moduli space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Yoneda lemma
+--{: .hide}
[[!include Yoneda lemma - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _rough idea_ is (but see the [caveat](#caveat) below) that the term **moduli space** is essentially a synonym for [[representable functor|representing object]] and for [[classifying space]]. People tend to say "classifying space" when in the context of [[topology]], and they tend to say _moduli space_ when in a context of [[complex geometry]] or [[algebraic geometry]].

More precisely, when a moduli space actually does exist as an ordinary space (or [[scheme]]), it is called for emphasis a **fine moduli space**. Fine here refers to the completeness of the description, not shared by coarse moduli below.

Those classifying "spaces" that are called moduli spaces are typically [[orbifold]]s, hence modeled not just as spaces but as [[groupoid]]s with extra structure. These are typically conceived as [[stack]]s: these are then called **[[moduli stack]]**s. Typically these are demanded to be [[Deligne-Mumford stack]]s.

So the term _fine moduli space_ mainly indicates that a given object that might be a [[Deligne-Mumford stack]] is actually just a plain [[scheme]]. But there is also the notion of **coarse moduli space**, which is a kind of conceptual hack designed to be able to keep thinking about what really wants to be a [[stack]] still as a plain [[sheaf]].

A coarse moduli space is one that at least has the right underlying [[set]] of points as the _right_ [[moduli stack]] has: as long as we don't look at families but just at single things, it does give the right information. From the point of view of [[derived algebraic geometry]], the coarse moduli spaces are 0-truncations of derived [[moduli stacks]] when they exist.

### Caveat {#caveat}

But one has to be **careful with this rough idea**: as there is usually some **implicit fine print** in the notion of moduli space:

while [[classifying space]] is the term typically used for a [[representable functor|representing object]] in a [[homotopy category]], a representing object is usually called a _moduli space_ only if it lives in the "original" category. For more on this see the chapter

* [... because they have automorphisms](#because)

below.

## History

The term possibly originates with Riemann, who was the first to study what are now called moduli spaces of (compact) [[Riemann surface]]s. A "modulus" here is meant to be a _parameter_ that parameterizes isomorphism classes of Riemann surfaces.

## Definition

...

### Coarse moduli space

A **coarse moduli space** for a [[presheaf]] $F:(Sch/\mathbb{C})^{op} \to Set$ on complex [[scheme]]s is a [[scheme]] $M \in Sch/\mathbb{C}$ equipped with a morphism

$$
  \Psi_M : F \to h_M
  \,,
$$

where $h$ denotes the [[Yoneda embedding]], such that

a) $F(Spec(\mathbb{C})) \to h_M(Spec \mathbb{C}) = hom(Spec \mathbb{C}, M)$ is a [[bijection]]

b) given $M'$ and $\Psi_{M'} : F \to h_{M'}$ then there exists unique $M \to M'$ such that $\array{ F && \stackrel{\Psi_{M'}}{\to}&& h_{M'} \\ & {}_{\Psi_M}\searrow && \nearrow \\ && h_M} $

So a coarse moduli space is one that at least has the right underlying set of points as the _right_ [[moduli stack]] has: as long as we don't look at families but just at single things, it does give the right information. From the point of view of [[derived algebraic geometry]], the coarse moduli spaces are 0-truncations of derived [[moduli stacks]] when they exist.

## "... because they have automorphisms." {#because}

A widespread slogan is

> **Slogan**: A type of objects that has nontrivial [[automorphism]]s cannot have a fine moduli space.

The typical example motivating and illustrating this slogan is the example of [[elliptic curve]]s: from a single [[elliptic curve]] $E$ with a non-trivial automorphism $\alpha : E \to E$ one may build, for instance, the family $\mathcal{E} \to S^1 \times S^1$ of elliptic curves over the torus which is obtained by starting with the trivial family $E \times S^1 \times [0,1] \to S^1 \times [0,1]$ over the cylinder and then by gluing the two ends by identifying the fiber using the automorphism $\alpha$. The resulting family is _locally trivial_ but globally nontrivial.

Such locally trivial but globally non-trivial families can in principle not be classified by a representing object in the category of spaces that we are working with, as long as the [[Grothendieck topology]] which we use to say what a locally trivial family is is [[subcanonical coverage|subcanonical]]:

for assume that a classifying object $K$ exists in our category and assume that at least one globally nontrivial but locally trivial family $\mathcal{E} \to X$ exists in the category.

Then by definition of local triviality there exists a covering [[sieve]] $\{U_i \to X\}$ such that the [[pullback]] of $\mathcal{E}$ to each $U_i$ is equivalent to the trivial family over $U_i$. Accordingly, it will be represented by the trivial element in $Hom(U_i, K)$. But if $\{U_i \to X\}$ is a covering [[sieve]] in a subcanonical topology, then it follows from the [[sheaf]] property of the [[representable functor|representable presheaf]] $Hom(-,K)$ that there is a unique element in $Hom(X,K)$ whose restriction to all the $U_i$ represents the trivial family. But the trivial family over $X$ has this property and hence must be that unique element. Accordingly, it would follow that there is no non-trivial family classified by $K$ over $X$, which is a contradiction and shows that the "fine moduli space" $K$ cannot exist under the given assumptions.

Notice well the two assumptions that were made to make this argument work:

1. the topology with which we glue families must be subcanonical: this is the assumption that fails for the situations in which one would speak of [[classifying space]]s instead of moduli spaces:

   For instance [[vector space]]s do certainly have nontrivial [[automorphism]]s. A (topological, say) family of vector space is a [[vector bundle]]. So a naive application of the above argument might lead one to conclude that there cannot be a classifying space of vector bundles!

   But of course it is a standard fact that there there are [[topological space]]s $\mathcal{B} U(n)$ such that [[homotopy]] classes of continuous maps $X \to \mathcal{B} U(n)$ classify isomorphism classes of rank-$n$ [[vector bundle]]s. But this means that the functor that assigns families of vector bundles to topological space is -- while not [[representable functor|representable]] in [[Top]] -- representable in the [[homotopy category]] $Ho(Top)$

   $$
     Rank n VectBund(X)/iso  \simeq Hom_{Ho(Top)}(X, \mathcal{B} U(n))
     \,.
   $$

   This might make the [[classifying space]] $\mathcal{B} U(n)$ look like something that deserves to be called a fine moduli space. But one should beware that the standard [[Grothendieck topology]] of [[Top]], which is [[subcanonical coverage|subcanonical]] in [[Top]] is not so in $Ho(Top)$: the functor $Hom_{Ho(Top)}(-, \mathcal{B} U(m))$ has a non-standard opinion about what it means to glue two topological spaces to a third one. That's how in $Ho(Top)$ the above argument that "automorphisms prevent a classifying object" breaks down.

   In fact, there is a theorem by [[Danny Stevenson]] and [[David Roberts]], extending a theorem by [[John Baez]] and Danny Stevenson  that shows that large classes of [[principal ∞-bundle]]s, even, do have classifying topological spaces in this sense. These are objects that not only have automorphisms, but have automorphisms of automorphisms, etc, and hence seem to contradict the above slogan in a maximal way.

   But these examples also indicate why it may be misguided to think of the "classifying spaces" such as $\mathcal{B} U(n)$ as being moduli spaces in the first place: the [[homotopy category]] $Ho(Top)$ in fact regards its objects -- even though they are presented as [[topological space]]s -- not really as spaces but as the [[∞-groupoid]]s that they are equivalent to under the [[homotopy hypothesis]]. So in that sense objects such as $\mathcal{B} U(n)$ are not classifying spaces, but are classifying [[groupoid]]s after all.

2. Another situation that makes the above argument break down is if the category of "spaces" that one works with is such that gluing trivial families by nontrivial automorphisms never produces globally nontrivial families. For instance because the category of "spaces" is such that no nontrivial families exist at all. This happens for instance in the case that we take a "space" to be simply a (finite, say) [[set]]. In the category [[Set]] of (finite) sets every fiber bundle $\mathcal{E} \to X$ (i.e. every surjection such that all fibers are isomorphic to a typical fiber $F$ ) are necessarily globally trivial in that they are isomorphic to $F \times X \to X$. And in fact, there _is_ a "fine moduli space" in this context (inside the category of non-necessarily finite sets), namely the set $\mathbb{N}$ of natural numbers: the family $(\mathcal{E} \to X) \in Set$ with typical fiber the $n$-element set is classified by the map that sends all $x \in X$ to $n \in \mathbb{N}$.

## Examples

[[!include moduli spaces -- contents]]

## Related concepts

* [[moduli]], [[moduli problem]], [[moduli stack]]

* [[classifying space]]. [[moduli stack]]

* [[classifying morphism]], [[modulating morphism]]

* [[representable functor]]
* [[classifying space]], [[classifying stack]], [[classifying topos]]
* [[moduli stack]], [[derived moduli space]], [[Deligne-Mumford stack]]
* [[geometric invariant theory]], [[Moduli Problems and DG-Lie Algebras]],
* [[universal principal bundle]], [[universal principal infinity-bundle]], * [[moduli space of curves]],
* [[Hilbert scheme]], [[Quot scheme]], [[Picard scheme]], [[moduli stabilization]]

## References
 {#References}

* [[David Ben-Zvi]], _Moduli spaces_ ([pdf](http://www.math.utexas.edu/users/benzvi/math/pcm0178.pdf))

* [[Ib Madsen]], _Moduli spaces from a topological viewpoint_, ICM 2006 ([pdf](http://www.icm2006.org/proceedings/Vol_I/20.pdf))

A bit of elementary exposition of these ideas is at

* [[basic ideas of moduli stacks of curves and Gromov-Witten theory]]

[[!redirects moduli spaces]]
[[!redirects space of moduli]]

[[!redirects fine moduli space]]
[[!redirects coarse moduli space]]

[[!redirects fine moduli spaces]]
[[!redirects coarse moduli spaces]]


moduli space of (higher) line bundles

#Contents#
* table of contents
{:toc}

## Idea

A [[moduli space]] of [[circle n-bundle with connection|higher line bundles (with connection)]]

[[!include moduli of higher lines -- table]]

## Related concepts

* [[F/M-theory on elliptically fibered Calabi-Yau 4-folds]]

[[!include moduli spaces -- contents]]

[[!redirects moduli spaces of (higher) line bundles]]


moduli space of bundles

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Analytic geometry
+--{: .hide}
[[!include analytic geometry -- contents]]
=--
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[moduli space|Moduli spaces]] and [[moduli stacks]] of [[vector bundles]] and of $G$-[[principal bundles]] for a complex [[algebraic group]] $G$ have been widely studied in geometry, with many deep results, especially for the case that the base space is a [[complex curve]] or [[algebraic curve]]. In a classical work of [[Grothendieck]] presented in [[FGA]] (see [[FGA explained]]), he studied moduli schemes of [[coherent sheaves]] with certain parameters fixed, so called [[Quot schemes]]. Later, geometric invariant theory defined other class of moduli spaces of bundles. Especially important is the case of the moduli space of [[stable bundles]] on a Riemann surface, which is deeply relevant for [[mathematical physics]], as the [[Narasimhan–Seshadri theorem]] relates it to the [[moduli space of flat connections]] (and to [[instanton|self-dual solutions of Yang-Mills equations]]; study of spaces of [[conformal blocks]]; [[representation theory]] of [[affine Lie algebras]] and [[loop groups]]; [[integrable systems]], esp. [[Hitchin systems]], etc.).

## Examples

### Bundles over curves and the Langlands correspondence
 {#OverCurvesAndTheLanglandsCorrespondence}

For $G$ some [[complex Lie group]] and $\Sigma$ some [[complex curve]], then the [[moduli stack]] of $G$-[[principal bundles]] on $\Sigma$ (which are equivalently [[holomorphic vector bundles]] when $G = \coprod_{n} GL(n,\mathbb{C})$) has a standard description as a [[double coset space]] [[quotient stack]] of the collection of [[formal power series]] around finitely many points in $\Sigma$ --  the _[[Weil uniformization theorem]]_. We frist disucss an easy toplogical version of this statement in

* [Over smooth real surfaces](#OverRealSurfaces)

and then we discuss the complex-analytic version

* [Over complex curves](#OverComplexCurves)

Notice here that the sub-moduli space of _[[stable vector bundles|stable]]_ $GL$-principal bundles is related via the [[Narasimhan-Seshadri theorem]] to that of [[flat connection|flat]] $GL$-[[principal connections]] which is the [[phase space]] of $G$-[[Chern-Simons theory]] and via the [[holographic principle|holographic]] relation of that to the [[WZW model]] an ingredient of the [[modular functor]] and of [[equivariant elliptic cohomology]] etc. This relation serves to explain to some extent why this object is of such interest.

