supergravity C-field

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###Context###
#### String theory
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[[!include string theory - contents]]
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#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The field content of 11-dimensional [[nLab:supergravity]] contains a [[higher U(1)-gauge field]] called the _supergravity C-field_ or _M-theory 3-form_ , which is locally a 3-form and globally _some_ variant of a [[circle n-bundle with connection|circle 3-bundle with connection]].

There have been several suggestions for what _precisely_ its correct global description must be, see at _[Models](#Models)_ below

## Consistency conditions
 {#ConsistencyConditions}

Several subtle consistency conditions ([[quantum anomaly cancellation]]-conditions) have been argued for the [[charge quantization]] of the supergravity C-field:

1. [Shifted flux quantization condition](#ShiftedCFieldFluxQuantizationCondition)

1. [C-Field tadpole cancellation condition](#CFieldTadpoleCancellationCondition)

### Shifted flux quantization condition
 {#ShiftedCFieldFluxQuantizationCondition}

The _[[shifted C-field flux quantization condition]]_ is a [[charge quantization]]-condition on the [[supergravity C-field]] expected in [[M-theory]]. It says that the [[real cohomology]] class of the [[flux density]] ([[field strength]]) [[differential 4-form]] $G_4 \in \Omega^4(X)$ on [[spacetime]] $X$ becomes integral after shifted by one quarter of the [[first Pontryagin class]], hence the condition that with the shifted 4-flux density defined as

\[
  \label{Shiftwed}
  \widetilde G_4
  \;\coloneqq\;
  G_4 + \tfrac{1}{4}p_1(\nabla_{T X})
  \;\in\;
  \Omega^4(X)
\]

(for $\nabla_{T X}$ any [[affine connection]] on [[spacetime]], in particular the [[Levi-Civita connection]]) we have (using the [[de Rham theorem]] to translate from [[de Rham cohomology]] to [[real cohomology]]) that $\widetilde G_4$ represents an [[integral cohomology]]-class:

$$
  [\widetilde G_4]
  \;\in\;
  H^4(X, \mathbb{Z})
  \overset{H^4(X, \mathbb{Z}\hookrightarrow \mathbb{R})}{\longrightarrow}
  H^4(X, \mathbb{R})
  \,.
$$

This condition was originally argued for in ([Witten 96a](#Witten96a), [Witten 96b](#Witten96b)) as a sufficient condition for ensuring that the [[prequantum line bundle]] for the [[7d Chern-Simons theory]] on an [[M5-brane]] [[worldvolume]] is divisible by 2.

Proposals for encoding this condition by a [[Wu class]]-shifted variant of [[generalized (Eilenberg-Steenrod) cohomology theory|stable]] [[ordinary differential cohomology]] were considered in [[Quadratic Functions in Geometry, Topology, and M-Theory|Hopkins-Singer 02]], [Diaconescu-Freed-Moore 03](#DiaconescuFreedMoore03), [FSS 12](#FSS12).

It turns out that the shifted flux quantization condition on the C-field is naturally implied ([FSS1 19b, Prop. 4.12](#FSS19b)) by the requirement that $G_4$ is the differential form datum underlying, via [[Sullivan model|Sullivan's theorem]], a [[cocycle]] in unstable [[J-homomorphism|J-]] [[twisted Cohomotopy]] in degree 4 ([[Hypothesis H]]).

### C-Field tadpole cancellation condition
 {#CFieldTadpoleCancellationCondition}

In [[M-theory]] [[KK-compactification|compactified]] on 8-dimensional [[compact topological space|compact]] [[fibers]] $X^{(8)}$ (see _[[M-theory on 8-manifolds]]_) [[tadpole cancellation condition for the supergravity C-field]] has been argued ([Sethi-Vafa-Witten 96](shifted+C-field+flux+quantization#SethiVafaWitten96), [Becker-Becker 96](shifted+C-field+flux+quantization#BeckerBecker96), [Dasgupta-Mukhi 97](shifted+C-field+flux+quantization#DasguptaMukhi97)) to be the condition

$$
  N_{M2}
  \;+\;
  \tfrac{1}{2} \big( G_4[X^{(8)}]\big)^2
  \;=\;
  \underset{
    I_8(X^8)
  }{
    \underbrace{
      \tfrac{1}{48}\big( p_2 - (\tfrac{1}{2}p_1)^2  \big)[X^{8}]
    }
  }
  \;\;\;\;
  \overset{!}{\in}
  \mathbb{Z}
  \,,
$$

where

1. $N_{M2}$ is the net number of [[M2-branes]] in the spacetime (whose [[worldvolume]] appears as points in $X^{(8)}$);

1. $G_4$ is the [[field strength]]/flux of the [[supergravity C-field]]

1. $p_1$ is the [[first Pontryagin class]] and $p_2$ the [[second Pontryagin class]] combining to [[I8]], all regarded here in [[rational homotopy theory]].

If $X^{8}$ has

* [[Spin(7)-structure]] (hence in particular if it is a [[Calabi-Yau manifold]], which has $SU(4) = $ [[Spin(6)]]-structure)

or

* [[Sp(2).Sp(1)-structure]]

then

$$
  \tfrac{1}{2}\big( p_2 - \tfrac{1}{4}(p_1)^2  \big)
  \;=\;
  \chi
$$

is the [[Euler class]] (see [this Prop.](Spin7-manifold#CharacteristicClassesForSpinStructure) and [this Prop.](quaternion-Kähler+manifold#CharacteristicClassesForSpin5Spin3Structure), respectively), hence in these cases the condition is equivalently

\[
  \label{DasguptaMukhiConditionInTermsOfEuler}
  N_{M2}
  \;=\;
  -
  \tfrac{1}{2} \big( G_4[X^{(8)}]\big)^2
  \;+\;
  \tfrac{1}{24}\chi[X^8]
  \;\;\;\;
  \in
  \mathbb{Z}
  \,,
\]

where $\chi[X]$ is the [[Euler characteristic]] of $X$.

## Models
 {#Models}

One proposal for a mathematical model of the C-field is as a [[cocycle]] in a [[Wu class]]-shifted variant of [[ordinary differential cohomology]] in degree 4:

* [The DFM model](#DFMmodel)

Another proposal is that the C-field is simply a [[cocycle]] in [[J-homomorphism|J-]] [[twisted Cohomotopy]] ([[Hypothesis H]]):

* [C-field charge quantization in J-twisted Cohomotopy](#ModelByTwistedCohomotopy)

### The DFM-model
 {#DFMmodel}

#### Construction via $E_8$ gauge fields

In ([DFM, section 3](#DiaconescuFreedMoore03)) the following definition is considered and argued to be a good model of the supergravity $C$-field.

+-- {: .num_note #NatureOfE8}
###### Note

The [[homotopy group]]s of the [[classifying space]] $B E_8$ of the [[Lie group]] [[E8]] satisfy

$$
  \pi_i B E_8 =
   \left\{
     \array{
       \mathbb{Z} | i = 4
       \\
       0 | i \neq 4, i \leq 15
     }
   \right.
  \,.
$$

Therefore for $X$ a [[manifold]] of [[dimension]] $dim X \leq 15$ there is a canonical [[morphism]]

$$
  H^1(X, E_8)
  \simeq
  H^4(X, \mathbb{Z})
  \,.
$$

=--

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

Let $X$ be a [[smooth manifold]] of [[dimension]] $dim X \lt 15$.
For each  $a \in H^4(X, \mathbb{Z})$. choose an [[nLab:E8]]-[[nLab:principal bundle]] $P \to X$ which represents $a$ under the above isomorphism.

Write then

$$
  \mathbf{E}(X) \in Grpd
$$

for the [[groupoid]] whose

* [[object]]s are triples $(P,\nabla,c)$ where

  * $P$ is one of the chosen $E_8$-bundles,

  * $\nabla$ is a [[nLab:connection on a bundle|connection]] on $P$;

  * $c \in \Omega^3(X)$ is a degree-3 [[nLab:differential form]] on $X$.

* [[morphism]]s $\omega : (P, \nabla_1, c_1) \to (P, \nabla_2, c_2)$
  are parameterized by their source and target triples together with a closed 3-form  $\omega \in \Omega^3_{\mathbb{Z}}(X)$ with integral [[period]]s, subject to the condition that

  $$
    c_2 -c_1 = CS(\nabla_1,\nabla_2) + \omega
    \,,
  $$

  where $CS(\nabla_1, \nabla_2)$ is the relative [[nLab:Chern-Simons form]] corresponding to the linear path of connections from $\nabla_1$ to $\nabla_2$

* the [[composition]] of morphisms

  $$
    (\omega_2 \circ \omega_1 )
     :
    (P,\nabla_1, C_1)
      \stackrel{\omega_1}{\to}
    (P, \nabla_2, C_2)
      \stackrel{\omega_2}{\to}
    (P, \nabla_3, C_3)
  $$

  is given by

  $$
    \omega_1 + \omega_2 + \langle (\nabla_2-\nabla_1)\wedge(\nabla_3-\nabla2) \rangle
   \,.
  $$

=--

See ([DFM, (3.22), (3.23)](#DiaconescuFreedMoore03)).

Here we think of $X$ as equipped with a [[pseudo-Riemannian manifold|pseudo]] [[Riemannian manifold|Riemannian structure]] and [[spin connection]] $\omega$ and think of each object $(P,\nabla,C)$ of $\mathbf{E}(X)$ as inducing an degree-4 cocycle in [[ordinary differential cohomology]] with [[curvature]] 4-form

$$
  \mathcal{G}_{\nabla,c} = tr F_\nabla \wedge F_\nabla - \frac{1}{2} tr R_\omega \wedge R_\omega + d c
  \,.
$$

Notice that with the normalization implicit here the second terms is one half of the image of something in integral cohomology. So this is not itself a differential character, but can be regarded as "shifted differential character": a trivialization of the trivial 5-character with global connection 4-form given by $\frac{1}{2} tr R_\omega \wedge R_\omega$. See [below](#DFMOrientationAndFractionalClasses) for more on this.

+-- {: .num_prop #DFModelHomotopyGroups}
###### Claim

The above groupoid has [[homotopy group]]s

* $\pi_0 \simeq H^4_{diff}(Y)$

* $\pi_1(-,(\nabla,c)) \simeq H^2(Y, U(1))$ .

=--

The first, the set of connected components (gauge equivalence classes of $C$-fields) is isomorphic to the set of [[ordinary differential cohomology]] in degree 4 of $X$. In fact $\pi_0$ is naturally a [[torsor]] over this abelian group: the torsor of $\frac{1}{2}tr R^2$-shifted differential characters.

The second, the [[fundamental group]], is that of flat [[circle bundle]]s.

#### Orientation and fractional classes
  {#DFMOrientationAndFractionalClasses}

Ordinarily, given a $Spin \times E_8$-bundle $P \to Y$ with first fractional [[Pontryagin class]]

$$
  \lambda := \frac{1}{2}p_1(P)
$$

and second [[Chern class]]

$$
  a := c_2(P)
$$

the $C$-field is supposed to have a [[curvature]] class in [[de Rham cohomology]] given by

$$
  a_{dR} + \frac{1}{2} \lambda_{dR} \in H_{dR}^4(Y)
  \,.
$$

Since in general $\lambda = \frac{1}{2}p_1(P)$ is not further divisible in integral cohomology, this means that this cannot be the curvature of any [[differential character]]/[[bundle 2-gerbe]]/[[circle n-bundle with connection|circle 3-bundle with connection]], since these are necessarily the images in de Rham cohomology of their integral classes.

See ([DFM 03, section 12.1](#DiaconescuFreedMoore03), [FSS 12](#FSS12))

#### Restriction to the boundary

By [DFM, section 12](#DiaconescuFreedMoore03) on a manifold $Y$ with boundary $X = \partial Y$ we are to impose $C|_{\partial Y} = 0$.

See the discussion [below](#RestrictionToBoundaryInChernWeil) for how this reproduces the [[Green-Schwarz mechanism]] for heterotic [[supergravity]] on the boundary.

### Description in $\infty$-Chern-Weil theory

Some remarks on ways to regard the $C$-field from the point of view of [[∞-Chern-Weil theory]].

#### Abstract definition

We shall consider the sum of two $C$ fields,  whose curvature is the image in de Rham cohomology of the proper integral class $2 a - \lambda $

Recall from the discussion at [[circle n-bundle with connection]]
that in the [[cohesive (∞,1)-topos]] $\mathbf{H} := $ [[Smooth∞Grpd]] the circle 3-bundles with local 3-form connection over an [[object]] $Y \in \mathbf{H}$ (for instance a [[smooth manifold]], or an [[orbifold]]) are [[object]]s in the [[3-groupoid]] $\mathbf{H}_{diff}(Y, \mathbf{B}^3 U(1))$ that is the [[(∞,1)-pullback]]

$$
  \array{
    \mathbf{H}_{diff}(Y, \mathbf{B}^3 U(1)) &\to& H^4_{dR}(Y)
    \\
    \downarrow && \downarrow
    \\
    \mathbf{H}(Y, \mathbf{B}^3 U(1))
    &\stackrel{curv}{\to}&
    \mathbf{H}(Y, \mathbf{\flat}_{dR} \mathbf{B}^4 U(1))
  }
$$

in [[∞Grpd]].

(Recall from the discussion there that if desired one may pass to the canonical presentation of this by the [[model structure on simplicial presheaves]] over [[CartSp]] and that in this explicit presentation we may replace $H^4_{dR}(Y)$ with the more familiar $\Omega^4_{cl}(Y)$. )

We consider now the analog of this definition for the universal curvature form on $\mathbf{B}^3 U(1)$ replaced by the difference of the differentially refined second [[Chern class]] of [[E8]] and the first fractional [[Pontryagin class]] of the [[spin group]]. The resulting $(\infty,1)$-pullback we tentatively call $C Field(Y)$, though we shall have to discuss to which extend this faithfully models the $C$-field, and which aspects of it.

