hep-th/0401167 = 1 Isometry of AdS And The c Matrix Model 2 4 Pei-Ming Ho 0 0 2 Department of Physics, Harvard University,Cambridge, MA 02138, USA 1 n Department of Physics, National Taiwan University,Taipei 106, Taiwan, R.O.C. a Physics Division, National Center for Theoretical Sciences, National Taiwan University,Taipei 106, Taiwan, J R.O.C. 3 2 pmho@phys.ntu.edu.tw 1 v 7 6 1 1 0 4 Abstract 0 / h Implications of the SL(2,R) symmetry of the c = 1 matrix models are explored. Based on -t theworkof deAlfaro, FubiniandFurlan,wenotethatwhentheFermiseaisdrained,thematrix p modelfor2dimensionalstringtheoryinthelineardilatonbackgroundisequivalenttothematrix e h model of AdS2 recently proposed by Strominger, for which SL(2,R) is an isometry. Utilizing its v: Lie algebra, we find that a topological property of AdS2 is responsible for quantizing D0-brane i charges in type 0A theory. We also show that the matrix model faithfully reflects the relation X between thePoincare patch and global coordinates of AdS2. r a 1OnleavefromNationalTaiwanUniversity. 1 Introduction MotivatedbytheproblemswithunstableD-branes,thec=1matrixmodel[1](forreviewssee[2,3,4]) has recently attracted a lot of attention [5] as the simplest string theory from which we might learn something useful. The c=1 matrix model is equivalent to a theory of free fermions Ψ in 1+1 dimensionalspacetime with the Hamiltonian 1 1 x2 H = ∂2+V(x), V(x)= . (1) −2 x − 2 It is dual to 2 dimensional bosonic string theory or type 0B theory [6, 7] depending on whether we fill one side or both sides of the potential. By filling the Fermi sea differently and an orbifolding, the same model was also conjectured to be dual to type IIBtheory [8]. A slight deformation of the potential x2 M V(x)= + . (2) − 2 2x2 was conjectured [9] to lead to bosonic string theory in 2 dimensions in the backgroundof a black hole of mass M. More recently it was also conjectured [6, 7] to be dual to type 0A string theory in the linear dilaton backgroundwith RR electric field proportional to q, where 1 M =q2 . (3) − 4 In the limit M (or x 0), we can scale x and ignore the quadratic term in the potential. → ∞ → Hence M V(x)= . (4) 2x2 This model was conjectured [10] to be dual to type 0A string theory in the AdS background. The 2 isometry group SL(2,R) of AdS is identified with a symmetry algebra of this matrix model. The 2 generators are 1 M 1 1 H = p2+ , K = x2, D = (xp+px), (5) 2 x2 2 −4 (cid:18) (cid:19) where p= i∂ is the conjugate momentum of x. They satisfy the SL(2,R) Lie algebra x − [H,D]=iH, [K,D]= iK, [H,K]=2iD. (6) − The ground states of matrix models are specified by a single parameter µ which is the energy at the surface of the Fermi sea. It is dual to the amplitude of a static tachyon background. In the AdS 2 model the potential (4) is bounded from below at zero, so µ 0. The ground states with µ > 0 ≥ spontaneouslybreaktheSL(2,R)symmetry. Onlythe uniquestatewithnofermion(µ=0)preserves the isometry, and is matched to the invariant vacuum of AdS . 2 One might be puzzled by the fact that the SL(2,R) symmetry exists also for other matrix models as part of the W algebra. But of course SL(2,R) can not be the isometry of the dual target spaces ∞ whichareasymptoticallyflat. Theresolutionis thattheSL(2,R)symmetrycanbe understoodasthe isometryofthe matrixmodelinawaywewillexplainlater,andthe isometryisspontaneouslybroken 1More precisely we should consider the field theory, and the Hamiltonian is dxΨ†HΨ, where Ψ(x) is the 2nd quantized fermion field. We will use the notation of quantum mechanics although everything can be extended to the R 2ndquantizedtheory. 