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\begin{document}

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\section*{modular equivariant elliptic cohomology}

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

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

[[!include elliptic cohomology -- contents]]

\hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory}

[[!include representation theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{General}{General}\dotfill \pageref*{General} \linebreak
\noindent\hyperlink{MotivationFromStringTheory}{Motivation from string theory}\dotfill \pageref*{MotivationFromStringTheory} \linebreak
\noindent\hyperlink{Definition}{Definition and construction}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The universal [[elliptic cohomology]] [[cohomology theory|cohomology]] called [[tmf]] may be realized (after [[local spectrum|localization]] at some primes) as the [[homotopy fixed points]] of an [[∞-action]] of the [[modular group]] modulo a [[congruence subgroup]] in direct analogy to how [[KO-theory]] arises as the $\mathbb{Z}_2$-homotopy fixed points of [[KU-theory]] (\hyperlink{MahowaldRezk09}{Mahowald-Rezk 09}, \hyperlink{LawsonNaumann12}{Lawson-Naumann 12}). This construction extends to equip [[tmf]] [[level structure on an elliptic curve|for level structure]] with -- almost -- the structure of a [[naive G-spectrum]] (hence a [[Bredon cohomology|Bredon equivariant cohomology theory]]) for pieces of the [[modular group]] (\hyperlink{HillLawson13}{Hill-Lawson 13, theorem 9.1}).

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

The way this works is roughly indicated in the following table (\hyperlink{LawsonNaumann12}{Lawson-Naumann 12}, \hyperlink{HillLawson13}{Hill-Lawson 13}):

[[!include moduli stack of curves -- table]]

For [[K-theory]] this relation induces the $\mathbb{Z}_2$-[[equivariant cohomology]]-version of [[KU]] which is [[KR-theory]]. In direct analogy to this one may hence consider $SL_2(\mathbb{Z}/n\mathbb{Z})$-equivariant versions of [[tmf]] (localized at some primes). This maybe deserves to be called \emph{modular equivariant elliptic cohomology}.

This hasn't been studied much yet (or not at all), but there is some motivation for this from [[string theory]] (see \hyperlink{MotivationFromStringTheory}{below}) and in this context a relevance of modular equivariant elliptic cohomology theory has been conjectured in (\hyperlink{KrizSati05}{Kriz-Sati 05, p.3, p. 17, 18}).

\hypertarget{MotivationFromStringTheory}{}\subsubsection*{{Motivation from string theory}}\label{MotivationFromStringTheory}

A relevant role of some modular equivariant elliptic cohomology theory in [[string theory]] has been conjectured in (\hyperlink{KrizSati05}{Kriz-Sati 05, p. 3, p. 17, 18}):

First of all, $\mathbb{Z}_2$-equivariant [[topological K-theory]], hence [[KR-theory]], is what controls the [[D-brane charges]] in general [[orientifold]] [[vacua]] of [[type II superstring theory]] (hence also of [[type I superstring theory]], which is thereby a special case).

Next, there are various hints (\hyperlink{KrizSati04a}{Kriz-Sati 04a}, \hyperlink{KrizSati04b}{Kriz-Sati 04b} \hyperlink{KrizSati05}{Kriz-Sati 05}, \hyperlink{Sati05}{Sati 05}) that lifting [[string theory]] to ``[[M-theory]]'' involves replacing K-theory by [[elliptic cohomology]]. Not the least of these is the [[String orientation of tmf]] (\hyperlink{AndoHopkinsStrickland01}{Ando-Hopkins-Strickland 01}, \hyperlink{AndoHopkinsRezk10}{Ando-Hopkins-Rezk 10}), which shows that where the [[partition function]] of a [[superparticle]] (such as that at the end of the [[type II superstring]] ending on a [[D-brane]]) is given by the [[Todd genus]] in [[KU-theory]], so the partition function of the [[superstring]] itself (possibly itself being the boundary of the [[M2-brane]] ending on a [[O9 plane]] ([[heterotic string]]) or on an [[M5-brane]] (self-dual ``M-string'')) is given by the refined [[Witten genus]] in [[elliptic cohomology]]/[[tmf]] (thus yielding ``O9-plane charge'' and [[M5-brane charge]] in [[elliptic cohomology]] (\hyperlink{Sati10}{Sati 10})).

