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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 26th 2016
   • (edited Aug 26th 2016)
   • PermaLink
   Author: Urs
   Format: MarkdownItexInstead of further scattering it over more or less related threads, I'll talk here about thoughts and progress on identifying the correct generalized cohomology theory for M-brane charges. Recall that the logic is as follows: F1/Dp-brane charges are famously argued to be not in ordinary cohomology, but in twisted K-theory, and we found that at the rational level this statement may be systematically derived via super Lie $n$-algebra cohomology. Moreover, it is an open problem how twisted K-theory lifts to M-theory; but now using the same super Lie $n$-algebra derivation, it is possible to systematically derive this: one finds that rationally the cohomology theory is represented by the quaternionic Hopf fibration, regarded as a twisted cohomology theory via $[S^4 \to B^3 U(1)] \in \mathbf{H}_{/\mathbf{B}^3 U(1)}$. The remaining question is what is going on away from the rational approximation. Given that twisted K-theory is argued to know a whole lot about the fine structure of type II/I string theory, the motivation here is that identifying the correct generalized cohomology theory for M-branes should go a long way towards finding the missing "microscopic degrees of freedom" that should define the theory. So what I am doing now is trying to fit as many hints as possible from the available physics story to deduce what the cohomology theory has to be. One big hint is ADE-equivariance, which is known to be crucial for the physics story, and which the quaternionic Hopf fibration naturally lends itself to. One observation here is that for the finite groups in the A-series inside $SU(2)$, the fixed points of the canonical action on the quaternionic Hopf fibration is the complex Hopf fibration. By mechanisms such as [[tom Dieck splitting]] we know that equivariant generalized cohomology will, roughly, split into contributions from fixed points of all closed subgroups. So this motivates the idea that with the quaternionic Hopf fibration regarded as a coefficient in equivariant cohomology theory somehow, then also the complex Hopf fibration should appear and play an analogous role on a smaller scale. Here is an observation that this indeed matches the physics story: First, observe that the quaternionic Hopf fibrations sees the M2- and M5-brane charges, but not the [[Kaluza-Klein monopole]] (the other BPS state in 11d sugra). Related to this, the free looping of the quaternionic Hopf fibration sees the type IIA D$p$-branes for $p \in \{0,2,4\}$, but not the D6-brane, which is the descendant of the KK-monopole. Hence the picture isn't complete until we find a generalized cohomology theory in 11d that also reflects the KK-monopole charge. (continued in the next comment)

   Instead of further scattering it over more or less related threads, I’ll talk here about thoughts and progress on identifying the correct generalized cohomology theory for M-brane charges.

   Recall that the logic is as follows: F1/Dp-brane charges are famously argued to be not in ordinary cohomology, but in twisted K-theory, and we found that at the rational level this statement may be systematically derived via super Lie $n$-algebra cohomology. Moreover, it is an open problem how twisted K-theory lifts to M-theory; but now using the same super Lie $n$-algebra derivation, it is possible to systematically derive this: one finds that rationally the cohomology theory is represented by the quaternionic Hopf fibration, regarded as a twisted cohomology theory via $[S^4 \to B^3 U(1)] \in \mathbf{H}_{/\mathbf{B}^3 U(1)}$.

   The remaining question is what is going on away from the rational approximation. Given that twisted K-theory is argued to know a whole lot about the fine structure of type II/I string theory, the motivation here is that identifying the correct generalized cohomology theory for M-branes should go a long way towards finding the missing “microscopic degrees of freedom” that should define the theory.

   So what I am doing now is trying to fit as many hints as possible from the available physics story to deduce what the cohomology theory has to be.

   One big hint is ADE-equivariance, which is known to be crucial for the physics story, and which the quaternionic Hopf fibration naturally lends itself to.

   One observation here is that for the finite groups in the A-series inside $SU(2)$, the fixed points of the canonical action on the quaternionic Hopf fibration is the complex Hopf fibration. By mechanisms such as tom Dieck splitting we know that equivariant generalized cohomology will, roughly, split into contributions from fixed points of all closed subgroups. So this motivates the idea that with the quaternionic Hopf fibration regarded as a coefficient in equivariant cohomology theory somehow, then also the complex Hopf fibration should appear and play an analogous role on a smaller scale.

   Here is an observation that this indeed matches the physics story:

   First, observe that the quaternionic Hopf fibrations sees the M2- and M5-brane charges, but not the Kaluza-Klein monopole (the other BPS state in 11d sugra). Related to this, the free looping of the quaternionic Hopf fibration sees the type IIA D$p$-branes for $p \in \{0,2,4\}$, but not the D6-brane, which is the descendant of the KK-monopole.

   Hence the picture isn’t complete until we find a generalized cohomology theory in 11d that also reflects the KK-monopole charge.

   (continued in the next comment)