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

For $Y \in $ [[Smooth∞Grpd]], let $C Field(Y) \in $ [[∞Grpd]] be the [[(∞,1)-pullback]]

$$
  \array{
    C Field(Y) &\to& H^4_{dR}(Y)
    \\
    \downarrow && \downarrow
    \\
    \mathbf{H}(Y, \mathbf{B} (Spin \times E_8))
    &\stackrel{(2\mathbf{c}_2)_{dR}- (\frac{1}{2}\mathbf{p}_1)_{dR}}{\to}&
    \mathbf{H}(Y, \mathbf{\flat}_{dR} \mathbf{B}^4 U(1))
  }
  \,.
$$

=--

+-- {: .num_note #PastingDecompOfAbstractDefOfCField}
###### Note

By its intrinsic definition we have that the differential characteristic class $(\mathbf{c}_2)_{dR}$ is the composite

$$
  (\mathbf{c}_2)_{dR} :
  \mathbf{B}E_8
   \stackrel{\mathbf{c}_2}{\to}
  \mathbf{B}^3 U(1)
    \stackrel{curv}{\to}
  \mathbf{\flat}_{dR} \mathbf{B}^4 U(1)
$$

of the smooth refinement of the second [[Chern class]] with the universal curvature form on $\mathbf{B}^3 U(1)$. Similarly for $(\frac{1}{2}\mathbf{p}_2)_{dR}$.

=--

Therefore we may either compute the [[(∞,1)-pullback]] in def. \ref{AbstractDefOfCField} directly, or in two consecutive steps. Both methods lead to their insights.

In

* [General properties](#CWPerspectiveGeneralProperties)

we consider general abstract consequences of the above definition, mainly making use of the factorization. In

* [Presentation by differential form data](#CWPresentationByDifferentialFormData)

we find a presentation by [[simplicial presheaves]] of the direct [[homotopy pullback]].

In the first approach [[connection on a bundle|connections]] on the [[E8]]-[[principal bundle]]s never appear explicitly. In the second approach they appear as [[pseudo-connection]]s, or as genuine connections whose morphisms are however allowed to shift them arbitrarily. This means that these connections are purely auxiliary data that serve to present the required homotopies. They do not survive in cohomology. This is as in the [DFM model](#DFMmodel) above.

Finally in

* [Restriction to the boundary](#RestrictionToBoundaryInChernWeil)

we comment how genuine $E_8$-connections may appear inside the second presentation of the $C$-model.

#### General properties
  {#CWPerspectiveGeneralProperties}

This implies by the [[pasting law]] for [[(∞,1)-pullback]]s that the $(\infty,1)$-pullback from def. \ref{AbstractDefOfCField} may be decomposed into two consecutive pullbacks of the form

$$
  \array{
    C Field(X) &\stackrel{\hat \chi}{\to}& \mathbf{H}_{diff}(X, \mathbf{B}^3 U(1))
     &\stackrel{\omega}{\to}&
     H^4_{dR}(X)
   \\
    \downarrow && \downarrow && \downarrow
   \\
   \mathbf{H}(X, \mathbf{B}(E_8 \times Spin(10,1)))
    &\stackrel{2\mathbf{c}_2- \frac{1}{2}\mathbf{p}_2}{\to}&
   \mathbf{H}(X, \mathbf{B}^3 U(1))
    &\stackrel{curv}{\to}&
   \mathbf{H}(X, \mathbf{\flat}_{dR} \mathbf{B}^4 U(1))
  }
  \,,
$$

where on the right we find the defining pullback for (the [[cocycle]] [[3-groupoid]] of) [[ordinary differential cohomology]].

=--

This implies the following structure and properties.

+-- {: .num_note #DifferentialCharacterOfCField}
###### Note/Definition

By the above there exists canonically a morphism

$$
  \hat \chi : C Field(X) \to \mathbf{H}(X,\mathbf{B}^3 U(1))
    \stackrel{\tau_{\leq 0}}{\to}
    H^4_{diff}(X)
$$

that maps $C$-field configurations to [[ordinary differential cohomology]] in degree 4, whose [[curvature]] $\omega(\hat \chi)$ is the image $(\mathbf{c}_2)_{dR}- (\frac{1}{2}\mathbf{p}_2)_{dR} :=  curv(2\mathbf{c}_2 - \frac{1}{2}\mathbf{p}_2)$ in de Rham cohomology of the second Chern-class of some $E_8$-bundle.

The differential cocycle $\hat \chi(C)$ has all the general properties that make its [[higher parallel transport]] over [[membrane]] worldvolumes be well-defined. (Apart from the coefficient of $\lambda$, this is the only requirement from which [DFM](#DiaconescuFreedMoore03) deduce their model.)

=--

The following proposition describes the first two [[homotopy group]]s of the [[3-groupoid]] $C Field(Y)$.

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

Over a fixed $Spin$-[[principal bundle]] $P_{Spin}$ we have a [[short exact sequence]] (of [[pointed object|pointed sets]])

$$
  * \to
   H^3(Y, U(1))
   \to
   \pi_0 C Field_{P_{Spin}}(Y)
   \to
   H^4_{dR}(Y)_{2 \mathbb{Z}}
   \to *
$$

and

$\pi_0 C Field_{P_{Spin}}(Y)$ is the group of pairs $([c], f) \in H^2(X, U(1)) \times C^\infty(X, E_8)$ where $f$ is a smooth refinement under $E_8 \simeq_{14} B^2 U(1) \simeq K(\mathbb{Z},3)$ of the integral image of $[c]$.

=--

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

Notice that we have the [[pasting]] [[diagram]] of [[(∞,1)-pullback]]s

$$
  \array{
    C Field_{\omega(\hat \chi(-)) = 0}(Y)
     &\to& \mathbf{H}(Y, \mathbf{\flat} \mathbf{B}^3 U(1)) &\to& {*}
    \\
    \downarrow && \downarrow && \downarrow^{\mathrlap{0}}
    \\
    C Field(Y)
     &\stackrel{\hat \chi}{\to}&
    \mathbf{H}_{diff}(Y, \mathbf{B}^3 U(1))
     &\stackrel{\omega}{\to}&
     H^4_{dR}(Y)
    \\
    \downarrow && \downarrow && \downarrow
   \\
   \mathbf{H}(Y, \mathbf{B}E_8)
    &\stackrel{2\mathbf{c}_2}{\to}&
   \mathbf{H}(Y, \mathbf{B}^3 U(1))
    &\stackrel{curv}{\to}&
   \mathbf{H}(Y, \mathbf{\flat}_{dR} \mathbf{B}^4 U(1))
  }
  \,,
$$

where the top right square is discussed at cohesive (∞,1)-topos --  Differential cohomology. By the discussion at smooth ∞-groupoid -- Flat cohomology we have that $\pi_0 \mathbf{H}(Y, \mathbf{\flat} \mathbf{B}^3 U(1)) \simeq H^3(Y,U(1))$, where on the right we have [[ordinary cohomology]] (for instance realized as [[singular cohomology]]). Finally observe that $\pi_0 \mathbf{H}(Y, \mathbf{E}_8) \simeq \pi_0 \mathbf{H}(Y.\mathbf{B}^3 U(1))$, by the [above remark](#NatureOfE8). Therefore after passing to [[0-connected|connected components]] by applying $\pi_0(-)$ we get on cohomology

$$
  \array{
    H^3(Y, U(1)) & \stackrel{\cdot 2}{\to}& H^3(Y, U(1)) &\to& {*}
    \\
    \downarrow && \downarrow && \downarrow^{\mathrlap{0}}
    \\
    \pi_0 C Field(Y)
     &\stackrel{\hat \chi}{\to}&
    \mathbf{H}_{diff}^4(Y)
     &\stackrel{\omega}{\to}&
     H^4_{dR}(Y)
    \\
    \downarrow && \downarrow && \downarrow
    \\
    H^1(Y, E_8)
    & \stackrel{\cdot 2}{\to}&
    H^4(Y, \mathbb{Z})
    &\stackrel{curv}{\to}&
    H^4_{dR}(Y)
  }
$$

by reasoning as discussed at [[fiber sequence]]. In parallel to the familiar [[short exact sequence]] for [[ordinary differential cohomology]]

$$
  * \to H^3(Y, U(1)) \to H^4_{diff}(Y) \to H^4_{dR}(Y)_{\mathbb{Z}} \to *
  \,.
$$

this therefore implies also the [[short exact sequence]]

$$
  * \to H^3(Y, U(1)) \to \pi_0 C Field \to H^4_{dR}(Y)_{2 \mathbb{Z}} \to *
  \,.
$$

Next we redo the entire discussion after applying the [[loop space object]]-construction to everything. Using that

$$
  \Omega \mathbf{H}(Y, \mathbf{B}Q) \simeq
 \mathbf{H}(Y, \Omega \mathbf{B}Q)
  \simeq
 \mathbf{H}(Y, Q)
$$

on general grounds (see [[fiber sequence]] for details) and that also

$$
  \Omega (\mathbf{\flat}\mathbf{B}^n U(1)) \simeq \mathbf{\flat}\mathbf{B}^{n-1} U(1)
$$

and

$$
  \Omega (\mathbf{\flat}_{dR}\mathbf{B}^n U(1)) \simeq \mathbf{\flat}_{dR}\mathbf{B}^{n-1} U(1)
$$

(since $\mathbf{\flat}$ and $\mathbf{\flat}_{dR}$ are [[right adjoint|right]] [[adjoint (∞,1)-functor]]s -- by the discussion at [[cohesive (∞,1)-topos]] -- and hence commute with the [[(∞,1)-pullback]] that defines $\Omega$), we have then the looped [[pasting]] [[diagram]] of [[(∞,1)-pullback]]s

$$
  \array{
    \Omega C Field(Y)_{P_{Spin}}
     &\stackrel{\Omega \hat \chi}{\to}&
    \mathbf{H}_{flat}(Y, \mathbf{B}^2 U(1))
     &\stackrel{\omega}{\to}&
     *
    \\
   \downarrow && \downarrow && \downarrow
   \\
   \mathbf{H}(Y, E_8)
    &\stackrel{2\Omega \mathbf{c}_2}{\to}&
   \mathbf{H}(Y, \mathbf{B}^2 U(1))
    &\stackrel{curv}{\to}&
   \mathbf{H}(Y, \mathbf{\flat}_{dR} \mathbf{B}^3 U(1))
  }
  \,.
$$

Observe that $E_8$ here is a smooth but [[0-truncated]] object: so that

$$
  \mathbf{H}(Y, E_8) \simeq H^0(Y, E_8) = C^\infty(Y, E_8)
$$

is the set of smooth functions $Y \to E_8$ (to be thought of as the the set of [[gauge transformation]]s from the trivial $E_8$-[[principal bundle]] on $Y$ to itself).

=--

#### Presentation by differential form data
  {#CWPresentationByDifferentialFormData}

In order to compute the $(\infty,1)$-pullback $C Field(X)$ more explicitly, we follow the discussion at  [[differential string structure]], where presentations of this pullback in terms of [[simplicial presheaves]] arising from  [[Lie integration]] is given.

Write now

$$
  \mathfrak{g} := \mathfrak{e}_8 \times \mathfrak{e}_8 \times \mathfrak{so}(10,1)
$$

for the [[Lie algebra]] of $G := E_8 \times E_8 \times Spin(10,1)$ and write

$$
  \mu := \mu_{\mathfrak{e}_8} + \mu_{\mathfrak{e}_8} - \mu_{\mathfrak{so}(10,1)}
$$

for the sum of the canonical [[Lie algebra cohomology|Lie algebra cocycles]] in [[transgression]] with the respective [[Killing form]] [[invariant polynomial]]s.

Write

$$
  \mathfrak{e}_8 \times \mathfrak{so}(10,1)
   \to \mathfrak{g}
$$

for the canonical [[diagonal]] embedding
Write

$$
  \array{
    \mathbf{c} :=
    &
    \mathbf{cosk}_3(
       \exp(\mathfrak{e}_8 \times \mathfrak{so}(10,1))
    )
    &\stackrel{\Delta}{\to}&
    \mathbf{cosk}_3(
       \exp(\mathfrak{g})
    )
     &
      \stackrel{\exp(\mu)}{\to}
     &
    \mathbf{B}^3 U(1)_c
    \\
    \downarrow^{\simeq}
     \\
     \mathbf{B}G
  }
$$

for the corresponding smooth [[characteristic class]]. See [[∞-Chern-Weil homomorphism]] for details. By the discussion there we present
$\hat \mathbf{c}$ by

$$
  \array{
    \mathbf{cosk_3} \exp(b \mathbb{R} \to \mathfrak{g}_\mu)_{diff}
    &\to&
    \mathbf{B}^4 \mathbb{R}_{dR}
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B}E_8
  }
  \,.
$$

By the discussion at [[differential string structure]] we have that the top morphism is a [[fibration]] in the global projective [[model structure on simplicial presheaves]] $[CartSp^{op}, sSet]_{proj}$ (there it is shown that the analogous morphism out of $\mathbf{cosk_3} \exp(b \mathbb{R} \to \mathfrak{e}_8)_{ChW}$ is a fibration, but then so is this one, because the components on the left are the same but with fewer conditions on them, so that the lifts that existed before still exist here).

