# One point functions for black hole microstates

**Part of the following topical collections:**

## Abstract

We compute one point functions of chiral primary operators in the D1–D5 orbifold CFT, in classes of states corresponding to microstates of two and three charge black holes. Black hole microstates describable by supergravity solutions correspond to coherent superpositions of states in the orbifold theory and we develop methods for approximating one point functions in such superpositions in the large *N* limit. We show that microstates built from long strings (large twist operators) have one point functions that are suppressed by powers of *N*. Accordingly, even when these microstates admit supergravity descriptions, the characteristic scales in these solutions are comparable to higher derivative corrections to supergravity.

## Keywords

Black holes Holography Black hole entropy## 1 Introduction

The microscopic origin of black hole entropy has been at the forefront of research ever since the discovery of Hawking radiation [1] and the formulation of the information loss paradox [2]. Twenty years ago, Strominger and Vafa showed that the entropy of a class of supersymmetric black holes in string theory could be understood microscopically by counting states in a dual conformal field theory [3]. These results, and their generalisations to other near supersymmetric black holes in string theory, were later understood to be part of the AdS/CFT correspondence discovered by Maldacena [4]. These black holes have anti-de Sitter regions in their interiors that are dual to conformal field theories.

AdS/CFT provides a microscopic explanation of the origin of black hole microstates in terms of states in the dual CFT. Holography also settles the longstanding information loss question: since the dual quantum field theory is unitary, the evolution of black holes must be unitary. However, neither of these answers is entirely satisfactory from the gravity perspective. In the field theory one can describe both individual black hole microstates and the thermal ensemble, and one can find computables that distinguish between individual states. The recovery of information in the quantum field theory is associated with the unitary evolution of pure states: the radiation emitted is not exactly thermal, but carries information about the specific state.

On the gravity side information loss is inextricably related with the causal structure of the black hole. This led Lunin, Mathur and collaborators to postulate the fuzzball proposal [5, 6, 7, 8, 9, 10, 11]: each individual black microstate should be described by a horizonless non-singular solution that differs from the black hole only at (sub)horizon scales. It is important to note that fuzzballs are not generically solutions of supergravity but rather solutions of the full quantum string theory. The fuzzball proposal directly addresses the issues of both black hole entropy and information loss: the entropy relates to the number of fuzzballs for a given black hole, while information is manifestly not lost as fuzzballs have no horizons and the radiation emitted depends on the specific microstate represented by the fuzzball.

Note that the fuzzball proposal is not the only proposal for black hole physics that postulates qualitative changes in the spacetime at or behind the black hole horizon. For example, firewalls at the horizon [12, 13] were proposed to resolve the black hole information loss paradox, while in the SYK duality it has been suggested that individual microstates are associated with shock waves behind the horizon [14].

The fuzzball proposal has been extensively explored in the context of near supersymmetric black holes, particularly in the D1–D5 system originally studied by [3]. There has been considerable work on constructing black hole microstate solutions of supergravity for this system. For 1/4 BPS black holes, namely the D1–D5 system with zero momentum, there are sufficient supergravity solutions to span all the microstates of the system [5, 15, 16, 17], although generically the solutions are actually only extrapolations to supergravity, as higher derivative corrections are non-negligible. The latter relates to the fact that the 1/4 BPS black holes do not have macroscopic event horizons; the event horizon is only manifest after taking into account higher derivative corrections to supergravity.

Nevertheless the 1/4 BPS black holes are an important arena for exploring the fuzzball proposal. The black hole microstates can be geometrically quantised and counted [18], giving rise to an entropy that matches the result from the corresponding D1–D5 CFT. The microstates have interior AdS\(_3\) regions and one can thus use the AdS\(_3\)/CFT\(_2\) duality to explore their properties. In particular, one can use the precise holographic dictionary to relate data in the asymptotically AdS\(_3\) region of the geometry to one point functions of chiral primary operators in the CFT [17, 19, 20]. This technique provides detailed matching between geometries and CFT states that goes beyond conserved charges, to complete Kaluza–Klein towers of operators.

Let \( | \Psi \rangle \) represent the microstate of interest and \(\mathcal{O}_s\) represent a specific single particle chiral primary operator of dimension \(\Delta \), dual to a supergravity mode. In general the microstate \(| \Psi \rangle \) is completely determined by the expectation values of all local operators in that state. The number of chiral primary operators (single plus multiple particle) grows exponentially as \(\sqrt{N}\); the spectral flow of these operators gives the 1/4 BPS black hole microstates, which are the Ramond ground states of the dual CFT. In a 1/4 BPS microstate only chiral primaries can acquire expectation values; operators preserving less supersymmetry or no supersymmetry cannot acquire expectation values. Thus knowledge of the expectation values of chiral primaries is enough to determine the state.

^{1}Here \(C_{\Psi }\) is dimensionless, and is the fusion coefficient of the associated three point function in the conformal vacuum.

*N*. Here \(N = N_1 N_5\) is the rank of the dual CFT, with \(N_1\) and \(N_5\) being the numbers of D1-branes and D5-branes, respectively. The interpretation of these results from the gravity side is the following. Fuzzball solutions representing long string microstates can be described only as extrapolations of supergravity solutions: the characteristic scales in the supergravity solutions are of order 1 /

*N*, and are hence comparable to higher derivative corrections to supergravity. Short string microstates are atypical but are well captured by supergravity solutions, as their characteristic scales are large compared to higher derivative corrections.

Now let us turn to the case of 1/8 BPS black holes, the D1–D5 system with momentum *P* studied in [3]. There has been a long history of constructing supergravity solutions corresponding to D1–D5-P microstates, see [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]. The precision holography dictionary has also been extended to 1/8 BPS black hole microstates in [52]. In contrast to the 1/4 BPS case, it has not been possible to carry out geometric quantisation and count explicitly the number of black hole microstates visible in supergravity.

However, one would not expect the supergravity solutions to account for a representative fraction of the black hole entropy. Just as in the case of a 1/4 BPS microstate, a 1/8 BPS microstate is characterised by the expectation values of all operators in that state. Both chiral primaries, single and multiple particle, and 1/8 BPS operators can acquire expectation values in 1/8 BPS microstates. Supergravity solutions can encode only the expectation values of the chiral primaries, not those of 1/8 BPS operators. The number of 1/8 BPS microstates with momentum *P* grows exponentially as \(\sqrt{N P}\) and one would not expect that information carried in the expectation values of chiral primaries (the number of which grows exponentially as \(\sqrt{N}\)) can suffice to capture all 1/8 BPS states.

The objective of this paper is to explore one point functions of chiral primaries in 1/4 BPS and 1/8 BPS states in the D1–D5 CFT. We work in the orbifold limit of the CFT. Since the three point functions of (single particle) chiral primaries are protected in this theory (see [53, 54, 55, 56]), the one point functions in 1/4 BPS states are not renormalised away from the orbifold point. While there are no proofs of non-renormalisation for three point functions involving two 1/8 BPS and one 1/4 BPS operators, the results of [52] suggest that there may also be non-renormalisation in this sector and our results will allow this issue to be explored further.

