JHEP02(2018)065 a Springer October 17, 2017 January 28, 2018 January 30, 2018 February 12, 2018 : : : : Revised Received and Pulastya Parekh Accepted Published c ) as the residual symmetry in I , Published for SISSA by https://doi.org/10.1007/JHEP02(2018)065 [email protected] Shankhadeep Chakrabortty , [email protected] b , . 3 1710.03482 Aritra Banerjee, The Authors. Conformal and W Symmetry, Superspaces, Superstrings and Heterotic a c

We construct a novel tensionless limit of Superstring theory that realises the , [email protected] [email protected] CAS Key Laboratory of TheoreticalChinese Physics, Academy Institute of of Sciences, Theoretical Beijing Physics, Indian 100190, Institute China of Technology Ropar, Rupnagar, Punjab 140001, India E-mail: Indian Institute of Technology Kanpur, Kanpur 208016, India b c a Open Access Article funded by SCOAP Keywords: Strings, Higher Spin Symmetry ArXiv ePrint: superstring, where a smalleras symmetry the algebra residual callednew gauge the tensionless symmetry Homogeneous theory intrinsically on SGCA asfeatures emerged the well of as worldsheet. the from tensile a theory.this systematic We We new limit obtain discuss of tensionless why various the limit it corresponding features and is the desirable of and larger the also algebra. natural to work with Abstract: Inhomogeneous Super Galilean Conformalthe Algebra analogue (SGCA of the conformal gauge, as opposed to previous constructions of the tensionless Arjun Bagchi, Inhomogeneous tensionless superstrings JHEP02(2018)065 ]. It was found that there 3 , 13 2 ]. Gross and Mende revisited 1 12 5 18 2 11 4 10 3 – 1 – 6 22 21 25 19 18 16 23 12 1 21 Analogously, it is of interest to look at a similar limit in string theory. This is the limit 6.1 A summary6.2 of the paper Comments and future directions 4.4 Mode expansions 5.1 Homogeneous scaling 5.2 Inhomogeneous scaling 4.1 Why bother? 4.2 Representations of4.3 the modified Clifford Constructing algebra inhomogeneous tensionless superstrings 1.3 Geometry of1.4 the tensionless worldsheet Outlook and outline: tensionless superstrings 1.1 Hagedorn physics1.2 and tensionless strings Tensionless worldsheet symmetries the spacetime in which they travel. where the tension of thetensionless string, strings i.e. was first the explored massthis by per limit Schild unit from in the length, the perspective 1970’s becomes of [ zero. scattering of The string theory states of in [ 1 Introduction The massless limit playsparticles. an In important a role quantum inthan field that understanding theory, of the the the nature limit incomingprocesses of explores or at processes the outgoing high at particle theory energies. energies states of Massless and much particles higher provides also insights help into understand scattering the causal structure of B Truncation of the algebra 6 Conclusions A Tensionless spinors 5 Tensionless Limits of the RNS Superstring 4 Inhomogenous tensionless superstrings 2 Quick look at tensile3 superstrings Homogenous tensionless superstrings Contents 1 Introduction JHEP02(2018)065 (1.2) (1.1) = 4 Supersym- N ]: 21 , 20 [ , string theory is thought to ] between Vasiliev higher spin H T 7 , T – 2 5  H T ) behaves like a theory of tensionless T . ]. When interactions are included, this N 0  α − 19 , ]. Beyond 1 √ 1 π 18 17 4 s – 2 – 0 = T H ]. T 16 ) = – T ( 11 eff T of a string at a temperature eff T break down, it just becomes different. The Hagedorn transition, is the usual string tension. So one would be able to access the physics 1 ] and this is an indication of an emergent very large, hitherto unknown, − 4 ) 0 does not πα ]. Other explorations between the connection of higher spin theories with ] and free field theories. It had been earlier speculated that free 10 = (4 , 8 9 0 T The common folklore regarding treating string theory near the very high energy Hage- There has been recent interest in understanding the tensionless limit of string the- , given by H the worldsheet metric degenerates andgeometry we to need the to language makeSo, of a let 2d transition us Carrollian from assert manifolds. 2dstring again Riemannian We that theory briefly near elaborate the onand Hagedorn this the temperature, below. physics the associated worldsheet with description it, of is extremely important to understand the fundamental where of the Hagedorn transition by understanding theto theory tensionless of tensionless string strings. theory Ourthe has approach been tensionless centred limit. aroundHagedorn the temperature, emergence One the of string of new worldsheetthe symmetries our does worldsheet in go not principle from break the claims down, tensile but version in to the the symmetries this tensionless on version. programme In the is tensionless limit, that near the less. The effective tension long string than topicture heat gets up more a complicated. gas of strings [ dorn phase revolves aroundstring worldsheet. the Near breakdown the of Hagedorn the temperature, the description strings of become theory effectively tension- in terms of the Rather than a limitingperature is temperature indicative beyond of whichtransition a the into phase the theory transition mysterious doeslittle [ Hagedorn firmly not established. phase, exist, It of has thisconsiders which been the tem- shown much that theory has near of the been free Hagedorn speculated temperature, strings, when but it one is thermodynamically favourable to form a single Another, albeit less explored,is avenue the of physics applicabilityfunction of of for the the free theory Hagedorn strings ofT transition. diverges above tensionless a strings It critical is temperature, the well-known Hagedorn that temperature the thermal partition metric Yang-Mills theory withstrings a gauge [ group SU( tensionless string theory include [ 1.1 Hagedorn physics and tensionless strings tensionless limit [ symmetry as we approach this very high energy sectorory of following string the theory. discoverytheories of [ holographic connections [ were infinitely many relations between string amplitudes of different string states in the JHEP02(2018)065 (1.5) (1.6) (1.7) (1.3) (1.4) is the 0 1 πα 4 = T , . dimensional Minkowski ) ) ] becomes n n 22 µν ), the tensionless action is D g − − ν 3 3 1.3 n n X . ( ( β µ 0 0 is the metric on the spacetime in ∂ X µ m, m, β , µν + + ∂ X g β n n µ α δ δ V ∂ X ¯ c c α 12 12 α . αβ V ∂ + + β αβ ), is invariant under worldsheet diffeomor- η h h → V m m α − + + = 1.3 αβ n n – 3 – √ ¯ L L . The most convenient choice of gauge for the αβ ) ) σ V h h h 2 µν m m d η − dσdτ − − √ Z ]. In terms of the action ( = , n n T Z 22 = 2 µν T V g − S ] = ( ] = ( ] = 0 m m m = , which satisfies: ¯ ¯ L L L n is the worldsheet metric and , , , P ¯ L n n n S ¯ αβ L L L [ [ [ h and n L is a vector density. It is clear that this replacement indicates that the metric α V degenerates in this limit. The Riemannian structure of the worldsheet thus undergoes The tensionless string action is obtained from the tensile theory by going to the Hamil- αβ a drastic transformation. The tensionless bosonic string action [ The above action, likephisms, and the hence tensile there action is ( gauge freedom and a need to fix gauge. In the analogue of the obtained by replacing where h This is, of course, thewhy symmetry string algebra theory of has a beenof 2d successfully conformal 2d formulated field CFT is theory can because (CFT). be the The used power reason extensively of to the quantisetonian techniques relativistic framework strings. of themultiplicative constant theory up front where [ in the action, the tension does not appear as a gauge does not completelyresidual fix symmetry the algebra gauge with redundancy whichspectrum. one of acts the The on residual system the gauge and Hilbert symmetrywith space we turns generators to out are extract to left the be with physical two copies a of the Virasoro algebra, quantisation of relativistic strings is the conformal gauge where If the choiceleft of with gauge the had simple completely job fixed of the quantising free gauge (worldsheet) symmetry, scalars. we would However, the have conformal been has gauge symmetries for which we needtension to of fix the a string, definite gauge.which In the the above, string moves. Wespacetime will from take the now string on. to move on So, flat to make substantial headway into this very essential1.2 problem. Tensionless worldsheet symmetries Consider tensile free bosonic closed strings. The Polyakov action structure of string theory. Armed with the new symmetry tools in our analysis, we hope JHEP02(2018)065 ]. 1) 38 and ], as − (1.8) (1.9) – µ 23 τ D 35 ], and also orthogonal 2 has ( µ 25 τ , µν 24 γ . The geometry of µ τ ]. , ) , 34 ) n n − − 3 3 n ( n 0 ( to infinity and hence the name. 0 m, c m, + and a time one-form n + δ n δ µν M 12 L γ c , 12 c 0) + + v, m m + + n n = ( – 4 – L M α ) ) V m m − − . n n ]. 33 dimensional Galilean spacetime, the metric ] = ( ] = ( ] = 0 m m m ]–[ D 26 ,L ,M ,M 0 limit has been christened the Carrollian limit after the mathematician Charles n n n L [ L → [ M [ c It turns out that in two spacetime dimensions, the non-relativistic and 1 = 0. In a ]. ) again has some residual gauge freedom which is given by the algebra µ τ 27 1.7 µν γ 0) [ ), where we systematically take the speed of light → 1) dimensional flat spatial fibres. 1.5 c When the speed of light is taken to zero instead of infinity, like in the limit from AdS Singular spacetime limits, like the non-relativistic limit, are characterised by the degen- , i.e. The rather peculiar For more details on this rather confusing statement, please have a look at [ − γ 1 2 D to flatspace, the spacetime manifoldA thus Carrollian obtained manifold is has called a a Carrollian structure manifold very [ similarDodgson, to who wrote a under Newton-Cartan“Alice’s the Adventures manifold, pseudonym in Lewis where Wonderland”. Carroll, and his astoundingly imaginative fantasy novel positive eigenvalues and one zeroat eigenvalue and the denotes spatial the slices non-dynamicalthe that spatial Newton-Cartan metric are manifold orthogonal is to( the that time of axis a defined fibre by bundle with a 1d base defined by eration of the metric descriptionno of longer the underlying remains spacetime. Riemannian due Herethe the to manifold. spacetime the manifold lack The of Galileanture a spacetime that non-degenerate consists manifold metric of is on ato endowed the contravariant with whole rank a of 2 Newton-Cartan tensor struc- infinite radius limit induces(i.e. a limit on theultra-relativistic field limit theory yield which thebecause is same the an symmetry spacetime algebra ultra-relativistic now from limit has the one parent contracted 2d and CFT. one uncontracted This direction. is 1.3 Geometry ofIn the tensionless the worldsheet context ofobtained as flat infinite holography, radius theobtainable limits from central of a AdS idea systematic spacetimes, is limit hence of the flat the holography following: well should established also flat AdS/CFT be spacetimes correspondence. are The cations. This appearsbra as ( a non-relativistic contractionThe GCA of has the been discussed relativistic inemergent the conformal symmetries context alge- of in the Galilean non-relativistic limit gaugeintriguingly of theories as AdS/CFT in [ the the symmetry conformaltotically regime algebra flat [ governing spacetimes putative [ holographic duals of 3d asymp- This algebra is the 2d Galilean Conformal Algebra and has found numerous recent appli- the action ( conformal gauge, which in this case becomes JHEP02(2018)065 , r . + is the (1.10) (1.11) 0 n I s, . H +  r 0  δ r s,  + − r 1 4 δ 2 n ] −   2 1 4 39 r −  ] = 2 r M 3 r c ]. Clearly, SGCA  ] was what we called the ,G n + 55 M 6 ). 42 s c ]. M + [ . This was given by 1.9 r + 45 H s M , + r r . + , 0 = 2 n 0 , M 0  } H m, → ), is given by s m, + ] is to consider tensionless superstrings.  , H αβ n r + 0 δ ,H , δ 40 n r ) 0 δ − m, ) n = ετ, ε ], and in the next sections, we briefly review G m, + 2 n n { } n + ) and also has interestingly appeared in the − 41 δ β s →  n − ) 3 – 5 – δ , 1.9 ) 3 n n ,Q 0 ( n n ] = α r s, ( − r σ, τ + M − Q 3 12 L r c { 3 n 12 c δ ,H → ( n n  + ( , where the fermionic generators were scaled differently + σ , M L 1 4 I 12 L [ r m c m 12 c + ] that there was another contraction of the two copies of + − + α n n + , n 2 + 42 r Q r ]. In two dimensions, Newton-Cartan and Carrollian manifolds L M m
+ m ) )  + r n 35 + n L m m n 3 G c − M L − −  ) ) 2 n + r ], following the lead of [ n n m m s  . The SGCA − I 42 + − − r 2 n L n n ] = ( ] = ( ] =  α r m m = 2 ,Q ,L ] = ( ] = ] = ( ,M n } n r n s ]. So, why another attempt at same problem? m m L L [ [ L [ ,G ,L ,G 44 ,M n , r n n We also mentioned in [ The symmetry algebra at the core of the construction of [ Let us return to the discussion of the tensionless string. It has been shown that the L L G [ 43 [ L { [ In the above, the suppressedin commutators the are context again of zero. supersymmetric extension This of algebra the was first 2d obtained GCA in [ generalisation of thestudy bosonic of 2d asymptotic symmetries GCA of ( flat space supergravity [ the Super Virasoro algebra whichAlgebra we or called SGCA the Inhomogeneous(as Super opposed Galilean to Conformal the homogeneous scaling of SGCA Here the suppressed commutators between the generators are zero. This is a natural is [ Homogeneous Super Galilean Conformal Algebra or SGCA 1.4 Outlook andA outline: very tensionless natural generalisation superstrings ofThis the was analysis done of in [ [ our previous construction. A non-exhaustive list of other earlier work in this direction This physically represents thezero. string In becoming terms long of andgoes the to floppy worldsheet, zero. this as The is its structure thusmanifold. tension which the The emerges is limit on algebra where taken the that the worldsheet to dictates is worldsheet the thus speed that theory of of is light a given 2d by Carrollian ( other in the above sensebecome [ identical to each other as the dimension oftensionless the limit base on and the fibre worldsheet become is the obtained same. by the following scaling [ the base and fibre of the fibre-bundle are interchanged. These structures are dual to each JHEP02(2018)065 (2.6) (2.2) (2.3) (2.4) (2.5) (2.1) . , we first discuss and switching to  4 ) ρ − ψ + ∂ . ·  µ − ψ ψ a ∂ . + a , we revisit the theory of tensile ρ ! + . i µ 0 2 ψ ¯ ψ i − 0 i = 0 , we review our earlier work on the . ∂ by means of systematic worldsheet

