superalgebra and (synthetic ) supergeometry
Discussion of superalgebra and supergeometry involves, by definition, crucial minus signs appearing in certain expressions. While there is a systematic rule for these, when superalgebra/supergeometry is combined with other topics that crucially involve their own signs, such as homological algebra, or ordering of terms, such as in noncommutative algebra, then the combined sign rule may appear intricate. This page here is meant to explicitly list some common expressions with their signs, for easy reference. We follow the article (Deligne-Freed 99) which has the same goal.
The main point to notice is that for algebraic terms that have both a homological degree $n \in \mathbb{Z}$ and a super-degree $\sigma \in \mathbb{Z}_2$ the “internalization sign rule” for the sign when exchanging two such elements is
(e.g. Castellani-D’Auria-Fré 91, (II.2.109), Bonora-Bregola-Lechner-Pasti-Tonin 87, beginning of section 2)
This we discuss below at The sign rule from internalization.
Parts of the literature however adops another sign rule, given by
This we discuss below in The super odd sign rule.
Here and in the following we adopt the perpective that homological algebra in superalgebra is concerned with the study of chain complexes internal to the symmetric monoidal category of super vector spaces (Deligne 90) or equivalently internal to the topos over superpoints (Schwarz 84). This means that there are two contributions to the sign when exchanging two terms, one from the cohomological grading, one from the super-grading. This is “the sign rule” also used for instance in (Castellani-D’Auria-Fré 91, equation (II.2.106), Bonora-Bregola-Lechner-Pasti-Tonin 87 Deligne-Freed 99, section 6).
There is however also a different sign rule use in the literature, the relation to which we discuss below.
We discuss the sign rules for the de Rham algebra of differential forms on a supermanifold (superdifferential forms), which is a differential graded-commutative superalgebra.
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 $\{ \theta^\alpha\}_{\alpha= 1}^q$
Accordingly, by the discussion at Kähler forms, the de Rham complex of 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 an $n_1$-forms past an $n_2$-form is $(-1)^{n_1 n_2}$;
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:
The above signs rules for differential forms on supermanifolds directly translate to the signs rule for super Lie algebra Lie algebra cohomology, by the identification of elements in the Chevalley-Eilenberg algebra with the left invariant differential forms on the corresponding super Lie group.
We discuss now the Chevalley-Eilenberg algebra of super Minkowski spacetime, regarded as a super translation Lie algebra, equivalently as the quotient of the quotient of the super Poincaré Lie algebra by the Lorentz Lie algebra.
This is the sub-algebra of the above de Rham complex of the left invariant differential forms:
With the above notation these are
With the above sign rules it follows that
More generally, since the super-Chevalley-Eilenberg algebra of a super Lie algebra is identified with the left invariant differential forms on the corresponding super Lie group, the signs from the super de Rham complex above induce the cotresponding sign rule for the computation of super-Lie algebra cohomology.
Write $Superpoint$ for the category of superpoints, the opposite category of that of real finitely-generated Grassmann algebras.
We list the terms with crucial signs that appear when regarding superalgebra as algebra internal to the presheaf topos over superpoints (see at super infinity-groupoid for background).
Given a super Lie algebra $\mathfrak{g}$, for each superpoint $\mathbb{R}^{0|q}$ write
for the vector space which is the even-degree part of the tensor product of the underlying vector spaces of $\mathfrak{g}$ and of the Grassmann algebra $C^\infty(\mathbb{R}^{0|q})$.
Here if $\{e_a, \psi_\alpha\}$ is a basis for $\mathfrak{g}$ with $\{e_a\}$ a basis for $\mathfrak{g}_{even}$ and $\{\psi_\alpha\}$ a basis for $\mathfrak{g}_{odd}$, and if we write $\{\epsilon^i\}_{i = 1}^q$ for the generators of $C^\infty(\mathbb{R}^{0|q})$, then $\mathbb{g}(\mathbb{R}^{0|q})$ is generated from elements of the form
and
The Lie bracket on the ordinary vector space $\mathfrak{g}(\mathbb{R}^{0|q})$ is defined on elements
by the formula
Some examples:
From this and from the ordinary skew-symmetry of the Lie bracket in $\mathfrak{g}(\mathbb{R}^{0|q})$ follows that also in the super Lie algebra we have
Next, from
and the fact that $\epsilon^1 \epsilon^2 = - \epsilon^2 \epsilon^1$ it follows that in the super Lie algebra
Given a super Lie algebra $\mathfrak{g}$ as above, the tensor product $\mathfrak{g}\otimes C^\infty(\mathbb{R}^{0|1})$ is the free $C^\infty(\mathbb{R}^{0|q})$-module generated from the super vector space underlying $\mathfrak{g}$.
