higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
superalgebra and (synthetic ) supergeometry
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.
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
This is cohesive over Super∞Grpd $\simeq Sh_\infty(SuperPoints)$
Let $\mathbb{K} \in \mathbf{H}$ be the canonical affine line object, whose underlying sheaf of sets assigns
By the discussion at superalgebra we have that $\Gamma(\mathbb{K})$-algebras in Super∞Grpd are, externally, superalgebras over the complex numbers.
Write
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
for the maximal braided 3-group inside this on the invertible objects.
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.
A superalgebra is invertible/Azumaya (see here) precisely if it is finite dimensional and central simple (see here).
This is due to (Wall).
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:
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.
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)).
Now we can analyse the super 2-stack $2\mathbf{sLine}$ of super 2-line 2-bundles.
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 $C^\infty(\mathbb{R}^p, \mathbb{C}^\times)$.
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
The homotopy groups of the geometric realization ${\vert 2\mathbf{sLine} \vert}$ of $2\mathbf{sLine}$ are
${\vert 2\mathbf{sLine}_{\mathbb{C}} \vert}$ | ${\vert 2\mathbf{sLine}_{\mathbb{R}} \vert}$ | |
---|---|---|
$\pi_3$ | $\mathbb{Z}$ | $\mathbb{Z}$ |
$\pi_2$ | 0 | 0 |
$\pi_1$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ |
$\pi_0$ | $\mathbb{Z}_2$ | $\mathbb{Z}_8$ |
Therefore we have a canonical morphism
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.
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
However in type II superstring theory the B-field is actually a super line 2-bundle, hence given by a morphism
in SmoothSuper∞Grpd. This observation (formulated in less stacky language) is due to the analysis of orientifold background fields in (Precis).
The first k-invariant of $\vert 2\mathbf{sLine}\vert$ is the essentially unique nontrivial
given by the Steenrod square. This is represented by the braiding equivalence on the tensor product of $Cl_1^{\mathbb{C}} \simeq \langle 1, e\rangle_{[e^2 = 1]}$
given by the algebra homomorphism
(exchange the tensor factors and introduce a sign when exchanging two odd graded ones).
For instance (Freed 12, 1.42).
The second k-invariant
is the delooping of that of super lines $\mathbf{sLine}$, being the image under the Bockstein homomorphism of
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).
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 $X \to \mathbf{B}\Omega (2\mathbf{sLine}) \to 2\mathbf{sLine}$.
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
See at ∞-Group of units – Augmented definition.
By (Sagave 11, theorem 12 and example 4.10) and comparing to the above discussion we have an equivalence
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
based on the old results about the Picard 2-groupoid of complex super algebras
and based on the discussion of twisted K-theory in
as refined in
and developing constructions in
See also
Similar comments appear earlier on p. 8 and following of
The above higher supergeometric story is made explicit 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.