As explained above, our primary motivation for exploring one point functions in such states is black hole physics. In cases where supergravity representations of the black hole microstates exist, our results will allow detailed tests of their identification in terms of states in the D1–D5 CFT, using the methods of [17, 19, 20]. Perhaps more importantly our results lead to a better understanding of which classes of 1/8 BPS microstates are accessible within supergravity: microstates which do not satisfy (1.3) cannot be captured in supergravity.

Black hole microstates represented by supergravity geometries are dual to certain coherent superpositions of states in the D1–D5 CFT. In the case of 1/4 BPS black holes, there is a direct relationship between the curves defining the supergravity geometries and the corresponding superpositions of Ramond ground states in the dual CFT, as discussed and tested in [17, 19]. For 1/8 BPS microstates, analogous maps between supergravity geometries and superpositions of CFT states were explored in [52].

Here we calculate one point functions of chiral primaries in superpositions of 1/4 BPS microstates, and corresponding 1/8 BPS microstates obtained by adding momentum excitations. The latter states have a similar structure to those explored in [52]. Our results build on our recent work [57], in which general correlation functions in the orbifold CFT were derived for processes involving *n* strands being joined by a twist *n* operator.

For 1/8 BPS states we do not work out exact results for the required combinatoric problems but instead develop approximation methods that make use of the large *N* limit, together with estimates of the dominant contributions to the correlation functions. These approximation methods would be applicable to other calculations of correlation functions in orbifold CFTs, and are hence of interest beyond the black hole microstate programme.

One of our main results is the suppression of one point functions in microstates built on long strings, relative to those built on short strings. Just as in 1/4 BPS black holes, the 1/8 BPS microstates associated with long strings have one point functions that are parametrically smaller than those in short string microstates; they are suppressed by factors of *N*. As we discuss in the conclusions, these results imply that such long string microstates have at best an extrapolation to supergravity: the characteristic scales in the supergravity solutions are comparable to higher derivative corrections to supergravity.

The plan of this paper is as follows. In Sect. 2 we summarise relevant features of the D1–D5 CFT. In Sect. 3 we calculate one point functions for representative single particle chiral primary operators in the short strand limit, namely in the limit where the strand lengths are of order one and the number of strands is of order *N*. We first give an overview of the procedure and then we calculate some examples explicitly for untwisted and twisted operators. Section 4 contains one point functions for the same operators as the previous section but in the long strand case (strand length of order *N*). We separate this section into two-charge states and three-charge states, as for the two-charge case we can give exact results whereas in the three-charge case we need to make approximations. In Sect. 5 we briefly discuss the results obtained for the one point functions. After that, we move to the calculation of multi particle one point functions, by considering products of twists operators. In Sect. 6.1 we present the state with which we calculate all results and we then derive the *n*-point function corresponding to the creation of a strand by joining strands two by two. At the end of this section, in Sect. 6.5, we comment on all other possible ways of joining *n* strands. We conclude with a summary of all results in Sect. 7 and a discussion of the implications of the results in Sect. 8.

## 2 D1–D5 orbifold CFT

Consider type IIB string theory compactified on \(X \times S^1\), with *X* being \({\mathbb {T}}^4\) or *K*3. Let \(N_5\) D5-branes wrap the five compact dimensions and \(N_1\) D1-branes wrap the \(S^1\). *X* is taken to be string scale and the scale of the \(S^1\) is assumed to be much larger (so that the circle can effectively be treated as non-compact). D1–D5 black hole solutions in the supergravity limit are asymptotic to \(M^{4,1}\times S^1 \times X\). The geometry of the decoupled near horizon limit is AdS\(_3\times S^3 \times X\), and there is supersymmetry enhancement (see [58] and references therein).

The CFT dual to the decoupling region geometry is a two-dimensional superconformal field theory (SCFT). In what follows, the focus is put on the theory for \(X = {\mathbb {T}}^4\), although much of the later analysis of this paper also holds for *K*3, i.e. it does not rely on features specific to \({\mathbb {T}}^4\). For toroidal compactifications, the SCFT is an \({\mathcal {N}} = (4,4)\) superconformal sigma model with central charges \(c = {{\tilde{c}}} = 6 N_1 N_5\); this theory can be viewed as a deformation of a free orbifold CFT with target space \(({\mathbb {T}}^4)^{N_1 N_5}/S( N_1 N_5)\), where *S*(*n*) is the symmetric group. Three point functions of (single trace) chiral primaries are protected in this theory (see [53, 54, 55, 56]), and thus agree with the corresponding three point functions calculated in supergravity.

*O*within them. Thus, in general the one point functions can be written as

*SO*(4) R-symmetry in the \({\mathcal {N}} = (4,4)\) superconformal algebra. The other

*SO*(4) symmetry of the field theory is identified with the \(SO(4)_I\) of the torus. In Sect. 2.2 an explicit index description with free fields is given.

*X*. Hence, chiral primaries in the NS sector are labelled as \({\mathcal {O}}_m^{(p,q)}\), where

*m*is the twist of that chiral primary and (

*p*,

*q*) refers to its associated cohomology class. The conformal weights \((h, {\tilde{h}})\) and the R charges \((j_3, \tilde{j}_3)\) of these chiral primaries are given by

*N*copies of the CFT implicit.

*c*is the central charge of the CFT. For chiral primaries of associated twist

*m*the central charge is \(c = 6m\); the central charge of the full theory is \(c = 6N_1 N_5\). Analogous expressions hold for the right moving sector. As we just said, NS chiral primaries are mapped by spectral flow to R ground state operators,

*C*(

*X*) is determined by the cohomology. \(C = 12\) for

*K*3 and \(C = 24\) for \(\mathbb {T}^4\). However, the corresponding black holes do not have macroscopic horizons. The famous 3-charge black holes with macroscopic horizons discussed in [3] are obtained by exciting the left moving sector with momentum

*P*, and they do have macroscopic horizons. The entropy is then

*P*. As discussed in early works such as [60], most of the 3-charge microstates are associated with excitations over maximal and near maximal twist ground states (“long strings”) as there are more ways to fractionate the momentum over such states. This is our motivation to calculate one point functions of chiral primaries for short and long strings and compare their results.

Throughout this paper we will be using nomenclature and results of the theory of integer partitions. We introduce all necessary concepts as they are needed. However, we include a short and introductory section on the topic, with the basic definition and references for further reading. The reader familiar with the concept can skip the next section.

### 2.1 Integer partitions

*n*be a positive integer. A partition of

*n*is a finite non-increasing sequence of positive integers \(m_1,\ldots , m_r\) such that they add up to

*n*,

*n*is denoted by

*p*(

*n*) and is called the partition function. Let us write the partitions for the first six natural numbers as an example.