· − ∓ ab . 3 4 η ψ = µ + ! 2  ψ 1 X µ µ ( − + − a i ψ ψ ∂ = µ +

, ∂ , ρ . We end with a section on discussions and } – 6 – X b X = ! 5 a i = 0 − ∂ µ , ρ 0 ∂ −  a ψ · and section X ρ σ ) takes the form i 0 2 { + 3 X d

∂ 2.1 + − = ∂ Z ∂ , which we discuss in detail later in our paper. 2 0 2  T H ρ σ − 2 d = Z S 2 T − = S are the spacetime coordinates of the string, which are scalars on the worldsheet, are their fermionic counterpart, represented by two component Majorana-Weyl µ are the worldsheet gamma matrices satisfying the two dimensional Clifford Algebra µ a X ψ ρ The following is an outline of the paper. In section This gives rise tootherwise we the will equations be of suppressing the motion Lorentz for indices the for simplicity. fields. Henceforth, unless mentioned This corresponds to athe fields Majorana respecting representation usual wherelight-cone Clifford derivatives, the the algebra. fermionic action After ( fields putting are in real forms with of An appropriate representation of these matrices is given by: spinors denoted by: The Here and future directions. 2 Quick look atWe tensile start superstrings with the Ramond-Neveu-Schwarz (RNS) action for a supersymmetric string to us in ourhomogeneous limit tensionless adding a constructions. few newwhy ingredients In along it the section is way. In desirable section inhomogeneous tensionless to superstring work in with an intrinsicWe the way starting recover inhomogeneous from our an algebra action answers andlimits principle. in go from on section the to tensile construct theory the in section superstring theory admits thiswe larger answer algebra this as questionsymmetry its algebra in worldsheet over the SGCA symmetry. affirmative. In There this paper aresuperstrings other in reasons order to to prefer set up this notation as and a highlight a few aspects that would be useful richer of the two algebras and an outstanding question is whether a tensionless limit of JHEP02(2018)065 )  ξ can 2.7 (2.8) (2.9) (2.11) (2.12) (2.7a) (2.10) Y (2.7b) (2.13a) 3 (2.13b) ). 0 iρ ) and super- ξ = X. . , C + ) + ∂   ψ + ψ σ   + (  2 + ξ ψ ) ξ −  . − + ¯ θθB ∂ − ∂ = ∂ . 1 2 1 2 1 2 − + ¯ ), the supersymmetry trans- . A general superfield θψ X. + a δ a δσ + = 0 ) + σ  ∂ 2.4  + − − a + yields ψ  ∂ σ a − iρ ( + . + − ∂ Y ψ ∂ − ∂ ξ a − X, δψ ) − ξ + ψ − − + ¯ = θψ ∂ ∂  ) back into the variation of the action, i = = δσ DYDY. + − + ( − −  matrices (  ∂  ψ ξ σ 2.8 2 iθ  − ) + + 2 ψ δσ ρ − ). Explicitly the following transformations ξ ξ − is also a two-component spinor. The spinor   ∂ + θd +  − σ 2 1  2 = ( ψ X , ∂ + d iθ – 7 – + ) ξ + ∂ − . − + + ¯ Z θδ − A ∂ = 0 i iθψ ∂ 2 − T X δσ − − ) + ( ξ ξ ¯ − , δψ θ δσ − ) + i + +
∂ +  ∂ ψ = + − − + ξ σ Now we will see how to realise these transformations ( + ξ ψ ( S + = ∂ + ∂ − iθδ + X + +  + ξ ξ + + X, δ δθψ +  − ∂ δσ a ¯ θ ∂ + ) = ∂ = 0 separately, gives rise to the following set of conditions on + X + i ¯ θ a ξ ξ ψ δ + ξ ψ, δ = ( S  + can be put to zero without loss of any generality if we consider on-shell  δ , θ, = = ¯ = = (  δY i B = σ X X Y (  ξ ξ 1) superspace associated to the superstring. We consider Grassmanian = ) along with the 2D worldsheet coordinates S δ δ δ ) is invariant under local diffeomorphism (parameterised by , ¯ ξ Y θ δ θ, 2.1 δX leave the action invariant: = (1 ) can be raised and lowered by the charge conjugation matrix ( ψ  N : and   The supersymmetry transformations particularly depend on the representation of the For details, please have a look at appendix 3 X The variation under diffeomorphisms on the superfield Integrating out the Grassmannianfor coordinates, the one superstring sans could the arrive auxiliarywork field at for term. the the To superstring usual see variables, how RNS the let action coordinate us transformations start by looking at an arbitrary variation: The auxiliary field supersymmetry. The total action can the be constructed as a superspace integral Realisation in superspace. in the coordinates ( then be constructed as, Since these transformations are local,and requiring plugging ( and The infinitesimal suspersymmetry paramter indices ( Dirac matrices. Using the representationformations of can the be written in the following form symmetry transformations (parameterised by of The action ( JHEP02(2018)065  is ξ Y ) and (2.20) (2.19) (2.14) (2.15) (2.16)  (2.18c) (2.18a) (2.17a) (2.18b) (2.17b) σ ( :  ξ Y Y  − .  ¯ θ∂ − + i . ψ ) − + iθ∂  . − , , ¯ θ  r r − ∂ + ( , . = ¯ ξ i . Q Q θ  + − ¯ θ −  ) − r + r ∂ ξ ξ + b b − + − ¯  ). It translates to θ∂ irσ + ) σ 1 2 e r r ( ¯ , δ θ∂ θ∂ ψ + X X 2.9  r ) − + 1 2 1 2 σ b =   + (  ¯ θ + + + ( r + ) = ) = =  i X + − − + Y.      + + ¯ θ Y )] ( ( = , δ ) and carefully imposing restrictions on ¯ ¯ − = =  θ + ¯ Q Q θ∂ ’s are the fermionic generators. Since the X   θ ξ ) ¯ − ( θ Y − ¯ θ∂ Q ¯ θ, δθ + − ¯ ψ ∂ θ∂ δ Q , ,  ) − ) σ as given by ( n n ,  ¯ ( θ∂ θ + – 8 – + ¯ i ¯ θδ   L L 1 2 − − i ¯ σ ) + θ, δθ θ, δ θ ξ − n + n ( = i − + − inσ + a a − = + ξ e ∂ − i i and ξ δθ∂ ( +  n + ( n n ¯ 1 2 + L ξ ψ + + a X X + ∂  i i X + ) realised on the superspace coordinates take the follow- 1 2 + − − n , δθ + − − ξ ξ ) + ∂ iθδ − X  ∂ + 2.7 θ, δσ + ∂ − ) ξ  ) = = + + = θ + ( − ) = ) = δσ ∂ = + i + −  Q X ) σ + −  i ξ ( +  ξ ξ + = δ δσ δσ ( ( − ) we see that the effect of the diffeomorphism transformations on σ + ¯ ξ ) + δσ ( + L L = = ( ∂ +  + + ξ δσ ξ 2.13 + Y (  δσ L δ  is equivalent to change of superspace coordinates = ( = = [ ) to ( Y δY 2.12 ’s are the bosonic generators and the , we get, ). Writing out the variation on the field L    σ (  We can define the Fourier modes of the generators in the following way: Here diffeomorphism and supersymmetry parameterswe can are write functions down of a Fourier worldsheet expansion coordinates, of the form  and Constructing the symmetry algebra Now we would liketo to keep find in the mind generator the of restrictions these on transformations. However we also need Together the transformations ( ing form: We see again thatequivalent to the change of effect superspace of coordinates: supersymmetry transformations on the superfield the superfield Furthermore, imposing the supersymmetry transformations on the superfield Equating ( JHEP02(2018)065 (2.27) (2.24) (2.25) (2.21) (2.22) (2.26a) (2.28a) (2.23a) (2.28b) (2.26b) (2.23b) . . s ) , . + σ ) r i 0 − ) L τ σ σ ( . . , − ir ) + ) − τ ( = 2 − J − + σ e , in } ( µ r ¯ s n θ∂ irσ − δ b ˜ i 0 iθ∂ e + Q 1 2 ) leads to the equations r µ n , + b αα + + r α δ · ¯ dσ e θ θ Z , 2.5 X r ∂ ∈ ∂ π + ˜ µν ( r {Q ( , 2 − ) η 0 0 b − + σ = 0 ) µν α Z + = , η 2 n . 