[[!include genera and partition functions - table]]

Finally, the [[M-theory]]-lift of [[type II string theory]] is (or is naturally identified as) \emph{[[F-theory]]}, which describes the [[axio-dilaton]] of the type II string [[vacua]] including the [[modular group|modular]] [[S-duality]]/[[U-duality]] acting on this as an [[elliptic fibration]] over [[spacetime]] (whose [[monodromy]] ``homology invariant'' is hence an $SL_2(\mathbb{Z})$-[[local system]]). It is hence natural to suspect that the combined [[worldsheet]]/[[target space]] $\mathbb{Z}_2$-equivariance of [[orientifold]] [[type II superstring]] backgrounds which is captured by [[KR-theory]] lifts in [[F-theory]] to some combined worldsheet/target space [[modular group]]-action. This is excatly what would be captured by modular [[equivariant cohomology|equivariant]] $tmf$ as indicated above (and as realized by theorem \ref{TheEquivariantConstruction} below), and and it is what is conjectured in (\hyperlink{KrizSati05}{Kriz-Sati 05, p.3}) (there the group $SL_2(\mathbb{Z}/2\mathbb{Z})$ is mentioned, the explicit construction \hyperlink{Definition}{below} does capture this but also includes equivariance under all other $SL_2(\mathbb{Z}/n\mathbb{Z})$).

So by extrapolation from the case of [[orientifolds]], where the [[target space]] $\mathbb{Z}_2$-[[involution]] (the ``[[real space]]''-structure) is accompanied by a $\mathbb{Z}_2$-action on the [[worldsheet]] (the [[worldsheet parity operator]]), this would suggest that in modular equivariant F-theory [[S-duality]] operations on the target space background would be accompanied by certain modular action on the worldsheet.

Mathematically this is precisely what happens in the equivariant version of [[tmf]] established by theorem \ref{TheEquivariantConstruction} below, which by prop. \ref{SystemOfModuliStacks} has the modular group action on the [[spectrum]] induced from the canonical action of the [[modular group]] on moduli of [[elliptic curves]] (hence: [[genus of a surface|genus]]-1 [[worldsheets]]).

Physically such an effect seems not to have been discussed much, but at least the following is knonw: first of all, under [[S-duality]] the [[worldsheet]] theory of the [[type IIB superstring]] certainly is affected: the superstring here is generally a [[(p,q)-string]], being a [[bound state]] of $p$ actual fundamental strings ([[F1-branes]]) with $q$ [[D1-branes]], and the [[S-duality]] [[modular group]] $SL_2(\mathbb{Z})$ does act canonically on these pairs of [[integers]]. Now, that this operation has to be accompanied with a worldsheet [[conformal transformation]] as the above mathematical story would suggest has been highlighted for instance in (\hyperlink{Bandos00}{Bandos 00}).

Generally, notice that the [[elliptic genus]] for [[type II superstrings]] lands in [[modular forms]] for the [[congruence subgroup]] $\Gamma_0(2) \hookrightarrow SL_2(\mathbb{Z})$ inside the full [[modular group]] (see at \href{Witten+genus#ModularityForTypeIISuperstring}{Witten genus -- Modularity -- For type II superstring}), the subgroup which fixes one of the NS-R [[spin structures]]. Therefore if this [[elliptic genus]] does have a ``topological'' (homotopy theoretic) lift to [[elliptic cohomology]] (as is known for the [[heterotic string]] via the [[string orientation of tmf]]) then not to plain [[TMF]], but to $TMF_0(2) \simeq \Gamma(\mathcal{M}_{ell}(2)_0,\mathcal{O}^{top})$ (this is implified for instance in \hyperlink{Stojanoska11}{Stojanoska 11, remark 6.2}), where $\mathcal{M}_{ell}(2)_0$ is the moduli stack of [[elliptic curves with level structure]] for $\Gamma_0(2)$, see at \emph{[[tmf0(2)]]}.

