We consider the closed string moving in the weakly curved background and its
totally T-dualized background. Using T-duality transformation laws, we find the
structure of the Poisson brackets in the T-dual space corresponding to the
fundamental Poisson brackets in the original theory. From this structure we
obtain that the commutative original theory is equivalent to the
non-commutative T-dual theory, whose Poisson brackets are proportional to the
background fluxes times winding and momenta numbers. The non-commutative theory
of the present article is more nongeometrical then T-folds and in the case of
three space-time dimensions corresponds to the nongeometric space-time with
$R$-flux.Comment: We add the Sec. 4. where we compared our results with previous ones.
  We also improved Abstract, Introduction and Conclusion as described above. In
  addition, we corrected all typos and grammatical errors we notice

Davidovic, Ljubica

Nikolic, Bojan

Sazdovic, Branislav

Facta Universitatis Series : Physics, Chemistry and Technology

English

arXiv

We consider the propagation of the closed bosonic string in the weakly curved background. We show that the closed string non-commutativity is essentially connected to the T-duality and nontrivial background. From the T-duality transformation laws, connecting the canonical variables of the original and T-dual theory, we nd the structure of the Poisson brackets in the T-dual space corresponding to the fundamental Poisson brackets in the original theory. We nd that the commutative original theory is equivalent to the non-commutative T-dual theory, in which Poisson brackets close on winding and momenta numbers and the coecients are proportional to the background uxes

