheless precisely related derives
from a certain `relativity principle for determinants', which
asserts that for preferred classes of unbounded operators, ratios
of $\z$-determinants can be written canonically in terms of
Fredholm determinants (Theorem 1). This is interesting because the
(usual) $\z$-determinant, on the other hand, does {\em not} define
an extension of the Fredholm determinant (operators with Fredholm
determinants do not have $\z$-determinants)\footnote{We consider
here infinite-dimensional Hilbert spaces.}; indeed there is no
extension of the Fredholm determinant to a (multiplicative)
determinant on general classes of elliptic pseudodifferential
operators ($\pdo$s).

This fact leads to the construction of `higher' differential
geometries for families of $\pdo$s, via regularized traces and
determinants, such as that defined by the spectral $\z$-function.
Our results indicate that  the `relativity principle' governs a
precise relation between relative higher differential geometries
and the (usual) canonical differential geometry associated to the
standard trace. Applied to families of EBVPs this means that
geometric anomalies, associated with the break down of symmetry
invariance in higher traces, are located on the boundary. Details
of the constructions here, which build on ideas in
\cite{Fo87,Gr99,GrSe96,LeTo98,Sc99,ScWo99}, will be presented
in \cite{Sc2000}.

%******************************************************************
%
\section{Relative zeta determinants}
 Let $A_1, A_2$ be invertible closed operators on a Hilbert space
$H$ with a common spectral cut $R_{\myth} = \{re^{i\myth}:r\geq 0\}$.
This means that there exists an $\e >0$ such that the resolvents
$(A_i - \la)\ii$ are holomorphic in the sector
\[\Lambda_{\myth,\e}
= \{z\in \C\backslash\{0\} \ | \ |\arg(z)-\myth| < \e\}\] and such
that the operator norms $\|(A_i - \la)\ii\|$ are $O(|\la|\ii)$ as
$\la\to\infty$ in $\Lambda_{\myth,\e}$. For $\Re(s)>0$ one has the
complex power operators
\begin{equation*}
  A_{i}\si = \frac{i}{2\pi}\int_{\Gamma}\la_{\myth}\si\, (A_i - \la)\ii \,
  d\la,
\end{equation*}
where $\la_{\myth}\si = |\la|\si e^{-is\arg(\la)}, \, \myth - 2\pi
\leq \arg(\la)\leq \myth$, is the branch of $\la\si$ defined by the
spectral cut $R_{\myth}$, and $\Gamma$ is the contour traversing
$R_{\myth}$ from $\infty$ to a small circle around the origin,
clockwise around the circle, and then back to $\infty$ along
$R_{\myth}$.


If we assume that the $A_{i}\si$ are trace class in some
half-plane $\Re(s)>r>0$, one then has spectral zeta functions
\begin{equation*}
  \z_{\myth}(A_{i},s) = \Tr A_i\si  = \sum_{\mu\in
  {\rm s p}(A_i)}\mu\si,\hskip 10mm \Re(s)>r.
\end{equation*}
We further assume that $\dd_{\la}^{m}(A_i - \la)\ii$ are trace
class for $m+1>r$ with asymptotic expansions  as $\la\to \myo$ along
the ray $R_{\myth}$
\begin{equation}\label{e:exp1}
  \Tr (\dd_{\la}^{m}(A_i - \la)\ii)   \sim
  \sum_{j=0}^{\infty}\sum_{k=0}^{n_j} a_{j,k}^{(i)}(-\la)^{-(\mya_j
  + m)}\log^k (-\la)
\end{equation}
with $0< \mya_j + m \nearrow +\infty$. From Seeley's work (see also
\cite{GrSe96}) it is well known that the expansion \eqref{e:exp1}
defines a meromorphic continuation of $\z_{\myth}(A_{i},s)$ to all
of $\C$, the term with coefficient $a_{j,k}^{(i)}$ corresponding
to a pole of $\G(s)\z_{\myth}(A_{i},s)$ at $s=\mya_j -1$ of order
$k+1$. In particular, if $a_{J,k}^{(1)} = a_{J,k}^{(2)} =0 $,
where $\mya_J = 1$  with $k\geq 1$, then there is no pole at $s=0$
and one has the $\z$-determinants
\begin{equation*}
 {\rm det}_{\z,\myth}A_1 = e^{-\z'_{\myth}(A_1,0)}, \hskip 15mm
 {\rm det}_{\z,\myth}A_2 = e^{-\z'_{\myth}(A_2,0)},
\end{equation*}
where $\z'_{\myth} = d/ds(\z_{\myth})$. In this case we refer to
each of $A_1, A_2$ as $\z$-{\it admissible}.

