Tensionless superstrings: view from the worldsheet

The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters.

Citation Bagchi, Arjun, Shankhadeep Chakrabortty, and Pulastya Parekh. “Tensionless Superstrings: View from the Worldsheet.” Journal of High Energy Physics 2016.10 (2016): n. pag.

As Published http://dx.doi.org/10.1007/JHEP10(2016)113

Publisher Springer Berlin Heidelberg

Version Final published version

Citable link http://hdl.handle.net/1721.1/106579

Terms of Use Creative Commons Attribution

Detailed Terms http://creativecommons.org/licenses/by/4.0/ JHEP10(2016)113 Springer arise in the August 12, 2016 October 14, 2016 October 21, 2016 gs, Higher Spin : : : b,d Received 10.1007/JHEP10(2016)113 Accepted Published at-spacetimes. This helps opted earlier for the closed doi: sionless superstring from the er-Sachs algebra, discussed in niversity of Groningen, [email protected] of the conformal gauge is (a trivial , f Technology, Published for SISSA by and Pulastya Parekh c [email protected] , . 3 1606.09628 Shankhadeep Chakrabortty The Authors. a,b c Conformal and W Symmetry, Superstrings and Heterotic Strin

In this brief note, we show that the residual symmetries that , [email protected] E-mail: Indian Institute of Technology Kanpur, Kanpur 208016, India Van Swinderen Institute for ParticleNijenborgh 4, Physics 9747 and AG Gravity, Groningen, U Indian The Institute Netherlands of ScienceDr Education Homi and Bhabha Research, Road, Pashan. Pune 411008, Iindia Center for Theoretical Physics, Massachusetts77 Institute Massachusetts o Avenue, Cambridge, MA 02139, U.S.A. b c d a Tensionless superstrings: view from the worldsheet Open Access Article funded by SCOAP Abstract: Arjun Bagchi, Keywords: the context of asymptoticus symmetries uncover a of limiting 3d approachpoint Supergravity to of the view in construction of fl the oftensionless worldsheet, the bosonic analogous ten string. to the one we had ad analysis of the tensionless superstringsextension in of) the equivalent the recently discovered 3d Super Bondi-Metzn Symmetry ArXiv ePrint: JHEP10(2016)113 1 4 8 14 16 19 (1.1) ] investigated 3 , 2 and the fundamental way to the very quantum e tension, instead of going ] and hence to holographic antum gravity. The funda- → ∞ 6 sed to capture the very high d that the string scattering d since the late 1970’s start- e only free parameter in the ′ , y, viz. the ultra-stringy limit, tal string which is given by ed out of usual quantum field limit and that there were in- 5 α its poorer cousin, supergravity. litudes hinting at the existence 0, the fundamental string shrinks to → ∞ ] play a central role. → ′ 10 ′ α . α ′ ]. More recently, the tensionless limit of 4 1 πα 4 – 1 – = T ]. Along a different angle, Gross and Mende [ 1 ] where higher spin Vasiliev theories [ 9 – 7 There exists a diametrically opposite limit of string theor The tensionless limit of string theory has been investigate ing with the work of Schild [ In this limit, we recovertheory. everything This we is could the have point-particle construct limit of string theory. a point-particle and superstring theory is approximated by string becomes long and floppy.energy This behaviour is the of limit stringnature that theory of is suppo gravity. and as a result offers a gate finitely many linear relations betweenof these a scattering higher amp symmetry structure in this limit [ which always been ato source infinity of as great in intrigue. the This point-particle is limit, where becomes th zero, or In the limit where the tension becomes infinite or amplitudes behave in a very simple way in the string theory has been linkeddualities to [ the higher spin structures [ the extreme high energy limit of string theory and discovere 1 Introduction String theory is themental most objects viable of of stringnon-interacting the theory theory current is are theories set vibrating of by strings the qu tension and of th the fundamen 6 Conclusions 3 Classical tensionless superstrings 4 Limit from the tensile5 superstring Quantum tensionless superstrings Contents 1 Introduction 2 Supersymmetrizing 2d GCA JHEP10(2016)113 s τ ]. ]. 14 35 → , ] and 34 13 , ǫσ, τ case of the 16 ] and hence → ]. For a non- 33 σ 13 { e [ mit and naming ] and Yang-Mills ], we would like to 37 36 . in non-relativistic he same algebra from ble in string theory is e equations of motion, prises as we briefly now lly tensionless theory are ldsheet. In the conformal istic one instead of a non- f the closed bosonic string g. in [ ski spacetimes [ ss string found in [ formal Algebra (GCA) [ ies, although the initial goal story. The intrinsic method solved when one understands e interested reader to have a stem. In the classical regime, chniques of 2d conformal field g earlier work [ tivistic limit on the worldsheet r)strings the reader is referred ts and higher spins (for work in ng which was now a fundamental cs in the quantum regime. The formal gauge. The residual sym- stic contraction ( a etc. could be derived by looking l steps in this direction were taken ion operators of the tensile string . This sends the worldsheet speed of Our perspective in this field has been } 0 → ǫ ], we found that there was an intrinsic method – 2 – 17 with In [ ǫτ → σ, τ ]). We have been interested in investigating the tensionles → 12 σ , { 11 ], one can see that there is a similar story that emerges in the ), viz. 13 ]. The apparent dichotomy between an ultra-relativistic li ], in the non-relativistic versions of Electrodynamics [ σ, τ 17 15 ) and the above mentioned ultra-relativistic limit yields t ], and surprising as the asymptotic symmetries of 3d flat spac , } ]. 0 38 14 27 → ]–[ ]. There were some expected results and some interesting sur ǫ The fact that the limit on the worldsheet is an ultra-relativ The above described limit is, interestingly, an ultra-rela One of the central reasons why a lot of progress has been possi Following [ 18 17 and hence the vacua for the tensile theory and the fundamenta relativistic one, has intriguingultra-relativistic consequences limit on mixes the the physi creation and annihilat light to zero. Forlook details at of [ this procedure, we encourage th the two copies of the Virasoro algebra of the tensile string. co-ordinates ( the residual symmetry the Galileanthat conformal in symmetry the is special re case of two dimensions, the non-relativi object and not aconstraints, mode derived expansions concept. and proceedeverything, One including to could mode quantize then expansions, the constraint sy writeat algebr down the appropriate th limit on the worldsheet. remind the reader. Quick recap: the bosonicto story. the theory ofwas tensionless the strings one and where thereassumed one that was started was the a with defining limiting feature the of action the tensionless of stri the tensionle theories [ has been used to constructTaking inspiration the from notion the of methods holographyaddress in of the Minkow tensionless 2d limit GCA of constructedin string e. theory. [ Some initia tensionless closed bosonic string inmetry the algebra analogue of that the emerges con inThe this GCA case is has theAdS/CFT 2d shown Galilean [ up Con in different contexts in recent years, viz to [ symmetries from the point ofexhaustive view list of the of worldsheet other followin relevant work on tensionless (supe with somewhat different from the recentwas focus to on concretize higher this spinthis connection theor between direction tensionless see limi e.g. [ Our point of view: worldsheet symmetries. the emergence of 2gauge, dimensional