Now the double quotient description is noteworthy because in this incarnation the moduli stack has, via the [[function field analogy]], an immediate analog in [[algebraic geometry]] and in fact in [[arithmetic geometry]] over any [[number field]]. This we discuss below in the section

* [Over algebraic curves](#OverAlgebraicCurves).

This parallel or analogy between the moduli stack of $G$-bundles over curves in analytic geometry and in arithmetic geometry is the underlying reason for the parallel between the [[number theory|number theoretic]] [[Langlands correspondence]] and the [[geometric Langlands correspondence]] (review includes [Frenkel 05, section 3.2](#Frenkel05)). It is also at the heart of the [[Weil conjecture on Tamagawa numbers]].

In summary/preview, the analogy is this:

[[!include function field analogy -- table]]

#### Over smooth real surfaces
 {#OverRealSurfaces}

Let

* $\Sigma$ be a [[smooth manifold|smooth]] [[closed manifold]] of [[dimension]] 2 -- a real [[surface]];

* $G$ a [[connected topological space|connected]] [[topological group]].

* $x \in X$ any point;

* $X^\ast \coloneqq X - \{x\}$

* $D \subset X$ a [[neighbourhood]] of $x$ [[homeomorphism|homeomorphic]] to a [[disk]];

* $D^\ast \coloneqq D - \{x\}$ the corresponding punctured disk around $x$.

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

There is a [[bijection]] between

$$
   [X^\ast,G] \backslash [D^\ast,G] / [D,G]
   \simeq
    G Bund(X)_{\sim}
$$

between the [[double coset space]] of [[topological groups]] as shown on the left and the set of [[equivalence classes]] of topological $G$-[[principal bundles]] on $X$.

=--

e.g. ([Sorger 99, prop. 4.1.1](#Sorger99))

+-- {: .proof}
###### Proof

The key observation is that in $X^\ast$ every $G$-bundle trivializes. Therefore

$$
  X^\ast \coprod D \longrightarrow X
$$

is a [[cover]] of $X$ which is good enough in that degree-1 [[nonabelian cohomology|nonabelian]] [[Cech cohomology]] on this cover with [[coefficients]] in $G$ classifies $G$-[[principal bundles]].

For this cover the group $[D^\ast, G]$ is precisely that of Cech cocycles, and $[D \coprod X^\ast, G]$ that of Cech coboundaries.

=--

#### Over complex curves
 {#OverComplexCurves}

(...)

e.g. [Frenkel 05, section 3.2](#Frenkel05)

(...)

#### Over algebraic curves
 {#OverAlgebraicCurves}

Let

* $k$ an [[algebraically closed field]];

* $G$ an [[affine variety|affine]] [[algebraic group]];

* $x\in X$ a [[closed point]];

* $X^\ast = X- \{x\}$;

* $D \coloneqq Spf(k[ [t_x] ])$;

* $D^\ast = \coloneqq Spf(k( (t_x)) )$;

+-- {: .num_prop}
###### Proposition
**(Weil uniformization theorem)**

There is an Equivalence of [[stacks]]

$$
   [X^\ast] \backslash [D^\ast,G]/[D,G]
   \simeq
   Bun_G(X)
$$

between the double [[quotient stack]] as shown on the left and the stack of algebraic $G$-[[principal bundles]] on $X$.

=--

e.g. ([Sorger 99, theorem 5.1.1](#Sorger99))

## Related concepts

* [[Narasimhan–Seshadri theorem]], [[Donaldson-Uhlenbeck-Yau theorem]], [[Kobayashi-Hitchin correspondence]]

[[!include moduli spaces -- contents]]

## References

### General

### Over complex curves

* [[Mudumbai Narasimhan]], [[C. Seshadri]], _Stable and unitary vector bundles on a compact Riemann surface_, Ann. Math. __82__, No. 3 (Nov., 1965), pp. 540-567, [jstor](http://www.jstor.org/pss/1970710), [doi](http://dx.doi.org/10.2307%2F1970710)

* A. Ramanathan, _Stable principal bundles on a compact Riemann surface_, Math Ann __213__, 129-152 (1975).

* {#AtiyahBott83} [[Michael Atiyah]], [[Raoul Bott]], _The Yang-Mills equations over Riemann surfaces_, Philos. Trans. Roy. Soc. London __308__ (1983), 523–615.

* A. Beauville, Y. Laszlo, _Conformal blocks and generalized theta functions_, Comm. Math. Phys. __164__ (1994), 385–419.

### Over algebraic curves

* [[Gerd Faltings]], _Vector bundles on curves_, 1995 Bonn lectures, write up by M. Stoll, [pdf](http://www.mathe2.uni-bayreuth.de/stoll/lecture-notes/vector-bundles-Faltings.pdf)

* [[Gerd Faltings]]], _Moduli-stacks for bundles on semistable curves_,   Math. Ann. __304__, 3 (1996) 489-515; _Stable $G$-bundles and projective connections_, J. Algebraic Geom. __2__, 3 (1993) 507-568, [doi](http://dx.doi.org/10.1007/BF01446303), _Algebraic loop groups and moduli spaces of bundles_, J. Eur. Math. Soc. __5__ (2003), 41-68.

* [[Gerd Faltings]], _Line-bundles on the moduli-space of G-torsors_, lecture at MSRI 2002, [video and pdf](http://www.msri.org/publications/ln/msri/2002/langlands/faltings/2/index.html)

* [[Günter Harder]], [[Mudumbai Narasimhan]], _On the cohomology groups of moduli spaces of vector bundle on curves_, Math Ann. __212__, 215-248 (1975).

* [[Nigel Hitchin]], _Stable bundles and integrable systems_,  Duke Math. J.  __54__  (1987)  pp. 91–114

* V. B. Mehta, [[C. Seshadri]], _Moduli of vector bundles on curves with parabolic structures_, Math. Ann. __248__ (1980), 205–239.

* P. E. Newstead, _Characteristic classes of stable bundles of rank 2 over an algebraic curve_, Trans. Amer. Math. Soc. __169__ (1972), 337–345.

*  S. Ramanan, _The moduli spaces of vector bundles over an algebraic curve_, Math. Ann. __200__ (1973), 69–84.

* [[Carlos Simpson]], _Higgs bundles and local systems_, Publ. Mathématiques de l'IHÉS __75__ (1992), p. 5-95, [numdam](http://www.numdam.org/item?id=PMIHES_1992__75__5_0)

* [[Constantin Teleman]], C. T. Woodward, _The index formula for the moduli
of G-bundles on a curve_, Ann. Math. __170__, 2, 495–527 (2009) [pdf](http://annals.princeton.edu/annals/2009/170-2/annals-v170-n2-p01-p.pdf)

* {#Sorger99} [[Christoph Sorger]], _Lectures on moduli  of principal $G$-bundles over algebraic curves_, 1999 ([pdf](http://users.ictp.it/~pub_off/lectures/lns001/Sorger/Sorger.pdf))

* {#Heinloth09} [[Jochen Heinloth]], _Uniformization of $\mathcal{G}$-bundles_ ([pdf](http://staff.science.uva.nl/~heinloth/Uniformization_17-8-09.pdf))

* [[Jonathan Wang]], _The moduli stack of $G$-bundles_, [arXiv:1104.4828](http://arxiv.org/abs/1104.4828v1).

* References for moduli spaces of bundles over *singular* curves are discussed at MathOverflow [here](http://mathoverflow.net/questions/20702/reference-request-moduli-spaces-of-bundles-over-singular-curves/)

### Over arithmetic curves

* {#Hoffmann02} Norbert Hoffmann, _On vector bundles over algebraic and arithmetic curves_, 2002 ([pdf](http://www.maths.mic.ul.ie/hoffmann/diss.pdf))

Review in the context of [[geometric Langlands duality]] is in

* {#Frenkel05} [[Edward Frenkel]], _Lectures on the Langlands Program and Conformal Field Theory_, in _Frontiers in number theory, physics, and geometry II_, Springer Berlin Heidelberg, 2007. 387-533. ([arXiv:hep-th/0512172](http://arxiv.org/abs/hep-th/0512172))

[[!redirects moduli spaces of bundles]]

[[!redirects moduli stack of bundles]]
[[!redirects moduli stacks of bundles]]

[[!redirects moduli space of G-principal bundles]]
[[!redirects moduli spaces of G-principal bundles]]
[[!redirects moduli stack of G-principal bundles]]
[[!redirects moduli stacks of G-principal bundles]]

[[!redirects moduli space of principal bundles]]
[[!redirects moduli spaces of principal bundles]]
[[!redirects moduli stack of principal bundles]]
[[!redirects moduli stacks of principal bundles]]

[[!redirects moduli stack of vector bundles]]
[[!redirects moduli stacks of vector bundles]]

[[!redirects moduli stack of line bundles]]
[[!redirects moduli stacks of line bundles]]


moduli space of Calabi-Yau spaces

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

#Contents#
* table of contents
{:toc}

## Idea

A [[moduli space]] of [[Calabi-Yau spaces]].

## Related concepts

* [[Weil-Petersson metric]]

* [[flop transition]]

* [[mirror symmetry]]

* [[Calabi-Yau cohomology]]

[[!include moduli spaces -- contents]]

## References

* [[Shing-Tung Yau]], _[Moduli of high dimensional Calabi-Yau manifolds](http://www.scholarpedia.org/article/Calabi-Yau_manifold#Moduli_of_high_dimensional_Calabi-Yau_manifolds)_

* [[Zhiqin Lu]], _Calabi-Yau moduli space_, 2014 ([pdf](http://www.math.uci.edu/~zlu/research/cy_moduli.pdf))

* [[Paul Aspinwall]], [[Brian Greene]], [[David Morrison]], _Calabi-Yau Moduli Space, Mirror Manifolds and Spacetime Topology Change in String Theory_, Nucl.Phys. B416 (1994) 414-480 ([arXiv:hep-th/9309097](http://arxiv.org/abs/hep-th/9309097))

Discussion of relation to [[Shimura varieties]] includes

* Alice Garbagnati, Bert van Geemen, _Examples of Calabi-Yau threefolds parametrised by Shimura varieties_ ([arXiv:1005.0478](http://arxiv.org/abs/1005.0478))

[[!redirects moduli spaces of Calabi-Yau spaces]]

[[!redirects moduli space of Calabi-Yau manifolds]]
[[!redirects moduli spaces of Calabi-Yau manifolds]]

[[!redirects moduli space of Calabi-Yau varieties]]
[[!redirects moduli spaces of Calabi-Yau varieties]]


moduli space of complex structures

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

#Contents#
* table of contents
{:toc}

## Idea

A _[[moduli space]]/[[moduli stack]] of [[complex structures]]_ on a given [[topological manifold]]/[[smooth manifold]].

For complex structures on 2-dimensional manifolds the moduli space of complex structures modulo [[homeomorphisms]] [[isotopy|isotopic]] to the identity is _[[Teichmüller space]]_, and the further [[orbifold quotient]] of that by the [[mapping class group]] is an [[orbifold]]/[[Deligne-Mumford stack]] called the Riemann [[moduli stack of complex curves]].

Specifically for once-punctured 2-dimensional manifolds of [[genus]] 1 ([[tori]]) this is equivalently the [[moduli stack of elliptic curves]] over the complex numbers.

## Related concepts

* [[Teichmüller theory]]

* [[Kodaira-Spencer theory]]

[[!include moduli spaces -- contents]]

## References

The _[[moduli space]] of [[complex structures]]_ on a [[manifold]] is discussed for instance from page 175 on of

* [[Yongbin Ruan]], _Symplectic topology and complex surfaces_ in _Geometry and analysis on complex manifolds_ (1994)

and in

* Yurii M. Burman, _Relative moduli spaces of complex structures: an example_ ([arXiv:math/9903029](http://arxiv.org/abs/math/9903029))

[[!redirects moduli spaces of complex structures]]

[[!redirects moduli stack of complex structures]]
[[!redirects moduli stacks of complex structures]]


moduli space of conformal structures

#Contents#
* table of contents
{:toc}

## Idea

a [[moduli space]] of [[conformal structures]]

(Typically a [[moduli stack]] and typically an [[orbifold]]/[[Deligne-Mumford stack]].)