We refer to $(A_1, A_2)$ as $\z$-{\it comparable} if
\begin{equation}\label{e:polescancel}
  a_{j,k}^{(1)} = a_{j,k}^{(2)} \hskip 5mm {\rm for}\ j < J,
\end{equation}
   and 
\[a_{J,k}^{(1)} = a_{J,k}^{(2)} \hskip 5mm {\rm for}\ k\geq
  1,\]
and if the relative resolvent $(A_1 - \la)\ii - (A_2 - \la)\ii$ is
trace class such that
\begin{equation*}
\Tr((A_1 - \la)\ii - (A_2 - \la)\ii) = -\frac{\dd}{\dd\la}\log{\rm
det}_{F}\Ss_{\la}.
\end{equation*}
Here the `scattering' operator $\Ss_{\la} = \Ss_{\la}(A_1,A_2)$ is
an operator of the form $Id + W_{\la}$ on a Hilbert space
$H'\subseteq H$ with $W_{\la}$ of trace class, so that
$\Ss_{\la}$ has a Fredholm determinant $\det_F\Ss_{\la} := 1 +
\sum_{k\geq 1}\Tr(\bigwedge^k W_{\la})$. The {\it relative spectral
$\z$-function}
\begin{equation*}
  \z_{\myth}(A_{1},A_{2},s) = \Tr (A_1\si - A_2\si)   =
 \frac{i}{2\pi}\int_{\Gamma}\la_{\myth}\si \, \Tr((A_1 - \la)\ii -
(A_2 - \la)\ii)  \,  d\la
\end{equation*}
is then well defined and holomorphic in $s$ for $\Re(s)>0$, and
equal to $\z_{\myth}(A_{1},s) - \z_{\myth}(A_{2},s)$.
 Using \eqref{e:polescancel}, we see that as
 $\la\to\myo$ in $\Lambda_{\myth,\e}$, there is an asymptotic expansion
\begin{equation*}
  \Tr((A_1 - \la)^{-1} - (A_2 - \la)^{-1})    \sim
  \sum_{j=J+1}^{\infty}\sum_{k=0}^{n_j} c_{j,k}
  (-\la)^{-\mya_j}\log^k (-\la) + \frac{c_{J,0}}{\la} \, .
\end{equation*}
The meromorphic extension of $\z_{\myth}(A_{1},A_{2},s)$ to $\C$ is
therefore regular at zero and we can define the {\it relative
$\z$-determinant} ${\rm det}_{\z,\myth}(A_1,A_2) =
e^{-\z^{'}_{\myth}(A_1,A_2,0)}  \, .$

The
{\em regularized limit} ${\rm LIM}^{\theta}_{\lambda\to \myo}$
of a function with an asymptotic expansion \[f(\la)\sim
\sum_{j=0}^{\infty}\sum_{k=0}^{m_j}
b_{j,k}(-\la)^{-\myb_j}\log^{k}(-\la)\] as $\la\to\myo$ in
$\Lambda_{\myth,\e}$, where  $\myb_j\nearrow +\infty$ and $\myb_N = 0$,
is defined to be the constant term in the expansion $b_{N,0}$. We
have (with $\Ss := \Ss_0$):

%Theorem 1
\begin{theorem}
For $\z$-comparable operators
$A_1,A_2$
\begin{equation}\label{e:prop1}
 {\rm det}_{\z,\myth}(A_1,A_2) = {\rm det}_F \Ss\, . \,
 e^{-{\rm LIM^{\myth}_{\la\to\infty}}\log\det_F\Ss_{\la}} \, .
\end{equation}
If $A_1,A_2$ are $\z$-admissible, $ {\rm det}_{\z,\myth}(A_1,A_2) =
{\rm det}_{\z,\myth}A_1 /{\rm det}_{\z,\myth}A_2 \, .$
\end{theorem}
\begin{proof}
Since $\la\si\log\det_F\Ss_{\la}\to 0$ at the ends of $\Gamma$ for
$\Re(s)>0$, we can integrate by parts in
\[ \z_{\myth}(A_{1},A_{2},s) =
 -\frac{i}{2\pi}\int_{\Gamma}\la_{\myth}\si \, \dd_{\la}\log{\rm
det}_{F}\Ss_{\la}  \,  d\la \, . \] Taking the $s$ derivative we
end up with $ \z_{\myth}'(A_{1},A_{2},s) = d/ds_{|s=0}(sg(s))$,
where \[ g(s) =
 -\frac{i}{2\pi}\int_{\Gamma}\la_{\myth}\si \, (\log{\rm
det}_{F}\Ss_{\la})/\la  \,  d\la \, \] has a simple pole at $s=0$.
From the Laurent expansion of $(\log{\rm det}_{F}\Ss_{\la})/\la$
around $0$, and its asymptotic expansion as $\la\to\infty$ in
$\Lambda_{\myth,\e}$, we can use the methods of \cite{GrSe96} to
obtain the full pole structure of $\z_{\myth}'(A_{1},A_{2},s)$ on
$\C$. Evaluation at $s=0$ then yields \eqref{e:prop1}. The final
statement follows from $\z_{\myth}(A_{1},A_{2},s) =
\z_{\myth}(A_{1},s) - \z_{\myth}(A_{2},s)$.
\end{proof}