conformal there symmetry is on awhich the residual is wor gauge 2d conformal symmetry invariance on andtheory this the to enables worldsheet great us o effect to use in te the quantisation of string theory. JHEP10(2016)113 . - 2 ] which was But there are 41 . Here we first 1 3 wo different and antum tensionless he study of Hagedorn ], where a worldsheet 28 considered in section nd the other contraction n section tively tensionless and we wo copies of the Virasoro e to the emergence of the et of the tensionless super- btained in [ iscussion of open directions ing to find the form of the There can be several ways ebras. Cartan structures arising in It is a priori not clear which re not trivially scaling away parts different algebras. We spend ing. Hence to begin with, in ut the thermal nature of the l contractions. e of the tensile world-sheet is d state in terms of the tensile . In this short report, we will e find that the algebra arising symmetries of 3d supergravity is thus a very highly energised ary to the usual folklore about et link the two sets of oscillators ldsheet [ lement on the string worldsheet. dsheet manifold. This Carrollian ties of the tensile superstring. In nity of the Hagedorn temperature, at least one Virasoro left over after the ] and fix a convenient gauge. We then do the – 3 – 18 , we show how one can obtain some of the aspects of 4 The obvious generalisation of this formulation is to the ten . In section 2 . 6 ], we remarked that our analysis was particularly suited to t , we take some preliminary steps in the construction of the qu , we visit the algebraic aspects of the Super GCA and mention t 5 2 17 In [ We then focus our attention on the tensionless superstring i This statement is somewhat ambiguous. What we mean is that we a 1 superstring. We conclude with ain summary of section our results and a d We also perform asame mode symmetry expansion structure. of theperformed We tensionless comment in on superstr section centralthe extensions tensionless a superstring from thesection corresponding quanti look at the tensionlessanalysis action of following residual gauge [ symmetriesfrom and the to our analysis surprise,obtained is w by a looking at trivial theon extension canonical flat analysis of of spacetimes. the asymptotic This one is recently one o of the two contractions that we one of these shouldsection be important to theacceptable tensionless contractions of superstr thesome Super time Virasoro on which the lead representation to theory aspects of the two alg non-unique ways to contract the supersymmetry generators. structure is very similarnon-relativistic (and systems. actually We dual) madeemergent to tensionless preliminary the vacuum remarks by Newton- invoking abo left-right entang Present considerations. equated the emergence offundamentally the tensionless long vacuum on string the inthe worldsheet. destruction the Contr of Hagedorn the string phas worldsheetwhat picture in our the analysis vici predictedreplaced is by that the the Carrollian Riemannian structuremetric structur of is no the longer tensionless the wor correct variable to describe the worl physics. Strings near the Hagedorn temperature become effec of the algebra.scaling. A concrete requirement is that there should be and the tensionless vacuum becomesoscillators expressible and as the a tensile squeeze vacuum.state The in tensionless terms vacuum of the variables of the tensile theory. very different. Bogoliubov transformations on the worldshe sionless superstring and thisshow is that the the aim residual of symmetrystring the that is current emerges paper a on the particularto worldshe supersymmetric supersymmetrize extension the bosonic ofalgebra GCA. the generates a The GCA. unique contraction answer when of we the look t at non-trivia JHEP10(2016)113 . (2.3) (2.1) (2.4) (2.2)  0 ] appeared ] following s, . + 0 52 48 r s, δ t [ +  r δ 1 4 1) superconformal  , − 4 1 2 r −  = (1 2 r tensile string action. The M 6 us dimensions. For general hat one could contract the
N  c ry arises on the worldsheet hat would play a central role n ntioned in the introduction, c 3 − . We already know that the + oing ahead, we would like to y of tensionless closed bosonic re interested in the tensionless ¯ c r L s stic contractions of the tensile lativistic fashion for consistency + heory. This is mentioned before, can be under- − + s + r ¯ Q 0 + n r M ǫ 0 m, L  L 0 + √ m, n ǫ m, + The residual symmetry algebra of the αβ δ = n ) = 2 δ + = δ n n − r ) } δ = n s ) n − } n Q replaced with ¯ = 2, details were worked out in [ 3 − β s , n r − ,Q c – 4 – 3 d ( r ,M 3 ,Q n ( n α r {Q Q n c 12 ( − ǫ Q ¯ M L with 12 L { c + , √ ) leads to the following algebra: 12 c ]. For r r − m ]. For a recent study of an extended version of SGCA, + + ¯ = 47 , + Q n n 2.2 + r – , 36 m n r + L + m Q n + L α n 45 ¯ Q +  L n ) = n Q r ]. m n M L  Here we will scale both the supercharges in a similar fashion ) ) − L r 49 − m m 2 n − n  − − 2 n n n  ] = ( ] = r m As this paper was being readied for submission, the pre-prin ] = ( ] = ( ] = Q L , α r , m m n n L ,Q ,L L [ ,M [ n n n L L [ [ L [ algebra This contraction, together with ( and similarly for generators the reader is referred to [ Two different algebrassuperstring from after contraction. fixing the conformal gauge is simply the 2d superstring, we willsuperstring focus algebra, our the attentionmention Super-Virasoro on previous algebra. work ultra-relativi on supersymmetrizingdimensions, Before the this GCA g was in done vario closely in [ the bosonic analysis of [ in the usual bosonicas a closed residual string symmetry theory,2d after conformal Galilean fixing Conformal symmet the Algebra conformalstrings. similarly gauge arises The in in tensionless the the limit onstood theor the as string an worldsheet, ultra-relativistic as contraction. So now that we a on the arXiv containing some overlap with the current2 paper. Supersymmetrizing 2d GCA We begin by looking at thein algebraic aspects the of construction the of symmetry t the tensionless superstring. As we me Homogenous scaling. From the point of viewfermionic of generators. the Below algebra, we there list two are options. several ways t Note added. with the analysis of the bosonic tensionless closed string t bosonic part of the algebra needs to contract in the ultra-re JHEP10(2016)113 ]. d 40 (2.5) , 39 [ 3 for Einstein /G ]. The two copies while the Ramond = 3 30 ] to be instrumental ime is the algebra of 2 1 limit on AdS M 41 + nce [ , c worthy being the Brown- Z al field theories and if the di-Metzner-Sachs algebras algebra is actually isomor- ional ASAs when one con- mmetric extension of a flat sidual gauge symmetries on our dimensional Minkowski d field theory, which in this ∈ 3 is often is just the isometries ts of [ ery useful in learning about = 0 tors depending on boundary A through a canonical formu- . Here it is important to note about the field theory just by s for 3d Minkowski spacetimes r mal Algebra, we shall call this L H = 1 Supergravity and proposed . c ) c N onnection between flat supergravity + ¯ = 1 Supergravity recently performed c ( N ǫ ]. . We shall exclusively be dealing with the = 43 0) conformal algebra would lead to same Z , . With indices on the supercharges removed, M 3 ∈ ], 2d field theories with this symmetry algebra – 5 – r 42 = (1 ¯ c, c N − c = ]. The central terms are ). The central terms remain the same as mentioned above L c 16 ] which leads to two copies of the Virasoro algebra. This can ) indices). This is the very same algebra that arises