superconvex space

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### Context
#### Analysis
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# Contents
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## Idea

The notion of "superconvex spaces" generalizes the idea of *[[convex spaces]]* by replacing [[finite set|finite]] affine [[sums]] with [[countable set|countable]] affine sums.

Let $\mathcal{G}(\mathbb{N})$ denote the set of all [[probability measures]] on the set of [[natural numbers]], hence every $\mathbf{p}$ can be represented as $\mathbf{p} = \sum_{i \in \mathbb{N}} p_i \delta_i$ where $\sum_{i \in \mathbb{N}} p_i=1$ with each $p_i \in [0,1]$.  Here $\mathcal{G}$ is the [[Giry monad]], but because we want to forget the $\sigma$-algebra associated with that measurable space we often write $\Delta_{\mathbb{N}}$ to denote the underlying set  of $\mathcal{G}(\mathbb{N})$. That set is the [[countable set|countably]] [[infinite]]-[[dimension|dimensional]] [[simplex]]. The set $\mathcal{G}(\mathbb{N})$ (= $\Delta_{\mathbb{N}}$) is the prototype space from which the axioms are abstracted.

## Definition

 Given any set $A$, a sequence $\mathbf{a}: \mathbb{N} \rightarrow A$, and any $\mathbf{p} \in \mathcal{G}(\mathbb{N})$, we refer to the formal expression $\sum_{i \in \mathbb{N}} p_i  a_i$  as a countably affine sum of elements of $A$, and for brevity we use the notation $\sum_{i\in \mathbb{N}} p_i a_i$  to refer to a countably affine sum dropping the explicit reference to the condition that the limit of partial sums $\sum_{i=0}^N p_i$ converges to one.  An alternative notation to the countable affine sum notation is to use the integral notation
 \begin{equation}
  \int_{\mathbb{N}} \mathbf{a} \, d\mathbf{p} := \sum_{i \in \mathbb{N}} p_i a_i.
 \end{equation}

We say a set $A$ has the structure of a superconvex space  if it comes equipped with a function
\begin{equation}
\begin{array}{ccccc}
st_A&:& \mathcal{G}{(\mathbb{N})} \times \Set(\mathbb{N}, A) & \rightarrow& A \\
&:& (\mathbf{p}, \mathbf{a}) & \mapsto & \int_{\mathbb{N}} \mathbf{a} \, d\mathbf{p}
\end{array}
\end{equation}
such that the following two axioms are satisfied:

Axiom 1. For every sequence $\mathbf{a}: \mathbb{N} \rightarrow A$ and every  $j \in \mathbb{N}$ the property
\begin{equation}
\int_{\mathbb{N}} \mathbf{a} \, \, d\delta_j = a_j
\end{equation}
holds.

Axiom 2. If $\mathbf{p} \in \mathcal{G}(\mathbb{N})$ and $\mathbf{Q}: \mathbb{N} \rightarrow \mathcal{G}(\mathbb{N})$ is a sequence of probability measures on $(\mathbb{N}, \mathcal{P}(\mathbb{N}))$   then
\begin{equation}
\int_{j \in \mathbb{N}} \big( \int_{\mathbb{N}} \mathbf{a} \, d\mathbf{Q}^j \big) \, d\mathbf{p} = \int_{\mathbb{N}} \mathbf{a} \,  d(  \int_{j \in \mathbb{N}} \mathbf{Q}^{\bullet} \, d\mathbf{p}  \big).
\end{equation}

The second axiom uses  the pushforward measure $\mathcal{G}(\mathbf{Q})\mathbf{p} \in \mathcal{G}^2{\mathbb{N}}$ and the natural transformation $\mu$ of the Giry monad at component $\mathbb{N}$, $\mu_{\mathbb{N}}: \mathcal{G}^2(\mathbb{N}) \rightarrow \mathcal{G}(\mathbb{N})$, which yields the probability measure on  the measurable space $(\mathbb{N}, \mathcal{P}{\mathbb{N}})$  whose value at the measurable set $\{j\}$ is given by the composite of measurable maps
\begin{centre}
\begin{tikzpicture}
      \node   (1)  at  (0,0)  {$1$};
      \node   (GN) at  (-3,1.5)  {$\mathcal{G}{\mathbb{N}}$};
      \node   (G2N) at  (0, 1.5)  {$\mathcal{G}^2{\mathbb{N}}$};
      \node   (GN2)  at   (2.5,1.5)  {$\mathcal{G}{\mathbb{N}}$};
      \node    (01)   at   (5,1.5)   {$[0,1]$};
      \draw[->,above] (GN2) to node {$ev_{\{j\}}$} (01);
      \draw[->,above] (G2N) to node {$\mu_{\mathbb{N}}$} (GN2);
      \draw[->,left] (1) to node [xshift=-2pt,yshift=-2pt]{$\mathbf{p}$} (GN);
      \draw[->,above] (GN) to node {$\mathcal{G}{\mathbf{Q}}$} (G2N);
      \draw[->,left] (1) to node [xshift=-2pt,yshift=3pt]{\small{$ \int_{j \in \mathbb{N}} \mathbf{Q}^{j} \, d \mathbf{p}$}} (GN2);
      \draw[->,right] (1) to node [xshift=0pt,yshift=-17pt]{$\begin{array}{lcl}
\mu_{\mathbb{N}}\big( \mathcal{G}(\mathbf{Q})\mathbf{p}\big)(\{j\}) &=& \int_{\{j\} }d\big(\mathcal{G}(\mathbf{Q})\mathbf{p}\big)  \\
&=& \int_{ \mathbb{N}} \mathbf{Q}^j \, d\mathbf{p} \\
&=& \sum_{i \in \mathbb{N}} p_i Q^j_i
\end{array}$} (01);
\end{tikzpicture}
\end{centre}
Note the second axiom alone is sufficient because by choosing the constant sequence $\mathbf{Q}: \mathbb{N} \rightarrow \mathcal{G}\mathbb{N}$ with value $\delta_j$ it follows that the second axiom implies the first axiom.

A morphism of superconvex spaces, called a countably affine map, is a set function $m: A \rightarrow B$ such that
\begin{equation}
m\big( \int_{\mathbb{N}} \mathbf{a} \, d\mathbf{p}\big) = \int_{\mathbb{N}} (m \circ \mathbf{a}) \, d\mathbf{p},
\end{equation}
where the composite  $m \circ \mathbf{a}$ gives the sequence in $B$  with component $m(a_i)$.
Composition of countably affine maps is the set-theoretical composition.
Superconvex spaces with morphisms the countably affine maps form a category denoted $\mathbf{SCvx}$.

## Probability Amplitudes

In physics, superconvex spaces have been referred to as *strong convex spaces*, and since [[probability amplitudes]] are employed there one makes use of the $\ell_2$-norm instead of the tradition ''$\ell_1$-norm'' which is used above. By using
 \begin{equation}
\mathcal{G}(\mathbb{N}) = \{  \mathbf{p}: \mathbb{N} \rightarrow  \mathbf{D}_2  |  \lim_{N \rightarrow \infty} \{ \sum_{i=1}^N p_i p_i^{\star} \}= 1 \}
 \end{equation}
where $\mathbf{D}_2 = \{r e^{\imath \theta} \in \mathbb{C} \, | r \in [0,1],  and  \theta \in [0,2 \pi) \}$ and $p_i^{\star}$ is the complex conjugate of $p_i$,
applied to the above axioms one obtains superconvex spaces useful for physics.

## Properties
The most basic property of superconvex spaces,  is
\begin{lemma} For $A$  any superconvex space  every countably affine map $m \in \SCvx(\Delta_{\mathbb{N}}, A)$ is uniquely specified by a sequence in $A$, hence we have $\SCvx(\Delta_{\mathbb{N}}, A) \cong  \Set(\mathbb{N},A)$.
\end{lemma}
\begin{proof}
Every element $\mathbf{p} \in \Delta_{\mathbb{N}}$ has a unique representation as a countable affine sum $\mathbf{p} = \sum_{i \in \mathbb{N}} p_i \delta_i$, and hence a countably affine map $m:\Delta_{\mathbb{N}} \rightarrow A$ is uniquely determined by where it maps each Dirac measure $\delta_i$.  Thus  $i \mapsto m(\delta_i)$  specifies a sequence in $A$.
\end{proof}
We denote the bijective correspondence $\SCvx(\Delta_{\mathbb{N}}, A) \cong  \Set(\mathbb{N},A)$ by
 $\langle \mathbf{a} \rangle \leftrightarrow \mathbf{a}$, where $\mathbf{a}: \mathbb{N} \rightarrow A$ specifies a sequence in $A$.

\begin{lemma} A function $f: \mathbb{N} \rightarrow \mathbb{N}$ is a countably affine map if and only if $f$ is monotone, $i \lt j$ implies $f(i) \le f(j)$.
\end{lemma}
\begin{proof}
Necessary condition.  Suppose that $f: \mathbb{N} \rightarrow \mathbb{N}$ is a countably affine map.  Let $i \lt j$. By the superconvex space structure on $\mathbb{N}$ it follows, for all $\alpha \in (0,1)$, that  $\alpha i + (1-\alpha) j = i$ .  If $f$ is not monotone then there exist a pair of elements $i,j \in \mathbb{N}$ such that $i \lt j$ with $f(j) \lt f(i)$.  This implies, for all $\alpha \in (0,1)$, that $f(j)= \alpha f(i) + (1-\alpha) f(j) \lt   f(\alpha i + (1-\alpha) j ) =f(i)$, which contradicts our hypothesis that $f$ is a countably affine map.

Sufficient condition. Suppose $f$ is a monotone function, and
 that we are given an arbitrary countably affine sum $\sum_{i \in \mathbb{N}} p_i i = n$ in $\mathbb{N}$, so that for all $i=0,1,\ldots,n-1$ we have $p_i=0$.  Since the condition defining the superconvex structure is conditioned on the property $p_i \ne 0$, the countably affine sum is not changed  by removing any number of terms $i$ in the countable sum whose coefficient $p_i=0$.  Hence
for all $j$ such that $n\lt j$ it follows that $f(n) \le f(j)$ so that
\begin{equation}
f( \sum_{i=0}^{\infty} p_i \, i) = f(n) = \sum_{i=n}^{\infty} p_i f(i)
\end{equation}
where the last equality follows from the definition of the superconvex space structure on $\mathbb{N}$.
\end{proof}

The category $\mathbf{SCvx}$ has all limits and colimits.  Furthermore it is a symmetric monoidal closed category under the tensor product. The proof of the latter condition follows the proof by Meng, replacing finite sums with countable sums.

The full subcategory consisting of the single object $\Delta_{\mathbb{N}}$ is dense in $\mathbf{SCvx}$, and hence we can employ the restricted Yoneda embedding to view superconvex spaces as the functors $\widehat{A}=\mathbf{SCvx}(\cdot,A) \in \mathbf{Set}^{\Delta_{\mathbb{N}}^{op}}$, where $\Delta_{\mathbb{N}}$ is viewed as a [[monoid]].

  An ideal in a superconvex space $A$ is a subset $\mathcal{I}$ such that whenever $a_0 \in \mathcal{I}$ and $\sum_{i \in \mathbb{N}}p_i a_i$ is a countable affine sum with the coefficient of $a_0$ nonzero then $\sum_{i \in \mathbb{N}}p_i a_i \in \mathcal{I}$.   Ideals are useful for defining functors to or from $\mathbf{SCvx}$.

## Examples
\begin{example}
A fundamental superconvex space is the set $\mathbb{N}$ with the superconvex space structure defined, for every sequence $\mathbf{s}: \mathbb{N} \rightarrow \mathbb{N}$  by
\begin{equation}
\int_{\mathbb{N}} \mathbf{s} d\mathbf{p} = \inf_i \{\mathbf{s}_i \, | \, p_i \gt 0 \}
\end{equation}
That structure on $\mathbb{N}$ shows that the function $\epsilon: \mathcal{G}(\mathbb{N}) \rightarrow \mathbb{N}$ defined by $\sum_{i \in \mathbb{N}} p_i \delta_i \mapsto \inf_i \{i | p_i\gt 0\}$ is a countably affine map.
\end{example}

\begin{example}
As an example of the utility of ideals in a superconvex space we note that for $\mathcal{G}$ the [[Giry monad]] the measurable space $\mathcal{G}(X)$, with the smallest $\sigma$-algebra such that the evaluation maps $ev_U: \mathcal{G}(X) \rightarrow [0,1]$ are measurable for every measurable set $U$ in $X$, has a superconvex space structure
defined on it pointwise.

