Strings with non-relativistic conformal symmetry and limits of the AdS/CFT correspondence

Harmark, Troels; Hartong, Jelle; Menculini, Lorenzo; Obers, Niels A.; Yan, Ziqi

Published in: Journal of High Energy Physics

DOI: 10.1007/JHEP11(2018)190

Publication date: 2018

Document version Publisher's PDF, also known as Version of record

Document license: CC BY

Citation for published version (APA): Harmark, T., Hartong, J., Menculini, L., Obers, N. A., & Yan, Z. (2018). Strings with non-relativistic conformal symmetry and limits of the AdS/CFT correspondence. Journal of High Energy Physics, 2018(11), [190]. https://doi.org/10.1007/JHEP11(2018)190

Download date: 27. sep.. 2021 JHEP11(2018)190 d Springer and Ziqi Yan a November 11, 2018 November 11, 2018 November 29, 2018 : : : Received Accepted Published Niels A. Obers a,c , , Published for SISSA by https://doi.org/10.1007/JHEP11(2018)190 [email protected] Lorenzo Menculini, , b . Spin Matrix theory arises as non-relativistic limits of 5 S [email protected] , × 5 . 3 Jelle Hartong, 1810.05560 a The Authors. c AdS-CFT Correspondence, Sigma Models, Bosonic Strings, Integrable Field

We find a Polyakov-type action for strings moving in a torsional Newton- , [email protected] [email protected] Dipartimento di Fisica eVia Geologia, Universit`adi Pascoli, Perugia, I-06123 and Perugia, INFN Italy Perimeter — Institute Sezione for di Theoretical Perugia, Physics, 31 Caroline St N, Waterloo,E-mail: ON N2L 6B9, Canada [email protected] The Niels Bohr Institute,Blegdamsvej University 17, of DK-2100 Copenhagen, Copenhagen, Denmark School of Mathematics andUniversity Maxwell of Institute Edinburgh, for Peter Mathematical Guthrie Sciences, Tait road, Edinburgh EH9 3FD, U.K. b c d a Open Access Article funded by SCOAP Keywords: Theories ArXiv ePrint: AdS/CFT correspondence and welimits show of that strings itthe on is AdS/CFT AdS correspondence realized close bystrings to the and BPS Spin ‘Spin Matrix bounds. theory Matrix provides Thea a Theory’ duality holographic framework between duality for of non-relativistic its more owndual tractable and to points holographic near towards dualities BPS whereby limits non-relativistic of the strings dual are field theory. space, we show thattarget the space world-sheet perspective theoryare these becomes still strings the relativistic. are Gomis-Ooguri non-relativisticsheet We action. theory show but becomes that their From non-relativistic one with a world-sheetby can an the theories infinite-dimensional take Galilean symmetry a algebra conformal scaling given algebra. limit This in scaling which limit also can the be world- taken in the context of the Abstract: Cartan geometry. This isfixing obtained the by momentum starting of with the the string relativistic along Polyakov a action non-compact and null isometry. For a flat target Troels Harmark, Strings with non-relativistic conformallimits symmetry of and the AdS/CFT correspondence JHEP11(2018)190 1 14 18 12 8 16 7 5 10 8 S 24 4 × 4 21 5 5 S 22 × – 1 – 5 5 S × 5 ] which presents an action principle for Newtonian gravity, but also goes 1 12 1 3) limit of strings on AdS | 20 2 , See however the recent work [ A.1 Dictionary fromA.2 string NC geometry Scaling limit of string NC geometry sigma model 4.4 SU(1 3.2 GCA symmetry of the non-relativistic sigma-model 4.1 Spin Matrix4.2 theory Limits of4.3 strings on AdS SU(2) limit and Polyakov Lagrangian for Landau-Lifshitz model 2.1 Polyakov action2.2 for strings on Relation TNC to geometry the Gomis-Ooguri non-relativistic string action 3.1 Polyakov action for strings with non-relativistic world-sheet theories 1 towards (relativistic) quantum gravity. Thispart third since path already has the been classical muchnor description less how of appreciated, such such a in a theory theory connects has to not string been theory. fully The understood, study ofbeyond non-relativistic by string including theory the effects of strong gravitational fields in non-relativistic gravity. dimension less, hasquantum provided gravity, a such further asparadox. the promising In microscopic arena these description to developments of theby address quest black approaching to it questions holes understand from and related quantum relativistic gravityan the classical to has alternative gravity information been route and/or guided is quantum to field first theory. consider However, the quantization of non-relativistic gravity as a step 1 Introduction One of the most appealingtum and gravity successful is approaches towards stringrelating a theory. consistent string theory The of theory celebrated quan- on discovery anti-de of the Sitter AdS/CFT backgrounds correspondence to conformal field theories in one 5 Discussion A Relation to string Newton-Cartan geometry 4 Spin Matrix theory limits of strings on AdS 3 Strings with non-relativistic world-sheet theories Contents 1 Introduction 2 Strings on torsional Newton-Cartan geometry JHEP11(2018)190 2 ]. ], ]. ]) / 2 2 1 4 19 , – 2 15 XXX = 4 SYM N ], showing that 5 ] (see also [ 14 = 4 spin chain exhibits non- N ] corresponds rather to a Nambu- ] is that of a periodic target space 5 5 ] of the 3 correspondence described by quantum me- 4 ]. – 2 – 20 field found in [ /CFT ]. One is thus led to consider SMT as a concrete and 5 η 4 this scaling limit is realized by the SMT limits of [ 5 S we find in this paper that the TNC geometry is extended with a periodic × 5 2.1 of these non-relativistic string theories. As part of this, we find 3 ] after a Legendre transform that puts the momentum conservation off- 13 – 6 [ . Indeed, these limits can in fact be thought of as a non-relativistic limit [ 3 2 S × Using the Polyakov-type formulation we show that the action for strings moving in In this paper we will present a general Polyakov-type formulation for the non- The first step towards uncovering this connection was taken in ref. [ A natural setting in which to expect a connection between non-relativistic gravity, As will be clear in section This Nambu-Goto form was also obtained in [ R 2 3 of this local Weylthen symmetry. show that When the going action to possesses a a flat symmetrytarget space corresponding space gauge to direction. on the the (two-dimensional) world-sheet, we Goto-type description that the correct interpretation ofdirection the on which the string has a winding mode. TNC geometry has a localfind Weyl that symmetry. the Interestingly, novel after class taking of the non-relativistic scaling sigma limit, models we exhibits a non-relativistic version Heisenberg spin chain. relativistic string action onsigma-model theory TNC obtained geometry from as thepresent well SMT work, scaling as the limit. formulation the In originally corresponding fact, presented non-relativistic as in also [ discussed in the applied to strings onThe AdS SMT limit isthat have in been turn studied in closelyplest connection example related to of to integrability the in non-relativistic limitsLandau-Lifshitz world-sheet AdS/CFT. of theory sigma-model In describes which spin particular, a is the chains covariant version the sim- [ of continuum the limit of the ferromagnetic geometry shell. In awhile second step rescaling a the furtherThis Newton-Cartan limit gives clock was a 1-form, performed, non-relativisticmoving sending so world-sheet in the as sigma-model a string to non-relativistic describing tension geometry, keep a to which the non-relativistic zero was string string dubbed U(1)-Galilean action geometry. finite. When strings moving in aby certain a type non-relativistic of world-sheetcorrespondence. non-relativistic action, target The are approach spacetime related taken geometry,the was to described to relativistic the first Polyakov SMT action consider limitstry, a with leading target of fixed to space the momentum a null of AdS/CFT covariant reduction the action of for string strings along moving the in null a isome- torsional Newton-Cartan (TNC) on since the relativistic magnonrelativistic dispersion features relation in [ the SMT limitwell-defined [ framework to formulate astring holograpic correspondence theory involving and non-relativistic corresponding non-relativistic bulk geometries. the realm of the AdS/CFT correspondence. string theory and holographyThese is are to tractable consider limits thechanical of theories, Spin obtained the Matrix by AdS zooming Theory into (SMT) the sector limits near unitarity of bounds [ of in this paper is motivated by pursuing this latter route, and in particular by applying it to JHEP11(2018)190 ] 22 ] for ] can 27 25 – 3) limit. | 2 25 , , 21 , ]. See also ref. [ ]. 