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\begin{document}

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\section*{elliptic cohomology}

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

\hypertarget{elliptic_cohomology}{}\paragraph*{{Elliptic cohomology}}\label{elliptic_cohomology}

[[!include elliptic cohomology -- contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{genera_the_elliptic_genus_and_relation_to_string_theory}{Genera, the elliptic genus and relation to string theory}\dotfill \pageref*{genera_the_elliptic_genus_and_relation_to_string_theory} \linebreak
\noindent\hyperlink{equivariant_elliptic_cohomology_and_loop_group_representations}{Equivariant elliptic cohomology and loop group representations}\dotfill \pageref*{equivariant_elliptic_cohomology_and_loop_group_representations} \linebreak
\noindent\hyperlink{chromatic_filtration}{Chromatic filtration}\dotfill \pageref*{chromatic_filtration} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{relation_to_quantum_field_theory}{Relation to quantum field theory}\dotfill \pageref*{relation_to_quantum_field_theory} \linebreak


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

An \emph{elliptic cohomology theory} is a type of \emph{[[generalized (Eilenberg-Steenrod) cohomology]]} theory associated with the datum of an [[elliptic curve]].

Even ([[weakly periodic cohomology theory|weakly]]) [[periodic cohomology theory|periodic]] [[multiplicative cohomology theory|multiplicative]] [[generalized (Eilenberg-Steenrod) cohomology]] theories $A$ are characterized by the [[formal group]] whose ring of functions $A(\mathbb{C}P^\infty)$ is the [[cohomology ring]] of $A$ evaluated on the complex projective space $\mathbb{C}P^\infty$ and whose group product is induced from the canonical morphism $\mathbb{C}P^\infty \times \mathbb{C}P^\infty \to \mathbb{C}P^\infty$ that describes the tensor product of complex [[line bundle]]s under the identification $\mathbb{C}P^\infty \simeq \mathcal{B} U(1)$.

There are precisely three types of such formal group laws:

\begin{itemize}%
\item the simple additive group structure -- this corresponds to standard integral cohomology given by the [[Eilenberg-MacLane spectrum]];


\item the multiplicative group that corresponds to complex [[K-theory]]


\item the formal group law on [[elliptic curve]].



\end{itemize}
An \textbf{elliptic cohomology} theory is an even [[periodic cohomology theory|periodic]] [[multiplicative cohomology theory|multiplicative]] [[generalized (Eilenberg-Steenrod) cohomology]] theory whose corresponding formal group is an elliptic curve, hence which is [[Brown representability theorem|represented]] by an [[elliptic spectrum]].

e.g. \hyperlink{Lurie}{Lurie, def. 1.2}, see also at \emph{[[elliptic spectrum]]}

Then [[Goerss-Hopkins-Miller-Lurie theorem]] shows that the assignment of [[generalized (Eilenberg-Steenrod) cohomology]] theories to [[elliptic curves]] lifts to an assignment of representing [[spectrum|spectra]] in a structure preserving way.

The [[homotopy limit]] of this assignment functor, i.e. the ``gluing'' of all spectra representing all elliptic cohomology theories is the [[spectrum]] that represents the cohomology theory called [[tmf]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{genera_the_elliptic_genus_and_relation_to_string_theory}{}\subsubsection*{{Genera, the elliptic genus and relation to string theory}}\label{genera_the_elliptic_genus_and_relation_to_string_theory}

\begin{quote}%
A properly developed theory of elliptic cohomology is likely to shed some light on what [[string theory]] really means. (\hyperlink{Witten87}{Witten 87, very last sentence})


\end{quote}
The following is \textbf{rough material} originating from notes taken live (and long ago), to be polished. See also at \emph{[[elliptic genus]]} and \emph{[[Witten genus]]}

Some topological invariants of [[manifolds]] that are of interest:

we restricted attention to [[closed manifold|closed]] connected [[smooth manifolds]] $X$

\begin{itemize}%
\item the [[Euler characteristic]] $e(X) \in \mathbb{Z}$

\begin{itemize}%
\item takes all values in $\mathbb{Z}$


\item is the obstruction to the existence of a nowhere vanishing [[vector field]] on $X$:

\begin{displaymath}
(e(X)= 0)
  \Leftrightarrow
  (\exists  v \in \Gamma(T X) : \forall x \in X : v(x) \neq 0)
\end{displaymath}


\end{itemize}

\item [[signature]] $sgn(X)$

this is the obstruction to $X$ being cobordant to a fiber bundle over the circle:

$X$ is [[bordant manifold|bordant]] to a [[fiber bundle]] over $S^1$ precisely if $sgn(X) = 0$


\item when $X$ has a [[spin structure]]

the index of the [[Dirac operator]] $D$:

\begin{displaymath}
ind D_X \in \mathbb{Z}
\end{displaymath}
\begin{displaymath}
\alpha(D) \in 
  \left\{
    \itexarray{
    \mathbb{Z} & dim X = 0 mod 4
    \\
    \mathbb{Z}_2 & dim X = 1, 2 mod 8
    \\
    0 & otherwise
    }
  \right.
\end{displaymath}
\textbf{theorem} (Gromov-Lawson / [[Stephan Stolz|Stolz]]) let $dim X \geq 5$ and

then $X$ admits a [[Riemannian metric]] of positive [[scalar curvature]] precisely when $\alpha(X) = 0$



\end{itemize}
These invariants share the following properties:

\begin{itemize}%
\item they are additive under [[disjoint union]] of [[manifold]]s


\item they are multiplicative under [[cartesian product]] of manifolds


\item $e(X) mod 2, sgn(X), ind(D_X)$ all vanish when $X$ is a boundary, $\exists W : X = \partial W$, which means that $X$ is [[cobordant manifold|cobordant]] to the empty manifold $\emptyset$.

in other words, these invariants are [[genus|genera]], namely [[ring]] homomorphisms

\begin{displaymath}
\Omega \to R
\end{displaymath}
form the [[cobordism ring]] $\Omega$ to some commutative [[ring]] $R$


\item good [[genus|genera]] are those which reflect geometric properties of $X$.


\item now for $X$ a [[topological space]] consider the [[cobordism ring]] \emph{over $X$}:

\begin{displaymath}
\Omega(X) := \{(M,f)| f : M \stackrel{cont}{\to X}\}/_{bordism}
\end{displaymath}
where addition and multiplication are again given by disjoint union and cartesian product.

this assignment of rings to topological spaces is a [[generalized homology]] theory: [[cobordism homology theory]]

\textbf{question} given a [[genus]] $\Omega \to R$, can we find a [[homology theory]] $R(-)$ with $R  = R(pt)$ its [[homology ring]] over the point and such that it all fits into a natural diagram

\begin{displaymath}
\itexarray{
   \Omega &\to& R
   \\
   \uparrow && \uparrow
   \\
   \Omega(X) &\stackrel{\rho}{\to}& R(X)
  }
\end{displaymath}
This would be a \emph{parameterized extension} $\rho = R(-)$ of $R$ .

Now let $X$ be a closed manifold.

consider $u_X : X \to K(\pi_1(X),1)$ (on the right an [[Eilenberg-MacLane space]]) which is the classifying map for the [[universal cover]]

\begin{displaymath}
u_* \pi_1(X) \stackrel{\simeq_{canon}}{\to}
  \pi_1(K(\pi_1(X), 1))
\end{displaymath}
then consider

\begin{displaymath}
\rho_X[X, u_X] \in R(K(\pi_1(X),1))
\end{displaymath}
\textbf{theorem (Julia Weber)}

take the [[Euler characteristic]] mod 2, $Eu(X)$

\begin{displaymath}
\itexarray{
    \Omega^0 &\stackrel{Eu(M)\cdot t^{dim M}}{\to}& \mathbb{Z}_2[t]
    \\
    \uparrow && \uparrow
    \\
    \Omega^0(X) &\to& Eu(X) &
    \simeq H_\bullet(X; \mathbb{Z}_2[t])
  }
\end{displaymath}
for $X$ smooth we have then:

\begin{displaymath}
Eu_X[X, id] = Poincare dual of total Stiefel-Whitney class
\end{displaymath}
\textbf{theorem} (Minalta)

something analogous for [[signature genus]]

\begin{displaymath}
\itexarray{
    \Omega_\bullet^{SO}
    &\to&
    Sig_\bullet(X)
  }
\end{displaymath}
$sign_X[X,u] \in sig_\bullet(K) \otimes \mathbb{Q}$

this is the Novikov higher signature

now the same for the $\alpha$-genus

\begin{displaymath}
\itexarray{
    \Omega_{X}^{Spin} &\stackrel{\alpha}{\to}&
    KO_\bullet(pt)
    \\
    \uparrow && \uparrow
    \\
    \Omega_\bullet^{Spin}
    &\to&
    KO_\bullet(X)
  }
\end{displaymath}


