superalgebra

and

supergeometry

# Contents

## Idea

### Basic idea

In the general sense, superalgebra is the study of (higher) algebra

equivalently

More specifically, an associative superalgebra is an associative algebra in the context of superalgebra. As in the ordinary case, this is often just called a superalgebra , too.

In the following we first discuss

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

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

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

### Abstract idea

We discuss the general abstract raison d’ être of super algebra. Readers looking for just the plain definition should probably skip to below on first reading.

Superalgebra is universal in the following sense. The crucial super-grading rule (the “Koszul sign rule”)

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

in the symmetric monoidal category of $ℤ$-graded vector spaces is induced from the subcategory which is the abelian 2-group of metric graded lines. This in turn is the free abelian 2-group (groupal symmetric monoidal category) on a single generator. (This point of view is amplified for instance in the first part of (Kapranov 13), whose second part is about super 2-algebra). Generally then super-grading and hence super-algebra arises from the 2-truncation (3-coskeleton) of the free abelian ∞-group on a single generator, which is the sphere spectrum $𝕊$. So the ${ℤ}_{2}$-grading of superalgebra comes from the stable homotopy groups of spheres ${\pi }_{n}\left(𝕊\right)$ in degree 1 and 2:

$n=$$0$$1$$2$$3$$4$$5$$6$$7$$\cdots$
${\pi }_{n}\left(𝕊\right)=$$ℤ$${ℤ}_{2}$${ℤ}_{2}$${ℤ}_{24}$$0$$0$${ℤ}_{2}$${ℤ}_{240}$$\cdots$
meaning:degreeboson/fermion super-degreespinstring$-$$-$??$\cdots$
free object on single generator:abelian groupabelian 2-groupabelian 3-groupabelian 4-groupabelian 7-groupabelian 8-groupabelian ∞-group

This suggests (as hinted at in (Kapranov 13)) that in full generality higher supergeometry is to be thought of as $𝕊$-graded geometry, hence dually as higher algebra with ∞-group of units augmented over the sphere spectrum.

But notice that this is canonically so for every E-∞ ring, see at ∞-group of units – Augmented definition. This would mean: In higher geometry/higher algebra supergeometry/superalgebra is intrinsic, canonically given.

Using this together with Sati’s Geometric and topological structures related to M-branes and the image of the J-homomorphism

$n$012345678910111213141516
homotopy groups of stable orthogonal group${\pi }_{n}\left(O\right)$${ℤ}_{2}$${ℤ}_{2}$0$ℤ$000$ℤ$${ℤ}_{2}$${ℤ}_{2}$0$ℤ$000$ℤ$${ℤ}_{2}$
stable homotopy groups of spheres${\pi }_{n}\left(𝕊\right)$$ℤ$${ℤ}_{2}$${ℤ}_{2}$${ℤ}_{24}$00${ℤ}_{2}$${ℤ}_{240}$${ℤ}_{2}\oplus {ℤ}_{2}$${ℤ}_{2}\oplus {ℤ}_{2}\oplus {ℤ}_{2}$${ℤ}_{6}$${ℤ}_{504}$0${ℤ}_{3}$${ℤ}_{2}\oplus {ℤ}_{2}$${ℤ}_{480}\oplus {ℤ}_{2}$${ℤ}_{2}\oplus {ℤ}_{2}$
image of J-homomorphism$\mathrm{im}\left({\pi }_{n}\left(J\right)\right)$0${ℤ}_{2}$0${ℤ}_{24}$000${ℤ}_{240}$${ℤ}_{2}$${ℤ}_{2}$0${ℤ}_{504}$000${ℤ}_{480}$${ℤ}_{2}$

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

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

• $d=1$ sigma-model

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

$\begin{array}{ccccccc}B\mathrm{BU}\left(1\right)& \simeq & BK\left(ℤ,2\right)& \to & B{\mathrm{gl}}_{1}\left(\mathrm{KU}\right)& \to & \mathrm{KU}\text{-}\mathrm{Mod}\end{array}$\array{ B BU(1) &\simeq& B K(\mathbb{Z},2) &\to& B gl_1(KU) &\to& KU \text{-}Mod }

for $\mathrm{KU}$ the complex K-theory spectrum E-∞ ring, and hence the characteristic twists are in degree 2 of the group of units, hence of the graded group of units

${\mathrm{gl}}_{1}^{\mathrm{gr}}\left(\mathrm{KU}\right)\to 𝕊$gl^{gr}_1(KU) \to \mathbb{S}

hence are graded by the second homotopy group

${\pi }_{2}\left(𝕊\right)\simeq {ℤ}_{2}$\pi_2(\mathbb{S}) \simeq \mathbb{Z}_2

of the sphere spectrum.

