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I have been alluding to the following in a bunch of recent edits on the $n$Lab. Here is finally a (still unfinished but) share-able version of:
If you are a mathematician, this article offers something precise in reply to “What is an M-brane?”, and it does so in terms of equivariant homotopy theory.
If you are a physicist, this article offers a proposal for “What are the missing M-theory degrees of freedom hidden at ADE-singularities?”
If you are both, then, well, enjoy.
M-theory starts to appear!
I suppose people will be wondering can the black brane scan be upgraded to a black brane bouquet? But then perhaps that’s just to wonder about inclusions of basic Kleinian shapes. On the other hand, there’s the suggestion of a fundamental/black brane correspondence p. 58.
Typo: furter.
There was a fairly elaborate D-brane theoretic explanation of ADE at MO. Should everything there be liftable to the M-theoretic account, e.g., something like pp. 58-59?
Yes, lots of territory opened to be explored now, such as what happens to the black brane scan as we move focus away from 11d spacetime to the rest of the bouquet. But this will have to be discussed elsewhere.
That MO reply just recalls a broad outline of the general folklore. One can expand on this endlessly, a few related articles appear each day. But the point is exactly that all of this ought to have an M-theory lift, and that only after this lift all the facets can be expected to unify.
p. 4 turns out encodes
p. 6 in that table at the top should say ’complex K-theory’ rather than ’real K-theory’.
p. 9 ’The orbit of y’ should be x; $g(x) =z y$, no $z$
Thanks! Will fix tomorrow.
When I heard Klein geometry was making an appearance, I was wondering whether we’d see some quotienting of higher groups. On p.3 there is orbifold Klein geometry with its double cosets, but no higher groups.
The appearance of double cosets got me wondering about their interpretation as invariant relations, e.g., for the Euclidean group in 2d, double cosets with $SO(2)$, the stabilizer of a point, and with the stabilizer of a line gives all the invariant relations between point and line in the plane. So in that case it’s the positive reals. When intersecting branes appear later, should we not expect double cosets to appear?
Back to the ’higher’ aspect, on p.4 it is stated that the Cartan form of orbifold Kleinian geometry is treated by higher Cartan geometry. But how? Where are the higher groups?
I did come across a paper recently – Matthew Young’s Real representation theory of finite categorical groups – which explains some $\mathbb{Z}_2$-equivariant phenomena with 2-groups. It claims work it develops, Ganter and Kapranov’s Representation and character theory in 2-categories, is motivated by equivariant homotopy theory. Perhaps the ’real’ of real equivariant cohomptopy relates to Young’s 2-group ideas.
on p.4 it is stated that the Cartan form of orbifold Kleinian geometry is treated by higher Cartan geometry. But how? Where are the higher groups?
Ah, remember that an orbifold is already a higher geometry! It’s a groupoid (with smooth structure).
Here is the fact that drives this:
Pick a local model space $V$ (a group object, or at least an “H-group object”, according to Wellen 17) and consider the definition of $V$-manifold in differential cohesion (geometry of physics – manifolds and orbifolds). Then when interpreted in the 1-topos this yields ordinary $V$-manifolds. When interpreted in the cohesive 2-topos, this yields V-orbifolds. Generally, it yields étale-infinity-stacks modeled on $V$. (prop. 6.5.60 in dcct v2).
when I heard Klein geometry was making an appearance, I was wondering whether we’d see some quotienting of higher groups.
Recall that the driving concept here is equivariant cohomology. There currently is very little known about $G$-equivariant homotopy for $G$ a higher group. (Some hints for String-2-group-equivariant cohomology are towards the end of A Survey of Elliptic Cohomology.)
I trust that all this will eventually find its place. But for the moment I am chasing phenomena, not concepts. It’s time for them to catch up.
More typos:
in the to row; ratioanl
Box on p. 24, the $G_{HM}$ should be $G_{HW}$ and are set too low.
