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A thought:
With $\mathbf{L}_{WZW}$ a higher connection on some group $V$, then there is the elementary concept (in differential cohesive HoTT) of a first-order integrable definite globalization $\mathbf{L}_{WZW}^X$ of $\mathbf{L}_{WZW}$ over some $V$-manifold $X$. This comes with an $\infty$-functor from such definite globalizations to $\mathrm{Stab}(\mathbf{L}_{WZW}^{inf})$-structures, on $X$, for $\mathbf{L}_{\mathrm{WZW}}^{inf}$ the restriction to the infinitesimal disk $\mathbb{D}^V \to V$.
When realized in the supergeometric model and with $V = \mathbb{R}^{10,1|N=1}$ and $\mathbf{L}_{\mathrm{WZW}} = \mathbf{L}_{M2}$ the GS-WZW term of the M2-brane, then this $\infty$-functor lands in the solutions of 11d supergravity with vanishing gravitino field strength and equipped with a genuine globalization of the M2 WZW term. So this means we may take the space of these definite globalizations as being the actual phase space of 11d SuGra. (Details are still here.)
An interesting question to ask then is which of the symmetries of $\mathbf{L}_{WZW}^{inf}$ covering $V$ carry over to $\mathbf{L}_{WZW}^X$, i.e. which common subgroups $Q$ there are of $Heis(\mathbf{L}_{WZW}) \coloneqq Stab_{V}(\mathbf{L}_{WZW})$ and $QuantMorph(\mathbf{L}_{\mathrm{WZW}}^X) \coloneqq Stab_{Aut(X)}(\mathbf{L}_{WZW}^X)$. In the above situation this means asking for BPS states of 11-d supergravity, namely spacetimes $X$ equipped with field configurations satisfying the SuGra equations of motion such that there are super-Killing vectors.
I should say this more suggestively, so that it has a chance to ring a bell in anyone:
so a first-order integrable definite globalization over a $V$-manifold $X$ is in particular a correspondence of the form
$\array{ && \flat^{rel} U \\ & \swarrow && \searrow \\ V && \swArrow_{\simeq} && X \\ & _{\mathllap{\mathbf{L}_{WZW}}}\searrow && \swarrow_{\mathrlap{\mathbf{L}_{WZW}^X}} \\ && \mathbf{B}\mathbb{G}_{conn} }$given this, it seems somewhat suggestive to consider correspondences of stabilizer $\infty$-groups
$\array{ && Q \\ & \swarrow && \searrow \\ Stab(\mathbf{L}_{WZW}) && && Stab(\mathbf{L}_{WZW}^X) }$and in the given realization in higher supergeometry, these capture the BPS states.
Dim question: What’s the $U$ of $\flat^{rel} U$?
Sorry, I am using notation as in the notes. So $U$ is a $V$-cover that exhbits $X$ as a $V$-manifold, hence it fits into a correspondence
$\array{ && U \\ & \swarrow && \searrow \\ V && && X }$where both maps are infinitesimally étale and the right one is also 1-epi. We could hence ask for extending this to a correspondence of the form
$\array{ && U \\ & \swarrow && \searrow \\ V && \swArrow && X \\ & {}_{\mathllap{\mathbf{L}_{WZW}}}\searrow && \swarrow_{\mathrlap{\mathbf{L}_{WZW}^X}} \\ && \mathbf{B}\mathbb{G}_{conn} }$That would make $\mathbf{L}_{WZW}^X$ an “integrable” globalization of $\mathbf{L}_{WZW}$. This is however a bit too strong in applications. One wants the two to coincide only on all infinitesimal disk, but not on full patches of the cover. Hence we restrict along the inclusion $\flat^{rel} U \to U$ of all infinitesimal disks in $U$ and just demands
$\array{ && \flat^{rel}U \\ & \swarrow && \searrow \\ V && \swArrow && X \\ & {}_{\mathllap{\mathbf{L}_{WZW}}}\searrow && \swarrow_{\mathrlap{\mathbf{L}_{WZW}^X}} \\ && \mathbf{B}\mathbb{G}_{conn} }$to thus get infinitesimal integrability.
