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\section*{bordism categories following Stolz-Teichner}

This is a sub-entry of [[geometric models for elliptic cohomology]] and [[A Survey of Elliptic Cohomology]]

See there for background and context.

This entry here is about the definition of [[cobordism category|cobordism categories]] for [[Riemannian cobordism]]s.

\begin{quote}%
\textbf{raw material}: this are notes taken more or less verbatim in a seminar -- needs polishing


\end{quote}
Previous:

\begin{itemize}%
\item [[Axiomatic field theories and their motivation from topology]].


\item [[(1,1)-dimensional Euclidean field theories and K-theory]]


\item [[(2,1)-dimensional Euclidean field theories and tmf]]



\end{itemize}
Next:

\begin{itemize}%
\item [[(2,1)-dimensional Euclidean field theories]]

\end{itemize}
see also

\begin{itemize}%
\item [[cobordism]]


\item [[Riemannian cobordism]]


\item [[cobordism category]]


\item [[(infinity,n)-category of cobordisms]]



\end{itemize}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{contents}{Contents}\dotfill \pageref*{contents} \linebreak
\noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak
\noindent\hyperlink{topological}{Part 1 (topological) bordism category}\dotfill \pageref*{topological} \linebreak
\noindent\hyperlink{Riemannian}{Part 2 Riemannian bordism category}\dotfill \pageref*{Riemannian} \linebreak
\noindent\hyperlink{onedim}{description for $d=1$}\dotfill \pageref*{onedim} \linebreak
\noindent\hyperlink{smoothversion}{smooth version / families version}\dotfill \pageref*{smoothversion} \linebreak
\noindent\hyperlink{fieldtheories}{Riemannian field theories}\dotfill \pageref*{fieldtheories} \linebreak
\noindent\hyperlink{invertible_field_theories}{Invertible Field Theories}\dotfill \pageref*{invertible_field_theories} \linebreak


\hypertarget{Idea}{}\section*{{Idea}}\label{Idea}

The goal here is to define a category of [[cobordism]]s that carry the structure of [[Riemannian manifold]]s. Where a functor on an ordinary [[cobordism category]] defines a [[TQFT]], the assignments of a functor on a category of [[Riemannian cobordism]]s do not only depend on the topology of a given cobordism, but also on its Riemannian structure. In physics terms such a functor is a \emph{Euclidean quantum field theory} .

\begin{quote}%
Notice however that the physicist's use of the word ``Euclidean'' is different from the way Stolz-Teichner use it: for a physicist it means that the Riemannian structure is not \emph{pseudo}-Riemannian. For Stolz-Teichner it means (later on) that the Riemannian metric is \emph{flat} .


\end{quote}
One central technical difference between plain topological cobordisms and those with Riemannian structure is that we want the functors on these to smoothly depend on variations of the Riemannian structure. This requires refining the bordism category to a \emph{smooth} category. By the logic of [[space and quantity]], one way to do this is to realize it as a [[stack]] on [[Diff]] with values in categories. This realization will be described here.

\hypertarget{topological}{}\section*{{Part 1 (topological) bordism category}}\label{topological}

\textbf{definition sketch}

the [[category]] $Bord_d$ has

\begin{itemize}%
\item as [[object]]s closed $(d-1)$-dimensional \emph{smooth} [[manifold]]s


\item and the [[morphism]]s are [[compact space|compact]] $d$-dimensional smooth [[manifold]]s with boundary, modulo [[diffeomorphism]] ``rel boundaries'' (i.e. those that restrict to the identy on the boundary)



\end{itemize}
The composition of [[morphism]]s is given by gluing of manifolds along their boundary

\hypertarget{Riemannian}{}\section*{{Part 2 Riemannian bordism category}}\label{Riemannian}

in all of the following

\begin{itemize}%
\item the symbol $Y$ denotes $d$-dimensional a [[Riemannian manifold]]

without boundary.


\item \textbf{note on boundaries} technically it is convenient to never ever work with manifolds with Riemannian or other structure with boundary. Instead, we always just mention manifolds without boundary and encoded the way in which they are still to be thouhgt of as [[cobordism]]s by injecting \emph{[[collars]]} into them. The manifolds with boundary could be obtained by cutting of at the \emph{core} of these collars (see the definition below) but, while this is morally the idea, in the construction this is never explicitly considered.

