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created stub for 6d (2,0)-supersymmetric QFT, mainly to record the references for the moment
There was a very accessible introductory lecture on this by Frenkel – What Do Fermat’s Last Theorem and Electro-magnetic Duality Have in Common?, which set me wondering in a way I’d never be able to answer myself as to whether there’s something in the number theoretic case which acts like the 6d QFT, i.e., two ways of ’compactifying’, or some equivalent, something which account for the arithmetic duality.
Yes, there must be some translation like this. I know almost nothing about the purely number-theoretic aspect of the story. Hopefully one day I’ll learn about it.
Briefly added statment and references for the holographic dual to the 6d (2,0)-superconformal QFT
Are there compactifications on the dual which correspond to the compactifications of the 6d (2,0)-superconformal QFT responsible for S-duality?
I am not sure yet. I had not been well aware even of the dual until it was pointed out to me a few minutes ago here (link won’t work until in a few days, probably)
I am being a bit silly here: of course this example is among those of the original article: section 3.1 in
The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2:231, 1998, hep-th/9711200;
Added pointer to Moore’s recent lecture to 6d (2,0)-supersymmetric QFT
I see Moore writes
This has led to many efforts to define a theory of nonabelian gerbes and nonabelian 7D Chern-Simons theory. At present there is no generally accepted version of what such a theory should be. For some examples of attempts see [91, 92, 59]. Other authors, notably Witten, [CITE GRAEME BIRTHDAY VOLUME] have expressed strong doubts that a sensible classical theory of nonabelian gerbes can be defined,
where 59 is one of your papers. Why would Witten suggest that?
I have already emailed Greg Moore about this. Something got mixed up here and I hope he will correct this.
On the last two pages of
which is the writeup of a talk in 2002 at Graeme Segal’s birthday conference, Witten talks about the 6d worldvolume theory of the M5-brane as a “nonabelian gerbe theory”. He concludes by making two points, which are entirely standard (by now, maybe):
The 6d worldvolume theory does contain a “nonabelian 2-form field” of sorts.
The 6d worldvolume theory fails to have a traditional “classical” Lagrangian description. It only exists as a quantum theory.
The second but last sentence of the article concludes:
Combining the two facts, this six-dimensional theory is a sort of quantum nonabelian gerbe theory. I doubt very much that this structure is accessible in the world of classical geometry; it belongs to the realm of quantum field theory.
This doubt is not only not in contradiction to what we say, but we are entirely following Witten’s own observations for how to deal with this: namely to invoke the holographic principle and instead describe “classically” (by an action functional) the holographically dual 7-dimensional quantum field theory. Witten famously did that for a single M5 brane (hence an abelian 2-form theory in 6d) in
by showing that it is holographically dual to 7d abelian Chern-Simons theory. Then in
he provides arguments that that abelian 7d Chern-Simons theory is precisely the compactification of the Chern-Simons term in 11-dimensional supergravity.
This was our starting point. Because it is also well known that this abelian 11-dimensional Chern-Simons terms receives nonabelian quantum corrections from an 11-dimensional Green-Schwarz mechanism
In view of this we did a re-analysis of the proposal in
of what the precise nature of the fields entering that 11-dimensional Chern-Simons term actually is. In
we showed that when dropping an artifical truncation to 1-stacks, that field, the supergravity C-field, is a twisted differential string structure (and its 1-truncation reproduces the above proposal) .
Based on this (but the appearance order of the two articles was reverse, which may be what is ill affecting their reception here) we argued in
that the proper and quantum-corrected 7-dimensional Chern-Simons term – the non-abelian refinement of the Chern-Simons term that Witten himself uses to describe, holographically, the abelian M5 brane theory – is a 7-dimensional nonabelian Chern-Simons action functional on String 2-connections.
This is all on the holographic dual side, where it is known and expected that “classical” Lagrangians do exist. It is a refinement of Witten’s very suggestion for how to deal with the intrinsic quantum nature of the M5 brane worldvolume theory.
