super Cartesian space

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

#Contents#
* table of contents
{:toc}

## Idea

The analog of a [[Cartesian space]] in [[supergeometry]], a [[supermanifold]] with a single canonical [[coordinate chart]].

For suitable even|odd dimension this is naturally equipped with the structure of a [[supergroup]] that makes it a [[super-translation group]]. For Minkowski signature this is [[super-Minkowski spacetime]].

## Definition

For $p,q \in \mathbb{N}$, let $\mathbb{R}^{p|q}$ the _super Cartesian space_ of [[dimension]]_ $(p|q)$. This is the [[supermanifold]] defined by the fact that its [[algebra of functions]] is freely generated, as a [[smooth superalgebra]] by even-graded [[coordinate]]-functions $\langle \{x^a\}_{1}^p$ and odd-graded coordinate function $\{ \theta^\alpha\}_{\alpha= 1}^q$. Equivalently this is the [[tensor product]] of the [[smooth functions]] on $\mathbb{R}^p$ with the real [[Grassmann algebra]] on $q$ generators:

$$
  \begin{aligned}
      C^\infty(\mathbb{R}^{p|q})
      & \coloneqq
      \langle \{x^a\}_{1}^p, \{ \theta^\alpha\}_{\alpha= 1}^q \rangle
      \\
      & = C^\infty(\mathbb{R}^p)\otimes_{\mathbb{R}} \wedge^\bullet \mathbb{R}^q
  \end{aligned}
  \,.
$$

## Details

The following is taken from _[[geometry of physics -- supergeometry]]_, see there for more:

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

For $q\in \mathbb{N}$, the real [[Grassmann algebra]]

$$
  C^\infty(\mathbb{R}^{0|q})
  \coloneqq
  \wedge^\bullet \langle \theta^1, \cdots, \theta^q\rangle
$$

is the $\mathbb{R}$-algebra [[free functor|freely]] generated from $q$ generators $\{\theta^i\}_{i = 1}^q$ subject to the relations

$$
  \theta^i \theta^j = - \theta^j \theta^i
$$

for all $i,j \in \{1,\cdots, q\}$.

For $p,q \in \mathbb{N}$, the [[super-Cartesian space]] $\mathbb{R}^{p|q}$ is the [[formal dual]] of the [[supercommutative superalgebra]] written $\mathcal{O}(\mathbb{R}^{p\vert q})$ or $C^\infty(\mathbb{R}^{p|q})$ whose underlying
$\mathbb{Z}/2\mathbb{Z}$-[[graded vector space]] is

$$
  C^\infty(\mathbb{R}^{p|q})
  \coloneqq
  C^\infty(\mathbb{R}^p) \otimes_{\mathbb{R}} \wedge^\bullet\langle\theta^1, \cdots, \theta^q\rangle
$$

with the product given by the relations

$$
  \begin{aligned}
  \left(
    f \theta^{i_1}\cdots \theta^{i_k}
  \right)
  \left(
    g \theta^{j_1}\cdots \theta^{j_l}
  \right)
  & =
  f \cdot g \; \theta^{i_1}\cdots \theta^{i_k} \theta^{j_1}\cdots \theta^{j_l}
  \\
  & =
  (-1)^{k l} g\cdot f \; \theta^{j_1}\cdots \theta^{j_l} \theta^{i_1}\cdots \theta^{i_k}
  \end{aligned}
$$

where $f \cdot g$ is the ordinary pointwise product of [[smooth functions]].

Write

$$
 SuperCartSp \hookrightarrow sCAlg_{\mathbb{R}}^{op}
$$

for the [[full subcategory]] of the [[opposite category]] of [[commutative superalgebras]] on those of this form. We write
$\mathbb{R}^{p|q} \in SuperCartSp$ for the [[formal dual]] of
$C^\infty(\mathbb{R}^{p|q})$.

=--

We write

$$
  CartSp \hookrightarrow SuperCartSp
$$

for the [[full subcategory]] on ordinary [[Cartesian spaces]] with [[smooth functions]] between them.
These are the "abstract coordinate charts" from the discussion at _[[geometry of physics -- smooth sets]]_,
and so we are evidently entitled to think of the objects in $SuperCartSp$ as **abstract super coordinate systems**
and to develop a geometry induced from these.

Recall the two magic algebraic properties of [[smooth functions]] that make the above algebraic description of
[[differential geometry]] work:

1. (**[[embedding of smooth manifolds into formal duals of R-algebras]]**) The functor that assigns [[algebra of functions|algebras of]] [[smooth function]] of [[smooth manifolds]]

   $$
     C^\infty(-) \;\colon\; SmthMfd \hookrightarrow CAlg_{\mathbb{R}}^{op}
   $$

   is [[fully faithful functor|fully faithful]].

1. (**[[smooth Serre-Swan theorem]]**) The functor that assigns smooth [[sections]] of smooth [[vector bundles]]
   of [[finite number|finite]] [[rank]]

   $$
     \Gamma_X(-) \;\colon\; SmthVectBund_{/X} \hookrightarrow C^\infty(X) Mod
   $$

   is [[fully faithful functor|fully faithful]] (its [[essential image]] being the [[finitely generated object|finitely generated]] [[projective modules]] over the $\mathbb{R}$-[[algebra of functions|algebra of]] [[smooth function]]).