from the 2.4 ]. The authors worked in the Chern-Simons formulation of the 29 [ 41 3 α, β . The suppressed commutators are all zero. We have also define ± = ]. In three dimensional spacetimes, the BMS 32 α, β , 31 ]. This can be obtained by looking at the ultra-relativistic 33 ]. The asymptotic symmetry algebra (ASA) of a certain spacet In asymptotically flat spacetimes, there are infinite dimens Interestingly, contraction of the We will see in the later sections that this is the algebra of re 41 in [ canonical analysis of asymptotic symmetries of 3d siders symmetries at theafter null the boundary. authors who Thesespacetime first [ are discovered them called in Bon the context of f algebra (stripped of the ( the tensionless superstring worldsheet. This intriguingand c the tensionless superstringboth would sides possibly of prove thisin to link. our be We future expect v explorations some of of tensionless the strings. physical insigh for the bosonic Einstein gravitywere case. constructed and In were [ shownlimit to of be Liouville equivalent theory, to earlier a constructed supersy in [ The supersymmetric version of thewas above recently asymptotic addressed analysi in [ simplest version of Supergravity in 3d, viz the minimal symmetry arguments. Henneaux analysis of AdS be looked at asof the the first Virasoro step algebra towardscase the dictates is AdS/CFT the a corresponde symmetries conformal ofsymmetry field the structure theory. dual is infinite The 2 dimensional, ASA a thus lot characterise can du be said allowed diffeomorphisms modded out by theof trivial the diffeos. vacuum. Th But there are famous exceptions, the most note phic to the 2d GCA given in [ lation and named the algebra the super-BMS a consistent set of boundary conditions. They derived the AS In the above, this is precisely the algebra ( NS sector through out this paper. gravity [ sector is characterized by integer labeling that there are twoconditions. different labeling The of Neveu-Schwarz sector the is fermionic characterized genera by Since the bosonic part ofSuper this Galilean algebra Conformal is Algebra the [Homogenous] Galilean or Confor SGCA JHEP10(2016)113 , (2.9) (2.8) (2.6) (2.7) . r e left- 0 + s, n ogenous] + r H δ   r vanishes, one 4 1 − M − c 2 n 2 r   , pearance of the factor 0 M 3 ] = 0 c r ]. We hope to comment m, he fermionic generators, is algebra is richer than er Virasoro algebra. The m, +
+ 17 ,G n + r o algebra when we consider s n δ n
]. − ) + δ n ions. In terms of spacetime, ¯ r M ) n 2d. The ultra-relativistic limit Q ¯ lativistic contraction described [ 36 L imit on the bosonic tensionless ork [ i n istic limit is a contraction of the M − , − r + − 3 in [ + is also reminiscent of another such r n 3 n = 2 n ( i L n Q ( } H M s , ǫ 12 L ǫ c  0 , 12 c r 0 = ,H m, = + r + − m, + n r n m G + m 2 n δ { n + ) + δ n  , n ) n – 6 – is that when the central term 0 ,H n ,M M L s, r I − ) ) ]. This isomorphism between the ultra-relativistic n ] = + − ¯ − 3 L r r m m ¯ 16 3 n δ Q One of the very surprising aspects of the contractions ( i + n − −  ,H ( n . M n 1 4 − n n 12 L c L r L 12 c [ − Q + = , 2 + ] = ( ] = 0 ] = ( r r n m = Here we scale the supercharges in a way similar to the bosonic + m m m m  + L n r + n L n 3 ,L G G c ,M ,M n M L n n  ) ) L + r ). This seems to indicate a fundamental difference between th [ L [ M m m s [ − + 2.6 − − r 2 n L ). We shall call this Super Galilean Conformal Algebra [Inhom n n  2.4 = 2 ( ] = ( ] = ( ] = } r s m m H ) and the non-relativistic contraction . I ,G ,L ,G ,M 2.2 n r n n L L G [ [ L An interesting feature of the SGCA We would like to mention that we are rather intrigued by the ap { in the scaling ( [ i can show that theanalysis algebra essentially truncates follows to the a bosonic single version copy outlined of the Sup and non-relativistic limitsthere is is a one curious contracted andis feature one a of non-contracted contraction direction of 2 in the dimens time direction while the non-relativ Non relativistic isomorphisms. of the two copies ofabove the ( Virasoro algebra is that the ultra-re the ultra-relativistic limit. This mysterious factor of on this in future work. of movers and right-movers in the SUSY extension of the Virasor or SGCA Again as before,the the one suppressed we commutators obtainedthe are above SGCA all by the zero. homogeneous contraction Th of t appearance of an imaginarystring factor oscillators, in discussed the in non-relativistic appendix l B of our earlier w both yield the 2d GCA generators. Inhomogenous scaling. This leads to the algebra This curious fact was first noticed in [ JHEP10(2016)113 o has H (2.10) (2.12) (2.13) (2.15) (2.16) . . 0 (2.14) = 0
r r p ¯ i ¯ Q algebras, there Q M ǫ − n, r > . W √ , h i r ∀ L ) (2.11) Q hat there exists iso- M = ¯ c h We remind the reader | ) is isomorphic to the r ne contracted direction − r , ǫ , h ] − 2 L H c Q 2.4 48 = ( h ( alogy with superconformal | = 0 ǫ r = arly given by only. We will first label the r p M ). p way with the non-relativistic ebras like tates by acting on them with i = o dimensions. The interested i h Q . M what is space and what is time. c under both contractions is an ǫ M M = 1 ,H 2.11 ras are thus different. When one r entations. , h √ i ion theory of what we call SGCA , h ¯ ≥ L Q L M = h n ]. h | | c, c + ∀ , h + r ± r r 51 r L + ¯ Q Q G h Q spatial directions. The algebras obtained c , | 0 = = d = ): =
p p 0 = 0 r n i i L ¯ i L c M M 0 – 7 – ,M ,M | i − 0 n , h , h ,G L n L L M
M h h L ) which are annihilated by the positive modes of the n | | , h p ¯ n n = i L L ǫ i h M M M | − 0 = | L n , h n n = = h L L L p p h = i i | We will briefly explore the representation theory of these tw ǫ i M M ,M = M n , h , h n ¯ ]. L L L , h h h L 50 | | + h n n | n Non-Relativistic limit: 0 L L ,M L n L ¯ L : : = I ) is isomorphic to the non-relativistic contraction [ H n + L n 2.9 +1 dimensions, the ultra-relativistic limit leads to only o L d SGCA = SGCA n In this work, our focus is on super-algebras and we discover t L We would here like to point out that recently the representat 2 are interesting departures fromreader this is isomorphism referred even to in [ tw Representation theory. different algebras, concentrating on highestthat weight repres we are going tostates with be the restricting Cartan ourselves of to the the algebras NS ( sector non-relativistic contraction The identification of the centralones charges being change identified in as the usual (time) while the non-relativistic limit contracts been looked at from the point of induced representations in [ indication that the procedure doesIn not higher distinguish between spatial direction and the fact that the algebra is isomorphi in the process byis contracting relativistic looking conformal to algeb generalize this procedure to more involved alg Now we construct the notion ofprimaries. SGCA These primary states are in states close ( an The identification of the central terms is again given by ( Similarly, ( algebras in question. morphism similar to that of the case of the Virasoro algebra. negative modded operators. The vacuum of the theory is simil As usual, the Hilbert space is constructed on these primary s JHEP10(2016)113 H (3.1) (3.2) (2.22) (2.17) (2.18) (2.23) (2.21) (2.19) (2.20) irσ − . e r 2 1 H of the relativistic 2 vector densities irσ / r r > − X e ∀ − r Q ction of the tensionless ) = , , r cyl ent algebras. The SGCA tra-relativistic contraction, inσ ¯ irw X inw = 0 J . − ibe. i − − the analysis of the constraints i e tensile theory by the following e = e 0 ) r + ess closed bosonic string action. inσ are weight 1 | n n ¯ r ¯ − Q cyl L a e ¯ cyl H J n V r J n ǫ ab . ( X = X M ǫ γ inτM i √ ab b = = 0 n = | + V gg = X r a 2 n cyl cyl 2 − G ¯ ¯ = L T J √ (
