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We are in the process of finalizing this article here:
Domenico Fiorenza, Hisham Sati, Urs Schreiber
Super Lie $n$-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields
Abstract. We formalize higher dimensional and higher gauge WZW-type sigma-model local prequantum field theory, and discuss its rationalized/perturbative description in (super-)Lie n-algebra homotopy theory (the true home of the “FDA”-language used in the supergravity literature). We show generally how the intersection laws for such higher WZW-type sigma-model branes (open brane ending on background brane) are encoded precisely in (super-)$L_\infty$-extension theory and how the resulting “extended (super-)spacetimes” formalize spacetimes containing $\sigma$-model brane condensates. As an application we prove in Lie $n$-algebra homotopy theory that the complete super $p$-brane spectrum of superstring/M-theory is realized this way, including the pure sigma-model branes (the “old brane scan”) but also the branes with tensor multiplet worldvolume fields, notably the D-branes and the M5-brane. For instance the degree-0 piece of the higher symmetry algebra of 11-dimensional spacetime with an M2-brane condensate turns out to be the “M-theory super Lie algebra”. We also observe that in this formulation there is a simple formal proof of the fact that type IIA spacetime with a D0-brane condensate is the 11-dimensional sugra/M-theory spacetime, and of (prequantum) S-duality for type II string theory. Finally we give the non-perturbative description of all this by higher WZW-type $\sigma$-models on higher super-orbispaces with higher WZW terms in higher differential geometry.
[Google+ might be better for this.]
Spacetime itself emerges from the abstract dynamics of 0-branes.
Are you close here to Liang Kong
The category of D-branes provides us a new way to look at geometry. From this new point of view, the classical set-theoretical geometries are not fundamental and should be viewed as emergent phenomena. For example, a set-theoretical target manifold can emerge as the moduli space of D0-branes,
and
our physical space is nothing but a network of structured stacks of information, from which spacetime can emerge?
Yes, it is that kind of statement! This has been circulating in the string theory community for a long time now, in one form or another. What we found striking here is that the form in which we find it is nicely systematic (“synthetic”) and fully rigorous.
So once one sees that the higher extension classified by a higher cocycle on spacetime is in a formally precise sense the original spacetime together with a condensate of branes defined as having this cocycle induce their sigma-model action functional, the statement that
“The 11d M-theory spacetime is the 10d type IIA string theory spacetime containing a condensate of D0-branes.”
becomes an actual theorem. This means (just to amplify) that there is no doubt about this statement, that it’s not a matter of “believing or not believing in string theory” or the like. It is just a fact in higher super L-infinity algebra homotopy theory.
That’s, to my mind, the point of that article we wrote here as a whole. It amplifies how the core ingredients of string/M-theory are structures that follow in a formally precise way from just the axioms of “Synthetic quantum field theory” as soon as one interprets these in the model of supergeometric cohesion. I find that noteworthy.
I have some study leave ahead ;), so at last a chance to see if I can get a handle on these discoveries. Something I see I would need but lacking on nLab is ’condensate’. Are you using it something like ’Bose–Einstein condensate’?
I like the way Liang went back to Descartes, Newton and Leibniz. Great if this is a stage in the story of what has flowed from their discussions.
Hi David,
yes, “condensate” here is derived from “Bose-Einstein condensate”. (I should create the nLab entries, but maybe a bit later when I have a free minute).
This is the terminology used in second quantization to refer to how a big number of quanta in the “thermodynamic limit” induces a background field for other quanta.
For instance one can consider gravitons perturbatively propagating on a Minkowski spacetime. Then one can “integrate out” a bunch of them and see how that affecs the propagation of one more graviton, perturbatively, in that “effective background” of the others zipping around. In the suitable limit the result is that of one single graviton propoagating on a curved spacetime. One says that the graviton’s have “condensed” to form a “background gravitational field” which is “sourced” by the condensed quanta.
This is how we identify the “condensates” in the article. We observe that the super-$L_\infty$-algebra extension of super-Minkowski space which is classified by a given cocycle is that space such that maps into it are plain maps to spacetime together with the field that is “sourced” by the boundaries of the open $p$-branes whose dynamics is encoded by that cocycle.
p.5 The codomain of $\mathcal{L}_{WZW}$ is shown as $\Omega_{n+1}^{cl}$, where this involves $d_{dR}$ above.
p.6 In diagram at bottom, $\phi_{bdr}$ goes other way from diagram above.
p.8
for the prensent purpose
equivalance classes
equivelence classes
p.9
the charged boundaries of the original the $\sigma$-model branes
Now a question, to allow the construction of a brane bouquet centered on a Lie $\infty$-algebra, what structure must be available?
p. 14 You have $\mu_{\mathfrak{String}_{IIA}}$, where you need $B$.
p. 15
a higher WZW-tpye
p. 16
Neveu-Shcwarz
this 7-cocycles
p. 17
Putting this together with discussed in
11d-super Poinceré algebra
Thanks, David, for all this! It’s nice that you are taking the time to look at the text so closely. Thanks. I think I have fixed all these typos now.
Concerning your question: there is no extra structure needed on an $L_\infty$-algebra for it to have a “brane bouquet”.
