_\Lambda := \int dg \prod_{k,x} d\phi_k(x) \int \prod_{\alpha,a,x} d\Psi_{\alpha a}(x)d\cj{\Psi_{\alpha a}(x)} \ \ P\exp(-S_{\opn tot }/\hbar). \Eeq Here the inner integral is Berezin integration along the fibres of the super vector bundle $M= M_{\opn bos }\times\L(\cV_r) \too\pr M_{\opn bos }$, producing an ordinary function on $M_{\opn bos }$. The $\phi_k(x)$ ($k=1,\dots,\dim \cV\Higgs$) are orthonormal coodinates on the $x$-component of ${\cV\Higgs}^\Lambda$; thanks to the exponential factor and our growth condition, the integral over them is finite. Finally, $dg$ is the normalized Haar measure on $G^L$. The sign ambiguity arising from the missing order of the $\phi_k(x)$ is resolved by fixing $\br<1>_\Lambda>0$. Now we suppose the existence of {\em time reflection} as a fixpoint-free involutive map $r:\Lambda\to\Lambda$ which respects the link structure, $(r\times r)(L^*)\seq L^*$. By permuting the factors, $r$ yields an involution $r: M_{\opn bos }\to M_{\opn bos }$. Also, we get a morphism \Beq r: M_{\opn ferm }\to M_{\opn ferm },\quad r^*(\Psi_{\alpha a}(x))= \sum_\beta \Psi_{\beta a}(rx)(\gamma_0)_{\alpha\beta}. \Eeq Altogether, we get an involutive morphism $r: M\to M$ and a new hermitian law \Beq \theta(P):= \cj{r^*(P)}. \Eeq (Actually, the notation is unlucky in view of \ref{HermStuff}, because it hides the skew-linearity.) We also need a decomposition of the lattice \Beq \Lambda=\Lambda_+ \cup \Lambda_-, \quad \Lambda_+ \cap \Lambda_-=\emptyset \Eeq such that $r(\Lambda_\pm)\seq\Lambda_\mp$ (cf. \cite{[Seiler]} for the background). We get a projection morphism $M=M_\Lambda \to M_{\Lambda_+}$, and hence an embedding $\O_b(M_{\Lambda_+})\seq \O_b(M)$. {\em Osterwalder-Schrader positivity} now states that \Beq \br

\ge0 \Eeq for all $p\in\O_b(M_{\Lambda_+})$. Also, the scalar product \Beq \O_b(M_{\Lambda_+})\times\O_b(M_{\Lambda_+})\to \mathbb C,\quad (P,Q)\mapsto\br

,
\Eeq
satisfies
\Beqn ScalPrd
\br };
\Eeq
hence, it equips $\O_b(M_{\Lambda_+})/\{P: \br =0\}$ with the
structure of a pre-Hilbert space the completion ${\mathcal H}$ of which is the
{\em Euclidian state space}. (Thus, $\O_b(M_{\Lambda_+})$ plays here the same
role as the space of polarized sections in geometric quantization.)
\Brm
(1) Note that working with a conventional, non-hermitian calculus would bring
trouble here since \eqref{ScalPrd} would acquire an additional factor
$(-1)^{\br|Q|\br|P|}$, and hence $\br $ would be for odd $P$
not positive but imaginary.
(2) It would be interesting to know something about the "supermanifold of
classical solutions" of the action $S$. That is, we may form the ideal sheaf
${\mathcal J}(\cdot)\seq\O_M(\cdot)$ generated by the "field equations"
\Beq
\frac \partial{\partial g_{xy}^i} S,\ \
\frac \partial{\partial \phi_k(x)}S,\ \
\frac \partial{\partial \Psi_{\alpha a}(x)}S,\ \
\frac \partial{\partial\cj{\Psi_{\alpha a}(x)}}S\ \in \O(M),
\Eeq
(here $g_{xy}^i$ are local coodinates on the $(x,y)$-component of $G^L$)
and take the factor space $M\sol := (\opn supp \O/{\mathcal J},\ \ \O/{\mathcal
J})$.