Note that if  $X$ is any [[measurable space]]  the maximal proper ideals of $\mathcal{G}{X}$ are of the form $ev_{U}^{-1}( (0,1])$ or $ev_U^{-1}([0,1))$ for $U\ne X$ a nonempty measurable set in $X$.

To prove this, note that  both $(0,1]$ and $[0,1)$ are ideals in the superconvex space $[0,1]$, with the natural superconvex space structure, and it follows that $ev_{U}^{-1}( [0,1))$ and $ev_U^{-1}((0,1])$ are ideals in $\mathcal{G}{X}$. (The preimage of an ideal under a countably affine map is an ideal in the domain space. The proof is the standard argument for ideals in any category.)  Consider the ideal $ev_U^{-1}( [0,1))$.  To show that this is a maximal proper ideal suppose that $\mathcal{I}$ is another ideal of $\mathcal{G}{X}$ such that $ev_U^{-1}( [0,1)) \varsubsetneqq  \mathcal{I}$.  Every element $P \in \mathcal{I}$ which is not in $ev_U^{-1}([0,1))$ has the defining  property that $P(U^c)=1$.    Now let $Q \in \mathcal{G}{X}$.  If  $Q \notin \mathcal{J}$ and $Q \notin ev_U^{-1}([0,1))$ then $Q(U^c) \ne 1$ which implies  $Q \in ev_U^{-1}( [0,1)) \varsubsetneqq J$ which is self-contradictory.  Thus $\mathcal{J}$ must be all of $\mathcal{G}{X}$ which shows $ev_U^{-1}([0,1))$ is a maximal (proper) ideal.
The argument that the ideal $ev_U^{-1}( (0,1])$ is a maximal ideal is similiar except we replace the condition $P(U^c)=1$ in the above proof with $P(U)=0$.

If we restrict to the category of standard measurable spaces then every object has a countable generating basis and it is then clear that every ideal in $\mathcal{G}(X)$ is a  countable intersection of maximal ideals, and hence measurable.  This implies, for example, that every countably affine map $k: \mathcal{G}(X) \rightarrow \mathbb{N}$ is also a  measurable function with $\mathbb{N}$ having the powerset $\sigma$-algebra. (The only ideals of $\mathbb{N}$ are the principal ideals $\downarrow \! 0 \subset \downarrow \! 1 \subset \ldots$.)
\end{example}

\begin{example}
The one point compactification of the real line $\mathbb{R}_{\infty}$, with one point adjoined, denoted $\infty$, which satisfies the property that any countably affine sum $\sum_{i \in \mathbb{N}} p_i r_i = \infty$ if either (1) $r_j = \infty$ and $p_j \gt 0$ for any index $j$, or (2) the sequence of partial sums does not converge, is a superconvex space.  The real line $\mathbb{R}$ is not a super convexspace since we could take $p_i = \frac{1}{2^i}$ and $r_i = 2^{i+1}$ and the limit of the sequence does not exist in $\mathbb{R}$.  Thus while $\mathbb{R}$ is a convex space it is not a superconvex space.

 The only nonconstant countably affine map $j: \mathbb{R}_{\infty} \rightarrow \mathbb{2}$ is given by $j(u)=1$ for all $u \in \mathbb{R}$ and $j(\infty)=0$ (for the superconvex space structure on $\mathbf{2}$ determined by $\frac{1}{2} \underline{0} + \frac{1}{2} \underline{1} = \underline{0}$).
\end{example}

\begin{example} A pathological space, useful for counterexamples, is given by the closed unit interval with the superconvex space structure defined by the infimum function, $\sum_{i \in \mathbb{N}} p_i u_i := inf_i \{ u_i | p_i \gt 0\}$.
\end{example}

\begin{example}  Consider the probability monad on compact Hausdorff spaces, where the algebras are precisely the compact convex sets $K$ in locally convex topological vector spaces together with the barycenter maps $\beta_K:\mathcal{P}K\rightarrow K$.

Given such a space $K$ we can endow it with a superconvex space structure by defining, for all $\mathbf{p} \in \mathcal{G}\mathbb{N}$, countable affine sums by  $\sum_{i \in \mathbb{N}} p_i k_i := \beta_K( \sum_{i \in \mathbb{N}} p_i \delta_{k_i})$ which, along with the pointwise superconvex space structure on $\mathcal{P}K$ makes the barycenter map $\beta_K$ a countably affine map.

To prove this endows $K$ with a superconvex space structure note that $\beta_K(\delta_{k_i})= k_i$ for all $k_i \in K$ to obtain
$$
\sum_{i \in \mathbb{N}} p_i \beta_K(\delta_{k_i}) = \sum_{i \in \mathbb{N}} p_i k_i := \beta_K(\sum_{i \in \mathbb{N}} p_i \delta_{k_i}).
$$
To prove $\beta_K$ is countably affine on $\mathcal{P}K$ use the property that $\beta_K \circ \mu_K = \beta_K \circ \mathcal{P}\beta_K$.

The method employed in this example is not restricted to locally convex compact Hausdorff spaces. It shows that the algebras of a probability monad are a superconvex space.
That implication is the motivation for the next example.
\end{example}

\begin{example}
 The standard *free space* construction can be applied to superconvex spaces to obtain  an adjoint pair $\mathcal{F}:\mathbf{Set} \leftrightarrows \mathbf{SCvx}: \mathcal{U}$
where $\mathcal{F}(A)$ consists of all formal countable affine sums, $\int_{\mathbb{N}} \mathbf{a} \, d\mathbf{p} := \sum_{i \in \mathbb{N}}p_i a_i$ for $\mathbf{p} \in \mathcal{G}{\mathbb{N}}$ and $\mathbf{a} \in \mathbf{Set}(\mathbb{N},A)$,
modulo the relations
\begin{equation}
\int_{j \in \mathbb{N}} \big( \int_{i \in \mathbb{N}} \mathbf{a} \, d\mathbf{Q}^i \big) \, d\mathbf{p} \cong  \int_{j \in \mathbb{N}} \mathbf{a} \, d\big( \int_{i \in \mathbb{N}} \mathbf{Q}^{\bullet} \,   d\mathbf{p} \big)  \quad \forall \, \mathbf{Q} \in \mathbf{Set}( \mathbb{N},  \mathcal{G}{\mathbb{N}})
\end{equation}
 so that the sole necessary axiom of a superconvex space is satisfied.

One important aspect of the free space adjunction is that it implies

*Lemma* Let $X$ be an arbitrary set and $\mathcal{F}X$ the free superconvex space on $X$.  Let the quotient map $\widehat{q}: \mathcal{F}{X} \rightarrow A$ be the coequalizer of  the parallel pair $\mathbf{1}_{\mathcal{F}{X}}, k: \mathcal{F}{X} \rightarrow \mathcal{F}{X}$.     Then the quotient of the free space is the free space of the quotient, $\mathcal{F}X / \widehat{q} \cong \mathcal{F}(X/q)$

\begin{centre}
\begin{tikzpicture}
    \node  (X)   at   (0,0)   {$X$};
    \node  (Xq)  at  (0, -1.5)  {$X/q$};
    \node  (UFX) at  (3,0)    {$\mathcal{U}(\mathcal{F}(X))$};
    \node  (UA)  at  (3, -1.5)  {$\mathcal{U}(A)$};
    \node  (c)   at  (.5, -2.5)   {in $\mathbf{Set}$};
    \draw[->, above] (X) to node {$\eta_X$} (UFX);
    \draw[->>,left] (X) to node [xshift=-3pt,yshift=-3pt]{$q$} (UA);
    \draw[->>,right] (UFX) to node {$\mathcal{U}(\widehat{q})$} (UA);

    \node  (FX)  at  (5.5,0)  {$\mathcal{F}{X}$};
    \node  (A)    at  (5.5, -1.5)  {$A \cong \mathcal{F}{X}/\widehat{q}$};
    \node   (d)   at   (5.5, -2.5)  {in $\mathbf{SCvx}$};
    \draw[->>,right] (FX) to node {$\widehat{q}$} (A);

    \draw[->>,left,dashed] (X) to node {$\pi$} (Xq);
    \draw[>->>,below,dashed] (Xq) to node {$\widetilde{q}$} (UA);

\end{tikzpicture}
\end{centre}
*Proof*
The quotient map $\widehat{q}:\mathcal{F}X \rightarrow A$  is, up to isomorphism, just $\widehat{q}:\mathcal{F}X \rightarrow \mathcal{F}X/\widehat{q}$.  Because the quotient map is countably affine it  is completely specified by where it sends the elements $1 x \in \mathcal{F}X$.  Because $q(x) = \widehat{q}(\eta_X(x)) = \widehat{q}(1x) = [1 x]_{\widehat{q}}=[k (1 x)]_{\widehat{q}}$ we have
\begin{equation}
\hat{q}(\sum_{i \in \mathbb{N}} p_i x_i)=\sum_{i \in \mathbb{N}} p_i
 [1 x_i]_{\hat{q}} = \sum_{i \in \mathbb{N}} p_i q(x_i).
\end{equation}
The function $q$ specifies an equivalence relation on $X$, yielding the set $X/q$. Because $q$ is surjective we can thus endow the set $X/q$ with the superconvex space structure of $A$ using the bijective map of sets $\widetilde{q}: X/q \rightarrow \mathcal{U}(A)$.  Hence the above equation extends (trivially) on the right  by  $\sum_{i \in \mathbb{N}} p_i q(x_i) = \sum_{i \in \mathbb{N}} p_i \, \widetilde{q}([x_i]_q) =  \widetilde{q}(\sum_{i \in \mathbb{N}} p_i [x_i]_q )$.
The map $\mathcal{F}X / \hat{q} \rightarrow \mathcal{F}( \mathcal{U}(A))$ specified by $\sum_{i \in \mathbb{N}}p_i [1 x_i]_{\widehat{q}} \mapsto \sum_{i \in \mathbb{N}}p_i \widetilde{q}( [x_i]_q )$ specifies  a $\mathbf{SCvx}$-isomorphism.  By the isomorphism $\widetilde{q}$ we can write this as $\mathcal{F}{X}/\widehat{q} \cong \mathcal{F}(X/q)$.  That completes the proof.

 Now let $X$ denote a standard measurable space, so $\mathcal{G}{X}$ is standard also. Forgetting the measurable structure on $\mathcal{G}{X}$ we can take the free space of that set, $\mathcal{F}( \mathcal{U}( \mathcal{G}{X}))$ which consist of all countable affine sums $\int_{\mathbb{N}} \mathbf{P} \, d\mathbf{p}$ for all $\mathbf{P} \in \mathbf{Set}(\mathbb{N}, \mathcal{U}(\mathcal{G}{X}))$ and all $\mathbf{p} \in \mathcal{G}{\mathbb{N}}$.  These countable affine sums are  defined pointwise $(\int_{\mathbb{N}} \mathbf{P} \, d\mathbf{p})U := \sum_{i \in \mathbb{N}} p_i P_i(U)$ for all $U \in \Sigma_X$.

 We observe that the free space $\mathcal{F}(\mathcal{U}(\mathcal{G}{X}))$ can also be viewed as the image of the functor $\mathcal{P}: \mathbf{Meas} \rightarrow \mathbf{SCvx}$ on $X$, where $\mathcal{P}$  is the Giry monad (functor) viewed as a functor into $\mathbf{SCvx}$.

 We would like to extend the free space adjunction $\mathcal{F} \dashv \mathcal{U}$ with an adjunction $\mathcal{P}: \mathbf{Std} \leftrightarrows \mathbf{SCvx}: \mathbf{\Sigma}$
such that the composite is the Giry monad on $\mathbf{Std}$. The category $\mathbf{SCvx}_{\star}$ denotes some subcategory of $\mathbf{SCvx}$ so that the adjunction can be constructed.  If such an adjunction exists the counit of the adjunction says every space $A$  is the quotient of a free space, i.e., $A$ is the coequalizer of a parallel pair of maps into a free space.    To illustrate this consider the following elementary example.