21 14 32 , 28 , called the SU(1 ]. However, the world-sheet 5 S 24 , × 5 23 ]) is non-relativistic in the sense that 25 – 3 – , paralleling the Virasoro algebra of relativistic string 4 ] from consistent low energy limits of relativistic string ]. These formulations are intimately connected to the non- ] for more recent work). Interestingly, the non-relativistic 3) limit. 26 | , ] to be related to strings that couple to the so-called string NC 28 2 33 , 25 – 32 28 , 20 , 5 ] appear to be of a different type. 24 , 23 ] and the present paper, by restricting the target space to flat NC spacetime. We For clarity, we include some words on nomenclature. The non-relativistic string theory There is considerable literature on non-relativistic strings. See e.g. [ By using the SMT limits to study the specific application to AdS/CFT of this non- The existence of such a general class of non-relativistic sigma models with GCA sym- 5 The GCA was also observed in earlier work on non-relativistic limits of AdS/CFT [ 4 field theories. theory obtained after thebut scaling is limit does also not governedremarked only above. by have non-relativistic a world-sheet non-relativistic symmetries, target leading space, tofor the useful GCA work as on representations of the GCA and aspects of non-relativistic conformal two-dimensional in [ furthermore discuss how our formulation relates to the oneon in TNC refs. geometry [ (andthe hence strings also move in the a theory non-relativisticis of target still [ space geometry. relativistic, The actual and world-sheet governed theory by a two-dimensional CFT. On the other hand, the geometry introduced in ref. [ relativistic strings on TNC geometryand that we subtleties) discuss be in this consideredIn paper particular, to and can we be (modulo will details differentalso show incarnations be that of obtained the the from Gomis-Ooguri same the non-relativistic overall action string structure. of action [ non-relativistic strings on TNC geometry discussed as special cases of the SU(1 earlier work and [ string theory obtained intheory was [ recently shown [ a four-dimensional non-relativistic targetPart of space this and, interpretation involves moreover, identifyingthe has the string periodic has the target a GCA space windingin symmetry. direction mode detail on as the which the most positionThis general on SMT SMT the limit limit Heisenberg admits spin of black chain. strings hole on We solutions. also AdS Further, treat the other SMT limits can be viewed relativistic sigma model, weknown will show physical that models. these world-sheet(called theories the In are SU(2) particular, realized limit) inThis the leads well- shows simplest to that example the a LL of Polyakov model version the can of SMT/scaling be the limit interpreted Landau-Lifshitz as (LL) a model. non-relativistic string theory with further exploration. Theseare are first non-trivial order interacting in two-dimensionalcouple time field to and theories a second type that of ordertwo-dimensional non-relativistic GCA in also target space appears space as geometry. derivatives the on(super)string It residual gauge is the in symmetry interesting world-sheet of the to the and note analogue tensionlesstheories that that closed of in the [ the conformal gauge [ theory. Thus our novel classin of SMT non-relativistic limits sigma of models,conformal the including two-dimensional AdS/CFT those field correspondence, that theories. represent appear a realization of non-relativistic metry, connected to non-relativistic strings, is expected to provide a fertile ground for Galilean Conformal Algebra (GCA) JHEP11(2018)190 , µ µ µν τ h dx (2.3) (2.4) (2.5) (2.6) (2.1) (2.2) µ . We µ m τ µ = X direction is α m ]. ) the embed- u ∂ 1 ] in agreement 36 – = , σ 13 and 0 – α , µ σ 34 τ , ( β 11 8 dx , , M A symmetric tensor µ ν 9  τ , X d η , 6 dx = α and exhibit a set of local u = µ ∂ τ u X M dx αβ β ), .  ]. Also in that case one works in a ∂ X µν β α h − , τ τ 37 u, µ , ν α τ αβ τ αβ µ ) + 35 = ( , αβ . γγ γγ m m 34 0 u − − M γγ − − P √ √ 1 − ν is a non-compact direction. The tensors ). The world-sheet has the topology of a T T 1 the world-sheet coordinates and we have du 1 √ m dσ ( , u − µ  − τ π , σ τ 2 T 0 = its determinant, 0 – 4 – = 0 αβ − σ Z = 2 − u ( γ ¯ h ) are independent of form α µν X M = N L , αβ αβ h α 2.1 ∂ α ¯ h X P dx σ γγ = ∂∂ αβ M − µν = ) = γγ √ dx ¯ h π α 2 u is − T P √ + 1)-dimensional manifold. The rank- MN u − + 2 null-reduced 2 d G T 1 = − = 1). We assume that L , σ 2 0 = σ ds ( ηA ,..., M + 2)-dimensional space-time with a null isometry. One can always put 1 L in the line element ( , d X µν +1)-dimensional diffeomorphisms. Thus these fields and their transformations h . For this reason, we focus on a sector of fixed null (light-cone) momentum d is the world-sheet metric with µν It is therefore convenient to find a dual formulation in which the conservation of h coordinates on a ( being a null Killing vector field and where and αβ 5 µ γ u µ x ∂ Our goal is to find an action for a closed string on the TNC geometry given by On this background, the Polyakov Lagrangian of a relativistic string is given by and m Note that this is analogous to the procedure of getting the action for a non-relativistic point-particle 5 is implemented off-shell. To this end consider the Lagrangian 6= 0. , µ µ on TNC geometry viasector null-reduction of of fixed a null momentum. massless This particle momentum [ becomes the mass of the non-relativistic particle. In the above formulation, the conservation of the momentumm is on-shell. P P The total momentum along have also defined which is invariant under localwinding, Galilean hence boosts. We arecylinder. considering The a world-sheet momentum closed current string of the without string’s momentum in the where ding coordinates of theperformed string pullbacks with of the target-space fields to the world-sheet, e.g. symmetries corresponding to Galilean (oralong Milne) with ( boosts and acorrespond U(1) to gauge those transformation, of torsional Newton-Cartanwith (TNC) the geometry known [ fact that null reductions give rise to TNC geometry [ with with has signature (0 τ 2.1 Polyakov action forWe consider strings a on TNC ( the geometry metric in the following 2 Strings on torsional Newton-Cartan geometry JHEP11(2018)190 v u ). 2.6 P/T (2.9) (2.7) (2.8) (2.12) (2.11) (2.10) 0). The is a scalar χ P > ]. 38 is periodic under ) the conservation where u 2.6 χ X α ∂ , . = b , α β = 0 e ) A 1 a 1 ). To this end, we introduce the α being periodic with period A . ) by demanding no winding, i.e. e , σ 1 β 0 v 2.6 ab τ needs to have non-zero winding to act as Lagrange multipliers in ( η . σ 2.2 dσ ). As we are in a sector with fixed η b , ( β 1 α v β π and using that . ∂ and hence the conservation of the mo- on the world-sheet and we employ the e 2 = . Write the world-sheet metric and its ba A α per X a 0 , σ 1  u η αβ η α 0 b α A Z β  β αβ β e e X σ A = ∂ + ( T e . To this end, write 0 1 − ab α ∂ u 1 η αβ (assuming here for simplicity αβ e and = = per σ  , γ e η η and = β η v e 1 αβ T  – 5 – η − τ  ηA 0 η P πT 1 b = 2 L = along = β αβ A = ) = ∂∂ ∂ e α a u η π a 1 e P α γγ β P α e = 0. This is solved by ) = is a non-compact null-direction and since we work in ∂ − e 1 dσ Note that + 2 β ab π √ u αβ , σ 1 2 A η 6  ). This is seen by first solving the equation of motion . 0 0 α σ Z , σ = P ∂ ( 2.2 0 η = σ αβ αβ (  v and its inverse γ P per ) as the target space direction a = 0. Thus, we see that with the Lagrangian ( η = 1 for the two-dimensional Levi-Civita symbols. This Lagrangian α e α 2.8 u 01 is zero . The constraints are now equivalent to P  . e ) are reminiscent of the procedure Roˇcek-Verlinde for T-duality [ α v χ − ∂ can be interpreted as the embedding field for a target-space direction = 2.6 1 are flat indices and = = , which follows from the fact that the string has no winding along the γ η , u π , giving that − in ( EOM which tells us that 01 η is periodic  X α √ = 0 η ), these constraints imply + 2 A . We emphasize that the above rewriting of the world-sheet Lagrangian does not 1 per u 2.4 or η σ a, b η is implemented off-shell. We now consider another form for the Lagrangian ( The field We notice that the two components of the field We will see below that some properties of the classically dual descriptions obtained by integrating out → 6 P , we interpret eq. ( 1 This gives either where inverse as using the σ direction before the change of variables to world-sheet zweibein and the string windingmomentum along one time around account for the non-zero momentum where P of dual to correspond to a T-dualitya since sector of fixed momentum They impose the constraints Using ( mentum current convention is classically equivalent(EOM) to for ( on the world-volume.identifying We get back the Lagrangian ( where we have introduced the new fields JHEP11(2018)190 ) ) 2.6 2.10 ) is a (2.18) (2.19) (2.13) (2.17) (2.14) (2.15) (2.16) ), the . ). Thus, i , is zero as 2.16 ) 2.16  η v 2.14 β are arbitrary . /f ∂ . Using (   α . = 1, which means − λ τ h 1 β  ! α τ ) = 0 h → e η αβ )( β  1 h = ∂ . α
