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\section*{(2,1)-dimensional Euclidean field theory}

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

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

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

\hypertarget{supergeometry}{}\paragraph*{{Supergeometry}}\label{supergeometry}

[[!include supergeometry - 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 is about the definition of $(2|1)$-dimensional [[super-cobordism]] categories where cobordisms are [[Euclidean supermanifold]]s, and about $the (2|1)$-dimensional [[FQFT]]s given by functors on these.

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]]


\item [[bordism categories following Stolz-Teichner]]



\end{itemize}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{contents}{Contents}\dotfill \pageref*{contents} \linebreak
\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{explicit_description_of_}{explicit description of $E Bord_{1}^{fam}$}\dotfill \pageref*{explicit_description_of_} \linebreak
\noindent\hyperlink{explicit_description_of__2}{explicit description of $E Bord_{2}$}\dotfill \pageref*{explicit_description_of__2} \linebreak
\noindent\hyperlink{explicit_description_of__3}{explicit description of $E Bord_{2}^{fam}$}\dotfill \pageref*{explicit_description_of__3} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Previously we had defined smooth categories of [[Riemannian cobordism]]s. Now we pass from [[Riemannian manifold]]s to [[Euclidean supermanifold]]s and define the corresponding smooth [[cobordism category]]. Then we define $(d|\delta)$-dimensional Euclidean field theories to be smooth representations of these categories.

As described at [[(2,1)-dimensional Euclidean field theories and tmf]], the idea is that $(2|1)$-dimensional Euclidean field theories are a geometric model for [[tmf]] [[cohomology theory]]. While there is no complete proof of this so far, in the next and final session

\begin{itemize}%
\item [[modular forms from partition functions]]

\end{itemize}
it will be shows that the claim is true at least for the [[cohomology ring]] over the point: the [[partition function]] of a $(2|1)$-dimensional EFT is a modular form. Hence $(2|1)$-dimensional EFTs do yield the correct [[cohomology ring]] of [[tmf]] over the point.

\hypertarget{details}{}\section*{{Details}}\label{details}

Let [[SDiff]] be the [[category]] of [[supermanifold]]s.

We will define a [[stack]]/[[fibered category]] on $SDiff$ called $E Bord_{2|1}$ whose morphisms are smooth families of (2|1)-dimensional [[super-cobordism]]s, and a [[stack]]/[[fibered category]] $sTV^{fam}$ of topological super vector bundles.

So recall

\begin{itemize}%
\item [[supergeometry - contents|supergeometry]].

\end{itemize}
\textbf{question}: What is the right notion of Riemannian or Euclidean structure on [[super-cobordism]]s?

\textbf{strategy}: From the [[path integral]] perspective we need some structure on $\Sigma$ such that the ``space'' of maps $Maps(\Sigma,X)$ naturally carries some measure that allows to perform a [[path integral]].

This perspective suggests certain generalizations of the notion of [[Riemannian manifold]] to [[supermanifold]]s which may be a little different than what one might have thought of naively.

We want to define [[Euclidean supermanifold]]s as a generalization of [[Riemannian manifold]] with \emph{flat} Riemannian metric.

notice that there is a canonical bijection between

\begin{itemize}%
\item flat [[Riemannian metric]]s on a $d$-dimensional [[manifold]] $X$


\item a maximal [[atlas]] on $X$ consisting of charts such that all transition functions belong the the \textbf{Euclidean group} or \textbf{Galileo group}

\begin{displaymath}
Eucl(\mathbb{R}^d)
  :=
  \mathbb{R}^d \rtimes O(\mathbb{R}^d)
\end{displaymath}
(rigid translations and rotations)



\end{itemize}
In analogy to that we define:

Similarly a \textbf{Euclidean structure} on a $(d|\delta)$-dimensional [[supermanifold]] is defined using the Euclidean [[super Lie group]] given by the [[semidirect product]]

\begin{displaymath}
Eucl(\mathbb{R}^{d|\delta})
  :=
  \mathbb{R}^{d|\delta}
  \rtimes
  Spin(\mathbb{R}^d)
\end{displaymath}
where the [[Spin]] group acts on the translations in $\mathbb{R}^{d|\delta}$ in a way to be specified.

first recall the notion of

\begin{itemize}%
\item [[complex supermanifold]]

\end{itemize}
\textbf{goal} replace the standard Euclidean group $(\mathbb{R}^d, Eucl(\mathbb{R}^d))$ by the [[super Euclidean group]] $(X,G)$ where $X$ is a suitable [[supermanifold]] and $G$ a suitable [[super Lie group]].

This leads to the notion of

\begin{itemize}%
\item [[Euclidean supermanifold]].

