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\begin{document}

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\section*{Axiomatic field theories and their motivation from topology}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory}

[[!include functorial quantum field theory - contents]]

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 indicates, generally, how [[FQFT]]s may be related to [[cohomology theory|cohomology theories]].

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


\end{quote}
next:

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


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



\end{itemize}

\vspace{.5em} \hrule \vspace{.5em}
All of the following assumes that the reader is well familiar with the basic ideas indicated at [[FQFT]].

In the following $d-RB$ or $R Bord_d$ and the like denotes a category of $d$-dimensional [[Riemannian cobordism]]s that are equipped with \emph{Riemannian structure}, i.e. with [[Riemannian metric]]. Similarly $d-E B$ or $E Bord_d$ or the like denotes a category of cobordisms with \emph{Euclidean structure}, by which is meant a \emph{flat} Riemannian metric.

The definition of this is discussed in detail in

\begin{itemize}%
\item [[bordism categories following Stolz-Teichner]]

\end{itemize}
\textbf{definition}

$d$-dimensional Riemannian field theories are symmetric monoidal functors $d-RB \to TV$ from $d$-dimensional Riemannian bordisms to [[topological vector space]]s.

A field theory is very similar to a [[representation]] of a group. Only where a representation of a [[group]] $G$ is a functor from the [[delooping]] $\mathbf{B}G = {*}//G$ of $G$ to [[Vect]], an [[FQFT]] is a representation of a more complicated domain category.

\textbf{how does [[topology]] enter?}

for $X$ some [[topological space]] there is also a [[symmetric monoidal category]]

\begin{displaymath}
d-RB(X)
\end{displaymath}
of Riemannian bordisms equipped with a continuous map to $X$.

Notice that $d RB(X)$ does depend covariantly on $X$. This means that $Fun^\otimes(d RB(X), TV)$ is contravariant in $X$.

When special structure is around, however, we also have a push-forward of such functors along morphisms.

\textbf{Example: push-forward to the point}: for $X$ as above and $X \to {*}$ the unique map to the [[point]] heuristically we want a map

\begin{displaymath}
d RFT(X) \stackrel{p_*}{\to} d RFT(pt)
\end{displaymath}
notice that this push-forward is not an [[adjoint functor]]. Instead, it is a map that comes from integration over fibers. In particular it will change the degree of [[cohomology theory|cohomology theories]].

heuristically the pushforward

\begin{displaymath}
d RFT(X) \stackrel{p_*}{\to} d RFT(pt)
\end{displaymath}
acts on field theories $E_X$ over $X$

\begin{displaymath}
E_X \mapsto E
\end{displaymath}
by the assignment

\begin{displaymath}
E(Y^{d-1}) \mapsto 
  \Gamma\left(
  \itexarray{
     E_X(Y)
     \\
     \downarrow
     \\
     Maps(Y,X)
  }
  regarded as a vector bundle
  \right)
\end{displaymath}
for instance when $E_X(Y) = \mathbb{C}$ then $E(Y) = \Gamma(Maps(Y,X))$.

For instance take $\Sigma$ to be the pair of pants with input $Y_0$ and output $Y_1$ and let $F : \Sigma \to X$ be a map.

Then $E(\Sigma)$ will take some section $\Psi \in E(Y_0)$ to

\begin{displaymath}
E(\Sigma)(\Psi) : (f_1 : Y_1 \to X)
  \mapsto
  \int_{\{F : \Sigma \to X\} | F/Y_1 = f_1} E_X(F)(\Psi(f_0))
  \frac{1}{Z}\exp(-S(F) dvol)
  \,.
\end{displaymath}
Here the expression $\frac{1}{Z}\exp(-S(F) dvol)$ denotes a would-be measure which is still to be defined.

Here $f_0 = F/Y_0$ is the restriction of $F$ to $Y_0$.

Contravariant functors with push-forward also arise as part of a [[cohomology theory]]

\begin{displaymath}
h^n : Diff^{op} \to Ab
\end{displaymath}
from the category [[Diff]] of smooth [[manifold]]s to the category [[Ab]] of abelian groups that satisfies the axioms of [[generalized (Eilenberg-Steenrod) cohomology]] theory.

These Eilenberg-Steenrod axioms are

\begin{enumerate}%
\item \textbf{homotopy axiom} for $h^n$


\item \textbf{Mayer-Vietoris axiom} for $h^n$


\item \textbf{suspension isomorphism} $h^n(X) \simeq h^{n+1}_{cvs}(X \times \mathbb{R})$



\end{enumerate}
here in the last axiom for [[topological space]]s we'd simply have the [[suspension]] $h^{n+1}(\Sigma X)$. Here in order to stay within [[manifold]]s we instead do as indicated, where ${}_{cvs}$ means ``compact vertical support''.

Concerning the \textbf{homotopy axiom}: given any [[functor]] that sends manifolds to a category of [[FQFT]]s over $X$

\begin{displaymath}
d-FTs := h : Diff^{op} \to C
\end{displaymath}
in $dRFT$s we can make it a homotopy functor by defining

$\omega_0, \omega_1 \in h(X)$ to be [[concordance|concordant]] if $\exists \omega \in h(X \times \mathbb{R})$ such that $\omega/(X\times \{i\}) \simeq \omega_i$ in $h(X)$, $i = 0,1$

then under this relation

\begin{displaymath}
X \mapsto  d-RFT(X)/\simeq
\end{displaymath}
is a homotopy invariant functor

\textbf{Homework}: for $h(X) = \Omega^n_{closed}(X)$

we have that $\omega_0$ concordant to $\omega_1$ exactly when $\omega_0 = \omega_1 + d \alpha$ for some $\alpha \in \Omega^{n-1}(X)$

Concerning the \textbf{Mayer-Vietoris axiom}: if $X = U \cup V$ then the functor should respect this gluing.

suppose $E \in 2-RFT(X)$ is a 2d Riemmanian field theory. let $\gamma : S^1 \to X$ be a loop in $X$. then $E(\gamma)$ is a [[vector space]].