0 irσ  irσ τ r n ( } − − ψ √ + µν ) + 1 + e e 0 in πα n  η r m 2 σ − s ∂ Q ( = = e · + 0 ) = (2 µ n r  r r = ν α mδ δ  r α ¯ r r Q Q h ψ X σ, τ , ψ = − ( Q , ) i 2 1 n ] = 4 1 µ σ − } ν n 2 n ( , , ν s + ˜ α + , 6=0 µ α = 0 b  ˜  ,  2 X n , – 9 – ¯ n ψ θ θ ) 0 µ r 0 µ m ++ { , ψ X + . b α ˜ ) ¯ α ] = θ∂ θ∂ r T  X { m 2 σ r + ∂ 2 2 α + , in √ in Q ·  = inσ ) m τ · i 0 , ( b ] = [˜ ˙ }  n σ + + · ir m X ν n + ν s ψ L − − − + [ − τ m dσ e e , b , α α ∂ ∂ µ 0 − = ( = µ r π µ r σ µ m 2 b   b α ( α ). These are: , 0 0 m α {  δ 1 2 − + [ X α Z m J  ). We now construct the mode expansions in the NS-NS sector + m 2 2.6 0 µν + 1 2 X T Z inσ inσ η n X √ ∈ − − 2.6 1 r L πα = = We have seen that varying the action ( 0 ) ie ie 2 r α + 2 n m )] = 2 0 = = L Q µ = σ − x √ n n ( n ¯ ν n L L L ˙ X ) = ) = , ) ] = ( σ ( τ, σ σ, τ m ( ( µ L µ µ + , X [ n ψ X L [ In terms of modes, the Super-Virasoro constraints are given by which are directly related to the constraints by the relations We can write down the components of the energy momentum tensor and the super current These lead us to the following brackets: The commutation relations are given by Mode expansions. of motion for the fieldsthat ( are compatible with ( sions above can be usedof to the determine Super-Virasoro that algebra: the symmetry algebra is nothing but two copies These are the generators of the residual symmetry for the tensile superstring. The expres- which leads to the following form of the generators: JHEP02(2018)065 ] 41 (3.6) (3.2) (3.3) (3.5) (3.1) (2.29) (3.4a) (3.4b) is fixed, to satisfy a = 0. This . a V 0 ρ S s,  + δ r δ ˙ . Once  ψ, I X. ) 1 4 a + a  ξ V − a − . = 0 and 2 ∂ = r ( i = a S 1 4  ψ ξ . X. ρ a + δ c 3 a + ∂ ∂ a a = 0 + ψ ρ a , · s iρ  . ) ∂ ] +  ¯ a ψ n ˙ r 0 − ˙ i ψ ξ 0). The Clifford algebra needs to be X, δψ L . · − , b − + = = 3  V iψ , ∂ n = 2 a ψ ψ X − (  b ξ = (1 } 0 + V ∂ s = = 0 2 a · m, ˙ Q 1 − V + X , : = 2 ) is then written as ξ [ X n r ). As mentioned in the introduction, in the 0  a δ σ becomes degenerate and is replaced by a product }  – 10 – ∂ 2 b 2.1 c b 2.1 {Q d 12 ab , ρ , δψ V η , ∂ ) a a Z + 1 and , ρ + ξ r V { 1 1 m ψ , = h + 0 ∂ + + n ) is simply reducing them to ξ σ n  S 2 Q = L d 3.2 + ]. A convenient choice of the gamma matrices ) 0  ξ r − m X, δ 42 Z 0 a ψ . The action ( ∂ − − ∂ = − O a 2  n n ψ, δ ξ ( S i  = = = ¯ 1 = ρ ] = ( ] = matrices, the supersymmetry transformation has the explicit structure X X r m  ξ ρ is a vector density. A convenient choice of gauge, which is analogous to δ δ δX Q L and , , a n n I V L L [ [ = ) the Super-Virasoro algebra is generated: 0 ρ 2.25 where b V a Now as done ingives the us case analogous conditions of on tensile string, we would require With the chosen as follows we have This action is invariant under the following restricted diffeomorphism and supersymmetry: We start with the so-called homogeneousdiscussed tensionless superstring, in which some has previously detail been the in modified [ Clifford Algebra ( tensionless limit, the worldsheet metric V the conformal gauge in theredefined tensile in theory, this is case: can now write down the supersymmetrised action for a fundamental tensionless string [ This will be the starting pointanalysis, of the inspired supersymmetric version by of our the fundamental tensionless RNS action ( quantise the theory. 3 Homogenous tensionless superstrings In this section we would discuss the residual symmetries of the tensionless superstring. We Here the central terms will arise from the proper normal ordering of the modes, when we Using ( JHEP02(2018)065 , ψ (3.7) (3.9) , that (3.8c) (3.10) (3.8a) (3.8b) (3.8d) and (3.11a)  (3.11b)  X , . and n  + τ . m , ¯ ¯ θ∂ f, g Q s M i 2 + ) r Q, n + in terms of M , ¯ θ − 0 τ . ¯ θ, ∂ ) ∂ Y αα − m  σ δ ( i Y, inσ are corresponding fermionic  irσ = )] −  Y. + − ¯ ) ] = ( Q − } 2 denotes ie e ¯ θ θ to get 0 n  = , ( α s + ¯ Q, = = θ∂ ¯   Q i δ  ,M ) in terms of generators as r n ,Q = 1 ¯ m α r Q + + ,  α 3.9 ) + L ) θ g Q [ and + σ {  ( ¯ θ, θ. + a ( 0 0 f δθ∂ ξ τ f f with Q 0 1 2 1 2 + = ,M f α 1 1 + + Q ) + ) on  g ) = + − – 11 – ( ¯ θ δσ∂ ¯   , ξ θ 3.7 ) − ¯ M θ∂ , + σ = =  r ( 0 2 , + a a + g + n ξ ξ ) + θ α m a a + θ + we could expand ( f δτ∂ Q ( θ∂ m + ¯ ¯ τ θ∂ θ∂ θ ] = 0 (3.12)  L  ) L ( 1 4 1 4 α r f, ) r + i σ = ( ( n τ 0 = [ + + − = , + ,Q and f − 0 1  + − τ∂ δY m 2 ξ ξ   m τ =  δY m M  0 θ∂ = = = = in ξ i 2 τ, σ, θ ¯ θ + ] = ( ] = ] = [ δθ δ δτ δσ + n n σ α r θ ∂ ∂  ,L are bosonic generators of the algebra and ,Q ,M  m ). We note that in the above m m inσ L M L [ irσ [ − M [ − 1.10 ie e ( and = = H L r n L Q 4 Inhomogenous tensionless superstrings In this section we willthe move theory on of to inhomogeneous the first superstringsless main from theory. result the of viewpoint this of paper, the that fundamental is tension- understanding which is theSGCA classical part of the homogeneous Super Galilean Conformal Algebra or These generators satisfy the following algebra: where generators. We can find afterthe similar generators Fourier can expansions be of the written functions in the following differential form, Given the changes in Following the same algorithm asthe before superfield in Y tensile and case, impose we conditions write ( out the transformation of and study the variation in superspace. It leads to the following superspace translations: phisms and supersymmetry transformations in general, We can see how toas realise we this did as for a the transformation tensile on case. the superspace We then by write similar methods down a superfield The solutions of the above equations gives us the allowed function space for the diffeomor- JHEP02(2018)065 ) 3.2 (4.1) (4.2) ). A generic form ]. ) one could generate 4.1 40 4.2 , . ) is the trivial solution of ! is the algebra which is the O 0 4.1 I 0 0 a = 0