### 2.2 Free field description

In this paper we do not use explicitly the free field description of the theory for most calculations, but it is necessary to introduce part of it to define the chiral primary operators with which we work. Details of this description and also a complete classification of all chiral primaries for this theory can be found in [62]. Our notation follows closely that of [52] and [41].

Let us first define our notation and conventions. Recalling that \(SO(4) \simeq SU(2)\times SU(2)\) we write the *SO*(4) symmetry associated with the torus as \(SU(2)_{{\mathcal {C}}} \times SU(2)_{{\mathcal {A}}}\). We label the \(SU(2)_{{\mathcal {C}}}\) group with \(\dot{A} = {\dot{1}, \dot{2}}\), and the \(SU(2)_{{\mathcal {A}}}\) with \(A = {1, 2}\). The R-symmetry *SO*(4) group also splits into two *SU*(2) subgroups, corresponding to the left and right R-symmetry. We use the labels \({\alpha }, {{\dot{{\alpha }}}}\) to identify them, with \({\alpha }= \{+, -\}\) and \({\dot{{\alpha }}} = \{\dot{+}, \dot{-}\}\). To refer to the copies of the torus we use a subindex (*r*), which runs from 1 to \(N_1 N_5\). Fields and operators corresponding to the right moving sector are denoted with a tilde.

*SU*(2) generators of the left R-symmetry current, which are defined as

### 2.3 Operators in the twisted sector

Let us now focus on the chiral primary operators defined in the twisted sector. All chiral primaries from single particle states of the SCFT are listed in [62], and we refer there for the exhaustive list. We calculate one point functions for a subset of them in this paper, but the methods can be easily extended to the rest.

*J*currents. For instance, we can consider the \(\kappa \)-twist operator

*J*operators this way, and also without the twist operator. The methods for computing one point functions showed in this paper extending the work of [52] can be easily applied to these cases. We will comment on this further in Sect. 3.9. Hence, it would be interesting to have a better understanding of the gravity duals of all these one point functions.

**Chiral primaries from single particle states with**\(h - {\tilde{h}} = 0\): The four chiral primaries with \(h - {\tilde{h}} = 0\) corresponding to the (1,1) cohomology are

^{2}we see that the twisted sector generalisation of the chiral primary \(\sum _{r = 1}^{\kappa } {\mathcal {O}}^{{\alpha }{\dot{{\alpha }}}}_{(r)}\) is

**Chiral primaries from single particle states with**\(h - {\tilde{h}} = 1\): In this case we have

**Heavy and light operators** In the context of one point functions in the D1–D5 system, and more generally in holographic CFT calculations, it is common in some literature to define heavy and light operators [64, 65, 66]. These are all chiral primaries, with different conformal weights. Light operators are operators with low conformal dimension relative to the central charge *c*, and heavy operators have large conformal dimension (of order *c*). With the definitions that we have given above, if we consider the operators alone it is natural to say that heavy operators are the ones we have constructed in the twisted case, using the gluing operator \(\Sigma \) and combinations of fermions. Light operators would then correspond to single trace operators in the untwisted sector.

When looking at the one point functions that we compute in Sect. 3 we need to take into account that, in the definition of our states (which we will see in Sect. 2.5), we have some operators inside the definition of the strands. As we will see in the following sections, the calculations that we make can be used to calculate both heavy and light one point functions. Now that we have defined all the strands and operators we will briefly review how to construct 1/8-BPS strands, that is, strands where we raise the left R symmetry charge. Further details can be found in [40, 41, 42, 52, 67].

### 2.4 Creating 1/8-BPS strands

*n*is an integer. These modes allow us to increase the R charge of a state by one unit while only raising the conformal dimension by \(n/\kappa \). Then, given the R ground states that we defined in Sect. 2.2, one can add momentum excitations by acting with these fractional modes,

We now have all the definitions for the strands and operators that we need to calculate all the one point functions that we consider explicitly in this paper. However, so far we have only considered building blocks of our states. Recalling equation (2.4), we see that any state of the full theory must have strands adding up to *N*. We call states which satisfy this condition physical states, and in the next section we construct all the ones that we use in this paper, and give their norm.

### 2.5 Physical states

This section is very closely related to section 3 of [52], but we include it here for completeness. Before we start constructing states let us introduce some notation. We denote by *N* the total winding number, and \(\left| gs\right\rangle _{(r)}\) denotes any of the ground states, i.e., \(\left| \pm , \pm \right\rangle _{(r)}\) or \(\left| 00\right\rangle _{(r)}\), on the copy *r* of the CFT. When instead of writing a number in parenthesis in the subindex we write a number \(\kappa \) then it denotes a strand of length \(\kappa \). We consider several copies of each strand, in order to satisfy the condition (2.4) and be able to have generic strand lengths. We denote the number of copies of each strand by \(N_\kappa ^{(gs)}\), where \(\kappa \) is the length of the strand \(\left| gs\right\rangle _\kappa \). To get the \(\frac{1}{8}\)-BPS states we use the R-symmetry current, as explained in Sect. 2.4. We denote by \(N_{\kappa }^{m_{\kappa }(00)}\) the number of copies of the three-charge strand, where the new index stands for the number of insertions of the \(J^+\) mode.

*N*we define

*N*. We denote by \({\mathcal {N}}(\{N_\kappa ^{(gs)}\})\) the norm of this state. This norm is taken to be the number of combinations in which one can produce \(N_\kappa ^{(gs)}\) strands \(\left| gs\right\rangle _\kappa \) starting from the state

*J*operators defined in 2.2, which are already normalised. Therefore we only need to consider the creation of the twisted sectors. Starting from the state (2.45) there are \(\frac{N!}{(N - \kappa )! \kappa }\) possible ways in which we can choose \(\kappa \) of these copies up to cyclic permutations. Taking this number into account every time we construct another twisted sector, we produce the following number of terms

*g*simply denotes the last term. The normalisation of the twist operator is calculated in this way as well, as we show explicitly in the next section. If we have several strands of the same type it does not matter in what order we got them, and thus we have to divide by an extra \(N_\kappa ^{(S)}!\). So, the norm of the physical state (2.44) is

### 2.6 Normalisations

*N*up to cyclic permutations. Formally, the twist operator can act in any of this combinations because when we write the operator \(\Sigma _{\kappa }\) this is shorthand notation for

*N*total ones up to cyclic permutations. The only non-trivial action its complex conjugate can perform is to undo that joining, and so the norm of the twist operator is given by

^{3}More concretely, if \({\mathcal {O}}\) is an operator for which we want to calculate the one point function and \(\left| O\right\rangle \) is the state we are interested in, the results that we will give will be

### 2.7 Other excited states

Now that we have given a description of the free field theory we are ready to start calculating the one point functions. In Sect. 3 we study one point functions in the short strand case, and in Sect. 4 we study them in the opposite limit; in the long strand one.