## Related concepts

* [[Teichmüller theory]]

[[!include moduli spaces -- contents]]

## References

See also the references at _[[moduli space of complex structures]]_.

[[!redirects moduli spaces of conformal structures]]

[[!redirects moduli stack of conformal structures]]
[[!redirects moduli stacks of conformal structures]]


moduli space of connections

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[moduli space]] of [[connections on bundles]] over some prescribed [[space]].

Often one considers [[flat connections]] only, see at _[[moduli space of flat connections]]_.

## Properties

### Compactness

If $\Sigma$ is a [[compact topological space|compact]] [[smooth manifold]], then the moduli space of flat connections over $\Sigma$ is compact.

### Over complex manifolds / complex varieties

Over a [[complex manifold]]/[[complex variety]], the [[Koszul-Malgrange theorem]] identifies holomorphic [[flat connections]] on [[complex vector bundles]] with [[holomorphic vector bundles]]. This identifies the moduli space of flat connections as a [[complex manifold]] with (a non-abelian version of) the first Griffiths [[intermediate Jacobian]]. See at _[intermediate Jacobian -- Examples -- k=0](http://nlab.mathforge.org/nlab/show/intermediate+Jacobian#ExamplePicard)_.

More specifically over a [[Riemann surface]] _[[Narasimhan–Seshadri theorem]]_ identifies the [[moduli spaces of flat connections]] with that of certain [[complex manifold|complex]] spaces of [[stable vector bundle|stable]] [[holomorphic vector bundles]].  This space appears as the [[phase space]] for [[Chern-Simons theory]] over that surface. See there for more.

More generally, the [[Donaldson-Uhlenbeck-Yau theorem]] similarly gives a [[Kähler structure]] on the moduli space of flat connections also over higher dimensional Kähler manifolds ([Scheinost-Schottenloher 96, corollary 1.16](#ScheinostSchottenloher96)).

## Examples

### Flat connections over a torus
  {#FlatConnectionsOverATorus}

Let $G$ be a [[compact Lie group]]. Assume either that $G$ is [[simply connected]] or is a [[torus]] (what we really need below is that any two commuting elements in $G$ sit jointly in one [[maximal torus]]).

The moduli space of $G$ [[flat connections]] on a 2-dimensional [[torus]] $A \simeq S^1 \times S^1$ (e.g. underlying a complex [[elliptic curve]]) has the following description:

first, the [[moduli stack]] of flat connections is

$$
  \begin{aligned}
    [\Pi(A), \mathbf{B}G]
    & \simeq [B [A, S^1], \mathbf{B}G]
    \\
    & \simeq Hom_{Grp}(\mathbb{Z} \times \mathbb{Z}, G)//_{ad} G
  \end{aligned}
$$

(see also the discussion at _[characters and fundamental groups of tori](http://ncatlab.org/nlab/show/group+character#CharactersAndFundamentalGroupsOfTori)_).

Here a single [[flat connection]] is just a choice of pair of two commuting elements in $G$, and $G$ acts on that by [[conjugation]]. Now any two commuting elements can be taken to sit in a [[maximal torus]] $T \hookrightarrow G$, and up to [[conjugation]] we can take this to be one fixed maximal torus. This means that the moduli space is actually

$$
  \pi_0 \left(
    Hom_{Grp}([A,S^1], G)//_{ad} G
  \right)
  \simeq
  Hom_{Grp}([A, S^1], T)/W
  \,,
$$

where $W$ is the [[Weyl group]] $W = N_G(T)/T$.

Moreover, by [[Pontryagin duality]] this may be re-expressed as

$$
  \cdots \simeq Hom_{Grp}([T,S^1], A)/W
  \,.
$$

where now $[T,S^1]$ is the [[character group]] of the [[maximal torus]].

In this form the moduli space of flat connections appears prominently for instance in the discussion of [[equivariant elliptic cohomology]]. But beware that the above interpretation in [[algebraic geometry]] is at least more subtle, see ([Lurie 15](#Lurie15)).

## Related concepts

* [[Donaldson-Uhlenbeck-Yau theorem]]

* [[Tamagawa numbers]]

* [[Hitchin connection]]

[[!include moduli spaces -- contents]]

## References

Original articles include

* [[Michael Atiyah]], [[Raoul Bott]], _The Yang-Mills equations over Riemann surfaces_, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615 ([jstor](http://www.jstor.org/stable/37156), [lighning summary](http://math.stackexchange.com/a/295505/58526))

* [[Nigel Hitchin]], _Flat connections and geometric quantization_, Comm.Math.Phys., 131 (1990) 347-380

* Nan-Kuo Ho, Chiu-Chu Melissa Liu, _On the connectedness of moduli spaces of flat connections over compact surfaces_, Canad. J. Math. 56(2004) 1228-1236 [doi](https://doi.org/10.4153/CJM-2004-053-3)

Reviews and lecture notes include, for the case of flat connections

* Remi Janner, _Notes on the moduli space of flat connections_, 2005 ([pdf](https://www.researchgate.net/publication/265223952_Notes_on_the_moduli_space_of_flat_connections))

* Daan Michiels, _Moduli spaces of flat connections_, Master Thesis Leuven 2013 ([pdf](http://www.staff.science.uu.nl/~Schat001/Daan_thesis.pdf))

* {#Teschner14} [[Jörg Teschner]], _Quantization of moduli spaces of flat connections and Liouville theory_, proceedings of the International Congress of Mathematics 2014 ([arXiv:1405.0359](http://arxiv.org/abs/1405.0359))

* {#FockGoncharov14} [[Vladimir Fock]], [[Alexander Goncharov]], _Symplectic double for moduli spaces of G-local systems on surfaces_ ([arXiv:1410.3526](http://arxiv.org/abs/1410.3526))

and for the case of general and [[logarithmic connections]]

* {#BiswasMunoz} Indranil Biswas, V.Munoz, _Moduli spaces of connections on a Riemann surface_  ([pdf](http://www.mat.ucm.es/~vmunozve/Moduli.pdf))

Detailed discussion of moduli space of flat connections also on higher dimensional base spaces is in

* {#ScheinostSchottenloher96} Peter Scheinost, [[Martin Schottenloher]], pp. 154 (11 of 76) of _Metaplectic quantization of the moduli spaces of flat and parabolic bundles_, J. reine angew. Mathematik, 466 (1996) ([web](https://eudml.org/doc/153753))

For more references see at _[[Hitchin connection]]_.

Discussion in [[algebraic geometry]] includes

* {#Lurie15} [[Jacob Lurie]], _[MO comment](http://mathoverflow.net/a/226716/381)_ 2015

[[!redirects moduli spaces of connections]]

[[!redirects moduli space of flat connections]]
[[!redirects moduli spaces of flat connections]]

[[!redirects moduli stack of connections]]
[[!redirects moduli stacks of connections]]

[[!redirects moduli stack of flat connections]]
[[!redirects moduli stacks of flat connections]]


moduli space of curves

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

#Moduli space of curves#
* table of contents
{:toc}

## Idea

The [[moduli space]]/[[moduli stack]] of [[algebraic curves]]/[[Riemann surfaces]] is a sort of space of parameters parametrizing [[algebraic curves]] of given [[genus]].

[Deligne-Mumford 69](#DeligneMumford69) a nontrivial [[compactification]] of a moduli space of Riemann surfaces of fixed genus which is a [[Deligne-Mumford stack]], since called the _[[Deligne-Mumford compactification]]_.

There is also a decorated version of curves with marked points, and
of the corresponding compactified moduli space of stable curves of genus $g$ with $n$ marked points $\mathcal{M}_{g,n}$ which plays an important role in the mathematical study of [[Gromov-Witten invariants]] and of [[conformal blocks]].

The special case of $g = 1$, $n =1$ is the [[moduli stack of elliptic curves]] $\mathcal{M}_{1,1}= \mathcal{M}_{ell}$.

## Properties

### Over the complex numbers
  {#OverTheComplexNumbers}

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

On 2-[[dimension|dimensional]] [[manifold]] every [[almost complex structure]] is integrable and hence is already a [[complex structure]] (see [this proposition](complex+structure#In2dAnyAlmostComplexStructureIsIntegrable)).

Hence the moduli space of complex surfaces is also equivalent to just the [[quotient orbifold]] of the space of [[almost complex structures]] by the orientation-preserving part of the [[diffeomorphism group]].

=--

This is reviewed for instance in ([Madsen 07, section 1.1](#Madsen07)).

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

We indicate how via remark \ref{EquivalentlyModuliOfAlmostComplexStructures} the moduli space of complex curves has a formulation in the [[homotopy type theory|homotopy-type theory]] $\mathbf{H}$ of [[smooth homotopy types]].

By the discussion at _[[almost complex structure]]_ (see [this remark](complex+structure#InTermsOfSmoothStacks)), if the [[tangent bundle]] of a $2n$-dimensional [[smooth manifold]] is modulated by a map

$$
  \tau_X \;\colon\;  X \longrightarrow \mathbf{B}GL(2n,\mathbb{R})
$$

to the [[delooping]] in [[smooth stacks]] of the [[general linear group]], then an [[almost complex structure]] on $X$ is equivalently a lift $J$ in

$$
  \array{
     X && \stackrel{J}{\longrightarrow} && \mathbf{B} GL(n,\mathbb{C})
     \\
     & {}_{\mathllap{\tau}}\searrow && \swarrow_{\mathrlap{almComp}}
     \\
     && \mathbf{B} GL(2n,\mathbb{R})
  }
  \,.
$$

This in turn is equivalently a map

$$
  J \;\colon\; \tau_X \longrightarrow almComp
$$

in the [[slice (∞,1)-topos]] $\mathbf{H}_{/\mathbf{B}GL(2n,\mathbb{R})}$, hence with the canonical empedding

$$
  \tau_{(-)} \;\colon\; SmthMfd_{2n}^{et} \hookrightarrow \mathbf{H}_{/\mathbf{B}GL(2n,\mathbb{R})}
$$

of the category of [[smooth manifolds]] with [[local diffeomorphisms]] between them
understood, then $almComp \in \mathbf{H}_{/\mathbf{B}GL(2n,\mathbb{R})}$ is the universal [[moduli stack]] of [[almost complex structures]].

Now $\tau_X$ carries a canonical [[∞-action]] by the [[diffeomorphism group]]. Using this one may canonically form the [[homotopy quotient]]

$$
  [\tau_X, almComp]//Diff(X)
$$

by a general abstract construction that is discussed in some detail at _[general covariant -- Formalization in homotopy type theory](general+covariance#InHomotopyTypeTheory)_. For $n =1$ this is hence the Riemann moduli space.

=--

### Homotopy type and cohomology
 {#HomotopyTypeAndCohomology}

The [[homotopy type]] of the Riemann moduli spaces $\mathcal{M}_g$ (i.e. of the [[geometric realization]] of the [[orbifold]], hence essentially of the delooping of the [[mapping class group]]) is essentially unknown.  Even its [[orbifold cohomology]] over the [[rational numbers]] is fully known only for $g \leq 4$.

However, in a stable range it has been fully computed in ([Madsen-Weiss 02](#MadsenWeiss02)), proving what was previously [[conjecture|conjectured]] by [[Mumford's conjecture]]. A review is in ([Madsen 07](#Madsen07)).

### Orbifold Euler characteristic
 {#OrbifoldEulerCharacteristic}

The [[orbifold Euler characteristic]] $\chi$ of $\mathcal{M}_{g,1}$ is given by the [[Riemann zeta function]] at negative integral values  as follows ([Zagier-Harer 86](#ZagierHarer86)):

$$
  \chi(\mathcal{M}_{g,1}) = \zeta(1-2g)
  \,.
$$

By the expression of the [[Riemann zeta function]] at negative integral values by the [[Bernoulli numbers]] $B_n$, this says equivalently that

$$
  \chi(\mathcal{M}_{g,1}) = -\frac{B_{2g}}{2g}
  \,.
$$

For instance for $g = 1$ (once punctured complex [[tori]], hence complex [[elliptic curves]]) this yields

$$
  \chi(\mathcal{M}_{1,1}) = -\frac{1}{12}
$$

for the [[orbifold Euler characteristic]] of the [[moduli space of elliptic curves]].

