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 $R^{-1}$.

(…)

References

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.