From Noncommutative String/Membrane to Ordinary Ones JHEP04(2001)015 1 String Theory Appeared Extremely Usefultive in Study ﬁeld of Theories

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This content was downloaded from IP address 128.194.86.35 on 02/10/2018 at 21:13 JHEP04(2001)015 April 10, 2001 Accepted: February 19, 2001, Received: BRST Quantization, p-branes, Non-Commutative Geometry. We discuss origin of equivalence between noncommutative and ordinary [email protected] HYPER VERSION Center for Theoretical Physics, TexasCollege A&M Station, University TX 77843,E-mail: USA Abstract: Yang-Mills from point of view of stringwork theory. first Working we in BRST/hamiltonian investigate frame- stringof model variables in and the applying decouplinggives limit commuting the coordinates and conversion on show of the thatin D-brane. constraints change general Also, of case we and decoupled discuss showto algebra string the of ways decoupling theory constraints of limit. having commutativeequivalent coordinates to It the without commutative could going model. being We on investigate argued the the that case M-5-brane of noncommutative inequivalence the constant string in membrane C-field end- this in and case. discuss B-field noncommutative/commutative is Keywords: Igor Rudychev From noncommutative string/membrane to ordinary ones JHEP04(2001)015 1 String theory appeared extremely usefultive in study field of theories. the propertieys Moreover,background of considering not non-commuta- strings only and helps to membranes investigatebut in properties constant also of the B-field gives low-energy new field insight theoriesels [1] on and the connecting string andand old M-theory [6]. ones itself for by From producing examplebetween this new non-commutative point into mod- and of the ordinary view frameworkthe Yang-Mills it appearing of string is as OM-theory theory interesting decopling in [2]–[4] commutative limit to and the of ask ordinary large cases if B-field, willand there be membranes. how is looking In the correspondence like this correspondence note frominvestigate we the the between study possible point first both ways of the of decoupling non- you disappearingexample limits of non-commutativity. of of strings First both decoupling we theories start limit and from malism of which bosonic is string. equivalent Here tofrom world-sheet we both treatment work and in points then the of discussYang-Mills hamiltonian view. comes the for- from procedure Correspondence considering between instantonssee non-commutative in that and both from ordinary point theories ofnon-commutativity [1]. view appears But of from how the interpreting can string timetwo-point we theory. functions ordering From as as the well operator world-sheet astreatment consideration ordering in non-commutativity in product arises of from vertexscribing operators modification open [7]. of string In theory the the by usual additional hamiltonian constraints constraint coming de- from modified boundary 1. Introduction. Contents 1. Introduction.2. String in constant B-field, decoupling3. limit and String constraints in constant B-field, general4. case Membrane in constant C-field, decoupling5. limit 3 Discussions and conclusion 9 6 1 12 JHEP04(2001)015 2 The organization of paper is the following: in section 2 we start from decoupling conditions due to presence of theing constant B-field from [8]–[13]. boundary, This is newrelations of constraints, com- between the the second target class spaceD-brane. and coordinate Apparently, for one and a of momenta the first onto ways sight the introduce to spoils world-volume struggle the of with commutational Dirac thetency brackets second between [9] class commutators constraints and is of [11]–[13].space coordinates On coordinates and these of momenta Dirac the disappear, brackets stringThere instead, inconsis- on is target the another surface way ofversion of D-brane procedure working do [14]–[16]. with not the Usually, commutevariables conversion second anymore. by means class introducing extending new constraints the degrees - of phaseclass to freedom constraints space use as in of such well the a as con- way modificationor that of they give the become another second of constraints. theto first This class, the i.e. new gauge either first symmetries commute classnew of constraints the gauge correspond, theory, symmetries i.e. as ofstood by usually, the introducing from generalized new BRST-quantization variables point action.variables we of acquire as view. This