I believe that I have figured it out. I am making notes at BPS state – Formalization in higher differential geometry.
Yes, this goes through in full detail. The key observation is that the Poisson bracket Lie n-algebra of a deDonder-Weyl pre-$n$-plectic form on a dual jet bundle of some field theory is, in its dg-Lie algebra incarnation, precisely the centrally extended Lie algebra of currents as in the traditional literature, enhanced by the gauge transformations of currents, which traditional literature (as far as I see) only deals with in a somewhat indirect hand-wavy way.
I have written out this translation now in section 3.3 of Classical field theory via Cohesive homotopy types (schreiber).
I should go an polish the entries BPS state and conserved current accordingly, but right now I may not have time and energy to do so. For the moment I have added pointers to that section 3.3. I’ll go and beautify these entries, I promise, but not right now.
Not sure if any of this means anything to anyone else reading here. I should amplify how neat this is from a general abstract perspective:
In the supergeometric differential cohesive $\infty$-topos, there are exceptional terms
$\mathbf{L}_{WZW} : G/H \longrightarrow \mathbf{B}\mathbb{G}_{conn}$arising from the differential refinement of a kind of Whitehad tower of some atomic type (a super-point). The definite globalizations of these over $G/H$-Cartan geometries $X$ to terms
$\mathbf{L}_{WZW}^X : X \longrightarrow \mathbf{B}\mathbb{G}_{conn}$imply orthogonal structure on $X$ satisfying a constraint that is generally implied by but a bit weaker than the equations of motion of supergravity.
There is one or two exceptional among these exceptional terms $\mathbf{L}_{WZW}$ (namely the “$\kappa$-symmetry WZW-term for the M2-brane” and the “$\kappa$-symmetry WZW-term for the M5-brane propagating in an M2-brane condensate”) and for that one a definite globalization $\mathbf{L}_{WZW}^X$ of over some $X$ is actually equivalent to the equations of motion of 11d supergravity (“low energy M-theory”).
Moreover, finally, the stabilizer $\infty$-group of any such definite globalization
$Stab_{\mathbf{Aut}(X)}(\mathbf{L}_{WZW}^X)$is the Lie integration of a homotopy-refinement (remebering the higher gauge-of-gauge transformation) of the M-theory Lie algebra of BPS brane charges of this super-spacetime $X$.
For instance for the G2-MSSM refinement of the standard model of particle physics we’d be after definite globalization such that this $Stab_{\mathbf{Aut}(X)}(\mathbf{L}_{WZW}^X)$ has precisely a 1-dimensional space of odd Lie generators.
I think it’s fair to say that it is neat how close the general abstract here gets to phenomenology…
Quite a stretch for most of us I guess to understand this.
I don’t really know what BPS states are. They came up in Vafa’s lecture we discussed the other day, as something to do with black holes that wouldn’t evaporate. And they feature in some clever piece of duality.
Does maybe your abstract treatment help with the latter?
BPS states are simply configurations which preserve some supersymmetry.
You know how a Riemannian manifold may have Killing vectors, witnessing that they have some translation and/or rotation symmetry. Now we pass to supergeometry, then on top of translation and rotation, there are the corresponding supersymmetry transformations. A supermanifold with the analog of Riemannian structure is “$1/k$-BPS” if it has $1/k$th of the maximumum possible number of such super-Killing vectors, aka Killing spinors.
Among all black brane spacetimes, those which are BPS are “extremal” in the sense of extremality of charged black holes, meaning they look like gravitational singularities carrying both math and charge in precisely such a way that the gravitational attraction and the charge repulsion compensate and that the horizon (as in black hole horizons) disappears (a “naked singularity”).
But at the heart of it, BPS states are simply states of a theory that preserve some of the theory’s symmetries, here: supersymmetry.
I should add a key point: the key point of BPS states is that they preserve part not of the plain supersymmetry Lie algebra (say the super translation Lie algebra), but part of its “polyvector extension”. It is these extensions which encode the charges mentioned above.