(…)
(…)
(…)
In all of the above we used the sign rule induced by setting up all mathematical concepts internal to the symmetric monoidal category of super vector spaces or internal the topos over superpoints.
There is another sign convention in the literature, for instance in (…). In this convention one considers the map
to induce a single $\mathbb{Z}_2$-grading and then lets all signs be induced by that.
This can motivated from regarding the $\mathbb{Z}_2$-graded de Rham complex of a supermanifold as the algebra of functions on its odd-shifted tangent bundle
In this convention a term such as $\mathbf{d}\theta^\alpha$ of bidegree $(1,1)$ has single degree $0$ and hence commutes with every other term.
In this other convention the signs in the de Rham complex of the super Cartesian space $\mathbb{R}^{p|q}$ are induced as follows:
The different choice of signs in the de Rham complex above affects those of any super-Chevalley-Eilenberg algebra, by the identification of elements in the CE-algebra with left invariant differential forms on the supegroup.
We show that as chain complexes the two CE-algebras are quasi-isomorphic and hence do have the same cochain cohomology, hence that the definition of super-Lie algebra cohomology is not affected by the different perspective on signs.
A chain-isomorphism can be obtained as follows:
Write
$CE^\bullet$ for the cochain complex underlying the Chevalley-Eilenberg algebra with the signs given by
write $CE^\bullet_{\mathbb{Z}_2}$ for the cochain complex obtained with the sign rule
Say that a product of generators representing an element in $CE^k$ is in normal order of all the $(\bullet,even)$-graded elements are left of the $(\bullet,odd)$-graded elements, schematically
To define a linear map $CE^k \longrightarrow CE^\bullet_{\mathbb{Z}_2}$ declare that elements in this normal form go to the elements given by the same expression.
This defines a linear map in each degree $k$. Notice that this is not a morphism of algebras, in any sense, but it is a chain map. To see this, notice that acting with the differential which is of degree $(1,even)$ on a term of the form $e \wedge \cdots e \wedge \psi \wedge \cdots \psi$ produces a sum of such terms where in each summand one generator is replaced by a term of the form $d e$ or by $d \psi$, respectively. These are in turn themselves sums of elements of the form $e \wedge \cdots e \wedge \psi \wedge \cdots \psi$. By commuting them to the middle we see that the differential of the whole term is a sum of terms of the form
and
Now observe:
for terms of the first type, the signs picked up by commuting $d e$ to the middle are by definition the same in $CE^\bullet$ and in $CE^\bullet_{\mathbb{Z}_2}$;
for terms of the second type there are a priori more signs picked up in $CE^\bullet$ than in $CE^\bullet_{\mathbb{Z}_2}$, namely one for every term $\psi$ that one commutes through. But as the differential first passes from the left into the string of $\psi$s and then $d \psi$ in turn is passed back to the left, these extra signs appear in even number and hence cancel out.
In summary this establishes a chain map $CE^\bullet \longrightarrow CE^\bullet_{\mathbb{Z}_2}$ which is an cochain isomorphism and hence in particular a quasi-isomorphism, hence an isomorphism on cochain cohomology.
The internalization into the topos over superpoints is due to
The internalization sign rule follows for instance from applying the construction of internal Grassmann algebras in the category of super vector space as discussed in
The corresponding internal sign rule appears for instance in
Loriano Bonora, M. Bregola, Kurt Lechner, Paolo Pasti, Mario Tonin, section 2 of Anomaly-free supergravity and super-Yang-Mills theories in ten dimensions, Nuclear Physics B Volume 296, Issue 4, 25 January 1988 (doi:10.1016/0550-3213(88)90402-6)
Leonardo Castellani, Riccardo D'Auria, Pietro Fré, equations (II.2.106) and (II.2.109) in Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)
Pierre Deligne, Dan Freed, Sign manifesto, 1999 (pdf)
which appears in
which in turn appears in
And with explicit acknowledgement that there is also the alternative super odd sign rule in:
…