That’s why it’s so nicely canonical. It just so happens that every cocycle on an $L_\infty$-algebra induces a “WZW-type $\sigma$-model brane” propagating on it, and every extension class induces a boundary brane for that. That’s not a choice we humans make. That’s how the world is organized. :-)
There are characteristic $p$ Lie algebras, so are there characteristic $p$ $L_\infty$-algebras with their brane bouquets?
You mean Lie algebra ($L\infty$-algebras) over a field of characeristic $p$, right?
So, let’s see, in order to regard Lie algebras or just vector spaces over a field other than the real or complex numbers as a target space for a sigma-model field theory, one needs to decide what the “worldvolume” of such a sigma-model is supposed to be.
For an ordinary sigma-model (in geometry over the real or complex numbers) the domain is a manifold $\Sigma$, hence its tangent spaces have the structure of real vector spaces, and hence we have a clear notion of linear maps $T_x \Sigma \to \mathfrak{g}$.
Now if $\mathfrak{g}$ here is a vector space over some other field, one would have to change the notion of $\Sigma$ such that $T_x \Sigma$ also has the strcucture of a vector space over that field.
So that means passing from smooth differential geometry to some kind of algebraic or arithmetic geometry.
But with that done properly, everything else goes through just as well. Yes.
We have developed that result a bit further:
Domenico Fiorenza, Hisham Sati, Urs Schreiber,
Rational sphere valued supercocycles in M-theory and type IIA string theory
Abstract. We show that supercocycles on super L-infinity algebras capture, at the rational level, the twisted cohomological charge structure of the fields of M-theory and of type IIA string theory. We show that rational 4-sphere-valued supercocycles for M-branes in M-theory descend to supercocycles in type IIA string theory to yield the Ramond-Ramond fields predicted by the rational image of twisted K-theory, with the twist given by the B-field. In particular, we derive the correspondence
M2/M5 ↔ F1/Dp/NS5
via dimensional reduction of sphere-valued L-∞ supercocycles in rational homotopy theory.
Four small typos:
M2/M5-brans; take values the 4-sphere; superstacetime; the rational of spaces.
Thanks! Fixed now.
We have been developing this result yet a bit further:
Domenico Fiorenza, Hisham Sati, Urs Schreiber:
T-Duality from super Lie n-algebra cocycles for super p-branes (schreiber)
Abstract We compute the L-infinity-theoretic double dimensional reduction of the F1/Dp-brane super L∞-cocycles with coefficients in rationalized twisted K-theory from the 10d type IIA and type IIB super Lie algebras down to 9d. We show that the two resulting coefficient L∞-algebras are naturally related by an L∞-isomorphism which we find to act on the super p-brane cocycles by the infinitesimal version of the rules of topological T-duality and inducing an isomorphism between $K^0$ and $K^1$, rationally. Moreover, we show that these L∞-algebras are the homotopy quotients of the RR-charge coefficients by the “T-duality Lie 2-algebra”. We find that the induced L∞-extension is a gerby extension of a 9+(1,1) dimensional (i.e. “doubled”) T-duality correspfondence super-spacetime, which serves as a local model for T-folds. We observe that this still extends, via the D0-brane cocycle of its type IIA factor, to a 10+(1,1)-dimensional super Lie algebra. Finally we show that this satisfies all the expected properties of a local model space for F-theory elliptic fibrations.
If we were expecting to see M-theory in that diagram of yours, would it be hovering dual to ’F-theory elliptic fibration’, at
principal 2-bundle for T-duality 2-group over 9d super-spacetime?
Mind you I recall you speaking of “M-theory-the-grandiose” and “M-theory-the-concrete”. Are there such variants for F-theory?
I remember there’s that idea (F-theory) of some larger theory behind the scenes
With a full description of M-theory available also F-theory should be a full non-perturbative description of type IIB string theory, but absent that it is some kind of approximation.
And that “doubled spacetime” is what Deser and Saemann are addressing in that article mentioned here?
That doesn’t seem right. M-theory is to do with the upper box ’11d N = 1 super-spacetime’, “Assume that a non-perturbative completion of type IIA string theory exists, given by a geometric theory on a 10+1 dimensional Riemannian circle fibration (M-theory)”
Presumably that “the composite diagonal morphism” on p. 28 should have ’9’ in place of that first ’8’.
Right, really better be doing something else – though it does seem quite a story.
There are moments dealing with IIA and IIB when we need to introduce a +1 dimension shift and moments when not. E.g, prop 6.2 on the one hand, but at F-theory there is “IIA string theory in $d = 9 \leftarrow$ T-duality $\rightarrow$ IIB string theory in $d=10$”
That $X_0$ twice in the diagram on p. 20 should be $X_9$?
Maybe the headline diagram misleads a little by giving the impression of a horizontal line of symmetry.
Thanks, David, for catching typos. I have fixed them now.
And I added citation for “double field theory”. My impression is that the observation that the doubling in “double field theory” is just the passage to the correspondence space in “topological T-duality” has not been voiced in the literature before. But if I am wrong about this, I’d be happy to receive pointers.
Regarding the symmetry of the diagram: yes, I know. It does extend to a fully symmetric diagram, but I could not fit that on a page.
Regarding the concrete and the grandiose: The article wants to be read as developing the concrete aspect further. We take what we know for sure (the GS-WZW terms) and see how far they take us, if we keep following them. The point is to show that they take us pretty far.
