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$\mathcal{I}$/$c$-Extremization in M/F-Duality
by Marieke van Beest, Sebastjan Cizel, Sakura Schäfer-Nameki, James Sparks
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Submission summary
| Authors (as registered SciPost users): | Sebastjan Cizel |
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| Preprint Link: | https://arxiv.org/abs/2004.04020v2 (pdf) |
| Date submitted: | 2020-04-28 02:00 |
| Submitted by: | Cizel, Sebastjan |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We study the holographic dual to $c$-extremization for 2d $(0,2)$ superconformal field theories (SCFTs) that have an AdS$_3$ dual realized in Type IIB with varying axio-dilaton, i.e. F-theory. M/F-duality implies that such AdS$_3$ solutions can be mapped to AdS$_2$ solutions in M-theory, which are holographically dual to superconformal quantum mechanics (SCQM), obtained by dimensional reduction of the 2d SCFTs. We analyze the corresponding map between holographic $c$-extremization in F-theory and $\mathcal{I}$-extremization in M-theory, where in general the latter receives corrections relative to the F-theory result.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 1) on 2020-6-23 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2004.04020v2, delivered 2020-06-23, doi: 10.21468/SciPost.Report.1778
Report
This is an interesting paper where the geometric counterpart of c-extremization is compared with the geometric counterpart of I-extremization for a class of dual F-theory and M theory backgrounds. The paper contains many interesting technical results and examples that certainly deserve to be published. Before publication I would like to ask the author to address a point and clarify a confusion I have.
The authors start with an AdS3 background in F-theory, T-dualize along a circle inside AdS3 and obtain an AdS2 background in M theory. I expect this background to be the horizon geometry of black hole. And indeed I-extremization has been introduced in this context. In particular, some of the examples in section 5, after reduction on the Sasaki-Einstein manifold, can be probably thought of as black strings in 5d with horizon AdS3 times a Riemann surface.
Now in this context there is a standard way of going from black strings (AdS3) to a black hole (adS2): compactify on a circle in AdS3 and add a momentum n along it. The entropy of the black hole is then related to the CFT central charge by the Cardy formula: S=Sqrt{n c}, which is obtained as the
Legendre transform of the logarithm of the partition function of the CFT:
S= log Z + beta n = c/beta + n beta -> n=c/beta^2 and S ~ Sqrt(n c) (*)
where beta ~ tau is the modular parameter of the torus.
How this construction is related with what the authors do? In particular, the author have an integer N0 on the AdS2 side with no counterpart in F theory, which is a bit strange. Can N0 be related to a construction similar to the above mentioned one? In particular, I notice analogies with (*) in the right-hand side of (1.1) and in (4.9) if we identify Delta phi=1/beta.
Author: Sebastjan Cizel on 2020-07-01 [id 868]
(in reply to Report 1 on 2020-06-23)We thank the referee for their report on our paper.
The referee makes an interesting observation, that perhaps there is an alternative interpretation of the flux parameter $N_0$ in terms of a momentum along a circle compactification of a black string. Although the expressions in (1.1) and (4.9) might suggest this, we are not convinced that this is a general interpretation of the parameter $N_0$. First of all, our analysis is not only restricted to black string solutions but, more importantly, as we explain in the paper, the parameter $N_0$ is a flux on the M-theory side, which to our knowledge does not (in general) have an interpretation as a momentum.
The referee seems to indicate that we are lacking an interpretation of the parameter $N_0$ on the F-theory side. We should emphasize that this is not the case: As explained around (4.9), and more explicitly in later examples, the flux quantization of $N_0$ fixes the period $\Delta \phi$ of the circle in F-theory. In this sense this is a complete dictionary, where there is no remaining mystery in terms of the supergravity analysis. We hope this clears up this confusion.
Anonymous on 2020-07-05 [id 873]
(in reply to Sebastjan Cizel on 2020-07-01 [id 868])The interpretation as a momentum would be on the F-theory side (essentially AdS3 replaced by BTZ) not on the M-theory one, but I understand that the black strings are only a subclass of solutions and I'm (partially) satisfied by the comment. I'm satisfied by the paper, in my opinion it can be published. I don't need to see it again.