This is at least a real superanalytic space which, however, might have
singularities. Note that, in contrast to the non-super situation, where
the singular locus of an analytic space has codimension $\ge1$,
it may on a superanalytic space be the whole space; it would be nice to show
that this does not happen here.
Also, one should prove that in the classical limit $\hbar\to0$,
the integral \eqref{MeanVal} becomes asymptotically equal
to an integral over $M\sol$ (or its non-singular part). Indeed, in the bosonic
sector, the exponential factor makes the measure accumulate on the
subspace $\widetilde{M\sol}$ of the minima of the action; however, in the
fermionic sector, the picture is less clear.
\Erm
%--------------------------------------------------------------
\section{Infinite-dimensional supermanifolds}\label{InfDSmfs}
An attentive reader of \cite{[CMP1]} will have noted that most of its material
(with the exception of local functionals, differential polynomials a.s.o.)
does not really depend on the fact that $E$ is a function space. In
fact, the theory can be developed in an abstract context, and one really
should do so in order to get conceptual clarity which may be useful if
a concrete situation does not fit into our framework. In that way, we
will also establish the connection with usual, finite-dimensional
supergeometry a la Berezin.
Some work has been done on infinite-dimensional supergeometry and its
application onto classical fields in quantum field theory.
Apart from the implicite appearance of infinite-dimensional supermanifolds
in \cite{[Ber2ndQ]} (cf. \cite{[Rapallo]}), the first work on the mathematical
side is Molotkov \cite{[Mol 84]}; however, his approach is not well suited
for physical purposes. Cf. below for a discussion.
\cite{[Abramov 86]} uses an ad-hoc definition of smooth Banach smfs in
order to describe mathematically the fermionic Faddeev-Popov ghost fields
used by physicists for quantization of the Yang-Mills field.
In \cite{[Kostant-Sternberg 87]}, infinite-dimensional supergeometry makes
an implicite appearance in a more general approach to the quantization
of systems with first-class constraints (cf. also the comment in
\cite{[Ewen-Schaller-Schwarz 1991]}).
\cite{[Schaller-Schwarz 1990]} and \cite{[Ewen-Schaller-Schwarz 1991]}
use, on a physical level of rigour, ad-hoc generalizations of the
Berezin-Kostant supergeometry framework for applying geometric
quantization in field theory, in particular to fermionic fields.
Finally, the present author constructed in \cite{[ISA]},
\cite{[IS1]}, \cite{[IS2]},
the predecessors of the present paper, a rather general theory of complex-
and real-analytic supermanifolds modelled over locally convex spaces.
Here, we will give an alternative description of this theory, using a
traditional treatment via ringed spaces and charts. Also, we
will treat only real-analytic supermanifolds with complete
model spaces.
We will give only a short account on the abstract variant since the
details should be clear from the material of \cite{[CMP1]}.
%------------------------------------------------------------
\subsection{Formal power series}\label{FPSer}
Let $E$, $F$ be complete \ztwo-lcs, and define the space
$\P^{k|l}(E;F)$ of {\em $F$-valued $k|l$-forms on $E$} as the space of
all $(k+l)$-multilinear continuous maps
\Beq
u_{(k|l)}:\prod^kE\seven\times\prod^lE\sodd\to F_{\mathbb C}
\Eeq
which satisfy the symmetry requirement
\Beq
u_{(k|l)}(e_{\sigma(1)},\dots,e_{\sigma(k)},e'_{\pi(1)},\dots,e'_{\pi(l)})
=\opn sign (\pi)u_{(k|l)}(e_1,\dots,e_k,e'_1,\dots,e'_l)
\Eeq
for all permutations $\sigma,\ \pi$. $\P^{k|l}(E;F)$ is a
\ztwo-graded vector space; note that we do not distinguish a
topology on it.