Starting this week and until the end of the year, I will be giving a weekly lecture series at CAS Prague on this work:
I am lagging behind with typing out my lecture notes for this, not the least because my laptop has died on Monday. But this means you will see me edit related nLab entries these days.
What would happen starting out from $\mathbb{R}^{0|3}$? That would have six 2-cocycles?
Does it perhaps just not encounter anything interesting like real spin representations?
For a talk that I will be giving later today, I have prepared detailed introductory talk notes on the whole story, at:
Super Lie n-algebra of Super p-branes (schreiber).
Today’s talk topic covers the first two sections of the notes “1. Spacetimes” and “2. Branes”.
The notes continue beyond that point for a later talk, but that later material is not stable yet.
$\beta$ on the right hand side?
Thanks! Fixed now.
With John Huerta we finally have the writeup of the next result in this series: the “root” of the brane bouquet in the superpoint. I keep the latest version of the file here:
Abstract. The “brane scan” classifies consistent Green–Schwarz strings and membranes in terms of the invariant cocycles on super-Minkowski spacetimes. The “brane bouquet” generalizes this by consecutively forming the invariant higher central extensions induced by these cocycles, which yields the complete brane content of string/M-theory, including the D-branes and the M5-brane, as well as the various duality relations between these. This raises the question whether the super-Minkowski spacetimes themselves arise as maximal invariant central extensions. Here we prove that they do. Starting from the simplest possible super-Minkowski spacetime, the superpoint, which has no Lorentz structure and no spinorial structure, we give a systematic process of consecutive maximal invariant central extensions, and show that it discovers the super-Minkowski spacetimes that contain superstrings, culminating in the 10- and 11-dimensional super-Minkowski spacetimes of string/M-theory and leading directly to the brane bouquet.
Page 18, you have
tyoe
and you also say
Below in section 3
when section 3 is earlier.
On p. 15, you now have the extension coming out of $\mathbb{R}^{5,1|\mathbf{8} + \bar{\mathbf{8}}}$ where you’ve always had it before, e.g., here as $\mathbb{R}^{5,1|\mathbf{8} + \mathbf{8}}$. In fact it’s clashing with the diagrams on pages 6 and 7.
I have been preparing some talk notes for later today:
Comments are welcome
Typos: homomorphim; we do nothing but analysing ; $\mathbb{R}^{10,1\mathbf{32}}$; we want to descent; this diagram commute; do descent;
$CE(\widehat{\mathfrak{g}}) \;\simeq\; CE(\mathfrak{g})[b_{p+1}]/(d b_{p+1} = \mu_{p+2})$, shouldn’t that be the dual $\mu^{\ast}$?
Typos
Thanks! Fixed now.
$CE(\widehat{\mathfrak{g}}) \;\simeq\; CE(\mathfrak{g})[b_{p+1}]/(d b_{p+1} = \mu_{p+2})$, shouldn’t that be the dual $\mu^{\ast}$?
There is a some intentional abuse of notation here. If we write $\mu$ for the map $\mathfrak{g} \to B^{p+1}\mathbb{R}$ and write $c_{p+2}$ for the chosen generator of $CE(B^{p+1}\mathbb{R})$, then that element on the right of that equation is strictly speaking $\mu^\ast(c_{p+1})$. But since both the map and that element “are” the cocycle in common parlance, I’ll suppress that heavy notation in favor of readability.
otber; [[nLab:super L-∞ algebra]
Thanks once more! Fixed now.
Any interesting questions after the talk?
There was little time today due to tight schedule. David Gepner asked about whether the bouquet allows to see what lifts KU to M-theory, whether it’s tmf-like. That’s of course exactly what motivated this analysis to some extent, and I briefly mentioned some of the stuff that one does see regarding the 4-sphere coefficients, but then we were interrupted. Tomorrow is more time with the conference hike. But I will probably have to miss it to get work done. Blessed those who have the leisure to go to a conference as if on vacation.
But I will probably have to miss it to get work done.
That’s a pity. Some good people to talk to in that line-up.
I am preparing slides for my talk at String Math 2017 end of the month.
In case anyone is interested, a first version is here.
Once this stabilizes, I will keep it at StringMath2017 (schreiber): “Super $p$-Brane Theory emerging from Super Homotopy Theory”.
Where you talk about $S^3$, should that really be $S^3_\mathbb{R}$? Since you are working rationally/with realifications.
Thanks. Right, so I try to announce this on slide 13: Entirely everything displayed happens in (super-) rational homotopy theory. (Hm, I should maybe replace all $\mathbb{R}$-s by $\mathbb{Q}$-s.) Also for instance where it says that $\mathrm{ku}/B^2 \mathbb{Z}$ is a direct summand of $\Sigma^\infty_{S^3}(S^4/S^1)$, this is crucially a statement in rational homotopy theory.
At one point I was wondering if I should give all the slides 14-75 a head or bottom-line saying “over $\mathbb{Q}$” or something like this. Do you think that would be good?
I think so.
Okay, I couldn’t find a satisfactory way to give an alert on each single slide, but I highlighted it more on slide 15.
The top right term $Cyc(KU/B^2 \mathbb{Z})$ changes at slide 75 (relative to the slides either side).