. Inserting this in ( 0 e 1 η  − β α ) where α τ , λ λ − e η β 0 ∂ is ) τ 0 0 α α τ − α α ∂ v ) )( for e e λ 1 1 is periodic. ( − η  α α α 0 u + e e η γ αβ α A 0 , .  ∂ τ X + and β − − ( is off-shell. γ β this constitutes a world-sheet Weyl β − λ 0 τ ). 0 α τ  ( η∂ λ 0 A α b τ α u 0 α + h 1 2 ), we can anyway set 0 β e e α f γγ P 2.7 αβ ( ( e α  ∂ = + ) + e where − 2 = that impose the constraints ( ab 2.17 η αβ T  0 0 f 1 η u αβ β = α −  α h ββ ∂ e = − f e λ X b → η  ) . It follows that the Lagrangian ( 0 1 ,  β α + 1 − η =  ∂ e ∂ αβ – 6 – α Pol α λ αα β a 1 0 η γ e  . τ L = α α α along which the string has a winding mode. The β η − − ∂∂ e ∂ e e )( ∂ 1 − α f + v b 1 ∂ + ab 0 A + 7 β α η = ) = 0 λ α f e e = β ( η e η . Apart from the TNC geometry the target space-time α ∂ v 1 1 2 a local Lorentz boost. Under this symmetry the world- β ab + α α  ∂ P → + µν e 1 0 , e + h m η − α + + α β e f αβ ) the momentum current along β αβ ( ∂ and ) γ τ  m 1 α = and α αβ α )( − τ ) is of Polyakov type and has a local Lorentz/Weyl symmetry ) imply that  = m e 2.16 1 µ −

this is equivalent to + = α f α αβ + m e λ ). Using this one finds that the total momentum along 0 b T  2.16 A 2.14 0 α , β on the world-sheet. For 2 + + e e µ α = τ a e 0 h  2.15 ( α α f e + e 2 NG T ( f ab L − αβ drop out and we obtain e  →  =  1 = ) using again the fact that h α given in ( e Pol αβ 2.9 L α +  A 0 The constraints ( For the Lagrangian ( The Lagrangian ( Consider the field redefinition α Note that using the local Lorentz/Weyl symmetry ( e 7 choosing a gauge where with in eq. ( functions on thefunctions world-sheet. If we substitute these expressions back into ( sheet metric transforms as two-dimensional conformal field theory. for any functions transformation and for This is ourgeometry proposal given for by ahas Polyakov-type one Lagrangian additional compact forLagrangian direction a has the string two Lagrange movingalso multipliers for in this the action the TNC conservation of the current With this, we trade thegives two independent components of These equations are equivalent to the constraints ( This corresponds to theand two equations adding and subtracting the two equations one gets the equivalent relations Using JHEP11(2018)190 ) we v 2.19 ]. To (2.22) (2.23) (2.21) (2.20) of the X . 25 8 = i ) is replaced η v ) and ( u X . ]. The Gomis- ) − , 2 2.16 0 26 + as the embedding i∂ X η )( + . Setting 0 1 πT λ ∂ ∂ , ( + = 1
1 2 , define furthermore ¯ ∂ ¯ γ , . . . , d β 2 ( = − ¯ β∂ iσ . Since the action is invariant , ¯ λ = 1 and to flat TNC spacetime by ∂ ) can be mapped to the non- ) we have to integrate out the v − + − ], omitting the instantonic part). a α )+ δ is an isometry. Furthermore, it is = , i, j v 25 ¯ 2.16 it will be also shown that contact 2.16 ∂γ , 0 ) v directions of the TNC manifold. = X β πT λ 2 ij σ A as a compact spacelike direction. We µ a α δ + i∂ + e = x 0 v i = − X 1 ij ¯ ) from ( )( ∂X ∂ h i , β 0 ( ∂ v 1 2 ] to describe strings on a TNC geometry given 2.19 − X – 7 – 5 1 = ∂ ] by performing a large speed of light limit of the + ( πT ∂X 0 25 2 + = 0 and λ X i , ∂ z ]. 0 . The importance of a compact target space direction + 2 2 = h i v 20 d iσ ¯ γ X ), in the sense that the two Lagrangians ( = that the Lagrangian ( 1 Z − ∂ 00 i , A π 1 ) to the Gomis-Ooguri non-relativistic string action [ 1 h 2.16 2 v σ X , the target space direction , 1 0 η X = = ∂ µ 2.16 δ ¯ − z + S i 0 = . We have shown here that it is the Nambu-Goto version X , X µ 0 2 τ µν ∂ = i h iσ γ X ), 0 + = 0, is not part of the TNC target space geometry but should rather be viewed as ] under certain conditions. In appendix ∂ 1 and µ v σ − 2.16 32 h µ m = 2 T m z − , µ It is shown in appendix We are thus able to match the relativistic string with a fixed momentum along a τ Similar observations were made in [ = 8 L is an entirely different way in which the samerelativistic result Polyakov can action be for obtained. ageometry [ string moving in what is called a string Newton-Cartan performing a duality transformation wherebyby the a non-compact compact null spacelike direction in direction the context ofOoguri non-relativistic string string action theory was obtainedrelativistic was string in also in [ emphasized the background in of [ a Kalb-Ramond 2-form. Here we thus see that there which agrees with the Gomis-Ooguri action (eq. (3.8) of [ non-compact null direction with the Gomis-Ooguri string action. This was possible by and, after Wick rotating to the Euclidean section by We then find the Euclidean action Now define It is possible todo relate that ( we specialize tochoosing the flat (conformal) gauge find from ( under a constant shift of wrapped by the stringstress and that thus we shouldan regard additional target space dimension added to the 2.2 Relation to the Gomis-Ooguri non-relativistic string action by Polyakov-type Lagrangian ( are classically equivalent andworld-sheet that zweibein. to get Moreover, ( wefunction have of found the string an on interpretation a of compact target-space direction This Lagrangian was previously found in [ JHEP11(2018)190 . v (3.1) (3.2) (3.3) , ˜ η , i β c τ 1 = α e differently. It is 1 + α , η η e . β ) does not scale. We 2 ∂ c 0 ψ ). αβ 2.5 2 and α ˜ h e 0   α 2.16 in ( = e 3 αβ c ω P αβ 2 ψ = + , h  β ] is τ 5 0 α = 0. String Newton-Cartan geometry α under this scaling. Clearly, this scaling ) (combined with a limit on the target e , λ m 1 α ) one gets the Lagrangian dτ αβ ˜ = ˜ e 1 2.19 ) for a string on a TNC geometry described 1 α ω α ˜ − e 2.16 c c + – 8 – 2.16 = = αβ , m 1 1 h ) of ( α 1 α e β α e 3.2 ˜ τ 1 2 , e α c and )–( 0 e e = 0 α 3.1 ). As we shall see below, this is the Lagrangian for a string α α ˜ e + ˜ e 2 η 2 − 3.2 c β c , τ ∂ = α = 0 0 m ] requires that we impose ˜ c α T α with one extra dimension added - the compact isometry parametrized e ) and ( e αβ 32  = , 2 ˜ µν e 3.1 h 3 h c 2 T ,T = − and e = µ m ] this conection will be explored further by including the Kalb-Ramond 2-form and , ] a zero tension limit of the Lagrangian ( → ∞ . In terms of the pullbacks, the scaling limit of [ Taking the scaling limit ( Our starting point is the Lagrangian ( µ 5 39 τ v NRPol c L where to avoid heavytilded notation fields we in have ( with removed the a tildes non-relativistic from world-sheetwhose the geometric theory properties tension propagating differ and from in those all of a the TNC non-relativistic geometry in target a space manner to be discussed Note that limit is a non-relativistic limitessentially a on limit the in world-sheet as which we it are scales sending the world-sheet speed of light to infinity. where the tilde quantitiessupplement are this the with rescaled the ones. following scaling Note of that the zweibeins and Lagrange multipliers by by In [ space geometry to ensureone finiteness obtains a of string thegeneralize with action this a scaling in non-relativistic limit the to world-sheet limit) our theory. Polyakov-type was Lagrangian In introduced ( the in following which we shall time that is closely relatedinfinite to dimensional TNC geometry. symmetry These algebras.dimensional sigma algebra models These of will local are be conformal shown the symmetries to analogue of admit of the3.1 relativistic the string classical theory. Polyakov infinite action for strings with non-relativistic world-sheet theories The string theories discussedspace-time in perspective. the However, the previous world-sheetCFTs. section theories are are In still non-relativistic described this from bythe section relativistic world-sheet the we theory target will becomes goobtain a first one non-relativistic order field step time theory. derivative further sigma We