Over some $U \in$ [[CartSp]] and $[k] \in \Delta$ we have that $\exp(b^\mathbb{R} \to \mathfrak{g}_\mu)_{diff}$ is given by differential form data

$$
  \left(
    \array{
      F_A =& d A + \frac{1}{2}[A \wedge A]
      \\
      C_3 =& \nabla B := d B + CS(A) - H_3
      \\
      \mathcal{G}_4  =& d H_3
      \\
      d F_A =& - [A \wedge F_A]
      \\
      d C_3 =&  \langle F_A \wedge F_A\rangle - \mathcal{G}_4
      \\
      d \mathcal{G}_4 =& 0
    }
  \right)_i
  \;\;\;\;
  \stackrel{
    \array{
       t^a & \mapsto A^a
       \\
       r^a & \mapsto F^a_A
       \\
       b & \mapsto B
       \\
       c & \mapsto C_3
       \\
       h & \mapsto H_3
       \\
       g & \mapsto \mathcal{G}_4
    }
  }{\leftarrow}|
  \;\;\;\;
  \left(
    \array{
       r^a  =& d t^a + \frac{1}{2}C^a{}_{b c} t^b \wedge t^c +
       \\
       c = & d b + cs - h
       \\
       g  =& d h
       \\
       d r^a  =&  - C^a{}_{b c} t^b \wedge r^a
       \\
       d c =& \langle -,-\rangle - g
       \\
       d g =&  0
    }
  \right)
$$

on $U \times \Delta^k$. Here, recall, $A$ takes values in $\mathfrak{g} = \mathfrak{e}_8 \times \mathfrak{e} \times \mathfrak{so}(10,1)$, so that for instance the $\mathcal{G}_4$-curvature is in detail given by

\[
  \label{TheFourCurvature}
  \mathcal{G}_4
  =
  d H_3
  =
  \langle F_{A^L_{\mathfrak{e}_8}} \wedge F_{A^L_{\mathfrak{e}_8}} \rangle
  +
  \langle F_{A^R_{\mathfrak{e}_8}} \wedge F_{A^R_{\mathfrak{e}_8}} \rangle
  -
  \langle F_{\omega} \wedge F_{\omega} \rangle
  -
  d C_3
  \,,
\]

where $\omega$ denotes the [[spin connection]].

Let $\{U_i \to X\}$ be a differentiably [[good open cover]]. We hit all connected components of $\mathbf{H}(X, \mathbf{B}G)$ by considering in

$$
  [CartSp^{op}, sSet](C(U_i), \exp(b \mathbb{R} \to \mathfrak{g}_\mu))_{diff}
$$

those cocycles that

* involve genuine $G$-[[connection on a bundle|connections]] (as opposed to the more general [[pseudo-connection]]s that are also contained);

* have a globally defined $C_3$-form.

Write therefore $(P, \nabla, C_3)$ for such a cocycle.

For gauge transformations between two such pairs, parameterized by the above form data patchwise on  $U \times \Delta^1$, the fact that $\mathcal{G}_4$ vanishes on $\Delta^1$ implies the infinitesmal gauge transformation law

$$
  \frac{d}{d t} C = d_U \omega_t
   + \iota_t \langle F_{\hat A} \wedge F_{\hat A}\rangle
  \,,
$$

where $\hat A\in \Omega^1(U \times \Delta^1, \mathfrak{e}_8)$ is the shift of the 1-forms. This integrates to

\[
  \label{CGaugeTransformation}
  C_2 = C_1 + d \omega + CS(\nabla_1,\nabla_2)
  \,,
\]

where

* $\omega := \int_{\Delta^1} \omega_t$

* $CS(\nabla_1, \nabla_2) = \int_{\Delta^1} \langle F_{\hat \nabla} \wedge F_{\hat \nabla}\rangle $ is the relative [[Chern-Simons form]] corresponding to the shift of $G$-connection.

#### Restriction to the boundary
  {#RestrictionToBoundaryInChernWeil}

We have seen that $C Field(Y)$ is the 3-groupoid of those [[Cech cohomology|Cech cocycles]] on $Y$ with coefficients in $\exp(b \mathbb{R} \to \mathfrak{g}_\mu)_{diff}$ such that the curvature 4-form $\mathcal{G}_4$ has a fixed globally defined value.

Consider the [[subobject]]

$$
  \exp(b \mathbb{R} - \mathfrak{g}_\mu)_{diff}^{C = 0}
  \hookrightarrow
  \exp(b \mathbb{R} - \mathfrak{g}_\mu)_{diff}
$$

of the [[simplicial presheaf]] $\exp(b \mathbb{R} \to \mathfrak{g}_\mu)$ on those [[object]]s and [[k-morphism]]s for which $C = 0$.

By the [[gauge transformation]] law (eq:CGaugeTransformation)

$$
  C_2 = C_1 + d \omega + CS(A_1, A_2)
$$

this means that this picks those [[morphism]]s for which the [[Chern-Simons form]] vanishes

$$
  CS(A_1,A_2) = \int_{\Delta^1} \langle F_{A} \wedge F_{A}\rangle
  = 0
  \,,
$$

where $A = A_U + \lambda d t \in \Omega^1(U \times \Delta^1, \mathfrak{g})$ is the 1-form datum (with $t$ the canonical coordinate on the 1-[[simplex]] $\Delta^1 = [0,1]$).

+-- {: .num_note #NatureOfGaugedCSForm}
###### Note

In the literature often the relative [[Chern-Simons form]] is considered for "ungauged" paths of connections: for $\lambda = 0$ in the above formula, hence for a $\mathfrak{g}$-valued 1-form on $U \times \Delta^1$ with no leg along the simplex (only depending on the simplex coordinate). Here, however, it is crucially important that we consider the general "gauged" paths.

=--

Notice that on the [[semisimple Lie algebra]] and [[compact Lie algebra]] $\mathfrak{e}_8$ the [[Killing form]] $\langle -,-\rangle$ is non-degenerate and positive definite (or negative definite, depending on convention). The latter condition means that this integral vanishes precisely if

$$
  \iota_{\partial_t} \langle F_A \wedge F_A \rangle = 0
  \,.
$$

This is the case on paths for which $\iota_t F_A = 0 $, but this are exactly the paths that induce genuine [[gauge transformation]]s between $A_1$ and $A_2$, where

$$
  \frac{d}{d t} A = d_U \lambda + [\lambda , A]
  \,.
$$

This means that cocycles with coefficients in this subobject for $C = 0$ are cocycles as described at [[differential string structure]], exhibiting the [[Green-Schwarz mechanism]] on the heterotic boundary, witnessed by the restriction of the curvature equation (eq:TheFourCurvature) to vanishing $C$-field

\[
  d H_3^L
  =
  \langle F_{A^L_{\mathfrak{e}_8}} \wedge F_{A^L_{\mathfrak{e}_8}} \rangle
  -
  \frac{1}{2}\langle F_{\omega} \wedge F_{\omega} \rangle
\]

### Model by twisted Cohomotopy
 {#ModelByTwistedCohomotopy}

It turns out that the [[shifted C-field flux quantization condition]] is naturally implied ([FSS1 19b, Prop. 4.12](#FSS19b)) by the requirement that $G_4$ is the differential form datum underlying, via [[Sullivan model|Sullivan's theorem]], a [[cocycle]] in unstable [[J-homomorphism|J-]] [[twisted Cohomotopy]] in degree 4 ([[Hypothesis H]]).

## Related concepts

* [[electromagnetic field]] ("$A$-field")

* [[Kalb-Ramond field]] ("$B$-field")

* **supergravity $C$-field**

  * [[C-field tadpole cancellation]]

* [[twisted differential c-structure]]

  * [[differential string structure]]

  * [[differential fivebrane structure]]

  * [[differential T-duality]]

[[!include table of branes]]

## References
 {#References}

### General

The C-field in [[D=11 supergravity]] originates as the $A_{\mu\nu\rho}$-field in

* {#CremmerJuliaScherk78} [[Eugene Cremmer]], [[Bernard Julia]], [[Joël Scherk]], _Supergravity in theory in 11 dimensions_, Phys. Lett. 76B (1978) 409 (doi:10.1016/0370-2693(78)90894-8)

Re-derivation in the [[D'Auria-Fré formulation of supergravity]]:

* {#DAuriaFre82}  [[Riccardo D'Auria]], [[Pietro Fré]], (5.11b) of: _[[GeometricSupergravity.pdf:file]]_, Nuclear Physics B201 (1982) 101-140 (doi:10.1016/0550-3213(82)90376-5,  [[GeometricSupergravityErrata.pdf:file]])

Review in:

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], III.8.53 of: _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

* [[André Miemiec]], [[Igor Schnakenburg]], Section 3.1.3 of: _Basics of M-Theory_, Fortsch. Phys. 54 (2006) 5-72 ([arXiv:hep-th/0509137](http://arxiv.org/abs/hep-th/0509137), [doi:10.1002/prop.200510256]( https://doi.org/10.1002/prop.200510256))

The [[shifted C-field flux quantization condition]] was originally proposed in

* {#Witten96a} [[Edward Witten]], _On Flux Quantization In M-Theory And The Effective Action_, J. Geom. Phys. 22:1-13, 1997 ([arXiv:hep-th/9609122](https://arxiv.org/abs/hep-th/9609122))

* {#Witten96b} [[Edward Witten]], _Five-Brane Effective Action In M-Theory_, J. Geom. Phys. 22:103-133, 1997 ([arXiv:hep-th/9610234](https://arxiv.org/abs/hep-th/9610234))

Proposals to model the condition by a [[Wu class]]-shifted variant of [[ordinary differential cohomology]] include the following:

* {#DiaconescuFreedMoore03} [[Duiliu-Emanuel Diaconescu]], [[Dan Freed]], [[Greg Moore]],  _The $M$-theory 3-form and $E_8$-gauge theory_, chapter in [[Haynes Miller]], [[Douglas Ravenel]] (eds.) _Elliptic Cohomology Geometry, Applications, and Higher Chromatic Analogues_, Cambridge University Press 2007 [[arXiv:hep-th/0312069](http://arxiv.org/abs/hep-th/0312069), [doi:10.1017/CBO9780511721489](https://doi.org/10.1017/CBO9780511721489)]

A related model of the C-field in terms of [[nonabelian bundle 2-gerbe|nonabelian]] [[bundle 2-gerbes]]:

* {#AschieriJurčo04} [[Paolo Aschieri]], [[Branislav Jurčo]], *Gerbes, M5-Brane Anomalies and $E_8$ Gauge Theory*, JHEP 0410:068 (2004) [[arXiv:hep-th/0409200](https://arxiv.org/abs/hep-th/0409200), [doi:10.1088/1126-6708/2004/10/068](https://doi.org/10.1088/1126-6708/2004/10/068)]

Further discussion ofthe [[quantum anomaly]] of the supergravity C-field, and its cancellation:

* {#FreedMoore04} [[Dan Freed]], [[Greg Moore]], _Setting the [[quantum integrand]] of M-theory_, Communications in Mathematical Physics **263** 1 (2006) 89-132, [[arXiv:hep-th/0409135](http://arxiv.org/abs/hep-th/0409135), [doi:10.1007/s00220-005-1482-7](https://doi.org/10.1007/s00220-005-1482-7)]

A summary and review of this:

* [[Greg Moore]], _Anomalies, Gauss laws, and Page charges in M-theory_ ([arXiv:hep-th/0409158](http://arxiv.org/abs/hep-th/0409158))

The discussion in [[twisted differential c-structures|twisted nonabelian differential cohomology]] is given in

* {#FSS12} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]],  _[[schreiber:The moduli 3-stack of the C-field]]_, Communications in Mathematical Physics **333** 1 (2015) 117-151, [[arXiv:1202.2455](http://arxiv.org/abs/1202.2455), [doi:10.1007/s00220-014-2228-1](http://link.springer.com/article/10.1007%2Fs00220-014-2228-1)]

* {#FiSaSc} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:7d Chern-Simons theory and the 5-brane|M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory]]_ Advances in Theoretical and Mathematical Physics, Volume 18, Number 2 (2014) p. 229?321   ([arXiv:1201.5277](http://arxiv.org/abs/1201.5277), [doi:10.4310/ATMP.2014.v18.n2.a1](https://dx.doi.org/10.4310/ATMP.2014.v18.n2.a1))

Discussion with [[Dirac charge quantization]] of the C-field in [[twisted Cohomotopy]] ([[schreiber:Hypothesis H]]):

* {#FSS19b} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation]]_, Communications in Mathematical Physics **377** (2020) 1961-2025 [[arXiv:1904.10207](https://arxiv.org/abs/1904.10207), [doi:10.1007/s00220-020-03707-2](https://doi.org/10.1007/s00220-020-03707-2)]

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Twisted Cohomotopy implies M5 WZ term level quantization]]_, Comm. Math. Phys. **384** (2021) 403–432  [[arXiv:1906.07417](https://arxiv.org/abs/1906.07417), [doi:10.1007/s00220-021-03951-0](https://doi.org/10.1007/s00220-021-03951-0)]

surveyed in

* {#FSS20} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], §12 of: _[[schreiber:The Character Map in Twisted Non-Abelian Cohomology|The Character Map in Nonabelian Cohomology --- Twisted, Differential, Generalized]]_, World Scientific (2023) [[arXiv:2009.11909](https://arxiv.org/abs/2009.11909), [doi:10.1142/13422](https://doi.org/10.1142/13422)]

and in the generality of [[orbifold]] [[spacetimes]], in [[equivariant Cohomotopy]]:

* [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Equivariant Cohomotopy implies orientifold tadpole cancellation]]_, J. Geometry and Physics, **156** (2020) 103775 [[arXiv:1909.12277](https://arxiv.org/abs/1909.12277), [doi:10.1016/j.geomphys.2020.103775](https://doi.org/10.1016/j.geomphys.2020.103775)]

* {#SatiSchreiber20} [[Hisham Sati]], [[Urs Schreiber]], pp. 99 of: _[[schreiber:Proper Orbifold Cohomology]]_ [[arXiv:2008.01101](https://arxiv.org/abs/2008.01101)]

Discussion of the dual 6-form field to the 3-form C-field (required notably in the context of [[exceptional generalized geometry]]):

* [[Eugene Cremmer]], [[Bernard Julia]], H. Lu, [[Christopher Pope]], _Dualisation of Dualities, II: Twisted self-duality of doubled fields and superdualities_, Nucl. Phys. B **535** (1998) 242-292 [[arXiv:hep-th/9806106](http://arxiv.org/abs/hep-th/9806106)]