ghosts framework In and could thispossess be new case new one under- gauge modified can invariance. actionin treat It ordinary in additional is BRST additional similar approach. to tobe But having previous in bosonic. new the symmetries There local case were fermionic ofFedosov interesting the symmetry quantization connections conversion [17]. new of symmetry BRSTconstraints could conversion We and procedure apply instead with of conversionall going to or to the crucial Dirac part system bracketsbrackets of and we with all constraints end target second of space up the coordinates class one with commute first is between the class. each able model other. In to with Inones this substitute some on sense case model the we with level don’t non-commutativetheory, of have coordinates it BRST Dirac by is quantization. the possibleconstraints It ordinary to only, is but use not we ordinary necessary needof Dirac to the the quantization BRST new former quantize of variables one the systemsspace and to as to with get a interpret rid first “ghosts”. these of class be extra Here ambiguity extended coordinates of we to of the mostly appearance supersymmetric the considerDescription case extended bosonic by of phase following part connection [23]–[25] of of and noncommutative the referencesconsidered Yang-Mills therein. theories. with in ordinary [18]. It one could was also limit of thefrom string hamiltonian in analysis constant ofconversion the B-field to constraints the and using system, thensection Dirac finding 3 see brackets. equivalent we one origin discuss Then withorigin general of we commutative of framework apply coordinates. non-commutativity non-commutativity of and the discuss In investigate bosonic possible decoupling ways string of limit in its of the removal.We the B-field, In discuss section membrane consider 4 analysis ending we of onnon-commutativity. constraints M-5-brane Discussions and in and ways large conclusion of C-field. could obtaining be equivalent found theory in without section 5. JHEP04(2001)015 (2.8) (2.6) (2.7) (2.3) (2.1) (2.5) (2.4) (2.9) (2.2) 2andtherest , nonzero only in , ab =1 b B 0 keeping fixed open . X t a, b ∂ ab → a , 0 X α ab B 0 ab , , , . . B where πα b B 1 2 ab ab 0 ab 2 ) =0 ab ) X t b ∂M − B πα b 1 B B ∂ 1 1 Z 1 2 a g X g 0 − t − − ∂ 1 X B =2 B 1 0 πα Bg g ab ab Bg ( 2 ab ( g 2 2 B 2 πα 0 ) ) . B 0 ) 0 3 − 0 B 1 dtB 0 ab 1 µ πα 2 +2 πα πα πα πα g 1 (2 ∂M 1 (2 B ∂X (2 +2 Z µ − − b 2 +2 − 2 1 ) 0 X g = = = ∂X ab n = g 2 ∂ πα ab ab ab ab S M ab θ (2 = = G G g Z − 0 ab ab = 1 G G πα 4 ab θ = S is nonzero only along Dp-brane and in general case includes B-field tois derivative normal to the boundary of the world-sheet. ab n B ∂ In this limit the kinetic term in (2.1) vanishes. The remaining part, which In the decoupling, zero slope, limit [1] one can take Here the effective metric seen by the open string is And noncommutativity parameter For simplicity and without loss of generality one can take of string coordinates do not give any contribution to (2.9). where two space directions on the D-brane, i.e. string parameters, i.e. on the world volume of D-brane governs the dynamics is theevolution second of term in the the boundary (2.1)D-brane. on which This in the part this string, of case i.e. the describes particle action the is living given on by the world-volume of The bosonic part ofdimensional the Minkowski world-sheet space in action constant of B-field the background string is ending given on by D-brane in ten 2. String in constant B-field, decoupling limit and constraints where gether with two-form field strengthD-brane on are the D-brane. The boundary conditions along JHEP04(2001)015 (2.17) (2.16) (2.11) (2.14) (2.12) (2.13) (2.10) (2.15) ). ξ P , and 1 ξ ]=1.Inthiscase ξ bX , − ] ξ,P b ) ξ P ,X 2 d φ √ [ b , cd + 2 2 . ] ab . c . b P b . ˙ b ab . 1 X ,φ 2 b a X a ab =( b 1 2 X ab 2 b X = X [ ˆ φ ab i b ab = with commutator [ − b 4 + ξ ] D (0) dt ]=2 b − ] b a b P b P , Z = ,φ ,X ) ,X b ,X a a a = ξ ) a t and φ 2 P a [ ( = X a ξ X φ √ [ S X =[ − h 2 D ] b X ( b ,X a + 1 X [ P = 1 ˆ φ Another way to treat constraints (2.12) is to extend the phase space of the So, the constraints that fully describe this model are All these constraints are of the second class, i.e. they don’t commute with each Going to Dirac brackets gives Now let us see how noncommutativity appears on the level of hamiltonian for- Canonical momentum is given by We see that coordinates of the string on the D-brane do not commute and in it is possible to convert