Lifting of superconformal descendants in the
D1-D5 CFT

Marcel R. R<EMAIL_ADDRESS>Samir D.
<EMAIL_ADDRESS>and Madhur<EMAIL_ADDRESS>
a,b,c Department of Physics, The Ohio State University,
Columbus, OH 43210, USA

Abstract

We consider D1-D5-P states in the untwisted sector of the D1-D5 orbifold CFT
where we excite one copy of the seed CFT with a left-moving superconformal
descendant. When the theory is deformed away from this region of moduli space
these states can ‘lift’, despite being BPS at the orbifold point. For
descendants formed from the supersymmetry $G^{\alpha}_{\\!\dot{A},-s}$ and
R-symmetry $J^{a}_{-n}$ current modes we obtain explicit results for the
expectation value of the lifts for various subfamilies of states at second
order in the deformation parameter. A smooth $\sim\sqrt{h}$ behaviour is
observed in the lifts of these subfamilies for large dimensions. Using
covering space Ward identities we then find a compact expression for the lift
of the above $J^{a}_{-n}$ descendant states valid for arbitrary dimensions. In
the large-dimension limit this lift scales as $\sim\sqrt{h}\,$, strengthening
the conjecture that this is a universal property of the lift of D1-D5-P
states. We observe that the lift is not simply a function of the total
dimension, but depends on how the descendant level is partitioned amongst
modes.

###### Contents

  1. 1 Introduction
    1. 1.1 Summary of results
  2. 2 The D1-D5 CFT and its free-orbifold point
    1. 2.1 Symmetries of the CFT
    2. 2.2 Superconformal primaries
    3. 2.3 Spectral flow
    4. 2.4 Deformation away from the orbifold point
    5. 2.5 Lift formulae
  3. 3 Lifting of descendent states via series method
    1. 3.1 Lifting of $|G\phi\rangle^{\alpha}_{\\!\dot{A}(h,s)}$
    2. 3.2 Lifting of $|J\phi\rangle^{+}_{(h,n)}$
    3. 3.3 Symmetries and checks
  4. 4 Lifting of descendants by method of Ward identities
    1. 4.1 Lifting of $|J\phi\rangle^{+}_{(h,n)}$ by Ward identity
    2. 4.2 Asymptotic behaviour
      1. 4.2.1 Large dimension limits
  5. 5 Discussion
  6. A The $\mathcal{N}=4$ superconformal algebra
    1. A.1 Commutation relations
    2. A.2 Hermitian conjugation
  7. B The mode correlator $C^{\beta\alpha}_{\\!\dot{B}\dot{A}}$
  8. C Lifting of $\mathcal{J}^{+,(m)}|\phi\rangle$

##  1 Introduction

One of the biggest ongoing problems in physics has to do with the study and
understanding of black holes beyond the classical predictions of general
relativity. A prototypical example black hole system that has lent itself well
to study from various angles is the extremal 3-charge Strominger-Vafa black
hole. The microscopic description of this classical black hole solution is
that of type IIB string theory on $\mathbb{R}^{1,4}\times
S^{1}_{y}\times\mathcal{M}^{4}$ (this compact manifold $\mathcal{M}^{4}$ can
be either $T^{4}$ or $K3$). On this background are placed D1-branes, D5-branes
and momentum (P) excitations wrapping the compact directions. This brane
system preserves $1/8$th of the 32 real supercharges of the underlying string
theory (referred to as $1/8$th BPS) and its many bound states are the
microstates of the classical black hole each with the same mass and charges.
The Bekenstein entropy of the Strominger-Vafa black hole $S_{\rm{bek}}=A/4G$
has been shown to be accounted for by the $e^{S_{\rm{bek}}}$ microstates of
the brane system in [1] and [2] for type IIB string theory compactified on
$K3\times S^{1}_{y}$ and $T^{4}\times S^{1}_{y}$ respectively. These are lower
bounds on the number of microstates due to the use of supersymmetric indices,
however, their leading-order behavior matches exactly the Bekenstein entropy
of the black hole.

Whilst the successful identification of the appropriate system in string
theory that describes the Strominger-Vafa black hole, with the correct count
of states, is a great achievement it is desirable to now ask more fine-grained
questions. In particular: what do the individual microstates of a black hole
look like, what are the dynamical properties of such states, and how does this
microscopic description give rise to the relatively simple structure and
properties of the classical solution in an appropriate coarse-grained
description. The explicit construction and study of such black hole
microstates is termed the fuzzball program. A long-standing effort to
construct microstate geometries (a special class of fuzzballs for which
suitably coherent behaviour leads to a semi-classical description) for the
D1-D5-P system has led to the discovery of large classes of horizonless
solutions to supergravity having the same asymptotic structure as the black
hole, but with different infrared behaviours encoding microscopic details of
the states (see [3, 4] for reviews of the current state of these efforts and
of the wider fuzzball proposal and also [5, 6, 7, 8, 9] for recent interesting
developments of non-BPS black hole microstate geometries).

Another reason for the richness of accessible information in the Strominger-
Vafa black hole system is the existence of a supersymmetric conformal field
theory (CFT) description that is holographically dual to the D1-D5-P system. A
large program of work has been devoted to understanding the holographic
dictionary between the black hole microstates and the spectrum of the dual
‘D1-D5 CFT’ [10, 11, 12, 13, 14, 15]. Despite the precise CFT dual to the
weakly coupled brane system being at strong coupling, it is conjectured that
there exists a locus of its moduli space at which the CFT has a description in
terms of a $(1+1)$-dimensional free symmetric group orbifold theory [16, 17,
18, 19, 20, 21, 22, 23] (this is analogous to free super Yang-Mills (SYM) in
the D3-brane system [24]). In this free theory we have much better control
over the spectrum and its observables, in particular one can ask what form
black hole microstates take in the dual field theory description. Since the
black hole is BPS the dual states should also be equally BPS, however, while
in the free orbifold theory any state with purely left-moving (or purely
right-moving) excitations is BPS, this simple classification no longer holds
at a general point in the D1-D5 CFT’s moduli space. Upon deforming the free
orbifold theory towards the strongly-coupled part of the moduli space, short
multiplets of the free theory (which contain the BPS states) can join together
into long multiplets of the theory with general moduli. This phenomena is
termed ‘lifting’ and is visible from an anomalous contribution to the
conformal dimension of these states under deformation of the free theory.
Therefore, only states that are globally BPS are seen by the index
calculations of [1, 2] and can correspond to microstates of extremal black
holes. Whilst the precise form of the globally BPS states is moduli dependent,
the number of such states (and some of their properties such as quantum
numbers and 3-point functions) is not.

In [25], the early work of Gava and Narain found that, as expected from the
gravity theory, almost all low-lying states of the CFT were lifted; the only
low-energy BPS states in the gravity picture describe supergravity quanta,
while the remainder are string states whose energies are lifted to the string
scale. However, the CFT spectrum should have a transition to a dense black
hole phase at high enough energies which should contain the large number of
expected black hole states seen by the index calculations111Some recent
progress has been made for the case of $1/16$th BPS states in $\mathcal{N}=4$
SYM with gauge group $SU(2)$ above the black hole threshold [26, 27, 28, 29]..
Whilst there has been progress in computing the lift of various families of
D1-D5-P states through the use of conformal perturbation theory, no simple
answer has been found to the question of what dramatic change occurs in
lifting calculations when above the black hole threshold.

Here we very briefly describe the families of states that have been considered
so far222See [30, 31, 32, 33, 34, 35, 36, 37, 38] for additional work about
the lifting of states using conformal perturbation theory in the D1-D5 CFT and
[39, 40] for some work related to the twist operators of the theory. Also of
note are [41, 42] in which for $S_{N}$ orbifold theories it is shown how to
relate correlators of Virasoro descendants (using both integer and fractional
modes) to correlators of a suitable set of ancestors using covering space
methods.. In [25], states on highly-wound component strings (with winding
$k\gg 1$) were considered with purely left-moving excitations of level $n$
above the NS vacuum satisfying $n\gg k$. All of these states were found to be
lifted bar those corresponding to the single-particle supergravity multiplet.
In [43] the currents of the higher-spin symmetry algebra of the free orbifold
theory were found to lift. This is expected since away from the free point in
moduli space the higher-spin symmetry is broken to the $\mathcal{N}=4$
superconformal algebra. In [44], a closed-form expression was found for the
lift of states where a subset of copies of the orbifold theory were excited
above the unique NS vacuum with an excitation formed from the modes of the
left-moving part of the $SU(2)$ R-symmetry current. This excitation amounts to
putting the maximal number of fermion excitations on top of the NS vacuum and
was studied using the particular property that this excited state is the
spectral flow of the vacuum. This 1-parameter family of states is labelled by
the dimension of the excitation. For the case of the $N=2$ theory, [45, 46]
constructed and computed the lifts for all D1-D5-P primaries up to level 4 on
a case-by-case basis. It was found that all such states were lifted for which
suitable groups of short multiplets exist to join up into long multiplets of
the deformed theory. The lift of untwisted-sector states in $(h,j)=(1,0)$
left-moving long multiplets and general $\bar{j}$ right-moving short
multiplets was studied in [47] where evidence was found for the undercounting
of unlifted states by the index computations. In [48], untwisted-sector states
were considered where all copies were in the NS vacuum bar one copy, which was
excited with a superconformal primary of the $c=6$ seed theory. A closed-form
expression was found for the lift of this family of states and it was shown
that explicit knowledge of the superconformal primary is not necessary, only
its conformal dimension. Most recently, [49] studied the lift of untwisted-
sector states where one copy was excited above the NS-NS vacuum by two left-
moving modes of the bosons and fermions of the free theory. Patterns in the
lift of this two-parameter family of states (labelled by the mode numbers $m$,
$n$ of the two modes) were studied. Since the computation method used
increased in complexity with the level of the state ($m+n$), this was done on
a state-by-state basis and thus no closed-form expressions were found. The
advantage of this method, however, was that lifts at much higher levels were
obtainable as compared with the low-level searches of [45, 46].

In these searches of the CFT spectrum, various patterns have emerged and it is
hoped that a general answer to the question of which states do not lift will
become clear given an understanding of the lift in enough special cases. One
of these patterns in the lift of states is that for large dimensions, the lift
appears to follow a $\sim\sqrt{h}$ growth independently of the particular
state chosen. Due to the majority of previous lifting calculations being done
on a state-by-state basis, this asymptotic growth has mostly been observed by
fitting the lift of 1-parameter families of states. In this paper we aim to
address this seemingly universal asymptotic behaviour of lifts analytically.

###  1.1 Summary of results

In this paper we find the following:

  * •

We study the second order lifting of states where one copy of the $c=6$ CFT is
excited above the NS vacuum by a descendent of a superconformal primary
$\phi$. Specifically, the single-copy descendants we discuss are

$G^{\alpha}_{\\!\dot{A},-s}|\phi\rangle\quad\text{ and }\quad
J^{+}_{-n}|\phi\rangle\ .$ (1.1)

The expectation value of lifts of these states is computed in Section 3 using
a new method that is well suited to having states where we do not specify
their form, i.e. the superconformal primary $|\phi\rangle$. This method, that
we dub the “series method” to distinguish it from the “fields method” of [49]
that relied on explicit free-field representations of the states being
considered. The lifts of the states (1.1) are found in terms of contour
integrals of various series, the complexity of which scales with the
descendant level ($s$ and $n$). Therefore, similarly to [49], we evaluate
these lifts on a state-by-state basis for various 1-parameter families of
states to much higher levels than has been previously possible. In this
lifting data we observe the same $\sim\sqrt{h}$ asymptotic growth that has
been previously observed for other families of states in [48, 49].

  * •

The lifting of the descendant state $J^{+}_{-n}|\phi\rangle_{h}$ on a single
copy is also studied using a method of Ward identities. In this method we use
Ward identities on the covering space of the amplitude, required for the
computation of the lifts, in order to remove the R-symmetry current modes.
This approach turns out to allow for an analytic expression for the
expectation value of the lift of these states given by

$\displaystyle
E^{(2)}\big{(}|J\phi\rangle^{+}_{(h,n)}\big{)}=E^{(2)}_{h}\\!\big{(}\phi\big{)}\Bigg{[}1+a_{n}\frac{P_{n}(h)}{(h+1)_{2n-2}}\Bigg{]}\
,$ (1.2)

where

$E^{(2)}_{h}\\!\big{(}\phi\big{)}=\pi^{\tfrac{3}{2}}\lambda^{2}\frac{\Gamma(h+\tfrac{1}{2})}{\Gamma(h)}\
,$ (1.3)

is the dimension-$h$ single-copy primary lift found in [48]. The $a_{n}$ are
rational coefficients listed in (4.39) and the $P_{n}(h)$ are polynomials in
$h$ of degree $2n-4$, explicit forms of the first few are given in (4.1). In
general these coefficients and polynomials are defined implicitly from the sum
of (4.1) and (4.1). Of note here is that the result (1.3) does not depend on
the form of the primary $\phi$, only its dimension $h$.

  * •

Having such a closed-form expression for the lift of this 2-parameter family
of states allows us to partially answer previously posed conjectures on the
universal behaviour of lifts. Namely, we discuss the following questions:

    1. (1)

Is the leading $\sim\sqrt{h}$ behaviour for large conformal dimension, derived
in [49] for single-copy primaries, a universal property of second order lifts?

    2. (2)

Are second-order lifts in general purely a function of the total dimension, as
in the case of (1.3), or do they depend on the details of the state.

With regards to point (1), we derive the asymptotic behaviour of the lift
(1.2) for large $h$ and fixed $n$ to the first subleading order to be

$\frac{E^{(2)}\\!\big{(}|J\phi\rangle^{+}_{(h,n)}\big{)}}{E^{(2)}_{h}\\!(\phi)}\approx
1+\frac{n(n^{2}-1)}{8h^{2}}\ ,$ (1.4)

and so we see that the leading behaviour is $\sim\sqrt{h}$ as this is the
leading behaviour of (1.3). However, with regards to point (2) above we argue
in Section 4.2.1 that the lift is not purely a function of the total dimension
$h+n$. A follow up to point (2) is then:

    1. (2a)

Do the second order lifts of level-$n$ descendants of a superconformal primary
of dimension $h$ depend on how $n$ is partitioned amongst descendant modes?

By comparing (1.4) the lift of states with the single-copy excitation
$J^{+}_{-(2m-1)}\cdots J^{+}_{-3}J^{+}_{-1}|\phi\rangle_{h}$ (whose descendant
level is $m^{2}$) which we compute in closed-form in Appendix C, we find that
the question of point (2a) is true.

### Paper layout

A summary of the present paper is as follows: Section 2 briefly reviews the
D1-D5 CFT, its symmetries, spectral flow and deformations away from the free
point in its moduli space. In particular, Section 2.2 defines superconformal
primaries in the D1-D5 CFT and gives a mechanism for the explicit construction
of an infinite family using the fields of the free orbifold theory. Section
2.5 briefly reviews a general method for, in principle, finding the
expectation value of the second order lift in energy. In Sections 3.1 and 3.2
we compute the lift of untwisted-sector states in which one copy is excited
with a superconformal descendent of a primary $\phi$ to high levels using a
new “series method”. Some patterns in the lifting data of these descendent
states are discussed in Section 3.3 including the observation of asymptotic
$\sim\sqrt{h}$ behaviour for large dimensions, inline with observations made
in [48, 49]. Section 4.1 derives a closed-form for the lift of single-copy
R-symmetry current descendants $\sim J^{+}_{-n}|\phi\rangle$ by a method of
covering space Ward identities. This analytic expression then allows for an
analysis of the large dimension behaviour of lifts in Sections 4.2. The lift
of states of the form $\sim J^{+}_{-(2m-1)}\cdots
J^{+}_{-3}J^{+}_{-1}|\phi\rangle$ is derived analytically in Section C in
order to strengthen conclusions about universal features of lifts in the D1-D5
CFT. We conclude with a discussion of the results in Section 5.

##  2 The D1-D5 CFT and its free-orbifold point

The setting is type IIB string theory on a background of

$M^{1,4}\times S^{1}\times T^{4}\ ,$ (2.1)

containing $n_{1}$ D1-branes and $n_{5}$ D5-branes that both wrap the $S^{1}$
and with the D5-branes also wrapping the $T^{4}$. The bound states of this
brane system generate the D1-D5 CFT – a $(1+1)$-dimensional conformal field
theory living on a cylinder made from the time direction and the spatial
$S^{1}$. This theory is thought to have a point in its moduli space, dubbed
the free-orbifold point, where there is a description of the theory in terms
of

$N=n_{1}n_{5}\ ,$ (2.2)

copies of a seed $c=6$ free CFT. This free CFT contains $4$ free bosons and
$8$ free fermions, 4 each in the left- and right-moving sectors. The Hilbert
space of these free fields are then subject to an orbifolding by the
permutation group $S_{N}$, leading to the Hilbert space factoring into twisted
sectors labelled by integers $1\leq k\leq N$. The different twisted sectors in
essence describe a CFT on a $k$-wound circle (sometimes called a component
string). This orbifold point of the D1-D5 CFT has been shown [50, 51, 52, 53,
54, 55] to be dual to the tensionless limit of a string in an AdS${}_{3}\times
S^{3}\times T^{4}$ background with one unit of NS-NS flux.

###  2.1 Symmetries of the CFT

Generically in the moduli space of the D1-D5 CFT there is $\mathcal{N}=(4,4)$
supersymmetry [56, 57], i.e. an ${\cal N}=4$ superconformal symmetry algebra
in both the left- and right-moving sectors. The chiral algebra generators are

$\Big{\\{}\,L_{n}\ ,\ \ G^{\alpha}_{\\!\dot{A},r}\ ,\ \ J^{a}_{n}\,\Big{\\}}\
,$ (2.3)

for the left movers associated with the stress-energy tensor, supercurrents
and $\mathfrak{su}(2)$ R-currents respectively. The right-moving sector has
analogous generators given by the set

$\Big{\\{}\,\bar{L}_{n}\ ,\ \ \bar{G}^{\bar{\alpha}}_{\\!\dot{A},r}\ ,\ \
\bar{J}^{a}_{n}\,\Big{\\}}\ .$ (2.4)

The various indices used are associated with fundamental representations of
the various $SU(2)$ factors of the symmetry algebra: $\alpha$ and
$\bar{\alpha}$ are indices of the $SU(2)_{L}$ and $SU(2)_{R}$ factors of the

$SO(4)_{E}\cong\big{(}SU(2)_{L}\times SU(2)_{R}\big{)}/\mathbb{Z}_{2}\ ,$
(2.5)

‘external’ R-symmetry, originating from rotations in the noncompact spatial
directions of the background (2.1). The broken ‘internal’ $SO(4)_{I}$ global
symmetry, coming from the $T^{4}$ factor of the background, provides a useful
organising principle for the spectrum of the orbifold theory. The doublet
indices $A,\dot{A}$ are from the $SU(2)_{1}$ and $SU(2)_{2}$ factors of

$SO(4)_{I}\cong\big{(}SU(2)_{1}\times SU(2)_{2}\big{)}/\mathbb{Z}_{2}\ .$
(2.6)

In (2.3) and (2.4) the index $a$ is a vector index of the $SO(4)_{E}$. The
small $\mathcal{N}=(4,4)$ superconformal algebra spanned by the currents (2.3)
and (2.4) and our conventions are outlined in Appendix A. At the free orbifold
point, where the theory has a realisation in terms of the free bosons and
fermions

$\Big{\\{}\,\partial X_{\\!A\dot{A}}\ ,\ \ \psi^{\,\alpha A}\ ,\ \
\bar{\psi}^{\,\bar{\alpha}\dot{A}}\,\Big{\\}}\ ,$ (2.7)

with their corresponding modes (see the mode expansions (A.4))

$\Big{\\{}\,\alpha_{A\dot{A},n}\ ,\ \ d^{\,\alpha A}_{s}\ ,\ \
\bar{d}^{\,\bar{\alpha}\dot{A}}_{s}\,\Big{\\}}\ ,$ (2.8)

this symmetry algebra is in fact enlarged to the contracted large ${\cal
N}=(4,4)$ superconformal symmetry [2, 57] (in fact exactly at the free point,
this symmetry is boosted to include also a $\mathcal{W}_{\infty}$ algebra
studied, for instance, in [58, 43]).

###  2.2 Superconformal primaries

A class of states that will play an important role in this paper are
superconformal primaries. Consider such a state $|\phi\rangle$ with conformal
dimension $h$ and $J^{3}_{0}$ charge $m$ can be defined by the mode conditions
(given in the NS sector)

$L_{n}|\phi\rangle=G^{\alpha}_{\\!\dot{A},s}|\phi\rangle=J^{a}_{n}|\phi\rangle=0\quad,\quad\text{
where }n>0\ ,\ s\geq\frac{1}{2}\ .$ (2.9)

If $\phi$ is to be a primary of the full $c=6N$ theory then these current
algebra modes should be global modes. For instance, the global Virasoro modes
$L^{(g)}_{n}$ in the untwisted sector are given by the diagonal sum

$L^{(g)}_{n}=\sum_{i=1}^{N}L^{(i)}_{n}\ ,$ (2.10)

where $L^{(i)}_{n}$ are the modes on the $i$th copy (in the rest of the paper
we will mostly drop the copy index since only single-copy modes are used).
Likewise, a superconformal primary on a single copy would satisfy (2.9) with
the modes on that particular copy of the $c=6$ seed theory. The superconformal
primary states $|\phi\rangle^{(1)}$ used in this paper are primaries of one
copy of the $c=6$ seed theory (written here as copy 1 for clarity), with all
other copies being in the unique NS vacuum (the right-moving part of the state
is in the NS vacuum on all copies). This type of state can be seen to be
constrained to have $J^{3}_{0}$ charge $m$ of333Whilst the constraint (2.11)
on single-copy superconformal primaries has been written in terms of the
eigenvalue of $J^{3}_{0}$, the action of the raising and lowering operators
$J^{\pm}_{0}$ for $SU(2)_{L}$ multiplets commutes with the definition (2.9).
Therefore, the constraint (2.11) must be satisfied by all members of a given
$SU(2)_{L}$ multiplet. By labelling these multiplets by the eigenvalue $j$ of
the quadratic Casimir of $SU(2)_{L}$ and since the range of $m$ in a multiplet
is $m=-j,-j+1,\dots,j-1,j$, we see that there are only two possibilities.
Single-copy superconformal primaries can either be in $j=\tfrac{1}{2}$ or
$j=0$ multiplets.