Thanks! Fixed now (here)
The $B^2 \mathbb{Z}$ appearing in various terms also differ in size on several slides.
On a more interesting note, has there been any progress on the full theory beyond the rational approximation?
has there been any progress on the full theory beyond the rational approximation?
Unfortunately not. I consider myself lucky that I found the time to prepare these slides. No time for research at the moment.
For the full story Snaith’s theorem will play a central role, I believe, in what should be its variant for twisted K-theory. That’s behind this switch from $ku / B^2 \mathbb{Z}$ to $KU / \mathbb{B}^2 \mathbb{Z}$. The former is rationally just $\Sigma^\infty_{B^3 \mathbb{Z}} B U(1)$, while the latter is truly $\Sigma^\infty_{B^3 \mathbb{Z}} B U(1) [\beta^{-1}_{B^3 \mathbb{Z}}]$. All I did meanwhile is make some more notes at Snaith’s theorem and advise a Bachelor thesis on the topic.
So the way (twisted) K-theory emerges in the full theory should be rather beautiful: The homotopy quotient $S^4/S^1$ looks like an $S^3$ with a copy of $\ast//S^1 = B U(1)$ attached to two opposite “poles”, and as we fiberwise stabilize this over $S^3$ and then make the result periodic, we obtain twisted K-theory inside something bigger.
Well, I can certainly say this already now, by taking the rational story and observing that this is clearly what it tells us to do. But what I don’t know yet is how things work in full super-homotopy theory if I step back and don’t “do” anything, but let the thing evolve by its own inner Proceß.
By the way, good to see you back. You and Vincent Schlegel seem to be the only other two people who see the big picture that this is getting at.
I only wish I understood things better.
On slide 54 you display the Ext-Cyc adjunction. By slide 75, there are two varieties of $Ext$, a type IIA and IIB. So they’re adjoint to different version of $Cyc$?
At slide 75, the units, $\eta$, might have subscripts A and B to differentiate?
And the downward diagonal arrow should presumably have B rather than A in the label.
And the fourth term down on the left hand side should be $CycExt_{IIB}(\mathbb{T}^{8,1|\mathbf{16}+\mathbf{16}})$, rather than 9,1.
On slide 54 you display the Ext-Cyc adjunction. By slide 75, there are two varieties of $Ext$, a type IIA and IIB. So they’re adjoint to different version of $Cyc$?
There is a single adjunction as shown on slide 54. But as seen there, one side of the adjunctions is homotopy types equipped with a map to $B S^1$, hence equipped with a cocycle for a circle extension. Now the space $\mathbb{T}^{8,1\vert \mathbf{16}+ \mathbf{16}}$ carries two different cocycles, one gives the IIA extension, the other the IIB extension.
If you want to understand this in more detail, the talk slides won’t help. But see the lecture notes on this adjunction here
At slide 75, the units, $\eta$, might have subscripts A and B to differentiate? And the downward diagonal arrow should presumably have B rather than A in the label.
Yes. okay, I have added that now.
And the fourth term down on the left hand side should be $CycExt_{IIB}(\mathbb{T}^{8,1|\mathbf{16}+\mathbf{16}})$, rather than 9,1.
Ah, thanks for catching that! Am fixing it now.
You still need to change A to B in the label on the southeast arrow, slides 75-79.
Oh. Thanks again. Fixed now.
Re #43,
…we obtain twisted K-theory inside something bigger.
Are you treating ’twisted K-theory’ as the less general twistings mentioned here:
From a homotopy theoretic viewpoint the twists of complex topological K-theory $KU$ are classified by the generalized cohomology theory $gl_1(KU)$ associated to its unit spectrum. Its first group $[X, BGL_1(KU)]$ contains $[X,BBU(1)] \cong H^3(X,Z)$ as a direct summand, which appears in the applications mentioned in the last paragraph. The existence of more general twistings was already pointed out in [3], but they were neglected since they had no obvious geometric interpretation in terms of bundles of operator algebras at that time,
or are you looking for something beyond these general twistings?
Something bigger methinks, since twisted K-theory should arise after dimensional reduction down to 10d.
Something bigger, because, rationally, it is just one direct summand in the fiberwise suspension spectrum of the homotopy quotient $S^4/S^1$ over $S^3$
$\Sigma^\infty_{S^3} (S^4/S^1) \;\simeq_{\mathbb{Q}}\; ku/B^2\mathbb{Z} \oplus \cdots \,.$This is the inclusion on slide 63. The formulas are at 4-sphere here.
The heuristic picture is (I mentioned this in #43) that the $S^1$-action on $S^4$ is free everywhere, except at two poles, where it is trivial. Hence the homotopy quotient $S^4/S^1$ looks like an $S^3$ with one copy of $\ast/S^1 = B U(1)$ attached to these two poles. The fiberwise suspension spectrum hence yields two copies of the suspension spectrum of $B U(1)$, roughly. Up to inversion of the Bott generator this is the complex K-theory spectrum.
Hence we find that as we come to 10d from 11d, the twisted KU-coefficients are but one summand in the Goodwillie linearization of the dimensionally reduced M-brane coefficients. So there is more in the M-brane coefficients, in some sense. The big open question is: what is this beyond the rational approximation.
what is this beyond the rational approximation
So instead of the rational approximation
$\Sigma^\infty_{S^3} (S^4/S^1) \;\simeq_{\mathbb{Q}}\; ku/B^2\mathbb{Z} \oplus \cdots \,,$there would be an equivalence (higher chromatic?) in the form of a sum?