the second class constraints (2.12) into the first one. where new constraints are ofof the degrees first of class freedom and tosecond they be class commute. sure constraints Let that eliminate us weone one count did number kills degree not two, of loose so freedom, anytherefore instead but information. we of each need Each two of of to second the the add class first two independent constraints class degrees we of have freedom two (i.e. first one now, which leads to noncommutative coordinates on the D-brane. variables and in the frameworkcanonically of BRST conjugated quantization variables to introduce additional pair of and other: malism. The string action in decoupling limit is the decoupling limitD-branes. one has It noncommutative is Yang-Millson possible on the to boundary the calculate of world-volume a the two-point of string function world-sheet. of And the it fields is given propagating by JHEP04(2001)015 a (2.21) (2.23) (2.22) (2.19) (2.20) (2.18) are not + ,X − ,thatcommute, P − ,X 1 ξ, ξ, X 2 . 2 , 1 , √ √ ) which are unchanged and i 1 + bX − 1 2 (0) 2 bX + 1 ,X X X 1 + + P ,X , = = P + ) 2 t + ˆ P − ( φ = ( 1 − 1 2 λ c ˆ φ X h + 5 1 )+ = ˆ ,X φ ,X − i ξ ξ λ , P P − bX 2 = (0) 2 − √ + √ bX H 1 − + + ,X P ) 2 ( 2 1 t c P ( P P 1 − − = = X h 1 = = Q ˆ φ + − P P are ghosts corresponding to each first class constraint. It is also are Lagrange multipliers. 1 1 c λ and and c λ and leads to commutative coordinates not only on the level of Poisson brackets b The hamiltonian of the system which is equivalent one given by (2.1) is The BRST charge is given by Those two constraint not only of the first class but also assume that propaga- Now we can identify new variables, they are: but on the level ofcould two-point be function on seen the using boundary correspondencetwo-point of between functions the operator are string and worldsheet, time that ordering [7]. So, new dynamical ones becausetaking they the don’t form: participate in the constraints. (2.17) now is and could be calculated usingthe path-integral noncommutativity BRST of approach. the Therefore,ables, we string and removed end-point applying coordinates the by conversiononly of introducing first the new class constraints. constraints vari- By wherenates this we commute way don’t on we have the ended to up level usenew with of coordinates Dirac the of brackets Poisson the and string brackets. all boundary coordi- Also, are after described field now redefinition by the because propagator is symmetric.tivity on So the using boundary timeby of using ordering string BRST product world-sheet. quantization gives of It commuta- the could theory be described more by appropriate the constraints showed (2.20). where where where they are consequently canonically conjugate. The variables tor (2.16) is notand antisymmetric anymore, but rather symmetric in interchange of and possible to include information thatin boundary [1], but action the comes main from resultredefinition string that gives conversion as together ordinary, was with commutative done change behaviorsense of variables for the and the fields conversion string procedure boundary. is equivalent In to some restoring gauge symmetry of theory JHEP04(2001)015 (2.24) (2.26) (2.25) . , a a ξ ξ + − a a . X X . − b = = + X a a − 2. Then generalizing (2.18) define X + − , P ab b − =1 6 + + a − X a ,X ,X + P ξ ξ a a P = P P where a + − ˆ a b φ are of the first class, i.e. commute. But, in difference δ Ψ= a a a P P ˆ φ ]= = = ξ b + − a a ,P P P a ξ . If one puts all new variables equal to zero the constraints (2.25) ,[ a ξ a P X and a ξ and doesn’t carry any new information. a In this Section we gave two examples of conversion that produced equivalent Then (2.12) could be modified into This choice in some sense reminds constraint that could be obtained from (2.12) Now all the constraints φ systems with commuting variables (i.e.ples commuting could be string of end-points). theonly Those strong on exam- suggestion the that level noncommutativityfunctions. could of be constraints/Dirac removed brackets, not but also on the level of two-point 3. String in constant B-field,Here general we case step asidesection from and decoupling limit analyze of completeis string more set which convenient of was to describedB-dependent constraints start in term coming not previous [1] from from and lagrangian the impose (2.1) action boundary but (2.1). condition rather (2.2) from as It one additional without boundary where it was broken. Forand example then if one introduce starts new