\end{itemize}
now towards elliptic genera: recall the notion of [[string structure]] of a [[manifold]] $X$: a lift of the structure map $X \to \mathcal{B}O(n)$ through the 4th connected universal cover $\mathcal{B}String(n) := \mathcal{B}O(n)\langle 4\rangle \to \mathcal{B} O$:

so consider [[string structure|String manifold]]s and the bordism ring $\Omega_\bullet^{String}$ of String manifold, let $M_\bullet$ be the ring of integral [[modular form]]s, then there is a [[genus]] -- the [[Witten genus]] $W$--

\begin{displaymath}
\itexarray{
    \Omega_\bullet^{String} &\stackrel{W}{\to}& M_\bullet
    \\
    \uparrow && \uparrow
    \\
    \Omega_\bullet^{String}(X)
    &\to&
    M_\bullet(X)
    \\
    &\searrow&
    \\
    && tmf_\bullet(X) 
  }
\end{displaymath}
where $\Omega_\bullet^{String}(pt) \to tmf_\bullet(pt)$ is surjective

\textbf{conjecture ([[Stolz conjecture]])}

If a String manifold $Y$ has a positive [[Ricci curvature]] [[Riemannian metric|metric]], then the [[Witten genus]] vanishes.

The attempted ``Proof'' of this is the motivation for the [[Stephan Stolz|Stolz]]-[[Peter Teichner|Teichner]]-program for [[geometric models for elliptic cohomology]]:

\textbf{``Proof''} If $Y$ is String, then the [[loop space]] $L Y$ is has [[spin structure]], so if $Y$ has positive [[Ricci curvature]] the $L Y$ has positive [[scalar curvature]] which implies by the above that $ind^{S^1} D_{L Y} = 0$ which by the index formula is the [[Witten genus]].

\hypertarget{equivariant_elliptic_cohomology_and_loop_group_representations}{}\subsubsection*{{Equivariant elliptic cohomology and loop group representations}}\label{equivariant_elliptic_cohomology_and_loop_group_representations}

The analog of the [[orbit method]] with [[equivariant K-theory]] replaced by [[equivariant elliptic cohomology]] yields (aspects of) the [[representation theory]] of [[loop groups]]. (\hyperlink{Ganter12}{Ganter 12})

\hypertarget{chromatic_filtration}{}\subsubsection*{{Chromatic filtration}}\label{chromatic_filtration}

[[!include chromatic tower examples - table]]

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

\begin{itemize}%
\item [[elliptic spectrum]]


\item [[elliptic genus]], [[equivariant elliptic genus]]


\item [[SO orientation of elliptic cohomology]], [[spin orientation of elliptic cohomology]]


\item [[elliptic Chern character]]


\item [[equivariant elliptic cohomology]]


\item [[sigma-orientation]]


\item [[tmf]]



\end{itemize}
[[!include moduli of higher lines -- table]]

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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

The concept of elliptic cohomology originates around

\begin{itemize}%
\item [[Peter Landweber]], [[Douglas Ravenel]], [[Robert Stong]], \emph{Periodic cohomology theories defined by elliptic curves}, in [[Haynes Miller]] et. al. (eds.), \emph{The Cech centennial: A conference on homotopy theory}, June 1993, AMS (1995) (\href{http://www.math.sciences.univ-nantes.fr/~hossein/GdT-Elliptique/Landweber-Ravenel-Stong.pdf}{pdf})

\end{itemize}
motivated by [[Serge Ochanine]]`s concept of [[elliptic genus]] and by [[Edward Witten]]'s [[QFT]]/[[string theory|string theoretic]] explanation of the [[Ochanine genus]] and of the [[Witten genus]] (as the [[partition functions]] of the [[type II superstring]] and the [[heterotic superstring]], respectively).