There is a third such magic algebraic property of smooth functions, which plays a role now:

+-- {: .num_defn #DerivationsOfSmoothFunctions}
###### Proposition
**([[derivations of smooth functions are vector fields]])**

Let $X$ be a [[smooth manifold]]. Write

$$
  der_X \;\colon\; \Gamma(T X) \longrightarrow Der(C^\infty(X))
$$

for the function that sends a smooth [[vector field]] $v \in \Gamma(T X)$ to the [[derivation]]
of the [[algebra of functions|algebra of]] [[smooth functions]] on $X$ given by forming
[[derivatives]]: $der_X(v)(f) \coloneqq v(f)$. This is a [[derivation]] by the [[chain rule]].

Then this function is a [[bijection]], hence every [[derivation]] of $C^\infty(X) \in CAlg_{\mathbb{R}}$
comes from [[differentiation]] along some smooth vector field, which is uniquely defined thereby.

=--

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

By the existence of [[partitions of unity]] we may restrict to the situation where $X = \mathbb{R}^n$ is a [[Cartesian space]].
By the [[Hadamard lemma]] every [[smooth function]] $f \in C^\infty(\mathbb{R}^n)$ may be written as

$$
  f(x) = f(0) + \sum_i x_i g_i(x)
$$

for [[smooth functions]] $\{g_i \in C^\infty(X)\}$ with $g_i(0) = \frac{\partial f}{\partial x_i}(0)$. Since any derivation $\delta : C^\infty(X) \to C^\infty(X)$ by definition satisfies the Leibniz rule, it follows that

$$
  \delta(f)(0) = \sum_i \delta(x_i) \frac{\partial f}{\partial x_i}(0)
  \,.
$$

Similarly, by translation, at all other points. Therefore $\delta$ is already fixed by its action of the coordinate functions $\{x_i \in C^\infty(X)\}$. Let $v_\delta \in T \mathbb{R}^n$ be the [[vector field]]

$$
  v_\delta \coloneqq \sum_i \delta(x_i) \frac{\partial}{\partial x_i}
$$

then it follows that $\delta$ is the derivation coming from $v_\delta$ under $\Gamma_X(T X) \to Der(C^\infty(X))$.
=--

Recall further from _[[geometry of physics -- superalgebra]]_ that the category of [[supercommutative superalgebras]]
is related to that of ordinary [[commutative algebras]] over $\mathbb{R}$ by an
[[adjoint cylinder]] ([this prop.](geometry+of+physics+--+superalgebra#InclusionOfCAlgIntosCAlg)):

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

The canonical inclusion of [[commutative algebras]] into [[supercommutative superalgebra]] is
part of an [[adjoint triple]] of the form

$$
  CAlg_k
    \underoverset
      {\underset{(-)_{even}}{\longleftarrow}}
      {\overset{(-)/(-)_{odd}}{\longleftarrow}}
      {\hookrightarrow}
  sCAlg_k
  \,.
$$

The [[formal dual]] of this statement is that [[affine superschemes]] are related to ordinary [[affine schemes]]
over $\mathbb{R}$ by an [[adjoint cylinder]] of this form

$$
  Aff(Vect_k)
    \underoverset
      {\underset
        {
          \phantom{AA}
          \overset
            {\phantom{A}}
            {\overset{\rightsquigarrow}{(-)}}
          \phantom{AA}
         }
         {\longleftarrow}
      }
      {\overset
         {
          \phantom{AA}
           \underset
            {\phantom{A}}
            { \overset{\rightrightarrows}{(-)} }
          \phantom{AA}
        }
        {\longleftarrow}
      }
      {\hookrightarrow}
  Aff(sVect_k)
  \,.
$$

(Beware that $\overset{\rightsquigarrow}{(-)}$ is the formal dual of $(-)/(-)_{odd}$ while $\overset{\rightrightarrows}{(-)}$ is the
formal dual of $(-)_{even}$. That they change position in the diagrams is because we always draw [[left adjoints]] on top of [[right adjoints]] and  the handedness of [[adjoints]] changes as we pass to [[opposite categories]].)

=--

The notation in prop. \ref{AdjointCylinderOnSuperAffines} is to serve as a convenient mnemonic for the nature of these functors:

In a [[Feynman diagram]]

1. a single [[fermion]] is denoted by a solid arrow "$\to$" and the functor
$\overset{\rightrightarrows}{(-)}$ produces a space whose [[algebra of functions]] is generated over
smooth functions by the product of two fermionic functions

1. a single [[boson]] is denoted by a wiggly arrow $\rightsquigarrow$ and the functor
   denoted by this symbol produces a spaces whose [[algebra of functions]] contains only the
   bosonic smooth functions, no odd-graded functions.

This highlights an important point: while the image of a [[super Cartesian space]] under $\rightsquigarrow$
is an ordinary [[Cartesian space]]

$$
   \overset{\rightsquigarrow}{\mathbb{R}^{p\vert q}}
   \simeq
   \mathbb{R}^p
$$

its image under $\overset{\rightrightarrows}{(-)}$ is a bosonic space, but not an ordinary manifold. For instance

$$
  \begin{aligned}
   \mathcal{O}\left( \overset{\rightrightarrows}{\mathbb{R}^{0\vert 2}}\right)
    & =
   \mathcal{O}(\mathbb{R}^{0\vert 2})_{even}
   \\
  & =
  (\wedge^\bullet(\mathbb{R}^2)^\ast)_{even}
  \\
  &=
  \left\{a_0 + a_1 \, \theta_1 + a_2 \, \theta_2 + a_{12} \theta_1 \theta_2 \,\vert a_\bullet \in \mathbb{R}\,\right\}_{even}
  \\
  &=
  \left\{a_0  + a_{12} \theta_1 \theta_2 \,\vert a_{12} \in \mathbb{R}\,\right\}
  \\
  &=
  \mathbb{R}[\epsilon]/(\epsilon^2 = 0)
  \end{aligned}
$$

where in the last line we renamed $\theta_1 \theta_2$ to $\epsilon$.