from variables,conversion, Maxwell that theory one were with fixed, get fixed i.e. abelian gauge nondynamical gaugemodels, before, symmetry with then on fixed use the gauge level and ofonly with effective on gauge action. invariance, classical are And but equivalent two almost to also the each same on other except quantum not that we level.theory start from could In theory be without our apparent interpreted case gauge asis fix, of one but just this interest with to the fixed gauge find situationthe and is equivalent the system extended main of and tasknoncommutativity of therefore first appearing conversion more class in general constraints initial theorytheory theory only. described but is by not This just a procedureis a fundamental possible helps some property to us particular ofway. the conduct to choice To conversion of argue do so for gauge that let stringvariables fixing us in start of from decoupling equivalent system limit model. of in constraints (2.12) more It and covariant then introduce new from previous example ofneed conversion, to degrees add of oneto freedom more be counting first equivalent tells class to us constraint, initial that one. otherwise we modified This constraint system could is be not chosen going in the form by projecting by transform to (2.12), andon (2.26) is becoming equal to zero, i.e. does not dependent JHEP04(2001)015 . ) σ ( (3.4) (3.3) (3.6) (3.1) (3.5) (3.2) (3.7) (3.8) 1 N − )= σ , − ( 1 are coordinates of =0 a . +1) X as in (3.7) except that µ k 0 a (2 X ,N P P , µ ) P σ = . ( , and = =0 . µ N = 0 leads to infinite number of 1 1 a a +1) m 0 c 0 k H X ∂X 1 B (2 X )= . µ c N σ a . All these constraints are equivalent P 0 σ − P + 0 ( ,H σ∂X 2 πα ,X 0 2 πα is the rest of coordinates. d ,X 7 X NH 0 +2 with hamiltonian gives either constraint or m =0 Z =2 a 1 0 0 a a =0 X πα ,N a b dσ 2 ) X ) ˙ 1 k B πα X σ b Z 4 (2 − 1 ( = ˙ 2 X a P = = P t Φ 0 + S a H 0 )= πα σ ,N X ] and taking the orbifold projection [19], [13] − ( =2 =0 π, π H − +1) k (2 ,P to [ ) N σ σ ( = 0 we have constraint (3.6). is full set of target-space coordinates for string, X k µ ) denotes k’s derivative in respect to k if X )= a σ Themomentuminthiscaseis The hamiltonian is given by If B-field is absent, the boundary condition As usually, we have first class constraints which follows from (3.1) Then the boundary condition (3.2) could be rewritten as constraint In the presence of constant B-field the secondary constraints appear, as usually, X − ( X and boundary constraint where condition for Lagrange multipliers.the Lagrange First multipliers of are the all,of same Φ’s in as leads constant in to B-field, (3.7) the conditions and same using for boundary the conditions linear for combinations where ( to extending for string endpoints on the D-brane and constraint. The starting action is constraints in the form from the fact that commutator of Φ JHEP04(2001)015 . π (3.9) (3.12) (3.11) (3.10) i.e. because a .Wesee ) =0or a ,andalso c 0 0 ( a P , P P ] b reminded ones σ, σ or ,X a ) and 2r additional +1) c 00 k b σ (2 a ( δ d P , 0 P ca , [ . If B-field is zero we see 1 ]= B b and ) = 0. One can ask why new c − cd c ( b P =0 C 0 P +3) a )] 0 k as in the case of string without ,P 00 a πα (2 a and Φ c σ 2 ( a X c ,[ − ,X ) Φ X c a ( , a c ,P a P ca = X [ . a 1 B 00 c 8 and P dσ 0 a =0 c ) Z πα c ( ,φ a − a ] +2 c ˆ b Φ commute with a 0 + ,X a X 0 a ,P = X X a = =0 =[ Φ 0 a a D ˆ P Φ )] 0 σ ( b ,X ) is inverse matrix of commutator coefficients between Φ and σ ( 1 a − cd X C [ Now, instead of 2r constraints we have 4r ones but we added 2r new variables, Because we start from (3.1) but not (2.1) in difference from [9] and [11]–[13] we where we have to use regularization for the endpoints (see [9, 11]) , i.e. for B-field and the restDirac of brackets. constraints (3.11), Procedure, (3.12)of which also all we don’t because used give conversion any here, assumessome contribution is that particular to different one values can from of obtainthat additional conversion. initial after variables. First constraints putting For after example all fixing Here in new it Section is variables not 2 equal the we to case.model showed zero is We can’t equivalent one fix to has c, oldthe otherwise one. initial it variables. It constraint gives could system. And be we argued that don’t what have we to did is fix just any redefining particular value of for the case ofappears zero because B-field. of the The nonzero noncommutativity commutator on of the level of Dirac brackets that coordinates of