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Number of closed strings emitted from a decaying D-brane Conboy, Steven Lyle 2007

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N u m b e r f r o m  o f  C l o s e d  a  S t r i n g s  D e c a y i n g  E m i t t e d  D - b r a n e  by Steven Lyle Conboy  B . S c , Queen's University, 2005 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F THE REQUIREMENTS FOR T H E DEGREE OF Master of Science  The Faculty of Graduate Studies (Physics)  The University Of British Columbia August, 2007 © Steven Lyle Conboy 2007  A b s t r a c t  We have calculated in the covariant gauge the total number of massless closed strings emitted from a decaying Dp-brane using fermionization technology on the time component of the boundary state. This verifies the computation carried out in the less well known temporal gauge despite differences in renormalization schemes. In addition we have attempted to use the fermionic technique to calculate the total number of closed strings emitted. Our result is difficult to interpret due to the ambiguity involved with rotating it back to Minkowski space.  ii  Table  o f  Contents  Abstract  ii  Table of Contents  iii  L i s t o f Tables  "  v  List of Figures  vi  Acknowledgements  vii  Dedication  viii  1  Introduction 1.1 .1.2 1.3 1.4  1  The Basics Tachyon Action Boundary State Formalism 1.3.1 Renormalization scheme comparison The boundary as a source of closed strings  2  F e r m i o n i z a t i o n o f X° 2.1 Compactification at the self-dual radius 2.2 The boundary state in fermion variables  3  C a l c u l a t i o n o f h{x°) 3.1 NS sector 3.1.1 3.1.2  3.2 3.3 4  1 2 3 6 7 10 13 13 16 17  '  (n n \BD) Calculation of (7i_ -KR|ai&i | B D ) N S L  R  17 19  NS  Ramond Sector Summary and Final Calculation  N u m b e r of Massless Strings E m i t t e d 4.1 1-point functions 4.2 Number of Massless Strings Emitted by Half Brane  19 20  i . .  23 23 26 iii  Table of Contents 5  Total N u m b e r of Particles E m i t t e d  28  6  Summary and Conclusion  32  Bibliography  33  Appendices A  C a l c u l a t i o n o f (-KL v r \ a i & i \ B D ) A . l (ir TT \a a \BD) A.l.l 7r >0,7r >0 A.1.2 7 r _ > 0 , 7 r < 0  34  R  L  R  1  1  34 35 36  NS  L  f l  fi  A.1.3  7r <0,7r >0 L  37  R  A . 1.4 7 r _ < 0 , 7 r < 0 A.1.5 7i-_=0,7r = 0 {ir n \a a \BD) A.2.1 7T_ > 0, TTR > 0 A.2.2 7T_ > 0,7r < 0 A.2.3 TT < O , ^ > 0 ft  R  A.2  L  R  1  1  :  R  fl  l  A.2.4  7r_<0,7r <0  38 39 39 40 41 42 43  f i  B  B R S T invariance of the M = 0 state  44  C  P o l a r i z a t i o n sums  46  2  iv  List  4.1  of  Tables  Number of particles emitted with M  2  = 0 for half S-brane.  List  1.1  of  Figures  Gluing process used to convert an open string world-sheet into a closed string world-sheet  4  vi  Acknowledgements First and foremost like to thank my supervisor Dr. Semenoff. This thesis would not exist without his help and guidance. I would also like to thank Dr. Karczmarek for taking the time to read and edit my thesis. M y thanks are also extended to my colleagues Lionel Brits, James Charbonneau and Benjamin Gutierrez for useful discussions as well as help with M _ X . I "acknowledge N S E R C for supporting my work the past two years.  vii  Dedication I would like to dedicate my thesis to my fiancee Carmen Romero as well as to thank her for her love and support over the past two years.  viii  Chapter  1  I n t r o d u c t i o n Perturbed slightly from the maximum of its potential an open string tachyon will undertake a rolling process towards the minimum of the potential, the closed string vacuum [5]. It has been shown [2, 7] that as the brane decays it radiates closed strings. The main goal of this thesis was to find the total number of strings emitted by the Dp-brane. This calculation has been previously carried out level by level in [2] in the temporal gauge. We have been motivated to attempt this same calculation in the familiar covariant gauge because there is some doubt that the use of the temporal gauge in [2] is properly justified. We find that our result for the total number of massless strings emitted agrees with [2]. This is despite the fact that the coupling constants g and g, defined below, are given by a different renormalization procedure in the fermionic approach than in the bosonic approach. For the special case of the half S-brane we find the result to be coupling independent, allowing us to find explicit values for the number of particles emitted. Our attempts at calculating the total number of closed strings emitted by an unstable Dp-brane in the covariant gauge have been unsuccessful. This is due to the fact that fermionization of the time component of the boundary state requires an analytic continuation to periodic Euclidean time. Thus any physical result requires the reinstatement of non-compact Euclidean time followed by a Wick rotation back to Minkowski time. As we will explain in chapter 5 there appears no clear way of rotating back to Minkowski space, preventing us from obtaining a meaningful result.  1.1  The Basics  The conformal field theory discussed in this paper has applications in many fields of physics [1], but we will be discussing it in the context of bosonic string theory. We will be dealing with three objects in bosonic string theory; the closed string, the open string and the D-brane. Oscillations of the strings are interpreted as bosons, for example the first excited state of the open and > closed strings are interpreted as spin 1 and spin 2 objects, respectively. 