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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*{Cartan geometry, Supergravity and Branes} \begin{quote}% under construction -- These are presently just notes to go along with a talk \href{http://muuk.karlin.mff.cuni.cz/cs/node/13}{here} \end{quote} This entry is going to become an exposition to the topics of \begin{enumerate}% \item [[Cartan geometry]]; \item higher dimensional [[supergravity]]; \item [[higher dimensional WZW models]]; \item the completed [[brane scan]]; \item anomaly cancellation in higher WZW models via higher string lifts. \end{enumerate} and an indication of how these all naturally flow out of a common source when considered in [[higher differential geometry|higher differential]] [[supergeometry]]. \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{cartan_geometry}{Cartan geometry}\dotfill \pageref*{cartan_geometry} \linebreak \noindent\hyperlink{higher_dimensional_supergravity}{Higher dimensional supergravity}\dotfill \pageref*{higher_dimensional_supergravity} \linebreak \noindent\hyperlink{higher_wesszuminowittentype_sigmamodels}{Higher Wess-Zumino-Witten-type sigma-models}\dotfill \pageref*{higher_wesszuminowittentype_sigmamodels} \linebreak \noindent\hyperlink{cohomology}{Cohomology}\dotfill \pageref*{cohomology} \linebreak \noindent\hyperlink{higher_wzw_terms}{Higher WZW terms}\dotfill \pageref*{higher_wzw_terms} \linebreak \noindent\hyperlink{brane_intersection_laws_and_brane_condensates_in_extended_superspacetime}{Brane intersection laws and Brane condensates in extended super-spacetime}\dotfill \pageref*{brane_intersection_laws_and_brane_condensates_in_extended_superspacetime} \linebreak \noindent\hyperlink{TheCompletedBrainScan}{The completed brane scan}\dotfill \pageref*{TheCompletedBrainScan} \linebreak \noindent\hyperlink{anomalyfree_higher_sigmamodels_via_higher_stringlifts_of_cartan_connections}{Anomaly-free higher sigma-models via higher string-lifts of Cartan connections}\dotfill \pageref*{anomalyfree_higher_sigmamodels_via_higher_stringlifts_of_cartan_connections} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{cartan_geometry}{}\subsection*{{Cartan geometry}}\label{cartan_geometry} [[smooth manifolds|Smooth manifolds]], such as [[spacetimes]], are modeled on the [[real line]] and its [[products]], the [[Cartesian spaces]] $\mathbb{R}^d$ (see also at \emph{[[geometry of physics -- coordinate systems]]}). It is clear that here it is crucial that $\mathbb{R}^n$ is regarded with its [[smooth structure]]. But $\mathbb{R}^d$ also has ([[smooth group|smooth]]) \emph{[[group]]} structure. As such it is the [[translation group]]. In [[Newtonian mechanics]] this group structure plays a key role, as it allows for instance to say what in means that a [[particle]] propagates along a straight [[trajectory]]. With the dawn of [[general relativity]] and its ``[[general covariance]]'' it was finally understood that this is a bit too naive: [[free field|free]] [[particles]] propagate on straight lines in $\mathbb{R}^n$ only [[infinitesimal object|infinitesimally]]. After every infinitesimal translation, the notion of ``straight'' needs to be adjusted, due to the ``[[force]]'' of [[gravity]]. This has a straightforward visualization in 2-dimensions: visualize a bumpy [[surface]] $X$ [[embedding|embedded]] in $\mathbb{R}^3$ (for instance a [[2-sphere]]). Pick a point $x \in X$ and visualize the [[tangent]] [[plane]] to that point, which is an $\mathbb{R}^2$. Now given a [[curve]] in $X$ that emanates from $x$, then one may visualize the plane to be ``rolling without sliding'' on $X$ such that the point where it touches $X$ follows the curve. Under this map from curves in the plane to curves in the $X$, straight lines in $\mathbb{R}^2$ now map to [[geodesics]] in $X$. These are the actual paths that free particles in $X$ follow. The most popular way to formalize this is the modern concept of the [[Levi-Civita connection]] of the [[Riemannian metric]] on $X$ (which in the above example is induced from the canonical one on the embedding space $\mathbb{R}^3$). But [[Élie Cartan]]`s original way of speaking about [[affine connections]] much closer resembles this picture of a ``local model space with group structure'' doing ``rolling without slipping'' over [[spacetime]]. This is now mostly called a \emph{[[Cartan connection]]}. Here one regards the [[Euclidean group]] $Iso(n)$ of all [[isometries]] of $\mathbb{R}^d$. Inside this is the [[rotation group]] $SO(d) \hookrightarrow Iso(d)$. The [[quotient group]] is [[Cartesian space]] \begin{displaymath} \mathbb{R}^d \simeq Iso(d)/SO(d) \,. \end{displaymath} The structure of ``rolling without sliding'' is then formalized in the concept of a [[Cartan connection]] by saying \begin{enumerate}% \item \textbf{rolling} -- there is an $Iso(d)$-[[principal connection]] on $X$ (the group $Iso(d)$, via its [[action]] on $\mathbb{R}^d$, rolls and slides the Cartesian space around); \item \textbf{without sliding} -- such that there is a [[reduction of the structure group]] to $SO(d)$ (this makes the original $Iso(n)$-bundle be have [[associated bundle|associated]] to it the actual $\mathbb{R}^d$-[[fiber bundle]]); \end{enumerate} and such that under this reduction the connection at each point infinitesimally identifies the [[tangent space]] of $X$ with the abstract copy of $\mathbb{R}^d \simeq Iso(d)/SO(d)$. This has an evident generalization where we consider any inclusion $H \hookrightarrow G$ of [[Lie groups]]. If one considers instead of [[Cartesian space]] $\mathbb{R}^d$ the [[Minkowski space]] $\mathbb{R}^{d-1,1}$, then $Iso(d-1,1)$ is the [[Poincaré group]]. and $SO(d-1,1)$ is the [[Lorentz group]]. Now the [[geodesics]] via [[parallel transport]] along a $(SO(d-1,1)\hookrightarrow Iso(d-1,1))$-[[Cartan connection]] reflect the [[force]] of [[gravity]] in the [[theory (physics)|theory]] of [[general relativity]]. Many other combinations $(H \hookrightarrow G)$ may be considered: [[!include local and global geometry - table]] \hypertarget{higher_dimensional_supergravity}{}\subsection*{{Higher dimensional supergravity}}\label{higher_dimensional_supergravity} In the contemporary [[physics]] literature this concept of [[Cartan connection]] may seem to play an orphaned role, at least if one compares the number of occurrences of the explicit term [[Cartan connection]] in the physics literature over that of ``[[Levi-Civita connection]]'', and certainly when compared to the number of contemporary articles that speak of [[geodesics]] without any concept of [[connection]] made explicit. However, this is partly an illusion. Examination of the literature shows that at least as soon as authors consider \emph{[[supergravity]]}, then everything is really secretly formulated in terms of [[supergeometry|super]] [[Cartan geometry]], as only in this formulation is the incorporation of [[fermions]] and of [[supersymmetry]] really natural. In fact, higher dimensional supergravity such as [[type II supergravity]], [[heterotic supergravity]] and [[11-dimensional supergravity]] crucially includes [[field (physics)|fields]] which are [[higher gauge fields]] -- the [[B2-field]], the [[B6-field]], the [[C3-field]], the [[C6-field]], and