Sullivan model of loop space

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###Context###
#### Rational homotopy theory
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[[!include differential graded objects - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

In [[rational homotopy theory]], given a [[rational topological space]] modeled by a [[Sullivan model]] [[dg-algebra]], there is an explicit description of the Sullivan model of its [[loop space]], ([[free loop space]] or [[based loop space]]).

This is a special case of [[Sullivan models of mapping spaces]].

## Construction

### For the free loop space
 {#ConstructionForFreeLoopSpace}

+-- {: .num_prop #SullivanModelForTheFreeLoopSpace}
###### Proposition
**([[Sullivan model]] for [[free loop space]])**

Let $(\wedge^\bullet V, d_X)$ be a [[semifree dg-algebra]] being a [[minimal Sullivan model]] of a [[rational space|rational]] [[simply connected space]] $X$. Then a Sullivan model for the [[free loop space]] $\mathcal{L} X$ is given by

$$
  \big(
    \wedge^\bullet( V \oplus s V )
    ,\,
    d_{\mathcal{L}X}
  \big)
  \,,
$$

where

* $s V$ is the [[graded vector space]] obtained from $V$ by shifting degrees down by one: $deg(s v) = deg(v)-1$;

* $d_{\mathcal{L}X}$ is defined on elements $v$ of $V$ by

  $$
    d_{\mathcal{L}X} v \coloneqq d v
  $$

  and on elements $s v$ of $s V$ by

  $$
    d_{\mathcal{L}X} s v \coloneqq - s ( d v )
    \,,
  $$

  where on the right $s \colon V \to s V$ is extended as a graded [[derivation]] $s \colon \wedge^2 V \to \wedge^\bullet (V \oplus s V) $.

=--

This is due to ([Vigué-Sullivan 76](#VigueSullivan76)). Review includes ([Felix-Halperin-Thomas 00, p. 206](#FelixHalperinThomas00), [Hess 06, example 2.5](#Hess06), [Félix-Oprea-Tanre 08, theorem 5.11](#FelixOpreaTanre08)).

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

The formula in prop. \ref{SullivanModelForTheFreeLoopSpace} is akin to that that for the [[Weil algebra]] of the [[L-infinity algebra|$L_\infty$-algebra]] of which $(\wedge^\bullet V,d_X)$ is the [[Chevalley-Eilenberg algebra]], except that here $s$ shifts _down_, whereas for the Weil algebra it shifts _up_.

=--

### For the based loop space
 {#ConstructionForBasedLoopSpace}

For $X$ a [[pointed topological space]] and for the [[circle]] $S^1$ regarded as [[pointed topological space|pointed]] by any base point $\ast \to S^1$
there is the following [[homotopy fiber sequence]] which exhibits the [[based loop space]] as the [[homotopy fiber]] of the [[evaluation map]] out of the [[free loop space]]:

$$
  \Omega X
  \overset{fib(ev_\ast)}{\longrightarrow}
  \mathcal{L}X
  \overset{ ev_\ast }{\longrightarrow}
  X
  \,.
$$

With the dgc-algebra model from Prop. \ref{SullivanModelForTheFreeLoopSpace} for $\mathcal{L}X$ it follows that the dgc-algebra model for the based loop space is the [[homotopy cofiber]] [[dgc-algebra]] $(\wedge^\bullet( s V ), d_{\Omega X})$ in

$$
  (\wedge^\bullet( s V ), d_{\Omega X})
  \overset{ cofib\big( (ev_\ast)^\ast \big) }{\longleftarrow}
  (\wedge^\bullet( V \oplus s V ), d_{\mathcal{L}X})
  \overset{ (ev_\ast)^\ast }{\longleftarrow}
  (\wedge\bullet V, d_X)
  \,.
$$

Since the inclusion on the right is manifestly a [[relative Sullivan algebra]], hence a [[cofibration]] in the projective [[model structure on dgc-algebras]], and since the latter is [[left proper model category|left proper]] (see [there](model+structure+on+dg-algebras#Properness)), the [[homotopy cofiber]] is represented by the ordinary [[cofiber]], which is readily read off:

\begin{prop}\label{SullivanModelForBasedLoopSpace}
**([[Sullivan model]] for [[based loop space]])**
\linebreak
For $X$ a [[connected topological space|connected]] and [[simply connected topological space]] with [[Sullivan model]] $(\wedge\bullet V, d_X)$, the Sullivan model
$(\wedge^\bullet( s V ), d_{\Omega X})$
of its [[based loop space]] $\Omega X$
is the dgc-algebra obtained from $(\wedge^\bullet V, d_X)$ by shifting down all generators in degree by 1, and by keeping only the co-unary component of the [[differential]].
\end{prop}

\begin{remark}\label{PontrjaginProductOnHomologyOfBasedLoopSpace}
  While Prop. \ref{SullivanModelForBasedLoopSpace} says that the rational [[Whitehead brackets]] of a loop space all vanish (as generally for any [[H-space]], see [there](H-space#RationalHomotopy)), the [[ordinary homology]] of a loop space inherits another product, namely the [[Pontrjagin product]], and this makes (see [there](Pontrjagin+ring#RelationToWhiteheadProduct)) the homology form the [[universal enveloping algebra]] of the binary [[Whitehead bracket]] [[super Lie algebra]] of the original space.
\end{remark}

## Properties

### Homotopy quotient by $S^1$

+-- {: .num_prop #ModelForS1quotient}
###### Proposition

Given a [[Sullivan model]] $(\wedge^\bullet (V \oplus s V), d_{\mathcal{L}X})$ for a [[free loop space]] as in prop. \ref{SullivanModelForTheFreeLoopSpace}, then a Sullivan model for the [[cyclic loop space]], i.e. for the [[homotopy quotient]] $\mathcal{L} X \sslash S^1$ with respect to the canonical [[circle group]] action that rotates loops (i.e. for the [[Borel construction]] $\mathcal{L}X \times_{S^1} E S^1$) is given by

$$
  \Big(
    \wedge^\bullet(
       V\oplus s V \oplus \langle \omega_2\rangle
    )
    ,\,
    d_{\mathcal{L}X/S^1}
  \Big)
$$

where

* $\omega_2$ is in degree 2;

* $d_{\mathcal{L}X/S^1}$ is defined on generators $w \in V\oplus s V$ by

  $$
    d_{\mathcal{L}X/S^1} w
      \;\coloneqq\;
    d_{\mathcal{L}X} w + \omega_2 \wedge s w
    \,.
  $$

Moreover, the canonical sequence of morphisms of [[dg-algebras]]

$$
  (\wedge \omega_2, d = 0)
    \longrightarrow
  (\wedge^\bullet(  V\oplus s V \oplus \langle \omega_2\rangle ), d_{\mathcal{L}X/S^1})
    \longrightarrow
  (\wedge^\bullet(  V\oplus s V ), d_{\mathcal{L}X})
$$

is a model for the [[rationalization]] of the [[homotopy fiber sequence]]

$$
  \mathcal{L}X
    \longrightarrow
  \mathcal{L}X \sslash S^1
    \longrightarrow
  B S^1
$$

which exhibits the [[infinity-action]] (by the discussion there) of $S^1$ on $\mathcal{L}X$.

=--

This is due to ([Vigué-Burghelea 85, theorem A](#VigueBurghelea85)).

\begin{example}\label{SullivanModelOfCyclicLoopOfEm}
**([[Sullivan model]] of [[cyclic loop space]] of [[Eilenberg-MacLane space|EM-space]])**
\linebreak
For $n \geq 1$ consider the [[Eilenberg-MacLane space]]
$X \,=\, B^{n+1} \mathbb{Q}$, whose [[Sullivan model of a classifying space]] is

$$
  CE\big(
    \mathfrak{l}
    B^{n+1} \mathbb{Q}
  \big)
  \;\;
  =
  \;\;
  \mathbb{Q}[c_{n+1}]
  \big/
  \big(
    \mathrm{d} \, c_{n+1}  \;=\; 0
  \big)
  \,.
$$

Notice -- from [this Prop](free+loop+space+of+classifying+space#FreeLoopSpaceOfClassifyingSpaceOfSimplicialAbelianGroup) at *[[free loop space of classifying space]]* -- that its [[free loop space]] is the [[product]]

$$
  \mathcal{L}
  \;
  B^{n+1} \mathbb{Q}
  \;\simeq\;
  B^{n} \mathbb{Q}
  \,\times\,
  B^{n+1} \mathbb{Q}
  \,.
$$

Now Prop. \ref{ModelForS1quotient} shows that the corresponding [[cyclic loop space]] is as in the middle item here:

$$
  \begin{array}{ccc}
  B^n \mathbb{Q}
  &\longrightarrow&
  \Big(
    \overset{
      B^{n} \mathbb{Q}
      \,\times\,
      B^{n+1} \mathbb{Q}
    }{
      \overbrace{
        \mathcal{L} \;B^{n+1} \mathbb{Q}
      }
    }
  \Big)
  \sslash S^1
  &\longrightarrow&
  B^n \mathbb{Q}
  \\
  \left(
  \begin{array}{lcl}
    d & c_{n} & = 0
  \end{array}
  \right)
  &
  \longleftarrow
  &
  \left(
  \begin{array}{lcl}
    d & c_{n+1} & = \omega_2 \wedge c_n
    \\
    d & c_{n} & = 0
    \\
    d & \omega_2 & = 0
  \end{array}
  \right)
  &
  \longleftarrow
  &
  \left(
  \begin{array}{lcl}
    d & c_{n} & = 0
  \end{array}
  \right)
  \end{array}
$$

Incidentally, as indicated by the full diagram, this readily shows that $\big(\mathcal{L}\, B^{n+1} \mathbb{Q}\big) \sslash S^1$ [[retraction|retracts]] onto $B^{n} \mathbb{Q}$.
\end{example}

### Relation to Hochschild- and cyclic-homology
  {#RelationToHochschildHomology}

Let $X$ be a [[simply connected topological space|simply connected]] [[topological space]].

The [[ordinary cohomology]] $H^\bullet$ of its [[free loop space]] is the [[Hochschild homology]] $HH_\bullet$ of its [[singular cohomology|singular chains]] $C^\bullet(X)$:

$$
  H^\bullet(\mathcal{L}X)
    \simeq
  HH_\bullet( C^\bullet(X) )
  \,.
$$

Moreover the $S^1$-equivariant cohomology of the loop space, hence the ordinary cohomology of the [[cyclic loop space]] $\mathcal{L}X \sslash S^1$ is the [[cyclic homology]] $HC_\bullet$ of the singular chains:

$$
  H^\bullet(\mathcal{L}X \sslash S^1)
    \simeq
  HC_\bullet( C^\bullet(X) )
$$

([Jones 87, Thm. A](cyclic+homology#Jones87), review in [Loday 92, Cor. 7.3.14](cyclic+homology#Loday92), [Loday 11, Sec 4](cyclic+homology#Loday11))

If the [[coefficients]] are [[rational numbers|rational]], and $X$ is of [[finite type]] then by prop. \ref{SullivanModelForTheFreeLoopSpace} and prop. \ref{ModelForS1quotient}, and the general statements at _[[rational homotopy theory]]_, the [[cochain cohomology]] of the above [[minimal Sullivan models]] for $\mathcal{L}X$ and $\mathcal{l}X/S^1$ compute the rational [[Hochschild homology]] and [[cyclic homology]] of (the cochains on) $X$, respectively.

In the special case that the [[topological space]] $X$ carries the structure of a [[smooth manifold]], then the singular cochains on $X$ are equivalent to the [[dgc-algebra]] of [[differential forms]] (the [[de Rham algebra]]) and hence in this case the statement becomes that

$$
  H^\bullet(\mathcal{L}X)
    \simeq
  HH_\bullet( \Omega^\bullet(X) )
  \,.
$$

$$
  H^\bullet(\mathcal{L}X \sslash S^1)
    \simeq
  HC_\bullet( \Omega^\bullet(X) )
  \,.
$$

This is known as _[[Jones' theorem]]_ ([Jones 87](#Jones87))

An [[(infinity,1)-category theory|infinity-category theoretic]] proof of this fact is indicated at _[Hochschild cohomology -- Jones' theorem](Hochschild+cohomology#JonesTheorem)_.

## Examples

### Free loop space of the 4-sphere
 {#4SphereAndTwistedDeRham}

We discuss the Sullivan model for the free and cyclic loop space of the [[4-sphere]]. This may also be thought of as the [[cocycle space]] for [[rational Cohomotopy|rational 4-Cohomotopy]], see [FSS16, Section 3](#FiorenzaSatiSchreiber16).