And what I am saying here is that these extensions come about from the fact that we look not just at the group of (super-)isometries of the super-spacetime $X$, but look at its homotopy stabilizer group of $\mathbf{L}_{WZW}$. Where an ordinary stabilizer group is always a subgroup, a homotopy stabilizer group instead may be bigge than the original group, and that’s what’s happening here.
In other words, we consider a pre-quantized higher symplectic geometry where “a geoemtry” is not just a manifold $X$ but is the pair $(X,\mathbf{L}_{WZW}^X)$, and the “higher prequantized symplectomorphisms” of this, they are what integrates those BPS algebras.
I’ll have a go at this later, but I just noticed that we have Killing vector and Killing vector field not talking to one another. The former has the latter as redirect, while the latter has the former but with incorrect spelling ’Kiling’.
[What happens when two pages have each other as redirects?]
Presumably, there’s a link between Killing vector fields and Cartan geometry.
Thanks for the alert. I have merged the two entries now.
Yes, Killing vector fields are just the infinitesimal automorphisms of $(O(\mathbb{R}^{p,q})\hookrightarrow Iso(\mathbb{R}^{p,q}) )$-Cartan geoemtries.
That’s why it is possible, as of more recently, to have a section “4.3.13 Isometries” in dcct. ;-)
@Urs: That file is 110 pages longer than the arXiv version from October 2013 and many section names are different (while the content mostly matches). Is there a place where you store the current version and do you not put a date on it on purpose? I like “Nature”. What’s different since - more emphasis on modalities?
Yes, I keep working on it. It was sent to the arXiv not because it was ready for eternity, but because I thought my vain mortal life would be eased if the document got an official time stamp at that time.
I’ll be keeping the current version at the above link. It does carry a date: “21st century”, that’s my estimate for the time frame for finishing it.
The main additions since are
section 4.4, incorporated from my last preprint last year before life kicked me out of the loop;
sections 4.3.10 –4.3.13 which give the formalization of manifolds (“derived schemes”, if you wish), their frame bundles, $G$-structures and Cartan geometry;
section 4.2.19.2 with a result on higher extensions of mapping class groups – we’ll have a preprint out on that in a few days with Domenico Fiorenza and Alessandro Valentino;
section 1.2.11.3 and 1.2.15.3.3 which give more details on higher conserved current algebras and BPS algebras;
sections 6.6.3, 6.6.7, 6.6.8 on applications of higher Cartan geometry to super $p$-brane geometry, which I incorporated from Obstruction theory for parameterized higher WZW terms (schreiber) following a request from a reader to provide a copy of this material where all the cross-references do work;
the section 2.2 on modalities which you noticed is mostly new, yes, but more a stub to remind myself where this discussion should eventually go. The whole section 2 is a stub. Strictly speaking this should be in an appendix, since I am just collecting some random stuff that I need to point to from the main text. But I keep it there as section 2 since that is the place where this stuff should be in the end.
Hisham and myself are finalizing a first article on this:
Abstract We uncover higher algebraic structures on Noether currents and BPS charges. It is known that equivalence classes of conserved Noether currents form a Lie algebra. We show that at least for target space symmetries of higher parameterized WZW-type sigma-models this naturally lifts to a Lie (p+1)-algebra structure on the Noether currents themselves. Applied to the Green-Schwarz-type action functionals for super p-brane sigma-models this yields super Lie (p+1)-algebra refinements of the traditional BPS brane charge extensions of supersymmetry algebras. We discuss this in the generality of higher geometry where it applies also to branes with (higher) gauge fields on their worldvolume. Applied to the M5-brane sigma-model we recover the M-theory super Lie algebra extension of 11-dimensional superisometries by 2-brane and 5-brane charges. Passing beyond the infinitesimal Lie theory we find cohomological corrections to these charges in higher analogy to the familiar corrections for D-brane charges as they are lifted from ordinary cohomology to twisted K-theory. This supports the proposal that M-brane charges live in a twisted cohomology theory.
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