## 3 One point functions: short strand case

In this section we compute one point functions for all the chiral primaries described in Sects. 2.2 and 2.3, for the two and three-charge cases. Some one point functions have been calculated in [52], where strands of length one and two were considered, and also strands of arbitrary length for the operator \(\Sigma _2^{- \dot{-}}\). This section extends the CFT calculation performed in that paper, by considering arbitrary strand length for all one point functions. In what follows, first we describe the approximation used in the short strand case, and then we go case by case calculating the one point functions for all different chiral primaries.

### 3.1 Approximation used

Now that we have all the ingredients needed we can start calculating the one point functions in this limit. Notice that for each chiral primary we can use a state which only has the strands that will come into play. With the approximation taken in this section the norm of the state will always cancel, and so having extra strands which do not play a role in the process does not affect the result. To see this clearly and to introduce the calculation and some simplified notation we start with a review example. Afterwards we will calculate one point functions in more general cases.

### 3.2 Review example: \(\Sigma _2^{+ \dot{-}}\) operator

*n*stands for the number of strands joined together. This coefficient was first calculated for \(n = 2\) in [69], and was recently generalised to arbitrary

*n*in [57].

### 3.3 \({\mathcal {O}}^{- \dot{-}}_{(r)}\) operator

*n*is an integer, multiple of

*N*. With this simplified notation we already took into account the constraint (2.4) for the lengths and number of strands. The sum over all the possible strand combinations is peaked in this case at, from Eq. (3.5),

*p*strands to act with the operator. For each strand, we are acting with

*n*terms, corresponding to the sum over copies. However, only the action on one copy is non-trivial and so there is no extra combinatorial factor. Another way to think about this is that once we have picked up the copy, the operator can act on any of the copies within the strand. Again, though, in this case this action is only non-trivial in one copy and so there is no extra factor. This will not be the case for more complicated operators. Therefore, the action of the operator on a ground state is straightforward, as the resulting combination is just the definition of a \(\left| 00\right\rangle _n\) ground state,

*c*coefficient, as we are not gluing strands. The action of the operator on the strands is thus

### 3.4 \({\mathcal {O}}^{+ \dot{-}}_{(r)}\) operator

### 3.5 \(\Sigma _n^{- \dot{-}}\) operator

*n*strands into a single one. We consider the case where we join strands of the same length, as then the formula for the result is much simpler. The generalisation to different strand lengths does not involve any extra subtleties. So, consider the state

*M*(and thus also

*n*) are small integers. The norm of this state is

*n*strands of length

*M*/

*n*, and at the end we have a single strand of length

*M*. The coefficient \(c_n\) was obtained in [57], and in the case of joining strands of the same length its expression is

### 3.6 \(\Sigma _n^{+ \dot{-}}\) operator

*M*/

*n*copies. However, there is a further subtlety in this case, as we explained in Sect. 3.2. Recalling that the twist operator acts up to cyclic permutations, we need to take the symmetrisation of the states over which it acts, as we did in that case. Notice that in the previous section we did not need to take this into account as all the strands were the same, but now we have two different kinds of strands and need to take that into account. So, the gluing process is

*M*copies in the strands is

### 3.7 \((J^+_{-1})^m\) operator

*m*-point function

*J*operator, which is

*m*-point function in the short strand case is

### 3.8 Twist sector \({\mathcal {O}}^{{\alpha }{\dot{{\alpha }}}}_n\) operators

*n*/2,

*n*/2). Therefore, we are raising the left and right R-symmetry charges by the same amount in both sides using this operator. The calculation of this one point function is completely analogous to the previous sections, taking into account that we now have a product of two operators and so in general we need to combine different results from previous sections. There is a \(c_n\) coefficient coming from the action of the twist operator on the \({\mathcal {O}}^{{\alpha }{\dot{{\alpha }}}}\) operator, which in this case turns out to be \(c_n = 1\) as there is only one choice and ordering of operators. Thus, taking into account that the norm of the state (3.65) is

*N*, as we did in the previous sections. However, to do so we need more insights on the supergravity side. More precisely, the formula for the relation comes from the mapping of the supergravity coefficients of the corresponding geometry. In the previous cases we have coherent superpositions of ground states, but in this case we are mixing supergravity operators in the CFT. The geometries associated to the state we consider have not been investigated, and so the relation between |

*C*| and

*N*needs more work to be obtained. We thus leave the final answer for this one point function as future work, and move to other cases.

### 3.9 Other chiral primaries

*n*copies of a unit length strand. That is, we can now easily calculate the process

*n*. As we increase

*n*, this answer becomes bigger close to \({\alpha }= 0\), and very close to zero as \({\alpha }\) approaches one. We present this second case in Fig. 2b.

So far, apart from this last example, we have illustrated with some examples how to obtain the CFT one point functions of chiral primary operators in the short strand case. The tools that we have shown can be used to calculate more general one point functions as well, for example for the \({\mathcal {O}}^{{\alpha }{\dot{{\alpha }}}, *}_{\kappa }\) operators defined in Eq. (2.32). We can also obtain one point functions for heavy and/or twisted *J* operators. Similarly, we can calculate one point functions for products of twist operators. The method is exactly the same, but the combinatorics will be more involved. In Sect. 6 we are interested in *n*-point functions for products of twist operators. There we explain the combinatorics involved, reviewing some integer partition theory, and use the results to give an upper bound and a lower bound for the \(\langle (\Sigma ^{- \dot{-}}_2)^n\rangle \) *n*-point function.

In the next section we focus on the same one point functions as the ones we have explicitly calculated in this section, but in the opposite limit; in the long strand case. Notice that some of the one point functions that we have calculated in this section, like the one above, are explicitly only possible in the short strand case. Generalisations of these can be considered also in the long strand case. We will not calculate all of them, as the examples provided should be enough to obtain these other cases.

## 4 One point functions: long strand case

In Sect. 3 we have calculated one point functions for chiral primaries in the short strand length case. That is, in the case where we have a large amount of copies of each strand, with the lengths being of order one. In this section we focus on the opposite limit, that is, in the limit where the strand length is large and the number of strands is small. As in the short strand section first we will explain the approximation that we use, and then we will go case by case calculating the results. As we will see, in this limit the one point functions give much smaller results than in the opposite one in general.

We consider now the case where the strand lengths are of order *N*, and the number of strands is of order one. In this section we separate the results for the two and three-charge cases, as the method used differs slightly in both. In the two-charge case we will be able to obtain exact results, whereas in the three-charge case we will strongly use the large *N* limit. We start with the 1/4-BPS states.