Box on p.27, the $G_W$ should be $G_{HW}$.
P. 30 Should $\mathbb{R}^{9, 1|\mathbf{N}}$ be $\mathbb{R}^{p, 1|\mathbf{N}}$?
everywhere: real -> Real
p38 fixed be every
p44 zero-cocylce
p48 the combinaotrics
p49 decomposition 49 (missing parentheses)
p50 Halmos for end of proof of (Trivialization of M2/M5-cocycle on black: MW, M2 and M5) runs into text
p50 coycle
p53 non-committal shorthand^{1011} - need to separate footnote symbols.
p55 every 2-cocyle; rational super homotopy theory… a 2-cocyle as above; homiotopyt
p56 homotpical
p57 equivaently
p59 black-hole like -> black hole-like
Re #7, sure higher Cartan geometry deals with orbifolds, but I thought you were suggesting something new was occurring here. Impressive though it is, is this just the story of how Klein geometry got localised to Cartan geometry, then extended to a higher Cartan geometry (which can deal with orbifolds) which must be the local form of some higher Klein geometry, so in this case it gets called ’orbifold Klein geometry’?
But orbifold Klein geometry is not a subpart of higher Klein geometry as we’ve written the latter, which is about homogeneous quotienting.
Of course, you could redefine ’Klein geometry’ to allow singularities, perhaps via double cosets. But then this is staying at the 1-group level, so why any temptation to say ’higher’?
Also, the diagram at the top of page 14 has two $P$s that should be $X$s, and $f(2)$ that should be $f_2$.
Brave to give a lightning review of M-theory.
p. 52 severeal
p.53 strengh; millenium
p. 54 furter
p. 58 diagram of given by
@David R.: thanks for catching all these typos! I think I have fixed them all now, and a few more. Thanks.
Also, there is now something more substantial in the beginning of section 4.2 “The black brane scan”.
but I thought you were suggesting something new was occurring here.
:-)
But orbifold Klein geometry is not a subpart of higher Klein geometry as we’ve written the latter, which is about homogeneous quotienting.
Ah, now I see what you are after. Right: In the formalization via $V$-manifolds in cohesive $\infty$-toposes, this means that we will not use $\Gamma \backslash G/H$ as $V$. In fact, $V$ needs to be a group (at least an $H$-group) but $\Gamma \backslash G/H$ in the application under consideration is not a group.
I am imagining the orbifold Cartan geometry modeled on $\Gamma \backslash G/H$ to be defined like so: 1. It is a higher Cartan geometry locally modeled on $V \coloneqq G/H$, and then 2. we demand that the orbifold singularities that this may feature must look like $\Gamma \backslash G/H$.
so why any temptation to say ’higher’?
Because an orbifold is a manifold in higher geometry!
I guess my point can be expressed by saying that it’s different enough that we might add an orbifold row to local and global geometry - table.
I see. So for good systematics, we might want to distinguish between two directions in which to go higher here:
On the one hand, the Kleinian model space may be an $n$-stack, for $n \in \mathbb{N} \cup \{\infty\}$. On the other hand, the Cartan geometry modeled on this may be a $k$-orbifold (étale $k$-stack) on that model for any
$k \geq n$So there is, say, “1.5 ways” of making Cartan geometry “higher”, not just one way.
Not sure how to fit that into the table though. We’d need the third dimension…
More:
p58 asymptoticall
p65 comprehevsive
p70 is called a quasi-isomorphisms
of a topological spaces
p73 cocycled
Re #17, ah good, I was just thinking on the way home this was something (1,2)-like happening.
P. 11 You have both ’Equivariant weak homotopy equivalence’ and ’weak equivariant homotopy equivalence’.
p. 54 The use of $d$ and $D$ is confusing. In the table there’s $d$ as in $d+1$-dimensions. Then in the description there is “$D$-dimensional super spacetimes”, so one might think $D = d+1$. However, later there is $D= 10$ for $\mu_{M2}$.