$|m|\leq\frac{1}{2}\ ,$ (2.11)

as seen by considering the non-negativity of the norm of the $J^{+(1)}_{-1}$
descendent of this state. It was shown in [48] that the states with
$m=\frac{1}{2}$ are forced to be chiral and as such do not lift. This leaves
single-copy superconformal primaries with $m=0$. While it may seem possible
that no such states exist, given the highly constraining conditions (2.9) and
(2.11), we now give a prescription for the explicit construction of an
infinite family in the free-field realisation of this theory. It should be
noted that by moving away from the untwisted sector or by exciting more than
one copy, these arguments are not as restrictive.

The key observation for constructing superconformal primaries of this theory
is that none of the generators (2.3) of the $\mathcal{N}=4$ superconformal
algebra (A.2) are charged under the $SU(2)_{1}$ of the ‘internal’ $SO(4)_{I}$
(2.6). By selecting states with the _minimal_ conformal dimension for a given
$SU(2)_{1}$ charge444More precisely, this charge $q_{A}$ is the eigenvalue of
the zero mode of the third component of the generators of $SU(2)_{1}$. This is
not to be confused with the R-symmetry charges discussed throughout for the
$SU(2)_{L}$ of (2.5). Since the $SU(2)_{L}$ is really a symmetry of the theory
we can use it to label states, however, the $SU(2)_{1}$ is a broken symmetry
and serves just to organise our discussion. $q_{A}$, it is guaranteed that the
positive current modes in the primary definition (2.9) (which all lower the
dimension without changing the value of $q_{A}$) will annihilate the state.
Trivially, the unique NS vacuum $|0_{\scriptscriptstyle{N\\!S}}\rangle$ is the
sole minimal dimension ($h=0$) state with charge $q_{A}=0$ and clearly
satisfies the conditions (2.9). In this section, for a superconformal primary
of dimension $h$ we use the notation $|\phi^{h}_{j_{A},q_{A}}\rangle$ with
$q_{A}$ and $j_{A}$ being the eigenvalues of the third component and quadratic
Casimir of $SU(2)_{1}$ respectively. The NS vacuum can then be labelled in
this notation as

$|0_{\scriptscriptstyle{N\\!S}}\rangle=|\phi^{0}_{0,0}\rangle\ .$ (2.12)

For $q_{A}=\frac{1}{2}$ the options are to add either one bosonic or one
fermionic mode of the free field realisation (2.8) to the NS vacuum. Both
options can yield the same $q_{A}$ charge, however, the minimal conformal
dimension for that charge555A bosonic zero mode $\alpha_{0}$ would add charge
without adding dimension, however, it would always commute through any other
modes to annihilate the NS vacuum so we do not consider it here. is from the
use of a $d_{-\frac{1}{2}}$. The $h=q_{A}=\frac{1}{2}$ superconformal primary
states are then

$|\phi^{\frac{1}{2}}_{\frac{1}{2},\frac{1}{2}}\rangle^{\alpha}=d^{\alpha+}_{-\frac{1}{2}}|0_{\scriptscriptstyle{N\\!S}}\rangle\
,$ (2.13)

which are precisely the chiral/anti-chiral primaries with $h=|m|=\frac{1}{2}$
discussed below (2.11). This state is the top member of an $SU(2)_{1}$ doublet
and so the bottom member

$|\phi^{\frac{1}{2}}_{\frac{1}{2},-\frac{1}{2}}\rangle^{\alpha}=d^{\alpha-}_{-\frac{1}{2}}|0_{\scriptscriptstyle{N\\!S}}\rangle\
,$ (2.14)

will also be a primary. In this paper we are interested in the primaries that
are not chiral (since they trivially do not lift), so we will restrict
ourselves to R-symmetry neutral states here. For $q_{A}=1$, the minimal
dimension state is

$|\phi^{1}_{1,1}\rangle=d^{++}_{-\frac{1}{2}}d^{-+}_{-\frac{1}{2}}|0_{\scriptscriptstyle{N\\!S}}\rangle\
,$ (2.15)

which is R-symmetry neutral666Since $J^{\pm}_{0}$ annihilate all states in the
triplet composed of the states in (2.15) and (2.16), they have R-symmetry
charges $m=j=0$ as expected. and the top member of an $SU(2)_{1}$ triplet that
is filled out by the states

$\displaystyle|\phi^{1}_{1,0}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\Big{(}d^{-+}_{-\frac{1}{2}}d^{+-}_{-\frac{1}{2}}+d^{--}_{-\frac{1}{2}}d^{++}_{-\frac{1}{2}}\Big{)}|0_{\scriptscriptstyle{N\\!S}}\rangle\
,$ (2.16) $\displaystyle|\phi^{1}_{1,-1}\rangle$
$\displaystyle=d^{--}_{-\frac{1}{2}}d^{+-}_{-\frac{1}{2}}|0_{\scriptscriptstyle{N\\!S}}\rangle\
.$ (2.17)

It should be noted that the singlet of the
$\mathbf{2}\otimes\mathbf{2}=\mathbf{1}\oplus\mathbf{3}$ decomposition given
by the state

$|\phi^{1}_{0,0}\rangle=\frac{1}{\sqrt{2}}\Big{(}d^{-+}_{-\frac{1}{2}}d^{+-}_{-\frac{1}{2}}-d^{--}_{-\frac{1}{2}}d^{++}_{-\frac{1}{2}}\Big{)}|0_{\scriptscriptstyle{N\\!S}}\rangle\
,$ (2.18)

is not a superconformal primary since $L_{1}|\phi^{1}_{0,0}\rangle\neq 0$ for
instance. For $q_{A}=\frac{3}{2}$, the minimal dimension states are obtained
by adding an $\alpha_{-1}$ mode on top of the $q_{A}=1$ states (2.15) to
obtain777A small subtlety here is that an $SU(2)_{1}$ charge of
$q_{A}=+\frac{1}{2}$ read off from an upper $A$ index is equivalent to a
charge of $q_{A}=-\frac{1}{2}$ from a lower $A$ index due to contractions
being made by Levi-Civita symbols.

$|\phi^{2}_{\frac{3}{2},\frac{3}{2}}\rangle_{\dot{A}}=\alpha_{-\dot{A},-1}d^{++}_{-\frac{1}{2}}d^{-+}_{-\frac{1}{2}}|0_{\scriptscriptstyle{N\\!S}}\rangle\
,$ (2.19)

which is the first state in our family of superconformal primaries that is
charged under $SU(2)_{2}$. Both states of the $SU(2)_{2}$ doublet indexed by
$\dot{A}$ are primaries. The $j_{A}=\frac{3}{2}$ multiplet can then be filled
out by the application of lowering operators of $SU(2)_{1}$ on the highest-
weight state (2.19). It turns out that by continuing this construction for
higher values of $q_{A}$ a pattern emerges: for $q_{A}\geq\frac{3}{2}$ the
minimal dimension states have

$h=2j_{A}-1\quad,\quad\text{ for }j_{A}\geq 1\ ,$ (2.20)

and the top members of the $j_{A}=q_{A}$ representations of $SU(2)_{1}$ are of
the schematic form

$|\phi^{h}_{q_{A},q_{A}}\rangle=\big{(}\alpha_{-,\dot{A},-1}\big{)}^{2q_{A}-2}d^{++}_{-\frac{1}{2}}d^{-+}_{-\frac{1}{2}}|0_{\scriptscriptstyle{N\\!S}}\rangle\
,$ (2.21)

where these states are in the $(\mathbf{2q_{A}-1})\subset\mathbf{2}^{\otimes
2(q_{A}-1)}$ representation of $SU(2)_{2}$ due to the requirement of symmetry
of the identical $\alpha$ modes. The total number of R-symmetry neutral
superconformal primaries $N_{q_{A}}$ in this family for a given value of
$q_{A}$ is then

$N_{q_{A}}=4q_{A}^{2}-1\quad\,\quad\text{ for
}q_{A}=\tfrac{1}{2},1,\tfrac{3}{2},\,\dots\ ,$ (2.22)

made by taking the state of the form (2.21) and filling out the suitable
$SU(2)_{1}$ and $SU(2)_{2}$ representations. If this family of states
represents the complete set of single-copy superconformal primaries, (2.22)
can be recast as the number of such states at a given level $h$

$N_{h}=h(h+2)\quad\,\quad\text{ for }h\geq 1\ .$ (2.23)

Clearly these states form an infinite family of superconformal primaries.

###  2.3 Spectral flow

A $1$-parameter family of global automorphisms of the $\mathcal{N}=4$
superconformal algebra in $(1+1)$ dimensions labelled by an angle $\pi\eta$ is
referred to as spectral flow. This map between equivalent algebras has the
effect of changing the periodicity of the fermionic currents. Here we very
briefly review the rules for spectral flow transformations [56] that will be
used in the main computations of Sections 3 and 4.1. Under spectral flow by
$\eta$ units888Here we have only mentioned spectral flow in the left-moving
sector of the $\mathcal{N}=(4,4)$ superconformal algebra, however, both the
left- and right-moving sectors can be spectral flowed independently by $\eta$
and $\bar{\eta}$ units respectively. See [59] for a smaller group of spectral
flows, dubbed ‘partial spectral flow’, which is related to the spectral flow
of an $\mathcal{N}=2$ algebra contained within the $\mathcal{N}=4$ algebra.,
the quantum numbers of a state with dimension $h$ and $J^{3}_{0}$ charge $m$
transforms as

$h\rightarrow h^{\prime}=h+m\eta+\frac{c\eta^{2}}{24}\quad\ ,\qquad
m\rightarrow m^{\prime}=m+\frac{c\eta}{12}\ ,$ (2.24)

where $c$ is the central charge of the theory. Note that if $\eta$ is an odd
integer, spectral flow maps between the NS- and R-sector boundary conditions
for the fermionic degrees of freedom of the theory. Whilst the transformations
of the currents $T(z)$ and $J^{3}(z)$ are more involved due to mixing in the
algebra and the central charge anomaly, the transformation properties of
currents under spectral flow required in this paper are straightforward: under
a spectral flow by $\eta$ units around the point $z_{0}$

$\displaystyle G^{\pm}_{\\!\dot{A}}(z)$
$\displaystyle\to(z-z_{0})^{\mp\frac{\eta}{2}}G^{\pm}_{\\!\dot{A}}(z)\ ,$
(2.25) $\displaystyle J^{\pm}(z)$
$\displaystyle\to(z-z_{0})^{\mp\eta}J^{\pm}(z)\ .$ (2.26)

In Appendix C of [48] it was shown that a superconformal primary field
$\phi(z)$ on the plane also transforms simply via

$\phi(z)\to(z-z_{0})^{-\eta m}\phi(z)\ .$ (2.27)

###  2.4 Deformation away from the orbifold point

It is thought that the free-orbifold theory can be deformed in the moduli
space of the D1-D5 CFT towards a strong coupling regime, at which the theory
would have a dual semiclassical gravity description. This deformation can be
done by adding a deformation operator $D$ to the action such that

$S\rightarrow S+\lambda\int d^{2}z\,D(z,\bar{z})\ ,$ (2.28)

where $\lambda$ is the deformation parameter and $D$ exactly marginal and so
has conformal dimensions $(h,\bar{h})=(1,1)$. In the theory there are $20$
exactly marginal operators to choose from; $16$ of these correspond to $T^{4}$
moduli, whilst the remaining $4$ are superdescendants of the order-2 twist
chiral/anti-chiral primaries $\sigma^{\alpha\bar{\alpha}}$. The set of $T^{4}$
moduli are in a sense ‘trivial deformations’, whereas the latter four break
the higher-spin symmetry found at the orbifold point and move the theory
towards the region with a semi-classical gravity description. The non-trivial
deformation operator that is a singlet under all of the $SU(2)$ factors of the
symmetry algebra discussed in 2.1 is given by

$D=\frac{1}{4}\epsilon^{\dot{A}\dot{B}}\epsilon_{\alpha\beta}\epsilon_{\bar{\alpha}\bar{\beta}}\,G^{\alpha}_{\dot{A},-{\frac{1}{2}}}\bar{G}^{\bar{\alpha}}_{\dot{B},-{\frac{1}{2}}}\sigma^{\beta\bar{\beta}}\
.$ (2.29)

Here $G$ and $\bar{G}$ are the supercharge modes at the _orbifold point_ ,
i.e. at $\lambda=0$. The remaining three non-trivial deformation operators
form the triplet projection of $SU(2)_{2}\otimes SU(2)_{2}$. The
$SO(4)_{I}\,$-invariant operator (2.29) corresponds to turning on the dual
string tension. I should be noted that the chiral/anti-chiral operators
$\sigma^{\alpha\bar{\alpha}}$ are formed by adding R-charge to the bare twist
operator $\sigma_{2}$ using a mixture of R-symmetry currents and spin fields.
In the particular case of $k=2$, this is done using only spin fields [60] to
give

$\sigma^{\alpha\bar{\alpha}}=S_{2}^{\alpha}\bar{S}_{2}^{\bar{\alpha}}\sigma_{2}\
.$ (2.30)

Spin fields change the boundary conditions of fermions around their insertion
points. The operator $\sigma_{2}$ twists together two copies, $i$ and $j$, of
the $c=6$ seed CFT, which in an $S_{N}$-invariant form is written as

$\sigma_{2}=\sum_{\begin{subarray}{c}i,j=1\\\
j<\>\\!i\end{subarray}}^{N}\sigma_{(ij)}\ .$ (2.31)

We will be considering the deformation of the orbifold theory at second order
in the parameter $\lambda$ since, as will be discussed below, this is the
first non-trivial order when it comes to the lifting of states.

###  2.5 Lift formulae

The underlying method for the computation of second order lifts of states
under the deformation discussed in Section 2.4 was developed in [44] by the
use of conformal perturbation theory. Here we do not give details of the
derivation of this method and instead use it as a tool to learn about the
behaviour of states under this deformation. At its core, this method requires
the computation of the second order lift from the integrated correlator

$A^{(2)}(T)=\frac{1}{2}\Big{<}\phi\big{(}\tfrac{T}{2}\big{)}\Big{|}\int
d^{2}w_{2}D(w_{2},\bar{w}_{2})\int
d^{2}w_{1}D(w_{1},\bar{w}_{1})\Big{|}\phi\big{(}\\!-\\!\tfrac{T}{2}\big{)}\Big{>}\
,$ (2.32)

where the factor of $\tfrac{1}{2}$ comes from the second order perturbation of
the action and the initial and final states are placed at finite Euclidean
time $\tau=\frac{T}{2}$ in order to regularise the calculation. We find it
convenient to break down these two insertions of the deformation operator
$D(w,\bar{w})$ into different equivalent forms

$D(w,\bar{w})=\epsilon^{\dot{A}\dot{B}}G^{-}_{\dot{A},-\frac{1}{2}}\bar{G}^{-}_{\dot{B},-\frac{1}{2}}\sigma^{++}_{2}(w,\bar{w})=\epsilon^{\dot{A}\dot{B}}G^{+}_{\dot{A},-\frac{1}{2}}\bar{G}^{+}_{\dot{B},-\frac{1}{2}}\sigma^{--}_{2}(w,\bar{w})\
,$ (2.33)

both of which are equal to (2.29) but with the $SU(2)$ singlet structure
obscured.

Figure 1: This figure depicts the correlator(2.36). In purple, we see the
initial $\ket{\Phi}$ and final states $\bra{\Phi}$ located at $\tau=-T/2$ and
$\tau=T/2$ respectively. The green contours are the supersymmetry current
modes from the two deformation operators placed at $(0,0)$ and $(w,\bar{w})$.

Fixing the insertion of the first deformation operator to the origin and
considering the large $T$ limit (since $T\to\infty$ at the end of the
calculation anyway) we can write

$A^{(2)}(T)\approx\frac{1}{2}(2\pi T)e^{-E^{(0)}T}\big{<}\Phi\big{|}\int
d^{2}w_{2}D(w_{2},\bar{w}_{2})\,D(0,0)\big{|}\Phi\big{>}\ .$ (2.34)

From conformal perturbation theory it was shown in [44] that the second order
lift in energy (the first order lift in energy is always vanishing by symmetry
arguments) can be obtained from the coefficient of $-Te^{-E^{(0)}T}$ in the
correlator $A^{(2)}(T\to\infty)$ and so we write

$E^{(2)}\big{(}|\Phi\rangle\big{)}=-\pi\lambda^{2}\lim_{T\to\infty}\frac{X(T)}{\langle\Phi|\Phi\rangle}\
,$ (2.35)

where the integrated amplitude $X(T)$ is given by

$X(T)\equiv\epsilon^{\dot{C}\dot{D}}\epsilon^{\dot{A}\dot{B}}\int
d^{2}w\,\big{<}\Phi\big{|}\big{(}G^{+}_{\dot{C},-\frac{1}{2}}\bar{G}^{+}_{\dot{D},-\frac{1}{2}}\sigma^{--}_{2}\big{)}(w,\bar{w})\,\big{(}G^{-}_{\dot{A},-\frac{1}{2}}\bar{G}^{-}_{\dot{B},-\frac{1}{2}}\sigma^{++}_{2}\big{)}(0,0)\big{|}\Phi\big{>}\
,$ (2.36)

with the initial and final states are located at $\tau=-\frac{T}{2}$ and
$\tau=\frac{T}{2}$ respectively. This cylinder amplitude is depicted in Figure
1. Since the right-moving part of the external states we consider is always
the unique NS vacuum on all copies, the right-moving part of the correlator in
(2.36) will be determined simply by the universal $D\times D$ OPE, thus
contributing a factor independent of the initial and final states of the form

$\langle
0_{\scriptscriptstyle{N\\!S}}|\big{(}\bar{G}^{+}_{\dot{D},-\frac{1}{2}}\bar{\sigma}^{-}_{2}\big{)}(\bar{w})\,\big{(}\bar{G}^{-}_{\dot{B},-\frac{1}{2}}\bar{\sigma}^{+}_{2}\big{)}(0)|0_{\scriptscriptstyle{N\\!S}}\rangle=\frac{-\epsilon_{\dot{D}\dot{B}}}{4\sinh^{2}(\frac{\bar{w}}{2})}\
.$ (2.37)

This right-moving amplitude was derived in [44]. It is then convenient to
write the integrated amplitude $X(T)$ as

$\displaystyle X(T)$
$\displaystyle=\epsilon^{\dot{C}\dot{D}}\epsilon^{\dot{A}\dot{B}}\int
d^{2}w\,A(w,0)\epsilon_{\dot{A}\dot{C}}\bigg{(}\frac{-\epsilon_{\dot{D}\dot{B}}}{4\sinh^{2}(\frac{\bar{w}}{2})}\bigg{)}$
$\displaystyle=-\frac{1}{2}\epsilon^{\dot{C}\dot{D}}\epsilon^{\dot{A}\dot{B}}\epsilon_{\dot{D}\dot{B}}\epsilon_{\dot{C}\dot{A}}\int
d^{2}w\,A(w,0)\,\partial_{\bar{w}}\Big{(}\coth(\tfrac{\bar{w}}{2})\Big{)}$
$\displaystyle=\frac{i}{2}\int_{C}dw\,A(w,0)\coth(\tfrac{\bar{w}}{2})$
$\displaystyle\equiv I_{C_{1}}+I_{C_{2}}+I_{C_{3}}\ ,$ (2.38)

where $A(w_{2},w_{1})$ is the non-integrated left-moving amplitude

$A(w_{2},w_{1})\equiv\big{<}\Phi\big{|}\Big{(}G^{+}_{-,-\frac{1}{2}}\sigma^{-}_{2}\Big{)}(w_{2})\Big{(}G^{-}_{+,-\frac{1}{2}}\sigma^{+}_{2}\Big{)}(w_{1})\big{|}\Phi\big{>}\
.$ (2.39)

The contour integrals $I_{C_{1}}$, $I_{C_{2}}$, $I_{C_{3}}$ in (2.5) are given
by

$\displaystyle I_{C_{1}}(T)$
$\displaystyle\equiv-\frac{1}{2}\int^{2\pi}_{0}\\!d\sigma\,A(w,0)\coth(\tfrac{\bar{w}}{2})\
,$ (2.40a) $\displaystyle I_{C_{2}}(T)$
$\displaystyle\equiv\frac{1}{2}\int^{2\pi}_{0}\\!d\sigma\,A(w,0)\coth(\tfrac{\bar{w}}{2})\
,$ (2.40b) $\displaystyle I_{C_{3}}$
$\displaystyle\equiv-\frac{i}{2}\oint_{|w|=\epsilon}\\!dw\,A(w,0)\coth(\tfrac{\bar{w}}{2})\
,$ (2.40c)

where the contours $C_{1}$, $C_{2}$, $C_{3}$ on the cylinder are at
$\tau=\frac{T}{2},-\frac{T}{2}$ and around $|w|=\epsilon$ respectively and the
complex coordinates on the cylinder have been written as $w=\tau+i\sigma$ and
$\bar{w}=\tau-i\sigma$. These contours are depicted in Figure 2.