Accounts include

\begin{itemize}%
\item [[Charles Thomas]], \emph{Elliptic cohomology}, Kluwer Academic, 2002

\end{itemize}
The concept of an [[elliptic spectrum]] [[Brown representability theorem|representing]] an elliptic cohomology theory is due to

\begin{itemize}%
\item [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], \emph{Elliptic spectra, the Witten genus, and the theorem of the cube}, Inventiones Mathematicae, 146:595--687, 2001, DOI 10.1007/s002220100175

\end{itemize}
Modern accounts of ([[equivariant elliptic cohomology|equivariant]]) elliptic cohomology in terms of [[stable homotopy theory]] include

\begin{itemize}%
\item [[David Gepner]], \emph{[[Homotopy topoi and equivariant elliptic cohomology]]}, 2005


\item [[Jacob Lurie]], \emph{[[A Survey of Elliptic Cohomology]]}


\item [[Jacob Lurie]], \emph{[[Elliptic Cohomology I]]: spectral abelian varieties}, 2018. 141pp (\href{http://math.harvard.edu/~lurie/papers/Elliptic-I.pdf}{pdf})


\item [[Jacob Lurie]], \emph{Elliptic Cohomology II: Orientations}, 2018. 288pp (\href{http://www.math.harvard.edu/~lurie/papers/Elliptic-II.pdf}{pdf})


\item [[Jacob Lurie]], \emph{Elliptic Cohomology III: Tempered Cohomology}, 2019. 286pp (\href{http://www.math.harvard.edu/~lurie/papers/Elliptic-III-Tempered.pdf}{pdf})


\item [[Jacob Lurie]], \emph{Elliptic Cohomology IV: Equivariant elliptic cohomology}, to appear.



\end{itemize}
Further discussion of [[equivariant elliptic cohomology]] and the relation to [[loop group]] [[representation theory]] is in

\begin{itemize}%
\item [[Nora Ganter]], \emph{The elliptic Weyl character formula} (\href{http://arxiv.org/abs/1206.0528}{arXiv:1206.0528})

\end{itemize}
Surveys include

\begin{itemize}%
\item [[Charles Rezk]], \emph{Elliptic cohomology and elliptic curves}, Felix Klein Lectures, Bonn 2015 (\href{http://www.hcm.uni-bonn.de/fkl-rezk/}{web})

\end{itemize}
Discussion of generalization to higher [[chromatic homotopy theory]] is discussed in

\begin{itemize}%
\item [[Douglas Ravenel]], \emph{Toward higher chromatic analogs of elliptic cohomology} \href{http://www.math.rochester.edu/people/faculty/doug/mypapers/high1.pdf}{pdf}


\item [[Douglas Ravenel]], \emph{Toward higher chromatic analogs of elliptic cohomology II}, Homology, Homotopy and Applications, vol. 10(1), 2008, pp.1\{36 (\href{http://www.math.rochester.edu/people/faculty/doug/mypapers/high2.pdf}{pdf}, \href{http://www.math.rochester.edu/people/faculty/doug/mypapers/halifax.pdf}{pdf slides})



\end{itemize}
\hypertarget{relation_to_quantum_field_theory}{}\subsubsection*{{Relation to quantum field theory}}\label{relation_to_quantum_field_theory}

The [[elliptic genus]] and [[Witten genus]] were understood as the [[large volume limit]] of the [[partition function]] of the [[superstring]] in

\begin{itemize}%
\item [[Edward Witten]], \emph{Elliptic Genera And Quantum Field Theory} , Commun.Math.Phys. 109 525 (1987) (\href{http://projecteuclid.org/euclid.cmp/1104117076}{Euclid})

\end{itemize}
The following reference discuss aspects of the construction of elliptic cohomology / [[tmf]] in terms of [[quantum field theory]]. See also at \emph{[[Witten genus]]}.

\begin{itemize}%
\item P Hu, [[Igor Kriz]], \emph{Conformal field theory and elliptic cohomology}, Advances in Mathematics, Volume 189, Issue 2, 20 December 2004, Pages 325--412 (\href{http://www.math.lsa.umich.edu/~ikriz/ell0311.pdf}{pdf})


\item [[Igor Kriz]], [[Hisham Sati]], \emph{M-theory, type IIA superstrings, and elliptic cohomology}, Adv. Theor. Math. Phys. 8 (2004), no. 2, 345--394 (\href{http://projecteuclid.org/euclid.atmp/1091543172}{Euclid}, \href{http://arxiv.org/abs/hep-th/0404013}{arXiv:hep-th/0404013})



\end{itemize}
A proposal for a construction via [[FQFT]] is discussed at

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

\end{itemize}
The case of elliptic cohomology associated with the [[Tate curve]] is discussed in

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

\end{itemize}
[[!redirects elliptic cohomologies]]

[[!redirects elliptic cohomology theory]] [[!redirects elliptic cohomology theories]]

[[!redirects elliptic generalized cohomology theory]] [[!redirects elliptic generalized cohomology theories]]



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