The giant graviton expansion in $AdS_5 \times S^5$

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.

1. Definition of the bulk functional integral We use the usual Feynman rules for traces as in any quantum field theory, where the thermal-like trace with an operator insertion is equal to the Euclidean functional integral with a time-circle, and the integrand is the action of the theory and the same operator insertion. For the Witten index, the Hamiltonian is taken to be {$Q, \overline{Q}$ } and the result is independent of the parameter multiplying it. To calculate the functional integral, we need to identify the space of fields, the operator, and the action. All of these are done by the standard rules of AdS/CFT.

-- 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.