 The coequalizer of the pair of points $\frac{1}{3}: \mathbf{1} \rightarrow [0,1]$ and $\frac{2}{3}: \mathbf{1} \rightarrow [0,1]$, where $[0,1] \cong \mathbf{P}{\mathbf{2}}$.  The coequalizer of those pair of points is the discrete space  $A=\{0,u,1\}$ with the structure defined by $p u + (1-p) 0 = u$ for all $p \in (0,1]$, $p u + (1-p) 1 = u$ for all $p \in (0,1]$, and $p 0 + (1-p) 1 = u$ for all $p \in (0,1)$.

If the functor $\mathbf{\Sigma}: \mathbf{SCvx}_{\star} \rightarrow \mathbf{Std}$ exists  it should recognize that that superconvex space arises as a quotient space of $\mathcal{P}{\mathbf{2}}$, and hence should have the barycenter map $\epsilon_A: \mathcal{P}{\mathbf{2}} \rightarrow A$ given by
the mapping $\delta_0 \mapsto 0$, $\delta_1 \mapsto 1$, and $r \delta_0 + (1-r) \delta_1 \mapsto u$ for all $r \in (0,1)$.

Let $\mathbf{\Omega}$ denote the full subcategory of $\mathbf{SCvx}$ consisting of the two objects $\mathbb{N}$ and $\Delta_{\mathbb{N}}$, and let $\iota:\mathbf{\Omega} \rightarrow \mathbf{SCvx}$ be the inclusion functor.

If we look at the category $A \downarrow \iota$ there are two countably affine maps,
\begin{equation}
\begin{array}{lcccc}
\phi_{0,u} &:& A & \rightarrow & \mathbb{N} \\
&:& a & \mapsto & \left\{ \begin{array}{ll} 0 & for \, a \in \{0,u\} \\ 1 & for \, a=1 \end{array} \right.
\end{array}
\end{equation}
and
\begin{equation}
\begin{array}{lcccc}
\phi_{1,u} &:& A & \rightarrow & \mathbb{N} \\
&:& a & \mapsto & \left\{ \begin{array}{ll} 0 & for \, a \in \{1,u\} \\ 1 & for \, a=0 \end{array} \right.
\end{array},
\end{equation}
from which every other countably affine map  $A \rightarrow \mathbb{N}$ can be obtained by composing either $\phi_{0,u}$ or $\phi_{1,u}$
 with a monotonic function $\phi: \mathbb{N} \rightarrow \mathbb{N}$.  Because $A$ is discrete there are no non-constant countably affine maps $A \rightarrow \Delta_{\mathbb{N}}$.

 The coequalizer of those two maps yields the discrete space $\mathbf{2}$ for which $\mathcal{P}{\mathbf{2}}$ is the space we are trying to find - it is the smallest free space such that there is a countably affine map  $\mathcal{P}{\mathbf{2}} \rightarrow A$.

 Whether probing a space $A$ with arrows to objects and products of objects  in $\mathbf{\Omega}$  is sufficient is unknown but a clue is given in the fact that every standard space is $\mathbf{Std}$-isomorphic to either $[0,1]$ or a countable discrete space. Hence $\mathbf{SCvx}_{\star}$ should intuitively consists of quotients of $\prod_{i \in \mathbb{N}} \Delta_{\mathbb{N}}$ (and smaller) - the countability condition being the critical condition. That condition rules out spaces such as $\prod_{i \in [0,1]}\Delta_{\mathbb{N}}$ and the pathological space of Example 5.4.

\end{example}

## Related concepts

* [[totally convex space]]
* [[convex space]]

## References

The notion of superconvex spaces originates with:

* [[Dieter Pumplün]], *Banach spaces and superconvex modules*, in *Symposia Gaussiana -- Conference A: Mathematics and Theoretical Physics*, De Gruyter (1995) [[doi:10.1515/9783110886726.323](https://doi.org/10.1515/9783110886726.323)]

The proof that $\mathbf{SCvx}$ has no cogenerator is due to:

* {#BörgerKemper94a} [[Reinhard Börger]], [[Ralf Kemper]], *There is no cogenerator for total convex spaces*, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome **35** 4 (1994) 335-338 [[numdam:CTGDC_1994__35_4_335_0](http://www.numdam.org/item/CTGDC_1994__35_4_335_0)]

The fact that $\mathbf{SCvx}$ is a symmetric monoidal argument can be proven the same way it is for [[convex spaces]] simply by replacing the finite affine sums with countable affine sums. That proof was given by

* {#Meng} Xiao-qing Meng, _Categories of convex sets and of metric spaces with applications to stochastic programming and related areas_, PhD thesis ([[Meng.djvu|djvu:file]])

Proposition 1.2 in

* {#BörgerKemper94b} [[Reinhard Börger]], [[Ralf Kemper]], *Cogenerators for convex spaces*, Applied Categorical Structures **2** (1994) 1-11 [[doi:10.1007/BF00878499](https://doi.org/10.1007/BF00878499)]

is particularly useful for viewing superconvex spaces as positively convex spaces which are somewhat easier to work with because the condition $\sum_{i \in \mathbb{N}}p_i=1$  is replaced by the inequality $\le 1$.

The fact that  the functor $\mathbf{\Sigma}: \mathbf{\Omega} \rightarrow \mathbf{Std}_2$ is a codense functor can be found  in

* {#Sturtz22} [[Kirk Sturtz]], _Giry algebras for standard measurable spaces_ $[$[arXiv:2202:10819](https://arxiv.org/abs/2202.10819)$]$

although several aspects, such as the construction with the right-Kan is incorrect.

For purposes of constructing models of complex systems using superconvex spaces the construction given in Example 6.1 of the following article applies equally well to superconvex spaces.

* [[Tobias Fritz]], Convex spaces I: definition and examples.
[arXiv/0903.5522](http://arxiv.org/abs/0903.5522)

The term *strong convex space* was employed in:

* [[George Mackey]], p. 68 of: *The Mathematical Foundations of Quantum Mechanics: a Lecture-note Volume*, Mathematical physics monograph series, Benjamin (1963), Dover (2004) [[google books](https://books.google.de/books?id=qlpb2mWYmfYC&printsec=frontcover&hl=de&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false)]

[[!redirects superconvex spaces]]

[[!redirects strong convex space]]
[[!redirects strong convex spaces]]

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

[[!redirects Super convex space]]
[[!redirects Super convex spaces]]


superdense coding

+-- {: .rightHandSide}
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###Context###
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[quantum information theory|quantum information]]-protocol, which is a kind of dual to [[quantum teleportation]].

## References

Textbook accounts:

* [[Michael A. Nielsen]], [[Isaac L. Chuang]], §2.3 in: *Quantum computation and quantum information*, Cambridge University Press (2000) [[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), [pdf](http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf), [[NielsenChuangQuantumComputation.pdf:file]]]

See also

* Wikipedia, *[Superdense coding](https://en.wikipedia.org/wiki/Superdense_coding)*


superdeterminant

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###Context###
#### Super-Algebra and Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _super determinant_ or _Berezinian_ is the generalization of the notion of [[determinant]] of a [[matrix]] from [[algebra]] to [[super algebra]].

## Related entries

* [[supergeometry]], [[quasideterminant]], [[supertrace]]

* [[Berezin integral]]

## References

Originally due to [[Felix Berezin]]. The first publication explaining the Berezin's determinant is the article of  Berezin's student V. F. Pakhomov, _Automorphisms of the tensor product of Abelian and Grassmannian algebras_, Mathematical Notes 1974, 16:1, 624–629 [mathnet.ru](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mzm&paperid=7436&option_lang=eng).

* Wikipedia, _[Berezinian](http://en.wikipedia.org/wiki/Berezinian)_
* Deligne's lectures in [[P. Deligne]], [[P. Etingof]], [[Daniel Freed|D.S. Freed]], L. Jeffrey, [[D. Kazhdan]], J. Morgan, D.R. Morrison and [[E. 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))
* I. L. Buchbinder, S. M. Kuzenko, _Ideas and methods of supersymmetry and supergravity; or A walk through superspace_
* [[Edward Witten]], _Notes on supermanifolds and integration_, [arxiv/1209.2199](http://arxiv.org/abs/1209.2199)

[[!redirects Berezinian]]
[[!redirects superdeterminants]]


superextensive site

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### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Conntents#
* table of contents
{: toc}

## Idea

A **superextensive site** is a [[site]] whose covering families can be reduced to single covers combined with stable coproducts.  Many naturally occuring sites are superextensive.  The prefix 'super-' is opposite to 'sub-' as in '[[subset]]' and indicates that the [[Grothendieck topology]] of the site contains the [[extensive topology]]; compare [[subcanonical site]].

## Definition

Let $C$ be a finitely [[extensive category]].  A [[Grothendieck topology]] on $C$ is (finitely) **superextensive** if it includes the (finitely) extensive topology (generated by families of injections into finite coproducts) and every covering family is generated by a finite one.

Likewise, if $C$ is infinitary-extensive, a topology is infinitary-superextensive if it includes the infinitary-extensive topology (generated by families of injections into small coproducts) and every covering family is generated by a small one.  (Of course, the last condition is vacuous if $C$ is small.)

## Examples

* Well known examples of infinitary superextensive sites include [[Top]] and [[Diff]] with their usual topologies, generated by the open covers.
* The [[coherent topology]] on any (finitely or infinitary) extensive coherent category is (finitely or infinitary) superextensive.  In particular, this includes (in the infinitary case) the [[canonical topology]] on a [[Grothendieck topos]].

## Properties

### Relationship to singleton covers

If $f$ is a single morphism in $C$ that generates a covering family, we call it a **cover**.  The basic property of a superextensive site is that one can replace any (finite or small, as appropriate) covering family $(j_i: U_i \to B)_i$ with the single map $[j_i]_i: \coprod_i U_i \to B$.  This is a cover, since the original covering family factors through it, and the original covering family can be written as the composite of this cover and the family of coproduct injections $U_i \to \coprod_i U_i$.

Note that covers are stable under pullback, whenever such pullbacks exist.  Even without pullbacks, they always satisfy the condition that if $f\colon U\to B$ is a cover and $g\colon A\to B$ is any morphism, then there is a cover $\bar{f}\colon V\to A$ such that $g \bar{f}$ factors through $f$.  For by the pullback condition on a Grothendieck topology, there is a covering family $(\bar{f}_j\colon V_j\to A)$ such that each $g \bar{f}_j$ factors through $f$, but then $[\bar{f}_j]\colon \coprod_j V_j \to A$ is the required cover.  Covers are also easily seen to be stable under composition, and generate another Grothendieck topology (which is a "singleton pretopology").  If the original topology was denoted $T$, we denote the topology generated by single covers by $T_{cov}$.

The main conclusion is:

+--{: .num_theorem}
###### Theorem
A superextensive Grothendieck topology $T$ on an extensive category $C$ is generated by the union of $T_{cov}$ and the extensive topology $T_{ext}$.
=--
+--{: .proof}
###### Proof
By definition, $T_{cov}$ and $T_{ext}$ are sub-topologies of $T$, so it suffices to observe that any covering family $(j_i: U_i \to B)_i$ in $T$ is the composite of the family of coproduct injections $(j_i: U_i \to \coprod_j U_j)_i$ with the cover $[j_i]_i: \coprod_j U_j \to B$.
=--

Beware, however, of the following common misconception: $T$ is *not* identical with $T_{cov}$.  That is, although a superextensive topology is *determined by* its singleton covers, in the sense that it can be recovered uniquely by knowing only them, it is not *generated by* its singleton covers as a Grothendieck topology; we need to also include $T_{ext}$ in the generating covering families.

An easy way to see that $T$ and $T_{cov}$ must be different is to notice that $T_{cov}$, like any singleton pretopology, is a [[locally connected site]], so that all constant presheaves are sheaves for $T_{cov}$.  But since the initial object is covered by the empty family in $T_{ext}$, no nontrivial constant presheaf can be a sheaf for $T_{ext}$.

### Compatibility of sheafification

In general, it is not true, for two Grothendieck topologies $T_1$ and $T_2$ on the same category, that $T_2$-sheafification preserves $T_1$-sheaves (and hence takes them to $T_1\cup T_2$-sheaves).  For example, let $C$ be the [[walking arrow]] $0 \to 1$, let $T_1$ be generated by the nonidentity morphism (so that $T_1$-sheaves are those for which the image of $0\to 1$ is an isomorphism), and let $T_2$ be generated by the empty family covering $0$ (so that $T_2$-sheaves are those for which the image of $0$ is terminal).