Non- rationally the whole thing is more complicated. Already for the identification of $\Sigma^\infty BU(1)$ with $ku$, non-rationally, one needs to first invert the Bott generator $\beta$: Snaith’s theorem says that, non-rationally:
$(\Sigma^\infty BU(1))[\beta^{-1}] \simeq KU$hence
$ConnectiveCover((\Sigma^\infty BU(1))[\beta^{-1}]) \simeq ku$(Inverting $\beta$ make the previously connective spectrum $\Sigma^\infty B U(1)$ non-connective, but it does more to it).
So for the full story to “flow” one will need to find an “abstract reason” why after forming the fiberwise suspension spectrum $\Sigma^\infty_{S^3} (S^4/S^1)$ one should next fiberwise invert the Bott generator $\Sigma^\infty_{S^3} (S^4/S^1) [\beta_{S^3}^{-1}]$.
Or maybe it’s even more interesting. Maybe there is not an abstract reason to do that, mabye one should leave it at that, or maybe do something else, and so maybe the statement that D-brane charge is in twisted $KU$ actually ends up receiving corrections.
I have been tossing around a bunch of potential ideas on this, but I still don’t have any asnwer.
(By the way, I am notationally suppressing adjoining of basepoints throughout here, bcause it becomes cumbersome in the parameterized case.)
Should we expect your story to continue so as to derive M5-brane charges?
That has been one of the motivations for a long time, yes.
I regard it as a substantial step forward that we see from first principles now how the D-brane charge arises rationally by a passage to cyclic loop spaces from M-brane charge.
Because this is what is expected from the higher chromatic Chern character/transchromatic Chern character (maybe these two pages deserve to be merged): A cohomology theory of some chromatic level is supposed to map into a cohomology theory of one chromatic level lower by a cyclic looping procedure.
So for instance elliptic cohomology, when evaluated on the Tate curve looks like a formal power series in K-theory (K-theory of the infinitesimal loop space). There is this rough idea that this is how D-brane charge in K-theory should come from M-brane charge in elliptic cohomology.
With the rational story of the double dimensional reduction via cyclic loop spaces, we have come a step closer to seeing where this idea should fit into the picture.
Of course, you were talking about M-brane charges last summer here.
I see transchromatic character begins
A kind of generalization of group characters to chromatic homotopy theory,
without any elaboration as to how this is a generalization of a group character: a homomorphism from a group to the circle group (or some other group of units).
Looking around, section 1.3 of Michael Hopkins, Nicholas Kuhn, Douglas Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000) explains the idea of ’generalized characters’, in terms of class functions, and then Stapleton generalizes further to allow maps with targets above level 0.
But then there is also higher chromatic Chern character which is generalizing Chern character. Now why the ’character’ in that concept? Is Chern character also a variant on group character?
I wonder if HoTT might provide some clarifying assistance in this area, as it does to the ext-cyc adjunction in Proposition 8.6 of geometry of physics – fundamental super p-branes.
Perhaps a couple of steps between ’group character’ and ’Chern character’ are Chern class and characteristic class. I guess it’s all just about homs in an $(\infty, 1)$-topos.
Do people do that, take a character, $G \to U(1)$, deloop it, $B G \to B U(1)$, and compose with classifying map for $G$-bundle, $X \to B G$?
A Chern character is more like a map to the rationalisation of $BU$, or at least the product of all the corresponding Eilenberg-Mac Lane spaces.
As it says at characteristic class
From the nPOV, where cocycles are elements in an (∞,1)-categorical hom-space, forming characteristic classes is nothing but the composition of cocycles.
So, Chern characters are just composition of $X \to B U(n)$ with things like $c_i: B U(n) \to B^{2i} \mathbb{Z}$.
The last two messages #59 and #60 are not really in contradiction, just a little on different aspects: #60 displays a Chern class, while the Chern character is indeed a map to a product of $B^{2n}\mathbb{Q}$s, whose components are built from the Chern classes.
I never intended otherwise. But still there’s a job of clarifying what I mentioned in #56.
Meanwhile, with Domenico and Hisham, we have some further developments:
Higher T-duality of super M-branes (pdf)
We establish a higher generalization of super L-∞-algebraic T-duality of super WZW-terms for super p-branes. In particular we demonstrate spherical T-duality of super M5-branes propagating on exceptional-geometric 11d superspacetimes. Finally we observe that this constitutes a duality-isomorphism relating a priori different moduli spaces for C-field configurations in exceptional generalized geometry.
There will be a conference, end of this year, dedicated in half to this super $L_\infty$-homotopy theory in M-theory:
Typos
underly (underlie); The subtlety is, that it (no comma); using relation to (missing word?); passing m2brane to (missing ’from’?)
Is there a way of knowing why the superpoint $\mathbb{R}^{0|32}$ is one which generates this exceptional geometry via maximal invariant central extension?
Thanks for spotting the typos, fixed now!