Davidović, Ljubica

Nikolić, Bojan

Sazdović, Branislav

University of Niš: Facta Universitatis (E-Journals) / Универзитет у Нишу

FACTA UNIVERSITATIS (NIS)Physics, Chemistry and Technology Vol. 12, No2 , Special Issue, 2014, pp. 101{110DOI: 10.2298/FUPCT1402101DCANONICAL APPROACH TO THE CLOSED STRINGNON-COMMUTATIVITY Ljubica Davidovic, Bojan Nikolic and Branislav SazdovicAbstract. We consider the propagation of the closed bosonic string in the weaklycurved background. We show that the closed string non-commutativity is essentiallyconnected to the T-duality and nontrivial background. From the T-duality transfor-mation laws, connecting the canonical variables of the original and T-dual theory, wend the structure of the Poisson brackets in the T-dual space corresponding to thefundamental Poisson brackets in the original theory. We nd that the commutativeoriginal theory is equivalent to the non-commutative T-dual theory, in which Poissonbrackets close on winding and momenta numbers and the coecients are proportionalto the background uxes.1. IntroductionRecently, in Refs. [1, 2] the non-commutativity of the closed string coordinates wasfound to exist in the presence of the nontrivial background elds uxes. In thesepapers the dierent T-dual backgrounds of the three dimensional torus were consid-ered. Refs. [1, 2] motivated us to investigate the closed string non-commutativity.Here we present a dierent approach, based on the canonical method and an analogywith the open string non-commutativity investigated in [3].The open string non-commutativity has a source in a Kalb-Ramond eld [4, 5].In the constant background only the open string endpoints attached to a Dp-braneare non-commutative, with the non-commutativity parameter proportional to theKalb-Ramond eld.For the open string described by the actionS(x) = Z(2G + "B)@x@x ;(1.1)Received June 20, 2014.UDC 537.6/.8:530.145This work is partially supported by Serbian Ministry of Education, Science and TechnologicalDevelopment under contract No.171031.101102 Lj. DAVIDOVIC, B. NIKOLIC and B. SAZDOVICthe minimal action principle in addition to the equation of motion produces theboundary conditions as well. Solving the boundary conditions, one obtains theexpression for the initial coordinate x, in terms of the eective coordinate q andthe eective momenta p (which are even parts of the original coordinates andmomenta)x = q   2Z d1p(1) :(1.2)Here, the non-commutativity parameter  is dened in terms of the eectivemetric GE by =   2(G 1E BG 1) ; GE = (G  4B2) :(1.3)Because the original coordinates x are the linear combination of both eectivecoordinates and momenta, using the relation fq(); p()g = 2s(; ) oneobtainsfx(0); x(0)g =  2 ; fx(); x()g = 2 :(1.4)So, the ends of the open string attached to Dp-brane become non-commutative ina presence of the constant Kalb-Ramond eld B . If the action does not containthe Kalb-Ramond eld then there is no non-commutativity, because the solutiondepends only on the eective coordinate q.The open string non-commutativity follows from the fact that original coordinatecan be expressed in terms of the eective coordinates and eective momenta. Thisrelation is a consequence of the boundary conditions at the open string endpoints.The closed string does not have endpoints. So, to follow the analogy with open stringwe are going to express coordinates of the closed string in terms of coordinates andmomenta of some other theory. We are going to show that for some backgroundelds we can take T-dual theory as that other theory.We consider the weakly curved background which depends on all the coordinates.In paper [6] we proposed the T-dualization procedure for such a background, whichis the generalization of the well known Buscher procedure [7]. T-dualising all thecoordinates we nd the T-dual theory, and the transformation laws between theoriginal and T-dual coordinates. We express these laws in the canonical form,and use them to nd the relation between Poisson brackets in the original and T-dual spaces. From the transformation laws we obtain that the original coordinatesdepend on both T-dual coordinates and T-dual momenta. Herefrom one obtainsthe closed string non-commutativity, and T-duality is a way to observe it. T-dualsof all the Poisson brackets between coordinates close on the winding numbers andmomenta of the T-dual background. The coecients are uxes introduced in [1, 2].The term of the action with constant part of the Kalb-Ramond eld b istopological and consequently it does not contribute to the equations of motion. Inthe open string case it contributes to the boundary conditions and it is a source of theopen string non-commutativity. In the closed string case it is absent from boundaryCanonical Approach to the Closed String Non-commutativity 103conditions as well. Classically we can gauge it away and Kalb-Ramond eld becomesinnitesimally small. But, if b = 0 one loses topological contributions. In orderto investigate the global structure of the theory with holonomies of the world sheetgauge elds in quantum theory we should preserve this term.2. T-dualityT-duality is the symmetry which exist only in string theories [8]. It is a consequenceof the fact that the fundamental objects in these theories are extended objects andnot the point particles. T-duality connects physically equivalent theories givingeectively dierent prescriptions