Protected string spectrum in AdS3/CFT2 from worldsheet integrability
Received: January
AdS3/CFT2 from
Marco Baggio 0 1 3 7 8
Olof Ohlsson Sax 0 1 3 5 8
Alessandro Sfondrini 0 1 3 6 8
Bogdan Stefan´ski 0 1 3 8
Jr. 0 1 2 3 8
Northampton Square 0 1 3 8
EC 0 1 3 8
V 0 1 3 8
HB London 0 1 3 8
U.K. 0 1 3 8
Open Access 0 1 3 8
c The Authors. 0 1 3 8
0 Guildford, GU2 7XH, U.K
1 Roslagstullsbacken 23 , SE106 91 Stockholm , Sweden
2 Centre for Mathematical Science , City , University of London
3 Celestijnenlaan 200D , B3001 Leuven , Belgium
4 Department of Mathematics, University of Surrey
5 Nordita, Stockholm University and KTH Royal Institute of Technology
6 Institut fu ̈r Theoretische Physik, ETH Zu ̈rich
7 Instituut voor Theoretische Fysica, KU Leuven
8 [40] J. de Boer, A. Pasquinucci and K. Skenderis, AdS/CFT dualities involving large 2
We derive the protected closedstring spectra of AdS3/CFT2 dual pairs with 16 supercharges at arbitrary values of the string tension and of the threeform fluxes. These follow immediately from the allloop Bethe equations for the spectra of the integrable worldsheet theories. Further, representing the underlying integrable systems as spin chains, we find that their dynamics involves lengthchanging interactions and that protected states correspond to gapless excitations above the BerensteinMaldacenaNastase vacuum. In the case of AdS3 × S3 × T4 the degeneracies of such operators precisely match those of the dual CFT2 and the supergravity spectrum. On the other hand, we find that for AdS3×S3×S3×S
worldsheet; integrability; AdSCFT Correspondence; Bethe Ansatz; Conformal Field Theory

there are fewer protected states than previous supergravity calculations had suggested. In
particular, protected states have the same su(2) charge with respect to the two
three
1 Introduction
2 BPS states in N = (4, 4) theories
2.2 (Modified) elliptic genus
3 The AdS3 × S3 × T4 Bethe equations
4 Spinchain interpretation 4.1 4.2 Massive excitations
Massless excitations
5 Fermionic zero modes and protected states
6 Protected states for AdS3 × S3 × S3 × S1
7 Protected states in mixedflux AdS3 backgrounds
Wrapping corrections
9 Conclusion
A The psu(1, 12) superalgebra
B Computation of the modified elliptic genus
C Multiplet joining and lengthchanging effects
Massive excitations
C.2 Massless excitations
D Dynamic spin chains
Introduction
Recent years have seen significant progress in the application of integrability techniques
to the planar sector of the AdS3/CFT2 correspondence.
Most of these advances have
stemmed from the study of superstrings on AdS3 × S
backgrounds, which preserve sixteen supercharges and can be supported by a mixture of
RamondRamond (RR) and NeveuSchwarzNeveuSchwarz (NSNS) fluxes. In particular,
it has been shown that the string nonlinear sigma models (NLSM) are classically integrable
for these backgrounds [1–3].1 An efficient way to check whether integrability persists at
the quantum level is to study the S matrix that scatters the excitations on the string
worldsheet, following the ideas pioneered by Zamolodchikov and Zamolodchikov for
twodimensional relativistic integrable QFTs [5]. Indeed, for the cases of AdS3 × S3 × T4 and
AdS3 × S3 × S3 × S1 supported by arbitrary threeform fluxes, the twobody S matrix can be
fixed by symmetry considerations up to “dressing factors” and turns out to automatically
satisfy the YangBaxter equation [6–11].2 The dressing factors are not fixed by symmetry,
rather they satisfy crossing equations. So far they have been found only for the case
of AdS3 × S3 × T4 supported by pureRR fluxes [13, 14]. Equipped with an integrable
S matrix, one may write down the Bethe equations that predict the asymptotic spectrum3
of closed strings in these backgrounds; this construction has been recently completed for
pureRR AdS3 × S3 × T4 [14, 17], and preliminary results are known for AdS3 × S3 × S3 × S
too [18, 19].
The integrability construction interpolates between the perturbative string NLSM
regime and the “perturbative CFT” regime. For this purpose, we introduce a coupling
constant h so that the largecoupling limit h
1 corresponds to large string tension4 (and
hence a perturbative description on the worldsheet) while h
1 corresponds to the ’t
Hooft coupling being small. The largeh expansion of the integrability construction has
been matched against a number of perturbative string NLSM computations in the
nearplanewave, nearflatspace and semiclassical regimes, see e.g. [20, 23–30]. On the other
hand, the smallh limit is more subtle. While it is known that the AdS3 × S3 × T4
background lives in the moduli space of the a semidirect product CFT (SymN (T4))nT4 [31, 32],
it is not clear how the symmetricproduct orbifold point is related to the h → 0 limit of
the integrability description nor how to find integrability there [33]. The first evidence of
integrability on the CFT2 side has instead been found in the IR limit of the Higgs branch
of the twodimensional superYangMills theory with matter [34] and deserves to be
exidentification of what the dual CFT2 might be remains a challenge [35].5
plored further. The scenario is even more mysterious for AdS3 × S3 × S3 × S1, where the
In this paper we use integrability to determine the protected closedstring spectra of
the AdS3/CFT2 models. We begin with the case of pureRR AdS3 × S3 × T4 for which we
derive in detail such protected states from the allloop Bethe equations, elaborating on the
results announced in [17]. Our findings are valid for generic values of the string tension
in the planar theory. The degeneracy we find matches precisely the spectrum of protected
operators [37] and the (modified) elliptic genus of the symmetricproduct orbifold CFT and
supergravity [38, 39]. Let us stress that the protected states’ Bethe roots are particularly
simple: they carry no worldsheet momentum. As such, determining the protected spectrum
from the Bethe equations is a remarkably straightforward exercise. Furthermore, it is also
1See also ref. [4] for some earlier progress in AdS3/CFT2 integrability.
2See also ref. [12] for a review of these constructions.
3This description is valid up to finitesize “wrapping” effects [15, 16].
4The precise relation between the coupling h and the string tension is at present only known
perturbatively [20, 21]. In the case of integrable AdS4/CFT3 backgrounds an analogous function h was determined
exactly by comparing integrability and localisation results [22].
5See however [36] for a recent proposal.
easy to show that finitesize “wrapping” corrections to the Bethe ansatz exactly cancel for
these protected states. In a small way, this simplicity indicates the power of integrability
in solving the spectral problem.
Based on these results, it becomes clear that protected states can be found for more
general AdS3 integrable backgrounds by classifying zeromomentum Bethe roots. In
particular, we find that the spectrum of protected operators for the pureRR AdS3 × S3 × S3 × S
− with respect to the
two threespheres; once again, this result is valid for generic values of the string tension in
the planar theory. We compare our findings with the results available in the literature. At
should exist, yielding a much larger degeneracy. On the other hand, at generic points of
the moduli space, it is expected [40, 41] that such multiplets should not be protected.6 Our
integrabilitybased findings give an explicit confirmation of this expectation, though this
does not address whether additional degeneracies might appear at the supergravity point.
The analysis of ref. [40] is not conclusive as it relies on the assumption that all
Kaluz˙aKlein (KK) modes sit in short representations and thus might overcount the BPS states. A
− exist is to study
pointlike string solutions for the AdS3 ×S3 ×S3 ×S1 background [1], see also [42, 43], which
are in onetoone correspondence with its bosonic supersymmetric ground states. We find
In addition, we show that our integrability analysis of protected states holds also for mixed
RR and NSNSflux AdS3 backgrounds. It is interesting to note that the restriction on
the allowed angular momenta J+ = J
− for AdS3 × S
1 was also found for BPS
giantgraviton Dstrings [44]. This suggests that this condition may well apply to the full
nonperturbative protected spectrum of the theory.