Figure 2: Here the three boundary contour integrals $I_{C_{1}},I_{C_{2}}$ and
$I_{C_{3}}$ as defined in equations (2.40) are shown on the $w$-cylinder. The
sum of these contour integrals gives the integrated amplitude in (2.5)
required in the lift (2.35).

In (2.40c) the cutoff $\epsilon$ should be taken to $0$ at the end of the
calculation. The computation thus boils down to evaluating the non-integrated
left-moving amplitude $A(w,0)$. We note that, as found in [44], the integral
$I_{C_{3}}$ for the states we consider will contribute only a universal
divergent piece $\sim\frac{1}{\epsilon}$ which can be removed by a
counterterm. Thus the integral $I_{C_{3}}$ does not contribute to the lift
(2.35) for us.

The computation of the lift via the method described above only sees a single
pair of copies at any given time999This is because of the order-2 twist fields
$\sigma_{2}$ in the deformation operators $D$ act on a pair of copies of the
seed theory and the lift being related to an expectation value. This forces
the second twist operator to act on exactly the same pair of copies as the
first. and as such throughout the calculation it is sufficient to consider
only a specific ordered pair of copies, with one copy being excited and one
copy in the NS-NS vacuum. As derived in Section (3.1) of [48] by combinatoric
arguments, the final $N$-dependent lifting result is then

$E^{(2)}_{N}\\!\big{(}|\Phi\rangle\big{)}=2(N-1)E^{(2)}\\!\big{(}|\Phi\rangle\big{)}\
,$ (2.41)

where $E^{(2)}(|\Phi\rangle)$ is the lift that we work with and that will be
quoted in this paper.

##  3 Lifting of descendent states via series method

In this section we will consider the lifting of descendants of superconformal
primaries on a single copy. More precisely, let the state $|\phi\rangle$ be a
superconformal primary on one copy of the seed $c=6$ CFT, then $|\phi\rangle$
satisfies the conditions (2.9). Descendants of this primary are then found by
acting with negative modes of the $\mathcal{N}=4$ superconformal algebra given
in Appendix A.1; namely acting with the current modes

$\Big{\\{}L_{-n}\ ,\ \ J^{a}_{-n}\ ,\ \ G^{\alpha}_{\\!\dot{A},-s}\Big{\\}}\
,$ (3.1)

for $n>0$ and $s\geq\frac{1}{2}$. In actual fact, it is enough to study the
lifting of descendants formed from acting with $J$ and $G$ modes. Specifically
we consider the following states

$\displaystyle|G\phi\rangle^{\alpha}_{\dot{A}(h,s)}$
$\displaystyle\equiv\frac{1}{\sqrt{2}}\Big{(}G^{\alpha(1)}_{\dot{A},-s}|\phi\rangle^{(1)}|0_{\scriptscriptstyle{N\\!S}}\rangle^{(2)}+G^{\alpha(2)}_{\dot{A},-s}|0_{\scriptscriptstyle{N\\!S}}\rangle^{(1)}|\phi\rangle^{(2)}\Big{)}\
,$ (3.2) $\displaystyle|J\phi\rangle^{+}_{(h,n)}$
$\displaystyle\equiv\frac{1}{\sqrt{2}}\Big{(}J^{+(1)}_{-n}|\phi\rangle^{(1)}|0_{\scriptscriptstyle{N\\!S}}\rangle^{(2)}+J^{+(2)}_{-n}|0_{\scriptscriptstyle{N\\!S}}\rangle^{(1)}|\phi\rangle^{(2)}\Big{)}\
,$ (3.3)

where $\phi$ has conformal dimension $h$ and $J^{3}_{0}$ eigenvalue $j=0$ and
the factor of $\frac{1}{\sqrt{2}}$ is a copy normalisation. The respective bra
states are defined as

$\displaystyle{}^{\quad\ \ \,\beta}_{(h,s)\dot{B}}\langle G\phi|$
$\displaystyle\equiv\frac{1}{\sqrt{2}}\bigg{(}\\!{}^{\ \,(2)}_{\
\;\scriptscriptstyle{N\\!S}}\langle
0|\,{}^{\,(1)}\langle\phi|G^{\beta(1)}_{\dot{B},s}+{}^{(2)}\langle\phi|{}^{\
\,(1)}_{\ \;\scriptscriptstyle{N\\!S}}\langle
0|G^{\beta(2)}_{\dot{B},s}\bigg{)}\ ,$ (3.4)
$\displaystyle{}^{\quad\,-}_{(h,n)}\langle J\phi|$
$\displaystyle\equiv\frac{1}{\sqrt{2}}\bigg{(}\\!{}^{\ \,(2)}_{\
\;\scriptscriptstyle{N\\!S}}\langle
0|\,{}^{\,(1)}\langle\phi|J^{-(1)}_{n}+{}^{(2)}\langle\phi|{}^{\ \,(1)}_{\
\;\scriptscriptstyle{N\\!S}}\langle 0|J^{-(2)}_{n}\bigg{)}\ ,$ (3.5)

where the Hermitian conjugate state of $|G\phi\rangle^{\alpha}_{\dot{A}(h,s)}$
should have opposite quantum numbers (see A.2 for our conventions). Whilst
these states are not currently normalised on an individual copy, this is taken
into account in the lift formula (2.35). Since the method of lifting used here
is for D1-D5-P states, the right-moving part of the states is necessarily the
NS vacuum on all copies. The lift of single-copy superconformal primaries was
previously computed in closed form in [48].

###  3.1 Lifting of $|G\phi\rangle^{\alpha}_{\\!\dot{A}(h,s)}$

In this section we compute the lift of the superdescendant
$|G\phi\rangle^{\alpha}_{\dot{A}(h,s)}$. From Section 2.5 the key quantity we
need to compute is the left-moving amplitude

$A^{(1)(1)}_{h,s}(w_{2},w_{1})\equiv\langle\phi|G^{\beta}_{\dot{B},s}\,\big{(}G^{+}_{-,-\frac{1}{2}}\sigma^{-}_{2}\big{)}(w_{2})\big{(}G^{-}_{+,-\frac{1}{2}}\sigma^{+}_{2}\big{)}(w_{1})\,G^{\alpha}_{\\!\dot{A},-s}|\phi\rangle\
,$ (3.6)

where we have taken the descendant excitation to be on a definite copy, namely
copy 1, for both the initial and final state. The general method of computing
this amplitude will be similar to that employed in [49] with some
modifications to account for the fact that we do not specify any explicit form
of the state $|\phi\rangle$. The steps are as follows:

  1. (1)

The amplitude $A^{(1)(1)}_{h,s}(w_{2},w_{1})$ is on a doubly-covered cylinder
due to the presence of twist operators $\sigma_{2}$ joining and then splitting
the two copies of the untwisted initial and final states. This effect of the
twist operators can be resolved by mapping the amplitude to the covering
space. As shown in [61] and used in the context of lifting of states in [44,
48, 49], the covering space map $z\to t$ for the case of two order-2 twist
operators is given by

$z(t)=\frac{(t+a)(t+b)}{t}\ ,$ (3.7)

where $z=e^{w}$ is the doubly-covered plane. The twist operator insertions on
the plane at $z_{1}=e^{w_{1}}$ and $z_{2}=e^{w_{2}}$ are mapped to
$t_{1}=-\sqrt{ab}$ and $t_{2}=\sqrt{ab}$ on the covering space. In our
conventions the points $z=0$ and $z=\infty$ on the first sheet of the doubly-
covered plane map to $t=-a$ and $t=\infty$ on the covering space. Under this
map the initial state transforms as

$\displaystyle G^{\alpha}_{\\!\dot{A},-s}|\phi\rangle$
$\displaystyle=\oint_{\tau\to-\infty}\frac{dw_{3}}{2\pi
i}\,e^{-sw_{3}}\,G^{\alpha}_{\\!\dot{A}}(w_{3})|\phi\rangle$
$\displaystyle=\lim_{\tilde{t}\to-a}\tilde{z}^{h}\bigg{(}\frac{d\tilde{t}}{d\tilde{z}}\bigg{)}^{\\!h}\oint_{\tilde{t}}\,\frac{dt_{3}}{2\pi
i}\,z_{3}^{-s+\frac{1}{2}}\bigg{(}\frac{dt_{3}}{dz_{3}}\bigg{)}^{\\!\frac{1}{2}}G^{\alpha}_{\\!\dot{A}}(t_{3})\phi(\tilde{t}\,)|0_{\scriptscriptstyle{N\\!S}}\rangle\
,$ (3.8)

and likewise the final state transforms as

$\displaystyle\langle\phi|G^{\beta}_{\dot{B},s}$
$\displaystyle=-\oint_{\tau\to\infty}\frac{dw_{4}}{2\pi
i}\,e^{sw_{4}}\langle\phi|G^{\beta}_{\dot{B}}(w_{4})$
$\displaystyle=-\lim_{t^{\prime}\to\infty}z^{\prime
h}\bigg{(}\frac{dt^{\prime}}{dz^{\prime}}\bigg{)}^{\\!h}\oint_{t^{\prime}}\,\frac{dt_{4}}{2\pi
i}\,z_{4}^{s+\frac{1}{2}}\bigg{(}\frac{dt_{4}}{dz_{4}}\bigg{)}^{\\!\frac{1}{2}}\langle
0_{\scriptscriptstyle{N\\!S}}|\phi(t^{\prime})G^{\beta}_{\dot{B}}(t_{4})\ ,$
(3.9)

where the negative sign accounts for the direction of the contour at $\infty$.
Since the covering space map (3.7) is not conformal there will be an
associated Liouville factor; this along with various conformal factors from
the twist operators and spin fields are wrapped up into a “base amplitude”
[61, 60, 44]

$f_{1}=\frac{a-b}{4\sqrt{ab}}\ ,$ (3.10)

that is independent of the external states.

  2. (2)

While this map resolves the geometric effect of the two twist operators, we
are left with insertions of the spin fields $S^{-}(t_{2})$ and $S^{+}(t_{1})$.
These can in turn be removed by performing a spectral flow of the amplitude by
$\eta=-1$ and $\eta=+1$ units around $t_{1}$ and $t_{2}$ respectively. Under
these spectral flows, the state at $t=-a$ transforms as

$G^{\alpha}_{\dot{A}}(t_{3})\phi(\tilde{t}\,)|0_{\scriptscriptstyle{N\\!S}}\rangle\rightarrow\bigg{(}\frac{t_{3}-t_{1}}{t_{3}-t_{2}}\bigg{)}^{\\!q_{\alpha}}G^{\alpha}_{\dot{A}}(t_{3})\phi(\tilde{t}\,)|0_{\scriptscriptstyle{N\\!S}}\rangle\
,$ (3.11)

where $q_{\alpha}$ is the $J^{3}_{0}$ charge of $G^{\alpha}_{\dot{A},-s}$ and
the state at $t=\infty$ transforms as

$\langle
0_{\scriptscriptstyle{N\\!S}}|\phi(t^{\prime})G^{\beta}_{\dot{B}}(t_{4})\rightarrow\bigg{(}\frac{t_{4}-t_{1}}{t_{4}-t_{2}}\bigg{)}^{\\!q_{\beta}}\langle
0_{\scriptscriptstyle{N\\!S}}|\phi(\tilde{t}\,)G^{\beta}_{\dot{B}}(t_{4})\ .$
(3.12)

While the $G$ modes from the deformation operators do transform under the
above maps, their transformations do not depend on the external states under
consideration and so they contribute a universal factor of (see Section (4.3)
of [44])

$f_{2}=ab(a-b)\ .$ (3.13)

  3. (3)

Following steps (1) and (2) the amplitude (3.6) is now given by

$\displaystyle
A^{(1)(1)}_{h,s}=-f_{1}f_{2}f_{3}\oint_{\infty}\frac{dt_{4}}{2\pi
i}\oint_{-a}\frac{dt_{3}}{2\pi
i}\,\frac{z_{4}^{s+\frac{1}{2}}}{z_{3}^{s-\frac{1}{2}}}\bigg{(}\frac{dt_{3}}{dz_{3}}\bigg{)}^{\\!\frac{1}{2}}\bigg{(}\frac{dt_{4}}{dz_{4}}\bigg{)}^{\\!\frac{1}{2}}\bigg{(}\frac{t_{3}-t_{1}}{t_{3}-t_{2}}\bigg{)}^{\\!q_{\alpha}}\bigg{(}\frac{t_{4}-t_{1}}{t_{4}-t_{2}}\bigg{)}^{\\!q_{\beta}}\widetilde{C}^{\beta\alpha}_{\\!\dot{B}\dot{A}}(t_{4},t_{3})\
,$ (3.14)

where

$f_{3}\equiv\lim_{\begin{subarray}{c}\tilde{t}\to-a\\\
t^{\prime}\to\infty\end{subarray}}\tilde{z}^{h}z^{\prime
h}\bigg{(}\frac{d\tilde{t}}{d\tilde{z}}\bigg{)}^{\\!h}\bigg{(}\frac{dt^{\prime}}{dz^{\prime}}\bigg{)}^{\\!h}=\bigg{(}\frac{a}{a-b}\bigg{)}^{\\!h}\
,$ (3.15)

and $\widetilde{C}^{\beta\alpha}_{\dot{B}\dot{A}}$ is the correlator

$\widetilde{C}^{\beta\alpha}_{\\!\dot{B}\dot{A}}(t_{4},t_{3})\equiv\langle\phi|G^{\beta}_{\dot{B}}(t_{4})G^{+}_{-}(t_{2})G^{-}_{+}(t_{1})G^{\alpha}_{\dot{A}}(t_{3})|\phi\rangle\
.$ (3.16)

  4. (4)

In [49], the equivalent correlator on the $t$-plane was computed by writing
all field insertions in terms of the bosons and fermions of the free-field
realisation of the orbifold CFT and then using Wick contractions. Here,
however, it is not convenient to utilise Wick contractions of free fields or
OPEs of currents since the $G\times\phi$ OPE will contain the superpartner of
$\phi$, which is not known generally. Instead, we expand each of the fields in
terms of modes around $t=-a$ using

$G^{\alpha}_{\dot{A}}(t)=\sum_{s\in\mathbb{Z}+\frac{1}{2}}\,(t+a)^{-s-\frac{3}{2}}G^{\alpha}_{\dot{A},s}\
,$ (3.17)

for the fields inserted at $t_{1}$ and $t_{2}$. The remaining $G$ fields are
on contours around $-a$ and $\infty$ respectively and their mode expansions
are more complicated. For these, the integrands of the contour integrals in
(3.14) have to be expanded in powers of $t_{3}+a$ and $t_{4}+a$ yielding

$\displaystyle
A^{(1)(1)}_{h,s}=f_{1}f_{2}f_{3}\sum_{k_{4}=0}^{s-1/2}\sum_{k_{3}=0}^{s-1/2}\sum_{s_{1},s_{2}}(t_{2}+a)^{-s_{2}-\frac{3}{2}}(t_{1}+a)^{-s_{1}-\frac{3}{2}}\,\mathcal{K}^{(G)}_{k_{3}}\widetilde{\mathcal{K}}^{(G)}_{k_{4}}\,C^{\beta\alpha}_{\\!\dot{B}\dot{A}}\
,$ (3.18)

where $C^{\beta\alpha}_{\\!\dot{B}\dot{A}}$ is the mode correlator

$C^{\beta\alpha}_{\\!\dot{B}\dot{A}}\equiv\langle\phi|G^{\beta}_{\dot{B},s-k_{4}}G^{+}_{-,s_{2}}G^{-}_{+,s_{1}}G^{\alpha}_{\dot{A},-s+k_{3}}|\phi\rangle\
.$ (3.19)

The expansion coefficients in (3.18) can be written as

$\displaystyle\mathcal{K}^{(G)}_{k_{3}}$
$\displaystyle=\sum_{k_{1}=0}^{k_{3}}\sum_{k_{2}=0}^{k_{3}-k_{1}}\frac{(-1)^{k_{1}}a^{n-k_{1}}(a-b)^{1-n-k_{2}}}{\big{(}a+\sqrt{ab}\big{)}^{1+k_{3}-k_{1}-k_{2}}}\,{}^{n}C_{k_{1}}\,{}^{k_{2}+n-2}C_{k_{2}}\
,$ (3.20a) $\displaystyle\widetilde{\mathcal{K}}^{(G)}_{k_{4}}$
$\displaystyle=\sum_{\tilde{k}_{1}=0}^{n}\sum_{\tilde{k}_{2}=0}^{k_{4}-\tilde{k}_{1}}(-1)^{\tilde{k}_{1}}a^{\tilde{k}_{2}}(a-b)^{\tilde{k}_{1}}\Big{(}a-\sqrt{ab}\Big{)}^{k_{4}-\tilde{k}_{1}-\tilde{k}_{2}}\;{}^{n}C_{\tilde{k}_{1}}\,{}^{n-2+\tilde{k}_{2}}C_{\tilde{k}_{2}}\
,$ (3.20b)

where we use the integer $n=s+\frac{1}{2}$.

  5. (5)

The mode correlator (B) can be calculated using the mode algebra of the
$\mathcal{N}=4$ superconformal algebra (A.2) and the definition of primaries
(2.9). We give the expression for $C^{\beta\alpha}_{\\!\dot{B}\dot{A}}$ in
Appendix B.

  6. (6)

Given the result of the mode correlator $C^{\beta\alpha}_{\\!\dot{B}\dot{A}}$
the first step is to resum the $s_{1}$ and $s_{2}$ series in (3.18) that came
from the expansion of the $G$ fields of the deformation operators. Each term
in the mode correlator contains either one or two Kronecker delta functions
which localise one or both of these series. These series only have to be
performed once for all mode numbers $s$ of the initial state.

  7. (7)

The next step is to resum the remaining finite sums in (3.18) which depend on
the choice of initial state. The number of terms in each of these sums scales
linearly with the level of the initial state $G$ mode so has to be done per
data point.

  8. (8)

The explicit dependence of the amplitude on $w_{1}$ and $w_{2}$ can then be
recovered by using the relations

$a=e^{S}\cosh^{2}\\!\Big{(}\frac{\Delta{w}}{4}\Big{)}\quad,\quad
b=e^{S}\sinh^{2}\\!\Big{(}\frac{\Delta{w}}{4}\Big{)}\ ,$ (3.21)

where we define

$S\equiv{\frac{1}{2}}(w_{1}+w_{2})\quad,\quad\Delta w\equiv w_{2}-w_{1}\ .$
(3.22)

This yields the amplitude $A^{(1)(1)}_{h,s}(w_{2},w_{1})$ where both the
initial and final state have descendent excitations placed on copy 1. The
left-moving copy symmetric amplitude required for the lift of the state (3.2)
is then given by

$\displaystyle A_{h,s}(w,0)$ $\displaystyle\equiv{}^{\quad\ \
\,\beta}_{(h,s)\dot{B}}\langle
G\phi|\big{(}G^{+}_{-,-\frac{1}{2}}\sigma^{-}_{2}\big{)}(w)\big{(}G^{-}_{+,-\frac{1}{2}}\sigma^{+}_{2}\big{)}(0)|G\phi\rangle^{\alpha}_{\dot{A}(h,s)}$
$\displaystyle=\frac{1}{2}\sum_{i,j=1}^{2}A^{(j)(i)}_{h,s}(w,0)$
$\displaystyle=A^{(1)(1)}_{h,s}(w,0)+A^{(2)(1)}_{h,s}(w,0)\ ,$ (3.23)

where we have used that $A^{(2)(2)}=A^{(1)(1)}$ and $A^{(1)(2)}=A^{(2)(1)}$ on
general symmetry grounds. The amplitude $A^{(2)(1)}$ can be obtained from
$A^{(1)(1)}$ via the continuation

$A^{(2)(1)}_{h,s}(w,0)=A^{(1)(1)}_{h,s}(w+2\pi i,0)\ ,$ (3.24)

which has the effect of interchanging the role of the two copies that the
upper $\sigma_{2}$ twists.

  9. (9)

The norm of our descendant state can be found from the correlator

${}^{\quad\ \ \,\beta}_{(h,s)\dot{B}}\langle
G\phi|G\phi\rangle^{\alpha}_{\dot{A}(h,s)}=\epsilon_{\dot{B}\dot{A}}\epsilon^{\beta\alpha}\Big{(}s^{2}-\frac{1}{4}+h\Big{)}\
,$ (3.25)

where the quantum numbers of the conjugate state need to be opposite to the
ket state and appropriate negative signs added as per the conjugation
conventions of Appendix A.2. This definition of the Hermitian conjugate state
should also be enforced on the final state of the amplitude ((8)) in order for
it to be related to the lift of a particular state. The lift is then given by

$E^{(2)}\Big{(}|G\phi\rangle^{\alpha}_{\dot{A}(h,s)}\Big{)}=\frac{\pi\lambda^{2}}{\big{|}|G\phi\rangle^{\alpha}_{\dot{A}(h,s)}\big{|}^{2}}\lim_{T\to\infty}\int_{0}^{2\pi}\\!d\sigma\,A_{h,s}(w,0)\coth\\!\big{(}\tfrac{\bar{w}}{2}\big{)}\
,$ (3.26)

where $w=\frac{T}{2}+i\sigma$ and $\bar{w}=\frac{T}{2}-i\sigma$ with
$T\to\infty$ on this contour as given in (2.40a). Note that of the three
integrals in (2.40) it was argued in Section 2.5 that $I_{C_{3}}$ gives a
vanishing contribution to the lift and for the states considered here we find
that $\lim_{T\to\infty}I_{C_{1}}(T)=\lim_{T\to\infty}I_{C_{2}}(T)$ and so
(3.26) contains only one integral.