1 by the ground state chosen for the other matrix models. Nevertheless, we will see that the SL(2,R) symmetry has important implications also for other matrix models. UtilizingtheSL(2,R)generators,theauthorsof[11]showedthatallquantumtheorieswithHamil- tonians of the form p2 a M H = +V(x), V(x)= x2+ (7) 2 2 2x2 with the same parameter M, but arbitrary coefficient a, are related to one another by coordinate transformations. This has many striking implications. For M = 0, the undeformed matrix model (a= 1) is in some sense equivalent to the simple harmonic oscillator (a=1), and also to the theory − without any potential (a = 0). For M > 0, the deformed matrix model (a = 1) is in some way the − same as type 0A theory in AdS (a = 0). Yet these theories have completely different asymptotic 2 behaviors! Asthepotentialsin(2)and(4)canbeviewedasthesametheorywrittenintwosetsofcoordinates, the µ limit of type 0A matrix model can be identified with the vacuum of the AdS matrix 2 → −∞ model. In other words, type 0A theory in AdS background should be identified with the result of 2 tachyon condensation from the linear dilaton background. 2 Due to the similarity between type 0A and 0B, we conjecture that the same is true for type 0B theory as well. The coordinate transformations relating theories with different a in (7) are not always bijective. Somecoordinatesystemsonlycoverasmallpartofthespacetimedefinedbyanothersetofcoordinates. Wewillseethatpropertiesofcoordinatetransformationsofthematrixmodelreflectthoseofthetarget space in string theory. It was proposed[10] that the Hamiltonian (7) with a=1 (i.e., H+K) is dual to AdS in global coordinates, while (4) is dual to the Poincare patch, which only covers half of the 2 former. Basedon this proposal,we will show that the RR flux backgroundq needs to be quantizedin order for the global time of AdS to be compactified. 2 In addition, we will show that, by adding a suitable (time-dependent) Fermi surface, we can use the Hamiltonian (7) with any a to describe string theory in the linear dilaton background. As the asymptoticbehaviorofparticlesfora>0,a=0anda<0aredrasticallydifferent,thephenomenology of each model appears to be very different, although the encoded information is (almost) equivalent. This reflects the observer dependence familiar in the context of general relativity. By analogy with AdS , there should exist a coordinate transformation (possibly combined with field redefinitions) of 2 string theory/supergravity in the linear dilaton background which extends the spacetime beyond the region manifest in the old description. In the appendix A we prove that (7) exhausts all possibilities of nontrivialisometries for 1 dimensional quantum mechanics with the standard kinetic term. The proof that all theories defined by (7) with the same M are equivalent at the quantum level are given in appendix B. 2 SL(2,R) Symmetry Thequantummechanicsdefinedby(4)wasextensivelystudied[11]asanexampleofquantummechan- icswithconformalsymmetry. Assumingthatthekineticenergyisstandard 1p2,bysimpledimensional 2 analysis one can see that (4) is the only potential consistent with conformal symmetry. (Conformal 2As the closed string tachyons are massless in 2 dimensions, tachyon condensation might never happen. What we meanhereistotunetheparameterµofthetachyon fieldbyhandto −∞. 2 quantum mechanics with more than one variables and general kinetic terms were discussed in [12].) The SL(2,R) symmetry (5) was found in [11] as the conformal symmetry. InthissectionwewillfirstreviewhowtheSL(2,R)symmetryisasymmetryofconformaltransfor- mations of the time variable together with a time-dependent scaling of the spatial coordinate for the matrix model [11]. This couldcome asa bit of surprisefor those who arenot familiarwith the results of [11], since the spacetime coordinates in the matrix model are nonlocally related to the coordinates of target space. (The nonlocal transformation are determined by the leg factors [4, 9].) Then we will use the Lie algebra generators, which are dual to Killing vectors in AdS , to define 2 new time coordinates, and find that the new Hamiltonians are all of the form (7) with the same M. 2.1 Isometry of Quantum Mechanics An infinitesimal general coordinate transformation is generated by a differential operator of the form =iA∂ +iB∂ +C, (8) t x D whereA,B,C arefunctions ofxandt. ClassicallyC canbe dropped,andtheinfinitesimalcoordinate transformation is δt=ǫA, δx=ǫB. (9) ThefinalresultwilljustifywhyC shouldbeallowedforquantummechanics. generatesasymmetry D if (EOM) = (EOM) (10) ′ D D forsomewelldefinedoperator sothatasolutionofthe Schrödingerequationisstillasolutionafter ′ D the transformation defined by , i.e., D (EOM)ψ =0 (EOM)eiǫDψ =0. (11) ⇒ Note that does not have to equal . The condition (10) is equivalent to ′ D D [(EOM), ˆ]=0, ˆ =AH Bp+C, (12) D D − where ˆ is with i∂ replaced by the Hamiltonian H. ( and ˆ are equivalent when acting on t D D D D solutions of the Schrödinger equation.) The generators (5) are to be identified with ˆ’s at t = 0 for D the matrix model. We will refer to the coordinate transformations which preserve the Schrödinger equation as the quantum mechanics isometry. It is naturalto ask when a quantum theory admits nontrivialisometry. Anytime-independentHamiltonianisanisometrygeneratoroftimetranslation. Weproveinappendix Aforwiththestandardkineticenergy 1p2 theonlytime-independentHamiltonianwithmorethanone 2 isometry generators is of the form (7). There are 3 isometry generators H,K,D (5) for a generic M. IfM =0,thereare2additionalgenerators,whichcanbeidentifiedwiththe creationandannihilation operators of the simple harmonic oscillator when a>0. In terms of the coordinates in the AdS theory (4), the SL(2,R) isometry is represented as the 2 projective transformation of time and a scaling of space [11] αt+β 1 t , x x. (13) → γt+δ → γt+δ 3 2.2 Killing Operators and Coordinate Transformations InanalogywithRiemanniangeometry,the symmetrygeneratorspreservingSchrödingerequationsare reminiscent of Killing vectors, which preserve the metric, and it is natural to use them to define new time coordinates. Of course, one can always perform arbitrary general coordinate transformations or change of variables to rewrite a theory. The special features of the time coordinates chosen by Killing operators are that their conjugate Hamiltonians are time-independent and of the standard form 1p2+V(x). 2 Take a generic SL(2,R) generator G=αH +βD+γK, (14) and we would like to introduce a new coordinate τ such that the old and new Schrödinger equations are equivalent (up to unitary transformation) ∂ ∂ i Ψ=HΨ i Ψ =GΨ, Ψ =UΨ. (15) ′ ′ ′ ∂t ⇐⇒ ∂τ It turns out that, remarkably, τ is simply a function of t [11] dt dτ = . (16) α+βt+γt2 For the range of t satisfying f(t)=(α+βt+γt2)>0, (17) τ is a legitimate reparametrization of time. Furthermore, after a simultaneous scaling of the spatial coordinate x σ = , (18) (α+βt+γt2)1/2 G is again of the form (7) [11] p2 ∆ M G= σ +V , V = σ2+ , (19) − 2 G G −8 2σ2 where ∆=β2 4αγ. (20) − The new coordinate system is as good as the old one as long as (17) is satisfied. The resultaboveis derivedinappendix B. Here we verify it in the classicaltheory. The actionfor the potential (4) is 1 M S = dt x˙2 . (21) 2 − 2x2 Z (cid:18) (cid:19) From (16) and (18), it follows that dx 1 =f−1/2(∂τσ+ (∂tf)σ). (22) dt 2 Plugging it in the action and integrating by parts, we find 1 τ1 1 d M S = (∂ f)x2 + dτ (∂ σ)2 σ2 . (23) 2 t 2 τ − 8 − 2σ2 (cid:20) (cid:21)τ0 Z (cid:18) (cid:19) This is exactly the action for the Hamiltonian G (19). Due to the boundary term in (23), a unitary transformation is involved in the quantum version. It is a very intriguing fact that all Hamiltonians of the form (7) are different descriptions of essentially the same theory related by simple coordinate transformations (16) and (18). 4 2.3 Coordinate Patches: S S S − 0 + ⊂ ⊂ Aredefinition(scaling)ofthecoordinates(σ λσ,τ λ2τ)hastheeffectofscalingthecoefficientof → → the σ2 term by λ4. So essentially we have 3 classes of G that are not relatedto each other in a trivial way. They are examplified by (α=1,β =0,γ =1), (α=1,β =0,γ =0) and (α=1,β =0,γ = 1). − For notation, we will equipvariableswith subscripts +,0, ,accordingto the values ofγ, andrefer to − the correspondingmatrix models by S ,S , andS , respectively. The Killing operatorsassociatedto + 0 − S ,S ,S are (H +K),H,(H K). Their potentials are + 0 − − 1 M M 1 M S : V = x2 + , S : V = , S : V = x2 + . (24) + + 2 + 2x2+ 0 0 2x20 − − −2 − 2x2 − The coordinate transformations are given by (16) and (18) t =tan(t )=tanh(t ), x =sec(t )x =sech(t )x . (25) 0 + 0 + + − − − It follows that the momenta are related by p =cos(t )p +sin(t )x =cosh(t )p sinh(t )x . (26) 0 + + + + − −− − − From these expressions we can see that the origins of all time coordinates coincide. At t = 0, the