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

Let $X = S^4$ be the [[4-sphere]]. The corresponding [[rational n-sphere]] has minimal Sullivan model

$$
  (\wedge^\bullet \langle g_4, g_7 \rangle, d)
$$

with

$$
  d g_4 = 0\,,\;\;\;\; d g_7 = -\tfrac{1}{2} g_4 \wedge g_4
  \,.
$$

Hence prop. \ref{SullivanModelForTheFreeLoopSpace} gives for the rationalization of $\mathcal{L}S^4$ the model

$$
  ( \wedge^\bullet \langle \omega_4, \omega_6, h_3, h_7 \rangle  , d_{\mathcal{L}S^4} )
$$

with

$$
  \begin{aligned}
    d_{\mathcal{L}S^4} h_3 & = 0
    \\
    d_{\mathcal{L}S^4} \omega_4 & = 0
    \\
    d_{\mathcal{L}S^4} \omega_6 & = h_3 \wedge \omega_4
    \\
    d_{\mathcal{L}S^4} h_7 & = -\tfrac{1}{2} \omega_4 \wedge \omega_4
    \\
  \end{aligned}
$$

and prop. \ref{ModelForS1quotient} gives for the rationalization of $\mathcal{L}S^4 \sslash S^1$ the model

$$
  ( \wedge^\bullet \langle \omega_2, \omega_4, \omega_6, h_3, h_7 \rangle  , d_{\mathcal{L}S^4 \sslash S^1} )
$$

with

\[
  \label{CERelationsForCyclicLoopSpaceOfS4}
  \begin{aligned}
    d_{\mathcal{L}S^4 \sslash S^1} h_3 & = 0
    \\
    d_{\mathcal{L}S^4 \sslash S^1} \omega_2 & = 0
    \\
    d_{\mathcal{L}S^4 \sslash S^1} \omega_4 & = h_3 \wedge \omega_2
    \\
    d_{\mathcal{L}S^4 \sslash S^1} \omega_6 & = h_3 \wedge \omega_4
    \\
    d_{\mathcal{L}S^4 \sslash S^1} h_7 & = -\tfrac{1}{2} \omega_4 \wedge \omega_4 + \omega_2 \wedge \omega_6
  \end{aligned}
  \,.
\]

=--

\begin{remark}\label{RelationOfCycS4ToTwistedDeRhamCohomology}
  **(relation to [[twisted de Rham cohomology]])** \linebreak
  The equations (eq:CERelationsForCyclicLoopSpaceOfS4) imply that [[dg-algebra]] [[homomorphisms]] of the form
  $$
    CE
    \Big(
      \mathfrak{l}
      \big(
        (\mathcal{L}S^4) \sslash S^1
      \big)
    \Big)
    \xrightarrow{\;\;AA\;\;}
    \Omega^\bullet_{dR}(X^f)
  $$
  into the [[de Rham dg-algebra]] of a [[smooth manifold]] $X^f$ of [[dimension of a manifold|dimension]] $\leq 7$ are equivalently [[cocycles]] in the degree-3 [[twisted de Rham complex]] of $X^7$ (together with any 7-form, if $dim = 7$), for 3-twist given by the image of the general $h_3$.

This suggests a relation between the cyclification of $S^4$ to the [[twisted Chern character]] on [[twisted K-theory]] (a relation further explored in [[schreiber:Gauge enhancement of Super M-Branes|BMSS 2019]]).
\end{remark}

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

Let $\hat \mathfrak{g} \to \mathfrak{g}$ be a [[central extension|central]] [[Lie algebra extension]] by $\mathbb{R}$ of a finite dimensional Lie algebra $\mathfrak{g}$, and let $\mathfrak{g} \longrightarrow b \mathbb{R}$ be the corresponding [[L-∞ algebra cohomology|L-∞ 2-cocycle]] with coefficients in the [[line Lie n-algebra|line Lie 2-algebra]] $b \mathbb{R}$, hence ([[schreiber:The brane bouquet|FSS 13, prop. 3.5]]) so that there is a [[homotopy fiber sequence]] of [[L-∞ algebras]]

$$
  \hat \mathfrak{g}
    \longrightarrow
  \mathfrak{g}
    \overset{\omega_2}{\longrightarrow}
  b \mathbb{R}
$$

which is dually modeled by

$$
  CE(\hat \mathfrak{g})
    =
  ( \wedge^\bullet ( \mathfrak{g}^\ast \oplus \langle e \rangle ), d_{\hat \mathfrak{g}}|_{\mathfrak{g}^\ast} = d_{\mathfrak{g}},\;  d_{\hat \mathfrak{g}} e = \omega_2)
  \,.
$$

For $X$ a space with [[Sullivan model]] $(A_X,d_X)$ write $\mathfrak{l}(X)$ for the corresponding [[L-∞ algebra]], i.e. for the $L_\infty$-algebra whose [[Chevalley-Eilenberg algebra]] is $(A_X,d_X)$:

$$
  CE(\mathfrak{l}X) = (A_X,d_X)
  \,.
$$

Then there is an [[isomorphism]] of [[hom-sets]]

$$
  Hom_{L_\infty Alg}( \hat \mathfrak{g}, \mathfrak{l}(S^4) )
  \;\simeq\;
  Hom_{L_\infty Alg/b \mathbb{R}}( \mathfrak{g}, \mathfrak{l}( \mathcal{L}S^4 / S^1 ) )
  \,,
$$

with $\mathfrak{l}(S^4)$ from prop. \ref{SullivanModelForTheFreeLoopSpace} and $\mathfrak{l}(\mathcal{L}S^4 \sslash S^1)$ from prop. \ref{ModelForS1quotient},
where on the right we have homs in the [[slice category|slice]] over the [[line Lie n-algebra|line Lie 2-algebra]], via prop. \ref{ModelForS1quotient}.

Moreover, this isomorphism takes

$$
  \hat \mathfrak{g}
    \overset{(g_4, g_7)}{\longrightarrow}
  \mathfrak{l}(S^4)
$$

to

$$
  \array{
    \mathfrak{g}
      &&
      \overset{(\omega_2,\omega_4, \omega_6, h_3,h_7)}{\longrightarrow}
      &&
    \mathfrak{l}( \mathcal{L}X / S^1 )
    \\
    &
    {}_{\mathllap{\omega_2}}\searrow
      &&
    \swarrow_{\mathrlap{\omega_2}}
    \\
    && b \mathbb{R}
  }
  \,,
$$

where

$$
  \omega_4 = g_4 - h_3 \wedge e
  \;\,,
  \;\;\;
  h_7 = g_7 + \omega_6 \wedge e
$$

with $e$ being the central generator in $CE(\hat \mathfrak{g})$ from above, and where the equations take place in $\wedge^\bullet \hat \mathfrak{g}^\ast$ with the defining inclusion $\wedge^\bullet \mathfrak{g}^\ast \hookrightarrow \wedge^\bullet \mathfrak{g}^\ast$ understood.

=--

This is observed in ([Fiorenza-Sati-Schreiber 16](#FiorenzaSatiSchreiber16), [FSS 16b](#FSS16b)), where it serves to formalize, on the level of [[rational homotopy theory]], the [[double dimensional reduction]] of [[M-branes]] in [[M-theory]] to [[D-branes]] in [[type IIA string theory]] (for the case that $\mathfrak{g}$ is type IIA [[super Minkowski spacetime]] $\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}$ and $\hat \mathfrak{g}$ is 11d [[super Minkowski spacetime]] $\mathbb{R}^{10,1\vert \mathbf{32}}$, and the cocycles are those of [[schreiber:The brane bouquet]]).

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

By the fact that the underlying graded algebras are free, and since $e$ is a generator of odd degree, the given decomposition for $\omega_4$ and $h_7$ is unique.

Hence it is sufficient to observe that under this decomposition the defining equations

$$
  d g_4 = 0
  \,,\;\;\;
  d g_{7} = -\tfrac{1}{2} g_4 \wedge g_4
$$

for the $\mathfrak{l}S^4$-valued cocycle on $\hat \mathfrak{g}$ turn into the equations for a $\mathfrak{l} ( \mathcal{L}S^4 / S^1 )$-valued cocycle on $\mathfrak{g}$. This is straightforward:

$$
  \begin{aligned}
    & d_{\hat \mathfrak{g}} ( \omega_4 + h_3 \wedge e ) = 0
    \\
    \Leftrightarrow \;\;\;\;
    &
    d_{\mathfrak{g}} (\omega_4 - h_3 \wedge \omega_2) = 0
    \;\;\; and \;\;\;
    d_{\mathfrak{g}} h_3 = 0
    \\
    \Leftrightarrow \;\;\;\;
    &
    d_{\mathfrak{g}} \omega_4 = h_3 \wedge \omega_2
    \;\;\; and \;\;\;
    d_{\mathfrak{g}} h_3 = 0
  \end{aligned}
$$

as well as

$$
  \begin{aligned}
    & d_{\hat \mathfrak{g}} ( h_7 - \omega_6 \wedge e )
      = -\tfrac{1}{2}( \omega_4 + h_3 \wedge e ) \wedge (\omega_4 + h_3\wedge e)
   \\
    \Leftrightarrow \;\;\;\;
    &
    d_\mathfrak{g} h_7 - \omega_6 \wedge \omega_2
    =
    -\tfrac{1}{2}\omega_4 \wedge \omega_4
    \;\;\; and \;\;\;
    - d_\mathfrak{g} \omega_6 = - h_3 \wedge \omega_4
    \\
    \Leftrightarrow \;\;\;\;
    &
    d_\mathfrak{g} h_7 = -\tfrac{1}{2}\omega_4 \wedge \omega_4  + \omega_6 \wedge \omega_2
    \;\;\; and \;\;\;
    d_\mathfrak{g} h_6 = h_3 \wedge \omega_4
  \end{aligned}
$$

=--

### Free loop space of the 2-sphere

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

Let $X = S^2$ be the [[2-sphere]]. The corresponding [[rational n-sphere]] has minimal Sullivan model

$$
  (\wedge^\bullet \langle g_3, g_2 \rangle, d)
$$

with

$$
  d g_2 = 0\,,\;\;\;\; d g_3 = -\tfrac{1}{2} g_2 \wedge g_2
  \,.
$$

Hence prop. \ref{SullivanModelForTheFreeLoopSpace} gives for the rationalization of $\mathcal{L}S^2$ the model

$$
  ( \wedge^\bullet \langle \omega^A_2, \omega^B_2, h_1, h_3 \rangle  , d_{\mathcal{L}S^2} )
$$

with

$$
  \begin{aligned}
    d_{\mathcal{L}S^2} h_1 & = 0
    \\
    d_{\mathcal{L}S^2} \omega^A_2 & = 0
    \\
    d_{\mathcal{L}S^2} \omega^B_2 & = h_1 \wedge \omega_2^A
    \\
    d_{\mathcal{L}S^2} h_3 & = -\tfrac{1}{2} \omega^A_2 \wedge \omega^A_2
  \end{aligned}
$$

and prop. \ref{ModelForS1quotient} gives for the rationalization of $\mathcal{L}S^2 \sslash S^1$ the model

$$
  ( \wedge^\bullet \langle \omega^A_2, \omega^B_2, \omega^C_2, h_1, h_3 , d_{\mathcal{L}S^2 \sslash S^1} )
$$

with

$$
  \begin{aligned}
    d_{\mathcal{L}S^2} h_1 & = 0
    \\
    d_{\mathcal{L}S^2} \omega^A_2 & = \omega^C_2 \wedge h_1
    \\
    d_{\mathcal{L}S^2} \omega^B_2 & = h_1 \wedge \omega_2^A
    \\
    d_{\mathcal{L}S^2} \omega^C_2 & = 0
    \\
    d_{\mathcal{L}S^2} h_3 & = -\tfrac{1}{2} \omega^A_2 \wedge \omega^A_2 + \omega^C_2 \wedge \omega^B_2
  \end{aligned}
  \,.
$$

=--

\linebreak

### Iterated based loop spaces of $n$-spheres
 {#IteratedLoopSpacesOfNSpheres}

By iterating the Sullivan model construction for the [[based loop space]] from Prop. \ref{SullivanModelForBasedLoopSpace} and using the [[Sullivan models of n-spheres]] we have that:

+-- {: .num_prop #SullivanModelsOfMapsFromSkToSnFornLargerk}
###### Proposition
**([[Sullivan models]] for [[iterated loop spaces]] of [[n-spheres]])**

The [[Sullivan model]] of the $k$-fold [[iterated based loop space]] $\Omega^k S^n $ of the [[n-sphere]] for $k \lt n$ is

$$
  CE\mathfrak{l}
  \big(
    \Omega^k S^n
  \big)
  \;=\;
  \left\{
  \array{
    \left(
      \array{
        d\,\omega_{n-k} & = 0
      }
    \right)
    &\vert&
    n \;\text{is odd}
   \\
    \left(
      \array{
        d\,\omega_{n-k} & = 0
        \\
        d\,\omega_{2n-1-k} & = 0
      }
    \right)
    &\vert&
    n \;\text{is even}
  }
  \right.
  \phantom{AAAA}
  \text{for}\; k \lt n
  \,.
$$

=--

(see also [Kallel-Sjerve 99, Prop. 4.10](#KallelSjerve99))

For the edge case $\Omega^D S^D$ the above formula does not apply, since $\Omega^{D-1} S^D$ is not [[simply connected topological space|simply connected]] (its [[fundamental group]] is $\pi_1\big( \Omega^{D-1}S^D \big) = \pi_0 \big(\Omega^D S^D\big) = \pi_D(S^D) = \mathbb{Z}$, the 0th [[stable homotopy group of spheres]]).