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeAug 26th 2016
   • (edited Aug 29th 2016)
   • PermaLink
   Author: Urs
   Format: MarkdownItexBut check out this: the quick reason that it is the 7-sphere and the 4-sphere which measure M2- and M5-brane charge is that the near-horizon geometry of an M2-brane is $\mathrm{AdS}_4 \times S^7$, and that of an M5-brane is $\mathrm{AdS}_7 \times S^4$. Moreover, with the ADE-orbifold structure taken into account then we are to think of $S^7$ here as $S(\mathbb{H}^2)$ and as the brane itself as sitting at the fixed point of the canonical $SU(2)$-action on $\mathbb{H}$. Analogously: what is the near-horizon geometry of a KK-monopole in 11d? Well, it's $contractible_{8} \times S^3$, and the singularity story says that the $S^3$ here is to be thought of as $S(\mathbb{C}^2)$ with the monopole sitting at the fixed point of the canonical $U(1)$-action. And the picture becomes stronger still: the relevant KK-compactification of the 11d theory is on a 6-dimensional fiber, to [[5d supergravity]] (to be the strongly coupled version of the actual 4d theory of ultimate interest), the KK-monopole in 11d is to "wrap" the 6d fiber and becomes the original KK-monopole in 5d. Whatever the charge of the KK-monopole in 11d is, it is to become the charge of the KK-monopole in 5d after this reduction. Now among all higher dimensional supergravity theories, the ones in 11d and in 5d stand out as having higher dimensional Chern-Simons terms that give non-linear equations of motion for the higher form field. In 11d the C-field strength $G_4$, locally of the form $G_4 = d C_3$ has a CS-term locally of the form $\propto \int G_4 \wedge G_4 \wedge C_3$, which leads to an equation of motion $$ d G_7 = G_4 \wedge G_4 $$ (with $G_7 = \star G_4$). This is of course the same relation as for the generators of the minimal Sullivan model for the 4-sphere, and this is how Hisham Sati originally conjectured that M-brane charge should rationally be in degree-4 cohomotopy, before we proved this systematically from the super Lie $n$-algebra cocycles. But now 5d supergravity stands out as exhibiting the analogous structure in lower dimension: it has a 2-form field strength $F_2$ locally of the form $F_2 = d A_1$ (the "graviphoton") and a Chern-Simons term of the form $\int F_2 \wedge F_2 \wedge A_1$ leading to an equation of motion $$ d F_3 = F_2 \wedge F_2 $$ (with $F_3 = \star F_2$). This now is of course precisely the relation on the generators in the minimal Sullivan model for the 2-sphere. Moreover, the $F_3$ here is what measures the KK-monopole charge in 5d, hence in 11d. In conclusion, we see that where the quaternionic Hopf fibration in 11d is exactly what measures (rationally) the M2- and M5-brane charge, the complex Hopf fibration in 5d is in direct analogy what measures the remaining KK-monopole charge. Moreover, the expected equivariance matches: just as the M2-brane in general sits at the $G_{ADE} \subset SU(2)$-orbifold singularity at the center of $\mathbb{H}^2$, so the KK-monopole sits at the $\mathbb{Z}/n\subset U(1)$-orbifold singularity at the center of $\mathbb{C}^2$. So this is good support of the idea that we have to consider the quaternionic Hopf fibration as a cohomology theory that somehow has substructure where it splits up into the complex (and then maybe also the real) Hopf fibration inside it. However, what puzzles me is that this picture, which clearly must be right, doesn't fit with my original idea that it is [[tom Dieck splitting]] which formalizes this "splitting up", with $S^7$ regarded with its $SU(2)$ action given by the conjugation action of quaternions. Because, while this does make the $A$-type fixed points of the quaternionic Hopf fibration become the complex Hopf fibration, it would have to correspond to the analogous passage to fixed points on the spacetime side. But this would be exactly wrong: the $U(1)$-fixed points on spacetime are exactly the locus of the KK-monopole/D6-brane, while for the argument to work we should _remove_ that fixed point locus to see the surrounding $S^3$ (in direct analogy to how Dirac in the 1930s removed the locus of the magnetic monopole to see a surrounding $S^2$). So I am left in a situation where part of my original idea is seen to work out right, but another part, which I thought was intimately connected, comes out wrong. Something of my original picture needs to be changed...

   But check out this: the quick reason that it is the 7-sphere and the 4-sphere which measure M2- and M5-brane charge is that the near-horizon geometry of an M2-brane is $\mathrm{AdS}_4 \times S^7$, and that of an M5-brane is $\mathrm{AdS}_7 \times S^4$. Moreover, with the ADE-orbifold structure taken into account then we are to think of $S^7$ here as $S(\mathbb{H}^2)$ and as the brane itself as sitting at the fixed point of the canonical $SU(2)$-action on $\mathbb{H}$.

   Analogously: what is the near-horizon geometry of a KK-monopole in 11d? Well, it’s $contractible_{8} \times S^3$, and the singularity story says that the $S^3$ here is to be thought of as $S(\mathbb{C}^2)$ with the monopole sitting at the fixed point of the canonical $U(1)$-action.

   And the picture becomes stronger still: the relevant KK-compactification of the 11d theory is on a 6-dimensional fiber, to 5d supergravity (to be the strongly coupled version of the actual 4d theory of ultimate interest), the KK-monopole in 11d is to “wrap” the 6d fiber and becomes the original KK-monopole in 5d. Whatever the charge of the KK-monopole in 11d is, it is to become the charge of the KK-monopole in 5d after this reduction.

   Now among all higher dimensional supergravity theories, the ones in 11d and in 5d stand out as having higher dimensional Chern-Simons terms that give non-linear equations of motion for the higher form field.

   In 11d the C-field strength $G_4$, locally of the form $G_4 = d C_3$ has a CS-term locally of the form $\propto \int G_4 \wedge G_4 \wedge C_3$, which leads to an equation of motion

   $$d G_7 = G_4 \wedge G_4$$

   (with $G_7 = \star G_4$). This is of course the same relation as for the generators of the minimal Sullivan model for the 4-sphere, and this is how Hisham Sati originally conjectured that M-brane charge should rationally be in degree-4 cohomotopy, before we proved this systematically from the super Lie $n$-algebra cocycles.

   But now 5d supergravity stands out as exhibiting the analogous structure in lower dimension: it has a 2-form field strength $F_2$ locally of the form $F_2 = d A_1$ (the “graviphoton”) and a Chern-Simons term of the form $\int F_2 \wedge F_2 \wedge A_1$ leading to an equation of motion

   $$d F_3 = F_2 \wedge F_2$$

   (with $F_3 = \star F_2$). This now is of course precisely the relation on the generators in the minimal Sullivan model for the 2-sphere. Moreover, the $F_3$ here is what measures the KK-monopole charge in 5d, hence in 11d.

   In conclusion, we see that where the quaternionic Hopf fibration in 11d is exactly what measures (rationally) the M2- and M5-brane charge, the complex Hopf fibration in 5d is in direct analogy what measures the remaining KK-monopole charge.

   Moreover, the expected equivariance matches: just as the M2-brane in general sits at the $G_{ADE} \subset SU(2)$-orbifold singularity at the center of $\mathbb{H}^2$, so the KK-monopole sits at the $\mathbb{Z}/n\subset U(1)$-orbifold singularity at the center of $\mathbb{C}^2$.

   So this is good support of the idea that we have to consider the quaternionic Hopf fibration as a cohomology theory that somehow has substructure where it splits up into the complex (and then maybe also the real) Hopf fibration inside it.

   However, what puzzles me is that this picture, which clearly must be right, doesn’t fit with my original idea that it is tom Dieck splitting which formalizes this “splitting up”, with $S^7$ regarded with its $SU(2)$ action given by the conjugation action of quaternions. Because, while this does make the $A$-type fixed points of the quaternionic Hopf fibration become the complex Hopf fibration, it would have to correspond to the analogous passage to fixed points on the spacetime side. But this would be exactly wrong: the $U(1)$-fixed points on spacetime are exactly the locus of the KK-monopole/D6-brane, while for the argument to work we should remove that fixed point locus to see the surrounding $S^3$ (in direct analogy to how Dirac in the 1930s removed the locus of the magnetic monopole to see a surrounding $S^2$).

