Contents

Ingredients

Concepts

Constructions

Examples

Theorems

supersymmetry

# Contents

## Idea

A super line 2-bundle is a line 2-bundle in (higher) supergeometry.

We discuss line 2-bundles in supergeometry and their relation to twisted K-theory. This follows the discussion in chapter 1 of (Freed), which in turn follows the classical text (Donovan-Karoubi) on twisted K-theory and (Wall) on Picard 2-groupoids of superalgebras. What we add to this here, following (Fiorenza-Sati-Schreiber 12) is that we make explit the incarnation of these constructions as the higher stack on supermanifolds $2\mathbf{sLine}$ of super line 2-bundles. This is a supergeometric refinement of the moduli 2-stack $\mathbf{B}^2\mathbb{C}^\times$ for bare complex line 2-bundles, $\mathbb{C}^\times$-principal 2-bundles.

## Definition

###### Definition

Let $\mathbf{H} \coloneqq$ SmoothSuper∞Grpd be the cohesive (∞,1)-topos of smooth super-∞-groupoids. With CartSp${}_{th}$ the site given by the full subcategory of the category of supermanifolds on those of the form $\mathbb{R}^{p|q}$ for $p,q \in \mathbb{N}$ this is the corresponding (∞,1)-category of (∞,1)-sheaves

$SmoothSuper\infty Grpd \simeq Sh_\infty(CartSp_{th})$

This is cohesive over Super∞Grpd $\simeq Sh_\infty(SuperPoints)$

$\Gamma \colon SmoothSuper\infty Grpd \to Super \infty Grpd \,.$
###### Definition

Let $\mathbb{K} \in \mathbf{H}$ be the canonical affine line object, whose underlying sheaf of sets assigns

$\mathbb{K} \colon \mathbb{R}^{p|q} \mapsto C^\infty(\mathbb{R}^p, \mathbb{C})\otimes (\wedge^\bullet \mathbb{C}^q)_{even} \,.$
###### Remark

By the discussion at superalgebra we have that $\Gamma(\mathbb{K})$-algebras in Super∞Grpd are, externally, superalgebras over the complex numbers.

###### Defintion

Write

$2\mathbf{sVect} \in SmoothSuper \infty Grpd$

for the object which over $\mathbb{R}^{p|q}$ is the 2-groupoid whose

• objects are semisimple $\mathbb{K}(\mathbb{R}^{p|q})$-algebras;

• 1-morphisms are invertible bimodules;

• 2-morphisms are invertible bimodule homomorphisms.

This is naturally a braided monoidal 2-category object. Write

$2 \mathbf{sLine} \in SmoothSuper \infty Grpd$

for the maximal braided 3-group inside this on the invertible objects.

## Properties

We now want to analyse the super 2-stack $2 \mathbf{sLine}$. In order to do so, first notice the following classical results about the Picard 3-group of superalgebras.

### The Brauer 3-group of superalgebras

###### Theorem

A superalgebra is invertible/Azumaya (see here) precisely if it is finite dimensional and central simple (see here).

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:

$sBr(\mathbb{C}) \simeq \mathbb{Z}_2$
$sBr(\mathbb{R}) \simeq \mathbb{Z}_8 \,.$

The non-trivial element in $sBr(\mathbb{R})$ is that presented by the superalgebra $\mathbb{C} \oplus \mathbb{C} u$ of the example here, 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 $sAlg^\times$ of Azumaya superalgebra are

$sAlg^\times_{\mathbb{C}}$$sAlg^\times_{\mathbb{R}}$
$\pi_2$$\mathbb{C}^\times$$\mathbb{R}^\times$
$\pi_1$$\mathbb{Z}_2$$\mathbb{Z}_2$
$\pi_0$$\mathbb{Z}_2$$\mathbb{Z}_8$

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

This is recalled for instance in (Freed 12, (1.38)).

### The homotopy type of the 2-stack of super 2-lines

Now we can analyse the super 2-stack $2\mathbf{sLine}$ of super 2-line 2-bundles.

###### Proposition

The object $2\mathbf{sLine} \in SmoothSuper\infty Grpd$ is equivalent to that which to $\mathbb{R}^{p|q}$ assigns the 2-groupoid whose

• objects are the algebra $C^\infty(\mathbb{R}^p, \mathbb{C})\otimes (\wedge^\bullet \mathbb{C}^q)_{even}$ and that algebra tensored with $Cl_1(\mathbb{C})$;

• morphisms are $C^\infty(\mathbb{R}^p, \mathbb{C})\otimes (\wedge^\bullet \mathbb{C}^q)_{even}$ regarded as a bimodule over itself and that bimodule tensored with $Cl_1(\mathbb{C})$;

• 2-morphisms form the group $C^\infty(\mathbb{R}^p, \mathbb{C}^\times)$.

###### Proof

First, the $\mathbb{K}$-algebras in the topos of supergeometry are externally the ordinary superalgebras, by the discussion at superalgebra – As algebras in the topos over superpoints.

With this the statement is a straightforward generalization of the discussion at superalgebra – Picard 2-groupoid from superalgebras over $\mathbb{C}$ to those over $C^\infty(\mathbb{R}^p, \mathbb{C})$.

While the invertible ordinary $\mathbb{C}^\infty(\mathbb{R})$-algebras are equivalent to that algebra itself (hence there is only one, up to equivalence); the invertible superalgebras are equivalent either to the ground field or to the complex Clifford algebra $Cl_1(\mathbb{C})$ (hence there are two, up to equivalence, the two elements in the Brauer group $\mathbb{Z}_2 = \pi_0(2\mathbf{sLine})$ ). Similarly for the invertible bimodules. Finally the invertible intertwiners are pointwise $\mathbb{C}^\times$.

