modular equivariant elliptic cohomology

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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Elliptic cohomology
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[[!include elliptic cohomology -- contents]]
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#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## 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]] ([Mahowald-Rezk 09](#MahowaldRezk09), [Lawson-Naumann 12](#LawsonNaumann12)). 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]] ([Hill-Lawson 13, theorem 9.1](#HillLawson13)).

### General
  {#General}

The way this works is roughly indicated in the following table ([Lawson-Naumann 12](#LawsonNaumann12), [Hill-Lawson 13](#HillLawson13)):

[[!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 _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 [below](#MotivationFromStringTheory)) and in this context a relevance of modular equivariant elliptic cohomology theory has been conjectured in ([Kriz-Sati 05, p.3, p. 17, 18](#KrizSati05)).

### Motivation from string theory
 {#MotivationFromStringTheory}

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

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 ([Kriz-Sati 04a](#KrizSati04a), [Kriz-Sati 04b](#KrizSati04b) [Kriz-Sati 05](#KrizSati05), [Sati 05](#Sati05)) 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]] ([Ando-Hopkins-Strickland 01](#AndoHopkinsStrickland01), [Ando-Hopkins-Rezk 10](#AndoHopkinsRezk10)), 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]] ([Sati 10](#Sati10))).

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

Finally, the [[M-theory]]-lift of [[type II string theory]] is (or is naturally identified as) _[[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 ([Kriz-Sati 05, p.3](#KrizSati05)) (there the group $SL_2(\mathbb{Z}/2\mathbb{Z})$ is mentioned, the explicit construction [below](#Definition) 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 ([Bandos 00](#Bandos00)).

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 [Witten genus -- Modularity -- For type II superstring](Witten+genus#ModularityForTypeIISuperstring)), 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 [Stojanoska 11, remark 6.2](#Stojanoska11)), where $\mathcal{M}_{ell}(2)_0$ is the moduli stack of [[elliptic curves with level structure]] for $\Gamma_0(2)$, see at _[[tmf0(2)]]_.

## Definition and construction
 {#Definition}

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

Write

$$
  G \coloneqq GL_2(\hat {\mathbb{Z}})
$$

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.[Greicius 09, (1.1)](#Greicius09))

$$
  GL_2(\widehat{\mathbb{Z}})
  \coloneqq
  GL_2(\underset{\leftarrow}{\lim})_n GL_2(\mathbb{Z}/n\mathbb{Z})
  \simeq
  \underset{\leftarrow}{\lim}_n GL_2(\mathbb{Z}/n\mathbb{Z})
  \,.
$$

Write

$$
  p \;\colon\; GL_2(\mathbb{Z}) \longrightarrow G
$$

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

$$
  p_n \;\colon\; G \longrightarrow GL_2(\mathbb{Z}/n\mathbb{Z})
$$

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]].

+-- {: .num_defn #LevelledOrbitCategory}
###### Definition

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

* [[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]];

* [[morphisms]] are  given by

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

=--

This is ([Hill-Lawson 13, def. 3.15](#HillLawson13)).

+-- {: .num_prop #SystemOfModuliStacks}
###### Proposition

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]]

$$
  \mathcal{M}_{\overline{ell}}(-)
  \;\colon\;
  \widetilde{Orb}_{GL_2(\hat{\mathbb{Z}})}
  \longrightarrow
  DMStack
$$

from the levelled orbit category of def. \ref{LevelledOrbitCategory} to the [[2-category]] of [[Deligne-Mumford stacks]], such that

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

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

   $$
     \mathcal{M}_{\overline{ell}}(n_1,\Gamma_1)\to \mathcal{M}_{\overline{ell}}(n_2,\Gamma_2)
   $$

   is a [[log-etale morphism]] [[covering]];

1. 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 $\Gamma/K$

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

=--

This is ([Hill-Lawson 13, prop. 3.16, prop. 3.17](#HillLawson13)).

+-- {: .num_theorem #TheEquivariantConstruction}
###### Theorem

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]]

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

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

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

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

1. 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}):

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

=--

This is ([Hill-Lawson 13, theorem 9.1](#HillLawson13)).

+-- {: .num_remark}
###### Remark

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]].

=--

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 ([Mahowald-Rezk 09, section 2](#MahowaldRezk09))) that for [[elliptic curve with level structure|level 3 structure]] we have [[congruence subgroups]]

$$
  \Gamma_1(3) \hookrightarrow \Gamma_0(3) \hookrightarrow GL(2,\mathbb{Z}/3\mathbb{Z})
$$

where the first inclusion is a [[normal subgroup]] of [[index of a subgroup|index]] 2.

+-- {: .num_prop}
###### Proposition

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]]

$$
  \array{
    \ast &\to& \mathcal{M}_{\overline{ell}}(3,\Gamma_1)
    \\
    \downarrow^{\mathrlap{\mathbb{Z}/2\mathbb{Z}}}
    &&
    \downarrow^{\mathbb{Z}/2\mathbb{Z}}
    \\
    \ast//(\mathbb{Z}/2\mathbb{Z})
    &\hookrightarrow&
    \mathcal{M}_{\overline{ell}}(3,\Gamma_0)
  }
$$

over $Spec(\mathbb{Z}[\tfrac{1}{3}])$ yields under theorem \ref{TheEquivariantConstruction} a diagram of the form

$$
  \array{
    ku[\frac{1}{3}] &\leftarrow& tmf_1(3)
    \\
    \uparrow && \uparrow
    \\
    ko[\frac{1}{3}] &\leftarrow& tmf_0(3)
  }
  \,.
$$

=--

([Hill-Lawson 13, theorem 9.3](#HillLawson13))

+-- {: .num_remark}
###### Remark

The spectrum

$$
  Tmf_1(3) \coloneqq Tmf(3,\Gamma_1)
$$

(first considered in ([Mahowald-Rezk 09](#MahowaldRezk09)), see at _[[congruence subgroup]]_ for the notation) is [[complex oriented cohomology theory|complex oriented]] ([Hill-Lawson 13, p.5](#HillLawson13)) (in contrast to [[Tmf]] $\simeq Tmf(1,1)$). This is one more way in which the inclusion

$$
  \array{
    Tmf_1(3)
    \\
    \uparrow
    \\
    Tmf
  }
$$

is analogous to the inclusion of [[KO]] into [[KU]]

=--

## Related concepts

* [[real-oriented generalized cohomology theory]]

* related but different: _[[equivariant elliptic cohomology]]_

## References

* {#MahowaldRezk09} [[Mark Mahowald]], [[Charles Rezk]], _Topological modular forms of level 3_, Pure Appl. Math. Quar. 5 (2009) 853-872 ([pdf](http://www.math.uiuc.edu/~rezk/tmf3-paper-final.pdf))

* {#Stojanoska11} [[Vesna Stojanoska]], _Duality for Topological Modular Forms_,  Documenta Math. 17 (2012), 271--311 ([arXiv:1105.3968](http://arxiv.org/abs/1105.3968))

* {#LawsonNaumann12} [[Tyler Lawson]], [[Niko Naumann]], _Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2_, Int. Math.
Res. Not. (2013) ([arXiv:1203.1696](http://arxiv.org/abs/1203.1696))

* {#HillLawson13} [[Michael Hill]], [[Tyler Lawson]], _Topological modular forms with level structure_ ([arXiv:1312.7394](http://arxiv.org/abs/1312.7394))

* {#AndoHopkinsStrickland01} [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], _Elliptic spectra, the Witten genus and the theorem of the cube_, Invent. Math. 146 (2001) 595–687 MR1869850

* {#AndoHopkinsRezk10} [[Matthew Ando]], [[Mike Hopkins]], [[Charles Rezk]], _Multiplicative orientations of KO-theory and the spectrum of topological modular forms_, 2010 ([pdf](http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf))

* {#KrizSati04a} [[Igor Kriz]], [[Hisham Sati]], _M Theory, Type IIA Superstrings, and Elliptic Cohomology_, Adv. Theor. Math. Phys. 8 (2004) 345-395 ([arXiv:hep-th/0404013](http://arxiv.org/abs/hep-th/0404013))

* {#KrizSati04b} [[Igor Kriz]], [[Hisham Sati]], _Type IIB String Theory, S-Duality, and Generalized Cohomology_, Nucl.Phys. B715 (2005) 639-664 ([arXiv:hep-th/0410293](http://arxiv.org/abs/hep-th/0410293))

* {#KrizSati05} [[Igor Kriz]], [[Hisham Sati]], _Type II string theory and modularity_, 	JHEP 0508 (2005) 038 ([arXiv:hep-th/0501060](http://arxiv.org/abs/hep-th/0501060))

* [[Igor Kriz]], Hao Xing, _On effective F-theory action in type IIA compactifications_, Int. J. Mod. Phys. A22:1279-1300, 2007 ([arXiv:hep-th/0511011](http://arxiv.org/abs/hep-th/0511011))

* {#Sati05} [[Hisham Sati]], _The Elliptic curves in gauge theory, string theory, and cohomology_, JHEP 0603 (2006) 096 ([arXiv:hep-th/0511087](http://arxiv.org/abs/hep-th/0511087))

* {#Sati10} [[Hisham Sati]], _[[Geometric and topological structures related to M-branes]]_ , part I, Proc. Symp. Pure Math. 81 (2010), 181-236 [arXiv:1001.5020](http://arxiv.org/abs/1001.5020)

* {#Bandos00} [[Igor Bandos]], _Superembedding Approach and S-Duality. A unified description of superstring and super-D1-brane_, Nucl.Phys.B599:197-227,2001 ([arXiv:hep-th/0008249](http://arxiv.org/abs/hep-th/0008249))

* {#Greicius09} [[Aaron Greicius]], _Elliptic curves with surjective adelic Galois representations_ ([arXiv:0901.2513](http://arxiv.org/abs/0901.2513))

[[!redirects modular equivariant tmf]]


modular form

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Elliptic cohomology
+-- {: .hide}
[[!include elliptic cohomology -- contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _modular form_  is a [[holomorphic function]] on the [[upper half-plane]] that satisfies certain transformation property under the [[action]] of the [[modular group]]. Abstractly this transformation property makes the function a [[section]] of a certain [[line bundle]] on the [[quotient]] of the [[upper half plane]] that makes it the [[moduli stack of elliptic curves]] (over the [[complex numbers]]) or more generally a [[modular curve]].

Modular forms are also often called _classical [[automorphic forms]]_, see [below](). As automorphic forms, they are related to Galois representations as part of the [[Langlands program]].

Modular forms appear as the [[coefficient]] ring of the [[Witten genus]] on manifolds with rational [[string structure]]. For manifolds with actual [[string structure]] this refines to [[topological modular forms]], which are the [[homotopy groups]] of the [[spectrum]] [[tmf]].