\hypertarget{Definition}{}\subsection*{{Definition and construction}}\label{Definition}

Write $\hat {\mathbb{Z}}$ for the [[profinite completion of the integers]].

Write

\begin{displaymath}
G \coloneqq GL_2(\hat {\mathbb{Z}})
\end{displaymath}
for the [[general linear group]] in [[dimension]] 2 with [[coefficients]] in $\hat{\mathbb{Z}}$.

Notice that this is the [[profinite group]] obtained as the [[limit]] over all general linear groups with coefficients in the [[cyclic groups]] (e.g.\hyperlink{Greicius09}{Greicius 09, (1.1)})

\begin{displaymath}
GL_2(\widehat{\mathbb{Z}})
  \coloneqq
  GL_2(\underset{\leftarrow}{\lim})_n GL_2(\mathbb{Z}/n\mathbb{Z})
  \simeq
  \underset{\leftarrow}{\lim}_n GL_2(\mathbb{Z}/n\mathbb{Z})
  \,.
\end{displaymath}
Write

\begin{displaymath}
p \;\colon\; GL_2(\mathbb{Z}) \longrightarrow G
\end{displaymath}
for the canonical projection from (the $\mathbb{Z}_2$-[[central extension]] $GL_2(\mathbb{Z}) \to PSL_2(\mathbb{Z})$ of) the [[modular group]].

For $n \in \mathbb{N}$ any [[natural number]] write

\begin{displaymath}
p_n \;\colon\; G \longrightarrow GL_2(\mathbb{Z}/n\mathbb{Z})
\end{displaymath}
for the corresponding [[projection]] to [[coefficients]] in the [[cyclic group]] of [[order of a group|order]] $n$.

Notice that for $\Gamma \hookrightarrow SL_2(\mathbb{Z}/n\mathbb{Z})$ then its [[preimage]] under $p_n$ is a ``profinite [[congruence subgroup]]''.

The following is a variant of the [[orbit category]] of $G = GL_2(\hat {\mathbb{Z}})$ which remembers the stage $n$ and consists only of orbits of the form of [[cosets]] by such [[congruence subgroups]].

\begin{defn}
\label{LevelledOrbitCategory}\hypertarget{LevelledOrbitCategory}{}
Write $\widetilde{Orb}_{GL_2(\hat{\mathbb{Z}})}$ for the [[category]] whose

\begin{itemize}%
\item [[objects]] are [[pairs]] $(n,\Gamma)$ with $n \in \mathbb{N}$ a [[natural number]] and $\Gamma \hookrightarrow GL_2(\mathbb{Z}/n\mathbb{Z})$ a [[subgroup]];


\item [[morphisms]] are given by

\begin{displaymath}
Hom\left(\left(n_1,\Gamma_1\right), \left(n_2,\Gamma_2\right)\right)
  \coloneqq
  \left\{
    \itexarray{
    Hom_{GL_2\left(\hat{\mathbb{Z}}\right)}
    \left(
    GL_2\left(\hat{\mathbb{Z}}\right)
    /
    p_{n_1}^{-1}\left(\Gamma_1\right), 
    \;
    GL_2\left(\hat{\mathbb{Z}}\right)
    /
    p_{n_2}^{-1}\left(\Gamma_2\right)
    \right)
      & if \; n'|n
      \\
      \emptyset & otherwise
    }
  \right.
  \,.
\end{displaymath}


\end{itemize}
\end{defn}
This is (\hyperlink{HillLawson13}{Hill-Lawson 13, def. 3.15}).