But:

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

The rational model for $\Omega^D S^D$ follows from [this Prop.](Sullivan+model+of+mapping+space#RationalHomotopyTypeOfMapsNSphereToNsphere) by realizing the pointed mapping space as the [[homotopy fiber]] of the [[evaluation map]] from the free mapping space:

$$
  \array{
    \mathllap{
      \Omega^D S^D
      \simeq
    \;}
    Maps^{\ast/\!}\big( S^D, S^D\big)
    \\
    \big\downarrow^{\mathrlap{fib(ev_\ast)}}
    \\
    Maps(S^D, S^D)
    \\
    \big\downarrow^{\mathrlap{ev_\ast}}
    \\
    S^D
  }
$$

This yields for instance the following examples.

In odd dimensions:

\begin{xymatrix}
    \mathrm{Maps}^{\ast/\!}
    \big(
      S^3, S^3
    \big)
    \ar[d]^-{ \mathrm{fib}_{(\mathrm{ev}_\ast)} }
    \ar@{}[r]|-{ \simeq_{\mathbb{Q}} }
    &
    \underset{
      n \in \mathbb{Z}
    }{\sqcup}
    \ast
    \ar@{^{(}->}[d]
    \\
    \mathrm{Maps}
    \big(
      S^3, S^3
    \big)
    \ar[d]^-{ \mathrm{ev}_\ast }
    \ar@{}[r]|-{ \simeq_{\mathbb{Q}} }
    &
    \underset{
      n  \in \mathbb{Z}
    }{\sqcup}
    S^3
    \ar[d]^-{ (\mathrm{id}_{S^3})_{n \in \mathbb{N}} }
    \\
    S^3
    \ar@{=}[r]
    &
    S^3
\end{xymatrix}

In even dimensions:

(In the following $h_{\mathbb{K}}$ denotes the [[Hopf fibration]] of the [[division algebra]] $\mathbb{K}$, hence $h_{\mathbb{C}}$ denotes the [[complex Hopf fibration]] and $h_{\mathbb{H}}$ the [[quaternionic Hopf fibration]].)

\begin{xymatrix}
    \mathrm{Maps}^{\ast/\!}
    \big(
      S^2, S^2
    \big)
    \ar[d]^-{ \mathrm{fib}_{(\mathrm{ev}_\ast)} }
    \ar@{}[r]|-{ \simeq_{\mathbb{Q}} }
    &
    \underset{
      n \in \mathbb{Z}
    }{\sqcup}
    S^1
    \ar@{^{(}->}[d]
    \\
    \mathrm{Maps}
    \big(
      S^2, S^2
    \big)
    \ar[d]^-{ \mathrm{ev}_\ast }
    \ar@{}[r]|-{ \simeq_{\mathbb{Q}} }
    &
    \big(
      S^2 \times S^1
    \big)
    \sqcup
    \big(
      \underset{
        n \neq 0  \in \mathbb{Z}
      }{\sqcup}
      S^3
    \big)
    \ar[d]^-{
      \big(
        p_1, (h_{\mathbb{C}})_{n \neq 0 \in \mathbb{N}}
      \big)
    }
    \\
    S^2
    \ar@{=}[r]
    &
    S^2
\end{xymatrix}

\begin{xymatrix}
    \mathrm{Maps}^{\ast/\!}
    \big(
      S^4, S^4
    \big)
    \ar[d]^-{ \mathrm{fib}_{(\mathrm{ev}_\ast)} }
    \ar@{}[r]|-{ \simeq_{\mathbb{Q}} }
    &
    \underset{
      n \in \mathbb{Z}
    }{\sqcup}
    S^3
    \ar@{^{(}->}[d]
    \\
    \mathrm{Maps}
    \big(
      S^4, S^4
    \big)
    \ar[d]^-{ \mathrm{ev}_\ast }
    \ar@{}[r]|-{ \simeq_{\mathbb{Q}} }
    &
    \big(
      S^4 \times S^3
    \big)
    \sqcup
    \big(
      \underset{
        n \neq 0  \in \mathbb{Z}
      }{\sqcup}
      S^7
    \big)
    \ar[d]^-{
      \big(
        p_1, (h_{\mathbb{H}})_{n \neq 0 \in \mathbb{N}}
      \big)
    }
    \\
    S^4
    \ar@{=}[r]
    &
    S^4
\end{xymatrix}

=--

## Related concepts

* [[Hochschild homology]]

* [[cyclic homology]]

* [[rational model of mapping spaces]]

* [[rational Cohomotopy]]

[[!include Sullivan models -- examples]]

## References

The original result is due to

* {#VigueSullivan76} [[Micheline Vigué-Poirrier]], [[Dennis Sullivan]], _The homology theory of the closed geodesic problem_, J. Differential Geometry 11 (1976) 633-644.

* {#VigueBurghelea85} [[Micheline Vigué-Poirrier]], Dan Burghelea, _A model for cyclic homology and algebraic K-theory of 1-connected topological spaces_, J. Differential Geom. Volume 22, Number 2 (1985), 243-253 ([Euclid](https://projecteuclid.org/euclid.jdg/1214439821))

Examples:

* Bitjong Ndombol & M. El Haouari, *The free loop space equivariant cohomology algebra of some formal spaces*, Mathematische Zeitschrift **266** (2010) 863–875 ([doi:10.1007/s00209-009-0602-z](https://doi.org/10.1007/s00209-009-0602-z))

* Kentaro Matsuo, *The Borel cohomology of the loop space of a homogeneous space*, Topology and its Applications **160** 12 (2013) 1313-1332 ([doi:10.1016/j.topol.2013.05.001](https://doi.org/10.1016/j.topol.2013.05.001))

Review:

* {#FelixHalperinThomas00} [[Yves Félix]], [[Steve Halperin]] and J.C. Thomas, _Rational Homotopy Theory_, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000.

* {#Hess06} [[Kathryn Hess]], example 2.5 of _Rational homotopy theory: a brief introduction_ ([math.AT/0604626](http://arxiv.org/abs/math.AT/0604626))

* {#FelixOpreaTanre08} [[Yves Félix]], John Oprea, [[Daniel Tanre]], _Algebraic models in geometry_, Oxford graduate texts in mathematics 17 (2008)

* A. Yu. Onishchenko and Th. Yu. Popelensky, _Rational cohomology of the free loop space of a simply connected 4-manifold_, J. Fixed Point Theory Appl. 12 (2012) 69–9 ([DOI 10.1007/s11784-013-0100-0](http://link.springer.com/article/10.1007/s11784-013-0100-0))

* [[Luc Menichi]], _Sullivan models and free loop space_, A short introduction to Sullivan models, with the Sullivan model of a free loop space and the detailed proof of Vigué-Sullivan theorem on the Betti numbers of free loop space. Workshop on free loop space à Strasbourg, November 2008 (scanned notes [pdf](http://math.univ-angers.fr/perso/lmenichi/Sullivan_models_free_loop_space.pdf))

* {#Menichi13} [[Luc Menichi]], *Rational homotopy -- Sullivan models*, in: *Free Loop Spaces in Geometry and Topology*, IRMA Lect. Math. Theor. Phys., EMS (2015) [[arXiv:1308.6685](https://arxiv.org/abs/1308.6685), [doi:10.4171/153](https://doi.org/10.4171/153)]

* {#FiorenzaSatiSchreiber16} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]]: _[[schreiber:Rational sphere valued supercocycles in M-theory|Rational sphere valued supercocycles in M-theory and type IIA string theory]]_, Journal of Geometry and Physics, Volume 114, Pages 91-108 (2017) ([arXiv:1606.03206](https://arxiv.org/abs/1606.03206))

* {#FSS16b} [[nLab:Domenico Fiorenza]], [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]], _[[schreiber:T-Duality from super Lie n-algebra cocycles for super p-branes]]_ ([arXiv:1611.06536](https://arxiv.org/abs/1611.06536))

General background on [[Hochschild homology]] and [[cyclic homology]] is in

* {#Jones87} [[John D.S. Jones]], _Cyclic homology and equivariant homology_, Invent. Math. __87__, 403-423 (1987) ([pdf](https://math.berkeley.edu/~nadler/jones.pdf))

* {#Loday98} [[Jean-Louis Loday]], _Cyclic homology_, Grundlehren Math.Wiss. __301__, Springer (1998)

* {#Loday11} [[Jean-Louis Loday]], _Free loop space and homology_ ([arXiv:1110.0405](https://arxiv.org/abs/1110.0405))

The case of [[iterated based loop spaces]] of [[n-spheres]] is discussed also in

* {#KallelSjerve99} [[Sadok Kallel]], [[Denis Sjerve]], _On Brace Products and the Structure of Fibrations with Section_, 1999 ([pdf](https://www.math.ubc.ca/~sjer/brace.pdf), [[KallelSjerv99.pdf:file]])

[[!redirects Sullivan models of loop spaces]]

[[!redirects Sullivan model of free loop space]]
[[!redirects Sullivan models of free loop space]]
[[!redirects Sullivan model of free loop spaces]]

[[!redirects Sullivan model of the free loop space]]
[[!redirects Sullivan models of the free loop space]]

[[!redirects Sullivan model of a free loop space]]
[[!redirects Sullivan models of a free loop space]]

[[!redirects Sullivan models of free loop spaces]]
[[!redirects Sullivan model for free loop space]]
[[!redirects Sullivan models for free loop space]]
[[!redirects Sullivan model for free loop spaces]]
[[!redirects Sullivan models for free loop spaces]]

[[!redirects Sullivan model of based loop space]]
[[!redirects Sullivan models of based loop spaces]]

[[!redirects Sullivan model for based loop space]]
[[!redirects Sullivan models for based loop spaces]]


Sullivan models -- examples

**Examples of [[Sullivan models]]** in [[rational homotopy theory]]:

* [[Sullivan model of an n-sphere]]

* [[Sullivan model of a spherical fibration]]

* [[Sullivan model of complex projective space]]

* [[Sullivan model of a classifying space]]

* [[Sullivan model of mapping space]]

* [[Sullivan model of free loop space]]

* [[Sullivan model of a suspension]]

* [[Sullivan model of a finite G-quotient]]


Sultan Catto

* [webpage](https://www.baruch.cuny.edu/wsas/academics/natural_science/scatto.htm)

## Selected writings

On [[hadron supersymmetry]] via [[quark]]/[[diquark]]-exchange:

* {#CattoGuersey85} [[Sultan Catto]], [[Feza Gürsey]], _Algebraic treatment of effective supersymmetry_, Nuov Cim A 86, 201–218 (1985) ([doi:10.1007/BF02902548](https://doi.org/10.1007/BF02902548))

* {#CattoGuersey85} [[Sultan Catto]], [[Feza Gürsey]], _New realizations of hadronic supersymmetry_, Nuov Cim A 99, 685–699 (1988) ([doi:10.1007/BF02730633](https://doi.org/10.1007/BF02730633))

* [[Sultan Catto]], _Algebraic Realization of Quark-Diquark Supersymmetry_ ([arXiv:hep-th/9811069](https://arxiv.org/abs/hep-th/9811069))

* [[Sultan Catto]], Yasemin Gürcan, A. Khalfan, L. Kurt, B. Nicolescu, E. Yu, _Hadronic Supersymmetry from QCD_, Ukrainian Journal of Physics, Vol. 64 No. 8 (2019) ([doi:10.15407/ujpe64.8.672](https://doi.org/10.15407/ujpe64.8.672))

* {#CGNY19} [[Sultan Catto]], Yasemin Gürcan, B. Nicolescu, E. Yu, _Effective Hadronic Supersymmetry from Quantum Chromodynamics_, Ukrainian Journal of Physics, Vol 64 No 12 (2019)  ([doi:10.15407/ujpe64.12.1096](https://doi.org/10.15407/ujpe64.12.1096))

category: people


sum

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

# Contents
* table of contents
{: toc}

## Idea

A _sum_ is a result of an operation called _addition_ and denoted $+$ (binary), $0$ (nullary), or $\sum$ (arbitrary).

* A [[magma]] $A$ is _written additively_ if its operation is a sum: $a,b \in A \;\vdash\; (a + b) \in A$.  Examples include addition of positive integers (an abelian [[semigroup]]), [[natural numbers]] (an abelian [[monoid]]) and the generalisation to other kinds of [[number]]s (most of which form abelian groups or at least abelian monoids, although the [[ordinal numbers]] form a nonabelian monoid).

