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I gave the brane scan table a genuine $n$Lab incarnation and included it at Green-Schwarz action functional and at brane.
I guess the pun is intentional? (sounds like brain scan)
I think so. That’s what it is being called since the 90s.
added to the brane scan now the type IIA D-branes and their characterizing super-$L_\infty$-algebra cocycles on the type II supergravity Lie 2-algebra.
hyperlinked one more item in the table: the super 4-brane in 9d
I have hyperlinked two more of the remaining entries in the brane scan:
and
Am creating the respective entries now…
What are pullbacks of D-branes supposed to correspond to? Say pb(d1brane->stringIIB<-d3brane) ?
As you know (probably that’s why you are asking) generally pullbacks of this form may serve in correspondences/dualities between one theory and the other.
For instance, if instead of D-branes we consider M2-branes, then the pullback in question features in the higher M-theoretic analog of T-duality (Prop. 4.4 on p. 32 here), at least rationally (further exposition of this example is in arXiv:1805.00233, although the pullback diagram is not displayed there):
$\array{ \mathfrak{m}2\mathfrak{brane} &&&& \mathfrak{m}2\mathfrak{brane} \\ & \searrow && \swarrow \\ && \mathbb{R}^{10,1\vert\mathbf{32}} }$But this requires that there are further cocycles satisfying conditions on the two sides of the pullback (the M5-banes in the above example) which should not be the case for D-branes.
(This last statement, that the D-brane extensions carry no further invariant extensions in the brane scan, remains without rigorous proof; but it would mean another brane species on which D-branes can end like M2s end on M5s, and folklore does not expect this to exist.)
So I don’t know that there is any significance to the fiber products of D-brane cocycles. But there is much room left to explore the idea of the brane bouquet further.
You want to expand on that?
Seems like the info about the brane scan/bouquet is scattered amongst different pages, e.g. brane scan, brane (which seems to have brane scan embedded on it), table of branes, and brane bouquet, which is actually part of your personal pages. Ideally, there should be a single entry with all this info but on the other hand brane scan is probably better reserved for the old brane scan. Any suggestions?
You could start a page brane bouquet on the main web, that would be useful.
Will do.
In M-theory from the Superpoint, what is the precise role of considering extensions invariant with respect to the ‘simple external automorphisms’ inside the automorphism group of g? Is this what allows us to get extensions that basically correspond to the coset space (in this case Minkowski spaces as coset space of, at the level of algebras, the super-Poincare algebra modulo Lorentz algebra)? What if one drops this assumption, can one get the super-Poincare algebra instead?
The reason I ask is the following. So far in these super- extensions of the Poincare algebra, we consider the anti-commutator of the supercharges to be proportional to momentum generator. In this case we can talk about R^{d-1,1|N} where the bracket is this pairing. However, in The twelve dimensional super (2+2)-brane, the main point is that one needs to consider different super- extensions of the Poincare algebra. In particular, the extensions that turns out to be the correct/useful one is that for which the anticommutator of the supercharges is not proportional to momentum but is a linear combination of momenta and rotations. So if I want my algebra extensions/cocycles to involve rotations, then it seems it cannot be an maximally invariant extension of some Minkowski space?
The motivation for restricting to the “external” automorphisms is meant to be illuminated by “Example 2” on that same p. 9:
The “internal” ones are the “R-symmetries” – in particular, “external”/”internal” is in the physics sense of “spacetime symmetry”/”internal DOFs symmetry”.
Since we are after the (ordinary) spacetime symmetry algebras, this makes it natural to discard all the internal symmetry and focus on the external one.
But in retrospect the definition is really justified by the result it yields.
This whole business of emergence of super-spacetimes here clearly feels just like the tip of an iceberg that remains to be uncovered. So if you want to look at variations and see what these yield, that could be most interesting.
Just a very basic question. In the Outlook section where you describe the (unique) central extension of the superpoint R^{0,0|1}, how does one see that the resulting extension is R^{1,0|1} and not R^{0,1|1}? Does this mean R^{0,1|1} is itself the root of a different bouquet that is not the one emanating from the superpoint?