ξ V : T 2 ,J I ,J , , n d cyl The Energy momentum tensor of the Super- – 8 – X ¯ = T irσ irσ irw b Z inw − = − + − e − V e ) e = e a r SGCA r cyl + r n cyl ¯ B V T Q T L Q S , − ǫ r r n version: irτH 0 X X X I cyl = 0 + → = T ǫ = = i r is the induced metric and 0 | = = lim G cyl cyl cyl ( ± r µν 1 2 T J J η T T Q r ν ǫ X X √ : = b H ∂ = µ cyl 1 ¯ X J J i a ∂ SGCA − = cyl J ab γ = 1 J On the other hand, the SGCA which are linked to the metricrelation of the string worldsheet in the along with quantum superstring. EM Tensor andVirasoro is its given by: superpartner. The important thing here isthe that vacua since of we the aretheory. looking SGCAs at would This the not ul is be going the to one we play find an from important the role limit in the constru where The superpartner of the EM tensor on the cylinder: 3 Classical tensionless superstrings We start this section by remindingThis the reader is of given the by tensionl The EM tensor and its superpartnerof will the play a tensionless crucial superstring role that in we will now go on to descr The supercurrents for the SGCAversion is are given different by: for the two differ The SGCA EM tensor is the same as the bosonic one: JHEP10(2016)113 ), 3.3 (3.3) (3.7) (3.6c) (3.5c) (3.4c) (3.4a) (3.6a) (3.5a) (3.4b) (3.5d) (3.6d) (3.6b) (3.5b) (3.4d) by the a χ om the above . i , , , . µ are ψ a a ξ = 0 ∂ µ ) = 0 ) = 0 ) = 0 µ ψ µ µ µ ψ a ψ ψ ψ a b b b tring. The action for the ∂ , iV χ χ χ are µ a Our interest in this present b a ψ ǫ 4. The fermionic partner of V − ciated with the action ( χ V venient choice where physical erized by ersymmetry transformations. wing the procedure outlined i ) / 2 ual tensile string theory. ) + , a y µ b 1 ) µ + ξ V s and one needs to replace it by X χ b − µ ( b µ ) + ∂ a ( µ X iχψ ψ i∂ a 2 1 i∂ ( ( ψ . ∂ b + µ ]. A similar process (without sending iǫ b + χ ψ µ 2 1 χ, ) + b µ 13 a ) = 0 iχ X µ ( a ψ V b − V b ξ a a X b ∂ a µ b b V χ ∂ , χ V ∂ a ∂ b ( X , V b ξ 0) − ) ( a V 4 1 , V · ∂ µ 2 – 9 – ǫ. + , a ǫχ a , , ) a + µ ( X a µ µ − µ V ∂ a a ξ χ ( ) = (1 a ǫV X ψ b ∂ a a b b µ ∂ µ a iǫψ − iV V ∂ ∂ ∂ ∂ b iχψ ψ b b a ψ V a b ξ V ξ ξ = = = = + ) is derived by taking the phase space action, integrating V − − − − iχ a ( µ µ µ χ a ǫ 3.1 , which is connected to the ordinary gravitino ψ V X X δ = = = = ǫ ǫ χ ǫ i∂ a + 2 δ δ δ a µ µ ∂ χ µ is ξ a ψ V X δ ξ ξ X ξ V a δ b δ ( δ ∂ V h µ ξ 2 X d a . It is straight-forward to obtain the equations of motion fr ∂ Z ( a b χ = a V V S and that of = µ χ ψ ], one can supersymmetrise the tensionless closed bosonic s is 18 µ action: In the above action, the fermions are densities of weight relation X and transformations under supersymmetry, parameterized b In light of these transformations, weinterpretations need to become fix clear a gauge. is the A con following gauge: which is invariant under worldsheetTransformations diffeomorphisms of and the sup fields under diffeomorphism, paramet in [ tensionless limit of the superstring is given by: out momenta and taking thethe tension tension to to zero zero) limit results [ in theAction, Polyakov equations action of for motion the andwork us residual is symmetries. to look at the tensionless closed superstring. Follo the vector density. The action ( In the case of the tensionless string, the metric degenerate As in the usual tensile case, there are gauge symmetries asso JHEP10(2016)113 (3.8) (3.9) (3.14) (3.10) (3.11) (3.12) (3.13c) (3.15c) (3.13a) (3.15a) (3.13b) (3.15b) along with (subject to needs to be becomes de- ¯ b θ V ab ab a η η and V θ = 2 s : } . ǫ b ) . σ , , ρ ( ¯ θ, a  and a ± = 0 ¯ θ ρ ǫ a { heet metric V ξ ffeomorphism and supersym- ˙ ρ X ]. To explicitly see the form − = r · . − ǫ erably. The equations of motion ǫ 21 i ± 2 i + = 0 ˙ + ψ θ , ψ α . ˙ . , ǫ · θ a ǫ b ) , . a ρ , . σ + − V iψ V + = 0 a a ( = 0 ξ ξ a , ǫ ξ ξ ′ + ), we need to consider tranformations in f + − µ + a a  ǫ V ψ ˙ i i 2 2 ψ to satisfy this algebra, is to replace them by 2 = · matrices in the fermionic part of the action. = 0 ¯ θ∂ θ∂ ξ, ǫ θ∂ θ∂ ˙ a 1 . This is also manifested by the choice of the ¯ X . A general superspace transformation on the 1 2 2 1 4 4 1 1 1 = 2 a ψ + + ˙ b ρ h a , ξ ρ i 2 a a } ξ σ V – 10 – + + + + b ξ 2 ξ a , ξ + d + − , + − = 0 ) , ρ ′ ′ + V + ǫ ǫ ǫ ǫ ) a σ µ ) become constraints the string is subject to. In this a Z a 1 ( ρ X + + + + ¨ ξ σ σ g { · X ] for further details. The transformation on superspace = 2 ¯ = ¯ θ θ θ θ 3.4 ˙ = } + = 18 X = ( S b = = = = a a τ ′ ′ ′ ′ 0 ′ ′ ) ˙ ¯ ¯ σ θ θ ξ θ θ σ , ,V σ ). Hence the Clifford Algebra is modified to a ( } ′ → → → V 0 → → → f = 0 , { ¯ is replaced by a product of the vector densities a ¯ θ θ a θ θ 1 2 = σ { σ ˙ assumes the role of X 0 ab , as ξ η a = a a V V V ), where 1) superspace. Thus let us consider Grassmanian coordiante 3.3 , = (1 action ( the vector density worldsheet is given by A convenient choice of the gamma matrices a gauge choice of changed. The metric the 2 dimensional worldsheet coordinates We recommend the reader to see [ N However, when we consider the tensionless limit, the worlds The equations of motion and thebecome constraints simplify consid thus becomes generate. Thus the interpretation of the Clifford Algebra: of the generators for such a transformation ( gauge, the action becomes: The last two equations of motion ( The constraints take the form: Then we are left with the following differential equations fo Solving for the parameters givemetry us transformations: the form for the allowed di This form of the parameters agree with the results of [ JHEP10(2016)113 ξ al (3.20) (3.19) (3.25) (3.16) (3.22) (3.18a) (3.18b) (3.17b) Φ (3.17a) to each of  ¯  θ τ ∂ inσ e ,  ¯ θ∂ ; (3.23) ¯ θ ± n i ′ 2 n . τ f Q +  + 1 2 ± n g∂ m τ . ¯ ; (3.24) ; (3.21) ζ θ s τ + ∂ M ¯ θ∂ inσ ∂ + ) n − i  2 ) = r e n ǫ X g ± n inσ ( M  + inσ ζ − ′ ring, the algebra would get . e ¯ ie θ M ) = + ∂ αα n m ± θ )Φ = δ = X  ¯ ǫ θ ∂ ; ( n − n − = =   ¯ ǫ ± θ∂ ) with the appropriate forms of ] = ( θ Q M  } δ ′ ± ′ n Q ¯ θ ǫ f α s + ) = 1 2 ¯ 3.16 ; θ∂ are fermionic generators. Since all the θ ,M ; −  ,Q ; ǫ n + + m ±  α r ( ¯ θ θ + L δθ∂ M Q − inσ Q ǫ n e { ¯ θ∂ Q θ∂ + b  n  b σ + + , n 2 1 ; θ σ X n we can do a fourier expansion in – 11 –  i X + δσ∂ )]Φ θ∂ τ f∂ σ − τ  − = + ǫ θ∂ + ( 2 ; g τ∂ τ i in 2 τ r −  ) = ;[ . ′ ∂ + g n ; + + Q f ; (  δτ∂ α m + τ  θ ¯ θ + τ Q m ∂ M 0), we could expand ( inσ − ] = 0 ) + ǫ , σ L e   α r θ∂ ) + i n r 2 + inτ∂ i ǫ ; 2 Φ = ( n a f∂ ǫ ( δ n ) would transform as ,Q − + + + ¯ = (1 θ − that we discussed in the previous section, without the centr + L n m θ σ 2 n Q θ a are functions of X m ) = ) = ∂ + m a ∂ M H , θ, ǫ V   f ± + = a  i ( ǫ ǫ 2 n ) + σ ( f L X g inσ ] = [ ] = ( ] = + ( inσ + i n n α r ie e ) Q g and − M are the bosonic generators, while ,L = = ,Q + ,M m 3.12 m n τ m ) = + n M f, g ) + ′ L L L f [ f f [ Q M ( ( [ L L and  from ( L = [ ǫ A superfield Φ( Φ = δ we will obtain the fourier modes of the generators them as follows: Defining the bosonic and fermionic generators such that Here parameters This is the 2d SGCA Once we have chosen satisfying the following algebra where the generators have the form and extensions. When we movecentral to extensions the as quantum indicated tensionless in superst the previous section. JHEP10(2016)113 ay, (3.29) (3.34) (3.27) (3.33) (3.35) (3.32) (3.31) (3.26) (3.30c) (3.30a) (3.30b) . , , s . +  