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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*{string structure} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology} [[!include cohomology - contents]] \hypertarget{higher_spin_geometry}{}\paragraph*{{Higher spin geometry}}\label{higher_spin_geometry} [[!include higher spin geometry - contents]] \hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory} [[!include string theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{topological_and_smooth_string_structures}{Topological and smooth string structures}\dotfill \pageref*{topological_and_smooth_string_structures} \linebreak \noindent\hyperlink{twisted_and_differential_string_structures}{Twisted and differential string structures}\dotfill \pageref*{twisted_and_differential_string_structures} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{choices_of_string_structures}{Choices of string structures}\dotfill \pageref*{choices_of_string_structures} \linebreak \noindent\hyperlink{ClassesOnTotalSpace}{String structures by gerbes on a bundle}\dotfill \pageref*{ClassesOnTotalSpace} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak A \emph{string structure} on a [[manifold]] is a higher version of a [[spin structure]]. A string structure on a [[manifold]] with [[spin structure]] given by a [[Spin group]]-[[principal bundle]] to which the [[tangent bundle]] is [[associated bundle|associated]] is a lift $\hat g$ of the classifying map $g : X \to \mathcal{B} Spin(n)$ through the third nontrivial step $\mathcal{B}String(n) \to \mathcal{B}Spin(n)$ in the [[Whitehead tower]] of $BO(n)$ to a [[String group]]-[[principal bundle]]: \begin{displaymath} \itexarray{ && \mathcal{B}String(n) \\ & {}^{\hat g}\nearrow & \downarrow \\ X &\stackrel{g}{\to}& \mathcal{B}Spin(n) } \end{displaymath} A lift one further step through the Whitehead tower is a [[Fivebrane structure]]. This has generalizations to the smooth context, where instead of the topological String-group one uses the [[String Lie 2-group]]. Let $X$ be an $n$-dimensional [[topological manifold]]. Its tangent [[bundle]] is canonically associated to a $O(n)$-principal bundle, which is in turn classified by a [[continuous function]] \begin{displaymath} X \to B O(n) \end{displaymath} from $X$ to the [[classifying space]] of the [[orthogonal group]] $O(n)$. \begin{itemize}% \item A \textbf{String structure} on $X$ is the choice of a lift of this map a few steps through the [[Whitehead tower]] of $BO(n)$. \item The manifold ``is string'' if such a lift exists. \end{itemize} This means the following: \begin{itemize}% \item there is a canonical map $w_1 : B O(n) \to B\mathbb{Z}_2$ from the [[classifying space]] of $O(n)$ to that of $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$ that represents the generator of the cohomology $H^1(B O(n), \mathbb{Z}_2)$. The classifying space of the group $SO(n)$ is the [[generalized universal bundle|homotopy pullback]] \begin{displaymath} \itexarray{ B SO(n) &\to& {*} \\ \downarrow && \downarrow \\ B O(n) &\stackrel{w_1}{\to}& \mathbb{B}\mathbb{Z}_2 } \end{displaymath} Namely using the [[homotopy hypothesis]] (which is a theorem, recall), we may identify $B O(n)$ with the one object [[groupoid]] whose space of morphisms is $O(n)$ and similarly for $B \mathbb{Z}_2$. Then the map in question is the one induced from the group homomorphism that sends orientation preserving elements in $O(n)$ to the identity and orientation reversing elements to the nontrivial element in $\mathbb{Z}_2$. \begin{itemize}% \item an \textbf{[[orientation]]} on $X$ is a