\end{itemize}
The morphisms of the category $E Bord_{(2|1)}$ will be [[cobordism]]s that are [[Euclidean supermanifold]]s.

\textbf{goal} define the [[fibered category]]

\begin{displaymath}
\itexarray{
    E Bord_{d|\delta}^{sfam}
    \\
    \downarrow
    \\
    cSDiff
  }
\end{displaymath}
where $cSDiff$ is the category of [[complex supermanifold]]s.

The objects of this fibered category are

\begin{displaymath}
\itexarray{
    Y^{\pm} &\to& Y &\leftarrow& Y^c
    \\
    & \searrow & \downarrow & \swarrow
    \\
    && S
  }
\end{displaymath}
where $Y \to S$ is a family of [[Euclidean supermanifold]]s of dimension $(d|\delta)$.

For the non-super, non-family version of \textbf{Euclidean bordism} we require that the core $Y^c$ is totally geodesic in $Y$.

now for the superversion we require that there exist charts (in the open atlas) of $Y \to S$ covering all of $Y^c$ such that

\begin{displaymath}
\itexarray{
    && S
    \\
    & \nearrow && \nwarrow
    \\
    Y \supset_{open} U &&\stackrel{\phi}{\to}&&
    V \subset S \times \mathbb{R}^{d|\delta}_{cs}
    \\
    \downarrow^{\supset} &&&& \downarrow^{\supset}
    \\
    Y^c \supset U \cap Y^c
    &&\stackrel{\simeq}{\to}&&
    V \cap S \times \mathbb{R}^{d-1|\delta}
    \subset S \times \mathbb{R}^{d-1|\delta}_{cs}
  }
\end{displaymath}
next, a \textbf{Euclidean superbordism} from $Y_0 \to S$ to $Y_1 \to S$ is a diagram

\begin{displaymath}
\itexarray{
    Y_1 
     &\stackrel{i_1}{\to}& 
    \Sigma 
     &\stackrel{i_0}{\leftarrow}& Y_1
    \\
     & \searrow & \downarrow & \swarrow
    \\
    && S
  }
\end{displaymath}
where $i_0, i_1$ are morphisms (of families of $(X,G)$-spaces) satisfying the (+)-condition and the (c)-condition described at [[bordism categories following Stolz-Teichner]].

Now a morphism in $E Bord^{sfam}_{d|\delta}$ from $Y_0 \to S_0$ to $Y_1 \to S_1$ is a bordism fitting into a diagram

\begin{displaymath}
\itexarray{
      \Sigma &\stackrel{i_1}{\leftarrow}&
      f^* Y_1 &\to& Y_1
     \\
     \uparrow^{i_0} &\searrow& \downarrow && \downarrow
     \\
     Y_1 &\to & S_0 &\stackrel{f}{\to}& S1
  }
\end{displaymath}
and we identify bordisms $\Sigma, \Sigma'$ if they are isometric -- namely isomorphic in the category of [[Euclidean supermanifold]]s -- ``rel boundary''.

\textbf{definition} A \textbf{$(d|\delta)$-dimensional Euclidean field theory} is a [[symmetric monoidal functor]]

\begin{displaymath}
E \in Fun_{csDiff}^\otimes(E Bord_{(d|\delta)}^{sfam},
  TV^{sfam})
\end{displaymath}
of [[symmetric monoidal category|symmetric monoidal]] [[fibered category|fibered cateories]] (i. [[symmetric monoidal functor]] as well as [[cartesian functor]] ) over the category $cSDiff$ of [[complex supermanifold]]s.

\textbf{Definition} (roughly) $TV^{sfam}$ is the category of families of topological vector spaces parameterized by [[complex supermanifold]]s.

Recall that the ordinary category $TV$ is the category of complete [[Hausdorff space|Hausdorff]], locally convex [[topological vector space]].

define the [[projective tensor product]] of two such $V, W \in TV$. This is a certain completion of the algebraic tensor product $V \otimes_{alg} W$ with respect to the projective topology on $V \otimes_{alg} W$.

This will be the coarsest [[topology]] (the one with the least open sets) making the following maps $f'$

\begin{displaymath}
\itexarray{
     V \otimes_{alg} W &\to& Z
     \\
     & \nearrow_{f'}
     \\
     V \times W
   }
\end{displaymath}
continuous.

\textbf{Remark}

\begin{displaymath}
\itexarray{
    C^\infty(M \times N)
    &\leftarrow&
    C^\infty(M) \otimes_{alg} C^\infty(N)
    \\
    & {}_{\simeq}\nwarrow & \downarrow^{\subset}
    \\
    && C^\infty(M) \otimes C^\infty(N)
  }
\end{displaymath}
\textbf{Definition} the objects of $TV^{sfam}$ are pairs $(S,V)$ for $S$ a [[supermanifold]] and $V$ is a [[sheaf]] of locally complex $\mathbb{Z}_2$-graded [[topological vector space]] with the structure of a sheaf of modules of the [[structure sheaf]] $O_S$.