One of the central lessons of holographic integrability in higher dimensions [45], see
also [46, 47] for a review, has been the appearance of an integrable spinchain in the smallh
regime. Such a description is very useful for enumerating the states of the theory, and in
particular for classifying the protected operators. As in the higherdimensional cases, all
the pureRR AdS3 backgrounds exhibit a latticelike dispersion relation
E(p) =
m2 + 4 h2 sin2
where m is the mass of a worldsheet excitation and p its momentum, suggesting that
the theory should also have a representation in terms of discretised degrees of freedom.
Motivated by this expectation, we construct a spin chain at h
1 and describe how the
protected states we found correspond to inserting gapless excitations at zero momentum
above the BerensteinMaldacenaNastase (BMN) [48] vacuum. This description is
particularly useful for enumerating the states of the theory and computing supersymmetric indices.
The paper is structured as follows. In section 2 we start by briefly reviewing the
6This follows from observing that the Liesuperalgebra shortening condition receives nonlinear
correcstructure of the Bethe equations for pureRR AdS3 × S3 × T4 strings at h
these, in section 4 we construct the weaklycoupled spin chain. It is then straightforward
1. Building on
to derive, in section 5, the spectrum of protected operators and to match that with the
results of section 2. We then describe in section 6 how to apply the same techniques to
the case of pureRR AdS3 × S
× S1, derive the spectrum of protected states and
corroborate it by a classical stringtheory calculation. In section 7 we argue on general
grounds that our results are unchanged in the presence of a mixture of RR and NSNS
threeform fluxes. Finally, in section 8 we argue that our results, which where derived from
the Bethe equations, remain valid even when wrapping corrections are taken into account.
We conclude in section 9, and relegate some technical details to four appendices.
Note added.
While we were in the final stages of preparing our manuscript, we
received [49] where the protected spectrum on AdS3 × S3 × S3 × S1 is also investigated both
at the WessZuminoWitten point by CFT techniques and at the supergravity one through
a direct analysis of the KK spectrum. The results perfectly agree with our integrability
findings for this background.
BPS states in N
= (4, 4) theories
of the psu(1, 12) algebra, acting on the left and rightmoving sectors respectively. As
reviewed in appendix A, representations are characterized by the (left and right) conformal
dimension (DL, DR) and the Rsymmetry quantum numbers (J L, J R) of the superconformal
primaries. The unitarity bound for the left copy of the algebra in these conventions is
DL ≥ J L,
and similarly for the right one. Multiplets that saturate this bound are called (left) chiral
primary multiplets. They contain 2J + 1 superconformal primaries differing in their J
and Q˙ 2.7 As a consequence, chiral primary multiplets are shorter than generic multiplets.
Multiplets that saturate the unitarity bound in both the left and right sector are 1/2BPS,
those that saturate it only in one sector are 1/4BPS. More details on the structure of
short and long representations of the psu(1, 12) algebra are presented in appendix A.
that representations are labelled by the conformal dimension D and the two su(2) spins
(J+, J−). The unitarity bound then reads8
conditions, in that certain combinations of their superconformal descendants vanish.
where the trace is taken over the space of highestweight states of the chiral primary
multiplets. This polynomial has finite degree, since the spectrum of chiral primaries is
bounded from above [51]
In the case of a sigma model with target space M, it can be shown that the Poincar´e
polynomial is given by [52]
interested in the case M
where hp,q are the Betti numbers of the target space M. In this paper, we are especially
The highestweight states of representations that saturate this bound are annihilated by
one supercharge, so such multiplets are shorter than generic ones, but preserve less
super1/8BPS multiplets, depending on whether they saturate the unitarity bound on both left
and right sectors or only on one sector. The BPS bound for the full superconformal algebra
not discuss this point further.
Poincar´e polynomial
usefully encoded in the Poincar´e polynomial
The Poincar´e polynomial of the symmetric orbifold SymN (M) can also be expressed in
terms of the Betti numbers of M [37, 53]:
X QN Pt,t¯(SymN (M)) =
m=1 p,q
where d is the dimension of M.
The largeN behaviour of Pt,t¯(SymN (M)) can be extracted using the relation [37]
Pt,t¯(Sym∞(M)) = lim (1 − Q) X
QN Pt,t¯(SymN (M)).
Pt,t¯ = Tr t2JLt¯2JR,
h ≤ 6
Pt,t¯ = X hp,qtpt¯q,
In the case of M = T4, this gives
Pt,t¯(Sym∞(T4)) =
(1 + t)2(1 + t¯)2 ∞
(1 + tJ t¯J−1)4(1 + tJ−1 t¯J )4
(1 − t t¯)5
J=2 (1 − tJ−2 t¯J )(1 − tJ t¯J−2)(1 − tJ t¯J )6
This exhibits the expected structure of a freely generated partition function: bosonic
gen1 + t2JLt¯2JR + t4JLt¯4JR + . . . =
(1 − t2JLt¯2JR) ,
while fermionic ones give rise to
• 6 bosonic generators with charges J2 , J2 ,
• 4+4 fermionic generators with charges J −21 , J2
• 1+1 bosonic generators with charges J
2 − 1, J2
J2 , J2 − 1 ,
and additionally, there are
• 2+2 fermionic generators with charges 12 , 0 and 0, 12
• 5 bosonic generators with charges 12 , 12 .
In section 5, we will derive this exact spectrum of protected states using the allloop
integrable Bethe Equations and show how such degeneracies appear from a spinchain
(Modified) elliptic genus
is defined in the NS sector as9
A more refined quantity that is constant over the moduli space is the elliptic genus. This
E (q, y) = TranythingiL⊗chiral primaryiR(−1)F q2DLy2JL.
ral algebra of the SymN (T4) theory, due to the additional u(1)4 symmetry associated to
algebra, so that its elliptic genus vanishes. Therefore, we consider the modified elliptic
genus [39], defined in the NS sector as
9This quantity is usually defined in the Ramond sector of the Hilbert space
TrRR(−1)F q2(DL−c/24)q¯2(DR−c/24)y2JL,
flow, it can be related to the quantity in (2.12) [38].
In analogy to the Poincar´e polynomial (2.7), it is useful to introduce the generating function
of elliptic genera for the symmetric orbifold:
E˜2(Q, q, y) = X QN E2(SymN (T4)).
Unlike the Poincar´e polynomial, the elliptic genus diverges in the large N limit, so the
comparison with the spinchain computation is more subtle. The same issue was
encountered in the computation of the elliptic genus in supergravity [38], where a suitable notion
of degree was introduced to take into account the stringy exclusion principle [54], see also
appendix B. In this case, the only terms that can be meaningfully compared are those that
correspond to states with dimension D < N/4 [39]
E˜2(Q, q, y) = X 2N QN + . . . .
We will reproduce this result in the spinchain picture after introducing an appropriate
notion of degree.
× T4 Bethe equations
Bethe equations [55] are a set of polynomial equations whose solutions — the Bethe roots—
determine the spectrum of an associated integrable system. In the simplest cases the Bethe
roots are in onetoone correspondence with the momenta of particles or magnons and the
Bethe equations are immediately recognisable as quantisation conditions for the momenta,
accounting for periodic boundary conditions and for phase shifts due to scattering of the
magnons. The energy of a given state is determined through the dispersion relation much
like in a free theory. In theories with extra global symmetries, like the ones we consider
here, the momentumcarrying roots are supplemented by auxiliary roots. The latter do not
directly affect the dispersion relation, but are necessary to reproduce states with different
Noether charges.