and models will describing take show strings a that moving scaling in one a can limit target whereby space- also is essentially Newton-Cartan geometryIn with [ one additional directionthe - dilaton. here denoted by 3 Strings with non-relativistic world-sheet theories with the work of [ JHEP11(2018)190 . ˜ v ] to X after (3.6) (3.7) (3.8) (3.4) (3.5) 5 | as the ) such = k ~ | c ). From and this η η 3.1 = π 2.15 ω + 2 1 ) to obtain ]. The resulting σ so that is closed. Indeed 3.1 28 α ) and ( , → . A 1 13 0 . P/T σ α 2.18 # . (where the fields are not a . ψ e αβ ) to the Nambu-Goto type λ η 1 2 h ) ) of ( α 0 a ν , = 0 β e ∂ + ) means then that the closed η ) τ 2.16 0 µ 3.2 1 0 u ( α γ = α τ 3.4 ∂ τ X , σ m )–( ) is consistent with the definition 1 0 γ with period 1 0 τ α ∂ 0 σ = 2 = ββ e v 1 ). Again, we can interpret 3.1 ( 3.2  1 0 γγ α 9  µν dσ per is periodic under 2 , σ αα . Eq. ( η π and direction. The momentum current along 0  u v 2 = 0 which gives that σ 0 v + α ,A ( X + α . This is analogous to the way that gauging Z τ v 1 σ η P σ per T σ , δh β β = α η ). From this we get that the momentum along µ ∂ – 9 – ∂ A 0 = ∂ α P πT α where 3.2 0 2 v αβ e m = ) = u P 1 π )–( µ T  αβ ] X  ) = 5 α dσ 1 = " 3.1 and a U(1) ∂ + 2 π T 2 , σ as [ 1 = ) respects the reduction of ( is the momentum vector. This follows from the Bargmann dispersion 0 0 η − , δm b ν Z σ Gal , σ α k ~ α e ). Thus this formula can be used also after the scaling limit. ( = 0 or from the relativistic massless dispersion relation 0 NRPol 3.2 = a µ A η = σ field is consistent with the scaling of the tension in ( ∂∂ e L = 0 ( m v ∂ )–( ab η 2.10 µ P δ per δτ = η 3.1 NRNG = ) and we can subsequently take the scaling limit ( α L v = 0 where ) is invariant under local transformations that act on the target space P 2 µν k ~ h 2.19 ). Indeed, if one puts 3.3 after setting this is seen to be conserved we have that and 2 2.19 . m η 2 k ~ 2 µ is periodic m = → ∞ , per ω c µ η τ The scaling limit ( The action ( The rescaling of the One way to see this is to note that for a free particle, massless Galilean symmetries, imply that the 9 is again zero is dispersion relation is relation sending without such a U(1) only admits spatial derivatives. Lagrangian ( rescaled) we get ( correspond to the gaugingthe of (massless) a Galilei spacetime algebra symmetrythe algebra Bargmann algebra consisting (massive of Galilei algebra) ageometry gives was direct TNC dubbed geometry sum [ U(1)-Galilean of geometry.is We note crucial that in the order U(1) factor to in allow this for algebra time derivatives in the theory. A massless Galilei symmetry fields along with target space diffeomorphisms. The former symmetries were shown in [ is unaffected by the scalingv limit ( This can also be obtainedthe by EOM taking the of scaling limitfrom ( section where embedding map for aTo periodic avoid target clutter space we directionstring now ˜ is also winding drop one the timev tilde around the on periodic of the inverse zweibeins ( that also after the scaling we have below. Note that the scaling limit of the zweibeins ( JHEP11(2018)190 ), m and and 3.8 (3.9) 0 (3.14) (3.12) (3.13) (3.10) (3.11) ω α . e 1 α δ = 1 ψ , α 1 e f → . We can then use the ) we do not need to 0 α δ ) is equivalent to the . 3.9 = . 3.3 1 0 α α . One obtains again ( = 0 in ( e e ψ , ψ , η 1 2  α ˆ α ψ f − β f ∂ τ f 10 , ], in analogy to what was done ˆ h τ 1 5 − = α α and → + e χ ω 1 1 1 η α 1 f ω α + α e α e η ) with local diffeomorphisms we have → β = h ∂ . ∂ , e and 0 3.9 a α = α δ α e 1 0 τ , ω  α = α ) impose the constraints e a = αβ – 10 – , ψ 2 0 ˆ α f 0 f α α e can be written as 3.3 , e α χ ˆ e fe 0 ψ ,  − is a world-sheet 1-form, we do not think of it as being α α 0 ]. Alternatively, one can first take the scaling limit + e α ) originally found in [ 5 α 1 e χ h τ and = on the world-sheet. These constitute local Galilei/Weyl 1 α 3.8 = 0 − = ˆ ω and subsequently the second diffeomorphism to set ω f fe f β ) to set 0 1 α τ α δ 0 → e → α 3.9 ∝ and 1 e 0 1 ) is invariant under the local symmetry α α α f αβ ), and then subsequently solve for the Lagrange multipliers e e  3.3 3.3 in the Lagrangian ( , e are the inverse vielbeins. The latter transform as limit. ψ 1 0 c α α e can be any functions on the world-sheet. In particular, we can use the local and fe . Thus, even though ˆ h α ω and χ → 0 0 and ) to obtain ( α α e h e The world-sheet geometry in the Polyakov-type formulation is described by The Lagrangian ( 2.16 We can use one diffeomorphism and the Weyl transformation to set . The Lagrange multipliers as we shall see below. 1 10 α Combining the localenough Galilei/Weyl symmetry symmetry to transform ( the zweibeins to the gauge local Galilean boost to set transforms under the localworld-sheet Galilean geometry boosts can that be actthe thought on geometry of the obtained as vielbeins). by a gauging 2-dimensional In the massless other massless Galilei words 2-dimensional3.2 geometry, the Galilei i.e. algebra. GCA symmetry of the non-relativistic sigma-model Hence in order totransform reproduce the transformationspart of of the geometry (like we would for example in the case of TNC geometry where e where Galilei/Weyl symmetries ( so this shows inNambu-Goto particular type that Lagrangian the ( before Polyakov-type the Lagrangian large ( The general solution to these constraints is where for arbitrary functions symmetries acting on themultipliers world-sheet vielbeins and Lagrange multipliers. The Lagrange of ( ψ This is the Lagrangian found in [ JHEP11(2018)190 ) . 3.15 ) and (3.15) (3.16) (3.19) (3.20) (3.21) (3.22) (3.23) (3.17) (3.18) σ ( i µ µ X X 0 ∂ µ τ . , ) . − m 0 ) + η σ σ 1 n ( (ˆ ∂ ˆ G M ψ  , 0 ) direction. The residual ψ + F m . +1 1 v , + n 1 σ 1 − ) ) µ ) has the GCA as its infinite σ ∂ 0 ) = 0 ) n 1 σ ) are σ 0 X σ σ ( ( , n 1 ( σ 0 n n 3.15 ( ∂ ) ) b 0 M µ F 0 ] = ( n 3.14 f σ n m . Let us define , ψ ωτ M X g n ) = and ) + b + ,M ) leave the flat gauge Lagrangian ( σ ) = 0 n ( ν σ n + 1)( σ + 1 (ˆ L and ( L X ˆ σ n n ˆ g [ ψ 3.17 1 ( f ) L ∂ 0 ) monotonically increasing. While = n → µ − 0 G a , g 1 1 0 σ ( . X – 11 – ( , ∂ + 1 1 +1 ) transform as F n ∂ 1 ) and ( ∂ m n +1 σ X ) σ ) takes the form , σ ( + n µν 0 +1 00 ) n ψ h = , ξ n 0 σ 3.16 F index does not include the 3.3 L ) ( ) ) σ α ) 0 ) are generated by the following infinitesimal + 0 n 0 ( ) with ∂ µ σ a σ σ m η 0 − α ( ( β σ ξ ) + ( f ) and n − 3.16 ( ∂ F σ = = ( X σ µ (ˆ n G ( = n n ˆ ω − X ω 0 0 ) = L α ξ M F σ = ∂ ] = ( ( 0 f ) and m αβ ˆ 0 σ  ) = ]. Here, we have found a general class of non-relativistic sigma models σ µ ,L σ ( to be analytic so that they can be analytically continued to the complex ( → 21 n m F ω g 0 2 L [ ) we find the algebra generators h σ 2 T and 3.18 − f = flat The transformations ( It is known that the GCA can be obtained from two copies of the Virasoro algebra L ) transform as scalars, σ ( via a contraction [ exhibiting this symmetry.the It GCA would be symmetry very of(including its interesting these central to sigma-models extension) study in plays in relativistic the string more same theory. detail role whether as the Virasoro algebra This algebra is theextensions. 2-dimensional The Galilei latter conformalWe are thus algebra conclude absent (GCA) that because without thedimensional we non-relativistic any symmetry treated sigma algebra. central model the ( world-sheet theory classically. They satisfy the algebra and if we furthermore define then from ( Assuming plane we can perform a Laurent expansion of It is straightforward to verifyinvariant. that ( diffeomorphisms for any functions η We remind the readergauge transformations, that i.e. the those obtainedworld-sheet from diffeomorphisms the local preserving Galilei/Weyl the transformations flat and gauge ( In this flat gauge, the Lagrangian ( JHEP11(2018)190 , ) is 1 1 Q J 0 3.3 = 4 ]. (4.2) (4.3) (4.1) E ) that 43 N ) 2.19 N is close to ) where 2 . On the gauge λ E 5 ( ] are realizations ]. One can equiv- S 2 O 2 × + 5 1 ] of the AdS/CFT cor- correspondence see [ , 2 ] 5 2 λE 2 for which S + and the three R-charges , 3 0 3 1 × S ) has certain unitarity bounds /CFT E 3 S 5 = fixed N × gE = ) that gives the Lagrangian ( on Q R + E 2 λ − Q S 2.16 E = and Q is a linear sum over the Cartan charges of , 1 Q, λ − S Q ≥ E – 12 – )) subject to a singlet constraint for the matrix 0 E = 4 SYM on to one. Taking a limit [ → is the one-loop correction, we see that only states N 3 lim λ 1 ) of the