   So I am left in a situation where part of my original idea is seen to work out right, but another part, which I thought was intimately connected, comes out wrong. Something of my original picture needs to be changed…

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 29th 2016
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI hope it goes without saying that even if no-one here's helping you out, it's always great to hear you thinking out loud (or 'thinking loudly', as Imre Lakatos used to say).

   I hope it goes without saying that even if no-one here’s helping you out, it’s always great to hear you thinking out loud (or ’thinking loudly’, as Imre Lakatos used to say).

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeAug 30th 2016
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI second David C's comments. I will pipe up if and when I can contribute something.

   I second David C’s comments. I will pipe up if and when I can contribute something.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeAug 30th 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, David and David. My last idea last weak was that maybe we should be considering cohomology of bundles $$ \array{ X_{11} \\ \downarrow \\ X_{5} } $$ with contributions from the total space and from the base separately $$ \array{ X_{11} &\longrightarrow& S^4 \\ \downarrow \\ X_{5} &\longrightarrow& S^2 } \,. $$ This may look a little weid, but a cocycle situation of this form we once proved does encode the local action functional of 3d Chern-Simons theory in the presence of Wilson loops (p. 26 of [arXiv:1301.2580](https://arxiv.org/abs/1301.2580) or else on the nLab at [orbit method -- Extended Chern-Simons and Wilson loops](https://ncatlab.org/nlab/show/orbit%20method#ExtendedChern-SimonsTheoryAndWilsonLoops)). One interesting aspect of this is that it would force the phenomenologically realistic dimensions on the 11d theory from just the structure of the Hopf-invariant 1 maps, since in this perspective the dimensions $$ \array{ X_{11} & X_{10} \\ \downarrow & \downarrow \\ X_{5} & X_4 } $$ are fixed by $11 = 7 + 4$ (from the quaternionic Hopf fibration) and $5 = 3+2$ (from the complex Hopf fibration). Apart from this it is interesting to notice that making brane charges in 5d appear manifestly makes the formalism directly connect to a hugely rich story: the "black branes" in 5d are black holes and [[black rings]] and the charges of these 5d structures are what governs all the existing analytical proofs of Bekenstein-Hawking entropy of 4d near extremal black holes coinciding with microscopic entropy of brane states. Nevertheless, I feel still puzzled that this perspective doesn't reflect the story of equivariant cohomology as I originally expected. But then I had this thought: We should possibly distinguish the brane charge coefficients for outside the support of the source branes from those that apply on the support. What I mean is this: In plain electromagnetism (charged 0-branes in 4d) the fact that magnetic charge has coefficients in $\mathbf{B}U(1)$ is true only outside the support of the magnetic charge (that's the classical Dirac charge quantization argument), while on the support of any magnetic current 3-form $J$ the first Maxwell equation $d F = J$ implies that $F$ cannot be the curvature 2-form of a $U(1)$-bundle, because that would imply $d F = 0$. But in the first section of _[[Dirac charge quantization and generalized differential cohomology]]_, Freed pointed out that the locus of $J$ support may be included in the theory if only we change the coefficients: $J$ itself should be classified by cohomology with compact support and with coefficients in $\mathbf{B}^2 U(1)$, while then $F$ would be (the curvature) of a _twisted_ $U(1)$-bundle, twisted by the gerbe of $J$. Now, all the above reasoning about M2- and KK/D6 brane charge are analogs of Dirac's old argument outside the support of the charge source, the M-brane analog of $F$. So maybe it's to be expeced that this does not show the equivariant cohomology phenomena that I expected to see at the charge source itself, and that this will instead be reflected by the M-brane analog of $J$. I wish I had a more concrete picture by now. But I need to let this rest now for a bit. Will get back to it later.

   Thanks, David and David.

   My last idea last weak was that maybe we should be considering cohomology of bundles

   $$\array{ X_{11} \\ \downarrow \\ X_{5} }$$

   with contributions from the total space and from the base separately

   $$\array{ X_{11} &\longrightarrow& S^4 \\ \downarrow \\ X_{5} &\longrightarrow& S^2 } \,.$$

   This may look a little weid, but a cocycle situation of this form we once proved does encode the local action functional of 3d Chern-Simons theory in the presence of Wilson loops (p. 26 of arXiv:1301.2580 or else on the nLab at orbit method – Extended Chern-Simons and Wilson loops).

   One interesting aspect of this is that it would force the phenomenologically realistic dimensions on the 11d theory from just the structure of the Hopf-invariant 1 maps, since in this perspective the dimensions

   $$\array{ X_{11} & X_{10} \\ \downarrow & \downarrow \\ X_{5} & X_4 }$$

   are fixed by $11 = 7 + 4$ (from the quaternionic Hopf fibration) and $5 = 3+2$ (from the complex Hopf fibration).

   Apart from this it is interesting to notice that making brane charges in 5d appear manifestly makes the formalism directly connect to a hugely rich story: the “black branes” in 5d are black holes and black rings and the charges of these 5d structures are what governs all the existing analytical proofs of Bekenstein-Hawking entropy of 4d near extremal black holes coinciding with microscopic entropy of brane states.

   Nevertheless, I feel still puzzled that this perspective doesn’t reflect the story of equivariant cohomology as I originally expected. But then I had this thought:

   We should possibly distinguish the brane charge coefficients for outside the support of the source branes from those that apply on the support.

   What I mean is this: In plain electromagnetism (charged 0-branes in 4d) the fact that magnetic charge has coefficients in $\mathbf{B}U(1)$ is true only outside the support of the magnetic charge (that’s the classical Dirac charge quantization argument), while on the support of any magnetic current 3-form $J$ the first Maxwell equation $d F = J$ implies that $F$ cannot be the curvature 2-form of a $U(1)$-bundle, because that would imply $d F = 0$.

   But in the first section of Dirac charge quantization and generalized differential cohomology, Freed pointed out that the locus of $J$ support may be included in the theory if only we change the coefficients: $J$ itself should be classified by cohomology with compact support and with coefficients in $\mathbf{B}^2 U(1)$, while then $F$ would be (the curvature) of a twisted $U(1)$-bundle, twisted by the gerbe of $J$.