M¯1, M¯2, M¯3.
The full set of Bethe equations for pureRR AdS3 ×S3 ×T4 strings was found in [14, 17].
It is convenient to work with the weakcoupling Bethe equations [14], which are valid
at small but nonvanishing h. Besides simplifying our formulae, this will allow us to
explicitly enumerate the states in a spinchain language, which is more natural at weak
coupling, and still reproduce the generic spectrum of protected states. The weakcoupling
Bethe equations are reproduced in equations (3.2)–(3.10) below. There are in total nine
types of Bethe roots, three momentumcarrying and six auxiliary. For the excitations
Finally, the massless sector is described by the momentumcarrying roots
zk± and the auxiliary roots r1,k and r3,k, with excitation numbers N0, N1 and N3.10 The
10The massless momentumcarrying roots are conveniently parametrised using two complex conjugate
parameters zk+ and zk− which satisfy zk+zk− = 1.
momentumcarrying Bethe roots are directly related to the momentum of the corresponding
excitations through the relations
uj − 2
u¯j − 2
u¯j + 2ii = eipj ,
zj−
j = eipj ,
where in the three equations above for simplicity of notation we indicate with the same pj
the momentum of a given leftmassive, rightmassive and massless excitation, respectively.
deduced via the weakcoupling limit of the dispersion relation (1.1), which in terms of the
Bethe roots is given below in equation (3.12) .
The massive leftmoving Bethe roots satisfy the psu(1, 12)L Bethe equations
j=1 v1,k − uj + 2i
j6=k
j=1 v3,k − uj + 2i
1 = YM¯2 v¯1,k − u¯j + 2i
j6=k
1 = YM¯2 v¯3,k − u¯j + 2i
uk + 2i !L−N0+N1+N3
= YM2 uk − uj + i YM1 uk − v1,j − 2i YM3 uk − v3,j − 2
i
j=1 uk − uj − i j=1 uk − v1,j + 2i j=1 uk − v3,j + 2i
while the massive rightmoving roots satisfy the psu(1, 12)R equations
¯ i i
j=1 uk − u¯j + i j=1 u¯k − v1¯,j − 2 j=1 u¯k − v3¯,j − 2
These two sets of equations differ from each other in two ways. Firstly, the equations for
the left movers are written in an su(2) grading (i.e. feature the Heisenberg su(2) S matrix)
while the equations for the right movers are written in an sl(2) grading.11 Secondly, the
lengths appearing in the driving terms on the left hand sides of equations (3.3) and (3.6)
are different whenever the excitation numbers Nj corresponding to the massless Bethe
roots are nonzero. As we will see below this feature is essential for obtaining a spinchain
interpretation of the full set of Bethe equations including the massless modes.
11At weak coupling the two sets of equations are completely decoupled and the grading of each can be
individually changed. However, at higher orders in the coupling the two sets of equations are coupled by
extra interaction factors and the two gradings need to be chosen in a consistent way. Hence, we find it
convenient to use different gradings also at weak coupling.
couple to the total momentum carried by the massive leftmoving roots uk.
Finally, the three types of momentumcarrying roots are coupled through the
levelmatching constraint
1 = YM2 uk + 2i YM¯2 u¯k + 2i YN0 z+
i i
k=1 uk − 2 k=1 u¯k − 2 k=1 zk−
and enter the dispersion relation as
j=1 1 + 4uj
j=1 1 + 4u¯j
+ − zj−
coupling limit, massless modes contribute one order earlier than massive ones, as expected
terms of the excitation numbers by
The Cartan charges corresponding to a solution to the Bethe equations are given in
The massless Bethe roots satisfy the equations
1 = YN0 r1,k − zj+ YM2 uj + 2i
j6=k
1 = YN0 r3,k − zj+ YM2 uj + 2i
z−
= YN0 z
j=1 zk− − zj
j=1 uj + 2i j=1 zk − r1,j j=1 zk − r3,j
is constrained to zj± = 1.
As discussed in [14] (see also [8, 9]) massless excitations are charged under an additional
Bethe equations and yields a 2N0 fold degeneracy of the spectrum. We can easily account
for it by arbitrarily assigning an eigenvalue ±1/2 to each massless momentumcarrying root.
of adding extra massive Bethe roots at infinity without affecting the Bethe equations or the
dispersion relation (3.12). Additionally, as discussed in [14] we can add an arbitrary number
D = DL + DR = L + M¯2 +
M1 + M3 + N1 + N3 − M¯1 − M¯3 − N0 + δD,
J = JL + JR = L − M2 +
M1 + M3 + N1 + N3 − M¯1 − M¯3 − N0 ,
S = DL − DR =
K = JL − JR =
− M¯2 +
− M2 +
M1 + M3 + N1 + N3 + M¯1 + M¯3 − N0 ,
M1 + M3 + N1 + N3 + M¯1 + M¯3 − N0 .
corresponding to shifts along T4 in target space. Adding a fermionic massless zero mode
in that case the fermionic zero modes precisely generate the protected states expected from
the discussion of section 2.
Spinchain interpretation
Let us now see how the structure of the weak coupling Bethe equations discussed above
can arise from a spinchain picture of local operators. Such a formulation is very helpful
for enumerating the solutions of the Bethe equations, and hence the states of the theory.
In particular, as we will see below, it will give us a concrete description of the protected
operators at weak coupling. The structure we find here follows directly from the Bethe
equations by introducing an appropriate set of fields that to describe local operators. This
notation is purposefully reminiscent of that introduced in [34] in the context of integrability
on the Higgs branch of the dual CFT.
Massive excitations
To start with we will consider the massive sector by setting the excitation numbers N0,
N1 and N3 of the massless Bethe roots to zero. This case was already discussed in [7, 18].
The massive spinchain is homogenous with the sites transforming in the 1/2BPS
portant in the following discussion let us review its construction.
The representation
as a doublet under the outer su(2)• automorphism of psu(1, 12). Additionally there are
bosons have su(1, 1) weight 1/2 while the fermions have weight 1. Explicit expressions for
the action of the generators on the states can be found in, e.g., appendix B of [18].
The symmetry of AdS3 × S
× T4 includes two copies of psu(1, 12) and the sites
of the massive spin chain transform under the same
representation under both
copies. However, only the diagonal outer automorphism su(2)• is a symmetry.13 The fields
appearing at the site of this spin chain are listed in table 1.
The Bethe equations are constructed with respect to a ground state of the form14
2 2 ⊗
conditions (2.1).
as part of the so(4) symmetry in the T4 directions in the NLSM target space; it is sometimes referred to as
“custodial” su(2), see for example [35].
14The states we write can be thought of as cyclic “singletrace” operators. However, since we are often
interested in more general operators that do not satisfy the level matching constraint we do not write out
an explicit trace.