Nambu-Goto-type Lagrangian ( = fixed N S g E with gauge group SU( is the coupling of the SMT. The global symmetry group 3.1 + 3 g S Q ,N ) (or U( × = survive. This gives the Hilbert space of the resulting SMT. N R 0 H ]. On SMT-type limit related to AdS Q is the energy and ). In this section we generalize this statement to the scaling 42 → = – E λ 3.8 0 40 E ) only states in 4.2 = 4 SYM reduces to the so-called spin group in the SMT limit. In table The AdS/CFT correspondence asserts a duality between SU( ) of the Polyakov-type Lagrangian ( = 4 SYM on N 11 3.2 N ] as well as [ 4 direction as the position of the spins in the spin-chain limit of Spin Matrix theory. . We have set the radius of )–( 4) of 4) being the two angular momenta = 4 SYM. In fact, we write v 3 | | 2 2 J , , 3.1 N ] it was found that the Spin Matrix theory limits introduced in [ In the limit ( In the course of this, we find a particularly nice interpretation of the limit of the 5 See also [ in and = 4 SYM simplifies greatly and is effectively described by a quantum mechanical theory 11 1 2 as the Hamiltonian of SMT where PSU(2 will survive. The restthe of classical the (tree-level) states energy decouple.with and Writing classical energy Moreover, the Hamiltonian ofE the resulting SMT corresponds to the one-loop correction indices. The Hamiltonian of SMTHilbert is space factorized by into removing a spin twoalently and excitations take a the and matrix limit creating part, in twocritical acting new the point. on ones grand the [ canonical ensemble by approaching a zero-temperature N called Spin Matrix theory (SMT). SMTshaving are a quantum Hilbert mechanical theories space characterizedtation made by of from a harmonic Lie oscillators groupadjoint with (called representation the both of spin an group) SU( index as in well as a matrix represen- indices corresponding to the where for a givenPSU(2 state J respondence. super Yang-Mills theory (SYM)theory and side, type IIB stringthat theory we on schematically AdS can write as compact 4.1 Spin MatrixLet theory us first briefly review the Spin Matrix theory (SMT) limits [ In [ of the zerogives tension the scaling Lagrangian limitlimit ( ( ( with a non-relativistic Weyl symmetry. 4 Spin Matrix theory limits of strings on AdS JHEP11(2018)190 . 2 1, / 1 1 (4.4) (4.5)  6 6 6 4 + 2 ). For 10 p ]). The d 4.2 4 ]. For the 2 0] , 1] 1] , , 1 0] = 4 SYM. for the algebra , , 0 0 s ] and [ , , 0 1 s [1] R , , 0 0 N 19 R , , 0 – [0 , [0 [0 15 [0 N ] 4 will find from taking SMT N is the length of the spin chain. , 2 N 4.2 J 1 ] (see also [ . This is described by the XXX + − 1 14 . defines a supersymmetric subsector

) gives [ −− −− −− −− ∞ J 2 p 2 p Q N N 2 −− −− −− −− −− 2 4.3 = = = N N sin Q N sin Cartan diagram −− 2 −− E 2 λ π g π −− 2 generalization of nearest-neighbor spin chains. 1 which one can think of as the strong coupling – 13 – 1 + = N 3) 2) 2)  | | | r ) we see that the SMT limit clearly can be thought Q 3) | 2 2 1 g s , , , = − = 4 SYM that give rise to SMTs in the limit ( 4.5 G ] Q SU(2) H , the Cartan diagram for the corresponding algebra and the N 3 SU(2 s SU(1 SU(1 SU(1 − SMT reduces to a nearest-neighbor spin chain [ G = 4 SYM where ]. This is even clearer for the small momentum limit . E ) and ( 5 2 3 J N S = 4 SYM. 4.4 Q → ∞ ). + × N 2 1 2 + 2 is the space-time dimension of the target space for the corresponding ≥ 3 5 N J J J 4.2 J d E 2 . Since the spin chain Hamiltonian defines uniquely the SMT also for = 4 SYM is [ + + + + J ]. Thus, one observes that the SMT limits in this sense correspond to s the spins of the spin chain are in the representation 1 2 1 2 4 G + N J J S J 1 of the algebra (in terms of Dynkin labels) that defines the Spin Matrix Theory for 1 + + + + s J 2 1 1 1 R = S J S S + Q = = = 1 is the momentum. Taking the SMT limit ( one can think of SMT as a finite- . Five unitarity bounds of Q Q Q S p N = 4 SYM, which means that SMT describes a near-BPS regime of = Unitarity bound Consider the SU(2) SMT in the planar limit Using coherent states on each spin chain site, one can find a semi-classical limit with The low energy excitations of spin chains are magnons. The dispersion relation of a In the planar limit N Q sigma-model is based on theThe coset related resulting to sigma-models the representation are oflimits what the of spin we strings group below on in table AdS in section ferromagnetic Heisenberg spin-chain. In this case to a Galilean onenon-relativistic [ limits of many magnon states oncan the furthermore spin find chain. adescription In long in the a wavelength limit limit. semi-classical in limit which This the of spin procedure the chain spin results is chain in long, [ a one sigma-model This corresponds to aThe magnon dispersion excitations relation dominate of for limit a of SMT. magnon Comparing in ( of a as nearest-neighbor a spin non-relativistic limit chain. corresponding [ to the pp-wave limit, in which one goes from a relativistic dispersion relation of the spin group finite single magnon in where we have listed five limitsof of five cases of table Table 1 each limit we listrepresentation the spin group a given limit. Moreover, sigma-model (see section JHEP11(2018)190 , 5 2 1 → J 1 that , ). (4.9) (4.6) (4.7) (4.8) 1 σ (4.10) ] (also J 3.2 14 = 4 SYM )–( → ∞ and N 5 Q 3.1 is the ’t Hooft given in table λ ) of Q 4.2 . i 2 5 where , Ω within AdS 4 ) with d /  . 3 1  . On the string theory side scaling limit ( + S λ 2 5 4.1 /R ). However, since the factor ) 2 3 0 2 s = φ Ω ( πl s θ → ∞ ρ d (2 2 . 2 / c R/l , and AdS N . The fields are periodic under s 5 1 + sin = 0 π and S . As we shall see, this corresponds to a , σ 0 πg 2 2 . Since the spin chain description arises 5 is the energy corresponding to the global + sinh . In the semi-classical limit ) 4 0 π k S Q 2 φ θφ θ E √ 1 ( π and λ/N × dρ ∂  with the position on the spin chain [ 5 = cos 0 1 4 are the string coupling and string length of the = is the length of the spin chain, thus explaining + 1 = = 2 1 σ – 14 – 2 s s s 2 σ 0 2 l 1 − 5 Q where φ πl dσ in the global patch as R ˙ σ πg φ S 2 is radius of π 5 ρ dt θ Q 2 S 2 and 0 R × = and Z ≥ , all measured in units of 1 s 5 cos with period 2 × 5 T g φ  5 E S 1 0 cosh ) the string tension is 1 π ∂ σ ], the cyclicity of the trace gives that the total momentum Q 4 − are the angular momenta on the 4.9 44 = h 2 = 2 ˙ S φ ) one identifies are functions of R LL φ = 4.6 L and 2 1 ds and S , θ t ]) 4 ) is uniform we can include it in the tension instead of the metric. With this, . Moreover, we can translate the unitarity bounds ( 1 and in a semi-classical regime, one obtains an effective sigma-model description 5 is a site on the spin chain and . We also defined S 4.9 k  π In terms of the metric ( We write the metric on AdS Q in ( are the angular momenta on + 2 translates to the flux of the self-dual Ramond-Ramond five-form field strength on AdS 2 1 3 R one gets an effective string tension N and on into corresponding BPS bounds time coordinate J The AdS/CFT dictionary states thatcoupling 4 of the gauge theorystring and theory side, respectively, and We now turn to thenary string of theory the side AdS/CFT of correspondence,as the we a can AdS/CFT limit formulate correspondence. a of Using SMTlimit type the limit of IIB dictio- ( the string sigma-model theory of on the AdS string that realizes the in the semi-classical limit. 4.2 Limits of strings on AdS where that the fields arefrom periodic single-trace in operators [ along the spin chain is zero, corresponding to the condition σ gives the Lagrangian ( reviewed in [ of the spin chain called the Landau-Lifshitz model. The Landau-Lifshitz model is where the fields For JHEP11(2018)190 . , 1 and (4.14) (4.15) (4.16) (4.12) (4.13) (4.17) (4.11) n 2 ,..., 2 , ) that are not 3 J = 1 external directions I limit. Writing , and , ) as . ) , 2 i . 