### Relation to Teichmüller space

Over the complex number then the moduli space of curves is a quotient of _[[Teichmüller space]]_. ([Hubbard-Koch 13](#HubbardKoch13))

## Examples

### $g = 0$, $n = 0$

By the [[Riemann mapping theorem]], $\mathcal{M}_{0,0}\simeq \ast$ is the point.

### $g = 1$, $n = 0$

$\mathcal{M}_{1,0} \simeq \mathbb{R}^2$

### $g = 1$, $n = 1$

$\mathcal{M}_{1,1} = \mathcal{M}_{ell}$, the [[moduli stack of elliptic curves]].

### $g \geq 2$, $n = 0$

For [[genus]] $g\geq 2$ the moduli stack of [[complex structures]] is equivalently that of [[hyperbolic metrics]]. This way a lot of [[hyperbolic geometry]] is used in the study of $\mathcal{M}_{g \geq 2, n}$.

## Related entries

* [[moduli space]], [[moduli stack]], [[mapping class group]], [[enumerative geometry]], [[Mumford class]], [[Deligne-Mumford stack]]

* [[moduli space of super Riemann surfaces]]

* [[moduli space of (higher) line bundles]]

* [[Witten conjecture]]

## References

Original articles:

* {#DeligneMumford69} [[Pierre Deligne]], [[David Mumford]], _The irreducibility of the space of curves of given genus_, Publications Mathématiques de l'IHÉS (Paris) 36: 75–109 (1969) [numdam](http://www.numdam.org/numdam-bin/fitem?id=PMIHES_1969__36__75_0)

* [[David Mumford]], _Towards an enumerative geometry of the moduli space of curves_, Arithmetic and geometry, Vol. II, Birkhäuser Boston, Boston, MA, 1983, pp. 271–328, [MR85j:14046](http://www.ams.org/mathscinet-getitem?mr=85j:14046)

On [[moduli spaces of curves]]/[[moduli spaces of Riemann surfaces]] with emphasis on their [[orbifold]]-structure:

* Lizhen Ji, [[Shing-Tung Yau]], _Transformation Groups and Moduli Spaces of Curves_, International Press of Boston 2011 ([ISBN:9781571462237](https://www.intlpress.com/site/pub/pages/books/items/00000353/index.php))

Discussion of the [[orbifold cohomology]] of the moduli stack is in

* John Harer, _The cohomology of the moduli space of curves_, Lec. Notes in Math. 1337, p. 138–221. Springer, Berlin, 1988.

The stable rational [[orbifold cohomology]] of the moduli stack was computed in

* {#MadsenWeiss02} [[Ib Madsen]], [[Michael Weiss]], _The stable moduli space of Riemann surfaces: [[Mumford's conjecture]]_,  Ann. of Math. (2) __165__ (2007), no. 3, 843--941, [MR2009b:14051](http://www.ams.org/mathscinet-getitem?mr=2009b:14051), [doi](http://dx.doi.org/10.4007/annals.2007.165.843), [math.AT/0212321](http://arxiv.org/abs/math.AT/0212321)

exposition of this is in

* {#Madsen07} [[Ib Madsen]], _Moduli spaces from a topological viewpoint_, Proceedings of the International Congress of Mathematics, Madrid 2006 (2007) ([pdf](http://www.icm2006.org/proceedings/Vol_I/20.pdf))

See also

* Gabriele Mondello, _Combinatorial classes on $\mathcal{M}_{g,n}$ are tautological_, Int. Math. Res. Not. __44__ (2004), 2329-–2390, [MR2005g:14056](http://www.ams.org/mathscinet-getitem?mr=2005g:14056), [doi](http://dx.doi.org/10.1155/S1073792804131462), [math.AG/0303207](http://arxiv.org/abs/math/0303207)

* Gabriele Mondello, _Riemann surfaces, ribbon graphs and combinatorial classes_, in: Handbook of Teichmüller theory. Vol. II,  151--215, IRMA Lect. Math. Theor. Phys., 13, Eur. Math. Soc., Zürich, 2009; draft with index: [pdf](http://www.mat.uniroma1.it/~mondello/me/papers/ober-definitive.pdf), arxiv version [math.AG/0705.1792](http://arxiv.org/abs/0705.1792), [MR2010f:32012](http://www.ams.org/mathscinet-getitem?mr=2010f:32012)

* [[Alastair Hamilton]], _Classes on compactifications of the moduli space of curves through solutions to the quantum master equation_, Lett. Math. Phys. __89__  (2009), no. 2, 115--130.

* _The moduli of curves_ [pdf](http://www2.imperial.ac.uk/~acorti/download/moduli.pdf)

The [[orbifold Euler characteristic]] of the moduli space of curves was originally computed in

* {#ZagierHarer86} [[Don Zagier]], John Harer, _The Euler characteristic of the moduli space of curves_, Inventiones mathematicae (1986) Volume: 85, page 457-486 ([EUDML](https://eudml.org/doc/143377))

The complex analytic structure and the relation to [[Teichmüller space]] is further discussed in

* {#HubbardKoch13} [[John Hubbard]], [[Sarah Koch]], _An analytic construction of the Deligne-Mumford compactification of the moduli space of curves_ ([arXiv:1301.0062](http://arxiv.org/abs/1301.0062))

Reviews include

* _Mathematical ideas and notions in quantum field theory -- 5. The Euler characteristic of the moduli space of curves_ ([[EulerCharacteristicOfSpaceOfCurves.pdf:file]])

[[etale homotopy type|Étale homotopy type]] of moduli stacks of curves is discussed in

* Paola Frediani, Frank Neumann, _Étale homotopy types of moduli stacks of algebraic curves with symmetries_, K-Theory 30: 315-340, 2003 ([arXiv:math/0404387](http://arxiv.org/abs/math/0404387))

[[!redirects moduli spaces of curves]]

[[!redirects moduli space of Riemann surfaces]]
[[!redirects moduli spaces of Riemann surfaces]]

[[!redirects moduli space of complex curves]]
[[!redirects moduli spaces of complex curves]]
[[!redirects moduli stack of complex curves]]
[[!redirects moduli stacks of complex curves]]

[[!redirects moduli stack of curves]]
[[!redirects moduli stacks of curves]]

[[!redirects moduli stack of Riemann surfaces]]
[[!redirects moduli stacks of Riemann surfaces]]

[[!redirects moduli stack of algebraic curves]]
[[!redirects moduli stacks of algebraic curves]]

[[!redirects Riemann moduli space]]
[[!redirects Riemann moduli spaces]]

[[!redirects moduli space of punctured curves]]
[[!redirects moduli spaces of punctured curves]]


moduli space of flat connections

[[!redirects moduli space of flat connection]]

#Contents#
* table of contents
{:toc}

## Idea

Given a [[space]] $X$ and a [[group]] $G$, a $G$-[[flat connection]] or $G$-[[local system]] is a map $X \to \flat \mathbf{B}G$, or equivalently a map $\Pi(X) \to \mathbf{B}G$. A _moduli space of flat connections_ is a [[moduli space]]/[[moduli stack]] of such structure.

## Examples

The moduli space of flat connections for suitable [[Lie groups]] over [[Riemann surfaces]] appears as the [[phase space]] of $G$-[[Chern-Simons theory]] over these surfaces. It carries itself a [[projectively flat connection]], the _[[Knizhnik-Zamolodchikov connection]]_ or _[[Hitchin connection]]_.

## Properties

The [[Narasimhan–Seshadri theorem]] asserts that the moduli space of [[flat connections]] on a [[Riemann surface]] is naturally a [[complex manifold]].

## Related concepts

* [[geometric Langlands correspondence]]

[[!include moduli spaces -- contents]]

## References

* {#Teschner14} [[Jörg Teschner]], _Quantization of moduli spaces of flat connections and Liouville theory_, proceedings of the International Congress of Mathematics 2014 ([arXiv:1405.0359](http://arxiv.org/abs/1405.0359))

For more see the references at _[[moduli space of connections]]_.

[[!redirects moduli spaces of flat connections]]

[[!redirects moduli stack of flat connections]]
[[!redirects moduli stacks of flat connections]]

[[!redirects moduli stack of local systems]]


moduli space of framed manifolds

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[moduli space]] $\mathcal{M}^{fr}_{\Sigma}$ of [[framed manifolds|framings]] on a given [[manifold]] $\Sigma$.

The [[homotopy type]] of the [[connected component]] of any fixed [[framed manifold|framing]] $\phi$ is

$$
  {\vert \mathcal{M}^{fr}_{\Sigma} \vert}_\phi \simeq B \Gamma^{fr}(\Sigma,\phi)
  \,,
$$

where $\Gamma^{fr}(\Sigma,\phi)\hookrightarrow \Gamma(\Sigma, or(\phi))$ is the [[subgroup]] of the [[mapping class group]] of $\Sigma$ (for the given [[orientation]]) which fixes the [[isotopy]] class of the framing $[\phi]$.

In particular this is a [[homotopy 1-type]] for every $\phi$, and so the whole

$$
  {\vert \mathcal{M}^{fr}_{\Sigma} \vert}
  =
  \underset{[\phi]}{\coprod} {\vert \mathcal{M}^{fr}_{\Sigma} \vert}_\phi
$$

is a homotopy 1-type.

([Randal-Williams 10, section 1.1, 2.2](#RandalWilliams10))

## Related concepts

* [[cobordism category]], [[cobordism hypothesis]]

[[!include moduli spaces -- contents]]

## References

### General

* Schumacher, Tsuji, section 8 of _Quasi-projectivity of moduli spaces of polarized varieties_ [pdf](http://annals.math.princeton.edu/wp-content/uploads/annals-v159-n2-p03.pdf)

### Moduli space of framed surfaces

*  {#RandalWilliams10} [[Oscar Randal-Williams]], _Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces_ ([arXiv:1001.5366](http://arxiv.org/abs/1001.5366))

* Daan Krammer, _A Garside like structure on the framed mapping class group_, 2007 ([pdf](http://homepages.warwick.ac.uk/~masbal/index_files/fr.pdf))

[[!redirects moduli space of framed surfaces]]


moduli space of monopoles

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Chern-Weil theory
+-- {: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

By the [[Nahm transform]], the [[moduli space]] of $x^4$-translation invariant [[self-dual Yang-Mills theory]] [[solitons]] on 4d [[Euclidean space]] $\mathbb{R}^{4}$ is equivalently the space of solutions to the [[Bogomolny equations]] on 3d [[Euclidean space]], which in turn may be thought of as [[magnetic monopoles]] in 3d [[Euclidean field theory|Euclidean]] [[Yang-Mills theory]] coupled to a charged [[scalar field]] (a "[[Higgs field]]"). Therefore this moduli space is traditionally referred to simply as the _moduli space of magnetic monopoles_ (e.g. [Atiyah-Hitchin 88](#AtiyahHitchin88)) or just the _moduli space of monopoles_.

## Definition

The moduli space

\[
  \label{ModuliSpaceOfkMonopoles}
  \mathcal{M}_k
  \;\coloneqq\;
  \cdots
\]

of $k$ monopoles is ... ([Atiyah-Hitchin 88, p. 15-16](#AtiyahHitchin88)).

## Properties

### Scattering amplitudes of monopoles
 {#ScatteringAmplitudesofMonopoles}

Write

\[
  \label{SpaceOfRationalFunctionsOfDegreek}
  Maps_{cplx\,rtnl}^{\ast/}\big( \mathbb{C}P^1, \mathbb{C}P^1 \big)_k
  \;\subset\;
  Maps_{cplx\,rtnl}^{\ast/}\big( \mathbb{C}P^1, \mathbb{C}P^1 \big)
  \;\subset\;
  Maps^{\ast/}\big( S^2, S^2 \big)
\]

for the [[space of maps|space of]] [[pointed topological space|pointed]] [[rational functions]] from the [[Riemann sphere]] to itself, of [[degree of a continuous function|degree]] $k \in \mathbb{N}$, inside the full [[Cohomotopy]] [[cocycle space]]. The [[homotopy type]] of this mapping space is discussed in [Segal 79](#Segal79), see *[[homotopy of rational maps]]*.