1  Chapter 1.  Introduction  A string is embedded in space-time via the parameterizations X ((r, T) where T is analogous to proper time in the case of the point particle. The string is an extended object which is why we require a second parameter a. The standard convention is to set the length of string to 2ir for the closed string and TT for the open string. The closed string obeys the condition X^(a + 27T,T) = X {a,T). The endpoints of the open string are forced to lie on an object known as a D-brane [4]. The D-brane is sometimes called a Dp-brane in the literature to signify that it extends in p spatial dimensions. As a string moves through space-time it traces out an object known as the world-sheet. Physics is parameterization invariant so we are free to use any parameterization of the world-sheet we wish. Two very useful parameterizations of the world-sheet are: the cylinder tl  )i  w =a+  IT iii = a — ir  (1-1)  and the disk z = ±e  iw  z = ±e~  (1.2)  iw  where + is used in the case of the closed string and — is used in the case of the open string. These parameterizations are used frequently throughout this thesis.  1.2  Tachyon Action  The tachyon action is described by the usual non-decaying world-sheet action plus an exactly marginal boundary operator, i poo pn — dr dadX°-dX° Air 4?r J-OO JO  poo  :  drS +...  (1.3)  [_  (1.4)  bdy  J-OO  given by Su  v  = ( f c « ° + f e " ^ ° + Ad X°) T  q  .  It is also possible to include a boundary interaction at the a = TT boundary with different couplings. When we do include an interaction at this boundary we take the couplings to be the same as on the <r = 0 boundary. We note that we have performed a wick rotation in (1.3), so that X° is time in Euclidean space, and the three dots indicate the usual action for the spatial directions and ghosts. Also in (1.3) we have set a' = 1 and we will keep this convention throughout the rest of the thesis. We are assuming that the  2  Chapter 1.  Introduction  tachyon condensate is space independent, resulting in the usual equation of motion (d -d )X»(a,T)=0 2  '  2  T  a  (1.5)  as well as a boundary condition given by  (  •  ^  f  -  '  h  L  -  '  < l 6 )  at the end of the string. Equation (1.3) defines a conformal field theory for all complex values of g and g, however, since the Minkowski space tachyon field is real g and g must also be real. We are interested in two tachyon profiles, the half Sbrane for which g = 0 and as the full S-brane first discussed by Sen in [6] for which g = g. The former corresponds to a tachyon decaying to a stable configuration and the latter to a tachyon coming up from the bottom of its potential and then falling back into it. The topological term associated with A in (1.3) does not affect the equations of motion or the boundary condition. For a non-compact boson it can be interpreted as a Bloch wave-number. Further discussion on this can be found in [1]. In this discussion the authors conclude that any value of A £ (0,1) allows us to treat X° as if it had the compactification X° —> X° + 2n. We will show that we need to treat X° as if it were compactified in order to use the fermionization technology developed in chapter 2. Since X° is not actually compactified we will wish at some point in our calculation to decompactify X° by integrating out the A dependence over all allowed values of A G (0,1). It is important to note that when we include an interaction at the a = IT end of the strip and wish to decompactify we must integrate over A = A — A o, is the difference between the parameters at each boundary. a=7r  1.3  a=  Boundary State Formalism  The boundary state technique is useful for calculating open string partition' functions Z\B] = Tre-P", (1.7) where H is the Hamiltonian obtained from (1.3). We can rewrite this partition function as a closed string correlation function by gluing together the ends of the open string world-sheet. This gluing procedure is illustrated in F i g 1.1. The incoming part of the world-sheet at r = 0 is connected to the outgoing part of the world-sheet at T = ir creating a cylinder and making 3  Chapter 1.  Introduction  T periodic. As a result the definitions of T and a can be interchanged expressing the correlator as a time evolution operator sandwiched between the boundary states, which encode information about the boundary conditions Z[(3] = (B \e—"\B ). 1  (1.8)  2  The action (1.3) is now that of a free boson living on a Euclidean cylinder •I  />Z7T  roo  5' = — / 41"  dr  da ((d X ) 0  + (drX ) )  2  0  T  2  + ...  '  (1.9)  JO  The operator H is the Hamiltonian of this action. The Euclidean time parameter is a = because we have used conformal invariance to rescale the closed string world-sheet to match the convention 0 < a < 2TT.  Out  A  In  Out  \1  In Figure 1.1: Gluing process used to convert an open string world-sheet into a closed string world-sheet  We can use the boundary state formalism to represent the interaction of closed strings with a decaying D-brane. The boundary state is a closed string state which is annihilated by the boundary condition that would be imposed on an open string embedding function. As described above this means interchanging the coordinates a <-> r and re-scaling the closed string world-sheet to match the usual convention 0 < a < 2ir. After this process the boundary condition (1.6) becomes the boundary state condition:  _L_ d X° + i ^ ° x  "2TT  T  -i°e-* °  \B) o  x  x  = 0.  (1.10)  r=0  4  Chapter 1.  Introduction  The spatial component \B)g and the ghost component \B) of the boundary state are given in [8]. The full boundary state of the tachyon is given by: bc  \B) = M \B) o <g> \B)x <g> \B) p  X  (1.11)  bc  where the normalization constant A/- = 7Tir(27r) 6  (1.12)  p  p  and the states \ )x  =J ( _ ^  B  \B)  bc  e  x  P  ( E E " ( - D ^ ^ - n J  + b-nC-S^j  = exp ^ - ^2(b- cn  n  1*11 =0.*x>  (1-13)  (c + co)ciCi |0)  (1-14)  0  are the same as for the non-decaying Dp-brane. A few comments on notation: if the direction of X is along the Dp-brane, it has a Neumann boundary condition with d = 1 and momentum denoted by fcy. Similarly a direction transverse to the Dp-brane has a Dirichlet boundary condition with d = 0 and momentum denoted by A_. The |fc|| = 0, A_) is the eigenstate of these momenta. The state |0) given in (1.14) is the SL(2,C) invariant ground state. In addition we remind the reader of the boson's commutation relations s  s  s  and the ghost's anticommutation relations  The boundary state in the time direction is \B) o x  =  i  I ^ I c  = ij^° °°  dX  elk  X  (f(k°)  ( / ( X  °  }  +  +  MfcW -! + -)  / l ( a ; 0 ) a  0  - " 1  0  1  + -)  1 0 , 0 ; k°)  ' °  |0  ;  '  fc0)  (  L  1  7  )  where x° is the position zero mode of X° and ... indicates higher order oscillations. The contour C is the real axis. Other authors, [2] for example, define the above boundary state without an integration over x ° and instead add a prescription to later integrate over an arbitrary contour. This gives rise to choices such as Hartle-Hawking contour discussed in [2]. 5  Chapter1.  Introduction  In [1] \B) o is expressed i n a basis of a; . We can also do this by integrating over k° in the second line of (1.17) to get 0  x  \B) o x  = ijdx°  (f(x°)\x°)  + h ^ a ^ a ^ x  0  )  +  ...)  (1.18)  where the ... indicates higher order oscillator numbers and \x°) is the position eigenstate. Using the method of fermionization, the function f(x°) was found in [1] to be: /(*")=  1.3.1  1  i*o + i  1 + nge  1  X  Renormalization  - -  1 + irge  M  tx  Ml  o -1  (1-19)  x  scheme comparison  Dependence on g and g of f{x°) and h(x°) differ from those found i n most other papers because of differences in renormalization prescriptions. The usual expression for f(x°) is given in [2] =  i^ •)  * / ^ /\ - o - 1  ,s o + I .  1 + sm(iTg')e  1 + sm(7rg')e  tx ix  C T  (1-20)  for the case of the full-brane(^ — g, g' = g') and  /  (  *  0  )  " T T i k w  (  L  2  1  )  for the half-brane (g = 0, g' = 0). In [2] g' and g' are the renormalized versions of the couplings in (1.3). In [1] the relation between g' and g' was found to be given by: s i n  2  7 r \ / ^  =  7 r  2  ^  ,  i- = 9 9  -  9  (1.22)  with the special case that g' = g when g = 0. Comparing (1.19) and (1.21) we find the fermionic approach agrees with the bosonic approach at least for the full S-brane and half S-brane. In this thesis we extend the work done i n [1] by calculating h(x°) h(x°)  (1.23)  = f(x°)-2(l-7r gg). 2  We can compare this to the values obtained i n [2]: h(x°)  = f{x°) - (1 + cos(27r ')) ff  (1-24) 6  Chapter 1.  Introduction  for the full S-brane and h(x°) = f(x°) - 2  (1.25)  for the half S-brane. Again these results agree at least for the cases of the full S-brane and half S-brane. 1  1.4  The boundary as a source of closed strings  In Q F T the average number N of particles emitted from a classical source j(x) is given by  2 J  (1.26)  (2n)D-'2E  where j(k) is the fourier transform of j(x) and E = k + m . Since we are interested in calculating the total number of massless strings emitted we should take our source to be the one point function for creation of massless strings from the D-brane 2  j(k) = W \c£\B).  2  2  (1.27)  B  Here CQ = ^ ^ ^B) is the B R S T invariant state at the massless level which we have derived in the appendix. As a result the total number of massless particles emitted should be given by 0  9  A N <  ^=0  = /  J^2E\^BK\B)\\  (1.28)  As was discussed in [2] the analogous form of equation (1.26) for the total number of closed strings emitted from the D-brane can be obtained from the optical theorem. On the Lorentzian cylinder it takes the form N = Im((B\  . \B))  (1.29)  where b£ = ^ °, CQ = °lf, Lo and Lo are the left and right moving V i rasoro zero modes respectively. The infinitesimal e is used for regularization purposes. We can prove the validity of our expression for the total number of massless particles emitted (1.28) by deriving it from (1.29). We begin by inserting a complete set of closed string states b0  h  c  l* )(*d = 1 c  (1-30)  ' W e have used the fact t h a t cP_\oP_\ — i > — a°_ia°_i under the W i c k r o t a t i o n X° >—* iX°.  7  Chapter 1.  (1.29). This conveniently splits (1.29) into a sum  after the c in equation over mass levels 0  N  = N2_ M  =  =  Introduction  +N2  4  M  + ...  =0  Im ( ( B | 6 + c - | * ) ( * | 0  M 2 =  _ (L + 4  U)- \B)) l  0  +/m((B|b+Co|*)(vI/| _ (L + Lo)- |B)) + . . . . M2  0  (1.31)  1  0  If we are interested in calculating the number of particles with M = n we need to know the explicit sum over states given by \*ff) (^\ 2 . In the covariant gauge, the most general from \fy) (ty\ i can take while ensuring a ghost number of 0 is 2  M  =n  M =0  f  d k 26  |*)(*| _ = y ^ 2 6 e ^ | 0 , 0 ; f c ) x ( 0 , 0 ; f e | . Ma  0  (1.32)  f f  The ghost contribution x is given by g  UXTT  D•  % = -i (i TTXii i -1 j +1 m m i -1 m e n (1-33) The states | J.J.),| |J.),| J.f) and | J.J.) are the four degenerate ground states of the ghost system. The state | J.J.) is defined to be annihilated by the ghosts bo and bo. The ghosts co and co can be used to construct the other ground states | fl) = co| J.J.) , | i t ) = co| | | ) and | t t ) = CQCQ\ IDThe properties of Grassman variables and hermiticity demand that the nonvanishing inner product is defined by (J.J. \C~OCQ\ J.J.) = i. Also worth noting is that the SL(2, C) ground state given in equation (1.14) is related to | [[) by the conformal transformation' from the disk to the cylinder, so that as  |0> - > b _ i 6 _ i | U>-  (1-34)  After inserting (1.32) into the expression for N 2 nipulation we obtain M  Nio M =  =0  and after some ma-  ^(/(08^K*B|cb|fl>| )  =  2  =  -/7||a^  =  /  0  2  -^)l(^|c -|B)|  £ w ^ ^ o \ B ) f  0  2  '  (1.35) 8  Chapter 1.  Introduction  where \^B) is the on-shell B R S T invariant state for a string with M — 0 level, the form of which is derived in the appendix. This calculation has put the particles on-shell so that E = k . We notice that the last line of (1.35) now justifies our conjecture (1.27) because it agrees with (1.28). A t first glance the calculation of the total number of closed strings emitted seems fairly straightforward. It was given in (1.29) as 2  2  N = Im((B\ \  2  °- ° . \B)). L + L - ie / 6  0  C  (1.36)  0  We remind the reader that - j — ! — = in5(x) + y  (1.37)  where y is the principal value. Since y is real, this allows us to express (1.29) as N = Im (in (B\6(L + L - ie) \B)j . (1.38) 0  0  We can now rewrite the total number of particles emitted as a product of partition functions by expanding the delta function in (1.38), giving N = Im(iixj  d\{B\{b+c^)q ~ °- '\B) j Lo+  L  ii  S  (1.39)  where q = e . The spatial and ghost partition functions are well known [4]. A t first glance fermionization seems promising for calculating the time partition function because it allows us to find an explicit expression for \B) ° and Lo + Lo after rotating to Euclidean space. There is, however, one drawback to this approach, finding a consistent way to rotate our final result back to Minkowski space. We will discuss this difficulty after we thoroughly develop the fermionic technology. We note that this problem is absent in the temporal gauge because no analytic continuation is required. lX  X  9  Chapter 2 Fermionization  o f X °  Here we will briefly summarize the process of fermionization and extend the work done in [1] and [3]. The goal is to express the boundary state in terms of fermion variables. For simplicity we label X° = X and ignore the spatial and ghost parts of the action. It was shown i n [3] that fermionizing X° directly leads to the identification X —> X + V2TT, which is inconsistent with the self-dual compactification scheme discussed in the introduction. A way to sidestep this is to first add an extra degree of freedom obeying a Dirichlet boundary condition. Introducing this new boson Y , the Euclidean action becomes: -i S=—  roo  rift dr da ({d X) J —oo J0  + (d X)  2  + (8 Y)  2  T  + (d Y) )  2  A  + ... (2.1)  2  T  a  The boundary state condition now becomes the set of equations (_J_ 8 V 2TT  + -e 2 9  tX  -  iX  l  ile~ ) / iX  2  \BD)  Y\ \BD} T  T=O  = 0  (2.2)  = 0  (2.3)  for Y which is thus uncoupled and hence exactly solvable. When calculating quantities such as physical amplitudes we can simply factor out the contribution of the Y boson. We construct the fields ^  = - ^ ( X + r)  ,  <h = ^=(X-Y)  (2.4)  which are used to define the mapping to the fermions Mz)  = CIL : e - V ^ M : , i>\ (z) =: e^ ^  : (\  ^2L(z)  =  : C\  1  L  4>IR{Z) •=  <2_:eV^ Gfl : e ^  1  R  (  2  z  i  W  -  )  : :  ,  ^(1)  =  ,  4R&  =  1>2R(z) = C2R : e - ^ * ( * " > : , 4 (z) R  :  e  " ^ W  (2.5)  L  (2.6)  L  : C\ '  :  R  =: ^ * ( * ) : e  (2.7)  (2.8)  10  Chapter 2. Fermionization of X  1  z and z are the complex co-ordinates corresponding to the conformal transformations z = e , z = e ~ and the CaL/R are co-cycles used to make the fermions anti-commute with each other. The co-cycles are constructed from the zero modes of the bosons T+I<7  T  w  V2<l>a = PaL + faR - ™  aL  In z - in  In z + ...  aR  (2.9)  and are given in reference [3] as ClL = ClR = exp (-1^ (7T1L + 7T1R + 27T2L + 27r fl)) 2  C.2L = C.2R = exp (-i^ (n  2L  + ^R)^ .  (2.10)  We now see the advantage of introducing the fields <f>i and ^ and of making the mappings (2.5)-(2.8): it compactifies X and Y at the self-dual .radius making them compatible with the compactification method discussed in the introduction. The fermion bilinear operators obtained from operator product expansions of (2.5)-(2.8) are given by  1>\ (*)Mz) L  := iV2d 4> (z) T  (2.11)  1L  V4»V<2L(2) := -iV2d 4>2L(z)  (2.12)  T  4>\ (Z)1>IR{Z) := -iy/2d <f> (z) r  r  (2.13)  1R  4R(^2R(Z) : = iV28 cj> {z). T  (2.14)  2R  The fermion action on the cylinder is poo S=—  J  p2n drj^  .  .  da(^l (d L  + id )ij  T  a  aL  + ^ (d -id )iP J. aR  T  a  (2.15)  aR  When quantized the fermions have mode expansions: i>La{z) = ^i>a,nZ~ n  n  n  ,  ^Ra(z) = X I 4'a,nZ~ . n n  n  where n runs over Z represents wayefunctions which are periodic on cylinder and Z + 1/2 represents wavefunctions which are antiperiodic. These correspond to Ramond(R) and Neveu-Schwarz (NS) sectors of the theory 11  Chapter 2. Fermionization of X  1  respectively. relations:  Quantization also gives the non-vanishing anticommutation  {^a.m.V'b-n} = <U<W , {V'a.m, i>\ _ „ } = S 6 ab  mn  (2.17)  Integrating equations (2.11)-(2.14) gives: r-2-n  TTIL = j  i  — : 1>\ (zW (z) L  lL  :  — : 4L(^L(Z) : ^H  =-J^:4R^)MZ):  *2R = ^  (2.18)  (2.19) (2.20)  ^ :V L W ^ M =  (2-21)  (^,-„tfa,» " V>a,-n<„)  (2-22)  (i>l-Ja,n  (2-23)  so that in the NS sector we have oo  VaL = ( - l ) " "  £  1  OO VTafl = ( - l ) £ a  ~  and in the R sector we have TTaL = ( - l ) ^  1  f j (<-n^a,n -  *aR = ( " I ) " £  (tl,-Ja,n  ^a,-nV4,„ +  ~ $ ,-J>l, a  n  ^,0^,0 - ^  + ^,0^,0 - ^  (2.24)  (2.25)  Equations (2.22)-(2.25) express the momentum of fa and fa in terms fermion number operators and imply that the momentum is discretized. Consequently X and Y are compactifled as discussed i n the introduction. As discussed in [3] the mapping to fermions described above is not oneto-one which requires us to investigate the compactification process. The goal of the next section will be to find the projection which makes the map one-to-one.  12  Chapter 2. Fermionization of X'  2.1  Compactification at the self-dual radius  As discussed above, when X and Y are compactified at the self dual radius they have momenta and winding numbers that take on discrete values Px  + Px  L  = V2m  PY  R  L  , px  x  + PY = V2m R  - PX  L  R  = V2w  (2.26)  x  , PY - PY = V2WY  Y  L  R  (2.27)  where m x . m y , W X , W Y € Z . Writing write these conditions in terms in terms of the momenta of 4>\ and <fo gives the following system of equations 7T1L + 7T2L + TT\R + 7T R = 2m 2  (2.28)  X  - 7T2L + TTlR ~ T^2R = 2mx  1~1L  7T1L + 7T2L - 'u  1R  - n  2 R  (2.29)  = 2m  (2.30)  x  TTlL - 7T2L - TTi + 7T /J = 2mx • R  (2-31)  2  In order for these equations to be satisfied the momenta 7ri_, ~K\R, t^ihi ~~2R must either all be integers or all be half-odd integers. A s a result using (2.22)-(2.25) the fermions ip\ and ip2 must either both be Neveu-Schwarz(NS) or Ramond(R). The final step to obtain all of the bosonic states is to project out states that do not satisfy one of (2.28)-(2.31). For example we can take the projection operator to be p - I[l  +  (_iyiL-*2L-KlR+*2R^  (2.32)  In the fermionic language this corresponds to all states in which the total fermion number is even.  2.2  The boundary state in fermion variables  We will write the double boundary state as \BD) to signify that it contains information about the boundary conditions of both the X and Y bosons. To find the fermionized version of \BD) we must first rewrite the boundary condition(2.2) in terms of the fermion variables. We find that the first term in (2.2) is given by d X{0,cr) T  where ip'  LR  1 r : Vl(0,cr)^_(0,a) : 2% and i/'L.fi 4 L.R  a  r  V4(0,ayV (0,<r) : f l  (2.33)  e  A;L,R  2;L,R  i>L,R  (2.34) 13  Chapter 2. Fermionization of X° We use the Dirichlet boundary condition on Y to find that iX{*P)\BD)  = _  e  ^ ^ ^ \ B D ) \/2i^iL(0,<7) v^i^iB(0,<7)|^g^ x  +  Y  e  e  _  e  2,  • V2i<t>i (0,a) .. \/2ict>iR(0,o-) .  2  e  L  =  \BD)  e  z ip\ (o,a)c; .<;l Mo,<r)\BD} 2  L  =  1L  ±Zm(0,a)(l  R  + o )ip (0,a)\BD)  (2.35)  3  R  where the infinite constant Z comes from normal ordering v 2» i iL(<V) j V2i<s>i (p,a) _ Absorbing Z into the coupling constant g is the only renormalization that is required. Note that the power of the cutoff makes g a marginal coupling. Through an analogous argument the last term in (2.2) is given by 2  /  < >  e  a n c  2  e  R  e  -iX(0»|  B £ ) )  -i(X(0,a)-Y(0,*))\  =  e  =  BD)  ±Z ^(0,a)(l-a )^(0,a)|BD) 2  3  (2.36)  so that the boundary condition written in fermion variables is given by [: i>l<r il> : - : ip ^ijj 3  :  ]  L  R  R  +7r<?4 (1 + a ) ip - itgtf ( l - a ) j> ]\BD) = 0. 3  3  R  L  R  (2.37)  It was shown in [1] that the state \BD) which satisfies (2.37) and the level matching condition ^L — L_„^ \BD) = 0 is given by n  = 2 - 4 T J exp ^_ U- ia i>- -^_ icj Unj)_ l  \BD)  l  l  r  NS  l  r  r  r  |0>  (2.38)  • + -+>  (2.39)  in the NS-NS sector and \BD)  R  2 4 Y[ exp n=l x exp  in the R - R sector. Since the Ramond sector contains eight zero modes, the ground state is 16-fold degenerate. Here we define | ) as 2  E q u a t i o n s (2.38) and (2.39) differ from their counterparts i n [1] b y a factor of 2 / because our definition of the normalization constant (1.12) already contains the factor of 2 / f the Xo boson state. T h e factor of 2 / t h a t remains i n these equations comes from the Y b o u n d a r y state. 