Threeparameter Solution, Tachon Condensation and ：三参数求解快子凝聚和

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1、Gaussian Brane and Open String Tachyon CondensationShinpei Kobayashi( RESCEU, The University of Tokyo )2005/02/17-19 Tateyama, ChibaYoshiaki Himemoto and Keitaro Takahashi( The University of Tokyo ) Tsuguhiko Asakawa and So Matsuura ( RIKEN ) MotivationlGravitational systems and string theorylBlack

2、holes = ?lOur universe = ?D-branelOpen string endpoints stick to a D-branelPropertiesX0XXiopen stringDp-brane)10(9, 1:., 1 , 0:DpiXpXiOpen string channel : D-braneiiixXpiXXpX00|:25, 10|:, 1 , 0X0XXiopen stringD-braneclosed stringD-braneopen stringboundary statetree graph of the closed string1-loop g

3、raph of the open stringclosed string channel : boundary stateBxBXpiXBXpXiii00|:25, 10|:, 1 , 0closed stringboundary stateMode expansionnnnnnnzizXzizX,2)(,2)(11iiwzexpexpBoundary state ( = D-brane) ,000cosexp)(,000sinexp)(,22)1(2/10)9(2/10)9(pdSdSxRRghostpbSbSxNSNSRRTNSNSTBRrNrMNMrnNnMNMniprNrMNMrnNn

4、MNMnippppmassRR-charge),(ijMNS direction where the D-brane extendsCoherent state of strings.1)7(21)(, 1)(),(,)()(7)8(1)(43)(81872ppppprpprjiijpppprpNTrfwhererferfedxdxrfdxdxrfdsString Field TheoryD-braneSupergravitylow energy limit 0classical solution( Black p-brane )low energy limitD-brane and Blac

5、k p-branexixMore general D-braneslBPS D-branelsupersymmetric, static BPS black holeTachyon CondensationDpDpDpNN D-branes and anti D-branes attracts together. Unstable multiple branes Open string tachyon denotes the instability. Stable D-branes are left. case)(NN NN -brane systemDDBoundary state for

6、DD-system )(),(,000cosexp)(,000sinexp)(,)(22)(2,;)1(2/10)9(2/10)9(2NNforSpdSdSxRRghostpbSbSxNSNSRRNNTNSNSeNNNTTNNBijMNRrNrMNMrnNnMNMniprNrMNMrnNnMNMnippTppmassRR-chargeconstant tachyonDuring the tachyon condensation1. D-branes, anti D-branes coincide with each other. ( t = 0 )2.During the tachyon co

7、ndensation ( t = t0 )tachyon vev is included in the mass.3.Final state ( t = )The mass will decrease via the closed string emission, and the total mass of the system will become coincident with the RR-charge.)(),(braneDaoftensionTNNTMpp2)(2TpeNNNTMQMNNTMp)(NN1.Tachyon CondensationDpDpGaussianDD99sys

8、temDD99The system extends to all directions.localized at braneDp )9 , 1(ppixi), 1 , 0(px)9 , 1(ppixi), 1 , 0(px0ixbraneDpGaussianGaussian in -directionixKraus-Larsen (01)iGiiprNrmMNMrmNmmMNMmGpGpppuppdbSbSuFuG, 08expexp)4(2;2292/1)(1)(290ijGGrMNijGGmMNururSumumS22)(22)(4141,4141,0, 0, 08exp229iiGiip

9、xpppuppd)0( .422constuwithuGGOrdinary Boundary Stateboundary stateclosed stringpipiipBxBXppiXBXpX00|: )9, 1(0|:), 1 , 0(source of closed stringsMode expansionlBoundary condition via oscillatorsinnnnnnnnnienixwwnixewwXX0000)(1212)()(), 0(innnnenpP00)(121), 0(2lBoundary condition by oscillators)0(0)(0

