JHEP02(2019)098 he BPS Springer ormal Field January 31, 2019 November 5, 2018 February 14, 2019 : : : which can escape to the Received Accepted onstrate directly from the Published One expects that both these e NS-NS flux background. We string theory backgrounds. At pure 3 Published for SISSA by https://doi.org/10.1007/JHEP02(2019)098 [email protected] , . 3 at a finite cost of energy. Related to this are certain gaps in t 1810.08621 3 The Authors. Conformal Field Models in String Theory, Long strings, Conf c

We revisit two related phenomena in AdS , [email protected] Institut Theoretische f¨ur Physik, ETH Zurich, CH-8093 Switzerland Z¨urich, E-mail: ArXiv ePrint: employ the hybrid formalism forworldsheet mixed that flux this backgrounds is to indeed dem the case. Keywords: Theory, Sigma Models spectrum one computes from theeffects RNS disappear worldsheet description. when perturbing slightly away from the pur NS-NS flux, the spectrumboundary contains of a AdS continuum of long strings Open Access Article funded by SCOAP Abstract: Lorenz Eberhardt and Kevin Ferreira formalism Long strings and chiral primaries in the hybrid JHEP02(2019)098 4 5 7 8 1 3 7 or 11 12 13 13 18 20 21 10 17 4 T = ures. In 4 M where 4 rent backgrounds are . 2.1 r’. This means that the × M ch can wrap various cycles 3 S ar-horizon limit of the D1-D5 he moduli space of the theory. × 3 ion in section ]. As such, the system is characterised ] that there is a codimension four locus 3 3 – 1 [ 2) | 4 1 backgrounds have typically a large number of , M 3 – 1 – (1 1 2) | psu 1 , -th level can be supported by either pure NS-NS flux, pure R-R flux, n 3 backgrounds exhibits a variety of rich and interesting feat 3 . The fluxes transform under the U-duality group, so that diffe 4 It was first discussed by Seiberg and Witten [ Provided that the charge vector is primitive, see the discuss 4.4 Missing chiral primaries 4.1 The spectrum4.2 at the A first unitarity level 4.3 bound Continuous representations 2.1 The D-brane2.2 setup The gauge theory description 3.1 The hybrid3.2 formalism The sigma-model on PSU(1 1 M classified by the norm of a charge vector. of in the moduli space in which the background becomes ‘singula by the values of the fluxes of D1-D5, F1-NS5 and D3-branes, whi or a mixture thereof. Because of this, AdS 1 Introduction String theory in AdS B The spectrum at the C Proof of the R-sector no-ghost theorem 5 Discussion A The (affine) Lie superalgebra 4 The spectrum of the sigma-model 3 The worldsheet description Contents 1 Introduction 2 The sigma-model description moduli and surprising phenomena occurIn at various this places paper, in we t will mostly discuss the backgrounds AdS K3. These backgrounds are realisedbrane in system string compactified theory on as the the manifold ne type IIB string theory, AdS JHEP02(2019)098 chiral ing theory with mixed 4 cription of the entum above a 2) and a sigma- | 1 rid formalism of ach the boundary × M , 3 S ingular locus, due to de of it. Since chiral ular locus. × her hand, the Coulomb , this locus is a wall of one indeed observes the ular locus. In particular, 3 f the D1- and D5-branes moduli space. The most ring theory side. The case fied with the sigma-model ely long tube. A Liouville following: some insight into the string he singular locus these two antum mechanically. In the dvolume CFT description of oup PSU(1 s itself in a variety of ways, and disappear from the Higgs normal to the singular locus. r, the string spectrum changes uum part of the spectrum. In rimaries in the spectrum. Nev- are otherwise disconnected. As , one might hope to understand free energy of the string. we described above. It is natural ory on pure NS-NS backgrounds ] in this context shows that the string 12 – 2 – ], they are not (yet) able to reproduce the quali- 11 ] to describe string theory on AdS 13 backgrounds [ 3 ], there is currently no exactly solvable description of str 2 -BPS operators, we will refer to this phenomenon as ‘missing 10 2 1 – 4 , coupled together by ghosts. This is an exact worldsheet des . These so-called long strings can have arbitrary radial mom 4 3 M ] these long strings were associated with divergences of the 4 primaries’. the fact that aof finite AdS amount of energy suffices for a string to re certain threshold. primaries are there are less BPS operators on the singular locus than outsi These expectations are hard to confirm explicitly from the st In this paper, we confirm the expectations above using the hyb We should mention that while integrability techniques give In [ 1. The string spectrum develops a continuum of states on the s 2. The spectrum of BPS operators is discontinuous on the sing 2 quantum theory, the twofield branches are associated connected with via thisthe an tube same infinit vein, is chiral responsible primaries canbranch, for get leading the swallowed to in contin the the missing tube chiral primaries. flows to a two-dimensional CFTon in the the Higgs IR branch limit,branch of describes which the the is worldvolume emission identi gauge ofbranches theory. meet branes classically from On in the the the system.we ot small shall On instanton t review limit, in but this paper, this statement is corrected qu These predictions can bethe understood D1-D5 by system considering the as worl follows. The theory on the intersection o flux. This formalism consists of a sigma-model on the supergr theory spectrum on AdS tative behaviour described above. Atheory recent spectrum study depends [ essentially only on the directions with mixed flux, which would allow us to move away from the sing theory, but it is exceedinglythis hard theory to just solve enough exactly. to However observe the qualitative features existence of a continuum of longertheless, strings while and there missing is chiral p via an WZW-models exact [ description of string the of pure NS-NS flux lies on the singular locus of the theory, and marginal stability ofwith the some system. drastic consequences The fordiscontinuously the as singularity theory. we manifest In moveextreme particula through changes in the the singular spectrum locus on the in singular the locus are the brane system can separate at no cost of energy. In other words model on Berkovits, Vafa and Witten [ JHEP02(2019)098 )- 2) | R 1 , , 2), on | ] these )-spins 1 R , 16 2). With , | ], and the (1 1 , 18 psu , ghost theorem 17 bra ]. llowed SL(2 ture on the SL(2 20 round with mixed flux , er look at the spectrum should not be derivable ase. We use the algebraic theorem [ 19 is turned on, i.e. as soon as cal limit. The Maldacena- ced worldsheet symmetries. and, we will explicitly show rantees the conformal sym- i bound arises in fact from , e Maldacena-Ooguri bound We discuss our findings in aic treatment of supergroup s representations describing orbids these representations ributable to the PSU(1 3 al, at the WZW-point (pure ow us to retrieve the missing hich results in the appearance or the long strings and missing re only the NS-sector no-ghost sing chiral primaries. After this, 2). This supergroup has the special | 1 , we review the arguments that lead to the 2 – 3 – ], which arises from demanding square integrability 7 . This involves in particular the computation of conformal 4 and explaining its application to the case of PSU(1 3 ] to analyse the spectrum of the supergroup sigma-model. In [ 16 – 14 spectrum of the supergroup sigma-model and the R-sector no- n . Three appendices with background on the affine Lie superalge 5 This paper is organised as follows. In section As a byproduct of our analysis, we fill a small gap in the litera in the literature. One is the unitarity bound of the no-ghost 2 The sigma-model description This section is mostly a review of the material appearing in [ complement the discussion. WZW model. Remarkably, there are two different bounds on the a the level weights of single-sidedsection excitations in the supergroup CFT. prediction of long strings, theirwe disappearance set and the the stage mis forsigma-models our in computations section by reviewing the algebr from unitarity alone.considering We discover the that R-sector the no-ghosttheorem Maldacena-Oogur theorem, was in discussed. thearises literatu Hence purely at from least unitarity. in the superstring, th other is the Maldacena-Ooguri boundof [ the respective harmonicOoguri functions bound in is the stronger, quantum and mechani it is somewhat puzzling that it we leave the singular locusthat in the the moduli conformal space. weightlong On of the strings other excited acquire h states afrom in appearing non-vanishing the in imaginary continuou a part. unitary string This theory. f methods were used tofrom derive a the large-charge full limit BMNNS-NS of spectrum flux) the the of spectrum in worldsheet of aHowever, theory. the this theory backg constraint In is is constrained gener absent by in enhan of the new case of representations mixed of flux, thechiral w primaries worldsheet as CFT. soon These as will an all infinitesimal amount of R-R flux to expect that the emergence of these features is largely att part of the hybridof formalism. the sigma-model With CFT this on in the mind, supergroup we PSU(1 take a clos property of having vanishing dualmetry of Coxeter the number, worldsheet which theory gua methods even away of from [ the pure NS-NS c these preparations at hand,chiral we analyse primaries their in implications section f JHEP02(2019)098 at a on 4 (2.1) (2.2) (2.3) 3 consider or K3. The 4 ]. T 22 , = ation. Locally, they 21 4 tem can break apart at M is way, for pure NS-NS ]. e boundary of AdS -branes can wrap any of x background lies on this ed. The remaining scalars ), under which the charge 21 e that the U-duality group n for pure NS-NS flux. Such ity should be reflected as a o cost of energy at any point vector [ ]. vector into the standard form 5+ fixed norm. Therefore we can ions. In the near-horizon limit , 7 hese considerations predict the 5 e following we will assume that theory, and are associated with alues in the even self-dual lattice denotes directions in which the iple of another charge vector. If , where = 16 for K3. The charge vector 4 n × M , , ) ) n n and ) ) 4 n n T 5 + 5 + O(5 + O(5 + , , – 4 – -current algebra [ × × k ) O(5 O(4 R = 0 for , n O(5) O(4) (2 0 1 2 3 4 5 6 7 8 9 ×× × × × × ∼ × ∼ × ∼ ∼ sl , where 4 = 1, with all other charges vanishing [ M ′ 5 ]. For a primitive charge vector, this happens on a codimensi Q D5-branes D1-branes 23 , 5 1 is located in the directions 6789, 3 denotes directions in which the brane is smeared. We can also Q Q 4 and 5 ∼ M Q 1 Q = . The U-duality group is the orthogonal group O(Γ + 6 two-cycles of N n Seiberg and Witten studied under what circumstances the sys The moduli space is provided by the scalars of the compactific n = 5+ , ′ 1 5 finite cost of energy. Theseexistence are of the a so-called long continuumstates strings. of indeed T states exist above incontinuous a the representations worldsheet certain of description threshold the of string in the moduli space, whichacts renders transitively the on dual the CFTalways set singular. apply of a Not primitive U-duality charge transformationQ to vectors bring of the a charge vector transforms in the fundamentalthis representation. charge In vector th isthis primitive, is i.e. not not so, the a brane non-trivial system mult can break into subsystems at n flux fundamental strings can leave the system and can reach th no cost of energysubspace [ of the modulisingularity space. in the On duallocus this CFT. sublocus, and In the is particular, instabil hence the a pure singular NS-NS region flu in the moduli space. In th parametrise the homogeneous space The manifold brane extends, D-branes are wrapped as follows: the the inclusion of F1-strings and NS5-branes, and moreover D3 and U-duality is reduced to the little group fixing the charge some of the moduli freezeparametrise out locally and the the moduli charge space vector becomes fix on which U-duality acts and which leads to global identificat parametrising different configurations of the systemΓ takes v We consider the D1-D5 system on compactified on 2.1 The D-brane setup JHEP02(2019)098 . 3 ], n d 5 24 4 Q (2.4) = 6 ), where c V n ]. − 25 [ H n ) gauge theory cou- 5 he Coulomb branch, ranch of the theory. = 6( Q c ingularity of the gauge is more subtle: since it gs branch breaks down (2)’s. orrected and develops a iption of the emission of a ulomb branches are gener- su a different set of variables. a U( hen . 1-strings nor NS5-branes are , ing the comparison of central ion of the D1-D5 system. For nt . In the low-energy limit, the 4) superconformal field theory ry which lives on the intersec- 4 . Nevertheless, the description nce the scalars transform non- , T al charge does not change since, superconformal field theories — ed by two decoupled SCFTs. ry, and the energy gaps can be seen to ts also a tube-like behaviour on hypermultiplets remain massless, e Brown-Henneaux central charge [ = = K3 = (4 V 4 4 two n N M M − , while all other fields are massive. The H 5 n Q , the number of vector multiplets. Evaluating ]. Hence the Coulomb branch moves infinitely V 3 – 5 – n + 1) , 5 massless gauge vector multiplets, that is 5 Q 5 Q 1 1 Q Q Q 6 6( ( ]. Indeed, these two SCFTs have different sets of massless = (2) R-symmetries and obtain non-trivial vacuum expectatio 19 c ], and which flows to an su 2 This implies that an instanton can travel through the tube an 5 ]. . The correction in the K3 case is a supergravity one-loop effect 5 3 Q 1 Q = 6 c ]. 20 , 3 These two branches meet classically at the small instanton s Let us have a closer look at the different central charges. On t is the number of hypermultiplets and There can of courseThe be same also mixed result branches. can be obtained semi-classically by using th This tube can be described by a Liouville field in the gauge theo 3 4 5 H one corresponding to the Coulomb branch and one to the Higgs b in the IR. In fact, the IR fixed-point is described by tion of the D1-D5 branes [ pling to the two-dimensional defectsdynamics given becomes by essentially the a D1-branes two-dimensional gauge theo n this number gives D1-brane, i.e. of the long strings. In this process the centr which yields of the Higgs branchnear SCFT the as singularities a ofIn sigma-model the those moduli on variables, space, the the and classicalthe small one Hig Higgs has instanton branch to singularity [ use come exhibi out on the Coulomb branch. This is the gauge theory descr theory. In thetube quantum near theory, the the small Coulombfar instanton branch singularity away metric from [ is theis c Higgs it hyperk¨ahler, branch. is not For the renormalised at Higgs the branch quantum the level story fields, and hence differenttrivially central under charges. the various Furthermore, si match [ We hence conclude that theically central charges different, on and the therefore Higgs the and IR Co fixed-point is describ There are a number ofcharges ways of and justifying R-symmetries this, [ the simplest be In this part we reviewsimplicity, the we gauge work theory in worldvolume descript thepresent. case The in worldvolume which theory neither of D3-branes, the F D5-branes is given by 2.2 The gauge theory description while all other fields become massive. The central charge is t On the other hand, on the Higgs branch only central charge is then given by the values, the R-symmetry is generically broken down to differe the gauge group is generically broken to U(1) JHEP02(2019)098 (2.5) . As cohomology . 