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super line 2-bundle

Context

Higher geometry

Super-Algebra and Super-Geometry

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 2sLine2\mathbf{sLine} of super line 2-bundles. This is a supergeometric refinement of the moduli 2-stack B 2 ×\mathbf{B}^2\mathbb{C}^\times for bare complex line 2-bundles, ×\mathbb{C}^\times-principal 2-bundles.

Definition

Definition

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

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

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

Γ:SmoothSuperGrpdSuperGrpd. \Gamma \colon SmoothSuper\infty Grpd \to Super \infty Grpd \,.
Definition

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

𝕂: p|qC ( p,)( q) even. \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 Γ(𝕂)\Gamma(\mathbb{K})-algebras in Super∞Grpd are, externally, superalgebras over the complex numbers.

Defintion

Write

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

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

  • objects are semisimple 𝕂( p|q)\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

2sLineSmoothSuperGrpd 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 2sLine2 \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() 2 sBr(\mathbb{C}) \simeq \mathbb{Z}_2
sBr() 8. sBr(\mathbb{R}) \simeq \mathbb{Z}_8 \,.

The non-trivial element in sBr()sBr(\mathbb{R}) is that presented by the superalgebra u\mathbb{C} \oplus \mathbb{C} u of the example here, with uu=1u \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 ×sAlg^\times of Azumaya superalgebra are

sAlg ×sAlg^\times_{\mathbb{C}}sAlg ×sAlg^\times_{\mathbb{R}}
π 2\pi_2 ×\mathbb{C}^\times ×\mathbb{R}^\times
π 1\pi_1 2\mathbb{Z}_2 2\mathbb{Z}_2
π 0\pi_0 2\mathbb{Z}_2 8\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 2sLine2\mathbf{sLine} of super 2-line 2-bundles.

Proposition

The object 2sLineSmoothSuperGrpd2\mathbf{sLine} \in SmoothSuper\infty Grpd is equivalent to that which to p|q\mathbb{R}^{p|q} assigns the 2-groupoid whose

  • objects are the algebra C ( p,)( q) evenC^\infty(\mathbb{R}^p, \mathbb{C})\otimes (\wedge^\bullet \mathbb{C}^q)_{even} and that algebra tensored with Cl 1()Cl_1(\mathbb{C});

  • morphisms are C ( p,)( q) evenC^\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()Cl_1(\mathbb{C});

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

Proof

First, the 𝕂\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 ( p,)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()Cl_1(\mathbb{C}) (hence there are two, up to equivalence, the two elements in the Brauer group 2=π 0(2sLine)\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 |2sLine|{\vert 2\mathbf{sLine} \vert} of 2sLine2\mathbf{sLine} are

|2sLine |{\vert 2\mathbf{sLine}_{\mathbb{C}} \vert}|2sLine |{\vert 2\mathbf{sLine}_{\mathbb{R}} \vert}
π 3\pi_3\mathbb{Z}\mathbb{Z}
π 2\pi_200
π 1\pi_1 2\mathbb{Z}_2 2\mathbb{Z}_2
π 0\pi_0 2\mathbb{Z}_2 8\mathbb{Z}_8
Remark

Therefore we have a canonical morphism

B 2 ×2Line2sLine \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 XX the background gauge field called the B-field is a line 2-bundle with connection given by a morphism

X2LineB 2 ×. 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

X2sLine 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 |2sLine|\vert 2\mathbf{sLine}\vert is the essentially unique nontrivial

2B 2 2 \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 1,e [e 2=1]Cl_1^{\mathbb{C}} \simeq \langle 1, e\rangle_{[e^2 = 1]}

Cl 1 Cl 1 Cl 1 Cl 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

11 11 ee ee 1e e1 e1 1e \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

B 2B 4 \mathbf{B}\mathbb{Z}_2 \to \mathbf{B}^4 \mathbb{Z}

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

B 2B 3U(1) \mathbf{B}\mathbb{Z}_2 \to \mathbf{B}^3 U(1)

which sends 2× 2× 2 2U(1)\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 “𝕋 ±\mathcal{B}\mathbb{T}^{\pm}” on the top of p. 14 there is Ω(2sLine)\Omega (2\mathbf{sLine}) here; a graded line bundle over some XX there is a map of stacks XΩ(2sLine)X \to \Omega (2\mathbf{sLine}) here.

For XX 1-truncated, hence a groupoid, a graded central extension in the sense there is a map X2sLineX \to 2\mathbf{sLine} which factors as XBΩ(2sLine)2sLineX \to \mathbf{B}\Omega (2\mathbf{sLine}) \to 2\mathbf{sLine}.

Relation to the unconnected delooping of the \infty-group of units of KUKU

In (Sagave 11) is introduced a “non-connected delooping” bgl 1 *(E)bgl_1^\ast(E) of the ∞-group of units gl 1(E)gl_1(E) of an E-∞ ring EE, fitting into a homotopy cofiber sequence

gl 1(E)gl 1 J(E)𝕊bgl 1 *(E). gl_1(E) \to gl_1^J(E) \to \mathbb{S} \to bgl_1^\ast(E) \,.

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

|2sLine|bgl 1 *(KU)0,..,4 {\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 KUKU.

hitting the Donovan-Karoubi twists of K-theory. This is what in (Freed-Distler-Moore, Freed) is written R 1R^{-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 KK-theory with local coefficients, Publications Mathématiques de l’IHÉS, 38 (1970), p. 5-25 (numdam)

as refined in

and developing constructions in

See also

  • Tadeusz Józefiak, Semisimple Superalgebras, Volume 1352 of the series Lecture Notes in Mathematics pp 96-113.

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 E_\infty-ring EE is introduced in

and the specific example for the case of E=KUE = KU is in example 4.10 there.

Revised on November 25, 2015 15:13:48 by Urs Schreiber (193.55.36.81)