However, if $T_1$ is the extensive topology and $T_2 = T_{cov}$ for some superextensive topology (which is then recovered as $T_1\cup T_2$), and coproducts preserve covers, then it is true.

+-- {: .num_theorem}
###### Theorem
Let $C$ be a finitary superextensive site, and suppose that finite coproducts of covers in $C$ are again covers.  If $X$ is a presheaf on $C$ which is a $T_{ext}$-sheaf, then the $T_{cov}$-sheafification of $X$ is also a $T_{ext}$-sheaf, and hence a $T$-sheaf.
=--
+-- {: .proof}
###### Proof
We show that if $X$ is a $T_{ext}$-sheaf, then so is $X^+$, where $(-)^+$ is the reflection into $T_{cov}$-separated presheaves.  Since $X^{++}$ is $T_{cov}$-sheafification, this will prove the theorem.

Since $T_{cov}$ is generated by single covers, for any $c\in C$ we have
$$X^+(c) = colim_{f\colon d \to c} gl(X,f)$$
where the colimit is over all covers with target $c$, and $gl(X,f)\subseteq X(d)$ consists of those $x\in X(d)$ such that $r^*(x) = s^*(x)$, where $(r,s)\colon d\times_c d \rightrightarrows d$ is the [[kernel pair]] of $f$.

The extensive topology is generated by the empty covering family of the initial object, and by the injections into binary coproducts.  First consider the initial object $0$.  $X$ being a sheaf for the empty family on $0$ means that $X(0)=1$ is [[terminal set|terminal]].  But since $C$ is extensive, its initial object is strict, so any morphism $f\colon d\to 0$ is an isomorphism.  Therefore, $X^+(0) = X(0) = 1$.

Now consider a binary coproduct $c = c_1 + c_2$.  $X$ being a sheaf for this means that $X(c) \to X(c_1)\times X(c_2)$ is an isomorphism (since coproducts are disjoint and $X(0)$ is terminal, there is no compatibility condition to worry about).  We want to show that $X^+(c) \to X^+(c_1)\times X^+(c_2)$ is also an isomorphism.

First suppose given $x,x'\in X^+(c)$.  Since covers are stable under pullback and composition, we can represent $x$ and $x'$ by some $\bar{x}, \bar{x'}\in X(d)$ for the same cover $f\colon d\to c$.  Now since $C$ is extensive and $c=c_1+c_2$, $f$ decomposes $d$ as $d = d_1+d_2$, and moreover $d_1$ and $d_2$ are the pullbacks of $f$ along the two inclusions $c_1 \to c \leftarrow c_2$.  Thus, since covers are stable under pullback, we have covers $f_1\colon d_1\to c_1$ and $f_2\colon d_2\to c_2$, and so the restrictions of $x$ and $x'$ to $c_1$ and $c_2$ can be represented by the restrictions of $\bar{x}$ and $\bar{x'}$ to $d_1$ and $d_2$ respectively.

Now if $x$ and $x'$ become equal in $X^+(c_1)$ and $X^+(c_2)$, there must be covers $g_i\colon b_i \to d_i$ for $i=1,2$ such that $g_i^*(\bar{x}) = g_i^*(\bar{x'})$ in $X(b_i)$ for $i=1$ and $2$.  Now since covers are stable under coproducts, the map $g_1+g_2\colon b_1+b_2 \to d_1+d_2 = d$ is a cover, and moreover $g_i^*(\bar{x})$ and $g_i^*(\bar{x'})$ can be obtained by first restricting to $b_1+b_2$ and then along the two coproduct injections.
$$\array{b_i & \overset{}{\to} & b_1+b_2\\
  ^{g_i}\downarrow && \downarrow^{g_1+g_2}\\
  d_i& \overset{}{\to} & d\\
  \downarrow && \downarrow\\
  c_i & \underset{}{\to} & c}$$
But since $X$ is a $T_{ext}$-sheaf and $g_i^*(\bar{x}) = g_i^*(\bar{x'})$ for $i=1,2$, we must have that $\bar{x}$ and $\bar{x'}$ become equal in $X(b_1+b_2)$.  Now since the composite $b_1+b_2 \to d \to c$ is a cover, it follows that $x$ and $x'$ are equal in $X^+(c)$.  Thus, $X^+(c) \to X^+(c_1)\times X^+(c_2)$ is injective, i.e. $X^+$ is $T_{ext}$-separated.

The case of surjectivity is easier.  Suppose given $x_i\in X^+(c_i)$, represented by $\bar{x_i}\in X(d_i)$ for some covers $f_i\colon d_i\to c_i$.  Then since covers are stable under coproduct, $f_1+f_2 \colon d_1+d_2 \to c_1+c_2 = c$ is a cover.  And since $X$ is a $T_{ext}$-sheaf, we have a $\bar{x}\in X(d)$ which restricts to $\bar{x_i}$ for $i=1,2$.  Hence $\bar{x}$ represents some $x\in X^+(c)$ which restricts to each $x_i$, as desired.  Thus, $X^+$ is a $T_{ext}$-sheaf.
=--

Of course, there is an analogous result for infinitary superextensive sites.  With a little more work we can also prove the same thing for [[stacks]] and stackification.

This result is especially interesting because sheaves and stacks for the extensive topology are easier to come by than those for some other topologies.  For instance, if $C$ is an extensive [[coherent category]], then any [[internal category]] in $C$ represents a functor $C^{op}\to Cat$ which is a stack for the extensive topology, but not usually for the [[coherent topology]].  However, the coherent topology is superextensive and its covers are precisely the regular epimorphisms, i.e. $T_{cov}$ is the [[regular coverage|regular topology]].  Thus, the stackification of an internal category in $C$ relative to the regular topology is still a stack for the extensive topology, and hence also for the coherent topology.

## References

* Chapter 1 of [[David Roberts]]' thesis uses superextensive sites in the study of [[anafunctors]]. The material is covered and updated in the paper _Internal categories, anafunctors and localisations_.

Some discussion about terminology in this entry is on the nForum [here](http://nforum.mathforge.org/discussion/486/superextensive-site/?Focus=47656#Comment_47656).


superfield

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###Context###
#### Supergeometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[physics]] one says "superfield" when referring to a [[field (physics)]] in the context of [[supersymmetry|supersymmetric]] field theory.

Technically, given a [[supermanifold]] $X$ ("[[superspace]]"), then the superspace of _superfields_ is the [[internal hom]] $\underline{Hom}(X,\mathbb{R})$ or $\underline{Hom}(X,\mathbb{C})$ of real or complex valued functions on $X$.

See at _[[geometry of physics -- supergeometry]]_ [this example](geometry+of+physics+--+supergeometry#Superfields).

## Related concepts

* [[superpotential]]

## References

The concept was introduced in physics (together with "[[superspace]]") in:

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

* [[Abdus Salam]] J.A. Strathdee, Physical Review D11, 1521-1535 (1975)

[[!redirects superfields]]

[[!redirects super field]]
[[!redirects super fields]]

[[!redirects super-field]]
[[!redirects super-fields]]


superfluid

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###Context###
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

#Contents#

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[Bose-Einstein condensate]]

* [[superconductivity]]

## References

* C. Lanczos, R. C. Clark, G. H. Derrick (eds.), Ch. 3 in: *Mathematical Methods in Solid State and Superfluid Theory*, Springer (1986) $[$[doi:10.1007/978-1-4899-6435-9](https://doi.org/10.1007/978-1-4899-6435-9)$]$

See also:

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

Discussion of superfluids via [[AdS-CFT in condensed matter physics]]:

* Yu-Kun Yan, Shanquan Lan, Yu Tian, Peng Yang, Shunhui Yao, Hongbao Zhang, *Towards an effective description of holographic vortex dynamics* [[arXiv:2207.02814](https://arxiv.org/abs/2207.02814)]

* Aristomenis Donos, Polydoros Kailidis, *Dissipative effects in finite density holographic superfluids* [[arXiv:2209.06893](https://arxiv.org/abs/2209.06893)]

* [[Mario Flory]], Sebastian Grieninger, Sergio Morales-Tejera, *Critical and near-critical relaxation of holographic superfluids* [[arXiv:2209.09251](https://arxiv.org/abs/2209.09251)]

On superfluidity/[[superconductivity]] *of* [[anyons]]:

* Yi-Hong Chen, [[Frank Wilczek]], [[Edward Witten]], [[Bertrand Halperin]], *On Anyon Superconductivity*,  International Journal of Modern Physics B **03** 07 (1989) 1001-1067 (reprinted in [Wilczek 1990](braid+group+statistics#Wilczek90)) $[$[doi:10.1142/S0217979289000725](https://doi.org/10.1142/S0217979289000725), [[CWWH-AnyonSuperfluidity.pdf:file]]$]$

Application of the [[conformal bootstrap]] to [[experiment|experimental]] [[superfluid]]-transition:

* [[Shai Chester]], Walter Landry, Junyu Liu, [[David Poland]], [[David Simmons-Duffin]], Ning Su, [[Alessandro Vichi]], _Carving out OPE space and precise $O(2)$ model critical exponents_, JHEP 06 (2020) 142 ([arXiv:1912.03324](https://arxiv.org/abs/1912.03324))

  exposition in:

  [[Slava Rychkov]], _Conformal bootstrap and the $\lambda$-point specific heat experimental anomaly_, Journal Club for Condensed Matter Physics recommendation 2020 ([pdf](https://www.condmatjclub.org/uploads/2020/01/JCCM_January_2020_02.pdf), [doi:10.36471/JCCM_January_2020_02](https://doi.org/10.36471/JCCM_January_2020_02))

Discussion of [[axion|axionic]] [[dark matter]] superfluids ([[fuzzy dark matter]]) on [[cosmology|cosmic]] scales:

* {#BerezhianiKhoury15} [[Lasha Berezhiani]], [[Justin Khoury]], _Theory of Dark Matter Superfluidity_, Phys. Rev. D 92, 103510 (2015) ([arXiv:1507.01019](https://arxiv.org/abs/1507.01019))

* {#Khoury16} [[Justin Khoury]], _Another Path for the Emergence of Modified Galactic Dynamics from Dark Matter Superfluidity_, Phys. Rev. D 93, 103533 (2016) ([arXiv:1602.05961](https://arxiv.org/abs/1602.05961))

[[!redirects superfluids]]

[[!redirects superfluidity]]


superfluous epimorphism

#Contents#
* table of contents
{:toc}

## Idea

Given a ring $R$ (or some analogue, say a Banach algebra),
a submodule $K$ of an $R$-module $M$ is called
__superfluous__ or small in $M$, written $K \ll M$,
if, for every submodule $L\subset M$ , the equality $K + L = M$ implies $L = M$.
An [[epimorphism]] $f : M\to N$ is called superfluous (or coessential) if $Ker f \ll M$.

Superfluous epimorphisms (submodules) are a notion [[formal duality|dual]] to essential monomorphisms
(submodules); their role in the study of [[projective covers]] is analogous to the role of essential monomorphisms in the study of [[injective envelopes]].

## References

* [[Robert Wisbauer]], _Foundations of module and ring theory_, [pdf](http://www.math.uni-duesseldorf.de/~wisbauer/book.pdf)

category: algebra

[[!redirects superfluous submodule]]


supergeometry

+-- {: .rightHandSide}
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###Context###
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Supergeometry is the ([[higher geometry|higher]]) [[geometry]] over the [[base topos]] on [[superpoint|superpoints]] modeled on the canonical [[line object]] $\mathbb{R}$ in there.

As ordinary [[differential geometry]] studies spaces -- [[smooth manifold|smooth manifolds]] -- that locally look like [[vector space|vector spaces]], supergeometry studies spaces -- [[supermanifold|supermanifolds]] -- that locally look like [[super vector space|super vector spaces]].

As ordinary [[algebraic geometry]] studies spaces -- [[scheme|schemes]] -- that locally look like [[affine space|affine spaces]], supergeometry studies [[superscheme|superschemes]].

From the point of view of [[noncommutative geometry]], the supergeometry is a very mild special case of noncommutativity in geometry: some coordinates commute, some anticommute.