endpointsthe do nature not of commute this butof noncommutativity. momenta second Here do. class we Let will constraints ussystem not into by investigate the use the conversion first way that of ones. second the class It system constraints is Φ more transparent to modify the The same contribution is given by considering higher odd derivatives of second class constraints. With modified initial ones we have that they commute and Diracthe brackets system are the of same constraintsmoment as to we Poisson will obtain ones. forget one Let aboutto us that incorporate higher modify is them derivative constraints, equivalent into but the to it whole initial is picture. system. straightforward We will For start a from additional constraints are notwhy they of can the be first solved class, algebraically to but produce rather initial second system. one Here and we that’s considered so number of degreesidentically equal of to freedom zero because remains the same. The Dirac brackets (3.9) are don’t have modification of theDirac higher brackets derivative contribution constraints but to rather the after noncommutativity going is to given only due to Φ where a runs from 1introduce to 2r r. new We have canonical 2r variables second class constraints here. To modify them JHEP04(2001)015 0 but → p l rather then a φ by the same way as in (3.11). are new variables playing role of ) c ˆ ( X P 9 ,where i b ˆ X , a ˆ X h 0) six-dimensional tensor multiplet. , is governed by (2 described by only Wess-Zumino membrane term; . And application of the same technique is straightforward. Now we see that 3. then, after decoupling, all dynamics of the world-volume theory of five-brane 1. bulk modes of the membrane must2. decouple string, and that disappear; lives on the boundary, i.e. on the surface of the five-brane is fully a 0 As was shown in [4]harmonics it and is [6] possible by to rewriting produce membrane decoupling kinetic explicitly term in the in limit terms when of Lorenz open membrane metric is fixed.Zumino membrane Following [4] term one in can themembrane find case has convenient of two form constant contributions of C-field. - Wess- The from Wess-Zumino pullback term for of eleven-dimensional three form X In this section we discussof membrane how ending to on remove the noncommutativity M-5between for branes. taking the decoupling First limit decoupling of of limit all, thein there string are and eleven some membrane crucial [4, dimensions differences 5].metric is The generated only the by constant the Plank five-branesconsider as constant. was stock was of explained five Also inThen branes [4]. we the and Moreover can’t decoupling probe we have limit membrane take to could ending flat be on background found one from of the the following five-branes. : 4. Membrane in constant C-field, decoupling limit simplified system comparing tois one of that [9]. in The the main later difference case in there sets were of taken constraints even higher derivatives of effective string coordinates on theand boundary, interpreting two-point time functions become orderingargue symmetric as that operator noncommutativity of orderingmovable string gives and endpoints commutativity. depends is Here on notconsider the we fundamental modified change but BRST of rather charge basis. re- andshow perform that It analysis by also of the possible mode sameproduce as decomposition way commutativity in and it of to the is the possible sectionbecoming string to 2 straightforward endpoints introduce to if new on algebraic one thein variables starts level [13] which from of and Dirac constraints modify for brackets. them by the It introducing string is c modes and given introducing new variables even without changingond the to nature of the constraints first (fromequal sec- to class) zero. produces Therefore thein we were Dirac new able brackets variables to for show commutenot that string between endpoints so endpoints of each straightforward the which other. as stringvariables are expressed for and Analysis field the of redefinition case two-point of functions the is decoupling limit but after change of JHEP04(2001)015 (4.6) (4.4) (4.9) (4.7) (4.3) (4.1) (4.8) (4.5) (4.2) (4.10) , where here , to membrane c (3) 0 (2) A X b B + ˙ X a (2) X . and dot - in respect to dB . abc σ ) = 0 =0 h. 3 h σ c . 