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\begin{document}

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\section*{(1,1)-dimensional Euclidean field theories and K-theory}

\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 1-dimensional [[FQFT]]s (the [[superparticle]]) may be related to [[topological K-theory|topological]] [[K-theory]].

\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]].

\end{itemize}
Next

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

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

\noindent\hyperlink{_efts}{$(1,1)d$ EFTs}\dotfill \pageref*{_efts} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


\hypertarget{_efts}{}\subsection*{{$(1,1)d$ EFTs}}\label{_efts}

recall the \textbf{commercial for supergeometry} with which we ended [[Axiomatic field theories and their motivation from topology|last time]]: the grading introduced by supergeometry makes it possible to have push-forward diagrams of the kind:

\begin{displaymath}
\itexarray{
    (0|1)TFTs^n(X)/\simeq &\leftarrow& H^n_{dR}(X)
    \\
    \downarrow && \downarrow
    \\
    (0|1)TFT^0(X)/\simeq &\leftarrow&  H^0_{dR}(pt)
  }
\end{displaymath}
\textbf{Example} of 1-EFT

\begin{displaymath}
\sigma_1(M^n) = E : 1-EB  \to tV
\end{displaymath}
\begin{displaymath}
pt \mapsto \Gamma M
\end{displaymath}
\begin{displaymath}
(pt \stackrel{[0,t]}{\to}) 
  \mapsto 
  e^{- t \Delta}
\end{displaymath}
\textbf{Example} of $(1|1)-EFT$ associated to a [[Spin structure|spin manifold]], there is the [[spinor bundle]]

\begin{displaymath}
S = S^+ \oplus S^-
\end{displaymath}
a $\mathbb{Z}/2$-graded [[vector bundle]] and on this there is the [[Dirac operator]]

\begin{displaymath}
D : \Gamma(S) \to \Gamma(S)
\end{displaymath}
where $\Gamma(S) = \Gamma(S^+) \oplus \Gamma(S^-)$. So we can write

\begin{displaymath}
D =
  \left(
    \itexarray{
       0 & D_-
       \\
       D+ & 0
    }
  \right)
\end{displaymath}
\begin{displaymath}
\sigma_{1|1}(M) : Bord_{1|1} \to TV
\end{displaymath}
\begin{displaymath}
\mathbb{R}^{0|1} \mapsto E(\mathbb{R}^{0|1}) = \Gamma(S)
\end{displaymath}
there is an involution $invol : \mathbb{R}^{0|1} \to \mathbb{R}^{0|1}$. It maps to

\begin{displaymath}
invol \mapsto grading involution
\end{displaymath}
we have the following [[moduli space]] of super [[interval]]s (super 1d-bordisms)

\begin{displaymath}
\mathbb{R}^{1|1}_+ \simeq
  \{super intervals I_{t,\theta}\}/\sim
\end{displaymath}
and these are mapped by the EFT as

\begin{displaymath}
I_{t,\theta} \mapsto e^{-t D^2 + \theta D}
\end{displaymath}
(here we are implicitly working in the [[topos]] of [[sheaf|sheaves]] on the category of [[supermanifold]]s and these equations have to be interpreted in that topos-logic, mapping [[generalized element]]s to [[generalized element]]s).

So we have for $E$ a $1|1$ EFT a \emph{reduced} non-susy field theory

\begin{displaymath}
\itexarray{
     (1|1)EBord &\stackrel{E}{\to}& TV
      \\
     \uparrow & \nearrow_{E_{red}}
      \\
      EBord_1^{spin}
  }
\end{displaymath}
\textbf{Definition} $E \in (1|1)EFT$, the [[partition function]] $Z_E$ of $E$ is the function

\begin{displaymath}
Z_E : \mathbb{R}_+ \to \mathbb{C}
\end{displaymath}
\begin{displaymath}
t \mapsto Z_{E_{red}}(t) = E_{red}(S^1_t)
\end{displaymath}
that sends a length to the value of the EFT on the circle of that circumferene.

\textbf{Example} Consider from above the EFT

\begin{displaymath}
E = \sigma_{1|1}(M)
\end{displaymath}
look at its reduced part

\begin{displaymath}
z_E(t) = E_{red}(S^1_t)
\end{displaymath}
notice that by the above this assigns

\begin{displaymath}
[0,t] \stackrel{E_{red}}{\mapsto} e^{-t D^2}
\end{displaymath}
\begin{displaymath}
S^1_t \mapsto str(e^{-t D^2})
  = 
  tr(e^{-t D^2})|_{even} - tr(e^{-t D^2})|_{odd}
\end{displaymath}
where on the right we have the [[super trace]].

This evaluates to

\begin{displaymath}
str(e^{-t D^2})
  =
  \sum_{\lambda \in Spec(D^2)}
  e^{-t \lambda}
   sdim E_{\lambda}
\end{displaymath}
where the [[super dimension]] of the [[eigenspace]] $E_\lambda$ is

\begin{displaymath}
dim E^+_\lambda - dim E^-_\lambda
\end{displaymath}
and this vanishes for $\lambda \neq 0$ since there $D : E_\lambda^+ \stackrel{\simeq}{\to} E_\lambda^-$

is an [[isomorphism]].

So further in the computation we have

\begin{displaymath}
\cdots = dim ker D_+ - dim coker D_+
  = 
  \hat A(M)
\end{displaymath}
where the last step is the [[Atiyah-Singer index theorem]].

So \textbf{due to supersymmetry} , the [[partition function]] has two very special properties:

\begin{itemize}%
\item it is constant -- in that it does not depend on $t$,


\item it takes integer values $\in \mathbb{N} \subset \mathbb{R}$.