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

Supergeometry may defined over an arbitrary commutative ring ([Schwarz and Shapiro 06](#SchwarzShapiro)).

## Related concepts

* [[signs in supergeometry]]

* [[superpoint]]

* [[superspace]]

  * [[supermanifold]], [[super scheme]]

  * [[spectral super-scheme]]

* [[supersymmetry]]

* [[super-spacetime]], [[super-Minkowski spacetime]], [[super Poincare group]]

* [[super-vielbein]]

* [[supergravity]]

* [[Super Gerbes]]

* [[bosonic modality]]

* [[super-Cartan geometry]]

  * [[superconformal geometry]]

[[!include geometries of physics -- table]]

$\,$

[[!include Isbell duality - table]]

## References
 {#References}

Some historically influential general considerations are in

* [[Yuri Manin]], _[[New Dimensions in Geometry]]_, talk at Arbeitstagung, Bonn 1984

Introductory lecture notes include

* [[Yuri Manin]], chapter 4 of _[[Gauge Field Theory and Complex Geometry]]_, Grundlehren der Mathematischen Wissenschaften 289, Springer 1988

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

* [[Gennadi Sardanashvily]], _Lectures on supergeometry_ ([arXiv:0910.0092](http://arxiv.org/abs/0910.0092))

Discussion via [[functorial geometry]]:

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

Discussion of traditional [[algebraic geometry]]  for [[super-schemes]]:

* [[Ugo Bruzzo]], Daniel Hernandez Ruiperez, [[Alexander Polishchuk]], _Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes_ ([arXiv:2008.00700](https://arxiv.org/abs/2008.00700))

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

* {#Schwarz84} [[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, [[Albert Schwarz]],

  _On $(k \oplus l|q)$-dimensional supermanifolds_ in [[Julius Wess]], V. Akulov (eds.) _Supersymmetry and Quantum Field Theory_ ([[Dmitry Volkov]] memorial volume), Lecture Notes in Physics **509**, Springer (1998) [[arXiv:hep-th/9706003](http://arxiv.org/abs/hep-th/9706003), [doi:10.1007/BFb0105247](https://doi.org/10.1007/BFb0105247)]

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

Generalization to supergeometry over an arbitrary commutative ring, in particular $p$-adic rings, is given in

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

A review of all this as geometry in the [[topos]] over the category of [[superpoints]] is in

* {#Sachse08} [[Christoph Sachse]], _A Categorical Formulation of Superalgebra and Supergeometry_ ([arXiv:0802.4067](http://arxiv.org/abs/0802.4067))

Formulation in terms of [[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))

For many more references see at _[[supermanifold]]_.

Plenty of discussion of supergeometry with an eye towards [[supersymmetry]] in [[quantum field theory]] is in

* [[Pierre Deligne|P. Deligne]], [[Pavel Etingof|P. Etingof]], [[Dan Freed|D.S. Freed]], L. Jeffrey, [[David Kazhdan|D. Kazhdan]], J. Morgan, D.R. Morrison, [[Edward Witten|E. 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))

especially in the contribution

* [[Pierre Deligne]], [[Daniel Freed]], _Supersolutions_ ([arXiv:hep-th/9901094](http://arxiv.org/abs/hep-th/9901094)).

The appendix there

* _Sign manifesto_ ([pdf](http://publications.ias.edu/sites/default/files/79_SignManifesto.pdf))

means to sort out various sign conventions of relevance.

Discussion of how supersymmetry is universally induced in [[higher category theory]]/[[homotopy theory]] by the [[free construction|free]] [[abelian ∞-group]] on the point -- the [[sphere spectrum]] -- is in

* {#Kapranov13} [[Mikhail Kapranov]], _Categorification of supersymmetry and stable homotopy groups of spheres_, April 2013 ([abstract](http://www.google.com/url?q=http%3A%2F%2Fwww.math.neu.edu%2F~bwebster%2FACRT%2Fcalendar-with-abstracts.pdf&sa=D&sntz=1&usg=AFQjCNHvbVTfzJvF0-HHfRuak0RlSZpC0A),  [video](http://208.52.189.22/zelevinsky/day2/2-kapranov.mov))

For more on this see at _[[superalgebra]]_.

Discussion related to [[G-structure]] and [[Killing spinors]] includes

* [[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 of [[classical field theory]] with [[fermions]] as taking place on supermanifolds is in the following references

* {#DeligneFreed99} [[Pierre Deligne]], [[Daniel Freed]], _Classical field theory_ (1999) ([pdf](https://publications.ias.edu/sites/default/files/79_ClassicalFieldTheory.pdf))

  this is a chapter in

  [[Pierre Deligne|P. Deligne]], [[Pavel Etingof|P. Etingof]], [[Dan Freed|D.S. Freed]], L. Jeffrey, [[David Kazhdan|D. Kazhdan]], J. Morgan, D.R. Morrison, [[Edward Witten|E. 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))

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

* {#GiachettaMangiarottiSardanashvily09} Giovanni Giachetta, Luigi Mangiarotti, [[Gennadi Sardanashvily]], chapter 3 of _Advanced classical field theory_, World Scientific (2009)

* {#Sardanashvily12} [[Gennadi Sardanashvily]], _Grassmann-graded Lagrangian theory of even and odd variables_ ([arXiv:1206.2508](https://arxiv.org/abs/1206.2508))

* {#Sardanashvily16} [[Gennadi Sardanashvily]], _Noether's Theorems: Applications in Mechanics and Field Theory_, Studies in Variational Geometry, 2016

Discussion on the context of [[superstring|super-]] [[string theory]] (especially regarding [[super Riemann surfaces]]):

* {#DHokerPhong05} [[Eric D'Hoker]], [[Duong Phong]], _Complex geometry and supergeometry_, Current Developments in Mathematics 2005 (2007): 1-40 ([euclid:1223654523](https://projecteuclid.org/euclid.cdm/1223654523))

Generalization of [[Artin's representability theorem]] to supergeometry:

* Nadia Ott, *Artin's theorems in supergeometry* ([arXiv:2110.12816](https://arxiv.org/abs/2110.12816))

[[!redirects supergeometries]]

[[!redirects super-geometry]]
[[!redirects super-geometries]]

[[!redirects super geometry]]
[[!redirects super geometries]]


supergeometry - contents

**[[superalgebra]]** and ([[synthetic differential supergeometry|synthetic]]
) **[[supergeometry]]**

## Background

* [[algebra]]

* [[geometry]]

* [[graded object]]

## Introductions

* [[geometry of physics -- superalgebra]]

* [[geometry of physics -- supergeometry]]

## Superalgebra ##

* [[super commutative monoid]]

* [[super abelian group]]

* [[super ring]]

  * [[supercommutative ring]]

  * [[exterior ring]]

  * [[Clifford ring]]

* [[super module]]

* [[super vector space]], [[SVect]]

* [[super algebra]]

  * [[supercommutative algebra]]

  * [[exterior algebra]]

  * [[Clifford algebra]]

* [[superdeterminant]]

* [[super Lie algebra]]

  * [[super Poincare Lie algebra]]

* [[chain complex of super vector spaces]] ([[model structure on chain complexes of super vector spaces|model structure]])

* [[differential graded-commutative superalgebra]] ([[model structure on differential graded-commutative superalgebras|model structure]])

* [[super L-infinity algebra]]

## Supergeometry ##

* [[superpoint]]

* [[super Cartesian space]]

* [[supermanifold]], [[SDiff]]

  * [[NQ-supermanifold]]

  * [[super vector bundle]]

  * [[complex supermanifold]]

  * [[Euclidean supermanifold]]

* [[super spacetime]]

  * [[super Minkowski spacetime]]

* [[integration over supermanifolds]]

  * [[Berezin integral]]

* [[super Lie group]]

  * [[super Lie group|super translation group]]

  * [[super Euclidean group]]

* [[super ∞-groupoid]]

* [[super formal smooth ∞-groupoid]]

* [[super line 2-bundle]]

## Supersymmetry

[[supersymmetry]]

* [[division algebra and supersymmetry]]

* [[super Poincare Lie algebra]]

* [[supermultiplet]]

* [[BPS state]]

* [[M-theory super Lie algebra]], [[type II super Lie algebra]]

* [[supergravity Lie 3-algebra]], [[supergravity Lie 6-algebra]]

## Supersymmetric field theory

* [[superfield]]

* [[supersymmetric quantum mechanics]]

  * [[superparticle]]

  * [[adinkra]]

* [[super Yang-Mills theory]]

* [[supergravity]]

  * [[gauged supergravity]]

* [[superstring theory]]

  * [[type II string theory]]

  * [[heterotic string theory]]

## Applications

* [[geometric model for elliptic cohomology]]

[Edit this sidebar](/nlab/edit/supergeometry+-+contents)


supergravity

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[quantum field theory]] of **supergravity** is similar to the theory of [[gravity]], but where (in [[first-order formulation of gravity|first order formulation]]) the latter is given by an [[action functional]] (the [[Einstein-Hilbert action]] functional) on the space of [[connection on a bundle|connection]]s (over [[spacetime]]) with values in the [[Poincare Lie algebra]] $\mathfrak{iso}(n,1)$, supergravity is defined by an extension of this to an action functional on the space of connections with values in the [[super Poincare Lie algebra]] $\mathfrak{siso}(n,1)$. One says that supergravity is the theory of _local_ (Poincaré) [[supersymmetry]] in the same sense that ordinary [[gravity]] is the theory of "local Poincaré-symmetry". These are [[gauge theories]] for the [[Poincare Lie algebra]] and the [[super Poincare Lie algebra]], respectively, in that the [[field (physics)]] is a [[Cartan connection]] for the inclusion $o(n,1) \hookrightarrow \mathfrak{siso}(n,1)$:

if we write $\mathfrak{siso}(n,1)$ as a [[semidirect product]] of the translation Lie algebra $\mathbb{R}^{(n,1)}$, the [[special orthogonal Lie algebra]] $\mathfrak{so}(n,1)$ and a [[spin group]] [[representation]] $\Gamma$, then locally a connection is a [[Lie algebra valued 1-form]]

$$
  A : T X \to \mathfrak{siso}(n,1)
$$

that decomposes into three components, $A = (E, \Omega, \Psi)$:

* a $\mathbb{R}^{n,1}$-valued 1-form $E$ -- the [[vielbein]]

  (this encodes the [[pseudo-Riemannian metric]] and hence the field of [[gravity]]);

* a $\mathfrak{so}(n,1)$-valued 1-form $\Omega$ -- called the [[spin connection]];

* a $\Gamma$-valued 1-form $\Psi$ -- called the [[gravitino]] field.

Typically in fact the field content of supergravity is larger, in that a field $A$ is really an [[∞-Lie algebra-valued differential form]] with values in an [[∞-Lie algebra]] such as the [[supergravity Lie 3-algebra]] ([DAuriaFreCastellani](#DAuriaFreCastellani)) $\mathfrak{sugra}(10,1)$. Specifically such a field

$$
  A : T X \to \mathfrak{sugra}(10,1)
$$

has one more component

* a 3-form $C$ -- the [[supergravity C-field]].

The [[gauge transformation]]s on the space of such connections that are parameterized by the elements of $\Gamma$ are called [[supersymmetries]].

[[!include local and global geometry - table]]

The condition of [[gauge invariance]] of an action functional on $\mathfrak{siso}$-connections is considerably more restrictive than for one on $\mathfrak{iso}$-connections. For instance there is, under mild assumptions, a _unique_ maximally supersymmetric supergravity extension of the ordinary [[Einstein-Hilbert action]] on a 4-dimensional manifold. This in turn is obtained from the _unique_ (under mild assumptions) maximally supersymmetric supergravity action functional on a (10,1)-dimensional [[spacetime]] by thinking of the 4-dimensional action function as being a [[dimensional reduction]] of the 11-dimensional one.

This uniqueness (under mild conditions) is one reason for interest in supergravity theories. Another important reason is that supergravity theories tend to remove some of the problems that are encountered when trying to realize [[gravity]] as a [[quantum field theory]]. Originally there had been high hopes that the maximally supersymmetric supergravity theory in 4-dimensions is fully [[renormalizable]]. This couldn't be shown computationally -- until recently: triggered by new insights recently there there has been lots of renewed activity on the renormalizability of maximal supergravity.