This algebra $\mathbb{R}[\epsilon]/(\epsilon^2)$
is known as the **[[algebra of dual numbers]]** over $\mathbb{R}$. It is to be thouhgt of as the [[algebra of functions]]
on a bosonic but [[infinitesimally thickened point]], a 1-dimensional neighbourhood of a point which is "so very small"
that the canonical [[coordinate]] function $\epsilon$ on it takes values "so tiny" that its square, which is bound to be
even tinier, is actually indistinguishable from zero.

In generalization of this we make the following definitions

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

Write

$$
  InfPoint \hookrightarrow CAlg_{\mathbb{R}}^{op}
$$

for the [[full subcategory]] of the [[opposite category]] of [[commutative algebras]] over $\mathbb{R}$ on [[formal duals]] of [[commutative algebras]] over the [[real numbers]] of the form $\mathbb{R}\oplus V$ with $V$ a [[nilpotent ideal]] of [[finite number|finite]]-[[dimension]]
over $\mathbb{R}$. We call this the category of _[[infinitesimally thickened points]]_.

In [[synthetic differential geometry]] these algebras ar called **[[Weil algebra]]**, while in [[algebraic geometry]] they are known as **local [[Artin algebras]]** over $\mathbb{R}$.

Write moreover

$$
  FormalCartSp \coloneqq CartSp \rtimes InfPoint \hookrightarrow CAlg_{\mathbb{R}}^{op}
$$

for the [[full subcategory]] on [[formal duals]] of those algebras which are [[tensor products]] of commutative $\mathbb{R}$-algebras of the form

$$
  C^\infty(\mathbb{R}^n) \otimes C^\infty(D)
$$

of algebras $C^\infty(\mathbb{R}^p)$ of [[smooth functions]] $\mathbb{R}^n$
with algebras corresponding to infinitesimally thickened points $D$ as above.

=--

This kind of construction is traditionally more familiar from the theory of [[formal schemes]], but the same kind of general abstract theory goes through in the context of [[differential geometry]], a point of view known as _[[synthetic differential geometry]]_.

The crucial property of [[infinitesimally thickened points]] (def. \ref{FormalCartSp}) is that they co-represent
[[tangent vectors]] and [[jets]]:

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

Write $\mathbb{D}^1 = Spec(\mathbb{R}[\epsilon]/(\epsilon^2))$ for the [[formal dual]] of the [[algebra of dual numbers]]. Then morphisms

$$
  \mathbb{R}^n \times \mathbb{D}\longrightarrow \mathbb{R}^n
$$

which are the identity after restriction along $\mathbb{R}^n \to \mathbb{R}^n \times \mathbb{D}$, are equivalently algebra homomorphisms of the form

$$
  (C^\infty(\mathbb{R}^n) \oplus \epsilon C^\infty(\mathbb{R}^n))
  \longleftarrow
  C^\infty(\mathbb{R}^n)
$$

which are the identity modulo $\epsilon$. Such a morphism has to take any function $f \in C^\infty(\mathbb{R}^n)$ to

$$
  f + (\partial f) \epsilon
$$

for some smooth function $(\partial f) \in C^\infty(\mathbb{R}^n)$. The condition that this assignment makes an algebra homomorphism is equivalent to the statement that for all $f_1,f_2 \in C^\infty(\mathbb{R}^n)$

$$
  (f_1 f_2 + (\partial (f_1 f_2))\epsilon )
  =
  (f_1 + (\partial f_1) \epsilon)
  (f_2 + (\partial f_2) \epsilon)
  \,.
$$

Multiplying this out and using that $\epsilon^2 = 0$ this in turn is equivalent to

$$
  \partial(f_1 f_2) = (\partial f_1) f_2 + f_1 (\partial f_2)
  \,.
$$

This in turn means equivalently that $\partial\colon C^\infty(\mathbb{R}^n)\to C^\infty(\mathbb{R}^n)$ is a [[derivation]].
But [[derivations of smooth functions are vector fields]] (prop. \ref{DerivationsOfSmoothFunctions}).

In particular one finds that maps

$$
  \mathbb{D} \longrightarrow \mathbb{R}^n
$$

are equivalently single [[tangent vectors]], hence for every $\mathbb{R}^n$ there is a [[natural bijection]]

$$
  Hom_{FormalCartSp}(\mathbb{D}^1, \mathbb{R}^n)
   \simeq
  \underset{\text{underlying} \atop \text{set}}{\underbrace{T \mathbb{R}^n}}
$$

between the [[hom-set]] from the formal dual of the [[ring of dual numbers]]
and the set of [[tangent vectors]].

=--

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

The canonical inclusion $i$ of the category of ordinary [[Cartesian spaces]] into that of [[formal manifold|formal]] [[Cartesian spaces]]
has a [[left adjoint]] $\Re$

$$
  CartSp
    \underoverset
      {\underset{\Re}{\longleftarrow}}
      {\overset{i}{\hookrightarrow}}
      {\phantom{AAAA}}
  FormalCartSp
$$

given by

$$
  \Re(\mathbb{R}^n \times \mathbb{D}) \coloneqq \mathbb{R}^n
  \,.
$$

Hence exhibits $CartSp$ as a [[coreflective subcategory]] of that of formal cartesian spaces.