10、nforBBpnn)0(0)(0nforBBxinini0001exp)(21)9(pSnxTBnNnMNMnipp0)(),(BSSNnMNMnijMNLongitudinal to the boundary stateTransverse to the boundary stateWe introduce to write both conditions MNSXiX 0piBX 0pBP0;uGPp0;)(uGpnn0;)4/(1)4/(10;41412222uGununuGnunupininpinin0;)2(2uGXuiPpiiLongitudinal to the boundary

11、 stateTransverse to the boundary state):(0;):0(0;DirichletuuGXNeumannuuGPpipi01exp;1)(NnnnMNMnoscpSnuGijnMNununSwhere)4/(1)4/(1,22)(ixuiipuizeropxedxpedpuGii0)(200)(810202202;Oscillator part which satisfies the boundary condition is 0-mode part is : Dp-brane)0(),()(),()(uuSijijnMN: D9-braneGaussianT

12、achyon CondensationDpDD) 1() 1(112) 1(2) 1(mmDDmDpHaussian braneAsakawa-SK-Matsuura,in preparation0, 0exp)4(2;2/1)(1)(20irNrmMNMrmNmmMNMmHHpxpbSbSuGuHijHHrMNijHHmMNururSumumS,4141,414122)(22)(How should we describe D-branes ?lNon-perturbative string theorylString Field TheorylMatrix Theorypoint part

13、icleclosed stringopen stringslStringsX),(XXX,BG,TAspacetimeworld-sheetsymmetry of world-sheetspacetime actionaaabXXhabhlFree motion of a one-dimensional objectlFlat background spacetime cf.) action for the free relativistic point particle S=0 eom of point-particle ,dsmSString in flat spacetime = -1

14、= -1 = 0 = 2 = 1 = 0 = 1 = 2 = 0 = world-line of point-particleworld-sheet of stringXXXXAction for free stringlIn the flat spacetimelanalogy to point-particle area of the world-sheet = action Nambu-Goto action S=0 eom ,det2121baXXhhddSbaababNGPolyakov action cf.) Nambu-Goto actionlWeyl invarianceS =

15、 0 lmode expansion of quantization state of string XXdSbaab22102XX0)(;,xeknnstatexiklilimode expansionwherennnzznnnzznnnnnnnnnnnnnnnnnnzLzTzTzLzTzTzczczczczbzbzbzbzizXzizX,)()(,)()(,)(,)(,)(,)(,2)(,2)(22112211iiwzexpexpStates of stringlfirst quantization0 ,) ()(),(nmnmipxiPXquantization ofcenter of

16、massquantization of oscillation mode0)(;,xeknnstatexiklilikgravitonge; 0.)1(1String in Curved SpacetimelString in curved background= non-linear sigma model are couplingslConformal inv. decides the behavior lThis can be reproduced by SUGRA action,BGXXBiGdSbaabab221,12142dBRGString with Boundary Inter

17、actionlIncluding the boundary interaction = Considering the D-branestringXXANon-linear sigma model with boundary interactionAdXiXXGzdS2210A)1()2(1)2det(dAFFGedSpBIeomEOM can be reproduced via the Born-Infeld actionString with tachyonic interactionlUnstable system has the tachyonic interactionAdXiSex

18、p)exp(int,)(exp)exp(intXMPTrSDXXAXTXTDXXAXM)()()()()(AATTDDKraus-Larsen (01)EOM99DDKraus-Larsen (01)kIIIDTFTxdTS1222109)(| 2exp2)()()0()()2(ln21)2(2)(4)(2/12xxOxxxOxxxxxFxIIGIIXuxT2/1)()(2exp()| 2exp(222IIXuTGaussian brane: linear tachyon)9 , 1(ppixi), 1 , 0(px)9 , 1(ppixi), 1 , 0(pxT)(TVT)(TVT)(TVa

19、ctionDBPSnonactionDD899)(99uT xdTuFuxdTuFxuxdTSDuDD99999992991092222)(2exp2989) 2(2DDTTactionDBPSactionDD799),(,9898uuTT xdTuFuuFuxdTuFuFxuxuxdTSDuuDD8922,998889982992881094222)()(2exp2989u792) 2(DDTTThree-parameter solution ( Zhou & Zhu (1999) )lSUGRA actionlansatz : SO(1, p)SO(9-p) ( D=10 ) 222321