ρ 1 ). So, it is definitely of interest to ρ or 0). With this, the modified Clifford + , 1.11 1 ! ρ a 0 ), ( = (1 0 0 0 a

, ρ 1.10 V O = 1 – 12 – = ]. The inhomogeneous tensionless superstring is 2 42 ) 1 , ρ ρ ( ! 1 , I − 1 0 0 = . The most obvious solution of (

2 ). However, due to this conjugacy structure the Dirac action ) that the algebra does not change form under a similarity ) S 0 a = ρ 4.1 ρ 4.1 ( 0 1 ρ − S → , which we have used for the homogeneous tensionless case. Now looking a O ρ = 1 ) as a symmetry algebra. We shall, however, comment on a rather interesting ) reduces to the matrix equations: . These matrices are disconnected from the trivial solution by any means of ρ R 3.2 1.11 ∈ and a I Apart from the above, and most importantly, as we will elaborate immediately, the = 0 with similarity transformations. Doingnew similarity sets transformations of on solutions ( to ( conjugacy class of solutionsthem by back itself. into the Taking abovethere some equation, exists general we a can form second easily ofof see unique matrices these that conjugacy and matrices apart class have putting from of the this curious solutions trivial structure satisfying solution, ( One could verifytransformation from ( ρ at this trivial solution, one can infer that this particular set of matrices is an isolated the degeneration of the metricon the on worldsheet. the As worldsheet. weone discussed This above, is modifies we the the need to analogue Clifford choosealgebra algebra of a ( ( gauge the and conformal a gauge very convenient inhomogeneous tensionless superstring issuperstring the theories, while general the solution homogeneous in one the is4.2 a space special of case. tensionless Representations ofAs the we modified have Clifford repeatedly algebra stressed, the tensionless regime of string theory is characterised by upcoming work. But thisrealise remains ( one ofdistinction our between principal the motivations homogeneous behind andwe attempting inhomogeneous investigate to some cases old later claims instrings the in in the section the tensionless where literature theory about building the on equivalence our of bosonic closed analysis and in open [ one attempts to look atevident the from quantum our regime of previousvery the analysis different tensionless in in superstring. [ this Some regardIn of and this this the is paper, fermions play weback an shall to active mostly the role quantum be in aspects, the concerned like tensionless about the regime. the structure of classical the analysis vacuum and of the would tensionless come theory, in We have already stressed inlarger the and introduction hence that richer oftry the the and SGCA two realise algebras this ( fermions as in symmetries the of theory the aremost somewhat tensionless uninteresting. of string. They the In scale analysis uniformly the turn and homogeneous hence out case, for to the the be essentially spectators. This is particularly true when 4.1 Why bother? JHEP02(2018)065 (4.6) (4.4) (4.3) (4.5c) (4.5a) (4.5b) . i ) 0 ψ, 0 = 1 without any loss of . 1 ψ 0 ρ a · 0 ψ ∗ 1 ρ ) , nor does a particular choice ) 0 I . ψ b ξ ξ 1 ! − a ∂ ∂ 1 ( ˙ b 0 0 1 0 a 1 2 ψ · E

( + ∗ , 1 . 1 4 0 1 = ψ ! ψ ψ 1 − + µ µ a a 0 1 ξ ξ 0 ψ ψ ψ ˙ a a ) ψ

∂ ∂ a , ρ · ξ 1 4 1 4 ∗ – 13 – 0 a = ! ∂ 1 ψ + + µ ) is invariant under the following set of restricted ( ( − 0 1 i ψ 1 4 , the homogeneous solution is just a point while the X, ψ ψ 1 4.4 1 0 0 + a a a ρ +