also the [[RR-field]] if one digs deeper. The only proposal in the physics literature for how to deal with such higher gauge higher supergravity theories \emph{geometrically} is the [[D'Auria-Fre formulation of supergravity]]. And this is secretly \emph{[[higher Cartan geometry]]}. Here [[super L-∞ algebra]] is the joint generalization of [[Lie algebra]] to [[super Lie algebra]] and to [[L-∞ algebra]]. \begin{displaymath} \itexarray{ && Lie\;algebras \\ &\swarrow && \searrow \\ super \;Lie \; algebras && && L_\infty-algebras \\ & \searrow && \swarrow \\ && super\; L_\infty-algebras } \,. \end{displaymath} Let $\mathbb{g}$ be a [[finite dimensional vector space]] and write $\wedge^\bullet$ for the [[Grassmann algebra]] of its [[dual vector space]]. A \emph{[[differential]]} in this algebra is a map $d_{CE} \colon \wedge^\bullet \mathfrak{g} \longrightarrow \wedge^\bullet \mathfrak{g}$ which is a graded [[derivation]] of degree 1 and squares to 0, $(d_{CE})^2 = 0$. One finds that choices of such differentials are equivalent to [[Lie algebra]] structures on $\mathfrak{g}$: a degree 1 derivation $d_{CE}$ on $\wedge^\bullet \mathfrak{g}^\ast$ is equivalently a skew bilinear bracket, and then the condition $(d_{CE})^2 = 0$ is equivalent to the [[Jacobi identity]], hence makes this bracket a [[Lie bracket]]. The resulting [[dg-algebra]] $(CE(\mathfrak{g}), d_{CE})$ is called the \emph{[[Chevalley-Eilenberg algebra]]} of this Lie algebra. If we her allow $\mathfrak{g}$ to be a [[super vector space]] so that $\wedge^\bullet \mathfrak{g}^\ast$ is now $(\mathbb{Z}, \mathbb{Z}/2)$-bigraded, and require $d_{CE}$ to be of degree $(1,even)$ (see at ) then in the same way we find that this is equivalent to the structure of a [[super Lie algebra]]. Now more generally, let $\mathfrak{g}$ be a $\mathbb{N}$-[[graded vector space|graded]] [[super vector space]] (degreewise finite dimensional, hence of [[finite type]]). Then choices of degree $(1,even)$ differentials $d_{CE}$ on $\wedge^\bullet \mathfrak{g}^\ast$ are equivalent to choices of [[super L-∞ algebra]] structures on $\mathfrak{g}$. Finally, still a bit more generally, let $\mathfrak{a}_0 = C^\infty(X)$ be the algebra of functions on some ([[supermanifold|super]]) [[manifold]] $X$, and let $\mathbb{a}$ be an $\mathbb{N}$-graded [[projective module]] over $\mathbb{a}_0$ which in degree 0 is $\mathfrak{a}_0$. Write now $Sym^\bullet_{\mathfrak{a}_0}(\mathfrak{a}^\ast)$ for the graded-symmetric algebra of the $\mathfrak{a}_0$-dual of $\mathfrak{a}$. Now a choice of differential $d_{CE}$ in this ($\mnathbb{R}$-linear, not necesssarily $\mathfrak{a}_0$-linear) gives the structure of a (super) [[L-∞ algebroid]]. We write again \begin{displaymath} CE(\mathfrak{a}) \coloneqq (Sym^\bullet_{\mathfrak{a}_0} \mathfrak{a}^\ast, d_{CE}) \end{displaymath} for this [[Chevalley-Eilenberg algebra]]. The [[full subcategory]] \begin{displaymath} s L_\infty Algd \hookrightarrow sdgAlg^{op} \end{displaymath} of that of super-[[dg-algebras]] whose underlying graded algebra is free on a graded super vector space in this way we call that of \emph{super $L_\infty$-algebroids}. examples [[super Poincaré Lie algebra]]\ldots{} [[super Minkowski spacetime]]: $\mathbb{R}^{d-1,1\vert N} = \mathfrak{sIso}(d-1,1\vert N)/\mathfrak{so}$ in fact for higher dimensional [[supergravity]] we need [[extended super Minkowski spacetimes]]\ldots{} To motivate these we now consider WZW models. \hypertarget{higher_wesszuminowittentype_sigmamodels}{}\subsection*{{Higher Wess-Zumino-Witten-type sigma-models}}\label{higher_wesszuminowittentype_sigmamodels} \hypertarget{cohomology}{}\subsubsection*{{Cohomology}}\label{cohomology} A