### 4.1 Two-charge states

#### 4.1.1 Method

*N*. Likewise, the number of strands \(N_{\kappa }^{(gs)}\) will also be a number of order one. Now, the important thing to notice is that the only

*N*-dependence of the norm of (4.1) is an

*N*! factor which comes out of the sum. To see this, we substitute in the norm the parametrisation (4.5), which gives

*N*then this will mean that, up to numbers of order one, the long strand one point functions will be suppressed by a factor of

*N*with respect to the short strand length ones. The expression of this numerator depends on the chiral primary we use, and so we will go one by one in the following sections. As we will see the

*N*dependence will also factor out in all the numerators, and so we will be able to obtain the exact result up to a polynomial which depends only on the \({\alpha }_i^{(gs)}\). We will give the polynomials for some cases as well, to see what is their behaviour for all values of the \({\alpha }_i^{(gs)}\). Let us start by calculating the numerator for the \(\Sigma ^{- \dot{-}}_n\) operator with the most general 1/4-BPS state.

#### 4.1.2 \(\Sigma _n^{- \dot{-}}\) operator

*gs*) superscript, as it is redundant in this case and will only complicate the notation. We start with the strands

*N*-dependence of the product of the two is \((N - 1)!\). Now, this is not the only part of the one point function that can give

*N*dependence. We also have the \(c_n\) and \({\alpha }\) coefficients, so we need to see what their product is. Let us start by calculating \({\alpha }\), as it is straightforward to obtain. Analogous to the previous section, we need to match the number of terms before and after, taking into account in how many ways can the gluing operator act. Therefore,

*N*behaviour, and so we give the result only for this case. The combinatorials needed for the exact result are explained in the previous section. Solving for \({\alpha }\) we obtain

*N*. Let us now consider the \(c_n\) coefficient. We have written its expression in Eq. (3.42) for strands of the same length, but let us recall its expression here in the most general case. It is

*N*. Keeping only the factors with

*N*we see that

*N*go to zero in the large

*N*limit. Let us check some simple cases first. As noted in [57] for \(n = 2,3\) we have simple expressions for the coefficient \(c_n\), and we also have a compact formula when all the \(m_i\) are equal. For \(n = 2\) the coefficient reads

*n*strands of length

*N*/

*m*into a single one of length

*Nn*/

*m*. Then the general expression (4.14) reduces in this case to Eq. (3.42), which with the current notation is

*N*factor can be read straight away and agrees with (4.17). We need to see what happens with the other factors. Let

*N*approximation we can drop some terms in the exponents and rewrite the coefficient as

*n*/

*Q*is a monotonically decreasing function, which satisfies

*n*. We show its behaviour in Fig. 3. Therefore,

*N*-independent polynomial which numerator is given by Eqs. (4.9), (4.10) and (4.11) and with the denominator given by (4.6). Notice that the one point function decreases exponentially with

*N*for \(n \ge 4\), as for \(n = 3\) we have \(n/Q = 1\). If we compare this result to the short strand case, Eq. (3.47), we see that the only

*N*dependence in the short strand case comes from the normalisation of the twist. Therefore, we have the following relations for the one point functions,

#### 4.1.3 \({\mathcal {O}}^{- \dot{-}}_{(r)}\) operator

*N*. Comparing to the result of the short strand case we see that

#### 4.1.4 Exact answers for the one point functions (examples)

We start with a couple of easy examples, and build up to more general ones. Let us start with the \(\Sigma _2^{- \dot{-}}\) operator in the easiest case possible.

**Simplest non-trivial case**Consider the state

*N*/ 4, and they each have

*N*/ 4 positions on which we can insert the gluing operator. Thus,

*f*is bounded by 2.15 for \(0 \le {\alpha }\le 1\), as we can see in Fig. 4.

**Joining strands of different length**Let us now do a similar calculation, but joining two strands of different lengths. The new feature in this example is that in this case we have two variables in the polynomial

*f*. Consider the state

*N*for both terms, just as in the previous case. As we can see, the polynomials that we obtain agree with the general Eqs. (4.9) and (4.6). Then, as we show in Fig. 5 both terms are small in the range of interest. The point \(({\alpha }, \beta ) = (0,0)\) is singular, but let us recall that point is not a physical state and thus is not under consideration. Let us now study the polynomial that results from the action of the \(\Sigma _2^{- \dot{-}}\) operator in the case where it joins any two strands of equal length.

**Joining two strands of the same length**Consider the state

*U*is a hypergeometric function. This function is of order one for most values of \(\alpha \). We present in Fig. 6 the plot of \(f({\alpha })\) for \(m = 17\). We see that we have an asymptote at \(\alpha = 0\), as it happens in the other cases. Also, setting \(m = 4\) in (4.61) we recover (4.47), as expected. Let us recall, from Eq. (4.28), that this one point functions also has a 1 /

*N*suppression with respect to the analogous short strand one.

**Joining any two strands**Consider the state

We observe again the asymptotes at the boundary values of \({\alpha }_1\) and \({\alpha }_2\). Let us recall once again that \({\alpha }_1\) and \({\alpha }_2\) are defined only in the open interval (0,1), as otherwise we would not have the strands necessary to have the process studied. Therefore, the polynomial is of order one in the range of interest, and so we confirm the behaviour described in the first line of Eq. (4.29). To finish this section let us study the case of the \({\mathcal {O}}^{- \dot{-}}_{(r)}\) operator.

**operator**For the \(\sum _{r = 1}^n {\mathcal {O}}^{- \dot{-}}_{(r)}\) operator consider the state

*m*) close to one. We can see it in Fig. 8 for \(m = 12\).

### 4.2 Three-charge states

*N*dependence of the norm is now more complicated. As in the two-charge case, we will first explain the approximation, and then we will go case by case giving the answers to the one point functions in this limit.

#### 4.2.1 Method

*N*, and \(m_\kappa \) can take any value between one and \(\kappa \). We work this out explicitly for each case in the following sections.

*N*-dependence inside of the sum in this case. In some cases we will still be able to obtain an exact result, as we will be able to perform the sums. However, in other cases we will not be able to do so. Them, since the resulting polynomial will have now powers of

*N*in the factors, what we will do is keep only the dominant term in the polynomial and compare the numerator and the denominator one. As shown in the previous cases, the denominator will always have a higher power of

*N*than the numerator, and so the

*f*function will be small. We start with a simple example to see explicitly how it works, and then we will give more general results for the rest of the operators.

#### 4.2.2 Elementary example

We start with a simple example, just as in the previous section, and then we build up to more general ones. For this initial example we consider the untwisted \({\mathcal {O}}^{+ \dot{-}}_{(r)}\) operator.

**operator**Consider the state

#### 4.2.3 Untwisted \({\mathcal {O}}^{- \dot{-}}_{(r)}\) operator

*N*behaviour except in the boundary values, as in the previous cases.