20、02|)!2(212121ppFepRgxdS.,10)(112)()(2)(22prppprjiijrBrAdxdxdxedFeedxdxedxdxedssame symmetry as the system DD.16)7)(1(7)8(2,1)(,)()(ln)(,)(sinh()(cosh()(sinh() 1(,)(sinh()(cosh(ln43)(16) 1)(7()(,)(sinh()(cosh(ln161)(64)3)(1()ln(71)(,)(sinh()(cosh(ln167)(64)3)(7()(217022/122)(212121cppppkrrrfrfrfrhrkh

21、crkhrkhcerkhcrkhprhcpprrkhcrkhprhcppffprBrkhcrkhprhcpprAprcharge ?mass ?tachyon vev ?New parametrization During the tachyon condensation, the RR-charge does not change its value. We need a new parametrization. .,4310012ppNQvckpvNM).0(11,2222070vvckvrp.12,223702/1227021pppprkNcQrNkccpMAsymptotic beha

22、vior of the solution .1)(,116)7)(1(143)(,1431811,1431871)7(270)7(27012)7(27012)(2)7(27012)(2pppppprBpprArrrCrrvkcppvprrrvkcpvperrvkcpvpelasymptotic behavior of the black p-brane = difference from the flat background = graviton, dilaton, RR-potential in SUGRAlmassless modes of the closed strings from

23、 the boundary state ( D-brane in closed string channel ) = graviton, dilaton, RR-potential in string theory ( string field theory )coincidentRelation between the D-brane ( the boundary state) and the black p-brane solution (Di. Vecchia et al. (1997)hg1sourceGravitational Fieldgraviton)()()(2rCrd)()(

24、)(2rkCrdisource,)(232222pprfee27878)7(223)7(223)(pppppprpTprpTprsourcepropagatorfieldmasslessWe can reproduce the leading term of a black p-brane solution ( asymptotic behavior ) via the boundary state.leading term at infinitye.g. ) asymptotic behavior of of black p-branecoincident2111)(1223; 0ippNM

25、MNkVTpBDk NSirNrmMNMrmNmmMNMmNSpxpbSbSCB0, 0exp22/1)(1)(0ijMNBAS)2/1(with1CBAordinary boundary statepppijpMNrBpApCNrrppCNBArh7)1(7)1(1)7() 1(41)(1) 1(,)7()(81)(ppijMNrvkcppvprrppvkcpvrh7120)1(7120)1(116)7)(1(1434)(1) 1(,)7(4318)(BpApCvBACvkc47411)(020100,)(QCBAMCompared with each other, we find and

26、the ADM mass and the RR-charge are 0, 0exp2122/1)(1)(|02irNrmMNMrmNmmMNMmTpxpbSbSeNNNBijMNS2|21, 1, 1TeNNNCBA2|212110TeNNNvcDpDpDpc1 does not correspond to the vev of tachyon !(as opposed to the result of hep-th/0005242)irNrmMNMrmNmmMNMmGpppbSbSxFuG, 0exp)(2;2/1)(1)(0ijrMNijmMNxrxrSxmxmS11,11,)()(.)

27、(4(2constuxGpxFCxxBA9)(,1212, 1DpDpGaussianDD99) 12(2) 12(4)()38()()7(4) 12)(1)(4()(27)8(22222221xxxFpxvxxFpxxFxFppcirNrmMNMrmNmmMNMmHpppbSbSyGuH, 0exp)(2;2/1)(1)(0ijrMNijmMNyryrSymymS,11,11)()(.)(42constuyH)(, 1,1212xGCByyADpDD) 1() 1() 12(2) 12(4)()38()() 1(4) 12)(1)(4()(27)8(22222221yyyGpyvyyGpyyGyGppcSummarylD-brane plays an important role in string theorylBlack hole, Universe, non-perturbative, lSymmetry of world-sheet spacetime actionFuture WorkslTime-dependent solutions