Also, later when we generalize [[manifold]]s to [[supermanifold]]s it will be very convenient not to have to talk about boundaries



\end{itemize}
$R Bord_d$ is defined using bicollars from the beginning

an object in $R Bord_d$ is a quintuple

consisting of

\begin{itemize}%
\item a $d$-dimensional [[Riemannian manifold]] $Y$;


\item a \textbf{core} $(d-1)$-manifold $Y^c$ sitting $Y^c \hookrightarrow Y$ in a thickening $Y^d$ -- being a $d$-manifold --


\item $Y^+, Y^- \hookrightarrow Y$ two disjointly embedded open $d$-dimensional manifolds such that

\begin{itemize}%
\item $Y^c$ is in the closure of both $Y^+$ and $Y^-$


\item that $Y^+ \coprod Y^- = Y \backslash Y^C$



\end{itemize}


\end{itemize}
\begin{quote}%
so the \emph{picture} of an object, which is missing in this writeup here for the moment, is a $d-1$-dimensional Riemannian manifold that is thickened a bit in one further othogonal direction


\end{quote}
\textbf{definition} A \textbf{Riemannian bordism} from $(Y_0,Y_0^c, Y_0^{\pm})$ to $(Y_1,Y_1^c, Y_1^{\pm})$ is a triple $(\Sigma, i_0, i_1)$ where

\begin{itemize}%
\item $\Sigma$ is a $d$-dimensional [[Riemannian manifold]] without boundary


\item for $i=0,1$ an open neighbourhood of the core $Y_i^c \hookrightarrow W_i \stackrel{open}{\hookrightarrow} Y_i$

this defines the intersections $W^\pm_k := W_k \cap Y^\pm_k$ with the two collars for each $k = 0,1$.


\item a smooth map $i_k : W_k \to \Sigma$

such that

\begin{itemize}%
\item $i_k : W^+_k \cup Y_k^c \to Z$ is a proper map;


\item (+) for $i^+_k := i_k/W^+_k$ are [[isometry|isometric]] embeddings into $\Sigma \backslash i_1(W^-_1 \cup Y^c_1)$

i.e. restricted to the (+)-collar the embedding of the thickened object into the would-be cobordisms is isomertric


\item the core $\Sigma^c := \Sigma \backslash (i_0(W^+_0) \cup i_1(W^-_1))$ is compact

i.e. cutting of the (+)-collar of the incoming object and the (-)-collar of the outgoing object yields a compact manifold



\end{itemize}


\end{itemize}
\textbf{Remark}. Notice that this builds in an asymmetry: the (+)-side is preferred. This is intentionally: also the category $TV$ of [[topological vector space]]s will have a similar asymmetry (from the fact that for $\infty$-dimensional vector spaces there is an evaluation map but not necessarily a coevaluation/unit for $V \otimes V^*$), similarly, with the above asymmetric definition we have a cobordims $Y \coprod Y^* \to \emptyset$ (where $Y^*$ is obtained from $Y$ by reversing orientation) but \emph{not} one going the other way round.

A big difference between [[TQFT]]s and the Riemannian QFTs is that for [[TQFT]]s the vector spaces assigned to objects are necessarily finite-dimensional. So this issue here with infinite-dimensional vector spaces and the asymmetry that this introduces is crucial for Riemannian QFTs.

\textbf{example} Given any [[isometry]]

\begin{displaymath}
\phi : W_0 \to W_1
\end{displaymath}
such that $\phi$ preserves the decomposition $W_k^\pm, Y_k^c$ we get a [[Riemannian cobordism]] using

\begin{displaymath}
\Sigma := W_1
\end{displaymath}
and

\begin{displaymath}
i_1 = Id_{W_1}\,,\;\;\;\;\; i_0 = \phi
\end{displaymath}
\textbf{definition (morphisms in $R Bord_d$)} morphisms from $Y_0$ to $Y$ in $R Bord_d$ (or $d-RB$ or whatever the notation is) are [[isometry]] classes \emph{rel. boundary} (see below) of [[Riemannian cobordism]]s from $Y_0$ to $Y_1$.