### As a gauge theory -- Super Cartan geometry
 {#SuperCartanGeometry}

Ordinary [[Einstein gravity]] has a natural formulation in terms of [[Cartan geometry]] for the inclusion of the [[Lorentz Lie algebra]] into the [[Poincaré Lie algebra]] $\mathfrak{o}(d-1,1) \hookrightarrow \mathfrak{Iso}(\mathbb{R}^{d-1,1})$. In this [[first order formulation of gravity]] a [[field (physics)|field]] configuration is a [[Cartan connection]] with such [[coefficients]].

This perspective directly generalizes to [[supergeometry]] and yields the _superspace formulation_ of theories of supegravity -- _[[super Cartan geometry]]_.

After picking a [[dimension]] $d\in \mathbb{N}$ and writing $\mathfrak{Iso}(\mathbb{R}^{d-1,1})$ for the [[Poincaré Lie algebra]], then a choice of "number of supersymmetries" is a choice of [real spin representation](spin+representation#RealIrreducibleSpinRepresentationInLorentzSignature) $N$. Then the [[direct sum]]

$$
  \mathfrak{Iso}(\mathbb{R}^{d-1,1|N})
  \coloneqq
  \underbrace{\mathfrak{Iso}(\mathbb{R}^{d-1,1})}_{even} \oplus \underbrace{N}_{odd}
$$

regarded as a [[super vector space]] with $N$ in odd degree becomes a [[super Lie algebra]] by letting the $[even,odd]$ bracket be given by the defining [[action]] and by letting the $[odd,odd]$ bracket be given by a canonically induced bilinear and $\mathfrak{o}$-equivariant pairing -- the [[super Poincaré Lie algebra]]. This still canonical contains the [[Lorentz Lie algebra]] $\mathfrak{o}(\mathbb{R}^{d-1,1})$ and the [[quotient]]

$$
  \mathbb{R}^{d-1,1|N}
  \coloneqq
  \mathfrak{Iso}(\mathbb{R}^{d-1,1|N})/\mathfrak{o}(\mathbb{R}^{d-1,1})
$$

is called [[super Minkowski spacetime]] (equipped with its [[super translation Lie algebra]] structure).

From this, a [[super-Cartan geometry]] is defined in direct analogy to the Cartan formulation of Riemannian geometry

| ([[higher Cartan geometry|higher]]-)[[Cartan geometry]] | $\mathfrak{g}$ | $\mathfrak{h}$ | $\mathfrak{g}/\mathfrak{h}$ |
|---------------------|----------------|----------------|-----------------------------|
| [[Einstein gravity]]  | $\mathfrak{Iso}(\mathbb{R}^{d-1,1})$ | $\mathfrak{o}(d-1,1)$ | $\mathbb{R}^{d-1,1}$ |
| [[supergravity]] | $\mathfrak{Iso}(\mathbb{R}^{d-1,1\vert N})$ | $\mathfrak{o}(d-1,1)$ | $\mathbb{R}^{d-1,1\vert N}$ |
| [[11-dimensional supergravity]] | $\mathfrak{Iso}(\widehat{\mathbb{R}}^{10,1\vert N=1})$ | $\mathfrak{o}(d-1,1)$ | $\widehat{\mathbb{R}}^{10,1\vert N=1}$ |

Indeed, all the traditional literature on supergravity (e.g. ([Castellani-D'Auria-Fré 91](#CastellaniDAuriaFre91))) is phrased, more or less explicitly, in terms of [[Cartan connections]] for the inclusion of the [[Lorentz group]] into the [[super Poincaré group]] this way, this being the formalization of what physicists mean when saying that they pass to "local supersymmetry".

One subtlety to take care of is that this makes [[spacetime]] a [[super-spacetime]] locally modeled on [[super Minkowski spacetime]]. But the resulting theory is supposed to be a field theory on an ordinary [[spacetime]] locally modeled on ordinary [[Minkowski spacetime]]. This is enforced by a further constraint on the super-Cartan connection which forces it to be determined by the bosonic manifold underlying the given supermanifold. This constraint is variously known as the _superspace constraints_ or as _[rheonomy](D%27Auria-Fre+formulation+of+supergravity#Rheonomy)_ .

The other subtlety to take care of is that a key aspect of higher dimensional [[supergravity]] theories is that their [[field (physics)|field]] content necessarily includes, in addition to the [[graviton]] and the [[gravitino]], higher [[differential n-form]] fields, notably the 2-form [[B-field]] of 10-dimensional [[type II supergravity]] and [[heterotic supergravity]] as well as the 3-form [[C-field]] of [[11-dimensional supergravity]].

This means that these higher dimensional supergravity theories are not in fact entirely described  by super-[[Cartan geometry]] -- but by super-[[higher Cartan geometry]].

This follows a key insight due to ([D'Auria-Fré-Regge 80](#DAuriaFreRegge80), [D'Auria-Fré 82](#DAuriaFre82)) -- the _[[D'Auria-Fre formulation of supergravity]]_ -- that the "tensor multiplet" fields of higher dimensional supergravity theories as above are naturally brought into the previous perspective if only one allows more general [[Chevalley-Eilenberg algebras]].

Namely, we may add to the above CE-algebra

* a single generator $c_3$ of degree $(3,even)$

and extend the differential to that by the formula

$$
  d_{CE} \, c_3 = \frac{1}{2}\bar \psi \Gamma^{a b} \wedge \psi \wedge e_a \wedge e_b
  \,.
$$

This still squares to zero due to the remarkable property of 11d [[super Minkowski spacetime]] by which $\frac{1}{2}\bar \psi \Gamma^{a b} \wedge \psi \wedge e_a \wedge e_b \in CE^4(\mathfrak{Iso}(10,1|N=1))$ is a representative of an exception [[super Lie algebra|super]]-[[Lie algebra cohomology]] class. (The collection of all these exceptional classes constitutes what is known as the _[[brane scan]]_).

In the textbook ([Castellani-D'Auria-Fré 91](#CastellaniDAuriaFre91)) a beautiful algorithm for constructing and handling higher supergravity theories based on such generalized CE-algebras is presented, but it seems fair to say that the authors struggle a bit with the right mathematical perspective to describe what is really happening here.

But from a modern perspective this becomes crystal clear: these generalized CE algebras are CE-algebras not of [[Lie algebras]] but of [[strong homotopy Lie algebras]], hence of [[L-infinity algebras]], in fact of [[Lie n-algebras|Lie (p+1)-algebras]] for $(p+1)$ the degree of the relevant differential form field.

Specifically, me may write the above generalized CE-algebra with the extra degree-3 generator $c_3$ as the CE-algebra $CE(\mathfrak{m}2\mathfrak{brane}) $

of the _[[supergravity Lie 3-algebra]]_ $\mathfrak{m}2\mathfrak{brane}$.

Now a morphism

$$
  \Omega^\bullet(U) \stackrel{}{\longleftarrow} CE(\mathfrak{m}2\mathfrak{brane}) \;\colon\; A
$$

encodes [[graviton]] and [[gravitino]] fields as above, but in addition it encodes a 3-form

$$
  C_3 \coloneqq A(c_3) \in \Omega^{(3,even)}(U)
$$

whose [[curvature]]

$$
  G_4 = \mathbf{d}C_3 + \frac{1}{2}\bar \Psi \Gamma^{a b} \wedge \Psi \wedge E_a \wedge E_b
$$

satisfies a modified [[Bianchi identity]], crucial for the theory of [[11-dimensional supergravity]] ([D'Auria-Fré 82](#DAuriaFre82)).

So this collection of differential form data is no longer a [[Lie algebra valued differential form]], it is an [[L-infinity algebra valued differential form]], with values in the [[supergravity Lie 3-algebra]].

The quotient

$$
  \widehat{\mathbb{R}}^{10,1|N=1}
  \coloneqq
  \mathfrak{g}/\mathfrak{h}
  =
  \mathfrak{m}2\mathfrak{brane} / \mathfrak{o}(\mathbb{R}^{10,1|N=1})
$$

is known as _[[extended super Minkowski spacetime]]_.

The [[Lie integration]] of this is a [[smooth infinity-group|smooth 3-group]] $G$ which receives a map from the [[Lorentz group]].

This means that a global description of the geometry which ([Castellani-D'Auria-Fré 91](#CastellaniDAuriaFre91)) discuss locally on [[charts]] has to be a higher kind of Cartan geometry which is locally modeled not just on [[cosets]], but on the [[homotopy quotients]] of ([[smooth infinity-group|smooth]], [[smooth super infinity-groupoid|supergeometric]], ...) [[infinity-groups]] -- [[higher Cartan geometry]].

### Solutions with global supersymmetry
 {#SolutionsWithGlobalSupersymmetry}

A solution to the bosonic [[Einstein equations]] of ordinary [[gravity]] -- some [[Riemannian manifold]] -- has a _global symmetry_ if it has a [[Killing vector]].

Accordingly, a configuration that solves the supergravity [[Euler-Lagrange equations]] is a _[[global supersymmetry]]_ if it has a [[Killing spinor]]: a [[covariantly constant spinor]].

Here the notion of [[covariant derivative]] includes the usual [[Levi-Civita connection]], but also in general [[torsion]] components and contributions from other [[background gauge fields]] such as a [[Kalb-Ramond field]] and the [[RR-field]]s in [[type II supergravity]] or [[heterotic string theory|heterotic supergravity]].

Of particular interest to phenomenologists around the turn of the millennium (but maybe less so today with new experimental evidence) has been in solutions of spacetime manifolds of the form $M^4 \times Y^6$ for $M^4$ the locally observed [[Minkowski spacetime]] (that plays a role as the background for all available particle accelerator experiments) and a small closed 6-dimensional [[Riemannian manifold]] $Y^6$.

In the absence of further fields besides gravity, the condition that such a configuration has precisely one [[Killing spinor]] and hence precisely one [[global supersymmetry]] turns out to be precisely that $Y^6$ is a [[Calabi-Yau manifold]]. This is where all the interest into these manifolds in [[string theory]] comes from. (Notice though that nothing in the theory itself demands such a compactification. It is only the phenomenological assumption of the factorized spacetime compactification together with $N = 1$ supersymmetry that does so.)

More generally, in the presence of other [[background gauge field]]s, the Calabi-Yau condition here is deformed. One also speaks of [[generalized Calabi-Yau space]]s. (For instance ([GMPT05](#GMPT))).

For more see

* [[supersymmetry and Calabi-Yau manifolds]]

## Properties

### As a background for Green-Schwarz $\sigma$-models

The [[equations of motion]] of those theories of supergravity which qualify as [[target spaces]] for [[Green-Schwarz action functional]] [[sigma models]] (e.g. 10d [[heterotic supergravity]] for the [[heterotic string]] and 10d [[type II supergravity]] for the [[type II string]]) are supposed to be equivalent to those $\sigma$-models being well defined (the [[WZW-model]] term being well defined, hence $\kappa$-symmetry being in effect). See at _[Green-Schwarz action -- References -- Supergravity equations of motion](Green-Schwarz+action+functional#ReferencesSupergravityBackgroundEquationsOfMotion)_ for pointers.

### Scalar moduli spaces and $U$-duality
 {#UDuality}

The [[compact space|compact]] [[exceptional Lie groups]] form a series

$$
  E_8, E_7, E_6
$$

which is usefully thought of to continue as

$$
  E_5 := Spin(10), E_4 := SU(5), E_3 := SU(3) \times SU(2)
  \,.
$$

Supergravity theories are controled by the corresponding
[[split real forms]]

$$
  E_{8(8)}, E_{7(7)}, E_{6(6)}
$$

$$
  E_{5(5)} := Spin(5,5), E_{4(4)} := SL(5, \mathbb{R}),
  E_{3(3)} := SL(3, \mathbb{R}) \times SL(2, \mathbb{R})
  \,.
$$

For instance the [[scalar fields]] in the field [[supermultiplet]] of $3 \leq d \leq 11$-dimensional supergravity have [[moduli spaces]] parameterized by the [[homogeneous spaces]]

$$
  E_{n(n)}/ K_n
$$

for

$$
  n = 11 - d
  \,,
$$

where $K_n$ is the [[maximal compact subgroup]] of $E_{n(n)}$:

$$
  K_8 \simeq Spin(16), K_7 \simeq SU(8), K_6 \simeq Sp(4)
$$

$$
  K_5 \simeq Spin(5) \times Spin(5),
  K_4 \simeq Spin(5),
  K_3 \simeq SU(2) \times SO(2)
  \,.
$$

Therefore $E_{n(n)}$ acts as a [[global symmetry]] on the supergravity fields.