∂ ∂ ∂ 2 are taken to be complex spinors written as a a a ˙ is not contained in SGCA ψ the more general symmetry algebra in the solution space. X ξ ξ ξ a = ψ h I and ∂ H 0 a = = = σ 0 ρ ξ 2 ρ 0 1 d X ψ ψ = ξ ξ ξ δ Z δ δ ψ ξ = δ S ) written in terms of components now reads: is a more general choice than the trivial representation and relatedly the I 2.1 . Of course, the SGCA H It is important to stress what we mean when we say that the above representation and Notice here the unique formform of under transformations. a This combination happens of because diffeomorphism the and fermions Weyl trans- transformations as pletely different footing altogether.diffeomorphisms: ( The action ( This action has a noticeable asymmetry to it, putting the two complex spinors in com- where fermions are scaledthe in a spinors non-trivial have way, to wecomplex be have in no nature. real. a The priori spinors In reason fact, to believe we that will see in a while that the spinors are indeed superstrings, making the SGCA 4.3 Constructing inhomogeneousWe tensionless should superstrings note here thatand hence in have real the components. tensile But theory, in the the spinors case of are inhomogeneous usually limit Majorana of the spinors, superstring, of the inhomogeneous gammawe matrices mean lead is that to when the oneparameter homogeneous looks space at gamma labelled the matrices. by space ofinhomogeneous What all one tensionless constitutes superstring theories, a withthe family the inhomogeneous of solution solutions. than It the is homogeneous thus solution more when likely considering to tensionless land up in this more general representation and analysesymmetry what structure differences of this makes the for tensionless the superstring. underlying hence SGCA SGCA generality and consider the forms: This is the choicemore of general gamma than the matrices trivial on choice which the led to worldsheet the of homogeneous case. the tensionless We shall strings work with that is and hence, the algebra would not change. We shall choose JHEP02(2018)065 = 0. (4.9) (4.7) (4.8c) 1 (4.8a) (4.8b) (4.10c) (4.11a) (4.10a) ρ (4.11b) (4.10b) (4.10d) ) as the 2.9 in terms of matrices, we . Y 0 ρ X 1  − : ): ˙  X . 0 )  4.8 a σ − and . ( χ, ∗ 1 1 , ξ , = 0 , ψ ) ) ξ 1 σ 1 σ ). The superfield ( = 0 iχψ = ψ ( ∂ ¯ θ  e f 1 0 1 2 ξ θ, , = 0 ) + = ∗ + by solving ( a 0 1 1  σ α χ. ˙ a 1 ( ) ) X, δ  0 ∂ 1 1 1 , ξ ξ  ) 1 1 = , ξ ,  − α ) ∂ ∂ ) iαψ and 1 1 = 0. Considering local transformations, the σ σ   = a ( + + ( 1 , ψ S ξ g 0 h 0 0 ∂ +  ) + – 14 – ξ ξ δ a ψ + χ 0 0  + = = 0 σ , ∂ ) as a transformation on this superspace by similar χ, 0 ∂ ∂ ( τ 1 1  τ 0 1 ( ( 0 0 ( ξ   ) ) 4 4 1 1 X i i 4.7 1 0 0 , δ σ σ ∂ ∂ ∂ ) ( ( + + + + 1 f e 0 1 ) = 0 1 = = = ψ = 0 and   ξ ξ ∗ = = 0 ∗ ∗ 0 1 0  ) and ( S ξ = = = = 0 0   ξ 0  ξ 0 0 , α, χ δ + ∂ is now written down as a 4.5 ∂ ∂ δτ 0 δσ δχ δα σ ( ψ ψ ∗ Y 1  ( and i = X ). In the next section, from the limiting perspective we will see how they . In the case of the homogeneous limit this term is also absent as  X ξ  are also complex. It is thus convenient to work with complex superspace α, χ δ  is the antisymmetric Levi-Civita tensor. It is interesting to note that the last b a E We would like to obtain the generators of the residual symmetry transformations, sub- We can use the appropriate expressions of We can see how tomethods realise ( as we did before: case. Now we have complexformations spinors and hence the parameterscoordinates of ( the supersymmetry trans- are related to theworldsheet conventional fields choice of coordinates ( about which we wouldMajorana elaborate representation more seems not from toof the be the limiting the tensionless side right strings. one of in the the theory. case ofject Indeed, inhomogeneous to limit the the above conditions, as we did before in the tensile and the homogeneous tensionless The last equation gives us a hint about the conjugate structure of the spinor components, As usual we wouldimposition require of invariance gives us the following conditions on term in the expression playsconditions no on role in theThis tensile extra transformations term as is weclosure crucial have of for ( the the residual invariance symmetrycan of algebra. write the the Plugging action form in under of the supersymmetry the explicit transformations transformations forms here of and where JHEP02(2018)065 and , r s (4.16) (4.13) (4.14c) (4.15c) (4.12c) (4.15a) (4.12a) (4.14a) each of + (4.14d) (4.15b) (4.15d) (4.12b) (4.12d) (4.14b) r ,G : n L I inσ e ,M = 2 n Y. } )] L s h ( ,G , r H ,   G α , α { i ) + τ , , e i r ( τχ∂ , τ τχ∂ + 2 n G 2 00 χτ∂ n + 2 m 0 , f m ie H χ − 0 ) + 2 rχτ∂ +  M g g +  ) r ieα, ( − .  we Fourier expand in τ α n s + τ α − + M + σ τ∂ − r τ∂ τ∂ 0 2 τχ m e 2 due to these transformations and find + M 00 m ) + ihχ irτ∂ +  f f χ Y ( χ + ) + = 2 + L σ ) + χ∂ ] = ( ] = χ∂ } σ r τχ n s α 2 0 + 0 χ∂ 2 = [ 2 + , χ∂ f ie α ,G ,  . ,H ,M + α – 15 – Y  α r i m + ) . + . + τ m α∂ τ α ) are ) when we quantize the theory. χ i G α∂ τ χ g L M h χ∂ 0 τ { [ [  2 are functions of α∂ χ∂  ieχ, f ( 0 2 + α∂ 0 + in in δχ∂ h 2 ( 4.11 1.11 f g ihχ∂ ie iχ∂ i τ + τ + 0 0 + + + + + f f e e + + + + σ τ α σ and α τ , , χ χ α ∂ ∂ = = = = r r ∂ ∂   from ( , f∂ g∂ + + e∂ h∂ h h n δα∂    h h m m δτ δσ δχ δα inσ inσ + irσ irσ + G H m f, g, e ie ie e e σ   L ) = ) = ) = ) = and ) r r = = = = e g h f ( ξ n ( ( ( r r − − δσ∂ n n G L − H M L G H M + 2 2 m m m τ   are Grassmann valued functions. The variations of the coordinates under h δτ∂ ] = ( ] = ] = r r n = ( and ,L ,G ,H e m m m δY L L L [ [ [ . This leads to the following vector fields: r All other suppressed (anti)commutators are zero.will Note turn that into this its is quantum a classical version algebra ( which The above generators satisfy the algebra, which we have been calling the SGCA Since all the parameters them as we did previouslyH and obtain the Fourier modes of the generators the corresponding generators Collecting the terms, we can see that the generators have the forms, Now we can find the variation of the superfield the residual symmetry transformation thatappropriate preserves forms the of invariance of the action, with the where JHEP02(2018)065 (4.17) ), on the 1.11 ( I ] was that we could pro- 40 . to a Virasoro algebra. µ 0 0 H ψ ) as its symmetry algebra, does not = µ 1 1.10 ˙ ψ ( H We have been looking at closed strings in , = 0 ]. The residual symmetries for the tensionless – 16 – µ 0 48 ˙ ψ , are: = 0 ψ µ ¨ = 0 thus reduces to a Virasoro algebra, which is also the sym- X and that indeed the truncation works in the expected way for the M ) and used the fact that when we are looking at theories where X c B 1.9 ]. We focused on the residual symmetry algebra of the tensionless bosonic string, to a single copy of the Virasoro algebra. This would be very counter-intuitive and 51 ) wirth respect to . We also find, rather satisfyingly, that through an analysis of null vectors it can I H 4.4 So, we see that much like in the case of the bosonic tensionless theory, in the inhomo- We show in appendix It is natural to expect that such an analysis would be readily generalisable to super- One of the interesting features of our bosonic analysis in [ = 0, the 2d GCA truncates to a single copy of the Virasoro algebra. This was shown M and in this subsection,subject we to will closed discuss stringing boundary the ( conditions. mode expansions The equations of of the motion equations obtained of by motion, vary- we are more interested in the inhomogeneous tensionless limit. 4.4 Mode expansions We now return to our analysis of the closed fundamental inhomogeneous tensionless string geneous superstring, the residualresidual symmetry symmetry on of the open worldsheet stringssimilar when of analysis we closed for take strings the theto go homogeneous tension come to to superstring back zero. the to is this The somewhat in obstruction more the to curious near a and future. we But hope this feature highlights one of the reasons why the bosonic one statedSuper-Virasoro algebra. above would lead to a truncationSGCA of the algebra downbe shown to that just there a is no consistent truncation of SGCA SGCA would somehow mean thatsuperstring by theory, looking we at somehow thepreserved. are tensionless To lead sector put of to thingsother a mildly, a hand, this perfectly sector does would fine have where be a tensile supersymmetry very Super-Virasoro peculiar. is sub-algebra. no The And longer it SGCA is expected that a sector like symmetric extensions of thethe 2d tensionless GCA closed and superstrings. wehomogeneous Here would tensionless we be superstring are able with faced SGCA to withhave make a a analogous rather Super-Virasoro peculiar claims subalgebra. problem. for Naively, it The seems a truncation would actually reduce the which is the 2dc GCA ( using an analysis ofclosed null bosonic vectors string with first inmetry [ governing an openclosed string strings worldsheet, and open suggesting strings. a connection between tensionless the tensionless limit andpoint will out some be potentially continuing interesting to connections do to open so. stringvide theory. But more here, hints as to astrings the [ brief claim that interlude, in we the tensionless limit, closed strings behave like open Remarks: connections