We require the commutativity of the following diagram

\begin{displaymath}
\itexarray{
    V_1 &\stackrel{i_1}{\to}& 
    X
    &\stackrel{i_0}{\leftarrow}&
     V_0
    \\
    \downarrow^{f_1} &&
    \downarrow &&
    \downarrow^{f_0}
    \\
    V'_1 &\stackrel{i'_1}{\to}&
    X'
    &\stackrel{i'_0}{\leftarrow}&
    V'_0
  }
\end{displaymath}
The [[isometry]] $(F,f_0, f_1)$ is ``rel. boundary'' if $f_0 = Id$ and $f_1 = Id$

so an isomorphism ``rel boundary'' in the sense here (more ``rel collars'', really) is an [[isometry]] $F$ sitting in a diagram

\begin{displaymath}
\itexarray{
    V_1 &\stackrel{i_1}{\to}& 
    X
    &\stackrel{i_0}{\leftarrow}&
     V_0
    \\
    \downarrow^{Id} &&
    \downarrow &&
    \downarrow^{Id}
    \\
    V'_1 &\stackrel{i'_1}{\to}&
    X'
    &\stackrel{i'_0}{\leftarrow}&
    V'_0
  }
\end{displaymath}
\hypertarget{onedim}{}\subsection*{{description for $d=1$}}\label{onedim}

we decribe $R Bord_1$ explicitly

it has at least the object

\begin{displaymath}
pt = 
  \left(
    \itexarray{
        pt^- & pt^c & pt^+
        \\
        -- & \bullet & --
    }
  \right)
  =
  (\mathbb{R}, \{0\}, \mathbb{R}_\pm)
\end{displaymath}
which is a point with collar all of $\mathbb{R}$.

\textbf{Lemma} every object in $R Bord_1$ which is \emph{connected} and not the empty set is [[isomorphism|isomorphic]] to this $pt$

now for $t \in \mathbb{R}_+$ consider the morphism

\begin{displaymath}
I_t \in R Bord_1(pt,pt)
\end{displaymath}
defined as the triple $(\mathbb{R}, i_0, i_1)$ where $i_0 : \mathbb{R} \to \mathbb{R}$ is the identity map, and where $i_1 : \mathbb{R} \to \mathbb{R}$ is translation by $t$.

This means that $i_0$ takes the core of in the incoming point to $0 \in \mathbb{R}$ while $i_1$ takes the core of the outgoing point to $t \in \mathbb{R}$. Everything in $\mathbb{R}$ outside of $[0,1]$ is hence ``collar'' and this describes what naively one would think of as just the [[interval]] $[0,1]$ regarded as a [[Riemannian cobordism]].

\textbf{Lemma} The composition of these cobordisms is given by

\begin{displaymath}
I_t \circ I_{t'} = I_{t+t'}
\end{displaymath}
There are also morphisms

\begin{displaymath}
L_+ : pt \coprod pt \to \emptyset
\end{displaymath}
and

\begin{displaymath}
R_+ : \emptyset \to pt \coprod pt
\end{displaymath}
which describe morally the same cobordisms as $I_t$ does, but where both boundary components are regarded as incoming or noth as outgoing, respectively.

Here $L_t$ is formall given exactly as $I_t$ only that the map $i_0 : \mathbb{R} \to \mathbb{R}$ is not the identity, but reflection at the origin. This encodes the orientation reversal at that end.

This is defined for $t \gt 0$. For $t= 0$ the morphism $L_0$ is still defined, but $R_0$ is not!! Exercise: check carefully with the above definition, keeping the asymmetry mentioned there in mind, to show that the obvious definition of $R_0$ does not satisfy the axioms above.

So this means that we have a cobordism of length 0 going $\emptyset \to pt \coprod pt$, but all cobordisms going the other way round $pt \coprod pt \to \emptyset$ will have to have non-vanishing length.

Another morphism in $R Bord_1$ is the morphism

\begin{displaymath}
\sigma : pt \coprod pt \to pt \coprod pt
\end{displaymath}
which just interchanges the two points, without having any length.