2  - 1  - 1  4  - 1  4  4  l o m  14  Chapter 2. Fermionization of X° the state which is annihilated by all positively moded operators as well as V'I.OI V'I.OJ "02,0 and ip2,o- The other 16 ground states are created by acting with v a r i o u s V ^ o d 4>lo I )• The phase convention we a  n  o  n  adopt gives creation operators ordered as ip\ , i/>j , , %j>\ o P sig For example the ground state given in (2.39) is given by | — | h) = a  0  0  m  s  11  0  ^1,0^2,0 I )• The matrix U has the property UU* — 1 for real values and is given by U  V - i~ gg —ing 1  2  e  \/l  27TlA  -iirg — ir gg 2  (2.40)  One may be surprised to discover that the parameter A denned in (1.4) reappears in this matrix despite the fact that is absent from the boundary state condition (1.10). The reason is that certain partition functions constructed from the boundary state depend on it. A thorough discussion of this can be found in [3].  15  Chapter 3 C a l c u l a t i o n  o f  / i ( x  u  We see from equation (1.18) that f(x°,x )  = {x ,x \B)  0  0  (3.1)  0  x0iJt0  and h{x°, x°) = {x°, x° \ ai\ ai  B) o  (3.2)  xox  where we have denned 1 x° = -j= (x  L  + x) R  1 , x° = -j= {x  - x ).  L  (3.3)  R  The new variable x° has been introduced because we have treated X° as if it were compactified. It is the variable whose conjugate momentum is the winding number. As a result we expect that once we decompactify \B), f and h should not depend on x°. The position eigenstate itself can be expressed in terms of the momentum eigenstate via a fourier transform  o^o, _  [x , x  ^ ^ *^*^XL^X )& e XL  (3.4)  X  R  | = PXL,PXR 2_,  so that the quantities we need are C = (PXL,PXR\B) and D = (pxL>PXR\ i^i\B). A t this point we will relate the inner product (J>XL,PXR\B) which we cannot calculate directly to the double boson boundary inner product (PXL,PXR,PYL,PYR\BD) which we can calculate using the method of fermionization. Since the Y boson was introduced by hand we can set PYL = PYR = 0. As discussed in [1] the inner product of the zero momentum states of the Y boson with its Dirichlet state gives a factor of 2 ~ 4 so that a  (PXL,PXR,PYL  = 0,p R Y  Remembering that p L Y  = ^  = 0\BD)  = 2? (p L,PXR\B).  (3.5)  X  (T~IL ~ ~~2L) and p R  allows us to unclutter our notation by defining TT_ =  Y  = ^  7ri_ =  (T~IR - ~~2R), =  -J^PXL  16  Chapter 3. and ir = -K\  =  (PXL,PXR,PYL  — 0,PYR  R  R  Similarly, the Y boson contribution to  = -^PXR-  7_fl  {PXL,PXR,PYL  Calculation of h(x°)  = 0\ai&i\BD)  is  2~5  so that we can relate  = 0 | a i d i | B D ) = 2*(pxL,PXR\a\ai\B)  = 0,PYR  (3.6)  where it is important to remember that the oscillators a\, oc\ are associated with the X° boson.  3.1  NS sector  In section 2.1 we concluded that in the NS sector (TT_, TT ) are both integers. The work done in [3] shows that the momentum eigenstate |7r_, ir ) corresponds to a fermion state that is filled up to a Fermi level. R  R  l >  7T_  0,7TR  >0  (TT^TTfll  __ WllM __ (^Ir)  (0|  =  l  1  r=7;  2 1 WL-2  7r >0,7r„<0  (TT^TTfll  L  _I  ~2 1 -^-2 n ^  ^ )  ^  JJ (V>U ) _ _ f > 2 r $ )  (3.9)  1  r  n  L  < 0,n  R  >0  (7r ,7T L  f i  |  =2  = (0|  3 7  r  (0| T J ( ^ V l r )  =  (')  1  1  2  r =  i  l  2r  r  l  l  1 -7T -2  1  • -HR-2 = (0| JJ ( V ^ r ) ( ^ D 1 _I 2 2 L  7T_<0,7T <0  <7r_,, 7 T « |  R  (" ) 3  10  r _  We give these states explicitly in the dual form for the convenience of our later calculations. 3.1.1  {IT IT \BD) S L  R  N  The boundary state in the NS sector was given in section 2.2 l = 2 " 4 YI exp ^-rU~ io i>-r l 0 0  \BD)  l  NS  l  - 4>l i<? Uil>-r\ l  r  |0)  (3.11)  =2  R  17  Chapter 3. Calculation of h(x°) and is repeated here for the convenience of the reader. Since the calculation of f(x°) was previously performed in [1] we will show only the technique used there by calculating (TTL,TTR\B, D)NS for 7TL > 0,TR > 0  " L  R  2  = TJ -(0\ (4<r fr)  (ir ,n \B,D)  (ft <r *j>r)  12  NS  12  r  \BD)  NS  TL-2  = 2" / 1  (Tr(a (if/-V )a  4  12  1  12t  (-icr (7)))' 1  (3.12)  = 2- / (-i(7 ) ^<5( r -^). 1  4  2  1 2  7  L  In this equation the matrix cr 12 is given by a  -a  1  >  2  (3.13)  C12  where a , i = 1,2,3 are the Pauli spin matrices. Using equation (3.5) we find z  1  (_i[/ )V2p 12  24(JPXL,PXR\B)  =  /  4  = (-iL7  C  1 2  XL  —V25(PXL-PXR)  )^^(p  X L  -p  X R  ).  (3.14)  In [1] this same technique was used to find  C=l  [U ]-^ -5{pxL  + PXR)  XI  N  [-iU i]~' S{pxL-PXR)  PXL<0,PXR>0  PXL < 0,PXR  /2pxL  2  0  \PXL\  (3.15)  < 0  \PXR\  for the other momentum ranges. The only case left to consider is PXL = PXR = 0, one can easily check that this gives (PXL  = 0  PXR  = 0\B)  = 1  (3.16)  18  Chapter 3.  3.1.2  Calculation of h(x°)  C a l c u l a t i o n of  7r \aiai\BD)^s R  The next goal is to calculate D — (pxL PXR\aia\\B). We must first find a* and a* in fermion variables. Notice that we have introduced the label X to emphasize that these are oscillators of the X boson. Matching up like powers of z and z on both sides of the equations in (2.11)-(2.14) gives the oscillator modes of fa and fa in terms of the fermion modes /  *a,n = ( - 1 ) °  1  £  (3.17)  Vl.mV'a.n-  m=Z+± (3.18) m=Z+i where, as a reminder, a = l gives oscillators associated with fa and a=2 gives oscillators associated with fa. Matching up like powers of z and z in (2.4) we get 1  1  71  (di i + d , i ) . 