A shallow matrix of lifts for the particular states
$|G\phi\rangle^{+}_{+(h,s)}$ is given by:

(3.29)

$s+\frac{1}{2}$ | $E^{(2)}\big{(}|G\phi\rangle^{+}_{+(1,s)}\big{)}/\lambda^{2}$ | $h$ | $E^{(2)}\big{(}|G\phi\rangle^{+}_{+(h,\frac{1}{2})}\big{)}/\lambda^{2}$ | $E^{(2)}\big{(}|G\phi\rangle^{+}_{+(h,h-\frac{1}{2})}\big{)}/\lambda^{2}$
---|---|---|---|---
1 | $\frac{\pi^{2}}{2}$ | 1 | $\frac{\pi^{2}}{2}$ | $\frac{\pi^{2}}{2}$
2 | $\frac{13\pi^{2}}{24}$ | 2 | $\frac{3\pi^{2}}{4}$ | $\frac{201\pi^{2}}{256}$
3 | $\frac{1045\pi^{2}}{1792}$ | 3 | $\frac{15\pi^{2}}{16}$ | $\frac{2035\pi^{2}}{2048}$
4 | $\frac{2047\pi^{2}}{3328}$ | 4 | $\frac{35\pi^{2}}{32}$ | $\frac{305375\pi^{2}}{262144}$
5 | $\frac{880799\pi^{2}}{1376256}$ | 5 | $\frac{315\pi^{2}}{256}$ | $\frac{11016927\pi^{2}}{8388608}$
6 | $\frac{167659\pi^{2}}{253952}$ | 6 | $\frac{693\pi^{2}}{512}$ | $\frac{1552560779\pi^{2}}{1073741824}$
7 | $\frac{61061485\pi^{2}}{90177536}$ | 7 | $\frac{3003\pi^{2}}{2048}$ | $\frac{13458915613\pi^{2}}{8589934592}$
8 | $\frac{82679905\pi^{2}}{119537664}$ | 8 | $\frac{6435\pi^{2}}{4096}$ | $\frac{1845612417915\pi^{2}}{1099511627776}$
9 | $\frac{220850992003\pi^{2}}{313532612608}$ | 9 | $\frac{109395\pi^{2}}{65536}$ | $\frac{250928035737055\pi^{2}}{140737488355328}$
10 | $\frac{139865669375\pi^{2}}{195421011968}$ | 10 | $\frac{230945\pi^{2}}{131072}$ | $\frac{8472249907603625\pi^{2}}{4503599627370496}$
11 | $\frac{11074140097459\pi^{2}}{15255723835392}$ | 11 | $\frac{969969\pi^{2}}{524288}$ | $\frac{71132256003286443\pi^{2}}{36028797018963968}$
12 | $\frac{13438244584645\pi^{2}}{18279380811776}$ | 12 | $\frac{2028117\pi^{2}}{1048576}$ | $\frac{38055140340356147281\pi^{2}}{18446744073709551616}$
13 | $\frac{4107842603142835\pi^{2}}{5523946417946624}$ | 13 | $\frac{16900975\pi^{2}}{8388608}$ | $\frac{1267812416041182611425\pi^{2}}{590295810358705651712}$
14 | $\frac{1209633084998329\pi^{2}}{1609685023064064}$ | 14 | $\frac{35102025\pi^{2}}{16777216}$ | $\frac{84214113560913762070065\pi^{2}}{37778931862957161709568}$
15 | $\frac{180249923867363485\pi^{2}}{237564880343793664}$ | 15 | $\frac{145422675\pi^{2}}{67108864}$ | $\frac{697387735039463877232775\pi^{2}}{302231454903657293676544}$
16 | $\frac{207717974301142693\pi^{2}}{271341877549072384}$ | 16 | $\frac{300540195\pi^{2}}{134217728}$ | $\frac{184381092183290428038099635\pi^{2}}{77371252455336267181195264}$
17 | $\frac{3887143893143114780555\pi^{2}}{5035961132122707591168}$ | 17 | $\frac{9917826435\pi^{2}}{4294967296}$ | $\frac{97300623587685344239590415359\pi^{2}}{39614081257132168796771975168}$
18 | $\frac{2202563353701091080685\pi^{2}}{2831575215314416173056}$ | 18 | $\frac{20419054425\pi^{2}}{8589934592}$ | $\frac{3203499112728575370103597789355\pi^{2}}{1267650600228229401496703205376}$
19 | $\frac{158637441680129531942191\pi^{2}}{202471462953036038537216}$ | 19 | $\frac{83945001525\pi^{2}}{34359738368}$ | $\frac{26326239008093207523832307852065\pi^{2}}{10141204801825835211973625643008}$
20 | $\frac{177415422817791031035205\pi^{2}}{224902703746666853302272}$ | 20 | $\frac{172308161025\pi^{2}}{68719476736}$ | $\frac{3456688488715760869871367416311599\pi^{2}}{1298074214633706907132624082305024}$
Table 1: We give the values of lifts of the states
$E^{(2)}\big{(}|G\phi\rangle^{+}_{+(h,s)}\big{)}/\lambda^{2}$ for three
1-parameter families: fixed small primary dimension but varying descendant
mode number $s$, fixed small descendant mode number but varying primary
dimension $h$, and a diagonal family with $s\sim h$. We choose to present the
first twenty lifts in each of these families.

Figure 3: Plot of the lifts
$E^{(2)}\big{(}|G\phi\rangle^{+}_{+(1,s)}\big{)}/\lambda^{2}$ for varying $s$.
The plot fits to the curve: $3.5943+1.45554\sqrt{s+1/2}-0.114676(s+1/2)$.

Figure 4: Plot of the lifts
$E^{(2)}\big{(}|G\phi\rangle^{+}_{+(h,1/2)}\big{)}/\lambda^{2}$ for varying
$h$. The plot fits to the curve: $-0.707422+5.7688\sqrt{h}-0.0167021h$.

Figure 5: Plot of the lifts
$E^{(2)}\big{(}|G\phi\rangle^{+}_{+(h,h-1/2)}\big{)}/\lambda^{2}$ for varying
$h$. The plot fits to the curve: $-1.52527+6.68922\sqrt{h}-0.104878h.$

###  3.2 Lifting of $|J\phi\rangle^{+}_{(h,n)}$

In this section we consider the lifting of $J^{+}$ descendants of a
superconformal primary state $|\phi\rangle$; the copy-symmetric state is
defined in (3.3). The lift of these descendant states is amenable to the same
method used in Section 3.1 and thus we will follow the same steps (1)-(9)
described there. Since this computation is simpler than that of $G$
descendants, as will be described presently, and that in Section 4.1 it will
turn out to be possible to obtain an exact analytic result for the lift using
a method of Ward identities, we give fewer details in this section.

From Section 2.5 the key quantity to be computed is the left-moving amplitude

$A^{(1)(1)}_{h,n}\equiv\langle\phi|J^{-}_{n}\,\big{(}G^{+}_{-,-\frac{1}{2}}\sigma^{-}_{2}\big{)}(w_{2})\big{(}G^{-}_{+,-\frac{1}{2}}\sigma^{+}_{2}\big{)}(w_{1})\,J^{+}_{-n}|\phi\rangle\
,$ (3.30)

with the initial and final state excitations placed on copy 1 for
concreteness. Using step (1) of the above-described method, we map this
amplitude to the covering space $t$ under which, analogously to ((1)) and
((1)), the initial state transforms as

$\displaystyle J^{+}_{-n}|\phi\rangle$
$\displaystyle=\oint_{\tau\to-\infty}\frac{dw_{3}}{2\pi
i}\,e^{-nw_{3}}\,J^{+}(w_{3})|\phi\rangle$
$\displaystyle=\lim_{\tilde{t}\to-a}\tilde{z}^{h}\bigg{(}\frac{d\tilde{t}}{d\tilde{z}}\bigg{)}^{\\!h}\oint_{\tilde{t}}\,\frac{dt_{3}}{2\pi
i}\,z_{3}^{-n}J^{+}(t_{3})\phi(\tilde{t}\,)|0_{\scriptscriptstyle{N\\!S}}\rangle\
,$ (3.31)

and the final state as

$\displaystyle\langle\phi|J^{-}_{n}$
$\displaystyle=-\oint_{\tau\to\infty}\frac{dw_{4}}{2\pi
i}\,e^{nw_{4}}\langle\phi|J^{-}(w_{4})$
$\displaystyle=-\lim_{t^{\prime}\to\infty}z^{\prime
h}\bigg{(}\frac{dt^{\prime}}{dz^{\prime}}\bigg{)}^{\\!h}\oint_{t^{\prime}}\,\frac{dt_{4}}{2\pi
i}\,z_{4}^{n}\langle
0_{\scriptscriptstyle{N\\!S}}|\phi(t^{\prime})J^{-}(t_{4})\ ,$ (3.32)

where the negative sign accounts for the direction of the contour at $\infty$.
The universal factor (3.10) will still be present since it is independent of
the external state under consideration. From step (2) the spectral flows that
remove spin field insertions on the covering space transform the state at
$t=-a$ as

$J^{+}(t_{3})\phi(\tilde{t}\,)|0_{\scriptscriptstyle{N\\!S}}\rangle\rightarrow\frac{t_{3}-t_{1}}{t_{3}-t_{2}}J^{+}(t_{3})\phi(\tilde{t}\,)|0_{\scriptscriptstyle{N\\!S}}\rangle\
,$ (3.33)

and the state at $t=\infty$ transforms as

$\langle
0_{\scriptscriptstyle{N\\!S}}|\phi(t^{\prime})J^{-}(t_{4})\rightarrow\frac{t_{4}-t_{2}}{t_{4}-t_{1}}\langle
0_{\scriptscriptstyle{N\\!S}}|\phi(t^{\prime})J^{-}(t_{4})\ .$ (3.34)

Once again, the transformations of the remaining $G$ field insertions at
$t_{1}$ and $t_{2}$ contribute the universal factor (3.13). The amplitude
(3.30) can then be written as

$\displaystyle
A^{(1)(1)}_{h,n}=-f_{1}f_{2}f_{3}\oint_{\infty}\frac{dt_{4}}{2\pi
i}\oint_{-a}\frac{dt_{3}}{2\pi
i}\,\frac{z_{4}^{n}}{z_{3}^{n}}\bigg{(}\frac{t_{3}-t_{1}}{t_{3}-t_{2}}\bigg{)}\bigg{(}\frac{t_{4}-t_{2}}{t_{4}-t_{1}}\bigg{)}\widetilde{C}^{+}(t_{4},t_{3})\
,$ (3.35)

where the factor $f_{3}$ is given in (3.15) and
$\widetilde{C}^{+}(t_{4},t_{3})$ is the correlator

$\widetilde{C}^{+}(t_{4},t_{3})\equiv\langle\phi|J^{-}(t_{4})G^{+}_{-}(t_{2})G^{-}_{+}(t_{1})J^{+}(t_{3})|\phi\rangle\
.$ (3.36)

As explained in step (4), this correlator will be computed by expanding all
currents in terms of their modes around the point $t=-a$ using (3.17) for the
$G$ fields at $t_{1}$ and $t_{2}$ and expansions of the integrand in (3.35)
for the remaining $J$ fields. This yields the form

$\displaystyle
A^{(1)(1)}_{h,n}=f_{1}f_{2}f_{3}\sum_{k_{4}=0}^{n-1}\sum_{k_{3}=0}^{n-1}\sum_{s_{1},s_{2}}(t_{2}+a)^{-s_{2}-\frac{3}{2}}(t_{1}+a)^{-s_{1}-\frac{3}{2}}\,\mathcal{K}^{(J)}_{k_{3}}\widetilde{\mathcal{K}}^{(J)}_{k_{4}}\,C^{+}\
,$ (3.37)

where $C^{+}$ is the mode correlator

$C^{+}\equiv\langle\phi|J^{-}_{n-k_{4}}G^{+}_{-,s_{2}}G^{-}_{+,s_{1}}J^{+}_{-n+k_{3}}|\phi\rangle\
.$ (3.38)

The expansion coefficients in (3.37) can be written as

$\displaystyle\mathcal{K}^{(J)}_{k_{3}}$
$\displaystyle=\sum_{k=0}^{1}\sum_{k_{1}=0}^{k_{3}-k}\sum_{k_{2}=0}^{k_{1}}\frac{(-1)^{1+k_{1}-k_{2}}\;{}^{n+k_{2}-1}C_{k_{2}}\;^{n}C_{k_{1}-k_{2}}(1-\delta_{k,0}(1+a-\sqrt{ab}))}{(a-b)^{k_{2}+n}a^{k_{1}-k_{2}-n}(a+\sqrt{ab})^{k_{3}-k-k_{1}+1}}\
,$ (3.39a) $\displaystyle\widetilde{\mathcal{K}}^{(J)}_{k_{4}}$
$\displaystyle=\sum_{\tilde{k}=0}^{1}\sum_{\tilde{k}_{1}=0}^{k_{4}+\tilde{k}-1}\sum_{\tilde{k}_{2}=0}^{\tilde{k}_{1}}\frac{(-1)^{\tilde{k}_{2}}\;{}^{n}C_{\tilde{k}_{2}}\;^{n+\tilde{k}_{1}-\tilde{k}_{2}-1}C_{\tilde{k}_{1}-\tilde{k}_{2}}(1-\delta_{\tilde{k},0}(1+a+\sqrt{ab}))}{(a-b)^{-\tilde{k}_{2}}a^{\tilde{k}_{2}-\tilde{k}_{1}}(a-\sqrt{ab})^{1+\tilde{k}_{1}-\tilde{k}-k_{4}}}\
.$ (3.39b)

Computing the mode correlator (3.38) using the $\mathcal{N}=4$ algebra (A.2)
and the primary definition (2.9) gives

$\displaystyle C^{+}$
$\displaystyle=\Big{(}s_{2}^{2}-\frac{1}{4}+h\Big{)}H\big{[}s_{2}-\tfrac{1}{2}\big{]}\bigg{(}H[k_{3}-k_{4}-1]-H\big{[}n-s_{1}-k_{3}-\tfrac{1}{2}\big{]}\bigg{)}\,\delta_{s_{1}+s_{2}+k_{3}-k_{4},0}$
$\displaystyle\quad+(n-k_{3})\Big{(}s_{2}^{2}-\frac{1}{4}+h\Big{)}H\big{[}\\!-\\!s_{1}-\tfrac{1}{2}\big{]}\delta_{s_{1}+s_{2},0}\,\delta_{k_{3},k_{4}}$
$\displaystyle\quad-\Big{(}s_{1}^{2}-\frac{1}{4}+h\Big{)}H\big{[}\\!-\\!s_{1}-\tfrac{1}{2}\big{]}H[k_{3}-k_{4}-1]\,\delta_{s_{1}+s_{2}+k_{3}-k_{4},0}$
$\displaystyle\quad+\Big{(}(s_{2}+n-k_{4})^{2}-\frac{1}{4}+h\Big{)}H\big{[}n-s_{1}-k_{3}-\tfrac{1}{2}\big{]}\,\delta_{s_{1}+s_{2}+k_{3}-k_{4},0}\
,$ (3.40)

where we use the discrete step function

$H[n]\equiv\begin{cases}\ 1\quad\text{if }n\geq 0\vspace{4pt}\\\ \
0\quad\text{otherwise}\end{cases}\ .$ (3.41)

Resumming the series in (3.37) as described in steps (6) and (7) can again be
done per choice of $h,n$ and an amplitude as a function of $w_{1}$, $w_{2}$
obtained by using the relations (3.21). As in ((8)) the copy symmetric
amplitude is given by

$\displaystyle A_{h,n}(w,0)$ $\displaystyle\equiv{}^{\quad\,-}_{(h,n)}\langle
J\phi|\big{(}G^{+}_{-,-\frac{1}{2}}\sigma^{-}_{2}\big{)}(w)\big{(}G^{-}_{+,-\frac{1}{2}}\sigma^{+}_{2}\big{)}(0)|J\phi\rangle^{+}_{(h,n)}$
$\displaystyle=\frac{1}{2}\sum_{i,j=1}^{2}A^{(j)(i)}_{h,n}(w,0)$
$\displaystyle=A^{(1)(1)}_{h,n}(w,0)+A^{(2)(1)}_{h,n}(w,0)\ ,$ (3.42)

with $A^{(2)(1)}$ given in terms of $A^{(1)(1)}$ by (3.24). Using the norm

${}^{\quad\,-}_{(h,n)}\langle J\phi|J\phi\rangle^{+}_{(h,n)}=n\ ,$ (3.43)

the lift is then given by

$E^{(2)}\Big{(}|J\phi\rangle^{+}_{(h,n)}\Big{)}=\frac{\pi\lambda^{2}}{\big{|}|J\phi\rangle^{+}_{(h,n)}\big{|}^{2}}\lim_{T\to\infty}\int_{0}^{2\pi}\\!d\sigma\,A_{h,n}(w,0)\coth\\!\big{(}\tfrac{\bar{w}}{2}\big{)}\
,$ (3.44)

where $w=\frac{T}{2}+i\sigma$ and $\bar{w}=\frac{T}{2}-i\sigma$ with
$T\to\infty$ on this contour as given in (2.40a) (noting again the comments
below equation (3.26)).

A shallow matrix of lifts for the states $|J\phi\rangle^{+}_{(h,n)}$ is given
by:

(3.47)

$n$ | $E^{(2)}\big{(}|J\phi\rangle^{+}_{(1,n)}\big{)}/\lambda^{2}$ | $h$ | $E^{(2)}\big{(}|J\phi\rangle^{+}_{(h,1)}\big{)}/\lambda^{2}$ | $E^{(2)}\big{(}|J\phi\rangle^{+}_{(h,h)}\big{)}/\lambda^{2}$
---|---|---|---|---
1 | $\frac{\pi^{2}}{2}$ | 1 | $\frac{\pi^{2}}{2}$ | $\frac{\pi^{2}}{2}$
2 | $\frac{9\pi^{2}}{16}$ | 2 | $\frac{3\pi^{2}}{4}$ | $\frac{51\pi^{2}}{64}$
3 | $\frac{77\pi^{2}}{128}$ | 3 | $\frac{15\pi^{2}}{16}$ | $\frac{1029\pi^{2}}{1024}$
4 | $\frac{645\pi^{2}}{1024}$ | 4 | $\frac{35\pi^{2}}{32}$ | $\frac{38505\pi^{2}}{32768}$
5 | $\frac{21367\pi^{2}}{32768}$ | 5 | $\frac{315\pi^{2}}{256}$ | $\frac{5547075\pi^{2}}{4194304}$
6 | $\frac{87857\pi^{2}}{131072}$ | 6 | $\frac{693\pi^{2}}{512}$ | $\frac{390395523\pi^{2}}{268435456}$
7 | $\frac{719069\pi^{2}}{1048576}$ | 7 | $\frac{3003\pi^{2}}{2048}$ | $\frac{6762666295\pi^{2}}{4294967296}$
8 | $\frac{11730309\pi^{2}}{16777216}$ | 8 | $\frac{6435\pi^{2}}{4096}$ | $\frac{115843402197\pi^{2}}{68719476736}$
9 | $\frac{1526945171\pi^{2}}{2147483648}$ | 9 | $\frac{109395\pi^{2}}{65536}$ | $\frac{125934725576295\pi^{2}}{70368744177664}$
10 | $\frac{6199001523\pi^{2}}{8589934592}$ | 10 | $\frac{230945\pi^{2}}{131072}$ | $\frac{2125128794403615\pi^{2}}{1125899906842624}$
11 | $\frac{50252720171\pi^{2}}{68719476736}$ | 11 | $\frac{969969\pi^{2}}{524288}$ | $\frac{35672711189099061\pi^{2}}{18014398509481984}$
12 | $\frac{406850316977\pi^{2}}{549755813888}$ | 12 | $\frac{2028117\pi^{2}}{1048576}$ | $\frac{4769806285786438533\pi^{2}}{2305843009213693952}$
13 | $\frac{13161421594619\pi^{2}}{17592186044416}$ | 13 | $\frac{16900975\pi^{2}}{8388608}$ | $\frac{635476448618853061493\pi^{2}}{295147905179352825856}$
14 | $\frac{53172571377153\pi^{2}}{70368744177664}$ | 14 | $\frac{35102025\pi^{2}}{16777216}$ | $\frac{21101411540283429819765\pi^{2}}{9444732965739290427392}$
15 | $\frac{429306030983453\pi^{2}}{562949953421312}$ | 15 | $\frac{145422675\pi^{2}}{67108864}$ | $\frac{349425974935218889182525\pi^{2}}{151115727451828646838272}$
16 | $\frac{13855330294144389\pi^{2}}{18014398509481984}$ | 16 | $\frac{300540195\pi^{2}}{134217728}$ | $\frac{5773130365885258168314605\pi^{2}}{2417851639229258349412352}$
17 | $\frac{7150471815953612587\pi^{2}}{9223372036854775808}$ | 17 | $\frac{9917826435\pi^{2}}{4294967296}$ | $\frac{48738566651126418399719594511\pi^{2}}{19807040628566084398385987584}$
18 | $\frac{28815168723125441183\pi^{2}}{36893488147419103232}$ | 18 | $\frac{20419054425\pi^{2}}{8589934592}$ | $\frac{802233196347055292535813677217\pi^{2}}{316912650057057350374175801344}$
19 | $\frac{232135596668140476407\pi^{2}}{295147905179352825856}$ | 19 | $\frac{83945001525\pi^{2}}{34359738368}$ | $\frac{13184066094015683210008550256055\pi^{2}}{5070602400912917605986812821504}$
20 | $\frac{1869337969360052852787\pi^{2}}{2361183241434822606848}$ | 20 | $\frac{172308161025\pi^{2}}{68719476736}$ | $\frac{432733110292413371373686195637405\pi^{2}}{162259276829213363391578010288}$
Table 2: We give the values of lifts of the states
$E^{(2)}\big{(}|J\phi\rangle^{+}_{(h,n)}\big{)}/\lambda^{2}$ for three
1-parameter families: fixed small primary dimension but varying descendant
mode number $n$, fixed small descendant mode number but varying primary
dimension $h$, and a diagonal family with $n=h$. We choose to present the
first twenty lifts in each of these families.