The space of {\em F-valued formal power series} on $E$ is defined by
\Beq
\P_f(E;F):=\prod_{k,l\ge0}\P^{k|l}(E;F);
\Eeq
thus, its elements are formal sums $u=\sum_{k,l\ge0}u_{(k|l)}$ where
$u_{(k|l)}\in\P^{k|l}(E;F)$. The rule
\Beq
\cj u_{(k|l)}(e_1,\dots,e_k,e'_1,\dots,e'_l):=
\cj{u_{(k|l)}(e_1,\dots,e_k,e'_l,e'_{l-1},\dots,f_1)}
\Eeq
\Beq
=(-1)^{\binom l2}\cj{u_{(k|l)}(e_1,\dots,e_k,e'_1,\dots,e'_l)}
\Eeq
turns $\P_f(E;F)$ into a hermitian vector space. The {\em product}
\Beq
\P^{k|l}(E;F)\times\P^{k'|l'}(E;F')\to\P^{k+k'|l+l'}(E;F\widehat\otimes F')
\Eeq
is defined by
\Bal
&(u\otimes v)_{(k+k'|l+l')}(e_1,\dots,e_{k+k'},e'_1,\dots,e'_{l+l'})=
\\
&\quad=\sum(\pm) \binom{k+k'}k^{-1} \binom{l+l'}l ^{-1}
u_{(k|l)}(e_{p_1},\dots,e_{p_k},e'_{q_1},\dots,e'_{q_l})\otimes
v_{(k'|l')}(e_{p'_1},\dots,e_{p'_k},e'_{q'_1},\dots,e'_{q'_l}).
\Eal
(the sign "$\otimes$" on the l.h.s. is somewhat abusive).
Here the sum runs over all $\binom{k+k'}k \binom{l+l'}l$ partitions
\Beq
\{1,\dots,k+k'\}=\{p_1,\dots,p_k\}\sqcup\{p'_1,\dots,p'_{k'}\},\quad
p_1\le\dots\le p_k,\quad
p'_1\le\dots\le p'_{k'}
\Eeq
\Beq
\{1,\dots,l+l'\}=\{q_1,\dots,q_l\}\sqcup\{q'_1,\dots,q'_{l'}\},\quad
q_1\le\dots\le q_l,\quad
q'_1\le\dots\le q'_{l'}
\Eeq
and $(\pm)$ is given by the sign rule:
\Beq
(\pm):=(-1)^{|v|(|q_1|+\dots+|q_l|)}\opn sign (\pi)
\Eeq
where $\pi$ is the permutation $(q_1,\dots,q_l$, $q'_1,\dots,q'_{l'})$
of $\{1,\dots,l+l'\}$ (cf. \cite[Prop. 2.3.1]{[IS1]}).
The product turns $\P_f(E;\mathbb R)$ into a \ztwo-commutative
hermitian algebra and each $\P_f(E;F)$ into a hermitian module over
that algebra. In both situations, we will usually write simply $uv$
instead of $u\otimes v$.
\Brm
An $F$-valued formal power series in the sense of \CMPref{2.3},
\Beq%n KlForm
K[\Phi|\Psi]=\sum_{k,l\ge0} \frac 1{k!l!} \sum_{I|J} \int dXdY K^{I|J}(X|Y)
\prod^k_{m=1} \Phi_{i_m}(x_m) \cdot \prod^l_{n=1} \Psi_{j_n}(y_n)
\ \in{\mathcal P}(F),
\Eeq
defines an $F$-valued formal power series $K\in\P(D;F)$ in the sense above,
$K=\sum_{k,l} K_{(k|l)}$ with
\Bmln TklMap
K_{(k|l)}: \prod^k D\seven\times \prod^l D\sodd\to F_{\mathbb C},\qquad
(\phi^1,\dots,\phi^k,\psi^1,\dots,\psi_l) \mapsto
\\
\frac 1{k!l!} (-1)^{l(l-1)/2} \int dXdY \sum_{I|J} K^{I|J}(X|Y)
\prod_{m=1}^k\phi^m_{i_m}(x_m)\cdot\prod_{n=1}^l\Pi\psi^n_{j_n}(y_n)
\Eml
where $D={\mathcal D}(\mathbb R^d)\otimes V$, and $V=V\seven\oplus V\sodd$ is
the
field target space. (The apparently strange parity shift $\Pi$ was motivated
by the wish to have $\Psi$ as an odd symbol.)