\textbf{goal} define the [[partition function]] of of a $(2|1)$-dimensional Euclidean field theory.

\textbf{definition} Let $E$ be an EFT as above.

We may think of $\mathbb{R}_+ \times h$ (positive axis times upper half plane) as moduli space of Euclidean tori, where for $(\ell, \tau) \in \mathbb{R}_+ \times h$ we get a torus (regarded as a [[cobordism]]) denoted $T_{\ell,\tau}$. This is the torus given by the lattice spanned by $(1,0)$ and $\ell Re(\tau) + Im(\tau)$ in the upper half plane. Then for the ordinary EFT we would define

\begin{displaymath}
Z_E : \mathbb{R}_+ \times h \to \mathbb {C}
\end{displaymath}
\begin{displaymath}
(\ell,\tau) \mapsto E(T_{\ell,\tau})
\end{displaymath}
For the superversion we put

\begin{displaymath}
Z_{E} := Z_{E_{red}}
\end{displaymath}
where

\begin{displaymath}
\itexarray{
    && E Bord_{2|1}^{sfam}
    \\
    & {}^{\rho}\nearrow
    &&
    \searrow^{E}
    \\
    E Bord_{2, Spin}^{fam}
    &\stackrel{E_{red}}{\to}&
    TV^{fam}
    &
     \hookrightarrow
    &
    TV^{sfam}
  }
\end{displaymath}
\hypertarget{examples}{}\section*{{Examples}}\label{examples}

\hypertarget{explicit_description_of_}{}\subsection*{{explicit description of $E Bord_{1}^{fam}$}}\label{explicit_description_of_}

See [[bordism categories following Stolz-Teichner]].

The category $E Bord_1^{fam}$ is generated from

\begin{itemize}%
\item the \emph{family of right elbows\_}

\begin{displaymath}
\itexarray{
  1-E Bord_{\mathbb{R}_+}^{fam}(\emptyset, pt \coprod pt)
  & \ni&  R  & :=  \mathbb{R}_+ \times \mathbb{R}
  \\
  && \downarrow
  \\
  && S := \mathbb{R}_+
 }
\end{displaymath}

\item the point-family of the left elbox

\begin{displaymath}
\itexarray{
    L_0
    \\
    \downarrow
    \\
    S := pt
  }
\end{displaymath}

\item the family of intervals in $E Bord^{fam}_{\mathbb{R}^+}(pt,pt)$

\begin{displaymath}
\itexarray{
    I
    \\
    \downarrow
    \\
    \mathbb{R}_{\geq 0}
  }
\end{displaymath}


\end{itemize}
Because:

$E \in Fun^{\otimes}_{Diff}(E Bord^{fam}, TV^{fam})$ \$

is determined by

\begin{displaymath}
E(pt) =: V \in TV
\end{displaymath}
\begin{displaymath}
E(L_0) =: \lambda : V \otimes V \to \mathbb{R}
\end{displaymath}
\begin{displaymath}
E(R) =:  \rho \in TV_{\mathbb{R}^+}(\mathbb{R}, V \otimes V) \simeq C^\infty(\mathbb{R}_+, V \otimes V)
\end{displaymath}
\begin{displaymath}
E(I) =: e^{-t H} \in C^\infty(\mathbb{R}_{\geq}, End(V))
\end{displaymath}
forms a \emph{smooth} semigroup under composition generated by

\begin{displaymath}
H \in End(V)
\end{displaymath}
(the [[Hamiltonian operator]])

\begin{displaymath}
\itexarray{
    V \otimes V &&\stackrel{\lambda}{\hookrightarrow}&&
    End(V)
    \\
    & {}_{\rho}\nwarrow && \nearrow_{e^{- t H}}
    \\
    && \mathbb{R}_+
  }
\end{displaymath}
so due to smoothness the data collapses to the infinitesimal data

\begin{displaymath}
(V, \lambda, H)
\end{displaymath}
\textbf{example -- ordinary quantum mechanics} Let $M$ be a [[Riemannian manifold]]. Then set

\begin{itemize}%
\item $H:= \Delta$ the corresponding [[Laplace operator]];


\item $V := C^\infty(M) \subset L^2(M)$;


\item $\lambda$ is the restriction of the $L^2(M)$ inner product to $V$



\end{itemize}
where $e^{-t H}$ is ``trace class'' in the non-standard sense described above in that it makes the above diagram commute.

So everything as known from standard [[quantum mechanics]] textbooks, except that we don't use the full [[Hilbert space]] of states, but just the [[Frechet space]] of smooth functions.