It follows that

###### Proposition

The homotopy groups of the geometric realization ${\vert 2\mathbf{sLine} \vert}$ of $2\mathbf{sLine}$ are

${\vert 2\mathbf{sLine}_{\mathbb{C}} \vert}$${\vert 2\mathbf{sLine}_{\mathbb{R}} \vert}$
$\pi_3$$\mathbb{Z}$$\mathbb{Z}$
$\pi_2$00
$\pi_1$$\mathbb{Z}_2$$\mathbb{Z}_2$
$\pi_0$$\mathbb{Z}_2$$\mathbb{Z}_8$
###### Remark

Therefore we have a canonical morphism

$\mathbf{B}^2 \mathbb{C}^\times \simeq 2\mathbf{Line} \to 2\mathbf{sLine}$

in SmoothSuper∞Grpd (a 2-monomorphism) from the moduli 2-stack of $\mathbb{C}^\times$-principal 2-bundles/bundle gerbes, which picks the “ordinary” super 2-line bundle (as opposed to its “superpartner”), ignores the odd auto-gauge transformations of that and keeps all the higher gauge transformations.

###### Remark

In bosonic string theory over a spacetime $X$ the background gauge field called the B-field is a line 2-bundle with connection given by a morphism

$X \to 2\mathbf{Line} \simeq \mathbf{B}^2 \mathbb{C}^\times \,.$

However in type II superstring theory the B-field is actually a super line 2-bundle, hence given by a morphism

$X \to 2\mathbf{sLine}$

in SmoothSuper∞Grpd. This observation (formulated in less stacky language) is due to the analysis of orientifold background fields in (Precis).

###### Proposition

The first k-invariant of $\vert 2\mathbf{sLine}\vert$ is the essentially unique nontrivial

$\mathbb{Z}_2 \to \mathbf{B}^2 \mathbb{Z}_2$

given by the Steenrod square. This is represented by the braiding equivalence on the tensor product of $Cl_1^{\mathbb{C}} \simeq \langle 1, e\rangle_{[e^2 = 1]}$

$Cl_1^{\mathbb{C}} \otimes_{\mathbb{C}} Cl_1^{\mathbb{C}} \stackrel{\simeq}{\to} Cl_1^{\mathbb{C}} \otimes_{\mathbb{C}} Cl_1^{\mathbb{C}}$

given by the algebra homomorphism

\begin{aligned} 1 \otimes 1 & \mapsto 1 \otimes 1 \\ e \otimes e & \mapsto - e \otimes e \\ 1 \otimes e & \mapsto e \otimes 1 \\ e \otimes 1 & \mapsto 1 \otimes e \end{aligned}

(exchange the tensor factors and introduce a sign when exchanging two odd graded ones).

For instance (Freed 12, 1.42).

###### Proposition

The second k-invariant

$\mathbf{B}\mathbb{Z}_2 \to \mathbf{B}^4 \mathbb{Z}$

is the delooping of that of super lines $\mathbf{sLine}$, being the image under the Bockstein homomorphism of

$\mathbf{B}\mathbb{Z}_2 \to \mathbf{B}^3 U(1)$

which sends $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \to \mathbb{Z}_2 \hookrightarrow U(1)$.

(…)

For instance (Freed 12, 1.44).

###### Remark

Comparison with the notation and terminology of (Freed-Hopkins-Teleman 07):

the “$\mathcal{B}\mathbb{T}^{\pm}$” on the top of p. 14 there is $\Omega (2\mathbf{sLine})$ here; a graded line bundle over some $X$ there is a map of stacks $X \to \Omega (2\mathbf{sLine})$ here.

For $X$ 1-truncated, hence a groupoid, a graded central extension in the sense there is a map $X \to 2\mathbf{sLine}$ which factors as $X \to \mathbf{B}\Omega (2\mathbf{sLine}) \to 2\mathbf{sLine}$.

### Relation to the unconnected delooping of the $\infty$-group of units of $KU$

In (Sagave 11) is introduced a “non-connected delooping” $bgl_1^\ast(E)$ of the ∞-group of units $gl_1(E)$ of an E-∞ ring $E$, fitting into a homotopy cofiber sequence

$gl_1(E) \to gl_1^J(E) \to \mathbb{S} \to bgl_1^\ast(E) \,.$

By (Sagave 11, theorem 12 and example 4.10) and comparing to the above discussion we have an equivalence

${\vert 2\mathbf{sLine}\vert} \simeq bgl_1^\ast(KU) \langle0,..,4\rangle$

of the geometric realization of the super-2-stack of super line 2-bundles with the 4-truncation of the connected delooping of the infinity-group of units of $KU$.

hitting the Donovan-Karoubi twists of K-theory. This is what in (Freed-Distler-Moore, Freed) is written $R^{-1}$.

(…)

Line 2-bundles in supergeometry as a model for the B-field and orientifolds are discussed (even if not quite explicitly in the language of higher bundles) in

• Daniel Freed, Lectures on twisted K-theory and orientifolds, lectures at ESI Vienna, 2012 (pdf)

based on the old results about the Picard 2-groupoid of complex super algebras

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

and based on the discussion of twisted K-theory in

• Peter Donovan, Max Karoubi, Graded Brauer groups and $K$-theory with local coefficients, Publications Mathématiques de l’IHÉS, 38 (1970), p. 5-25 (numdam)

as refined in

and developing constructions in

The “unconnected delooping” of the infinity-group of units of an $E_\infty$-ring $E$ is introduced in
and the specific example for the case of $E = KU$ is in example 4.10 there.