In the case $\z_{\myth}(A_1,A_2,0)=0$  the regularized LIM can be
replaced by the usual $\lim$, and $\z_{\myth}(A_{1},A_{2},s)$
exists at $0$ without continuation. In particular, this applies to
determinant class operators, that is, for $A_1,A_2$ with Fredholm
determinants. The $\z_{\myth}(A_i,s)$ are then undefined for all
$s$, but $A_1,A_2$ are always $\z$-comparable and from
\eqref{e:prop1}
\begin{equation}\label{e:relFred}
{\rm det}_{\z,\myth}(A_{1},A_{2}) = {\rm det}_{F}(A_2\ii A_1) =
\frac{{\rm det}_{F}A_1}{{\rm det}_{F} A_2}.
\end{equation}
Thus for any $A$ of determinant class, ${\rm det}_{\z,\myth}(A,Id) =
{\rm det}_{F}(A)$. This is independent of $\myth$, equivalent to the
fact that $\Gamma$ can be closed at $\myo$ and replaced by a bounded
contour.

\subsection{Elliptic boundary value problems}

Let $X$ be a compact connected Riemannian manifold with boundary
$\dd X = Y$ and let $E^1,E^2$ be Hermitian vector bundles over
$X$. Let $A: \Ci(X,E^1)\to \Ci(X,E^2)$ be a first-order elliptic
operator of Dirac type. By this we mean that there is a collar
neighborhood $U = [0,1)\times Y$ of the boundary in which $A$ has
the form
\begin{equation}\label{e:collar}
  A_{|U} = \sigma\left(\frac{\dd}{\dd u} + B + R \right),
\end{equation}
where $B : \Ci(Y,E_{|Y}^1)\to\Ci(Y,E_{|Y}^1)$ is a first-order
selfadjoint elliptic operator over the closed manifold $Y$, $R$
is an operator of order $0$, and $\sigma : E_{|U}^1 \to E_{|U}^2$
a unitary isomorphism constant in $u$.

For each real $s > 1/2$,  restriction to the boundary defines a
continuous operator $\gamma :H^s (X,E^1)\to H^{s-1/2}(Y,E_{|Y}^1)$
on the Sobolev completions, and we have the Cauchy data space
$H(A,s) = \gamma\Ker(A,s)$, where 
\[
\Ker(A,s) = \{\psi\in H^s(X,E^1)\mid A\psi = 0\}.
\] 
Because of the Unique Continuation Property,
$\gamma:\Ker(A,s)\to H(A,s) $ is a bijection while the {\it
Poisson operator} $ \Kk_{A} : H^{s-1/2}(Y,E_{|Y}^1) \too \Ker(A,s)
\subset H^s (X,E^1)$ of $A$ defines a left inverse to $\gamma$.

The classical pseudodifferential operator ($\pdo$) $P(A) :=
\gamma\Kk_A$ of order $0$ is a projection on
$H^{s-1/2}(Y,E_{|Y}^1)$ with range $H(A,s)$, called the {\it
Calder\'{o}n projection}. Associated to $P(A)$ we have the {\it
pseudodifferential Grassmannian} $Gr_{-1}(A)$ parameterizing
(orthogonal) projections $P$ on $H_Y = L^2 (Y,E_{|Y}^1)$ such that
$P-P(A)$ is a $\pdo$ of order $-1$. The {\it smooth Grassmannian}
$Gr_{-\myo}(A)$ is the infinite-dimensional dense submanifold of
$Gr_{-1}(A)$ of those $P$ such that $P-P(A)$ is a smoothing
operator. Each $P\in Gr_{-1}(A)$ defines an EBVP
\begin{equation*}
  A_P = A : \dom(A_P) \too L^2 (X,E^2),
\end{equation*}
with $\dom(A_P) = \{\psi\in H^1 (X,E^1) \ | \ P\g\psi=0 \}.$ As a
Fredholm operator, $A_P$ is modeled by the boundary operator $S_A
(P) = P\circ P(A) : H(A) \to W = \ran(P),$ in so far as the
Poisson operator effects canonical isomorphisms $\Ker A_P \cong
\Ker S_{A}(P)$, $\Cok A_P \cong \Cok S_{A}(P)$, leading to the
relative-index formulas
\begin{equation*}
 \ind (A_{P_1}) -  \ind (A_{P_2})= \ind (P_2,P_1), \hskip 10mm
  \ind (A_P) = \ind(S_{A}(P)),
\end{equation*}
where $(P_i,P_j) := P_j\circ P_i :\ran(P_i)\to \ran(P_j)$.
Moreover, if $A_P$ is invertible, then so is $S_{A}(P)$ and we can
then define the Poisson operator of $A_P$ by
\begin{equation*}
  \Kk_{A}(P) := \Kk_A S_{A}(P)\ii P : H^{s-1/2}(Y,E_{|Y}^1)
  \too H^s (X,E^1).
\end{equation*}
From the identities $ A_P\ii A = I - \Kk_{A}(P)\g$ and $A_P A_P
\ii = Id_{L^2}$ we obtain the relative-inverse formula
\begin{equation}\label{e:relinv}
A_{P_1}\ii = A_{P_1}\ii A_{P_2}A_{P_2} \ii = (A_{P_1}\ii A)A_{P_2}
\ii  = A_{P_2} \ii   - \Kk_{A}(P_1)\g  A_{P_2} \ii.
\end{equation}
For references to details of these facts we refer the reader to
\cite{Gr99,GrSe96,ScWo99}.