of a string, described while moving in the dierentbackground elds and therefore dierent space-times.If one coordinate in original theory is compactied on the circle of radius R andone coordinate in the T-dual theory is compactied on the circle on radius ~R, thenthe mass squared of any stateM2 =n2R2+m2R202+ oscillators ;(2.1)is invariant under transformationn$ m; R$ ~R  0=R :(2.2)Here the numbers n and m are integers which denotes momentum and windingmodes. The complete spectrums of the T-dual theories are the same.It can be shown that the T-dual action ?S has the same form as initial one butwith dierent background elds. The T-dual metric and T-dual Kalb-Ramond eld?G  (G 1E ) ; ?B   ;(2.3)are proportional to the eective metric and non-commutativity parameter fromthe open string case. One can compare the original and T-dual Hamiltonian andobtain the canonical T-dual transformation laws. When string moves in the constantbackground then these laws are of the following form = y0 ; ? = x0 :(2.4)The momenta are T-dual to the sigma derivatives of the dual coordinates and viceversa. Note that coordinates do not depend on dual coordinates but only on dualmomenta. So, because momenta commute for constant background there is noclosed string non-commutativityf; g = 0 =) fy; yg = 0 :(2.5)104 Lj. DAVIDOVIC, B. NIKOLIC and B. SAZDOVIC3. Weakly curved backgroundWe have learned that T-duality is not enough to establish closed string non-commu-tativity. We should generalize the background elds. The weakly curved back-ground is the simplest coordinate dependent solution of the space time equationsfor the background elds [9]. It consists of the constant metric G = const andthe coordinate dependent Kalb-Ramond eldB = b +13Hx  b + h(x);(3.1)where b and H are constant and H is innitesimal.The standard Buscher's T-dualization procedure [7] is not applicable for thecoordinate dependent backgrounds which depend on all the space-time coordinates.So, we generalized it.The generalized Buscher's construction [6] has three steps. First is to gauge theglobal symmetry x = , which is a symmetry even if B is linear in coordinate,and substitute the ordinary derivative with the covariant one@x ! Dx = @x + v ;(3.2)where v are gauge elds. Second is to substitute the coordinate in the argumentof the background elds with its invariant extensionx ! xinv =ZPdDx;which is the line integral of the covariant derivatives of the original coordinate.With this substitutions one nds the gauged action. In order to obtain equivalenttheories we require that gauge elds should be nonphysical. So, third step is toadd a new term to the lagrangian yF where y is Lagrange multiplier and Fis a eld strength of the gauge elds v. The T-dual action is obtained on thesolution of the equations of motion for the gauge elds. It is dened on the doubledgeometry with coordinates (y; ~y), where _y = ~y0 and _~y = y0. So, the coordinatefrom the original space is replaced by two coordinates x ! (y; ~y). The T-dualbackground elds equal?G = (G 1E )(V ) ; ?B =2(V );(3.3)and the T-dual action is of the form?S =22Zd2@+y  (V )@ y ;(3.4)whereV  =  0 y + (g 1E ) ~y ;(3.5)Canonical Approach to the Closed String Non-commutativity 105and (x) =  2 G 1E (x)(x)G 1 ;  = B(x) 12G :(3.6)T-dual transformation laws are obtained comparing the solutions of the equa-tions of motions for the gauge xed actions with respect to the Lagrange multipliersand gauge elds@x =   (V )@y  20 (V ) ;(3.7)@y =  2(x)@x   (x) ;(3.8)where (x) = 16H@xx :(3.9)Expression for  comes from the termZd2v+B(V )v  =Zd2 (V )v :(3.10)The transformation laws can be presented in the canonical form asx0 =1?   0 0(V )  (g 1E )1(V ) ;(3.11)y0 =1   0(x) :(3.12)The innitesimal quantities  are an improvement in comparison to the at spacecase. Also, as we will see, they are the source of a non-commutativity.4. Non-commutativity of the closed string canonical variablesFinally, we are ready to consider closed string non-commutativity, as we introducedboth necessary components, the T-duality and the weakly curved background. Thecanonical T-dual transformation law (3.12) express the dual coordinates y in termsof the original coordinates x and momenta . Let us take the following Poissonbracket in the original theoryfx(); ()g =  (   ); fx(); x()g = 0; f(); ()g = 0;(4.1)and nd the corresponding Poisson brackets of the T-dual theory.Using T-duality transformation law (3.12), we search for the corresponding Pois-son structure in T-dual theory i.e. the expressions for the Poisson brackets betweenthe T-dual string coordinates y(), ~y() and momenta?