We say that $\mathbb{R}^n$ is the **[[reduced scheme]]** of $\mathbb{R}^n \times \mathbb{D}$.

=--

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

We check the [[natural isomorphism]] on [[hom-sets]] that characterizes a pair of [[adjoint functors]]:

By definition, a morphism of the form

$$
  f \;\colon\; i(\mathbb{R}^{n_1}) \longrightarrow \mathbb{R}^{n_2} \times \mathbb{D}
$$

is equivalently a [[homomorphism]] of [[commutative algebras]] of the form

$$
  f^\ast
   \;\colon\;
  C^\infty(\mathbb{R}^{n_1})
    \longleftarrow
  C^\infty(\mathbb{R}^{n_2}) \otimes_{\mathbb{R}} (\mathbb{R} \oplus V)
$$

where all elements $v \in V$ are nilpotent, in that there exists $n_v \in \mathbb{N}$
such that $(v)^{n_v} = 0$. Every algebra homomorphism needs to preserve this
equation, and hence needs to send nilpotent elements to nilpotent elements. But the
only nilpotent element in the ordinary function algebra $C^\infty(\mathbb{R}^n)$
is the zero-function, and so it follows that the above homomorphism has to vanish on all of $V$, hence has
to factor (necessarily uniquely) through a homomorphism of the form

$$
  \tilde f^\ast
   \;\colon\;
  C^\infty(\mathbb{R}^{n_1})
    \longleftarrow
  C^\infty(\mathbb{R}^{n_2})
  \,.
$$

This is dually a morphism of the form

$$
  \tilde f \;\colon\; \mathbb{R}^{n_1} \longrightarrow \mathbb{R}^{n_2}
$$

in $CartSp$. This establishes a natural bijection $f \leftrightarrow \tilde f$.

=--

The above discussion following prop. \ref{AdjointCylinderOnSuperAffines} means that in passing to commutative superalgebras there are _two_ stages of generalizations of plain differential geometry involved:

1. [[Cartesian spaces]] are generalized to [[formal manifold|formal]] [[Cartesian spaces]];

1. [[formal manifold|formal]] [[Cartesian spaces]] are further generalized to [[super formal Cartesian spaces]].

In order to make this explicit, it is convenient to introduce the following slight generalization of
[[super Cartesian spaces]] (def. \ref{SuperCartesianSpace}), which are simply [[Cartesian products]]
of ordinary [[Cartesian spaces]] with an [[infinitesimally thickened point]] that may have both even and
odd graded elements in its [[algebra of functions]].

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

Write

$$
  SuperFormalCartSp
    \hookrightarrow
  sCAlg_{\mathbb{R}}^{op}
$$

for the [[full subcategory]] of that of the [[opposite category]] of [[supercommutative superalgebras]]
on those of the form

$$
  C^\infty(\mathbb{R}^{p|q}) \otimes_{\mathbb{R}} (\mathbb{R} \oplus V)
$$

where $V$ is a [[nilpotent ideal]] of [[finite number|finite]] [[dimension]] over $\mathbb{R}$.

=--

One place where such super formal Cartesian spaces are made explicit is in [Konechny-Schwarz 97](#KonechnySchwarz97)

In conclusion we have the following situation:

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

The [[coreflective subcategory]] inclusion of [[Cartesian spaces]] into
[[formal manifold|formal]] [[Cartesian spaces]] from prop. \ref{CartSpCoreflectiveInclusion}
and the coreflective as we all [[reflective subcategory]] inclusion
of affine schemes into [[affine superschemes]] from prop. \ref{AdjointCylinderOnSuperAffines}
combine to give the following system of [[adjoint functors]] on our local model spaces

$$
  \array{
   {\text{} \atop {\text{differential} \atop \text{geometry}}}
   &&
   {\text{formal} \atop {\text{differential} \atop {geometry}}}
   &&
   {\text{super} \atop {\text{differential} \atop \text{geometry}}}
   \\
   \\
  CartSp
   &
    \underoverset
      {\underset{\Re}{\longleftarrow}}
      {\overset{}{\hookrightarrow}}
      {}
   &
  FormalCartSp
  &
    \underoverset
      {\underset
        {
          \phantom{AA}
          \overset
            {\phantom{A}}
            {\overset{\rightsquigarrow}{(-)}}
          \phantom{AA}
         }
         {\longleftarrow}
      }
      {\overset
         {
          \phantom{AA}
           \underset
            {\phantom{A}}
            { \overset{\rightrightarrows}{(-)} }
          \phantom{AA}
        }
        {\longleftarrow}
      }
      {\hookrightarrow}
  &
  SuperFormalCartSp
  }
  \,.
$$

=--

## Examples

* [[superpoint]], [[odd line]]

* [[super Minkowski spacetime]]

## Properties

### De Rham complex of differential forms

We discuss the [[de Rham complex]] of [[super differential forms]] on a super Cartesian space. (See also at _[[signs in supergeometry]]_.)