* In the case of numbers or more generally any [[topological abelian group]] or [[topological vector space]] (and generalizations), we can consider sums of [[infinite series]], and more generally [[integrals]]. (There are however also noncommutative integrals when the order of summation/multiplication of noncommuting quantities is taken into account.)

* In a [[category]] the term _sum_ may refer to the [[coproduct]] of two [[objects]]; in particular, the sum of two [[abstract sets]] is their [[disjoint union]].  The sum of objects in a [[preadditive category]] may refer to the [[biproduct]] or [[direct sum]].

* In a [[poset]] this becomes the notion of [[join]]; in particular, the [[union]] of two [[concrete sets]] ([[material set theory|material sets]] or [[subsets]] of a fixed set $U$) was once often called their _sum_.

* A more general notion of sum in [[category theory]] and [[type theory]] is that of the _[[dependent sum]]_ of a family of objects.

## Related concepts

* [[product]]

[[!redirects sum]]
[[!redirects sums]]
[[!redirects additive operation]]
[[!redirects addition]]


sum type

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

# Contents
* table of contents
{: toc}

## Idea

In [[type theory]] a _sum type_ of two [[types]] $A$ and $B$ (really: [[coproduct]]) is the type whose [[terms]] are either terms $a\colon A$ or terms $b\colon B$.

In a [[model]] of the type theory in [[categorical semantics]] this is a [[distributive coproduct]]. In [[set theory]], it is a [[disjoint union]].

## Definition

Like all type constructors in type theory, to characterize sum types we must specify how to build them, how to construct elements of them, how to use such elements, and the computation rules.

The way to build sum types is easy:

$$ \frac{A\colon Type \qquad B\colon Type}{A+B \colon Type} $$

### As a positive type

Sum types are most naturally presented as [[positive types]], so that the constructor rules are primary.  These say that we can obtain an element of $A+B$ from an element of $A$, or from an element of $B$.

$$ \frac{a\colon A}{inl(a)\colon A+B} \qquad \frac{b\colon B}{inr(b)\colon A+B} $$

The eliminator is derived from these: it says that in order to use an element of $A+B$, it suffices to specify what should be done for the two ways in which that element could have been constructed.

$$ \frac{p\colon A+B \qquad x\colon A\vdash c_A\colon C \qquad y\colon B \vdash c_B\colon C}{match(p, x.c_A, y.c_B) \colon C}$$

The terms $c_A$ and $c_B$ can have [[free variables]] $x$ and $y$ respectively, but those variables become bound in the $match$ expression.  In [[dependent type theory]], we must generalize the eliminator to allow $C$ to depend on $A+B$.

The [[beta reduction]] rules for a constructor followed by an eliminator:

$$
\begin{aligned}
  match(inl(a), x.c_A, y.c_B) &\to_\beta c_A[a/x]\\
  match(inr(b), x.c_A, y.c_B) &\to_\beta c_B[b/y]
\end{aligned}
$$

The [[eta reduction]] rule for the opposite composite says that for any term $c\colon C$ in the context of $p\colon A+B$,

$$match(p, x.c[inl(x)/z], y.c[inr(y)/z]) \to_\eta c[p/z].$$

This says that if we unpack a term of type $A+B$, but only use the resulting term of type $A$ or $B$ by way of packing them back into $A+B$, then we might as well not have unpacked them to begin with.  Note that choosing $C\coloneqq A+B$ and $c \coloneqq z$, we obtain a simpler form of $\eta$-conversion:

$$match(p, x.inl(x), y.inr(y)) \to_\eta p.$$

The positive presentation of sum types can be regarded as a particular sort of [[inductive type]].  In [[Coq]] syntax:

    Inductive sum (A B:Type) : Type :=
    | inl : A -> sum A B
    | inr : B -> sum A B.

Coq implements the beta reduction rule, but not the eta (although eta equivalence is provable for the inductively defined [[identity types]], using the dependent eliminator mentioned above).

### As a negative type

It is possible to present sum types as [[negative types]] as well, but only if we allow [[sequents]] with multiple conclusions.  This is common in [[sequent calculus]] presentations of [[classical logic]], but not as common in type theory and almost unheard of in [[dependent type theory]].

The two definitions are provably equivalent, but only using the [[contraction rule]] and the [[weakening rule]].  Thus, in [[linear logic]] they become distinct; the positive sum type is "plus" $A\oplus B$ and the negative one is "par" $A \parr B$.

### With typal computation and uniqueness rules

Assuming that [[identification types]], [[function types]] and [[dependent product types]] exist in the type theory, the sum type of types $A$ and $B$ is the [[inductive type]] generated by a [[function]] from $A$ to $A + B$ and a function from $B$ to $A + B$:

Formation rules for sum types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash A + B \; \mathrm{type}}$$

Introduction rules for sum types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash \mathrm{in}_A:A \to A + B} \qquad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash \mathrm{in}_B:B \to A + B}$$

Elimination rules for sum types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A + B \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_{\mathrm{in}_A}:\prod_{x:A} C(\mathrm{in}_A(x)) \quad \Gamma \vdash c_{\mathrm{in}_B}:\prod_{y:B} C(\mathrm{in}_B(y)) \quad \Gamma \vdash z:A + B}{\Gamma \vdash \mathrm{ind}_{A + B}^C(c_{\mathrm{in}_A}, c_{\mathrm{in}_B}, z):C(y)}$$

Computation rules for sum types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A + B \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_{\mathrm{in}_A}:\prod_{x:A} C(\mathrm{in}_A(x)) \quad \Gamma \vdash c_{\mathrm{in}_B}:\prod_{y:B} C(\mathrm{in}_B(y)) \quad \Gamma \vdash a:A}{\Gamma \vdash \beta_{A + B}^{\mathrm{in}_A}(c_{\mathrm{in}_A}, c_{\mathrm{in}_B}, a):\mathrm{Id}_{C(\mathrm{in}_A(a))}(\mathrm{ind}_{A + B}^C(c_{\mathrm{in}_A}, c_{\mathrm{in}_B}, \mathrm{in}_A(a)), c(a))}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A + B \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_{\mathrm{in}_A}:\prod_{x:A} C(\mathrm{in}_A(x)) \quad \Gamma \vdash c_{\mathrm{in}_B}:\prod_{y:B} C(\mathrm{in}_B(y)) \quad \Gamma \vdash b:B}{\Gamma \vdash \beta_{A + B}^{\mathrm{in}_B}(c_{\mathrm{in}_A}, c_{\mathrm{in}_B}, b):\mathrm{Id}_{C(\mathrm{in}_B(b))}(\mathrm{ind}_{A + B}^C(c_{\mathrm{in}_A}, c_{\mathrm{in}_B}, \mathrm{in}_B(b)), c(b))}$$

Uniqueness rules for sum types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A + B \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c:\prod_{x:A + B} C(x) \quad \Gamma \vdash z:A + B}{\Gamma \vdash \eta_{A + B}(c, z):\mathrm{Id}_{C(z)}(\mathrm{ind}_{A + B}^C(\lambda x:A.c(\mathrm{in}_A(x)), \lambda y:B.c(\mathrm{in}_B(y)), z), c(z))}$$

The elimination, computation, and uniqueness rules for the sum type of $A$ and $B$ state that the sum type of $A$ and $B$ satisfy the **dependent universal property of the sum type** of $A$ and $B$. If the dependent type theory also has [[dependent sum types]] and [[product types]], allowing one to define the [[uniqueness quantifier]], the dependent universal property of the sum type of of $A$ and $B$ could be simplified to the following rule:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A + B \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_{\mathrm{in}_A}:\prod_{x:A} C(\mathrm{in}_A(x)) \quad \Gamma \vdash c_{\mathrm{in}_B}:\prod_{y:B} C(\mathrm{in}_B(y))}{\Gamma \vdash \mathrm{up}_{A + B}^C(c_{\mathrm{in}_A}, c_{\mathrm{in}_B}):\exists!c:\prod_{x:A + B} C(x).\left(\prod_{a:A} \mathrm{Id}_{C(\mathrm{in}_A(a))}(c(\mathrm{in}_A(a)), c_{\mathrm{in}_A}(a))\right) \times \left(\prod_{b:B} \mathrm{Id}_{C(\mathrm{in}_B(b))}(c(\mathrm{in}_B(b)), c_{\mathrm{in}_B}(b))\right)}$$

### In terms of dependent sum types and booleans

The sum type can be defined in terms of the [[boolean domain]] and the [[dependent sum type]]. Given types $A$ and $B$, the sum type $A + B$ is defined as

$$A + B \coloneqq \sum_{x:\mathrm{Bool}} ((x =_\mathrm{Bool} 1) \to A) \times ((x =_\mathrm{Bool} 0) \to B)$$

## Related concepts

* [[copy]]

* [[product type]]

* [[dependent sum type]]

## References

A textbook account in the context of [[programming languages]] is in section 12 of

* [[Robert Harper]], _[[Practical Foundations for Programming Languages]]_

For sum types in [[homotopy type theory]], see:

* *Homotopy Type Theory: Univalent Foundations of Mathematics*, The [[Univalent Foundations Project]], Institute for Advanced Study, 2013. ([web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf))

* [[Egbert Rijke]], *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press ([arXiv:2212.11082](https://arxiv.org/abs/2212.11082))

[[!redirects sum type]]
[[!redirects sum types]]

[[!redirects universal property of the sum type]]
[[!redirects universal property of the sum types]]

[[!redirects universal property of sum type]]
[[!redirects universal property of sum types]]

[[!redirects dependent universal property of the sum type]]
[[!redirects dependent universal property of the sum types]]

[[!redirects dependent universal property of sum type]]
[[!redirects dependent universal property of sum types]]

[[!redirects disjoint union type]]
[[!redirects disjoint union types]]
[[!redirects coproduct type]]
[[!redirects coproduct types]]


Sumeet Khatri

* [personal page](https://sumeetkhatri.com/)

* [GoogleScholar page](https://scholar.google.com/citations?user=CcJ9FpcAAAAJ&hl=en)

## Selected writings

On [[quantum information theory]]:

* [[Sumeet Khatri]], [[Mark M. Wilde]], *Principles of Quantum Communication Theory: A Modern Approach* [[arXiv:2011.04672](https://arxiv.org/abs/2011.04672)]

category: people


Sumio Watanabe

* [personal page](http://watanabe-www.math.dis.titech.ac.jp/users/swatanab/)

* [GoogleScholar page](https://scholar.google.com/citations?user=_KUAdxcAAAAJ&hl=en)

## Selected writings

On [[machine learning]], [[singular learning theory]] and [[Bayesian statistics]]:

* [[Sumio Watanabe]], _Algebraic geometry and statistical learning theory_, CRC Press (2009) [[doi:10.1017/CBO9780511800474]( https://doi.org/10.1017/CBO9780511800474)]

* [[Sumio Watanabe]], _Mathematical theory of Bayesian statistics_, Cambridge University Press (2018) [[ISBN:9780367734817](https://www.routledge.com/Mathematical-Theory-of-Bayesian-Statistics/Watanabe/p/book/9780367734817), [pdf](http://196.189.45.87/bitstream/123456789/36917/1/Sumio%20Watanabe_2018.pdf)]

category: people


Summer 2018 math problem solving

[Studip](https://www.studip.uni-goettingen.de/dispatch.php/course/overview?cid=3ce65c7babac83800a87ccf6249da099)

1. Christian

2. Geoffrey

3. exam

4. Tobias Reis

5. exam

6.


Sumner Byron Myers

[[!redirects S. B. Myers]]
Sumner Byron Myers was a mathematician at the University of Michigan.
He got his PhD degree in 1932 at Harvard University, advised by [[Marston Morse]].