(). This is doneconsidering the brackets betweenY(; 0) =Z 0d Y 0() = Y()  Y(0);(4.2)106 Lj. DAVIDOVIC, B. NIKOLIC and B. SAZDOVICY = (y; ~y) and calculating the equal time commutators. The fact that T-dualcoordinates under T-duality transform to both coordinate and momenta dependentexpressions, enables non-commutativity. The relation of the formfX 0(); Y 0()g = K 0()(   ) + L()0(   );(4.3)implies the following relation between coordinatesfX(; ); Y(; )g =   [K() K() + L()] (   ) ;(4.4)where () is the periodic step function() =8<: 0 if  = 01=2 if 0 <  < 2;  2 [0; 2]:1 if  = 2(4.5)Using the transformation law (3.12), we can calculate Poisson brackets fy0; y0g,fy0(); ~y0()g and f~y0(); ~y0()g. We can re-express them in terms of uxes:Christoel connection corresponding to the eective metric GE E; =12@GE + @GE   @GE=  43B(G 1b) +B(G 1b);(4.6)and the coecient of the dual Kalb-Ramond eldQ =  13h(g 1)(g 1)   20 0iB ;(4.7)dened by the relation ?B(V ) = ?b +QV. We have1. fy0; y0gK [x] =3h [x] =1Bx; L = 0;(4.8)2. fy0; ~y0gK [x; ~x] =3h [~x]  6hh[x]G 1b+ bG 1h[x]i=1B~x   32 E;x;L [x] =1g   6hh[x]G 1b+ bG 1h[x]i=1g   32 E;x;(4.9)with~x0 =1(G 1) + 2(G 1B)x0 :(4.10)Canonical Approach to the Closed String Non-commutativity 1073. f~y0; ~y0gK [x] =3h [x] +24hbh[x]bi+6hh[~x]b  bh[~x]i=   1hB   6gQgix+h  32 E;    E;+4B(G 1b)i~x:L = 0 :(4.11)For the above values of K and L, the relation (4.4) givesfy(); y()g =   1Bx()  x()(   );(4.12)fy(); ~y()g =  n 1B~x()  ~x()  32 E;x()  x()+1g   32 E; x()o(   );(4.13)f~y(); ~y()g =  n  1hB   6gQgix()  x()+h  32 E;    E;+4B(G 1b)i~x()  ~x()o (   ) :(4.14)After two-dimensional reparametrizations, the  dependent part takes the formX(f()) X(f())[f()  f()];where f() is monotonically increasing function with properties f(0) = 0 andf(2) = 2. Therefore, Poisson bracket between dierent points is not repara-metrization invariant. For xed points, it can be t to be arbitrary small, by theappropriate choice of the function f(). So, only Poisson brackets at the same pointare physically signicant.Taking  =  we obtain that all Poisson brackets vanish, and consequently, co-ordinates commute. But, taking  = +2, in the non-commutativity relation be-tween the dual coordinates y's (4.12), we obtain the closed string non-commutativityrelationfy( + 2); y()g =  2BN;(4.15)whereN =12[x( + 2)  x()] ;(4.16)108 Lj. DAVIDOVIC, B. NIKOLIC and B. SAZDOVICis winding number for the initial coordinate x. This result is in agreement withRef.[2]. Similarly, from (4.13) and (4.14), we obtainfy( + 2); ~y()g+ fy(); ~y( + 2)g = 42Bp + 3 E;   8BbN;(4.17)andf~y( + 2); ~y()g = 2h B   6gQg + 2Bg+ 3  E;    E;biN+23  E;    E;p   8Bbp :(4.18)Using (4.10) and integrating from  to  + 2 we have12[~x( + 2)  ~x()] = 1(G 1)p + 2(G 1)bN ;(4.19)wherep =12Z +2d() :(4.20)To complete the algebra we add the following relationsfy(); ?()g = (   ) + h[x()]0 (   )+ h[x0()]0 (   ) ;(4.21)f~y(); ?()g =h  2bG 1   3h[x()]G 1   2bh[x()]0i (   ) h3h[x0()]G 1 + 2bh[x0()]0i (   );(4.22)f?(); ?()g = 0:(4.23)Because the T-dual momenta ? are bilinear in original coordinates, theirPoisson bracket vanishes. The Poisson bracket between T-dual coordinates andmomenta however gains the additional term linear in coordinates.In doubled space the additional coordinate ~y appears. It consists of the termlinear in original momenta and the other terms bilinear in original coordinates. So,it produces nontrivial Poisson brackets with all variables (y; ~y;?), (4.13), (4.14)and (4.22).5. Concluding remarksWe showed that we need two ingredients in order to have closed string non-commu-tativity: T-duality and nontrivial background. The T-dual transformation lawsCanonical Approach to the Closed String Non-commutativity 109connect the world-sheet derivatives of the coordinates and momenta in the originaland T-dual theory. The T-dual coordinates y has two terms: one linear in originalmomenta and the other bilinear in original coordinates. This produces the nontrivialPoisson bracket fy; yg (4.12) which is linear in coordinate. Consequently, thenontrivial innitesimal expression 0, which exists only in the coordinate dependentbackgrounds, is the source of the closed string non-commutativity. Note that in thecase of open string moving in the at background coordinate is linear function inboth eective momenta and coordinates. So, the corresponding Poisson bracket isconstant.The general structure of the non-commutativity relations isfY(); Y()g = fF [x()  x()] + ~F [~x()  ~x()]g(   ) ;(5.1)where Y = (y; ~y) and F and ~F are constant and innitesimal uxes. At thesame points, for  = , all Poisson brackets are zero. In the important particularcase for  =  + 2 we getfY( + 2); Y()g = 2(F + 2 ~Fb )N +1~Fp;(5.2)where N and p are winding numbers and momenta of the original theory. Wecan rewrite it in the formfY( + 2); Y()g =ICFdx +I~C~Fd~x ;(5.3)where C and ~C are cycles around which the closed string is wrapped. This gen-eralizes the conjecture of Ref.[10] between closed string non-commutativity anduxes.REFERENCES1. D. Lust, JHEP 12 (2010) 084 [arxiv:1010.1361]; arxiv:1205.0100 [hep-th].2. D. Andriot, M. Larfors, D. Lust, P. Patalong, arXiv:1211.6437 [hep-th]; D. Lust,arxiv:1205.0100 [hep-th]; arxiv:1010.1361 [hep-th].3. Lj. Davidovic, B. Nikolic and B. Sazdovic, arxiv: 1307.6158.4. F. Ardalan, H. Arfaei and M. M. Sheikh-Jabbari, JHEP 02 (1999) 016; C. S. Chu andP. M. Ho, Nucl. Phys. B550 (1999) 151; N. Seiberg and E. Witten, JHEP 09 (1999)032; F. Ardalan, H. Arfaei and M. M. Sheikh-Jabbari Nucl. Phys. B576 (2000) 578;C. S. Chu and P. M. Ho, Nucl. Phys. B568 (2000) 447; T.Lee, Phys. Rev. D62 (2000)024022.5. B. Sazdovic, Eur. Phys. J. C44 (2005) 599; B. Nikolic and B. Sazdovic, Phys. Rev.D 74 (2006) 045024; B. Nikolic and B. Sazdovic, Phys. Rev. D 75 (2007) 085011; B.Nikolic and B. Sazdovic, Adv. Theor. Math. Phys. 14 (2010) 1, Lj. Davidovic and B.Sazdovic, Phys. Rev. D 83 (2011) 066014; JHEP 08 (2011)112.110 Lj. DAVIDOVIC, B. NIKOLIC and B. SAZDOVIC6. Lj. Davidovic and B. Sazdovic, arXiv:1205.1991 [hep-th]; F. Ardalan, H. Arfaei and M.M. 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CANONICAL APPROACH TO THE CLOSED STRING NON-COMMUTATIVITY