choice of lift of the structure group through $B SO(n) \to B O(n)$\begin{displaymath} \itexarray{ && B SO(n) \\ & {}^{orientation}\nearrow& \downarrow \\ X &\stackrel{}{\to}& B O(n) } \,. \end{displaymath} \end{itemize} \item there is a canonical map $w_2 : B SO(n) \to B^2 \mathbb{Z}_2$ representing the generator of $H^2(B SO(n), \mathbb{Z}_2)$. The classifying space of the group $Spin(n)$ is the [[generalized universal bundle|homotopy pullback]] \begin{displaymath} \itexarray{ B Spin(n) &\to& {*} \\ \downarrow && \downarrow \\ B SO(n) &\stackrel{w_2}{\to}& \mathbb{B}^2\mathbb{Z}_2 } \end{displaymath} \begin{itemize}% \item a \textbf{[[spin structure]]} on an oriented manifold $X$ is a choice of lift of the structure group through $B Spin(n) \to B SO(n)$\begin{displaymath} \itexarray{ && B Spin(n) \\ & {}^{spin structure}\nearrow& \downarrow \\ X &\stackrel{}{\to}& B SO(n) } \,. \end{displaymath} \end{itemize} \item there is a canonical map $B Spin(n) \to B^3 U(1)$ The classifying space of the group $String(n)$ is the [[generalized universal bundle|homotopy pullback]] \begin{displaymath} \itexarray{ B String(n) &\to& {*} \\ \downarrow && \downarrow \\ B Spin(n) &\stackrel{\frac{1}{2}p_1}{\to}& \mathbb{B}^3 U(1) } \end{displaymath} \begin{itemize}% \item a \textbf{string structure} on an oriented manifold $X$ is a choice of lift of the structure group through $B String(n) \to B Spin(n)$\begin{displaymath} \itexarray{ && B String(n) \\ & {}^{string structure}\nearrow& \downarrow \\ X &\stackrel{}{\to}& B Spin(n) } \,. \end{displaymath} \end{itemize} \item there is a canonical map $B String(n) \to B^7 U(1)$ The classifying space of the group $Fivebrane(n)$ is the [[generalized universal bundle|homotopy pullback]] \begin{displaymath} \itexarray{ B Fivebrane(n) &\to& {*} \\ \downarrow && \downarrow \\ B String(n) &\stackrel{\frac{1}{6}p_2}{\to}& \mathbb{B}^7 U(1) } \end{displaymath} \begin{itemize}% \item a \textbf{[[fivebrane structure]]} on an string manifold $X$ is a choice of lift of the structure group through $B Fivebrane(n) \to B String(n)$\begin{displaymath} \itexarray{ && B Fivebrane(n) \\ & {}^{fivebrane structure}\nearrow& \downarrow \\ X &\stackrel{}{\to}& B String(n) } \,. \end{displaymath} \end{itemize} \end{itemize} \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \hypertarget{topological_and_smooth_string_structures}{}\subsubsection*{{Topological and smooth string structures}}\label{topological_and_smooth_string_structures} Let the ambient [[(∞,1)-topos]] by $\mathbf{H} =$ [[ETop∞Grpd]] or [[Smooth∞Grpd]]. Write $X$ for a [[topological manifold]] or [[smooth manifold]] of [[dimension]] $n$, respectively. Write $String(n)$ for the [[string 2-group]], a [[1-truncated]] [[∞-group]] object in $\mathbf{H}$. \begin{defn} \label{PlainStringStructures}\hypertarget{PlainStringStructures}{} The [[2-groupoid]] of (topological or smooth) \textbf{string structures} on $X$ is the hom-space of [[cocycle]]s $X \to \mathbf{B}String(n)$, or equivalently that of (topological or smooth) $String(n)$-[[principal 2-bundle]]s: \begin{displaymath} String(X) := String(n) Bund(X) \simeq \mathbf{X}(X,\mathbf{B}String) \,. \end{displaymath} \end{defn} Write $\frac{1}{2} \mathbf{p}_1 : \mathbf{B} Spin(n) \to \mathbf{B}^3 U(1)$ in $\mathbf{H}$ for the topological or smooth refinement of the first fractional Pontryagin class (see [[differential string structure]] for details on this). \begin{prop} \label{StringStructuresAsObstructions}\hypertarget{StringStructuresAsObstructions}{} The [[2-groupoid]] of string structure on $X$ is the [[homotopy fiber]] of $\frac{1}{2}\mathbf{p}_1^X$: the [[(∞,1)-pullback]] \begin{displaymath} \itexarray{ String(X) &\to& * \\ \downarrow && \downarrow \\ \mathbf{H}(X, \mathbf{B}Spin(n)) &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}& \mathbf{H}(X, \mathbf{B}^3 U(1)) } \,. \end{displaymath} \end{prop} \begin{proof} By definition of the [[string 2-group]] we have the [[fiber sequence]] $\mathbf{B} String \to \mathbf{B}Spin \stackrel{\frac{1}{2}} \mathbf{p_1}{\to} \mathbf{B}^3 U(1)$. The [[hom-functor]] $\mathbf{H}(X,-)$ preserves every [[(∞,1)-limit]], hence preserves this [[fiber sequence]]. \end{proof} \begin{defn} \label{FiberOverSpinStructure}\hypertarget{FiberOverSpinStructure}{} Given a [[spin structure]] $S : X \to \mathbf{B} Spin(n)$ we say that the \textbf{string structures extending this spin-structure} is the [[homotopy fiber]] $String_S(X)$ of the projection $String(X) \to Spin(X)$ from observation \ref{StringStructuresAsObstructions}: \end{defn} \hypertarget{twisted_and_differential_string_structures}{}\subsubsection*{{Twisted and differential string structures}}\label{twisted_and_differential_string_structures} (\ldots{}) The [[2-groupoid]] of string structures is the [[homotopy fiber]] of \begin{displaymath} \frac{1}{2}p_1 : Top(X, \mathcal{B}Spin) \to Top(X, \mathcal{B}^4 \mathbb{Z}) \end{displaymath} over the trivial cocycle. Followowing the general logic of [[twisted cohomology]] the 2-groupoids over a nontrivial cocycle $c : X \to \mathcal{B}^4 \mathbb{Z}$ may be thought of as that of \emph{twisted} string structures. The [[Pontryagin class]] $\frac{1}{2}p_1$ refines to the [[smooth first fractional Pontryagin class]] $\frac{1}{2}\mathbf{p}_1 : \mathbf{B}Spin \to \mathbf{B}^3 U(1)$. That leads to [[differential string structure]]s. (\ldots{}) \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{choices_of_string_structures}{}\subsubsection*{{Choices of string structures}}\label{choices_of_string_structures} \begin{prop} \label{ChoicesOfSpinStructures}\hypertarget{ChoicesOfSpinStructures}{} The space of choices of string structures extending a given spin structure $S$ are as follows \begin{itemize}% \item if $[\frac{1}{2}\mathbf{p}_1(S)] \neq 0$ it is empty: $String_S(X) \simeq \emptyset$; \item if $[\frac{1}{2}\mathbf{p}_1(S)] = 0$ it is $String_S(X) \simeq \mathbf{H}(X, \mathbf{B}^2 U(1))$. In particular the set of [[equivalence class]]es of string structures lifting $S$ is the [[cohomology]] set \begin{displaymath} \pi_0 String_S(X) \simeq H^3(X, \mathbb{Z}) \,. \end{displaymath} \end{itemize} \end{prop} \begin{proof} Apply the [[pasting law]] for [[(∞,1)-pullback]]s on the [[diagram]] \begin{displaymath} \itexarray{ String_S(X) &\to& String(X) &\to& * \\ \downarrow && \downarrow && \downarrow \\ * &\stackrel{S}{\to}& \mathbf{H}(X, \mathbf{B} Spin(n)) &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}& \mathbf{H}(X, \mathbf{B}^3 U(1)) } \,. \end{displaymath} The outer diagram defines the [[loop space object]] of $\mathbf{H}(X, \mathbf{B}^3 U(1))$. Since $\mathbf{H}(X,-)$ commutes with forming loop space objects (see [[fiber sequence]] for details) we have \begin{displaymath} String_S(X) \simeq \Omega \mathbf{H}(X, \mathbf{B}^3 U(1)) \simeq \mathbf{H}(X, \mathbf{B}^2 U(1)) \,. \end{displaymath} \end{proof} \hypertarget{ClassesOnTotalSpace}{}\subsubsection*{{String structures by gerbes on a bundle}}\label{ClassesOnTotalSpace} One can reformulate an \begin{itemize}% \item [[orientation]]- \item [[spin structure|Spin]]- \item [[string structure|String]]- \item [[fivebrane structure|Fivebrane]]- \end{itemize} structure in terms of the existence