## Definition

### In components

+-- {: .num_defn}
###### Definition

An **(integral) [[modular form]]** of weight $w$ is a [[holomorphic function]] on the [[upper half-plane]]

$$
  f : (\mathbb{R}^2)_+ \hookrightarrow \mathbb{C}
$$

(complex numbers with strictly positive imaginary part)

such that

1. if $A = \left( \array{a & b \\ c& d}\right) \in SL_2(\mathbb{Z})$ acting by $A : \tau \mapsto = \frac{a \tau + b }{c \tau + d}$ we have

   $$
     f(A(\tau)) = (c \tau + d)^w f(\tau)
   $$

   (notice that for $A = \left( \array{1 & 1 \\ 0& 1}\right)$ then  $f(\tau + 1) = f(\tau)$)

1. $f$ has at worst a pole at $\{0\}$ (for _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.

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

=--

More generally there is such a definition for $SL(2,\mathbb{Z})$ replaced by any other arithmetic subgroup $\Gamma \subset SL(2,\mathbb{R})$ (e.g. [Litt, def.1](#Litt)), giving modular forms on [[modular curves]].

### As functions on lattices

The reference for this definition  is [Calegari13](#Calegari13).

A modular form $f$ of weight $k$ over $\mathbb{C}$ is a function on lattices $\Lambda=\mathbb{Z}+\tau\mathbb{Z}\subset \mathbb{C}$ such that

1. $f(\mathbb{Z}+\tau\mathbb{Z})$ is holomorphic as a function of $\tau$

2. $f(\mu \Lambda)=\mu^{-k} f(\Lambda)$ for all $\mu\in\mathbb{C}^{\times}$

3. $f(\mathbb{Z}+\tau\mathbb{Z})$ is bounded as $\tau\to i\infty$.

If $f(\mathbb{Z}+\tau\mathbb{Z})\to 0$ as $\tau\to i\infty$, we say that $f$ is a [[cusp form]].

### As functions on elliptic curves together with a choice of differential

Again the reference for this definition  is [Calegari13](#Calegari13).

Since [[elliptic curves]] over $\mathbb{C}$ can be defined in terms of lattices, the previous definition can also be expressed as follows.

A modular form $f$ of weight $k$ over $\mathbb{C}$ is a function on pairs $(E,\omega)$ where $E$ is an elliptic curve and $\omega\in H^{0}(E,\Omega_{E}^{1})$, such that

$$f(E,\mu\omega)=\mu^{-k}f(E,\omega)$$

and such that $f(\mathbb{C}/\mathbb{Z}+\tau\mathbb{Z})$ is bounded as $\tau\to i\infty$.

### As sections of a line bundle over the moduli stack
 {#AsSections}

More abstractly, for $\mathcal{M}_{ell}$ the [[moduli stack of elliptic curves]] (or rather its [[Deligne-Mumford compactification]]) and $A \to \mathcal{M}_{ell}$ the corresponding universal bundle, write $\Omega^1_{A/S}$ for the [[line bundle]] of fiberwise [[Kähler differential forms]]. Write $e$ for the 0-[[section]] of this line bundle. Then

$$
  \omega \coloneqq e^\ast \Omega^1_{A/S}
$$

is a [[line bundle]] over the [[moduli stack of elliptic curves]]. A modular form of weight $k$  is a [[section]] of $\omega^{\otimes k}$. Using the Kodaira-Spencer isomorphism one can show that the space of [[cusp forms]] of weight $2$ are the differentials on the corresponding modular curve.

### For congruence subgroups
 {#ForCongruenceSubgroups}

Similarly one considers modular forms for [[congruence subgroups]] of the full [[modular group]], hence on the space of [[elliptic curves with level structure]].

### As automorphic forms
 {#AsAutomorphicForms}

Instead of regarding, as [above](#AsSections), modular forms as [[sections]] of a [[line bundle]] on a [[quotient]] of the [[upper half plane]], one may regard them alternatively as plain functions, but on the ([[Möbius group|projective]]) [[special linear group]] $SL(2,\mathbb{R})$. (e.g. [Martin 13, section 2](Martin13), [Litt, section 2](#Litt)).

As such these functions are then invariant under the [[action]] of the [[modular group|modular]] [[subgroup]] $SL(2,\mathbb{Z})\hookrightarrow SL(2,\mathbb{R})$ and hence are really functions on the [[coset space]] $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$ (for forms on moduli of elliptic curves) or more generally $\Gamma \backslash SL(2,\mathbb{Z})$ (for forms on more general [[modular curves]]).

This generalizes to the case of other [[congruence subgroups]] (as [above](ForCongruenceSubgroups)). Generally such functions on [[coset spaces]] like this are called _[[automorphic forms]]_. See there for more.

For the history of the terminology "modular form"/"automorphic form" see also [this MO comment](http://mathoverflow.net/a/124785/381).

## Properties

### Action by Hecke operators

Modular forms can be acted on by Hecke operators (related to [[Hecke correspondence]]). Viewing a modular form as a function on lattices, the most common kind of Hecke operator $T_{n}$ acts on a modular form $f$ as follows:

$$T_{n}f(\Lambda)=\sum_{\Lambda'}f(\Lambda')$$

where the sum runs over all lattices $\Lambda'$ which are index $n$ subgroups of $\Lambda$. A modular form which is a simultaneous eigenvector is also called a _Hecke eigenform_ or simply _eigenform_.

A modular form which vanishes at the cusps (equivalently, whose zeroth Fourier coefficient is equal to $0$) is called _cuspidal_, or a [[cusp form]]. A cusp form is _normalized_ if its first Fourier coefficient is equal to $1$. For normalized cuspidal eigenforms of integer weight, the eigenvalue of the Hecke operator $T_{n}$ is exactly the $n$-th Fourier coefficient.

Hecke operators were first developed by [[Louis Mordell]] to study the conjectures of [[Srinivasa Ramanujan]] on the tau function (the Fourier coefficients of the modular discriminant). They were later developed by (and named after) [[Erich Hecke]] who used them to study [[L-functions]].

### Analytic continuation of L-functions

Given a cusp form $f$ of weight $k$ with Fourier expansion

$$f(z)=\sum_{n=1}^{\infty}a_{n}e^{2\pi i n z}$$

we can associate to it its L-function $L(f,s)$, which is given by

$$L(f,s)=\sum_{n=1}^{\infty}a_{n}n^{-s}$$

Alternatively the L-function $L(f,s)$ can be defined via the [[Mellin transform]] of $f$:

$$\Lambda(f,s)=(2\pi)^{-s}\Gamma(s)L(f,s)=\int_{0}^{\infty}f(iy)y^{s}d^{\times}$$

The L-function $L(f,s)$ has an analytic continuation to all of $\mathbb{C}$.

This is important for instance to formulate conjectures such as the [[Birch and Swinnerton-Dyer conjecture]], which involves the special value of the [[Hasse-Weil L-function]] $L(E,s)$ of an elliptic curve at a value of $s$ where it is originally not defined (the series only converges for $s \gt 3/2$). But the [[modularity theorem]] tells us that this Hasse-Weil L-function $L(E,s)$ is equal to the L-function of some cusp form $f$, which we know has an analytic continuation to the entire complex plane.

### Relation to Galois representations

Modular forms can be used to construct [[Galois representations]]. For instance in the case of weight $2$
normalized eigenforms this is the Eichler-Shimura construction discussed in [DiamondShurman05, chapter 9](#DiamondShurman05). This was generalized to higher weight in [Deligne71](#Deligne71), and the case of weight $1$ comes from the work of [DeligneSerre74](#DeligneSerre74).

The problem of whether a Galois representation comes from some modular form is part of the problem of _modularity_. Known cases include the [[modularity theorem]] of [TaylorWiles95](#TaylorWiles95) and [BreuilConradDiamondTaylor2001](#BreuilConradDiamondTaylor2001). The former is notably an important part of [[Wiles' proof of Fermat's last theorem]].

On a more general level, as modular forms are a special case of automorphic forms, their relation to Galois representations is part of the [[Langlands program]].

### Relation of $MF_0(2)$ to the elliptic genus

Write $\Gamma_0(2) \hookrightarrow SL_2(\mathbb{Z})$ for the [[subgroup]] of the [[modular group]] on those elements $\left(\array{a & b \\ c & d}\right)$ for which $c = 0\, mod\, 2$.

A modular function for $\Gamma_0(2)$ is a [[meromorphic function]] on the [[upper half plane]] which transforms as a modular form under the action of $\Gamma_0(2) \hookrightarrow SL_2(\mathbb{Z})$. Write $MF_\bullet(\Gamma_0(2))$ for the ring of these.

There is a natural isomorphism

$$
  MF_\bullet(\Gamma_0(2)) \simeq \mathbb{C}[\epsilon, \delta]
$$

(see at [[elliptic genus]]) for the notation.

([Landweber-Ravenel-Stong 93, theorem 1.5 and sections 5.3, 5.8](#LandweberRavenelStong93))

### Relation to elliptic cohomology

For $E$ the [[elliptic cohomology theory]] associated to the [[elliptic curve]] $C$, then

$$
  E_{2n} \simeq \omega(C)^{\otimes n}
$$

(where $\omega$ is the line bundle from [above ](#AsSections))

and

$$
  E_{2n+1}\simeq 0
  \,.
$$

## Examples

* [[Eisenstein series]]

* [[j-invariant]]

* [[Weierstrass sigma-function]]

* [[Dedekind eta function]]

## Related concepts

* [[elliptic curve]], [[moduli stack of elliptic curves]]

* [[Hecke correspondence]]

* [[Jacobi form]]

* [[elliptic genus]], [[Witten genus]]

* [[topological modular form]], [[tmf]]

* [[p-adic modular form]]

* generalization to functions on moduli of higher dimensional [[abelian varieties]]: [[Hilbert modular form]], [[Siegel modular forms]]

* [[modularity theorem]]

* [[Rankin-Cohen bracket]]

## References

A basic and handy reference is

* [[Pierre Deligne]], _Courbes elliptiques: formulaire d'apres J. Tate_, In _Modular functions of one variable_, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pages 53{73. Lecture Notes in Math., Vol. 476. Springer, Berlin, 1975 ([web](http://modular.math.washington.edu/Tables/antwerp/deligne/))

Textbook accounts:

* N. Koblitz, _Introduction to Elliptic Curves and Modular Forms_, Graduate Texts in Math. **97** Springer (1984, 1993)

* {#HirzebruchGergerJung92} [[Friedrich Hirzebruch]], Thomas Berger, Rainer Jung, Appendix I of: *Manifolds and Modular Forms*, Aspects of Mathematics **20**, Viehweg (1992), Springer (1994) [[doi:10.1007/978-3-663-10726-2](https://doi.org/10.1007/978-3-663-10726-2), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/hirzjung.pdf)]

* {#DiamondShurman05} [[Fred Diamond]], [[Jerry Shurman]], *A First Course in Modular Forms*, GTM **228**, Springer (2005) $[$[doi:10.1007/978-0-387-27226-9](https://doi.org/10.1007/978-0-387-27226-9), ISBN-13: 978-0387232294$]$

Lecture notes and reviews:

* {#Hain08} Richard Hain, section 4 of _Lectures on Moduli Spaces of Elliptic Curves_ ([arXiv:0812.1803](http://arxiv.org/abs/0812.1803))

* [[Charles Rezk]], section 10 of [pdf](http://www.math.uiuc.edu/~rezk/512-spr2001-notes.pdf)

* {#Litt} [[Daniel Litt]], _Automorphic forms notes, part I_ ([pdf](http://math.stanford.edu/~dlitt/Talks/automorphicformspt1.pdf))

* Jan Hendrik Bruinier, Gerard van der Geer, [[Günter Harder]], Don Zagier, _The 1-2-3 of modular forms_, Lectures at a Summer School 2004 in Nordfjordeid, Norway; Universitext, Springer 2008.