∇L
∇R
∂ ⊗ 1
1 ⊗ ∂
The rest of the table shows how the fields transform under the bosonic subalgebra as well as under
writing F a˙ b˙ = Da˙ b˙ + a˙ b˙F , where Da˙ b˙ is symmetric.
the two extra symmetries su(2)• and su(2)◦. The derivatives in the last two lines of the table are
not fields but are included to indicate the quantum numbers carried by su(1, 1) descendants of the
fields. Note that the field F a˙ b˙ can be decomposed into a triplet and a singlet under su(2)• by
one of the other fields in table 1. This can be interpreted as acting on the field at those
A generic state will contain several excitations and transform as a long representation
leftmoving excitation, so that the state takes the form15
If the excitation has nonzero momentum this is a highest weight state with charges16
L2 , L2 − 1; L2 , L2 0
up to anomalous corrections of order h2 given by eq. (3.12). Since the
eigenvalues. In fact, as long as the momentum of the excitation is nonvanishing, we expect
resolution of this puzzle is that several such 1/4BPS multiplets join to form long multiplets.
and S˙ R 1, as well as by the raising operators. However, this representation can join up with
15When we write a sum over permutations as in (4.2) this is meant to indicate linear combination of
important for this discussion, we only indicate the field content.
states where the excitation appears at different sites. In general we consider a state of definite spin chain
momentum p, such as PL
16We use the notation (DL, JL; DR, JR)J• to denote the eigenvalues of a state under the Cartan elements
DL, JL, DR, JR of psu(1, 12) ⊕ psu(1, 12) ⊕ su(2)•.
2 , 2 − 1;
L−1 , L−1
2 , 2 − 1;
L−1 , L−1
2 , 2 − 1;
2 − 1,
2 − 1
we obtain a single representation with charges
2 − 1;
− 2
three more 1/4BPS representations with highest weight states carrying charges
This deformed representation is long under both psu(1, 12)L and psu(1, 12)R.
Using the field content of table 1 we can construct the multiplets (4.3) explicitly. Their
highestweight states are
respectively. As all these states must be part of a long multiplet of psu(1, 12)R, they must
be related to the action of some rightmoving supercharges; namely, in this case, by Q˙ 1R and
QR2. We see here that these supercharges change the length of the spinchain groundstate,
as proposed in [7]. Clearly we could derive a similar picture starting from a rightmoving
excitation in equation (4.2), see appendix C.1 where we also further detail the
lengthchanging action of the supercharges on such multiplets. Furthermore, in appendix D we
first few horders in a more general subsector. These lengthchanging effects are then seen
to follow from the closure of the algebra much like in AdS5/CFT4 [56].
Massless excitations
Having understood the structure of the massive spin chain, let us now turn to the massless
excitations. To start with we consider a simple configuration consisting of one leftmoving
roots these massive roots satisfy the free momentum quantisation conditions
= 1,
= 1,
u + 2i !L−1
= 1,
= 1.
The new equations still describe two free massive excitations. However, the effective length
of the spin chain along which the excitations propagate is different in the left and right
sectors. It is natural to interpret this as the presence of a chiral site in the spin chain:
at the site where the massless excitation sits the representation
replaced by 1˜a
of the worldsheet excitations studied in [8, 9] we know that the massless highest weight
excitation is fermionic. Hence we have included a tilde on the singlet to indicate that it
has an odd grading.
When we add the massless root, we also get an additional equation from (3.9),
describing the quantisation of the momentum of the massless excitation. In the simple case
of N0 = 1 and M2, M¯2 ≥ 0 this equation reads
We see that the massless excitation behaves almost like a free excitation propagating on a
chain of length L, except it feels an extra twist which depends on the total momentum of
the massive leftmoving excitations, cf. equation (3.1).
Let us now consider an even simpler system consisting of only a single massless
excitation inserted above a ferromagnetic ground state of length L. In order to simplify the
description of such states we label the extra fields appearing in the chiral spin chain as in
z−
= YM2 uj + 2i
and carries psu(1, 12)L ⊕ psu(1, 12)R charges
L − 1 , L − 1 ; L , L
the free theory it transforms in a short representation which is annihilated by the creation
operators QL 2, Q˙ 1, S2R and S˙ R 1. Should its dimension to be protected even when the
L
coupling constant h is nonzero? This cannot be the case in general: firstly it would result
in a glut of protected operators [57]. Secondly, when the excitation has nonzero momentum
and h > 0, the dispersion relation in equation (3.12) shows that the state should receive
corrections comes once again through multiplet joining and length changing.
In order to construct the additional states that are needed to complete a long
representation we need one additional ingredient: a bosonic field T aa˙ that transforms as a singlet
emerges naturally in the nearBMN analysis of the string spectrum [8], represents a
spinchain site that transforms trivially in both the left and the right sector. When we allow for
states that include this massless bosonic excitation we can construct a long representation
as described in appendix C.2.
If the massless excitation in (4.9) has vanishing momentum, its anomalous dimension
vanishes even for nonzero coupling. In fact, it is easy to see that this is the case even
under this extra symmetry.
17As discussed above there is also an additional su(2)◦ symmetry acting only on the massless modes. We
take this into account by letting the psu(1, 12) singlet transform as a doublet (denoted by the index a)
∇LT aa˙
∇RT aa˙
∇L
∇R
∂ ⊗ 1
1 ⊗ ∂
as a doublet under su(2)◦.
in the allloop dispersion relation (1.1). These states are solutions of the Bethe equations
that correspond to 1/2BPS multiplets at arbitrary values of the string tension. We will see
in the next section that they precisely reproduce the spectrum of protected closedstring
states that we expect from supergravity and from the dual CFT2.
Fermionic zero modes and protected states
As we have just seen, it is interesting to consider massless excitations with zero momentum
on top of the BMN ground state. The resulting states solve the allloop Bethe equations,
the ferromagnetic BMN vacuum. Since there are four different massless fermionic
excita
The Bethe equations ensure that these 1/2BPS multiplets exist for arbitrary values of
the coupling h [17]. It is particularly transparent to construct their highestweight states in
the spinchain description that emerges at small but nonvanishing h. The ferromagnetic
ground state of length L is
and in the Bethe equations its excitation numbers are all set to zero. Turning on a single
(φ++)L−1χR+±i + symmetric permutations,
depending on which su(2)◦ spin we assign to the root. Since the excitation has vanishing
momentum, the above states are completely symmetric, cf. footnote 15. The corresponding
to one. Similarly we can get a leftmoving fermion
the two auxiliary roots, r1 = r3 = 1.
L−1
L−2
L−1
inserting the four fermionic zero modes, and the corresponding excitation numbers and charges
under su(2)L, su(2)R and su(2)◦.
Note that only the fermions χ+± and χ+± give a new state with the same energy as
R L
would no longer satisfy the 1/2BPS condition, because the fermion would give a positive
contribution to the energy.18 Hence the next states we can obtain contain two fermionic
excitations. For example, to consider a state containing two rightmoving massless fermions,
(φ++)L−2χR++χR+−i + symmetric permutations,
states is shown in table 4.
that appear. For J ≥ 2 we find
spin.19 Continuing in this fashion we obtain the sixteen states shown in table 3.
In the grading we use it is natural from the spinchain perspective to group the BPS
states according to their su(2)R charge. However, since the states sit in different irreducible
more transparent picture of the BPS states emerges if we instead reorganise the states
It is now straightforward to count the total number of states of a given charge (JL, JR)
18From the Bethe equation point of view, such a state can be interpreted as a multiexcitation state, as
discussed at the end of section 4.2.
19In terms of the Bethe equations this exclusion principle is encoded in the usual requirement that
regular solutions must have distinct Bethe roots. This can be seen explicitly by constructing the Bethe
the four fermionic zero modes. The length L of the corresponding spinchain ground state and the
charges under su(2)L, su(2)R and the su(2)◦ representation are also given.
• 6 bosonic states with charges J2 , J2 ,
• 4+4 fermionic states with charges J −21 , J2
• 1+1 bosonic states with charges J
2 − 1, J2
J2 , J2 − 1 .
Additionally, there are
• 2+2 fermionic states with charges 12 , 0 and 0, 12
• 5 bosonic states with charges 12 , 12 .
Summing over all the singleparticle and multiparticle states with the terms weighted as
in the Poincar´e polynomial in equation (2.3), we find that the partition function in the
1/2BPS sector of the spin chain perfectly reproduces the result for the symmetric orbifold
SymN (T4) presented in equation (2.9).