1 , the whole diamond 1 4 pace of the theory. hen slightly perturbed M -singular point in the s branch. omology cycles associated rs from the moduli space, ee figure Higgs branch chiral primary , e spectrum. From a string all instanton singularity will 5 ion. Note that this happens d by the long strings should gular locus. This means that cohomology classes which are s the tube forms, these cycles Q s branch and the Coulomb branch on, this means that these chiral ally in figure diamonds of ral primary states obtained from eorem makes the Higgs branch flat. he Higgs branch. a continuum in the spectrum. When + 6 (right-hand picture), the moduli space ling constant blowing up in the middle. 5 ral primaries can escape to the Coulomb Q 1) − 1 Q – 6 – = 6( 5 Q It is hard to say which cycles are these from a gauge the singular locus 1 6 Q perturbation away from = 6 tot c , 4 chiral primary T = 4 M Higgs branch Coulomb branch Coulomb branch . The structure of the moduli space on the singular locus and w emission of a D1-brane Let us slowly move away from the singular locus in the moduli s There is one related phenomenon occurring. Starting at a non Furthermore, since the chiral primaries always come in Hodge 6 Figure 1 for example for Associated with the tubeslightly are perturbing long the strings, system which awayapproximation from give becomes the rise good singular to locus andThe tube the disappears non-renormalization and th all chiral primaries are confined to t away from it. On thebranch singular and locus are (left-hand emitted picture), asare chi D1-branes connected from by an the infinitely system. long tube, The with Higg the string coup where we have used the central charge for the Coulomb and Higg primaries are missing onobtained the singular by locus. multiplication In intheory this point the way, of all chiral view, this ringa means given vanish that chiral all from multi-particle primary th chi are missing. classes correspond to chiral primaries in the CFT descripti will be missing. From the gauge theorysince picture we the learn sigma-model that descriptionimmediately the is at tube always the disappea a slightestwhen perturbation good perturbing away descript from the the theorycompletely sin slightly, disappear. the The continuum situation provide is depicted schematic moduli space, as we slowlyform approach at the some singular places locus onwith the the Higgs sm the branch. instanton shrinks The support towill of move zero down the size the coh in tube and this disappear process from and, the a Higgs branch, s JHEP02(2019)098 ] ) 4 3 × ve 3 3.2 S × M 3 S tion exists. × 3 anton moduli ]. Note also that 26 , 3 orth only focus on , see the action ( f cated. However, we are = 1 theory, and in which the was not yet realised when [ 1). However, more chiral ll argue in the worldsheet its main constituent: the tries manifest, but the ex- vel of the sigma-model on 5 3 − and Q . Since the hybrid formalism WZW models. Furthermore, 5 k d to the background fluxes as k s outlined in the previous sec- Q 1) (2.6) tring theory in AdS ( to be quantised. In the sigma- ≡ 1 2 tion of the instanton moduli space − k n of AdS NS 5 5 = Q Q or K3. We start by briefly describing ¯ h tement and the one in [ 4 2). = T | + 1) 1 , i.e. h , = k w 2). ( 4 | corresponds to the spectral flow parameter on , 4) CFT. This can be a sigma-model on either 1 1 , , } M – 7 – coupled to the remaining fields of the theory. 1 − σ 5 = (4 Q N and ,...,Q 1 ρ + 1) , 0 = 6 w c 2, since only for these values a complete worldsheet descrip (( ∈ { ≥ w 5 ] gives a covariant formalism describing string theory on Ad Q ] it was argued that the first missing chiral primary should ha 13 1), i.e. conformal weights 26 , − 3 5 It would be interesting to confirm this directly from the inst ] for a recent proposal on how to make sense of the 7 ,Q 28 1 , 27 − 5 and has the following ingredients: 2), and we will see that this is indeed the case. We will hencef 2) sigma-model. or K3. | | Q 4 1 1 4 or K3. In [ , , T 4 The sigma-model is characterised by two parameters The importance of spectral flow in the worldsheet descriptio T 3. Two additional ghost fields 1. A sigma-model on the supergroup PSU(1 2. A topologically twisted × M 7 3 The hybrid formalism makes halfistence of of the ghost spacetime couplings supersymmeonly interested makes in the the theorytion. emergence much of We more expect the compli PSU(1 these qualitative feature features to be already present at the le the sigma-model on the supergroup PSU(1 S See however [ missing chiral primaries are present. the hybrid formalism, andPSU(1 then concentrate our attention on 3.1 The hybridThe formalism hybrid formalism [ was published, which explains theour differences statements between are our true sta for below. In the hybridfollows. formalism, The these amount parameters of are NS-NS relate flux is given by 3 The worldsheet description In this section we introduce the worldsheet description of s space side. degree ( the worldsheet. primaries are expected todescription be that missing all from cohomology the classes spectrum. of We degrees wi with mixed NS-NS and R-R fluxes, where is a perturbative string theory description, we expect are in fact missing, where on theory perspective, since there is no good explicit descrip model, this follows from the usual topological argument for JHEP02(2019)098 ¯ a f n ¯ P , (3.2) (3.3) (3.1) (3.4) ¯ a n ¯ Q , a ing theory n P , — which for , , 9 ] , n , n n a n + 0, we would have n + + 29 b ¯ ¯ b c ¯ a m Q + , c m b m b ¯ < a m A A the embedding co- 16 ¯ c c – b , ¯ 5 NS g ¯ ab a ] kmA Q g 14 f f [ kmA − 1 was further studied and , ¯ − ∂g . the following, the indices WZ 2 1 ] = ] = i ] = i ] =
S b b b ¯ ¯ ¯ − a n n e pure NS-NS background is b n b n ] for sigma-models on super- 0. For as ¯ nticommutators for two fermionic ¯ ∂gg g k P RR 5 2 2 , ,P 15 > ,A ,A + ic, nor anti-holomorphic, except Q a m 2 2 b m , ¯ ¯ a m a m 2
kf kf G symmetry, acting by left- and ¯ f f 2 1 Q ¯ Q [ Q Q 14 [ 2 2 g kf − − [ [ 1 in the perturbative treatment, × 1 1 + + , , ¯ ≪ 2 − − ∂gg n ,
n , 1 ) the conserved currents associated with + g = = + n z n − NS 5 ( ¯ ¯ c m c + ¯ ¯ m z z + j ¯ ¯ j ¯ Q This corresponds to the WZW-point for the c m c Q P m ∂gg ¯ ¯ c c ), b b ¯ ¯ Q P 8 , j ¯ ¯ a a z c c – 8 – q ( 1 f f ab ab 2) j Tr − | ∂g , = f f 1 = 1. z 1 + i + i 2 2 − , -indices. We will refer to this algebra as the mode 0, and therefore to 2 2) — with action 1 0 0 + i + i d ∂gg g | f (in units of string length). Solving the supergravity g 1 n, n, 2 2 0 0 > kf 3 , + + Z = n, n, 2 2 kf kf m m 2 + + f f NS 5 δ δ 2 2 m m b ¯ − Q 1 δ δ ¯ πf 2 ab a AdS ′ 1 + 1 4 R ab ab α ] denotes the standard WZW-term, and 2 − − − g [ π kmκ kmκ . We denote by = = = 1. From these currents we can define modes 4 g kmκ kmκ − − ] = z z WZ 2 = 0, that is j j ¯ g S [ denote adjoint kf ] = ] = ] = ] = S RR 5 b b ¯ b n n n b n ¯ ) below) whose (anti)commutation relations are [ ¯ ¯ P Q Q ,P , ,Q , ], where it was used to compute the plane-wave spectrum of str 3.6 b, . . . ¯ ¯ a m a m ¯ a m m a ] the two-parameter family of sigma-models on a supergroup G 16 ¯ a, ¯ Q Q Q Q [ [ [ [ 16 , G symmetry, whose components are is the string coupling constant. Since backgrounds with mixed flux. ) and ( and ¯ 15 1. 3 g × − 3.5 is equal to the radius of AdS In [ = We will always write commutators. These are understood to be a In the following we restrict to 2 9 8 2 − characterised by effectively becomes a continuous parameter of the theory. Th was analysed. Here, generators. this G kf Importantly, the currents are in generalat neither the holomorph WZW-point where groups G with vanishingextended dual in Coxeter [ number.in This AdS formalism our purposes is taken to be PSU(1 sigma-model. 3.2 The sigma-model on PSU(1 equations of motion, this is related to the background fluxes In this section, we review the formalism developed in [ ordinate on the supergroup.right-multiplication on The model possesses G f (see ( with all other commutationa, b, . relations . . vanishing. Here and in JHEP02(2019)098 . . a f n ¯ Y (3.7) (3.5) (3.9) (3.8) (3.6) , a n and ¯ X k , . m , + m n ) ] that this energy- , + ,