Accordingly, by the discussion at _[[Kähler forms]]_, the [[de Rham complex]] of [[super differential forms]] on $\mathbb{R}^{p|q}$ is freely generated as a super-[[module]] over the [[smooth superalgebra]] $C^\infty(\mathbb{R}^{p|q})$ by expressions of the form $\mathbf{d}x^{a_1} \wedge \cdots \wedge \mathbf{d}x^{a_k} \wedge \mathbf{d}\theta^{\alpha_1} \wedge \cdots \wedge \mathbf{d}\theta^{\alpha_l}$.

$$
  \Omega^\bullet(\mathbb{R}^{p|q})
  =
  C^\infty(\mathbb{R}^{p|q})
  \left[
   \left\{
     \mathbf{d}x^{a_1} \wedge \cdots \wedge \mathbf{d}x^{a_k}
     \wedge
     \mathbf{d}\theta^{\alpha_1} \wedge \cdots \wedge \mathbf{d}\theta^{\alpha_l}
   \right\}
  \right]
  \,.
$$

This [[de Rham complex]] now carries the structure of a

* differential graded-commutative graded commutative superalgebra

which should be thought of as bracketet as follows

* differential graded-commutative graded (commutative superalgebra).

This means in effect that elements of $\Omega^\bullet(\mathbb{R}^{p|q})$ carry a $\mathbb{Z} \times \mathbb{Z}_2$-[[graded object|grading]], where we may say that

* $\mathbb{Z}$ corresponds to the "cohomological grading";

* $\mathbb{Z}_2$ corresponds to the super-grading.

We write

$$
  (n,\sigma)\in \mathbb{Z} \times \mathbb{Z}_2
$$

for elements in this grading group.

In this notation the grading of the elements in $\Omega^\bullet(\mathbb{R}^{p|q})$ is all induced by the fact that the de Rham differential $\mathbf{d}$ itself is a [[derivation]] of degree $(1,even)$.

| generator | bi-degree |
|--|--|
| $x^a$ | (0,even) |
| $\theta^\alpha$ | (0,odd) |
| $\mathbf{d}$ | (1,even) |

Here the last line means that we have

| generator | bi-degree |
|--|--|
| $x^a$ | (0,even) |
| $\theta^\alpha$ | (0,odd) |
| $\mathbf{d}x^a$ | (1,even) |
| $\mathbf{d}\theta^\alpha$ | (1,odd) |

The formula for the "cohomologically- and super-graded commutativity" in $\Omega^\bullet(\mathbb{R}^{p|q})$ is

$$
  \alpha \wedge \beta
  =
  \;
  (- 1)^{n_\alpha n_\beta + \sigma_\alpha \sigma_\beta}
  \;
  \beta \wedge \alpha
$$

for all $\alpha,  \beta \in \Omega^\bullet(\mathbb{R}^{p|q})$ of homogeneous $\mathbb{Z}\times \mathbb{Z}_2$-degree. Hence there are _two_ contributions to the sign picked up when exchanging two super-differential forms in the wedge product:

1. there is a "cohomological sign" which for commuting a $p_1$-forms past a $p_2$-form is $(-1)^{p_1 p_2}$;

1. in addition there is a "super-grading" sich which for commuting a $\sigma_1$-graded coordinate function past a $\sigma_2$-graded coordinate function (possibly under the de Rham differential) is $(-1)^{\sigma_1 \sigma_2}$.

Some examples:

$$
  x^{a_1} (\mathbf{d}x^{a_2}) = + (\mathbf{d}x^{a_2}) x^{a_1}
$$

$$
  \theta^\alpha (\mathbf{d}x^a) = + (\mathbf{d}x^a) \theta^\alpha
$$

$$
  \theta^{\alpha_1} (\mathbf{d}\theta^{\alpha_2})
   =
  - (\mathbf{d}\theta^{\alpha_2}) \theta^{\alpha_1}
$$

$$
  \mathbf{d}x^{a_1}
  \wedge
  \mathbf{d} x^{a_2}
  =
  -
  \mathbf{d} x^{a_2}
  \wedge
  \mathbf{d} x^{a_1}
$$

$$
  \mathbf{d}x^a
  \wedge
  \mathbf{d} \theta^{\alpha}
  =
  -
  \mathbf{d}\theta^{\alpha}
  \wedge
  \mathbf{d} x^a
$$

$$
  \mathbf{d}\theta^{\alpha_1}
  \wedge
  \mathbf{d} \theta^{\alpha_2}
  =
  +
  \mathbf{d}\theta^{\alpha_2}
  \wedge
  \mathbf{d} \theta^{\alpha_1}
$$

## Related concepts

* [[Euclidean space]]

* [[Euclidean G-space]]

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

* [[super-Cartan geometry]]

## References

* {#KonechnySchwarz97} 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

[[!redirects super Cartesian spaces]]

[[!redirects Cartesian superspace]]
[[!redirects Cartesian superspaces]]

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

[[!redirects super-Cartesian space]]

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

[[!redirects super formal Cartesian space]]
[[!redirects super formal Cartesian spaces]]

[[!redirects SuperCartSp]]

[[!redirects SuperFormalCartSp]]
[[!redirects SuperFormalCartSpace]]


super Chern-Simons theory

#Contents#
* table of contents
{:toc}

## Idea

The [[supersymmetry|supersymmetric]] version of [[Chern-Simons theory]]...