miracle happens when one passes from [[Lorentzian geometry]] to Lorentzian [[supergeometry]]. While the [[cohomology]] of [[Cartesian space]] and [[Minkowski spacetime]] is fairly trivial\ldots{} \ldots{}the [[cohomology]] of [[super-Minkowski spacetime]] $\mathbb{R}^{d-1,1\vert N}$ turns out to have ``exceptional'' cohomology classes of degree $(p+2)$ for some special combinations of \begin{enumerate}% \item the [[dimension]] $d$ of [[spacetime]]; \item the real [[spin representation]] $N$; \item the degree $(p+2)$ (where, see in a few lines below, $p$ turns out to be the [[dimension]] of a [[super p-brane]] propagating through spacetime). \end{enumerate} More precisely, it is the $SO(d-1,1)$-invariant [[group cohomology]] of $Iso(\mathbb{R}^{d-1,1\vert N})$ which has these exceptional cocycles, and hence the [[super Lie algebra|super]] [[Lie algebra cohomology]] of [[super Minkowski spacetime]]. We come to this \hyperlink{TheCompletedBrainScan}{below}. But the thing is: ever since [[Paul Dirac]] it is known that given a [[cocycle]] in [[differential cohomology]] of degree $(p+2)$ on [[spacetime]] $X$ -- such as a [[line bundle with connection]] representing the [[electromagnetic field]] -- then this serves as the [[background field]] for a [[sigma-model]] describing the propagation of a [[p-brane]] in $X$ which is [[charge|charged]] under this [[higher gauge field]] and feels its [[forces]] -- such as, for $p = 0$, an [[electron]] subject to the [[Lorentz force]]. Moreover, the consideration of [[Dirac monopoles]] and other [[instantons]] and [[black branes]] shows that given any such [[higher gauge field]], it automatically \emph{induces} [[p-branes]] which are charged under it: the [[p-brane]] [[sigma model]] turns out to give the [[perturbation theory]] for which the [[black branes]] are the [[non-perturbative effects]]. This is how most of the [[p-branes]] in [[superstring theory]] and [[M-theory]] were originally found by looking at [[black brane]] solutions in higher dimesnional [[supergravity]]. More in detail, the higher dimensional analog of the [[Lorentz force]], felt by these [[p-branes]] is given by [[interaction]] [[action functional]] which is the [[higher parallel transport]] ([[higher volume holonomy]]) of the [[background gauge field]] over the [[worldvolume]] of the [[p-brane]]. This is also known as the \emph{[[WZW term]]} in \emph{[[higher dimensional WZW theory]]}. Locally this is a simple phenomenon, and this local picture is what most of the physics textbook will ever consider: there is a [[differential form|differential]] [[curvature form|curvature (p+2)-form]] $\omega$ on the local model space $G/H$, and if that is like [[Minkowski space]] then it is [[contractible space]] hence there is a ``higher [[vector potential]]'' $A \in \Omega^{p+1}(G/H)$ such that $\omega = \mathbf{d}A$, and the [[action functional]] in question is just that given by [[integration of differential forms]]. However, already in local model spaces such as [[super Minkowski spacetime]], each $\omega$ may not have such a potential that is $H$-invariant. Worse, once the [[p-brane]] leaves a given local model space $G/H \hookrightarrow X$ of [[spacetime]], then one needs a [[higher gauge transformation]] to connect the interaction terms on the two patches. Still worse, these gauge transformation need to glue (need to ``[[descent]]'' along the [[cover]] $\coprod_i G/H \to X$) to a globally defined (``non-perturbative'', free of [[classical anomalies]]) higher gauge field on all of $X$. For the traditional 2d [[WZW model]] this