We consider the propagation of the closed bosonic string in the weakly curved
   background. We show that the closed string non-commutativity is essentially
   connected to the T-duality and nontrivial background. From the T-duality
   transformation laws, connecting the canonical variables of the original and
   T-dual theory, we find the structure of the Poisson brackets in the T-dual
   space corresponding to the fundamental Poisson brackets in the original
   theory. We find that the commutative original theory is equivalent to the
   non-commutative T-dual theory, in which Poisson brackets close on winding and
   momenta numbers and the coefficients are proportional to the background
   fluxes.</jats:p

Ljubica Davidovic

Bojan Nikolic

Branislav Sazdovic

Crossref

Facta universitatis - series Physics Chemistry and Technology

Canonical approach to the closed string non-commutativity

Lj. Davidović

B. Nikolić

B. Sazdović

The European Physical Journal C

We consider the closed string moving in a weakly curved background and its totally T-dualized background. Using T-duality transformation laws, we find the structure of the Poisson brackets in the T-dual space corresponding to the fundamental Poisson brackets in the original theory. From this structure we see that the commutative original theory is equivalent to the non-commutative T-dual theory, whose Poisson brackets are proportional to the background fluxes times winding and momentum numbers. The non-commutative theory of the present article is more nongeometrical than T-folds and in the case of three space-time dimensions corresponds to the nongeometric space-time with $$R$$-flux

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Canonical approach to the closed string noncommutativity

Canonical approach to the closed string noncommutativity

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