r ) (3.28) n ′ = 0 + ] = 0 σ M = 0 − r n β − inσ αβ inσ · n . irσ δ − − σ ,M r + e − e ( µν inσ ) − = e m − n δ m η µ ′ − n β ± r s } B e M M [ β s + · αα  Q + r ) δ δ n n n m ′ r ,Q X inτB , µν + + − X α r ): ′ n btained by using the commu- + − r r η αα ′ c straints can be worked out by − B + Q β β rs are not in a simple harmonic c δ . Writing them down explicitly { · · = µ n m σ 3.29 m = r r = 4 } A X , = 4 M ) + − ( − − } ′ ) ′ β β 1 n σ n inσ and  ( irσ = 0 inτ να s ′ ) + − − − ν 6=0 α τ e n n e − X n , β  ′ , inσ  + m , ψ n n + + c r r µα − ) r + + 2 β r + e β σ · m i m ( ± { m √ (2 n i ] = ( ] = 0 µ r α B β B ) and using mode expansions. The solutions in n α r + − · · ψ r · irσ – 12 – + , { β X − 3.4 m m m τ ,Q e ,M )( µν 1 4 − − − µ 0 inτM m η ± n , m µ A B B ) n r B + ′ ′ L − M + β [ + [ c n n σ r n m m m 2 r m + + X X X + r L − √ m m (2 X ±  h   m ′ mδ σ , β B B c r + ( r r · · · n n n , 2 δ X + µ X X X X n ′ ′ ′ ′ m m m 2 1 x √ α m µν + ] = 2 c c c c − − − η Q ν + n m B A B  L ) = ) = We will now look to derive the above symmetries in a different w ) = 2 = 4 = 2 = 2 r ,B m m m n ′ )] = 2 µ m ˙ ′ X X X − X ˙ ψ τ, σ σ, τ σ A − 1 2 2 2 1 1 X · · ( ( [ ( 2 m µ ν µ ± ¯ ± m ψ ˙ = = =  ψ X i X ψ 2 , n n ± r ) + L ] = ] = ( σ Q M ′ ( n α r µ X · ,L ,Q X [ ˙ m m X L L [ [ We note here that the commutationoscillator relations basis. of the This oscillato wouldtaking be the of appropriate importance derivatives later. with The respect con to gives us and that of the modes: The commutation relations follows by considering: the NS-NS sector are of the following form The classical algebra of thetation modes and of anti-commutation the relations constraints of can the be oscillators o ( viz. by solving the equations of motion ( Mode expansions. In the above, we have made the following definitions: JHEP10(2016)113 . ) I ]. ǫ 1 14 3.30 fit into . 3 the Virasoro ). If we start c ) take the form = 0, the SGCA + ¯ onless theories as ), then there is a c ( M ǫ c 3.30 in the parent theory , there is the obvious 5.20 I = he equations of ( c o make the connection tended. These central M = ¯ as the residual symmetry c ve, however, been unable , with bove ( c I 2 which does not scale like ion between the closed and nsions there may emerge in sis of asymptotic symmetries t any central extensions and central terms in the limit [ s theory, one needs to rework ionless superstring. and ent gauge, but as if often with on ra. It is conceivable that there s to a bit obscured and harder es SGCA s theory. The arguments above ¯ c ons would be zero hinting at the ny number of dimensions. Let us closed string and the open string, Like in the bosonic case, the limit − c ) indices, this above symmetry algebra = ] and is called the Super BMS L 41 c α, β . We would be able to organize the physics of H – 13 – ], that when we considered closed tensionless strings 17 ]. If we are able to come up with a choice of gauge such that from the modes of the solutions of the equations of motion. 23 H counts the number of bosonic worldsheet fields in the tensile c ) to see how the algebraic considerations considered before 2.22 = 0. Again, L c ) and ( = 0, the residual gauge symmetry truncated to a single copy of = 1 Supergravity recently performed in [ = 0. As before, we may wish to make a distinction between tensi M 2.20 N c M As mentioned before, when stripped off the ( We saw in the bosonic case in [ What we have seen here is that if we use the choice of gauge ( It is also of interest to see that the classical constraints a c is the very same algebraof that 3d arises from the canonical analy the usual tensile methodsthis to case. check what restrictions on dime derived from a tensile theoryhold or for a a fundamentally derived tensionles theory and for a fundamentally tensionles So, algebra. This will again be a preferred choice of gauge. algebra indicating a deep relation betweenpreviously the anticipated tensionless e.g. in [ from a parent theory free from diffeomorphism anomaly, then the tensionless superstring inexists terms another of “good” this choice symmetry of algeb gauge which lands us on SGCA with in the tensionless superstring the residual symmetry becom residual gauge symmetry which is the SGCA required from the algebra,with viz. ( EM tensorthis = particular 0. example. We can compare t The algebra has two distinct central extensions: reduces to a singleto copy find of this the gaugebetween Super-Virasoro closed yet. algebra. and open It tensionless We isgauge strings ha choices, also even the in possible physics the thatto of curr one interpret this in may particular the be property current seem able form. t and hence We have rederived the SGCA advantage of making theopen supersymmetric strings even version here. of As the we have relat mentioned earlier in secti theory and hence in any healthy theory, this is a finite number from the tensile string suggestsfact that that tensionless these superstrings central would extensi just be remind consistent the reader in of a the argument of the vanishing of the As stressed previously, thiswhen is we the look symmetryextensions algebra to would withou quantize then determine the the theory, consistency this of the algebra tens wouldCentral get terms ex and the other symmetry algebra. JHEP10(2016)113 (4.9) (4.1) (4.7) (4.4) (4.8) (4.2) (4.6) (4.3a) (4.3b) ) s σ + i − r ) τ σ L ( − ir τ ( − = 2 e in µ r } − erated: b ˜ s ]. Starting from the e 1 2 n µ n Q 44 + , + α r r Z rrent are directly related b X i ∈ + ˜ r) are listed below: · r µ ) {Q ′ r ψ σ inking of the tensionless su- α − a e of the mapping between the by + b µν 2 = 0 (4.5) ∂ , τ ) η ( a r ion on the worldsheet. We now ous properties of the tensionless √ n + ρ + 0 in . µν µ ψ ativistic contraction of the tensile − m + n, η ¯ ψ e ± ) = + r i Q 0 = 0 µ n ∂ m s, ·  α (2 − µ ± r h + σ, τ ± r ψ µ r ( mδ 1 n δ ψ − X ∓ µ − X i 2 ∂ a 1 4 = 6=0 2 m ] = ∂ X n } + ν n µ ′ + = 0  ν s , ψ ˜ 2 α α b ˜ ) n ) X , ′ , 2 – 14 – σ a X + . µ r µ m = 0; ] = + r ∂ √ ± X m b ˜ r τ α h i + ∂ µ ( { α σ · ± Q m ir · + 2 X , b = ˙ − ± d a · τ m X ] = [˜ m e } ∂ ψ µ 0 ν n − µ r a L ν s m Z [ b α α ∂ ′ − = ( = , b , α 1 2 2 T α α µ r + µ m m , 2 ± b − X Z n α { J ±± X [ √ m ∈ + 2 1 X T r = ′ m S α + 2 = = L 2 ) µ r n n x √ L Q − ) are applicable, and gives us the following brackets: Before that, let us briefly review the tensile case [ ) = ) = m 3.28 τ, σ σ, τ ( ( ] = ( µ n µ + ) the Super-Virasoro Algebra (without central terms) is gen L ψ X , 4.4 m L [ The relations ( The equations of motions are: tensile action: The components of the energyto momentum tensor the and constraints: the super cu Using ( The bosonic and fermionc mode expansions (in the NS-NS secto string worldsheet. Tensile case. In our analysis soperstring far, as we an have intrinsic been object andsuperstring essentially we theory confined have derived without to the taking th vari wish recourse to to venture the into the contract othertensile direction and where tensionless we make theory full i.e. us by taking the ultra-rel 4 Limit from the tensile superstring In terms of modes, the Super-Virasoro constraints are given JHEP10(2016)113 . g π inσ − + 2 e . The (4.11) (4.12) (4.15) (4.10) ′  ǫ (4.13a) (4.13b) c σ ) i n ) ∼ µ − → ′ α σ ). Therefore α iǫnτ + ˜ µ n e parent Super − as 3.34 α ′ irσ ( ǫ (1 irσ α − dentify e ators, the fermionic − √ r e inσ , the fermionic scaling µ r µ − e to see how to obtain e , b b ˜ ) µ n sically, it can be easily inτ H n α r r − µ − ld scale ) (4.14) X X ) α ′ ′ n ow the fermions would scale n ) + ˜ c c − s opposed