was eventually fairly widely appreciated, for [[higher dimensional WZW models]] this is still much of an open secret. Before proceeding to the explanation of the global higher WZW terms, we consider some [[boundary field theory]] now. \hypertarget{higher_wzw_terms}{}\subsubsection*{{Higher WZW terms}}\label{higher_wzw_terms} Observation: $L_\infty$-cocycle is \begin{displaymath} \mathfrak{g} \longrightarrow \mathbf{B}^{} \end{displaymath} \begin{uprop} These have [[Lie integration]] to global WZW term \begin{displaymath} \mathbf{L}_{WZW} \;\colon\; \tilde G \longrightarrow \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn} \end{displaymath} \end{uprop} (\hyperlink{FSS13}{FSS 13, section 5}) \hypertarget{brane_intersection_laws_and_brane_condensates_in_extended_superspacetime}{}\subsubsection*{{Brane intersection laws and Brane condensates in extended super-spacetime}}\label{brane_intersection_laws_and_brane_condensates_in_extended_superspacetime} We discuss now a way (\hyperlink{FSS13}{FSS 13}) to find the [[boundary conditions]] and brane condensates via [[homotopy theory]] of [[super L-∞ algebras]] and via the [[cobordism hypothesis]] for [[local prequantum field theory|local prequantum]] [[boundary field theory]]. So given a $(p+2)$-[[infinity-Lie algebroid cohomology|cocycle]] on [[super Minkowski spacetime]], which is just a [[homomorphism]] of [[super L-∞ algebras]] of the form \begin{displaymath} \mathbb{R}^{d-1,1 \vert N} \stackrel{\mu_{p+2}}{\longrightarrow} \mathbf{B}^{p+1} \mathbb{R} \end{displaymath} then we discussed how this is the local (``rational'') expression for a higher WZW term for a [[sigma-model]] of a [[p-brane]]. A [[field (physics)|field]] configuration of that sigma-model is hence a map as on the left of \begin{displaymath} \itexarray{ \Sigma_{p+1} \stackrel{\phi}{\longrightarrow} \mathbb{R}^{d-1,1 \vert N} \stackrel{\mu_{p+2}}{\longrightarrow} \mathbf{B}^{p+1} \mathbb{R} } \end{displaymath} and the composite is (the [[curvature]] of) the [[local Lagrangian]] of the [[gauge field|gauge]] [[interaction]]. In (\hyperlink{FSS13}{FSS 13}) it is show how to use [[Lie integration]] to produce from this the full higher [[WZW term]]/[[prequantum n-bundle]]/[[interaction]] [[local Lagrangian]] \begin{displaymath} \mathbf{L}_{WZW} \;\colon\; \mathbb{R}^{d-1,1 \vert N} \longrightarrow \mathbf{B}^{p+1} (\mathbb{R}/\Gamma)_{conn} \end{displaymath} But for the moment it is helpful for simplicity to stay at the rational/infinitesimal/$L_\infty$-algebraic level. All what we find here lifts as expected under [[Lie integration]]. So, now according to the [[cobordism hypothesis]] for [[local prequantum field theory|local prequantum]] [[boundary field theory]], a [[boundary condition]] for this Lagrangian is a diagram of the form \begin{displaymath} \itexarray{ && Q &\longrightarrow& \ast \\ && \downarrow &^{\mathllap{\simeq}}\swArrow_{\mathrlap{\kappa}}& \downarrow^{\mathrlap{0}} \\ \Sigma_{p+1} &\stackrel{\phi}{\longrightarrow}& \mathbb{R}^{d-1,1 \vert N} &\stackrel{\mu_{p+2}}{\longrightarrow}& \mathbf{B}^{p+1} \mathbb{R} } \,, \end{displaymath} hence is some space $Q$ inside spacetime such that the pullback of the higher [[WZW term]]/[[prequantum n-bundle]]/[[interaction]] [[local Lagrangian]] for the [[bulk field theory|bulk]] [[field (physics)|fields]] to this space is equipped with a choice of gauge trivialization $\kappa$. Moreover, given this then the [[field (physics)|fields]] of the [[sigma-model]] on a [[worldvolume]] $\Sigma_{p+1}$ with [[boundary]] $(\partial \Sigma)_p \hookrightarrow \Sigma_{p+1}$ is a diagram as on the left of \begin{displaymath} \itexarray{ (\partial \Sigma)_{p+1} &\stackrel{\phi_{bdr}}{\longrightarrow}& Q &\longrightarrow& \ast \\ \downarrow && \downarrow &^{\mathllap{\simeq}}\swArrow_{\mathrlap{\kappa}}& \downarrow^{\mathrlap{0}} \\ \Sigma_{p+1} &\stackrel{\phi}{\longrightarrow}& \mathbb{R}^{d-1,1 \vert N} &\stackrel{\mu_{p+2}}{\longrightarrow}& \mathbf{B}^{p+1} \mathbb{R} } \,, \end{displaymath} We want to understand what kind of object this $Q$ is that the boundary of the $p$-brane may stick to. To that end, observe that by the [[universal property]] of [[homotopy pullbacks]], we may decompose the diagram on the right into two diagrams, where the intermediate stage is the [[extended super Minkowski spacetime]] ${\widehat{\mathbb{R}}}^{d-1,1 \vert N}$ which, as a [[super L-∞ algebra]], is the [[homotopy fiber]] of $\mu_{p+2}$ \begin{displaymath} \itexarray{ (\partial \Sigma)_{p+1} &\stackrel{\phi_{bdr}}{\longrightarrow}& Q &\stackrel{\Phi}{\longrightarrow} & {\widehat{\mathbb{R}}}^{d-1,1 \vert N} &\longrightarrow& \ast \\ \downarrow && \downarrow &{}^{\mathllap{\simeq}}\swArrow& \downarrow &^{\mathllap{\simeq}}\swArrow& \downarrow^{\mathrlap{0}} \\ \Sigma_{p+1} &\stackrel{\phi}{\longrightarrow}& \mathbb{R}^{d-1,1 \vert N} &\stackrel{id}{\longrightarrow}& \mathbb{R}^{d-1,1 \vert N} &\stackrel{\mu_{p+2}}{\longrightarrow}& \mathbf{B}^{p+1} \mathbb{R} } \,. \end{displaymath} This is the higher [[infinity-Lie algebra cohomology|L-∞ extension]] classified by the [[cocycle]] in generalization to the familiar fact that 2-coycles classify plain [[Lie algebra extensions]]. \begin{uprop} The [[Chevalley-Eilenberg algebras]] of these ${\widehat{\mathbb{R}}}^{d-1,1 \vert N}$ are precisely the dg-algebras used at least since (\hyperlink{DAuriaFre82}{D'Auria-Fr\'e{} 82}) in the [[D'Auria-Fré formulation of supergravity]]. \end{uprop} This is shown in (\hyperlink{FSS13}{FSS 13}) using a characterization of [[homotopy pullbacks]] in a [[model structure for L-infinity algebras]] derived in (\hyperlink{FRS13b}{FRS 13b}). But now from this we see that on $Q$ there is itself a [[sigma-model]] [[field (physics)|field]] $\Phi$ that exhibits $Q$ itself as a [[brane]], propagating in this [[extended super Minkowski spacetime]]. Or rather: that would exhibit this if there were an [[action functional]] for this sigma model. By repeating the reasoning, this is given in turn by (if it exists nontrivially) a higher WZW term given by a higher [[L-∞ cocycle]] $\mu_{\tilde p + 2}$ of some further degree $\tilde p + 2$. \begin{displaymath} \itexarray{ (\partial \Sigma)_{p+1} &\stackrel{\phi_{bdr}}{\longrightarrow}& \Sigma_{\tilde p + 1} &\stackrel{\Phi}{\longrightarrow} & {\widehat{\mathbb{R}}}^{d-1,1 \vert N} &\stackrel{\mu_{\tilde p + 2}}{\longrightarrow}& \mathbf{B}^{\tilde p + 1} \\ \downarrow && \downarrow \\ \Sigma_{p+1} &\stackrel{\phi}{\longrightarrow}& \mathbb{R}^{d-1,1 \vert N} &\stackrel{id}{\longrightarrow}& \mathbb{R}^{d-1,1 \vert N} &\stackrel{\mu_{p+2}}{\longrightarrow}& \mathbf{B}^{p+1} \mathbb{R} } \,. \end{displaymath} \hypertarget{TheCompletedBrainScan}{}\subsection*{{The completed brane scan}}\label{TheCompletedBrainScan} This gives a curious direct identification between [[L-∞ algebra cohomology]] and brane intersection laws: every $L_\infty$-extension classified by an $L_\infty$-cocycle together with a further cocycle on the extension gives a [[higher WZW sigma-model]] for some [[super p-brane]] wich may end on a super-$\tilde p$-brane. It turns out that this reasoning reproduces the \emph{completed} [[brane scan]] of [[superstring theory]]/[[M-theory]], including notably the [[D-branes]] of [[type