\begin{prop}\label{p:prop2}
Let $A_z$ be a $1$-parameter family of Dirac-type operators
depending smoothly on a complex parameter $z$. Let $\dot{A}_z =
\frac{d}{d z}A_z$ and let $P_1,P_2\in Gr_{-1}(A)$ such that $P_1 -
P_2$ has a smooth kernel and such that $A_{z,P_i}$ are invertible
for each $z$. Then $\dot{A}_z (A_{z,P_1}\ii - A_{z,P_2}\ii)$ is a
trace class operator on $L^{2}(X,E_2)$ with
\begin{equation}\label{e:variation}
\Tr(\dot{A}_z (A_{z,P_1}\ii - A_{z,P_2}\ii))  =  \frac{d}{d
z}\log{\rm det}_{F}\left(\frac{S_z (P_1)}{P_1 S_z (P_2)}\right).
\end{equation}
\textup{[}For simplicity we assume here that $ (P_2,P_1)$ is invertible.
For the general case see \cite{Sc2000}.\textup{]}
\end{prop}
\begin{proof}
 We compute that
\begin{equation}\label{e:fact1}
\frac{d}{d z}A_{z}.\Kk_{z}(P_1) = - A_{z,P_1}\frac{d}{d
z}\Kk_{z}(P_1),
\end{equation}
\begin{equation}\label{e:fact2}
P_1\gamma A_{z,P_2}\ii A_{z,P_1} = -\Kk_{z}(P_1)\ii
\Kk_{z}(P_2)P_2\gamma = -S_{z}(P_1)S_{z}(P_2)\ii P_2\g,
\end{equation}
\begin{equation}\label{e:fact3}
P_2\frac{d}{d z}\g \Kk_{z}(P_1) = \frac{d}{d z}\left(
\Kk_{z}(P_2)\ii\Kk_{z}(P_1)P_1\right) = \frac{d}{d z}\left(
S_{z}(P_2)\ii\ S_{z}(P_1)P_1\right),
\end{equation}
where $S_z := S_{A_z}$. Using \eqref{e:relinv} and \eqref{e:fact1}
we obtain
\begin{equation*}
\Tr(\dot{A}_z (A_{z,P_1}\ii - A_{z,P_2}\ii)) = \Tr(P_1 \g
A_{z,P_2}\ii A_{z,P_1}\frac{d}{d z}\Kk_{z}(P_1) ),
\end{equation*}
while \eqref{e:fact2}, \eqref{e:fact3} reduce this to the right
side of \eqref{e:variation}.
\end{proof}

We assume now that $E^1 = E^2$ and $A$ is a first-order elliptic
operator of Dirac type. Let $P_1,P_2\in Gr_{-1}(A)$ with $P_1 -
P_2$ smoothing. Assume further that $A_{P_1}, A_{P_2}$ are
invertible, $\z$-admissible, with spectral cut $R_{\myth}$. Then by
setting $A_{\la} = A - \la$ for $\la\in\Gamma$, an application of
Theorem 1 yields the following formula:

%Theorem 2
\begin{theorem}
 With the above assumptions (putting
$S_A = S$)
\begin{equation}\label{e:firstreldet}
 \frac{{\rm det}_{\z,\myth}(A_{P_1})}{{\rm
det}_{\z,\myth}(A_{P_2})} = {\rm det}_{F}\left(\frac{S(P_1)}{P_1 S
(P_2)}\right) \, \cdot \,
 e^{-{\rm LIM}^{\myth}_{\la\to\infty}\log\det_F ((P_1 S_{\la} (P_2))\ii S_{\la}
 (P_1))}\, .
\end{equation}
 Further, the regularized limit is independent of the operator
 $A$, and depends only on the boundary conditions $P_1
 ,P_2$.
\end{theorem}

 The final statement follows from the fact that the left side of
\eqref{e:variation} is the logarithmic derivative of the left side
of \eqref{e:firstreldet} (\cite{Fo87}, Prop 1.3).