## Related concepts

* [[ABJM model]]

* [[Rozansky-Witten theory]]

## References

* Victor Mikhaylov, _Aspects of Supergroup Chern-Simons Theories_, [[spire:1466706](https://inspirehep.net/record/1466706)]

* Victor Mikhaylov, *Analytic Torsion, 3d Mirror Symmetry, And Supergroup Chern-Simons Theories* [[arXiv:1505.03130](https://arxiv.org/abs/1505.03130)]

* [[Nathan Geer]], [[Matthew B. Young]], *Three dimensional topological quantum field theory from $U_q(\mathfrak{gl}(1\vert1))$ and $U(1\vert 1)$ Chern--Simons theory* [[arXiv:2210.04286](https://arxiv.org/abs/2210.04286)]

Review:

* [[Matthew B. Young]], *Non-semisimple TFT and $U(1\vert 1)$ Chern-Simons theory*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023)[[web](Center+for+Quantum+and+Topological+Systems#YoungMar2023), video: [YT](https://www.youtube.com/watch?v=m73KQCwXOrY)]

On [[integration over supermanifolds]] and [[picture changing operators]] in view of super Chern-Simons theory:

* [[C. A. Cremonini]], [[Pietro Grassi]], _Pictures from Super Chern-Simons Theory_ ([arXiv:1907.07152](https://arxiv.org/abs/1907.07152))


super commutative monoid

[[!redirects super commutative monoids]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Super-Algebra and Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### Monoid theory
+-- {: .hide}
[[!include monoid theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

A $\mathbb{Z}/2\mathbb{Z}$-[[graded object|graded]] [[commutative monoid]].

## Definition ##

An **super commutative monoid** is a [[commutative monoid]] $M$ with decomposition functions $\mathcal{D}_0:M \to M$ and $\mathcal{D}_1:M \to M$, such that

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

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

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**.

## See also ##

* [[commutative monoid]]

* [[graded commutative monoid]]

* [[super abelian group]]


super Euclidean group

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

#Contents#
* table of contents
{:toc}

## Idea

The ordinary [[Euclidean group]] of $\mathbb{R}^n$ is the group generated from the rigid translation action of $\mathbb{R}^n$ on itself and rotations about the origin.

The _super Euclidean group_ is analogously the [[supergroup]] of translations and rotations of the [[supermanifold]] $\mathbb{R}^{p|q}$.

Its [[super Lie algebra]] should be the [[super Poincare Lie algebra]] (up to the signature of the metric).

## Details

> **incomplete** for the moment, to be finished off tomorrow

The following description of the super Euclidean group (once it is finished, and polished) is due to [[Stephan Stolz]] and [[Peter Teichner]].

The data needed to define the super Euclidean group is

1. $V$ a $d$-dimensional inner product space

1. a [[spinor representation]] $\Delta^*$ of $Spin(V)$

1. a $Spin(V)$-equivariant map

   $$
      \Gamma : \Delta^* \otimes_{\mathbb{C}}
       \Delta^* \to V \otimes_{\mathbb{R}} \mathbb{C}
   $$

where $Spin(V)$ is the [[Spin group]] (see [[Clifford algebra]] for the moment).

Here is the construction of $Eucl(\mathbb{R}^{d|\delta})$ for

* $d = dim_{\mathbb{R}} V$

**remark** $\delta$ is a multiple of $2^{[\frac{d-1}{2}]}$

set

$$
  X = \Pi
  \left(
    \array{ V \times \Delta^* \\ \downarrow \\ V}
  \right)
  =
  V \times \Pi \Delta^*
$$

is a [[complex supermanifold]] of dimension $(d|\delta)$

$$
  C^\infty(V \times \Pi \Delta^*)
  =
  C^\infty(V) \otimes
  \wedge^\bullet \Delta
  =
  C^\infty(V, \wedge^\bullet(\Delta))
$$

for $\delta = 1$ this is

$$
  \cdots = C^\infty(C, \wedge^\bullet \Delta)^{ev}
    \oplus
   C^\infty(C, \wedge^\bullet \Delta)^{odd}
$$

$$
  \cdots \simeq
    C^\infty(V) \oplus C^\infty(V, \wedge^\bullet \Delta)
$$

where the last factor is $\simeq C^\infty(V; \Delta) \simeq C^\infty(V, S^+)$ where $S^+$ is the [[spinor bundle]]

now define the multiplication

$$
  (V \times \Pi \Delta^*)
  \times
  (V \times \Pi \Delta^*)
  \stackrel{\mu}{\to}
  (V \times \Pi \Delta^*)
$$

by sayin what it does on sets of probes by $S$

$$
  (V \times \Pi \Delta^*)(S)
  \times
  (V \times \Pi \Delta^*)(S)
  \stackrel{\mu(S)}{\to}
  (V \times \Pi \Delta^*)(S)
$$

here on the left we have the sets of sections

$$
  C^\infty(S)^{ev} \otimes V
  \times
  C^\infty(S)^{ev} \otimes \Delta^*
$$

so we can map these as

$$
  ((v_1, \theta_1),
   (v_2, \theta_2)
  )
  \mapsto
  (v_1 + v_2 + \Gamma(\theta_1 \otimes \theta_2),
   \theta_1 + \theta_2)
$$

**Remark**

if the data $(V, \Delta^*, \Gamma)$ and
$(V', (\Delta^*)', \Gamma')$ is [[isomorphism|isomorphic]] we get compatible notions of structures

But if $d = 0,1,2$ and $\delta = 1$ then there is a _unique_ such triple with non-degenerate pairing $\Gamma$ up to isomorphism.

**Definition**

The structure of a  **[[Euclidean supermanifold]]** on a $(d|\delta)$-dimensional [[supermanifold]] $Y$ is a $(V \times \Pi \Delta^*, End(V, \Delta^*, \Gamma))$-structure. See there for details.