II string theory]] together with the information that the [[fundamental string]] may end on them, as well as the sigma-model for the [[M5-brane]] with its tensor multiplet fields (\hyperlink{FSS13}{FSS 13}) and the information that [[M2-brane]] may end on it. These maps out much of the key statements about [[M-theory]] (and does so in a precise/rigorous way). \begin{displaymath} \itexarray{ && &\mathfrak{D}(2p)\mathfrak{brane} &&& \mathfrak{D}(2p+1)\mathfrak{brane} \\ &&& \downarrow && & \downarrow \\ && & \mathfrak{string}_{IIA} && & \mathfrak{string}_{IIB} \\ && & \searrow && \swarrow \\ \mathfrak{sdstring} & && \mathbb{R}^{10 \vert N=(1,1)} & & \mathbb{R}^{10 \vert N=(2,0)} &&& \mathfrak{string}_{het} \\ & \searrow & && \downarrow && & \swarrow \\ && \mathbb{R}^{6 \vert N=(2,0)} && \mathbb{R}^{10 \vert N=2} && \mathbb{R}^{10 \vert N=(1,0)} \\ && & \searrow & \downarrow & \swarrow \\ && && \mathbb{R}^{0\vert N} &\leftarrow& \mathbb{R}^{11\vert N=1} &\leftarrow& \mathfrak{m}2\mathfrak{brane} &\leftarrow& \mathfrak{m}5\mathfrak{brane} } \end{displaymath} [[!include brane scan]] \textbf{Proposition} We may construct the [[prequantum n-bundle|prequantum (p+1)-bundle]] \begin{displaymath} \mathbf{L}_{WZW} \;\colon\; {\widehat \mathbb{R}}^{d-1,1\vert N} \stackrel{\mathbf{L}_{WZW}}{\longrightarrow} \mathbf{B}^{p+1} (\mathbb{R}/\Gamma)_{conn} \end{displaymath} for all [[super p-brane]] [[sigma-models]] via a kind of [[Lie integration]]. (\hyperlink{FSS13}{FSS13}) \hypertarget{anomalyfree_higher_sigmamodels_via_higher_stringlifts_of_cartan_connections}{}\subsection*{{Anomaly-free higher sigma-models via higher string-lifts of Cartan connections}}\label{anomalyfree_higher_sigmamodels_via_higher_stringlifts_of_cartan_connections} All would be done and said if [[spacetime]] were fixed to be a give [[extended super Minkowski spacetime]]. But of course the key now is that in [[supergravity]] instead spacetime $X$ only locally looks this way, and globally is a [[Cartan connection]] for the [[super Poincaré group]] or its higher analogs acting on an [[extended super Minkowski spacetime]]. What we need to solve the gluing problem for the higher WZW term and hence cancel its [[classical anomaly]] is that on $X$ itself there is a WZW term \begin{displaymath} \mathbf{L}_{WZW}^{global} \colon X \longrightarrow \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn} \end{displaymath} such that for a given [[cover]] by local model space as given by the [[Cartan connection]] [[background gauge field]] of [[gravity]] and [[B-field]], [[C-field]], etc; this restricts to the canonical one described above, i.e. we need a [[commuting diagram]] of the form \begin{displaymath} \itexarray{ {\widehat{\mathbb{R}}}^{d-1,1\vert N} \\ \downarrow & \searrow^{\mathrlap{\stackrel{\mathbf{L}_{WZW}}} \\ X & \stackrel{\mathbf{L}_{WZW}^{global}}{\longrightarrow} & \mathbf{B}^{p+1} (\mathbb{R}/\Gamma)_{conn} } \,. \end{displaymath} \begin{quote}% I claim that the solution to this globalization problem works as follows, though I have not written this down in full detail yet. Notice that while I write super-Minkowski spacetimes here just for the heck of it, this may be considered much more generally in [[higher Cartan geometry]]. \end{quote} Consider the restriction of the WZW term to any [[formal disk]] (this is [[synthetic geometry|synthetically]] the $L_\infty$-algebra, really), in the sense discussed at \emph{[[Lie differentiation]]}. \begin{displaymath} \mathbf{L}_{WZW}^{formal} \;\colon\; {\widehat{\mathbb{D}}}^{d-1,1\vert N} \hookrightarrow {\widehat \mathbb{R}}^{d-1,1\vert