• $d=2$ sigma-model

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

$\begin{array}{ccccccc}B{B}^{2}U\left(1\right)& \simeq & BK\left(ℤ,3\right)& \to & B{\mathrm{gl}}_{1}\left(\mathrm{tmf}\right)& \to & \mathrm{tmf}\text{-}\mathrm{Mod}\end{array}$\array{ B B^2 U(1) &\simeq& B K(\mathbb{Z},3) &\to& B gl_1(tmf) &\to& tmf \text{-}Mod }

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

${\mathrm{gl}}_{1}^{\mathrm{gr}}\left(\mathrm{tmf}\right)\to 𝕊$gl^{gr}_1(tmf) \to \mathbb{S}

hence are graded by the third homotopy group

${\pi }_{3}\left(𝕊\right)\simeq {ℤ}_{24}$\pi_3(\mathbb{S}) \simeq \mathbb{Z}_{24}

of the sphere spectrum.

• $d=5$ sigma-model

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

$\begin{array}{ccccccc}B{B}^{5}U\left(1\right)& \simeq & BK\left(ℤ,6\right)& \to & B{\mathrm{gl}}_{1}\left(K\left(5\right)\right)& \to & K\left(5\right)\text{-}\mathrm{Mod}\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ B B^5 U(1) &\simeq& B K(\mathbb{Z},6) &\to& B gl_1(K(5)) &\to& K(5) \text{-}Mod } \,,

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

${\mathrm{gl}}_{1}^{\mathrm{gr}}\left(K\left(5\right)\right)\to 𝕊$gl^{gr}_1(K(5)) \to \mathbb{S}

hence are graded by the sixth homotopy group

${\pi }_{6}\left(𝕊\right)\simeq {ℤ}_{2}$\pi_6(\mathbb{S}) \simeq \mathbb{Z}_{2}

of the sphere spectrum.

## Associative superalgebras

### Definition

#### Superalgebras

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

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

###### Definition

Write SVect for the symmetric monoidal category of super vector spaces over $k$. This is the category of ${ℤ}_{2}$-graded vector spaces equipped with the unique non-trivial symmetric braided monoidal structure.

Objects are vector spaces with a direct sum decomposition

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

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

$V\otimes W=\left({V}_{\mathrm{even}}\otimes {W}_{\mathrm{even}}\oplus {V}_{\mathrm{odd}}\otimes {W}_{\mathrm{odd}}\right)\oplus \left({V}_{\mathrm{even}}\otimes {W}_{\mathrm{odd}}\oplus {V}_{\mathrm{odd}}\otimes {W}_{\mathrm{even}}\right)$V \otimes W = (V_{even} \otimes W_{even} \oplus V_{odd}\otimes W_{odd}) \oplus (V_{even} \otimes W_{odd} \oplus V_{odd} \otimes W_{even})

but equipped with the non-trivial braiding morphism

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

that is the usual braiding isomorphism of Vect on ${V}_{\mathrm{even}}\otimes {W}_{\mathrm{even}}$ and on ${V}_{\mathrm{even}}\otimes {W}_{\mathrm{odd}}\oplus {V}_{\mathrm{odd}}\otimes {W}_{\mathrm{even}}$ but is $\left(-1\right)$ times this on ${V}_{\mathrm{odd}}\otimes {W}_{\mathrm{odd}}$.

###### Definition

A super (associative) algebra over $K$ is a monoid in the symmetric monoidal category SVect of super vector spaces.

A (graded)-commutative (associative) algebra over $K$ is a monoid in the symmetric monoidal category SVect of super vector spaces.

This means that a commutative superalgebra is a super vector space

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

equipped with a morphism of super vector spaces

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

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

$a\cdot b=-b\cdot a\phantom{\rule{thinmathspace}{0ex}}.$a \cdot b = - b \cdot a \,.

Whereas if either of $a$ or $b$ is in ${A}_{\mathrm{even}}$ we have

$a\cdot b=b\cdot a\phantom{\rule{thinmathspace}{0ex}}.$a \cdot b = b \cdot a \,.