   Now, all the above reasoning about M2- and KK/D6 brane charge are analogs of Dirac’s old argument outside the support of the charge source, the M-brane analog of $F$. So maybe it’s to be expeced that this does not show the equivariant cohomology phenomena that I expected to see at the charge source itself, and that this will instead be reflected by the M-brane analog of $J$.

   I wish I had a more concrete picture by now. But I need to let this rest now for a bit. Will get back to it later.

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 30th 2016
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexFrom #2, >...the idea that we have to consider the quaternionic Hopf fibration as a cohomology theory that somehow has substructure where it splits up into the complex (and then maybe also the real) Hopf fibration inside it. A dimension $2 = 1+ 1$ could appear?

   From #2,

   …the idea that we have to consider the quaternionic Hopf fibration as a cohomology theory that somehow has substructure where it splits up into the complex (and then maybe also the real) Hopf fibration inside it.

   A dimension $2 = 1+ 1$ could appear?

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeAug 30th 2016
   • (edited Aug 30th 2016)
   • PermaLink
   Author: Urs
   Format: MarkdownItexFrom extrapolating the role of the quaternionic and the complex Hopf fibration it might seem so. However, that these two appear is intimately related to the non-linear structure of the Sullivan models for the even dimensional spheres $S^4$ ($d g_7 = g_4 \wedge g_4$) and $S^2$ ($d g_3 = g_2 \wedge g_2$), which control the higher Chern-Simons terms in 11d SuGra and 5d SuGra. But for the real Hopf fibration with its odd dimensional base sphere this pattern breaks down. So I am not sure. In my original picture the real Hopf fibration inside the quaternionic Hopf fibration was crucial, being the fixed point locus of the adjoint action of any finite group in the $D$-series or $E$-series. But in the picture from #2 it seems not to play a role. So we are back to the issue in #5, needing to understand how these two pictures fit together -- if they do.

   From extrapolating the role of the quaternionic and the complex Hopf fibration it might seem so. However, that these two appear is intimately related to the non-linear structure of the Sullivan models for the even dimensional spheres $S^4$ ($d g_7 = g_4 \wedge g_4$) and $S^2$ ($d g_3 = g_2 \wedge g_2$), which control the higher Chern-Simons terms in 11d SuGra and 5d SuGra. But for the real Hopf fibration with its odd dimensional base sphere this pattern breaks down. So I am not sure.

   In my original picture the real Hopf fibration inside the quaternionic Hopf fibration was crucial, being the fixed point locus of the adjoint action of any finite group in the $D$-series or $E$-series. But in the picture from #2 it seems not to play a role.

   So we are back to the issue in #5, needing to understand how these two pictures fit together – if they do.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeSep 7th 2016
   • (edited Sep 7th 2016)
   • PermaLink
   Author: Urs
   Format: MarkdownItexHere is another idea: Replace $S^4$ for the moment simply by the topological groupoid $S^4//G_{\mathrm{ADE}}$. Then when forming the free loop space (in topological stacks) $\mathcal{L}(S^4 //G_{\mathrm{ADE}})//S^1$ (to pass from M-brane charges to type IIA brane charges) we pick up "twisted sector" contributions even from constant loops $$ \mathcal{L}_{const}(S^4 //G) = [ \ast//\mathbb{Z}, S^4 //G ] \hookrightarrow [S^1, S^4 //G] \,. $$ Concretely, $$ \mathcal{L}_{const}(S^4 //G) = \underset{[g]}{\coprod} (S^4)^g // C(g) $$ is the disjoint union over conjugacy classes $[g]$ of the $g$-fixed point subspaces quotiented by the remaining action of the centralizer of $g$. Of course the full $\mathcal{L} (S^4//G)$ contains this and more, but focusing on $\mathcal{L}_{const}(S^4 //G)$ may help to make plausibility checks as to whether this structure captures effects that we want to see. Specifically $\mathcal{L}_{const} (S^4//G)//S^1$ is the "twisted loop space" whose rational cohomology (for $t = 0$) is considered in [Stapleton 13](https://ncatlab.org/nlab/show/transchromatic+character#Stapleton13) as the target of a "[[transchromatic character]]" map from Morava $E_n$-cohomology of $S^4 // G$. This is just the kind of effect that we are after here, a natural lift of the (rational) cohomology of IIA brane charges to a non-rational generalized cohomology theory of M-brane charges. So this might be interesting to pursue. As a first consistency check then, we'd have to compute the rational cohomology of $\mathcal{L}_{const} (S^4//G)//S^1$ and see if the contributions by the $G$-quotients make it pick up classes that would reflect the previously missing D6-brane charges at $G$-fixed points in spacetime. If we do consider on $S^4$ the $G$-action induced by the _conjugation_ action of quaternions (which I first thought was the relevant one, but now I am no longer sure) then $\mathcal{L}_{const} (S^4//G)$ has a direct summand $S^4//G$ (the $e$-fixed points) which is $S^1 \times B G$. So we'd be seeing extra contributions to the rational cohomology of $\mathcal{L}S^4 // S^1$ from the rational group cohomology of the ADE supgroup $G$. However, I suppose that the _rational_ group cohomology of $G$ vanishes in positive degree. Hm.

   Here is another idea:

   Replace $S^4$ for the moment simply by the topological groupoid $S^4//G_{\mathrm{ADE}}$. Then when forming the free loop space (in topological stacks) $\mathcal{L}(S^4 //G_{\mathrm{ADE}})//S^1$ (to pass from M-brane charges to type IIA brane charges) we pick up “twisted sector” contributions even from constant loops

   $$\mathcal{L}_{const}(S^4 //G) = [ \ast//\mathbb{Z}, S^4 //G ] \hookrightarrow [S^1, S^4 //G] \,.$$

   Concretely,

   $$\mathcal{L}_{const}(S^4 //G) = \underset{[g]}{\coprod} (S^4)^g // C(g)$$

   is the disjoint union over conjugacy classes $[g]$ of the $g$-fixed point subspaces quotiented by the remaining action of the centralizer of $g$.

   Of course the full $\mathcal{L} (S^4//G)$ contains this and more, but focusing on $\mathcal{L}_{const}(S^4 //G)$ may help to make plausibility checks as to whether this structure captures effects that we want to see. Specifically $\mathcal{L}_{const} (S^4//G)//S^1$ is the “twisted loop space” whose rational cohomology (for $t = 0$) is considered in Stapleton 13 as the target of a “transchromatic character” map from Morava $E_n$-cohomology of $S^4 // G$.