Figure 6: Plot of the lifts
$E^{(2)}\big{(}|J\phi\rangle^{+}_{(1,n)}\big{)}/\lambda^{2}$ for varying $n$.
The plot fits to the curve: $3.79158+1.39824\sqrt{n}-0.109782n$.

Figure 7: Plot of the lifts
$E^{(2)}\big{(}|J\phi\rangle^{+}_{(h,1)}\big{)}/\lambda^{2}$ for varying $h$.
The plot fits to the curve: $-0.820134+5.85066\sqrt{h}-0.0292201h$.

Figure 8: Plot of the lifts
$E^{(2)}\big{(}|J\phi\rangle^{+}_{(n,n)}\big{)}/\lambda^{2}$ for varying $n$.
The plot fits to the curve: $-1.35675+6.61605\sqrt{h}-0.0935374h$.

###  3.3 Symmetries and checks

There are a few immediate checks of the results of Sections 3.1 and 3.2 and
therefore, on the method used. Firstly and most simply, the lift of states in
the same $SU(2)$ multiplet – be it the $SU(2)_{L}$ or $SU(2)_{2}$ under which
the states $|G\phi\rangle$ and $|J\phi\rangle$ are charged – should be the
same. The calculation of Section 3.1 in steps (1)-(9) is written for generic
$\alpha$ and $\dot{A}$ charges on the $G$ mode and so this can be immediately
checked. The cases of $J^{-}$ and $J^{3}$ work similarly.

Figure 9:  The general tactic in deriving these relations between lifts will
be to start with the correlator (3.3) containing descendants of $|\phi\rangle$
as initial and final states (a), deform the contour of the $G_{-1/2}$ mode
through the insertions of the deformation operators (b)-(d) and then commute
the $G_{-1/2}$ mode with the $G_{+/2}$ mode from the final state. This leaves
(up to constant coefficients) a correlator that computes the lift of the state
$|\phi\rangle$ (e) and one that computes the lift of the state
$G_{1/2}|\phi\rangle$ (f) which vanishes due to $|\phi\rangle$ being a
superconformal primary. We have suppressed the trivial right-moving part
throughout. See [49] for a complete derivation.

The second check against known results emerges by considering an explicit
realisation of the state $|J\phi\rangle^{+}_{(1,1)}$ in terms of the free
boson and fermion realisation of the theory. A prescription for finding
explicit realisations of (at least an infinite family of) superconformal
primaries at arbitrary levels was given in Section 2.2, however, here we
simply need a level-1 example. Taking the $(h,m)=(1,0)$ superconformal primary
with minimal charge under $SU(2)_{1}$

$|\phi^{1}_{1,-1}\rangle=d^{--}_{-\frac{1}{2}}d^{+-}_{-\frac{1}{2}}|0_{\scriptscriptstyle{N\\!S}}\rangle\
,$ (3.48)

the following is then a realisation of its level-1 $J$ descendant

$J^{+}_{-1}|\phi^{1}_{1,-1}\rangle=d^{+-}_{-\frac{3}{2}}d^{+-}_{-\frac{1}{2}}|0_{\scriptscriptstyle{N\\!S}}\rangle\
.$ (3.49)

The lift of this two-mode state was found in [49] to be (using the notation of
that paper)

$E^{(2)}\big{(}|dd\rangle^{+-+-}_{(3/2,1/2)}\big{)}=\frac{\pi^{2}\lambda^{2}}{2}\
,$ (3.50)

matching the first element in the lifting matrix (3.47). Similarly,
considering an explicit realisation of the descendant state
$|G\phi\rangle^{+}_{+(1,1/2)}$ using (3.48), giving the two-mode state

$G^{+}_{+,-\frac{1}{2}}|\phi^{1}_{1,-1}\rangle=i\,\alpha_{++,-1}d^{+-}_{-\frac{1}{2}}|0_{\scriptscriptstyle{N\\!S}}\rangle\
,$ (3.51)

whose lift was found in [49] to be

$E^{(2)}\big{(}|\alpha
d\rangle^{+-}_{++(1,1/2)}\big{)}=\frac{\pi^{2}\lambda^{2}}{2}\ ,$ (3.52)

in agreement with the first element of the lifting matrix (3.29).

The third check is for the families of states
$|G\phi\rangle^{\alpha}_{\\!\dot{A}(h,1/2)}$ and relating their lifts to that
of the single-copy superconformal primary $|\phi\rangle$. Here we only briefly
describe the key steps in deriving such relations, however, full details can
be found in Section (4) of [49]. We start with the initial state
$G^{\alpha(1)}_{\\!\dot{A},-\frac{1}{2}}|\phi\rangle^{(1)}|0_{\scriptscriptstyle{N\\!S}}\rangle^{(2)}$
in the amplitude (3.6) on the $w$-cylinder. The single-copy mode
$G^{\alpha(1)}_{\\!\dot{A},-\frac{1}{2}}$ can be trivially made into a global
mode $G^{\alpha}_{\\!\dot{A},-\frac{1}{2}}$, as defined around Equation
(2.10), since it annihilates the NS vacuum state on all but one copy. This
starting point is depicted in Figure 9(a). By writing this global mode as a
contour of a $G$ field it can be passed through the two deformation operator
insertions to the final state (see Figure 9(b)-(c)), yielding the equality
between amplitudes

$\displaystyle A^{\alpha}_{\\!\dot{A}}\big{(}G\phi\big{)}$
$\displaystyle\equiv\langle
0_{\scriptscriptstyle{N\\!S}}|\langle\phi|\big{(}G^{\alpha}_{\\!\dot{A},-\frac{1}{2}}\big{)}^{\dagger}\int\\!d^{2}w_{2}\,D(w_{2},\bar{w}_{2})\int\\!d^{2}w_{1}\,D(w_{1},\bar{w}_{1})\,G^{\alpha}_{\\!\dot{A},-\frac{1}{2}}|\phi\rangle|0_{\scriptscriptstyle{N\\!S}}\rangle$
$\displaystyle=\langle
0_{\scriptscriptstyle{N\\!S}}|\langle\phi|\big{(}G^{\alpha}_{\\!\dot{A},-\frac{1}{2}}\big{)}^{\dagger}G^{\alpha}_{\\!\dot{A},-\frac{1}{2}}\int\\!d^{2}w_{2}\,D(w_{2},\bar{w}_{2})\int\\!d^{2}w_{1}\,D(w_{1},\bar{w}_{1})|\phi\rangle|0_{\scriptscriptstyle{N\\!S}}\rangle\
.$ (3.53)

The initial state $G$ mode can then be anti-commuted through the final state
$G$ mode to annihilate both the superconformal primary $\langle\phi|$ (using
the defining relations (2.9)) and the NS vacuum $\langle
0_{\scriptscriptstyle{N\\!S}}|$, leaving a term with an $L_{0}$ insertion from
the anti-commutator (see Figure 9(d)-(f)). The amplitude (3.3) is then given
by

$\displaystyle
A^{\alpha}_{\\!\dot{A}}\big{(}G\phi\big{)}=K^{\alpha}_{\\!\dot{A}}\,\langle
0_{\scriptscriptstyle{N\\!S}}|\langle\phi|\int\\!d^{2}w_{2}\,D(w_{2},\bar{w}_{2})\int\\!d^{2}w_{1}\,D(w_{1},\bar{w}_{1})\,|\phi\rangle|0_{\scriptscriptstyle{N\\!S}}\rangle=K^{\alpha}_{\\!\dot{A}}\,A\big{(}\phi\big{)}\
,$ (3.54)

where the factor $K^{\alpha}_{\\!\dot{A}}$ depends on the dimension of $\phi$
and the $G$ mode chosen. This relation between amplitudes can then be
converted to a simple relation between lifts

$E^{(2)}\big{(}|G\phi\rangle^{\alpha}_{\dot{A}(h,1/2)}\big{)}=E^{(2)}\big{(}|\phi\rangle\big{)}\
,$ (3.55)

for all $\alpha$ and $\dot{A}$ values. The data for
$E^{(2)}(G^{+}_{+,-\frac{1}{2}}|\phi\rangle)$ for the first $20$ values of $h$
is given in the first column of Table 1 which can be seen to match the
analytic expression found in [48] for the lift of single-copy superconformal
primaries

$E^{(2)}_{h}\big{(}|\phi\rangle\big{)}=\pi^{\frac{3}{2}}\lambda^{2}\frac{\Gamma(h+\tfrac{1}{2})}{\Gamma(h)}\
.$ (3.56)

Likewise, from the second column of Table 2 we see that the lifts of the
states $|J\phi\rangle^{+}_{(h,1)}$ also equal those of the single-copy
superconformal primary $|\phi\rangle$.

##  4 Lifting of descendants by method of Ward identities

In the previous section the lift of descendent states was found using a method
that required the computation of a 6-point function of fields on the covering
space. This was done by expanding the four currents in modes, evaluating the
mode correlator using the $\mathcal{N}=4$ algebra, and resumming all of the
series. Due to the complexity of the functions involved, this resummation and
thus the lifts were done on a state-by-state basis. This is unfortunately not
well-suited for understanding the asymptotic behaviour of lifts for states of
large dimension, comparing primaries and descendants, and testing the
dependence of the second order lift on the partition of the dimension between
current modes generating the descendant state. In order to study these
aspects, analytic expressions for lifts are required. In this section we again
study the case of the state $J^{+}_{-n}|\phi\rangle$ and find an analytic
formula for its lift in terms of a handful of finite sums.

###  4.1 Lifting of $|J\phi\rangle^{+}_{(h,n)}$ by Ward identity

The starting point here will be the covering space correlator (3.36) given by

$\widetilde{C}^{+}(t_{4},t_{3})=\langle\phi(\infty)J^{-}(t_{4})G^{+}_{-}(t_{2})G^{-}_{+}(t_{1})J^{+}(t_{3})\phi(-a)\rangle\
,$ (4.1)

that is, we follow steps (1)-(3) for $J^{+}_{-n}|\phi\rangle$ as in Section
3.2, however, instead of expanding fields in terms of modes we evaluate (4.1)
by a Ward identity. We do this by first trivially rewriting the field
$J^{+}(t_{3})$ as a contour integral centred around $t_{5}=t_{3}$

$\displaystyle J^{+}(t_{3})=\oint_{t_{3}}\frac{dt_{5}}{2\pi
i}\frac{J^{+}(t_{5})}{t_{5}-t_{3}}\ ,$ (4.2)

followed by unwrapping this contour to get a sum of contours around each
insertion on the $t$-plane

$\displaystyle\widetilde{C}^{+}(t_{4},t_{3})=\widetilde{C}^{+}_{t_{1}}+\widetilde{C}^{+}_{t_{2}}+\widetilde{C}^{+}_{t_{4}}+\widetilde{C}^{+}_{-a}+\widetilde{C}^{+}_{\infty}\
.$ (4.3)

Figure 10: The general idea in deriving $|J\phi\rangle^{+}_{(h,n)}$ lifts
using Ward identities is be to start with the correlator (4.1) containing
different insertions on the $t$-plane. Moving clockwise in the figure,
trivially rewrite the field $J^{+}(t_{3})$ as a contour integral centred
around $t_{5}=t_{3}$. Unwrapping this contour yields a sum of contours around
each insertion on the $t$-plane. Different terms in the sum (4.3)evaluate to
different correlators one of which is shown as a 4-point insertion. Hence the
descendant $J^{+}$ can be removed using this method leading to a different
method of lifting computation.

The term $\widetilde{C}^{+}_{t_{2}}$ where the $t_{5}$ contour encircles the
point $t_{2}$ will vanish due to the OPE $G^{+}_{\\!\dot{A}}(t_{2})\times
J^{+}(t_{5})$ being purely regular. Likewise the term $\widetilde{C}^{+}_{-a}$
where the $t_{5}$ contour wraps $\phi(-a)$ can also be seen to vanish due to

$\displaystyle-\oint_{-a}\frac{dt_{5}}{2\pi
i}\frac{J^{+}(t_{5})\phi(-a)}{t_{5}-t_{3}}$
$\displaystyle=\sum_{m=0}^{\infty}\oint_{-a}\frac{dt_{5}}{2\pi
i}\frac{(t_{5}+a)^{m}}{(t_{3}+a)^{m+1}}J^{+}(t_{5})\phi(-a)$
$\displaystyle=\sum_{m=0}^{\infty}(t_{3}+a)^{-m-1}J^{+}_{m}\phi(-a)=0\ ,$
(4.4)

where the properties of a superconformal primary (2.9) and the fact that
$\phi$ has zero $J^{2}$ charge were used. Thus $\widetilde{C}^{+}_{-a}=0$ and
by similar arguments one sees that also $\widetilde{C}^{+}_{\infty}=0$. The
contour around $t_{1}$ gives

$\displaystyle-\oint_{t_{1}}\frac{dt_{5}}{2\pi
i}\frac{J^{+}(t_{5})G^{-}_{+}(t_{1})}{t_{5}-t_{3}}=-\oint_{t_{1}}\frac{dt_{5}}{2\pi
i}\frac{G^{+}_{+}{(t_{1})}}{(t_{5}-t_{3})(t_{5}-t_{1})}=-\frac{G^{+}_{+}{(t_{1})}}{(t_{1}-t_{3})}\
,$ (4.5)

using the current OPEs (A.8) and so

$\widetilde{C}^{+}_{t_{1}}=-\frac{1}{t_{1}-t_{3}}\langle\phi(\infty)J^{-}(t_{4})G^{+}_{-}(t_{2})G^{+}_{+}(t_{1})\phi(-a)\rangle\
.$ (4.6)

Lastly, the contour around $t_{4}$ yields

$\displaystyle-\oint_{t_{4}}\frac{dt_{5}}{2\pi
i}\frac{J^{+}(t_{5})J^{-}{(t_{4})}}{t_{5}-t_{3}}$
$\displaystyle=-\oint_{t_{4}}\frac{dt_{5}}{2\pi
i}\bigg{(}\frac{1}{(t_{5}-t_{3})(t_{5}-t_{4})^{2}}+\frac{2J^{3}(t_{4})}{(t_{5}-t_{3})(t_{5}-t_{4})}\bigg{)}$
$\displaystyle=\frac{1}{(t_{4}-t_{3})^{2}}-\frac{2J^{3}(t_{4})}{(t_{4}-t_{3})}\
,$ (4.7)

and so the correlator $\widetilde{C}^{+}_{t_{4}}$ becomes

$\widetilde{C}^{+}_{t_{4}}=\frac{A^{+-}_{-+}(t_{2},t_{1};h)}{(t_{4}-t_{3})^{2}}-\frac{2}{t_{4}-t_{3}}\langle\phi(\infty)J^{3}(t_{4})G^{+}_{-}(t_{2})G^{-}_{+}(t_{1})\phi(-a)\rangle\
,$ (4.8)

where $A^{+-}_{-+}(t_{2},t_{1};h)$ is defined from

$A^{\alpha\beta}_{\\!\dot{A}\dot{B}}(t_{2},t_{1};h)\equiv\langle\phi|G^{\alpha}_{\dot{A}}(t_{2})G^{\beta}_{\dot{B}}(t_{1})|\phi\rangle=\epsilon_{\dot{A}\dot{B}}\epsilon^{\alpha\beta}\frac{a+b(2h-1)}{4(a-b)(ab)^{3/2}}\
.$ (4.9)

Next we trivially write the field $J^{-}(t_{4})$ in
$\widetilde{C}^{+}_{t_{1}}$ as the contour

$\displaystyle J^{-}(t_{4})=\oint_{t_{4}}\frac{dt_{6}}{2\pi
i}\frac{J^{-}(t_{6})}{t_{6}-t_{4}}\ ,$ (4.10)

and again unwrap this $t_{6}$ contour yielding non-zero terms around $t_{2}$
and $t_{1}$. Using

$\displaystyle\frac{1}{(t_{1}-t_{3})}\oint_{t_{2}}\frac{dt_{6}}{2\pi
i}\frac{J^{-}(t_{6})G^{+}_{-}(t_{2})}{t_{6}-t_{4}}=\frac{1}{(t_{1}-t_{3})}\oint_{t_{2}}\frac{dt_{6}}{2\pi
i}\frac{G^{-}_{-}(t_{2})}{(t_{6}-t_{4})(t_{6}-t_{2})}=\frac{G^{-}_{-}(t_{2})}{(t_{1}-t_{3})(t_{2}-t_{4})}\
,$ (4.11)

and

$\displaystyle\frac{1}{(t_{1}-t_{3})}\oint_{t_{1}}\frac{dt_{6}}{2\pi
i}\frac{J^{-}(t_{6})G^{+}_{+}(t_{1})}{t_{6}-t_{4}}=\frac{1}{(t_{1}-t_{3})}\oint_{t_{1}}\frac{dt_{6}}{2\pi
i}\frac{G^{-}_{+}(t_{1})}{(t_{6}-t_{4})(t_{6}-t_{1})}=\frac{G^{-}_{+}(t_{1})}{(t_{1}-t_{3})(t_{1}-t_{4})}\
,$ (4.12)

we get

$\widetilde{C}^{+}_{t_{1}}=\frac{A^{-+}_{-+}(t_{2},t_{1};h)}{(t_{1}-t_{3})(t_{2}-t_{4})}+\frac{A^{+-}_{-+}(t_{2},t_{1};h)}{(t_{1}-t_{3})(t_{1}-t_{4})}\
.$ (4.13)

Likewise, the $J^{3}(t_{4})$ field in $\widetilde{C}^{+}_{t_{4}}$ is treated
similarly and one finds

$\widetilde{C}^{+}_{t_{4}}=\frac{A^{+-}_{-+}(t_{2},t_{1};h)}{(t_{4}-t_{3})^{2}}+\frac{A^{+-}_{-+}(t_{2},t_{1};h)}{(t_{2}-t_{4})(t_{4}-t_{3})}-\frac{A^{+-}_{-+}(t_{2},t_{1};h)}{(t_{1}-t_{4})(t_{4}-t_{3})}\
.$ (4.14)

All in all the amplitude (4.1) is given in terms of (4.1) as

$\displaystyle\widetilde{C}^{+}(t_{4},t_{3})$
$\displaystyle=\frac{A^{-+}_{-+}(t_{2},t_{1};h)}{(t_{1}-t_{3})(t_{2}-t_{4})}+A^{+-}_{-+}(t_{2},t_{1};h)\bigg{(}\frac{1}{(t_{1}-t_{3})(t_{1}-t_{4})}+\frac{1}{(t_{4}-t_{3})^{2}}$
$\displaystyle\hskip
170.71652pt+\frac{1}{(t_{2}-t_{4})(t_{4}-t_{3})}-\frac{1}{(t_{1}-t_{4})(t_{4}-t_{3})}\bigg{)}\
.$ (4.15)

From this the amplitude required for the lift is then

$A^{(1)(1)}_{h,n}(w_{2},w_{1})\equiv\langle\phi|J^{-}_{n}\,\big{(}G^{+}_{-,-\frac{1}{2}}\sigma_{2}^{-}\big{)}(w_{2})\big{(}G^{-}_{+,-\frac{1}{2}}\sigma_{2}^{+}\big{)}(w_{1})\,J^{+}_{-n}|\phi\rangle=f_{1}f_{2}f_{3}\,A^{+-}_{-+}(t_{2},t_{1};h)\,\mathcal{I}_{n}\
,$ (4.16)

in terms of factors $f_{1}$, $f_{2}$, $f_{3}$ given in (3.10), (3.13) and
(3.15) that do not depend on the presence of the $J$ modes and some integrals
$\mathcal{I}_{n}$ originating from the $J$ modes. The $n$-dependent integrals
are

$\displaystyle\mathcal{I}_{n}$
$\displaystyle\equiv\oint_{\infty}\frac{dt_{4}}{2\pi
i}\oint_{-a}\frac{dt_{3}}{2\pi
i}\,z_{4}^{n}z_{3}^{-n}\,\frac{t_{3}-t_{1}}{t_{3}-t_{2}}\frac{t_{4}-t_{2}}{t_{4}-t_{1}}\;\widetilde{C}^{+}(t_{4},t_{3})$
$\displaystyle=\oint_{\infty}\frac{dt_{4}}{2\pi i}\oint_{-a}\frac{dt_{3}}{2\pi
i}\,z_{4}^{n}z_{3}^{-n}\frac{1}{(t_{4}-t_{3})^{2}}\ ,$ (4.17)

where the result of the Ward identity (4.1) was used in the second line and
$z_{3}$ and $z_{4}$ are functions of $t_{3}$ and $t_{4}$ respectively as seen
from (3.7). The integrand has $n$th order poles in both $t_{3}$ and $t_{4}$;
consider the $t_{3}$ integral first

$\mathcal{I}_{n}^{(3)}\equiv\oint_{-a}\frac{dt_{3}}{2\pi
i}\,z_{3}^{-n}\frac{1}{(t_{4}-t_{3})^{2}}=\oint_{-a}\frac{dt_{3}}{2\pi
i}\,(t_{3}+a)^{-n}\frac{t_{3}^{\,n}}{(t_{3}+b)^{n}(t_{4}-t_{3})^{2}}\ .$
(4.18)

By expanding the part of the integrand that is regular at $t_{3}=-a$ as a
power series around $-a$ we find