* [[Eric Bergshoeff]], [[Mees de Roo]], [[Olaf Hohm]], _Can dual gravity be reconciled with E11?_, 	Phys. Lett.  B **675** (2009) 371-376 [[arXiv:0903.4384](http://arxiv.org/abs/0903.4384)]

Discussion of [[discrete torsion]] ([[orbifold]] [[equivariant cohomology|equivariance]]) for [[circle n-bundle|circle 3-bundles]] describing the [[supergravity C-field]] is discussed in

* [[Eric Sharpe]], _Analogues of Discrete Torsion for the M-Theory Three-Form_, Phys.Rev. D68 (2003) 126004 ([arXiv:hep-th/0008170](https://arxiv.org/abs/hep-th/0008170))

* Shigenori Seki, _Discrete Torsion and Branes in M-theory from Mathematical Viewpoint_, Nucl.Phys. B606 (2001) 689-698 ([arXiv:hep-th/0103117](https://arxiv.org/abs/hep-th/0103117))

and applied to discussion of [[black brane|black]] [[M2-brane]] [[worldvolume]] [[field theory]] ([[ABJM model]]) in

* [[Ofer Aharony]], Oren Bergman, Daniel Louis Jafferis, _Fractional M2-branes_, JHEP 0811:043,2008 ([arXiv:0807.4924](https://arxiv.org/abs/0807.4924))

* [[Mauricio Romo]], _Aspects of ABJM orbifolds with discrete torsion_, J. High Energ. Phys. (2011) 2011 ([arXiv:1011.4733](https://arxiv.org/abs/1011.4733))

### Duality-symmetric formulation

[[!include self-duality for pregeometric C-field -- references]]

[[!include C-field gauge algebra -- references]]

[[!redirects C3-field]]
[[!redirects C-field]]

[[!redirects supergravity C-fields]]


supergravity Lie 6-algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _supergravity Lie 6-algebra_ ([D'Auria & Fré 1982, p. 18](#DauriaFre82)) is (with hindsight, following [FSS15](#FSS15)) a [[super L-∞ algebra]] such that [[L-infinity algebra valued differential forms|$L_\infty$-algebra valued differential forms]] (local [[connection on an ∞-bundle|∞-connections]]) with values in it encode [[field history|field histories]] of

1. the [[vielbein]] field with a [[spin connection]],

   hence the [[first order formulation of gravity]] for a [[graviton]] field

   in 10+1 dimensions;

1. the [[gravitino]] field;

1. the [[supergravity C-field]]

   _and_ its [[electric-magnetic duality|magnetic dual]].

This is such that the [[field strengths]] and [[Bianchi identities]] of these fields are governed by certain fermionic [[L-∞ algebra cohomology|super L-∞ algebraic cocycles]] as suitable for [[11-dimensional supergravity]].

## Definition

> Our normalization conventions entirely follow [Castellani, D'Auria & Fré 1991, §III.8.3](#CastellaniDAuriaFre), but we denote the [[super-vielbein]] 1-forms by $\big\{e^a\big\}_{a=0}^{10}$ instead of by $\big\{V^a\big\}_{a=0}^{10}$.

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

The [[supergravity Lie 3-algebra]] $\mathfrak{sugra}_3(10,1)$ carries an [[L-∞ algebra cohomology|L-∞ algebra cocycle]] $\mu_7 \in CE\big(\mathfrak{sugra}_3(10,1)\big)$ of degree 7, given in the standard generators $\{e^a\}$ ([[vielbein]]), $\{\omega^{a b}\}$ ([[spin connection]]) $\{\psi^\alpha\}$ ([[gravitino]]) and $\{c_3\}$ ([[supergravity C-field]]) by

$$
  \mu_7
   \;\coloneqq\;
   \frac{i}{2}\bar \psi \wedge \Gamma^{a_1 \cdots a_5} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_5}
   +
   15 \mu_4 \wedge c_3
  \,,
$$

where

$$
  \mu_4
  \;=\;
  \frac{1}{2}
  \overline{\psi}
    \Gamma^{a_1 a_2}
  \psi \wedge e_{a_1} \wedge e_{a_2}
$$

is the 4-cocycle which defines $\mathfrak{sugra}_3(10,1)$ as an extension of $\mathbb{R}^{10,1\vert \mathbf{32}}$, and where $c_3$ is the generator that cancels the class of this cocycle, $d_{CE} c_3 \,=\, \mu_4$.

=--

In other words, we have

$$
  \begin{array}{l}
    \mathrm{d}
    \,
    \mu_4 \;=\; 0
    \\
    \mathrm{d}
    \,
    \mu_7
    \;=\;
    15\,
    \mu_4 \wedge \mu_4
    \mathrlap{\,.}
  \end{array}
$$

which has the same structure as the [[equations of motion]] of the [[field strength]] $G_4$ of the [[supergravity C-field]] and its [[Hodge dual]] $G_7 = \ast G_4$ in [[11-dimensional supergravity]] (to which it is related [below](#eq:SuperMaxwellEquation))

This appears (in the dual language of [[Chevalley-Eilenberg algebras]]) in [DAuria & FrFré 1982, page 18](#DAuriaFre) and
[Castellani, D'Auria & Fré 1991, §III.8.3](#CastellaniDAuriaFre).

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

One computes

$$
  \begin{aligned}
    d_{{}_{CE}} \mu_7
    \;=\;
    &
     - \frac{5}{4} \bar \psi \wedge \Gamma^{a_1 \cdots a_4 b} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_4} \wedge \bar \psi \wedge \Gamma_b \psi
  \\
  & - i 15 \wedge \Gamma^{a b} e_a \wedge \bar \psi \wedge \Gamma_b \psi \wedge c_3
   \\
   & +
    \frac{15}{4} \bar \psi \wedge \Gamma_{a b} \psi \wedge e^a \wedge e^b \wedge \bar \psi \wedge \Gamma_{c d} \psi \wedge e^c \wedge e^d
  \end{aligned}
  \,.
$$

This expression vanishes due to the [[Fierz identities]]

$$
  \bar \psi \wedge
  \Gamma^{a_1 \cdots a_4 b} \psi \wedge \bar \psi \wedge \Gamma_b \psi
  \;=\;
  3 \bar \psi \wedge \Gamma^{[a_1 a_2} \psi \wedge \bar \psi \wedge \Gamma^{a_3 a_4 ]} \psi
$$

and

$$
  \bar \psi \wedge \Gamma^{a b} \psi \wedge \bar \psi \wedge \Gamma_b \psi
  \;=\; 0
  \,.
$$

=--

+-- {: .num_def #TheDefinition}
###### Definition

The _supergravity Lie 6-algebra_ $\mathfrak{sugra}_{7}(10,1)$ is the [[super L-∞ algebra|super Lie 7-algebra]] that is the $b^6 \mathbb{R}$-[[L-∞ algebra cohomology|extension]] of $\mathfrak{sugra}_3(10,1)$ classified by the cocycle $\mu_7$ from def. \ref{TheSevenCocycle}.

$$
  b^5 \mathbb{R} \to \mathfrak{sugra}_6 \to \mathfrak{sugra}_3
  \,.
$$

=--

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

This means that the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{sugra}_6)$ is generated from

* $\{e^a\}$ ([[vielbein]]) in degree $(1,even)$

* $\{\omega^{a b}\}$ ([[spin connection]]) in degree $(1,even)$;

* $\{\psi^\alpha\}$ ([[gravitino]]) in degree $(1,odd)$

* $\{c_3\}$ ([[supergravity C-field]]) in degree $(3,even)$

* $\{c_6\}$ (magnetic dual [[supergravity C-field]]) in degree $(6,even)$

with [[differential]] defined by

$$
  \begin{array}{ccl}
  \omega^{a b}
  &\mapsto&
  \omega^{a c} \wedge \omega_c{}^b
  \\
  e^a
    &\mapsto&
  -\omega^{a b} e_b  - \frac{1}{2}i \bar \psi \wedge  \Gamma^a \psi
  \\
  \psi &\mapsto&
   - \frac{1}{4}\omega^{a b} \Gamma^{a b}
  \\
  c_3
  &\mapsto&
  \frac{1}{2} \bar \psi \wedge \Gamma^{a b} \psi \wedge e_a \wedge e_b
  \\
  c_6
  &\mapsto&
  - \frac{1}{2} \bar \psi \wedge \Gamma^{a_1 \cdots a_5} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_5}
  -
  \frac{13}{2} \bar \psi \Gamma^{a_1 a_2} \psi \wedge e_{a_1} \wedge e_{a_2} \wedge c_3
  \,.
  \end{array}
$$

=--

This appears in [Castellani, D'Auria & Fré, (III.8.18)](#CastellaniDAuriaFre).

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

According to [Castellani, D'Auria & Fré 1991, comment below (III.8.18)](#CastellaniDAuriaFre): "no further extension is possible".

=--

## Relation to $D = 11$ supergravity
  {#RelationToSupergravity}

The supergravity Lie 6-algebra is something like the gauge $L_\infty$-algebra of [[11-dimensional supergravity]], in the sense discussed at _[[D'Auria-Fré formulation of supergravity]]_ .

\begin{definition}
\label{ModifiedWeilAlgebra}

Write $W\big(\mathfrak{sugra}_6(10,1)\big)$ for the [[Weil algebra]] of the supergravity Lie 6-algebra.

Write $g_4$ and $g_7$ for the shifted generators of the Weil algebra corresponding to $c_3$ and $c_6$, respectively.

Define an [[adjusted Weil algebra]] $\tilde W(\mathfrak{sugra}_6(10,1))$ by declaring it to have the same generators and differential as before, except that the differential for $c_6$ is modified to

$$
  d_{\tilde W} \, c_6
    \coloneqq
  d_{W} \, c_6
    +
  15 \, g_4 \wedge c_3
$$

and hence the differential of $g_7$ is accordingly modified in the unique way that ensures $d_{\tilde W}^2 = 0$ (yielding the modified [[Bianchi identity]] for $g_7$).

\end{definition}

This ansatz appears in [Castellani, D'Auria & Fré 1991 (III.8.24)](#CastellaniDAuriaFre).

Note that this amounts simply to a redefinition of [[curvatures]]

$$
  \tilde g_7
    \;\coloneqq\;
  g_7 + 15 g_4 \wedge c_3
  \,.
$$

\begin{proposition}
\label{SugraEOM}

A field configuration of [[11-dimensional supergravity]] is given by [[∞-groupoid of ∞-Lie algebra valued forms|L-∞ algebra valued differential forms]] with values in $\mathfrak{sugra}_6$. Among all of these the solutions to the equations of motion (the points in the [[covariant phase space]]) can be characterized as follows:

A field configuration

$$
  \Omega^\bullet(X)
     \leftarrow
  \tilde W(\mathfrak{sugra}_6)
  \;\colon\;
  \Phi
$$

solves the equations of motion precisely if

1. all [[curvatures]] sit in the ideal of differential forms spanned by the 1-form fields $E^a$ ([[vielbein]]) and $\Psi$ ([[gravitino]]);

   more precisely if we have

   * $\tau = 0$

     (super-[[torsion]]);

   * $G_4 = (G_4)_{a_1, \cdots a_4} E^{a_1} \wedge \cdots E^{a_4}$

     ([[field strength]] of [[supergravity C-field]])

   * $G_7 = (G_7)_{a_1, \cdots a_7} E^{a_1} \wedge \cdots E^{a_7}$

     ([[electric-magnetic duality|dual]] field strength)

   * $\rho = \rho_{a b} E^a \wedge E^b  + H_a \Psi \wedge E^a$

     ([[Dirac operator]] applied to [[gravitino]])

   * $R^{a b} = R^{a b}_{c d} E^c \wedge E^d + \bar \Theta^{a b}{}_c \Psi \wedge E^c + \bar \Psi \wedge K^{a b} \Psi$

     ([[Riemann tensor]]: [[field strength]] of [[gravity]])

1. such that the coefficients of terms containing $\Psi$s are polynomials in the coefficients of the terms containing no $\Psi$s. ("rheonomy").

