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The giant graviton expansion in $AdS_5 \times S^5$
by Giorgos Eleftheriou, Sameer Murthy, Martí Rosselló
Submission summary
| Authors (as registered SciPost users): | Sameer Murthy |
| Submission information | |
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| 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 | |
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| Academic field: | Physics |
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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.
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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.
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