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The giant graviton expansion in $AdS_5 \times S^5$
by Giorgos Eleftheriou, Sameer Murthy, Martí Rosselló
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Submission summary
| Authors (as registered SciPost users): | Sameer Murthy |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2312.14921v3 (pdf) |
| Date submitted: | 2024-05-13 19:34 |
| Submitted by: | Murthy, Sameer |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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Abstract
The superconformal index of $\frac12$-BPS states of $N=4$ U(N) super Yang-Mills theory has a known infinite $q$-series expression with successive terms suppressed by $q^N$. We derive a holographic bulk interpretation of this series by evaluating the corresponding functional integral in the dual $AdS_5 \times S^5$. The integral localizes to a product of small fluctuations of the vacuum and of the collective modes of an arbitrary number of giant-gravitons wrapping an $S^3$ of maximal size inside the $S^5$. The quantum mechanics of the small fluctuations of one maximal giant is described by a supersymmetric version of the Landau problem. The quadratic fluctuation determinant reduces to a sum over the supersymmetric ground states, and precisely reproduces the first non-trivial term in the infinite series. Further, we show that the terms corresponding to multiple giants are obtained precisely by the matrix versions of the above super-quantum-mechanics.
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Author comments upon resubmission
The responses to the minor comments are contained in the List of changes.
List of changes
Major comments:
Our reply to the referee contains the responses to the referee report. Correspondingly, the introductory part of Section 3 has been expanded and Footnotes 10 and 11 have been added to clarify and emphasize some of the points.
Minor comments:
1. « On page 9, “in these coordinate" should be “in these coordinates” ».
Response: We did not find “in these coordinate” on Page 9. The phrase “in these coordinates” does appear just above Eqn 2.23, and it seems to be correct.
2. « In several occasions, e.g. eq.(2.41), (3.25), L is introduced as a momentum operator, but it may cause a confusion with the radius of AdS5 for which it was firstly used. »
Response: We have changed the angular momentum operator to \widehat{L}.
3. « While μ is used for mass of particle in eq.(2.31), it again appears as moduli in eq.(3.1) as the moduli of m branes. If they are not related with each other, it would be better to use a different notation. »
Response: We have changed the notation of moduli to μ_m (since they depend on m anyway). We have also cleaned up the text below Equation 3.1.
4. « On page 13, it is mentioned “this is the 2d chiral N = 4 supersymmetry algebra”. Why is the supersymmetry algebra associated with 2d spacetime? The effective Lagrangian is 1d quantum mechanics (rather than 2d field theory) whose fields only depend on time coordinate. On the other hand, if it is N = (4, 4) discussed below, it will be non-chiral supersymmetry rather than chiral supersymmetry. »
Response: This was simply an observation. Indeed, as we had already remarked just below that sentence: “However, it is misleading to try to identify the brane theory as having a 2d (4, 4) algebra, especially because of the existence of the central extension H − R, which is the same in both lines.” We have modified the text accordingly by explicitly inserting the word “observation" in order to avoid a confusion along the above lines.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2024-6-17 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2312.14921v3, delivered 2024-06-17, doi: 10.21468/SciPost.Report.9262
Report
This paper computes the expansion of the index of the 1/2 BPS operators in $\mathcal{N}=4$ SYM , known as the Giant Graviton expansion, from the bulk point of view.
The main object is the Euclidean functional integral over the configuration space of 1/2 BPS giants and dual giants. The integral localizes to a sum over fluctuations around arbitrary number of maximal giant gravitons. An interesting observation is that the excitations of one maximal giant are governed by the supersymmetric version of the Landau problem. The authors compute the index of these excitations which recovers the first term in the expansion $Z_N/Z_\infty$ and extend this to multiple giants.
The paper is an interesting contribution since it illustrates how each term (including the signs) in the expansion of the 1/2 BPS index directly arises from giant gravitons in the bulk and opens potential to computing the expansion on the bulk side in other examples. Therefore, I am happy to recommend this paper for publication in SciPost.
There are a few small typos:
- footnote 5: construction "of the" Hilbert space
- above (2.36): "take" the form
- appendix C title: "a" giant graviton
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Report #1 by Anonymous (Referee 1) on 2024-6-9 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2312.14921v3, delivered 2024-06-09, doi: 10.21468/SciPost.Report.9209
Report
The authors improved the manuscript. However, there is the unclear point which I am still confused with.
If the method which authors propose is correct, one should be able to derive the indices of the half-BPS M-brane giants. As the authors discuss in footnote 9, the Lagrangian of the half-BPS M2-brane giant and that of the half-BPS M5-brane giant take the same form as that of the half-BPS D3-brane giant and the only different thing is the prefactor of the $\rho^2\dot{\varphi}$. So it is mentioned that the same procedure can be performed in a unified way. Then their method will lead to the essentially same indices of the half-BPS M-brane giants. However, this does not seem to be consistent since the indices of the half-BPS M-brane giants will be significantly different from the index of the half-BPS D3-brane giant. As the half-BPS D3-brane giant index was already known in the literature, the potential significance of this work will be to provide a new pathway to giant graviton indices in other setup. Hence I would like the authors to explain how their method can lead to the correct indices of the half-BPS M-brane giants in a transparent manner. If their method does not work for the M-brane giants, their method or interpretation may contain a fatal flaw.
Recommendation
Ask for major revision
Author: Sameer Murthy on 2024-07-15 [id 4619]
(in reply to Report 1 on 2024-06-09)
This is a very useful comment for which we thank the referee. Indeed, Footnote 9 in the paper says that the small fluctuations of the half-BPS branes in the maximally supersymmetric cases (AdS5 x S5, AdS4 x S7, and AdS7 x S4) are all essentially the same, up to the coefficient of the term corresponding to the B-field that the fluctuations couple to. The referee asks about the interpretation in the dual field theory.
In fact, the result agrees with the half-BPS indices for the M2 and M5 brane theories found in the literature. The half-BPS index is shown to coincide with the one for N=4 SYM for ABJM theory in https://arxiv.org/pdf/0904.4605 (see Section 2.2 and, in particular, Equation (2.25) and the text surrounding it) and for the M5-brane theory in https://arxiv.org/pdf/1210.0853 (see Equation (4.4)).
We are happy to include this in the paper with a bit more detail (appropriately citing the referee for the question).
Author: Sameer Murthy on 2024-07-31 [id 4669]
(in reply to Report 2 on 2024-06-17)We thank the referee for the report. We will make the small corrections that have been listed.
-- The authors.