* [[Nora Ganter]], _[Topological modular forms literature list](http://www.ms.unimelb.edu.au/~nganter/talbot/index.html)_

* {#Martin13} Kimball Martin, _A brief overview of modular and automorphic forms_, 2013 [pdf](http://www2.math.ou.edu/~kmartin/papers/mfs.pdf)

* {#Calegari13} Frank Calegari, _Congruences between modular forms_, 2013 [pdf](https://swc-math.github.io/aws/2013/2013CalegariLectureNotes.pdf)

* Wikipedia, _[Modular form](http://en.wikipedia.org/wiki/Modular_form)_

Introduction in relation to [[string theory]]:

* [[Eric D'Hoker]], [[Justin Kaidi]], *Lectures on modular forms and strings* [[arXiv:2208.07242](https://arxiv.org/abs/2208.07242)]

The work of Erich Hecke:

* [[Erich Hecke]], *Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. I.*, Mathematische Annalen **114** (1937)  1–28 [[doi:10.1007/BF01594160](https://doi.org/10.1007/BF01594160)]

* [[Erich Hecke]], *Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. II.*, Mathematische Annalen (in German), **114** (1937) 316–351 [[doi:10.1007/BF01594180](https://doi.org/10.1007/BF01594180)]

The construction of Galois representations associated to normalized eigenforms of weight $\geq 2$ comes from

* {#Deligne71}[[Pierre Deligne]], _Formes modulaires et représentations l-adiques_, Sém. Bourbaki 1968/69, exp. 355, Springer Lecture Notes 179 (1971), 139–172

The construction of Galois representations from weight $1$ modular forms comes from

* {#DeligneSerre74}[[Pierre Deligne]] and [[Jean-Pierre Serre]] _Formes modulaires de poids 1_ Ann. Sci. Ecole Norm. Sup. (4), 7:507–530 (1975), 1974.

References related to the modularity theorem are

* {#TaylorWiles95} [[Richard Taylor]], [[Andrew Wiles]], *Ring-Theoretic Properties of Certain Hecke Algebras*, Annals of Mathematics Second Series **141** 3 (1995) 553-572 $[$[doi:10.2307/2118560 ](https://doi.org/10.2307/2118560 )$]$

* {#BreuilConradDiamondTaylor2001} [[Christophe Breuil]], [[Brian Conrad]], [[Fred Diamond]], [[Richard Taylor]], *On the modularity of elliptic curves over $\mathbf{Q}$: Wild 3-adic exercises*, Journal of the American Mathematical Society **14** 4 (2001) 843–939 $[$[doi:10.1090/S0894-0347-01-00370-8](https://doi.org/10.1090/S0894-0347-01-00370-8)$]$

Original discussion in the context of [[elliptic genera]] and [[elliptic cohomology]] includes

* {#LandweberRavenelStong93} [[Peter Landweber]], [[Douglas Ravenel]], [[Robert Stong]], _Periodic cohomology theories defined by elliptic curves_, in [[Haynes Miller]] et. al. (eds.), _The Cech centennial: A conference on homotopy theory_, June 1993, AMS (1995) ([pdf](http://www.math.sciences.univ-nantes.fr/~hossein/GdT-Elliptique/Landweber-Ravenel-Stong.pdf))

* {#Landweber88} [[Peter Landweber]], *Elliptic Cohomology and Modular Forms*, in: _Elliptic Curves and Modular Forms in Algebraic Topology_, Lecture Notes in Mathematics **1326** (1988) 55-68 [[doi:10.1007/BFb0078038](https://doi.org/10.1007/BFb0078038)]

Reviewed in:

* [[Kefeng Liu]], _Modular forms and topology_, 1996 ([citeseer](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.130.9779))

[[!redirects modular forms]]

[[!redirects integral modular form]]
[[!redirects weak modular form]]
[[!redirects modular function]]
[[!redirects integral modular function]]
[[!redirects weak integral modular form]]

[[!redirects integral modular forms]]
[[!redirects weak modular forms]]
[[!redirects modular functions]]
[[!redirects integral modular functions]]
[[!redirects weak integral modular forms]]


modular forms from partition functions

<div class="rightHandSide toc">

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

***

[[!include supergeometry - contents]]

</div>

+-- {: .standout}

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 fact and its derivation that the [[partition function]] of a [[(2,1)-dimensional Euclidean field theories]] is a [[modular form]].

=--

Previous:

* [[Axiomatic field theories and their motivation from topology]]

* [[(1,1)-dimensional Euclidean field theories and K-theory]]

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

* [[bordism categories following Stolz-Teichner]]

* [[(2,1)-dimensional Euclidean field theories]]

#Contents#

* automatic table of contents goes here
{:toc}

#Idea#

As described at  [[(2,1)-dimensional Euclidean field theories and tmf]], the idea is that [[(2,1)-dimensional Euclidean field theories]] are a [[geometric models for elliptic cohomology|geometric model for tmf]] [[cohomology theory]].

While there is no complete proof of this so far, here
we discuss the construction and proof -- due to [[Stephan Stolz]] and [[Peter Teichner]]  --
for the situation 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.

#  first top-level headline#

...

## first second-level headline ##

...

## second second-level headline ##

...

#  second top-level headline#

...


modular functor

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Functorial quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The local data for a [[CFT]] in [[dimension]] $d$ allows to assign to each $d$-[[dimension|dimensional]] [[cobordism]] $\Sigma$ a [[vector space]] of "possible [[correlators]]": those functions on the space of [[conformal structures]] on $\Sigma$ that have the correct behaviour (satisfy the conformal [[Ward identities]]) to qualify as the (chiral) [[correlator]] of a CFT. This is called a space of _[[conformal blocks]]_ $Bl(\Sigma)$. This assignment is [[functor|functorial]] under [[diffeomorphism]]. The corresponding functor is called a **modular functor**. ([Segal 89](#Segal89), [Kriz 03](#Kriz03), [Segal 04, def. 5.1](#Segal04)).

To get an actual collection of correlators one has to choose from each space of conformal blocks $Bl(\Sigma)$ an element such that these choices glue under composition of [[cobordism]]: such that they solve the [[sewing constraints]], see for instance at _[[FRS-theorem on rational 2d CFT]]_.

Dually, under a [[holographic principle]] such as [[AdS3-CFT2 and CS-WZW correspondence|CS3/WZW2]] the space of [[conformal blocks]] on $\Sigma$ is equivalently the [[space of quantum states]] of the [[TQFT]] on $\Sigma$. See at _[[quantization of 3d Chern-Simons theory]]_ for more on this.

## Definition

+-- {: .num_defn #CategoryOfRiemannSurfaces}
###### Definition

For $\Phi$ any [[finite set]] ("of labels") write $\mathcal{S}_{\Phi}$ for the [[category]] whose [[objects]] are [[Riemann surfaces]] with [[boundary]] [[circles]] labeled by elements of $\Phi$, and whose [[morphisms]] are [[holomorphic maps]] $X \to X_{sewed}$, where $X_{sewed}$ is obtained from $X$ by [[sewing constraint|sewing]] along boundary circles carrying the same labels.

=--

([Segal 04, section 4, section 5](#Segal04))

This category, being essentially the total [[Riemann moduli space]] is naturally a [[complex analytic stack]].

For $\Sigma$ a [[topological manifold|topological]] [[surface]] write $\mathcal{S}_\Sigma$ for the component of $\mathcal{S}$ on Riemann surfaces whose underlying topological surface is $\Sigma$.

+-- {: .num_remark}
###### Remark

If $\Sigma$ has at least one hole (boundary component), then the [[fundamental group]] $\pi_1(\mathcal{S}_\Sigma)$ is the [[mapping class group]] of $\Sigma$.

=--

+-- {: .num_defn #ModularFunctor}
###### Definition

A _modular functor_ is a holomorphic functor

$$
  E \colon \mathcal{S}_\Phi \longrightarrow sVect
$$

(i.e. a morphism of [[complex analytic stacks]] from the [[Riemann moduli space]] to the stack of ([[finite number|finite]]-[[rank of a vector bundle|rank]]) [[holomorphic vector bundles]], in general with [[super vector space]]-[[fibers]])

such that

1. $E$ is [[strong monoidal functor|strong monoidal]]: $E(X \coprod Y) \simeq E(X)\otimes E(Y)$;

1. $E$ respects sewing: if $X_\phi$ is obtained from $X$ by cutting along a circle and giving the same label $\phi \in \Phi$ to both resulting boundaries, then the [[natural transformation]]

   $$
     \underset{\phi \in \Phi}{\oplus} E(X_\phi) \longrightarrow E(X)
   $$

   is a [[natural isomorphism]].

1. normalization: for $X = S^2$ the [[Riemann sphere]] we have $E(S^1) = 1$ (the[[tensor unit]] vector space).

=--

([Segal 04, def.  5.1](#Segal04))

## Properties

### Relation to genus-zero conformal blocks
 {#RelationToGenusZeroConformalBlocks}

For a [[rational CFT|rational]] [[2d CFT]] the modular functor is fully determined by the [[conformal blocks]] on the [[Riemann sphere]] (the *genus=0 conformal blocks*) -- this is proven in [Andersen & Ueno 2012](#AndersenUeno12).

Closely related is the statement that the [[braided monoidal category|braided monoidal structure]] on the [[modular tensor category|modular]] [[representation category]] of the corresponding [[vertex operator algebra]] is fully determined by the genus=0 conformal blocks, a statement that seems to be [[folklore]] (highlighted in [EGNO 15, p. 266](modular+tensor+category#EGNO15), [Runkel, Sec. 4.3](modular+tensor+category#Runkel)).