#### Related notions

###### Definition

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

###### Definition

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

${a}_{1}{\cdot }_{{A}^{\mathrm{op}}}{a}_{2}≔\left(-1{\right)}^{\mid {a}_{1}\mid \mid {a}_{2}\mid }{a}_{2}{\cdot }_{A}{a}_{1}\phantom{\rule{thinmathspace}{0ex}}.$a_1 \cdot_{A^{op}} a_2 \coloneqq (-1)^{{\vert a_1\vert}{\vert a_2\vert}} a_2 \cdot_{A} a_1 \,.
###### Definition

A superalgebra $A$ is called central simple if

1. its center, def. 3 is the ground field;

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

###### Definition

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

###### Remark

By the discussion at 2-vector space this is equivalently the 2-category of super 2-vector spaces. Equivalence in $2\mathrm{sVect}\simeq \mathrm{sAlg}$ is also called Morita equivalence of super-algebras.

###### Definition

A superalgebra is an Azumaya algebra if it is an invertible object in the monoidal 2-category $s2\mathrm{Vect}\simeq \mathrm{sAlg}$, def. 6.

###### Remark

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

### Examples

#### Endomorphisms algebras, matrix algebras

###### Definition

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

###### Proposition

A matrix superalgebra, def. 9 is central simple, def. 5.

#### Clifford algebra

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

In fact, let $V$ be a vector space equipped with symmetric inner product $⟨-,-range$. Write ${\wedge }^{•}V$ be the Grassmann algebra on $V$. The inner product makes this a super Poisson algebra. The Clifford algebra $\mathrm{Cl}\left(V,⟨rangl\right)$ is the deformation quantization of this.

###### Example

There is a superalgebra over the complex numbers of the form

$A=ℂ\oplus ℂ⟨u⟩\phantom{\rule{thinmathspace}{0ex}},$A = \mathbb{C} \oplus \mathbb{C}\langle u\rangle \,,

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

### Properties

#### General

###### Proposition

A superalgebra is isomorphic to a matrix algebra, def. 9 precisely if it is equivalent in $2\mathrm{sVect}\simeq \mathrm{Alg}$, def. 6, (Morita equivalent) to the ground field super algebra.

#### Picard 3-group, Brauer group

We discuss the Picard 3-group of $2\mathrm{sVect}\simeq \mathrm{sAlg}$, def. 6, hence the corresponding Brauer group. See also at super line 2-bundle.

###### Theorem

A superalgebra is invertible/Azumaya, def. 8, precisely if it is finite dimensional and central simple, def. 5.

This is due to (Wall).

###### Theorem

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

$\mathrm{sBr}\left(ℂ\right)\simeq {ℤ}_{2}$sBr(\mathbb{C}) \simeq \mathbb{Z}_2
$\mathrm{sBr}\left(ℝ\right)\simeq {ℤ}_{8}\phantom{\rule{thinmathspace}{0ex}}.$sBr(\mathbb{R}) \simeq \mathbb{Z}_8 \,.

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

This is due to (Wall).

The following generalizes this to the higher homotopy groups.

###### Proposition

The homotopy groups of the braided 3-group ${\mathrm{sAlg}}^{×}$ of Azumaya superalgebra are

${\mathrm{sAlg}}_{ℂ}^{×}$${\mathrm{sAlg}}_{ℝ}^{×}$
${\pi }_{2}$${ℂ}^{×}$${ℝ}^{×}$
${\pi }_{1}$${ℤ}_{2}$${ℤ}_{2}$
${\pi }_{0}$${ℤ}_{2}$${ℤ}_{8}$

where the groups of units ${ℂ}^{×}$ and ${ℝ}^{×}$ are regarded as discrete groups.

This appears in (Freed, (1.38)).

## Algebra in the topos over superpoints

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

### The topos

###### Definition

Write $\mathrm{SuperPoint}$ for the site of superpoints. Write

$\mathrm{SuperSet}:=\mathrm{Sh}\left(\mathrm{SuperPoint}\right)$SuperSet := Sh(SuperPoint)

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

$\mathrm{Super}\infty \mathrm{Grpd}:={\mathrm{Sh}}_{\left(\infty ,1\right)}\left(\mathrm{SuperPoint}\right)$Super \infty Grpd := Sh_{(\infty,1)}(SuperPoint)

for the (∞,1)-sheaf (∞,1)-topos over this site: the (∞,1)-topos of super ∞-groupoids.

### The line object $ℝ$

###### Definition

Write

$j:\mathrm{SuperSmthMfd}↪\mathrm{Sh}\left(\mathrm{SuperPoint}\right)$j \colon SuperSmthMfd \hookrightarrow Sh(SuperPoint)

for the restricted Yoneda embedding of supermanifolds given by the canonical inclusion $\mathrm{SuperPoint}↪\mathrm{SuperSmoothManifold}$.