Also in the sentence “Similarly, the old brane scan sees that…” (perhaps better ’shows that’), it could be clearer which part of the scan shows this.
there is $D= 10$ for $\mu_{M2}$.
Could you give me the page number where this appears? (Yes, I am trying to stick to $D = d + 1$, to be the “space-filling” version of the worldvolume dimension $p + 1$ of $p$-branes.)
As I said, p. 54.
Thanks for all the comments, to both of you! Much appreciated. I have now implemented all of this.
Also, section 4.2 is now more well-rounded, contentwise.
You seem not to have updated my final correction in #9 or those in #13 (page numbers have now shifted).
Some new typos:
P. 60 there is zoo of P. 61 back p-brane; determing P. 62 datat; site at
I was having a look about for any other work along the lines of an orbifold Klein geometry. I can’t see much out there. There’s Thurstonian geometrization of orbifolds, but I can’t see anyone classifying invariant “basic shapes”.
Sorry, fixed now locally. (But not public yet, am at the Gate in Istanbul…) Except that last item in #9: I don’t understand what this is referring to.
Thanks!
In definition 3.9, you don’t want to specify 9 in $\mathbb{R}^{9, 1|\mathbf{N}}$.
Would it be worth explaining why you are looking at real ADE-singularities? You gave an explanation here, but why did people before you think to look for them?
Thanks again! New version now finally uploaded.
Also comparison to the literature in Example 4.2 ($\mathrm{M}5$), Example 4.6 ($\mathrm{M}5_{ADE}$) and Example 4.7 ($\tfrac{1}{2}\mathrm{M}5$) improved and expanded.
Would it be worth explaining why you are looking at real ADE-singularities? You gave an explanation here, but why did people before you think to look for them?
Yes, I should still add a paragraph on the choice of action on the $S^4$, and the aspect of “Real cohomology”.
But are you just suggesting that it would be good for exposition, or are you actually asking why the real structure appears?
The main real structure results from the fact that the fundamental M2-cocycle $\mu_{M2}$ picks up a sign when pulled back along an orientation reversing element of $Pin(10,1)$. Hence in order to get an equivariant enhancement, there must be a corresponding involution on the coefficient space that accordingly changes the coefficiens by a minus sign. Since the relevant coefficient here is the degree-4 element in the minimal dgc-model for the 4-sphere, and since this comes from the volume element of the 4-sphere under the Sullivan functor, the reuqired action on the 4-sphere is an orientation-reversing isometry.
This is a slightly fancy perspective on a famous – if traditionally somewhat vague – statement in Horava-Witten theory, saying that the C-field is “odd” under Horava-Witten involution.
Curious though to see how the sphere coefficients govern these signs. See also prop. 3.35, where the Sullivan model for the 2-sphere works some magic.
I meant for purposes of exposition to explain why equivariance under those particular groups. Since you’re including generous help for the mathematician with no background in M-theory, perhaps there could be something more concise in section 4 about why gauge enhancement should occur to anyone to investigate.
So imagine you can agree that some orbifold Klein geometry with a nice homogeneous superspace could be fun. You learn in section 4 that the superspace in the paper can be thought to arise naturally mathematically by extensions of the superpoint, and in any case physicists are interested. You see that an extension is given by a rational cocycle to $S^4$. Now you might want to know why we should choose to look at $G_{ADE} \times \mathbb{Z}_2$-equivariance. The first clues in terms of physics begin at p. 60. ADE gets a mention, but I think it’s not very obvious there why $\mathbb{R}^{10, 1|\mathbf{32}}$ equipped with cocycle is asking to have its ADE-singularities examined.
I see Acharya here gives an explanation (section 3) in terms of M-theory-heterotic duality. There $SU(2)$-holonomy turns out to be required (bottom of p. 15).
