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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{super line 2-bundle} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry} [[!include higher geometry - contents]] \hypertarget{superalgebra_and_supergeometry}{}\paragraph*{{Super-Algebra and Super-Geometry}}\label{superalgebra_and_supergeometry} [[!include supergeometry - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{TheBrauer3Group}{The Brauer 3-group of superalgebras}\dotfill \pageref*{TheBrauer3Group} \linebreak \noindent\hyperlink{HomtopyTypeOfSuper2Lines}{The homotopy type of the 2-stack of super 2-lines}\dotfill \pageref*{HomtopyTypeOfSuper2Lines} \linebreak \noindent\hyperlink{RelationToUnconnectedDeloopingOfUnitsOfKU}{Relation to the unconnected delooping of the $\infty$-group of units of $KU$}\dotfill \pageref*{RelationToUnconnectedDeloopingOfUnitsOfKU} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A \emph{super line 2-bundle} is a [[line 2-bundle]] in ([[higher geometry|higher]]) [[supergeometry]]. We discuss line 2-bundles in [[supergeometry]] and their relation to [[twisted K-theory]]. This follows the discussion in chapter 1 of (\hyperlink{Freed12}{Freed}), which in turn follows the classical text (\hyperlink{DonovanKaroubi}{Donovan-Karoubi}) on [[twisted K-theory]] and (\hyperlink{Wall}{Wall}) on \emph{\href{super+algebra#Picard2Groupoid}{Picard 2-groupoids of superalgebras}}. What we add to this here, following (\hyperlink{FSS}{Fiorenza-Sati-Schreiber 12}) is that we make explit the incarnation of these constructions as the [[∞-stack|higher stack]] on [[supermanifolds]] $2\mathbf{sLine}$ of super line 2-bundles. This is a supergeometric refinement of the [[moduli infinity-stack|moduli 2-stack]] $\mathbf{B}^2\mathbb{C}^\times$ for bare complex line 2-bundles, $\mathbb{C}^\times$-[[principal 2-bundles]]. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \begin{defn} \label{}\hypertarget{}{} Let $\mathbf{H} \coloneqq$ [[SmoothSuper∞Grpd]] be the [[cohesive (∞,1)-topos]] of [[SmoothSuper∞Grpd|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]] \begin{displaymath} SmoothSuper\infty Grpd \simeq Sh_\infty(CartSp_{th}) \end{displaymath} This is cohesive over [[Super∞Grpd]] $\simeq Sh_\infty(SuperPoints)$ \begin{displaymath} \Gamma \colon SmoothSuper\infty Grpd \to Super \infty Grpd \,. \end{displaymath} \end{defn} \begin{defn} \label{}\hypertarget{}{} Let $\mathbb{K} \in \mathbf{H}$ be the canonical [[affine line|affine]] [[line object]], whose underlying sheaf of sets assigns \begin{displaymath} \mathbb{K} \colon \mathbb{R}^{p|q} \mapsto C^\infty(\mathbb{R}^p, \mathbb{C})\otimes (\wedge^\bullet \mathbb{C}^q)_{even} \,. \end{displaymath} \end{defn} \begin{remark} \label{}\hypertarget{}{} By the discussion at \emph{[[superalgebra]]} we have that $\Gamma(\mathbb{K})$-algebras in [[Super∞Grpd]] are, externally, superalgebras over the [[complex numbers]]. \end{remark} \begin{defn} \label{}\hypertarget{}{} Write \begin{displaymath} 2\mathbf{sVect} \in SmoothSuper \infty Grpd \end{displaymath} for the object which over $\mathbb{R}^{p|q}$ is the [[2-groupoid]] whose \begin{itemize}% \item objects are semisimple $\mathbb{K}(\mathbb{R}^{p|q})$-algebras; \item [[1-morphisms]] are invertible [[bimodules]]; \item [[2-morphisms]] are invertible bimodule homomorphisms. \end{itemize} This is naturally a [[braided monoidal 