###### Definition

Write

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

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

$ℝ\in \mathrm{Ring}\left(\mathrm{Sh}\left(\mathrm{SuperPoint}\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathbb{R} \in Ring(Sh(SuperPoint)) \,.
###### Remark

By the discussion at supermanifold (in the section As locally ringed spaces - Properties) $ℝ$ sends the formal dual of a Grassmann algebra to its even subalgebra

$ℝ:\mathrm{Spec}{\wedge }^{•}V↦\left({\wedge }^{•}V{\right)}_{\mathrm{even}}\phantom{\rule{thinmathspace}{0ex}}.$\mathbb{R} : Spec \wedge^\bullet V \mapsto (\wedge^\bullet V)_{even} \,.

This is canonically equipped with the structure of a (unital) commutative ring in $\mathrm{SuperSet}$.

In (Sachse) this appears around (21).

### $ℝ$-Modules

###### Definition

Write ${\mathrm{Mod}}_{ℝ}\left(\mathrm{SuperSet}\right)$ for the category of modules over $ℝ$ of def. 12 in $\mathrm{SuperSet}$.

###### Proposition

The restriction of the embedding of def. 11 to supermanifolds which are super vector spaces is a functor

$j:{\mathrm{SVect}}_{ℝ}={\mathrm{Mod}}_{ℝ}\left(\mathrm{Set}\right)↪{\mathrm{Mod}}_{𝕂}\left(\mathrm{SuperSet}\right)$j : SVect_{\mathbb{R}} = Mod_{\mathbb{R}}(Set) \hookrightarrow Mod_{\mathbb{K}}(SuperSet)

from real super vector spaces to internal modules over $ℝ$ that sends $V\in {\mathrm{SVect}}_{ℝ}$ to

$j\left(V\right):\mathrm{Spec}\Lambda ↦\left(\Lambda {\otimes }_{ℝ}V{\right)}_{\mathrm{even}}=\left({\Lambda }_{\mathrm{even}}{\otimes }_{ℝ}{V}_{\mathrm{even}}\right)\oplus \left({\Lambda }_{\mathrm{odd}}{\otimes }_{ℝ}{V}_{\mathrm{odd}}\right)\phantom{\rule{thinmathspace}{0ex}}.$j(V) : Spec \Lambda \mapsto (\Lambda \otimes_\mathbb{R} V)_{even} = (\Lambda_{even} \otimes_\mathbb{R} V_{even}) \oplus (\Lambda_{odd} \otimes_\mathbb{R} V_{odd}) \,.

This is a full and faithful functor.

This appears as (Sachse, corollary 3.2).

###### Proof

The proof is a variation on the Yoneda lemma.

This means that ordinary super vector spaces are embedded as a full subcategory in $𝕂$-modules in the topos over super points.

### Associative and Lie Superalgebras

###### Proposition

The functor $j$ from prop 4 induces a full and faithful functor

${\mathrm{SAlg}}_{ℝ}\left(\mathrm{Set}\right)↪{\mathrm{Alg}}_{ℝ}\left(\mathrm{SuperSet}\right)$SAlg_{\mathbb{R}}(Set) \hookrightarrow Alg_{\mathbb{R}}(SuperSet)

of superalgebras over $ℝ$ as in def. 2 and internal associative algebras over $ℝ$ in $\mathrm{SuperSet}$.

Similarly we have a faithful embedding

${\mathrm{SLieAlg}}_{ℝ}\left(\mathrm{Set}\right)↪{\mathrm{LieAlg}}_{ℝ}\left(\mathrm{SuperSet}\right)$SLieAlg_{\mathbb{R}}(Set) \hookrightarrow LieAlg_{\mathbb{R}}(SuperSet)

of ordinary super Lie algebras over $ℝ$ into the internal Lie algebras over $ℝ$.

This appears as (Sachse, corollary 3.3).

(…)

## References

### Definition

Discussion of superalgebra as algebra in certain symmetric monoidal tensor categories is in

Lecture notes include

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

and in

• V. Molotkov., Infinite-dimensional ${ℤ}_{2}^{k}$-supermanifolds , ICTP preprints, IC/84/183, 1984.

A summary/review is in the appendix of

• Anatoly Konechny and Albert Schwarz,

On $\left(k\oplus l\mid q\right)$-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 $\left(k\oplus l\mid q\right)$-dimensional supermanifolds Sel. math., New ser. 6 (2000) 471 - 486

• Albert Schwarz, I. Shapiro, Supergeometry and Arithmetic Geometry (arXiv:hep-th/0605119)

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

### Brauer groups and Picard 2-groupoid

Brauer groups of superalgebras are discussed in

• C. T. C. Wall, Graded Brauer groups, J. Reine Angew. Math. 213 (1963/1964), 187-199.