So will it be from ’lifting’ or ’duality’ arguments that one sees one wants finite subgroups of $SU(2)$, or can you see it directly on the M-theoretic side?
But the appearance of SU(2)xZ_2 on R^10,1vert 32 is the content of section 3.2: The classification result shows that finite subgroups of this are those that act, fixing the same bosonic subspace as some involution and having fermionic subspace at least 1/4th the 32.
Def 3.1 vi) has an $SU(2)_{Delta}$
Re #32, hmm, I’m not sure that’s clear in the order of exposition. The reader arrives at section 3 and is told that real ADE-space structure will be found for domain space and coefficient space. No motivation for choice of group yet. Then you set about the 4-sphere in 3.1. A decomposition is given, which allows for some specific actions. Aren’t there other possible actions by other groups? Is the choice made because we know we want to look for real ADE-actions already?
Then we reach 3.2 where it starts out as though we know already that we’re looking for real ADE-actions. But you’re saying that the explanation is because of theorem 3.11? That it’s because of this result, that we are to care about real ADE-actions?
OK, that might have been mentioned earlier, but anyway what’s special about $\geq 1/4$-BPS states? Won’t $\lt 1/4$ BPS singularities also produce forms of black brane?
Perhaps none of this matters if the real audience is someone who wants to make better sense of the zoo of M-branes already discovered.
Two small things:
Prop 2.4, why the ’+1’ in $\mathbb{R}^{d+1,1|\mathbf{N}_{irr}}$? And later in the line you have $\mathbb{N}_{irr}$.
Why in Proposition 3.23 does it speak of the two actions in 3.15, where five are listed.
Thanks for insisting, the order of 3.1 and 3.2 is swapped now.
There is a hint on p. 4 on what determines the particular action on the 4-sphere. I have expanded that now to a full discussion, now pages 59-60.
The restriction to the stratum $\geq \frac{1}{4}$-BPS is in order to have a classification. For lower BPS the classification is not known (to us).
For a more serious observation, is it the case that the $SU(2)\times SU(2)$ is really $Spin(4)$ (I mean, it is, but does it arise naturally from the representations theory), and, more conjecturally, the 4-sphere as the representing object for $\pi^4$ is some representation sphere for it?
That’s looking much more comprehensible now. Regarding low BPS states, I see people do look for them, e.g., for type IIB Supergravity – 1/8 BPS States in Ads/CFT. Is there any reason to see them as any less fundamental? Maybe that depends on finding simple singularities there.
Something I meant to ask, what would happen if, in the context of the $(n, k)$-geometry of #17, one carried out some $(6, k)$-geometry in relation to the supergravity Lie 6-group associated to
$\mathfrak{sugra}_6 \to \mathfrak{sugra}_3 \to \mathfrak{siso}(10,1) ? \,.$Typos:
Line 5 of p. 59 you have $H_{\mathbb{H}}$, but $H_{\mathbb{Q}}$ elsewhere.
P. 6 You have Real K-theory in the box, where the surrounding discussion is about the complex form.
For a more serious observation, is it the case that the $SU(2)\times SU(2)$ is really $Spin(4)$ (I mean, it is, but does it arise naturally from the representations theory),
I know what you mean, but I am not sure if it “really” wants to be $Spin(4)$. For one, ultimately the action of $SU(2)\times SU(2)$ is through $Spin(8)$ (for the M2), while for the M5 it is really strictly just one copy of $SU(2)$, so that it seems to really be more about identifying small subgroups, than about unifying them in bigger ambient groups. I suppose part of the magic here is that it is really just the finite ADE-groups that control all this, sitting (or hiding), as they do, inside various bigger groups.
and, more conjecturally, the 4-sphere as the representing object for $\pi^4$ is some representation sphere for it?
I suppose that’s the point you once usefully made here, when I first started thinking about this stuff, and I suppose your observation still applies: The $S^4$ with its actions, as we consider it, is indeed a representation sphere. I am sure once we get back to discussing the stable equivariant version of cohomotopy (as back then) this will be relevant.