$\displaystyle\mathcal{I}_{n}^{(3)}$
$\displaystyle=\oint_{-a}\frac{dt_{3}}{2\pi
i}\sum_{k_{5}=0}^{\infty}\sum_{k_{3}=0}^{\min(k_{5},n)}\sum_{k_{1}=0}^{k_{5}-k_{3}}\frac{(-1)^{k_{3}}(k_{1}+1)a^{n-k_{3}}(t_{3}+a)^{k_{5}-n}}{(t_{4}+a)^{k_{1}+2}(a-b)^{n+k_{5}-k_{1}-k_{3}}}\,{}^{n+k_{5}-k_{1}-k_{3}-1}C_{k_{5}-k_{1}-k_{3}}\,{}^{n}C_{k_{3}}$
$\displaystyle=\sum_{k_{3}=0}^{n-1}\sum_{k_{1}=0}^{n-k_{3}-1}\frac{(-1)^{k_{3}}(k_{1}+1)a^{n-k_{3}}}{(t_{4}+a)^{k_{1}+2}(a-b)^{2n-k_{1}-k_{3}-1}}\,{}^{2n-k_{1}-k_{3}-2}C_{n-1}\,{}^{n}C_{k_{3}}\
,$ (4.19)

where the integral picks out the term $k_{5}=n-1$. The $t_{4}$ integral of
(4.1) is then given by

$\mathcal{I}_{n}^{(4)}\equiv\oint_{\infty}\frac{dt_{4}}{2\pi
i}\frac{z_{4}^{n}}{(t_{4}+a)^{k_{1}+2}}=\oint_{\infty}\frac{dt_{4}}{2\pi
i}\,t_{4}^{n-k_{1}-2}\,\Big{(}1+\frac{a}{t_{4}}\Big{)}^{n-k_{1}-2}\Big{(}1+\frac{b}{t_{4}}\Big{)}^{n}\
.$ (4.20)

By converting this to a contour integral around $t_{4}=0$ and expanding the
regular part of the integrand as a power series around $0$ we find

$\displaystyle\mathcal{I}_{n}^{(4)}$
$\displaystyle=-\oint_{0}\frac{d\tilde{t}_{4}}{2\pi
i}\,\tilde{t}_{4}^{\,k_{1}-n}\,\Big{(}1+a\tilde{t}_{4}\Big{)}^{n-k_{1}-2}\Big{(}1+b\tilde{t}_{4}\Big{)}^{n}$
$\displaystyle=-\oint_{0}\frac{d\tilde{t}_{4}}{2\pi
i}\sum_{j_{3}=0}^{2n-k_{1}-2}\sum_{j_{2}=0}^{\min(n,j_{3})}a^{j_{3}-j_{2}}\,b^{j_{2}}\,\tilde{t}_{4}^{\,k_{1}-n+j_{3}}\;{}^{n-k_{1}-2}C_{j_{3}-j_{2}}{}^{n}C_{j_{2}}$
$\displaystyle=-\sum_{j_{2}=0}^{n-k_{1}-1}a^{n-k_{1}-1-j_{2}}\,b^{j_{2}}\;{}^{n-k_{1}-2}C_{j_{2}-1}\,{}^{n}C_{j_{2}}\
,$ (4.21)

where the integral picks out the term $j_{3}=n-k_{1}-1$. The total integral
$\mathcal{I}_{n}$ is then

$\displaystyle\mathcal{I}_{n}=\Big{(}\frac{a}{a-b}\Big{)}^{\\!2n-1}\\!\sum_{k_{3}=0}^{n-1}\sum_{k_{1}=0}^{n-k_{3}-1}\sum_{j_{2}=0}^{n-k_{1}-1}\frac{(-1)^{k_{3}+1}(k_{1}+1)b^{j_{2}}}{(a-b)^{-k_{1}-k_{3}}a^{k_{1}+k_{3}+j_{2}}}\,{}^{2n-k_{1}-k_{3}-2}C_{n-1}{}^{n}C_{k_{3}}{}^{n}C_{j_{2}}{}^{n-k_{1}-2}C_{j_{2}-1}\
,$ (4.22)

and the amplitude (4.16) is then given by

$\displaystyle A^{(1)(1)}_{h,n}$
$\displaystyle=\frac{a+b(2h-1)}{16b}\bigg{(}\frac{a}{a-b}\bigg{)}^{\\!h+2n-2}\sum_{k_{1}=0}^{n-1}\sum_{k_{1}=0}^{n-1-k_{3}}\sum_{j_{2}=0}^{n-k_{1}-1}(-1)^{k_{3}+1}(k_{1}+1)\bigg{(}\frac{b}{a}\bigg{)}^{\\!j_{2}}\bigg{(}\frac{a}{a-b}\bigg{)}^{\\!\\!-k_{1}-k_{3}}$
$\displaystyle\hskip
199.16928pt\times{}^{2n-k_{1}-k_{3}-2}C_{n-1}{}^{n}C_{k_{3}}{}^{n}C_{j_{2}}{}^{n-k_{1}-2}C_{j_{2}-1}\
.$ (4.23)

After replacing $a$ and $b$ using (3.21) this is an analytic function of $w$
and the copy-symmetric amplitude

$\displaystyle A_{h,n}(w,0)$ $\displaystyle\equiv{}^{\ \ \ \,-}_{(h,n)}\langle
J\phi|\big{(}G^{+}_{-,-\frac{1}{2}}\sigma^{-}_{2}\big{)}(w)\big{(}G^{-}_{+,-\frac{1}{2}}\sigma^{+}_{2}\big{)}(0)|J\phi\rangle^{+}_{(h,n)}$
$\displaystyle=\frac{1}{2}\sum_{i,j=1}^{2}A^{(j)(i)}_{h,n}(w,0)$
$\displaystyle=A^{(1)(1)}_{h,n}(w,0)+A^{(2)(1)}_{h,n}(w,0)\ ,$ (4.24)

where we have used that $A^{(2)(2)}=A^{(1)(1)}$ and $A^{(1)(2)}=A^{(2)(1)}$ by
copy symmetry. The amplitude $A^{(2)(1)}$ can be obtained from $A^{(1)(1)}$
via the continuation

$A^{(2)(1)}_{h,n}(w,0)=A^{(1)(1)}_{h,n}(w+2\pi i,0)\ .$ (4.25)

The lift is then obtained by integrating over $w$, for which we will consider
the contour integral $I_{C_{2}}$ where $C_{2}$ is at $\real w\to-\infty$ as
defined in (2.40b). The right-moving part of the computation is simple ends up
contributing a factor of $-1$ and the integral then picks out the order
$(e^{w})^{0}$ term from $A^{(1)(1)}$ and $A^{(2)(1)}$. It turns out that the
contribution to this integral from $A^{(1)(1)}$ and $A^{(2)(1)}$ are equal in
this case, with the contribution from (4.1) being

$\displaystyle A^{(1)(1)}_{h,n}(w,0)\Big{|}_{(e^{w})^{0}}$
$\displaystyle=\sum_{k_{3}=0}^{n-1}\sum_{k_{1}=0}^{n-1-k_{3}}\sum_{j_{2}=0}^{n-k_{1}-1}{}^{2n-k_{1}-k_{3}-2}C_{n-1}\,{}^{n}C_{k_{3}}\,{}^{n}C_{j_{2}}\,{}^{n-k_{1}-2}C_{j_{2}-1}\frac{(-1)^{k_{3}+1}(k_{1}+1)}{2^{2(h+2n-k_{1}-k_{3})-1}}$
$\displaystyle\quad\times\frac{\sum_{\nu=0}^{2}c_{\nu}x^{\nu}}{x^{h+2n-k_{1}-k_{3}-2}}\sum_{\ell_{3}=0}^{\infty}\sum_{\ell_{2}=0}^{\min(q,\ell_{3})}(-1)^{\ell_{3}-\ell_{2}}\;{}^{2(j_{2}-1)}C_{\ell_{3}-\ell_{2}}\,{}^{q}C_{\ell_{2}}\,x^{\ell_{3}}\Bigg{|}_{x^{0}}$
$\displaystyle=\sum_{k_{3}=0}^{n-1}\sum_{k_{1}=0}^{n-1-k_{3}}\sum_{j_{2}=0}^{n-k_{1}-1}\sum_{\nu=0}^{2}\sum_{\ell_{2}=0}^{\min(q,L_{3})}\frac{c_{\nu}(-1)^{h+k_{1}+\ell_{2}+\nu}(k_{1}+1)}{2^{2(h+2n-k_{1}-k_{3})-1}}$
$\displaystyle\hskip
56.9055pt\times{}^{2n-k_{1}-k_{3}-2}C_{n-1}\,{}^{n}C_{k_{3}}\,{}^{n}C_{j_{2}}\,{}^{n-k_{1}-2}C_{j_{2}-1}\,{}^{2(j_{2}-1)}C_{L_{3}-\ell_{2}}\,{}^{q}C_{\ell_{2}}\
.$ (4.26)

where the shorthand $x=e^{w/2}$ was used and the projection onto $x^{0}$
picked out the term $\ell_{3}=L_{3}$ where

$L_{3}=h+2n-k_{1}-k_{3}-2-\nu\quad,\quad q=2(h+2n-k_{1}-k_{3}-j_{2}-2)\ ,$
(4.27)

and $c_{\nu}$ are coefficients such that

$\sum_{\nu=0}^{2}c_{\nu}\,x^{\nu}=hx^{2}-2(h-1)x+h\ .$ (4.28)

From (2.35) the lift is then given by

$\displaystyle E^{(2)}\big{(}|J\phi\rangle^{+}_{(h,n)}\big{)}$
$\displaystyle=\frac{4\pi\lambda^{2}}{\big{|}|J\phi\rangle^{+}_{(h,n)}\big{|}}\,A^{(1)(1)}_{h,n}(w,0)\Big{|}_{(e^{w})^{0}}$
$\displaystyle=\frac{8\pi^{2}\lambda^{2}}{n}\\!\sum_{k_{3}=0}^{n-1}\sum_{k_{1}=0}^{n-1-k_{3}}\sum_{j_{2}=0}^{n-k_{1}-1}\sum_{\nu=0}^{2}\sum_{\ell_{2}=0}^{\min(q,L_{3})}\frac{c_{\nu}(-1)^{h+k_{1}+\ell_{2}+\nu}}{2^{2(h+2n-k_{1}-k_{3})}}(k_{1}+1)$
$\displaystyle\hskip
56.9055pt\times{}^{2n-k_{1}-k_{3}-2}C_{n-1}\,{}^{n}C_{k_{3}}\,{}^{n}C_{j_{2}}\,{}^{n-k_{1}-2}C_{j_{2}-1}\,{}^{2(j_{2}-1)}C_{L_{3}-\ell_{2}}\,{}^{q}C_{\ell_{2}}\
.$ (4.29)

It is useful to note that if $2(j_{2}-1)<L_{3}-\ell_{2}$ then the binomial
${}^{2(j_{2}-1)}C_{L_{3}-\ell_{2}}$ in (4.1) vanishes and the ranges of
$L_{3}$ and $q$ are

$\displaystyle h-2\leq L_{3}\leq h+2(n-1)\quad,\quad 2h\leq q\leq 2h+4(n-1)\
.$ (4.30)

While the expression (4.1) for the lift of $|J\phi\rangle^{+}_{(h,n)}$ is not
particularly compact, it is valid for all values of $h$ and $n$ and as will be
seen in the following section some information about the large dimension
behaviour will be extractable. One aspect of the form (4.1) that makes the
large dimension regime difficult to analyse is the dependence of the range of
the $\ell_{2}$ sum on $h$. It turns out that we can resum the $\nu$ and
$\ell_{2}$ sums by considering the $j_{2}=0$ and $j_{2}\geq 1$ terms
separately. We find that for $j_{2}=0$

$\displaystyle F_{0}$
$\displaystyle\equiv\sum_{\nu=0}^{2}c_{\nu}(-1)^{\nu}\sum_{\ell_{2}=0}^{L_{3}}(-1)^{\ell_{2}}\>{}^{-2}C_{L_{3}-\ell_{2}}\,{}^{2L_{3}+2\nu}C_{\ell_{2}}$
$\displaystyle=\sum_{\nu=0}^{2}c_{\nu}(-1)^{h-k_{1}-k_{3}}\sum_{\ell_{2}=0}^{L_{3}}(L_{3}-\ell_{2}+1)\>{}^{2L_{3}+2\nu}C_{\ell_{2}}$
$\displaystyle=\frac{2(h+L_{3})\,\Gamma(2L_{3})}{\Gamma(L_{3})\,\Gamma(L_{3}+1)}\bigg{|}_{\nu=0}\
,$ (4.31)

and for $J=j_{2}-1\geq 0$

$\displaystyle F_{J}$
$\displaystyle\equiv\sum_{\nu=0}^{2}c_{\nu}(-1)^{\nu}\sum_{\ell_{2}=0}^{L_{3}}(-1)^{\ell_{2}}\>{}^{2J}C_{L_{3}-\ell_{2}}\,{}^{2L_{3}+2\nu-2J-2}C_{\ell_{2}}$
$\displaystyle=\frac{(-1)^{L_{3}+J}2^{2L_{3}-1}\big{(}L_{3}-h(2J+1)\big{)}\,\Gamma(J+\tfrac{1}{2})\,\Gamma(L_{3}-J-\tfrac{1}{2})}{\pi\,\Gamma(L_{3}+1)}\bigg{|}_{\nu=0}\
.$ (4.32)

The term (4.1) in the lift (4.1) can be fully resummed to give

$\displaystyle E_{1}$
$\displaystyle=\frac{\pi^{2}\lambda^{2}}{n\,2^{2h+4n-3}}\\!\sum_{k_{1},k_{3}=0}^{n-1}\frac{(-1)^{h+k_{1}}(k_{1}+1)}{2^{2(-k_{1}-k_{3})}}\,{}^{2n-k_{1}-k_{3}-2}C_{n-1}\,{}^{n}C_{k_{3}}\,\delta_{k_{1},n-1}\,\widetilde{F}_{0}$
$\displaystyle=\frac{1}{2}\pi^{\frac{3}{2}}\lambda^{2}\,\frac{(2h+n-1)\,\Gamma(h+n-\tfrac{1}{2})}{(h+n-1)\,\Gamma(h+n-1)}\
,$ (4.33)

where the terms $k_{1}=n-1$ and $k_{3}=0$ were picked out by the various
binomials. We find that the lift (4.1) can be written as

$\displaystyle E^{(2)}\big{(}|J\phi\rangle^{+}_{(h,n)}\big{)}$
$\displaystyle=E_{1}+E_{2}\ ,$ (4.34)

with $E_{2}$ being given by

$\displaystyle E_{2}$
$\displaystyle=\frac{\pi\lambda^{2}}{4n}\sum_{K=1}^{n}\sum_{k_{1}=1}^{\min(K,n-1)}\sum_{J=0}^{n-k_{1}-1}\frac{(-1)^{K-k_{1}+J}k_{1}}{\Gamma(h+2n-K)}\,{}^{n}C_{K-k_{1}}\,{}^{2n-K-1}C_{n-1}\,{}^{n}C_{J+1}\,{}^{n-k_{1}-1}C_{J}$
$\displaystyle\hskip
99.58464pt\times(2n-K-1-2hJ)\,\Gamma(J+\tfrac{1}{2})\,\Gamma(h+2n-K-J-\tfrac{3}{2})\
.$ (4.35)

where $K=k_{1}+k_{3}$ was used. We note that the upper limit of the $k_{1}$
sum can be written as $K$ since the one term where $\min(K,n-1)\neq K$ is
$K=k_{1}=n$ and then the binomial ${}^{n-k_{1}-1}C_{J}$ vanishes.

A simple check of this exact result for the lift is that, as seen in the
series method data of Section 3.2, the case of $n=1$ should reduce to the lift
of a superconformal primary, for which an analytic form was found in [48] and
rewritten in (3.56). Considering the case of $n=1$ in (4.1) we see that
$E_{2}=0$ and so

$\displaystyle
E^{(2)}_{h,1}\big{(}J^{+}|\phi\rangle\big{)}=E_{1}\big{|}_{n=1}=\pi^{\frac{3}{2}}\lambda^{2}\,\frac{\Gamma(h+\tfrac{1}{2})}{\Gamma(h)}=E^{(2)}_{h}\big{(}|\phi\rangle\big{)}\
.$ (4.36)

Despite the complicated form of (4.1) it turns out that the lift is polynomial
in $h$ and we find that for a given value of $n$

$E^{(2)}\big{(}|J\phi\rangle^{+}_{(h,n>1)}\big{)}=E^{(2)}_{h}\\!\big{(}\phi\big{)}\bigg{[}1+a_{n}\frac{P_{n}(h)}{(h+1)_{2n-2}}\bigg{]}\
,$ (4.37)

where $P_{n}(h)$ is a polynomial in $h$ of degree $2n-4$ and $(x)_{n}$ is the
Pochhammer symbol

$(x)_{n}\equiv\frac{\Gamma(x+n)}{\Gamma(x)}\ .$ (4.38)

The rational coefficients $a_{n}$ in (4.37) for the first few values of $n$
are

$\big{\\{}a_{n\geq
2}\big{\\}}=\bigg{\\{}\,\frac{3}{4},\,\frac{3}{8},\,\frac{15}{32},\,\frac{15}{128},\,\frac{105}{256},\,\frac{21}{512},\,\dots\bigg{\\}}\
,$ (4.39)

and the respective polynomials are

$\displaystyle P_{2}$ $\displaystyle=1\ ,$ $\displaystyle P_{3}$
$\displaystyle=8h^{2}+16h+41\ ,$ $\displaystyle P_{4}$
$\displaystyle=16h^{4}+96h^{3}+578h^{2}+798h+1305\ ,$ $\displaystyle P_{5}$
$\displaystyle=128h^{6}+1536h^{5}+16976h^{4}+64608h^{3}+237920h^{2}+268512h+352107\
,$ $\displaystyle P_{6}$
$\displaystyle=64h^{8}+1280h^{7}+22592h^{6}+163520h^{5}+1077776h^{4}+3208160h^{3}+9205198h^{2}$
$\displaystyle\hskip 255.22125pt+9091050h+10377045\ .$ (4.40)

Given the knowledge of the form (4.37) for the lift of the
$|J\phi\rangle^{+}_{(h,n)}$ states, we were able to find that the series
method data of Section 3.1 for the lift of $|G\phi\rangle^{+}_{+(h,n)}$
follows a similar pattern of

$E^{(2)}\big{(}|G\phi\rangle^{+}_{+(h,n>1)}\big{)}=E^{(2)}_{h}\\!\big{(}\phi\big{)}\bigg{[}1+b_{n}\frac{\widetilde{P}_{n}(h)}{(h+1)_{2(n-1)}(h+n^{2}-n)}\bigg{]}\
,$ (4.41)

where the rational coefficients are

$\big{\\{}b_{n\geq
2}\big{\\}}=\bigg{\\{}\,\frac{3}{4},\,\frac{15}{16},\,\frac{21}{16},\,\frac{45}{256},\,\frac{165}{256},\,\frac{273}{1024},\,\dots\bigg{\\}}\
,$ (4.42)

and $b_{1}=0$ and $\widetilde{P}_{n}(h)$ are degree $2n-3$ polynomials in $h$
of the form

$\displaystyle\widetilde{P}_{2}$ $\displaystyle=h+1\ ,$
$\displaystyle\widetilde{P}_{3}$ $\displaystyle=4h^{3}+16h^{2}+51h+78\ ,$
$\displaystyle\widetilde{P}_{4}$
$\displaystyle=8h^{5}+72h^{4}+601h^{3}+2193h^{2}+3711h+4905\ ,$
$\displaystyle\widetilde{P}_{5}$
$\displaystyle=128h^{7}+2048h^{6}+31952h^{5}+219392h^{4}+979664h^{3}+2908064h^{2}+3700845h+4296180\
,$ $\displaystyle\widetilde{P}_{6}$
$\displaystyle=64h^{9}+1600h^{8}+39952h^{7}+446320h^{6}+3690736h^{5}+19756240h^{4}+64904013h^{3}\
,$ $\displaystyle\hskip 192.62527pt+164470575h^{2}+176337585h+185479875\ .$
(4.43)

We do not, however, have a derivation of this form for the lift of
$|G\phi\rangle^{+}_{+(h,n)}$.

###  4.2 Asymptotic behaviour

In [48] the lift of a single-copy superconformal primary was found in a closed
form – displayed in (3.56) – and it was observed that in the limit of large
dimension the leading behaviour of the lift was of the form

$E^{(2)}_{h}\big{(}\phi\big{)}\approx\pi^{3/2}\lambda^{2}\,\sqrt{h}\ .$ (4.44)

This $\sim\sqrt{h}$ behaviour has also been hinted at in computations of lifts
for which closed form expressions are not known, such as in [49] where the
lift of states with excitations on one copy from two bosonic or fermionic
modes was computed state by state. Looking at the lifts of states where one
mode had a fixed and small mode number and the other was varied to, say, level
30 a square root behaviour was observed in the data. Likewise, in the state-
by-state computations of Section 3 for descendants, similar behaviour was
observed. It is natural to then ask more nuanced questions: for a level-$n$
descendant of a primary of dimension $h$, is the asymptotic behaviour of lifts
the same as that of the primary and is the important quantity the total
dimension $h_{tot}=h+n$ or are the dimensions of the primary and descendant
excitation treated differently? If the latter case is true, it is interesting
to then ask whether the partitioning of the descendant excitation dimension
amongst different modes is important when it comes to the lift of the state.
In this section we hope to answer exactly these questions.

####  4.2.1 Large dimension limits

Since the lifting formula of the descendant state $|J\phi\rangle^{+}_{(h,n)}$
is known exactly, albeit in terms of finite sums, we may derive a large
dimension limit. Namely, consider the large $h$ regime

$h\gg n\sim O(1)\ .$ (4.45)

In this regime the dominant part of $E_{2}$ in (4.1) comes from the function

$\displaystyle\frac{h\,\Gamma(h+2n-K-J-\tfrac{3}{2})}{\Gamma(h+2n-K)}\sim
h^{-\frac{1}{2}-J}\ ,$ (4.46)

which for all values of $J\in[0,n-2]$ is subleading compared with the leading
order behaviour of $E_{1}$ in (4.1) and so we find

$\displaystyle E^{(2)}\big{(}|J\phi\rangle^{+}_{(h,n)}\big{)}\approx
E_{1}\approx\pi^{\frac{3}{2}}\lambda^{2}\,\sqrt{h}\ ,$ (4.47)

which exactly matches the leading asymptotic behaviour of the lift of
superconformal primaries given in (4.44) and notably (4.47) is independent of
the descendant excitation level $n$.