\textbf{Lemma} We have the following composition laws:

\begin{itemize}%
\item $L_t \circ \sigma = L_t$


\item $R_t = \sigma \circ R_t$


\item $R_t \circ_{L_0} R_{t'} = R_{t+t'}$



\end{itemize}
where in the last line we have the composition that is obvious once you draw the corresponding picture, which in full beuaty is

\begin{displaymath}
(Id_{pt} \otimes L_0 \otimes Id_{pt}) \circ (R_t \otimes R_{t'})
\end{displaymath}
where the [[tensor product]] $\otimes$ is given by disjoint union.

\textbf{theorem} the [[symmetric monoidal category]] $R Bord_1$ is \emph{generated} as a symmetric monoidal category by

\begin{itemize}%
\item the object $pt$


\item the morphisms $L_0$, $\{R_t\}_{t \gt 0}$



\end{itemize}
subject to the relations

\begin{displaymath}
L_0 \circ \sigma = L_0
\end{displaymath}
\begin{displaymath}
\sigma \circ R_t = R_t
\end{displaymath}
\begin{displaymath}
\forall t,t' \gt 0 : R_t \circ_{L_0} R_{t'} = R_{t + t'}
\end{displaymath}
\textbf{corollary} symmetric monoidal functors

\begin{displaymath}
E \in Fun^\otimes(R Bord_1, TV)
\end{displaymath}
to the category $TV_\mathbb{R}$ of [[topological vector space]]s are specified by their imagges of these generators. We have

\begin{itemize}%
\item $E : pt \mapsto V$


\item $E : L_0 \mapsto (\lambda :  V\otimes V \to \mathbb{R})$


\item $E : R_t \mapsto \rho_t \in V \otimes V$



\end{itemize}
The map $\lambda : V \otimes V \to \mathbb{R}$ is necessarily a \emph{nondegenerate} and \emph{symmetric} bilinear form and thus may be used to produce and fix an isomorphism $V \simeq V^*$.

This isomorphism is used to get an embedding

\begin{displaymath}
V \otimes V to V \otimes V^* \hookrightarrow End(V)
  \,.
\end{displaymath}
The image of this embedding is the set of what in this context will be called ``trace class'' operators.

With respect to this identification the map $\rho$ is to be understood. For varying $t$ the $\rho_t$ form a [[semigroup]] (for instance a typical example would be $V = \Gamma(E)$ a space of sections of a vector bundle and $\rho_t = e^{-t \Delta}$ for $\Delta$ a [[Laplace operator]] on $E$).

\textbf{note} for $\lambda : V \otimes V \to \mathbb{R}$ to be continuous, one cannot use the Hilbert tensor product $\otimes_H$

the reason is that we have the folloing possible mpas out of the following possible tensor products

\begin{displaymath}
\mathbb{R}
  \stackrel{\lambda}{\leftarrow}
  V \otimes_{algebraic} V \stackrel{finite rank}{\hookrightarrow}
  End(V)
\end{displaymath}
\begin{displaymath}
\mathbb{R}
  \stackrel{\lambda}{\leftarrow}
  V \otimes V \stackrel{trace class}{\hookrightarrow}
  End(V)
\end{displaymath}
\begin{displaymath}
\mathbb{R}
  \stackrel{\lambda}{\leftarrow}
  V \otimes_H V \stackrel{Hilbert Schmitdt}{\hookrightarrow}
  End(V)
\end{displaymath}
(so here the middle is the projective tensor product, the one that we are actually using)

\hypertarget{smoothversion}{}\subsection*{{smooth version / families version}}\label{smoothversion}

We now refine the definition of the categories $R Bord_d$ and $TV$ such that they remember smooth stucture.

\begin{quote}%
Effectively, what the following implicitly does is to refine these categories to [[stack]]s with values in categories over [[Diff]]. The [[fibered category|fibred categories]] that appear in the following, $R Bord_d^{fam} \to Diff$ and $TV^{fam} \to Diff$ are the [[Grothendieck construction]] of these stacks.