\end{proposition}

This is, in paraphrase, the content of [Castellani, D'Auria & Fré, section III.8.5](#CastellaniDAuriaFre).

\begin{remark}
\label{EOMOfSuperCField}

In particular, the [[Bianchi identity]] for the super-form enhancement of the [[supergravity C-field]] [[flux density]] is (by [III.8.23j, 34, 35, 36](#CastellaniDAuriaFre) and using hupf):

\[
  \label{SuperMaxwellEquation}
  \begin{array}{l}
  \mathrm{d}
  \big(
    \underset{
      G_7^{super}
    }{
      \underbrace{
        F_{a_1 \cdots a_4}
        e^{a_1} \wedge \cdots \wedge e^{a_4}
        \,+\,
        \mathrm{i}
        \overline{\psi} \Gamma_{a_1 \cdots a_5} \psi
        \,
        e^{a_1} \wedge \cdots \wedge e^{a_5}
      }
    }
  \big)
  \\
  \;=\;
  -15
  \big(
    \underset{G_4^{super}}{
    \underbrace{
      F_{a_1 \cdots a_4}
      e^{a_1} \wedge \cdots \wedge e^{a_4}
      +
      \tfrac{1}{2}
      \overline{\psi}\Gamma_{a_1 a_2}\psi
      \,
      e^{a_1} \wedge e^{a_2}
    }
    }
  \big)
  \wedge
  \big(
    \underset{G_4^{super}}{
    \underbrace{
      F_{a_1 \cdots a_4}
      e^{a_1} \wedge \cdots \wedge e^{a_4}
      +
      \tfrac{1}{2}
      \overline{\psi}\Gamma_{a_1 a_2}\psi
      \,
      e^{a_1} \wedge e^{a_2}
    }
    }
  \big)
  \mathrlap{\,}
  \end{array}
\]

and its rheonomic solution implies for the "actual" [[flux densities]] $F_4 \coloneqq F_{a_1 \cdots a_4} e^{a_1} \wedge \cdots \wedge e^{a_4}\vert_{\psi=0}$ and $G_4 \coloneqq  G_{a_1 \cdots a_7} e^{a_1} \wedge \cdots e^{a_7} \vert_{\psi = 0}$ that

1. the CJS [[higher gauge theory|higher]] [[Maxwell equation]] holds

   $$
     \mathrm{d}\, G_7 \;=\; - 15 \, F_4 \wedge F_4
   $$

   ([III.8.53](#CastellaniDAuriaFre))

1. the [[Hodge duality|Hodge duality]] holds

   $$
     G_7 \;=\; \star F_4
   $$

   ([III.8.52](#CastellaniDAuriaFre))

\end{remark}

## Related concepts

**supergravity Lie 6-algebra** $\to$ [[supergravity Lie 3-algebra]] $\to$ [[super Poincaré Lie algebra]]

* [[4d supergravity Lie 2-algebra]]

* [[type II supergravity Lie 2-algebra]]

* [[M-theory super Lie algebra]]

* [[adjusted Weil algebra]]

## References

The supergravity Lie 6-algebra originates in:

* {#DAuriaFre82} [[Riccardo D'Auria]], [[Pietro Fré]], p. 18 of: _[[GeometricSupergravity.pdf:file]]_, Nuclear Physics B201 (1982) 101-140 [doi:10.1016/0550-3213(82)90376-5,  [[GeometricSupergravityErrata.pdf:file]]]

Brief review is in:

* {#CFGPN83} [[Leonardo Castellani]], [[Pietro Fré]], F. Giani, [[Krzysztof Pilch]], [[Peter van Nieuwenhuizen]], §4 of: *Gauging of $d = 11$ supergravity?*, Annals of Physics **146** 1 (1983) 35-77 [[spire:11998](https://inspirehep.net/literature/11998), doi:10.1016/0003-4916(83)90052-0]

* [[Pietro Fré]], [[Pietro Antonio Grassi]], §3 of: *Free Differential Algebras, Rheonomy, and Pure Spinors* [[arXiv:0801.3076](http://arxiv.org/abs/0801.3076), [inspire:777785](https://inspirehep.net/literature/777785)]

* [[Pietro Fré]], §6.4.1 in: *Gravity, a Geometrical Course*, Volume 2: *Black Holes, Cosmology and Introduction to Supergravity*, Springer (2013) [[doi:10.1007/978-94-007-5443-0](https://doi.org/10.1007/978-94-007-5443-0)]

A monograph account is in:

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], vol 2, III.8.3 of: *[[Supergravity and Superstrings - A Geometric Perspective]]*, World Scientific (1991) [[doi:10.1142/0224](https://doi.org/10.1142/0224), [epdf](https://epdf.pub/supergravity-and-superstrings-a-geometric-perspective-vol-2-supergravity.html), ch III.8: [[CastellaniDAuriaFre-ChIII8.pdf:file]]]

It was (apparently) rediscovered in:

* {#CAIB99} C. Chryssomalakos, [[José de Azcárraga]], [[José M. Izquierdo]], C. Pérez Bueno, around equation (8.8) in: _The geometry of branes and extended superspaces_, Nucl. Phys. B **567** (2000) 293-330  [[arXiv:hep-th/9904137](http://arxiv.org/abs/hep-th/9904137), doi:10.1016/S0550-3213(99)00512-X]

where a detailed discussion is given.

These authors all consider the [[Chevalley-Eilenberg algebra]] ("FDA") of the actual [[super L-infinity algebra|super $L_\infty$-algebra]]. The latter and its relation to [[smooth super ∞-groupoids]] is considered in:

* {#FSS15} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]_, Int. J. Geom. Meth. Modern Physics **12** 02 (2015) 1550018 [[arXiv:1308.5264](http://arxiv.org/abs/1308.5264), [doi:10.1142/S0219887815500188](http://www.worldscientific.com/doi/abs/10.1142/S0219887815500188)]

* [[Urs Schreiber]], last section of: _[[schreiber:differential cohomology in a cohesive topos]]_


supergroup

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

#Contents#
* table of contents
{:toc}

## Idea

A _super-group_ is the analog in [[supergeometry]] of [[Lie groups]] in [[differential geometry]].

## Algebraic super groups

An [[affine scheme|affine]] algebraic super group is the [[formal dual]]  of a [[superalgebra|super]]-[[commutative Hopf algebra]].

## Super Lie groups

### Definition

A _super Lie group_ is a [[group object]] in the [[category]] [[SDiff]] of [[supermanifolds]], that is a super [[Lie group]].

#### In terms of generalized group elements

One useful way to characterize [[group object]]s $G$ in the
[[category]] $SDiff$ of [[supermanifold]] is by first sending
$G$ with the [[Yoneda embedding]] to a [[presheaf]] on $SDiff$
and then imposing a lift of $Y(G) : SDiff^{op} \to Set$
through the [[forgetful functor]] [[Grp]] $\to$ [[Set]]
that sends a (ordinary) [[group]] to its underlying [[set]].

So a group object structure on $G$ is a diagram

$$
  \array{
    && Grp
    \\
    & {}^{(G,\cdot)}\nearrow & \downarrow
    \\
    SDiff^{op} &\stackrel{Y(G)}{\to}& Set
  }
  \,.
$$

This gives for each [[supermanifold]] $S$ an ordinary group
$(G(S), \cdot)$, so in particular a product operation

$$
  \cdot_S : G(S) \times G(S) \to G(S)
  \,.
$$

Moreover, since morphisms in $Grp$ are group homomorphisms, it follows
that for every morphism $f : S \to T$ of [[supermanifold]]s we get a
commuting diagram

$$
  \array{
    G(S) \times G(S) &\stackrel{\cdot_S}{\to}& G(S)
    \\
    \uparrow^{G(f)\times G(f)}  && \uparrow^{G(f)}
    \\
    G(T) \times G(T) &\stackrel{\cdot_T}{\to}& G(T)
  }
$$

Taken together this means that there is a morphism

$$
  Y(G \times G)  \to Y(G)
$$

of representable presheaves. By the [[Yoneda lemma]], this uniquely
comes from a morphism $\cdot : G \times G \to G$, which is the
product of the group structure on the object $G$ that we are after.

etc.

This way of thinking about supergroups is often explicit in
some parts of the literature on supergeometry: some authors
define a supergroup or [[super Lie algebra]] as a
rule that assigns to every [[Grassmann algebra]] $A$
over an ordinary [[vector space]] an ordinary [[group]]
$G(A)$
or [[Lie algebra]] and to a morphism of [[Grassmann algebra]]s
$A \to B$
_covariantly_ a morphism of groups $G(A) \to G(B)$.
But the [[Grassmann algebra]] on an $n$-dimensional [[vector space]]
is naturally isomorphic to the function ring on the [[supermanifold]]
$\mathbb{R}^{0|n }$. So the definition of supergroups in terms of
Grassmann algebras is secretly the same as the above definition
in terms of the [[Yoneda embedding]].

### Examples

#### The super-translation group
  {#SuperTranslationGroup}

also called the **[[super-Heisenberg group]]**

The additive group structure on $\mathbb{R}^{1|1}$ is given on [[generalized element]]s in (i.e. in the logic internal to) the [[topos]] of [[sheaf|sheaves]] on the category [[SCartSp]] of [[cartesian space|cartesian]] superspaces by

$$
  \mathbb{R}^{1|1} \times \mathbb{R}^{1|1} \to
  \mathbb{R}^{1|1}
$$

$$
  (t_1, \theta_1), (t_2, \theta_2)
  \mapsto
  (t_1 + t_2 + \theta_1 \theta_2, \theta_1 + \theta_2)
  \,.
$$

Recall how the notation works here: by the [[Yoneda embedding]] we have a [[full and faithful functor]]

> [[SDiff]] $\hookrightarrow$ $Fun(SDiff^{op}, Set)$

and we also have the theorem, discussed at [[supermanifold]]s, that maps from some $S \in SDiff$ into $\mathbb{R}^{p|q}$ is given by a tuple of $p$ even section $t_i$ and $q$ odd sections $\theta_j$. The above notation specifies the map of supermanifolds by displaying what map of sets of maps from some test object $S$ it corresponds to under the [[Yoneda embedding]].

Now, for each $S \in $ [[SDiff]] there is a [[group]] structure on the [[hom-set]] $SDiff(S, \mathbb{R}^{1|1}) \simeq C^\infty(S)^{ev} \times C^\infty(S)^{odd}$ given by precisely the above formula for this given $S$

$$
  \mathbb{R}^{1|1}(S) \times \mathbb{R}^{1|1}(S) \to
  \mathbb{R}^{1|1}(S)
$$

$$
  (t_1, \theta_1), (t_2, \theta_2)
  \mapsto
  (t_1 + t_2 + \theta_1 \theta_2, \theta_1 + \theta_2)
  \,.
$$

where $(t_i, \theta_i) \in C^\infty(S)^{ev} \times C^\infty(S)^{odd}$ etc and where the addition and product on the right takes place in the function [[super algebra]] $C^\infty(S)$.

Since the formula looks the same for all $S$, one often just writes it without mentioning $S$ as above.

#### The super Euclidean group

The super-translation group is the $(1|1)$-dimensional case of the [[super Euclidean group]].

#### General linear supergroup

[[general linear supergroup]]

...

#### Orthosymplectic supergroup

[[orthosymplectic supergroup]]

...

## Finite super-groups

There is a finite analog for super-groups that does not quite fit in the framework presented here:

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

A finite super-group is a tuple $(G, z \in G)$, where $G$ is a finite group and $z$ is central and squares to $1$.

The representations of a finite super-group are $\mathbb{Z}_2$-graded:
An irreducible representation has odd degree if $z$ acts by negation, and even degree if it acts as the identity.

This definition is found e.g. in:

* Paul Bruillard, Cesar Galindo, Tobias Hagge, Siu-Hung Ng, Julia Yael Plavnik, Eric C. Rowell, Zhenghan Wang, _Fermionic Modular Categories and the 16-fold Way_ ([pdf] (https://arxiv.org/pdf/1603.09294v2))

=--

## Properties

### Representations, Tannaka duality and Deligne's theorem

[[Deligne's theorem on tensor categories]] (see there for details) says that every suitably well-behave linear [[tensor category]] is the [[category of representations]] of an algebraic supergroup. In particular the [[Hopf algebra]] of functions on an affine algebraic supergroup is a [[triangular Hopf algebra]].

[[!include structure on algebras and their module categories - table]]

## References

Discussion via [[functorial geometry]]:

* [[Claudio Carmeli]], Lauren Caston, [[Rita Fioresi]], _Mathematical Foundations of Supersymmetry_, EMS Series of Lectures in Mathematics Volume: 15; 2011; 263 pp; ( [ISBN:978-3-03719-097-5](https://bookstore.ams.org/emsserlec-15/), [arXiv:0710.5742](https://arxiv.org/abs/0710.5742))

See also

* Katsumi Yagi, _Super Lie Groups_, Adv. Stud. Pure Math. Progress in Differential Geometry (1993) 407 - 412 ([euclid:1534359537](https://projecteuclid.org/euclid.aspm/1534359537))

* [[Dennis Westra]], _Superrings and supergroups_, 2009 ([pdf](http://www.mat.univie.ac.at/~michor/westra_diss.pdf))

* Groeger, _Super Lie groups and super Lie algebras_, lecture notes 2011 ([pdf](http://www.mathematik.hu-berlin.de/~groegerj/teaching_files/lecture12.pdf))

* [[Veeravalli Varadarajan]], section 6.1 of: _[[Supersymmetry for mathematicians]]: An introduction_

Discussion of [[group extensions]] of supergroups includes

* {#Drupieski14} [[Christopher Drupieski]], _Strict polynomial superfunctors and universal extension classes for algebraic supergroups_ ([arXiv:1408.5764](http://arxiv.org/abs/1408.5764))

Discussion as Hopf-superalgebras includes

* Nicolás Andruskiewitsch, Iván Angiono, Hiroyuki Yamane, _On pointed Hopf superalgebras_, Contemp. Math. vol. 544, pp. 123--140, 2011 ([arXiv:1009.5148](https://arxiv.org/abs/1009.5148))

[[!redirects supergroups]]

[[!redirects super Lie group]]
[[!redirects super Lie groups]]

[[!redirects super group]]
[[!redirects super groups]]

[[!redirects Lie supergroup]]
[[!redirects Lie supergroups]]

[[!redirects super-group]]
[[!redirects super-groups]]


superisometry group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The analog of the concept of [[isometry group]] in [[supergeometry]].