That’s looking much more comprehensible now.
Thanks for the feedback.
Regarding low BPS states, I see people do look for them, e.g., for type IIB Supergravity – 1/8 BPS States in Ads/CFT. Is there any reason to see them as any less fundamental? Maybe that depends on finding simple singularities there.
Of course people discuss them, and they are interesting, but more involved. We are sitting on the tip of an iceberg, and the lower BPS black branes are part of the big bulk of the iceberg, waiting to be mapped. Broadly speaking, the lower the BPS degree the more of the “basic” branes (“simple singularities”) are intersecting. Eventually it is certainly these intersecting branes that will be of interest, such as notably in intersecting D-brane models.
P. 6 You have Real K-theory in the box, where the surrounding discussion is about the complex form.
Right, you had mentioned this before. I should say that we really do mean real K-theory, not in the sense of “$KO$” opposite to $KU$, but, on the contrary, in the sense of $KR$, unifying the two. This real K-theory (KR-theory) is the right K-theory for orientifolds, hence in the presence of O-planes, and we are highlighting this because it is exactly the analogous phenomenon that leads to Real cohomotopy in the presence of the MO9-plane.
Re #40, is it worth a word then to explain this, since at the moment the box comes right after:
in direct analogy to how the generalized (co-)homology theory complex K-theory is understood
and before
degree-4 cohomotopy transmutes into the cohomology theory complex K-theory
Right, the real structure was not amplified well yet.
There is now on p. 2: “…or rather real twisted K-theory on real orbifolds (“orientifolds”) … See example A.25”.
And on p. 4: “In addition to the ADE-singularities captured by ADE-equivariance, the real structure (see Example A.25) reflects the presence of M-theoretic O-planes, such as the MO9.
And Example A.25 now mentions explicitly real cohomology in general and real K-theory in particular.
I see the intriguing suggestion of a relation between cohomotopy and bordism theory has disappeared.
You are an attentive reader!
The page breaks in the introduction are carefully arranged and break easily with a new edit, as they did when I added the words on real structure. Here I thought that a clarifying remark about something that we do discuss in the article is more worthwhile than a vague allusion to something we hope to discuss in the future, so I removed the latter.
But it’s still true. I came to believe that the reason that cohomotopy is indeed the right cohomology theory for brane charges is because, via its isomorphism to cobordism in complementary degree, it is fully true to the real idea of Dirac charge quantization: The statement will be that equivariant cohomotopy in degree 4 on an 11-manifold precisely counts (cobordism classes) of submanifolds at singularities that look like black 5-branes, and hence precisely measures the charge carried by these.
But this really must wait for its own article. This one is too long already.
Do we have the “isomorphism to cobordism in complementary degree” written anywhere?
With a Dirac charge quantization understanding could you approach this?:
Right, so the fact that the 4-form $G_4$ which the M2-brane couples to is to be the Hodge-dual of the 7-form $G_7$ which the M5-brane couples to
$G_7 = \star G_4$is imposed by the sugra equations of motion. This is not seen by the cocycles themselves, but comes from imposing Cartan geometry: When we go beyond speaking of the cocycles themselves and instead ask for their “definite globalization” (here) and ask that to be torsion-free, then this implies the SuGra equations of motion (here) and part of these equations is the “electric/magnetic self-duality” of the M-brane charges.
To my mind that’s where the program is headed: figure out what these cocycles are non-rationally, then M-theory ought to be the quantization of their definite globalizations. (I.e. the space of definite globalizations of these cocycles ought to make a pre-n-plectic phase space of sorts, and that’s the thing to be quantized).
Do we have the “isomorphism to cobordism in complementary degree” written anywhere?