to open strings. JHEP02(2018)065 (4.19) (4.20c) (4.21c) (4.18c) (4.20a) (4.21a) (4.18a) ). This (4.20b) (4.20d) (4.21b) (4.21d) (4.18b) 4.16 ,  , n , , + r = 0 β = 0 inσ · − inσ r e irσ ) . − − µ − n e γ µν e i , i η n . + r s n , + inτB r = 0 ) which gives us ( n + = 0 δ r + − γ r irτH inτM inσ · β µ n irσ 2.24 which allows consistency of both = 2 · − r − − − A e 1 } r − ( r e n n ν r s − β − 1 n L G β  h M h H , β ) ) . , 6=0 µ r ) n n r r n n X n r γ irσ 0 X X X X { + + + c 0 0 0 0 − c c c c m 2 r r e ] γ , √ µ (2 (2 r · i = 4 = 4 = 4 = 4 µν r r – 17 – , m + η X X 0 i ˙ ˙ − ) we get the non-zero commutations of the modes: n 0 1 X X irτβ τ ψ irσ 1 4 1 4 B + · · µ 0 ψ · − − m · 1 0 . e 0 0 + + B + 2.24 0 ) µ r 0 ψ ψ µ r ψ c r r n n γ β ψ mδ i 2 [ 2 + + + + + 0 m m m m √ + r r + β β X X 1 B B X 2 + · · 0 0 ] = 2 · · · ψ ˙ c c ν n µ · X 0 m m m m √ x √ 0 0 ψ − − − − ,B ψ A B A B µ m h ( ( i ) = ) = ) = 4 A [ m m m m algebra as we found from the symmetry analysis. X X X X + σ, τ σ, τ τ, σ 0 . Writing them down explicitly gives us I 1 2 1 2 1 2 1 2 ( ( ( σ µ µ µ 0 1 X is a constant of dimensions of (mass) = = = = · ψ ψ 0 X c ˙ r r n n X and L G H M τ The constraints can be worked out by finding the components of the energy-momentum The classical algebra of thetation modes and of anti-commutation the relations constraints of canis the be the oscillators obtained same by ( SGCA using the commu- Using these and the modefor expansions the for generators: the fields, we can have the following expressions tensor and the supercurrent fromrespect the to action, then taking the appropriate derivatives with We note here that the commutationoscillator relations of basis. the oscillators This areof would not in our be a present an simple construction harmonic importantleave and that fact attempt analysis when to for we understand our look the upcoming underlying at work. Hilbert the space. quantum version We In the above, sides of the equations. Considering ( The solutions in the NS-NS sector are of the following form JHEP02(2018)065 . ) ) 0 εξ 3.5 (5.2) (5.3) (5.4) (5.1) 3.11 → ( 0 ξ ) on the H 5.1 denotes the homogenous : t ψ SGCA ) also scales in the ¯ θ θ, . We have to ensure that our ε ), consistently obtaining ( ). It turns that to preserve the . t ¯ θ ε 2.17 2.10 1 σ √ , ,t = → .  ) to ( h  t σ ¯ θ  2.8 ε ε ψ 1 √ √ and and . In this method we can make a comparison with = = – 18 – ) and obtain the generators for t /ε  h ,h ετ 0 θ  c  ε 2.21 1 ψ → √ → ), which is the action for the Homogeneous tensionless ’s can be scaled in two different ways: the τ 0 = ψ α 3.3 h θ denotes the homogeneous scaling and the subscript h scaling. As the name suggests, in the homogeneous scaling we treat ) gives us the ( 2.5 ) does not blow up while taking this limit. It turns out that in order to do this, We can also apply this scaling to the generators of the residual symmetry trans- : ), we scale the supersymmetry parameters as should also transform as is a small dimensionless parameter. We are looking at the tensionless limit and  4 2.5 0  ε ). 2.8 α inhomogeneous 3.8 We also need to consider that the time component of the diffeomorphism parameter scales as 4 We can apply thisto scaling ( all theformation way of from the ( tensile superstring ( We can also take thesuperspace limit of under the such superfield a asway scaling, given as we in have ( to scale the superspace coordinates in the same tensile theory. This alongtensile with action the ( contraction ofsuperstring. the worldsheet To obtain coordinates a ( meaningfultions contraction ( of the tensile supersymmetry transforma- Let us mention at first about the homogeneous contraction of the fermions where the subscript can be taken atthe each superfield level, all starting the from wayconsistent the down with action, to the the the analysis constraint supersymmetry done algebra, transformations, in thus the showing above5.1 how sections. this scaling is Homogeneous scaling action ( the fermionic degrees ofand freedom both the fermions in thescaling same of way while the in fermions the also lattersame implies we manner that scale inorder the the to superspace fermions have differently. coordinates a The ( meaningful superfield expansion. This specific contraction where hence the quantities from the fundamental side and relate them with In this section we studytensile the string limiting and approach then wherethe we take worldsheet start speed an from of ultra-relativisitc theintroduction, light contraction theory this effectively on of entails to taking a the zero. particular closed worldsheet, limits As i.e. on we take the have worldsheet mentioned coordinates, previously namely in the 5 Tensionless Limits of the RNS Superstring JHEP02(2018)065 (5.7) (5.8) (5.9) (5.5) (5.6) (5.11) (5.10) X. 0 ∂ algebra. A more 1 ,  ) H n − . − ¯ L X )] . 0 0 0 r ∂ + ψ − ). Once again we take the 0 · ¯ n  . Q 0 ]. L ε 4.8 − . − ( ψ ε √ , − = iψ 47 ,t 2 − , . i 1 = = 1 2 ¯ θ 0 − √ ˙ ψ i r n + 2 ψ √ . 46 , + , ¯ 0 · 1 t Q + = √ +   ψ 1  42 θ ,i ψ = 1 = = = X, δψ = ψ ∗ 1 i + 1 0 1 1 1 ∂ ˙ 1 ψ  ,  · and , χ ), this scaling on the tensile action gives us the ,  ,  , ψ 0 t − 0 1 – 19 –  ¯ θ − −  i ψ ε 1 ,t 2 5.1 = ( i 0 i iψ 2 = − 0 2 √ = − ∗ √ εψ θ + + √ 0 i 0  + 2  + =  = ˙ ,M ψ X ,i , δψ [ n α 0 = ) σ − 1 = ψ 2 0 ¯ , L ψ  d 0 r 1 ψ  − Q denotes the inhomogeneous case. Here ) to find out the limiting transformations. Needless to say the Z ε n i + L √ 2.8 = 0 ψ = = S 0 ) provided we use the last equation of (  r n ( i L Q 4.4 = δX ) provided there is a complex conjugate relationship between the components of the 4.7 This strengthens our idea offollows these that new we can conjugate perform spinorscomplex a which superspace similar we scaling coordinates have on noticed in the before. the superspace following It coordinates. way: We define the One notices here that thein form ( of the abovesupersymmetry supersymmetry parameter, transformation i.e. is the same as where The limiting form of the supersymmetry for inhomogeneous contraction has the form proper contraction on ( structure of the diffeomorphismfollowing form remain exactly the same. These contractions have the the course of this work.form of Together with the ( action for the Inhomogeneous tensionless superstring: This is same as ( where now the subscript This ansatz for the scaling is natural given the hints and conditions we picked up through 5.2 Inhomogeneous scaling As we discussed previously,would scale in differently. the We inhomogeneous do this limit, in the the components following way: of the spinors It follows once againdetailed exposition that of these this scaled scaling can generators be follow found the in [ SGCA using the following redefinitions: JHEP02(2018)065 ) and (5.14) (5.13) (5.12c) (5.15a) (5.16a) (5.12a) (5.16b) (5.15b) 5.6 . . irσ − inσ  , e ) −  e irσ ) of the worldsheet  r , − ) , r ) e µ − iεnτ (5.12b) n ) r ¯ b ˜  ) Q n i , µ − − ) 1 , − . ) − ¯ α r ) ψ Q n ), thereby showing the + ¯ r i L ... (1 µ − Q + ˜ ( µ r µ − − 2 α b b + + ˜ µ n / 0 ( i Let us now apply the inho- inσ 4.15 r 3 n ε α ( + ˜ e ψ ε ( and + L Q µ n √ ε µ n )( r ( ( µ r α n ε ε α χ √ b ¯ H ( ) + L ( irτ ε r , = = − ε ) + ˜ µ − n inτ − √ r n b √ α ˜ L i ( ), the expression is modified as and r − = i 2 = µ − + r ) iεnτ b ˜ µ n µ r n i ε µ r G 4.13 − b µ − ) + √ ( − , ˜ α 1 ε n (1 µ r τ ψ b − ) √ M µ 0   + inσ µ ,B n , β , – 20 – α − 0 α 1  − n r r  e ( ψ n r X X + ˜ L µ n ε ¯ µ − 0 0 θψ µ − )( 1 α µ 0 i c c iχψ b ˜ ˜ √ ετ α χ  i α µ 0  √ √ ( + + 1 n − α + ε − 1 n ,H 0 0 + = = µ n ,M r c µ r √ α ε 6=0 0 n α b ( 2 − 6=0 X n   c − i ¯ 2 X n   iαψ iθψ 0 2 Q µ µ − − ¯ 0 r c i L ε ε ε c √ 2 1 1 + + + ). Now this superfield is invariant under the scaling ( iψ iψ 2 √ √ − − + + 2 √ r X X X r − + n 4.9 i i µ µ = = L Q µ µ + + + x + x = = = µ µ n r ψ ψ = = γ A = Y   r n . Under these limits, the bosonic modes transform under this scaling as 2 2 ) = 0 L G ε 1 ε c √ √ τ, σ ( → µ 0 ) = ) = α X σ, τ σ, τ as . We can apply the scaling to the generators ( ( 0 µ µ 0 1 α εα ψ ψ → seen that If we compare these modes with the one that we obtained intrinsically, it can be easily The fermionic modes scale as Contraction of the modemogeneous expansion scaling and on the thetensionless algebra. mode ones. expansions of We need thescale tensile be string careful to here see and how should to keep obtain in the mind that we also have to and obtain theconsistency correct of expressions this of proecdure. and we end upα getting ( residual symmetry transformations, as donegenerators for in the the homogeneous following case. way We can define new Starting from the definition of the superfield in ( JHEP02(2018)065 ) be- 5.13 ). Therefore this 4.16 has interesting implications for the quantum – 21 – I are characterised by a residual symmetry algebra ) from which the inhomogeneous tensionless string derives its name. . I I For the homogeneous tensionless superstring, the non-participation of the fermions in The emergence of the larger SGCA After a brief recapitulation to the usual tensile superstrings and the tensionless limit conjectured this to bethe the Hagedorn worldsheet transition. signature of this long string phasethe and tensionless the limit onset of meantsome that unnatural the differences analogous between story the bosonic was not and