z n c ) , 15
 )( z ¯ present at the WZW- c P b ( ¯ z 2 b  ¯ j a ¯ ¯ z k P ¯ a 2 z b Q ¯ g ¯ a are defined as kf j ¯ n s motivates our (slightly ¯ ¯ z j ( Q ¯ heory, and hence can mix Hilbert space spanned by a Y 1 kf × a n b ¯ n fficult to impose a highest ations of the mode algebra bc ¯ a a − ¯ k ¯ c Y Q 1 + − b n ¯ κ , , f in the adjoint representation. g d modes) does not commute ¯ a , z 2 1 2 n n a f a g n − ) ) to express the Virasoro modes f omorphic nor anti-holomorphic, z 2 + + fact prevents us from computing ¯ i 2 P R 2 − d f ¯ a a m m i 2 , , 3.8 − ¯ 2 a kf P ¯ a f n n kf R
− nP n 2 m ¯ P ¯ a = a n I X − kf | 4 4 − + , m z a | n f f X g t a n 1 2 2 + (1 2 2 a ¯ a ≡ n t Q 1 + k k  ¯ nP 1 kf kf P a 2 n 2  4 4 is still present in the form of the subalgebra − − − n ) = 2 g 2 k 1 1 z ), which give rise to the operators kf kf g 2 2 z )( as usual. It was shown in [ kf – 9 – ( kf + − b z − 2 − × a j n j ¯ 1 k ,Y m m a z L = = 2 g j ) + + 1 + ( = STr − z ¯ a n a n ¯ a a n n ). ( ¯ ¯ ab ¯ a − Q P a z m . The modes a κ j n nQ + 3.6 m ¯ a m n A 2 2 2 a n , ,P + ¯ ) z Q ¯ a a ¯ 2 a n n n z kf kf ¯ ¯ nQ 2 2 2 Y Y R Q d 2 kf f − n + + and 2 2 R 1 1 + ) has quantum conformal symmetry for any values of kf ¯ a a n n = I a m | − − kf − ¯ z (1 + X X | 3.2 Q ) will be denoted ≡ = = ] = ] = ] = ] = z a ¯ a a ) = n ¯ ( a n a n ¯ a n a n n n z ¯ ¯ ¯ T P Q X Q Q ( , ,P ,Q , T m m m m L L L L [ [ [ [ are the modes of a bi-adjoint field defined as are the generators of each of the two copies of ¯ a ¯ a a n t , A a t The sigma-model ( This field has vanishingwith conformal the dimension other operators. in the Sincethere the quantum is currents t are no neither hol unusual) sense uniform in conventions in which ( they are left- or right-moving. Thi Finally, where spanned by the modes where The energy-momentum tensor is given by point. In fact, the algebra Kaˇc-Moody Evidently the Virasoro tensorthe does modes. not act Furthermore,with diagonally the the on left-algebra right-algebra the (i.e. (theweight condition the barred for modes), unbarre both the which left- and makes right-algebra. it This di momentum tensor is indeed holomorphic.in One terms can of use bilinears ( to in derive the the mode following algebra, commutation and relations: then use the rel and the modes of and analogously for the right-currents algebra. This algebra replaces the usual algebr Kaˇc-Moody JHEP02(2019)098 , i ⊕ ) Φ and | R n , 0 (3.11) (3.10) ) ) zero- ] these ] b (2 ], which R P a m< 16 16 , sl a 30 Q , (2 Since these Q ra 15 sl ,( . tions as usual: i 10 0 . Φ 2 | ≥ p n a balgebra generated Q rposes in this paper. + 2 , where the parameter , m s. p 2 1 ]. More precisely, repre- 7 g, they give rise to long + i nflowed representations. = 0 contracts and the action of ) fall in two categories: gative modes of 2 1 i r highest weight representa- red oscillators. In [ . It then follows that their states obtained by the action R ntations may appear in the Φ itations. These are important sheet spectrum for any values 1) = 0 guri [ sentations of the | , = 2) are defined as [ arising in the spectrum of the a | R m j − , they are commonly referred to (2 1 . The associated lowest weight p 0 of the lowest weight state. These , 0 sl j asimir. ( R j , R (1 j 2 ) 0 − in psu R ( , such as for example g ,P = of i C i 2 Φ 0 C Φ | (2) are the finite-dimensional ones, and are f | R a . 2 1 su t p – 10 – = ) = i i Φ Φ | | a 0 ( in the representation h g , these can be derived by a mini-superspace analysis [ ,Q ] k in a representation 0 16 i , and others. In the following we will be able to compute the , . The relevant representations of i Φ ℓ | 2) are induced from representations of its bosonic subalgeb Φ 15 | | 1 n , m > ) , ¯ a a (1 . For this subalgebra, we can define lowest weight representa A = 0 a m i psu . These ‘chiral’ representations will be sufficient for our pu ) denotes the quadratic Casimir of ,( P i Φ are the generators of 0 i | , by looking only at one half of the mode algebra, namely the su Φ Φ 0 R | a m ]. The relevant representations of | f ( determines the quadratic Casimir as a R n and Q C t 30 ) on tions. They can be viewed as representations of spin representations give rise to the so-calledto short understand string exc the phenomenon of the missing chiral primarie mode algebra, and they are labelled by the spin representations depend on a continuous parameter strings states with radial momentum p as continuous representations. In the string theory settin ¯ a a m 0 ¯ Despite this, it is still possible to access part of the world It is useful to recall the possible representations and Q Note the additional factor of two in our conventions for the C Q 2. Continuous representations. These are neither lowest no 1. Discrete representations. These are lowest weight repre a k (2) [ 10 a m< Q affine primary states by model. For large values of P of labelled by their spin where essentially gives the spectrum proposed by Maldacenasentations of and Oo problems were solved in athe BMN-like Virasoro limit modes in can which be the algebra diagonalised. conformal weights of excitations with both barred and unbar In general we would like toof determine normal-ordered the products conformal on weight a of primary state Additionally, the spectrallyspectrum. flowed In images this paper of we these will be represe mostly concerned4 with the u The spectrum of the sigma-model representation can then be constructed by acting with the ne where conformal weight is [ ( su JHEP02(2019)098 ). 3.9 (4.1) (4.2) (4.4) (4.5) (4.6) (4.3) using 2 i Φ | . .  i i 1 Φ − | Φ |
 c  b t P n i b ns introduced in the 1 b ¯ -field, infinitely many n c tion of affine primary − Q ¯ ¯ , a P P a i -sided states of the type n A bc bc Φ =1 a a | not been able to overcome. n M Y ble to derive strong results i f f  2 2 1 thods, but we have not man- n i f f a ed oscillators or solely barred − i i ¯ n a r the ion of the conformal weights of this way its eigenvalues can be P ¯ rum described in section mixes only finitely many states Q − − 4 . 0 1 1 f n we will retrieve the chiral primaries 2 c =1 2 a a L − − . t N . n Y k i c i P P i ng states of the type kf a ) ) 4.4 ) 4 ) and the commutation relations ( Φ  − Φ 2 2 | | are the generators in the adjoint repre- 1 n 1 d ad c =1 t a a kf kf ∞ − − a Y n ( 3.10 + ) − − bd 1 ad b κ a − t or we will derive a unitarity bound on the values (1 (1 ,P ,P − Q 1 2 2 1 – 11 – i i ) i , and its eigenvalues cannot be extracted with a 2 Φ Φ = ( = 0 4.2 | | + + Φ | 1 as follows: b n a L 1 1 t kf a −  a − 0 bc a − a − we will argue that the continuous representations can- bc n n f Q L Q i a Q Q b − f 2 2 i (1 + P 4.3 2 1 kf kf n =1 n M Y i n n i a − (Φ) + (Φ) + (Φ) + h h h . The reason for this is that Q ¯ a    a m n =1 N A n Y = = = i i i  Φ Φ | | 1 1 =1 ∞ a Y − a − n P ) associated with the ‘chiral’ lowest weight representatio Q 0 0 L can take, in subsection L ). In particular, in subsection 3.10 2 4.2 kf We are then interested in the conformal weights of the single finite amount of calculation. This is a difficulty which we have For simplicity, in the followingsuch we states illustrate using single-oscillator the excitations, computat i.e. usi states get mixed under the action of The multi-oscillator states can beaged treated to using find the a sameconcerning closed the me form expected solution. qualitative behaviour Nevertheless, of we the will spect be a We have used the definition of affine primary ( Note that the structure constants i The states in the spectrum at the first level are not be part of thethat CFT spectrum, are and missing in at subsection the pure NS-NS point. 4.1 The spectrum at the first level only ( that previous section. When including also barred oscillators o oscillators, and no conformal weight of a state containing either solely unbarr constructed using solely unbarredcomputed. oscillators, say, In and this in states case, ( we will be able to make use of the defini sentation and hence They mix under the application of JHEP02(2019)098 1 ) is ℓ 12 2) · | 2. and + a 1 = 0. (4.9) (4.8) (4.7) , h i ion. In (4.10) ) that ≥