As an example, applied to {\it selfadjoint boundary problems} for
a Dirac operator over an odd-dimensional spin manifold, equation
\eqref{e:firstreldet} yields the relative determinant formula of
\cite{ScWo99} as a special case.


\section{Geometry of the determinant line bundle}

\subsection{Families of EBVPs}
Let $\pi : Z\too B$ be a smooth Riemannian fibration of manifolds
with fibre $X_b$ diffeomorphic to a compact manifold $X$ with
boundary $ Y$, and let $\Ee^i \to Z, i=0,1$, be vertical Clifford
bundles with compatible connection.  Then we have a family of
Dirac operators $\Df =\{ D_{b} \ | \ b\in B
\}:\Ff^{0}\too\Ff^{1}$, with $\Ff^{i}$ the infinite-dimensional
Fr\'{e}chet bundle on $B$ with fibre $C^{\infty}(X_{b},E^{i}_{b})$,
where $E_{b}^{i}= \Ee^{i}_{|X_b}$. (Here $\Df$ can be a family of
total or `chiral' Dirac operators.)

The corresponding structures are inherited on the boundary
fibration $\dd \pi : \dd Z\!\too B$ of closed manifolds with fibre
$Y_b = \dd X_b$.  We assume a product geometry in a collar
neighborhood $U$ of $\dd Z$ so that in $U_{b} = [0,1)\times \dd
X_{b} $ one has $(D_{b})_{|U_b} = \sigma_{b}(\dd/\dd u +
D_{Y_b})$, with $\Df_{\dd Z} = \{ D_{Y_b} \ | \ b\in B\} :
\Ff_{\dd Z}\to \Ff_{\dd Z}$  the family of selfadjoint boundary
Dirac operators defined by $\dd \pi$. Thus we have a smooth
fibration of Grassmannians $Gr(\Df)\to B$ with fibre
$Gr_{-\myo}(D_b)$. A {\it Grassmann section} for $\Df$ is defined to
be a smooth section $\Pf = \{P_b\in Gr_{-\myo}(D_b) \ | \ b\in B\}$
of $Gr(\Df)$, any such section differing from the Calder\'{o}n section
$P(\Df) = \{P(D_b) \ | \ b\in B\}$ by a smooth family of smoothing
operators.

The primary role of a Grassmann section is to define a smooth
family of EBVPs $(\Df,\Pf)$ parameterizing the operators $D_{P_b}
= D_b : \dom(D_{P_b}) \too L^2 (X_b,E^{1}_b),$  and hence
 an associated determinant line bundle\footnote{For 
a smooth family of Fredholm operators $\Af = \{A_b \mid
 b\in B\}$ parameterized by $B$, the complex lines $\Det(A_b) =
\bigwedge^{\operatorname{max}}\Ker(A_b)^* \otimes 
\bigwedge^{\operatorname{max}}\Cok(A_b)$ fit
together to define the determinant line bundle $\DET(\Af)\to B$
endowed with a canonical section $b\mto \det(A_b)$ (see
\cite{BiFr86,Qu85,Sc99}).} 
$\DET(\Df,\Pf)$ with determinant section
$b\mto\det(D_{P_b})$. On the other hand, a Grassmann section
defines a smooth infinite-rank Hermitian subbundle $\Ww\to B$ of
$\Ff_{\dd Z}$ with fibre $W_b = \ran (P_b) \subset L^2 (Y_b,
E^{1}_{b|Y})$. For each pair of Grassmann sections $\Pf_1 , \Pf_2$
we therefore have a smooth family of boundary Fredholm operators
\begin{equation*}
(\Pf_2 , \Pf_1) = \{P_{1,b} \circ P_{2,b} : W^{2}_b\to W^{1}_b\}
\in \Ci(B,\Hom(\Ww^2,\Ww^1)),
\end{equation*}
with determinant line bundle $\DET(\Pf_2 , \Pf_1)$, and from
 \cite{Sc99} there is a canonical isomorphism
\begin{equation}\label{e:isom2}
\DET(\Df,\Pf_1)\cong \DET(\Df,\Pf_2)\otimes\DET(\Pf_2,\Pf_1),
\end{equation}
where $\Sf(\Pf_i) : = (P(\Df),\Pf_i)$. This means that the {\em
abstract} relative determinant is a canonical section of the
relative determinant line bundle $\DET(\Pf_2,\Pf_1)$. By choosing
$\Pf_2 = P(\Df), \Pf_1 = \Pf$ this becomes a section of
\begin{equation}\label{e:isom3}
\DET(\Sf(\Pf)) \cong \DET(\Df,\Pf).
\end{equation}