fermionic quite oscillators true when and mapping there were of the usual tensileintriguing theory of oscillators. which was This that led theretheory to were two and some distinct very vacua, one interesting onetransformations corresponding to physics, to the and the the most the tensile tensionless tensionlessof theory. vacuum the turned tensile These vacuum outconnected and were to the this linked be tensile to by a oscillators, the (singular) squeezed and appearance hence state Bogoliubov of a in the very terms long high string energy near state. the We Hagedorn temperature and aspects of the tensionlesswas theory. such that In this practically the left homogeneous thewe fermions case, looked as the spectator at fields. scaling the on Inidentified intrinsic the the oscillators bosonic analysis on worldsheet story, when of the the worldsheet which tensionless turned case, out the to be mode Bogoliubov expansions transformations naturally (homogeneous) tensionless limit, weresidual symmetry find algebra that which is thelook larger analysis at than the the of mode one symmetries expansions arrivedreproduce which at brings helps our in us us answers earlier arrive of at work. at the thescaling intrinsic a We symmetries then analysis the in by fermionic a constructing fields different the way. and inhomogeneous We corresponding limit by superspace coordinates. ways. One ingeneralisation an of intrinsic the methodtensionless bosonic that limit, starts tensionless the off strings worldsheet metric withworldsheet to degenerates fermions an the and modifies action hence supersymmetric to which theof incorporate version. Clifford is this this algebra the modified feature. for In Clifford “natural” the algebra With the that a is choice different of from representation and more general than the previous work. This richer algebra(SGCA is called the Inhomogeneous Super Galilean Conformal Algebra that was previously constructed (whichwe we approach call the construction the of homogeneous the tensionless inhomogeneous superstrings), tensionless superstrings in two different 6 Conclusions 6.1 A summaryIn of this the paper, paper weinhomogeneous tensionless have constructed superstrings aon new the tensionless string worldsheet limit which of is superstring larger than theory. the residual These symmetries considered in earlier tween the tensile and theof tensionless constraints. the The Virasoro scaling, together Algebra,analysis with agrees corresponds the with two to copies the inhomogeneousSGCA the scaling algebra of obtained the Super-Virasoro in algebra ( to arrive at Plugging the above relations back into the constraints gives us the connection ( JHEP02(2018)065 5 ]. the 45 viz. ). It is thus O ) by similarity = 4.2 1 ρ and I ], which the authors rather = in the tensionless limit (with 56 0 I ρ ), the inhomogeneous tensionless = 2 supergravity. Our previous dis- 3.2 ]. N 45 could change from ( = 1 supergravity in 3d flat space [ I N ] that even for the homogeneous tensionless 42 0). Recently, spectral flows in these algebras , – 22 – also has recently been realised as the asymptotic I = (1 = 2 flat supergravity [ a N V ]. It would be of interest to understand whether similarity 58 , 57 and the usual (untwisted) flat H 0), these two are the only sectors. We mentioned that the representative , was actually a trivial extension of the asymptotic symmetry algebra of 3d flat supergravity, = 2 flat supergravities, the despotic dominates over the democratic theories. H N = (1 a V One of the interesting co-incidences of the bosonic tensionless theory was that the There are, of course, five distinct tensile superstring theories and it would be of impor- In the parameter space of tensionless superstring theories, which are characterised by The SGCA 5 cussions about the naturalness of thereference emergence to of the the SGCA solutionsspace of of 3d the modified Clifford algebra) seems towhich indicate was called that the in super Bondi-Metzner-Sachs the algebra in [ superstring, the underlying algebra boreof a a close canonical resemblance asymptotic to symmetryIt the analysis symmetries is arising of of out interest tosymmetries note of that a the particularwonderfully SGCA twisted call “despotic” supergravity asleads opposed to to the the SGCA usual “democratic” limit which theories are also linked byilluminating a web in of the dualities. tensionless A limit study and of may this tell web us ofsymmetry dualities something algebra, could important viz. be in the very M-theory. bosonic3d flat 2d space GCA, at coincided null with infinity. the We asymptotic observed in symmetries [ of would also be of interestof to the understand tensionless whether theories anytwo transformation can disjoint or lead sectors deformation to of of the the one other. tensile theory, At albeit the onetance moment, larger to it than understand the seems what other. that the these tensionless limit are means in all these different theories. These transformations, but thethese algebra are remains the unaffected. onlywith two a However, sectors general in which gaugehave order emerge and been not in to studied just the prove intransformations tensionless that [ limit, of one these needs modified to gamma start matrices can be linked to spectral flows. It led us to two different symmetry algebrashomogeneous and and two inhomogeneous. different tensionless The superstrings, naturalother question choices is of whether these this is modifiedgauge exhaustive gamma or matrices, by we couldelement find of even more the versions. matrices In that the lead to the SGCA is just a specialclear, case apart (corresponding from to the ourstudy trivial motivations this solution for class the of quantum tensionless superstring theory theories. stated6.2 above, why we Comments need and to We future have directions seen that the choice of different gamma matrices in the modified Clifford algebra extension for our bosonicthis analysis in an to upcoming the publication. superstring case. Wethe will representations have of more the detailssuperstring modified of thus Clifford corresponds algebra to ( a generic class of solutions and the homogeneous version from the tensile to the tensionless phases. In the inhomogeneous case, we find a natural JHEP02(2018)065 M h (A.1) ] was explored 61 , which acts on the , one would need to This meant that the C M c 6 ] where the connections of 60 = 0. For more details of what = 0. ] about scattering amplitudes M 3 h , M 2 c . T ¯ ψ ], it was claimed that the null string and C 59 = c ψ – 23 – → ψ . B ] for further related developments. 62 ] noted that the theory had 60 One of the principal objectives of our project of understanding tensionless strings is A rather fascinating direction in which the theory of tensionless strings has found recent The authors also show that for the theory they are interested in, 6 spinor to produce conjugate spinors via the operation is and on the truncation, see appendix A Tensionless spinors In this appendix, wein comment on this the paper. spinorsimportant and The conventions concept first related here issue to would the to be spinors note used the is charge about conjugation the operator spinor indices used in this work. An Academy of Sciences (CAS)Frontier Sciences, Hundred-Talent CAS, Program, and by by Project the 11647601Kanpur supported for Key by hospitality NSFC. Research during SC a Program acknowledges part IIT Institute of of at this work. Vienna, PP ULB thanks and IIT the Ropar, University Erwin Schr¨odinger of Groningen. of Arjun Bagchi (AB)Department is of Science supported and partially Technology (India). byand AB the Physics, thanks Centre Max the for Simon Planck Centre TheoreticalVienna Society for Physics Geometry University (Germany), at the of MIT, Technologyvarious Universite and de stages Libre of the Brussels this University (ULB), work. of Southampton Aritra for Banerjee hospitality (ArB) at is supported in part by the Chinese Acknowledgments It is aAditya pleasure Mehra, to V. P. thank Nair, Marika Rudranil Taylor and Basu, Chi Rajesh Zhang for Gopakumar, helpful discussions. Yang The Lei, work Wout Merbis, string, the authorssymmetry in algebra [ truncates to aCFT single vertex copy of operators. the Virasorofind But algebra for vertex and a one operators can general ofstring use theory amplitudes. usual the with full a GCA non-zero and use them for the construction of tensionless to ultimately link up toin the the results of very Grossthe high and formalism Mende energy [ that limitable we of to have string address built theory thisoperators question in with in in the our methods the 2d GCA previous near on and future. its papers the supersymmetric The versions. and worldsheet. initial In this crucial the context With one, step of is the we ambitwister defining hope vertex to be application is the theorythe of ambitwistor ambitwistors. string In are one [ of and the the null same string. whenthis There one null also or considers has ambitwistor a been string particular to somein quantisation the recent further Cachazo-He-Yuan advances (CHY) detail. [ formalism See [ also [ JHEP02(2018)065 (A.9) (A.6) (A.7) (A.8) (A.2) (A.3) (A.4) (A.5) and show that the 0 iρ , = 0, so that the projection ! does not satisfy the relevant ρ 0 1 0 ρ 0 0 −  , .