\begin{prop}
\label{SystemOfModuliStacks}\hypertarget{SystemOfModuliStacks}{}
The construction for each $(n,\Gamma) \in \widetilde{Orb}_{GL_2(\hat{\mathbb{Z}})}$ of the [[Deligne-Mumford compactification|compactified]] [[moduli stack]] $\mathcal{M}_{\overline{ell}}(\Gamma)$ over $Spec(\mathbb{Z}[\frac{1}{n}])$ of [[elliptic curves with level structure]] determined by $\Gamma$ (a [[modular curve]]) extends to a (lax) [[2-functor]]

\begin{displaymath}
\mathcal{M}_{\overline{ell}}(-)
  \;\colon\;
  \widetilde{Orb}_{GL_2(\hat{\mathbb{Z}})}
  \longrightarrow
  DMStack
\end{displaymath}
from the levelled orbit category of def. \ref{LevelledOrbitCategory} to the [[2-category]] of [[Deligne-Mumford stacks]], such that

\begin{enumerate}%
\item $\mathcal{M}_{\overline{ell}}(1,1) \simeq \mathcal{M}_{\overline{ell}}$ is the standard compactified [[moduli stack of elliptic curves]] over $Spec(\mathbb{Z})$


\item for each morphism $(n_1,\Gamma_1)\to (n_2,\Gamma_2)$ the induced morphism

\begin{displaymath}
\mathcal{M}_{\overline{ell}}(n_1,\Gamma_1)\to \mathcal{M}_{\overline{ell}}(n_2,\Gamma_2)
\end{displaymath}
is a [[log-etale morphism]] [[covering]];


\item for each $n$ and each [[normal subgroup]] inclusion $K \hookrightarrow \Gamma \hookrightarrow GL_2(\mathbb{Z}/n\mathbb{Z})$ the induced map exhibits the [[homotopy quotient]] [[projection]] by $\Gamma/K$

\begin{displaymath}
\mathcal{M}_{\overline{ell}}(n,K)\to \mathcal{M}_{\overline{ell}}(n,\Gamma)
  \simeq 
  \mathcal{M}_{\overline{ell}}(n,K)//(\Gamma/K)
  \,.
\end{displaymath}


\end{enumerate}
\end{prop}
This is (\hyperlink{HillLawson13}{Hill-Lawson 13, prop. 3.16, prop. 3.17}).

\begin{theorem}
\label{TheEquivariantConstruction}\hypertarget{TheEquivariantConstruction}{}
It is possible to extend the [[Goerss-Hopkins-Miller theorem]] to the [[Deligne-Mumford compactification|compactified]] [[moduli stacks]] of [[elliptic curves with level-n structure]] $\mathcal{M}_{\overline{ell}}[n]$ in prop. \ref{SystemOfModuliStacks}, such that taking [[global sections]] produces an [[(∞,1)-presheaf]] on the levelled [[orbit category]] of def. \ref{LevelledOrbitCategory} with values in [[E-∞ rings]]

\begin{displaymath}
Tmf
  \;\colon\;
  (\widetilde{Orb}_{SL_2(\hat{\mathbb{Z}})})^{op}
  \longrightarrow
  CRing_\infty
\end{displaymath}
which is such that

\begin{enumerate}%
\item for $n = 1$ (where $SL_2(\mathbb{Z}/\mathbb{Z}) = 1$ and hence $\Gamma = 1$ necessarily) one recovers

$Tmf(1,1)\simeq$ [[Tmf]];


\item the morphism induced by any morphism of the form $(n k ,P_k(\Gamma))\to (n,\Gamma)$ is $k$-[[localization of a spectrum|localization]];