to shrinking into a + ˜ µ − 2 2 SGCA ¯ us the connection between the uper-Virasoro algebra to arrive L ˜ α µ n der is the ultra-relativistic limit √ √ r α iǫnτ . + − − ( r ≈ ≈ ¯ ǫ n Q µ n − µ − ) ) ǫ b L ˜ √ α ( ( (1 √ ǫ = ǫ = iǫrτ iǫrτ 1 = = inσ √ µ n − µ − − − r  n − r e 1 n µ n Q (1 (1 α ,B 6=0 h . The physical insight at play here is that in irσ irσ  X n ,M – 15 – 1 n n σ − + ′ n e e c µ − − µ r µ r 2 ˜ α 6=0 ¯ b b → ˜ L X n √ r ] and hence the limit to consider would be one where i − ′ σ r r − c Q ǫ µ n 14 + X X ǫ 2 ; n ′ ′ α τ . √ L ) α α  ǫτ r ), corresponds to the algebra obtained in ( µ 0 2 2 i ǫ , β ǫψ = = 1 α µ r The tensionless limit on the worldsheet is obtained by takin √ √ √ √ + b → 4.9 ǫ ǫ n + r + ˜ τ √ √ L = = ǫτ → 0 limit) [ µ 0 Q µ 0 α µ n + ψ ( α → µ r ′ ǫ A ) = ) = c ǫ β α √ 2 ′ c σ, τ σ, τ r 2 ( ( µ µ + − √ ψ ψ + 2 + µ µ ) don’t get scaled at all in this particular contraction of th x x . It is interesting to note here that at the level of the oscill H = ) = . But we are on the closed string worldsheet and would like to i 4.13b τ, σ Let us now apply this on the mode expansions of the tensile cas ( µ → ∞ X the tensionless limit, thepoint string particle will in grow the long and floppy (a tensile and the tensionless constraints. Plugging the above relations back into the constraints give The fermionic ones scale as would be homogenous: If we compare theseseen modes that with the one that we obtained intrin this analysis agrees with the “homogeneousat scaling” SGCA of the S Virasoro algebra to the homogeneous SGCA. σ bosonic modes transform under this scaling as ones ( the following contraction Tensionless contraction. This scaling, together with ( just mentioned above. Alongin with the this, case one of needs the to tensionless specify superstring. h Since we have the So, in the case of the closed string, the contraction to consi the tensionless ones. We need to keep in mind that we also shou JHEP10(2016)113 (5.6) (5.2) (5.7) (5.1) (5.3) (5.4) , vacuum: = 0 i i 0 | . S phy s | = 0 2 − b best formulated in ˜ J i | ′ ... . phy ature. 1 | aces of the limiting and phy s hed. In both cases, the h ± r − in between physical states pose the above constraints = 0 b eation operators acting on ˜ ory and then imposing the Q | nic tensionless string, there . i ′ , e tensionless subsector. This ss the tensionless superstring 0 e tensionless superstring. We | ert space. For the tensionless R r = 0 (5.5) r he physical states. The mass is g theory, the Hilbert space that b ˜ = 0 phy responding Hilbert space. These − µ i h i φ 0 = | n i B , . . . b ′ 0 phy B 2 | c 1 | r r · 1 µ b = 0 − J r n ˙ | X ′ i − ′ . b = B 1 µ πc M phy phy | 2 h n = 0; m n X − i – 16 – = dσP 0 M ˜ , α | | ⇒ π µ ′ of the string n 2 P 0 µ α ... = 0 Z p 1 phy i = ˜ h ) have been transformed to their quantum counterparts m = = 0 i − i 0 µ ˜ phy | , α | p φ 3.30 n | . 2 0 α T | = 0 N ′ M n i : − i phy h 0 phy | | in this Hilbert space is built up of oscillators acting on the constraint: n . . . α i , 1 L 0 φ | n | ′ M − = 0 α i phy h . In terms of the modes of the EM tensor, this boils down to Y i ′ phy | = 1 i T phy | φ | ′ | , i The crucial point of difference is the fact that the Hilbert sp As in the bosonic case, the theory of quantum superstrings is phy h phy which leads to the total momentum we would start with ison that the of states the of tensile thatHilbert theory theory space and to is we obtain will built thethis im allowed out vacuum. states of The of the vacuum th is tensile defined vacuum by and all the cr the fundamental theories are different.theory When as we wish a to sub-sector discu of a well-behaved tensile superstrin by elevating them to quantum| operators and sandwiching them superstring, in close analogyare with two the distinct treatment methodstensionless constraints of in are the imposed which on boso thisconstraints the states are process of the can cor be approac We have so far dealtnow want exclusively to with make classical some aspects preliminary of remarks about th itsthe quantum covariant n approach byconstraints quantising as the physical theory conditions as on a the free states the of the Hilb 5 Quantum tensionless superstrings where the classical conditions ( So a general state derived from the The tensionless constraints acting on these states give us t Now we have, JHEP10(2016)113   i do 0 | (5.9) S (5.13) (5.10) (5.11) (5.12) s − A, B b ˜ .. 1 . s  − . ǫ b ˜ † k . 1 √ R . r k a − ± √ ǫ ..b ] = 0 1 = √ r m ) (5.8) k  − ˜ µ n n C − 1 2 om the spacetime index .b ˜ ensionless sector. nnihilation operators and α − , n bov transformation can be n M B = θ ctor which is characterised µ − , α C m heet. Redefining coefficients ˜ rbitrary spins and hence this k gebra. [ α st as in the bosonic case, the ± + − f θ ˜ y of massless higher spins. It is n α ionless creation-annihilation op- k a , ionless. The oscillators .. − 0 The fundamental tensionless the- 1 θ, √ A + cosh i m ( m, sinh φ n − = 2 1 | + µ − − ˜ n n α where k . − µ n B α N nδ · = sinh θ α = n , n n − θ α − n ˜ − ] = α ˜ C ) and hence we need to change to a different sinh ..α + B m . These are defined as 1 f ˜ C n − ˜ θ, f where C – 17 – 6=0 , − + X n 3.29 n = cosh = , α ′ n ˜ C, ) C , 2 c [ − n i iG iG α n Y e e B = ˜ a ) = cosh − µ µ n n , . i f n 0 ˜ + α α does not get any contribution from the fermionic modes, n + φ − a | n f m, n = iG iG 0 α + A − − − n B n ( M + ˜ e e ˜ · † n C 1 2 n ˜ a nδ 0 = = . α = † n B , )( µ µ ′ n n a n 2 n n ] = c ˜ h C C − α C θ m − ˜ α + ˜ − =1 ,C ∞ = n f X n n − i i C φ α + [ | ( = n µ ǫ . p α G µ 6=0 + p X n f = 0 −   i ]. We will call these operators ′ = 2 φ c | = 17 n 2 0 i C → φ m ǫ | 2 ⇒ = lim m These new oscillators now do obey the harmonic oscillator al not obey a harmonic oscillator algebra ( structure (which we have suppressedis above), a also link can between carry theinteresting a tensionless to superstrings note and that the since theor So we get thattensionless in the limit case from ofby the the states tensile tensionless superstrings, which theory ju have lands zero up mass. in a These subse states, as is obvious fr ory is built out of oscillators which are intrinsically tens the mass formula does not change fromBogoliubov the transformations case on of the the bosonic worldsheet. t In terms of theerators tensile are oscillators, given the by fundamentally tens basis [ We see that these newhence operators these are define a Bogoliubov mixturein transformations of a on creation more and the familiar a worlds form: we find that with thisrecast new as, definition of coefficients the Bogoliu The mass of the state is given by The generator of the above transformation can be written as, JHEP10(2016)113 (5.14) (5.16) (5.17) (5.19) (5.20) (5.18) (5.15) s. isfy . i 00 | S s − − . . l † 0 c . . . β attention purely to the aling with the fermionic 1 kl ed string theory the fun- s u . − − nsformations to be canoni- , to the physical state condi- + ¯ he tensile vacuum does not sformations in much greater t. Let us remind the reader . β e now linked by the fermionic j he above brackets. ctor is more intricate and will = 0 e that even though there is no c ich leads from the usual tensile = 0 ace is given by ion operators and the fermionic e tensile theory. This obviously R ormations before specifying our } , n, r > rs with tension, as is clear from r kj † j superstring Hilbert space is built . T + v − , c ! . = 0 † k = ¯ r VU c i − k † { . . . β 0 0 0 1 0 limit, the map between the fermionic b ˜ c ˜ + 00 1