N} \stackrel{\mathbf{L}_{WZW}}{\longrightarrow} \mathbf{B}^{p+1} (\mathbb{R}/\Gamma)_{conn} \end{displaymath} Consider then the [[quantomorphism n-group]] of this formal WZW term (\hyperlink{FRS13a}{FRS 13a}), defined by sitting in the [[homotopy fiber sequence]] \begin{displaymath} QuantMorph(\mathbf{L}_{WZW}^{formal}) \longrightarrow \mathbf{Aut}(\mathbb{D}^{d-1,1\vert N}) \simeq GL(d\vert N) \stackrel{\mathbf{L}_{WZW}^{forma} \circ (-)}{\longrightarrow} (\mathbf{B}^{p+1} (\mathbb{R}/\Gamma))\mathbf{Conn}(\mathbb{D}^{d-1,1\vert N}) \end{displaymath} where the rightmost term is the [[differential concretification]] of the [[mapping stack]] $[\mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn}$, hence the higher [[moduli stack of connections]] on the given [[formal disk]] inside [[extended super Minkowski spacetime]]. Intuitively this [[smooth super ∞-group|smooth super]] [[∞-group]] looks as follows (\hyperlink{FRS13a}{FRS 13a}): \begin{displaymath} QuantMorph(\mathbf{L}_{WZW}^{formal}) \;=\; \left\{ \itexarray{ \mathbb{D}^{d-1,1\vert N} && \stackrel{\simeq}{\longrightarrow} && \mathbb{D}^{d-1,1\vert N} \\ & {}_{\mathllap{\mathbf{L}_{WZW}^{formal}}}\searrow &\simeq& \swarrow_{\mathrlap{\mathbf{L}_{WZW}^{formal}}} \\ && \mathbf{B}^{p+1} (\mathbb{R}/\Gamma)_{conn} } \right\} \end{displaymath} An element in this group is precisely the datum needed to change [[tangent spaces]] in a [[Cartan connection]] while carrying also the WZWZ term along! See also at \emph{[[parameterized WZW model]]}. By the discussion in (\hyperlink{FRS13a}{FRS 13a}), this is the general higher and super-analog of [[string 2-group]], [[fivebrane 6-group]], [[ninebrane 10-group]]. Hence it makes sense to give it a name like so: \begin{displaymath} p Brane(d\vert N) \coloneqq QuantMorph(\mathbf{L}_{WZW}^{formal}) \end{displaymath} Now the claim is that the [[obstruction]] to globally anomlay free super $p$-brane WZW models with respect to a given supergravity Cartan backround is a ``super $p$-brane''-structure, hence a lift \begin{displaymath} \itexarray{ && \mathbf{B}(p Brane(d\vert N)) \\ & {}^{\mathllap{\widehat {\tau}_X}}\nearrow & \downarrow \\ X &\stackrel{\tau_X}{\longrightarrow} & \mathbf{B}GL(d,N) } \end{displaymath} of the map that classifies the [[frame bundle]] as discussed at \emph{[[differential cohesion]]} in the section \emph{\href{differential+cohesion#GLnTangentBundles}{Differential cohesion -- Frame bundles}}. By the above picture of $p Brane(d\vert N)$ it should be plausbile that this is precisely the data needed to carry the WZW turn around with Cartan's ``rolling without slipping''. \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item [[Riccardo D'Auria]], [[Pietro Fré]] \emph{Geometric supergravity in $D = 11$ and its hidden supergroup}, Nuclear Physics B201 (1982) 101-140 ([[GeometricSupergravity.pdf:file]]) \item [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], \emph{[[Supergravity and Superstrings - A Geometric Perspective]]}, World Scientific (1991) \item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]] \emph{[[schreiber:Higher geometric prequantum theory]]} \item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:L-∞ algebras of local observables from higher prequantum bundles]]}, Homology, Homotopy and Applications, Volume 16 (2014) Number 2, p. 107 -- 142 (\href{http://arxiv.org/abs/1304.6292}{arXiv:1304.6292}) \item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]] \emph{[[schreiber:The brane bouquet]]}, International Journal of Geometric Methods in Modern Physics, Vol. 12 (2015) 1550018 (\href{http://arxiv.org/abs/1308.5264}{arXiv:1308.5264}) \end{itemize} \end{document}