To each configuration $ c \in \mathcal{M}_k$ of $k \in \mathbb{N}$ magnetic monopoles is associated a [[scattering amplitude]]

\[
  \label{ScatteringAmplitudes}
  S(c)
    \in
  Maps_{cplx\,rtnl}^{\ast/}\big( \mathbb{C}P^1, \mathbb{C}P^1 \big)_k

\]

([Atiyah-Hitchin 88 (2.8)](#AtiyahHitchin88))

\linebreak

### Charge quantization in Cohomotopy

+-- {: .num_prop #ModuliSpaceOfKMonopolesIsSpaceOfComplexRationalFunctions}
###### Proposition
**([[moduli space of monopoles|moduli space of k monopoles]] is [[space of maps|space of]] [[degree of a continuous function|degree]] $k$ [[complex number|complex]]-[[rational functions]] from [[Riemann sphere]] to itself)**

The assignment (eq:ScatteringAmplitudes) is a [[diffeomorphism]]
identifying the moduli space (eq:ModuliSpaceOfkMonopoles) of $k$ magnetic monopoles with the space (eq:SpaceOfRationalFunctionsOfDegreek) of complex-[[rational functions]] from the [[Riemann sphere]] to itself, of [[degree of a continuous function|degree]] $k$ (hence the [[cocycle space]] of complex-rational 2-[[Cohomotopy]])

$$
  \mathcal{M}_k
  \;
  \underoverset{
    \simeq_{diff}
  }{
    S
  }{
    \;\;\longrightarrow\;\;
  }
  \;
  Maps_{cplx\,rtnl}^{\ast/}
  \big(
    \mathbb{C}P^1, \mathbb{C}P^1
  \big)_k
$$

=--

(due to [Donaldson 84](#Donaldson84), see also [Atiyah-Hitchin 88, Theorem 2.10](#AtiyahHitchin88)).

+-- {: .num_prop #SpaceOfComplexRationalMapsOnRiemannSpherekEquivalentToCohomotopyCocycleSpace}
###### Proposition
**([[space of maps|space of]] [[degree of a continuous function|degree]] $k$ [[complex number|complex]]-[[rational functions]] from [[Riemann sphere]] to itself is [[n-equivalence|k-equivalent]] to [[Cohomotopy]] [[cocycle space]] in degree $k$)**

The inclusion of the complex rational self-maps maps of [[degree of a continuous function|degree]] $k$ into the full based [[space of maps]] of [[degree of a continuous function|degree]] $k$ (hence the $k$-component of the second [[iterated loop space]] of the [[2-sphere]], and hence the plain [[Cohomotopy]] [[cocycle space]]) induces an [[isomorphism]] of [[homotopy groups]] in degrees $\leq k$ (in particular a [[n-equivalence|k-equivalence]]):

$$
  Maps_{cplx\,rtnl}^{\ast/}
  \big(
    \mathbb{C}P^1, \mathbb{C}P^1
  \big)_k
  \;
  \underset{
    \simeq_{\leq k}
  }{
    \;\;\hookrightarrow\;\;
  }
  \;
  Maps^{\ast/}\big( S^2, S^2 \big)_k
$$

=--

([Segal 79, Prop. 1.1](#Segal79), see at *[[homotopy of rational maps]]*)

Hence, Prop. \ref{ModuliSpaceOfKMonopolesIsSpaceOfComplexRationalFunctions} and
Prop. \ref{SpaceOfComplexRationalMapsOnRiemannSpherekEquivalentToCohomotopyCocycleSpace} together say that the moduli space of $k$-monopoles is $k$-equivalent to the Cohomotopy cocycle space $\mathbf{\pi}^2\big( S^2 \big)_k$.

$$
  \mathcal{M}_k
  \;
  \underoverset{
    \simeq_{diff}
  }{
    S
  }{
    \;\;\longrightarrow\;\;
  }
  \;
  Maps_{cplx\,rtnl}^{\ast/}
  \big(
    \mathbb{C}P^1, \mathbb{C}P^1
  \big)_k
  \;
  \underset{
    \simeq_{\leq k}
  }{
    \;\;\hookrightarrow\;\;
  }
  \;
  Maps^{\ast/}\big( S^2, S^2 \big)_k
$$

This is a [[non-abelian group|non-abelian]] analog of the [[Dirac charge quantization]] of the [[electromagnetic field]], with [[ordinary cohomology]] replaced by [[Cohomotopy]] [[generalized cohomology theory|cohomology theory]]:

{#Illustration} $\,$

\linebreak

### Relation to braid groups

+-- {: .num_prop #ModuliSpaceOfkMonopolesStablyWeakHomotopyEquivbalentToClassifyingSpaceOfBraids}
###### Proposition
**([[moduli space of monopoles]] is [[stable weak homotopy equivalence|stably weak homotopy equivalent]] to [[classifying space]] of [[braid group]])**

For $k \in \mathbb{N}$ there is a [[stable weak homotopy equivalence]] between the [[moduli space of k monopoles]] (eq:ModuliSpaceOfkInstantons) and the [[classifying space]] of the [[braid group]] $Braids_{2k}$ on $2 k$ strands:

$$
  \Sigma^\infty \mathcal{M}_k
  \;\simeq\;
  \Sigma^\infty Braids_{2k}
$$

=--

([Cohen-Cohen-Mann-Milgram 91](#CohenCohenMannMilgram91))

\linebreak

### Geometric engineering by D$p$-D$(p+2)$-brane intersections
 {#GeometricEngineeringByIntersectingBranes}

Generally [[Dp-D(p+2)-brane intersections]] [[geometric engineering of QFT|geometrically engineer]] [[Yang-Mills monopoles]] in the [[worldvolume]] of the higher dimensional D$(p+2)$-branes.

Specifically for $p = 6$, i.e. for [[D6-D8-brane intersections]], this fits with the [[Witten-Sakai-Sugimoto model]] [[geometric engineering of QFT|geometrically engineering]] [[quantum chromodynamics]], and then gives a [[geometric engineering of QFT|geometric engineering]] of the [[Yang-Mills monopoles]] in actual [[QCD]] ([HLPY 08, p. 16](D6-D8-brane+bound+state#HLPY08)).

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

Here we are showing

1. the [[color charge|color]] [[D4-branes]];

1. the [[flavour (particle physics)|flavor]] [[D8-branes]];

   with

   1. the [[5d Chern-Simons theory]] on their [[worldvolume]]

   1. the corresponding [[4d WZW model]] on the boundary

   both exhibiting the [[meson fields]]

1. the [[baryon]] [[D4-branes]]

   (see below at _[WSS -- Baryons](Witten-Sakai-Sugimoto+model#Baryons)_)

1. the [[Yang-Mills monopole]] [[D6-branes]]

   (see at _[[D6-D8-brane bound state]]_)

1. the [[NS5-branes]].

\linebreak

## Related concepts

* [[Nahm transform]], [[Bogomolny equation]]

[[!include moduli spaces -- contents]]

## References
 {#References}

### General
 {#ReferencesGeneral}

* [[Michael Atiyah]], [[Nigel Hitchin]], J. T. Stuart and M. Tabor, _Low-Energy Scattering of Non-Abelian Magnetic Monopoles_, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 315, No. 1533, New Developments in the Theory and Application of Solitons (Aug. 13, 1985), pp. 459-469 ([jstor:37546](https://www.jstor.org/stable/37546))

* {#GibbonsManton86} [[Gary Gibbons]], [[Nicholas Manton]], _Classical and Quantum Dynamics of BPS Monopoles_, Nucl. Phys. B274 (1986) 183-224 ([spire:18322](http://inspirehep.net/record/18322), doi:10.1016/0550-3213(86)90624-3)

* {#AtiyahHitchin88} [[Michael Atiyah]], [[Nigel Hitchin]], _The geometry and dynamics of magnetic monopoles_, M. B. Porter Lectures. Princeton University Press, Princeton, NJ, 1988 ([jstor:j.ctt7zv206](https://www.jstor.org/stable/j.ctt7zv206))

* [[Paul Sutcliffe]], *BPS Monopoles*, Int. J. Mod. Phys. A12:4663-4706 (1997) ([arXiv:hep-th/9707009](https://arxiv.org/abs/hep-th/9707009), [doi:10.1142/S0217751X97002504](https://doi.org/10.1142/S0217751X97002504))

* [[Yakov Shnir]], Part II of: *Magnetic Monopoles*, Springer 2005 ([ISBN:978-3-540-29082-7](https://www.springer.com/gp/book/9783540252771))

See also:

* Wikipedia, _[Monopole moduli space](https://en.wikipedia.org/wiki/Monopole_moduli_space)_

On the [[ordinary cohomology]] of the moduli space of YM-monopoles:

* [[Graeme Segal]], Alex Selby, *The cohomology of the space of magnetic monopoles*, Commun. Math. Phys. 177, 775–787 (1996) ([doi:10.1007/BF02099547](https://doi.org/10.1007/BF02099547))

The special case of [[gauge group]] [[SU(3)]]:

* F. A. Bais, [[Bernd Schroers]], *Quantisation of Monopoles with Non-abelian Magnetic Charge*, Nucl. Phys. B512 (1998) 250-294 ([arXiv:hep-th/9708004](https://arxiv.org/abs/hep-th/9708004))

[[!include Yang-Mills monopoles as rational maps -- references]]

### As transversal D$p$/D$(p+2)$-brane intersections

In [[string theory]] [[Yang-Mills monopoles]] are [[geometric engineering of QFT|geometrically engineeted]] as transversally [[intersecting brane|intersecting]] [[Dp-D(p+2)-brane bound states]]:

For transversal [[D1-D3-brane bound states]]:

* {#Diaconescu97} Duiliu-Emanuel Diaconescu,  _D-branes, Monopoles and Nahm Equations_, Nucl. Phys. B503 (1997) 220-238 ([arxiv:hep-th/9608163](https://arxiv.org/abs/hep-th/9608163))

* [[Amihay Hanany]], [[Edward Witten]], _Type IIB Superstrings, BPS Monopoles, And Three-Dimensional Gauge Dynamics_, Nucl. Phys. B492:152-190, 1997 ([arxiv:hep-th/9611230](https://arxiv.org/abs/hep-th/9611230))

* Jessica K. Barrett, Peter Bowcock, _Using D-Strings to Describe Monopole Scattering_ ([arxiv:hep-th/0402163](https://arxiv.org/abs/hep-th/0402163))

* Jessica K. Barrett, Peter Bowcock, _Using D-Strings to Describe Monopole Scattering - Numerical Calculations_ ([arxiv:hep-th/0512211](https://arxiv.org/abs/hep-th/0512211))

For transversal [[D2-D4 brane intersections]] (with an eye towards [[AdS/QCD]]):

* Alexander Gorsky, Valentin Zakharov, Ariel Zhitnitsky, _On Classification of QCD defects via holography_, Phys. Rev. D79:106003, 2009 ([arxiv:0902.1842](https://arxiv.org/abs/0902.1842))

For transversal [[D3-D5 brane intersections]]:

* {#GaiottoWitten08} [[Davide Gaiotto]], [[Edward Witten]], _Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory_, J Stat Phys (2009) 135: 789 ([arXiv:0804.2902](https://arxiv.org/abs/0804.2902))

For transversal [[D6-D8-brane intersections]] (with an eye towards [[AdS/QCD]]):

* {#HLPY08} Deog Ki Hong, Ki-Myeong Lee, Cheonsoo Park, Ho-Ung Yee, Section V of: _Holographic Monopole Catalysis of Baryon Decay_, JHEP 0808:018, 2008 ([https:arXiv:0804.1326](https://arxiv.org/abs/0804.1326))

With emphasis on [[half NS5-branes]] in [[type I' string theory]]:

* {#HananyZaffaroni99} [[Amihay Hanany]], [[Alberto Zaffaroni]], _Monopoles in String Theory_, JHEP 9912 (1999) 014 ([arxiv:hep-th/9911113](https://arxiv.org/abs/hep-th/9911113))

The moduli space of monopoles appears also in the [[KK-compactification]] of the [[M5-brane]] on a complex surface ([[AGT-correspondence]]):

* Benjamin Assel, [[Sakura Schafer-Nameki]], Jin-Mann Wong, _M5-branes on $S^2 \times M_4$: Nahm's Equations and 4d Topological Sigma-models_, J. High Energ. Phys. (2016) 2016: 120 ([arxiv:1604.03606](https://arxiv.org/abs/1604.03606))