   This is just the kind of effect that we are after here, a natural lift of the (rational) cohomology of IIA brane charges to a non-rational generalized cohomology theory of M-brane charges. So this might be interesting to pursue.

   As a first consistency check then, we’d have to compute the rational cohomology of $\mathcal{L}_{const} (S^4//G)//S^1$ and see if the contributions by the $G$-quotients make it pick up classes that would reflect the previously missing D6-brane charges at $G$-fixed points in spacetime.

   If we do consider on $S^4$ the $G$-action induced by the conjugation action of quaternions (which I first thought was the relevant one, but now I am no longer sure) then $\mathcal{L}_{const} (S^4//G)$ has a direct summand $S^4//G$ (the $e$-fixed points) which is $S^1 \times B G$. So we’d be seeing extra contributions to the rational cohomology of $\mathcal{L}S^4 // S^1$ from the rational group cohomology of the ADE supgroup $G$. However, I suppose that the rational group cohomology of $G$ vanishes in positive degree.

   Hm.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeSep 7th 2016
   • (edited Sep 7th 2016)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIt just occurs to me that what I demand in #5 is naturally provided by Bredon cohomology. [...] [edit: something wrong removed here] [...] So I suppose what we need is an $SU(2)$-action on $S^4$ with a single fixed point. The result of section 8.3 in * Paul de Medeiros, José Figueroa-O'Farrill, _Half-BPS M2-brane orbifolds_ ([arXiv:1007.4761](http://arxiv.org/abs/1007.4761)) suggests that the action to consider is the action given by canonically embedding $S^4$ in $\mathbb{R}^5 \simeq \mathbb{R} \oplus \mathbb{H}$ and then acting by left quaternion multiplication. This does have a single fixed point, doesn't it namely $x_1 = 1, x_{2 \leq i \leq 5} = 0$. So this is looking good.

   It just occurs to me that what I demand in #5 is naturally provided by Bredon cohomology.

   […]

   [edit: something wrong removed here]

   […]

   So I suppose what we need is an $SU(2)$-action on $S^4$ with a single fixed point. The result of section 8.3 in

   • Paul de Medeiros, José Figueroa-O’Farrill, Half-BPS M2-brane orbifolds (arXiv:1007.4761)

   suggests that the action to consider is the action given by canonically embedding $S^4$ in $\mathbb{R}^5 \simeq \mathbb{R} \oplus \mathbb{H}$ and then acting by left quaternion multiplication. This does have a single fixed point, doesn’t it namely $x_1 = 1, x_{2 \leq i \leq 5} = 0$.

   So this is looking good.

10. • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 7th 2016
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexProbably not helpful, but I remember playing with something like 'relative fixed points' [earlier this year](https://nforum.ncatlab.org/discussion/4489/induced-representation/?Focus=57680#Comment_57680) as arising in the form of modalities induced by group homomorphisms.

    Probably not helpful, but I remember playing with something like ’relative fixed points’ earlier this year as arising in the form of modalities induced by group homomorphisms.

11. • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 7th 2016
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItex> ... the action to consider is the action given by canonically embedding $S^4$ in $\mathbb{R}^5 \simeq \mathbb{R} \oplus \mathbb{H}$ and then acting by left quaternion multiplication. This does have a single fixed point, doesn't it namely $x_1 = 1, x_{2 \leq i \leq 5} = 0$. what about $(-1,0,0,0,0)$? One could perhaps think of $S^4 = \mathbb{HP}^1$, and acting as $a\cdot[p;q] = [ap;q]$.... hmm, this seems to have two fixed points. (in fact this is probably the same action :-)

    … the action to consider is the action given by canonically embedding $S^4$ in $\mathbb{R}^5 \simeq \mathbb{R} \oplus \mathbb{H}$ and then acting by left quaternion multiplication. This does have a single fixed point, doesn’t it namely $x_1 = 1, x_{2 \leq i \leq 5} = 0$.

    what about $(-1,0,0,0,0)$? One could perhaps think of $S^4 = \mathbb{HP}^1$, and acting as $a\cdot[p;q] = [ap;q]$…. hmm, this seems to have two fixed points. (in fact this is probably the same action :-)

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2016
    • (edited Sep 7th 2016)
    • PermaLink
    Author: Urs
    Format: MarkdownItex> what about $(-1,0,0,0,0)$? Ah, sorry, of course.

    what about $(-1,0,0,0,0)$?

    Ah, sorry, of course.

13. • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 7th 2016
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexDid you want an answer to your (now deleted) question? Was the $S^0$ the two-point space, or the reduced one (aka $pt$)?

    Did you want an answer to your (now deleted) question? Was the $S^0$ the two-point space, or the reduced one (aka $pt$)?

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2016
    • (edited Sep 7th 2016)
    • PermaLink
    Author: Urs
    Format: MarkdownItexSorry, I deleted the question and the bulk of #9 because there was an obvious silliness in there. I was thinking in terms of differential forms, but in terms of non-differential cohomology my previous statements of course were empty. But let me try to fix it, and then I'll ask the (fixed) question again.

    Sorry, I deleted the question and the bulk of #9 because there was an obvious silliness in there.

    I was thinking in terms of differential forms, but in terms of non-differential cohomology my previous statements of course were empty.

    But let me try to fix it, and then I’ll ask the (fixed) question again.

15. • CommentRowNumber15.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 7th 2016
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexok :-)

    ok :-)

16. • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2016
    • (edited Sep 7th 2016)
    • PermaLink
    Author: Urs
    Format: MarkdownItexSecond attempt. So we are considering a $p$-brane carrying charge and sitting at $\mathbb{R}^{p,1}\times \{0\}$ in $\mathbb{R}^{p,1}\times \mathbb{R}^{d-p-1}$. If the $p$-brane is really sharply localized at $\{0\}$ then we have to take out that singular locus. On the complement $$ \mathbb{R}^{p,1}\times \mathbb{R}^1 \times S^{d-p-2} $$ there is then a closed $(d-p-2)$-form $\omega$, whose integral $\int_{S^{d-p-2}} \omega = Q$ is the total quarge carried by the $p$-brane, i.e. $$ \omega = \frac{Q}{vol(S^{d-p-2})} dvol_{S^{d-p-2}} $$ But now (as in #5) we are after a generalization where we do not remove the locus of the $p$-brane. The idea is that then we have to realize that the $p$-brane has some finite extension in radial direction (the factor $\mathbb{R}^1$ appearing above). For example if $p = 0$ then we are replacing a charged black hole spacetime with point singularity by the spacetime of a charged star of finite extension, without singularity. In this case there is still a $(d-p-1)$-form whose integral over any $(d-p-1)$-sphere measures the total charge inside that sphere. But now this will depend on the radius $r$ (the coordinate along the factor of $\mathbb{R}^1$ above), as $$ \omega = \left\{ \array{ \frac{Q}{vol(S^{d-p-2})}\frac{ r^{d-p-1} }{r_0^{d-p-1}} dvol_{S^{d-p-1}} & for \; r \lt r_0 \\ \frac{Q}{vol(S^{d-p-2})} dvol_{S^{d-p-1}} & for \; r \geq r_0 } \right. $$ In particular $\omega$ together with its first few $r$-derivatives vanish at the original sharp locus of the $p$-brane. To model this in terms of relative cohomology now, thicken the locus $\mathbb{R}^{p,1} \times \{0\}$ by the infinitesimal disk $\mathbb{D}^{d-p-1}$ of infinitesimal order $d-p-1$. Then a commuting diagram $$ \array{ \mathbb{R}^{p,1} \times \mathbb{D}^{d-p-1} &\longrightarrow& 0 \\ \downarrow && \downarrow \\ \mathbb{R}^{p,1} \times \mathbb{R}^{d-p-1} &\longrightarrow& \mathbf{\Omega}^{d-p-2} } $$ expresses the fact that we are dealing with a kind of differential cocycles on a contractible space, but subject to the constraint that they vanish in a formal neighbourhood of the origin. This is an incomplete picture, we need to replace the plain 1-sheaf of $(d-p-2)$-forms here with some higher homotopy type containing the relevant gauge transformations and classifying some differential cohomology theory, such as maybe one of the models for differential rational degree-4 cohomtopy that we have built. Then it _ought_ to be true (I was thinking) that the above relative cohomology -- of differential cocycles on $\mathbb{R}^{d-p-1}$ subject to the constraint that in a formal neighbourhhod of the origin everything has to vanish -- is equivalently given by some cocycles on the complement of that formal neighbourhood, hence on something that looks like an $S^{d-p-2}$. But let me leave that open for the moment. My point that I had tried to express in #9 had been that this is beginning to look suggestive of Bredon cohomology. Namely if we now assume, as we want to, that for $p = 2$ and $d = 11$ the locus $\mathbb{R}^{2,1} \times \{0\}$ is the fixed point locus of a $G_{ADE}$-action on $\mathbb{R}^{p,1} \times \mathbb{H}\times \mathbb{H}$, and if we regard that space with that action as a presheaf on the orbit category $Orb_G$ given by assigning _infinitesimally thickened_ fixed point loci, then the above diagram for relative cohomology looks like part of the diagram that exhibits a morphisms of presheaves over the orbit category $$ \array{ G/G & && \mathbb{R}^{2,1} \times \mathbb{D}^{8} &\longrightarrow& 0 \\ \uparrow &&& \downarrow && \downarrow \\ G/1 & && \mathbb{R}^{2,1} \times \mathbb{R}^{8} &\longrightarrow& \mathbf{\Omega}^{7} } $$ In summary, this is possibly an argument for further details as to how genuine (Bredon) $G_{ADE}$-equivariant cohomology theory may capture those orbifold aspects that we are expecting to see for M-brane charges.

    Second attempt.

    So we are considering a $p$-brane carrying charge and sitting at $\mathbb{R}^{p,1}\times \{0\}$ in $\mathbb{R}^{p,1}\times \mathbb{R}^{d-p-1}$.

    If the $p$-brane is really sharply localized at $\{0\}$ then we have to take out that singular locus. On the complement

    $$\mathbb{R}^{p,1}\times \mathbb{R}^1 \times S^{d-p-2}$$

    there is then a closed $(d-p-2)$-form $\omega$, whose integral $\int_{S^{d-p-2}} \omega = Q$ is the total quarge carried by the $p$-brane, i.e.

    $$\omega = \frac{Q}{vol(S^{d-p-2})} dvol_{S^{d-p-2}}$$

    But now (as in #5) we are after a generalization where we do not remove the locus of the $p$-brane. The idea is that then we have to realize that the $p$-brane has some finite extension in radial direction (the factor $\mathbb{R}^1$ appearing above). For example if $p = 0$ then we are replacing a charged black hole spacetime with point singularity by the spacetime of a charged star of finite extension, without singularity.

    In this case there is still a $(d-p-1)$-form whose integral over any $(d-p-1)$-sphere measures the total charge inside that sphere. But now this will depend on the radius $r$ (the coordinate along the factor of $\mathbb{R}^1$ above), as

    $$\omega = \left\{ \array{ \frac{Q}{vol(S^{d-p-2})}\frac{ r^{d-p-1} }{r_0^{d-p-1}} dvol_{S^{d-p-1}} & for \; r \lt r_0 \\ \frac{Q}{vol(S^{d-p-2})} dvol_{S^{d-p-1}} & for \; r \geq r_0 } \right.$$

    In particular $\omega$ together with its first few $r$-derivatives vanish at the original sharp locus of the $p$-brane.

    To model this in terms of relative cohomology now, thicken the locus $\mathbb{R}^{p,1} \times \{0\}$ by the infinitesimal disk $\mathbb{D}^{d-p-1}$ of infinitesimal order $d-p-1$. Then a commuting diagram

    $$\array{ \mathbb{R}^{p,1} \times \mathbb{D}^{d-p-1} &\longrightarrow& 0 \\ \downarrow && \downarrow \\ \mathbb{R}^{p,1} \times \mathbb{R}^{d-p-1} &\longrightarrow& \mathbf{\Omega}^{d-p-2} }$$

    expresses the fact that we are dealing with a kind of differential cocycles on a contractible space, but subject to the constraint that they vanish in a formal neighbourhood of the origin.

    This is an incomplete picture, we need to replace the plain 1-sheaf of $(d-p-2)$-forms here with some higher homotopy type containing the relevant gauge transformations and classifying some differential cohomology theory, such as maybe one of the models for differential rational degree-4 cohomtopy that we have built.

    Then it ought to be true (I was thinking) that the above relative cohomology – of differential cocycles on $\mathbb{R}^{d-p-1}$ subject to the constraint that in a formal neighbourhhod of the origin everything has to vanish – is equivalently given by some cocycles on the complement of that formal neighbourhood, hence on something that looks like an $S^{d-p-2}$.