### Central charge and Central extensions

Any modular functor defines a [[central extension]] of the [[semigroup]] of conformal [[annuli]].

These correspond precisely to [[group extensions]] of $Diff^+(S^1)$ by $\mathbb{C}^\times$.

([Segal 04, prop. 5.6](#Segal04))

These in turn are classified by $(c,h) \in \mathbb{C} \times \mathbb{C}/\mathbb{Z}$.

([Segal 04, prop. 5.8](#Segal04))

+-- {: .num_defn #CentralCharge}
###### Definition

Here in terms of standard [[2d CFT]] terminology

*$c$ is the _central charge_

*$h$ is the [[eigenvalues]] of $L_0$.

=--

### The Knizhnik–Zamolodchikov-Hitchin connection

+-- {: .num_prop #TheProjFlatConnection}
###### Proposition

Given a modular functor $E$ as in def. \ref{ModularFunctor} and given a non-[[closed manifold|closed]] [[topological manifold|topological]] labelled [[surface]] $\Sigma$ with $E_\Sigma \to \mathcal{S}_\Sigma$ the resulting [[vector bundle]], then this bundle carries a canonical [[projectively flat connection]] $\nabla_\Sigma$ compatible with the [[sewing constraint|sewing operation]] of def. \ref{CategoryOfRiemannSurfaces}.

=--

([Segal 04, prop. 5.4](#Segal04), see also at [[Knizhnik-Zamolodchikov equation]])

+-- {: .num_remark}
###### Remark

When thinking of the modular functor $E$ as the functor of [[conformal blocks]] of a [[2d CFT]] then the projectively flat connection of prop. \ref{TheProjFlatConnection} would often be called the _[[Knizhnik-Zamolodchikov connection]]_. Thining of $E$ dually as the functor assigning [[spaces of quantum states]] of [[Chern-Simons theory]] then it would typically be called the [[Hitchin connection]]. (see also [Segal 04, p. 44, p. 84](#Segal04)).

=--

+-- {: .num_pro #FlatConnectionForVanishingCentralCharge}
###### Proposition

The connection of prop. \ref{TheProjFlatConnection} is a genuine [[flat connection]] (not projective) precisely if the central charge, \ref{CentralCharge}, vanishes.

=--

([Segal 04, below prop. 5.4](#Segal04))

### The Verlinde fusion algebra

+-- {: .num_defn}
###### Definition

For $\phi,\chi,\psi \in \Phi$ three labels, write $P_{\phi,\chi,\psi}$ for the three-holed sphere ("pair of pants", "trinion") with inner circles labeled by $\phi$ and $\chi$ and outer circle labeled by $\psi$.

For $E$ a modular functor as in def. \ref{ModularFunctor}, write

$$
  n_{\phi, \chi,\psi} \coloneqq dim(E(P_{\phi,\chi,\psi}))
$$

for the [[dimension]] of the vector space that it assigns to this surface.

Then the [[free abelian group]] $\mathbb{Z}[\Phi]$ on the set of labels inherits the structure of an [[associative algebra]] via

$$
  (\phi,\chi) \mapsto \underset{\psi}{\sum}  n_{\phi,\chi, \psi} \psi
  \,.
$$

The _[[Verlinde algebra]]_.

=--

([Segal 04, section 5, p. 36-37](#Segal04))

### Deprojectivization, Cancelling of central charge, topological modular functor
 {#TopologicalLift}

By prop. \ref{FlatConnectionForVanishingCentralCharge} and prop. \ref{CentralChargeOfDeterminantLine}, if $E$ is a modular functor of central charge $c$ then the [[tensor product]]

$$
  \tilde E \coloneqq E \otimes Det^{\otimes c/2}
$$

with a possibly fractional power of the [[determinant line bundle]], def. \ref{PowerOfDeterminantLine}, produces a modular functor with vanishing central charge.

To make sense of this however one needs to consistently define the fractional power. For that one needs to pass to surfaces equipped with a bit more structure.

+-- {: .num_defn #RiggedSurfaces}
###### Definition

The category of _rigged surfaces_ $\hat {\mathcal{S}}_\Phi$ is the [[central extension]] of that of [[smooth manifold]] [[surfaces]] such that for [[genus of a surface|genus]] $\gt 1$ it gives the universal central extension of the [[diffeomorphism group]].

For instance the category of surfaces equipped with a choice of [[universal covering space]] of the [[circle group]]-[[principal bundle]] underlying the [[determinant line bundle]] over $\mathcal{S}_\Sigma$.

=--

([Segal 04, def. (5.10) and following](#Segal04), also [Bakalov-Kirillov, def. 5.7.5](#BakalovKirillov))

+-- {: .num_prop}
###### Proposition

Given a [[modular functor]] $E$, def. \ref{ModularFunctor} of central charge $c$, def. \ref{CentralCharge}, then the tensor product $\tilde E \coloneqq E \otimes Det^{\otimes c/2}$ is well defined on the category $\hat S_{\phi}$ of rigged surfaces, def. \ref{RiggedSurfaces}.

=--

Of course if one has an extension of the diffeomorphism group by a multiple of the universal extension in def. \ref{RiggedSurfaces}, then this still trivializes the conformal anomaly for all modular functors whose central charge is a corrsponding multiple. In particular:

+-- {: .num_prop #SurfacesWithAtiyah2Framing}
###### Proposition

The category of smooth surfaces equipped with "[[Atiyah 2-framing]]" (hence with a trivialization of the spin lift of the double of their tangent bundle) provides an extension of the diffeomorphic group of level 12.

=--

([Segal 04, p. 46](#Segal04))

+-- {: .num_remark}
###### Remark

There is a natural functor from smooth surfaces $\Sigma$ equipped with [[3-framing]] (trivialization of $T \Sigma\oplus \underline{\mathbb{R}}$) to that equipped with [[Atiyah 2-framing]] in prop. \ref{SurfacesWithAtiyah2Framing}.

=--

> thanks to [[Chris Schommer-Pries]] for highlighting this point.

## Examples

### Powers of the determinant line

+-- {: .num_defn #PowerOfDeterminantLine}
###### Definition

For $n \in \mathbb{Z}$ let $E = Det^{\otimes n}$ be the functor which sends a [[Riemann surface]] to the $c$th power of its [[determinant line]] (i.e. that of its [[Laplace operator]]).

=--

Super-line, see ([Kriz-Lai 13](#KrizLai13))...

+-- {: .num_defn }
###### Proposition

The determinant lines of def. \ref{PowerOfDeterminantLine} constitute precisely the modular functors, def. \ref{ModularFunctor}, for which $dim(E(X)) = 1$ for all $X$.

=--

([Segal 04, corollary (5.17)](#Segal04))

+-- {: .num_prop #CentralChargeOfDeterminantLine}
###### Proposition

The central charge, def. \ref{CentralCharge}, of the determinant line $E = Det$, def. \ref{PowerOfDeterminantLine}, is

$$
 (c,h ) = (-2,0)
  \,.
$$

=--

([Segal 04, p.43](#Segal04))

## Related concepts

* [[quantization of Chern-Simons theory]]

## Properties

### Relation to equivariant elliptic cohomology / equivariant $tmf$

The modular functor for $G$-[[Chern-Simons theory]] restricted to [[genus of a surface|genus-1 surfaces]] ([[elliptic curves]]) is essentially what is encoded in the universal $G$-[[equivariant elliptic cohomology]] (equivariant [[tmf]]). In fact equivariant elliptic cohomology remembers also the [[local prequantum field theory|pre-quantum]] incarnation of the modular functor as a systems of [[prequantum line bundles]] over Chern-Simons [[phase spaces]] (which are [[moduli stacks of flat connections]]) and remembers the [[quantization]]-process from there to the actual [[space of quantum states]] by forming holomorphic [[sections]]. See at _[equivariant elliptic cohomology -- Idea -- Interpretation in Quantum field theory](equivariant+elliptic+cohomology#InterpretationInQuantumFieldTheory)_ for more on this.

## References

Original formulations:

* [[Graeme Segal]], *Two-dimensional conformal field theories and modular functors*, in: *IXth International Congress on Mathematical Physics*, Swansea 1988, Adam Hilger, Bristol and New York (1989) 22–37 [[pdf](https://dmitripavlov.org/scans/segal-two-dimensional-conformal-field-theories-and-modular-functors.pdf), [[Segal-2dCFTandModularFunctors.pdf:file]]]

* [[Graeme Segal]], *The definition of conformal field theory*, in: [[Ulrike Tillmann]] (ed.), *Topology, geometry and quantum field theory* , London Math. Soc. Lect. Note Ser. **308**, Cambridge University Press (2004) 421-577 [[doi:10.1017/CBO9780511526398.019](https://doi.org/10.1017/CBO9780511526398.019), [pdf](https://people.maths.ox.ac.uk/segalg/0521540496txt.pdf), [[SegalDefinitionCFT.pdf:file]]]

* {#Kriz03} [[Igor Kriz]], _On spin and modularity in conformal field theory_, Ann. Sci. ANS (4) 36 (2003), no. 1, 57112 ([numdam:ASENS_2003_4_36_1_57_0](http://www.numdam.org/item?id=ASENS_2003_4_36_1_57_0))

* {#KrizLai13} [[Igor Kriz]], Luhang Lai, _On the definition and K-theory realization of a modular functor_, ([arxiv/1310.5174](http://arxiv.org/abs/1310.5174)).