We did not and do not claim nor think that the nonabelian 2-form on the M5-brane is given by a classical action functional on nonabelian 2-form connections. Instead we follow the standard lore that the quantum theory of this 2-form is to be defined holographically by a 7-dimensional theory. We use Witten’s argument above that in the simple abelian case this 7d dual is entirely controled by a Chern-Simons functional originating in 11-dimensional supergravity, and then we include known corrections in this old argument and present a refined argument.
So the above “doubt” does not apply to our argument, in fact we follow precisely Witten’s idea for how to deal with the situation in the justified presence of this doubt.
Of course, since this is physics, what we formally prove is only that a certain extended 7-dimensional Chern-Simons action functional on String 2-form connections exists and has a bunch of properties that fit all the consistency checks that it should satisfy based on physics arguments. There is no way, in the present state of affairs in the field, to rigorously prove that this 7d action functional is indeed the holographic dual to the intrinsically quantum nonabelian 2-form theory on the 5-brane. But, as our argument proceeds along the very lines of established predecessors, only refining where necessary, I think this argument is at least as robust as these generally accepted predecessor arguments are.
I have tried to indicate that to Greg Moore in an email yesterday.
Any reply from Moore?
Yes, he did reply, 5 days ago, saying that he will make changes as soon as his two weeks of “many lectures” is over.
I’ll let you know when I hear more.
Added quick pointer to the realization via geometric engineering.
added pointer to Monnier 17
Oh, maybe they shouldn’t be here but at some $(1,0)$ page.
Changed the page name to something more systematic, to align with other entries.
Added a bunch of references on the KK-compactification of the D=6 N=(1,0) SCFT to D=4 N=1 super Yang-Mills:
Ibrahima Bah, Christopher Beem, Nikolay Bobev, Brian Wecht, Four-Dimensional SCFTs from M5-Branes (arXiv:1203.0303)
Shlomo S. Razamat, Cumrun Vafa, Gabi Zafrir, $4d$ $\mathcal{N} = 1$ from $6d (1,0)$, J. High Energ. Phys. (2017) 2017: 64 (arXiv:1610.09178)
Ibrahima Bah, Amihay Hanany, Kazunobu Maruyoshi, Shlomo S. Razamat, Yuji Tachikawa, Gabi Zafrir, $4d$ $\mathcal{N}=1$ from $6d$ $\mathcal{N}=(1,0)$ on a torus with fluxes (arXiv:1702.04740)
Hee-Cheol Kim, Shlomo S. Razamat, Cumrun Vafa, Gabi Zafrir, E-String Theory on Riemann Surfaces, Fortsch. Phys. (arXiv:1709.02496)
Hee-Cheol Kim, Shlomo S. Razamat, Cumrun Vafa, Gabi Zafrir, D-type Conformal Matter and SU/USp Quivers, JHEP06(2018)058 (arXiv:1802.00620)
Hee-Cheol Kim, Shlomo S. Razamat, Cumrun Vafa, Gabi Zafrir, Compactifications of ADE conformal matter on a torus, JHEP09(2018)110 (arXiv:1806.07620)
Shlomo S. Razamat, Gabi Zafrir, Compactification of 6d minimal SCFTs on Riemann surfaces, Phys. Rev. D 98, 066006 (2018) (arXiv:1806.09196)
Jin Chen, Babak Haghighat, Shuwei Liu, Marcus Sperling, 4d N=1 from 6d D-type N=(1,0) (arXiv:1907.00536)
added pointer to this approach to construction of candidate Lagrangian densities for D=6 N=(2,0) SCFTs:
I have expanded (here) the list of references on the suggestion to regard the $D=6$, $\mathcal{N}=(2,0)$-SCFT as an (extended) functorial QFT.
I think I’ll give this its own little bare entry now, so that we can !include
this elsewhere, such as at extended TQFT.
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