### As Coulomb branches of $D=3$ $\mathcal{N}=4$ SYM

Identification of the [[Coulomb branch]] of [[D=3 N=4 super Yang-Mills theory]] with the [[moduli space of monopoles]] in [[Yang-Mills theory]]:

* {#SeibergWitten96} [[Nathan Seiberg]], [[Edward Witten]], _Gauge Dynamics And Compactification To Three Dimensions_, In: J.M. Drouffe, J.B. Zuber (eds.) _The mathematical beauty of physics: A memorial volume for Claude Itzykson_ Proceedings, Conference, Saclay, France, June 5-7, 1996 ([arXiv:hep-th/9607163](http://arxiv.org/abs/hep-th/9607163), [spire:420925](http://inspirehep.net/record/420925))

* N. Dorey, V. V. Khoze, M. P. Mattis, [[David Tong]], S. Vandoren, _Instantons, Three-Dimensional Gauge Theory, and the Atiyah-Hitchin Manifold_, Nucl. Phys. B502 (1997) 59-93 ([arXiv:hep-th/9703228](https://arxiv.org/abs/hep-th/9703228))

* [[David Tong]], _Three-Dimensional Gauge Theories and ADE Monopoles_, Phys. Lett. B448 (1999) 33-36 ([arXiv:hep-th/9803148](https://arxiv.org/abs/hep-th/9803148))

* [[Mathew Bullimore]], [[Tudor Dimofte]], [[Davide Gaiotto]], _The Coulomb Branch of 3d $\mathcal{N}=4$ Theories_, Commun. Math. Phys. (2017) 354: 671 ([arXiv:1503.04817](https://arxiv.org/abs/1503.04817))

* [[Alexander Braverman]], [[Michael Finkelberg]], [[Hiraku Nakajima]], *Coulomb branches of 3d $\mathcal{N}=4$ quiver gauge theories and slices in the affine Grassmannian* (with appendices by Alexander Braverman, Michael Finkelberg, Joel Kamnitzer, Ryosuke Kodera, Hiraku Nakajima, Ben Webster, and Alex Weekes), Advances in Theoretical and Mathematical Physics Volume 23 (2019) Number 1 ([arXiv:1604.03625](https://arxiv.org/abs/1604.03625))

### Rozansky-Witten invariants

Discussion of [[Rozansky-Witten invariants]] of [[moduli spaces of monopoles]]:

* {#RozanskyWitten96} [[Lev Rozansky]], [[Edward Witten]], p. 36 of: _Hyper-Kähler geometry and invariants of 3-manifolds_, Selecta Math., New Ser. __3__ (1997), 401–458 ([arXiv:hep-th/9612216](https://arxiv.org/abs/hep-th/9612216), [doi:10.1007/s000290050016](https://doi.org/10.1007/s000290050016), [MR98m:57041](http://www.ams.org/mathscinet-getitem?mr=98m:57041))

### Relation to braids

Relation to [[braid groups]]:

* {#CohenCohenMannMilgram91} [[Fred Cohen]], [[Ralph Cohen]], [[B. M. Mann]], [[R. James Milgram]], _The topology of rational functions and divisors of surfaces_, Acta Math (1991) 166: 163 ([doi:10.1007/BF02398886](https://doi.org/10.1007/BF02398886))

* [[Ralph Cohen]], [[John D. S. Jones]], _Monopoles, braid groups, and the Dirac operator_, Comm. Math. Phys. Volume 158, Number 2 (1993), 241-266 ([euclid:cmp/1104254240](https://projecteuclid.org/euclid.cmp/1104254240))

Relation of [[Dp-D(p+2)-brane bound states]] ([hence](Dp-Dp+2-brane+bound+states#ReferencesRelationToMonopoles)
[[Yang-Mills monopoles]]) to [[Vassiliev braid invariants]] via [[chord diagrams]] computing [[radii]] of [[fuzzy spheres]]:

* [[Sanyaje Ramgoolam]], [[Bill Spence]], S. Thomas, Section 3.2 of: _Resolving brane collapse with $1/N$ corrections in non-Abelian DBI_, Nucl. Phys. B703 (2004) 236-276 ([arxiv:hep-th/0405256](https://arxiv.org/abs/hep-th/0405256))

* [[Simon McNamara]], [[Constantinos Papageorgakis]], [[Sanyaje Ramgoolam]], [[Bill Spence]], Appendix A of: _Finite $N$ effects on the collapse of fuzzy spheres_, JHEP 0605:060, 2006 ([arxiv:hep-th/0512145](https://arxiv.org/abs/hep-th/0512145))

* [[Simon McNamara]], Section 4 of: _Twistor Inspired Methods in Perturbative FieldTheory and Fuzzy Funnels_, 2006 ([spire:1351861](http://inspirehep.net/record/1351861), [pdf](https://strings.ph.qmul.ac.uk/sites/default/files/Mcnamaraphd.pdf), [[McNamara06.pdf:file]])

* [[Constantinos Papageorgakis]], p. 161-162 of: _On matrix D-brane dynamics and fuzzy spheres_, 2006 ([[Papageorgakis06.pdf:file]])

[[!redirects moduli spaces of monopoles]]

[[!redirects moduli space of Yang-Mills monopoles]]
[[!redirects moduli spaces of Yang-Mills monopoles]]

[[!redirects moduli space of k monopoles]]
[[!redirects moduli spaces of k monopoles]]

[[!redirects Atiyah-Hitchin charge quantization]]

[[!redirects Yang-Mills monopole]]
[[!redirects Yang-Mills monopoles]]

[[!redirects monopole in Yang-Mills theory]]
[[!redirects monopoles in Yang-Mills theory]]


moduli space of Riemannian metrics

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[moduli space]] of ([[pseudo-Riemannian metric|pseudo]])-[[Riemannian metric]]s $g$ on a given [[space]] ([[manifold]]) $X$.

## Definition

On the [[site]] [[CartSp]] of [[smooth manifold|smooth]] [[Cartesian space]]s consider the [[sheaf]]

$$
  Met : CartSp^{op} \to Set
$$

which sends each $U \in CartSp$ to the set of [[Riemannian metric]]s on $U$.

Then let $X$ be a [[smooth manifold]], or more generally a [[diffeological space]], or more generally a [[Lie groupoid]] or more generally a [[smooth ∞-groupoid]], all regarded in $\mathbf{H} = $ [[Smooth∞Grpd]].

Write

$$
  Met(X) := Conc [X,Met]
$$

for the [[concrete sheaf|concretification]] of the [[internal hom]]: the space of metrics on $X$.
(Various variations and extensions of this statement are of interest and can easily be written out. The above direct statement works for possibly degenerate metrics.)

A point in this space is a single (pseudo-)Riemannian metric on $X$.

The group $Aut(X)$ of [[automorphism]]s of $X$ [[action|acts]] on this by precomposition in the natural way

$$
  Aut(X) \times Met(X) \to Met(X)
$$

$$
  ((X \stackrel{\phi}{\to} X), g)
  \mapsto
  \phi^* g
  \,.
$$

If $X$ is a [[smooth manifold]] then $Aut(X) = Diff(X)$ is the group of [[diffeomorphism]]s of $X$.

The [[quotient]] ([[action groupoid]], [[moduli stack]])

$$
  Met(X)//Diff(X) \in Smooth\infty Grpd
$$

is the _moduli space of (pseudo-)Riemannian metrics_ on $X$.

Various variations of this are of interest. For instance there one consider the [[Einstein-Hilbert action]]

$$
  S : Met(X)//Diff(X) \to \mathbb{R}
  \,.
$$

The [[critical locus]] of this function is the moduli space of [[Einstein metric]]s.

## Properties

For $X$ a smooth manifold, $Met(X)$ itself is a [[contractible space]].

## Applications

In the context of the theory of [[gravity]], the moduli space of pseudo-Riemannian metrics on $X$ is the [[configuration space]] of the [[quantum field theory|field theory]] of gravity ([[general relativity]]). The moduli space of Einstein metrics is then called the [[covariant phase space]]: this is the subspace of solutions of the [[Einstein equations]].

## Related concepts

[[!include moduli spaces -- contents]]

## References

Textbook references include

* [[Mikhail Gromov]], _Metric structures for Riemannian and non-Riemannian spaces_ Birkhäuser (1999)

chapter 4 of

* A. L. Besse, _Einstein Manifolds_ , Ergeb. Math. Grenzgeb. 10, Springer-Verlag, New York,
(1987) MR 88f:53087 Zbl 0613.53001

* M. E. Shanks, _The Space of Metrics on a Compact Metrizable Space_ , American Journal of Mathematics  Vol. 66, No. 3, Jul. (1944) ([JSTOR](http://www.jstor.org/stable/2371909))

* Halldor Eliasson, _On variations of metrics_ Math. Scand. 29 (1971) 317-327 ([pdf](http://www.mscand.dk/article.php?id=2047))

* F. Farrell, Pedro Ontaneda, _The Teichmüller space of pinched negatively curved metrics on a hyperbolic manifold
is not contractible_ ([pdf](http://annals.math.princeton.edu/wp-content/uploads/annals-v170-n1-p02-p.pdf))

* Boris Botvinnik, Bernhard Hanke, [[Thomas Schick]], and Mark Walsh, _Homotopy groups of the space of metrics of positive scalar curvature_ ([pdf](http://www.math.uni-augsburg.de/de/prof/diff/arbeitsgruppe/hanke/Arbeiten/bhsw-06.pdf))

[[!redirects moduli spaces of Riemannian metrics]]


moduli space of super Riemann surfaces

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Supergeometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[supergeometry|super]]-[[moduli space]] of [[super Riemann surfaces]].

Just like the [[moduli space of Riemann surfaces]] is an [[orbifold]], so the [[moduli space]] [[super-Riemann surfaces]] is a super-orbifold (e.g. [Rabin 87](#Rabin87), [LeBrun-Rothstein 88](#LeBrunRothstein88), [Witten 12](#Witten12), [Codogni-Viviani 17](#CodogniViviani17)).

## Properties

[[!include non-projected super Riemann surface moduli -- section]]

## References

[[!include super-moduli of super Riemann surfaces -- references]]


moduli spaces -- contents

**[[moduli spaces]]**

* [[moduli space of complex structures]]

* [[moduli space of conformal structures]]

* [[moduli space of curves]]

* [[moduli space of Riemannian metrics]]

* [[moduli stack of elliptic curves]]

* [[moduli space of Calabi-Yau spaces]]

* [[moduli stack of tori]]

* [[moduli stack of formal groups]]

* [[moduli space of connections]], [[moduli space of flat connections]]

* [[Hitchin moduli space]]

* [[moduli space of monopoles]]

* [[moduli space of framed manifolds]]

* [[moduli space of bundles]]

* [[moduli space of (higher) line bundles]]

* [[quiver variety]]


moduli stabilization

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[physics]], _moduli stabilization_ refers to the problem of rendering [[Kaluza-Klein compactifications]] stable.

A [[Kaluza-Klein compactification]] is a [[model (physics)|model]] of [[gravity]] where [[spacetime]] is assumed to be a higher dimensional [[fiber bundle]], with [[compact topological space|compact]] [[fibers]] of tiny extension, such that the resulting physics looks effectively lower-dimensional, but inheriting extra [[field (physics)|fields]]. Namely the size and shape of the compactified extra dimension is encoded in the [[Riemannian metric]], hence in the field of gravity, hence are themselves dynamical fields. Since these fields parameterize the [[moduli space]] of the KK-compactification, they are called _[[moduli fields]]_.

The problem of _moduli stabilization_ is the problem of identifying mechanisms or conditions that ensure that as these fields dynamically evolve, the compact spatial dimensions remain stably so, neither opening up nor collapsing. For [[phenomenology|phenomenologically]] realistic KK-compactifications the compact volume has to stably be a tiny but finite value ("volume stabilization").

Equivalently, since fast varying moduli appear as light or massless [[particles]] in the low-dimensional [[effective field theory]] which would show up in accelerator [[experiments]] (such as the [[LHC]]) but don't, the problem is to identify mechanisms or conditions that would render these moduli fields massive.

### For pure vacuum gravity compactifications
 {#ForPureVacuumGravity}

In pure [[classical field theory|classical]] [[gravity]] KK-compactifications have been suggested ([Penrose 03, section 10.3](#Penrose03)) to generically be unstable due to the [[Penrose-Hawking singularity theorem]].