This is no longer quite true for their [[UV-completion]] by the corresponding [[Kaluza-Klein mechanism|compactifications]] of [[string theory]] (e.g. [[type II string theory]] for [[type II supergravity]], etc.). Instead, on these a [[discrete group|discrete subgroup]]

$$
  E_{n(n)}(\mathbb{Z}) \hookrightarrow E_{n(n)}
$$

acts as global symmetry. This is called the **[[U-duality]]** group of the supergravity theory (see there for more).

It has been argued that this pattern should continue in some way further to the remaining values $0 \leq d \lt 3$,
with "[[Kac-Moody groups]]" corresponding to the [[Kac-Moody algebras]]

$$
  \mathfrak{e}_9, \mathfrak{e}_10, \mathfrak{e}_{11}
  \,.
$$

Continuing in the other direction to $d = 10$ ($n = 1$) connects to the [[T-duality]] group $O(d,d,\mathbb{Z})$ of [[type II string theory]].

See the references ([below](#UDualityReferences)).

[[!include U-duality -- table]]

### Exceptional geometry

For the moment see the remarks/references on supergravity at _[[exceptional geometry]]_ and _[[exceptional generalized  geometry]]_.

## Examples

The usual [[folklore]] is that for supergravity Lagrangians "of ordinary type" it turns out that

* $N = 1$ [[11-dimensional supergravity]]

is the highest dimension possible, but see also at

* [[12-dimensional supergravity]]

for some qualification.

All lower dimensional theories in this class appear as [[KK-compactifications]] of this theory or are [[deformations]] of such:

* [[10-dimensional supergravity]]

  * [[type II supergravity]]

    * [[type IIA supergravity]]

      * [[massive type IIA supergravity]]

    * [[type IIB supergravity]]

  * [[heterotic supergravity]]

* [[9-dimensional supergravity]]

* [[8-dimensional supergravity]]

* [[7-dimensional supergravity]]

* [[6-dimensional supergravity]]

* [[5-dimensional supergravity]]

* [[4-dimensional supergravity]]

* [[3-dimensional supergravity]]

* [[2-dimensional supergravity]]

In dimension $(1+0)$ supergravity coupled to [[sigma-model]] fields is the [[spinning particle]].

In dimension $(1+1)$ supergravity coupled to [[sigma-model]] fields is the [[spinning string]]/[[NSR superstring]].

In non-Lorenzian signature it is also possible to consider

* [[12-dimensional supergravity]]

## Phenomenology

Discussion of evidence for supergravity from [[experiment]]/[[phenomenology]] includes the following:

in  ([Dalianis-Farakos 15](#DalianisFarakos15)) it is argued that the [[Starobinsky model of cosmic inflation]], which is strongly preferred by experiment, further improves after embedding into supergravity.

## Related concepts

* [[gravitino condensation]]

* [[gauged supergravity]]

* [[magic supergravity]]

* [[topologically twisted supergravity]]

* [[Rarita-Schwinger field]]

* [[duality in physics]], [[duality in string theory]]

* [[special geometry]]

* [[torsion constraints of supergravity]]

* [[D'Auria-Fre formulation of supergravity]]

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

* [[cosmic inflation]] and [[supersymmetry breaking]] via [[higher curvature corrections]] of supergravity are discussed in the context of the [[Starobinsky model of cosmic inflation]]

[[!include table of branes]]

## References

### General

Early review:

* {#Nieuwenhuizen81} [[Peter van Nieuwenhuizen]], _Supergravity_, Physics Reports, **68,** (1981) 189-398 [doi:10.1016/0370-1573(81)90157-5]

* {#DuffNilssonPope86} [[Mike Duff]], [[Bengt Nilsson]], [[Christopher Pope]], *Kaluza-Klein supergravity*, Physics Reports **130** 1–2 (1986) 1-142 [[spire:229417](https://inspirehep.net/record/229417), doi:10.1016/0370-1573(86)90163-8]

  > (emphasis on [[Kaluza-Klein compactification]])

Textbook accounts:

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

* [[Steven Weinberg]], *Supergravity*, Section 31 in: *The quantum theory of fields. Vol. 3: Supersymmetry*, Cambridge University Press (2000) [[ISBN:9781139632638](https://www.cambridge.org/ae/universitypress/subjects/physics/theoretical-physics-and-mathematical-physics/quantum-theory-fields-volume-3?format=AR&isbn=9780521670555),  [spire:527189](https://inspirehep.net/literature/527189), [pdf](http://www.stat.ucla.edu/~ywu/research/documents/weinberg3.pdf)]

  > "Gravity exists, so if there is any truth to supersymmetry then any realistic supersymmetry theory must eventually be enlarged to a supersymmetric theory of matter and gravitation, known as supergravity. Supersymmetry without supergravity is not an option, though it may be a good approximation at energies far below the Planck scale."

* [[Antoine Van Proeyen]], [[Daniel Freedman]], _Supergravity_, Cambridge University Press (2012) [[doi:10.1017/CBO9781139026833]( https://doi.org/10.1017/CBO9781139026833)]

* [[Pietro Fré]], Ch 6 in: *Black Holes, Cosmology and Introduction to Supergravity*, volume 2 of: *Gravity, a Geometrical Course*, Springer (2013) [[doi:10.1007/978-94-007-5443-0](https://doi.org/10.1007/978-94-007-5443-0)]

Collection of original articles:

* [[Abdus Salam]], [[Ergin Sezgin]] (eds.), *Supergravities in Diverse Dimensions*, Elsevier & World Scientific (1990) [[doi:10.1142/0277](https://doi.org/10.1142/0277)]

Survey:

* Florian Domingo, Michel Rausch de Traubenberg
*Supergravity: Application in Particle Physics*, Handbook of Quantum Gravity (2023) [[arXiv:2209.12541](https://arxiv.org/abs/2209.12541)]

* [[Ergin Sezgin]], *Survey of supergravities*, in: *[[Handbook of Quantum Gravity]]* [[arXiv:2312.06754](https://arxiv.org/abs/2312.06754)]

Lecture notes:

* P. Binetruy, G. Girardi, R. Grimm, _Supergravity couplings: a geometric formulation_, Phys.Rept.343:255-462,2001 ([arXiv:hep-th/0005225](http://arxiv.org/abs/hep-th/0005225))

* [[Friedemann Brandt]], _Lectures on supergravity_ ([arXiv:hep-th/0204035](http://arxiv.org/abs/hep-th/0204035))

* [[Bernard de Wit]], _Supergravity_ ([arXiv:hep-th/0212245](http://arxiv.org/abs/hep-th/0212245))

* [[Antoine Van Proeyen]], _Structure of supergravity theories_ ([arXiv:hep-th/0301005](http://arxiv.org/abs/hep-th/0301005))

* [[Joachim Gomis]], _Three lectures on Supergravity_ ([pdf slides](http://www.fis.puc.cl/~jalfaro/supergravity/Three%20Lectures%20on%20Supergravity.pdf))

Further surveys:

* {#Duff04} [[Michael Duff]], *The status of local supersymmetry*, in: *From Quarks to Black Holes: Progress in understanding the logic of Nature*, World Scientific (2005) 60-116 [[arXiv:hep-th/0403160](http://arxiv.org/abs/hep-th/0403160), [doi:10.1142/9789812701794_0004](https://doi.org/10.1142/9789812701794_0004)]

A fair bit of detail on [[supersymmetry]] and on supergravity is in

* [[Pierre Deligne|P. Deligne]], [[Pavel Etingof|P. Etingof]], [[Dan Freed|D.S. Freed]], L. Jeffrey, [[David Kazhdan|D. Kazhdan]], J. Morgan, D.R. Morrison, [[Edward Witten|E. 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))

The original article that introduced the [[D'Auria-Fre formulation of supergravity]] is

* [[Riccardo D'Auria]], [[Pietro Fré]], _[[GeometricSupergravity.pdf:file]]_, Nuclear Physics B201 (1982) 101-140

The underlying [[super Cartan geometry]] is made fully explicit in:

* [[Konstantin Eder]], _Super Cartan geometry and the super Ashtekar connection_ ([arXiv:2010.09630](https://arxiv.org/abs/2010.09630))

A compendium, of relevant [[action functionals]] and [[equations of motion]] is in

* M. J. D. Hamilton, _The field and Killing spinor equations of M-theory and type IIA/IIB supergravity in coordinate-free notation_ ([arXiv:1607.00327](http://arxiv.org/abs/1607.00327))

On [[black hole]]-solutions:

* [[Riccardo D'Auria]], [[Pietro Fre]], _BPS black holes in supergravity_, ([hep-th/9812160](http://arxiv.org/abs/hep-th/9812160))

* Antonio Gallerati, _Constructing black hole solutions in supergravity theories_ ([arXiv:1905.04104](https://arxiv.org/abs/1905.04104))

On [[pure spinor]]-techniques:

* [[Martin Cederwall]], *Pure spinors in classical and quantum supergravity*, in *[[Handbook of Quantum Gravity]]* (2023) [[arXiv:2210.06141](https://arxiv.org/abs/2210.06141)]

On [[higher curvature corrections]]:

* Mehmet Ozkan, Yi Pang, [[Ergin Sezgin]], *Higher Derivative Supergravities in Diverse Dimensions* [[arXiv:2401.08945](https://arxiv.org/abs/2401.08945)]

### Renormalization

* S. Deser, J.H. Kay, K.S. Stelle, _Renormalizability Properties of Supergravity_, Phys Rev Lett 38, 527 (1977) (reproduced as [arXiv:1506.03757](http://arxiv.org/abs/1506.03757))

### U-duality
 {#UDualityReferences}

Some basic facts are recalled in

* [[Jacques Distler]], _Split real forms_ ([blog post](http://golem.ph.utexas.edu/~distler/blog/archives/001213.html)).

The $E_{7(7)}$-symmetry was first discussed in

* [[Bernard de Wit]], [[Hermann Nicolai]], _$D = 11$ Supergravity With Local $SU(8)$ Invariance_, Nucl. Phys.
B 274, 363 (1986)

and $E_{8(8)}$ in

* [[Hermann Nicolai]], _$D = 11$ Supergravity with Local $SO(16)$ Invariance_ , Phys. Lett. B 187, 316 (1987).

* K. Koepsell, [[Hermann Nicolai]], [[Henning Samtleben]], _An exceptional geometry for $d = 11$
supergravity?_, Class. Quant. Grav. 17, 3689 (2000) ([arXiv:hep-th/0006034](http://arxiv.org/abs/hep-th/0006034)).

The discrete quantum subgroups were discussed in

* [[Chris Hull]], [[Paul Townsend]], _Unity of Superstring Dualities_ Nucl.Phys.B438:109-137 (1995) ([arXiv:hep-th/9410167](http://arxiv.org/abs/hep-th/9410167))

which also introduced the term "U-duality".

Review and further discusssion is in

* Shun'ya Mizoguchi, Germar Schroeder, _On Discrete U-duality in M-theory_, Class.Quant.Grav. 17 (2000) 835-870 ([arXiv:hep-th/9909150](http://arxiv.org/abs/hep-th/9909150))

A careful discussion of the topology of the U-duality groups is in

* [[Arjan Keurentjes]], _The topology of U-duality (sub-)groups_ ([arXiv:hep-th/0309106](http://arxiv.org/abs/hep-th/0309106))

* [[Arjan Keurentjes]], _U-duality (sub-)groups and their topology_ ([arXiv:hep-th/0312134](http://arxiv.org/abs/hep-th/0312134))

A discussion in the context of [[generalized complex geometry]] / [[exceptional generalized complex geometry]] is in

* Paulo Pires Pacheco, [[Daniel Waldram]], _M-theory, exceptional generalised geometry and superpotentials_ ([arXiv:0804.1362](http://arxiv.org/abs/0804.1362))

* Nicholas Houston, _Supergravity and Generalized Geometry_ Thesis (2010) ([pdf](https://workspace.imperial.ac.uk/theoreticalphysics/Public/MSc/Dissertations/2010/Nicholas%20Houston%20Dissertation.pdf))

The case of "$E_{10}$" is discussed in

* [[Thibault Damour]], [[Marc Henneaux]], [[Hermann Nicolai]], _$E(10)$ and a 'small tension expansion' of M
theory_, Phys. Rev. Lett. 89, 221601 (2002) ([arXiv:hep-th/0207267](http://arxiv.org/abs/hep-th/0207267));

* [[Axel Kleinschmidt]], [[Hermann Nicolai]], _$E(10)$ and $SO(9,9)$ invariant supergravity_, JHEP 0407,
041 (2004) ([arXiv:hep-th/0407101](http://arxiv.org/abs/hep-th/0407101))

and that of "$E_{11}$" in

* [[Peter West]], _$E_{11}$ and M-theory_, Class. Quant. Grav. 18, 4443 (2001) ([arXiv:hep-th/0104081](http://arxiv.org/abs/hep-th/0104081)).