#### 4.2.4 \(\Sigma _2^{+ \dot{-}}\) operator

*n*, 1/8-BPS one point function we need an extra commutator to obtain it, which we are not giving in this paper. Thus, we calculate the twist two case, as we did in Sect. 3.2 for the short strand case. Consider the state

*N*is enough. Since the sums only have factors of order one within them, that is, since they do not have any

*N*’s, their ratio will be a number much smaller than

*N*, away from the boundaries of \({\alpha }\) and \(\beta \). Therefore, we can approximate this one point function by

#### 4.2.5 Untwisted \({\mathcal {O}}^{+ \dot{-}}_{(r)}\) operator in full generality

*p*(

*n*) terms, that is, the number of integer partitions of

*n*terms. Each term will have as many sums as elements has every partition of

*n*, and the binomial coefficient’s powers correspond to the numbers in each partition (to its parts). Take the last parenthesis in (4.111) for instance. We have four sums in the state, and so we have \(p(4) = 5\) terms in the norm. And the exponents of the binomials coefficients are given by the five possible partitions of four [see Eq. (2.14) or Fig. 16 for the partitions]. Let us also recall that the coefficients in the state satisfy

*N*yields

*N*limit, all coefficients except the first ones must be very small. Therefore, we can work with the parametrisation

*N*of these parameters in the \(\beta _i\). The running parameters labelled by the Greek letters taking values between zero and one, as usual. As in previous cases we use the commutator (3.31) to see that

*p*refers to the sum index in Eq. (4.110). Clearly

*N*limit. We need to keep in mind that the operator \({\mathcal {O}}^{+ \dot{-}}_{(r)}\) only generates states with one insertion of the \(J^+\) mode. For \(p = 3\) we have

*N*limit to approximate the second term in the parenthesis by zero. Similarly, for \(p = 2\) we find

*f*is the polynomial described above. Notice that, maybe except for boundary values of the \({\alpha }\), \(\beta _i\) variables, the polynomial

*f*is a very small number, as the denominator has terms with higher powers of

*N*than the numerator. Therefore, we see once again that the long strand one point function is parametrically smaller in

*N*than the short strand one, i.e. at most

#### 4.2.6 Untwisted \(\left( J^+_{-1}\right) ^m\) operator

*m*, even with the combinatorial factor of the equation above, as this factor will just increase the power of the corresponding factor in the norm of the state. The exact result for this one point function is thus

*N*, we see that in this case the long strand result is much bigger than the short strand one, for any value of the parameters.

## 5 Brief discussion of the results

Now that we have calculated some exemplary one point functions for short and long strands we can compare the results we obtained. We give a summary of the results and of the relations between them in Sect. 7, but let us outline the main highlights here.

*O*be a chiral primary of the theory. The main result that we have obtained so far in this paper is that

*N*with respect to the short strand ones.

## 6 Different ways of joining strands

*n*strands (of any length) together at the same time, with one step. The second one consists in joining the

*n*strands in \(n-1\) steps, namely, joining them two by two. Both cases represented in Fig. 11.

As is obvious from the figure, in the first case, Fig. 11a, the number of ways in which the joining can be done is much smaller, as it only depends where within each strand the operator is inserted. However, in the other case, Fig. 11b, there are many ways to join, as the second application of the gluing operator could have joined the pair which was joined in the previous step to one of the two strands that was alone for instance. Hence, when joining the strands two by two the combinatorics are more complicated and will give a non-trivial contribution. We present work in this section which aims to compare both cases. Normalisations are the key point for this calculation, and they are related to integer partitions. The different countings are obtained in Sect. 6.3.

### 6.1 Setting up

*n*-point functions, as both describe the same process at the end. Namely, all the calculations will be exactly the same, except for the \({\alpha }\) and \(c_n\) coefficients. Furthermore, the two results will only differ in factors proportional to

*n*and

*M*, and so all other contributions will cancel. We give more details and the results at the end of the section, after we have done all the combinatorics. In what follow we assume \(n \gg 1\) but not necessarily comparable to

*N*. The final result will be valid for any \(n \gg 1\).

*n*strands of length

*M*/

*n*and end up with a single strand of length

*M*. In both cases we need the \(\Sigma ^{- \dot{-}}\) operator, so that the final state has the correct charges. In this section we write out the normalisations of the twist operators explicitly, as they play an important role. Let us now compute the \({\alpha }\) coefficient and the

*c*coefficient for both cases, to be able to compare both results.

### 6.2 \(\langle \Sigma _n^{- \dot{-}}\rangle \) coefficients

### 6.3 Calculation of \(\langle (\Sigma _2^{- \dot{-}})^{n-1}\rangle \)

*n*/ 2 strands of length 2

*M*/

*n*, then we joined these in pairs again to have

*n*/ 4 strands of length 4

*M*/

*n*, then to have

*n*/ 8 strands of length 8

*M*/

*n*, and so on until the end, where we have only one strand of length

*M*. Clearly, this is only one way of doing the gluing process. Instead, we could have joined all new strands to the same one, creating an increasingly long strand, and leaving the rest untouched. We depict this process in Fig. 13, again for \(n = 4\).

The final result for the coefficients may depend on the way in which we join the strings. We calculate the two limit cases, which are the ones depicted in Figs. 11b and 13. In what follows we find which one gives the leading contribution, we check in how many ways the gluing can be done for general *n* and then we approximate the result for the \((n - 1)\)-point function, obtaining a lower bound and an estimate for the upper bound.

#### 6.3.1 \(c_{n2b2}\) coefficient

*c*coefficient. First let us recall what the coefficient for two strands is. As we said in the previous sections, it was first computed in [69], and in our case at hand is

*c*coefficient depends on the length of both strands that are being joined together, and so it might different for each case. Let us do first the case depicted in Fig. 13 which we denote by \(c_{2b2t}\). In this case we simply need to apply that coefficient repeatedly, with one of the strands increasing in length one by one at each step. The product gives

*n*/ 2 times, that same coefficient for strands of double length

*n*/ 4 times and so on. More concretely, we have

*c*coefficient is the same for both joining processes. So, to simplify notation from now on let us define

*c*coefficients are the same all the difference in the result will come from the comparison of the number of terms; from the \({\alpha }\) coefficient. Let us calculate it.

#### 6.3.2 Counting the number of terms

*n*-point function. We need to take into account all the possibilities for the application of each \(\Sigma _2\) operator, and also the different ways in which we can join the strings. As before, we calculate the contribution for the two limit cases, Figs. 11b and 13, and see which one is dominant. Then we calculate the total number of ways in which we can join the strands two by two in order to get the final strand of length

*M*, and then we approximate the result. As is being hinted, all this also involves some integer partition theory.

*p*strands of length

*M*/

*n*. In the first step we join two of these, so we need to pick two of them, \(\left( {\begin{array}{c}p\\ 2\end{array}}\right) \). Then, each of the two strands has

*M*/

*n*insertion points for the operator. Thus, the combinatorial factor for the first step is

*M*/

*n*and join it to the 2

*M*/

*n*strand, giving

*n*. We will now calculate the opposite case, the one shown in Fig. 11b. In this case we will need to assume

*n*to be a power of 2, to simplify the expressions. As we will comment later in Sect. 6.3.4, nothing qualitatively new happens when

*n*is not a power of 2.