But a rigorous analysis in [Anderson-Blue-Wyatt-Yau 20](#AndersonBlueWyattYau20) claims to show that, in contrast, even vacuum spacetimes are stable under KK-compactification (for $\geq 10$-dimensions with a [[Killing spinor]] on the [[compact topological space|compact]] [[fiber]] and for [[Schwarzschild spacetime|Schwarzschild]]-asymptotics).

### For Freund-Rubin flux compactifications
 {#FluxCompactfications}

If in addition to pure [[gravity]] extra [[gauge fields]] or [[higher gauge field]] beyond pure gravity are admitted in the higher dimensions, then stable compactifications may exist if there is "magnetic [[flux]]" in the compact fiber spaces. These are called _[[Freund-Rubin compactifications]]_, or _[[flux compactifications]]_.

A well-studied example is 6-dimensional [[Einstein-Maxwell theory]] with magnetic [[flux]] on a 2-dimensional [[fiber]] spaces over a 4-dimensional base space ([Freund-Rubin 80](#FreundRubin80), [RDSS 83](#RDSS83)).

(On the other hand, Freund-Rubin compactifications usually have fibers the site of the curvature radius of the base, and hence not "small".)

Similarly, in [[string theory]] it is argued that the extra fields and further string theoretic effects may stabilize the compact dimensions, namely a combination of [[flux compactification]] and [[non-perturbative effect|non-perturbative]] [[brane]] effects ([Acharya 02](#Acharya02), [KKLT 03](#KKLT03)). However, these arguments typically focus on fluctuations that preserve given [[special holonomy]] ([[supersymmetry and Calabi-Yau manifolds|Calabi-Yau 3-folds]] in type II or [[M-theory on G2-manifolds|G2-manifolds in M-theory]]). There is also a more generic argument for volume compactification by string winding modes ("[[Brandenberger-Vafa mechanism]]" [Brandenberger-Vafa 89](#BrandenbergerVafa89), [Watson-Brandenberger 03](#WatsonBrandenberger03)) and the claim ([Kim-Nishimura-Tsuchiya 12](#KimNishimuraTsuchiya12)) that in the non-perturbative [[IKKT model]] computer simulations show a spontaneous stable compactification to 3+1 dimensions.

### For string theory compactifications

The issue of stabilization of compact dimensions arises notably in [[string theory]] [[Kaluza-Klein compactifications]].

In the context of [[type II string theory]] one way to design the [[model (in theoretical physics)|model]] such that the moduli fields are massive is to consider the case where [[higher gauge field|higher]] [[background gauge fields]] [[vacuum expectation value|vacuum expectation values]] (VEVs) $F_p$ are present on the compactification space.  Since these fields are characterized by their higher [[field strength]]/[[curvature]] forms which are referred to as "[[flux]]" terms in physics, these models are called **[[flux compactification]]** models ([KKLT 03](#KKLT03)).

Because the standard [[kinetic action]] term

$$
  S_{kin} \propto \int F_p \wedge \star_g F_p
$$

couples the flux VEV to the metric $g$ (via the [[Hodge star operator]]) and hence to the moduli, it generically induces an effective [[potential energy]] for these, which may stabilize them (when including [[non-perturbative effects]]).

Similarly in [[M-theory on G2-manifolds]] the 4-form flux of the [[supergravity C-field]] leads to potentials for the moduli, which is argued to generically stabilize them ([Acharya 02](#Acharya02)).

Since for these flux compactifications only the [[periods]] of the form fields on the compact space matter, under a bunch of further assumptions on the nature of the compactification, one can reduce the number of possible such compactifications to a combinatorial problem. The resulting space of possibilities is also known  as the _[[landscape of string theory vacua]]_.

\linebreak

A widely studied but non-rigorous scenario of moduli stabilization in string theory is due to ([KKLT 03](#KKLT03)). More recently, the assumptions of the KKLT scenario have been called into question ([Danielsson-Van Riet 18](#de+Sitter++spacetime#DanielssonVanRiet18), see also at [[swampland conjectures]]):

The moduli stabilization in ([KKLT 03](#KKLT03)) was (informally) argued in two steps. First, all moduli were stabilized at a fixed minimum with a negative [[cosmological constant]]. This was achieved by combining [[flux compactification|fluxes]] with [[non-perturbative effects]]. Second, the minimum was lifted to a metastable vacuum with a positive cosmological constant. This was accomplished by adding anti D-branes and using previous results, obtained in ([Kachru-Pearson-Verlinde 01](#KachruPearsonVerlinde01)), that the flux-anti D-brane system can form a metastable bound state with positive energy. In ([KKLT 03](#KKLT03)) it was also shown that one can fine tune various parameters to make the value of the [[cosmological constant]] consistent with the observed amount of [[dark energy]].

## Related concepts

* [[flux compactification]]

* [[G2-MSSM]]

* [[landscape of string theory vacua]]

* [String Theory FAQ -- Do the extra dimensions lead to instability of 4 dimensional spacetime?](string+theory+FAQ#StabilityOfKKCompactification)

## References

### In pure gravity

The problem of generic in-stability of moduli of pure gravity KK-compactifications is highlighted in

* {#Penrose03} [[Roger Penrose]], section 10.3 in _On the stability of extra space dimensions_ in Gibbons, Shellard, Rankin (eds.) _The Future of Theoretical Physics and Cosmology_, Cambridge (2003) ([spire:608935](https://inspirehep.net/record/608935))

A rigorous [[proof]] that, in contrast, even vacuum spacetimes are stable under KK-compactification is claimed (for $\geq 10$-dimensions with a [[Killing spinor]] on the [[compact topological space|compact]] [[fiber]] and for [[Schwarzschild spacetime|Schwarzschild]]-asymptotics) in

* {#AndersonBlueWyattYau20} [[Lars Andersson]], [[Pieter Blue]], [[Zoe Wyatt]], [[Shing-Tung Yau]], _Global stability of spacetimes with supersymmetric compactifications_ ([arXiv:2006.00824](https://arxiv.org/abs/2006.00824))

### Freund-Rubin flux compactifications
 {#ReferencesFreundRubinCompactificationo}

[[Freund-Rubin compactification|Freund-Rubin]] [[flux compactifications]] are due to

A class of stable compactifications of 6d Einstein-Maxwell theory down to four dimensions is due to

* {#FreundRubin80} [[Peter Freund]] and M. A. Rubin, _Dynamics Of Dimensional Reduction_, Phys. Lett. B 97, 233 (1980) (doi:10.1016/0370-2693(80)90590-0, [spire:154579](http://inspirehep.net/record/154579))

and the special case of compactifications of 6d Einstein-Maxwell theory to 4d is in

*  {#RDSS83} S. Randjbar-Daemi, [[Abdus Salam]] and J. A. Strathdee, _Spontaneous Compactification In Six-Dimensional Einstein-Maxwell Theory_, Nucl. Phys. B 214, 491 (1983) (doi:10.1016/0550-3213(83)90247-X, [spire:182427](https://inspirehep.net/record/182427/))

Further discussion of these models as toy models for [[flux compactifications]] in [[string theory]] is in

* {#DouglasKachru07} [[Michael Douglas]], [[Shamit Kachru]], section II.D.1 of _Flux compactification_, Rev. Mod. Phys. 79, 733 (2007) ([arXiv:hep-th/0610102](https://arxiv.org/abs/hep-th/0610102))

* [[Frederik Denef]], [[Michael Douglas]], [[Shamit Kachru]], _Physics of String Flux Compactifications_, Ann. Rev. Nucl. Part. Sci. 57:119-144, 2007, [arXiv:hep-th/0701050](https://arxiv.org/abs/hep-th/0701050)

### In string theory

Review:

* [[Liam McAllister]], [[Fernando Quevedo]], *Moduli Stabilization in String Theory*, in: *[[Handbook of Quantum Gravity]]* [[arXiv:2310.20559](https://arxiv.org/abs/2310.20559)]

#### In type II string theory

A generic argument for stabilization of compact dimensions in [[type II string theory]] via string winding modes at the self-[[T-duality]] radius is the [[Brandenberger-Vafa mechanism]], see e.g.

* {#BrandenbergerVafa89} [[Robert Brandenberger]], [[Cumrun Vafa]], _Superstrings In The Early Universe_, Nucl. Phys. B 316, 391  (1989) ([spire](http://inspirehep.net/record/263348))

* {#WatsonBrandenberger03} Scott Watson, [[Robert Brandenberger]], _Stabilization of Extra Dimensions at Tree Level_, JCAP 0311 (2003) 008 ([arXiv:hep-th/0307044](http://arxiv.org/abs/hep-th/0307044))

* [[Brian Greene]], Daniel Kabat, Stefanos Marnerides, _On three dimensions as the preferred dimensionality of space via the Brandenberger-Vafa mechanism_,   10.1103/PhysRevD.88.043527  ([arXiv:1212.2115](http://arxiv.org/abs/1212.2115))

Discussion of moduli stabilization via [[flux compactification]] of and [[non-perturbative effects]] in [[type II string theory]]/[[F-theory]] originates with the influential article ("KKLT")

* {#KKLT03} [[Shamit Kachru]], [[Renata Kallosh]], [[Andrei Linde]], [[Sandip  Trivedi]], _de Sitter Vacua in String Theory_, Phys. Rev. D68:046005, 2003 ([arXiv:hep-th/0301240](http://arxiv.org/abs/hep-th/0301240))

which led to a little burst of discussion of the [[landscape of string theory vacua]]. The analysis there relies on

* {#KachruPearsonVerlinde01} [[Shamit Kachru]], J. Pearson, [[Herman Verlinde]], _Brane/Flux Annihilation and the String Dual of a Non-Supersymmetric Field Theory_, JHEP 0206 (2002) 021 ([arXiv:hep-th/0112197](http://arxiv.org/abs/hep-th/0112197))

Further developments include

* [[Frederik Denef]], [[Michael Douglas]], Bogdan Florea, Antonella Grassi, [[Shamit Kachru]], _Fixing All Moduli in a Simple F-Theory Compactification_, Adv.Theor.Math.Phys.9:861-929, 2005 ([arXiv:hep-th/0503124](http://arxiv.org/abs/hep-th/0503124))

* [[Vijay Balasubramanian]], Per Berglund, [[Joseph Conlon]], [[Fernando Quevedo]], _Systematics of Moduli Stabilisation in Calabi-Yau Flux Compactifications_,  JHEP 0503:007,2005 ([arXiv:hep-th/0502058](http://arxiv.org/abs/hep-th/0502058))

A variant via Kähler uplifting is

* [[Alexander Westphal]], _de Sitter String Vacua from Kähler Uplifting_, JHEP 0703:102,2007 ([arXiv:hep-th/0611332](https://arxiv.org/abs/hep-th/0611332))

Review includes

* [[Renata Kallosh]], _Stabilization of moduli in string theory_, lectures 2005 ([part I pdf ](http://web.stanford.edu/~rkallosh/Talks/LectureI.pdf), [part II pdf ](http://web.stanford.edu/~rkallosh/Talks/LectureII.pdf))

* [[Joseph Conlon]], _Moduli Stabilisation and Applications in IIB String Theory_, Fortsch.Phys.55:287-422,2007 ([arXiv:hep-th/0611039](http://arxiv.org/abs/hep-th/0611039))

* Sibasish Banerjee, _Calabi-Yau compactification of type II string theories_ ([arXiv:1609.04454](http://arxiv.org/abs/1609.04454))

Analogous discussion in [[type IIA string theory]] includes ([Acharya 02](#Acharya02)) and

* Oliver DeWolfe, Alexander Giryavets, [[Shamit Kachru]], [[Washington Taylor]], _Type IIA Moduli Stabilization_ ([arXiv:hep-th/0505160](http://arxiv.org/abs/hep-th/0505160))

Discussion of volume stabilization of compact dimensions in the context of [[cosmic inflation]] is in

* Jonathan P. Hsu, [[Renata Kallosh]], _Volume Stabilization and the Origin of the Inflaton Shift Symmetry in String Theory_, JHEP 0404 (2004) 042 ([arXiv:hep-th/0402047](http://arxiv.org/abs/hep-th/0402047))

In

* {#KimNishimuraTsuchiya12} S.-W. Kim, J. Nishimura, and A. Tsuchiya, _Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions_, Phys. Rev. Lett. 108, 011601 (2012), ([arXiv:1108.1540](https://arxiv.org/abs/1108.1540)).