As anticipated in section 2, the computation of the (modified) elliptic genus requires
more care, since it diverges in the largeN limit. Following [38, 39], we assign to each state
In appendix B we show that if we consider states with (left) conformal dimension DL < d/4,
the result is given by
E˜2(Q, q, y) =
in perfect agreement with (2.15).
Protected states for AdS3 × S3
Another interesting string theory background is AdS3 × S3 × S3 × S1, which also preserves
sixteen supercharges. Strings in this background are expected to be dual to a CFT with
related to the ratio of the radii of the two threespheres
The allloop integrable S matrix for AdS3 × S
1 was found in [11]. The
worldsheet spectrum contains, among other excitations, two massless fermions. It is
straightforward to generalize the allloop derivation of the protected spectrum carried out
in section 5 to the present background. As before, the Bethe equations have a (trivial)
BMN vacuum solution above which there are excitations that are obtained by acting with
ZamolodchikovFadeev creation operators in the conventional way. In general, their
momenta are determined via nontrivial solutions to the Bethe equations and so will have
nonzero energies. However, just as in section 5, zeromomentum solutions of the allloop
Bethe equations corresponding to massless fermionic excitations above the BMN vacuum
zero momentum. This demonstrates that to all orders in the string tension, the massless
fermionic excitations are gapless just as in the case of the AdS3 × S3 × T4 background.
Let us discuss the protected AdS3 × S3 × S3 × S1 spectrum in more detail from a
spinchain perspective. The spin chain describing the massive sector of the pureRR AdS3 ×
S3 × S3 × S1 integrable system was constructed in [18]. The basic building blocks of the
sl(2) ⊕ su(2) ⊕ su(2) charges (D, J+, J−) =
they saturate the bound discussed in section 2
α2 , 12 , 0 and 1−2α , 0, 12 , and are 1/4BPS since
states then take the form
than the eight that are broken in AdS3 × S3 × T4 [1].
Note that the massive spin chain is alternating, with even and odd sites transforming in
different representations of the symmetry algebra. Each doublet representation preserves
half of the 16 supersymmetries, but the tensor product of the two representations only
preserves four supersymmetries so that the spinchain ground states are 1/4BPS. This is
to be expected: while string theory on AdS3 × S3 × S3 × S1 preserves 16 supersymmetries
× T4, fixing lightcone gauge here breaks twelve supersymmetries, rather
The two massless fermionic excitations correspond to sites transforming in
representations of the form 1 ⊗ 21 , 21 , 21
⊗ 1.20 Denoting the highest weight state of
These states saturate (6.2) and have equal angular momentum on the two threespheres,
J+ = J .
multiplet of the large N = (4, 4) algebra.
This prediction, valid for generic string tension, is at odds with the proposal for the
—an assumption that may indeed lead to overestimating the number of protected states.21
This additional degeneracy appeared unlikely to exist at generic points of the moduli space,
as emphasised in ref. [35]. In fact, states with J+ 6= J
− must satisfy different shortening
it difficult to believe that they would remain protected. Our derivation of the protected
spectrum explicitly confirms the reasoning of [35]. It is however still interesting to see
whether such an additional degeneracy is there at all at the supergravity point.
A simple yet effective way to investigate the protected supergravity spectrum is to
consider pointlike string solutions in AdS3 × S3 × S
× S1. These are in onetoone
corso must such classical solutions. Pointlike string solutions for AdS3 × S3 × S3 × S
briefly discussed in [1] (see also [42, 43]), and we review them here. The strings can be
taken to move only along the time direction t of AdS3 and the great circles ϕ
threespheres. The equations of motion are then solved by
± are related to the energy D and angular momenta J
The Virasoro constraint gives
D = q
that saturate the bound, together with the two fermionic zero modes that we obtained
position of the tensor product of the two doublet representations
21We would like to thank Jan de Boer for detailed discussions related to this point.
above, then generate the spectrum of protected operators,22
[J, J ; J, J ]s ⊕ J + , J + ; J + , J +
⊕ J, J ; J + , J +
⊕ J + , J + ; J, J
For J+ 6= J
long representations.
These solutions correspond to supergravity modes with classical
energy (6.7). In addition, since they sit in long representations, their energy will receive
corrections in the string tension.
Protected states in mixedflux AdS3 backgrounds
So far we have focused on string theory on AdS3 backgrounds supported by a RR
threeform. Both AdS3 × S3 × S3 × S1 and AdS3 × S3 × T4 remain classsically integrable when
additionally the NSNS threeform is turned on [3]. Moreover, up to the dressing factors,
the integrable S matrices can also be fixed entirely by symmetry in these cases [10, 11]. In
fact, for both families of backgrounds the structure of the worldsheet fluctuations and the
symmetries of the lightcone gaugefixed theory remain the same when the NSNS flux is
turned on. The only modification is that the representation of the lightcone gauge
symmetries gets deformed in a way that can be accounted for by a suitable deformation of the
Zhukovski parameters.23 This means that the form of the Bethe equations is independent
of the flux, which only enters in the relation between the Bethe roots and the worldsheet
energy and momentum. The dispersion relation takes the form [10, 58]
E(p) =
where k is the integervalued coupling of the WessZumino term of the bosonic string NLSM
action, and the coupling h is suitably rescaled [10, 11].
Without needing to resort to constructing the spin chain, we immediately see that
in the pureRR case, we find four fermionic zero modes over the BMN vacuum for the
AdS3 × S3 × T4 background and two for AdS3 × S3 × S
× S1. Their quantisation leads
to a tower of sixteen 1/2BPS states for each ground state in the former case, and of four
1/4BPS states in the latter, so that the structure of the chiral ring is the same for any flux.
It is worth remarking that, even if the structure of the Bethe equations is not modified
There are two reasons for this.
Firstly, the mixedflux dispersion relation (7.1) is no
longer a periodic function of the worldsheet momentum p, which might suggest that the
discretised lattice picture should be altered. Secondly, we expect the spinchain picture to
23The crossing equations are also modified in the presence of NSNS flux.
appear when the coupling constant is small. However, in the mixed flux case, the string
tension is bounded from below due to the presence of the WessZumino term in the NLSM
action, and hence there is no genuine weakcoupling limit in the dual CFT. It would be
very interesting to investigate this further.
Wrapping corrections
It is wellknown [15] that the Bethe equations for AdS/CFT integrability yield an
incomplete description of the spectrum, as they do not account for finitesize effects of the type
first described by Lu¨scher for relativistic systems [59, 60]. These corrections can be
interpreted as arising from particles wrapping the worldsheet cylinder one or more times,
and can be in principle be understood as part of a systematic expansion around large
volume [61, 62]. The finitevolume description that resums all these effects can be expressed
by employing more refined formalisms such as the mirror thermodynamic Bethe ansatz
(TBA) [63–67] or the quantum spectral curve [68–70]. It is interesting to note that
generically such wrapping corrections may lift degenerate multiplets [71]. Therefore, we should
ask ourselves whether the spectrum of protected states we found from the Bethe equations
is robust when wrapping effects are taken into account.24
As a mirrorTBA formalism for AdS/CFT pairs involving gapless excitation has not
yet been developed, discussing the exact finitevolume spectrum is beyond our reach.
Interestingly, we will nonetheless be able to see exactly what happens to protected states.
Let us start by recalling the form of a singlewrapping correction25
are considering is
25See also [72, 73].
dp˜ e−LE˜ X(−1)Fa Sab1 (p˜, p1) · · · Sabk (p˜, pk) .