0 (3) σ = , 2 − . X C (2) c , d 0 b ) σ (3) c 456 B ( 0 . X X (2) µν 3 Z δ c b a 0 c a X 0 ˙ b X P + X ∂M , + a X b k X Z a 0 abc X X P abc ,C (2) µν abc + X a 2 3 0 h j are five-brane coordinates normal to the abc B abc h h which gives zero pullback to the five-brane ˙ 3 h (3) X 6 10 X p 2 3 a + i l 3 h 3 h A 2 (3) + X σ h 3 X =3( − 2 a A M d 1+ λ Ψ= )] = P ijk 0 = Z . Then first class constraint is X σ p 2 a Z = ( = h b P ,P a − ,P 6 p 0 (3) µνλ = l φ ,φ C = ) WZ S h =0 σ S ( 1+ a (3) X, X 012 a P C φ p a [ 3 by X σ a 2 φ d . Excluding part of Z (2) = da S lies on the membrane and = i (3) X A is pullback to five-brane world-volume. The notion of nonlinear self-duality for to membrane surface and from pullback of five-brane two-form Consider the second part of the (4.4) With constraints The action of the membrane ending on M-5-brane in decoupling limit is given The constant C-field on the five-brane is given by Among three constraints two are of the second class and one is the first class. Themomentumisgivenby The components of constant C-field could be chosen in the form [1] (3) (3) where 5-brane world-volume was decomposedwhere into the two three-dimensional pieces, and the second ones and Poisson brackets boundary, i.e. A membrane and prime denotestime. differentiation in respect to A world-volume and using the fact that C is constant one has by two terms [4] C-field on the five-branewrite is extremely important here [20]. At least locally we can To see that project JHEP04(2001)015 ). 0 σ − (4.14) (4.13) (4.12) (4.15) (4.11) σ ( δ a b δ )] = = 0 gives initial 0 ξ σ . ( P ) ξ ( b =0 . , ,P a a ) ) ) 0− ξ ξ σ ( ( X ( a =0and P P − ξ , ξ P − + + a a c + 0 P P + X with [ . + = = ) X b ξ ( + b − + 0 X a a =0 P c abc 0 11 X 3 h and b a ˙ ,P + X ,P ,P ξ a a − a abc ξ ξ =0 P + − − = a a X a X X ˆ − φ P = = − + − a a + X X X + P Then the modified constraints (4.7) could be rewritten as the following first class In this example we see that even for the nonlinear case of membrane boundary It is not unique but convenient choice. Now all the constraints (4.14) and (4.15) This BRST charge describes theory equivalent to initial one not only on the Here, because we have a mixture of first and second class constraints it is more The equations of motion are and they identically commute.thing Now constraints. we need It two is additional possible commuting to with choose every- them in the form constraints theory. In some sense onegauge can symmetries think that , model which beforeBy could conversion the be was same one restored way with on one fixed the can level proceed of for BRST theit invariant first is action. term possible in (4.4). procedure to is change close variables to tomembrane one one end of in up the theconversion with constraints directly end commutative was in of coordinates. of the second mixture the of section first This two class except types and of that constraints. we for were the forced case to of have But if we wantof to degrees convert of two freedom secondsix. tells class us The constraints that resolution into weindependent the of and need first this that’s only why one problem two wekills counting extra is need four variables to that and degrees impose not we two ofthat all extra added freedom we of first needed. leaving class the constraints as It above that is with new convenient only to variables two proceed are extra as follows. independent Let variables us define are of theconstraints first multiplied by class. corresponding ghosts plus The contribution BRST from commutators. chargeclassical is but also now on given the by quantum level. sum Fixing of gauge the first class convenient to convert themvariables directly which give in us all the constraints mixture,in of mixture the i.e. first of we class. first need and Thein second discussions to on class [21]. find conversion constraints additional and To referencesvolume do let therein us could it introduce be new covariantly found variables in three dimensions orthogonal to membrane world- JHEP04(2001)015 12 0) six-dimensional theory. Therefore there is a connection to , Here we presented only analysis of bosonic part of theories. It is interesting to Hamiltonian systems with boundaries were also extensively studied in [28] from It could be noted that membrane boundary string in decoupling limit and critical It is interesting to ask what are the other nonlinearFirst models of which all, because posses action the (4.5) is pure Wess-Zumino term of the membrane it consider whole supersymmetric case extendingmetric case approach it of is [23]–[25]. also importantsolutions For to supersym- understand and situation Dp-Dq on the brane level of systems supersymmetric [26, 27]. the similar point of viewconditions as well as as constraints in [29, thecontribution. 