*M*/

*n*and join them in another pair of length 2

*M*/

*n*. We can do this in

*M*/

*n*less, as we joined two in the previous step. Repeating this process, we see that the factor that we obtain from creating

*n*/ 2 strands of length 2

*M*/

*n*is

*n*/ 2 strands of length 2

*M*/

*n*and join them in pairs to create strands of length 4

*M*/

*n*. We can look for all the coefficients again and multiply them, or just use the equation above and change it accordingly. In any case, the result for moving our state from

*n*/ 2 2

*M*/

*n*strands to

*n*/ 4 4

*M*/

*n*strands is

*M*/ 2, and then the final step is just to join them. We have also included this last coefficient in Fig. 15. Let us recall that we are assuming

*n*to be a power of two, and so we end up using all the initial strands following this process. If we write all the contributions together we obtain

*i*is straightforward to compute, but the product in

*j*is more involved. After doing the

*i*multiplication, we separate the product above in three products, which we will calculate separately. They are

*p*as far as we are aware, as there is no formula that gives its result. We will consider the case \(p = n\), as that one can be summed. Notice that in this case the product reduces to the product of factorials of all the powers of 2 until

*n*,

^{4}Let us rewrite, for convenience, the product above as

*n*as

*n*. Then, the third factor gives

*n*and

*n*being a power of 2,

*n*limit. This means that there are many more ways to join the strands by creating pairs of equal size than accumulating them all together in one big strand, which was to be expected. However, we are not finished with the counting. We just saw that the counting is bigger when we join them in pairs, and in the previous section we saw that the

*c*coefficient is the same for both cases. However, these are the two limit cases. There are many ways in which we can join all the strands. For instance, we could create three pairs of length 2

*M*/

*n*, and then join everything together in one big strand by accumulating them, just as we did above. That is, we can have a process which is a combination of both limit processes that we just calculated.

*n*strands can be joined. The number of ways may in fact change the scaling with

*n*of the result, and so we need to calculate it.

*n*1, ..., 1 to the partition

*n*by adding numbers in pairs. For example, for four we have only two ways, as we show in Fig. 16. Finding this for arbitrary

*n*is again a very difficult problem in number theory. And again, as far as we are aware there is no exact formula for this counting. However, the scaling with

*n*is known for the case of large

*n*. Paul Erdős and collaborators found that, if

*f*(

*n*) is the number we want to find and \(c_1\) and \(c_2\) are constants, then [70]

*n*being large is consistent. We do include a figure inspired by that paper to illustrate the problem better, and show that

*f*(

*n*) grows very fast by studying the case \(n = 7\); Fig. 17. For our case at hand we will ignore the constants, as we are only concerned about the scaling with

*n*and the main contribution comes from the other factor. Therefore, we approximate it by saying that the number of ways in which we can join

*n*strands two by two to get a single strand is \(n^{\frac{n}{2}}\).

We now have all the coefficients that we need to compute the one point function, so let us put them all together.

#### 6.3.3 Result

*p*. Therefore, we will use the coefficients that we obtained for the other case. Since we know that is the smallest case, the result that we obtain is a lower bound. First let us find \({\alpha }\), which corresponds to the matching of number of terms. We calculated the number of ways in which we can apply the \(n - 1\) \(\Sigma _2\) operators in the case where we keep gluing the single length strands to the same one in (6.16). Therefore, we have

#### 6.3.4 *n* not being a power of two

We have now computed both *n*-point functions. As we saw in Sect. 6.2, the result of the one point function of \(\Sigma _n^{- \dot{-}}\), Eq. (3.47) is valid for any *n*. However, as we have just seen, (6.34) is only valid when *n* is a power of two, because we join the strands in pairs. This does not mean that we can only calculate this \(n - 1\)-point function in that case, though. If *n* is not a power of two we simply need to look for the biggest power of two smaller than *n*, do the process for that subset, and do the same for the smaller subset which is not a power of two. Clearly the result will be longer to write, but the calculation is exactly the same, and we will just end up with a product of results of the form (6.34). For tidiness we keep assuming that *n* is a power of two.

### 6.4 Comparison of results

*n*of this one point function is higher than what we use in this section. Also, notice that to compare both results we do not need to use the answers we obtained in the short strand limit, nor we need to take the long strand limit or worry about the \(f({\alpha })\) polynomial. The process described by both operators is the same, and so the calculations are exactly the same except for the \({\alpha }\) and

*c*coefficients. Since these are independent of the sum index (except for a term in the \({\alpha }\) coefficient, which is the same in both cases), all factors which are different for both calculations come out of the sums, and thus the sums cancel. Therefore, to compare both answers we only need to divide the

*c*and \({\alpha }\) factors and the normalisations of the twists. By doing so we obtain

*M*and

*n*the fraction above will either go to zero or infinity in the large

*N*limit. Namely, for large values of

*M*it will go to infinity. To see this more clearly, let us simplify the result. Let us recall that we are assuming \(n \gg 1\), however we can have the case \(1 \ll n \ll N\). Let us assume this is the case. Then, the equation above simplifies to

*M*is of order

*N*then \(\langle (\Sigma _2^{- \dot{-}})^{n - 1}\rangle \) will be bigger. If

*M*is orders of magnitude smaller than

*N*then the one point function \(\langle \Sigma _n^{- \dot{-}}\rangle \) has a bigger value. Thus, the joining of strands with multiple twist operators should also need to be considered, as the expectation values of both calculations can be of the same order of magnitude depending on the case.

As we have seen there are many contributions that play a role here, but the main difference comes from combinatorics. There are many ways in which all the \(\Sigma _2\) operators can be inserted, and that gives a very big contribution, whereas for \(\Sigma _n\) the process is much more restricted, the number of ways in which the gluing can be done is much smaller, and so the coefficients also are. Let us recall that we have done the comparison with a lower bound of \(\langle (\Sigma _2^{- \dot{-}})^{n - 1}\rangle \). By looking at Eqs. (6.21) and (6.29) we see that its upper bound will have higher powers of *n* and *M*. Therefore, the qualitative comparison done above holds in the same way for the upper bound as well.

There is an obvious extension to this result to obtain a stronger one, which we consider in the next section. Up until now we have been concerned about two ways of joining strands: all at the same time, or by pairs. However, there are many more ways to join strands. So far we have only used the \(\Sigma _2\) twist and the \(\Sigma _n\), but there are also twist operators \(\Sigma _i\), for \(2 \le i \le n\). Let us see how this is translated in terms of *n*-point functions.