indeed (1 Φ ) + 1 at | k i C from the 1 3.1 ad psu a − . Φ | − × C (  Q h 1 . Restricting 2 h ). We do not ) 2) | C ad = 1 ng that 4.2 ,  ⊗ + d (∆ (1 , t 0 4 , h ], except that 2( b c f t R a psu ! 16 δ bd + C C κ has no affine representation mixes only C ∆ ∆ 1 tion vanishes 0 2 − 2 -th level ( ∆ 2) f L 4 f d be the only effect of the n | 2 1 d ad 1 1 2 t it for the sake of completeness with the wrong mode number. , ] that only the solution es of the algebra, the solution kf − 2 4 (1 e model away from the WZW-point. . b ad 16 on of conformal weights was already ) + t Thus 2 = − la found in [ kf psu bd c 4 κ a kf ⊂ 11 ]. δ p , − 1 −  31 we find )  ) ). Classically, we know from ( ± , 0
d (2) (1 t 0 28 1 2 R su ad ( , R 4.10 + + 2 ) ) is therefore consistent with the one found ⊗ 2 27 C d ad 4 0 − C t f 2 ∆ kf − C ) – 12 – 2 4.10 reduces to the correct result 2 )( 1 k kf
b f 4 − t + ⊂ R R  1 ( h ad 1 (1 + 4 1 + C 1 2 R ⊗  + b ad

2 (2), the theory cannot have a field in the adjoint representat 0 t
f ( i su + R 1 2 bd Φ 1 | C κ − ) takes the form does not transform in a valid representation of i   h ¯ a = 0 1 2 2 1 Φ the difference of Casimirs. a | b L . It is easy to confirm that in the large-charge limit ∆ = ( A t ) transform in the (reducible) representation C = = C h
bc i d a = 1. Hence we will discard the state with eigenvalue t = 4.3 Φ f | ). The associated eigenvalues are 2 0 2 1 b ad t f a L − i kf bd 3.11 κ ), and the exact formula ( ,P ℓ . i · 1, and we will see that also holds at the quantum level, assumi B Φ | ]. a 1 ≤ 15 = 1 theory behaves quite differently. Since a − 2 k Q , and in this basis i kf = 1 theory at the WZW-point is discussed in [ ± Φ A similar analysis can be performed for the spectrum at k | h ≤ The pertinence of the difference ofThe Casimirs to the computati can be interpreted as the application of a barred oscillator 1 12 11 a 1 − − at the WZW-point.The Hence it is not clear whether we can deform th physical spectrum. Itphysical constraints is on not the clear one-sided worldsheet at spectrum this point if this shoul the WZW-point based on the adjoint representation of particular, the biadjoint field Notice that this result is similar to the large-charge formu where we used ( On the other hand, only the solution physical. In fact, dueh to the identifications between the mod has been replaced by ∆ becomes 2( to an irreducible subrepresentation 4.2 A unitarityThere bound is one very− interesting consequence of ( make use of it in thein following appendix analysis, but we have included in the large-charge limit. Furthermore, it was argued in [ P where we denoted by ∆ Note that the states ( noticed in [ This can be expressed as a difference of Casimirs: where we have used that the Casimir of the adjoint representa JHEP02(2019)098 and 1 the ℓ (4.14) (4.12) (4.11) (4.13) . The − 4 2 k 2) should T | = 1 ), we realise × , ) appearing. ℓ 3 (1 (2)-spin R S , ) we obtain the 4.10 su (2 psu × te representations sl 3 4.10 )-spin and o this statement is the hat the energy momen- enomenon in the world- ts are real. We see that R licitly on the difference of , + 1 of hat are missing from the ar from the spectrum in a nd AdS h a single oscillator depend j (2 . 2). In this way, we can choose p . ntation with sl ce already the ‘chiral’ contin- + 1). At first excitation level, | . 1 ℓ he full representations must be ht at level one ( ad  , ( and 4 ese representations are forbidden ℓ (1 ⊗ f j 2 is the + 2  k 1, j psu 1 . 2 − p 1 − − 1 and 2 + 4i j ≤ 2 k + 2 p 2 2 1 j, , , and inserting this into ( , where  k
0 kf 1 – 13 – ⊂ ≤ − + 1 + = 2 1 2 k  2 p , C + 1) = 2 k ) part transforms in a continuous representation. Its − 4i = ℓ kf R ( j,  − 2 the spectrum of the sigma-model on , ℓ 1 2  between the ground state representation and the repre- j, ℓ 6= 0 generated by charged oscillators become generically (2 ≥ C = p = sl for the conventions of = 2 k 1 = h 1) + 2 C A 0 R ∆ − R j ( j , i.e. the 2 p − + i = 2 1 , eq. (6.3)], for C = 30 j . This result should continue to hold once we consider comple C ) spin R Since the appearance of complex conformal weights implies t , (2)-spin, see also appendix (2 Plugging this result into the formula for the conformal weig The respective differences of Casimirs are su that the conformal weights for following conformal weight of the excited state: we have states in the representations with spin this is only the case provided that An obvious requirement of any CFT is that the conformal weigh complex. According to [ Casimir is then This choice of representations yields ∆ We found that theon conformal weight the of difference states of constructed Casimirs wit ∆ 4.3 Continuous representations sl sentation of the state. Consider then a ground state represe contain the representation Casimirs ∆ tum tensor is not self-adjointand in hence these cannot representations, be th WZW-point, part where the of conformal the dimensions spectrum. do not depend The exp only exception t 4.4 Missing chiralWe primaries are alsospectrum in at the the position WZW-point.sheet to In description. retrieve the the following For we chiral simplicity, review primaries this we t ph focus on the backgrou of the mode algebra,uous and representations not contain just complex of conformalruled its weights, out. ‘chiral‘ t version. Hence wemixed-flux Sin confirm background. the fact that long strings disappe JHEP02(2019)098 .
5 14 1 2 and and ): − (4.16) (4.15) (4.17) (4.18) and is A 2 2 +1 +1 Z ). Note k k 2 ¯ k δ ussion , with the 2.2 + 4 \ This can be T j < Z kw . They have the only treat the 2 1 i < j < + ]. These are the , ℓ 1 2 13 1 2 34 δ, ℓ – − + . 32 j
0) | , , ty restriction, the state kw ) with 3 , ℓ kw ed by the unitarity bounds + same quantum numbers. restriction to 2 1 ge-diamond of ℓ is is then the most stringent hilate the state and so it lies is minimal. ring the no-ghost theorem in − ) . It has the following form: j, ℓ, w ℓ 2 1 0) takes values in j , h , + 1 + − 2 1 kw  ℓ (1 ( j kw k + × × ℓ + + , ℓ l representations, each of which yielding one  ,  , 1 2 } ) (1 2 1 2 kw − 2) BPS bound and are therefore atypical represen- 2 − , | 2 + 1 k 1) 0) 2 1 j 1 (2) spin + , , , ( ) for which k  1 2 su (1 (1 (0 2 – 14 – × ≤ A.8 + ,..., , since at level zero there is a state with the same quantum psu − j i 1 2 ℓ j | ,   1 ) 2 0 − − 2 1 1) 2 is the spectral flow number. Every discrete represen- = × , J , i 1 2 2 ∈ { (2) representation with spin ( ] discrete representations ( , whose norm at the WZW-point is (see appendix Z ( (0 i ℓ su j, ℓ 30 ∈ | × × , ) yields four chiral primary states [ j, ℓ ⊕ |