If $\gamma$ sits neither entirely in $U$ or in $V$, then there is no way that the vector space $E(\gamma)$ can be reconstructed by knowing just the restriction of $R$ to $U$ and $V$.

So this is a \textbf{problem} for the definition of field theories so far.

The \textbf{proposed solution} (from [[What is an elliptic object?]]) is to use \emph{extended} [[FQFT]]s instead. This introduces \textbf{locality} into [[FQFT]]s, at the expense of working with [[n-category|n-categories]].

This will however not be studied here for the moment.

Concerning the \textbf{suspension isomorphism}: for that first we need for $n \in \mathbb{Z}$ the notion of a field theory over $X$ of \textbf{degree $n$}, i.e.

\begin{displaymath}
X \mapsto d-RFTs^n(X)
\end{displaymath}
such that for $n=0$ this is an ordinary Riemannian field theory, in $d-RFT(X)$.

This requires to replace [[manifold]]s by [[supermanifold]]s.

\textbf{Example} let $d = 0$ and consider 0-dimensional TFTs over $X$.

so consider

\begin{displaymath}
Fun^\otimes(0 Bord(X), TV)
\end{displaymath}
in $0 Bord(X)$ there is only a single [[object]]: the [[empty set]] $\emptyset$ which has a unique map $\emptyset \to X$

the collecton of morphisms is $\{finite set \to X\}$ in there is $Hom_{Diff}(pt,X)$.

here composition of morphisms is the same as tensor product of objects: both comes from the disjoint union of these finite set domains.

\begin{displaymath}
E : 0 Bord(X) \to TV_{\mathbb{R}}
\end{displaymath}
\begin{displaymath}
\emptyset \mapsto \mathbb{R}
\end{displaymath}
\begin{displaymath}
x \in X \mapsto E(X) \in \mathb{R}
\end{displaymath}
so a priori we have

\begin{displaymath}
Fun^\otimes(0 Bord(X), TV) \simeq Maps(X, \mathbb{R})
  \,,
\end{displaymath}
where on the right we have \emph{all} maps.

This is not quite what is intended. We want to see \emph{smooth} maps on the right. To get that, we need to talk about \emph{smooth functors} on the left. This is the topic of later discussion, which will yield

\begin{displaymath}
SmoothFun^\otimes(0 Bord(X), TV) \simeq C^\infty(X, \mathbb{R})
  \,.
\end{displaymath}
But one other ting goes wrong the: the corresponding homotopy functor is $C^\infty(X)/\simeq = \{0\}$. So turning this into an Eilenberg-Steenrod theory yields the trivial theory.

The way out to that will be to go to [[supermanifold]]s.

\textbf{Punchline} of this session here:

\begin{displaymath}
(0|1)-TFTs(X)
  \simeq
  SmoothFun^\otimes((0|1)Bord(X), TV)
\end{displaymath}
and then it is a $lemma^G$ that

\begin{displaymath}
C^\infty(SuperDiff(\mathbb{R}^{0|1}, X))^G
  \simeq 
  \Omega^\bullet(X)^G
  = (\oplus_{n = 0}^{\infty} \Omega^n(X))^G
\end{displaymath}
where $G$ is the [[supergroup]] $G = Diff(\mathbb{R}^{0|1}) \wim42 \mathbb{R}^{0|1} \rtimes \mathbb{R}^\times$.

The degree decomposition on forms then turns out to be the eigenvalue decompositon of the $\mathbb{R}^\times$-part, while the deRham differential is the $\mathbb{R}^{0|1}$-action (cite Kontsevich, Severa here \ldots{})

so then from that one finds that the $G$-invariant bit is the \emph{closed} 0-forms

\begin{displaymath}
\Omega^\bullet(X)^G
  \simeq
  \Omega^0_{closed}(X)
\end{displaymath}
So we get $n$-\textbf{Lemma} :

\begin{displaymath}
(0|1)-TFT^n(X) \simeq \Omega^n_{cl}(X)
\end{displaymath}
and

\begin{displaymath}
(0|1)-TFT^n(X)_{concord} \simeq H^n_{dR}(X)
\end{displaymath}
\textbf{push-forward}

Now with the super-directions included, there is a notion of push-forward of the TFTs that does shift the degree, and we get the following

\textbf{Theorem (Stolz-Teichner-Kreck-Hohnhold)}

when $X$ is oriented

\begin{displaymath}
(0|1)TFT^n(X) \stackrel{push}{\to} (0|1)TFT^0
  \simeq \mathbb{R}
\end{displaymath}
\begin{displaymath}
\Omega^n_{cl}(X) \stackrel{\int_{X_n}}{\to}
  \Omega^0_{cl}(pt)
  \simeq
  \mathbb{R}
\end{displaymath}
and similarly after dividing out concordance, the push-forward becomes the push-forward in [[deRham cohomology]].

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Stefan Stolz]] (notes by Arlo Caine), \emph{Supersymmetric Euclidean field theories and generalized cohomology} Lecture notes (2009) (\href{http://www.nd.edu/~jcaine1/pdf/Lectures_complete.pdf}{pdf})

\end{itemize}


\end{document}