## Related concepts

* [[Myers–Steenrod theorem]]

[[!redirects Sumner Byron Myers]]
[[!redirects Sumner B. Myers]]
[[!redirects Sumner Myers]]
[[!redirects S. Myers]]
[[!redirects S. B. Myers]]


Sundararaman Ramanan

* [Wikipedia entry](https://en.wikipedia.org/wiki/S._Ramanan)

## Selected writings

On [[universal connections]] (as used in [[Chern-Weil theory]]):

* [[Mudumbai Narasimhan]], [[Sundararaman Ramanan]], _Existence of Universal Connections_, American Journal of Mathematics Vol. 83, No. 3 (Jul., 1961), pp. 563-572 ([jstor:2372896 ](https://www.jstor.org/stable/2372896 ))

* [[Mudumbai Narasimhan]], [[Sundararaman Ramanan]], _Existence of Universal Connections II_, American Journal of Mathematics Vol. 85, No. 2 (Apr., 1963), pp. 223-231 ([jstor:2373211](https://www.jstor.org/stable/2373211))

category: people


Sunil Gadhia

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=128119)

## Selected writings

On the [[ADE classification]] of [[black brane|black]] [[BPS state|BPS]]-[[M2-brane]] solutions to [[D=11 supergravity]] (via [[spherical space forms]] of the [[7-sphere]]):

* {#Gadhia07} [[Sunil Gadhia]], _Supersymmetric quotients of M-theory and supergravity backgrounds_, PhD thesis, School of Mathematics, University of Edinburgh, 2007 ([spire:1393845](http://inspirehep.net/record/1393845/))

* {#MFFGME09} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Sunil Gadhia]], [[Elena Méndez-Escobar]], _Half-BPS quotients in M-theory: ADE with a twist_, JHEP 0910:038,2009 ([arXiv:0909.0163](http://arxiv.org/abs/0909.0163), [pdf slides](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf))

## Related $n$Lab entries

* [[group action on an n-sphere]]

* [[spherical space form]]

* [[M2-brane]]

* [[ABJM model]]

category: people


Sunil Mukhi

* [Wikipedia entry](https://en.wikipedia.org/wiki/Sunil_Mukhi)

## Selected writings

On [[orientifolds]]:

* {#Mukhi97} [[Sunil Mukhi]], _Orientifolds: The Unique Personality Of Each Spacetime Dimension_, [Workshop on Frontiers of Field Theory, Quantum Gravity and String Theory, Puri, India, 12 - 21 Dec 1996](http://cds.cern.ch/record/323857) ([arXiv:hep-th/9710004](https://arxiv.org/abs/hep-th/9710004), [cern:335233](http://cds.cern.ch/record/335233))

and their lift to [[M-theory]] ([[MO5]], [[MO9]]):

* Keshav Dasgupta, [[Sunil Mukhi]], _Orbifolds of M-theory_, Nucl. Phys. B465 (1996) 399-412 ([arXiv:hep-th/9512196](https://arxiv.org/abs/hep-th/9512196))

On [[C-field tadpole cancellation]]:

* {#DasguptaMukhi97} Keshav Dasgupta, [[Sunil Mukhi]], equation (1) in _A Note on Low-Dimensional String Compactifications_, Phys. Lett. B398:285-290, 1997 ([arXiv:hep-th/9612188](https://arxiv.org/abs/hep-th/9612188))

On the [[WZW model]] [[2d CFT]] ar fractional [[level (Chern-Simons theory)|level]]:

* [[Sunil Mukhi]], Sudhakar Panda, *Fractional-level current algebras and the classification of characters*, Nuclear Physics B **338** 1 (1990) 263-282 $[$doi:10.1016/0550-3213(90)90632-N$]$

On the [[pp-wave spacetime]] [[Penrose limit]] of the [[AdS5-CFT4 correspondence]]:

* N. Itzhaki, [[Igor Klebanov]], [[Sunil Mukhi]], _PP Wave Limit and Enhanced Supersymmetry in Gauge Theories_, JHEP 0203 (2002) 048 ([arXiv:hep-th/0202153](https://arxiv.org/abs/hep-th/0202153))

On multiple [[M2-branes]] and their worldvolume [[BLG model]]/[[ABJM model]]:

* {#BaggerLambertMukhiPapageorgakis13} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], _Multiple Membranes in M-theory_, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 ([arXiv:1203.3546](http://arxiv.org/abs/1203.3546), [doi:10.1016/j.physrep.2013.01.006](https://doi.org/10.1016/j.physrep.2013.01.006))

## Related $n$Lab entries

* [[M2-brane]]

  * [[3d superconformal gauge field theory]]

    [[BLG model]], [[ABJM model]]

category: people


super 1-brane in 3d

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## Idea

By the _[[brane scan]]_ there exists a [[classical field theory|classical]] ([[prequantum field theory|pre-quantum]]) [[Green-Schwarz super p-brane sigma model]] for $p =1 $ ([[superstrings]]) with [[target space]] being $d = 3$ dimensional [[super Minkowski spacetime]] or more generally a [[superspacetime]] which is a [[super Cartan geometry]] solving the [[equations of motion]] of $N=1$ or $N=2$ [[3d supergravity]]. (The literature traditionally knows this as the "3d superstring".)

For $N=(1,1)$ this is the [[double dimensional reduction]] of the [[super 2-brane in 4d]] (e.g. [Mezincescu-Townsend 11, p. 44](#MezincescuTownsend11)).

These superstrings in 3d target spacetime play a role in the [[AdS3-CFT2 and CS-WZW correspondence]] (e.g. ([YuZ-Zhang 98](#YuZhang98))) and at least after a [[topological twist]]  as exhibiting [[3d Chern-Simons theory]] as a [[second quantization]] (see at [TCFT -- Effective background theories](TCFT#ActionFunctionals)).

## Quantization

While the  [[light-cone gauge quantization]] of the [[Green-Schwarz super p-brane|Green-Schwarz superstrings]] [[broken symmetry|breaks]] [[Lorentz group]]-symmetry for $d = 4$ and $d = 6$, it preserves Lorentz symmetry not only for $d = 10$ (the "critical string" of [[heterotic string theory]] or [[type II string theory]]) but also for $d = 3$ ([Mezincescu-Townsend 10](#MezincescuTownsend10), [Mezincescu-Townsend 11](#MezincescuTownsend11)). This quantization turns out to be equivalent to that of the RR-sector of the worldsheet supersymmetric [[spinning string]] in $d = 3$ ([Mezincescu-Routh-Townsend 13](#MezincescuRouthTownsend13)).

## Related concepts

[[super 2-brane in 4d]]

[[!include brane scan]]

## References
 {#References}

* {#YuZhang98} Ming Yu, Bo Zhang, _Light-Cone Gauge Quantization of String Theories on AdS3 Space_, Nucl.Phys. B551 (1999) 425-449 ([arXiv:hep-th/9812216](http://arxiv.org/abs/hep-th/9812216))

* {#MezincescuTownsend10} [[Luca Mezincescu]], [[Paul Townsend]], _Anyons from Strings_, Phys.Rev.Lett.105:191601,2010 ([arXiv:1008.2334](http://arxiv.org/abs/1008.2334))

* {#MezincescuTownsend11} [[Luca Mezincescu]], [[Paul Townsend]], _Quantum 3D Superstrings_, PhysRevD.84.106006 ([arXiv:1106.1374](http://arxiv.org/abs/1106.1374))

* {#MezincescuRouthTownsend13} [[Luca Mezincescu]], Alasdair J. Routh, [[Paul Townsend]], _Equivalence of 3D Spinning String and Superstring_, J. High Energ. Phys. (2013) 2013: 24 ([arXiv:1305.5049](http://arxiv.org/abs/1305.5049))

The [[F-theory]]-lift of the 3d superstring is discussed in

* William D. Linch III, [[Warren Siegel]], _F-theory Superspace_ ([arXiv:1501.02761](http://arxiv.org/abs/1501.02761))

[[!redirects superstring in 3d]]


super 1-brane in 4d

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{:toc}

## Idea

By the old [[brane scan]] there is a [[Green-Schwarz sigma-model]] for a [[superstring]] on 4d [[super-Minkowski spacetime]].

Due to the 4d [[Fierz identity]]

$$
  \Gamma_a \psi \left( \overline{\psi} \Gamma^a \psi\right)
  \;= 0\;
$$

(e.g. [Ravera 18, (A.17)](#Ravera18))

Being of [[codimension]]=2, this is a [[defect brane]].

[[!include brane scan]]

## Related concepts

* [[super 2-brane in 4d]]

* [[super 1-brane in 3d]]

## References

Discussion relating to the analog of the [[DF-algebra]]/[[M-theory super Lie algebra]] in [[4d supergravity]] is in

* {#Ravera18} [[Lucrezia Ravera]], _Hidden Role of Maxwell Superalgebras in the Free Differential Algebras of D=4 and D=11 Supergravity_ ([arXiv:1801.08860](https://arxiv.org/abs/1801.08860))


super 2-algebra

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#### Higher category theory
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=--

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* table of contents
{:toc}

## Idea

The notion of _[[symmetric monoidal category|symmetric monoidal]] [[tensor category]]_ may be thought of as a [[categorification]] of the notion of  [[commutative ring]], a _2-ring_ or [[2-algebra]].

Now the notion of _super 2-algebra_ is accordingly supposed to be the notion categorifying [[super algebra]].

## Related concepts

* [[spectral super-scheme]]

## References
 {#References}

The definition of super 2-algebra starts at 33:10 in:

* {#Kapranov13} [[Mikhail Kapranov]], _Categorification of supersymmetry and stable homotopy groups of spheres_, talk at _[Algebra, Combinatorics and Representation Theory: in memory of Andrei Zelevinsky (1953-2013)](https://web.archive.org/web/20130617191515/http://mathserver.neu.edu/~bwebster/ACRT/)_ (April 2013) [[[Webster-ACRTAbstracts2013.pdf:file]],  video:[YT](https://www.youtube.com/watch?v=WBsZrcYOrxI)]

  > **Abstract:**. The "minimal sign skeleton" necessary to formulate the [[signs in supergeometry|Koszul sign rule]] is a certain [[Picard category]], a [[symmetric monoidal category]] with all [[invertible object|objects]] and [[invertible morphism|morphisms invertible]]. It can be seen as the free Picard category generated by one object and corresponds, by [[Grothendieck]]'s dictionary, to the [[n-truncation|truncation]] of the [[sphere spectrum|spherical spectrum]] $S$ in degrees 0 and 1, so that $\{\pm 1\}$ appears as the [[first stable homotopy group of spheres]] $\pi_{n+1}(S^n)$. This suggest a "[[higher structures|higher]]" or [[categorification|categorified]] versions of [[super-algebra|super-mathematics]] which utilize deeper structure of [[sphere spectrum|$S$]]. The first concept on this path is that of a supersymmetric monoidal category which is [[categorification|categorified]] version of the concept of a [[supercommutative algebra]].

(while the first 30 minutes are concerned with discussion of how super-grading is implied by grading ([[augmented infinity-groups]]) over the [[sphere spectrum]], see  at [superalgebra - Abstract idea](super+algebra#AbstractIdea)).

The example of the super 2-algebra of superalgebras starts at 39:30,

the example of the exterior 2-algebra of a super-linear category starts at 47:30.


super 2-brane in 4d

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## Idea

By the _[[brane scan]]_ there exists a [[classical field theory|classical]] ([[prequantum field theory|pre-quantum]]) [[Green-Schwarz super p-brane sigma model]] for $p =2 $ ([[supermembrane]]) with [[target space]] being $d = 4$ dimensional [[super Minkowski spacetime]] with [[spin representation|N=1]] [[supersymmetry]],  or more generally a [[superspacetime]] which is a [[super Cartan geometry]] solving the [[equations of motion]] of [[d=4 N=1 supergravity]]. (The literature traditionally knows this as the "4d supermembrane".)

Its [[double dimensional reduction]] is the [[super 1-brane in 3d]] (see there) for $N=(1,1)$ [[3d supergravity]].

## Related entries

* [[super 1-brane in 3d]]

* [[super 1-brane in 4d]]

[[!include brane scan]]

## References

The action functional is apparently originally due to

* A. Achúcarro, [[Jerome Gauntlett]], K. Itoha, [[Paul Townsend]], _World-volume supersymmetry from spacetime supersymmetry of the four-dimensional supermembrane_, Nuclear Physics B Volume 314, Issue 1, 20 February 1989, Pages 129–157 (doi:10.1016/0550-3213(89)90115-6)

Further dicussion of the coupling to the [[d=4 N=1 supergravity]] background:

* [[Burt Ovrut]], [[Daniel Waldram]], _Membranes and Three-form Supergravity_, Nucl.Phys. B506 (1997) 236-266 ([arXiv:hep-th/9704045](http://arxiv.org/abs/hep-th/9704045))

* [[Maria P. Garcia del Moral]], J. M. Pena, [[Alvaro Restuccia]], _$\mathcal{N}=1$ 4D Supermembrane from $11D$_, JHEP0807 039 (2008) [[arXiv:0709.4632](http://arxiv.org/abs/0709.4632)]

* M. Huebscher, P. Meessen, T. Ortin, _Domain walls and instantons in $N=1$, $d=4$ supergravity_, JHEP 1006:001,2010 ([arXiv:0912.3672](http://arxiv.org/abs/0912.3672))

* [[Igor Bandos]], Carlos Meliveo, _Supermembrane interaction with dynamical $D=4$ $N=1$ supergravity. Superfield Lagrangian description and spacetime equations of motion_ ([arXiv:1205.5885](http://arxiv.org/abs/1205.5885))

* Carlos Meliveo, _On Supermembrane in $D=4$, multiple M0-brane in $D=11$ and Supersymmetric Higher Spin Theories_ ([arXiv:1312.3512](http://arxiv.org/abs/1312.3512))

See also

* [[Igor Bandos]], [[Fotis Farakos]], Stefano Lanza, Luca Martucci, [[Dmitri Sorokin]], _Three-forms, dualities and membranes in four-dimensional supergravity_, JHEP07(2018)028 ([arXiv:1803.01405](https://arxiv.org/abs/1803.01405))

Discussion of coupling of the [[super 2-brane in 4d]] to [[D=4 N=1 super Yang-Mills theory]]:

* [[Igor Bandos]], Stefano Lanza, [[Dmitri Sorokin]], _Supermembranes and domain walls in $\mathcal{N}=1$, $D=4$ SYM_ ([arXiv:1905.02743](https://arxiv.org/abs/1905.02743))

On [[membranes]] in [[D=4 supergravity]] ([[super 2-brane in 4d]]):

* [[Igor Bandos]], [[Fotis Farakos]] Stefano Lanza, L. Martucci,  [[Dmitri Sorokin]], _Higher Forms and Membranes in 4D Supergravities_, in: Proceedings of _[[Higher Structures in M-Theory 2018]]_, Fortschritte der Physik, Special Issue Volume 67, Issue 8-9 ([arXiv:1903.02841](https://arxiv.org/abs/1903.02841), [doi:10.1002/prop.201910020](https://doi.org/10.1002/prop.201910020))

Discussion of strongly coupled 4d physics akin to 11d [[M-theory]] with its [[M2-brane]] includes

* [[Edward Witten]], _Strong coupling and the cosmological constant_, Mod.Phys.Lett.A10:2153-2156,1995 ([arXiv:hep-th/9506101](https://arxiv.org/abs/hep-th/9506101))

(on possible solutions to the [[cosmological constant]] problem).