2 5 Q J 4.9 Ω one can find coordinates dx , → ∞ d + i 1 1 , . . . , d c β J 1 + E ( , , 2 3 + 1 2 = fixed 2 , that we call 0 Ω , S I ˜ x = 0 y = Q , ρ d ) the SMT limit of type IIB string of table s 1 u µ 2 F d − ), we identify = 0 g 1 S 2 i∂ i Q c , . E 0 commuting angular momenta that are 3.2 − u h , = n N 0 = i s , )–( = + sinh ˜ x 2 1 , where 2 directions of table dx πg c 00 = 0. This gives an effective reduction of the 5 3.1 i 4 n h dρ β S I Q = √ – 15 – y rotation planes that drives the strings to sit at = + + 0 × x 0 2 0 = , are held fixed in the n 5 ) to the string theory side. To implement this, we ) one has a confining potential with effective mass = fixed Q,P c m (with 4.2 ρ dt − F dx 2 4.12 Q E , . . . , d = ,N ≥ = µ = 0 and one can furthermore put the metric restricted to cosh ) for AdS = 1 0 E 0 dx x i − uu µ , ) with g i∂ τ i → = 4.11 x s 2 = 2.1 , g . u τ ds is for each of these 5 s S /g are Killing vector fields with coordinate as for each case. , . . . , d × u 2 0 is defined as the momentum along rotation planes associated to the 5 n , ∂ x 2 = 0 one has n P I − , such that = 1 we have furthermore recorded the number of surviving space-time dimensions y for the metric ( and n i = 0 in a form ( 2 is the number of angular momenta (out of I 1 0 x I y − for where For y ∂ n , To make contact with the scaling limit ( For a given BPS bound µ • • x = 8 + 2 = 10 , and scale the while the coordinates d not included. Inproportional the to SMT 1 limit ( the minimum of thenumber potential of spatial located dimensions at in theIn SMT limit table for four out of the five limits listed in table Here included in the unitarity bound. Thisthat gives realize 2 d This is the translationuse of that the given limit a ( u particular BPS bound defined by theory on AdS In using this as the string tension, we should rescale the metric ( JHEP11(2018)190 . ) 1 . To 4.11 u (4.21) (4.22) (4.23) (4.18) (4.19) (4.20) (4.24) (4.25) . This 2 J 2 . and + )  2 1 do not scale . Write the 0 5 which means J du x S 1 2 2 αβ θdφ = c h 2 × − ). This should be Q 5 0 , is held fixed. Thus, i ≥ and 3.1 dx 2 + sin ) u α  E 2 A ) m θdφ . α dθ + direction becomes ( and 2 4 1 ) for AdS u ˜ η cos dγ c = 1 2 u . + sin 4.11 = + ( ν 1 2 = ρ 2 η ) dx 2 1 as linear functions of + µ Σ 0 , γ dx d along the x ,A 2 ( (sinh u , h µν = . 0 α ) − i∂ and ˜ x 2 2 π α dβ 1 i t Q. − 2 limit the first term goes like scales like ν 2 ) when we do not fix a gauge for the zweibeins F d = 0. Using this, one can write the metric ( = = θdφ η = dx – 16 – ) plane is in fact a strip. This does not affect the non- 2 α = 3.2 ]. , + cos µ P P T 5 u , γ dφ , h , u 2 + sin τ 2 → ∞ = 0 dx 1 2 θ 2 x c ρ 2 µν + sin − dα α dβ h 2 cos 0 2 + x − dθ . Write now ) is a realization of the scaling limit ( 2 3 ( ) + γ = = 1 4 Ω = 0 for m i∂ t + sin 4.12 = − ρ d 2 − uu 2 2 g ) = dα du 1 is compact the ( ( Σ Q = τ γ 12 d 2 2 5 ( having the correct scaling, one finds also that h + sinh Ω θdφ , m limit described above the effective string tension becomes α τ 2 and d = 2. Our starting point is the metric ( 2 t (defined below) after the limit. dρ d cos limit. As seen in section i∂ 0 + as 1 4 x → ∞ 5 ) we need = = cos ) we see that the momentum c + S 2 0 E → ∞ × 4.13 ds 4.13 dx + 2 = 4 and 5 c = 3 and d = In the In addition to n Note here that since has the correct scaling. Taking the limit one finds τ 12 α with compactness of ˜ of AdS We have satisfy ( This, in turn, ensures that five-sphere part as with 4.3 SU(2) limitOur and first Polyakov example Lagrangian is for thehas SMT/scaling Landau-Lifshitz limit model towards the BPS bound Using ( in the in this way the SMTsupplemented with limit the ( zweibein scalings ( before the limit. Below weThese carry are out the this limit same more cases explicitly considered in in two of [ the cases of table τ where we removed the tildes on the l.h.s. . before the limit, one sees that in the JHEP11(2018)190 . ) ]. µ 4 X = 0. ). )[ 1 4.27 (4.26) (4.28) (4.29) (4.30) (4.27) ∂ α µ 4.6 = 4.12 that con- ρ s T m ) and ( ) then gives direction has /g ). This shows = ) v and the coordi- . α ) we find 4.26 4.16 0 v ) 2 β 2 4.28 P , and furthermore , . The momentum η 4.20 α v α θdφ , + sin ∂ ρ 2 ) and ( ρ ) and ( = 2 ) for the Landau-Lifshitz ) and ( 1 4.15 + sin 4.8 α 4.27 2 e = 0 corresponds to , ), ( 4.19 ) = 0 we find dθ I ), ( 1 ( y = 0 in the SMT limit ( µ 1 4 and ), ( 4.12 . , σ I 4.7 , 0 X y α = 1 ) on the background ( 3.4 0 σ , τ is ∂ ( ν = 0 σ 1 ) and finding a coordinate transforma- µ v 2 3.3 = ), the four-dimensional target space in- τ φ per dx Q 1 Qσ η µ 0 direction is identified with the position on = α , θ, φ θ∂ + v dx e 0 , θ, φ 6, such that 0 ) = 1 ) = 0 x . Using ( 1 1 µν 1 v x cos e – 17 – 1 , σ Qσ ) of zero momentum along X , σ = ( 0 0 ,..., = (˜ σ µ dσ = 2 σ 3.6 ( ) combined with ( ( , x µ π ) = 0 η η 2 1 can be seen to correspond to the above-mentioned x 0 X 3.2 Z = 1 dφ , h 5 , σ ). Since ) with the U(1)-Galilean background given by η 0 θ S I )–( as well. The closed string on this background has winding = 0 corresponding to σ 2 , ( 3.5 3.3 ) with I × v η cos α 3.1 y 5 − 3.4 = = ρ = 0 in the following. α as in ( ) that the sigma-model reduces to the Landau-Lifshitz model ( = , m v ρ is given by ( 3.8 0 v ˜ x d . = ) coordinates by identifying τ πQ I , y µ As part of this, we see that the compact The above shows that the general Lagrangian ( We consider now the condition ( Fixing the gauge on the world-sheet to The SMT/scaling limit ( u, x interpreted as a Polyakov versionthe of Landau-Lifshitz the model Landau-Lifshitz as model arelativistic non-relativistic in target string which space. theory we with can a interpret four-dimensional non- the Heisenberg spin chain. This is seen by combining ( This is seen tosigma-model. correspond to the zero momentum condition ( is an equivalent description of the Landau-Lifshitz model. This general description can be one finds using ( current along Therefore, the condition of zero momentum along as well as the static gauge choice as we shall see belowperiod this 2 has an interesting interpretation. Note thatchoosing the the following gauge for In addition to thecludes three the compact directions direction number one along Therefore we set the sigma-model Lagrangian ( ( tion for the external directionsnates between the for six the coordinates three-sphere given to by The six external directions havefines a them potential to proportional the to (sinh point These coordinates for AdS JHEP11(2018)190 ] 45 = 0, given (4.35) (4.36) (4.32) (4.33) (4.34) (4.31) of [ n Q 5 direction S 3) case of , | v 2 2 × ) , . Obviously, with 5 1 . Q dζ 3 , 2 2.1 on the spin chain ξ 2 ≥ dζ ) k 3 . ξ C E 3 direction, both before J + + cos cos v direction in the general 2 + , the compact 1 2 dχ v 2 direction is a null isometry . v dζ 3) SMT limit on the string background. The condition J ( . We parametrize the three- | ψ = u X 1 5 + ( 5 2 ξ + i∂ , S 2 3) SMT. This is the SMT with S = can be obtained as subsectors 2 | ) 1 − 1 2 × η J × 1 , 5 ,C Σ 5 cos = 2, given by = d , 1 ξ 2 2 = ( = 1 dζ 5 ,J 2 3 ,J sin k S  with the position on the spin chain. Ω 1 , 1 4 ) for AdS χ ξ k v 2 × 2 S i∂ P 5 , d ) + 4.11 C − + 2 sin 1 3) SMT limit is the only SMT limit with – 18 – | 1 1 2 = 2 dζ 2 S ) 2 , ξ − A = 2 1 3 ). This is seen to perfectly correspond to the fact that + ,S 16 BPS supersymmetric black hole in AdS  / 3.2 = 4 SYM one gets SU(1 dψ . t + + sin leads to zero momentum along the ) 1 2 and it survives thus the SU(1 )–( i∂ 2 N 2 3 + ( u dζ J,S dξ with the position on the spin chain is further verified by the = 2 Q dζ 1 3.1 ( ) 3 ξ ) of 1 2 + v ξ E = ξ 2 2 Σ . The origin of this is the fact that the 2 S 4.2 ). d E sin π = limit ( sin 2 3.1 ξ (2 = ( 4 1 + sin Q 2 5 2 3 v/ + Ω cos dξ d → ∞ 2 1 3) limit of strings on AdS 2 1 ( 3) SMT, both with respect to the Hilbert space and the Hamiltonian. Con- | are Fubini-Study metrics for | c 1 4 2 dξ − 2 2 , , ) = = k = 2 2 Σ ). The starting point is the metric ( ) ) A d 2 1 with = 4 SYM). Moreover, the other four SMTs of table Σ Σ 1 Below we consider the SMT/scaling limit towards the BPS bound The identification of d d 4.31 ( ( N with the one-forms We have where ( by ( and five-sphere as hence one does not decouple anyside. directions This when means taking the that SMT thealso limit resulting interesting on target-space to the geometry note string is that theory obeys ten-dimensional. the the 1 Finally, BPS it is bound theory side of the correspondence. Taking the SMT limit ( the largest possible Hilbert space andof global symmetry group (of theof ones obtained the as SU(1 limits nected to this is the fact that the SU(1 4.4 SU(1 The most interesting SMT limittable of the AdS/CFT correspondence is the SU(1 and hence cannot bethis periodic is in and particular have theof winding, case having when as zero starting winding explained with along inand the after AdS section the the spin chain description alsochain, dictates in a zero accordance momentum with condition the along identification the of periodic spin gives the position on theis spin identified chain. with In detail, one has that a givenconnection site to thedescription requirement of section of zero momentum along the that in the general situation in which we do not gauge-fix JHEP11(2018)190 . is ). w 3). 