\subsection{The $\z$ and $\Cc$ metrics}

The bundle isomorphism \eqref{e:isom3} means that $\DET(\Df,\Pf)$
inherits an Hermitian metric from $\DET(\Sf(\Pf))$. This is the
{\it canonical metric} $\|\cdot  \|_{\Cc}$, given over the open
subset $\Omega\subset B$ where the operators $D_P$ are invertible
by
\begin{equation*}
\|{\rm det}\, D_P \|_{\Cc}^2  :=  {\rm det}_F (S(P)^* S(P)) =  {\rm
 det}_F (P(D)\cdot P \cdot P(D)).
\end{equation*}
Here $\Delta_P = D^* D : \dom(\Delta_P)\to L^2 (X,E^0)$ is the
Dirac Laplacian with domain
\begin{equation*}
\dom(\Delta_P) = \{\psi\in H^2 (X,E^0): P\g \psi =0, P\st\g D\psi
= 0 \},
\end{equation*}
where $P\st = \sigma (I - P) \sigma\ii$ is the adjoint boundary
condition for $D^*$ (thus $(D_P)^* = D^{*}_{P\st}$).

On the other hand,   from recent work of Grubb \cite{Gr99} we know
that $\z(\Delta_P,s)$ is regular at $s=0$ for $P\in Gr_{-\myo}(D)$.
The resulting Quillen metric on $\DET(\Df,\Pf)$ is defined over
$\Omega$ by $\|{\rm det} D_P \|_{\z}^2 = {\rm det}_{\z}\Delta_P.$

%Theorem 3
\begin{theorem} Let $\Pf_1$, $\Pf_2$ be Grassmann
sections for $\Df$ and let $D_{P_1}\in (\Df,\Pf_1), D_{P_2}\in
(\Df,\Pf_2)$ be invertible at $b\in B$. Then
\begin{equation}\label{e:metrics1}
\frac{\|{\rm det}(D_{P_1})\|_{\z}}{\|{\rm det}(D_{P_2})\|_{\z}} =
\frac{\|{\rm det}(D_{P_1})\|_{\Cc}}{\|{\rm det}(D_{P_2})\|_{\Cc}}.
\end{equation}
That is,
\begin{equation}\label{e:metrics2}
\frac{{\rm det}_{\z}(\D_{P_1})}{{\rm det}_{\z}(\D_{P_2})} = \frac{
{\rm det}_F (S(P_1)^* S(P_1))}{ {\rm det}_F (S(P_2)^* S(P_2))}.
\end{equation}
Or, from \eqref{e:relFred},
\begin{equation}\label{e:metrics3}
{\rm det}_{\z}(\D_{P_1},\D_{P_2}) = {\rm det}_{\z}( S(P_1)^*
S(P_1) \  ,\  S(P_2)^* S(P_2) ).
\end{equation}
 Equivalently, since $S(P(D)) = Id$,
\begin{equation}\label{e:metrics4}
{\rm det}_{\z}(\D_{P}) = {\rm det}_{\z}(\D_{P(D)})\cdot{\rm det}_F
(S(P)^* S(P)) \ .
\end{equation}
\end{theorem}

\begin{rem} In fact, this holds for $P_1 - P_2$ differing just by a
$\pdo$ of order less than $-\dim(X)$. Equations 
\eqref{e:metrics1}--\eqref{e:metrics3} say that the {\em relative $\z$-metric} and
{\em relative $\Cc$-metric} on $\DET(\Df,\Pf_1)\otimes
\DET(\Df,\Pf_2)^*$ coincide. Similarly, \eqref{e:metrics4}
corresponds to the isomorphism \eqref{e:isom3}.
\end{rem}

\begin{proof}
We study $\D_P$ by embedding in a fully elliptic first-order
system. Associated to $\D$ we have the first-order elliptic
operator acting on sections of $E^0 \oplus E^1$
\begin{equation*}
  \wD = \begin{pmatrix}
    0 & D^* \\
    D & -I \
  \end{pmatrix} : H^1 (X, E^0 \oplus E^1 )\to L^2 (X, E^0 \oplus E^1
  ) \ .
\end{equation*}
In the collar $U_b \cong [0,1)\times Y$, $\wD$ has the form
  $\wD_{|U} = \ws\left(\frac{\dd}{\dd u} + \wB + \wR\right),$ where
\begin{equation*}
  \ws = \begin{pmatrix}
    0 & \sigma\ii \\
    \sigma & 0 \
  \end{pmatrix}, \,\,\,\,\,\wB = \begin{pmatrix}
    B &  0\\
    0 &  -\sigma B\sigma\ii \
  \end{pmatrix},
  \,\,\,\,\,\wR = \begin{pmatrix}
    0 & -\sigma\ii \\
    0 & 0 \
  \end{pmatrix} \ ,
\end{equation*}
satisfying the relations $\ws^2 = - I, \,\,\ws^* = - \ws, \,\,
\ws\wB + \wB\ws = 0,
  \,\,\ws\wR + \wR\ws = -I.$
Hence $\wD$ is of Dirac type with Calder\'{o}n projection $P(\wD)$ on
$L^2 (X, E^{0}_{|Y}\oplus E^{1}_{|Y})$ (see \S2), and for each
$\tilde{P}\in Gr_{-1}(\wD)$ we have a first-order EBVP
\[
\wD_{\tilde{P}} = \wD :\dom(\wD_{\tilde{P}}) \to L^2(X,E^0 \oplus
E^1 ).
\]