[[!redirects supermembrane in 4d]]


super 2-brane in 5d

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{:toc}

## Idea

According to the [[brane scan]] of [[super p-branes|super $p$-branes]] there exists a [[supermembrane]] propagating on 4+1 dimensional [[super Minkowski spacetime]] (being of [[codimension]]=2, this is a [[defect brane]]):

[[!include brane scan]]

## References

* Minkyu Park, [[Masaki Shigemori]], _Codimension-2 Solutions in Five-Dimensional Supergravity_, JHEP 1510 (2015) 011 ([arXiv:1505.05169](http://arxiv.org/abs/1505.05169))

* [[Masaki Shigemori]], *Interpolating between multi-center microstate geometries*, JHEP 09 (2021) 010 ([arXiv:2105.11639](https://arxiv.org/abs/2105.11639))

* S. Bellucci, N. Kozyrev, S. Krivonos, A. Yeranyan, *Supermembrane in $D=5$: component action*, Journal of High Energy Physics volume **2014** 142 (2014)  ([arXiv:1312.0231](https://arxiv.org/abs/1312.0231), doi:10.1007%2FJHEP05%282014%29142)


super 3-brane in 8d

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## Idea

According to the [[brane scan]] of [[super p-branes|super $p$-branes]] there exists a 3-[[brane]] propagating on 5+1 dimensional [[super Minkowski spacetime]]:

[[!include brane scan]]

## References

* S. Bellucci, N. Kozyrev, S. Krivonos, A. Yeranyan, *Component on-shell actions of supersymmetric 3-branes: II. 3-brane in D = 8*, Class. Quantum Grav. **32** 115008 (2015) ([doi:10.1088/0264-9381/32/11/115008](https://iopscience.iop.org/article/10.1088/0264-9381/32/11/115008))


super 4-brane in 9d

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{:toc}

## Idea

The [brane scan](Green-Schwarz+action+functional#BraneScan) of [[Green-Schwarz sigma-model]] for [[super p-branes]] shows a 4-brane in 9d.

## Related entries

[[!include brane scan]]

## References

On regularized [[quantization]] of [[Green-Schwarz sigma-models]] for [[super p-branes]], specifically of the super 4-brane in 9d, in generalization of the [matrix model of the M2-brane](M2-brane#QuantizationViaBFSSMatrixModel):

* [[Meer Ashwinkumar]], [[Lennart Schmidt]], [[Meng-Chwan Tan]], Section 2 of: _Matrix Regularization of Classical Nambu Brackets and Super $p$-Branes_, J. High Energ. Phys. **2021** 172 (2021) [[arXiv:2103.06666](https://arxiv.org/abs/2103.06666), doi:10.1007/JHEP07(2021)172]

reviewed in:

* [[Meer Ashwinkumar]], *Matrix Quantization of Classical Nambu Brackets and Super $p$-Branes*, talk at *[M-Theory and Mathematics 2023](/nlab/show/M-Theory+and+Mathematics#2023)*, New York University, Abu Dhabi (2023) [[web](/nlab/show/M-Theory+and+Mathematics#Ashwinkumar2023)]

[[!redirects super 4-brane in 9 dimensions]]


super abelian group

[[!redirects super abelian groups]]

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## Idea ##

A super abelian group is an $\mathbb{Z}$-[[super module]], or a .

## Definition ##

### $\mathbb{Z}/2\mathbb{Z}$-graded abelian group ##

A **$\mathbb{Z}/2\mathbb{Z}$-graded** is an [[abelian group]] $G$ with decomposition functions $\mathcal{D}_0:G \to G$ and $\mathcal{D}_1:G \to G$, such that

* for all $a:G$, $a = \mathcal{D}_0(a) + \mathcal{D}_1(a)$
* for all $a:G$, and $b:G$, $\mathcal{D}_0(a + b) = \mathcal{D}_0(a) + \mathcal{D}_0(b)$
* for all $a:G$, and $b:G$, $\mathcal{D}_1(a + b) = \mathcal{D}_1(a) + \mathcal{D}_1(b)$
* for all $a:G$, $\mathcal{D}_0(\mathcal{D}_0(a)) = \mathcal{D}_0(a)$
* for all $a:G$, $\mathcal{D}_0(\mathcal{D}_1(a)) = 0$
* for all $a:G$, $\mathcal{D}_1(\mathcal{D}_0(a)) = 0$
* for all $a:G$, $\mathcal{D}_1(\mathcal{D}_1(a)) = \mathcal{D}_1(a)$

As a result, the [[image]] of the two decomposition functions $\im(\mathcal{D}_0)$ and $\im(\mathcal{D}_1)$ are abelian groups and there exists a [[group]] [[isomorphism]] $i:V \cong \im(\mathcal{D}_0) \otimes \im(\mathcal{D}_1)$, where $A \otimes B$ is the [[tensor product of abelian groups]].

The elements of $\im(\mathcal{D}_0)$ are called **even elements** or **bosonic elements**, and the elements of $\im(\mathcal{D}_1)$ are called **odd elements** or **fermionic elements**.

### Super abelian group ###

The tensor product of $\mathbb{Z}/2\mathbb{Z}$-graded abelian groups $A \otimes B$ for $a:A$, $b:B$ is defined as the following:

$$\mathcal{D}_0(a \otimes b) = \mathcal{D}_0(a) \otimes \mathcal{D}_0(b) + \mathcal{D}_1(a) \otimes \mathcal{D}_1(b)$$

$$\mathcal{D}_1(a \otimes b) = \mathcal{D}_0(a) \otimes \mathcal{D}_1(b) + \mathcal{D}_1(a) \otimes \mathcal{D}_0(b)$$

This plus the group homomorphisms $\mathcal{D}_0$ and $\mathcal{D}_1$ result in the category of $\mathbb{Z}/2\mathbb{Z}$-graded abelian groups to be a [[monoidal category]].

A **super abelian group** is an object of the category of $\mathbb{Z}/2\mathbb{Z}$-graded abelian groups with the braiding for the tensor product $A \otimes B$:

$$t_{A, B}:(A \otimes B) \to (B \otimes A)$$

such that

$$\mathcal{D}_0(t_{A, B}(a, b)) = \mathcal{D}_0(a) \otimes \mathcal{D}_0(b) - \mathcal{D}_1(a) \otimes \mathcal{D}_1(b)$$

$$\mathcal{D}_1(t_{A, B}(a, b)) = \mathcal{D}_0(a) \otimes \mathcal{D}_1(b) + \mathcal{D}_1(a) \otimes \mathcal{D}_0(b)$$

## See also ##

* [[abelian group]]

* [[graded abelian group]]

* [[super commutative monoid]]

* [[super module]]

* [[super ring]]

* [[supercommutative ring]]


super algebra

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> from [Grassmann 1844, p. 61 and 84](Ausdehnungslehre#Grassmann44)

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

### Basic idea

In the general sense, _superalgebra_ is the study of ([[higher algebra|higher]]) [[algebra]]

* [[internalization|internal]] to the [[symmetric monoidal category]] of $\mathbb{Z}_2$-[[graded vector spaces]] ([[super vector spaces]]);

equivalently

* over the [[base topos]] on [[superpoints]].

More specifically, a _[[supercommutative superalgebra]]_ is an [[commutative algebra]] in the context of superalgebra.

See at _[[geometry of physics -- superalgebra]]_ for more on this.

In the following we first discuss

* _[Associative superalgebras](#AssociativeSuperalgebras)_

as [[monoids]] in the [[symmetric monoidal category]] of [[super vector spaces]]. Then we pass to the perspective of

* _[Algebra in the topos over superpoints](#AlgebraOverSuperpoints)_

and consider systematically [[algebra]] in the [[sheaf topos]] over the [[site]] of [[superpoints]] and show how this reproduces and generalizes the previous notions.

See ([Sachse](#Sachse)) and the references at _[[super ∞-groupoid]]_ for some history of the topos-theoretic perspective on superalgebra.

### Abstract idea
 {#AbstractIdea}

We discuss the general abstract _raison d' être_ of super algebra. Readers looking for just the plain definition should probably [skip to below](#AssociativeSuperalgebras) on first reading.

One way to understand the relevance of [[supercommutative superalgebra]] is [[Deligne's theorem on tensor categories]], which states that well-behaved [[tensor categories]] over the [[complex numbers]] are [[equivalence of categories|equivalent]] to [[categories of representations]] of [[supergroups]]. From this perspective the crucial sign rule is related to the [[symmetric monoidal category|symmetric]] [[braided monoidal category|braiding]] in [[tensor categories]]. This in turn may itself be understood from a more general perspective as follows.

Superalgebra is [[universal property|universal]] in the following sense. The crucial super-grading rule (the "Koszul sign rule", [Grassmann 1844, §37, §55](#Grassmann1844))

$$
  a \otimes b = (-1)^{deg(a) deg(b)} b \otimes a
$$

in the [[symmetric monoidal category]] of $\mathbb{Z}$-[[graded vector spaces]] is induced from the [[subcategory]] which is the [[abelian 2-group]] of metric graded [[lines]]. This in turn is the [[free construction|free]] [[abelian 2-group]] (groupal [[symmetric monoidal category]]) on a single generator.  (This point of view is amplified in the first part of ([Kapranov 13](#Kapranov13)), whose second part is about [[super 2-algebra]], more details in [Kapranov 15](#Kapranov15)). Generally then super-grading and hence super-algebra arises from the [[truncated object|2-truncation]] (3-[[coskeleton]]) of the free [[abelian ∞-group]] on a single generator, which is the [[sphere spectrum]] $\mathbb{S}$. So the $\mathbb{Z}_2$-grading of superalgebra comes from the [[stable homotopy groups of spheres]] $\pi_n(\mathbb{S})$ in degree 1 and 2:

| $n = $ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $\cdots$ |
|--------|-----|-----|-----|-----|-----|-----|-----|-----|--|
| $\pi_n(\mathbb{S}) = $ | $\mathbb{Z}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | $\mathbb{Z}_{24}$ | $ 0 $ | $0$ | $\mathbb{Z}_2$ | $\mathbb{Z}_{240}$ | $\cdots$ |
| meaning: | degree | [[boson]]/[[fermion]] super-degree | [[spin geometry|spin]] | [[string geometry|string]] | $-$ | $-$ | ? | ? | $\cdots$ |
| [[free construction|free object]] on single [[generators and relations|generator]]: | [[abelian group]] | [[abelian 2-group]] | [[abelian 3-group]] | [[abelian 4-group]] |  |  | [[abelian 7-group]] | [[abelian ∞-group|abelian 8-group]] | [[abelian ∞-group]] |

This suggests (as indicated in ([Kapranov 13](#Kapranov13), [Kapranov 15](#Kapranov15))) that in full generality [[higher geometry|higher]] [[supergeometry]] is to be thought of as $\mathbb{S}$-[[graded object|graded]] geometry, hence [[Isbell duality|dually]] as [[higher algebra]] with [[∞-group of units]] [[augmented ∞-group|augmented]] over the [[sphere spectrum]].

But notice that this is canonically so for every [[E-∞ ring]], see at _[∞-group of units -- Augmented definition](infinity-group+of+units#AugmentedDefinition)_. This would mean: In [[higher geometry]]/[[higher algebra]] supergeometry/superalgebra is intrinsic, canonically given.