3 | w c 2 , 4.16 + (4.42) (4.43) (4.37) (4.40) (4.41) (4.38) (4.39) u 3 b . Assuming 3 + , Demanding b , 0 2 ) and ( − ) 2 x 2 ) direction with 13 = that is periodic . 2 = 0 and that = Σ u 2 v Σ 4.15 d b w ψ uu d g − , + ( u . ), ( and = + ( and Q 1 2 2 ) 1 2 0 b − 1 ) w ) with x + 4.12 , 2 1 Σ c E 0 Σ d 2.1  x ( + d = ν ( ρ u 0 = 2 ρ X x 2 A, b 1 2 . 1 2 i∂ ∂ gives that 1 + µ σ 0 . w x X ) + 1 + sinh ρ A , and a new variable = C + sinh ∂ ρ A , 2 2 = 0 for the world-sheet zweibeins and the and w , ψ to be large. ) u 2 2 χ − + µν µ ) η , ψ − , A + h 0 Q α A , w X for the target space embedding, we obtain 1 2 x ∂ 1 ) combined with ( χ u 1 − dw c cosh ( 0 − ∂ 1 2 cosh + ρ ) with the (8 + 1)-dimensional U(1)-Galilean C σ µ = . + 3.2 − C 2 2 1 2 µ + − u m 1 ) + is periodic in – 19 – Q 3.3 1 1 0 ) + α )–( by demanding only X = b C x e 0 C 3 sinh u dσ dw + ∂ per b ). The zero-momentum condition is + 3.1 ( dw , = ) = η µ 1 2 0 π + in the null-reduced form ( ( ρ 1 2 x ρ m and 5 0 2 and + dw 4.41 0 2 , σ = Z ( S  dw 0 0 ˜ 0 α x ( ρ t x τ σ π ) fixes the coefficients in front of ρ × )–( 2 Q ( 2 2 ρ d 5 0 = ρ dx ) where 2 u , χ − + tanh 1 we get 2 X 4.13 0 + tanh 4.40 1 2 2 tanh 0. We choose we get α = 2 S , σ tanh as linear functions of e 0 − > − dρ Q L − dρ = σ and 3 ψ 0 . Demanding periodicity of ( b − 2 3 = cosh = = x 1 w b = cosh = = = E ν per τ = i∂ m ν 2 = τ and η Qσ b m t = given in ( − dx 2 2 b χ dx + µ 0 = = 1. One can check that this background has global isometry group µ x , . Then eq. ( 1 µν 1 = t ) = b dx π i∂ h 1 2 1 dx − b . The closed string has winding number one on this background, so that Qσ µν πT , σ µν h 0 h and πQ σ ) = 3) as well. The full target space geometry is ten-dimensional, as the (8 + 1)- ( µ | 1 η 2 m , , σ We take the SMT/scaling limit ( Choosing the gauge 0 we find = 0 gives 0 One can derive the coefficients of σ 13 ( uu To have a semi-classical sigma-model one needs periodic. From g P > with η gauge the Lagrangian and with 2 SU(1 dimensional U(1)-Galilean geometryperiod is 2 supplemented by the compact background given by corresponding to an (8 + 1)-dimensional TNC background with isometryThis group SU(1 gives the sigma-model Lagrangian ( This brings the metric of AdS with period 2 furthermore that We write now JHEP11(2018)190 ] ]. 5 19 – 15 3) SMT limit. | 2 , ]. For this we have shown that the spin 14 – 20 – )). Another central result of the paper is that this new 2.16 ] of the AdS/CFT correspondence. The latter are tractable limits of 2 can thus be considered as specific string theory realizations relevant 4 ], extended with a periodic target space direction. )) of the non-relativistic world-sheet sigma-model action, which was recently [ 5 . Since these models have the GCA as a symmetry group, and hence an underly- 3 3.3 The residual gauge symmetry in flat gauge is the Galilean Conformal Algebra. The target space is aetry type [ of non-relativistic geometry, namely U(1)-Galilean geom- With these sigma-models in hand, we can start to address the question of how non- In this connection already the simplest case, being the SU(2) SMT limit, provides a Importantly, the non-relativistic world-sheet sigma-model action is concretely realized • • SMT, and subsequently more generallyof for section the entire class ofing non-relativistic (non-relativistic) sigma conformal models symmetry, itpossible would to be compute very the analogue interesting of toThis the see standard would whether beta-functions point of it relativistic is the string theory. way towards uncovering the underlying low-energy non-relativistic relativistic geometry. The same holdsintegrable for theories more which general are SMTIn generalizations limits particular, of and we the the discussed Landau-Lifshitz corresponding the sigma most model symmetric [ and mostrelativistic general gravity SU(1 (with the gauge symmetries of U(1)-Galilean geometry) emerges from tion. Theyquantum represent theory. a natural starting point fornew further perspective studies on of thelong the wavelength Landau-Lifshitz corresponding limit sigma of integrable model,chain spin which direction chains is has [ known an to interpretation appear as as the the periodic target space direction of the non- Moreover, following the two bulletstarget space above, that our these non-relativistic description stringstheories i) move in, are elucidates and the ii) non-relativistic shows nature two-dimensionalmodels that of these conformal of world-sheet the field section theories.to the The non-relativistic SMT sector sigma- of quantum gravity/holography advocated in the introduc- non-relativistic string theory. in the SMT limitsthe [ AdS/CFT correspondence, and wenon-relativistic thus string obtain theories, a covariant being form well-defined for the and corresponding quantum mechanically consistent. Thus this novel class of non-relativistictwo-dimensional sigma-models, field describing non-relativistic theories, conformal provides an interesting setting for further exploration of type formulation (see eq.non-relativistic world-sheet ( sigma model has the following properties: One of the main results of(see this eq. paper ( is that wefound have in presented a Nambu-Goto Polyakov-type form. formulation describing This action strings is with obtained from Poincar´eworld-sheet(TNC) a symmetry target scaling but limit spacetime, moving of for a in string which a action we non-relativistic also have obtained the corresponding Polyakov- 5 Discussion JHEP11(2018)190 ] 31 are , 4 29 , specializing to A.1 scaling limit (in the c ] was considered in detail. 28 and discuss the difference between 16 ], it is interesting to study the precise 5 ]. 1 ] and briefly comment on the result. ]. – 21 – 32 39 and ref. [ ]. Moreover, the inclusion of supersymmetry, D- we also consider the large 2 39 A.2 The backgrounds described for the Polyakov-type actions in section ) of the action given in [ 3 14 ] for the SU(2) SMT limit of the non-abelian Born-Infeld action for D-branes in the AdS/CFT and many of the other standard features of relativistic string theory are natural 46 15 ] the theory for strings moving in a string-NC geometry [ To complete this program, one should obviously include all the string theory back- It would furthermore be interesting toSee [ find a string theory connectionA more to general the analysis will action be for included in the [ non- 32 14 15 16 spirit of section relativistic limit of Einstein gravity, recently found in [ correspondence. Being a theory ofrespect strings to moving the in one arelation presented non-relativistic between in the target section two. geometry, We hence workbackgrounds akin out with the zero in precise Kalb-Ramond this dictionary field in andthe section dilaton two frameworks. Moreover, in A Relation to string Newton-CartanIn geometry [ Research at Perimeter InstituteDepartment is of supported Innovation, by Science theOntario through Government and the of Economic Ministry Canada of DevelopmentInstitute, through Research, and the Perugia Innovation and by University Science. the andPerimeter LM Province INFN Institute thanks of for Niels of Bohr hospitality PerugiaBohr during for Institute the for support. early hospitality. phases JH of and this work. NO thank ZY thanks the Niels supported by the Royal Society Universityin Research Fellowship “Non-Lorentzian Holography” Geometry (grantpart number by UF160197). the projectholography” The “Towards of a the work Independent deeper of Research understanding FundThe Denmark TH of (grant research number black and of DFF-6108-00340). holes ZY NO with was is non-relativistic supported supported in part in by Perimeter Institute for Theoretical Physics. Acknowledgments We thank Eric Bergshoeff and Gerben Oling for useful discussions. The work of JH is extensions to consider.models More with GCA generally, symmetry understandingthere would the are be tantalizing general connections a properties withthat further doubled are of important field worthwhile direction. sigma- to theory examine. andner We doubled Finally, of also one geometry string note could theory [ that speculate andstring that its field the relation theory. non-relativistic to cor- SMT could be useful towards understanding closed then naturally expected to be