2-category]] object. Write \begin{displaymath} 2 \mathbf{sLine} \in SmoothSuper \infty Grpd \end{displaymath} for the maximal [[braided 3-group]] inside this on the invertible objects. \end{defn} \hypertarget{properties}{}\subsection*{{Properties}}\label{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. \hypertarget{TheBrauer3Group}{}\subsubsection*{{The Brauer 3-group of superalgebras}}\label{TheBrauer3Group} \begin{theorem} \label{AzumayaAreCentralSimple}\hypertarget{AzumayaAreCentralSimple}{} A superalgebra is invertible/Azumaya (see \href{super+algebra#Azumaya}{here}) precisely if it is finite dimensional and central simple (see \href{super+algebra#CentralSimple}{here}). \end{theorem} This is due to (\hyperlink{Wall}{Wall}). \begin{theorem} \label{}\hypertarget{}{} 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: \begin{displaymath} sBr(\mathbb{C}) \simeq \mathbb{Z}_2 \end{displaymath} \begin{displaymath} sBr(\mathbb{R}) \simeq \mathbb{Z}_8 \,. \end{displaymath} The non-trivial element in $sBr(\mathbb{R})$ is that presented by the superalgebra $\mathbb{C} \oplus \mathbb{C} u$ of the example \href{super+algebra#ComplexCl1}{here}, with $u \cdot u = 1$. \end{theorem} This is due to (\hyperlink{Wall}{Wall}). The following generalizes this to the higher [[homotopy groups]]. \begin{prop} \label{}\hypertarget{}{} The [[homotopy groups]] of the [[braided 3-group]] $sAlg^\times$ of Azumaya superalgebra are \begin{tabular}{l|l|l} &$sAlg^\times_{\mathbb{C}}$&$sAlg^\times_{\mathbb{R}}$\\ \hline $\pi_2$&$\mathbb{C}^\times$&$\mathbb{R}^\times$\\ $\pi_1$&$\mathbb{Z}_2$&$\mathbb{Z}_2$\\ $\pi_0$&$\mathbb{Z}_2$&$\mathbb{Z}_8$\\ \end{tabular} where the [[groups of units]] $\mathbb{C}^\times$ and $\mathbb{R}^\times$ are regarded as [[discrete groups]]. \end{prop} This is recalled for instance in (\hyperlink{Freed12}{Freed 12, (1.38)}). \hypertarget{HomtopyTypeOfSuper2Lines}{}\subsubsection*{{The homotopy type of the 2-stack of super 2-lines}}\label{HomtopyTypeOfSuper2Lines} Now we can analyse the super 2-stack $2\mathbf{sLine}$ of super 2-line 2-bundles. \begin{prop} \label{}\hypertarget{}{} The object $2\mathbf{sLine} \in SmoothSuper\infty Grpd$ is equivalent to that which to $\mathbb{R}^{p|q}$ assigns the 2-groupoid whose \begin{itemize}% \item objects are the algebra $C^\infty(\mathbb{R}^p, \mathbb{C})\otimes (\wedge^\bullet \mathbb{C}^q)_{even}$ and that algebra tensored with $Cl_1(\mathbb{C})$; \item 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 $Cl_1(\mathbb{C})$; \item 2-morphisms form the group $C^\infty(\mathbb{R}^p, \mathbb{C}^\times)$. \end{itemize} \end{prop} \begin{proof} First, the $\mathbb{K}$-algebras in the topos of supergeometry are externally the ordinary [[superalgebras]], by the discussion at \emph{\href{super+algebra#AlgebraOverSuperpoints}{superalgebra -- As algebras in the topos over superpoints}}. With this the statement is a straightforward generalization of the discussion at \emph{\href{super+algebra#Picard2Groupoid}{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 $\mathbb{C}^\times$. \end{proof} It follows that \begin{prop} \label{}\hypertarget{}{} The [[homotopy groups]] of the [[geometric realization]] ${\vert 2\mathbf{sLine} \vert}$ of $2\mathbf{sLine}$ are \begin{tabular}{l|l|l} &${\vert 2\mathbf{sLine}_{\mathbb{C}} \vert}$&${\vert 2\mathbf{sLine}_{\mathbb{R}} \vert}$\\ \hline $\pi_3$&$\mathbb{Z}$&$\mathbb{Z}$\\ $\pi_2$&0&0\\ $\pi_1$&$\mathbb{Z}_2$&$\mathbb{Z}_2$\\ $\pi_0$&$\mathbb{Z}_2$&$\mathbb{Z}_8$\\ \end{tabular} \end{prop} \begin{remark} \label{}\hypertarget{}{} Therefore we have a canonical morphism \begin{displaymath} \mathbf{B}^2 \mathbb{C}^\times \simeq 2\mathbf{Line} \to 2\mathbf{sLine} \end{displaymath} in [[SmoothSuper∞Grpd]] (a [[n-monomorphism|2-monomorphism]]) from the [[moduli ∞-stack|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]]. \end{remark} \begin{remark} \label{}\hypertarget{}{} In [[bosonic string theory]] over a [[spacetime]] $X$ the [[background gauge field]] called the [[B-field]] is a line 2-bundle [[circle n-bundle with connection|with connection]] given by a morphism \begin{displaymath} X \to 2\mathbf{Line} \simeq \mathbf{B}^2 \mathbb{C}^\times \,. \end{displaymath} However in [[type II superstring theory]] the [[B-field]] is actually a super line 2-bundle, hence given by a morphism \begin{displaymath} X \to 2\mathbf{sLine} \end{displaymath} in [[SmoothSuper∞Grpd]]. This observation (formulated in less stacky language) is due to the analysis of [[orientifold]] background fields in (\hyperlink{Precis}{Precis}). \end{remark} \begin{prop} \label{FirstKInvariant}\hypertarget{FirstKInvariant}{} The first [[k-invariant]] of $\vert 2\mathbf{sLine}\vert$ is the essentially unique nontrivial \begin{displaymath} \mathbb{Z}_2 \to \mathbf{B}^2 \mathbb{Z}_2 \end{displaymath} given by the [[Steenrod square]]. This is represented by the [[braided monoidal category|braiding]] equivalence on the [[tensor product]] of $Cl_1^{\mathbb{C}} \simeq \langle 1, e\rangle_{[e^2 = 1]}$ \begin{displaymath} 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}} \end{displaymath} given by the algebra homomorphism \begin{displaymath} \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} \end{displaymath} (exchange the tensor factors and introduce a sign when exchanging two odd graded ones). \end{prop} For instance (\hyperlink{Freed12}{Freed 12, 1.42}). \begin{prop} \label{SecondKInvariant}\hypertarget{SecondKInvariant}{} The second [[k-invariant]] \begin{displaymath} \mathbf{B}\mathbb{Z}_2 \to \mathbf{B}^4 \mathbb{Z} \end{displaymath} is the delooping of that of super lines $\mathbf{sLine}$, being the image under the [[Bockstein homomorphism]] of \begin{displaymath} \mathbf{B}\mathbb{Z}_2 \to \mathbf{B}^3 U(1) \end{displaymath} which sends $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \to \mathbb{Z}_2 \hookrightarrow U(1)$. (\ldots{}) \end{prop} For instance (\hyperlink{Freed12}{Freed 12, 1.44}). \begin{remark} \label{}\hypertarget{}{} Comparison with the notation and terminology of (\hyperlink{FreedHopkinsTeleman}{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 $X \to \mathbf{B}\Omega (2\mathbf{sLine}) \to 2\mathbf{sLine}$. \end{remark} \hypertarget{RelationToUnconnectedDeloopingOfUnitsOfKU}{}\subsubsection*{{Relation to the unconnected delooping of the $\infty$-group of units of $KU$}}\label{RelationToUnconnectedDeloopingOfUnitsOfKU} In (\hyperlink{Sagave11}{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]] \begin{displaymath} gl_1(E) \to gl_1^J(E) \to \mathbb{S} \to bgl_1^\ast(E) \,. \end{displaymath} See at \emph{\href{infinity-group+of+units#AugmentedDefinition}{∞-Group of units -- Augmented definition}}. By (\hyperlink{Sagave11}{Sagave 11, theorem 12 and example 4.10}) and comparing to the above