It is $\mathbb{R}^{0, 1\vert 1}$. Maybe I was careless in dropping the “0” in that Outlook section, or maybe I was meaning to indicate that in this degenerate case the distinction is not relevant.
In previous lecture notes here I did write $\mathbb{R}^{0, 1\vert 1}$.
This is the super Lie algebra of supersymmetric quantum mechanics, where the single supercharge squares to the Hamiltonian, hence to the time-translation generator.
On the other hand, if you’d thought of it as a supersymmetric 1-spatial translation algebra it would look no different, whence one could argue that keeping the extra “0” around is meaningless.
Sounds good, thanks Urs.
In 1308.5264 we talk of the heterotic string $L_{\infty}$-algebra $\mathfrak{string}_{\text{het}}$. Where does the discussion of $E_8\times E_8$ and $\text{Spin}(32)/Z_2$ enter the picture?
Yes, the brane scan, as is, only produces the heterotic string in its trivial background fields (flat space, no background gauge fields). This is actually no different from the other strings in the scan, only that for these it’s not as notable, since they have no nonabelian background fields anyways.
The superspace formulation of heterotic sugra with background fields, from which the super $L_\infty$-algebra of the heterotic string would be derivable if it were there, is in the references listed here. We once tried but never managed to connect this to the brane bouquet.
The closest we came to seeing the heterotic string is in arXiv:2008.08544, where it certainly shows up in some guise, but rather more indirectly than one may have expected.
Incidentally, something similar happens in the $E_{10}$/$E_{11}$-approach to M-theory. There, the heterotic string also does not show up.
“Well, too bad for the heterotic string, then!”, as Hermann Nicolai once said (here). :-)
But isn’t it weird that $\mathfrak{string}_{\text{het}}$ just appears from a cocycle in $\mathbb{R}^{9,1\vert N=1}$ without any reference to the left-moving bosonic string modes? For instance, $\mathfrak{string}_{\text{IIB}}$ arises from an extension of $\mathbb{R}^{9,1\vert 16+\overline{16}}$, which we can regard as a pullback of a span where one leg is $\mathbb{R}^{9,1\vert 16}$ and the other is $\mathbb{R}^{9,1\vert \overline{16}}$. Can one read this as a left-moving, right-moving statement? If so, then shouldn’t the full heterotic algebra arise as a similar pullback where now one leg is $\mathbb{R}^{9,1\vert 16}$ and the other is $\mathbb{R}^{25,1\vert 1}$? There some of the stuff appearing here would appear, I’d think.
I see where you are coming from, but you are thinking of the NSR superstring-type worldsheet formulation, while the brane scan sees the Green-Schwarz superstring formulation with target space supersymmetry.
Specifically those cocycles in the brane scan are WZ-terms made entirely from target space super-fields. In this target space formulation, the left-moving worldsheet bosons of the NSR-type formulation of the heterotic string appear as the non-abelian target space background field, and hence don’t appear if no such background field is considered.
You can see this on the first pages of Shapiro & Taylor 1987 .
Of course in this case the GS-form of the heterotic string essentially coincides with that of the Type-I string, and maybe that’s a more suggestive way of speaking about it.
But if you feel adventurous, maybe you can discover a new rule and a previously unrecognized branch of the bouquet.
If the content of non-Lorentzian type II string theory is meaningful then one should be able to use say 1806.01115 to get some of those things starting from M theory. In particular, (72) is where we specify the circle fiber is space-like and not time-like, right? Then to get $IIA_E$ one first should express $\mathbb{R}^{10,1\vert \mathbf{32}}$ as an extension of some Euclidean $\mathbb{R}^{10,0\vert ?}$ (with some ? spin representation). Of course, the ideal situation would be to first discover such Euclidean space itself as some extension, this is where I think something like the split- or para-algebras become relevant. In fact, I would think these bring up another branch stemming from $\mathbb{R}^{3,1}$, since only for the real numbers their para- variant is isomorphic to itself, but not for complex numbers and beyond.
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