2 2 w ++ . i − 0 ++ kw j (2) | , ℓ, w S − 0 1) 4 + 1 su + , S − − ℓ 1 + 1 −− 0 (0 J − − S ℓ + J , where 1 = 0, for comments on the spectrally flowed sectors see the Disc } J ++ (2) representations in ( -th Hodge diamond is missing. This was alluded to in section 2 − 0 w k 2 − S su −− k + 0 has to be physical in string theory, which is in fact the case. (2)-spins: ⊕ WZW-model has [ S ) | i su ) denotes an k j R ¯ | δ ,..., , whose origin we will briefly review in the following. We will , j, ℓ 2) 1 2 2 | h δ, , ++ 2 (2 1 − 0 , k − 0 sl The absence of the chiral primaries on the worldsheet is caus S In fact, these representations saturate the This would not be true for the state (1 1 ≤ ∈ { 13 14 − − that because of the restriction where ( constraining the worldsheet theory. The main bounds are the lowest state having left- and right-moving tation of the form ( ℓ unflowed sector Consider the state what we mean by ‘missing chiral primary’. ℓ four followings seen from the fact that there is no state at level zero with the and thus every Combining with the right-movers, we obtain the complete Hod bound possible. In the RNS formalism, it arises from conside which is the Maldacena-OoguriJ bound. For this to beHence a all unitari positive modes ofin uncharged particular operators in have the to BRST-cohomology anni of physical states. Th This norm is non-negative if psu numbers, namely tations. Thus theBPS representation state. splits up into four atypica JHEP02(2019)098 , 2 4, . 2 − k −
2 j (4.21) (4.23) (4.24) (4.22) (4.19) (4.20) ≤ − = 1. We , where -currents ℓ ℓ i = 4 2 J 2 Φ | C − ) kf 1 k ange. For this a − P ± for the ± b with ∆ = P J 2 + . ±
b 1 4 sponding state for the . at the WZW-point. We ZW-point, but nothing 2 , a − f i be non-negative gives the C 2 − Q ) ℓ j, ℓ C 2 dered state is again physical. | from the spectrum. Now we + 2 the theorem in the R-sector. 2 dix k 1 elds a stronger bound than the e in the literature. Thus, to fill − , 2 − i +++ k k < + (∆ -modes and Q, 4 0 1 = j, ℓ Q | S q ), and reduces to it at ℓ − kf C 4 ++ ++ f − − 4.18 , 2 2 4∆ 2 Q, Q, 0 k 0 f 2 to be non-negative. This yields 2 S S − kf 2 ) + ) − i 4 4 − 4 + 15 (2) representations can be obtained by requir- k 1 1 f P, P, p j, ℓ − − ), respectively. As noted before, only the state su – 15 – | WZW-point ≤ J −−−−−−−→± has negative norm and is therefore unphysical, as argued p K ) and the explicit form of ± ± ℓ ± -currents of the − 2 b 1) − b 4.10 h +++ 3.4 J 0 + 1 2 − kf S + at the first level, which are ( 2 ℓ − + 1 + 0 ++ 1 − Q, − − L Q, − − 0 as in ( 2 J k S for the ( f K 2 + 1 1 ± ( C Q + − is part of the physical spectrum. The analogue of the state k h J ∆ K + 1)( + ≤ i ≡ ℓ h = j Ψ | 4( ± b − 4 − to have positive norm led at the WZW-point to the constraint eigenvalues f in the mixed flux case is i 0 p i L Ψ | ± j, ℓ | = -modes. Using the algebra ( i P ++ Ψ − 0 | The situation is entirely different when looking at the corre Let us move away from the WZW-point and see how these bounds ch Similarly, the unitarity constraint for S Notice the state with conformal weight Ψ 1 (2)-spin bound h 15 − − of the where we use the notation J su Asking for see that the boundspectacular changes happens. slightly when going away from the W which is less constraining than the usual bound ( find that the norm of this state is in general which is the familiar bound fromThese the RNS are formalism. the The bounds consi we mentioned above. which in turn excluded the missing chiral primary at with conformal weight explain the very small difference which occurs in the proof of These have NS-sector version was to our knowledgethis not gap, considered we befor review the proof of the no-ghost theorem in appen the R-sector. The fact that the R-sector no-ghost theorem yi the norm of this stateconstraint can be computed. Requiring this norm to before. we first find the eigenvectors of ing the norm of the state JHEP02(2019)098 . is 0 j N 1 2 ∈ ℓ 1 maries are − j = (2)-spin ℓ per bound on , which now allows su j ositive sign. The two n ‘avoided crossing’ and t square at the WZW- s for the chiral primaries. nt. ations with nch has always positive norm d be chosen in the last equality, . On the other hand, the bound 2 +2 k 2 2 . We see that away from the WZW-point, − k 2 , there is such a bound on = ℓ . At the WZW-point, the two branches intersect at i ℓ – 16 – Ψ | . Combining this with spectral flow, it precisely fills ! 1 ℓ 2 − k = . Thus, we see that there is one new chiral primary compared ℓ } 1 2 − k

,..., etre solution perturbed 2 perturbed solution 1 ,

0 positive norm WZW solution solution WZW norm negative ) in the BPS spectrum. We conclude that the missing chiral pri i ∈ { Ψ ℓ 2.6 | . The two branches of the norm of Ψ h . For a slight perturbation away from the WZW-point, we have a 2 2 completely disappears. This has the following consequence − In summary, away from the pure NS-NS point we found that the up k ℓ = As we discussed above, chiral primaries come from represent While there is no longer an upper bound on ℓ the gaps ( indeed reinstated by any perturbation away from the WZW-poi and there is no unitarity bound on on the two branches no longer cross. In particular, the first bra to the WZW-point, namely slightly shifted upwards, but always strictly less than As indicated, thepoint. term From under this description the itbut is square from not root clear the which WZW-description becomes signbranches we shoul a for know the perfec that norm we of should are take plotted the in p figure the first branch has always positive norm. Figure 2 the values JHEP02(2019)098 g to extend backgrounds. 3 = 0 continuous w s a left and right ) to exist. However, ms continue to hold m of string theory on 3.4 tent of the supergroup ged to surmount in this ess, we believe this to be the same conclusions hold rally flowing the retrieved -sided excitations. Clearly, mary states, which behave hand, unitarity of the CFT arred oscillators, so that in ated and involves also the l primary spectrum, and all epresentation content as the tates which are excited both heory in AdS tive amount of spectral flow. m, and more precisely on the e are several intriguing open oup CFT, a sigma-model on = 0 sector of the worldsheet hts slightly, it will not modify ed qualitative behaviour of the not a physical state of the full ctrum. These relatively simple ], and it is far more complicated s, and of avoided crossing as in ion from the physical spectrum. w perturbation the phenomenon of pectrally flowed continuous repre- 16 (2)-spins. Once perturbing slightly (2)-spin disappears completely and su su ) the existence of a single non-unitary , since the unflowed w 3.4 – 17 – 2) sigma-model. We found that continuous representa- | 1 , when leaving the singular locus. It would be very interestin 4 × M 3 2) symmetry, which is all one needs for the mode algebra ( | S 1 , will not disappear. Thus, we believe that the same mechanis , 2 × Our arguments were limited in that they involved only single While we have explained the change in the representation con Our computations were performed in the unflowed The complete worldsheet CFT (including the ghosts) still ha We have explained the two most drastic changes in the spectru 3 , and the ghost couplings of the two constituents. Neverthel 4 = 0 missing chiral primary will fill the other gaps in the chira a single-sided excitation isstring not theory. level-matched However, and due to hence the is mode algebra ( than at the WZW-point. Inorder particular, to it understand mixes spectral barred flowon and one the unb first has left to and understandpaper. right. s An This exception is to ain this difficulty a statement which is simple we given have mannerw by not under the mana spectral affine flow. pri For this reason, spect state in one representation excludes the whole representat CFT. Spectral flow of the mode algebra was discussed in [ their analytical structure. Inimaginary particular, conformal under weights a of generic figure continuous representation in the full model. the construction of theadditional fields. Virasoro While tensor this is may correct then the more conformal complic