discussion we have an [[equivalence]] \begin{displaymath} {\vert 2\mathbf{sLine}\vert} \simeq bgl_1^\ast(KU) \langle0,..,4\rangle \end{displaymath} of the [[geometric realization]] of the super-2-stack of super line 2-bundles with the [[n-truncated object in an (infinity,1)-category|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 (\hyperlink{Precis}{Freed-Distler-Moore}, \hyperlink{Freed12}{Freed}) is written $R^{-1}$. (\ldots{}) \hypertarget{references}{}\subsection*{{References}}\label{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 \begin{itemize}% \item [[Daniel Freed]], \emph{Lectures on twisted K-theory and orientifolds}, lectures at ESI Vienna, 2012 ([[FreedESI2012.pdf:file]]) \end{itemize} based on the old results about the [[Picard 2-groupoid]] of complex [[super algebras]] \begin{itemize}% \item [[C. T. C. Wall]], \emph{Graded Brauer groups}, J. Reine Angew. Math. 213 (1963/1964), 187-199. \end{itemize} and based on the discussion of [[twisted K-theory]] in \begin{itemize}% \item [[Peter Donovan]], [[Max Karoubi]], \emph{Graded Brauer groups and $K$-theory with local coefficients}, Publications Math\'e{}matiques de l'IH\'E{}S, 38 (1970), p. 5-25 (\href{http://www.numdam.org/item?id=PMIHES_1970__38__5_0}{numdam}) \end{itemize} as refined in \begin{itemize}% \item [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], \emph{Loop groups and twisted K-theory I}, Journal of Topology (2011) 4 (4): 737-798 (\href{http://arxiv.org/abs/0711.1906}{arXiv:0711.1906}) \end{itemize} and developing constructions in \begin{itemize}% \item [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], \emph{Orientifold Pr\'e{}cis} in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) \emph{[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]} Proceedings of Symposia in Pure Mathematics, AMS (2011) (\href{http://arxiv.org/abs/0906.0795}{arXiv:0906.0795}, \href{http://www.ma.utexas.edu/users/dafr/bilbao.pdf}{slides}) \end{itemize} See also \begin{itemize}% \item [[Tadeusz Józefiak]], \emph{Semisimple Superalgebras}, Volume 1352 of the series Lecture Notes in Mathematics pp 96-113. \end{itemize} Similar comments appear earlier on p. 8 and following of \begin{itemize}% \item [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], \emph{Consistent Orientation of Moduli Spaces} (\href{http://arxiv.org/abs/0711.1909}{arXiv:0711.1909}), chapter XIX, pages 395-419 in: Oscar Garcia-Prada, Jean Pierre Bourguignon, Simon Salamon (eds.) \emph{The Many Facets of Geometry: A Tribute to Nigel Hitchin}, Oxford University Press 2010 (\href{http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199534920.001.0001/acprof-9780199534920}{doi:10.1093/acprof:oso/9780199534920.001.0001}) \end{itemize} The above higher supergeometric story is made explicit in \begin{itemize}% \item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:A higher stacky perspective on Chern-Simons theory]]} \end{itemize} The ``unconnected delooping'' of the [[infinity-group of units]] of an $E_\infty$-ring $E$ is introduced in \begin{itemize}% \item [[Steffen Sagave]], \emph{Spectra of units for periodic ring spectra} (\href{http://arxiv.org/abs/1111.6731}{arXiv:1111.6731}) \end{itemize} and the specific example for the case of $E = KU$ is in example 4.10 there. [[!redirects super line 2-bundles]] [[!redirects super 2-line bundle]] [[!redirects super 2-line bundles]] \end{document}