of a certain class in abelian cohomolgy on the total space of the given [[principal bundle]]. This decomposition is a special case of th general [[Whitehead principle of nonabelian cohomology]]. \begin{defn} \label{StringStrucDefinitionByTotalSpace}\hypertarget{StringStrucDefinitionByTotalSpace}{} Let $X$ be a manifolds with [[spin structure]] $S : X \to \mathbf{B}Spin$. Write $P \to X$ for the corresponding [[spin group]]-principal bundle. Then a \textbf{string structure} lifting $S$ is a cohomology class $H^3(P,\mathbb{Z})$ such that the restriction of the class to any fiber $\simeq Spin(n)$ is a generator of $H^3(Spin(n), \mathbb{Z}) \simeq \mathbb{Z}$. \end{defn} This kind of definition appears in (\hyperlink{Redden}{Redden, def. 6.4.2}). \begin{prop} \label{DirectStringStructureImpliesIndirectOne}\hypertarget{DirectStringStructureImpliesIndirectOne}{} Every string structure in the sense of def. \ref{FiberOverSpinStructure} induces a string structure in the sense of def. \ref{StringStrucDefinitionByTotalSpace}. \end{prop} \begin{proof} Consider the [[pasting]] diagram of [[(∞,1)-pullback]]s \begin{displaymath} \itexarray{ String &\to& \hat P &\to& {*} \\ \downarrow && \downarrow && \downarrow \\ Spin &\to& P &\to& B^2 U(1) &\to& {*} \\ \downarrow && \downarrow && \downarrow && \downarrow \\ {*} &\to& X &\to& B String(n) &\to& B Spin(n) } \end{displaymath} This uses repeatedly the [[pasting law]] for $(\infty,1)$-pullbacks. The map $P \to B^2 U(1)$ appears by decomposing the homotopy pullback of the point along $X \to B Spin(n)$ into a homotopy pullback first along $B String(n) \to B Spin(n)$ and then along $X \to B String(n)$ using the given String structure. This is the [[cocycle]] for a $\mathbf{B}U(1)$-[[principal 2-bundle]] on the total space $P$ of the $Spin$-principal bundle: a [[bundle gerbe]]. The rest of the diagram is constructed in order to prove the following: \begin{itemize}% \item The class in $H^3(P, \mathbb{Z})$ of this bundle gerbe, represented by $P \to B^2 U(1)$ has the property that restricted to the fibers of the $Spin(n)$-principal bundle $P$ it becomes the generating class in $H^3(Spin(n), \mathbb{Z})$. \end{itemize} \end{proof} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{itemize}% \item On a 3-[[dimension|dimensional]] [[orientation|oriented]] [[manifold]] string structures are equivalently [[2-framings]]. \end{itemize} \hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries} \begin{itemize}% \item [[orientation]] \item [[spin structure]], [[twisted spin structure]] \item [[p1-structure]], [[Atiyah 2-framing]] \item \textbf{string structure}, [[differential string structure]] \item [[fivebrane structure]], [[differential fivebrane structure]] \end{itemize} [[!include higher spin structure - table]] \hypertarget{references}{}\subsection*{{References}}\label{references} The relevance of String structures (like that of Spin structures half a century before) was recognized in the physics of spinning strings, therefore the name. The article \begin{itemize}% \item Killingback, \emph{World-sheet anomalies and loop geometry} Nuclear Physics B Volume 288, 1987, Pages 578-588 \end{itemize} was (it seems) the first to derive the Green-Schwarz anomaly cancellation condition of the effective background theory as the [[quantum anomaly]] cancellation condition for the worldsheet theory of the heterotic string's [[sigma-model]] by direct generalization of the way the condition of a [[spin structure]] may be deduced from anomaly cancellation for the superparticle. String stuctures had at that time been discussed in terms