## Related concepts

* [[Killing spinor]]

* [[BPS state]]

* [[super-diffeomorphism]]

* [[supersymmetry]]

## References

Discussion identifying [[twistor spinors]] (hence in particular [[Killing spinors]]) as infinitesimal superisometries is in

* [[Dmitri Alekseevsky]], [[Vicente Cortés]], [[Chandrashekar Devchand]], [[Uwe Semmelmann]], _Killing spinors are Killing vector fields in Riemannian Supergeometry_ ([arXiv:dg-ga/9704002](http://arxiv.org/abs/dg-ga/9704002))

Discussion regarding the conceptualization of Killing spinors in [[super-Cartan geometry]] is in

* {#EgeilehChami13} Michel Egeileh, Fida El Chami, _Some remarks on the geometry of superspace supergravity_, J.Geom.Phys. 62 (2012) 53-60 ([spire](http://inspirehep.net/record/1333125))

Discussion in [[string theory]] includes

* [[Paul Townsend]], _Killing spinors, supersymmetries and rotating intersecting branes_, in _Novelties in string theory_ (Göteborg, 1998), pp. 177–182. World Sci.
Publishing, River Edge, NJ, 19 ([arXiv:hep-th/9901102](http://arxiv.org/abs/hep-th/9901102))

and specifically for [[type II supergravity]]:

* [[José Figueroa-O'Farrill]], Emily Hackett-Jones, George Moutsopoulos, _The Killing superalgebra of ten-dimensional supergravity backgrounds_, Class.Quant.Grav.24:3291-3308,2007 ([arXiv:hep-th/0703192](http://arxiv.org/abs/hep-th/0703192))

and [[11-dimensional supergravity]]

* [[José Figueroa-O'Farrill]], Andrea Santi, _On the algebraic structure of Killing superalgebras_ ([arXiv:1608.05915](http://arxiv.org/abs/1608.05915))

Discussion in the context of [[superconformal group|superconformal symmetry]] is in

* P. Claus, [[Renata Kallosh]], _Superisometries of the $adS \times S$ Superspace_, JHEP 9903:014,1999 ([arXiv:hep-th/9812087](http://arxiv.org/abs/hep-th/9812087))

[[!redirects superisometry supergroupd]]

[[!redirects superisometry]]
[[!redirects superisometries]]

[[!redirects super-isometry]]
[[!redirects super-isometries]]

[[!redirects super-isometry group]]
[[!redirects super-isometry groups]]


supermanifold

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

#Contents#
* table of contents
{:toc}

## Idea

A [[supermanifold]] is a [[space]] locally modeled on [[Cartesian space|Cartesian spaces]] and [[superpoints]].

There are different approaches to the definition and theory of supermanifolds in the literature. The definition

* [As locally ringed spaces](#AsLocallyRingedSpace)

is popular. The definition

* [As manifolds over superpoints](#OverSuperpoints)

has been argued to have advantages, see also the references at [[super ∞-groupoid]].

## As locally representable sheaves on super Cartesian spaces

See at _[[geometry of physics -- supergeometry]]_ the section _[Supermanifolds](geometry+of+physics+--+supergeometry#Supermanifolds)_.

## As locally ringed spaces
 {#AsLocallyRingedSpace}

We discuss a description of supermanifolds that goes back to
([BerezinLeites](BerezinLeites)).

### Definition

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

A **supermanifold** $X$ of dimension $p|q$ is a
[[ringed space]] $(|X|, O_X)$ where

* the [[topological space]] $|X|$ is [[second countable space]], [[Hausdorff space]],

* $O_X$ is a [[sheaf]] of commutative [[super algebras]] that is locally on small enough open subsets $U \subset |X|$ isomorphic to one of the form $C^\infty(\mathbb{R}^p) \otimes \wedge^\bullet \mathbb{R}^q$.

A [[morphism]] of supermanifolds is a [[homomorphism]] of [[ringed spaces]] (...).

=--

Forgetting the graded part by projecting out the [[nilpotent ideal]] in $O_X$ (i.e. applying the [[bosonic modality]]) yields the underlying ordinary  [[smooth manifold]] $X_{red}$.

One just writes $C^\infty(X)$ for the [[super algebra]] $O_X(X)$ of global sections.

With the obvious morphisms of [[ringed space]] this forms the [[category]] [[SDiff]] of supermanifolds.

### Examples

* [[super Cartesian space]]

  * [[superpoint]]

    * [[odd line]]

+-- {: .num_example }
###### Example

For $E \to X$ a smooth finite-rank [[vector bundle]] the [[manifold]] $X$ equipped with the [[Grassmann algebra]] over $C^\infty(X)$ of the sections  of the dual bundle

$$
  O_X(U) := \Gamma (\wedge^\bullet(E^*))
$$

is a supermanifold. This is usually denoted by $\Pi E$.

=--

+-- {: .num_example }
###### Example

In particular, let $\mathbb{R}^{p+q} \to \mathbb{R}^p$ be the trivial rank $q$ [[vector bundle]] on $\mathbb{R}^p$ then one writes

$$
  \mathbb{R}^{p|q} := \Pi (\mathbb{R}^{p+q} \to \mathbb{R}^p)
$$

for the corresponding supermanifold.

=--

### Properties
 {#AsLocallyRingedSpacesProperties}

+-- {: .num_theorem #BatchelorTheorem}
###### Theorem
**(Batchelor's theorem)**

Every supermanifold is [[isomorphism|isomorphic]] to one of the form $\Pi E$ where $E$ is an ordinary smooth [[vector bundle]].

=--

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

Nevertherless, the category of supermanifolds is far from being [[equivalence of categories|equivalent]] to that of [[vector bundles]]: a morphism of vector bundles translates to a morphism of supermanifolds that is strictly homogeneous in degrees, while a general morphism of supermanifolds need not be of this form.

=--

But we have the following useful characterization of morphisms of supermanifolds:

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

* There is a [[natural bijection]]

  $$
    SDiff(X,Y) \simeq SAlgebras(C^\infty(Y), C^\infty(X)),
  $$

  so the contravariant embedding of supermanifolds into superalgebra is a [[full and faithful functor]].

* Composition with the standard coordinate functions on $\mathbb{R}^{p|q}$ yields an [[isomorphism]]

  $$
    SDiff(X, \mathbb{R}^{p|q})
    \simeq
    \underbrace{
    (C^\infty(X)^{ev} \times \cdots
    \times C^\infty(X)^{ev})}_{p\; times}
    \times
    \underbrace{
      (C^\infty(X)^{odd} \times \cdots
      \times C^\infty(X)^{odd})}_{q\; times}
  $$

=--

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

The first statement is a direct extension of the classical fact that [[smooth manifolds embed into formal duals of R-algebras]].
=--

## As manifolds modeled on Grassman algebras
 {#ModeledOnGrassmannAlgebra}

We discuss a description of supermanifolds that goes back to
([DeWitt 92](#DeWitt92)) and ([Rogers](#Rogers)).

(...)

## As manifolds over the base topos on superpoints
 {#OverSuperpoints}

Let $SuperPoint$ be the [[category]] of [[superpoint|superpoints]]. And $Sh(SuperPoint) = PSh(SuperPoint)$ its [[presheaf topos]].

We discuss a definition of supermanifolds that characterizes them, roughly,  as manifolds over this [[base topos]]. See ([Sachse](#Sachse)) and the references at [[super ∞-groupoid]].

See also [this post](http://theoreticalatlas.wordpress.com/2013/07/26/john-huerta-supermanifolds/) at Theoretical Atlas.

### Definition

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

Let

$$
  SuperSet := Sh(SuperPoint)
$$

be the [[sheaf topos]] over [[superpoints]]. Let

$$
  \mathbb{R} \in Ring(SuperSet)
$$

be the canonical [[continuum]] [[real line]] under the restricted [[Yoneda embedding]] of supermanifolds and equipped with its canonical internal algebra structure, hence by prop. \ref{MorphismsOfSupermanifoldsByAlgebras} the presheaf of algebras which sends a Grassmann algebra to its even subalgebra, as discussed at [[superalgebra]].

=--

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

A **superdomain** is an open subfunctor (...) of a [[locally convex vector space|locally convex]] $\mathbb{K}$-module.

=--

This appears as ([Sachse, def. 4.6](#Sachse)).

We now want to describe supermanifolds as [[manifold|manifolds]] in $SuperSet$ modeled on superdomains.

Write [[SmoothMfd]] for the [[category]] of ordinary [[smooth manifold|smooth manifolds]].

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

A _supermanifold_ is a functor $X : SuperPoint^{op} \to SmoothMfd$ equipped with an equivalence class of [supersmooth atlases](#).

A [[morphism]] of supermanifolds is a [[natural transformation]] $f : X \to X'$, such that for each pair of [[chart|charts]] $u : U \to X$ and $u' : U' \to X'$ the [[pullback]]

$$
  \array{
    U \times_{X'} U' &&\stackrel{p'}{\to}&& U'
    \\
    {}^{\mathllap{p_1}}\downarrow & & && \downarrow^{\mathrlap{u'}}
    \\
    U &\stackrel{u}{\to}& X &\stackrel{f}{\to}& X'
  }
$$

can be equipped with the structture of a [Banach superdomain](#) such that $p_1$ and $p_2$ are supersmooth (...)

=--

This appears as ([Sachse, def. 4.13, 4.14](#Sachse)).

### Properties

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

The categories of supermanifolds defined as locally ringed spaces,
def. \ref{SupermanifoldLocallyRingedSpace} and as manifolds over superpoints,
def. \ref{SupermanifoldOverSuperpoint} are
[[equivalence of categories|equivalent]].

=--

This appears as ([Sachse, theorem 5.1](#Sachse)). See section 5.2 there for a discussion of the relation to the [DeWitt-definition](#ModeledOnGrassmannAlgebra).

## Related concepts

* [[super-scheme]], [[spectral super-scheme]]

* [[integration over supermanifolds]]

* [[super vector bundle]]

* [[complex supermanifold]], [[super Riemann surface]]

* [[superconnection]]

* [[superdifferential form]]

* [[superalgebra]], [[smooth superalgebra]]

* [[NQ-supermanifold]]

* [[super-orbifold]]

## References

### Via functorial geometry

Discussion from the point of view of [[functorial geometry]]:

* [[Claudio Carmeli]], Lauren Caston, [[Rita Fioresi]], _Mathematical Foundations of Supersymmetry_, EMS Series of Lectures in Mathematics Volume: 15; 2011; 263 pp; ( [ISBN:978-3-03719-097-5](https://bookstore.ams.org/emsserlec-15/), [arXiv:0710.5742](https://arxiv.org/abs/0710.5742))

* {#HohnholdSolzTeichner11} [[Henning Hohnhold]], [[Stephan Stolz]], [[Peter Teichner]], _Super manifolds: an incomplete survey_, Bulletin of the Manifold Atlas (2011) 1–6 ([pdf](https://people.mpim-bonn.mpg.de/teichner/Math/ewExternalFiles/Survey-Journal.pdf))

* [[Henning Hohnhold]], [[Matthias Kreck]], [[Stephan Stolz]], [[Peter Teichner]], Sections 2-3 of: _Differential forms and 0-dimensional supersymmetric field theories_, Quantum Topology Volume 2, Issue 1 (2011) pp. 1–41 ([doi:10.4171/QT/12](http://dx.doi.org/10.4171/QT/12))

### As locally ringed spaces

* {#BerezinLeites} [[Felix Berezin]], D. A. Leĭtes, _Supermanifolds_, (Russian) Dokl. Akad. Nauk SSSR __224__ (1975), no. 3, 505--508; English transl.: Soviet Math. Dokl. __16__ (1975), no. 5, 1218--1222 (1976).

* I. L. Buchbinder, S. M. Kuzenko, _Ideas and methods of supersymmetry and supergravity; or A walk through superspace_

A more general variant of this in the spirit of [[locally algebra-ed topos|locally algebra-ed toposes]] is in

* Alexander Alldridge, _A convenient category of supermanifolds_ ([arXiv:1109.3161](http://arxiv.org/abs/1109.3161))

### As manifolds over superpoints
 {#ReferencesOverSuperpoints}

The observation that the study of super-structures in mathematics is usefully regarded as taking place over the [[base topos]] on the [[site]] of [[super points]] has been made around 1984 in

* [[Albert Schwarz]], _On the definition of superspace_, Teoret. Mat. Fiz. (1984)  Volume 60,  Number 1, Pages 37–42, ([russian original pdf](http://www.mathnet.ru/links/b12306f831b8c37d32d5ba8511d60c93/tmf5111.pdf))

* [[Alexander Voronov]], _Maps of supermanifolds_ , Teoret. Mat. Fiz. (1984)  Volume 60,  Number 1, Pages 43–48

and in

* V. Molotkov., _Infinite-dimensional $\mathbb{Z}_2^k$-supermanifolds_ , ICTP
preprints, IC/84/183, 1984.

A summary/review is in the appendix of

* {#KonechnySchwarz98} Anatoly Konechny, [[Albert Schwarz]],

  _On $(k \oplus l|q)$-dimensional supermanifolds_, in: [[Julius Wess]], V. Akulov (eds.) _Supersymmetry and Quantum Field Theory_ (D. Volkov memorial volume) Lecture Notes in Physics, 509, Springer 1998 ([arXiv:hep-th/9706003](http://arxiv.org/abs/hep-th/9706003))

  Anatoly Konechny, [[Albert Schwarz]], _Theory of $(k \oplus l|q)$-dimensional supermanifolds_, Sel. math., New ser. 6 (2000) 471 - 486 ([doi:10.1007/PL00001396](https://doi.org/10.1007/PL00001396))

* [[Albert Schwarz]], I. Shapiro, _Supergeometry and Arithmetic Geometry_ ([arXiv:hep-th/0605119](http://arxiv.org/abs/hep-th/0605119))

A review with more emphasis on the relevant [[category theory]]/[[topos theory]] is in

* {#Sachse} [[Christoph Sachse]], _A Categorical Formulation of Superalgebra and Supergeometry_ ([arXiv:0802.4067](http://arxiv.org/abs/0802.4067))

The site of formal duals not just to [[Grassmann algebras]] but to all super-[[infinitesimally thickened points]] is discussed in ([Konechny-Schwarz](#KonechnySchwarz)) above and also in

* L. Balduzzi, C. Carmeli, R. Fioresi, _The local functors of points of Supermanifolds_, Expositiones Mathematicae Volume 28, Issue 3, 2010, Pages 201-217 ([doi:10.1016/j.exmath.2009.09.005](https://doi.org/10.1016/j.exmath.2009.09.005), [arXiv:0908.1872](http://arxiv.org/abs/0908.1872))

### As manifolds modelled on Grassmann algebras

* {#DeWitt92} [[Bryce DeWitt]], _Supermanifolds_, Cambridge Monographs on Mathematical Physics, 1984

* {#Rogers} [[Alice Rogers]], Supermanifolds: Theory and Applications, World Scientific, (2007)

Alice Rogers claims, in Chapter 1, that the smooth-manifold-of-(infinite-dimensional)-Grassmann-algebras approach (the "concrete approach") is identical to the sheaf-of-ringed-spaces approach (the "algebro-geometric" approach) and that this equivalence is shown in Chapter 8. DeWitt seems unsure of this, but is writing more than 20 years earlier, before the ringed-space approach has been fully developed.