We recover the resolvent $(\D_P -\la)\ii$ via the canonical
embeddings
\begin{equation*}
  Gr_{-1}(D) \too Gr_{-1}(\wD),\hskip 10mm P\mtoo \wP := P \op
P\st,
\end{equation*}
\begin{equation*}
  \widehat{i}: H^2 (X,E^0 ) \too H^2 (X, E^0 ) \op H^1(X, E^1 ), \hskip 10mm
  \psi\mtoo (\psi, D\psi) \ .
\end{equation*}
More precisely, setting $
  \wDla = \bigl(\begin{smallmatrix}
    -\la & D^* \\
    D & -I \
  \end{smallmatrix}\bigr)$,
 $\widehat{i}$ restricts to an isomorphism $\Ker(\D - \la)
\cong \Ker(\wDla)$ and to an inclusion $\widehat{i} :
\dom(\D_P)\to \dom( \wD_{\la,\wP})$ into the domain of the
first-order selfadjoint (note that $\ws(I-\wP)\ws\ii =
  \wP$) local-elliptic boundary problem
$\wD_{\la,\wP}$, and one has
\begin{equation}\label{e:matrixinv}
\wD_{\la,\wP}\ii = \begin{pmatrix}
  (\D_P -\la)\ii &  D^{*}_{P\st} (\tilde{\D}_{P\st} - \la)\ii   \\
 D_P (\D_P - \la)\ii & \la(\tilde{\D}_{P\st} - \la)\ii
\end{pmatrix} \ ,
\end{equation}
where $\tilde{\D}_{P\st} =  D_P D^{*}_{P\st}$. The relative
resolvent is trace class for $P_1, P_2 \in Gr_{-\myo}(D)$, and we use
\eqref{e:matrixinv} and \eqref{e:variation} to compute that
\begin{equation*}
\Tr\left((\D_{P_1} - \la)\ii - (\D_{P_2 }- \la)\ii\right) =
 -\frac{\dd}{\dd\la}\log{\rm
det}_F \left(\frac{S_{\la}(\wP_1)}{\wP_1 S_{\la}(\wP_2)}\right),
\end{equation*}
where $S_{\la}(\wP) := S_{\wDla}(\wP)$. This determines the
scattering operator for $(\D_{P_1},\D_{P_2})$. Combined with
results from \cite{Gr99} we find that
 $\D_{P_1},\D_{P_2}$ are $\z$-comparable. Applying
Theorem 1 with $\myth = \pi$ we obtain
\begin{equation}\label{e:relLapdet}
 \frac{{\rm det}_{\z}(\D_{P_1})}{{\rm
det}_{\z}(\D_{P_2})} = {\rm det}_{F}\left(\frac{S(\wP_1)}{\wP_1 S
(\wP_2)}\right) \cdot
 e^{-{\rm LIM}_{\la\to +\infty}\log\det_F ((\wP_1 S_{-\la} (\wP_2))\ii S_{-\la}
 (\wP_1))}.
\end{equation}
On the other hand, for a 1-parameter family $ \D^r = D_{r}^*
D_{r}$  the identity \eqref{e:variation} leads to
\begin{equation*}\label{e:rvar}
\frac{d}{d r}\log \frac{{\rm det}_{\z}(\D^{r}_{P_1})}{{\rm
det}_{\z}(\D^{r}_{P_2})}  = \frac{d}{d r}\log \frac{{\rm
det}_{\Cc}(\D^{r}_{P_1})}{{\rm det}_{\Cc}(\D^{r}_{P_2})} \ .
\end{equation*}
Combined with \eqref{e:relLapdet} we find that
\begin{equation*}
\frac{{\rm det}_{\z}(\D_{P_1})}{{\rm det}_{\z}(\D_{P_2})} =
\frac{{\rm det}_{\Cc}(\D_{P_1})}{{\rm det}_{\Cc}(\D_{P_2})} \cdot
\mya(P_1 , P_2),
\end{equation*}
where $\mya(P_1 , P_2)$ depends only on $P_1, P_2$. Finally, by
studying the `vertical' boundary data variation 
\[ \frac{d}{dr}\log {\rm det}_{\z}(\D_{P_{r}}) = \Tr\left(S(\wP_r)\ii
\frac{d}{d r}S(\wP_r) \right)  - \underset{\la\to +\myo}{\rm
LIM}\Tr\left(S_{-\la}(\wP_r)\ii \frac{d}{d r}S_{-\la}(\wP_r) \right),
\] 
the homogeneous (boundary symmetry group) structure of the
Grassmannian forces $\mya(P_1 , P_2) =1$. (Note: It is not generally
true that $\mya =1$ for other classes of boundary conditions.)
\end{proof}

