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\begin{document}

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\section*{(2,1)-dimensional Euclidean field theories and tmf}

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

\hypertarget{functorial_quantum_field_theory}{}\paragraph*{{Functorial quantum field theory}}\label{functorial_quantum_field_theory}

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

\hypertarget{supergeometry}{}\paragraph*{{Super-Geometry}}\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 indicates how 2-dimensional [[FQFT]]s may be related to [[tmf]].

\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]]



\end{itemize}
recall the big diagram from the end of the [[(1,1)-dimensional Euclidean field theories and K-theory|previous entry]].

The \textbf{goal} now is to replace everywhere [[topological K-theory]] by [[tmf]].

previously we had assumed that $X$ has [[spin structure]]. Now we assume [[String structure]].

So we are looking for a diagram of the form

\begin{displaymath}
\itexarray{
    1
    && 
    (2|1)EFT^0(X)/\sim
    && \stackrel{\simeq conjectural}{\leftarrow}&&
    tmf^0(X)
    && \ni 1
    \\
    \downarrow && \downarrow^{quantization}
    &&&&
    \downarrow^{\int_X}
    &&
    \downarrow
    \\
    \sigma_{(2|1)(X)}&& (2|1)EFT^{-n}(X)/\sim
    &&\stackrel{\simeq conjectural }{\leftarrow}&&
    tmf^{-n}(pt)
    &&
    \\
    &\searrow & \searrow &&& \swarrow& \swarrow
    \\
    &&&& mf^{-n}
    \\
    &&&&
    index^{S^1}(D_{L X})
    = W(X)
  }
\end{displaymath}
the vertical maps here are due to various theorems by various people -- except for the ``physical quantization'' on the left, that is used in physics but hasn't been formalized

the \textbf{horizontal maps are the conjecture we are after} in the Stolz-Teichner program: The top horizontal map will involve making the notion of $(2|1)$EFT \emph{local} by refining it to an \emph{extended} [[FQFT]]s. This will not be considered here.

we will explain the following items

\begin{itemize}%
\item the [[ring]] $mf^\bullet$ of [[integral modular form]]s

\begin{displaymath}
mf^\bullet \simeq \mathbb{Z}[c_4, c_6, \Delta, \Delta^{-1}]/(c_4^{3}- c_6^{2} - 1728 \Delta)
\end{displaymath}
one calls $w = -\frac{n}{2}$ the \emph{weight} . We have degree of $\Delta$ is $deg(\Delta) = -24$, hence $w(\Delta) = 12$.


\item $W(X)$ is the [[Witten genus]]

\begin{displaymath}
W(X) = \sum_{k \in \mathbb{Z}} a_k \cdot q^k
  \,,
  a_k \in \mathbb{Z}
\end{displaymath}
where $a_k = index(D_X \otimes E_k)$ where $E_k$ is some explicit vector bundle over $X$.



\end{itemize}
\hypertarget{modular_forms}{}\subsection*{{modular forms}}\label{modular_forms}

\textbf{definition} An \textbf{(integral) [[modular form]]} of weight $w$ is a [[holomorphic function]] on the [[upper half plane]]

\begin{displaymath}
f : (\mathbb{R}^2)_+ \hookrightarrow \mathbb{C}
\end{displaymath}
(complex numbers with strictly positive imaginary part)

such that

\begin{enumerate}%
\item if $A = \left( \itexarray{a & b \\ c& d}\right) \in SL_2(\mathbb{Z})$ acting by $A : \tau \mapsto = \frac{a \tau + b }{c \tau + d}$ we have

\begin{displaymath}
f(A(\tau)) = (c \tau + d)^w f(\tau)
\end{displaymath}
\textbf{note} take $A = \left( \itexarray{1 & 1 \\ 0& 1}\right)$ then we get that $f(\tau + 1) = f(\tau)$


\item $f$ has at worst a pole at $\{0\}$ (for \emph{weak} modular forms this condition is relaxed)

it follows that $f = f(q)$ with $q = e^{2 \pi i \tau}$ is a meromorphic funtion on the open disk.


\item \textbf{integrality} $\tilde f(q) = \sum_{k = -N}^\infty a_k \cdot q^k$ then $a_k \in \mathbb{Z}$



\end{enumerate}
by this definition, modular forms are not really functions on the upper half plane, but functions on a [[moduli space]] of [[torus|tori]]. See the following definition:

if the weight vanishes, we say that modular form is a \textbf{[[modular function]]} .

\textbf{definition (2|1)-dim [[partition function]]}

Let $E$ be an EFT

\begin{displaymath}
(2|1)EFT^0 \stackrel{S}{\to}
  2 EFT \ne E
\end{displaymath}
\begin{displaymath}
E \mapsto E_{red}
\end{displaymath}
then the [[partition function]] is the map $Z_E : \mathbb{C} \to \mathbb{R}$

\begin{displaymath}
Z_E : \tau \mapsto E_{red}(T_\tau)
\end{displaymath}
where

\begin{displaymath}
T_\tau := \mathbb{C}/{\mathbb{Z} \times \mathbb{Z} \cdot \tau}
\end{displaymath}
is thee standard torus of modulus $\tau$.

then the central \textbf{theorem} that we are after here is

\textbf{therorem (Stolz-Teichner)} (after a suggestion by [[Edward Witten]])

There is a precise definition of $(2|1)$-EFTs $E$ such that the [[partition function]] $Z_E$ is an integral [[modular function]]

(so this is really four theorems: the function is holomorphic, integral, etc.)

moreover, every [[integral modular function]] arises in this way.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

A concrete relation between [[2d SCFT]] and [[tmf]] is the lift of the [[Witten genus]] to the [[string orientation of tmf]]. See there fore more.

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

\begin{itemize}%
\item [[Stephan Stolz]], [[Peter Teichner]], \emph{[[What is an elliptic object?]]} in \emph{Topology, geometry and quantum field theory} , London Math. Soc. LNS 308, Cambridge Univ. Press (2004), 247-343. (\href{http://web.me.com/teichner/Math/Reading_files/Elliptic-Objects.pdf}{pdf})


\item Pokman Cheung, \emph{Supersymmetric field theories and cohomology} (\href{http://arxiv.org/abs/0811.2267}{arXiv:0811.2267})


\item [[Stefan Stolz]], [[Peter Teichner]], \emph{Supersymmetric Euclidean field theories and generalized cohomology} , in [[Hisham Sati]], [[Urs Schreiber]] (eds.), \emph{[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]}



\end{itemize}
A hint supporting the conjectured relation of [[2d SCFT]] to [[tmf]], vaguely in line with the lift of the [[Witten genus]] to the [[string orientation of tmf]]:

\begin{itemize}%
\item [[Davide Gaiotto]], [[Theo Johnson-Freyd]], \emph{Holomorphic SCFTs with small index} (\href{https://arxiv.org/abs/1811.00589}{arXiv:1811.00589})

\end{itemize}


\end{document}