Lectures and reviews:

* {#BakalovKirillov} [[Bojko Bakalov]], [[Alexander Kirillov]], chapter 5 of _Lectures on tensor categories and modular functor_ ([web](http://www.math.sunysb.edu/~kirillov/tensor/tensor.html), [[BakalovKirillovChapter5.pdf:file]])

* {#Gawedzki99} [[Krzysztof Gawedzki]], section 5.6 of _Conformal field theory: a case study_ ([arXiv:hep-th/9904145](http://arxiv.org/abs/hep-th/9904145))

* {#Looijenga10} [[Eduard Looijenga]], _From WZW models to Modular Functors_ ([arXiv:1009.2245](http://arxiv.org/abs/1009.2245))

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Simon Wood]], [[Yang Yang]], p. 11 of: *Algebraic structures in two-dimensional conformal field theory*, Encyclopedia of Mathematical Physics [[arXiv:2305.02773](https://arxiv.org/abs/2305.02773)]

Proof that the values of a modular functor at [[genus of a surface|genus]]=0 (ie. the [[conformal blocks]] on the punctured [[Riemann sphere]]) determine the full modular functor:

* {#AndersenUeno12} [[Jørgen Ellegaard Andersen]], [[Kenji Ueno]], *Modular functors are determined by their genus zero data*, Quantum Topology **3** 3/4 (2012) 255-291 $[$[doi:10.4171/QT/29](https://doi.org/10.4171/qt/29)$]$

Discussion in the context of [[(2,1)-dimensional Euclidean field theories and tmf]] is in

* {#StolzTeichner11} [[Stefan Stolz]], [[Peter Teichner]], section 5.2 of _Supersymmetric field theories and generalized cohomology_, in [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[Mathematical foundations of Quantum field theory and String theory](http://ncatlab.org/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory#ContributionStolzTeichner)_, Proceedings of Symposia in Pure Mathematics, Volume 83, AMS (2011)

Discussion in the context of the [[cobordism hypothesis]]:

* {#FiorenzaValentino14} [[Domenico Fiorenza]], [[Alessandro Valentino]], *Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors*, Commun. Math. Phys. **338** (2015) 1043–1074  $[$[arXiv:1409.5723](http://arxiv.org/abs/1409.5723), [doi:10.1007/s00220-015-2371-3](https://doi.org/10.1007/s00220-015-2371-3)$]$

Constructing [[modular functors]] from [[pivotal category|pivotal]] [[bicategories]] using [[string net models]]:

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Yang Yang]], *String-net models for pivotal bicategories* [[arXiv:2302.01468](https://arxiv.org/abs/2302.01468)]

[[!redirects modular functors]]


modular ideal

## Definition

A left (right) ideal $I$ in a [[nonunital ring|(non necessarily unital) ring]] $R$ is __modular__ if  $\exists e\in R$ such that $\forall r \in R$ $r e - r\in I$ (respectively $e r - r\in I$).

## Examples

* Every left ideal in a unital ring is modular, since we can take $e = 1$.

* Every left ideal containing a modular ideal is modular. In particular, any maximal ideal containing given proper modular ideal is modular.

## Properties

* For a two-sided ideal $I$ in $R$ the quotient $R/I$ is unital iff $I$ is modular as both a left and a right ideal.

* The [[Jacobson ideal]] of a ring coincides with the intersection of all maximal modular ideals.

## References

* EoM: [modular ideal](https://encyclopediaofmath.org/wiki/Modular_ideal)

* Wikipedia: [Modular ideal](https://en.wikipedia.org/wiki/Modular_ideal)

category: algebra


modular integral lattice

> This entry is about a notion in [[quadratic form]]-theory. For an unrelated notion of a similar name in _[[order theory]]/[[logic]]_ see at _[[modular lattice]]_.

***

#Contents#
* table of contents
{:toc}

## Idea

A _modular integral lattice_ is an [[integral lattice]] which is similar to its dual.

## Related concepts

* [[unimodular integral lattice]]

## References

* Gabriele Nebe, _[Modular lattices](http://www2.research.att.com/~njas/lattices/modular.html)_

[[!redirects modular integral lattices]]


modular invariance

## Idea

Modular invariance is one of the consistency conditions on the data specifying a [[2d CFT]]. It says that the [[partition function]] associated to the [[torus]] is invariant under [[modular transformations]].

## Related concepts

* [[Cardy condition]]

## References

* Wikipedia, _[Modular invariance](http://en.wikipedia.org/wiki/Modular_invariance)_


modular lattice

> This is about a notion in _[[order theory]]/[[logic]]_. For an unrelated notion of a similar name in [[group theory]]/[[quadratic form]]-theory see at _[[modular integral lattice]]_.

***

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## Idea

A _modular lattice_ is a lattice where "opposite sides" of a "diamond" formed by four points $x \wedge y$, $x$, $y$, $x \vee y$ are "congruent".

## Definition

A modular lattice is a [[lattice]] which satisfies a modular law, which we introduce after a few preliminaries.

In any lattice $L$, given two elements $x, y \in L$ with $x \leq y$, let $[x, y]$ denote the [[interval]] $\{z \colon x \leq z \leq y\}$. Then, given any two elements $a, b \in L$, there is an [[adjunction|adjoint pair]]

$$a \vee (-) \colon [a \wedge b, b] \leftrightarrows [a, a \vee b] \colon (-) \wedge b$$

where $a \vee (-)$ is left adjoint to $(-) \wedge b$. Indeed, for any $w \in [a \wedge b, b]$, we have a [[adjunction|unit]]

$$w \leq (a \vee w) \wedge b,$$

whereas for any $z \in [a, a \vee b]$, we have dually a [[adjunction|counit]]

$$a \vee (z \wedge b) \leq z.$$

+-- {: .num_defn}
###### Definition
A lattice $L$ is **modular** if for any $a, b \in L$, the adjoint pair

$$a \vee (-) \dashv (-) \wedge b \colon [a, a \vee b] \to [a \wedge b, b]$$

is an [[adjoint equivalence]].
=--

This is perhaps the most memorable definition for a category theorist: it is a precise expression of the slogan given in the Idea section.

It is immediate that the concept of modular lattice is **[[duality|self-dual]]**, i.e., if $L$ is modular, then so is $L^{op}$.

## Alternative formulations

In the lattice-theoretic literature, modularity is usually formulated somewhat differently. Here are three alternative conditions on a lattice, all equivalent to that of Definition 1.

1. The **modular law** is the universal [[Horn theory|Horn sentence]]
$$a \leq b \vdash (a \vee z) \wedge b = a \vee (z \wedge b).$$

1. The **modular identity** is the universal equation
$$(a \vee z) \wedge (a \vee b) = a \vee (z \wedge (a \vee b))$$

1. "**Freyd's modular law**" (for lack of  better term; see [[allegory]]) is the universal inequality
$$(a \vee z) \wedge b \leq a \vee (z \wedge (a \vee b)).$$

### Proofs of equivalence

#### Derivation of modular identity

To see that the modular identity follows from Definition 1, observe that for any $z \in L$ we have

$$a \leq (a \vee z) \wedge (a \vee b) \leq a \vee b$$

Let $w = (a \vee z) \wedge (a \vee b)$. Under $(-) \wedge b \colon [a, a \vee b] \to [a \wedge b, b]$, this element $w$ is sent to

$$(a \vee z) \wedge (a \vee b) \wedge b = (a \vee z) \wedge b.$$

Under Definition 1, this last element is sent back to $w$ by $a \vee (-)$. Therefore we have

$$(a \vee z) \wedge (a \vee b) = w = a \vee ((a \vee z) \wedge b)$$

and since this is true for all $a, b, z$, we can interchange $z$ and $b$ and rearrange by commutativity to get

$$(a \vee z) \wedge (a \wedge b) = a \vee (z \wedge (a \vee b))$$

which is the modular identity.

#### Modular law $\Leftrightarrow$ modular identity

To get the modular law from the modular identity, just use the fact that the hypothesis $a \leq b$ is equivalent to $a \vee b = b$, and use this to substitute $b$ for $a \vee b$ in the modular identity. Conversely, from the tautology $a \leq a \vee b$, we can instantiate the modular law to derive the modular identity.

#### Freyd's modular law $\Leftrightarrow$ modular identity

From the tautology $(a \vee z) \wedge b \leq (a \vee z) \wedge (a \vee b)$, it is clear that Freyd's modular law follows from the modular identity. Conversely, by substituting $a \vee b$ for $b$ in Freyd's modular law, we derive the special case

$$(a \vee z) \wedge (a \vee b) \leq a \vee (z \wedge (a \vee b))$$

whereas the opposite inequality

$$a \vee (z \wedge (a \vee b)) \leq (a \vee z) \wedge (a \vee b)$$

holds in any lattice, so the modular identity follows from Freyd's modular law.

#### Modular identity $\Rightarrow$ definition 1

Finally, we derive the adjoint equivalence of Definition 1 from the modular identity. One half of the adjoint equivalence states that if $a \leq z \leq a \vee b$, then $z = a \vee (z \wedge b)$; if this holds, then the other half follows because it is the dual statement. If $a \leq z \leq a \vee b$, then

$$z = (a \vee b) \wedge z = (a \vee b) \wedge (a \vee z)$$

just by the laws of a lattice. By the modular identity (again switching $b$ and $z$), the right side equals $a \vee (b \wedge (a \vee z))$. But since $a \vee z = z$, this equals $a \vee (b \wedge z) = a \vee (z \wedge b)$, as was to be shown.

## Examples

* Every [[distributive lattice]], e.g., a [[Heyting algebra]], is modular. Indeed, if $a \leq b$ in a distributive lattice, we have
$$(a \vee z) \wedge b = (a \wedge b) \vee (z \wedge b) = a \vee (z \wedge b)$$
which proves the modular law.

* For any [[Mal'cev variety]] or Mal'cev [[algebraic theory]], the lattice of internal [[equivalence relations]] of an algebra is a modular lattice. The equivalence classes often arise as cosets of kernels; for example, for a [[vector space]] $V$, equivalence relations correspond to subspaces of $V$, and form a modular lattice. Other examples include the lattice of [[normal subgroups]] of a [[group]], the lattice of two-sided [[ideals]] of a [[ring]], etc.

* In fact, any lattice of commuting equivalence relations on a [[set]] is a modular lattice (being a suballegory of the [[allegory]] of sets, one in which composition provides the join).

* Every abstract [[projective plane]] gives rise to a modular lattice $L$ whose underlying set is the disjoint union
$$\{0\} \cup \{1\} \cup \{points\} \cup \{lines\}$$
where $0$ is taken as bottom, $1$ as top, the points are atoms, and the lines are coatoms, ordered by the incidence relation. The projective plane need not be Desarguesian.

* [[Young–Fibonacci lattice]]

## Characterization

The smallest non-modular lattice has 5 elements and is called the _pentagon_, denoted $N_5$. It can be described as the lattice $\{\bot, a, b, c, \top\}$ where $b \leq c$ and $a$ is incomparable with $b$ and $c$.

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###### Theorem
**(Dedekind)**
A lattice $L$ is modular if and only if there is no injective function $f \colon N_5 \to L$ that preserves meets and joins.
=--

(Notice we are leaving out the condition of preservation of the top and bottom elements.) The direction $\Rightarrow$ is easy enough: $L$ being modular is incompatible with an injective $f: N_5 \to L$ preserving meets and joins, since we contradict the modular law in $L$ by applying $f$ to $(b \vee a) \wedge c = \top \wedge c = c$ and $b \vee (a \wedge c) = b \vee \bot = b$.

This is reminiscent of forbidden minor characterizations of certain classes of graphs; see [[graph minor]]. There is a similar "forbidden sublattice" characterization of [[distributive lattices]] -- see this [comment](#Lein) by Tom Leinster at the $n$-Category Café.