Moreover, if $E$ is an admissible function space in the sense of
\CMPref{3.1}, i.~e. $E$ is a \ztwo-graded complete locally convex space
with continuous inclusions $D\seq E\seq{\mathcal D}'(\mathbb R^d)\otimes V$,
then
we have $K\in\P_f(E;F)$ iff for all $k,l\ge0$, \eqref{TklMap}
extends to a continuous map
$\bot^k E\seven\otimes \bot^l E\sodd\to F_{\mathbb C}$. In that way, we get
a natural identification between $\P_f(E;F)$ as defined in \CMPref{3.1} and
the $\P_f(E;F)$ defined here.
(In \CMPref{3.1}, we had also assigned to $K_{(k|l)}[\Phi|\Psi]$ the linear
map
\Beq
\bot^k D\seven\otimes \bot^l\Pi D\sodd\to F_{\mathbb C},\qquad
\bot_{m=1}^k \phi^m \otimes \bot_{n=1}^l \Pi\psi^n \mapsto
(-1)^{l(l-1)/2}\cdot(\text{second line of \eqref{TklMap}}).
\Eeq
Note, however, that the parity of this map differs from that of
$K[\Phi|\Psi]$ by the parity of $l$.)
\Erm
\subsection{Analytic power series}
Let be given continuous seminorms $p\in\CS(E)$, $q\in\CS(F)$.
We say that $u\in\P_f(E;F)$ {\em satisfies the $(q,p)$-estimate} iff
we have for all $k,l$ with $k+l>0$ and all $e_1,\dots,e_k\in E\seven$,
$e'_1,\dots,e'_l\in E\sodd$ the estimate
\Beq
q(u_{(k|l)}(e_1,\dots,e_k,e'_1,\dots,e'_l))
\le p(e_1)\cdots p(e_k)p(e'_1)\cdots p(e'_l)
\Eeq
(we extend every $q\in\CS(F)$ onto $F_{\mathbb C}$ by
$q(f+\opn i f'):=q(f)+q(f')$). We call $u$ {\em analytic} iff for each
$q\in\CS(F)$ there exists a $p\in\CS(E)$ such that $u$ satisfies the
$(q,p)$-estimate.
Now every $k|l$-form $u_{(k|l)}\in\P^{k|l}(E;F)$ is analytic, due
to its continuity property, and analyticity of a formal power
series is just a joint-continuity requirement onto its coefficients
$u_{(k|l)}$.
The analytic power series form a hermitian subspace $\P(E;F)$
of $\P_f(E;F)$. Moreover, tensor product of analytic power series
and composition with linear maps in the target space produce
analytic power series again.
For $e"\in E$, the {\em directional derivative} $\partial_{e"}$
is defined by
\Beq
(\partial_{e"}u)_{(k|l)}(e_1,\dots,e_k,e'_1,\dots,e'_l)
:=\begin{cases}
(k+1)u_{(k+1|l)}(e",e_1,\dots,e_k,e'_1,\dots,e'_l)
& \text{for $|e"|=0$}\\
(-1)^{|u|}(l+1)u_{(k|l+1)}(e_1,\dots,e_k,e"_1,e'_1,\dots,e'_l)
& \text{for $|e"|=1$.}
\end{cases}
\Eeq
$\partial_{e"}$ maps both $\P_f(E;F)$ and $\P(E;F)$ into themselves,
and it acts as derivation on products:
\Beq
\partial_e(u\otimes v) = \partial_eu\otimes v
+ (-1)^{\br|e|\br|u|}u\otimes \partial_ev.
\Eeq
The abstract analogon of the functional derivative of $K$ is the
linear map
\Beq
E\to\P_f(E;F),\qquad e\mapsto\partial_eK.
