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The wealth of $n$Lab edits of the last weeks on configuration spaces of points and on chord diagrams go along with a new article that we are finalizing:
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Abstract: We introduce a differential refinement of Cohomotopy cohomology theory, defined on Penrose diagram spacetimes, whose cocycle spaces are unordered configuration spaces of points. First we prove that charge quantization in this differential 4-Cohomotopy theory implies intersecting p/(p+2)-brane moduli given by ordered configurations of points in the transversal 3-space. Then we show that the higher observables on these brane moduli, conceived as the cohomology of the Cohomotopy cocycle space, reflects a multitude of effects expected in the quantum theory of Dp-D(p+2) brane intersections: condensation to stacks of coincident branes and their Chan-Paton factors, fuzzy funnel shapes and BLG 3-algebra structures, the Hanany-Witten rules, and the Chern-Simons sector of AdS3/CF2 duality. We discuss this in the context of the Hypothesis H that the M-theory C-field is charge-quantized in Cohomotopy theory.
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See there for the latest version. Comments are welcome.
Very nice! No substantive comments yet.
Impressive! Is there any way to apply some kind of related technology to the Dp-D(p+4)-brane bound state?
More typos:
Insersecting; the the; diragrams
Thanks! Typos fixed now.
Is there any way to apply some kind of related technology to the Dp-D(p+4)-brane bound state?
The maths in the article applied not to 4-Cohomotopy but to 6-Cohomotopy immedately yields the analogous conclusions: the Cohomotopy cocycle space now is the ordered configuration space of points in $\mathbb{R}^5$ instead of $\mathbb{R}^3$, and the cohomology of these two configuration spaces is in fact isomorphic (as is hence that of their corresponding loop spaces) up to, only, a shift in degree.
But it’s not clear why 6-Cohomotopy should come in. So I am not sure yet.
Random thoughts: at least $S^6$ has an interesting coset structure, $G_2/SU(3)$; one could look to lift 4-Cohomotopy to 6-Cohomotopy through $\pi_6(S^4) = \mathbb{Z}_2$.
There are some evident mathematical niceties about 6-Cohomotopy, true, but a priori it cannot appear directly in charge-quantization of the C-field. It might appear indirectly, but this needs thinking. For instance Dp/D(p+2) may be T-dual to Dp/D(p+4)…
On a trivial point, if you strip out the references from your opening sentence, it seems to me to be not grammatically well-formed:
The rich physics expected on coincident and intersecting branes, which geometrically engineer non-perturbative quantum gauge field theories close to quantum chromodynamics and modelling quantum microstates accounting for black hole entropy, has come to be center stage in string theory – or rather in the “theory formerly known as strings.
True, thanks. I am changing “modelling” to “model”
When you write that
The case $d=3$ in Def. 16, hence $p=6$, is singled out as being the mathematically exceptional one,
is it not the case that Def. 16 is symmetric in the sense that $d=1$ gives the same result?
Or is somehow $(\mathbb{R}^3)^{cpt} \wedge (\mathbb{R}^1)_+ \union (\mathbb{R}^3)_+ \wedge (\mathbb{R}^1)^{cpt}$ different from $(\mathbb{R}^1)^{cpt} \wedge (\mathbb{R}^3)_+ \union (\mathbb{R}^1)_+ \wedge (\mathbb{R}^3)^{cpt}$?
True, that’s the same. I’ll make that clearer, thanks.
Well it’s not just clarity. You say that unlike for $d=3$, for $d \in \{0, 1, 2,4\}$ the cohomology is fairly uninteresting, when in fact it’s the same for $d=1$. And you give a reason why only $d=3$ is interesting.
All right, I have fixed it. :-)
Being fussy, I guess the sentence beginning
It is only in the case $d=3$…,
could be modified to say that $d=1$ is (equally) interesting.
By the way, just how uninteresting is $d =2$?
Re #4, I’m not sure I quite got whether we should be pleased that it’s 4-cohomotopy and then $d = 1 or 3$ that is singled out, so that $D6-D8$ intersections are accorded special importance.
So with #12 I had changed the wording to “The case $d=3$ (equivalently $d=1$)…”
The three remarks 2.12, 2.13, 2.14 go together (and maybe originally were a single remark…), meaning to point out that the dimensions conspire in such a way that the monopole physics seen pertains to monopoles of actual 4d QCD, not just of some SYM in any other dimension.
I admit not to have computed the full cohomology of the cocycle space for $d=2$. But the thing is it retains some symmetrization over the ordering of the points in the configuration, and by the degrees in the graph complex, that kills the trivalent interaction vertex and leaves only the free propagators (the chords).
I have a file sitting here with lots of computations of graph cohomology in the underordered case, until it dawned on us that this all just descibes free field theories, and that somehow the ordering needs to be retained if there is any interacting field theory to come out of this.
Interestingly then, the linear ordering on the points/branes, hence their numbering $a = 1, 2, \cdots$ is seen to be their Chan-Paton labels. And so then everything fell into place…
So with #12 I had changed the wording to “The case d=3 (equivalently d=1)…”
Yes, but I’m talking about two sentences later:
It is only in the case $d=3$ that the remaining codimension $1 = 4-3$ induces the linear ordering on the branes via Prop. 2.4, which then leads to the rich observables found in §4.
which might say
It is only dimension $d=1$, or equivalently $d=3$ with remaining codimension $1=4-3$, that induces the linear ordering on the branes via Prop. 2.4, which then leads to the rich observables found in §4.
Oh, thanks. I’ll change that, too. Thanks.
expanded expository slides to go with my talk at M-Theory and Mathematics tomorrow are now here
comments are welcome!
(if you check it out, please grab the latest version here, I’ll sure be fixing little things for a little bit longer…)
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