It turns out that more can be said about the asymptotic forms of the lifts of
the descendant states $J^{+}_{-n}|\phi\rangle$ and
$G^{\alpha}_{\dot{A},-s}|\phi\rangle$ that will shine a light on the questions
raised at the beginning of this section. Firstly, and most simply, we have
already argued in Section 3.3 that the lifts of the lowest-level descendants
($n=1$ and $s=\frac{1}{2}$ respectively) are exactly equal to the lift of the
primary, i.e.

$E^{(2)}\big{(}|J\phi\rangle^{+}_{(h,1)}\big{)}=E^{(2)}\big{(}|G\phi\rangle^{+}_{+(h,1/2)}\big{)}=E^{(2)}_{h}\\!\big{(}\phi\big{)}=\pi^{\frac{3}{2}}\lambda^{2}\frac{\Gamma(h+\tfrac{1}{2})}{\Gamma(h)}\
.$ (4.48)

In the case of $J^{+}_{-1}|\phi\rangle$ this statement was derived from the
exact result in (4.36). The large $h$ behaviour of these families is thus
trivially also equal to that of $\phi$, given in (4.44).

Although the expressions (4.37) and (4.41) are not in closed form, it is
possible to find a closed form for the subleading large $h$ behaviour

$\displaystyle\frac{E^{(2)}\\!\big{(}|J\phi\rangle^{+}_{(h,n)}\big{)}}{E^{(2)}_{h}\\!(\phi)}$
$\displaystyle\approx 1+\frac{n(n^{2}-1)}{8h^{2}}\ ,$ (4.49)
$\displaystyle\frac{E^{(2)}\\!\big{(}|G\phi\rangle^{+}_{+(h,n)}\big{)}}{E^{(2)}_{h}\\!(\phi)}$
$\displaystyle\approx 1+\frac{n(n-1)(2n-1)}{8h^{2}}\ ,$ (4.50)

where again $n=s+\frac{1}{2}$ for the $G$ descendant. The subleading term
carries information about the descendant excitation, however, if the
asymptotic behaviour for large total dimension $h_{tot}=h+n$ was
$\sim\sqrt{h_{tot}}$ as might be expected, then the subleading corrections
should appear at order $1/h$ and not $1/h^{2}$ since for $h\gg n$

$\sqrt{h+n}\approx\sqrt{h}\bigg{(}1+\frac{n}{2h}+O\big{(}h^{-2}\big{)}\bigg{)}\
.$ (4.51)

It therefore appears that the dimension of the primary and that of the
descendant excitation are treated differently in the lift. In Appendix C we
present a special two parameter family of states for which we have complete
control of the lifting calculation and can obtain a closed form expression for
the lift. These untwisted sector states are in the NS vacuum on all copies bar
one, on which there is the following excited state

$\mathcal{J}^{+,(m)}|\phi\rangle\equiv J^{+}_{-(2m-1)}\cdots
J^{+}_{-3}J^{+}_{-1}|\phi\rangle\ ,$ (4.52)

where $\phi$ is again a single-copy superconformal primary of dimension $h$.
This state has a total dimension $h^{(m)}$ and $J^{3}_{0}$ charge $j^{(m)}$ of

$h^{(m)}=h+m^{2}\quad,\quad j^{(m)}=m\ .$ (4.53)

The excitation $\mathcal{J}^{+,(m)}$ is special because it is the spectral
flow of the NS vacuum under a flow by $\eta=2m$ units, a fact that enables the
simple calculation of the lift of (4.52) to be

$\displaystyle
E^{(2)}\big{(}\mathcal{J}^{+,(m)}|\phi_{h}\rangle\big{)}=\frac{\pi^{3/2}\lambda^{2}}{2}\frac{(2h+m^{2}-1)\,\Gamma(h+m^{2}-\frac{1}{2})}{\Gamma(h+m^{2})}\
.$ (4.54)

This result is the first closed-form expression we have obtained (that does
not contain sums as in (4.1)) for a two-parameter family of states, albeit a
very special one. We can then use (4.54) to understand the questions posed at
the start of Section 4.2. Firstly, in the regime $h\gg m^{2}\sim O(1)$
(analogous to the regime (4.45)) we find the asymptotic growth

$E^{(2)}\big{(}\mathcal{J}^{+,(m)}|\phi_{h}\rangle\big{)}\approx\pi^{3/2}\sqrt{h}\bigg{(}1-\frac{1}{8h}+\frac{16m^{4}-16m^{2}+1}{128h^{2}}\bigg{)}\
,$ (4.55)

which should be compared against the large $h$ expansion of the lift of $\phi$
(3.56)

$E^{(2)}_{h}\big{(}|\phi\rangle\big{)}\approx\pi^{3/2}\sqrt{h}\bigg{(}1-\frac{1}{8h}+\frac{1}{128h^{2}}\bigg{)}\
,$ (4.56)

and so of interest is the asymptotic behaviour of the ratio

$\frac{E^{(2)}\big{(}\mathcal{J}^{+,(m)}|\phi_{h}\rangle\big{)}}{E^{(2)}_{h}(|\phi\rangle)}\approx
1+\frac{m^{4}-m^{2}}{8h^{2}}\ .$ (4.57)

This once again shows that information of the excitation on top of a
superconformal primary comes only at order $h^{-2}$ in the large $h$ limit, as
seen in (4.49) and (4.50), so once again we see that even asymptotically the
lift cannot be a function of the total dimension. However, in contrast to
those asymptotic results, (4.57) displays a different dependence on the
excitation dimension $h_{ex}$. This was $\sim h_{ex}^{3}=n^{3}$ in the cases
of (4.49) and (4.50) and $\sim h_{ex}^{2}=m^{4}$ in the case of (4.57), thus
suggesting that the lift depends not only on $h_{ex}$ but on how this
excitation dimension is partitioned amongst modes. Considering instead the
regime of large excitation

$m^{2}\gg h\sim O(1)\ ,$ (4.58)

then the lift (4.54) has the leading behaviour

$E^{(2)}\big{(}\mathcal{J}^{+,(m)}|\phi_{h}\rangle\big{)}\approx\frac{\pi^{3/2}m}{2}\bigg{(}1+\frac{12h-5}{8m^{2}}\bigg{)}\
,$ (4.59)

where information of the primary comes in already at the first subleading
order, in contrast to (4.57). From the exact result (4.54) we can also look at
the regime of large total dimension

$h,m^{2}\gg 1\ ,$ (4.60)

in which the lift behaves as

$E^{(2)}\big{(}\mathcal{J}^{+,(m)}|\phi_{h}\rangle\big{)}\approx\frac{\pi^{3/2}\lambda^{2}}{2}\frac{2h+m^{2}}{\sqrt{h+m^{2}}}\
,$ (4.61)

which is the first time that we have had an expression valid in this part of
parameter space. From this one can then see exactly why there is no $\sim
h^{-1}$ term if $h\gg m^{2}$ since

$\frac{2h+m^{2}}{2\sqrt{h+m^{2}}}\approx\sqrt{h}\bigg{(}1+\frac{m^{2}}{2h}\bigg{)}\bigg{(}1-\frac{m^{2}}{2h}\bigg{)}\approx\pi^{3/2}\lambda^{2}\sqrt{h}\Big{(}1+O(h^{-2})\Big{)}\
.$ (4.62)

##  5 Discussion

Understanding the BPS spectrum of the D1-D5 CFT is relevant to identifying and
studying the states dual to microstates of certain extremal black holes. BPS
states are classified very simply at the locus of moduli space where the
theory has a description in terms of a free orbifold theory, in terms of
states with purely left- or right-moving excitations above the NS-NS vacuum.
However, away from this special region their classification is unknown. This
is because multiple short multiplets of the free theory can group together
into a long multiplet of the general theory and so ‘lift’. Thus, members of
those free-theory short multiplets are not in fact globally BPS. Progress
towards identifying the globally BPS states that can be the duals of black
hole microstates can be made by looking at the lifting of various families of
states of the free orbifold theory in order to understand general patterns in
which states lift and by how much. In this paper we have studied new families
of states in the untwisted sector where one copy of the $c=6$ seed theory is
excited above the NS-NS vacuum with a descendant of a superconformal primary.
More precisely, we have considered the single-copy states

$\Big{\\{}\,G^{\alpha}_{\dot{A},-s}|\phi\rangle\ ,\ \
J^{+}_{-n}|\phi\rangle\,\Big{\\}}\ ,$ (5.1)

where $|\phi\rangle$ is a primary of the $c=6$ theory with dimension $h$. In
Section 3 the expectation value of the lift of states in the 2-parameter
families (5.1) was found in terms of an integral of a fixed number of nested
sums on a given summand; this summand depends on the descendant level of the
state (i.e. on $s$ and $n$ of the states (5.1)) and the dimension of the
primary $h$. Explicit lifting matrices for mode numbers and primary dimensions
up to order $7$ were presented for examples of the two types of descendants in
(3.29) and (3.47). We evaluate these sums to obtain the explicit value of the
lifts for various 1-parameter subfamilies of states: (i) with the descendant
mode number held to be small and $h$ allowed to grow large, (ii) with $h$ held
fixed and the descendant level allowed to grow large, and (iii) with both the
descendant mode number and primary dimension taken to have the same value. In
each case we presented the first $20$ lifts with this explicit data found in
tables 1 and 2. The new “series” method used to compute these lifts is
different to the “fields method” used in [49] and is well suited to states of
the form (5.1) where we do not specify the form of the superconformal primary.
We chose to present explicit lifts for these states purely for space and
computational reasons; similarly to the method used in [49], in principle the
method described in this paper can scale arbitrarily given the resources. By
having values for the lift of these states up to a suitably high level allows
us to observe what seems to be $\sim\sqrt{h}$ growth for large $h$ in the
various subfamilies considered. This is similar to the behaviour observed in
the two-modes states of [49] and the asymptotic $\sqrt{h}$ behaviour seen in
the closed-form expression obtained in [48] for the lift of single-copy
superconformal primaries.

We stress that the lifts found in this paper are in the theory for a general
$N$ number of copies of the $c=6$ seed CFT. The method used to compute lifts
sees only two copies of the $c=6$ theory at any given time and so it is
sufficient to consider a particular pair of copies in intermediate steps. As
written in (2.41), the existence of the $N$ total copies is easily reinstated
using combinatorics.

In Section 4.1 we study the expectation value of lifts again for the
$J^{+}_{-n}$ descendants of $|\phi\rangle$, however, this time using a method
of Ward identities on the covering space. This method allows us to find a
compact form for these lifts valid for arbitrary $h$ and $n$. Because of this
exact expression for the lift we are able to derive a leading order
$\sim\sqrt{h}$ growth from it in Section 4.2.1. This leading asymptotic
behaviour matches that seen in the closed-form expression of [48] for the lift
of single-copy primaries. In that paper it was conjectured that this
$\sim\sqrt{h}$ behaviour may be a universal feature of the lift of D1-D5-P
states and it has since been hinted at in explicit values of lifts for various
families of states in [49] and Section 3. However, what was not known is
whether expectation values of lifts are functions purely of the total
dimension of the state. The result of [48] was that for single-copy
superconformal primaries the lift did not depend of the form of the state
under consideration, only its total conformal dimension. From the results of
Section 2.5 we find that for single-copy superconformal descendant states, the
lift depends differently on the primary’s dimension $h$ and the descendant
excitation $h_{ex}$. Not only this, but by comparison of the lift of
$J^{+}_{-n}|\phi\rangle_{h}$ from Section 4.1 and the lift of
$J^{+}_{-(2m-1)}\cdots J^{+}_{-3}J^{+}_{-1}|\phi\rangle_{h}$ from Appendix C
we find that the lift depend on the partitioning of $h_{ex}$ amongst
descendant modes. This is seen even at the first subleading order in the large
$h$ regime, with the $O(h^{-2})$ term scaling as $\sim h_{ex}^{3}=n^{3}$ and
$\sim h_{ex}^{2}=m^{4}$ for $J^{+}_{-n}|\phi\rangle_{h}$ and
$J^{+}_{-(2m-1)}\cdots J^{+}_{-3}J^{+}_{-1}|\phi\rangle_{h}$ respectively.

These results have strengthened the conjecture that the second order
expectation value of lifts of D1-D5-P states universally grow as
$\sim\sqrt{h}$ in the large $h$ limit. It would be interesting to understand
whether this seemingly universal behaviour gives insight into the nature of
these very stringy states in the dual string theory. It would also be
interesting to see if any lessons can be brought over from the recent
successes of identifying candidate black hole states [26, 27, 28, 29] in
$\mathcal{N}=4$ SYM with gauge group $SU(2)$.

## Acknowledgements

We would like to thank Bin Guo for discussions. This work is supported in part
by DOE grant DE-SC0011726.

##  Appendix A The $\mathcal{N}=4$ superconformal algebra

The conventions and notation that we follow is that of appendix A of [44]. In
our conventions the indices $\alpha=(+,-)$ and $\bar{\alpha}=(+,-)$ correspond
to the fundamental representations of the subgroups $SU(2)_{L}$ and
$SU(2)_{R}$ arising from rotations on the $S^{3}$, whereas the indices
$A=(+,-)$ and $\dot{A}=(+,-)$ correspond to the subgroups $SU(2)_{1}$ and
$SU(2)_{2}$ arising from rotations in the internal $T^{4}$. We use the
convention

$\epsilon_{+-}=1\quad,\quad\epsilon^{+-}=-1\ .$ (A.1)

###  A.1 Commutation relations

The current algebra of the small $\mathcal{N}=4$ superconformal algebra are

$\displaystyle\big{[}L_{m},L_{n}\big{]}$
$\displaystyle=\frac{c}{12}m(m^{2}-1)\delta_{m+n,0}+(m-n)L_{m+n}\ ,$ (A.2a)
$\displaystyle\big{[}J^{a}_{m},J^{b}_{n}\big{]}$
$\displaystyle=\frac{c}{12}m\,\delta^{ab}\delta_{m+n,0}+i\epsilon^{ab}_{\,\,\,\,c}\,J^{c}_{m+n}\
,$ (A.2b)
$\displaystyle\big{\\{}G^{\alpha}_{\dot{A},r},G^{\beta}_{\dot{B},s}\big{\\}}$
$\displaystyle=\epsilon_{\dot{A}\dot{B}}\bigg{[}\epsilon^{\alpha\beta}\frac{c}{6}\Big{(}r^{2}-\frac{1}{4}\Big{)}\delta_{r+s,0}+\big{(}\sigma^{aT}\big{)}^{\alpha}_{\gamma}\>\epsilon^{\gamma\beta}(r-s)J^{a}_{r+s}+\epsilon^{\alpha\beta}L_{r+s}\bigg{]}\
,$ (A.2c) $\displaystyle\big{[}J^{a}_{m},G^{\alpha}_{\dot{A},r}\big{]}$
$\displaystyle={\frac{1}{2}}\big{(}\sigma^{aT}\big{)}^{\alpha}_{\beta}\,G^{\beta}_{\dot{A},m+r}\
,$ (A.2d) $\displaystyle\big{[}L_{m},G^{\alpha}_{\dot{A},r}\big{]}$
$\displaystyle=\Big{(}\frac{m}{2}-r\Big{)}G^{\alpha}_{\dot{A},m+r}\ ,$ (A.2e)
$\displaystyle\big{[}L_{m},J^{a}_{n}\big{]}$ $\displaystyle=-nJ^{a}_{m+n}\ ,$
(A.2f)

where $\sigma^{aT}$ are the transpose of the Pauli sigma matrices and with the
right-moving currents satisfying an analogous algebra. A different basis for
the $J$ currents is also often used; instead of $J^{a}$ we have
$J^{3},J^{\pm}$ where

$J^{\pm}=J^{1}\pm iJ^{2}\ .$ (A.3)

The full contracted large $\mathcal{N}=4$ superconformal algebra of the D1-D5
CFT also includes the modes $d^{\alpha A}_{r}$ and $\alpha_{A\dot{A},n}$ of
the free fermions $\psi^{\alpha A}$ and bosons $\partial X_{\\!A\dot{A}}$
respectively. The mode expansions of these free fields are

$\displaystyle\partial X_{A\dot{A}}(z)$
$\displaystyle=-i\sum_{n}z^{-n-1}\,\alpha_{A\dot{A},n}\ ,$ (A.4a)
$\displaystyle\psi^{\alpha A}(z)$ $\displaystyle=\sum_{r}z^{-r-1/2}\,d^{\alpha
A}_{r}\ ,$ (A.4b)

and likewise the inverse relations are

$\displaystyle\alpha_{A\dot{A},n}$ $\displaystyle=i\oint\frac{dz}{2\pi
i}z^{n}\partial X_{A\dot{A}}(z)\ ,$ (A.5a) $\displaystyle d^{\alpha A}_{s}$
$\displaystyle=\oint\frac{dz}{2\pi i}z^{n-\frac{1}{2}}\psi^{\alpha A}(z)\ .$
(A.5b)

The brackets of the $\alpha$ and $d$ modes are

$\displaystyle\big{[}\alpha_{A\dot{A},n},\alpha_{B\dot{B},m}\big{]}$
$\displaystyle=-n\frac{c}{6}\epsilon_{AB}\,\epsilon_{\dot{A}\dot{B}}\,\delta_{n+m,0}\
,$ (A.6a) $\displaystyle\big{\\{}d^{\alpha A}_{r},d^{\beta B}_{s}\big{\\}}$
$\displaystyle=-\frac{c}{6}\epsilon^{\alpha\beta}\epsilon^{AB}\delta_{r+s,0}\
,$ (A.6b) $\displaystyle\big{[}\alpha_{A\dot{A},n},d^{\beta B}_{s}\big{]}$
$\displaystyle=0\ ,$ (A.6c)

and similarly for the right-moving modes. The remaining parts of the full
contracted large $\mathcal{N}=4$ superconformal algebra are the current-free
field brackets

$\displaystyle\big{[}L_{n},\alpha_{\\!A\dot{A},m}\big{]}$
$\displaystyle=-m\,\alpha_{\\!A\dot{A},n+m}\ ,$ (A.7a)
$\displaystyle\big{[}L_{n},d^{\alpha A}_{s}\big{]}$
$\displaystyle=-\Big{(}\frac{n}{2}+s\Big{)}d^{\alpha A}_{n+s}\ ,$ (A.7b)
$\displaystyle\big{[}J^{a}_{n},d^{\alpha A}_{s}\big{]}$
$\displaystyle=\frac{1}{2}\big{(}\sigma^{Ta}\big{)}^{\alpha}_{\
\beta}\,d^{\beta A}_{s+n}\ ,$ (A.7c)
$\displaystyle\big{[}G^{\alpha}_{\dot{A},s},\alpha_{B\dot{B},n}\big{]}$
$\displaystyle=-in\,\epsilon_{AB}\epsilon_{\dot{A}\dot{B}}\,d^{\alpha
A}_{s+n}\ ,$ (A.7d) $\displaystyle\big{\\{}G^{\alpha}_{\dot{A},r},d^{\beta
B}_{s}\big{\\}}$
$\displaystyle=i\epsilon^{\alpha\beta}\epsilon^{AB}\alpha_{\\!A\dot{A},r+s}\
.$ (A.7e)

In the computation of Section 4.1 we will need the current OPEs which are
given below

$\displaystyle T(z)T(w)$
$\displaystyle\sim\frac{c/2}{(z-w)^{4}}+\frac{2T(w)}{(z-w)^{2}}+\frac{\partial
T(w)}{z-w}\ ,$ (A.8a) $\displaystyle J^{-}(z)J^{+}(w)$
$\displaystyle\sim\frac{c/6}{(z-w)^{2}}-\frac{2J^{3}(w)}{z-w}\ ,\ \
J^{3}(z)J^{-}(w)\sim-\frac{J^{-}(w)}{z-w}\ ,\ \
J^{3}(z)J^{+}(w)\sim\frac{J^{+}(w)}{z-w}\ ,$ (A.8b) $\displaystyle
T(z)J^{a}(w)$ $\displaystyle\sim\frac{J^{a}(w)}{(z-w)^{2}}+\frac{\partial
J^{a}(w)}{z-w}\ \ ,\quad
T(z)G^{\alpha}_{\dot{A}}(w)\sim\frac{3}{2}\frac{G^{\alpha}_{\dot{A}}(w)}{(z-w)^{2}}+\frac{\partial
G^{\alpha}_{\dot{A}}(w)}{z-w}\ ,$ (A.8c) $\displaystyle
G^{\alpha}_{\dot{A}}(z)G^{\beta}_{\dot{B}}(w)$
$\displaystyle\sim\epsilon_{\dot{A}\dot{B}}\bigg{[}\epsilon^{\alpha\beta}\frac{c/3}{(z-w)^{3}}+\big{(}\sigma^{aT}\big{)}^{\alpha}_{\,\gamma}\epsilon^{\gamma\beta}\bigg{(}\frac{2J^{a}(w)}{(z-w)^{2}}+\frac{\partial
J^{a}(w)}{z-w}\bigg{)}+\epsilon^{\alpha\beta}\frac{T(w)}{z-w}\bigg{]}\ ,$
(A.8d) $\displaystyle J^{3}(z)G^{\alpha}_{\dot{A}}(w)$
$\displaystyle\sim\frac{1}{2}\frac{(\sigma^{3})^{\alpha}_{\,\beta}\,G^{\beta}_{\dot{A}}(w)}{z-w}\
\ ,\quad J^{+}(z)G^{-}_{\dot{A}}(w)\sim\frac{G^{+}_{\dot{A}}(w)}{z-w}\ \
,\quad J^{-}(z)G^{+}_{\dot{A}}(w)\sim\frac{G^{-}_{\dot{A}}(w)}{z-w}\ ,$ (A.8e)

where $\sim$ here is equality up to regular terms.