## Free modular lattices

Free modular lattices tend to be complicated. [Dedekind](#Ded) showed that the free modular lattice on 3 elements has 28 elements; its [[Hasse diagram]] can be seen in these [lecture notes](http://www.math.hawaii.edu/~jb/math618/os9uh.pdf) by J.B. Nation (chapter 9, page 100).

N.B.: this notion of lattice is meant with respect to the signature $(\wedge, \vee)$; if we include top and bottom constants in the signature, then the free modular lattice on three elements has 30 elements. A compelling illustration (in gif form) which exhibits [[triality]] of this lattice is given in this $n$-Category Café [post](https://golem.ph.utexas.edu/category/2015/09/the_free_modular_lattice_on_3.html#c049649), as part of a larger discussion which explores the connection with linear representations of the [[quiver]] $D_4$ (the [[Coxeter diagram]] of $SO(8)$).

For $n \geq 4$, the free modular lattice generated by $n$ elements is infinite and in fact has an undecidable word problem (Freese, Herrmann).

## See also

* [[allegory]]

* [[orthomodular lattice]]

* [[distributive lattice]]

* [[geometric lattice]]

## References

* [[Richard Dedekind]], "Über die von drei Moduln erzeugte Dualgruppe gemeinsamen Teiler", Math. Annalen 53 (1900), 371–403, reprinted in _Gesammelte mathematische Werke_, Vol. 2, pp. 236–271, Chelsea, New York, 1968.
{#Ded}

* C. Herrmann, "On the word problem for the modular lattice with four free generators", Mathematische Annalen 265 (1983), 513-527. ([Springerlink](http://www.springerlink.com/content/w862594668m2n375/))

* J.B. Nation, _Revised Notes on Lattice Theory_. Available here: ([web](http://www.math.hawaii.edu/~jb/))

* [[Tom Leinster]], Comment on Solèr's Theorem, December 4, 2010. ([link](http://golem.ph.utexas.edu/category/2010/12/solers_theorem.html#c035900))
{#Lein}

[[!redirects modular lattices]]


modular operad

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## Idea

A _modular operad_ is an [[operad]] that has the kind of extra structure which is exhibited by the operad whose $n$-ary operations are [[Riemann surface]]s with $(n+1)$-punctures, and whose composition is sewing of such surfaces. Every modular operad has a structure of a [[cyclic operad]] and also some additional structure.

## Motivations

* [[2d TQFT]]
* constructing [[graph complex]]es of Kontsevich as Feynman transforms of suitable [[twisted modular operad]]s

## References

* [[Ezra Getzler]], [[Mikhail Kapranov]], _Modular operads_,  Compositio Math.  110  (1998),  no. 1, 65--126, [doi](http://dx.doi.org/10.1023/A:1000245600345), [arXiv:dg-ga/9408003](http://arxiv.org/abs/dg-ga/9408003), [MR99f:18009](http://www.ams.org/mathscinet-getitem?mr=99f:18009)

* [[Ezra Getzler]], _The semi-classical approximation for modular operads_, [arxiv:alg-geom/9612005](http://arxiv.org/abs/alg-geom/9612005)
* [[Ezra Getzler]], J. D. S. Jones, _Operads, homotopy algebra and iterated integrals for double loop spaces_, [hep-th/9403055](http://arxiv.org/abs/hep-th/9403055)

* [[André Joyal]], [[Joachim Kock]], _Feynman graphs, and nerve theorem for compact symmetric multicategories_, proceedings "Quantum Physics and Logic VI", [arxiv/0908.2675](http://arxiv.org/abs/0908.2675)

* Joseph Chuang, Andrey Lazarev, _Dual Feynman transform for modular operads_, [arxiv/0704.2561](http://arxiv.org/abs/0704.2561)

* Serguei Barannikov, _Modular operads and Batalin-Vilkovisky geometry_, Int. Math. Res. Not. IMRN  2007,  no. 19, Art. ID rnm075, 31 pp. [arxiv/0912.5484](http://arxiv.org/abs/0912.5484)
* Martin Doubek, [[Branislav Jurčo]], Lada Peksová, Ján Pulmann, _Connected sum for modular operads and Beilinson-Drinfeld algebras_, [arXiv:2210.06517](https://arxiv.org/abs/2210.06517)
* [[Jeffrey Giansiracusa]], _Moduli spaces and modular operads_, Morfismos __17__:2, 101--125 [journal](https://issuu.com/morfismos/docs/v17n2dic2013) [author pdf](https://cronfa.swan.ac.uk/Record/cronfa18942)

[[!redirects modular operads]]


modular polynomial

\tableofcontents

## Definition

Given a [[field]] $K$ and a [[polynomial]] $p \in K[x]$, a **polynomial modulo $p$** is an element of the [[quotient ring]] $q \in K[x]/p K[x]$. The quotient ring itself is called the **ring of polynomials modulo $p$**.

## Examples

* The [[Gaussian rationals]] are the rational polynomials $\mathbb{Q}[x]$ modulo $x^2 + 1$.

* The [[complex numbers]] are the real polynomials $\mathbb{R}[x]$ modulo $x^2 + 1$.

* The [[dual numbers]] are the real polynomials $\mathbb{R}[x]$ modulo $x^2$.

## Properties

If $p$ is an [[irreducible element]] in $K[x]$, then the quotient ring $K[x]/p K[x]$ is a [[field]].

If $p^n$ is a power of an [[irreducible element]] $p \in K[x]$, then the quotient ring $K[x]/p^n K[x]$ is a [[local ring]].

Given an [[irreducible element]] $p \in K[x]$, one could construct the [[completion of a ring]]

$$K[x]_p \coloneqq \underset{n \to \infty}\mathrm{colim} K[x]/p^n K[x]$$

which is a [[discrete valuation ring]], a [[local integral domain]] with a [[Dedekind-Hasse norm]]. In particular, if $p = x - a$ is a [[monic polynomial]] of [[degree of a polynomial|degree]] one with $a \in K$ a constant, then $K[x]_{x - a} \coloneqq K[[x - a]]$ is the ring of [[formal power series]] expanded around $a$.

If $p$ is a [[square-free element]] of $K[x]$, then the quotient ring $K[x]/p K[x]$ is a [[reduced ring]].

In general, the quotient ring $K[x]/p K[x]$ is a [[prefield ring]], whose [[monoid of regular elements]] are the [[equivalence class]] of polynomials $q$ modulo $p$ such that the [[greatest common divisor]] is in the [[group of units]] of the polynomial ring.

$$\gcd(p, q) \in K[x]^\times$$

## See also

* [[integers modulo n]]

## References

* Garrett Sobczyk, *New Foundations in Mathematics: The Geometric Concept of Number*. (doi:10.1007/978-0-8176-8385-6)


modular tensor category

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## Idea ##

A _modular tensor category_ is, roughly, a [[braided monoidal category]] that encodes the _topological_ structure underlying a [[rational CFT|rational]] 2-dimensional [[conformal field theory]]. In other words, it is a basis-independent formulation of _Moore-Seiberg data_.

It is in particular a [[fusion category]] that is also a [[ribbon category]] such that the "modularity operation" is non-degenerate (this is what the name "modular tensor category" comes from):

this means that for $i,j \in I$ indices for representatives of simple objects $U_i$, $U_j$, the matrix
$$
  (s^{i j}) = (U_i\text{-circle threading through the} \, U_j\text{ -circle})
$$
is non-degenerate.

Here on the right what is meant is the diagram in the modular tensor category made from the identity morphisms, the duality morphisms and the braiding morphism on the objects $U_i$ and $U_j$ that looks like a figure-eight with one circle threading through the other, and this diagram is interpreted as an element in the endomorphism space of the tensor unit object, which in turn is canonically identified with the [[ground field]].

In the description of 2-dimensional [[conformal field theory]] in the [[FFRS-formalism]] it is manifestly this kind of modular diagram that encodes the torus partition function of the CFT. This explains the relevance of modular tensor categories in the description of [[conformal field theory]].

Since 2-dimensional conformal field theory is related by a [[holographic principle]] to 3-dimensional [[TQFT]], modular tensor categories also play a role there, which was in fact understood before the full application in conformal field theory was: in the [[Reshetikhin-Turaev model]].

## Definition ##

A **modular tensor category** is a [[category]] with the following long list of extra structure.

> needs to be put in more coherent form, just a stub

* it is an [[abelian category]], $\mathbb{C}$-linear (i.e. $Vect_{\mathbb{C}}$ [[enriched category]]), [[semisimple category]] [[monoidal category]] ([[tensor category]])

* the tensor unit is a simple object, $I$
  a finite set of representatives of isomorphism classes of simple objects

* [[fusion category]]

* [[braided monoidal category]]

* [[ribbon category]], in particular objects have [[dual objects|duals]]

* **modularity** a non-degeneracy condition on the [[braiding]] given by an [[isomorphism]] of algebras
  $$
    K(C) \otimes_{\mathbb{Z}} \stackrel{\simeq}{\to} End(Id_C)
  $$
  where
  $$
    [U] \mapsto \alpha_U
  $$
  where the transformation $\alpha_U$ is given on the simple object $V$ by
  $$
    \alpha_U(V) = \text{straight } V\text{-line encircled by } U\text{-loop}
  $$
  (on the right we use string diagram notation)

## Examples

### Rep categories of VOAs
 {#RepCategoriesOfVOAs}

#### General

Many modular tensor categories arise as [[representation]] categories of [[vertex operator algebras]] ([Huang 2005, Sec. 1](#Huang05); [Huang 2008](#Huang08); [Huang, Lepowski & Zhang 2014](#HuangLepowskiZhang14) see [EGNO 15, Sec. 8.27.6](#EGNO15)), hence of chiral [[field (physics)|fields]] of [[2d conformal field theories]].

(For [[logarithmic CFTs]] one still gets [[braided monoidal category|braided]] [[tensor categories]], see [Creutzig, Lentner & Rupert 2021](#CreutzigLentnerRupert21)).

#### Relation to conformal blocks
{#RelationToConformalBlocks}

 In this case the [[monoidal category|monoidal]]- and the [[braided monoidal category|braided]] [[structure]] (hence the modular tensor structure) on the underlying [[representation category]] is entirely fixed by the space of [[conformal blocks]] of the [[2d CFT]] on the [[Riemann sphere]] (the "genus zero conformal blocks").

This may be found highlighted in [EGNO 15, p. 266](#EGNO15), [Runkel, Sec. 4.3](#Runkel). The essentially equivalent fact that the genus=0 conformal blocks already determine the [[modular functor]] of the CFT is proven in [Andersen & Ueno 2012](modular+functor#AndersenUeno12).