Here we consider a physical state consisting of k particles with fixed momenta p1, . . . pk
and flavours b1, . . . bk. Moreover, we consider the scattering of a virtual particle in the the
socalled mirror kinematics [63] with mirror momentum p˜, mirror energy E˜, and flavour a.
We will have to sum over all possible particle types a and integrate over momenta p˜. Note
that we have assumed that the scattering is always given by a pure transmission process,
for reasons that will become clear soon. Let us now take the the case in which the index
Tρ(p˜; {p1, . . . pk}) = X(−1)Fa Sab1 (p˜, p1) · · · Sabk (p˜, pk).
well as bound states. Let us now specialise the case in which all particles of flavours b1, . . . bk
singlets of the lightcone symmetry algebra psu(11)c4.e. [9]. Hence the scattering that we
24We thank the anonymous referee for raising this point, and giving us the opportunity to report on the
i.e. it can only result in pure transmission. Furthermore, invariance under psu(11)c4.e. means
independently from a.26 All in all we find
Tρ(p˜; {p1, . . . pk}) = exp iϕb1 (p˜) + · · · iϕbk (p˜)
X(−1)Fa = 0,
tion process is completely analogous to the one that guarantees that the BMN vacuum is
protected from wrapping corrections. In fact, while it is subtle to systematically define
multiplewrapping formulae, cf. also [16, 62, 74], it is seems natural that the wrapping
corrections for the states that we consider here should follow the same fate as those for
the BMN vacuum — this can be explicitly checked for certain wrapping contributions for
which explicit formulae are available [16, 62].
As we have seen, supersymmetry forces the wrapping corrections to vanish order by
order for the protected states that we constructed out of fermionic zeromodes. In fact, our
argument applies to both massless fermionic and bosonic zeromodes. In the latter case,
it shows that the shift isometries along the flat directions are unbroken as expected. The
argument holds for both the AdS3 × S3 × T4 and the AdS3 × S
in presence of arbitrary threeform fluxes. Therefore, this condition may be taken a useful
guiding principle in the formulation of a set of mirror TBA equations for the finitevolume
spectrum: each protected state should yield a sector for the mirror TBA equations. The
exact Bethe roots will be fixed in a manner similar to what happens in ref. [75] for
“exceptional” operators; furthermore, and in contrast to the case of exceptional operators, the
vacuum energy should be quantised in each sector.
In this paper we investigated the spectrum of closedstring excitations in integrable
AdS3/CFT2 backgrounds. We developed a spin chain for the weakcoupling limit of the
allloop Bethe equations [17] of string theory on AdS3 ×S3 ×T4 supported by RR fluxes,
emphasizing the role that lengthchanging interactions play in it. The spinchain constructed
here is very similar to the one appearing in largeN perturbative calculations in the
Higgsbranch CFT [34]. It would be interesting to extend the perturbative analysis carried out
there to the complete weakcoupling theory and match with the spin chain proposed here.
Further, constructing a framework for incorporating wrapping interaction [15, 16] into AdS3
holography, such as the mirror TBA [63–67] and quantum spectral curve [68–70] remains
an important task.
We showed that protected closedstring states are determined by classifying
zeromomentum Bethe roots. This can be done straightforwardly at all values of the string
tension in the planar theory. We found that, in addition to the BMN vacuum, a number of
further protected states exist. For example, in the weaklycoupled spinchain description
in refs. [9, 11]; Smatrix fusion then yields all scattering phases.
of the pureRR AdS3 × S3 × T4 theory, these are obtained by inserting up to four fermionic
zero modes, yielding a 16fold degeneracy for each ferromagnetic vacuum.27 We have shown
that our allloop integrabilitybased analysis of protected states matches precisely the
results of [37] where the supergravity spectrum was found to agree with the large N limit of
the SymN (T4) orbifold, and with the modified elliptic genus [39]. This indicates that the
weaklycoupled spin chain lives in the same moduli space as the symmetricproduct orbifold
CFT. It would be important to pinpoint where exactly the weaklycoupled spin chain point
is, and its relation with the symmetricproduct orbifold. It is intriguing to note that the
generators of the chiral ring for the weaklycoupled spin chain collected in table 4 closely
resemble their counterparts in the symmetricproduct orbifold CFT. While this guarantees
that the spectrum of 1/2BPS operators is identical, it is hard to say whether we should
expect a onetoone matching of the generators. Lessons from supergravity [76] suggest
that we should expect a mixing between single and multiparticle (or, in our language,
single and multitrace) states. A way to unravel this question would be to match the
threepoint functions of protected operators [76–81], which are also constant on the moduli
space [82, 83].
It is also interesting to notice that this analysis, valid for generic values of the tension,
would be significantly different in the strict h → 0 limit. Here the anomalus dimension
BPS states. While it is hardly surprising that a “free” point of the moduli space might
is a qualitative difference with the symmetricproduct orbifold CFT, where no such extra
degeneracy exists. Furthermore, from a sigmamodel perspective, BPS states correspond to
cohomology classes of the target manifold; this suggests that the weaklycoupled spin chain
corresponds to some singular limit of the target space geometry. It would be interesting to
explore this in more detail.
We then turned to the AdS3 × S3 × S3 × S1 background. An analysis of the
zeromomentum Bethe roots shows that the protected closedstring spectrum of this theory is
constructed out of BMN vacua and two gapless fermionic excitations. The results are again
valid for any value of the string tension in the planar theory. The fermionic zero modes
give rise to a 4fold degeneracy for each ferromagnetic BMN vacuum which precisely yields
classical pointlike string solutions, only states with equal su(2) charge with respect to the
two threespheres are protected. This is a significantly smaller degeneracy than the one
proposed in supergravity in [40], which was derived under the assumption that all KK
modes sit in short multiplets. The potential for a smaller protected spectrum than the one
proposed in [40] was already argued for in [35] based on the different shortening conditions
in the superLie and superVirasoro algebras. It would be interesting to determine whether
this new understanding of the protected spectrum can shed any light on the longstanding
problem of identifying the dual of AdS3 ×S3 ×S3 ×S1. In this regard, it is worth noting that
27Our results represent a derivation of the proposal for incorporating massless modes into the integrable
the spectrum that we found is compatible with that of symmetricproduct orbifold CFTs
Finally, the AdS3 backgrounds considered here can be supported by a mixture of
RR and NSNS flux all of which are known to be integrable. Based on the integrability
results we have argued that the spectrum of protected operators remains the same as
one deforms away from the pureRR case. This is quite interesting, as exchanging these
background fluxes amounts to an Sduality transformation, which in general would affect
rather drastically the spectrum. The pure NSNS backgrounds can be understood using
WZW techniques [86, 87] and it would be interesting to see what the relation between
these and the integrable methods used here is.
We would like to thank Lorenz Eberhardt, Matthias Gaberdiel, Rajesh Gopakumar and
Wei Li for sharing with us their manuscript [49] before it appeared on the arXiv, and for
their comments on our results. We would like to thank Riccardo Borsato for interesting
discussions on many aspects of integrability in AdS3/CFT2, Jan de Boer for explanations
and discussions of the supergravity analysis of protected spectra in AdS3 backgrounds,
Andrea Prinsloo for discussions of giant gravitons in AdS3 backgrounds, and Kostya Zarembo
for numerous discussions. Finally, we would like to thank the Galileo Galilei Institute
in Florence for their kind hospitality during part of this work. A.S. would like to thank
Andrea Dei, Lorenz Eberhardt, Matthias Gaberdiel, Edi Gava, Kumar Narain, and Ida
Zadeh for interesting discussions on AdS3/CFT2. B.S. would like to thank Kelly Stelle for
enlightening discussions on KK supergravity modes. The work of M.B. was supported in
part by the European Research Council grant no. ERC2013CoG 616732 HoloQosmos and
in part by the FWO and the European Union’s Horizon 2020 research and innovation
programme under the Marie SklodowskaCurie grant agreement no. 665501. M.B. is an FWO
[PEGASUS]2 Marie SklodowskaCurie Fellow. O.O.S. was supported by ERC Advanced
grant No. 341222. A.S.’s research was partially supported by the NCCR SwissMAP, funded
by the Swiss National Science Foundation. B.S. acknowledges funding support from an
STFC Consolidated Grant “Theoretical Physics at City University” ST/J00037X/1. A.T.
thanks the EPSRC for funding under the First Grant project EP/K014412/1 and the STFC
under the Consolidated Grant project nr. ST/L000490/1.