* S.-W. Kim, J. Nishimura, and A. Tsuchiya, _Late time behaviors of the expanding universe in the IIB matrix model_, JHEP 10, 147 (2012), ([arXiv:1208.0711](https://arxiv.org/abs/1208.0711)).

it is claimed that computer simulation shows that the [[IKKT matrix model]] description of, supposedly, non-perturbative type II string theory exhibits spontanous decompactification of 3+1 large dimensions, with the other 6 remaining tiny.

See also:

* [[David Prieto]], *Moduli Stabilization and Stability in Type II/F-theory flux compactifications* [[arXiv:2401.13339](https://arxiv.org/abs/2401.13339)]

#### In M-theory
 {#ReferencesInMTheory}

Discussion of moduli stabilization in [[M-theory on G2-manifolds]] for stabilization via "[[flux]]" (non-vanishing bosonic  [[field strength]] of the [[supergravity C-field]]) is in

* {#Acharya02} [[Bobby Acharya]], _A Moduli Fixing Mechanism in M theory_ ([arXiv:hep-th/0212294](http://arxiv.org/abs/hep-th/0212294))

and [[moduli stabilization]] for fluxless compactifications via [[nonperturbative effects]], claimed to be sufficient and necessary to solve the [[hierarchy problem]], is discussed in

* {#AcharyaBobkovKaneKumarVaman06} [[Bobby Acharya]], Konstantin Bobkov, [[Gordon Kane]], [[Piyush Kumar]], Diana Vaman, _An M theory Solution to the Hierarchy Problem_, Phys.Rev.Lett.97:191601,2006 ([arXiv:hep-th/0606262](http://arxiv.org/abs/hep-th/0606262))

* {#AcharyaBobkovKaneKumarShao07} [[Bobby Acharya]], Konstantin Bobkov, [[Gordon Kane]], [[Piyush Kumar]], Jing Shao, _Explaining the Electroweak Scale and Stabilizing Moduli in M Theory_, Phys.Rev.D76:126010,2007 ([arXiv:hep-th/0701034](http://arxiv.org/abs/hep-th/0701034))

* {#AcharyaKumarBobbkovKaneShaoWatson08} [[Bobby Acharya]], [[Piyush Kumar]], Konstantin Bobkov, [[Gordon Kane]], Jing Shao, Scott Watson, _Non-thermal Dark Matter and the Moduli Problem in String Frameworks_,JHEP 0806:064,2008 ([arXiv:0804.0863](http://arxiv.org/abs/0804.0863))

and specifically for the [[G2-MSSM]] in

* [[Bobby Acharya]], Konstantin Bobkov, [[Gordon Kane]], [[Piyush Kumar]], Jing Shao, _The $G_2$-MSSM - An $M$ Theory motivated model of Particle Physics_ ([arXiv:0801.0478](http://arxiv.org/abs/0801.0478))

Discussion of moduli stabilization in [[M-theory on 8-manifolds]] for the [[product manifold]] of two [[K3]]s:

* [[Paul Aspinwall]], [[Renata Kallosh]], _Fixing All Moduli for M-Theory on $K3 \times K3$_, JHEP 0510:001, 2005 ([arXiv:hep-th/0506014](https://arxiv.org/abs/hep-th/0506014))

#### In heterotic string theory

Discussion of moduli stabilization in [[heterotic string theory]] includes

* {#BuchbinderOvrut04} [[Evgeny Buchbinder]], [[Burt Ovrut]], _Vacuum Stability in Heterotic M-Theory_, Phys.Rev. D69 (2004) 086010 ([arXiv:hep-th/0310112](http://arxiv.org/abs/hep-th/0310112))

* [[Sergei Gukov]], [[Shamit Kachru]], Xiao Liu, [[Liam McAllister]], _Heterotic Moduli Stabilization with Fractional Chern-Simons Invariants_,  Phys.Rev.D **69** (2004) 086008 [[arXiv:hep-th/0310159](http://arxiv.org/abs/hep-th/0310159)]

[[!redirects moduli stabilizations]]

[[!redirects Brandenberger-Vafa mechanism]]
[[!redirects Brandenberger-Vafa mechanisms]]


moduli stack

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
#### Yoneda lemma
+--{: .hide}
[[!include Yoneda lemma - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[stack]] $A$ is called a **moduli stack** for certain [[structures]], if for any other object $X$ the [[groupoid]] of [[morphisms]] $X \to A$ into $A$ is [[equivalence of categories|equivalent]] to the groupoid of these kinds of structures on $X$.

This is in contrast to the notion of _[[moduli space]]_, which is only about [[equivalence classes]] of structures and loses the information about the [[gauge equivalences]]/ [[automorphism]] groups of these structures.

There is an evident generalization of the concept of moduli stacks in the more general context of [[higher topos theory]], to **[[moduli ∞-stacks]]**.

Notice that every stack is the moduli stack of _something_ and in fact in general of different things at the same time (see below). So to some extent saying "moduli stack" is redundant. It is usually used to indicate, roughly, that there are some [[spaces]]/[[stacks]] that one is working on/over, and then there are apart from this stacks, the moduli stacks, that one is mapping _into_.

This distinction however easily disappears. For instance a historically famous moduli stack is the [[moduli stack of elliptic curves]] which started out as an object used to classify [[bundles]] of elliptic curves over other spaces. Later in the study of [[elliptic cohomology]] and [[tmf]] the "moduli stack" of elliptic curves came to be regarded as a space interesting in itself for the geometry _on_ it, specifically since this is naturally a [[derived algebraic geometry]].

Analogous comments apply to other moduli stacks. For instance for $G$ a [[topological group]], the moduli stack $\mathbf{B}G \simeq \ast //G$ for topological $G$-[[principal bundles]] is itself interesting for its own geometry. Notably it is the base stack of the [[universal principal bundle]] which as such may be equipped with [[differential geometry]] such as a [[connection on a bundle]] etc.

Generally, what one needs for a stack to classify bundles in this way is a _universal bundle_ over it, for then what the stack _modulates_ are precisely the [[pullbacks]] of this universal bundle. A stack with a prescribed universal bundle over it may be regarded as a stack equipped with an [[atlas]].

In conclusion then "moduli stack" pretty much means "stack" or more precisely "stack with specified universal bundle over it or [[atlas]] into it", with the implicit implication that we say "moduli stack" to indicate that we care about pulling back that atlas/bundle along maps into the stack.

(Compare this to how one says "[[presheaf]]" for what is really just a [[functor]] in order to indicate a certain attitude, namely that one will be interested in asking which presheaves are [[sheaves]].)

## Examples

### Of smooth principal bundles

Let $\mathbf{H} := Sh_\infty(SmthMnfd)$ be collection of [[differentiable stack]]s and generally of [[stacks]] and [[∞-stack]]s over the [[site]] of [[smooth manifold]]s (see [[Smooth∞Grpd]] for details).

Then every [[Lie group]] $G$ is canonically a [[∞-group|group object]] in $\mathbf{H}$ -- a "group stack" -- and its [[delooping]] in $\mathbf{H}$ produces a stack denoted $\mathbf{B}G$. This is simply the ([[stackification]] of) the [[Lie groupoid]] $*//G$ with a single object and $G$ worth of automorphisms on this object.

Let the $X$ be any [[smooth manifold]], also regarded as a stack, via the [[Yoneda embedding]]. Then one finds that morphisms of stacks

$$
  X \to \mathbf{B}G
$$

are the same as smooth $G$-principal bundles over $X$. More precisely the [[groupoid]] $G Bund(X)$ of smooth $G$-principal bundles and smooth [[gauge transformation]]s between them is canonically equivalent to the [[derived hom-space|hom-groupoid]] of maps from $X$ to $\mathbf{B}G$:

$$
  G Bund(X) \simeq \mathbf{H}(X, \mathbf{B}G)
  \,.
$$

This is discussed in some detail at _[[principal bundle]]_.

The statement immediately generalizes to higher degrees and to other notions of ([[higher geometry|higher]]) [[geometry]]. This is discussed at _[[principal ∞-bundle]]_.

### Of flat connections

* [[moduli stack of flat connections]]

### Of formal groups

* [[moduli stack of formal groups]]

### Of elliptic curves

A famous moduli stack is that of [[elliptic curves]]. See _[[moduli stack of elliptic curves]]_ for more on this.

### Of Line bundles

* [[Picard stack]]

### Of L-parameters

* [[moduli stack of L-parameters]]

## Related concepts

* [[moduli]], [[moduli problem]]

* [[classifying space]], [[moduli space]], [[derived moduli space]], [[geometric invariant theory]]

* [[algebraic stack]]

* [[rigidification of a stack]]

* [[Yoneda lemma]], [[(∞,1)-Yoneda lemma]]

## References

Lecture notes:

* [[Daniel Halpern-Leistner]], *Moduli theory*, Lecture notes (2020) [[pdf](http://pi.math.cornell.edu/~danielhl/moduli_theory_notes.pdf), [[Halpern-Leistner_ModuliTheory.pdf:file]]]

For more see also the references at *[[algebraic stack]]*.

[[!redirects moduli stack]]
[[!redirects classifying stack]]
[[!redirects moduli stacks]]
[[!redirects classifying stacks]]

[[!redirects moduli ∞-stack]]
[[!redirects classifying ∞-stack]]
[[!redirects moduli ∞-stacks]]
[[!redirects classifying ∞-stacks]]

[[!redirects moduli infinity-stack]]
[[!redirects classifying infinity-stack]]
[[!redirects moduli infinity-stacks]]
[[!redirects classifying infinity-stacks]]

[[!redirects moduli 2-stack]]
[[!redirects moduli 2-stacks]]

[[!redirects higher moduli stack]]
[[!redirects higher moduli stacks]]

[[!redirects modulus]]


moduli stack of curves -- table

**Substructure of the [[moduli stack of curves]] and the ([[equivariant cohomology|equivariant]]) [[cohomology theory]] associated with it via the [[Goerss-Hopkins-Miller-Lurie theorem]]:**

| |  |  |  |  |  |  |  |  |  |
|--|--|--|--|--|--|--|--|--|--|
| covering  |  |  |  |  | by [[level structure on an elliptic curve|of level-n structures]] ([[modular curve]]) |  |  |  |  |
| | $\ast = Spec(\mathbb{Z})$  | $\to$ | $Spec(\mathbb{Z}[ [q] ])$  | $\to$ | $\mathcal{M}_{\overline{ell}}[n]$  |  |  |  |  |
| [[structure group]] of covering | $\downarrow^{\mathbb{Z}/2\mathbb{Z}}$ |  | $\downarrow^{\mathbb{Z}/2\mathbb{Z}}$ |  | $\downarrow^{SL_2(\mathbb{Z}/n\mathbb{Z})}$ ([[modular group]]) |  |  |  |  |
| [[moduli stack]] | $\mathcal{M}_{1dTori}$ | $\hookrightarrow$ |  $\mathcal{M}_{Tate}$ | $\hookrightarrow$ | $\mathcal{M}_{\overline{ell}}$ ([[moduli stack of elliptic curves|M_ell]]) | $\hookrightarrow$ | $\mathcal{M}_{cub}$  | $\to$  | $\mathcal{M}_{fg}$ ([[moduli stack of formal groups|M_fg]])
| of  | [1d tori](torus#InAlgebraicGeometry)  |  | [[Tate curves]] | | [[elliptic curves]] |  | [[cubic curves]] | |  [[moduli stack of formal groups|1d commutative formal groups]] |
| value $\mathcal{O}^{top}_{\Sigma}$ of [[structure sheaf]] over curve $\Sigma$ | [[KU]] |  | $KU[ [q] ]$ |  | [[elliptic spectrum]] |  |  |  | [[complex oriented cohomology theory]] |
| [[spectrum]]   $\Gamma(-, \mathcal{O}^{top})$ of [[global sections]] of [[structure sheaf]] | ([[KO]] $\hookrightarrow$ [[KU]]) = [[KR-theory]] |  | [[Tate K-theory]] ($KO[ [q] ] \hookrightarrow KU[ [q] ]$) |  |  ([[tmf|Tmf]] $\to$ [[Tmf(n)]]) ([[modular equivariant elliptic cohomology]])  |  |  [[tmf]]  |  |  $\mathbb{S}$ |