Not really, yet. There is a brief indication on p. 1 and p. 3 of Kirby-Melvin-Teichner 12. It comes down to a direct consequence of Pontryagin-Thom collapse, but it should be written out in more detail. Not discussed in the literature seems to be the equivariant refinement, which we really need here. Daniel Grady has something on this, but I don’t know when this will see the light of day.
With a Dirac charge quantization understanding could you approach this?:
What I said there still applies, both in what is known, what is to be expected, and what remains to be done.
What has been changing since is that what initially looked just like a vague (rational) shadow of the 4-sphere, which potentially still could have been all sorts of other things, has come more into focus and is looking more and more like the actual 4-sphere.
Now we are working on finalizing the writeup of “gauge enhancement”, which shows how, rationally, twisted K-theory of D-brane charges arises from the cohomology of M-brane charges. After that, possibly the next step should be to pass beyond the rational approximation on that front:
The way twisted K-theory arises, rationally, from the 4-sphere is by a kind of rational and twisted version of Snaith’s theorem. Roughly, the A-type orbispace of the 4-sphere $S^4 \sslash S^1$ looks like an $S^3$ with two copies of $B S^1$ attached to a pair of antipodal points, and under fiberwise stabilization this looks like an $S^3$ with two copies of $\Sigma^\infty_+ B S^1$ attached. Rationally, these are already connective K-theory, and analysis shows that as a parameterized spectrum over $S^3 \simeq_{\mathbb{Q}} K(\mathbb{Z},3)$, this is twisted connective K-theory, rationally. Hence it seems clear enough that the non-rational lift of “gauge enhancement of M-branes” to twisted K-theory is exactly along these lines, just with a fiberwise inversion of the Bott element thrown in, and appealing fiberwise to Snaith’s theorem.
Re #46, I see there is in IX Framed Manifolds:
This chapter discusses a construction that associates to every map from a manifold $M^{k+n}$ to a sphere $S^n$ a submanifold $V^k$ of $M$ and a framing of its normal bundle.
The book is available here.
Thanks! That’s a nicely explicit reference. I am adding this to cohomotopy, Pontrjagin-Thom construction, cobordism ring and wherever else it belongs.
You’ll probably guess that my look at Baas-Sullivan theory was inspired by things to do with cobordism and singularities, but I don’t know if it helps in your case.
I see. While details remain to be worked out, the emerging picture of the equivariant version of the isomorphism
$[X,S^k] \simeq \Omega^{n-k}(X)$was maybe different in spirit: Here the equivariant version gives submanifolds sitting at $G$-fixed points in $X$, hence at “singularities” in the ambient space, without the submanifolds themselves, or the cobordisms between them, being singular, necessarily.
But I only have a vague idea about this. Daniel Grady seemed to have a theorem and proof almost worked out.
But then, the idea is evident enough that I wouldn’t be surprised if this has been discussed in the literature before. If you see anything related to equivariant cobordism and equivariant cohomotopy, drop me a note!
It makes me think there might be a half-way version, for stratified spaces. I mean, given a stratification of the sphere by lower-dimensional spheres, one gets a stratification of the submanifold given by a regular value of a map to the sphere… The equivariant version is then a richer version of this.
That sounds right. I suppose this is suggesting that, under Elmendorf’s theorem, the equivariant version of the isomorphism $[X,S^k] \simeq \Omega^{n-k}(X)$ should become some kind of bi-natural isomorphism
$[(X)^H,(S^k)^H] \simeq \Omega^{dim\left((X)^H\right)-dim\left((S^k)^H\right)}((X)^H)$If you see anything related to equivariant cobordism and equivariant cohomotopy, drop me a note!
On p. 168 of Fixed point theory and framed cobordism, Prieto is discussing an equivariant cohomology theory, FIX-theory, which is isomorphic to equivariant stable cohomotopy,
Since FIX-theory admits a symmetric version for the action of any compact Lie group $G$, our approach provides an equivariant alternative to (framed) cobordism.
Thanks! Interesting. Will have a look.
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