The psu(1, 12) superalgebra
commutation relations as28
28Up to a change in the convention for the indices this follows the notation of appendix D of [85].
where R is an arbitrary generator. The su(1, 1) algebra takes the form
[K, P] = 2D,
[D, P] = +P,
[D, K] = −K.
The action of P and K on the supercharges is given by
and the nontrivial anticommutators by
{Qα, Sβ} = +Jβα + δαβ(D − C),
{Q˙ α, S˙ β} = −Jαβ + δβα(D + C).
The (conformal) supercharges carry dimension (D)
dim(Q) = + ,
dim(Q˙ ) = + ,
dim(S) = − 2 ,
dim(S˙ ) = − 2 .
If we restrict ourselves to representations with a vanishing central charge C the resulting
algebra is psu(1, 12).
su(2)• with generators J• and J•± which satisfy
The algebra psu(1, 12) has an automorphism which we will refer to as
[J•, J•±] = ±J•±,
1 Q˙ α,
Grading. In a superalgebra the choice of simple roots and corresponding Dynkin diagram
is not unique. Here we will mainly consider two different gradings of psu(1, 12). In the
first one, which we refer to as the su(2) grading, the simple roots are given by
while the sl(2) grading, the simple roots are given by
The corresponding Dynkin diagrams are shown in figure 1.
Representations. In this paper we are interested in unitary highest weight
representations of psu(1, 12). Such a representation can be parametrised by the charges h and
j that the highest weight state29 carries under the Cartan elements D and J11, and by
the eigenvalue b under the Cartan element of the su(2)• automorphism. We will denote a
generic such module by (h; j)b. The representation can be decomposed into a number of
irsupercharges act as creation operators on the highest weight state we in general obtain
sixteen such submodules.
29The assignment of weights to the states of the representation depends on the choice of grading, as
discussed above. For concreteness we will always write the weights corresponding to the su(2) grading.
(a) su(2) grading
(h+b12+, j12− 21 )
(h+ 12 , j1− 21 )
b − 2
b − 2
b − 1
(h+b32+, j12− 21 )
b − 2
(h+ 32 , j1− 21 )
b − 2
(h + 1, j − 1)
this figure the charges of these submodules are depicted together with their eigenvalues under J .
The solid red (blue) arrows indicate the action of the generators Q1 (Q˙ 2) and the dashed red (blue)
arrows indicate the action of generators Q2 (Q˙ 1). Note that not all such actions are depicted. For
highest weight states corresponding to the states in the double boxes.
Computation of the modified elliptic genus
In this appendix we compute the elliptic genus for the spin chain associated to AdS3 × S3 ×
T4, defined in equation (5.5). We deal with the zero mode insertion by first considering
the “partition function”
The quantum number d is the degree introduced in section 5, following [38, 39].
We first consider the partition function in the singletrace sector Zs.t.(Q, q, y, y¯). By
looking at tables 1 and 2 we see that the fields that can be used to build 1/4BPS states are
and we can also act with an arbitrary number of left derivatives. Consider a singletrace
spectrum necessarily contains the four operators
It is evident that these operators have the same DL, J L and d quantum numbers. Therefore,
they contribute to the partition function as
Zs.t.(Q, q, y, y¯) = . . . + Qdq2DLy2JLy¯2JR y¯ − 2 +
where we have taken into account the effect of spectral flow on the right sector.31 Given a
singletrace partition function of the form
from which the modified elliptic genus can be computed using30
E˜2(Q, q, y) = y¯∂y¯(y¯∂y¯Z(Q, q, y, y¯))y¯=1.
the full partition function including multitrace contributions is given by
Zs.t.(Q, q, y, y¯) =
c(m, n, j, k)Qmqnyj y¯k,
Z(Q, q, y, y¯) =
m,n,j,k (1 − Qmqnyj y¯k)c(m,n,j,k) .
In summary, we have shown that
Z(Q, q, y, y¯) =
m,n,j,k (1 − Qmqnyj y¯k+1)(1 − Qmqnyj y¯k−1) ,
(1 − Qmqnyj y¯k)2
where the prime on the product symbol indicates that we multiply over the (m, n, j, k)
E˜2(Q, q, y) =
m,n,j,k (1 − Qmqnyj )2
30We remind the reader that the elliptic genus is typically defined in the Ramond sector, where it receives
contributions only from the rightmoving Ramond ground states (see footnote 9). Ground states are then
mapped under spectral flow to the highestweight component of chiral primary multiplets in the NS sector,
that is states such that DR = JR.
31Spectral flow is implemented in a sector of a given degree d with an “effective” central charge equal to
6d, as in [39]. Of course this also changes the “overall” charge JR in (B.5), but we will see that, thanks to
the double logarithmic derivative in (B.2), the elliptic genus is independent of overall powers of y¯ in the
which together with DL < d/4 implies k < 1. This means that the only singletrace
operators that satisfy our criterion are
As a consequence, from (B.9) we obtain
Since we are interested in the modified elliptic genus only up to powers of q such that
DL < d/4 [38, 39], we see from the previous formula that we need to consider only
singleDL ≥ 2
d ≤ k + 1,
E˜2(Q, q, y) =
where the ellipses denote terms that correspond to states with large conformal dimension
compared to N . This reproduces the modified elliptic genus of [39].
Multiplet joining and lengthchanging effects
We have seen in section 4 that when states that saturate the BPS bound (2.1) receive an
anomalous dimension, we expect short multiplets to join into long ones. Below we detail
Massive excitations
Let us first consider the case of the accidentallyshort multiplet with highestweight state
given by (4.2). We now need to understand how to obtain states with the charges given
in (4.3). The leftmoving supercharges QL 2 and Q˙ 1L schematically act on it as on a standard
long multiplet, as illustrated in the following figure32
( L2 , L2 − 1; L2 , L2 )0
32See also figure 2 in appendix A for a more complete illustration of a long psu(1, 12) multiplet.
However, the rightmoving supercharges QR 2 and Q˙ 1R to leading order annihilate the
1/4BPS multiplet. For them to have a nontrivial action on the state (4.2) we need to let them
insert extra sites into the spinchain state,
( L2 , L2 − 1; L2 , L2 )0
Note that the rightmoving supercharges at each step increase the length of the state by
one, and precisely yields the highestweight states (4.5).
Similarly, a highest weight state with a single rightmoving excitation
(φ++)L−1∇Rφ++i + permutations
has charges ( L , L2 ; L2 + 1, L2 )0. This gives a short representation under the psu(1, 12)L
2
algebra, but a long one under psu(1, 12)R. For this representation to be deformed by an
anomalous dimension it needs to join up with three other states with charges
L+1 L+1 L
L+1 L+1 L
to form a long psu(1, 12)L representation. Again this is made possible by lengthchanging
actions of the supercharges, which is summarised in the following figure
( L+21 , L+21 ; L2 + 1, L2 )−1/2
( L2 , L2 ; L+21 , L−21 )+1/2
( L2 , L2 ; L+21 , L−21 )−1/2
Here the leftmoving supercharges increase the length by one at each step. Note however
that these lengthchanging effects are not visible in the leading order Bethe equations, but
only show up at higher orders in the coupling constant h.