    But let me leave that open for the moment. My point that I had tried to express in #9 had been that this is beginning to look suggestive of Bredon cohomology.

    Namely if we now assume, as we want to, that for $p = 2$ and $d = 11$ the locus $\mathbb{R}^{2,1} \times \{0\}$ is the fixed point locus of a $G_{ADE}$-action on $\mathbb{R}^{p,1} \times \mathbb{H}\times \mathbb{H}$, and if we regard that space with that action as a presheaf on the orbit category $Orb_G$ given by assigning infinitesimally thickened fixed point loci, then the above diagram for relative cohomology looks like part of the diagram that exhibits a morphisms of presheaves over the orbit category

    $$\array{ G/G & && \mathbb{R}^{2,1} \times \mathbb{D}^{8} &\longrightarrow& 0 \\ \uparrow &&& \downarrow && \downarrow \\ G/1 & && \mathbb{R}^{2,1} \times \mathbb{R}^{8} &\longrightarrow& \mathbf{\Omega}^{7} }$$

    In summary, this is possibly an argument for further details as to how genuine (Bredon) $G_{ADE}$-equivariant cohomology theory may capture those orbifold aspects that we are expecting to see for M-brane charges.

17. • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2016
    • (edited Jul 31st 2017)
    • PermaLink
    Author: Urs
    Format: MarkdownItexCuriously, a 4-form $\omega_4$ on $\mathbb{R}^8$ of just the right form such that $\omega_4 \wedge \omega_4 = d \omega_7$ with $\omega_7 \propto r^8 \wedge d_{vol_{S^7}}$ appears in the standard way to induce (nearly parallel) $G_2$-structure on $S^7$ from the 4-form on $\mathbb{R}^8$ defining $Spin(7)$. See at _[7-sphere -- G2-structure](https://ncatlab.org/nlab/show/7-sphere#G2Structure)_. It's given by $$ \omega_4 = r^3 d r \wedge \phi + r^4 \star_{S^7} \phi $$ (with $\phi$ the 3-form on $S^7$ corresponding to the $G_2$-structure). Hence its square is proportional to $$ \omega_4 \wedge \omega_4 \propto r^7 dvol_{S^7} $$ hence this is the differential of a 7-form proportional to $$ \omega_7 \propto r^8 dvol_{S^7} \,, $$ just as it should be for the above discussion.

    Curiously, a 4-form $\omega_4$ on $\mathbb{R}^8$ of just the right form such that $\omega_4 \wedge \omega_4 = d \omega_7$ with $\omega_7 \propto r^8 \wedge d_{vol_{S^7}}$ appears in the standard way to induce (nearly parallel) $G_2$-structure on $S^7$ from the 4-form on $\mathbb{R}^8$ defining $Spin(7)$. See at 7-sphere – G2-structure.

    It’s given by

    $$\omega_4 = r^3 d r \wedge \phi + r^4 \star_{S^7} \phi$$

    (with $\phi$ the 3-form on $S^7$ corresponding to the $G_2$-structure). Hence its square is proportional to

    $$\omega_4 \wedge \omega_4 \propto r^7 dvol_{S^7}$$

    hence this is the differential of a 7-form proportional to

    $$\omega_7 \propto r^8 dvol_{S^7} \,,$$

    just as it should be for the above discussion.

18. • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2016
    • (edited Sep 8th 2016)
    • PermaLink
    Author: Urs
    Format: MarkdownItexThis seems to work out well: Notice that for any such $\omega_4$ on $\mathbb{R}^8$ to be well defined when we pass (as we do) to the orbifold $\mathbb{R}^8 // G_{ADE}$, we need it to be $G_{ADE}$-invariant. For $\gt 1/2$ BPS black M2-brane solutions then the $SU(2)$-action on $\mathbb{R}^8 \simeq \mathbb{H}^2 \simeq (\mathbb{C}^2)^2$ is via the diagonal of the canonical action (by section 3 of [MFFGME 09](https://ncatlab.org/nlab/show/ABJM+theory#MFFGME09)). Now by item (ii) on p.18 of [Lotay 12](https://ncatlab.org/nlab/show/7-sphere#Lotay12), that diagonal $SU(2)$ action factors through the $Spin(7)$-action on $\mathbb{R}^8$, with $Spin(7)$ identified as the stabilizer of $\omega_4$ in #17 above. Hence in particular this $\omega_4$ on $\mathbb{R}^8$ indeed is $G_{ADE}$-invariant, and hence well defined. It would now be interesting to see how many more 4-forms there are on $\mathbb{R}^8$ which are (not invariant under $Spin(7)$ but) invariant under this $SU(2) \subset Spin(7)$.

    This seems to work out well:

    Notice that for any such $\omega_4$ on $\mathbb{R}^8$ to be well defined when we pass (as we do) to the orbifold $\mathbb{R}^8 // G_{ADE}$, we need it to be $G_{ADE}$-invariant.

    For $\gt 1/2$ BPS black M2-brane solutions then the $SU(2)$-action on $\mathbb{R}^8 \simeq \mathbb{H}^2 \simeq (\mathbb{C}^2)^2$ is via the diagonal of the canonical action (by section 3 of MFFGME 09).

    Now by item (ii) on p.18 of Lotay 12, that diagonal $SU(2)$ action factors through the $Spin(7)$-action on $\mathbb{R}^8$, with $Spin(7)$ identified as the stabilizer of $\omega_4$ in #17 above.

    Hence in particular this $\omega_4$ on $\mathbb{R}^8$ indeed is $G_{ADE}$-invariant, and hence well defined.

    It would now be interesting to see how many more 4-forms there are on $\mathbb{R}^8$ which are (not invariant under $Spin(7)$ but) invariant under this $SU(2) \subset Spin(7)$.

19. • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2016
    • (edited Jul 31st 2017)
    • PermaLink
    Author: Urs
    Format: MarkdownItexFor ease of reading, I have typed up these last thoughts more comprehensively into a pdf, [here](https://ncatlab.org/schreiber/files/ADEInterior.pdf).

    For ease of reading, I have typed up these last thoughts more comprehensively into a pdf, here.

20. • CommentRowNumber20.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 31st 2017
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexIf you still want this pdf to be available, note that it currently isn't because of Dropbox's new policy on Public folders.