Using this together with [[Hisham Sati|Sati]]'s _[[Geometric and topological structures related to M-branes]]_ and the [[image of the J-homomorphism]]

[[!include image of J -- table]]

we can derive the terminology in the above table as indicated now.

> The following uses the notions of _[[motivic quantization]]_ as indicated there, to be expanded.

* **$d = 1$ [[sigma-model]]**

  The [[local coefficients]] for [[quantization|quantizing]] the ([[spinning particle|spinning]]) [[particle]] on the [[boundary]] of the [[string]] ending on a [[D-brane]] (by K-theoretic [[geometric quantization by push-forward]]/[[D-brane charge]]) are

  $$
    \array{
      B BU(1)
      &\simeq&
      B K(\mathbb{Z},2)
      &\to&
      B gl_1(KU)
      &\to&
      KU \text{-}Mod
    }
  $$

  for $KU$ the [[complex K-theory spectrum]] [[E-∞ ring]], and
  hence the characteristic twists are in degree 2 of the group of units, hence of the [graded ∞-group of units](infinity-group%20of%20units#AugmentedDefinition)

  $$
    gl^{gr}_1(KU) \to \mathbb{S}
  $$

  hence are graded by the second [[homotopy group]]

  $$
    \pi_2(\mathbb{S}) \simeq \mathbb{Z}_2
  $$

  of the sphere spectrum.

* **$d = 2$ [[sigma-model]]**

  The [[local coefficients]] for [[quantization|quantizing]] the [[string]] (on the [[boundary]] of the [[M2-brane]] ending on an [[M9-brane]]) are

  $$
    \array{
      B B^2 U(1)
      &\simeq&
      B K(\mathbb{Z},3)
      &\to&
      B gl_1(tmf)
      &\to&
      tmf \text{-}Mod
    }
  $$

  for the [[tmf]] [[E-∞ ring]], and
  hence the characteristic twists are in degree 3 of the group of units, hence of the graded group of units

  $$
    gl^{gr}_1(tmf) \to \mathbb{S}
  $$

  hence are graded by the third [[homotopy group]]

  $$
    \pi_3(\mathbb{S}) \simeq \mathbb{Z}_{24}
  $$

  of the sphere spectrum.

* **$d = 5$ [[sigma-model]]**

  The [[local coefficients]] for [[quantization|quantizing]] the [[Yang monopole]] (on the [[boundary]] of the [[M5-brane]] ending on an [[M9-brane]]) are

  $$
    \array{
      B B^5 U(1)
      &\simeq&
      B K(\mathbb{Z},6)
      &\to&
      B gl_1(K(5))
      &\to&
      K(5) \text{-}Mod
    }
    \,,
  $$

  and
  hence the characteristic twists are in degree 6 of the group of units, hence of the graded group of units

  $$
    gl^{gr}_1(K(5)) \to \mathbb{S}
  $$

  hence are graded by the sixth [[homotopy group]]

  $$
    \pi_6(\mathbb{S}) \simeq \mathbb{Z}_{2}
  $$

  of the sphere spectrum.

## Associative superalgebras
 {#AssociativeSuperalgebras}

### Definition

#### Superalgebras

An ordinary [[associative algebra]] (a [[vector space]] with a linear and associative and unital product operation) is a [[monoid]] in the [[monoidal category]] [[Vect]] of [[vector spaces]].

Throughout, fix a [[field]] $k$ of [[characteristic]] 0.

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

Write [[SVect]] for the [[symmetric monoidal category]] of [[super vector space]]s over $k$. This is the [[category]] of $\mathbb{Z}_2$-[[graded vector space]]s equipped with the unique non-trivial symmetric [[braided monoidal category|braided monoidal structure]].

Objects are vector spaces with a [[direct sum]] decomposition

$$
  V = V_{even} \oplus V_{odd}
$$

and the [[tensor product]] is given in terms of that on vector spaces by

$$
  V \otimes W =
  (V_{even} \otimes W_{even} \oplus V_{odd}\otimes W_{odd})
  \oplus
  (V_{even} \otimes W_{odd} \oplus V_{odd} \otimes W_{even})
$$

but equipped with the non-trivial braiding morphism

$$
  b_{V, W} : V \otimes W \to W \otimes V
$$

that is the usual braiding isomorphism of [[Vect]] on $V_{even} \otimes W_{even}$ and on $V_{even} \otimes W_{odd} \oplus V_{odd} \otimes W_{even}$ but is $(-1)$ times this on $V_{odd}\otimes W_{odd}$.

=--

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

A **super (associative) algebra** over $K$ is a [[monoid]] in the [[symmetric monoidal category]] [[SVect]] of [[super vector space]]s.

A **[[graded commutative algebra|(graded)-commutative (associative) algebra]]** over $K$ is a [[commutative monoid in a symmetric monoidal category | commutative monoid]] in the [[symmetric monoidal category]] [[SVect]] of [[super vector space]]s.

=--

This means that a commutative superalgebra is a [[super vector space]]

$$
  A = A_{even} \oplus A_{odd}
$$

equipped with a morphism of super vector spaces

$$
  (-)\cdot (-) : A \otimes A \to A
$$

that is associative and commutative in the usual sense. Specifically for the commutativity this means that with $a,b  \in A_{odd}$ we have

$$
  a \cdot b = - b \cdot a
  \,.
$$

Whereas if either of $a$ or $b$ is in $A_{even}$ we have

$$
  a \cdot b = b \cdot a
  \,.
$$

#### Related notions
 {#RelatedNotions}

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

The **[[center]]** of a superalgebra $A$ is the sub-superalgebra $Z(A) \hookrightarrow A$ spanned by all those elements $z \in A$ of homogeneous degree which graded-commute with all other homogeneois elements $a$.

=--

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

For $A$ a superalgebra, its **opposite** $A^{op}$ is the superalgebra with the same underlying [[super vector space]] as $A$, and with multiplication defined on homogeneous elements by

$$
  a_1 \cdot_{A^{op}} a_2 \coloneqq (-1)^{{\vert a_1\vert}{\vert a_2\vert}} a_2 \cdot_{A} a_1
  \,.
$$

=--

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

A superalgebra $A$ is called **central simple** if

1. its [[center]], def. \ref{Center} is the ground field;

1. its only 2-sided graded [[ideals]] are $0$ and $A$ itself.

=--

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

Write $2sVect \simeq sAlg$ for the [[2-category]] equivalent to the one whose [[objects]] are superalgebra, [[1-morphisms]] are [[bimodules]] and [[2-morphisms]] are intertwiners. This is naturally a [[monoidal 2-category]].

=--

+-- {: .num_defn #2sVect}
###### Remark

By the discussion at _[[2-vector space]]_ this is equivalently the 2-category of **super 2-vector spaces**. [[equivalence|Equivalence]] in $2sVect \simeq sAlg$ is also called _[[Morita equivalence]]_ of super-algebras.

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

A superalgebra is an [[Azumaya algebra]] if it is an invertible object in the [[monoidal 2-category]] $s2Vect \simeq sAlg$, def. \ref{AlgWithBimodules}.

=--

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

The group of [[equivalence classes]] of Azumaya super algebras is called the super _[[Brauer group]]_, see there for more details.

=--

### Examples

#### Endomorphisms algebras, matrix algebras

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

For $V \in SVect$ a [[super vector space]], its [[endomorphism ring]] is canonically a super-algebra. Superalgebras isomorphic to ones of this form, are also called **matrix super algebras**.

=--

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

A matrix superalgebra, def. \ref{EndomorphismSuperalgebra} is central simple, def. \ref{CentralSimple}.

=--

#### Grassmann algebra

* For $V$ a [[vector space]] or [[graded vector space]] the [[Grassmann algebra]] $\wedge^\bullet V$ is a super algebra. This are the _free_ superalgebras.

#### Clifford algebra

An class of examples of non-(graded)-commutative superalgebra
are [[Clifford algebra]].

In fact, let $V$ be a [[vector space]] equipped with symmetric [[inner product]]
$\langle -,- \rangle$. Write $\wedge^\bullet V$ be the [[Grassmann algebra]] on $V$. The inner product makes this a super [[Poisson algebra]]. The [[Clifford algebra]] $Cl(V, \langle -,- \rangle)$ is the [[deformation quantization]] of this.

+-- {: .num_example #ComplexCl1}
###### Example

There is a superalgebra over the [[complex numbers]] of the form

$$
  A = \mathbb{C} \oplus \mathbb{C}\langle u\rangle
  \,,
$$

where the single odd generator satisfies $u \cdot u = 1$.

=--

### Properties
 {#Properties}

#### Relation to ordinary commutative algebras
 {#RelationToOrdinaryCommutativeAlgebra}

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

Given some [[ground field]] $k$, write

$$
  \iota \colon CAlg_k \hookrightarrow SCAlg_k
$$

for the [[full subcategory]] of ordinary [[commutative algebras]]
over $k$ into [[supercommutative superalgebras]] (as those having trivial odd part).

=--

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

The inclusion $\iota$ of def. \ref{InclusionOfCommutativeSuperalgebras}
has

1. a [[right adjoint]] $(-)_{even}$ given by restricting a superalgebra
to its even part;

1. a [[left adjoint]] $(-)/((-)_{odd})$ given by forming the "[[body]]", the [[quotient]] by the ideal generated by the odd part (by the "[[soul]]").

=--

This is immediate, but conceptually important, it is made explicit for instance in ([Carchedi-Roytenberg 12, example 3.18](#CarchediRoytenberg12)).

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

Prop. \ref{AdjointsToInclusionOfPlainAlgebra}
gives an [[adjoint triple]] of the form

$$
  CAlg_k \stackrel{\overset{(-)/((-)_{odd})}{\longleftarrow}}{\stackrel{\hookrightarrow}{\underset{(-)_{even}}{\longleftarrow}}}
  SCAlg_k
$$

and hence an [[adjoint cylinder]], which induces a pair of [[adjoint modalities]] ([[fermionic modality]] $\dashv$ [[bosonic modality]]).
See at _[[super smooth infinity-groupoid]]_ for more on this.

=--

#### Relation to matrix algebras

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

A superalgebra is isomorphic to a matrix algebra, def. \ref{EndomorphismSuperalgebra}, precisely if it is [[equivalence|equivalent]]  in  $2 sVect \simeq Alg$, def. \ref{AlgWithBimodules}, ([[Morita equivalence|Morita equivalent]]) to the ground field super algebra.

=--

#### Picard 3-group, Brauer group
 {#Picard2Groupoid}

We discuss the [[Picard 3-group]] of $2sVect \simeq sAlg$, def. \ref{AlgWithBimodules}, hence the corresponding [[Brauer group]]. See also at _[[super line 2-bundle]]_.

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

A superalgebra is invertible/Azumaya, def. \ref{Azumaya}, precisely if it is finite dimensional and central simple, def. \ref{CentralSimple}.

=--

This is due to ([Wall](#Wall)).

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

The [[Brauer group]] of superalgebras over the [[complex numbers]] is the [[cyclic group of order 2]]. That over the [[real numbers]] is cyclic of order 8:

$$
  sBr(\mathbb{C}) \simeq \mathbb{Z}_2
$$
$$
  sBr(\mathbb{R}) \simeq \mathbb{Z}_8
  \,.
$$

The non-trivial element in $sBr(\mathbb{R})$ is that presented by the superalgebra $\mathbb{C} \oplus \mathbb{C} u$ of example \ref{ComplexCl1}, with $u \cdot u = 1$.

=--

This is due to ([Wall](#Wall)).

The following generalizes this to the higher [[homotopy groups]].

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

The [[homotopy groups]] of the [[braided 3-group]] $sAlg^\times$ of Azumaya superalgebra are

| | $sAlg^\times_{\mathbb{C}}$ | $sAlg^\times_{\mathbb{R}}$ |
|--|--|--|
| $\pi_2$ | $\mathbb{C}^\times$ | $\mathbb{R}^\times$
| $\pi_1$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$
| $\pi_0$ | $\mathbb{Z}_2$ | $\mathbb{Z}_8$

where the [[groups of units]] $\mathbb{C}^\times$ and $\mathbb{R}^\times$ are regarded as [[discrete groups]].

=--

This appears in ([Freed, (1.38)](#Freed)).

## Algebra in the topos over superpoints
 {#AlgebraOverSuperpoints}

We now consider [[higher algebra]] in the [[(∞,1)-topos]] over [[super points]]: the [[cohesive (∞,1)-topos]] $\mathbf{H} = $ [[Super∞Grpd]].

### The topos

+-- {: .num_defn}

###### Definition

Write $SuperPoint$ for the [[site]] of [[superpoint]]s. Write

$$
  SuperSet := Sh(SuperPoint)
$$

for the [[sheaf topos]] (a [[presheaf topos]]) over this site.
Write

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

for the [[(∞,1)-sheaf (∞,1)-topos]] over this site: the [[(∞,1)-topos]] of [[super ∞-groupoid]]s.