\item for any $n \in \mathbb{N}$ and every [[normal subgroup]] $K \hookrightarrow \Gamma \hookrightarrow GL_2(\mathbb{Z}/n\mathbb{Z})$, we have the $(\Gamma/K)$-[[homotopy fixed points]] of $Tmf(n,\Gamma)$ (induced by action of $\Gamma/K$ on $\mathcal{M}_{\overline{ell}}(\Gamma)$ given by prop. \ref{SystemOfModuliStacks}):

\begin{displaymath}
Tmf(n,\Gamma) \stackrel{\simeq}{\longrightarrow} Tmf(K)^{(\Gamma/K)}      \,.
\end{displaymath}


\end{enumerate}
\end{theorem}
This is (\hyperlink{HillLawson13}{Hill-Lawson 13, theorem 9.1}).

\begin{remark}
\label{}\hypertarget{}{}
The system of spectra in theorem \ref{TheEquivariantConstruction} is essentially a [[spectrum with G-action]] (see there) for $G$ the ``[[profinite completion of the integers|profinite]] [[modular group]]'' $GL_2(\hat {\mathbb{Z}})$, except that the parameterization is not quite over the [[orbit category]] of this $G$, but just to the subcategory on objects which are [[coset spaces]] just by [[congruence subgroups]] and subject to that divisibility constraint on the $n$s, the ``levelling''. So $Tmf(-)$ defines a ``levelled'' kind of genuine $GL_2(\hat {\mathbb{Z}})$-[[equivariant cohomology]] version of [[Tmf]].

\end{remark}
The following proposition gives one way how the modular equivariance of [[tmf]] as in theorem \ref{TheEquivariantConstruction} restricts to the $\mathbb{Z}_2$-equivariance of [[KU]] (hence [[KR-theory]], which is known to be the precise form of [[type II string theory]] [[orientifolds]]).

First observe (see also (\hyperlink{MahowaldRezk09}{Mahowald-Rezk 09, section 2})) that for [[elliptic curve with level structure|level 3 structure]] we have [[congruence subgroups]]

\begin{displaymath}
\Gamma_1(3) \hookrightarrow \Gamma_0(3) \hookrightarrow GL(2,\mathbb{Z}/3\mathbb{Z})
\end{displaymath}
where the first inclusion is a [[normal subgroup]] of [[index of a subgroup|index]] 2.

\begin{prop}
\label{}\hypertarget{}{}
The inclusion of the [[nodal cubic|nodal elliptic curve]] with its $\mathbb{Z}/2\mathbb{Z}$-worth of [[automorphisms]] (the [[inversion involution]]) as the [[cusp]] of the [[Deligne-Mumford compactification|compactified]] [[moduli stack of elliptic curves]]

\begin{displaymath}
\itexarray{
    \ast &\to& \mathcal{M}_{\overline{ell}}(3,\Gamma_1)
    \\
    \downarrow^{\mathrlap{\mathbb{Z}/2\mathbb{Z}}}
    &&
    \downarrow^{\mathbb{Z}/2\mathbb{Z}}
    \\
    \ast//(\mathbb{Z}/2\mathbb{Z})
    &\hookrightarrow&
    \mathcal{M}_{\overline{ell}}(3,\Gamma_0)
  }
\end{displaymath}
over $Spec(\mathbb{Z}[\tfrac{1}{3}])$ yields under theorem \ref{TheEquivariantConstruction} a diagram of the form

\begin{displaymath}
\itexarray{
    ku[\frac{1}{3}] &\leftarrow& tmf_1(3)
    \\
    \uparrow && \uparrow
    \\
    ko[\frac{1}{3}] &\leftarrow& tmf_0(3)
  }
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{HillLawson13}{Hill-Lawson 13, theorem 9.3})

\begin{remark}
\label{}\hypertarget{}{}
The spectrum

\begin{displaymath}
Tmf_1(3) \coloneqq Tmf(3,\Gamma_1)
\end{displaymath}
(first considered in (\hyperlink{MahowaldRezk09}{Mahowald-Rezk 09}), see at \emph{[[congruence subgroup]]} for the notation) is [[complex oriented cohomology theory|complex oriented]] (\hyperlink{HillLawson13}{Hill-Lawson 13, p.5}) (in contrast to [[Tmf]] $\simeq Tmf(1,1)$). This is one more way in which the inclusion

\begin{displaymath}
\itexarray{
    Tmf_1(3)
    \\
    \uparrow
    \\
    Tmf
  }
\end{displaymath}
is analogous to the inclusion of [[KO]] into [[KU]]