\hypertarget{explicit_description_of__2}{}\subsection*{{explicit description of $E Bord_{2}$}}\label{explicit_description_of__2}

The category $E Bord_{2}_{oriented}^{fam}$ has the following generators:

objects are generated from

\begin{itemize}%
\item the \textbf{circle} $K_\ell := \mathbb{R}^2/\mathbb{Z}\cdot \ell$ of length $\ell \gt 0$ (with collars!! that's why it looks like a cylinder of circumference $\ell$)

notice that we may think of $\ell$ as parameteriing translation by $\ell$ in $\mathbb{R}^2 \rtimes SO(2) = Eucl_{or}(\mathbb{R}^2)$

and the circle with $(+)$/$(-)$-collars reversed



\end{itemize}
morphisms are generated from

\begin{itemize}%
\item \textbf{cylinders} $C_{\ell,\tau}$ which as a manifold is $\simeq \mathbb{R}^2/\mathbb{Z}\cdot \ell$ where $\tau$ parameterizes the embedding of the outgoing circle: the incoming circle is embedded in the canonical way (the identity map on the cylinder, really), while the outgoing circle is embedded by translating the cylinder by $\ell \cdot Re(\tau)$ upwards and rotated by $\ell \cdots Im(\tau)$.


\item \textbf{right elbows} which are the same as the cylinder, except that now the second circle is embedded after reflection so that it becomes an ingoing circle.


\item the \textbf{thin left elbow} $L_\ell$, similar to the above, with arbitrary $\ell$ but $\tau = 0$


\item the \textbf{torus} $T_\tau$ obtained from the cylinder by gluing incoming and outgoing



\end{itemize}
\textbf{notice} the \textbf{pair of pants} is not a morphism in the category at all! since, recall, we require all bordisms to be \emph{flat} and all boundaries to be \emph{geodesics} . There is no way to put such a flat metric on the trinion.

satisfying the relations

\begin{displaymath}
L_\ell \circ R_{\ell, \tau}
  =
  T_{\ell, \tau}
\end{displaymath}
as for the non-family version, but now also with the new relations

\begin{displaymath}
T_{\ell', \tau'} = T_{\ell, \tau}
\end{displaymath}
whenever $\ell' = \ell \cdot|c \tau + d|$ and $\tau' = \frac{a \tau + b }{c \tau + d}$

for $\left(\itexarray{a & b \\ c & d }\right) \in SL_2(\mathbb{Z})$.

Notice that $SL_2(\mathbb{Z})$ is generated by

\begin{itemize}%
\item translation $(\ell, \tau) \mapsto (\ell, \tau + 1)$


\item S-matrix $(\ell, \tau) \mapsto (\ell \cdot |\tau|, - \frac{1}{\tau})$



\end{itemize}
and now there is \textbf{one more relation}

\begin{displaymath}
T_{\ell, \tau}
  =
  T_{\ell |\tau|, - \frac{1}{\tau}}
\end{displaymath}
as usual write $q := e^{2 \pi  i \tau}$ which is on the pointed unit disk since $\tau$ is half plane since $\tau$

\hypertarget{explicit_description_of__3}{}\subsection*{{explicit description of $E Bord_{2}^{fam}$}}\label{explicit_description_of__3}

thwe category $E Bord_{2, oriented}^{fam}$ is generated from

objects:

\begin{itemize}%
\item $\itexarray{K \\ \downarrow \\ S = \mathbb{R}_+}$


\item morphisms

\begin{displaymath}
\itexarray{
    L \\ \downarrow \\ \mathbb{R}_+
  }
\end{displaymath}
\begin{displaymath}
\itexarray{
    R \\ \downarrow \\ \mathbb{R}_+ \times h/\mathbb{Z}
  }
\end{displaymath}
\begin{displaymath}
\itexarray{
    C \\ \downarrow \\ \mathbb{R}_+ \times (h \cup \mathbb{R})/\mathbb{Z}
  }
\end{displaymath}


\end{itemize}
subject to the relations

\ldots{} as before (homework 3, problem 4).. and the furhter one:

for

\begin{displaymath}
\itexarray{
    T  && \alpha^* T
    \\ 
    \downarrow && \downarrow
    \\ 
    \mathbb{R}_+ \times h
    &\stackrel{\alpha}{\leftarrow}&
    \mathbb{R}_+ \times h
    \\
    \\
    (\ell\cdot |\tau|, -\frac{1}{\tau})
    &\stackrel{}{\lt\leftarrow}&
    (\ell, \tau)
  }
\end{displaymath}
the relation is

\begin{displaymath}
\alpha^* T  = T
  \,.
\end{displaymath}
\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}
[[!redirects (2,1)-dimensional Euclidean field theories]]



\end{document}