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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 |
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| Preprint Link: | https://arxiv.org/abs/2312.14921v2 (pdf) |
| Date submitted: | 2024-02-20 16:55 |
| Submitted by: | Murthy, Sameer |
| Submitted to: | SciPost Physics |
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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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Report
The authors study the index of the half-BPS giant gravitons. While it has been already given in the literature, they propose an alternative derivation from the bulk string theory. They claim that as the effective Lagrangian of the giant gravitons obtained from the D3-brane action can be viewed as the 1d supersymmetric quantum mechanical Lagrangian for the Landau problem describing a particles in a two-dimensional plane with a constant magnetic flux, the index can be evaluated as its Witten index which has contributions from the lowest Landau level. Though the result of the index is already known, their approach would be potentially interesting and useful for experts working on the holographic duality in the context of string theory and supersymmetric gauge theory. However, there are several parts with which I am confused in the draft and I would like the authors to carefully improve.
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Sameer Murthy on 2024-04-23 [id 4442]
We thank the referee for all the useful comments. Our responses are written below, and we hope this removes the confusions that are mentioned in the report. -- The authors.
Main comments 1. In Bullet 1 the referee states that the various quantities in the functional integral (3.1) are unclear. 2. In Bullets 2 and 1, the referee asks how the calculation of the above functional integral is done in practice, and questions the use of the Hamiltonian formalism. 3. In Bullets 3 and 4, the referee questions our treatment of fermions and fermion number.
-- The space of $\frac12$-BPS configurations in the bulk string theory on AdS$_5 \times S^5$ has been studied in the references quoted in the paper. When the bulk gravitational coupling is weak, these configurations are identified with $\frac12$-BPS graviton fluctuations around empty AdS5, fluctuations of $\frac12$-BPS D3-branes in AdS$_5 \times S^5$, and multiple combinations of them.
-- The functional integral (3.1) consists of a sum over an arbitrary number $m=0,1,2,...$ of $\frac12$-BPS D3-branes, with an integral over their moduli spaces, and with a further integral over $\frac12$-BPS field fluctuations around any given point in moduli space. For $m$ branes one has a combination of $m$ such objects with the usual identifications of identical particles in the quantum theory.
-- The charge $R$ is the $R$-charge used in the index (1.1), and acts on the above space of $\frac12$-BPS brane configurations and their fluctuations in the bulk string theory. This is the usual identification of R-charge between the bulk and the boundary in AdS/CFT, which identifies it with a certain rotation of the $S^5$. The parameter $q$ is an external parameter with $|q|<1$, which can be regarded as a background value for the gauge field in AdS$_5 \times S^5$ that couples to $R$.
-- We agree that, a priori, the non-perturbative definition and convergence of the functional integral (3.1) in the string theory is not completely clear. The point of the paper is that we can calculate the integral at weak gravitational coupling using localization, where the rules are well-defined. The agreement of the answer with the boundary result, as expected from AdS/CFT, indicates that we are doing the correct thing.
2a. Calculation of the bulk functional integral -- For one brane, the moduli space is the possible configurations of $\frac12$-BPS giants or dual giants. The space of giants is labelled by the size of $S^3 \subset S^5$, or equivalently, the angle on $S^2$. The space of dual giants is labelled by the size of $S^3 \subset AdS_5$. This has been well-studied in the references, e.g. Ref.[2]. The Euclidean functional integral localizes onto the space of maximal giants.
-- Following the statement of localization, we now have to calculate the action and determinant of quadratic fluctuations around the space of $m$ maximal giants and sum them up. The fluctuations include the fluctuations of the gravitational fields, as well as the collective coordinates of the brane or, equivalently, the open-string fields. In general, the fluctuation analysis is non-trivial in general where the supergravity fields are coupled to the fluctuations of the branes. However, the nice thing about localization is that it allows us to analyze this in a weakly-coupled limit in which the gravitons and the collective coordinates can be separately diagonalized.
2b. Hamiltonian vs functional integral -- Finally, we need to calculation of the determinant of the quadratic operator. For the reason explained above, this factorizes into the determinant of the supergravity fields and the determinant of the collective coordinates. Each of these two determinants is a separately well-defined quantum problem (without gravity!). Therefore, we can calculate these determinants as a functional integral or as a Hamiltonian problem on a well-defined Hilbert space of fluctuations. We have chosen to perform the Hamiltonian version since it is easier.
-- The first problem of integrating over the fields of supergravity is equivalent to quantizing a gas of gravitons. The corresponding index is the multi-graviton index. This is well-known and we simply quote the result in the paper. The second problem is the supersymmetric Landau problem which we discuss in detail in the paper.
-- We agree that it is an interesting problem to perform the same determinants (and reproduce our answer consistent with AdS/CFT!) by functional integral methods---either by explicitly calculating the eigenmodes, or perhaps by (again) using localization. This would be a nice follow-up to our paper, but not a logical necessity to complete the calculation.
3a. Treatment of fermionic action The referee's point here is a good one. We agree that a fully first-principles derivation of the action should begin by analyzing the $\kappa$-symmetry fixed D3-brane action in the curved space (or perhaps, depending on one's point of view, from string field theory). Here we do something more modest. We analyze the bosonic problem in curved space, analyze it in the Lagrangian and the Hamiltonian formalisms, and then supersymmetrize it minimally according to the symmetries of the problem. We believe that, at the level of quadratic fluctuations that we need in the paper, the supersymmetric terms should be universal. This is a fairly standard approach to constructing supersymmetric actions from bottom-up. In our opinion, the $\kappa$-symmetry analysis that the referee wishes to do is at the level of a separate paper (which, of course, would be interesting!).
3b. Treatment of fermion number -- By the AdS/CFT correspondence, the fermion number in the bulk follows from the fermion number in the boundary. Once we have constructed a brane theory, the fermion number in the free limit is naturally defined. In our context, it should simply be the fermion number of the theory of fluctuations of the branes. Since we have an essentially free theory at quadratic order, with extended supersymmetry, we actually have a fermion number current from which follows the $\mathbb{Z}_2$ symmetry.
-- One nice point about the whole above discussion is that the identifications that we make following the standard rules of AdS/CFT and the known space of branes leads to the correct giant graviton formula---with a very simple identification of the negative signs, i.e. as the bulk fermion number. In particular, we do not need any auxiliary construct to explain it.