=--

### The line object $\mathbb{R}$
 {#AlgebraInToposOverSuperPoints-TheLineObject}

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

Write

$$
  j \colon SuperSmthMfd \hookrightarrow Sh(SuperPoint)
$$

for the restricted [[Yoneda embedding]] of [[supermanifolds]] given by the canonical inclusion $SuperPoint \hookrightarrow SuperSmoothManifold$.

=--

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

Write

$$
  \mathbb{R} := j(\mathbb{R}) \in Sh(SuperPoint)
$$

for the presheaf represented by the [[real line]], regarded as a [[supermanifold]]. Equipped with its canonical [[internalization|internal]] [[ring]] structure this is

$$
  \mathbb{R} \in Ring(Sh(SuperPoint))
  \,.
$$

=--

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

By the discussion at [[supermanifold]] (in the section [As locally ringed spaces - Properties](http://ncatlab.org/nlab/show/supermanifold#AsLocallyRingedSpacesProperties)) $\mathbb{R}$ sends the formal dual of a [[Grassmann algebra]] to its even subalgebra

$$
  \mathbb{R}  : Spec \wedge^\bullet V \mapsto (\wedge^\bullet V)_{even}
  \,.
$$

This is canonically equipped with the structure of a (unital) [[commutative ring]] in $SuperSet$.

=--

In ([Sachse](#Sachse)) this appears around (21).

### $\mathbb{R}$-Modules
 {#SuperModulesAsbbKModules}

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

Write $Mod_{\mathbb{R}}(SuperSet)$ for the [[Mod|category of modules]] over $\mathbb{R}$ of def. \ref{TheRealLine} in $SuperSet$.

=--

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

The restriction of the embedding of def. \ref{EmbeddingOfSupermanifoldsIntoSheavesOnSuperpoints} to supermanifolds which are [[super vector spaces]] is a [[functor]]

$$
  j : SVect_{\mathbb{R}} = Mod_{\mathbb{R}}(Set) \hookrightarrow Mod_{\mathbb{K}}(SuperSet)
$$

from real [[super vector spaces]]  to internal modules over $\mathbb{R}$ that sends $V \in SVect_{\mathbb{R}}$ to

$$
  j(V) : Spec \Lambda \mapsto (\Lambda \otimes_\mathbb{R} V)_{even}
  =
  (\Lambda_{even} \otimes_\mathbb{R} V_{even})
    \oplus
  (\Lambda_{odd} \otimes_\mathbb{R} V_{odd})
  \,.
$$

This is a [[full and faithful functor]].

=--

This appears as ([Sachse, corollary 3.2](#Sachse)).

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

The proof is a variation on the [[Yoneda lemma]].

=--

This means that ordinary [[super vector spaces]] are embedded as a [[full subcategory]] in $\mathbb{K}$-modules in the topos over [[super points]].

### Associative and Lie Superalgebras

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

The functor $j$ from prop \ref{EmbeddingOfSVectIntoKMod} induces a [[full and faithful functor]]

$$
  SAlg_{\mathbb{R}}(Set) \hookrightarrow Alg_{\mathbb{R}}(SuperSet)
$$

of superalgebras over $\mathbb{R}$ as in def. \ref{SuperAlgebras} and internal [[associative algebra]]s over $\mathbb{R}$ in $SuperSet$.

Similarly we have a faithful embedding

$$
  SLieAlg_{\mathbb{R}}(Set) \hookrightarrow LieAlg_{\mathbb{R}}(SuperSet)
$$

of ordinary [[super Lie algebras]] over $\mathbb{R}$ into the internal [[Lie algebras]] over $\mathbb{R}$ in $SuperSet$.

=--

This appears as ([Sachse, corollary 3.3](#Sachse)).

## Properties

(...)

## Related concepts

* [[super module]], [[super vector space]]

* [[supercommutative algebra]], [[differential graded-commutative superalgebra]]

  [[model structure on differential graded-commutative superalgebras]]

* [[smooth superalgebra]]

* [[supermanifold]]

* [[super 2-algebra]]

## References

### Definition

The concept of [[Grassmann algebra]] and the super-sign rule is due to

* {#Grassmann1844} [[Hermann Grassmann]], _[[Ausdehnungslehre]]_, 1844

Review of basic superalgebra includes

* [[Yuri Manin]], chapter 3 of _[[Gauge Field Theory and Complex Geometry]]_, Grundlehren der Mathematischen Wissenschaften 289, Springer 1988

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

Discussion of superalgebra as algebra in certain [[symmetric monoidal category|symmetric monoidal]] [[tensor categories]] is in

* [[Pierre Deligne]], _Catégorie Tensorielle_ ([pdf](https://www.math.ias.edu/files/deligne/Tensorielles.pdf))

(see also at [[Deligne's theorem on tensor categories]]).

Lecture notes include

* [[Daniel Freed]], _[[Five lectures on supersymmetry]]_

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

* {#KonechnySchwarz} Anatoly Konechny and [[Albert Schwarz]],

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

  _Theory of $(k \oplus l|q)$-dimensional supermanifolds_ Sel. math., New ser. 6 (2000) 471 - 486

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

For more along these lines see also the references at _[[supermanifold]]_ and at _[[super infinity-groupoid]]_.

Discussion in terms of [[smooth algebras]] ([[synthetic differential supergeometry]]) is in

* {#CarchediRoytenberg12} [[David Carchedi]], [[Dmitry Roytenberg]], _On theories of superalgebras of differentiable functions_, Theory and Applications of Categories, Vol. 28, 2013, No. 30, pp 1022-1098. ([arxiv:1211.6134](http://arxiv.org/abs/1211.6134), [TAC](http://www.tac.mta.ca/tac/volumes/28/30/28-30abs.html))

### Brauer groups and Picard 2-groupoid

[[Brauer groups]] of superalgebras are discussed in

* [[Daniel Freed]], Lectures on twisted K-theory and orientifolds ([pdf](http://www.ma.utexas.edu/users/dafr/ESI.pdf))
 {#Freed}

* [[C. T. C. Wall]], _Graded Brauer groups_, J. Reine Angew. Math. 213 (1963/1964), 187-199.
 {#Wall}

* [[Pierre Deligne]], _Notes on spinors_ in _[[Quantum Fields and Strings]]_

* [[Peter Donovan]], [[Max Karoubi]], _Graded Brauer groups and K-theory with local coefficients_, Publications Math. IHES 38 (1970), 5-25 ([pdf](http://www.math.jussieu.fr/~karoubi/Donavan.K.pdf))

See also at _[[super line 2-bundle]]_ for more on this.

Discussion of superalgebra as induced from free groupal symmetric monoidal categories ([[abelian 2-groups]]) and hence ultimately from the [[sphere spectrum]] is in

* {#Kapranov13} [[Mikhail Kapranov]], _Categorification of supersymmetry and stable homotopy groups of spheres_, talk at _[Algebra, Combinatorics and Representation Theory: in memory of Andrei Zelevinsky (1953-2013)](https://web.archive.org/web/20130617191515/http://mathserver.neu.edu/~bwebster/ACRT/)_ (April 2013) [[[Webster-ACRTAbstracts2013.pdf:file]],  video:[YT](https://www.youtube.com/watch?v=WBsZrcYOrxI)]

  > **Abstract:**. The "minimal sign skeleton" necessary to formulate the [[signs in supergeometry|Koszul sign rule]] is a certain [[Picard category]], a [[symmetric monoidal category]] with all [[invertible object|objects]] and [[invertible morphism|morphisms invertible]]. It can be seen as the free Picard category generated by one object and corresponds, by [[Grothendieck]]'s dictionary, to the [[n-truncation|truncation]] of the [[sphere spectrum|spherical spectrum]] $S$ in degrees 0 and 1, so that $\{\pm 1\}$ appears as the [[first stable homotopy group of spheres]] $\pi_{n+1}(S^n)$. This suggest a "[[higher structures|higher]]" or [[categorification|categorified]] versions of [[super-algebra|super-mathematics]] which utilize deeper structure of [[sphere spectrum|$S$]]. The first concept on this path is that of a supersymmetric monoidal category which is [[categorification|categorified]] version of the concept of a [[supercommutative algebra]].

  > (cf. *[[super 2-algebra]]* and *[[spectral super-scheme]]*)

* {#Kapranov15} [[Mikhail Kapranov]], _Supergeometry in mathematics and physics_, in [[Gabriel Catren]], [[Mathieu Anel]], (eds.) _[[New Spaces for Mathematics and Physics]]_ ([arXiv:1512.07042](http://arxiv.org/abs/1512.07042))

* {#Kapranov15b} [[Mikhail Kapranov]], _Super-geometry_, talk at _[New Spaces for Mathematics & Physics](http://ercpqg-espace.sciencesconf.org/program)_, IHP Paris (Oct-Sept 2015) [[video recording](https://www.youtube.com/watch?v=bjsNwKYT8JE)]

[[!redirects super algebras]]

[[!redirects superalgebra]]
[[!redirects super-algebra]]

[[!redirects superalgebras]]

[[!redirects even rules]]

[[!redirects commutative superalgebra]]
[[!redirects commutative superalgebras]]
[[!redirects commutative super-algebra]]
[[!redirects commutative super-algebras]]
[[!redirects super-commutative superalgebra]]
[[!redirects super-commutative superalgebras]]
[[!redirects super-commutative super-algebra]]
[[!redirects super-commutative super-algebras]]


super anti de Sitter spacetime

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

#Contents#
* table of contents
{:toc}

## Idea

A [[supergeometry|supergeometric]] analog of [[anti-de Sitter spacetime]]. By the discussion at _[supersymmetry -- Classification -- Superconformal and super anti de Sitter symmetry](supersymmetry#ClassificationSuperconformal)_ this includes the following [[coset]] [[superspacetimes]] ([[super Klein geometries]]):

| $\phantom{A}$$d$$\phantom{A}$ | $\phantom{A}$[[super anti de Sitter spacetime]]$\phantom{A}$ |
|-----|---|
| $\phantom{A}$4$\phantom{A}$ | $\;\;\;\;\frac{OSp(8\vert4)}{Spin(3,1) \times SO(7)}\;\;\;\;$ |
| $\phantom{A}$5$\phantom{A}$ | $\;\;\;\;\frac{SU(2,2 \vert 5)}{Spin(4,1)\times SO(5)}\;\;\;\;$ |
| $\phantom{A}$7$\phantom{A}$ | $\;\;\;\;\frac{OSp(6,2 \vert 4)}{Spin(6,1) \times SO(4)}\;\;\;\;$ |

(Here $Osp(-\vert-)$ denotes [[orthosymplectic super Lie groups]].)

Notice that the [[de Sitter spacetime]] (not anti-de Sitter) does not have a standard extension to [[supergeometry]], but see [arXiv:1610.01566](https://arxiv.org/abs/1610.01566).

## Related concepts

* [[super anti de Sitter group]]

* [[super Minkowski spacetime]]

* [[AdS/CFT correspondence]], [[AdS/QCD correspondence]]

## References

General discussion includes

* {#CastellaniDAuriaFre91} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], sections II.2.6, II.3.2 and II.3.3 in _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

* [[Antoine Van Proeyen]], sections 4.5, 4.7 of _Tools for supersymmetry_ ([arXiv:hep-th/9910030](https://arxiv.org/abs/hep-th/9910030))

The super 3-cocycle for the [[Green-Schwarz superstring]] on $\frac{SU(2,2 \vert 5)}{Spin(4,1)\times SO(5)}$ is originally due to

* {#MetsaevTseytlin98} [[Ruslan Metsaev]], [[Arkady Tseytlin]], _Type IIB superstring action in $AdS_5 \times S^5$ background_, Nucl. Phys.B533:109-126, 1998 ([arXiv:hep-th/9805028](https://arxiv.org/abs/hep-th/9805028))

However, a supersymmetric trivialization of this cocycle seems to have been obtained in

* {#RoibanSiegel00} Radu Roiban, [[Warren Siegel]], _Superstrings on $AdS_5 \times S^5$ supertwistor space_, JHEP 0011:024, 2000 ([arXiv:hep-th/0010104](https://arxiv.org/abs/hep-th/0010104))

(according to [arxiv:1808.04470, p. 5 and equation (5.5)](https://arxiv.org/abs/1808.04470), but check).

See also the references at _[Green-Schwarz sigma-model -- References -- AdS backgrounds](Green-Schwarz+action+functional#ReferencesAdSBackgrounds)_

[[!redirects super anti de Sitter spacetimes]]

[[!redirects super anti-de Sitter spacetime]]
[[!redirects super anti-de Sitter spacetimes]]

[[!redirects super anti de Sitter super Lie algebra]]
[[!redirects super anti de Sitter super Lie algebras]]