T −±
Massless excitations
Let us consider the multiplet identified by (4.9). If the excitation has nonvanishing
momentum, we expect the short multiplets to join and the supercharges to act according to
the following diagrams:
( L−21 , L−21 ; L2 , L2 )0
Note that neither the left nor the right supercharges now preserve the length of the spin
chain state. The structure of the dynamic spin chain involving massless modes is further
discussed in appendix D.
we need four states containing a single massless excitation. Let us see how to obtain these
from the Bethe equations. We already know that we can obtain
sponding to the massive auxiliary Bethe roots v1,k, v3,k, v¯1,k and v¯3,k. The only remaining
possibility for constructing massless single particle states is hence to turn on the massless
auxiliary roots r1,k and r3,k. By reading off the corresponding charges from equation (3.13)
we can identify the configurations displayed in table 5. As discussed, the su(2)◦ index on
these excitations is not directly encoded in the Bethe roots but needs to be kept track of
externally. Note that in the above spinchain description the left and right parts of the
spinchain sites enter in a nonsymmetric fashion. This is because of our choice of grading.
Dynamic spin chains
As discussed above, several short multiplets can combine into a long multiplet through
lengthchanging effects. In order to see this more explicitly we can restrict the spinchain
to include only massive excitations, so that the spin chain becomes homogenous with all
consider a closed subsector consisting of states that saturate the bound
By performing a set of fermionic dualities we can change the grading in such a way that
the role of the left and right copies of psu(1, 12) are exchanged [9].
So far we have described the fundamental excitations of the spin chain. However, the
a state containing this field is a multiexcitation state. It can be obtained from a state
containing the field χ++ by further turning on a massive momentumcarrying root u. This
L
leads to the mixing of the states
Similarly there is a mixing between the states
which means that the states can be built from the three fields
D ≥ 2JL + J•,
together with the bosonic generators
33Note that at weak coupling these states do not mix with
L = 2JL,
(φ++)L−1∇Rφ++T +±i + permutations,
which has M¯2 = N0 = N1 = 1.
S−
− 2
− 2
− 2
The generator L measures the length of the spin chain, which is a preserved quantity in
this sector.34 We further note that in this sector
− 2
− 2
− 2
− 2
− 2
−1
The algebra then takes the form
{Q˙ , S˙ } =
(D − J• − L),
The charges of the fields and generators under the various u(1):s are shown in table 6.
decrease it by one, respectively. The last column in the table simply shows that that the
algebra leaves the sector closed.
To leading order the generators acting on the fields take the simple form
J• =
34As we will see below, the spinchain length L commutes with the spinchain Hamiltonian. However, the
supercharges Q˙ and S˙ change the length and hence the full symmetry algebra acts on a dynamic spin chain.
{Q˙ , S˙ } =
{Q2, S2} =
H − M .
To the leading order we then have
H = −M =
(D + J•) − J11 = φ2ihφ2 + ψihψ ,
We now want to find higher order corrections to this representation, by writing the
generators as a series expansion in the coupling constant h
J = J(0) + h J(1) + h2 J(2) + · · · ,
imposing the commutation relations, the reality conditions and that the algebra preserves
supercharges as well as the dilatation operator D do. The latter we will express in terms of
the spinchain Hamiltonian H. As we will see, the expansion of the charges of su(12) only
contain terms that are of even order in the coupling constant, while the su(11) supercharges
only come in at odd orders.
The first correction we find is a nontrivial contribution to the su(11) super charges
to introduce the Hamiltonian H and mass operator M which are defined such that35
Q˙ (1) = + 2 α1e−i(β0−β1) αβ φαφβihψ ,
S˙ (1) = − 2 α1e+i(β0−β1) αβ ψihφαφβ .
can be absorbed in a rescaling of the coupling constant h. At the second order we find
corrections to the Hamiltonian
and to the supercharges
H(2) =
21 α12 φαφβihφαφβ − φβφαihφαφβ + φαψihφαψ − ψφαihφαψ
+ ψφαihψφα − φαψihψφα + 2 ψψihψψ ,
18 α12eiβ0 φβψihφβφα − φβψihφαφβ + ψφβihφαφβ − ψφβihφβφα
+ iγ1eiβ0 φβψihφβφα + ψφβihφαφβ − ψψihφαψ + ψψihψφα
+ iγ2eiβ0 φβψihφαφβ + ψφβihφβφα + ψψihααψ − ψψihψφα
2 γ3eiβ0 φβψihφβφα + ψφβihφαφβ + ψψihφαψ − ψψihψφα ,
T3 =
It is straightforward to continue this expansion to higher orders in the coupling.
However, let us now instead focus on how the eigenstates described by the above spinchain
the coupling, all states in the subsector are annihilated by the leftmoving supercharges Q˙
in a short representation of psu(1, 12)L. When the coupling is turned on a generic state
no longer saturates the 1/4BPS bound, since the state receives an anomalous dimension.
Hence, the state has to transform in a long psu(1, 12)L representation, and, indeed, is no
longer annihilated by the su(11)L generators Q˙ and S˙ . Instead, these generators act on
the state by adding or removing a spinchain site.
this sector the Hamiltonian, as well as the full su(12)R algebra, preserves the length of the
spin chain. In a more generic sector, the leftmoving and the rightmoving supercharges
and the Hamiltonian will all contain terms that relate states of different lengths.
A natural subsector for studying the massless modes is the 1/2BPS sector which
and by the spinchain Hamiltonian H and the mass operator M. These charges generate the
algebra psu(11)4, which is centrally extended by H and M, which is the same symmetry
algebra that was used in [8, 17] to determine the worldsheet S matrix.
We will now show that the above generators do not have a welldefined expansion in
the interaction length. For simplicity, let us consider only the three fields
T ≡ T ++.
Following the structure of the multiplets discussed in section 4.2, we can write an ansatz
for the first few orders of the expansions of the supercharges when acting on the above
fields. For Q˙ 1R and S˙ 1R we have
where we have introduced an auxiliary coupling constant g to keep track of the orders in
the expansion. Similarly, the leading expansion of Q2L and S2L is given by
Q2L = g β1 φT ihχ + β2 T φihχ) + · · · ,
S2L = g β¯1 χihφT  + β¯2 χihT φ + · · · ,
These expressions are very similar to what we had in the massive sector discussed above.
However, the massless excitations are all annihilated by the mass generator M. Hence, the
supercharges now satisfy the algebra
{Q2L, S2L} =
{Q˙ 1R, S˙ 1R} =
with exactly the same generator on the right hand side of both commutation relations.
The expansion of the rightmoving supercharges give a leading contribution to H of order
g0, which takes one field to one field. However, such a term can not appear from the
antiNow, the left relation above starts at order g2, while the right one starts at order g4, which
means that we also need to set the order g coefficients of the leftmoving supercharges to
zero. Continuing in this fashion, we see order for order that there is no way to perturbatively
deform the representations of the above algebra. Instead, the spinchain Hamiltonian for
the massless excitations is longrange even at the first nontrivial order.
A hint of the form of the Hamiltonian can be obtained from the massless dispersion
Let us consider a single massless excitation on a spinchain of infinite length. The above
dispersion relation can be written as a Fourier sum as
E(p) = 2h sin
E(p) =
The exponential in the sum can be interpreted as a hopping term in the Hamiltonian, where
the excitation jumps n sites. From this expression we see that the Hamiltonian involves
interactions involving fields an arbitrary distance apart.
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