of their transgressions to loop spaces \begin{itemize}% \item [[Edward Witten]], \emph{The Index of the Dirac Operator in Loop Space} Proc. of Conf. on Elliptic Curves and Modular Forms in Algebraic Topology, Princeton, N.J., Sep 1986. \end{itemize} \begin{itemize}% \item [[Edward Witten]], \emph{Elliptic Genera and Quantum Field Theory} Commun.Math.Phys.109:525,1987 \end{itemize} later it was reformulated in terms of the classes down on base space just mentioned in \begin{itemize}% \item [[Stefan Stolz]], [[Peter Teichner]], \href{http://math.ucr.edu/home/baez/qg-winter2007/Oxford.pdf}{What is an elliptic object}. \end{itemize} The relation between the two pictures is analyzed for instance in \begin{itemize}% \item A. Asada, \emph{Characteristic classes of loop group bundles and generalized string classes} , Differential geometry and its applications (Eger, 1989), 33--66, Colloq. Math. Soc. J\'a{}nos Bolyai, 56, North-Holland, Amsterdam, 1992. ([[Asada.pdf:file]]) \end{itemize} A precise formulation of Killingbacks original argument in [[differential K-theory]] appeared in \begin{itemize}% \item [[Ulrich Bunke]], \emph{String structures and trivialisations of a Pfaffian line bundle}, Commun. Math. Phys. (2011) 307: 675 (\href{http://arxiv.org/abs/0909.0846}{arXiv:0909.0846}, \href{https://doi.org/10.1007/s00220-011-1348-0}{doi:10.1007/s00220-011-1348-0}) \end{itemize} A review of that is in \begin{itemize}% \item [[Konrad Waldorf]], \emph{Geometric string structure and supersymmetric sigma-models} (\href{http://www.konradwaldorf.de/docs/cardiff.pdf}{pdf}) \end{itemize} The definition of string structures by degree-3 classes on the total space of the spin bundle is used in \begin{itemize}% \item [[Corbett Redden]], \emph{Canonical metric connections associated to string structures} PhD Thesis, (2006)(\href{https://www.math.sunysb.edu/~redden/Thesis.pdf}{pdf}) \end{itemize} For discussion of String-structures using 3-classes on total spaces see for instance the work by Corbett Redden and Konrad Waldorf described at \begin{itemize}% \item [[Oberwolfach Workshop, June 2009 -- Friday, June 12]] \end{itemize} Discussion of the [[moduli stack]] of [[twisted differential string structures]] is in \begin{itemize}% \item [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{[[schreiber:Twisted Differential String and Fivebrane Structures]]}, Communications in Mathematical Physics, October 2012, Volume 315, Issue 1, pp 169-213 (\href{http://arxiv.org/abs/0910.4001}{arXiv:0910.4001}) \end{itemize} An explicit cocycle construction of the essentially unique [[string 2-group]]-[[principal 2-bundle]] lift of the [[tangent bundle]] of the [[sphere|5-sphere]] is given in \begin{itemize}% \item [[David Roberts]] \emph{Explicit string bundles}, talk at \emph{\href{http://www.christiansaemann.de/higherworkshop2014/index.html}{Workshop on Higher Gauge Theory and Higher Quantization}} 2014 ([[RobertsStringS5.pdf:file]]) \end{itemize} Discussion for indefinite (Lorentzian) signature is in \begin{itemize}% \item Hyung Bo Shim, \emph{Indefinite string structure}, thesis 2013 (\href{http://d-scholarship.pitt.edu/19620/}{web}) \end{itemize} More discussion of relation to spin structures on loop spaces is in \begin{itemize}% \item Alessandra Capotosti, \emph{[[From String structures to Spin structures on loop spaces]]}, Ph.D. thesis, Universit\`a{} degli Studi Roma Tre, Rome, April 2016 \end{itemize} [[!redirects String structure]] [[!redirects String-structure]] [[!redirects string-structure]] [[!redirects string structures]] [[!redirects String structures]] \end{document}