spatial coordinates and conjugate momenta also coincide. For later use, we list here the solutions of the classical equations of motion x = A2 cos(2(t T ))+ A4 +M, (27) + + +− + + r q M x = 2E (t T )2+ , (28) 0 0 0 0 − 2E r 0 x = A2 cosh(2(t T )) A4 M, (29) − r − −− − ± −− q where we assumed that M > 0, and so the particle stays on the positive side of the real line. The parameters are related by 2 A2 = E (T2 1)+ M +4E2T2, T = tan 1 2E0T0 , (30) + r(cid:16) 0 0 − 4E0(cid:17) 0 0 + − − (cid:18)E0(T02−1)+4ME0(cid:19) 2 A2− =r(cid:16)E0(T02+1)+ 4ME0(cid:17) −4E02T02, T− =−tanh−1(cid:18)E0(T202E+01T)0+4ME0(cid:19). (31) For M =0, the solutions are x = A(sin(t ) T cos(t )), (32) + + 0 + − x = A(t T ), (33) 0 0 0 − x = A(sinh(t ) T cosh(t )). (34) 0 − − − − These solutions are related to each other through (25). But they are not the M 0 limit of (27-29), → since the M =0 theory is not continuously connected to finite M theories. The coordinate transformations (25) are not 1-1 mappings. Roughly speaking, the coordinate systems S ,S ,S forms a cascade of patches 0 + − S S S . (35) 0 + − ⊂ ⊂ 5 The whole range R of time for a smaller patch is mapped to a finite interval in a larger patch. More precisely, 0,the classicaldescriptionofS has aperiodofπ instead + of 2π as shownby (27). 3 Hence the classicaltheory of S is completely equivalent to that of S . We + 0 need to examine the wave functions of S in order to see that the period of t is actually 2π. On + + the other hand, from the viewpoint of dual theories, it is a classicalstatement that the Poicare patch is half of the global AdS . This is hence an example of the fact that sometimes AdS/CFT duality 2 matches classical effects to quantum effects. 4 Fermi Sea The ground state of a matrix model is a Fermi sea filled to an energy µ. In another theory this state is mapped to a time-varying Fermi sea, which can not be viewed as a small fluctuation over a static 3ForM =0,theclassicalperiodoft+ isalready2π. 8 Figure 1: Phase space of S for M =0 Figure 2: Phase space of S for M >0 + + Figure 3: Phase space of S for M =0 Figure 4: Phase space of S for M >0 + 0 Fermi sea in view of the new Hamiltonian. A configuration easy to discuss in one theory may be complicated for another. A generic Fermi surface can be described by its boundary in the phase space f(x,p,t) = 0. As every point on the boundary has to follow the trajectories determined by the equation of motion (27- 29) and (32-34), one can always rewrite f(x,p,t) as a function of two variables. The two variables can be any two constants of motion, such as (E ,T ) in (28). Or instead we can use the coordinate 0 0 and momentum at t = 0. Whichever variables we choose, we can use (25) and (26) to switch the descriptions of a Fermi sea. The benefit of using the phase space variables at t = 0 is that they are identical in all theories. For M = 0, the phase space evolution for S is simply the circular periodic motion of the simple + harmonic oscillator. (See Fig.1.) 4 It is a periodic motion along deformed circles if M > 0. (See Fig.2.) Classically the period for M =0 is twice that for M >0. The phase space evolution for S is straight horizontal motion along the x axis if M = 0. (See 0 0 Fig.3.) For M > 0 a point in the phase space turns around at the infinite potential wall. The trajectories are asymptotically horizontal lines. (See Fig.4.) For S , a point in the phase space always moves along a hyperbolic curve. (See Fig.5 and Fig.6.) − It is then easy to visualize the Fermi surface in a different coordinate system. The phase space configuration of a Fermi surface at t = 0 in one coordinate system is identical in another coordinate system. As long as we knowhow eachpoint in the phase space evolveswith time, we caneasily figure out the evolution of the Fermi surface in any theory. 4The lengths of curves are different because they are particle trajectories for the same period of time, i.e., faster particles leave alonger curve. ForM =0thephase spaces have left-rightsymmetryand onlythe righthalfisplotted. ForM >0onlyx>0isallowed. 9

Isometry of AdS2 And The c=1 Matrix Model

Similar Isometry of AdS2 And The c=1 Matrix Model