In a more comprehensive story we find $\mathbb{R}^{0\vert 32}$ by first climbing up to $\mathbb{R}^{10,1\vert \mathbf{32}}$ as before, and then applying the “cyclification” adjunction (double dimensional reduction) all the way back down. This also gives another way to see the “exceptional” coordinates appear, as they are the double dimensional reduction, this way, of the M2- and M5-brane cocycle.
viewed an entry (missing ’as’); (Z,(Z/2) (missing second bracket); like to be to account
One more for now
corresponding the irreducible
Re #65, could one not think of forming the bottom row as superpoints of dimension $2^n$? Couldn’t one double dimension reduce from other positions, such as $\mathbb{R}^{5, 1|8}$ down to the 8-dimensional superpoint?
With John Huerta we already though a little about the maximal invariant extension of $\mathbb{R}^{0|4}$. Even if it doesn’t result in a spacetime, John had suggested
$R^{10|4}$ is like a superspace built from (3,3) spinors and either self-dual 3-forms or anti-self-dual 3-forms.
Re #65, could one not think of forming the bottom row as superpoints of dimension $2^n$?
Absolutely!
Couldn’t one double dimension reduce from other positions, such as $\mathbb{R}^{5, 1|8}$ down to the 8-dimensional superpoint?
Yes. Somebody should explore this!
refinement the exceptional; there k different; necessaril; viewed as M5-brane
Definition 2.9:
defines differential of degree 1 mod 6
missing ’a’, and do you mean ’mod 2t’?
Introduction to section 4, the Roman numerals don’t line up with the sections, and there’s no description of 4.8.
Thanks! These and some more typos have been fixed now.
Also, there is now on p. 5 (here) an actual informal explanation of how one sees spherical T-duality from the M5-brane WZW term.
Regarding the items in the lead-in of section 4: Maybe the numerals don’t necessarily have to match? Also I am undecided as to the status of section 4.8, as it is somewhat tangential to the higher aspect of T-duality, being instead another example of ordinary super-topological T-duality. Maybe it should go to an appendix, or something.
genralised; regarded a fibered (prop 4.12); we the two summands; as a special case. difficulty; the two object
In the middle of the diagram in section 4.6, that should presumably be $\mathbb{R}^{6, 1|\mathbf{16}}$.
In Example 4.24, won’t $B E_8$ and $K(\mathbb{Z}, 4)$ agree up to dimension 16 rather than 14?
Thanks! This didn’t make it into the the arXiv version anymore, but I’ll be keeping an updated pdf here.
Hmm, I see the change to my last point of #72 in your last version of the paper, but it still sounds odd. You seem to use “up to dimension 15” twice in different ways in 4.25 - up to but not including 15, up to and including 15.
And my point before that about the term in the diagram (which is now appearing in section 4.7)?
There are different authors editing here, which requires some harmonization. But it looks consistent to me now, reading “up to” as “up to and including”. But I see why it reads confusingly: first the second nontrivial homotopy group of $E_8$ is included in the statement, next it is not.
Fixed that middle 7d spacetime now. Thanks for it all!
Sorry to persist, but isn’t it that $E_8$ has homotopy first in dimension 3 and then in dimension 15. Then $B E_8$ has homotopy first in dimension 4 and then in dimension 16. Then $K(\mathbb{Z}, 3)$ has homotopy only in dimension 3, and $K(\mathbb{Z}, 4)$ has homotopy only in dimension 4.
So $E_8$ and $K(\mathbb{Z}, 3)$ agree up to dimension 14 and then disagree, while $B E_8$ and $K(\mathbb{Z}, 4)$ agree up to 15 and then disagree.
So why does it say both pairs agree up to dimension 15?
I think the first “up to” referred not to the pair of “$K(\mathbb{Z},3)$ and $E_8$”, but to the pair “$(3,15, \cdots)$ and $E_8$”, which is why I said it is consistent, if maybe a little confusingly stated. I have rephrased it now in the file.
Notice that this paragraph is a side remark, and that its author is offline today, since Friday here is like Sunday on your side of the Bosporus. (While Saturday is Saturday, but Sunday is Monday :-)
How might we think of what in this extended brane scan is driving things towards exceptional generalized geometry? In #65 there’s a continuation of the ’extending the superpoint’ story. In this area, there’s an ’E11 is pulling the strings’ story too, as told by Peter West. Is $E_{11}$ lurking behind your paper somewhere?
Is $E_{11}$ lurking behind your paper somewhere?
The problem with the $\mathfrak{e}_{11}$-proposal is (still) that it makes a huge extrapolation from a comparativley small amount of data. That data is
the fact that one mathematically natural continuation of the sequence of U-duality Lie algebra $\mathfrak{u}(5,5)$, $\mathfrak{e}_6$, $\mathfrak{e}_7$, $\mathfrak{e}_8$, $\mathfrak{e}_9$ (whose appearance is fairly well understood) continues with hyperboilic Kac-Moody algebras as …$\mathfrak{e}_{10}$, $\mathfrak{e}_{11}$. The trouble here is that this does not seem to be the only possible way to naturally continue this sequence of mathematical structures.
the fact that the level-expansion of the “fundamental representation” of $\mathfrak{e}_{11}$ (here) has the first three (of infinitely many) summands of the right form: $\mathbb{R}^{10,1} \oplus \wedge^2(\mathbb{R}^{10,1})^\ast \oplus \wedge^5(\mathbb{R}^{10,1})^\ast \oplus \cdots$.