+ − = , ψ ! a = iψ 1 ρ . − . ! − = − ! 1 = ! 1 0 0 − = − −

1 0 1 0 1 1 0 0 , A, B 0 1 0 0 − − − =

, ψ C

. BA 1

, ψ – 24 – + , which picks out the left/right spinor component, ρ T C − ! ) ¯ 0 = ρ = a iψ B ψ ρ 2  ¯ ρ ρ 1 C χ 0 0 1 0 ( = = =

C = , the supersymmetry parameter. For the homogeneous = ¯ ρ + + .  A ψ ψ ! χ 1 L/R P − 1 0 0

in four dimensions. So the chiral projection operators in two makes a good choice for a charge conjugation operator. This has 0 5 γ iρ matrices for the inhomogeneous case, with some tedium, we can propose such are the usual Dirac matrices. For example, in case of tensile strings it can be ρ a ρ We can also comment on the chirality of the theory via defining proper projection but for homogeneous case,operators the are concept just proportional completely to fails unity as matrices. ¯ In the inhomogeneous case however, So also in this case the concept of chiral projection of fermions is not at all clear. the matrix This is analogousdimensions to will have the form Which has same structurereaffirms as the hints the we charge haveof conjugation picked tensile up matrix and throughout for inhomogeneous the Dirac the work theory. about tensile the string. structuraloperators similarity This on the spinors. For example, in the case of tensile strings, we can write down we can instead find outthe another matrix set that of satisfies thea criterion better. matrix with Starting the out with form one under the propera contraction. charge For conjugation the inhomogeneous matrix.criterion. case, But, As we curiously could also in define this such case and it could becontraction, done we similarly can for again chooserequired the identity charge works conjugation operator out as fine. But now the spinor indices are raised/loweredThis as is very useful in seeing how the homogeneous supersymmetry matches with the tensile Now we can see that we can raise/lower the indices on tensile spinors as Where explicitly shown that an important consequence asvia we action know of that the the charge conjugation spinor operator indices in can the be following raised way, and lowered It can be shown that the operator should have the following symmetry by definition, JHEP02(2018)065 L L h h (B.1) (B.2) (B.3) , which 0 L states, which = 0, provided have only half L s i i modes. Now if we N | ,H , r M n . Thus we only need = 0, we find that we 0) M L G , h M L c h | r > N M n, r > at this level by considering a h or i = in the theory such that the 2 n . / i . p = 0 ( 1 i . We can solve for the coefficients i N | p M i M φ p M = | h , h Let us consider representations of the φ 1 2 where | r L , h r − h r L | H h or H 0 H anhilate this state. This will give us some 2 n = a 1 2 i H + p and i ≡ | ,M i φ p – 25 – i r | p φ r = 0. It is sufficient to perform this analysis upto | φ after solving the conditions we obtain by imposing M and | G ,G eigenvalue. i 1 2 M n , h 1 2 L and the weight − h = N L | M G h i 2 h G | p 1 a L a φ , | h 1 is zero. This is done by analysing null vectors (states that n etc.) acting on the primary state labeled by the weights a ] = 0, this means that these states are further labeled under = ). We will label the states with the eigenvalues of n 0 M M i − c in the NS-NS sector. = M ,M 1.11 i 0 ,M I p , h n L φ L − | h n L | 0 L L SGCA . Since [ L we can write down the most general state h . Then we impose the conditions that all the positive modes anhilate this state, . The same is applicable for M ], it was shown how GCA can be reduced to a single copy of Virasoro Algebra, provided h Below we make a list of a general state at an arbitrary integral or half-integral level We are going to attempt to do the same analysis of null states for the two versions The recipe is as follows: we define general states by considering a linear combinations 48 . So, and the corresponding null state 0 combination of the fermionic generators only: Now we impose the conditionconditions that on the coefficients N that all positive modesn > annihilate N them.to calculate It upto is number trivial of conditions to where Level see 1/2: that while the negative modes raise the Since we are ininteger the modes NS-NS and sector,construct we the the don’t supersymmetry notion have generators ofvalue any is primary extra bounded states labels from below. compared The to positive modes the annihilate bosonic the case. primary state: We levels (as we are dealing with statesNull in states the in NS-NS sector). inhomogeneous algebra ( we will call M remove the null statesbelong from to the the Hilbert representation Space, of a we single are copy left ofof with the the only Virasoro Supersymmetric the Algebra. GCAtime that we we have need in to connection consider to fermionic tensionless generators superstrings. as This well as general states with half-integral and which fixes the coefficients for thebetween linear the combination of weights the and nullhave states the along non-trivial central with null charges. a states relation Since onlylevel we for 2. consider We see that the null states at each level consists only the In [ one of the centralare charges, orthogonal to all states in the Hilbert spaceof including the themselves) descendants ( of the GCFT. B Truncation of the algebra JHEP02(2018)065 2 / null = 0. (B.8) (B.9) (B.4) (B.5) (B.6) (B.7) 3 2 L (B.10) h 3 b along with . i i p + 2 . 2 . φ i / | i 2 p p i 1 | φ 3 2 φ | , b | i − 3 2 1 2 H − − 1 2 = 0 1 2 H − H − 6 1 M d H G . h − 1 2 i 9 3 i = 0. Then the level + p f b − p M . 5 i φ 4 φ | i p G + d | d = p 1 2 1 . φ 1 2 1 2 | φ 2. So if we set − − = | + − 3 2 M / −  L = 0 along with their descendants, 1 H h  = 0 − G 6 1 2 H d 1 2 2 i 3 2 1 f 3 2 b G − 1 . − − M | − 5 i − 2. + H h d p G = 2 null states from the Hilbert Space), / H M 1 1 2 = G 2 3 φ ) 3 / 1 | 8 b + − i − L d b 1 2 f i 2 h G + p M − / + , a + 5 i L 1 φ i | | H p f 1 1 2 h p 2 1 2 = 0, and therefore the null state is = 0, and the level 1 null state is = 0, we will find that to get a non-trivial state φ 2 − | a (1 + = 0 φ + − | 1 – 26 – M M M 1 H 1 2 − − − = H M c 2 1 2 h h − − 1 1 i 1 2 h M − − − 2 2 H M 2 − a / 1 G 4 b M M 1 1 G f − | 4 1  + − + d L 2 − + i 2 L b M = 0. This would mean p 1 d + h 7 M as the null state at level 1. φ 1 − = f | M + a i 1 + i M h p i + 1 − 1 1 2 | p φ | L − φ − in the null state at level 3 1 1 | L b i 1 2 H − 3 p 1 − f φ M − | G + 3 2 L 1 term, all the other terms are descendants of the null state at level 1 1  − − 2 2 − L rd H d 1 L h 2 d f = the most general state: i + 2 at level 2, the general state is given by a linear combination of 9 terms the most general state: 2 / term is the descendant of the null state at level 1 − 3 | L nd 1 f h Level 2: Except for the 3 and 1. As seen before,we if are we left set the with null states If we concentrate only on theat case this where level, we needstate to would set be Level 3/2: The 2 is descendants to zerothen (in we order are left to with remove level 1 We impose the anhilation conditionsTo get and a obtain non-trivial state we must have Level 1: To get a non-trivial state we must have to obtain the null state at this level. It is easy to see that we need to simply solve for JHEP02(2018)065 0 = i . r 2 and and f + i L (B.12) (B.13) (B.11) p = 0 = h φ m 9 | 1 | f 2 f − = + 2 , M , 8 M , where f h i   = 0 i i , ) = 0 s at integer levels. M , p + 2 p 8 . M φ f φ , h | | h M M 0) 2 L 1 = 0 9 0 + h ) = 0 2 − h − f , 7 9 L | L f f M M , M h = 2 , h = acting on a state . = 0  L 7 0) i n, r > 0) s − M h f L 9 0) p ¯ ) + 2 h f 3 h Q 2 φ 8 + M L | 4 (3 + 2 f n > h h 4 . We can thus set n > 9 ( r > 7 + 2 f i 3 + 2 ( ( f = 0 ( = f p i and i 2 +  2 i p i φ + p s p | p φ M L  (1 + 3 i − 2 | φ − + 2( φ 6 c φ 4 Let us look at a similar analysis for the | h | p | n − f 2 ,Q f 3 3 3 + n r r φ − 3 n 2 s, leaving us with a single copy of Super | + i − α − ¯ M Q 1 2 p G L M L M Q φ ) ) − ) , f , f , , , f h | = , n n r 3 2 H n i  after imposing the anhilation conditions. Just 1 2 p 8 − and L + + + = 0 = 0 = 0 = 0 i f φ − – 27 – | H p L L L ) = 0 2 8 9 L | r 1 2 + G M f f f h h h L 1 Q − h h M ) − = G h 9 = ( = ( = ( + 2 = = 0 at all levels in order to get a non-trivial null state. 7 1 . Once again, we can list the null states level by level f M , the null state at this level is: ) + 2 i i i 6 i f f r  9 2 p p p f s at all half-integer levels and the p L M , and define their weights under 7 i − correspondingly. We can also write down the descendants φ φ φ i (3 + 2 φ h + + 3 p f h | | | | 6= 7 p 0 + 2 r H φ n n 8 n + 3 f | φ + α − f L 6 | − − M 1 2 c i f 4 M + M L Q (1 + p − 5 M 0 0 4 5 h on 0 + 5 φ in the NS-NS sector. = ) H | s > m f f L L and 7 7 L 1 3 2 i 2 ¯ Q f f p H ) + 0 − − or L φ + 2 L | + G M h 3 n n + 6 1 , all the remaining terms are descendants of the null states at lower 6= 0 and  5 f and i ). We consider a similar primary state 8 = 0, these are: L f 6 − p L f f L Q φ h M 3 | SGCA + , c 2 f 3.12 (1 + 2 + 4 4 − ) + 2( M f 3 = L f , M i . For + h L 2 9 (6 | 3 f f (1 + = 2 3 are the weights of 3 f f 2 M Now we proceed tomention again do that the it nullgives is state zero, trivial provided analysis to see asand that we analyse the did corresponding for nullas the state before, previous will be case. required to We set can The anhilation conditions are other algebra ( h by acting means we can throw awayThis the essentially truncates the algebraVirasoro to algebra. Null states in Except for levels. So, theits only descendants new to null zero state and at carry level out 2 the is same exercise for arbitrary higher levels. This Demanding a non-trivial solution, we0, can simplify the above equations by level. If we set We find a set of equations which we can solve to obtain the non-trivial null state at this JHEP02(2018)065 to i = 0: m (B.18) (B.19) (B.20) (B.14) (B.15) (B.16) (B.17) | M h . . i p . i φ | p i = 1) φ | 3 2 3 2 L − 1 2 h − ¯ ( Q ¯ − Q 1 2 = 0, and therefore the ¯ 6 Q − i d 1 2 . . p M i Q i − i φ p + 9 p h | p 2 f φ i Q φ  | φ | p a 1  . | 1 2 1 2 1 2 i . + 3 2 φ − 1 2 respectively. i | p − − − = 1 2 − L p − 3 2 i ¯ φ ¯ ¯ Q 6 | ¯ Q . We then impose the conditions Q φ − 2 = 0) we can write down the null Q − . Q | 1 2 f | i 1 i M 1 2 ¯ 1 2 1 1 i p Q 1 2 − Q − h − − 1 2 − 3 2 − + 6= 1) M φ 5 1 or − | c Q M d − 2 Q a − s. Also, to get from null state Q ¯ 2 1 L M M 1 i 4 Q 3 ¯ − is just the descendant of the null state 3 1 2 Q h 2 Q 1 2 − − b d 1 2 + ab ( 2 d 8 / = 0, and the level 1 null state is ¯ i δ i − M a − Q a f 3 M − + + 1 p 5 | 2 + | 7 Q Q M i i f = 1 2 i φ a + + f 3 i | p p p h b p − = 1, after some coefficients drop out: } 1 2 1 2 1 2 2 . It is interesting to note that while descending + φ φ φ + + | | 1 2 ¯ φ | i b Q − | L 1 − − + 1 i 1 2 i 1 1 – 28 – b − 3 2 1 2 ¯ 2 | h p ¯ − − 1 2 Q 1 2 Q = Q − − − 1 φ − 1 1 2 − − | M ,Q Q M ¯ − a M Q 1 2 M ¯ Q 4 − 1 2 1 2 Q h 4 M 1 f b 1 2 h − − 1 2 M 2 f Q a − 3 − 4 b 1 s as = b − Q + + Q − h M d L + 1 2 − 1 i 3 { i 2 1 Q 1 2 a 6= 1 and = 2 a − p d = 0, some coefficients would vanish after setting 1 d + = − M / − φ 1 − 7 L Q i | 1 b ¯ + = M | f M h Q 1 1 doesn’t appear in | 1 M c i i 1 2 − 7 2 p + − 2 − f L − φ / L | 1 3 Q 3 b 1 2 | + M 1 f , we operate only with − 1 − i is the descendant of 2 + − 2 L Q i 1 1 / and  2 M = 0, and the null state is − − 2 6 4 / anhilate this state. This gives us 3 2 6 f f L 3 L | + 1 a − 1 1 2 d 2 h  d f ¯ Q Q m = = = | the general state is the general state is given by + i 5 2 2 d the general state is given by the general state is given by | to − and we act twice with the i = L 1 2 1 i 2 m f | d Q . Similarly h i + 1 2 = / 1 m 1 We note that from | An analysis of the| null states will reveal that By similar techniques asstate done at for this level the for above 2 cases cases ( Level 2: Considering the case where d Level 3/2: The conditions obtained are: that null state is Level 1: Level 1/2: JHEP02(2018)065 i p D φ | 07 3 2 , Phys. , , − (2002) (2004) Q 52 JHEP B 303 Phys. Lett. , , s in the case Phys. Rev. n B 644 , M (1988) 1229 ]. (2014) 044 60 ]. 11 Nucl. Phys. to a single copy of the ]. ]. , vector model Fortsch. Phys. SPIRE , we cannot eradicate all Nucl. Phys. H ) , I IN , SPIRE N [ ( JHEP IN SPIRE SPIRE from the worldsheet , O ][ IN IN 3 ]. ’s, it would have been expected ][ ][ M Phys. Rev. Lett. , ABJ triality: from higher spin fields to SPIRE JHS/60: Conference in Honor of John (1977) 1722 IN ][ ]. dual for minimal model CFTs ]. Stringy AdS 3 D 16 hep-th/0103247 ]. 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