| r = T = → ± + r − → } = β ǫ j − r k † . β , c = k i M c , c ,UV , β ,V { – 18 – m k † r 00 | c − b ! n ˜ = 1 ik C , ˜ = v C † 1 0 0 0 kj + δ ... ]. r + =

β 1 j i which is defined by VV 53 = c m = i } ij 00 + − | † j ˜ u U † 00 C n | , c . ]. C = k c i UU N 53 { ˜ c n : are the matrix representations of the above transformation − be a set of fermionic annihilation operators. Thus, they sat i → j ij i 00 c v c | ...C = 1 n − ,V C ij u Y ). = = U 5.2 i ψ In the superstring, the two sets of fermionic oscillators ar It is important to note here again that we have restricted our The main point, as stressed above, is that in the bosonic clos For our tensionless case, we find that in the | damental tensionless Hilbert space isalso very holds different true from for th the superstringfrom theory. the The tensionless tensionless vacuum version of the Bogoliubovbriefly transformations of the on general the theoryparticular of worldshee case. fermionic Let Bogoliubov transf NS-NS sector of the tensionlessbe superstring addressed theory. in The detail R elsewhere se [ Now we consider canonical transformations which preserve t A general state in the fundamentally tensionless Hilbert sp Here complex conjugate are denotedcal, by we bars. need For the above tra generators and the fermionicdetail version in of upcoming work the [ Bogoliubov tran Since the C’s are agenerators mixture of are tensile given creation and byland annihilat us the onto above, the the tensionlessscaling vacuum. tensionless of It the limit is fermionic of interestingthe creation to t above, and not there annihilation is still operato theory a non-trivial to mixing a of theory the with modes a wh different vacuum state. We would be de Hence we see that the transformation matrices become oscillators are given by where The allowed states aretions the ( ones in this Hilbert space subject JHEP10(2016)113 ]. ]. 53 41 gauge in the he different vacua . This (stripped of H of the two approaches ion theory of the two attering amplitudes in erent. We would like to ciple which will help us ressed and some of it is also remarked about the ally tensionless vacuum. erstrings following earlier s and showed two different ons before we close. et description still existed ructure of the Bogoliubov ring worldsheet. When we SGCA e for what is a much longer l would be to look at the d that the usual tensionless strings residual symmetries tions of motion using mode ored in upcoming work [ gested the use of Carrollian he supersymmetric case, we . We would obviously need iz. the fundamental and the mmetries of the worldsheet. totically flat spacetimes [ lgebra obtained recently in the ] was the connection to Hage- n structures to emerge in the ng amplitudes there. 17 ymmetry, the algebra of which is ic and the limit procedure gave us gs by appealing to the symmetries upersymmetrizations of the parent ]. It is very likely that a similar story 17 – 19 – ]. We have seen that when we fix the equivalent of the conformal 18 Most immediately, we would like to expand on our comments on t It is important to emphasise that this is just a stepping ston In the process, we also visited some algebraic aspects of GCA One of the principal goals of this project is to understand sc One of the most intriguing connections outlined in [ We performed an intrinsic analysis of the tensionless super Like in the bosonicsystematically solve case, the we theory have ofon now tensionless the found superstrin string an worldsheet.already organising work prin There in progress. are numerous Let us things comment to on be some open add directi project of trying to understand tensionless strings from sy ultra-relativistic contractions leading tobosonic two different GCA. s We commentedalgebras. on some aspects of the representat the tensionless limit fromto the first symmetries understand onsupersymmetric the the tensionless bosonic string worldsheet calculation, theory and but the the scatteri final goa limiting one. We haveextend pointed our out analysis that to thetransformations include NS-NS between vacua the are the R diff thermal sector two nature and sets analyse of of the the oscillators. st tensionless vacuum We in had [ of the two different approaches to the tensionless theory, - v supersymmetric version. dorn physics. Itnear was remarkable the that Hagedorn the temperaturewish notion defying to of earlier investigate a notions. this worldshe manifolds in In for more t detail. this application. The bosonic We case expect sug Super-Carrollia would hold true in the supersymmetric case. This will be expl were looking at classical aspectsthe of same the theory, answers. the intrins Weand briefly showed that commented the on vacua of thelimit the quantum two on were aspects the markedly different tensile an vacuum does not land one on the fundament and rederived the same byexpansions. looking We at obtained the the solutions same of from the a equa contraction on the st tensionless superstring action, theregiven is by a the residual homogenous gauge Super s Galilean Conformal Algebra or the indices on the supercharges) isanalysis actually of the asymptotic same analysis as the of a 3d Supergravity in asymp In this paper, wework [ have looked at the theory of tensionless sup 6 Conclusions JHEP10(2016)113 , , ommons ] is also very 52 (1988) 1229 ]. 60 1), less higher spins ]. , SPIRE ull strings [ pagates. Hence a singular hanks the Fulbright Foun- gular is also happening to IN hailesh Lal for discussions. pacetime in this tensionless search. We also gratefully m should admit tensionless Stony Brook (AB and PP), [ SPIRE redited. vestigating the consequences in the tensionless limit. The IN s of our methods. to be examined in each case to rsities where parts of this work els (AB and SC), LPTHE Paris n would be to consider the case lly and also understand the fate ent theories. ][ posed to be background indepen- f Technology (PP), Saha Institute Phys. Rev. Lett. , (1977) 1722 . D 16 hep-th/0103247 [ JHS/60: Conference in Honor of John Schwarz’s – 20 – ]. ]. , in SPIRE Phys. Rev. SPIRE , IN [ IN [ (2001) 113 ), which permits any use, distribution and reproduction in The High-Energy Behavior of String Scattering Amplitudes String Theory Beyond the Planck Scale 102 (1988) 407 (1987) 129 CC-BY 4.0 This article is distributed under the terms of the Creative C Stringy gravity, interacting tensionless strings and mass , California Institute of Technology, Pasadena U.S.A. (200 B 303 High-Energy Symmetries of String Theory B 197 Spacetime Reconstruction Classical Null Strings ]. SPIRE IN 60th Birthday Nucl. Phys. Proc. Suppl. http://theory.caltech.edu/jhs60/witten/1.html Phys. Lett. Nucl. Phys. [ Another of our current avenues of exploration is the fate of s The recently discovered link between the ambitwistors and n There are five consistent superstring theories and all of the [1] A. Schild, [5] B. Sundborg, [6] E. Witten, [3] D.J. Gross and P.F. Mende, [4] D.J. Gross, [2] D.J. Gross and P.F. Mende, any medium, provided the original author(s) and source areReferences c Center for Theoretical Physics, Massachusetts Instituteof o Nuclear Physics, Kolkata (PP), Universite(AB) Libre and de Bruss the Albert Einstein Institute, PotsdamOpen (AB). Access. It is a pleasureWe to acknowledge thank the Aritra support Banerjee, ofdation, Rudranil various Basu PP funding and thanks agencies: S theacknowledge AB the Council t warm of hospitality Scientific ofwere many and carried institutes/unive out: Industrial the Re Simon Center for Geometry and Physics, intriguing and opens completely new avenues of application Acknowledgments