It seems that people have eventually agreed that the next summand somehow reflects the “dual graviton”. Beyond that, nobody seems to have an idea what to make of this.
So what we have is a derivation of $\mathbb{R}^{10,1} \oplus \wedge^2(\mathbb{R}^{10,1})^\ast \oplus \wedge^5(\mathbb{R}^{10,1})^\ast$ “from first principles”, but presently I don’t see how or why this should continue. (Which is not to say that it doesn’t, just that I don’t see any reason, except the above two analogy items.)
The operations of actual interest, namely the U-duality kind of transformations can already be considered on $\mathbb{R}^{10,1} \oplus \wedge^2(\mathbb{R}^{10,1})^\ast \oplus \wedge^5(\mathbb{R}^{10,1})^\ast$ itself. This is highlighted in Baraglia 11, section 2.3.
I see. Maybe your first principles provide the way then.
It would be nice to fill in other point expansions. Was there something special about $\mathbb{R}^{0|32}$ that made the maximal invariant central extension easier to establish? One might guess the first extension of the $2^n$-dim point would map to the entry of the same odd dimension in the original tower over $\mathbb{R}^{0|2}$.
Was there something special about $\mathbb{R}^{0|32}$ that made the maximal invariant central extension easier to establish?
No, the first maximal invariant extension of a superpoint $\mathbb{R}^{0 \vert N}$ is the one that is always immediate: the maximal tuple of cocyles consists simply of all symmetric $N \times N$-matrices.
The thing about the maximal extension of $\mathbb{R}^{0\vert 32}$ is that it connects to something else we understood, as explained in the article, and so there was something to say. For the other superpoints it is just as immediate to write down their first maximal invariant extension, but presently I don’t know anything interesting to say about them.
Presently there are enough interesting loose ends to be explored for which we have some idea what they mean and how to approach them. Next we are after writing up the phenomenon of gauge enhancement of M-branes. Your implicit suggestion of considering more of the same already previously established could be a good strategy for a thesis topic, or else something to be picked up later.
Oh yes, we already discussed somewhere that the even dimension of the first extension is $N(N + 1)/2$.
Since the decomposition for $N = 32$ is
N = 32: 528 = 11 + 55 + 462,
corresponding to the decomposition
$\mathbb{R}^{10, 1} + \wedge^2(\mathbb{R}^{10, 1})^{\ast} + \wedge^5(\mathbb{R}^{10, 1})^{\ast},$a glance at Pascal’s triangle might suggest
N = 16: 136 = 1 + 9 + 126
where the 9 is $\mathbb{R}^{8, 1}$, and 126 is the dimension of $\wedge^4(\mathbb{R}^{8, 1})^{\ast}$.
Similarly,
N = 8: 36 = 1 + 35
N = 4: 10 = 10
N = 2: 3 = 3
This last one is merely $\mathbb{R}^{2, 1}$.
Yes, these are famous numerological coincidences, which control the invariant central extensions of supersymmetry algebras “by brane charges”. A mathematical classification of these “polyvector extensions” of susy-algebras is in section 6 here.
What one would need to repeat in other dimensions the particular story of our article is that these extended supersymmetry algebras admit a further fermionic extension over which a given higher brane cocycle decomposes. In the D’Auria-Fré-school these are called the “hidden” symmetries. After the original result in $d = 11$, they recently considered the case $d = 7$, references are listed here.
Next we are after writing up the phenomenon of gauge enhancement of M-branes.
Today, for an event called “SuperPhysMatics2018”, I had occasion to expand and update my previous slides on this, now here.
The full theory emerges once passing beyond the rational approximation in full super-geometric homotopy theory.
Is it your impression that there’s plenty still to find out about the rational approximation? I guess Gauge enhancement suggests so.
If physicists got there early with FDAs, are there any indications from that literature as to relevant algebras beyond the rational approximation?
There seems to be quite a concentration of minds on higher chromatic versions of Sullivan/Quillen models of rational homotopy, such as in Lie algebras and vn-periodic spaces. What then of $v_n$-periodic superspaces?
Is it your impression that there’s plenty still to find out about the rational approximation?
Notice all but a micro-fraction of existing physics/string literature ignores torsion group phenomena. Where it does not, it is typically based but on guesswork. For instance the claim that D-brane charge is, non-rationally, in K-theory is based on but a bunch of plausibility arguments. And some of these actually fail. For instance the assumption that D-brane charge is in K-theory is, at least at the face of it, incompatible with the assumption of S-duality in type II string theory. The only way to clear this up is to first get the rational situation fully under control, such as to see which actual pattern the integral lift will have to follow.
It’s tempting to invoke grand speculation regarding modern homotopy theory here: We are seeing the unstable spheres as the true coefficients in M-theory. From there we may think of passage to stabilization as just the first order approximation in the Goodwillie-Taylor tower, and then the identification of K-theory as just the first order approximation of that in the chromatic filration. If M-theory is what it promises to be, it might just be plausible that it is or involves at the same time the unification of all homotpy theory this way. But what is necessary for progress is concrete ideas for little steps forward, otherwise we get lost in grand speculation.