\end{quote}
\textbf{definition of $TV^{fam}$}

recall that $TV$ denotes the category of locally convex Hausdorff [[topological vector space]]

now let $TV^{fam}$ be the [[fibred category]] over [[Diff]] whose fiber over $X \in Diff$ is the category of topological [[vector bundle]]s over $X$. This has as objects [[vector bundle]]s of [[topological vector space]]s, and the morphisms are fiberwise linear $C^\infty$-morphisms of bundles in the following sense:

let $V, W \in TV$

Then a linear map $F : V \to W$ is -- for any inclusion $U \hookrightarrow V$

\begin{displaymath}
\itexarray{
    V &\stackrel{f}{\to}& W
    \\
    \uparrow^\subset & \nearrow
    \\
    U
  }
\end{displaymath}
-- called $C^1$ at $u \in U$ in the direction $v \in V$ if

\begin{displaymath}
\lim_{t \to 0} \frac{F(u+t v) - F(u)}{t}
\end{displaymath}
exists in $W$ and

\begin{displaymath}
U \times V \to W
\end{displaymath}
\begin{displaymath}
(u,v) \mapsto d F_u(v)
\end{displaymath}
is continuous.

Iteratively one defines $C^n$ and then $C^\infty$. The morphsims of $TV$-bundles are supposed to be $C^\infty$ maps in this sense (linear in the fibers, of course)

\begin{displaymath}
\itexarray{
    V' &\stackrel{\tilde f}{\to}& V
    \\
    \downarrow &&\downarrow
    \\
    S' &\stackrel{f}{\to}& S
  }
\end{displaymath}
\textbf{definition of $R Bord_d^{fam}$}

Similarly $R Bord_d^{fam}$ has as objects submersions $Y \to S$ and $Y^c \to S$ (not necessarily surjective) with a smooth rank-2 tensor on $Y$ that fiberwise induces the structure of a [[Riemannian manifold]] (so these are $S$-families of [[Riemannian manifold]]s) such that

\begin{displaymath}
\itexarray{
    Y &\leftarrow^\subset& Y^c
    \\
    \downarrow^{submersion} & \swarrow_{proper subm.}
    \\
    S
  }
\end{displaymath}
recall that a map is a [[proper map]] if inverse images of compact sets are compact.

\textbf{remark} Notice that if we fix the topology of the fibers in $Y \to S$, then what varies as we vary the fibers is the [[Riemannian metric]] on the fibers, so here each $S$ can be thought of as a (subspace of a) moduli space of Riemannian metrics on a given topological space. Don't confuse this with the role the space always called $X$ here will play as a kind of ``moduli space of field theories''.

a morphism in $R Bord_d^{fam}$ in

\begin{displaymath}
R Bord_d^{fam}\left(
    \itexarray{
       Y_0 \\ \downarrow \\ S_0
    },
    \;\;
    \itexarray{
       Y_1 \\ \downarrow \\ S_1
    }
  \right)
\end{displaymath}
are [[isometry|isometric]] rel boundary classes of submersions $\Sigma \to S_0$ such that

\begin{displaymath}
\itexarray{
    \Sigma
    &\stackrel{i_1}{\leftarrow}&f^* Y_1 &\to&  Y_1
    \\
    \uparrow^{i_0} &&\downarrow && \downarrow
    \\
    Y_0& \to&S_0 &\stackrel{f}{\to}&
    S_1
  }
\end{displaymath}
so here $\Sigma$ is an $S_0$-family of cobordisms.

\hypertarget{fieldtheories}{}\subsection*{{Riemannian field theories}}\label{fieldtheories}

\textbf{definition}

A \textbf{$d$-dimensional Riemannian quantum field theory} is a [[symmetric monoidal functor]]

\begin{displaymath}
E \in Fun^\otimes_{Diff}(R Bord_d^{fam}, TV^{fam})
\end{displaymath}
such that

\begin{displaymath}
\itexarray{
      R Bord_d^{fam} &&\stackrel{}{\to}&&
      TV^{fam}
      \\
      & \searrow && \swarrow 
      \\
      && Diff
  }
\end{displaymath}
and such that it preserves [[pullback]]

(so its a [[cartesian functor]] between these [[fibered category|fibered categories]] that is also symmetric monoidal)

\hypertarget{invertible_field_theories}{}\subsection*{{Invertible Field Theories}}\label{invertible_field_theories}

In study of invertible field theories, let $\mid \mid Bord^G_n \mid \mid$ be the fundamental groupoid of the classifying space of $Bord^G_n$. Invertible field theories factor through this and the Picard subgroupoid $\mathcal{C}^*$ of the target.

Kreck, S. Stolz, and P. Teichner.Invertible topological field theories are SKK invariants.



\end{document}