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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 20th 2019
   • (edited Dec 20th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThe 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: $\,$ * [[schreiber:Differential Cohomotopy implies intersecting brane observables|Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams]] $\,$ **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. $\,$ See [[schreiber:Differential Cohomotopy implies intersecting brane observables|there]] for the latest version. Comments are welcome.

   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:

   $\,$

   • Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams

   $\,$

   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.

   $\,$

   See there for the latest version. Comments are welcome.

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeDec 20th 2019
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexVery nice! No substantive comments yet.

   Very nice! No substantive comments yet.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeDec 21st 2019
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexImpressive! 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

   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

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeDec 22nd 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks! 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.

   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.

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeDec 22nd 2019
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexRandom 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 $\pi_6(S^4) = \mathbb{Z}_2$.

   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 $\pi_6(S^4) = \mathbb{Z}_2$.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeDec 23rd 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexThere 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)...

   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)…

7. • CommentRowNumber7.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 1st 2020
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexOn 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.

   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.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJan 1st 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexTrue, thanks. I am changing "modelling" to "model"

   True, thanks. I am changing “modelling” to “model”

9. • CommentRowNumber9.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 9th 2020
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexWhen 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 $(\mathbb{R}^1)^{cpt} \wedge (\mathbb{R}^3)_+ \union (\mathbb{R}^1)_+ \wedge (\mathbb{R}^3)^{cpt}$?

   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 $(\mathbb{R}^1)^{cpt} \wedge (\mathbb{R}^3)_+ \union (\mathbb{R}^1)_+ \wedge (\mathbb{R}^3)^{cpt}$?

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJan 9th 2020
    • PermaLink
    Author: Urs
    Format: MarkdownItexTrue, that's the same. I'll make that clearer, thanks.

    True, that’s the same. I’ll make that clearer, thanks.

11. • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 9th 2020
    • (edited Jan 9th 2020)
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexWell 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.

    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.

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJan 9th 2020
    • PermaLink
    Author: Urs
    Format: MarkdownItexAll right, I have fixed it. :-)

    All right, I have fixed it. :-)

13. • CommentRowNumber13.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 15th 2020
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexBeing 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.

    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.

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJan 15th 2020
    • PermaLink
    Author: Urs
    Format: MarkdownItexSo 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 $d=1$)…”

    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…

15. • CommentRowNumber15.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 15th 2020
    • (edited Jan 15th 2020)
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItex> 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.

    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.

16. • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJan 15th 2020
    • PermaLink
    Author: Urs
    Format: MarkdownItexOh, thanks. I'll change that, too. Thanks.

    Oh, thanks. I’ll change that, too. Thanks.

17. • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJan 26th 2020
    • PermaLink
    Author: Urs
    Format: MarkdownItexexpanded expository slides to go with my talk at _[[M-Theory and Mathematics]]_ tomorrow are now [here](https://ncatlab.org/schreiber/show/Microscopic+Brane+Physics+from+Cohomotopy) comments are welcome! (if you check it out, please grab the latest version [here](https://ncatlab.org/schreiber/show/Microscopic+Brane+Physics+from+Cohomotopy#ExpandedTalkSlides), I'll sure be fixing little things for a little bit longer...)

    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…)