Next I think I see now finally, using some computations that John Huerta has been doing meanwhile, how to invoke ADE-equivariant rational homotopy theory to explain from first principles the expected KK-monopole/D6-brane appearing at the ADE-fixed points. This is (or would be) already a little step away from the plain rational approximation, since the finite ADE groups are not visible in plain rational homotopy theory.
I am preparing talk slides which, towards the end, mention the results about ADE-singularities that I have been alluding to in discussion with David C. in other threads. They are not completely finalized yet (references are missing, for instance) but fairly polished and should be readable. A link to the pdf (“”Structured Homotopy Theory from String Theory”) is on my web here.
A naive question no doubt, but how would one know one had found the elusive M-theory? Do the checks go beyond that it has the 5 string theories and 11-d supergravity as some kinds of limiting case?
For instance, the gauge enhancements shown in slides 43-45 come out “as predicted by the folklore on M-theoretic gauge enhancement”, and this folklore derives from gauge enhancement in (some of) the string theories?
how would one know one had found the elusive M-theory?
The established method is to see if the computations, drawn on a blackboard, invoke thunder and lightning.
Seriously, to complete this program, one will delve more deeply into dynamics. Presently what we have is a fair bit of indications that lifting 11d SuGra to M-theory means to lift its description as the ordinary torsion-free super Cartan geometry modeled on $\mathbb{R}^{10,1\vert \mathbf{32}}$ to some kind of torsion-free equivariant higher super-Cartan geometry, because we show that super tangent space wise this produces pretty much all the expected “topological” objects and dualities.
In a naive picture this gives a higher kind of differential equations, generalizing those of torsion-freedom for $\mathbb{R}^{10,1\vert \mathbf{32}}$-Cartan geometry, and one might consider perturbative quantization of these differential equations and try to compare the resulting scattering matrices to those of the various superstring theories.
But I suspect that as we keep understanding the mathematical structure appearing here better, we will find that it wants to do something a little different than what this naive picture suggests. We have to see.
Some further talk slides in this project:
So far Part I is available:
Part I reviews the derivation of M-brane charge in equivariant cohomotopy, and explains how bringing in anti-M-branes should enhance this to equivariant stable cohomotopy.
Nice!
Is there an absolute topological K-theory, whose spectrum is the shape of some smooth sphere spectrum, coefficients for differential cohomotopy?
No hits for “smooth sphere spectrum”, but one (arXiv:1306.0247) for “differential sphere spectrum”. Seems to be in the right sense.
“Differential stable cohomotopy” gets a mention by you here. And one mention for “differential stable homotopy”.
Very elusive for some reason.
$[$ differential cohomotopy is$]$ Very elusive for some reason.
The sphere spectrum is “differentially simple” (Bunke-Nikolaus 14, example 9.4) which means that it is straightforward to at least define (“construct”) some version of differential cohomotopy. (Of course computing differential cohomotopy groups generally won’t be easier than computing bare cohomotopy groups, which, famously, can be arbitrarily hard.)
But for the moment the issue is still to avoid jumping to conclusions with guessing/postulating flavours of cohomology theories just because they are mathematically suggestive; the issue is to figure out what the actually correct flavour of cohomology theory for various branes is, or to first of all figure out by which rules this is to be systematically decided.
On the other hand, so far ADE-equivariant stable cohomotopy does quite look like going in the right direction – further consistency checks on that are the topic of part II of the talk.
In addition we have some further results from Dan Grady (not yet public) showing that already unstable smooth equivariant cohomotopy sees non-trivial fine-print of M-brane charge.
So maybe next we get to combine this and explore more of the smooth/differential equivariant stable cohomotopy of M-brane charge. But it needs more thinking.
Is there an absolute topological K-theory, whose spectrum is the shape of some smooth sphere spectrum, coefficients for differential cohomotopy?
By the most common way of speaking, a “smooth sphere spectrum” would by design/definition have as its shape the plain sphere spectrum, hence the “absolute algebraic K-theory spectrum”.
I was just thinking that slide 11 was suggestive of a second way of going from cohomotopy to topological K-theory. Rather than, as with equation (1) here, extension of scalars, $K \mathbb{F}_1$ to $K \mathbb{C}$, and then the comparison map between algebraic and topological K-theory, there should be another way around a square, first taking $K \mathbb{F}_1$ to a topological image of $K \mathbb{F}_1$, then extension of scalars to $KU$.
first taking $K \mathbb{F}_1$ to a topological image of $K \mathbb{F}_1$, then extension of scalars to $KU$.
Hm, not sure what these two steps should be(?)
Now Part II of the talk slides is available at
Part II presents computations of putative M-brane charge in equivariant stable cohomotopy and comparison to the folklore of perturbative string theory.
Re #95: certainly one could try to construct topological K-theory over $\mathbb{F}_{1}$ by looking at $\mathbb{F}_1$-vector bundles over a topological space. The question of course is what this means. I guess it would mean the same as usual but looking at fibres as pointed sets. So one could try to see whether can one represent such a thing, and if so, that would be topological $\mathbb{F}_{1}$. My guess would be that it is possible, and if so it seems reasonable to imagine it would be isomorphic to the algebraic $K$-theory.
On slide 18 you have the image of $\beta$ inside $RO(2I)_{|irr}$, is this correct?
David, thanks, I mean in that line to restrict also the image of beta to the set of irreps. Will fix.