### 6.5 All possible ways of joining the strands

*n*-point function when we join strands two by two, and when we join them all at the same time. However, we could have also joined them all three by three, if

*n*was a power of three. Or with any combination of gluing operators, up to \(\Sigma _n\). That is, for every

*n*we have as many possibilities for joining them as possible combinations of twist operators are there that will join them all together. Rephrasing this in terms of integer partitions, there are as many ways of joining the

*n*strands as there are partitions of

*n*that do not contain 1 as a part. This is another hard problem in number theory related to integer partitions. The sequence that results from this counting is recorded in the On-Line Encyclopedia of Integer Sequences (OEIS). It is the sequence A002865 [71]. Again, there is no exact formula for this counting, but it grows exponentially fast with

*n*. There is an approximate formula for this counting, which is

*N*. Thus, just as it happened in the previous case, Eq. (6.35), in all the intermediate cases the corresponding one point function will have a value comparable to \(\langle \Sigma _n \rangle \) depending on how big

*M*(and

*n*) are.

Calculating the *n*-point functions in the middle by the same methods we used would not be straightforward, as we would need to count all the possible ways of joining. For \(\Sigma _2\) we used the result from Erdős’ paper [70], but that counting has not been studied for any other integer as far as we are aware. To finish this section let us give an example to illustrate what the counting problem is.

Assume that *n* is a power of *m*, where *m* is a natural number bigger than two. Thus, let \(n = m^k\), for some positive integer *k*. It is straightforward to see that if we want to join all the strands using only \(\Sigma _m\) we will need to do \((n-1)/(m-1)\) steps, that is, that the *n*-point function that we would want to calculate is \(\langle (\Sigma _m)^{\frac{n - 1}{m - 1}} \rangle \). However, to calculate it we would need to know in how many ways we can join the *n* strands by joining *m* in each step. As we said above, this has only been studied for \(m = 2\) so far, which is the calculation we did in the previous section. Just to get an idea, the number of possible ways grows very fast for \(m > 2\) as well, but slower than for two, as would be expected. For example, if \(m = 3\), then we have one combination for \(k = 1\), five combinations for \(k = 2\) and 5,026,161 combinations for \(k = 3\). Also, we would need to consider all combinations of twist operators that add up to *n*, which again results in problems within the theory of integer partitions that have not been solved, as far as we are aware.

These countings need to be studied carefully in all their limits, as they might point towards other relevant subclasses of microstates. The bounds given for the results might also help in the holographic calculations, as some of these multi-particle one point functions are also relevant in supergravity.

## 7 Review of results

In this section we give a summary of all the results presented in this paper. Let us remind that this is not supposed to be a comprehensive list of all possible one point functions for chiral primaries. Rather, we rewrite the results we have obtained for some exemplary cases, which can be easily extended to calculate many more one point functions.

### 7.1 Short strand one point functions

### 7.2 Long strand one point functions

*N*, with the \({\alpha }_i\) parameters taking values in the (0, 1) interval. The polynomials are of order one for all values of the \({\alpha }_i\), and they diverge at the (excluded) boundary values. We have given explicit examples of the polynomials in Sect. 4.1.4.

### 7.3 Comparison between short and long strand results

*N*with respect to the short ones, except for the R-symmetry current mode. As we said, this indicates that one point functions for long strand states are comparable to supergravity corrections.

### 7.4 Different ways of joining strands

*n*-point function for the \(\Sigma _2^{- \dot{-}}\) twist operator. We have obtained a lower bound for it, Eq. (6.34), which is

*n*-point functions of products of twist operators and commented the result. In the last section we have found the exact result, using the lower bound for \(\langle (\Sigma _2^{- \dot{-}})^{n-1}\rangle \),

*n*-point functions which, depending in the lengths of the strands we join, will have values comparable (or bigger) to \(\langle \Sigma ^{- \dot{-}}_n\rangle \).

## 8 Conclusions and outlook

In this paper we have calculated one point functions of chiral primary operators in the D1–D5 orbifold CFT, in classes of two and three charge black hole microstates. Three charge microstates are obtained by adding momentum excitations to the Ramond ground states, as discussed in Sect. 2.4. The typical structure of the three charge microstates that we consider in this paper is shown in (4.75): these states involve adding excitations with integer momentum to Ramond ground states. As reviewed in Sect. 2.4, the majority of three charge microstates are obtained by acting with fractional modes on long string states and it would be interesting to extend the calculations of this paper to generic states involving fractional modes.

Black hole microstates of the type (4.75) have been explored before, as dual descriptions for the classes of supergravity solutions analysed in [40, 41, 42, 43, 52]. The results of this paper indicate that when such microstates involve excitations of long string Ramond ground states the one point functions of chiral primaries are suppressed by factors of *N*. Holographically this implies that the characteristic scales in the supergravity solutions are very small: the supergravity solutions differ from the naive black hole geometry at horizon scales, by contributions that are comparable to the scale of higher derivative corrections to supergravity. Thus, even when such microstates have a representation in supergravity, the supergravity description may just be an extrapolation of a string background.

Another important issue is distinguishability of microstates: the only information encoded in supergravity geometries is the expectation values of single particle chiral primary operators in the CFT. Complete information about the microstates requires the expectation values of all operators: multi particle chiral primaries and (for three charge black hole microstates) 1/8 BPS operators. In this paper we have computed expectation values of multi particle chiral primaries and it would be interesting to explore how this information can be encoded into the holographic description of a black hole microstate.

The results derived in this paper have applications beyond the black hole microstate programme. A recent paper [72] proposed a worldsheet dual for the symmetric product CFT itself and the correlation functions calculated here could be used to test this duality. Efficient methods to compute correlation functions holographically were developed in [73]; it would be interesting to use these methods to calculate the correlation functions discussed here from dual \(AdS_3 \times S^3\) backgrounds.

## Footnotes

- 1.
The explicit factor of \(\mathcal{N}_{\Psi }\) is included as the normalisation of the operators in supergravity differs by a factor of

*N*from the standard CFT normalisation. - 2.
Note that the operators given in (2.37) correspond only to the case \({\alpha }= +\), \({\dot{{\alpha }}} = \dot{+}\) of the \({\mathcal {O}}^{{\alpha }{\dot{{\alpha }}}}_{\kappa }\) operator. That is because, as is usual in the literature, in [62] only the bottom component of the multiplet is given explicitly. We obtain the other \({\mathcal {O}}^{{\alpha }{\dot{{\alpha }}}}_{\kappa }\) operators that we have written above by taking other components of that same multiplet.

- 3.
By in state we mean the state for which we calculate the one point function. We will call out state to the state after we act on it with the operator for which we are calculating the one point function.

- 4.
Using Stirling’s approximation with the exact product (not setting \(p = n\)) also results in a sum which cannot be performed by any methods we are aware of.

## Notes

### Acknowledgements

This work is funded by the STFC Grant ST/P000711/1. This project has received funding and support from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant agreement No. 690575.

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