## Examples

recall the [[Clifford algebra]] table:

$$
  \array{
      d & Cl(\mathbb{R}^d)^{ev} & Spin(\mathbb{R}^d)
      \\
      \\
      1 & \mathbb{R} & \{\pm 1\}
      \\
      2 & (\mathbb{R} 1 \oplus \mathbb{R} e_1 e_2, e_i^2 = 1) \simeq \mathbb{C} & S^1
  }
$$

the group structure on $V \times \Pi \Delta^*$ is that of the "translations" and "rotations"

it will be defined on [[generalized element]]s with domain $S$ by maps of sets

$$
  \mu:
  (V \times \Pi \Delta^*)(S)
  \times
  (V \times \Pi \Delta^*)(S)
  \to
  (V \times \Pi \Delta^*)(S)
$$

$$
  (v_1, \theta_1) , (v_2, \theta_2)
  \mapsto
  (v_1+ v_2 + \Gamma(\theta_1 \otimes \theta_2),
  \theta_1 + \theta_2)
$$

## $d = 1$

$\Delta^* = \mathbb{C}$

$$
  \Gamma : \Delta^* \otimes \Delta^* \to V \otimes_{\mathbb{R}} \mathbb{C}
$$

$$
  \mathbb{C} \otimes \mathbb{C}
  \to
  \mathbb{R} \otimes_{\mathbb{R}} \mathbb{C}
$$

$$
  1 \otimes 1 \mapsto 1 \otimes 1
$$

so here this is the [[supergroup|super translation group]].

## $d = 2$

$\Delta^* = \mathbb{C}$

$$
  u \in S^1 \simeq U(1) \simeq Spin(\mathbb{R}^2)
$$

$$
  \Delta^* \otimes_{\mathbb{C}}
  \Delta^*
  \stackrel{\Gamma}{\to}
  \mathbb{R}^2 \otimes \mathbb{R}
  \simeq \mathbb{C} \oplus \mathbb{C}
$$

the first map is multiplication by $u^{-1}$ and then the isomorphism on the right sends

$$
  (x,y)\otimes 1 \mapsto (z, \bar z)
$$

where $z = x + i y$

translation group $V \times \Pi \Delta^* \simeq \mathbb{R}^{2|1}$

multiplication on $S$-elements

$$
  \mathbb{R}^{2|1}(S) \times
  \mathbb{R}^{2|1}(S)
  \to
  \mathbb{R}^{2|1}(S)
$$

given by

$$
  (z_1,\bar z_1, \theta_1),
  (z_2,\bar z_2, \theta_2)
  \mapsto
  (z_1 + z_2, \bar z_1 + \bar z_2 + \theta_1 \theta_2, \theta_1 + \theta_2)
$$

## Related concepts

* [[Euclidean group]]

* [[general linear supergroup]], [[orthosymplectic supergroup]]

* [[Lorentz group]], [[Poincare group]]

* [[super Poincaré group ]]

[[!redirects Euclidean supergroup]]


super formal smooth infinity-groupoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The concept of _super smooth $\infty$-groupoid_ or _super smooth [[geometric homotopy type]]_ is the combination of _[[super ∞-groupoid]]_ and _[[smooth ∞-groupoid]]_. The [[cohesive (∞,1)-topos]] of smooth super-$\infty$-groupoids is a context that realizes [[higher geometry|higher]] [[supergeometry]].

Super smooth $\infty$-groupoids include
[[supermanifolds]], [[super Lie groups]] and their [[deloopings]] etc. Under [[Lie differentiation]] these map to [[super L-∞ algebras]].

## Definition

We consider one of at least two possible definitions, that differ (only) in some fine technical detail. The other is at _[[smooth super infinity-groupoid]]_.

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

Write

* [[CartSp]] for the site of [[Cartesian spaces]];

* $InfPoint \coloneqq WAlg^{op}$ for the category of first-order [[infinitesimally thickened points]] (i.e. the [[abstract duality|formal duals]] of [[commutative algebras]] over the [[real numbers]] of the form $\mathbb{R}\oplus V$ with $V$ a finite-dimensional  square-0 [[nilpotent ideal]]).

* $SuperPoint \coloneqq WAlg_{super}^{op}$ for the category of [[superpoints]], by which we here mean the [[formal duals]] to commutative [[superalgebras]] which
are super-[[Weil algebras]].

There are then "semidirect product" sites $CartSp \rtimes InfinPoint$ and $CartSp \rtimes SuperPoint$ (whose objects are [[Cartesian products]] of the given form inside [[synthetic differential supergeometry]] and whose morphisms are all morphisms in that context (not just the product morphisms)).

Set then

$$
  FormalSmooth\infty Grpd \coloneqq Sh_\infty(CartSp \rtimes InfPoint)
$$

for the collection of [[formal smooth ∞-groupoids]] (see there) and finally

$$
  SuperSmooth\infty Grpd \coloneqq Sh_\infty(CartSp \rtimes SuperPoint)
$$

for that of super smooth $\infty$-groupoid

=--

## Properties

### Cohesion
 {#Cohesion}

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

The [[sites]] in question are alternatingly (co-)[[reflective subcategories]] of each other (we always display [[left adjoints]] above their right adjoints)

$$
  \ast
  \stackrel{\longleftarrow}{\hookrightarrow}
  CartSp
  \stackrel{\hookrightarrow}{\longleftarrow}
  CartSp\rtimes InfPoint
  \stackrel{\longleftarrow}{\stackrel{\hookrightarrow}{\longleftarrow}}
  CartSp \rtimes SuperPoint
  \,.
$$

Here

* the first inclusion picks the [[terminal object]] $\mathbb{R}^0$;

* the second inclusion is that of [[reduced objects]]; the coreflection is [[reduction]], sending an algebra to its [[reduced algebra]];

* the third inclusion is that of even-graded algebras, the reflection sends a $\mathbb{Z}_2$-graded algebra to its even-graded part, the co-reflection sends a $\mathbb{Z}_2$-graded algebra to its quotient by the ideal generated by its odd part, see at _[superalgebra -- Adjoints to the inclusion of plain algebras](super+algebra#AdjointsToInclusionOfPlainAlgebra)_.