### Other

Discussion with an eye towards [[supergravity]] is in

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], section II.2.4 of _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

Discussion with an eye on [[integration over supermanifolds]] is in

* [[Edward Witten]], _Notes On Supermanifolds and Integration_ ([arXiv:1209.2199](http://arxiv.org/abs/1209.2199))

Global properties are discussed in

* [[Louis Crane]], Jeffrey M. Rabin, _Global properties of supermanifolds_, Comm. Math. Phys. Volume 100, Number 1 (1985), 141-160. ([Euclid](http://projecteuclid.org/euclid.cmp/1103943340))

See also:

* [[Yuri Manin]], _Topics in noncommutative geometry_, Princeton Univ. Press 1991.

* [[Pierre Deligne]], P. Etingof, [[Daniel Freed]], L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison and [[Edward Witten]] (eds.)  _[[Quantum Fields and Strings]], A course for mathematicians_, 2 vols. Amer. Math. Soc. Providence 1999. ([web version](http://www.math.ias.edu/qft))

* V. S. Varadarajan, _Supersymmetry for mathematicians: an introduction_, AMS and Courant Institute, 2004.

* [[Alberto S. Cattaneo]], Florian Schaetz, _Introduction to supergeometry_, [arxiv/1011.3401](http://arxiv.org/abs/1011.3401)

There are many books in [[physics]] on [[supersymmetry]] (most well known is by Wess and Barger from 1992), but they emphasise more on the [[supersymmetry algebra|supersymmetries]] rather than on (the superspace and) _supermanifolds_. They should therefore rather be listed under entry [[supersymmetry]].

See also [pdf](http://www.math.uni-hamburg.de/home/sachse/handoutbatchelor.pdf)

[[!redirects supermanifolds]]

[[!redirects super manifold]]
[[!redirects super manifolds]]

[[!redirects super-manifold]]
[[!redirects super-manifolds]]


supernova

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## Idea

(...)

## Related concepts

* [[neutron star]]

* [[black hole]]

* [[gravitational wave]]

## References

### General

See also

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

Computer simulation:

* R. Bollig et al., _Self-consistent 3D Supernova Models From -7 Minutes to +7 Seconds: a 1-bethe Explosion of a ~19 Solar-mass Progenitor_ ([arXiv:2010.10506](https://arxiv.org/abs/2010.10506))

### As standard candles for cosmology
 {#ReferencesAsStandardCandles}

Original apparent evidence, from observation of type IA supernovae, for positive [[dark energy]] (as in [[de Sitter spacetime]]) and hence for the "[[standard model of cosmology]]"  (but notice that the conclusion has been debated, see the references [here](cosmological+constant#NonObservation) and see at *[[inhomogeneous cosmology]]*):

* {#Riess1998} [[Adam G. Riess]] et al., *Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant*,  The Astronomical Journal, **116** 3 (1998) [[doi:10.1086/300499](https://iopscience.iop.org/article/10.1086/300499)]

* {#PerlmutterEtAl1999} [[Saul Permutter]] et al. *Measurements of Omega and Lambda from 42 High-Redshift Supernovae*, Astrophys. J. **517** (1999) 565-586 [[arXiv:astro-ph/9812133](https://arxiv.org/abs/astro-ph/9812133), [doi:10.1086/307221](https://doi.org/10.1086/307221)]

Early amplification and review:

* {#Goldhaber2001a} [[Gerson Goldhaber]], *Evidence for Dark Energy from a Study of Type Ia Supernovae*, in: *Dark Matter in Astro- and Particle Physics*, Springer (2001) [[doi:10.1007/978-3-642-56643-1_13](https://doi.org/10.1007/978-3-642-56643-1_13)]

* {#Goldhaber2001b} [[Gerson Goldhaber]], *Large Evidence for Dark Energy from a Study of Type Ia Supernovae*, in: *Recent Developments in Particle Physics and Cosmology*, Springer (2001) [[doi:10.1007/978-94-010-0676-7_12](https://doi.org/10.1007/978-94-010-0676-7_12)]

* [[Gerson Goldhaber]], *The Acceleration of the Expansion of the Universe: A Brief Early History of the Supernova Cosmology Project (SCP)*, AIP Conference Proceedings **1166** 53 (2009) [[arXiv:0907.3526](https://arxiv.org/abs/0907.3526), [doi:10.1063/1.3232196](https://doi.org/10.1063/1.3232196)]

Later improved measurements started to be in tension with results of the [[Planck Collaboration]]:

* {#RiessEtAl16} [[Adam G. Riess]] et al., _A 2.4% Determination of the Local Value of the Hubble Constant_,  The Astrophysical Journal **826** 1 (2016) [[arXiv:1604.01424](https://arxiv.org/abs/1604.01424), [doi:10.3847/0004-637X/826/1/56](https://iopscience.iop.org/article/10.3847/0004-637X/826/1/56)]

[[!redirects supernovae]]


superoperator

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## Idea

In [[quantum physics]] and in particular in the context of [[quantum computing]] a [[linear map]] between spaces of [[linear operators]] (hence an "operator on operators") is sometimes called a _superoperator_.

Notably [[density matrices]] representing [[quantum states]] are given by linear operators and a [[quantum operation]] is a superoperator that preserves the defining properties of density matrices. (Beware that sometimes different or no distinctions between "[[quantum operation]]" and "superoperator" are made.)

## Details
 {#Details}

Specifically, in a [[symmetric monoidal closed category]] $(\mathcal{C}, \otimes, \mathbb{1})$ with [[internal hom]] denoted $(\text{-})\multimap (\text{-})$, a superoperator is a morphism of a form like

\[
  \label{OperatorOnInternalHoms}
  (\mathscr{H} \multimap \mathscr{H})
  \longrightarrow
  (\mathscr{K} \multimap \mathscr{K})
  \,.
\]

If $(\mathcal{C}, \otimes, \mathbb{1})$ is also [[compact closed category|compact closed]] then (eq:OperatorOnInternalHoms) is [[isomorphism|isomorphic]] to a [[morphism]] of the form

\[
  \label{SuperoperatorOnMatrices}
  \mathscr{H} \otimes \mathscr{H}^\ast
  \longrightarrow
  \mathscr{K} \otimes \mathscr{K}^\ast
\]

(whence one also speaks of operators on spaces of ([[density matrix|density]]) [[matrices]])

which in turn is [[adjunct]], in particular, to a morphism of the form

\[
  \label{PartialAdjunctOfSuperoperator}
  \mathscr{K}^\ast \otimes \mathscr{H}
  \longrightarrow
  \mathscr{K}^\ast \otimes \mathscr{H}
  \mathrlap{\,.}
\]

Sometimes (eq:PartialAdjunctOfSuperoperator) is the more transparent incarnation of superoperators:

For example, if $(\mathcal{C}, \otimes, \mathbb{1})$ is even a [[dagger-compact category]], then a superoperator in the form (eq:OperatorOnInternalHoms) or (eq:SuperoperatorOnMatrices) is a *[[completely positive map]]* (and hence restricts to a "[[quantum operation]]" on [[density matrices]]) iff its adjunct (eq:PartialAdjunctOfSuperoperator) is a [[positive operator]] [[Selinger 2005 Cor. 4.13](#Selinger05)].

## Related entries

* The [[Geometry of Interaction]]-construction is a genral abstract construction of a category of superoperators. See there for more.

## References

Traditional discussion:

* {#ŻyczkowskiBengtsson04} [[Karol Życzkowski]], [[Ingemar Bengtsson]], Section 4 of: *On Duality between Quantum Maps and Quantum States*, Open Systems & Information Dynamics **11** 01 (2004) 3-42 [[doi:10.1023/B:OPSY.0000024753.05661.c2](https://doi.org/10.1023/B:OPSY.0000024753.05661.c2)]

* {#Kuperberg05} [[Greg Kuperberg]], Section 1.4 of: _A concise introduction to quantum probability, quantum mechanics, and quantum computation_ (2005) [[pdf](http://www.math.ucdavis.edu/~greg/intro-2005.pdf), [[Kuperberg-ConciseQuantum.pdf:file]]]

See also:

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

In terms of [[string diagrams]] in [[dagger-compact categories]] (cf. [[quantum information theory via dagger-compact categories]]):

* {#Selinger05} [[Peter Selinger]], *Dagger-compact closed categories and completely positive maps*, Electronic Notes in Theoretical Computer Science, **170** (2007) 139-163 [[doi:10.1016/j.entcs.2006.12.018](https://doi.org/10.1016/j.entcs.2006.12.018), [[SelingerPositiveMaps.pdf:file]]]

Formalizing superoperators as [[arrows (in computer science)]]:

* J. K. Vizzotto, [[Thorsten Altenkirch]], A. Sabry, *Structuring quantum effects: superoperators as arrows*, Mathematical Structures in Computer Science **16** 3 (2006)  453-468 [[arXiv:quant-ph/0501151](https://arxiv.org/abs/quant-ph/0501151), [doi:10.1017/S0960129506005287](https://doi.org/10.1017/S0960129506005287)]

[[!redirects superoperators]]

[[!redirects super-operator]]
[[!redirects super-operators]]


superparticle

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## Idea

The _superparticle_ is a [[sigma-model]] [[quantum field theory]] analogous to the ordinary [[relativistic particle]] but where either the worldline or the target space are taken to be [[supermanifolds]]. Described by [[supersymmetric quantum mechanics]].

## Definitions

One way to define what this means classically is to give an [[action functional]] on spaces of maps from the worldline into super target space. One such is the

* [[Green-Schwarz action functional]].

The [[quantization]] of this action functional to a 1-dimensional worldvolume [[QFT]] is still largely an open problem.

An axiomatic description of the worldvolume QFT of the superparticle can be given in terms of [[FQFT]] as a representation of a category of super-[[cobordisms]]. One such approach is described at

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

The algebraic data encoded by such a representation is essentially that of a

* [[spectral triple]].

See there for more details.

## Related concepts

* [[super p-brane]]

* [[kappa-symmetry]]

* [[sigma model]], [[brane]]

  * [[particle]], [[spinning particle]], [[fermion]], **superparticle**

  * [[string]]

  * [[membrane]]

* [string theory FQA -- Does string theory predict supersymmetry?](string+theory+FAQ#DoesSTPredictSupersymmetry)

## References

Analysis in [[classical field theory]] in terms of [[supergeometry]] is in

* {#Freed01} [[Daniel Freed]], lecture 4 of _Classical field theory and Supersymmetry_, IAS/Park City Mathematics Series Volume 11 (2001) ([pdf](https://www.ma.utexas.edu/users/dafr/pcmi.pdf))

The [[pure spinor]] superparticle:

* [[Nathan Berkovits]], Eduardo Casali, [[Max Guillen]], Lionel Mason, _Notes on the $D=11$ pure spinor superparticle_ ([arXiv:1905.03737](https://arxiv.org/abs/1905.03737))

In a [[superfield|super]]-[[background field]]:

* Yu.A. Markov, M.A. Markova, A.I. Bondarenko, N.Yu. Markov, *Superparticle in an external chiral matter superfield* &lbrack[https://arxiv.org/abs/2209.11565](https://arxiv.org/abs/2209.11565)]

[[!redirects superparticles]]

[[!redirects super-particle]]
[[!redirects super-particles]]

[[!redirects super particle]]
[[!redirects super particles]]


superpoint

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## Idea

A _superpoint_ is an [[infinitesimally thickened point]] whose infinitesimal extension is _odd_ in the sense of [[supergeometry]]. A [[super Cartesian space]] of vanishing ordinary [[dimension]].

## Definition

A _superpoint_ is a [[supermanifold]] of the form $\mathbb{R}^{0|q}$.

For more see at _[[geometry of physics -- superalgebra]]_.

The object $\mathbb{R}^{0|1}$ is also called the [[odd line]].

The [[category]] of superpoints

$$
  SuperPoint \hookrightarrow SuperMfd
$$

is the [[full subcategory]] of the category of [[supermanifold]]s on the superpoints.

## Properties

### Formal duals

The algebra of functions on superpoints are precisely the [[Grassmann algebras]] (regarded as $\mathbb{Z}/2$ [[graded algebra]]s).

We have an [[equivalence of categories]]

$$
  SuperPoint \simeq GrAlg^{op}
$$

of the category of superpoints with the [[opposite category]] of [[Grassmann algebra]]s.

### The site of superpoints

Regard $SuperPoint$ as a [[site]] with trivial [[coverage]]. Much of [[superalgebra]] and [[supergeometry]] can be usefully understood as taking place over the [[base topos]] $Sh(SuperPoint)$ -- the [[sheaf topos]] over superpoints -- or rather the [[(∞,1)-sheaf (∞,1)-topos]]

$$
  Super\infty Grpd := Sh_{(\infty,1)}(SuperPoint)
$$

of [[super ∞-groupoid]]s. See there for more details.

### Relation to super-translations and super-Minkowski spacetime

A [[super translation Lie algebra]] and hence [[super Minkowski spacetime]], is a [[Lie algebra extension]] of a superpoint, with the latter regarded as an abelian [[super Lie algebra]]. See at _[[super translation Lie algebra]]_ for more on this.

## Related concepts

* [[infinitesimally thickened point]]

* [[super smooth infinity-groupoid]]

* [[super Cartan geometry]]

* [spectral superpoint](spectral+super-scheme#SpectralSuperpoint)

## References

For instance

* Christoph Sachse, section 2.2.1 of _A Categorical Formulation of Superalgebra and Supergeometry_ ([arXiv:0802.4067](http://arxiv.org/abs/0802.4067))

[[!redirects super point]]
[[!redirects super points]]

[[!redirects superpoints]]


superposition

> This page is concerned with the general superposition of solutions of [[linear differential equations]]. For the special case of *[[quantum superposition]]* see there.

