superalgebra and (synthetic ) supergeometry
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.
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:
The following is taken from geometry of physics -- supergeometry, see there for more:
For $q\in \mathbb{N}$, the real Grassmann algebra
is the $\mathbb{R}$-algebra freely generated from $q$ generators $\{\theta^i\}_{i = 1}^q$ subject to the relations
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
with the product given by the relations
where $f \cdot g$ is the ordinary pointwise product of smooth functions.
Write
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
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:
(embedding of smooth manifolds into formal duals of R-algebras) The functor that assigns algebras of smooth function of smooth manifolds
is fully faithful.
(smooth Serre-Swan theorem) The functor that assigns smooth sections of smooth vector bundles of finite rank
is fully faithful (its essential image being the finitely generated projective modules over the $\mathbb{R}$-algebra of smooth function).
There is a third such magic algebraic property of smooth functions, which plays a role now:
(derivations of smooth functions are vector fields)
Let $X$ be a smooth manifold. Write
for the function that sends a smooth vector field $v \in \Gamma(T X)$ to the derivation of the 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.
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
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
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
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.):
The canonical inclusion of commutative algebras into supercommutative superalgebra is part of an adjoint triple of the form
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
(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. 1 is to serve as a convenient mnemonic for the nature of these functors:
In a Feynman diagram
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
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
its image under $\overset{\rightrightarrows}{(-)}$ is a bosonic space, but not an ordinary manifold. For instance
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
Write
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-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
for the full subcategory on formal duals of those algebras which are tensor products of commutative $\mathbb{R}$-algebras of the form
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. 3) is that they co-represent tangent vectors and jets:
Write $\mathbb{D}^1 = Spec(\mathbb{R}[\epsilon]/(\epsilon^2))$ for the formal dual of the algebra of dual numbers. Then morphisms
which are the identity after restriction along $\mathbb{R}^n \to \mathbb{R}^n \times \mathbb{D}$, are equivalently algebra homomorphisms of the form
which are the identity modulo $\epsilon$. Such a morphism has to take any function $f \in C^\infty(\mathbb{R}^n)$ to
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 $f_1,f_2 \in C^\infty(\mathbb{R}^n)$
Multiplying this out and using that $\epsilon^2 = 0$ this in turn is equivalent to
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. 2).
In particular one finds that maps
are equivalently single tangent vectors, hence for every $\mathbb{R}^n$ there is a natural bijection
between the hom-set from the formal dual of the ring of dual numbers and the set of tangent vectors.
The canonical inclusion $i$ of the category of ordinary Cartesian spaces into that of formal Cartesian spaces has a left adjoint $\Re$
given by
Hence exhibits $CartSp$ as a coreflective subcategory of that of formal cartesian spaces.
We say that $\mathbb{R}^n$ is the reduced scheme of $\mathbb{R}^n \times \mathbb{D}$.
We check the natural isomorphism on hom-sets that characterizes a pair of adjoint functors:
By definition, a morphism of the form
is equivalently a homomorphism of commutative algebras of the form
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
This is dually a morphism of the form
in $CartSp$. This establishes a natural bijection $f \leftrightarrow \tilde f$.
The above discussion following prop. 1 means that in passing to commutative superalgebras there are two stages of generalizations of plain differential geometry involved:
Cartesian spaces are generalized to formal Cartesian spaces;
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. 1), 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.
Write
for the full subcategory of that of the opposite category of supercommutative superalgebras on whose of the form
where $V$ is a nilpotent ideal of finite dimension over $\mathbb{R}$.
One place where such super formal Cartesian spaces are made explicit is in Konechny-Schwarz 97
In conclusion we have the following situation:
The coreflective subcategory inclusion of Cartesian spaces into formal Cartesian spaces from prop. 2 and the coreflective as we all reflective subcategory inclusion
of affine schemes into affine superschemes from prop. 1 combine to give the following system of adjoint functors on our local model spaces
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 $\mathbf{d}x^{a_1} \wedge \cdots \wedge \mathbf{d}x^{a_k} \wedge \mathbf{d}\theta^{\alpha_1} \wedge \cdots \wedge \mathbf{d}\theta^{\alpha_l}$.
This de Rham complex now carries the structure of a
which should be thought of as bracketet as follows
This means in effect that elements of $\Omega^\bullet(\mathbb{R}^{p|q})$ carry a $\mathbb{Z} \times \mathbb{Z}_2$-grading, where we may say that
$\mathbb{Z}$ corresponds to the “cohomological grading”;
$\mathbb{Z}_2$ corresponds to the super-grading.
We write
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
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:
there is a “cohomological sign” which for commuting a $p_1$-forms past a $p_2$-form is $(-1)^{p_1 p_2}$;
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:
Anatoly Konechny and Albert Schwarz,
On $(k \oplus l|q)$-dimensional supermanifolds in Supersymmetry and Quantum Field Theory (D. Volkov memorial volume) Springer-Verlag, 1998, Lecture Notes in Physics, 509 , J. Wess and V. Akulov (editors)(arXiv:hep-th/9706003)
Theory of $(k \oplus l|q)$-dimensional supermanifolds Sel. math., New ser. 6 (2000) 471 - 486 A summary/review is in the appendix of