weig questions and interesting future directions. additional chiral primaries appearcomputations in give the a string mechanism explaining theorysingular the spe locus change in of the the r moduli space is crossed. However, ther PSU(1 the missing chiral primaries are retrieved.sentations are Furthermore, s not allowed in the spectrum for any In this paper we proposed anspectrum explicit argument at for the the expect singular locus of the moduli space of string t 5 Discussion away from the WZW-point, the upper bound on the To perform our computations we reliedalgebraic on structure the of hybrid the formalis PSU(1 representation can be obtained from these by applying a nega true for the following reasons. at the WZW-point introduces bounds on the allowed tions are only allowed to exist at the WZW-point. On the other WZW-model, we have not presented ain convincing the argument that complete worldsheetM theory consisting of the supergr AdS JHEP02(2019)098 k × 2) 3 | S 1 , × (1 3 to string looking at psu ]). Therefore, 35 , respectively. The a m sily confirms that the K erdiel, Juan Maldacena, ) can be used to describe . In particular, the indices α sations and Matthias Gab- and derstand this better. he background AdS . Having this at hand, one s however far more intricate, of a wall-crossing formula. related to a BPS state in the way from the singular locus. nce of chiral primaries seems h is supported by the NCCR 1; a m iptic genus of [ g of the (dis)appearance of all s. We use these commutation ous at the singular locus, the , hanks Imperial College London J (2 d n function infinitesimally far away ]. There is no hybrid formalism for 2) 34 | 1 . The bosonic subalgebra of , (1 {±} – 18 – psu continue to hold true for this superalgebra and hence the 4 . We display here the commutation relations for the affine 4.4 , whose modes we denote by k (2) su 2) plays a major when rˆole applying the algebraic formalism ⊕ | . The commutation relations for the global algebra follow by 1 k k , ) 2) R (1 | , 1 , (2 psu sl (1 psu , whose spectrum exhibits similar features [ denote spinor indices and take values 1 S Finally, it would be interesting to repeat our analysis for t × 3 S consists of calculations presented in section the string propagation in a background with mixed flux. One ea the zero-modes only. Weα, use β, γ a spinor notation for the algebra The algebra algebra A The (affine) Lie superalgebra SwissMAP, funded by the Swiss National Science Foundation. We thank Minjae Cho,Alessandro Scott Sfondrini, Collier, Xi Andrea Yin Dei,erdiel and for Matthias reading Ida Gab the Zadeh manuscriptfor for prior hospitality to useful where publication. conver part LE of t this work was done. Our researc Acknowledgments mechanism for the disappearance ofto long be the strings same. and appeara Theand structure in of particular the not chiral every primaryunflowed BPS spectrum sector state i by in spectral the flow. twisted sector It is would be interesting to un one should be able to quantify this discontinuity in the form the background, but one expects that the superalgebra relations explicitly in section theory, so we recall here the relevant commutation relation from the singular locus.states This in would entail the the theory,could understandin not compute just protected quantities theObviously, which special since remain ones the constant we a chiralsame considered primary should be spectrum true is for the discontinu elliptic genus (or the modified ell these results to obtain the complete string theory partitio JHEP02(2019)098 3 0 ∈ 16 J , ]: (2), . ℓ n (A.7) (A.2) (A.1) (A.3) (A.4) (A.5) (A.6) 36 and 1 su + , a m − ) = 1 30 ∓ K = , βν , a ,K ) 2), which can 1 , , respectively. a n | ± ℓ 1 − σ + ( , K 3 m ( = 2 = γρ (1 1 for , K ǫ and + − n , − ++ . αµ j psu , κ + ) ) . 0 + 2 ǫ 0 ανγ m 1 2 + + n, 0 , S + σ pplications. The discrete σ + n, ators are chosen to be ( ν ( n n m β + equals , e-norm states imposes + ) = ) = 0 ) + i m 3 a a ± m a a . m . orm on c σ ng strings are continuous rep- f of the supercurrents: J K kmδ ( j, ℓ , , m > ,K ,K | ). We have chosen the signature 2 2 1 1 , , ± kmδ 3 ℓ αµ (2)-part, but are neither highest, ) + 1) R K a , αβγ ℓ = = 0 = 0 su ( σ ( ∈ {±} ] = ] = ] = ] = = 1 = 1 S i i i ( ℓ (2 3 n ( ± − n n a + − sl c ] to accommodate the fact that one direction is αβγ j, ℓ j, ℓ j, ℓ n + + | | | ,K α, β ) ,K ) ,K )-oscillators, and half-infinite. For γρ 16 3 0 + , κ , κ a 3 3 3 0 ǫ m m ,S 3 + m m R 1) + 2 σ σ a , m K ( βν ( K K [ ǫ [ [ (2) or − K (2 are spacelike. This is important in the main [ ,K j ) = 1 ) = 0 − , su sl – 19 – ( 0 a − ∓ 0 j n 2 = 0 for J + n, ,J , , . They satisfy the commutation relations [ ,J i 3 − + m ,K n 1 ± , J , m i + αβγ J − 2 δ j, ℓ αβγ n 0 and ( | µβγ m S ) = S n, j, ℓ γρ ,K , m > ( − − = = 1 S | ǫ + + µ j 0 j, ℓ αβ J − + 0 α m , βν ( n, ) − − S ǫ = = 0 = 0 C , κ n a + ) ) i i i 3 3 + σ m αµ ( γρ σ σ kmδ ± m ǫ ( a ( j, ℓ j, ℓ j, ℓ 1 2 J c | | | βν 3 1 2 0 , κ kmǫ kmδ ± − a − m 0 ǫ J J J 2 1 = αµ ǫ − ] = ] = ] = ] = } , , 3 n − ± n n denote adjoint indices of αβγ n ,J µνρ n ) = ) = ,J ,J } = 2 = 1 is timelike, but 3 3 m 3 3 + m m ,S ,S , J − 3 µνρ J J [ a ,J m [ [ + −− J 3 J ) ) αβγ [ ,S m J is not quantised. The Casimir of such a representation reads − − ( S , and we denote their eigenvalues throughout the text as ∈ {± σ j σ κ 3 { 0 ( ( αβγ . a K S 0 We consider two kinds of representations for string theory a ( We changed our conventions slightly with respect to [ ≥ κ 16 Z 2 1 text, where we compute the norm of states. The constant otherwise. The sigma-matrices read explicitly such that Here and all other components are vanishing. The two Cartan gener timelike and the others are spacelike. and fermionic generators are denoted The other important classresentations of which are representations still describing finite-dimensional lo for the Furthermore, there is a unique (up to rescaling) invariant f they are finite dimensional. Hence they are characterised by be read off from the central terms: representations are lowest weight for the Furthermore, the highest weight state is annihilated by hal Requiring that the zero-mode representation has no negativ JHEP02(2019)098 can (2)- ince (B.4) (B.3) (B.2) (B.1) (B.6) (B.5) i (A.8) su n i b − ⊕ ) J R , , n (2 the number of ]. This implies sl 0 ). The first type 29 ), each . L i 3.9 Φ B.2 | . m )-representation is then n m  excitations of the form b − R 1 2 , J , + 1 is saturated, and it is n , ssion ( lations ( mber of modes. The second m ℓ ± (2 ecause of the second relation b ··· 1 sl , ℓ = 0 entation, see [ ≥ − creases. +1 1 2 m i c j b j . +1 n 2 b i i i ± b − to level − b . Φ m j J f | i a i  n m m f 4.1 Φ b 2 n − | m ) ··· b − n , d 1 ··· n . a J − b )-part. The consists of the following 16 a P 2 enters the representation by imposing that 1) i c R a = 2 , ··· i b α Q ± 1 1 ( j, ℓ ). There will be two types of terms appearing, n ,P (2 a | , g n 1 . 1 – 20 – i f 0 b sl − − 1 j, ℓ i m =1 of the form Φ 1 ≥ J ( B.2 i − | n X a i = 0 b 1 these states mix with multi-oscillator states such as n g . m R − , b b j a − a J ) α 0 b ··· ∈ i f together with its Casimir. The Casimir is commonly Q 1 L b , ℓ + b p ··· 1 κ a = -th level Z behaves as follows: under the action of 1 g Z m / ± n n b 1 m 0 for b b − j R ··· . Moreover, we require that the state is at level L ( 1 J i ) is not relevant: the oscillators can freely be reordered, s ··· 2 b 1 n i p is atypical if the BPS bound ∈ , a b cd b i − on the state ( a g ) b i ) has the property B.2 g α P 0 f + 2 j, ℓ L j, ℓ m 2 1 B.3 or | b 4( i ··· 1 n = i b b − a . However, C i g Q Φ | is an invariant tensor of 1 c − m P b )-spins take values in ··· R +1 1 , b n ). We compute a b − (2 g To prove this assertion, we start with a state of the form A representation Q sl B.5 bc a The invariant tensor ( oscillators either increases or stays the same, but never de up to possible permutations of the free indices. In the expre Here f multiplets: stand either for As we will see, under the action of otherwise typical. A typical representation in ( corresponding to the two typesof of terms terms are in linear the in commutationtype the re of modes terms and yields obviously the preserve following the expression: nu that normal ordering in ( the commutator produces structure constants. They vanish b In this section we generalise the analysis of section nor lowest weight representations for the B The spectrum at the specified by an element parametrised by thanks to the vanishing of all Casimirs of the adjoint repres the JHEP02(2019)098 (B.8) (B.9) (B.7) (C.1) .  ), either both 2 ) ). In the former C B.6 i , Φ | (∆ o the computation in ( 4 states which can be ! i f C n n ), so we may still freely ∆ . + − 2 ) C d f B.5 le. However, once one tries              ihilates 1 2 P . . . ∆ 0 c with positive mode vanishes, n f the physical spectrum. This + r. 4 . . . n Q ⋆ ⋆ number of oscillators can never 2 ions, the computations become n ) kf n to the eigenvalues with respect . . . 0 0 0 0 0 0 0 0 0 ⋆ ⋆ 0 2 m ( 4 y do not contribute to the eigen- on single-oscillator excitations, we on all level , . − kf 0 . 0 ), we do not have to consider other . ··· 2 ··· ··· ··· ··· L int L − n . . . B.2 4 0 (1 1 2 p . . . CFT 0 0 ⋆ ⋆ n kf ± ⊕ . . . ) ⋆ ⋆ 0 0 0 0 0 0 0 ⋆ 2 n n ] for values of the charges. Furthermore, at the              4 (1) k – 21 – f 2 ) kf as follows: 2 16 + 2 k R i kf 4 , C − Φ 1 | ∆ (2 (1 + n 2 2 1 a sl − f still has the same property as (

P  . 0 closes on the set ( 1 4 . . . cd + +1 oscillators, whereas in the latter case, the zero mode on i L b 0 and hence the number of oscillators remains the same. Also, 1 = and f m ) L d 1 oscillators oscillators i
i t m i , we hence get the following block structure: b 1 oscillator Φ 3 oscillators 2 oscillators n Φ − 0 | | ··· Φ or ( | 1 n L n = 1 we retrieve the WZW result. b h n c a − a t a 2 − g Q = kf 0 ,P i L Φ | n acts on a − 0 Q L ± ): δh When computing the matrix-representation of This result makes it seem as if the structure is always so simp With this at hand, the computation is completely analogous t gives a generator 4.1 i Φ C Proof of theWe R-sector consider the no-ghost following theorem CFT as a worldsheet theory: to compute the conformalquickly weight very of complicated. multioscillator excitat where we ignored all multi-oscillatorto terms. the The ground correctio state is given by reduces again to thepure BMN-like limit NS-NS point of [ We again expect only the positive sign eigenvalue to be part o invariant tensors. This proves thebe above decreased assertion by that the the action of reorder the oscillators. In theoscillators normal-ordered have product negative ter modes orsince one we is can a commute zero-mode it (a through term to the right, where it then ann we note that the action of case, we obtain a term| with The invariant tensor mixed by the action of can simply ignore multi-oscillatorvalue. excitations, They since do the however contribute to the precise eigenvecto Thus for the purpose of computing the spectrum of in ( JHEP02(2019)098 . 0 to L ] in and , 7 i (C.2) (C.3) (C.4) (C.7) (C.5) (C.6) n ) with f ], a no- ∈ H | L m φ 18 C.5 µ , ) = 1 SCFT, m ]. In parallel 17 3 − N 38 J ( = 0, i.e. ε the corresponding ··· 1 3 n µ ψ = 15. In [ ) 1 c 3 − . N, ], one finds the following es of the form ( , we will only explain the J 0 acena-Ooguri bound [ = 0 and ( ≤ 18 a the end. ually yield a different bound . We define a state λ δ ) L . Here, we want to fill the gap N a is some internal is therefore null [ . + 3 l charge − )-algebra, 0 s s ψ int s ( R φ ) with , , can be written as a spurious states µ , n, r > φ ··· , (2 s 1 C.5 sl = 1 superconformal algebra by X δ = 0 ) by the requirements = 0 1 χ . + φ N 3 − 3 , n, r > r r φ + + 2. CFT r ψ 0 ψ s ( δ k G . The complete Hilbert space of the worldsheet φ m = 0 = m r Z – 22 – and λ − = F ⊂ H = X φ φ L r 3 ∈ } ) is n φ , the states φ + 1 J 0 G r R , c ··· , 2 , L +2 0 c = = 1 k n 1 δ (2 λ − φ φ sl ∈ { r c L n with grade less or equal to a b X G a L ε − H , δ + < j < = b G . It enjoys as usual a natural grading by the eigenvalue of b ε b φ H , ε n ··· L 1 L 1 b ε − X . are separately physical states and G ) N χ ( := = 15 and 0 ] for our conventions. Then analogously to [ H i c and 37 : , f is at grade ) s } N φ For ( 6= 0. By the Lemma, every physical states is the Cartan-generator of the bosonic H ε ∈ F 3 n f J denotes the subspace of ], we have then = 15, ε, λ, δ, µ We define furthermore the subspace Since the proof is almost identical to theWe denote NS-sector in version the following the worldsheet We call a state spurious, if it is a linear combination of stat ) c 18 |{ plus a linear combination of states of the form ( . Since we focus on the R-sector, N 6= 0 or ( r s be physical if it satisfies the physical state conditions H theory will be denoted by In string theory, a physical state has to satisfy in addition and the GSO-projection. G fermion. See [ Here, to [ λ basis for ghost theorem for these theoriesand was prove proven the in no-ghost the NS-sector theoremand in explains the the R-sector. somewhat Thisthe mysterious will literature. appearance act of the Mald Lemma. For as usual the level of the bosonic which has the correct central charge to give the total centra where φ form a basis for strategy and point out the small, but important difference in JHEP02(2019)098 1 2 + = 0, (C.8) j ε ommons , ]. , and apply ed in detail 1 2 . As explained . This state is = 0 and − (2) in the case of i at the R-moded . (1) is unitary in SPIRE eigenvalue 1 2 λ j field theories, 2 +2 su u IN k 2 3 / +1 0 ⊕ − ) N k ][ J ) j differs by a spurious R R , ≤ is bound. Let us also , ) with = φ j (2 wing this bound to the (2 < j < Large sl sl sion proof. The only small ]. C.5 t eigenvalue es have j, m redited. | Oz, 3 0 ound 1 ) of . Thus, we have: nt state. This state is exactly J − − 2 2 1 , J SPIRE 2 k . ( IN · hep-th/9905111 [ + ][ i ≡ j  Φ 1 2 | = 2 spectral flow, but under the spectral − j N in the supergroup language. For this, we look  get shifted by , every physical state (2000) 183 2 j – 23 – − 2 +1 4.4 k 323 unit. Indeed, when spectrally flowing the formulas = hep-th/9711200 [ We pick a state with i 2 1 . The fermionic zero-modes in the R-sector construct j j Φ 17 | . = 1 Virasoro modes, since there is no state at level zero j 1 2 is a physical state of the form ( < j < limit of superconformal field theories and supergravity . Consequently, the norm of every physical state is non- Φ ), which permits any use, distribution and reproduction in h N − χ F 1 2 N j Phys. Rept. , if , (1999) 1113 2 +2 k 38 CC-BY 4.0 The large = 15 and eigenvalue This article is distributed under the terms of the Creative C c . The resulting state is denoted -eigenvalue. Hence it is physical and its norm is < j < 3 0 1 3 0 )-spin gets shifted by (1) is unitary. The NS-moded version of this coset was analys − J − J u R J For . / , )-representation of spin 4 ) ], one sees that the bounds on For 0 . T R (2 R , , By the previous two lemmas, the proof boils down to showing th 39 sl × (2 3 (2 ∈ F S sl sl ], it was found that in that case it is unitary provided that 0 χ × Int. J. Theor. Phys. string theory and gravity Strictly speaking, we have demonstrated the sufficiency of th So far, everything is exactly the same as in the NS-sector ver 3 For example, the zero modes generate the representation 2 39 17 [1] J.M. Maldacena, [2] O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. 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