30] Poisson where It instead is brackets of interesting were interpreting to boundary modified compare to results coming includeC-field from boundary should two approaches. be not onlythat tensionless reminds but a also Little doesn’tworld-sheet String formulation have Theory gravity of concept, in tensionless it’s string seetogether spectrum, [22] [31] with has and null kinetic references vectors term therein.times and quadratic in in world-sheet Usual literature fields of it is such calledcomplete a null-string. spectrum string Unfortunately is is this not theory a knownless is yet. null string anomalous surface. and It could could be Some- beBut described interesting because by to (4.5) equations assume describes of thatabove motion tension- membrane tensionless (4.7) boundary string and in doesn’t by decoupling containlimit lagrangian graviton limit is (4.5). in the described it’s mentioned by spectrum (2 and it’s low energy same equations of motion asin (4.11) the and decoupling carry limit. some resemblance with the membrane is useful to start from known nonlinear Wess-Zumino models in two dimensions. Let In this notethe we phase showed space that of the byto models appropriate be of change noncommutative equivalent string of toof and variables string/membrane membrane ordinary they endpoints, and ones. appear appearing extensionpends fundamental, It of is on is rather field removable possible andjob, content to de- and as argue change for that ofof the noncommutativity variables. two case phase of Not spaces string/membraneIt without only in is changing conversion type decoupling important is of limit, toand doing constraints ordinary investigate but it’s is one the also giving from correspondence equivalence equivalence the worldsheet works between same point for result. noncommutative of the string view.mutative product algebra. of It vertex is operators, not which quite apriori clear obey yet noncom- how this Little String Theory. Modifiedcould system be of quantized the in first theoperator class BRST framework. constraints BRST (4.14) At quantization. least and itwell (4.15) It is as straightforward is build to interesting conduct vertex operators to and find study a this spectrum model of from world-sheet this point model of as view. 5. Discussions and conclusion JHEP04(2001)015 ) 5 (5.2) (5.3) (5.1) ,X 4 ,X 3 X it is possible J. High Energy ]. J. High Energy , , ]. g. A noncommutative ν , )with( φ, θ, ψ ∂ OM theory in diverse 1 ) 4 − 0 ψ. X gg ν 5 µ ˙ cosθ, ψ ∂ θ∂ X − µ µν − hep-th/0005026 1 φ, 5 0 − hep-th/0006062 X ˙ sinθ∂ gg 4 1 Strings in background electric field, ˙ µν − X ( 3 ξg ξφ 2 2 d d (2000) 173 [ ξX 2 13 ]. Z ]. d (2000) 008 [ Z π dr Z k 08 4 π B 590 Z k 4 = (2) in terms of Euler‘s angles π k = 96 SU it is possible to identify ( String theory and noncommutative geometry = a WZNW − X hep-th/9908142 hep-th/0005040 WZNW Nucl. Phys. − WZ , S WZNW WZ − S WZ J. High Energy Phys. S , (1999) 032 [ (2000) 021 [ 09 06 Phys. dimensions M-theory five-brane space/time noncommutativity and a new noncritical string theory Phys. At least for compact Using parametrization of [2] R. Gopakumar, S. Minwalla, N. Seiberg and A. Strominger, [1] N. Seiberg and E. Witten, [3] N. Seiberg, L. Susskind and N. Toumbas, [4] E. Bergshoeff, D.S. Berman, J.P. van der Schaar and P. Sundell, and it is always possiblesame to equations go of to free-field motionendpoints realization on as of five-brane (4.11). (5.3). and Wess-Zumino Thislent So, part action on of two gives the SU(2) the level models, WZNW ofof string model, lagrangians equations are describing are of equiva- different. motion, membrane between but Therefore, description not it of could the evolution be action ofand because useful the to the topological boundary find symmetries part a string connectionmembrane of on if end-points. the SU(2) any surface WZNW of model five-brane at least for compact coordinates of consequently. In this case WZ part of WZNW SU(2) model could be rewritten as I would like toE. thank Sezgin A. for Sadrzadeh, fruitful T.A. discussions Tran and for for discussions constant and encouragement. especiallyReferences thank Acknowledgments us consider Wess-Zumino term of SU(2) WZNW model. It is given by the action: to rewrite (5.1) as JHEP04(2001)015 , B 06 Mod. Phys. ]. (1989) , , (2000) 014 (2000) 193 Phys. Lett. ]. , (1984) 81; Critical fields B 314 07 Nucl. Phys. , B 492 B 145 hep-th/9906192 Nucl. 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