Attribution License ( limits. It would be ofof interest the to examine web all of ofdepartures dualities these of carefu that the classical link andsee the quantum if parent superstrings there tensile needs is theories further substructure emerging in these differ of the superstring. limit. String theory (in itsdent second and quantised avatar) is is expected sup limit to determine on the the geometry stringthe in worldsheet spacetime which where would it it mean pro propagates.of that We this are something for in sin the the bosonic process of theory in and the obvious generalisatio JHEP10(2016)113 , , , , ]. ] SPIRE ]. , , IN ][ (2013) 141 , , (2014) 044 SPIRE 05 ]. ries and higher spins IN 11 ][ ]. , hep-th/0205131 SPIRE JHEP (2013) 214006 , IN , ]. , [ JHEP ]. SPIRE , Classical and quantized IN ]. hep-th/0305098 ]. A 46 [ ]. ]. ][ s, SPIRE IN SPIRE [ (2003) 403] [ Strings IN SPIRE ]. SPIRE ABJ Triality: from Higher Spin Fields to hep-th/9307108 SPIRE SPIRE ][ & [ ]. IN IN IN IN The Zero Tension Limit of the Spinning J. Phys. ][ Quantization of the Null String and Absence ][ , ][ ][ Dual for Minimal Model CFTs (2004) 178 B 660 Quantum Null (super)strings Null spinning strings SPIRE 3 arXiv:1006.3354 SPIRE Tensionless Strings from Worldsheet Symmetries IN ]. (1986) 326 , IN ][ – 21 – ][ ]. (1994) 122 ]. B 584 arXiv:1207.4485 [ . An AdS Higher Spins SPIRE B 182 AdS dual of the critical O(N) vector model IN [ SPIRE arXiv:1011.2986 SPIRE [ The Classical Bosonic String in the Zero Tension Limit B 411 hep-th/0305155 IN Erratum ibid. Galilean Conformal Algebras and AdS/CFT Tensionless strings, WZW models at critical level and [ [ [ IN hep-th/0210114 hep-th/0311257 hep-th/0401177 [ [ [ [ On higher spins and the tensionless limit of string theory ]. ]. ]. Phys. Lett. (1991) 331 , Massless higher spins and holography Phys. Lett. (2013) 214009 (1986) L73 , SPIRE SPIRE SPIRE arXiv:0902.1385 arXiv:1507.04361 [ [ 3 IN IN IN (2004) 83 (2003) 159 (1990) 143 (2002) 303 B 258 Nucl. Phys. (1989) 335 (2002) 213 ][ ][ ][ (2004) 702 , (2011) 066007 A 46 52 Higher spin gauge theories in various dimensions Notes on Strings and Higher Spins B 682 B 669 B 338 B 644 B 225 B 550 On the tensionless limit of bosonic strings, infinite symmet D 83 Tensionless Strings and Galilean Conformal Algebra The BMS/GCA correspondence (2009) 037 (2016) 158 ]. J. Phys. Physl. Lett. 07 01 , , SPIRE arXiv:1112.4285 arXiv:1303.0291 arXiv:1406.6103 IN Nucl. Phys. Nucl. Phys. massless higher spin fields Nucl. Phys. [ Class. Quant. Grav. of Critical Dimensions Phys. Lett. JHEP String [ JHEP [ Phys. Rev. Fortsch. Phys. Strings Phys. Lett. Nucl. Phys. [ tensionless strings [9] M.R. Gaberdiel and R. Gopakumar, [8] E. Sezgin and P. Sundell, [7] I.R. Klebanov and A.M. Polyakov, [25] G. Bonelli, [26] U. and Lindstr¨om M. Zabzine, [22] J. Gamboa, C. Ramirez and M. Ruiz-Altaba, [23] A. Sagnotti, [24] A. Sagnotti and M. Tsulaia, [20] F. Lizzi, B. Rai, G. Sparano and A. Srivastava, [21] J. Gamboa, C. Ramirez and M. Ruiz-Altaba, [17] A. Bagchi, S. Chakrabortty and P. Parekh, [18] U. Lindstrom, B. Sundborg and G. Theodoridis, [19] A. Karlhede and U. Lindstr¨om, [15] A. Bagchi and R. Gopakumar, [16] A. Bagchi, [12] M.R. Gaberdiel and R. Gopakumar, [13] J. Isberg, U. B. Lindstr¨om, Sundborg and G. Theodoridi [14] A. Bagchi, [10] M.A. Vasiliev, [11] C.-M. Chang, S. Minwalla, T. Sharma and X. Yin, JHEP10(2016)113 ] ] ]. ull , , , SPIRE IN ]. ][ gr-qc/0610130 ]. (2010) 004 [ , , SPIRE arXiv:1204.3288 (1962) 2851 [ , IN 08 , , ][ ical solutions ]. 128 SPIRE invariant boundary theory IN 3 ][ JHEP ]. , SPIRE (2007) F15 IN ]. [ arXiv:1407.4275 [ 24 Liouville theory , ]. (2012) 024020 SPIRE Phys. Rev. , IN , SPIRE Gravitational waves in general Asymptotic symmetries and dynamics Super-BMS ][ Holography of 3D Flat Cosmological IN GCA in 2d SPIRE D 86 arXiv:1203.5795 ]. [ ][ IN ]. ]. ]. ]. (2014) 071 ][ ]. . Galilean Yang-Mills Theory arXiv:1208.4372 The Flat limit of three dimensional Three-dimensional Bondi-Metzner-Sachs [ Conformal Carroll groups SPIRE 08 SPIRE SPIRE IN SPIRE SPIRE Superstring Theory. Vol. 1: Introduction IN IN SPIRE ][ IN IN arXiv:1510.08824 Phys. Rev. – 22 – [ ][ ][ (2012) 092 IN , , [ Class. Quant. Grav. ][ JHEP , hep-th/9711200 , 10 [ Galilean Conformal Electrodynamics arXiv:1210.0731 [ (2013) 141302 Central Charges in the Canonical Realization of Asymptotic limit of superconformal field theories and supergravity arXiv:1403.4213 (1962) 21 [ JHEP Classical central extension for asymptotic symmetries at n BMS/GCA Redux: Towards Flatspace Holography from ]. , N (1986) 207 110 On the tensionless limit of gauged WZW models (1999) 1113 104 A 269 SPIRE arXiv:1408.0810 arXiv:1512.08375 arXiv:1208.4371 hep-th/0403165 [ [ [ [ 38 IN ][ (2013) 124032 The Large- (2014) 335204 Cambridge Monogr. Math. Phys. Entropy of three-dimensional asymptotically flat cosmolog D 87 (2014) 061 (2016) 051 (2012) 095 (2004) 049 Phys. Rev. Lett. Asymptotic symmetries in gravitational theory A 47 , ]. ]. ]. 11 04 10 06 SPIRE SPIRE SPIRE IN arXiv:0912.1090 IN IN from three-dimensional flat supergravity of three-dimensional flat supergravity Horizons infinity in three spacetime dimensions invariant two-dimensional field theoriesPhys. as Rev. the flat limit of submitted to: [ JHEP Non-Relativistic Symmetries asymptotically anti-de Sitter spacetimes [ JHEP [ JHEP Proc. Roy. Soc. Lond. [ Commun. Math. Phys. Int. J. Theor. Phys. relativity. 7. Waves from axisymmetric isolated systems JHEP J. Phys. Symmetries: An Example from Three-Dimensional Gravity [41] G. Barnich, L. Donnay, J. Matulich and R. Troncoso, [34] A. Bagchi, S. Detournay, R. Fareghbal and J. Sim´on, [33] G. Barnich and G. Compere, [43] G. Barnich, A. Gomberoff and H.A. Gonz´alez, [44] M.B. Green, J.H. Schwarz and E. Witten, [42] G. Barnich, L. Donnay, J. Matulich and R. Troncoso, [39] A. Bagchi and R. Fareghbal, [40] G. Barnich, A. Gomberoff and H.A. Gonzalez, [36] A. Bagchi, R. Gopakumar, I. Mandal and A. Miwa, [37] A. Bagchi, R. Basu and A. Mehra, [38] A. Bagchi, R. Basu, A. Kakkar and A. Mehra, [35] G. Barnich, [32] R. Sachs, [30] J.M. Maldacena, [31] H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, [28] C. Duval, G.W. Gibbons and P.A. Horvathy, [29] J.D. Brown and M. Henneaux, [27] I. Bakas and C. Sourdis, JHEP10(2016)113 ] , ]. SPIRE IN ][ , arXiv:1606.05636 , , arXiv:1512.03353 [ (2010) 018 ]. , ]. 11 ]. ]. BMS Modules in Three Rotating Higher Spin Partition SPIRE super-Virasoro arXiv:1603.03812 SPIRE IN [ SPIRE (2016) 034 JHEP IN 2) SPIRE ][ IN , , IN ][ 04 ][ ][ = (2 N Tensionless Superstrings: Vacuum structure, JHEP , – 23 – (2016) 1650068 Galilean Superconformal Symmetries arXiv:0905.0188 [ arXiv:0905.0580 [ A 31 , in preperation. arXiv:1607.02439 arXiv:0905.0141 [ [ ]. On the null origin of the ambitwistor string Super-GCA from Supersymmetric Extension of Galilean Conformal Algebras SPIRE IN (2016) 195 (2009) 411 (2010) 042301 ][ (2009) 086011 51 Super Galilean conformal algebra in AdS/CFT Int. J. Mod. Phys. , B 754 B 678 D 80 Supersymmetric Extension of GCA in 2d ]. ]. SPIRE SPIRE IN IN arXiv:1003.0209 Critical dimensions and all that [ Dimensions Functions and Extended BMS Symmetries [ [ Phys. Lett. J. Math. Phys. Phys. Lett. Phys. Rev. [53] A. Bagchi, S. Chakrabortty and P. Parekh, [52] E. Casali and P. Tourkine, [51] A. Campoleoni, H.A. Gonzalez, B. Oblak and M. Riegler, [48] I. Mandal, [49] I. Mandal and A. Rayyan, [50] A. Campoleoni, H.A. Gonzalez, B. Oblak and M. Riegler, [46] J.A. de Azcarraga and J. Lukierski, [47] A. Bagchi and I. Mandal, [45] M. Sakaguchi,