solutions of such gravityground theories. fields in thebe analysis considered performed in an inbranes upcoming this work paper. [ The first steps towards this will gravity theory. JHEP11(2018)190 , i (A.4) (A.5) (A.6) (A.7) (A.1) (A.2) (A.3) ) 1 β τ transverse − , = 0 using Λ 0 B = 0 (ensuring β 0 and τ v M τ µ are the pullbacks )( τ . ξ 1) and transverse 1 v B , α ∂ αβ A e AB σ . η H − ) 1) Lorentz transforma- , + 0 B α = 1 A M e and 1 longitudinal σ ( v m describe string Galilei boost 17 M A αβ , such that . We can set D A A N ¯ 0 λ B τ α , τ A + M , A and m 0 ξ  λ + B ) + , B A 1 B A M β M M N = 0 , α ) are of different nature: those on the zweibein = Λ τ m τ E m 0 0 m . . On the world-sheet we have diffeomor- B v + B , 0 B A.1 A ), 0 A = 0 0 A A B AB 0 A M β A M τ τ A λ ] M , τ τ 2.11 δ a (spatial) longitudinal direction. We choose 1 in ( + Λ )( ω N , describes longitudinal SO(1 + 0 1 τ + ( , while those on the clock form select the longitudinal – 22 – η A , 0 α ) are those of string-NC geometry, given by a 0 A M e [ B and B B M 0 = 0 = 0 + 1 respectively denote M ) with Λ D + τ A A.1 E A M 1 v δ λ A 0 τ A µ 0 AB are only constrained to be traceless, in order that the 0 ] 0 A.4 α X δ B B A A e B A 32 ( λ λ N , . . . , d A , τ and E αβ + σ − 0 0 + Λ B A A λ = 2 µ A ) with A A 0 τ M  M + M M A E form string Bargmann-type (non-central) extensions of the alge- = τ E m v, µ is introduced as in ( ξ ξ ξ αβ 0 = A = Λ a µ L L L H σ τ α = ( which are taken to satisfy e B = = = 1 and αβ MN , A 0 A A A M H A γγ ) remains invariant, but they will play no role in what follows. The indices M M M τ − = 0 M δτ √ A.1 A δm δE h on the target spacetime tangent space. are raised/lowered with 2 T A 0 − A ) transformations, respectively. The parameters = Let us split The target space symmetries of ( d It is important to notice that the superscripts 0 , and where Λ L + 2)-dimensional geometry is [ and 17 M d symmetry transformation here isand ( subsequently perform a diffeomorphism choosing specify the flatdirections world-sheet tangent directions tangent space. to set The last two conditions can be shown to be a gauge choice. The relevant infinitesimal transformations and bra. Finally, the parameteres Lagrangian ( A phisms, Weyl transformations and local Lorentz transformations acting on the world-sheet where the target space-timeξ diffeomorphisms are generatedSO( by the Lie derivatives along where the derivative is covariant withtions, respect so to it longitudinal includes SO(1 a spin-connection field directions of the manifold’stwo tangent one-forms space. The two-dimensional foliation is specified by where the zweibein on the world-sheet of the spacetime tensors The indices The Polyakov Lagrangian for( the sigma model describing a closed string in a string-NC A.1 Dictionary from string NC geometry JHEP11(2018)190 . ], A 39 = 0 M ) we τ (A.8) (A.9) where (A.10) (A.11) (A.12) A dτ 0 µ A.3 A τ . For the = 0. This = 1. The λ ν 1 0 σ 1 m A = . Using this, v v v τ 0 v , − D E A ) and ( ). Doing so one β ν τ , = M m µ ρ ) A.7 1 A.7 1 ξ v δE m v α is a symmetry if and e ∂ . − 0 σ δm ρ µ + it cannot be put to zero τ µν ∂ . 0 0 = 0 1 h α µ µ v = e τ τ = 0. ξ ( = = 0 µ µ ) we also need to assume that 1 − 0 ∂ 1 αβ = 0. From ( µν v A v  do not depend on δm ξ λ + e A.1 1 H µ µ ∂ AB µν + τ + h µ , m M Λ + − ξ ω µ 1 λ v τ ). µ . ∂ , 0 and 0 τ turns out to be ). = v v be closed. The necessity of setting A = 0 and A.3 = 0. Nevertheless, as will be shown in [ µ 1 ∂ ¯ λ µ m = 0 µ µ v 1 = 0 ) in which we substitute ( m vµ τ τ E ] 2.16 ξ µ in ( = v ν , τ , H τ ξ v µ + 1 – 23 – A.1 β v µ τ [ µ ρ = = ∂ m ξ ) µ N vv 1 ∂ , ∂ ) is then that α 1 H , m e ρ and τ µ A.3 − = 0 ) becomes ( . With the redefinition µ m ) are those respecting 0 + 0 µ = B 1 α ξ other than the gauge choices already discussed. The residual 0 = A.1 of ( = 0 to obtain a relation with the string NC description. In e µ v A A.7 ( τ 0 A M ∂ 6= 0 the TNC transformation δ 1 one instead finds that similarly to ρ 0 µ µ µ 18 µ 1 αβ ∂ τ τ B τ µ  µ ρ = 0 is not a gauge choice but rather a truncation of the theory. m ν τ ξ e 1 0 1 m E v 0 µ = Λ + + τ A m 1 µ + µ E − λ δτ 1 = = µ being a longitudinal direction, it makes sense to also impose λ m µν v = 0. h 1 X v is a Killing vector, i.e. the fields δτ v ∂ = We end this section with some comments. It turns out that in order to correctly retrieve From the above it follows that We furthermore choose With Instead, setting This can be seen by putting 0 = v 18 Thus we have the additionalcan condition be that seen by studyingcan the see Lagrangian that ( whenever the target space NC symmetriesV starting from those ofan ( important consequence then obtain we defined one then easily verifies that ( The last of thegauge above choices) can by be using imposedcombination for without example loss the of extension generalityby transformation (and gauge respecting fixing. previous is easily done throughResidual a transformations string must Galilei then boost, respect infinitesimally given by impose any conditions on transformations preserving ( and we do not havethe enough situation freedom to is set improveddo when not we need include toother the truncate words Kalb-Ramond in 2-form the as presence in of that the case we Kalb-Ramond 2-form we do not need to arbitrarily residual gauge transformations are givenδτ by the subset of gauge transformations respecting Indeed the residual gauge transformation of preservation of the first condition). This still leaves us the freedom to set JHEP11(2018)190 . i 19 ) 3 0 , ) on . In ), so β τ 2 4 ). The (A.14) (A.15) (A.13) 1 c ψ 3.9 2 α . . A.14 e 1 ) ,  1 2.10 α + β 3 ˜ 1 m c ω m β c 2 1 τ 0 + = 0 by hand we = = α 0 1 e β 1  ( µ α m m αβ )( 1 ψ α e , λ , m + 1 0 0 + α . α β 0 ˜ e τ τ α c m 0 e α ( = = ˜ e αβ 0 1 αβ  α α e 1 for any ω ]. + σ ), similarly to what is done in section 39 µ − ) + ∂ , e λ 0 ), resulting from the contraction ( 0 A.1 , m β 6= 0 (before the SMT limit is taken) for all ], a fundamental distinction on the allowed = α = ˜ e m ⊥ αβ µ 32 1 ¯ λ 2 dτ A.15 ˜ c H α – 24 – An action principle for newtonian gravity τ δm , = = − ) 0 1 1 ⊥ α αβ β β ], we refer to [ = 0. m m 28 0  − ), which permits any use, distribution and reproduction in α λ ]. 0 ,H τ A ( β 1 σ δ α ˜ m αβ T , e ˜ τ  c 1 )( c SPIRE 1 , i.e. α IN = = + 2 [ A e 1 σ . CC-BY 4.0 α ⊥ − αβ 4 0 H This article is distributed under the terms of the Creative Commons α 1 e β ( , τ e 0 1 αβ α α ,T  ˜ τ e 1 2 = 0. This same calculation also shows that when we set e e c h + ) dτ 2 → ∞ = T + c 0 λ − arXiv:1807.04765 α Therefore, though we showed that the novel Polyakov string action we propose in A.11 D. Hansen, J. Hartong and N.A. Obers, This shift of the multipliers has the nice property of making them invariant under extension transfor- is essential in order to link the non-relativistic string theory described in this paper τ = = 19 [1] λ L References mations parametrized by Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. where we removed tildes to avoid clutter.above Lagrangian The inverse is zweibein invariant is under stillthe defined the by world-sheet theory ( local also Galilei/Weyl symmetry hasfull ( non-relativistic symmetry (Galilean) algebra conformal that symmetry. emergesthe from string For ( Galilei the algebra study of of ref. the [ The outcome is the Lagrangian The limit is The limit is non-relativisticsimplicity, we both first redefine on the theof Lagrange multipliers ( target according spacetime to and a more on general the version world-sheet. For to Spin Matrix Theoryfact limits one of can the straightforwardly AdS/CFTexamples verify in correspondence that section analyzed in section A.2 Scaling limitWe of record string here NC how to geometry take sigma a model scaling limit of ( recover the usual TNC gauge symmetry this paper is closelybackgrounds related is to to be the made.dτ one The in relevance [ of this is underlined by the fact that a nonzero only if JHEP11(2018)190 ] ]. , (2006) ] 04 ]. SciPost SPIRE , ]. IN ]. string theory SPIRE ][ B 741 JHEP ]. 5 ]. IN , S ]. ]. (2004) 3 ][ SPIRE SPIRE × IN IN (2008) 945 5 SPIRE arXiv:1311.6471 ][ SPIRE ][ IN [ SPIRE SPIRE 12 IN ][ AdS B 692 IN IN ][ (2004) 161602 arXiv:1503.02682 Nucl. Phys. ][ ][ [ , spin chain and spinning 93 ]. ]. arXiv:1409.1522 (2) [ (2014) 057 ]. Boundary stress-energy tensor and Torsional Newton-Cartan geometry SL arXiv:1311.4794 01 [ Nucl. Phys. 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