General discussion of the [[Kac-Moody groups]] arising in this context is for instance in

* Philipp Fleig, [[Axel Kleinschmidt]], _Eisenstein series for infinite-dimensional U-duality groups_ ([arXiv:1204.3043](http://arxiv.org/abs/1204.3043))

### Gauged supergravity

* Natxo Alonso-Alberca; and Tomáas Ortín, _Gauged/Massive supergravities in diverse dimensions_ ([pdf](http://digital.csic.es/bitstream/10261/38952/1/ARTICULOS302400%5B1%5D.pdf))

### Chern-Simons supergravity

A survey of the [[Chern-Simons gravity]]-style action functionals for supergravity is in

* [[Jorge Zanelli]], _Lecture notes on Chern-Simons (super-)gravities_ ([arXiv:0502193](http://arxiv.org/abs/hep-th/0502193))
{#Zanelli}

### History
 {#History}

The idea of supergravity was proposed by

* [[Dmitry Volkov]], V.P. Akulov, _Possible universal neutrino interaction_, ZhETF Pis. Red. (JETP Letters, AIP translation), 16, n.11 (1972) 621 ([pdf](http://www.jetpletters.ac.ru/ps/1766/article_26864.shtml))

followed up by the first model of supergravity (in nonlinear realization) constructed in

* [[Dmitry Volkov]], [[Vyacheslav Soroka]], _Higgs effect for Goldstone particles with spin 1/2_, ZhETF Pis. Red. (JETP Letters, AIP translation), 18, n.8 (1973) 529 ([pdf](https://www.jetpletters.ac.ru/ps/1568/article_24038.shtml))

However, the term "supergravity" was coined later by
[Freedman, Nieuwenhuizen, Ferrara 76](#FreedmanNieuwenhuizenFerrara76), whose work on supergravity is regarded as foundational for the subject.

This early history is discussed in

* [[Steven Duplij]], _Supergravity was discovered by D.V. Volkov and V.A. Soroka in 1973, wasn't it?_, East Eur. J. Phys., v3, p. 81-82 (2019) ([arXiv:1910.03259](https://arxiv.org/abs/1910.03259))

* {#Kuzenko21} Sergei M. Kuzenko, *Local supersymmetry: Variations on a theme by Volkov and Soroka* ([arXiv:2110.12835](https://arxiv.org/abs/2110.12835))

Supergravity, in the guise of [[4d supergravity]], was first found (constructed) in

* {#FreedmanNieuwenhuizenFerrara76} [[Daniel Freedman]], [[Peter van Nieuwenhuizen]], [[Sergio Ferrara]], _Progress toward a theory of supergravity_, Phys. Rev. D13 (1976) 3214 ([doi.org/10.1103/PhysRevD.13.3214](https://doi.org/10.1103/PhysRevD.13.3214))

Accounts of the early history include the following:

* [[Sergio Ferrara]], _Supergravity and the quest for a unified theory_ ([arxiv:hep-th/9405065](https://arxiv.org/abs/hep-th/9405065))

* [[Dmitry Volkov]], _Supergravity before 1976_,
International Conference on the History of Original Ideas and Basic Discoveries in Particle Physics, Erice, 1994 ([Springer, Chapter](http://link.springer.com/chapter/10.1007/978-1-4613-1147-8_34) or [arXiv:hep-th/9410024](http://arxiv.org/abs/hep-th/9410024))

* R. Arnowitt, [[Ali Chamseddine]], Pran Nath, _The Development of Supergravity Grand Unification: Circa 1982-85_ ([arXiv:1206.3175](http://arxiv.org/abs/1206.3175))

* [[Dmitry Volkov]], _Supergravity before and after 1976. The story of goldstonions_, in  [Concise Encyclopedia of Supersymmetry,
S. Duplij, J. Bagger, W. Siegel (Eds.), Springer, 2004](http://www.springer.com/gp/book/9781402013386) or [arXiv:hep-th/9404153](http://arxiv.org/abs/hep-th/9404153)

* David Appell, _When supergravity was born_, 2012 ([pdf](http://www.davidappell.com/articles/PWSep12appell-supergravity.pdf))

* [[Peter van Nieuwenhuizen]], _Aspects of supergravity_, 2014 ([pdf](http://media.scgp.stonybrook.edu/presentations/20140109_vanNieuwenhuizen.pdf))

* [[Vyacheslav Soroka]], _The Sources of Supergravity_ in [The Supersymmetric World. The Beginnings of the Theory, G. Kane and M. Shifman (Eds.), World Scientific, 2000](http://www.worldscientific.com/doi/10.1142/9789812385505_0011) or [arXiv:hep-th/0203171](https://arxiv.org/abs/hep-th/0203171)

* [[Sergio Ferrara]], A. Sagnotti, _Supergravity at 40: Reflections and Perspectives_, Based in part on the talk delivered by S. F. at the "Infeld Colloquium and Discrete", in Warsaw, on December 1 2016, and on a joint CERN Courier article. Dedicated to [[John Schwarz]] on the occasion of his 75-th birthday ([arXiv:1702.00743](https://arxiv.org/abs/1702.00743))

* [[Stanley Deser]], _A brief history of Supergravity: the first three weeks_ ([arXiv:1704.05886](https://arxiv.org/abs/1704.05886))

See also _[supersymmetry -- History](supersymmetry#ReferencesHistory)_.

### Related

Further physics monographs on supergravity include

* I. L. Buchbinder, S. M. Kuzenko, _Ideas and methods of supersymmetry and supergravity; or A walk through superspace_, [googB](http://books.google.com/books?isbn=0750305061)

* [[Julius Wess]], Jonathan Bagger, _Supersymmetry and supergravity_, 1992

* [[Steven Weinberg]], _Quantum theory of fields_, volume III: supersymmetry

* [Concise Encyclopedia of Supersymmetry, S. Duplij, J. Bagger, W. Siegel (Eds.), Springer, 2004](http://www.springer.com/gp/book/9781402013386), [SUSY story](http://ivv5hpp.uni-muenster.de/u/douplii/susy/SUSYEnc_Story.pdf) narrated by its founders

The [[Cauchy problem]] for classical solutions of simple supergravity has been discussed in

* [[Yvonne Choquet-Bruhat]], _[[The Cauchy Problem in Classical Supergravity]]_, Letters in Mathematical Physics 7 (1983) 459-467. 0377

A canonical textbook reference for the role of Calabi-Yau manifolds in compactifications of 10-dimensional [[supergravity]] is volume II, starting on page 1091 in

* [[P. Deligne]], [[P. Etingof]], [[D. Freed]], L. Jeffrey, [[D. Kazhdan]], J. Morgan, D. R. Morrison and [[E. 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))

Discussion of solutions with $N = 1$ global supersymmetry left and their relation to Calabi-Yau compactifications are for instance in

* {#GMPT} [[Mariana Graña]], [[Ruben Minasian]], Michela Petrini, [[Alessandro Tomasiello]], _Generalized structures of $N=1$ vacua_, [arXiv:hep-th/0505212](http://arxiv.org/abs/hep-th/0505212)

### Phenomenology

Discussion of implications of supergravity for [[phenomenology]]/[[cosmology]] includes

* {#DalianisFarakos15} Ioannis Dalianis, [[Fotis Farakos]], _On the initial conditions for inflation with plateau potentials: the $R + R^2$ (super)gravity case_ ([arXiv:1502.01246](http://arxiv.org/abs/1502.01246))

[[!redirects supergravity]]
[[!redirects supergravities]]
[[!redirects Supergravity]]

[[!redirects super-gravity]]
[[!redirects super-gravities]]


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This entry contains material related to the textbook

* [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]],

  _Supergravity and Superstrings - A Geometric Perspective_

  World Scientific, 1991

  [doi:10.1142/0224](https://doi.org/10.1142/0224)

  [epdf: vol 1](https://epdf.pub/supergravity-and-superstrings-a-geometric-perspective-vol-1-mathematical-foundat.html), [epdf: vol 2](https://epdf.pub/supergravity-and-superstrings-a-geometric-perspective-vol-2-supergravity.html), [epdf: vol 3](https://epdf.pub/supergravity-and-superstrings-a-geometric-perspective-vol-3-superstrings.html)

on [[supergravity]] and [[string theory]] with an emphasis on the [[D'Auria-Fré formulation of supergravity]], based on

* {#DAuriaFre}  [[Riccardo D'Auria]], [[Pietro Fré]] _[[GeometricSupergravity.pdf:file]]_, Nuclear Physics B201 (1982) 101-140

The book is out of print, unfortunately, but check $gen.lib$. The newer book

* [[Pietro Fré]], _Gravity, a Geometrical Course: Volume 2: Black Holes, Cosmology and Introduction to Supergravity_, Springer 2012

is much more brief in comparison, but has more history and exposition, and treats more recent developments, such as aspects related to [[AdS-CFT]].

## Description

This book focuses on the discussion of [[supergravity]]-aspects of [[string theory]] from the point of view of the [[D'Auria-Fré formulation of supergravity]], which is implicitly a formulation via [[supergeometry|supergeometry]] [[higher Cartan geometry]]. Therefore, while far, far from being written in the style of a mathematical treatise, this book stands out as making a consistent proposal for what the central ingredients of a mathematical formalization might be: as explained at the above link, secretly this book is all about describing supergravity in terms of [[connection on an infinity-bundle|infinity-connections]] with values in [[super L-infinity algebra]]s such as the [[supergravity Lie 3-algebra]].

See also [[higher category theory and physics]].

## Further references

The original article that introduced the D'Auria-Fré-formalism is

* **geSuGra** R. D'Auria, P. Fré _Geometric supergravity in $D = 11$ and its hidden supergroup_ [[GeometricSupergravity.pdf:file]]

The _geometric perspective_ discussed there is both the emphasis of working over base [[supermanifold]]s and combined with that the the approach that here we call the _D'Auria-Fré-formalism_ .

The interpretation of the D'Auria-Fré-formalism in terms of [[∞-Lie algebra valued forms]] together with a discussion of the [[supergravity Lie 3-algebra]] in the context of [[String Lie 2-algebra|String Lie n-algebra]]s was given in

* [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _$L_\infty$-algebra connections_ (web)

Apart from that the first vague mention of the observation that the "FDA"-formalism for supergravity is about higher categorical Lie algebras (as far as [[Urs Schreiber|I]] am aware, would be grateful for further references) is page 2 of

* [[Pietro Fré]], [[Pietro Antonio Grassi]], _Free differential algebras, rheonomy and pure spinors_ ([arXiv:0801.3076](http://arxiv.org/abs/0801.3076))

An attempt at a comprehensive discussion of the formalism in the context of [[cohesive (∞,1)-topos]]-theory for
[[smooth super ∞-groupoid]]s is in the last section of

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

Here are some more references:

* [[Pietro Fré]], _M-theory FDA, twisted tori and Chevalley cohomology_ ([arXiv] (http://www.arxiv.org/abs/hep-th/0510068))

* [[Pietro Fré]], [[Pietro Antonio Grassi]],
_Pure spinors, free differential algebras, and the supermembrane_ ([arXiv:hep-th/0606171] (http://www.arxiv.org/abs/hep-th/0606171))

* [[Pietro Fré]], [[Pietro Antonio Grassi]], _Free differential algebras, rheonomy, and pure spinors_ ([arXiv:0801.3076](http://www.arxiv.org/abs/0801.3076))

category: reference