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

\begin{itemize}%
\item [[real-oriented generalized cohomology theory]]


\item related but different: \emph{[[equivariant elliptic cohomology]]}



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

\begin{itemize}%
\item [[Mark Mahowald]] [[Charles Rezk]], \emph{Topological modular forms of level 3}, Pure Appl. Math. Quar. 5 (2009) 853-872 (\href{http://www.math.uiuc.edu/~rezk/tmf3-paper-final.pdf}{pdf})


\item [[Vesna Stojanoska]], \emph{Duality for Topological Modular Forms}, Documenta Math. 17 (2012), 271--311 (\href{http://arxiv.org/abs/1105.3968}{arXiv:1105.3968})


\item [[Tyler Lawson]], [[Niko Naumann]], \emph{Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2}, Int. Math. Res. Not. (2013) (\href{http://arxiv.org/abs/1203.1696}{arXiv:1203.1696})


\item [[Michael Hill]], [[Tyler Lawson]], \emph{Topological modular forms with level structure} (\href{http://arxiv.org/abs/1312.7394}{arXiv:1312.7394})


\item [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], \emph{Elliptic spectra, the Witten genus and the theorem of the cube}, Invent. Math. 146 (2001) 595--687 MR1869850


\item [[Matthew Ando]], [[Mike Hopkins]], [[Charles Rezk]], \emph{Multiplicative orientations of KO-theory and the spectrum of topological modular forms}, 2010 (\href{http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf}{pdf})


\item [[Igor Kriz]], [[Hisham Sati]], \emph{M Theory, Type IIA Superstrings, and Elliptic Cohomology}, Adv.Theor.Math.Phys. 8 (2004) 345-395 (\href{http://arxiv.org/abs/hep-th/0404013}{arXiv:hep-th/0404013})


\item [[Igor Kriz]], [[Hisham Sati]], \emph{Type IIB String Theory, S-Duality, and Generalized Cohomology}, Nucl.Phys. B715 (2005) 639-664 (\href{http://arxiv.org/abs/hep-th/0410293}{arXiv:hep-th/0410293})


\item [[Igor Kriz]], [[Hisham Sati]], \emph{Type II string theory and modularity}, JHEP 0508 (2005) 038 (\href{http://arxiv.org/abs/hep-th/0501060}{arXiv:hep-th/0501060})


\item [[Igor Kriz]], Hao Xing, \emph{On effective F-theory action in type IIA compactifications} (\href{http://arxiv.org/abs/hep-th/0511011}{arXiv:hep-th/0511011})


\item [[Hisham Sati]], \emph{The Elliptic curves in gauge theory, string theory, and cohomology}, JHEP 0603 (2006) 096 (\href{http://arxiv.org/abs/hep-th/0511087}{arXiv:hep-th/0511087})


\item [[Hisham Sati]], \emph{[[Geometric and topological structures related to M-branes]]} , part I, Proc. Symp. Pure Math. 81 (2010), 181-236 \href{http://arxiv.org/abs/1001.5020}{arXiv:1001.5020}


\item [[Igor Bandos]], \emph{Superembedding Approach and S-Duality. A unified description of superstring and super-D1-brane}, Nucl.Phys.B599:197-227,2001 (\href{http://arxiv.org/abs/hep-th/0008249}{arXiv:hep-th/0008249})


\item [[Aaron Greicius]], \emph{Elliptic curves with surjective adelic Galois representations} (\href{http://arxiv.org/abs/0901.2513}{arXiv:0901.2513})



\end{itemize}
[[!redirects modular equivariant tmf]]



\end{document}