=--

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

Passing to [[(∞,1)-categories of (∞,1)-sheaves]], this yields, via [[(∞,1)-Kan extension]], a sequence of [[adjoint quadruples]] as follows:

$$
  \array{
    & && && &\longleftarrow&
    \\
    & && &\hookrightarrow& &\hookrightarrow&
    \\
    & &\longleftarrow& &\longleftarrow& &\longleftarrow&
    \\
    \Delta \colon
    &
    \infty Grpd
    &\hookrightarrow&
    Smooth \infty Grpd
    &\hookrightarrow&
    FormalSmooth \infty Grpd
    &\hookrightarrow&
    SuperFormalSmooth \infty Grpd
    \\
    & &\longleftarrow& &\longleftarrow&
    \\
    & &\hookrightarrow&
  }
$$

=--

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

Passing to the [[adjoint triples]] of [[idempotent monads]] and [[idempotent comonads]] which this induces, then yields

* on the left the [[shape modality]] $\int$, [[flat modality]] $\flat$ and [[sharp modality]] $\sharp$,

* in the middle yields the [[reduction modality]] $\Re$, the [[infinitesimal shape modality]] $\Im$ and the [[infinitesimal flat modality]] $\&$.

* on the right we get an adjoint triple whose whose middle bit $\rightsquigarrow$ is the [[bosonic modality]] and whose left piece $\rightrightarrows$ produces _super-even_ components, containing all the "[[fermion]] [[currents]]" if one wishes , which in this [[unity of opposites]] hence deserves to be called the _[[fermionic modality]]_. The further right adjoint $Rh$ is the [[rheonomy modality]].

Hence we get a process of [[adjoint modalities]] of the form

$$
  \array{
    && id &\dashv& id
    \\
    && \vee && \vee
    \\
    &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{}
    \\
    && \bot && \bot
    \\
    &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{}
    \\
    && \vee && \vee
    \\
    &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{}
    \\
    && \bot && \bot
    \\
    &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{}
    \\
    && \vee && \vee
    \\
    &\stackrel{contractible}{}& ʃ &\dashv& \flat & \stackrel{discrete}{}
    \\
    && \bot && \bot
    \\
    &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{differential}{}
    \\
    && \vee && \vee
    \\
    && \emptyset &\dashv& \ast
  }
$$

where "$\vee$" denotes inclusion of [[modal types]]. The first level is [[cohesion]], the second is [[differential cohesion]] ([[elasticity]]), the third is a further refinement given by supergeometry, which takes further "square roots" of all infinitesimal generators.

=--

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

All the sites are [[∞-cohesive sites]], which gives that we have an [[cohesive (infinity,1)-topos]]. The composite inclusion on the right is an [∞-cohesive neighbourhood site](differential+cohesive+%28infinity%2C1%29-topos#PresentationOnInfinitesimalNeighbourhoodSites), whence the inclusion $Smooth\infty Gpd\hookrightarrow SuperFormalSmooth\infty Grpd$ exhibits [[differential cohesion]].

With this the rightmost adjoint quadruple gives the [[Aufhebung]] of $\Re \dashv \Im$ by $\rightsquigarrow \dashv Rh$ and the further opposition $\rightrightarrows \dashv \rightsquigarrow$.

=--

## Related concepts

* [[super-Cartan geometry]]

* [[higher Cartan geometry]]

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

## References

* [[Urs Schreiber]], section 4.5 of: _[[schreiber:differential cohomology in a cohesive topos]]_

* [[Hisham Sati]], [[Urs Schreiber]], Section 3.1.3 of: _[[schreiber:Proper Orbifold Cohomology]]_ ([arXiv:2008.01101](https://arxiv.org/abs/2008.01101))

* [[Urs Schreiber]], _[[schreiber:Introduction to Higher Supergeometry]]_, lecture at _[[Higher Structures in M-Theory 2018]]_, [Durham Symposium](http://www.maths.dur.ac.uk/lms/)

  published in parts as: _Higher Structures in M-Theory_ (with [[Branislav Jurčo]], [[Christian Saemann]], [[Martin Wolf]]), Fortschritte der Physik (2019) ([arXiv:1903.02807](https://arxiv.org/abs/1903.02807), [doi:10.1002/prop.201910001]( https://doi.org/10.1002/prop.201910001))

[[!redirects super formal smooth infinity-groupoids]]

[[!redirects super formal smooth ∞-groupoid]]
[[!redirects super formal smooth ∞-groupoids]]

[[!redirects super smooth infinity-groupoid]]
[[!redirects super smooth infinity-groupoids]]

[[!redirects super smooth ∞-groupoid]]
[[!redirects super smooth ∞-groupoids]]

[[!redirects SuperFormalSmooth∞Grpd]]
[[!redirects SuperFormalSmooth?Type]]

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

[[!redirects super differential homotopy type]]
[[!redirects super differential homotopy types]]

[[!redirects SuperFormalSmooth∞Groupoid]]
[[!redirects SuperFormalSmooth∞Groupoids]]