\end{itemize}
\textbf{recall} from $V \to X$ a [[vector bundle]] [[connection on a bundle|with connection]] $\nabla$ we get a 1d EFT

\begin{displaymath}
E_{(V,\nabla)} \in 1d EFT(X)
\end{displaymath}
given by the assignment

\begin{displaymath}
E_{(V,\nabla)} : 1s EB(X) \to TV
\end{displaymath}
\begin{displaymath}
(x : pt \to X) \mapsto V_x = fiber of V over x
\end{displaymath}
a morphism is an [[interval]] $[0,t]$ of length $t$ equipped with a map $\gamma : [0,t] \to X$, this is sent to the [[parallel transport]] associated with the [[connection on a bundle]]

\begin{displaymath}
\gamma \mapsto (V_{\gamma_x} \to V_{\gamma_y})
\end{displaymath}
Now refine this example to super-dimension $(1|1)$:

\textbf{example} of a $(1|1)$-EFT over $X$ consider

\begin{displaymath}
EBord_{(1|1)} \to EBord_{1}(X) \stackrel{E_{(V,\nabla)}}{\to}
  TV
\end{displaymath}
given by the assignment

\begin{displaymath}
(\Sigma^{(1|1)} \to X)(
  \mapsto 
  (\Sigma^{(1|1)}_{red} \to X)
  \mapsto 
  parallel transport as before
\end{displaymath}
so we just forget the super-part and consider the same [[parallel transport]] as before.

now to [[K-theory]]:

$KO^0(X) =$ [[Grothendieck group]] of real [[vector bundle]]s over $X$

\begin{displaymath}
KO^{-n}(pt) = 
  \left\{
    \itexarray{
         \mathbb{Z} & n = 0 mod 4
         \\
         \mathbb{Z}_2 & n = 1,2 mod 8
         \\
         0 & otherwise
    }
  \right.
\end{displaymath}
there is a [[Bott element]] $\beta \in KO^{-8}(pt)$

such that

\begin{displaymath}
KO^*(pt) \stackrel{\simeq_{\mathbb{Q}}}{\to}
  \mathbb{Z}[u,u^{-1}]
\end{displaymath}
\begin{displaymath}
\beta \mapsto u^2
\end{displaymath}
now the \textbf{push-forward in [[topological K-theory]]}

\begin{displaymath}
p : X^n \to pt
\end{displaymath}
for $X$ a closed [[spin structure]] manifold

then there exists an embedding $X \hookrightarrow S^{n+m}$. Let $\nu$ be the [[normal bundle]] to this embedding.

then we define

\begin{displaymath}
\int_X : KO^k(X) \to KO^{k-n}(pt)
\end{displaymath}
as follows:

let $D(\nu)$ be the [[disk bundle]] and $S(\nu)$ be the [[sphere bundle]] of $\nu$. Then the [[Thom bundle]] is

\begin{displaymath}
T(\nu) := D(\nu)/S(\nu)
\end{displaymath}
we get a map

\begin{displaymath}
S^{n+m} \stackrel{C}{\to} T(\nu) := D(\nu)/S(\nu)
\end{displaymath}
involving the [[Thom isomorphism]]

\begin{displaymath}
C(X) = \left\{
   \itexarray{
      X & if x \in D(\nu)
      \\
     * & otherwise
   }
  \right.
\end{displaymath}
then we set

\begin{displaymath}
\itexarray{
      KO^k(X)
      && \stackrel{\int_X}{\to}&& KO^{k-n}(pt)
      \\
      & {}_{Thom iso}\searrow
      &&&
      \downarrow^{\simeq}_{suspension}
      \\
      && \tilde KO^{k+m}(T(\nu))
      &\stackrel{C^*}{\to}&
    
  }
\end{displaymath}

\vspace{.5em} \hrule \vspace{.5em}
now start with $X^n$ again a [[spin structure|spin]] [[manifold]]

then

\textbf{theorem} (Stolz-Teichner): we have the horizontal isomorphism in the following diagram:

\begin{displaymath}
\itexarray{
     && [E_{(V,\nabla)}]&& \stackrel{}{\leftarrow}
     && [V^+ - V^-]
     \\
     1 \in 
     &&(1|1)EFT^0(X)/_{conc}
     &&\stackrel{\simeq}{\to}&& KO^0(X) && \ni 1 
     \\
     \downarrow &&\downarrow^{quantization}
     &&&& \downarrow^{\int_X} && \downarrow
     \\
     \sigma_{(1|1)}(X)
     &&EFT^{-n}(pt)/_{conc}
     &&\stackrel{\simeq}{\to}&& KO^{-n} && \alpha(X)
     \\
     &\searrow&&{}_{partition func}\searrow&& 
    \swarrow_{\simeq} && \swarrow_{Atiyah's \alpha invariant}
     \\
     &&&&
     (\mathbb{Z}[u,u^{-1}])^{-n}
     \\
     &&&&
     index D = \hat A(X) u^{n/4}
  }
\end{displaymath}
\textbf{question} if we don't divide out [[concordance]], do we get [[differential K-theory]] on the right?

\textbf{answer} presumeably, but not worked out yet

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

\begin{itemize}%
\item [[supersymmetric quantum mechanics]]


\item [[Euclidean quantum field theory]]


\item [[spectral triple]]


\item [[spectral action]]


\item [[higher category theory and physics]]: 


\item \textbf{(1,1)-dimensional Euclidean field theories and K-theory}


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


\item [[2-spectral triple]]



\end{itemize}
\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]] (notes by Arlo Caine), \emph{Supersymmetric Euclidean field theories and generalized cohomology} Lecture notes (2009) (\href{http://www.cpp.edu/~jacaine/pdf/Lectures_complete.pdf}{pdf})


\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]]} Proceedings of Symposia in Pure Mathematics, AMS (2011)



\end{itemize}


\end{document}