###  A.2 Hermitian conjugation

Contractions between $su(2)$ indices are done using antisymmetric tensors such
as $\epsilon_{\alpha\beta}$ and this gives rise to certain extra negative
signs in the definitions of Hermitian conjugates. For the left-moving cylinder
supercharge fields we use the following conjugation rules

$\Big{(}G^{+}_{+}(\tau,\sigma)\Big{)}^{\dagger}=-G^{-}_{-}(-\tau,\sigma)\quad,\quad\Big{(}G^{+}_{-}(\tau,\sigma)\Big{)}^{\dagger}=G^{-}_{+}(-\tau,\sigma)\
,$ (A.9)

which leads to the conjugation rules for the modes

$\big{(}G^{+}_{+,s}\big{)}^{\dagger}=-G^{-}_{-,-s}\quad,\quad\big{(}G^{+}_{-,s}\big{)}^{\dagger}=G^{-}_{+,-s}\
,$ (A.10)

with the right-moving supercharges obeying similar conjugation conventions.
For the degree-2 twist operators our conjugation conventions are

$\big{(}\sigma^{--}(\tau,\sigma)\big{)}^{\dagger}=-\sigma^{++}(-\tau,\sigma)\quad\
,\quad\big{(}\sigma^{-+}(\tau,\sigma)\big{)}^{\dagger}=\sigma^{+-}(-\tau,\sigma)\
.$ (A.11)

Likewise, in terms of the free fields of the orbifold theory the conjugation
conventions we use are

$\displaystyle\big{(}\alpha_{++,n}\big{)}^{\dagger}$
$\displaystyle=-\alpha_{--,-n}\quad,\quad\big{(}\alpha_{+-,n}\big{)}^{\dagger}=\alpha_{-+,-n}\
,$ (A.12a) $\displaystyle\big{(}d^{++}_{s}\big{)}^{\dagger}$
$\displaystyle=-d^{--}_{-s}\quad,\quad\big{(}d^{+-}_{s}\big{)}^{\dagger}=d^{-+}_{-s}\
.$ (A.12b)

##  Appendix B The mode correlator $C^{\beta\alpha}_{\\!\dot{B}\dot{A}}$

In Section 3.1 the lift of the state $G^{\alpha}_{\dot{A},-s}|\phi\rangle$ is
found in terms of the $t$-plane mode correlator

$C^{\beta\alpha}_{\\!\dot{B}\dot{A}}\equiv\langle\phi|G^{\beta}_{\dot{B},s-k_{4}}G^{+}_{-,s_{2}}G^{-}_{+,s_{1}}G^{\alpha}_{\dot{A},-s+k_{3}}|\phi\rangle\
.$ (B.1)

Using the shorthand $\Delta\equiv\delta_{s_{1}+s_{2}+k_{3}-k_{4},0}$ and the
discrete step function

$H[n]\equiv\begin{cases}\ 1\quad\text{if }n\geq 0\vspace{4pt}\\\ \
0\quad\text{otherwise}\end{cases}\ ,$ (B.2)

this correlator can be found straightforwardly using the $\mathcal{N}=4$
superconformal algebra (A.2) and the definition of a superconformal primary in
this theory (2.9). We give the result here for reference

$\displaystyle C^{\beta\alpha}_{\\!\dot{B}\dot{A}}$
$\displaystyle=\epsilon_{+\dot{A}}\epsilon^{-\alpha}\epsilon_{\dot{B}-}\epsilon^{\beta+}\Big{(}s_{1}^{2}-\frac{1}{4}\Big{)}\Big{(}(s-k_{4})^{2}-\frac{1}{4}+h\Big{)}\delta_{s_{1}-s+k_{3},0}\,\delta_{s_{2}+s-k_{4},0}\,H\big{[}\\!-\\!s_{2}-\tfrac{1}{2}\big{]}$
$\displaystyle\ \ \
-\frac{1}{2}\epsilon_{+\dot{A}}\big{(}\sigma^{aT}\big{)}^{\\!-}_{\gamma}\epsilon^{\gamma\alpha}\epsilon_{\dot{B}-}(s_{1}+s-k_{3})\Delta\Bigg{[}\big{(}\sigma^{aT}\big{)}^{+}_{\delta}\epsilon^{\beta\delta}\Big{(}(s-k_{4})^{2}-\frac{1}{4}+h\Big{)}H\big{[}s-k_{4}-\tfrac{1}{2}\big{]}$
$\displaystyle\ \ \
+\big{(}\sigma^{aT}\big{)}^{\beta}_{\delta}\epsilon^{\delta+}\Big{(}(s_{1}+k_{3}-k_{4})^{2}-\frac{1}{4}+h\Big{)}H\big{[}\\!-\\!s_{2}-\tfrac{1}{2}\big{]}\Bigg{]}H[s-s_{1}-k_{3}-1]$
$\displaystyle\ \ \
-\frac{1}{2}\epsilon_{+\dot{A}}\epsilon^{-\alpha}\epsilon_{\dot{B}-}\epsilon^{\beta+}\Delta\Bigg{[}(s_{1}-s+k_{3}-2s_{2})\Big{(}(s-k_{4})^{2}-\frac{1}{4}+h\Big{)}H\big{[}s-s_{1}-s_{2}-k_{3}-\tfrac{1}{2}\big{]}$
$\displaystyle\ \ \
+(s_{1}-3s+k_{3}+2k_{4})\Big{(}(s_{1}+k_{3}-k_{4})^{2}-\frac{1}{4}+h\Big{)}H\big{[}\\!-\\!s_{2}-\tfrac{1}{2}\big{]}\Bigg{]}H[s-s_{1}-k_{3}]$
$\displaystyle\ \ \
+\epsilon_{+\dot{A}}\epsilon^{-\alpha}\epsilon_{\dot{B}-}\epsilon^{\beta+}h\Big{(}(s-k_{4})^{2}-\frac{1}{4}+h\Big{)}\delta_{s_{2}+s-k_{4},0}\,\delta_{s_{1}-s+k_{3},0}\,H[s-s_{1}-k_{3}]H\big{[}\\!-\\!s_{2}-\tfrac{1}{2}\big{]}$
$\displaystyle\ \ \
-\epsilon_{-\dot{A}}\epsilon^{+\alpha}\epsilon_{\dot{B}+}\epsilon^{\beta-}\Big{(}s_{2}^{2}-\frac{1}{4}\Big{)}\Big{(}(s-k_{4})^{2}-\frac{1}{4}+h\Big{)}\delta_{s_{2}-s+k_{3},0}\,\delta_{s_{1}+s-k_{4},0}\,H\big{[}\\!-\\!s_{1}-\tfrac{1}{2}\big{]}$
$\displaystyle\ \ \
-\frac{1}{2}\epsilon_{-\dot{A}}\big{(}\sigma^{aT}\big{)}^{\\!+}_{\gamma}\epsilon^{\gamma\alpha}\epsilon_{\dot{B}+}(s_{2}+s-k_{3})\Delta\Bigg{[}\big{(}\sigma^{aT}\big{)}^{-}_{\delta}\epsilon^{\beta\delta}\Big{(}(s-k_{4})^{2}-\frac{1}{4}+h\Big{)}H\big{[}s-k_{4}-\tfrac{1}{2}\big{]}$
$\displaystyle\ \ \
+\big{(}\sigma^{aT}\big{)}^{\beta}_{\delta}\epsilon^{\delta-}\Big{(}(s-k_{4})^{2}-s_{1}^{2}\Big{)}H[s-s_{2}-k_{3}-1]H[s_{1}+s-k_{4}-1]\Bigg{]}H\big{[}\\!-\\!s_{1}-\tfrac{1}{2}\big{]}$
$\displaystyle\ \ \
-\frac{1}{2}\epsilon_{-\dot{A}}\epsilon^{+\alpha}\epsilon_{\dot{B}+}\epsilon^{\beta-}\Delta\Bigg{[}(s_{2}-s+k_{3}-2s_{1})\Big{(}(s-k_{4})^{2}-\frac{1}{4}+h\Big{)}H\big{[}s-k_{4}-\tfrac{1}{2}\big{]}$
$\displaystyle\ \ \
+(s_{1}+s-k_{4})\Big{(}(s_{1}+s-k_{4})^{2}-1\Big{)}H[s-s_{2}-k_{3}]H[s+s_{1}-k_{4}]$
$\displaystyle\ \ \
+2h(s_{1}-s_{2}+2s-k_{3}-k_{4})H[s-s_{2}-k_{3}]H[s+s_{1}-k_{4}]\Bigg{]}H\big{[}\\!-\\!s_{1}-\tfrac{1}{2}\big{]}$
$\displaystyle\ \ \
-\epsilon_{-\dot{A}}\epsilon^{+\alpha}\epsilon_{\dot{B}+}\epsilon^{\beta-}\bigg{(}h\Big{(}(s-k_{4})^{2}-\frac{1}{4}\Big{)}+h^{2}\bigg{)}\delta_{s_{1}+s-k_{4},0}\,\delta_{s_{2}-s+k_{3},0}\,H\big{[}\\!-\\!s_{1}-\tfrac{1}{2}\big{]}$
$\displaystyle\ \ \
+\epsilon_{\dot{B}\dot{A}}\epsilon^{\beta\alpha}\Big{(}(s-k_{4})^{2}-\frac{1}{4}\Big{)}\Big{(}s_{2}^{2}-\frac{1}{4}+h\Big{)}\delta_{k_{3}-k_{4},0}\,\delta_{s_{1}+s_{2},0}\,H\big{[}\\!-\\!s_{1}-\tfrac{1}{2}\big{]}$
$\displaystyle\ \ \
-\frac{1}{2}\epsilon_{\dot{B}\dot{A}}\big{(}\sigma^{aT}\big{)}^{\\!\beta}_{\gamma}\epsilon^{\gamma\alpha}(2s-k_{3}-k_{4})\Delta\Bigg{[}\big{(}\sigma^{aT}\big{)}^{+}_{\delta}\epsilon^{\delta-}(s_{2}-s_{1})(k_{3}-k_{4})H[-s_{1}-s_{2}-1]$
$\displaystyle\ \ \
+\big{(}\sigma^{aT}\big{)}^{-}_{\delta}\epsilon^{\delta+}\Big{(}(s_{1}+k_{3}-k_{4})^{2}-\frac{1}{4}+h\Big{)}H\big{[}\\!-\\!s_{2}-\tfrac{1}{2}\big{]}\Bigg{]}H[k_{3}-k_{4}-1]H\big{[}\\!-\\!s_{1}-\tfrac{1}{2}\big{]}$
$\displaystyle\ \ \
+\epsilon_{\dot{B}\dot{A}}\epsilon^{\beta\alpha}H\big{[}\\!-\\!s_{1}-\tfrac{1}{2}\big{]}\bigg{(}h\Big{(}s_{2}^{2}-\frac{1}{4}\Big{)}+h^{2}\bigg{)}\delta_{s_{1}+s_{2},0}\,\delta_{k_{3}-k_{4},0}$
$\displaystyle\ \ \
+\frac{1}{2}\epsilon_{\dot{B}\dot{A}}\epsilon^{\beta\alpha}H\big{[}\\!-\\!s_{1}-\tfrac{1}{2}\big{]}H[k_{3}-k_{4}]\,\Delta\Bigg{[}(k_{3}-k_{4})\Big{(}(k_{3}-k_{4})^{2}-1\Big{)}H[-s_{1}-s_{2}]$
$\displaystyle\ \ \
+2h(k_{3}-k_{4}-s_{1}-s_{2})H[-s_{1}-s_{2}]-(k_{3}-k_{4}-2s_{1})\Big{(}s_{2}^{2}-\frac{1}{4}+h\Big{)}H\big{[}\\!-\\!s_{2}-\tfrac{1}{2}\big{]}\Bigg{]}\,,$
(B.3)

##  Appendix C Lifting of $\mathcal{J}^{+,(m)}|\phi\rangle$

In this appendix we consider lift of the family of untwisted sector states
with the NS vacuum on all copies bar one, on which there us the excited state

$\mathcal{J}^{+,(m)}|\phi\rangle\equiv J^{+}_{-(2m-1)}\cdots
J^{+}_{-3}J^{+}_{-1}|\phi\rangle\ ,$ (C.1)

where $\phi$ is a single-copy superconformal primary of dimension $h$. The
Hermitian conjugate of this state is given by

$\langle\phi|\mathcal{J}^{-,(m)}\equiv\langle\phi|J^{-}_{-1}J^{-}_{-3}\cdots
J^{-}_{-(2m-1)}\ ,$ (C.2)

and its quantum numbers are

$(h^{(m)},j^{(m)})=(h+m^{2},m)\ .$ (C.3)

This particular family of states has been chosen for the fact that it is the
unique spectral flow of the single-copy superconformal primary $|\phi\rangle$
under a flow by $\eta=2m$ units. Since $m\in\mathbb{Z}^{+}$ here the spectral
flow maps the NS sector theory back to the NS sector. The mapping of quantum
numbers under spectral flow can be found in (2.24). This fact can then be
leveraged in the lifting computation in a manner very similar to that used in
[44]: once on the covering space of the key amplitude required for the lift,
the inverse spectral flow can be used to effectively remove the
$\mathcal{J}^{+,(m)}$ excitations. This roughly reduces the problem to the
calculation of the lift of $|\phi\rangle$ which was done in [48], albeit with
a different method than used here.

As explained in Section 2.5, we need to compute the $w$-cylinder left-moving
amplitude

$A^{(1)(1)}_{h,m}(w_{2},w_{1})\equiv\langle\phi|\mathcal{J}^{-,(m)}\,\big{(}G^{+}_{-,-\frac{1}{2}}\sigma_{2}^{-}\big{)}(w_{2})\big{(}G^{-}_{+,-\frac{1}{2}}\sigma_{2}^{+}\big{)}(w_{1})\,\mathcal{J}^{+,(m)}|\phi\rangle\
,$ (C.4)

which upon mapping to the covering space as per the steps (1)-(3) of Section
3.1 is given by

$\displaystyle A^{(1)(1)}_{h,m}(w_{2},w_{1})$
$\displaystyle=f_{1}f_{2}f_{3}\bigg{(}\frac{dt}{dz}\bigg{|}_{-a}\bigg{)}^{\\!m^{2}}\bigg{(}\frac{-a-t_{1}}{-a-t_{2}}\bigg{)}^{\\!m}\,C^{(m)}_{h}\
,$ (C.5)

where the factors $f_{1}$, $f_{2}$ and $f_{3}$ are given in (3.10), (3.13) and
(3.15) respectively and $C^{(m)}_{h}$ is the $t$-plane correlator

$C^{(m)}_{h}\equiv\big{<}\big{(}\mathcal{J}^{-,(m)}\phi\big{)}(\infty)\,G^{+}_{-}(t_{2})\,G^{-}_{+}(t_{1})\,\big{(}\mathcal{J}^{+,(m)}\phi\big{)}(-a)\big{>}\
.$ (C.6)

In (C.5) the factors of $\big{(}\frac{dt}{dz}\big{|}_{-a}\big{)}^{\\!m^{2}}$
and $\big{(}\frac{-a-t_{1}}{-a-t_{2}}\big{)}^{\\!m}$ come respectively from
the transformation of the initial state $\mathcal{J}^{+,(m)}$ modes under the
map to the covering space (3.7) and the spectral flows used to remove spin
fields (see step (2)). As explained above, the $t$-plane correlator (C.6) can
be evaluated by spectral flowing away the $\mathcal{J}^{+,(m)}$ by a spectral
flow by $\eta=-2m$ units around $t=-a$, which also automatically removes the
$\mathcal{J}^{-,(m)}$ excitation at $t=\infty$. Under this spectral flow, the
$G$ field insertions transform as per (2.25) (the $\phi$ do not since they are
not charged under $J^{3}_{0}$) yielding

$\displaystyle
C^{(m)}_{h}=\bigg{(}\frac{t_{2}+a}{t_{1}+a}\bigg{)}^{\\!m}A^{+-}_{-+}(w_{2},w_{1};h)\
,$ (C.7)

with $A^{+-}_{-+}(w_{2},w_{1};h)$ defined in (4.9). The amplitude $A^{(1)(1)}$
is then

$\displaystyle
A^{(1)(1)}_{h,m}(w_{2},w_{1})=\frac{a+b(2h-1)}{16\,b}\bigg{(}\frac{a}{a-b}\bigg{)}^{\\!h+m^{2}-1}\
.$ (C.8)

Replacing $a$ and $b$ as functions of $w$ as per (3.21) we get

$\displaystyle
A^{(1)(1)}_{h,m}(w_{2},w_{1})=\frac{hx^{2}-2(h-1)x+h}{2^{2h+2m^{2}+1}x^{h+m^{2}-1}}\frac{(x+1)^{2(h+m^{2}-1)}}{(x-1)^{2}}\
,$ (C.9)

where we use the shorthand $x=e^{w/2}$. The amplitude with the other
independent copy structure is obtained by $x\to-x$ as

$A^{(2)(1)}_{h,m}(w_{2},w_{1})=(-1)^{h+m^{2}+1}\frac{hx^{2}+2(h-1)x+h}{2^{2h+2m^{2}+1}x^{h+m^{2}-1}}\frac{(x-1)^{2(h+m^{2}-1)}}{(x+1)^{2}}\
.$ (C.10)

Considering the contour integral $I_{C_{2}}$ as defined in (2.40b), the right-
moving part contributes a factor of $-1$ and we then project onto the $x^{0}$
part of the amplitude, giving

$\displaystyle I^{(1)(1)}_{C_{2}}$
$\displaystyle\equiv-A^{(1)(1)}_{h,m}(w_{2},w_{1})\Big{|}_{x^{0}}$
$\displaystyle=-\sum_{\nu=0}^{2}\frac{c_{\nu}}{2^{2h+2m^{2}+1}x^{h+m^{2}-1-\nu}}\sum_{\ell_{3}=0}^{\infty}\sum_{\ell_{2}=0}^{\min(\ell_{3},2(h+m^{2}-1))}(\ell_{3}-\ell_{2}+1)\,{}^{2(h+m^{2}-1)}C_{\ell_{2}}\,x^{\ell_{3}}\bigg{|}_{x^{0}}$
$\displaystyle=-\sum_{\nu=0}^{2}\sum_{\ell_{2}=0}^{h+m^{2}-1-\nu}\frac{c_{\nu}(h+m^{2}-\nu-\ell_{2})}{2^{2h+2m^{2}+1}}\,{}^{2(h+m^{2}-1)}C_{\ell_{2}}$
$\displaystyle=-\frac{(2h+m^{2}-1)\,\Gamma(h+m^{2}-\frac{1}{2})}{8\sqrt{\pi}\,\Gamma(h+m^{2})}\
,$ (C.11)

where the term $\ell_{3}=h+m^{2}-1-\nu$ was picked out and $c_{\nu}$ are
coefficients such that

$\sum_{\nu=0}^{2}c_{\nu}\,x^{\nu}=hx^{2}-2(h-1)x+h\ .$ (C.12)

For $A^{(2)(1)}$ the integral results in

$\displaystyle I^{(2)(1)}_{C_{2}}$
$\displaystyle\equiv-A^{(2)(1)}_{h,m}(w_{2},w_{1})\Big{|}_{x^{0}}$
$\displaystyle=\sum_{\nu=0}^{2}\frac{c_{\nu}(-1)^{h+m^{2}+\nu}}{2^{2h+2m^{2}+1}}\sum_{\ell_{3}=0}^{\infty}\sum_{\ell_{2}=0}^{\min(\ell_{3},2(h+m^{2}-1))}(-1)^{\ell_{3}}\,(\ell_{3}-\ell_{2}+1)\,{}^{2(h+m^{2}-1)}C_{\ell_{2}}\,x^{\ell_{3}}\bigg{|}_{x^{0}}$
$\displaystyle=\sum_{\nu=0}^{2}\sum_{\ell_{2}=0}^{h+m^{2}-1-\nu}\frac{(-1)^{2\nu+1}c_{\nu}}{2^{2h+2m^{2}+1}}(h+m^{2}-\nu-\ell_{2})\,{}^{2(h+m^{2}-1)}C_{\ell_{2}}$
$\displaystyle=I^{(1)(1)}_{C_{2}}\ .$ (C.13)

From (2.35) the lift is then

$\displaystyle E^{(2)}\big{(}\mathcal{J}^{+,(m)}|\phi_{h}\rangle\big{)}$
$\displaystyle=\pi^{3/2}\lambda^{2}\frac{(2h+m^{2}-1)\,\Gamma(h+m^{2}-\frac{1}{2})}{2\,\Gamma(h+m^{2})}\
.$ (C.14)

The asymptotic behaviour of this lift for large dimension is discussed in
Section 4.2.1. We note that for $m=1$ the state is a single-copy descendant
state of $\phi$ and, as expected from Section 3.3, the lift obeys the relation

$E^{(2)}\big{(}\mathcal{J}^{+,(1)}|\phi_{h}\rangle\big{)}=\pi^{3/2}\lambda^{2}\frac{\Gamma(h+\frac{1}{2})}{\Gamma(h)}=E^{(2)}\big{(}|\phi_{h}\rangle\big{)}\
.$ (C.15)

## References

  * [1] A. Strominger and C. Vafa, _Microscopic origin of the Bekenstein-Hawking entropy_ , _Phys. Lett. B_ 379 (1996) 99 [hep-th/9601029].