\Eeq
\medskip
Suppose that $F$ is a \ztwo-graded Banach space, and fix
$p\in\CS(E)$. Set
\Beq
\|u_{(k|l)}\|_p:=\opn inf \{c>0:\ \
\text{$u_{(k|l)}$ satisfies a ($c^{-1}\|\cdot\|,p)$-estimate} \}
\Eeq
for $k+l>0$, and define $\|\cdot\|$ on $0|0$-forms to be the norm
in $F_{\mathbb C}$. Then
\Beq
\P(E,p;F)=\{u\in\P(E;F):\ \ \|u\|_p:=\sum_{k,l\ge0}\|u_{(k|l)}\|<\infty\}
\Eeq
is a Banach space. Moreover, for any $E$ we have
\Beq
\P(E;F)=\bigcup_{p\in\CS(E)}\P(E,p;F).
\Eeq
Now if $F'$ is another \ztwo-graded Banach space then %Checked
\Beq
\|u\otimes v\|_p \le\|u\|_p \|v\|_p
\Eeq
for $u\in\P(E;F)$, $v\in\P(E;F')$. In particular, $\P(E,p;\mathbb R)$
is a Banach algebra.
Note that for every $p\in\CS(E)$, $c>1$, the directional derivative
$\partial_e$ maps $\P(E,p;F)\to\P(E,cp;F)$ for any $e\in E$.
\Brm
(1) A $k|l$-form $u_{(k|l)}$ lies in $\P(E,p;F)$
iff it factors through $\hat E_p$; in that case, $\|u_{(k|l)}\|$
is just its supremum on the $k+l$-fold power of the unit ball of this space.
(2) Remark 3.2.2 of \cite{[CMP1]} carries over, linking the approach here
with \cite{[IS1]}: Fixing $u\in\P_f(E;F)$ and $k,l\ge0$ we get a
continuous map
\[
\S^k E_{\even,\mathbb C}\cdot \S^l E_{\odd,\mathbb C}\to F_{\mathbb C},\qquad
e_1\cdots e_k e'_1\cdots e'_l
\mapsto
k!l! u(e_1,\dots,e_k,e'_1,\dots,e'_l)
\]
(using notations of \cite{[IS1]}; the topology on the l. h. s. is induced
from the embedding into $\S E_{\mathbb C}$).
Using Remark 2.1.(2) of \cite{[IS1]} we get a bijection
\[
\P_f(E;F)\to \prod_{k\ge0} {\mathcal L}(\S^k E_{\mathbb C},F_{\mathbb C})
\]
(symmetric algebra in the super sense). The r. h. s. is somewhat
bigger than ${\mathcal L}(\S E_{\mathbb C},F_{\mathbb C})={\opn P }(E;F)$,
due to the absence of growth conditions. Having identified both sides,
one shows for $u\in\P_f(E;F)$ the estimates
\[
\Vert u\Vert\dn{U_{p/2}} \le \Vert u\Vert_p \le \Vert u\Vert\dn{U_{2p}}
\]
(cf. \cite[2.5]{[IS1]} for the notations
$\Vert\cdot\Vert\dn U,\ {\opn P }(E,U;F)$), and hence
\[
{\opn P }(E,U_{p/2};F) \seq \P(E,p;F)\seq{\opn P }(E,U_{2p};F),
\qquad \P(E;F) = {\opn P }(E;F).
\]
*
\Erm
\subsection{Insertions}\label{Insts}
On the level of formal power series,
we will define $u[v]\in\P_f(E';F)$ with the data
\Beq
u\in\P_f(E;F),\qquad
v\in\P_f(E';E)\seven,\qquad
v_{(0|0)}=0.
\Eeq
We split $v=v_{(\even)}+v_{(\odd)}$ with $v_{(\mathbf i)}
\in\P_f(E';E_{\mathbf i})\seven$ $(\mathbf i=\mathbf 0,\mathbf 1)$, and
we set
\Beqn InsertedSer
u[v]:=\sum_{k,l\ge0}\Bigl = \cj{\br