\subsection{The $\z$- and $\Cc$-connection and curvature}

We briefly describe the extension of these methods to the
regularized connection forms. To define a connection on
$\DET(\Df,\Pf)$  we make a choice of splitting $T Z = T(Z/B)
\oplus T^{H}Z$. This induces a natural connection $\nabla^{Z}$ on
$\Ff^i$ (see \cite{BiFr86}). Similarly, we obtain a connection
$\nabla^{\dd Z}$ on $\Ff_{\dd Z}$, and we assume that
$\nabla^{Z}_{|U} = \gamma^{*}\nabla^{\dd Z}$. Given Grassmann
sections $\Pf_i$, $i=1,2$, the associated subbundles $\Ww^i$ of
$\Ff_{\dd Z}$ inherit a Hermitian metric with compatible
connection $\nabla^{i} = \Pf_i\cdot\nabla^{\dd Z}\cdot \Pf_i$ ,
and hence induce a connection $\nabla^{1,2}$ on
$\Hom(\Ww^1,\Ww^2)$. The $\Cc$-connection on $\DET(\Df,\Pf)$ is
defined over $\Omega$ by $\nabla^{\Cc}\det(D_{P})/\det(D_{P}) =
\Tr(\Ss(P)\ii\nabla^{1,2}\Ss(P))$, with $\Pf_1 = P(\Df)$ and
$\Pf_2 = \Pf$. (See \cite{Sc99} for details.)

On the other hand, with a suitable boundary modification defined
using  $\Pf\cdot \nabla^{\dd Z}\cdot \Pf$, $\nabla^{Z}$ descends
to a connection $\nabla^{\Pf}$ on the subbundle $\Ff_{\Pf}$ with
fibre $\dom(D_{P_{b}})$ at $b\in B$.  For $\Re(s)>>0$, let
$\omega_{P}(s) = -\Tr(\Delta_{P}^{-s}D_{P}\nabla^{\Pf}D_{P}\ii)$.
This has a continuation to $\mathbb{C}$ with a simple pole at
$s=0$. Following \cite{BiFr86}, the $\z$-connection form  on
$\DET(\Df,\Pf)$ over $\Omega$ is defined by $(d/ds)_{|s=0}(s\omega
(s)).$ The $\Cc$- and $\z$-connections are compatible with their
metrics and we obtain the following result:


%Theorem 4
\begin{theorem} Let $\Pf_1$, $\Pf_2$ be choices of
Grassmann sections. Let $\RR^{\Pf_i}_{\Cc}, \RR^{\Pf_i}_{\z}$ be
the curvature $2$-forms of the canonical and zeta connection on
$\DET(\Df,\Pf_i)$. Then one has
\begin{equation}\label{e:curvature}
\RR^{\Pf_1}_{\z} -  \RR^{\Pf_2}_{\z} = \RR^{\Pf_1}_{\Cc} -
\RR^{\Pf_2}_{\Cc}.
\end{equation}
Equivalently,
\begin{equation}\label{e:curvature2}
\RR^{\Pf}_{\z}  =  \RR^{P(\Df)}_{\z} + \RR^{\Pf}_{\Cc}.
\end{equation}
\end{theorem}

Equation \eqref{e:curvature} holds in $\Omega^{2}(B)$,
since the endomorphism bundle of a complex line bundle is
canonically trivial.  The second identity \eqref{e:curvature2},
which says that the $\z$ curvature consists of an interior
(cohomologically trivial) part plus a boundary correction term,
follows from \eqref{e:curvature} because $\nabla^{\Cc}$ on
$\DET(\Df,P(\Df))$ is the trivial connection.


As an example, consider the `universal' family $(\Df,\Pf) = \{D_P
\ | \ P\in Gr_{-\myo}(D)\}$ of EBVPs parameterized by $Gr_{-\myo}(D)$
relative to a {\em fixed} operator $D$. Let $\RR_{\z}$ be the $\z$
curvature of the corresponding determinant bundle. Then
$\DET(\Sf(\Pf))$ is the determinant bundle of \cite{PrSe86}, used
to construct loop group representations, $\nabla^{Y}$ is just the
trivial connection, $\RR^{P(\Df)}_{\z} =0$, and $\nabla^{\Cc}$ the
connection used in \cite{PrSe86}, and we obtain:
\begin{cor}
\[ \RR_{\z} = i\omega_{Gr},\]
where $\omega_{Gr}$ is the K\"{a}hler form on the Grassmannian.
\end{cor}

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\end{document}