#### Examples

A database of examples is given by ([Gannon & Höhn](#GannonHoehn)).

## Related concepts

* [[tensor category]]

## References

### General

Original article:

* [[Vladimir Turaev]], *Modular categories and 3-manifold invariants*,  International Journal of Modern Physics B Vol. 06, No. 11n12, pp. 1807-1824 (1992) ([doi:10.1142/S0217979292000876](https://doi.org/10.1142/S0217979292000876))

Review in the context of the [[Reshetikhin-Turaev construction]] of [[modular functors]]:

* [[Bojko Bakalov]], [[Alexander Kirillov]], *Modular Tensor Categories*, Ch. 3 in: *Lectures on tensor categories and modular functors*, University Lecture Series **21**, Amer. Math. Soc. (2001) [[[BakalovKirillov-Ch3-ModularTensorCategories.pdf:file]],[web](http://www.math.stonybrook.edu/~kirillov/tensor/tensor.html), [ams:ulect/21](https://bookstore.ams.org/view?ProductCode=ULECT/21)]

Review in the context of [[2d CFT]]/[[VOA]]

* [Fuchs-Runkel-Schweigert 02, Sec. 2.1](#FuchsRunkelSchweigert02)

* {#Runkel} [[Ingo Runkel]], *Algebra in Braided Tensor Categories and Conformal Field Theory* ([pdf](https://www.math.uni-hamburg.de/home/runkel/PDF/alg.pdf), [[Runkel-AlgebraInBTCAndCFT.pdf:file]])

and with focus on relation to [[braid representations]]:

* [[Colleen Delaney]], *Lecture notes on modular tensor categories and braid group representations*, 2019 ([pdf](http://web.math.ucsb.edu/~cdelaney/MTC_Notes.pdf), [[DelaneyModularTensorCategories.pdf:file]])

Construction of modular tensor categories from [[vertex operator algebras]]:

* {#Huang05} [[Yi-Zhi Huang]], *Vertex operator algebras, the Verlinde conjecture and modular tensor categories*, Proc. Nat. Acad. Sci. **102** (2005) 5352-5356 $[$[arXiv:math/0412261](http://arxiv.org/abs/math/0412261), [doi:10.1073/pnas.0409901102](https://doi.org/10.1073/pnas.0409901102)$]$

* {#Huang08} [[Yi-Zhi Huang]], *Rigidity and modularity of vertex tensor categories*,  Communications in Contemporary Mathematics **10** supp01 (2008) 871-911   $[$[arXiv:math/0502533](https://arxiv.org/abs/math/0502533), [doi:10.1142/S0219199708003083](https://doi.org/10.1142/S0219199708003083)$]$

* {#HuangLepowskiZhang14} [[Yi-Zhi Huang]], [[James Lepowsky]], Lin Zhang, *Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: Introduction and strongly graded algebras and their generalized modules*,  In: Bai, Fuchs, Huang, Kong, Runkel, Schweigert (eds.), *Conformal Field Theories and Tensor Categories* Mathematical Lectures from Peking University. Springer (2014) $[$[arXiv:1012.4193](https://arxiv.org/abs/1012.4193), [doi:10.1007/978-3-642-39383-9_5](https://doi.org/10.1007/978-3-642-39383-9_5)$]$

* {#EGNO15} [[Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], Section 8.27.6 of: *Tensor Categories*, AMS Mathematical Surveys and Monographs **205** (2015) [[ISBN:978-1-4704-3441-0](https://bookstore.ams.org/surv-205)]

brief survey:

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Simon Wood]], [[Yang Yang]], pp. 5 of: *Algebraic structures in two-dimensional conformal field theory*, Encyclopedia of Mathematical Physics [[arXiv:2305.02773](https://arxiv.org/abs/2305.02773)]

and generalization to [[logarithmic CFT|logarithmic]] VOAs:

* {#CreutzigLentnerRupert21} [[Thomas Creutzig]], [[Simon Lentner]], [[Matthew Rupert]], *Characterizing braided tensor categories associated to logarithmic vertex operator algebras* $[$[arXiv:2104.13262](https://arxiv.org/abs/2104.13262)$]$

A list of examples (with an emphasis on [[representation categories]] of rational [[vertex operator algebras]]) is in

* {#GannonHoehn} [[Terry Gannon]], [[Gerald Höhn]], Hiroshi Yamauchi, _[The online database of Vertex Operator Algebras and Modular Categories](http://www.math.ksu.edu/~gerald/voas/)_

Classification results

* [[Eric Rowell]], [[Richard Stong]], [[Zhenghan Wang]], *On classification of modular tensor categories*, Comm. Math. Phys. **292** 2 (2009) 343--389 [[arXiv:0712.1377](https://arxiv.org/abs/0712.1377), [doi:10.1007/s00220-009-0908-z](https://doi.org/10.1007/s00220-009-0908-z)]

On [[number theory|number theoretic]] aspects of modular tensor categories:

* {#DavidovichHaggeWang2013} [[Orit Davidovich]], [[Tobias Hagge]], [[Zhenghan Wang]], _On Arithmetic Modular Categories_, arXiv preprit, 2013 [[arXiv:1305.2229](https://arxiv.org/abs/1305.2229)]

### Relation to 3dCS/2dWZW quantum field theory

Discussion of modular tensor categories in [[quantum field theory]] ([[3d TQFT]] and [[2d CFT]], as well as their relation via the [[AdS3-CFT2 and CS-WZW correspondence|CS/WZW correspondence]]) includes the following.

A general survey of the literature is in

* [[Eric Rowell]], _From quantum groups to Unitary modular tensor categories_, Contemporary Mathematics 2005 ([arXiv:math/0503226](http://arxiv.org/abs/math/0503226))

See also

* [[Victor Ostrik]], _Tensor categories in conformal field theory_, talk notes 2011  ([pdf slides](http://pages.uoregon.edu/vostrik/talks/beijing.pdf))

More specific discussion in the context of [[2d CFT]] is in

* {#FuchsRunkelSchweigert02} [[Jürgen Fuchs]], [[Ingo Runkel]], [[Christoph Schweigert]], _TFT construction of RCFT correlators I: partition functions_ ([arXiv:hep-th/0204148](http://arxiv.org/abs/hep-th/0204148))

  (for more along these lines see at _[[FRS formalism]]_)

Review of construction of MTCs from [[vertex operator algebras]] is in

* [[James Lepowsky]], _From the representation theory of vertex operator algebras to modular tensor categories in conformal field theory_ ([pdf](http://www.pnas.org/content/102/15/5304.full.pdf))

Discussion from the point of view of the [[cobordism hypothesis]] (see also the discussion at [[fusion category]]) is in

* [[Bruce Bartlett]], [[Christopher Douglas]], [[Chris Schommer-Pries]], [[Jamie Vicary]], _Modular categories as representations of the 3-dimensional bordism 2-category_ ([arXiv:1509.06811](http://arxiv.org/abs/1509.06811))

[[!include anyonic topological order via braided fusion categories -- references]]

[[!redirects modular tensor categories]]


modular theory

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## Idea ##

This page is about the modular theory introduced by Tomita for [[von Neumann-algebras]]. It is important both for the structure theory of von Neumann-algebras and in the [[Haag-Kastler approach]] to [[AQFT]], one important example is the [[Bisognano-Wichmann theorem]]. It is often called Tomita-Takesaki theory, because the first presentation beyond a preprint is due to [[Masamichi Takesaki]].

## nPOV definition

The modern approach to defining the modular automorphism group
is through the theory of noncommutative [[L_p-spaces]].
This was pioneered by [Haagerup in 1979](#Haagerup) and [Yamagami in 1992](#Yamagami).

In this approach,
given a [[von Neumann algebra]] $M$, a faithful semifinite normal [[weight]] $\mu$ on $M$, and an [[imaginary number]] $t$,
the modular automorphism associated to $M$, $\mu$, and $t$ is
$$\sigma_\mu^t\colon M\to M,\qquad m\mapsto \mu^t m \mu^{-t}.$$

This approach makes it easy to deduce various properties of the modular automorphism group.

For more details, see a [MathOverflow answer](https://mathoverflow.net/questions/386464/making-sense-of-every-non-commutative-algebra-has-its-own-internal-time-evoluti/386481#386481).

## Traditional definition ##
Let $\mathcal{H}$ be a [[Hilbert space]], $\mathcal{M}$ a [[von Neumann-algebra]] with commutant $\mathcal{M}'$ and a separating and cyclic vector $\Omega$. Then there is a **modular operator** $\Delta$ and a **modular conjugation** $J$ such that:

1. $\Delta$ is self-adjoint, positive and invertible (but not bounded).

2. $\Delta\Omega = \Omega$ and $ J\Omega = \Omega$

3. $J$ is antilinear, $J^* = J, J^2 = \mathbb{1}$, $J$ commutes with $\Delta^{it}$. This implies
$$
Ad(J) \Delta = \Delta^{-1}
$$

4. For every $A \in \mathcal{M}$ the vector $A\Omega$ is in the domain of $\Delta^{\frac{1}{2}}$ and
$$
J \Delta^{\frac{1}{2}} A \Omega = A^* \Omega =: SA \Omega
$$

5. The unitary group $\Delta^{it}$ defines a group automorphism of $\mathcal{M}$:
$$
Ad(\Delta^{it}) \mathcal{M} = \mathcal{M} \; \; \text{for all} \; t \in \mathbb{R}
$$

6. $J$ maps $\mathcal{M}$ to $\mathcal{M}'$.

## References

* {#Haagerup} [[Uffe Haagerup]], [$L^p$-spaces associated with an arbitrary von Neumann algebra](https://dmitripavlov.org/haagerup.pdf).  Algèbres d'opérateurs et leurs applications en physique mathématique.  Colloques Internationaux du Centre National de la Recherche Scientifique 274, 175–184.

* {#Yamagami} [[Shigeru Yamagami]], _Algebraic aspects in modular theory_, Publications of the Research Institute for Mathematical Sciences 28:6 (1992), 1075-1106.  [doi](http://dx.doi.org/10.2977/prims/1195167738).

* [[Shigeru Yamagami]], _Modular theory for bimodules_, Journal of Functional Analysis 125:2 (1994), 327-357.  [doi](http://dx.doi.org/10.1006/jfan.1994.1127).

### General

Introduction:

* Stephen J. Summers: "Tomita-Takesaki Modular Theory" ([arXiv] (http://xxx.uni-augsburg.de/abs/math-ph/0511034))

Role in [[algebraic quantum field theory]]:

* [[Hans-Jürgen Borchers]], *On Revolutionizing of Quantum Field Theory with Tomita's Modular Theory*, ESI Preprint 773 (1999) [[pdf](https://www.mat.univie.ac.at/~esiprpr/esi773.pdf)]

Many textbooks on operator algebras contain a chapter about modular theory.