***

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## Idea

The [[solutions]] to [[linear equations]] and [[linear differential equations]] may be [[linear combination|linearly combined]] to produce further solutions. Such linear combinations, in particular the [[sums]] of two solutions, is also called *superposition* of solutions, in particular if it concerns solutions to [[wave equations]].

## Examples

### Classical waves

If two [[waves]] are described by a [[linear map|linear]] [[wave equation]], then the superposition of the two waves exist and the [[amplitude]] of the superposition of the two waves is the [[sum]] of the amplitudes of the two waves.

### Quantum states

In [[quantum physics]], [[quantum states]] in the form of [[wavefunctions]] are solutions to [[linear differential equations]] such as the [[Schrödinger equation]], [[Dirac equation]], [[Klein-Gordon equation]], [[Tomonaga-Schwinger equation]], etc. Their superposition is also known as *[[quantum superposition]]*, a concept that is at the heart of peculiar features of quantum physics, notably in the context of [[entanglement]].

## See also

* [[wave]]
* [[wave equation]]
* [[linear map]]
* [[quantum superposition]]
* [[Fourier analysis]]

## References

* Wikipedia, _[Superposition principle](https://en.wikipedia.org/wiki/Superposition_principle)_

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

[[!redirects superpositions]]

[[!redirects superposition principle]]


superselection theory

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## Idea ##

Superselection theory is about identifying superselection sectors in [[quantum field theory]]. In [[quantum field theory]] there are [[conservation laws]] that specify that in an isolated system the total value of certain [[charge|charges]] (e.g. [[electric charges]], [[magnetic charges]]) cannot change due to any interactions taking place in the system. A superselection sector can be specified by prescribing a number to each of these charges, it is the space of all [[state]]s of the system with these values for the charges.

There are two complementary viewpoints about superselection sectors:

In the picture of ([Wick-Wigner-Wightman 52](#WickWignerWightman52)) all possible [[states]] of a system form one big [[Hilbert space]] and  superselection sectors are subspaces of this Hilbert space, and no measurement, that is no [[observable]], can map a state in one superselection sector to another. Further, [[superpositions]] of states in different sectors do not exist in physical reality.

Hence, mathematically: superselection sectors are [[irreducible representations]] of a given [[algebra of observables]] inside the [[space of quantum states]].

An example would be a superposition of a state containing one electron with a state containing two electrons.

In the [[AQFT]] approach to [[quantum field theory]] the observables of the theory are selfadjoint elements of a [[C-star algebra]]. A concrete physical system is a [[state]] of this algebra, which is accompanied via the [[GNS construction]] with a representation of the algebra. A superselection sector from this viewpoint is an equivalence class of unitarily equivalent representations, see [[representation of a C-star algebra]].

## Examples ##

Usually, in [[AQFT]] not _all_ representations of the algebra of observables are considered to be _physically relevant_, so that superselection theory starts with the statement of conditions that representations have to fulfill in order to be **admissible**, and only those are considered.
Example:

* [[DHR superselection theory]]

## Related concepts

* [[quantum parallelism]]

* [[quantum superposition]]

* [[quantum interference]]

* [[entanglement]]

## References ##

For the Wick--Wightman--Wigner picture of superselection sectors see

* [[Gian-Carlo Wick]], [[Eugene Wigner]], [[Arthur Wightman]], _The intrinsic parity of elementary particles_ Phys. Rev. 88, 101-105, 1952 ([pdf](http://dieumsnh.qfb.umich.mx/archivoshistoricosmq/ModernaHist/Wick.pdf))
 {#WickWignerWightman52}

A brief survey of that is in [section 2.3](http://iopscience.iop.org/0034-4885/38/7/001/pdf/0034-4885_38_7_001.pdf#page=781) of

* R. F. Streater, _Outline of axiomatic relativistic quantum field theory_ ([pdf](http://iopscience.iop.org/0034-4885/38/7/001/pdf/0034-4885_38_7_001.pdf))

See also [[AQFT]], [[QFT]] and [[Haag-Kastler axioms]].

* [[Klaas Landsman]], _Quantization and superselection sectors II. Dirac Monopole and Aharonov-Bohm effect_, Review in mathematical physics volume 2, No 1 (1990) ([pdf](http://www.math.ru.nl/~landsman/landsman2.pdf))

Discussion in the context of [[holographic entanglement entropy]]:

* Horacio Casini, Marina Huerta, Javier M. Magan, Diego Pontello, _Entanglement entropy and superselection sectors I. Global symmetries_ ([arXiv:1905.10487](https://arxiv.org/abs/1905.10487))

[[!redirects superselection theory]]
[[!redirects superselection sector]]
[[!redirects superselection sectors]]

[[!redirects superselection rule]]
[[!redirects superselection rules]]


superset > history

< [[superset]]


supersingular elliptic curve

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## Idea

An [[elliptic curve]] over a [[field]] of [[positive characteristic]] whose [[formal group law]] has [[height of a formal group]] equal to 2 is called a _supersingular elliptic curve_. Otherwise the height equals 1 and the elliptic curve is called _ordinary_.

## References

* Wikipedia, _[Supersingular elliptic curve](https://en.wikipedia.org/wiki/Supersingular_elliptic_curve)_

[[!redirects supersingular elliptic curves]]


superspace

This entry is about the concept in [[supergeometry]]. For the concept in [[gravity]]/[[cosmology]] see at _[[Wheeler superspace]]_.

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## Idea

Physicists often refer to [[spaces]] in [[supergeometry]], such as [[supermanifolds]] or [[super schemes]], as _superspaces_. Hence a superspace can be an affine superspace (the affine counterpart of the [[super vector space]] over real or complex numbers), superscheme, supermanifold, etc.

Mostly however "superspace" is used for [[superspacetimes]], and here mostly for [[super Minkowski spacetimes]].

## Related concepts

* [[super-manifold]], [[super-orbifold]]

* [[super Lie group]], [[super Lie algebra]], [[super L-infinity algebra]]

* [[supersymmetry]]

* [[adinkra]]

## References

The concept of superspace in physics (together with that of [[superfields]]) is due to

* [[Abdus Salam]], [[John Strathdee]], _Supergauge Transformations_, Nucl.Phys. B76 (1974) 477-482 ([spire:89208](http://inspirehep.net/record/89208))

* [[Abdus Salam]], [[John Strathdee]], _Superfields and Fermi-Bose symmetry_, Physical Review D11, 1521-1535 (1975) ([doi:10.1142/9789812795915_0051](https://doi.org/10.1142/9789812795915_0051))

(which considered superspace of dimension $d = 4$ with number of supersymmetries $N = 2$, hence the [[supermanifold]] $\mathbb{R}^{4\vert \mathbf{4}+ \mathbf{4}}$, or rather the [[super Minkowski spacetime]] $\mathbb{R}^{3,1\vert \mathbf{4}+ \mathbf{4}}$)

Textbook accounts:

* {#GatesGrisaruRocekSiegel83} [[Jim Gates]] Jr, [[Marcus Grisaru]], [[Martin Roček]], [[Warren Siegel]], _Superspace, or One thousand and one lessons in supersymmetry_, Front.Phys. 58 (1983) 1-548  (1983) ([arXiv:hep-th/0108200](https://arxiv.org/abs/hep-th/0108200), [spire:195126](inspirehep.net/record/195126))

specifically for [[D=4 supergravity]]:

* S. M. Kuzenko, E. S. N. Raptakis, G. Tartaglino-Mazzucchelli, *Superspace approaches to $\mathcal{N} = 1$ supergravity*, *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.17088](https://arxiv.org/abs/2210.17088)]

* S. M. Kuzenko, E. S. N. Raptakis, G. Tartaglino-Mazzucchelli, *Covariant superspace approaches to $\mathcal{N}=2$ supergravity*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2211.11162](https://arxiv.org/abs/2211.11162)]

Further review:

* I. L. Buchbinder, S. M. Kuzenko, _Ideas and methods of supersymmetry and supergravity; or A walk through superspace_

* [[Albert Schwarz]], _On the definition of superspace_ Teoret. Mat. Fiz., 1984,  Volume 60,  Number 1, Pages 37–42   (Mi tmf5111), ([russian original](http://www.mathnet.ru/links/b12306f831b8c37d32d5ba8511d60c93/tmf5111.pdf))

* [[Urs Schreiber]], _[[geometry of physics -- supergeometry]]_

Discussion of superspace within [[philosophy of physics]]:

* Tushar Menon, _Taking up superspace -- the spacetime structure of supersymmetric field theory_ ([philsci:14682](http://philsci-archive.pitt.edu/14682/), [pdf](https://core.ac.uk/download/pdf/157868366.pdf),)

[[!redirects superspaces]]

[[!redirects super-space]]
[[!redirects super-spaces]]

[[!redirects super space]]
[[!redirects super spaces]]


supersphere

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## Idea

In the context of [[supergeometry]], the supersphere $S^{n \vert m}$ is the supersubmanifold of the superspace $\mathbb{R}^{n+1 \vert m}$ of points at a fixed distance from the origin.

## Coset representations

* The supersphere $S^{2|2}$ is the super coset space $UOSp(1|2)/U(1)$.

* The supersphere $S^{r-1|2s}$ is the super coset space $OSp(r|2s)/OSp(r-1|2s)$ of orthosymplectic  groups ([GJS 18](#GJS18)).

## Related concepts

* [[super Klein geometry]]

## References

* {#GJS18} Etienne Granet, Jesper Lykke Jacobsen, Hubert Saleur, _Spontaneous symmetry breaking in 2D supersphere sigma models and applications to intersecting loop soups_, ([ arXiv:1810.07807](https://arxiv.org/abs/1810.07807))


superstring

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## Idea

The _superstring_ 2-dimensional [[sigma-model]] is the analog of the [[string]] $\sigma$-model but with local [[worldsheet]] [[supersymmetry]].

This is the higher dimensional analog of how the [[superparticle]] is related to the ordinary [[relativistic particle]].

There are several different incarnations of the superstring.

* the **[[spinning string]]** or **[[NSR string]]** has in fact local [[worldsheet]] [[supersymmetry]]

  depending on how the [[GSO projection]] is chosen, this yields the [[type II string theory|type II superstring]] or [[type I string theory|type I superstring]] or [[type 0 string theory|type 0 superstring]];

* the **[[Green-Schwarz superstring]]** instead has manifest [[target space]] supersymmetry.

* a mixture between the two has been proposed: the **[[Berkovits superstring]]** .

[[!include worldvolume-target supersymmetry of brane sigma-models]]

## Related concepts

* [[sigma-model]], [[supersymmetric sigma-model]], [[brane]]

  * [[relativistic particle]], [[spinning particle]], [[superparticle]]

  * [[string]], [[spinning string]],

  * **superstring**

    * [[super Riemann surface]]

    * [[Goddard-Thorn no-ghost theorem]]

    * [[2d SCFT]]

      * [[2-spectral triple]], [[Dirac-Ramond operator]]

      * [[Gepner model]], [[flop transition]]

    * [[type II superstring]], [[type I superstring]], [[heterotic superstring]]

    * [[superstring in 3d]]

    * [[Freed-Witten anomaly cancellation]]

    * [string theory FAQ -- Does string theory predict supersymmetry?](string+theory+FAQ#DoesSTPredictSupersymmetry)

  * [[membrane]]

* [[string theory]], [[supersymmetry]]

* [[string scattering amplitude]]

* [[2d supergravity]]

[[!include worldvolume-target supersymmetry of brane sigma-models]]

## References

### General

The relationship between the NRS [[spinning string]] and the [[Green-Schwarz superstring]] [[sigma-models]] is discussed in

* Rafael I. Nepomechie, _Nonabelian bosonization, triality, and superstring theory_ Physics Letters B
Volume 178, Issues 2-3, 2 October 1986, Pages 207-210

* I. Bars, D. Nemschansky and S. Yankielowicz, SLACPub-
3758.

This involves the relation between the [[octonion]] algebra and [[spinor]]s in 2+8 dimensions (see also [[division algebra and supersymmetry]]):

* H. Tachibana, K. Imeda, _Octonions, superstrings and ten-dimensional spinors_ , Il nuovo cimento, Vol 104 B N.1

### Superspace and superembedding formalism

Discussion of the superstring [[equations of motion]] in [[superspace]] is due to

* [[Igor Bandos]], [[Dmitrij Sorokin]], [[Dmitrij Volkov]], _On the generalized action principle for superstrings and supermembranes_,  	Phys. Lett. B352:269-275, 1995 ([arXiv:hep-th/9502141](https://arxiv.org/abs/hep-th/9502141))

and discussion via the [[superembedding formalism]]

* {#BPSTV95} [[Igor Bandos]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], [[Dmitry Volkov]], Chapter 4 of _Superstrings and supermembranes in the doubly supersymmetric geometrical approach_, Nucl. Phys. B446:79-118, 1995 ([arXiv:hep-th/9501113](https://arxiv.org/abs/hep-th/9501113))

Discussion of the superstring [[Green-Schwarz sigma-model]] in the broader context of [[schreiber:The brane bouquet]]:

* {#CAIB99} C. Chryssomalakos, [[José de Azcárraga]], [[José M. Izquierdo]], C. Pérez Bueno, _The geometry of branes and extended superspaces_, Nucl. Phys. B **567** (2000) 293-330  [[arXiv:hep-th/9904137](http://arxiv.org/abs/hep-th/9904137), doi:10.1016/S0550-3213(99)00512-X]

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]_, International Journal of Geometric Methods in Modern Physics, Volume 12, Issue 02 (2015) 1550018 ([arXiv:1308.5264](http://arxiv.org/abs/1308.5264))

[[!redirects superstrings]]

[[!redirects super-string]]
[[!redirects super-strings]]