    If you still want this pdf to be available, note that it currently isn’t because of Dropbox’s new policy on Public folders.

21. • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeJul 31st 2017
    • (edited Jul 31st 2017)
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks for the alert! I have moved the file to a stable place here: [pdf](https://ncatlab.org/schreiber/files/ADEInterior.pdf). In fact I had entirely forgotten about this. Re-reading it now, it strikes me that my former self had had a good idea there. :-) Now that I have a little bit of time again, I should come back to this. Meanwhile John Huerta has made much progress with understanding AD-type fixed point BPS super-sub-algebas of the 11d super Minkowski super Lie algebra. These two stories should be two facets of a unified phenomenon. I'll try to get back to this. Thanks again for recalling my attention to this.

    Thanks for the alert! I have moved the file to a stable place here: pdf.

    In fact I had entirely forgotten about this. Re-reading it now, it strikes me that my former self had had a good idea there. :-)

    Now that I have a little bit of time again, I should come back to this. Meanwhile John Huerta has made much progress with understanding AD-type fixed point BPS super-sub-algebas of the 11d super Minkowski super Lie algebra. These two stories should be two facets of a unified phenomenon. I’ll try to get back to this.

    Thanks again for recalling my attention to this.

22. • CommentRowNumber22.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 31st 2017
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItex>my former self Not even a year ago! >John Huerta has made much progress... Did he also follow up on the idea of basing the bouquet on $\mathbb{R}^{0|3}$?

    my former self

    Not even a year ago!

    John Huerta has made much progress…

    Did he also follow up on the idea of basing the bouquet on $\mathbb{R}^{0|3}$?

23. • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeJul 31st 2017
    • (edited Jul 31st 2017)
    • PermaLink
    Author: Urs
    Format: MarkdownItex>> John Huerta has made much progress... > Did he also follow up on the idea of basing the bouquet on $\mathbb{R}^{0|3}$? No, we didn't further talk about this. What he worked out is AD-type actions on $\mathbb{R}^{10,1\vert \mathbf{32}} $ and how their fixed point sub-algebras are just those super-sub-algebras whose fermionic space has dimension 16. Unsurprisingly but very satisfactorily, he finds precisely the subalgebras for the black M-branes, the "M-wave", the black M2, the M6 (aka KK6), the M5 and also an M9. I had suggested that this ought to be the case from what I gathered ought to be true by reading between the lines of [Sorokin 99](https://ncatlab.org/nlab/show/Green-Schwarz+action+functional#Sorokin99), but it seems interesting because this is _not_ the way that these black BPS states are usually derived. Usually one solves the non-linear equations of motion of 11d supergravity subject to the BPS condition. In the course of this the computations that John did do appear as an intermediate step, slightly hidden, but John's computations show that this tangent-space wise BPS condition already fixes the full situation globally. I feel this must have some formulation in terms of something like "Cartan geometry with singularities" where we pre-specify not just one Klein geometry, but also a subgroup of its automorphism group and then ask for spacetimes locally modeled on the full Klein geometry with local singularities governed by fixed points of that subgroup. Maybe now I find some time to think about this. But now there is this backlog of referee reports to be written and the next lecture to be prepared, and that after that, and Vincent Schlegel's thesis to be finalized. We'll have to see...

    John Huerta has made much progress…

    Did he also follow up on the idea of basing the bouquet on $\mathbb{R}^{0|3}$?

    No, we didn’t further talk about this. What he worked out is AD-type actions on $\mathbb{R}^{10,1\vert \mathbf{32}}$ and how their fixed point sub-algebras are just those super-sub-algebras whose fermionic space has dimension 16. Unsurprisingly but very satisfactorily, he finds precisely the subalgebras for the black M-branes, the “M-wave”, the black M2, the M6 (aka KK6), the M5 and also an M9.

    I had suggested that this ought to be the case from what I gathered ought to be true by reading between the lines of Sorokin 99, but it seems interesting because this is not the way that these black BPS states are usually derived. Usually one solves the non-linear equations of motion of 11d supergravity subject to the BPS condition. In the course of this the computations that John did do appear as an intermediate step, slightly hidden, but John’s computations show that this tangent-space wise BPS condition already fixes the full situation globally. I feel this must have some formulation in terms of something like “Cartan geometry with singularities” where we pre-specify not just one Klein geometry, but also a subgroup of its automorphism group and then ask for spacetimes locally modeled on the full Klein geometry with local singularities governed by fixed points of that subgroup.

    Maybe now I find some time to think about this. But now there is this backlog of referee reports to be written and the next lecture to be prepared, and that after that, and Vincent Schlegel’s thesis to be finalized. We’ll have to see…

24. • CommentRowNumber24.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 1st 2017
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItex>“Cartan geometry with singularities” where we pre-specify not just one Klein geometry, but also a subgroup of its automorphism group and then ask for spacetimes locally modeled on the full Klein geometry with local singularities governed by fixed points of that subgroup. So choose a [[Klein geometry]], $G/H$ and then a subgroup of $Aut(G/H)$? The choice of $H$ as such a subgroup yields just a point of $G/H$ as fixed point, so presumably choosing other groups acting on $G/H$ would yield all singularities. This ought to have a nice [general abstract](https://ncatlab.org/nlab/show/stabilizer+group#GeneralAbstract) formulation.

    “Cartan geometry with singularities” where we pre-specify not just one Klein geometry, but also a subgroup of its automorphism group and then ask for spacetimes locally modeled on the full Klein geometry with local singularities governed by fixed points of that subgroup.

    So choose a Klein geometry, $G/H$ and then a subgroup of $Aut(G/H)$? The choice of $H$ as such a subgroup yields just a point of $G/H$ as fixed point, so presumably choosing other groups acting on $G/H$ would yield all singularities.

    This ought to have a nice general abstract formulation.

25. • CommentRowNumber25.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 1st 2017
    • (edited Sep 25th 2017)
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexFrom that pdf, > the inclusion $Spin(7) \hookrightarrow GL(8)$ is defined as the stabilizer of a 4-form... Perhaps that sort of thing would make a good example of a [stabilizer of a coshape](https://ncatlab.org/nlab/show/stabilizer+group#StabilizersOfCoShapes), which is sparse at the moment.

    From that pdf,

    the inclusion $Spin(7) \hookrightarrow GL(8)$ is defined as the stabilizer of a 4-form…

    Perhaps that sort of thing would make a good example of a stabilizer of a coshape, which is sparse at the moment.