MathOverflow question [tomita-takesaki-versus-frobenius-where-is-the-similarity](http://mathoverflow.net/questions/68270/tomita-takesaki-versus-frobenuis-where-is-the-similarity)

* [[Alain Connes]], [[Carlo Rovelli]], _Von Neumann algebra automorphisms and time-thermodynamics relation in general covariant quantum theories_, [arXiv:gr-qc/9406019](http://arxiv.org/abs/gr-qc/9406019), [pdf](http://www.alainconnes.org/docs/carlotime.pdf)

Discussion in terms of [[topos theory]] is in

* {#Henry14} [[Simon Henry]], _From toposes to non-commutative geometry through the study of internal Hilbert spaces_, 2014  ([pdf](http://www.math.jussieu.fr/~henrys/Thesis.pdf))

See also

* Wikipedia, _[Tomita–Takesaki theory](https://en.wikipedia.org/wiki/Tomita%E2%80%93Takesaki_theory)_

### Modular flow

On [[Tomita-Takesaki modular flow]] as emergent [[time]] evolution in [[quantum physics]] ([[AQFT]]):

* {#Longo19} [[Roberto Longo]], _The emergence of time_ ([arxiv:1910.13926](https://arxiv.org/abs/1910.13926))

[[!redirects modular conjugation]]
[[!redirects modular conjugations]]
[[!redirects modular flow]]
[[!redirects modular flows]]
[[!redirects modular automorphism]]
[[!redirects modular automorphisms]]
[[!redirects modular automorphism group]]
[[!redirects modular automorphism groups]]

[[!redirects Tomita theory]]
[[!redirects Tomita modular theory]]
[[!redirects Tomita modular flow]]

[[!redirects Tomita-Takesaki theory]]
[[!redirects Tomita-Takesaki modular theory]]
[[!redirects Tomita-Takesaki modular flow]]

[[!redirects Tomita–Takesaki theory]]
[[!redirects Tomita–Takesaki modular theory]]
[[!redirects Tomita–Takesaki modular flow]]


modular transformations

See:

* [[modular group]]

* [[modular tensor category]]

[[!redirects modular transformation]]


modularity theorem

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## Statement

The **modularity theorem** states that given an [[elliptic curve]] $E$ over the [[rational numbers]] with a [[conductor]] $N_E$, there is a [[cusp form]] $f \in S_2(\Gamma_0(N_E))$ such that $a_1(f) = 1$ and $a_p(f) = a_p(E)$ for all [[primes]] $p$.

The "semistable" case of the modularity theorem was proven in [Wiles 1995](#Wiles95), [Taylor & Wiles 1995](#TaylorWiles95). The full theorem was proved by [Breuil, Conrad, Diamond & Taylor 2001](#BreuilConradDiamondTaylor2001).

## Generalizations

The modularity theorem was extended to the case of elliptic curves over real quadratic fields by Freitas-Le Hung-Siksek in [FLS15](#FLS15), and work in progress by Caraiani-Newton extends this to the case of elliptic curves over imaginary quadratic fields.

More generally, the term "modularity" is often applied to mean the question of whether a [[Galois representation]] comes from a modular form or one of its higher-dimensional generalizations such as [[Hilbert modular forms]] or [[Siegel modular forms]].

## Related concepts

* [[elliptic curve]], [[modular curve]], [[modular form]]

* [[Fermat's last theorem]]

* [[Ribet's theorem]]

* [[Wiles' proof of Fermat's last theorem]]

## References
 {#References}

* {#Wiles95} [[Andrew Wiles]], *Modular Elliptic Curves and Fermat's Last Theorem*, Annals of Mathematics Second Series, **141** 3 (1995) 443-551 $[$[doi:10.2307/2118559](https://doi.org/10.2307/2118559)$]$

* {#TaylorWiles95} [[Richard Taylor]], [[Andrew Wiles]], *Ring-Theoretic Properties of Certain Hecke Algebras*, Annals of Mathematics Second Series **141** 3 (1995) 553-572 $[$[doi:10.2307/2118560 ](https://doi.org/10.2307/2118560 )$]$

* {#BreuilConradDiamondTaylor2001} [[Christophe Breuil]], [[Brian Conrad]], [[Fred Diamond]], [[Richard Taylor]], *On the modularity of elliptic curves over $\mathbf{Q}$: Wild 3-adic exercises*, Journal of the American Mathematical Society **14** 4 (2001) 843–939 $[$[doi:10.1090/S0894-0347-01-00370-8](https://doi.org/10.1090/S0894-0347-01-00370-8)$]$

A generalization to elliptic curves over real quadratic fields is in

* {#FLS15} Nuno Freitas, Bao V. Le Hung, Samir Siksek _Elliptic Curves Over Real Quadratic Fields Are Modular_, Invent. Math. 201 (2015), no. 1, 159–206. ([arXiv:1310.7088](https://arxiv.org/abs/1310.7088))

Review:

* [[Gerd Faltings]], *The Proof of Fermat’s Last Theorem by R. Taylor and A. Wiles*, Notices of the AMS (1995) $[$[pdf](https://www.ams.org/notices/199507/faltings.pdf)$]$

* [[Fred Diamond]], [[Jerry Shurman]], *A First Course in Modular Forms*, GTM **228**, Springer (2005) $[$[doi:10.1007/978-0-387-27226-9](https://doi.org/10.1007/978-0-387-27226-9), ISBN-13: 978-0387232294$]$

* R. van Dobben de Bruyn, *The Modularity Theorem*, ([pdf](https://math.leidenuniv.nl/scripties/DobbendeBruynBach.pdf))

See also:

* Wikipedia, _[Modularity theorem](https://en.wikipedia.org/wiki/Modularity_theorem)_

[[!redirects modularity theorem]]

[[!redirects modularity conjecture]]

[[!redirects Taniyama-Shimura conjecture]]

[[!redirects Taniyama-Shimura modularity conjecture]]


modulated Cauchy complete space

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## Idea ##

A notion of Cauchy space where only certain Cauchy nets with a particular modulus of convergence converges in the space.

## Definition ##

Let $S$ be a [[Cauchy space]] (such as a [[uniform space]] or a [[metric space]]). Given a [[directed set]] $I$, $S$ is **$I$-modulated Cauchy complete** if every [[Cauchy net]] in $S$ with index set $I$ and with a $I$-[[modulus of convergence]] [[converges]].

A Cauchy space is said to be **sequentially modulated Cauchy complete** if every [[Cauchy sequence]] in $S$ with a $\mathbb{N}$-modulus of convergence converges.

## Examples ##

* The [[Dedekind real numbers]] are sequentially modulated Cauchy complete.
* The [[HoTT book real numbers]] are sequentially modulated Cauchy complete.

## See also ##

* [[Cauchy space]]
* [[complete space]]
* [[sequentially Cauchy complete space]]

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[[!redirects modulated-complete space]]
[[!redirects modulated complete spaces]]
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modulating morphism

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## Idea

On the one hand, a map into a [[classifying space]] _classifies_ some kind of [[object]], it is a _[[classifying morphism]]_, meaning that it characterizes the object (only) up to [[equivalence]].

On the other hand, the more refined concept of a [[moduli stack]] is such that [[morphisms]] into it characterize the object itself. Hence it makes sense to say that such maps into moduli stacks not just classify, but **modulate** the given object.

Indeed, that is the idea that originally gave rise to the name _[[moduli]]_.
Because, more specifically, a [[moduli stack]] $\mathbf{F}$ of a certain kind of objects is such that morphisms $X \stackrel{\chi}{\longrightarrow} \mathbf{F}$ into it determine a [[bundle]]

$$
  \array{
     P
     \\
     \downarrow
     \\
     X
  }
$$

of $\mathbf{F}$-like objects over $X$. It is this bundle which is "being modulated (by $\chi$) as one moves around in $X$", much as in the language of electronics a waveform is being modulated as one moves around in time.

## Examples

### Metric structure

To see that the difference really matters, consider the map of [[classifying spaces]]

$$
  B O(n) \longrightarrow B GL(n)
$$

for the [[orthogonal group]] and the [[general linear group]]. These classify, respectively, $O(n)$-[[principal bundles]] and $GL(n)$-[[principal bundles]]. While as geometric (e.g. topological or smooth) [[bundles]] these are different, their [[equivalence classes]] are the same. Accordingly the above map is in fact a [[homotopy equivalence]] and accordingly for $\Sigma$ any $n$-dimensional [[manifold]] whose [[tangent bundle]] is classified by $\iota \tau_\Sigma \colon \Sigma \longrightarrow B GL(n)$, then the space of lifts $\iota \hat \tau_\Sigma$ in

$$
  \array{
     && B O(n)
     \\
     & {}^{\mathllap{\iota \hat \tau_\Sigma}} \nearrow & \downarrow
     \\
     \Sigma &\stackrel{\iota \tau_\Sigma}{\longrightarrow}& B GL(n)
  }
$$

is [[contractible space|contractible]]. Hence classifying maps see no difference here.

However, there is an important difference which the modulating morphisms do see. Write

$$
  \mathbf{B}O(n) \longrightarrow \mathbf{B}GL(n)
$$

for the corresponding morphism of [[smooth infinity-groupoid|smooth]] [[moduli stacks]] (see at [[looping and delooping]] for more on this). Then a lift $e$ of the modulating map $\tau_\Sigma \colon \Sigma \longrightarrow \mathbf{B}GL(n)$ to one that modulates an actual orthogonal bundle

$$
  \array{
     && \mathbf{B} O(n)
     \\
     & {}^{\mathllap{e}} \nearrow & \downarrow
     \\
     \Sigma &\stackrel{\tau_\Sigma}{\longrightarrow}& \mathbf{B} GL(n)
  }
$$

is genuine data: this is a choice of [[orthogonal structure]]/[[vielbein]] on $\Sigma$ and hence a [[Riemannian metric]] on $\Sigma$.

### $G$-Structure for maximal compact subgroup inclusions

An analogous situation is obtained for any inclusion of a [[maximal compact subgroup]] into a given [[Lie group]] and the corresponding notion of [[G-structure]]. All $G$-structures arising this way are invisible to classifying maps, but are seen by modulating maps. See at _[[twisted differential c-structure]]_ for a list of further examples

## References

The terminology "modulating" in the context of "moduli stacks" would seem to be inevitable but is not used much in practice. One place where it is used consistently is

* {#BenZvi} [[David Ben-Zvi]], _Moduli Spaces_ ([pdf](https://www.ma.utexas.edu/users/benzvi/math/pcm0178.pdf))

[[!redirects modulating morphisms]]

[[!redirects modulates]]
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