2  (3.19)  Using this with (3.17) and (3.18) we can write  J2  (3.20)  ^y^m=Z+i  The rest of the calculation is somewhat involved and given in the appendix, the results are  [-iU ] * 6(p L y/  - PXR)  PXL  12  X  -[U22r^ S(pXL+PXR) PXL  D  -[U ]- ~ 8(J>XL V2  >0 <0  X  + PXR)  PXL  U  [-iU ]-^ 5(j>xL l-2f7 (7  - PXR)  PXL  21  n  0  3.2  PXL > 0,PXR PXL > 0,p R  2 2  PXL<0,PXR>0  PXL < 0,PXR  = 0  PXL=PXR \PXL\  (3.21)  < 0  ¥=  \PXR\  Ramond Sector  In the Ramond sector iti and TTR both take half-odd integer values. The calculations of C = (pxL PXR\B) and D = (J>XL PXR\&\a\\B) in this sector 19  Chapter 3. Calculation of h(x°) are almost identical to that of the NS sector except that one must take the degenerate ground states into consideration. In [1] (pxL PXR\B) was calculated to be the same as in the N S sector { [-iUn}^ S{p R-PXR)  PXL > 0,PXR  PXL  X  [U ]^H(p 22  C=  {  +  XR  [Un}-^ X(PXL  PXL)  + PXR)  PXL  [-iU l]-^ -S(pxL+PXR) 0  PXL  <  PX  PXL <0,PXR  2  (3.22)  <0  X  ^  \PXL\  0  > 0  PXL < 0,p R  xl  > 0  > 0, R  \PXR\  The calculation of (7r_ -K \a\a\\BD) R is once again quite involved so we reserve it for the appendix and state the results here R  [-iUi ] 6{pxL  ~ PXR)  PXL > 0,p R  >0  -[U 2]^ S(pxL  + PXR)  PXL > 0, R  <0  V2pxL  2  PXL  2  D=l  -[U ]-^PXH(PXL+PXR) [-iU i}~^ S(pxL 0 2  (3.23)  PXL<0,PXR>0  U  PXL  3.3  X  PX  - PXR)  PXL  0,PXR  <  ^  \PXL\  <  0  \PXR\  Summary and Final Calculation  In this section we summarize our results and use them to complete the calculation. In the preceding subsections we found that [-iUn]' ~ 6(pxR-PXR) /2  [U ]^H( R 22  C=l  Pn}~^ S(pxL  + PXR)  PXL  21  1 0 a n  d  -J^PXR  PXL>0,PXR<0 (324.)  PXL<0, R>0 PX  + PXR)  XL  > 0  X  + PXL)  PX  [-iU ]-^H{p  where -^PXL  PXL > 0,p R  PXL  PXL<0,p R<0 = PXR = 0 X  PXL \PXL\  ^  \PXR\  are either both integers or both half-odd integers.  Now we can calculate the matrix element f(x°,x°)  = (x°,x \B) 0  where  20  Chapter 3. (x°,x°\  Calculation of h(x°)  was given by (3.4). We find that oo  (x°,x°\B)  =  l  £[( -V  +  e  1  1  r+(e-^V  2  r  2  n=l •  +(e * (-iU )r 1 i  +  0  12  1 - Ue 1  ixo  n  +  U e~ ° 1 ix  22  l +  ixo  The final step is to decompactify f(x°,x°) values of A, giving  f Jo  n  12  1  2  =  i  1-  +-1 + iUi e  f(x°)  (e- *°(-iU )) }  - 3  iU ie- ° ix  (3.25)  2  by integrating over all allowed  dA(x°,x°\B)  1  1 + nge °  1 + nge  lx  - 1.  °  IX  (3.26)  We see that the integration over A eliminates the coordinate x° and gives the expected value of / ( x ° ) The calculation of h(x°) is almost the same. The matrix element D = (pxL,PXR\ i®i\B) was calculated in the previous sections to be a  f [-iU ] * S(pxL V  ~  PXL  12  -[U 2] S(PXL 2  D  PXR)  PXL  + PXR)  V2pxl  -[UU]-^ -5(PXL  + PXR)  XI  R >  PXL > 0 ,  P X  R < 0  < 0 ,  P  X  R >  0  (3.27)  PXL < 0,p R < 0 = p =0  PXL  21  n  P X  PXL  [-iU ]-^ 6{pxL-PXR) l-2U U  0  > 0,  X  22  P  0  x  L  X  R  \PXL\  \PXR\  We can now calculate h(x°, x°) = (a; , x° | a j d i | B) 0  XQ  h(x°,x°)  =  l +  x  0  J2[-(e °Un) -(e- U ) iA  n  ti0  n  22  n=l + W (-iUi2)T + (e~ {-iU )T] -1 -1 + 1 - U e~ ° 1 - Ue ° X  ix  n  ix  ix  n  +  Y  T  22  ^  +  i  +  l  u  2  l  e^  +  l  -  2  U  ^ -  ( 3  '  2 8 )  21  Chapter 3. Calculation of h(x°) Once again we can decompactify by integrating out A which gives h(x°) = f(x )-2(1-n gg). 0  2  (3.29)  22  \  C h a p t e r  4  N u m b e r  o f  M a s s l e s s  S t r i n g s  E m i t t e d  4.1  1-point functions  Following the discussion in section 1.4, the 1-point function for the production of a closed string field with M = 0 from the boundary \B) is given by 2  (*B\%\B),  ,  (4.1)  where \^B) is the on-shell B R S T invariant state for the M  = 0 level  2  | * B ) =e^a1 a1 |0,0;fc)®|U) 1  (4.2)  1  and we remind the reader that we are in Minkowski space. The polarization tensor e „ has 24 x 24 components coming from restrictions e fc' = e^ k = Q , ^ e^K and e „ ^ k^e . A full review of how to obtain this state and its restrictions is given in the appendix. The calculation of the 1-point function can be split into two inner products M  i  /J1/  v  v  M  <*B|^=-^|B)  =  u  Ar ^(0,0;fc|aK|S)x |B)i p e  o  x(U|^^|B) where we can find \B) ° space X  \B)  X0  (4.3)  6 c i  by analytically continuing (1.18) back to Minkowski = Jdt  (f(t)\t) - h(t)a°_,at \t) t  + ...)  (4.4)  where /ffl = T -  L  ^ +  1 + Ttge  1  1  , / - -  1 + irge  t  1  - l  (4-5)  and h(t) = f(t)-2{l-ir gg). 2  .  (4.6) 23  Chapter 4. Number of Massless Strings  Emitted  The ghost part of (4.3) is evaluated to be (II |^-=-^|B)  f c c  =  i a i K c o - ^ ^ + co^Ci^fe-ilU)  =  (11 ICQCQI II)  (4.7) so that (4.3) simplifies to = iM e ^ ( 0 , 0 ; f c K d r | B ) | S )  ^ B \ ^ ^ \ B )  p  X o  ;  25  ='• iM (0,0; k\ p  =  -iM  Xo  x  J dte {2Trf6 (k )F{t) iEt  p  \B) \B)  ^ e o o a K + E e a a i d i J  p  where F(t) is given by  (4.8)  i  (  25  p  ^2eni=i  \ -  £  ejj  i= +i P  (4.9) /  and E = \k±_\ is the energy of the closed strings. We now see that the 1 point function is in the form given in equation (1.35) (* \c£\B)(k) B  (  = -iA^(27r)^(fc ) <  P  25  i=\  j=p+i '(4.10)  where the integrals If and Ih are given by  oo  dtf(t)e  '  iEt  /  -oo oo  / =  -oo  Aft,(i)e  (4.11)  ijEi  I -2(1-Tr gg)6(E).  (4.12)  2  f  We can never move to the rest frame of a particle with M = 0 therefore the delta function on the right hand side of (4.12) always gives 0 allowing us to set f = Ih- This is another way of saying that h(x°) = f(x°) up to terms that are non-singular in x° 2  24  Chapter 4. Number of Massless Strings  Emitted  The integral If can be calculated I  =  f  E In ng _ ^iE In ng  -i(e  (4.13)  sinh(7TjE) '  Using equation (1.26) with (4.10) as our j(k) we can write down an expression for the number per volume of massless closed string modes emitted by the brane  (2n)D-i N  f  2  V  p p  J  2E  d ~Pk d*-r  \I \I\ \ ^  25  2  f  k  (27T)25-P  25  2  f  2E  E «-  eoo  E  e  ~33  (4.14)  j=p+l  In the above equation the sum is over a basis of polarization tensors. This sum is well known [9] and we provide a brief review of how to evaluate it in the appendix. We can gain information by splitting the sum into three parts, a sum over the polarizations of the g r a v i t o n S c dilaton 5$ and antisymmetric tensor SB 25  E  eoo  +E «-  So + SB + S$  E  e  (4.15)  with S  G  = i p ( 2 4 - p), S  = 0, 5$ = 24 - i p ( 2 4 - p).  B  (4.16)  We can use this result to find out how many gravitons and dilatons are emitted from a decaying Dp-brane 3  ^  =  ^  =  (4.17)  N \p(24-p)I 2  p  M (24-\p{24-p))I 2  p  (4.18)  as well as the total number of particles emitted N_ v„ = 3  24M/I  P  (4.19)  5 a = 0 gives no antisymmetric tensors being emitted b y the Dp-brane  25  Chapter 4. Number of Massless Strings  Emitted  where d ^k 2  h  =  2  (2TT) -P 25  d -i>k 25  -iE\n(ixg)  (2TT) -P  _  iEln(-ng)  2Esmh TiE  25  2  (4.20)  In conclusion the total number of massless closed strings emitted is N  of  25  f  2  ~  d ~Pk d ~k 25  J  p  TT  p  2  iE\n(ng)  _  iEln(-irg)  (2n)^-P2Esmh 7rE 2  (4.21)  We see that for the case of the full brane we can proceed no further, our solution will depend on our choice of g. For the case of the half S-brane equation (4.11) is evaluated to be /  /  _  =  i  (  e  - ^ > -  9  _ ^ _ sinh(-7rS)"  (4.22)  )  Upon inserting this into the first line of (4.21) we see that N_ _ V  p  ~  f  2 2  4  7  V  d -Pk 25  (4.23)  J (2TT)25-Z> 2 £ s i n h ^ 7 r £ l  p  is independent of g. In the next section we explicitly calculate the number of closed strings emitted by a decaying half S-brane.  4.2  Number of Massless Strings Emitted by Half Brane  Inserting (4.22) into the first line of (4.20) gives d ~k 25  p  (2TT) -P 2 £ s i n h 7 r £ 2  25  27t(25-P)/2  r  23-p  2 r((25 - P)/2)(2TT) -P J  dx-  25  (4.24)  sinh (7ra;) 2  0  which can be integrated numerically. We expect this integral to diverge when p = 22,23,24,25 by inspection. Below we give a table showing the number of gravitons and dilations emitted per volume for various values of P26  Chapter 4. Number of Massless Strings  Emitted  N /V 0 3.39 x 10"-3 5.16 x 10-- 4 7.07 x 10"- 5 8.06 x 10"- 6 8.90 x 10 - 7  N^/V 0.262 .0179 1.28 x 10"-3 9.09 x 10"-5 6.48 x 10"-6 4.61 x 10"-7  N/V .262 .0213 1.84 x 10"-3 1.65 x 10"- 4 1.45 x 10"-5 1.35 x 10"- 6  2.23 x 10"-11 1.04 x 10"-10 5.04.x 10--10  1.63 x 10--12 1.94 x 10"-13 2.37 x 10"-14  1.14 x 10- 14 0 1.65 x 10--16  1.64 x 10"-12 1.94 x 10--13 2.38 x 10"-14  2.46 x 10"- 4  4.33 x 10'-20 1.16 x 10"-20  3.47 x 10--20 1.49 x 10- 20  7.79 x 10"-20 2.66 x 10--20  p 0 1 2 3 4 5  9.47 3.14 1.07 3.72 1.32 4.85  11 12 13 20 21  h x x x x x x  G  10" 18 10" 17 10- 16 10- 16 10- 15 10- 15  1/967T  P  p  Table 4.1: Number of particles emitted with M  2  p  = 0 for half S-brane.  27  Chapter 5  Total Number of Particles Emitted We begin by reminding the reader the total number of particles emitted can be expressed as a product of partition functions N = Im (iix j°°  d\(B\  (b^Co)q °- \B)) Lo+L  .  it  (5.1)  We call these partition functions is because they were obtained from open string partition functions via the gluing procedure described in the introduction. The spatial and ghost partition functions are straight forward to complete giving Z  =  NN  (N\q o+ i\N) L  L  oo  = «- n -—^ i/12  1  n=l  -  2nm  (5 2)  7  in the Neumann directions, with 2ir5(0) interpreted as a volume factor and ZDD  =  (D\q ° o\D) L  +L  -.-•'"nnWs«*  -  (5 3)  in the Dirichlet directions, and Zghost =  bc(B\(b+c^)q o "\B) L  +L  bc  OO  =  iq^X\^-d ?  (5.4)  2n  n=l  for the ghosts. The only non-trivial part left to calculate is the partition function in the time direction. After analytic continuation to Euclidean space it seems as simple as computing Z(bd\bd) = {BD\q ° °\BD) L  +L  + (BD\q ° °\BD) . L  NS  +L  R  (5.5) 28  Chapter 5. Total Number of Particles Emitted and factoring out the contribution from the Y boson. We remind the reader that |0>  = 2-3 FT exp  \BD)  NS  \BD)R =  2"4  f[  (5.6)  exp xexpf^t/- ^ ^! 1  +  1  (5-7)  and  J2 {f-r^r + </>-rV4 + $l j>r r  (L(j + L ° ) 7 v s  r  ^ - r # ) ~ g(5.8)  +  r=l/2 oo £ r  ^ (Vlrl^r + V » - r $ + ^ l 1 r V P  + l£-r$)  +  3-  (5-9)  r=l  The boundary state ) 5 Z ) | which satisfies the dual form of (1.10) is given by  1 (BD\  = 2~4 (0| JJ exp [ ^ L ? - ^ ^ , . - ^ V t t y r ] 1  NS  {BD\  0 0  1  = 2 " 4 (- + - + I JJ exp [ ^ f f - ^ ^ 1  R  1  x exp  "  (5-10)  ft i* Ufl>. l  n  (5.11)  -i>\io iJ^ l  The matrices £/ and [7 are of the same form as (2.40) but differ only by the definition of the parameter A. We will label A as the parameter living at the r = 0 boundary of the cylinder and A\ as the parameter living at the T = — boundary. As discussed in the introduction we integrate over A = A\ — A to decompactify the boson. In [3] it was shown that (5.5) depends only on the eigenvalues of the matrix (—io- U)(Uia^). These eigenvalues are given by 2  2  l  C = cos(27rA)(l - 7r gg) + n gg ± i\J\ - (cos(27r 4)(l - ir gg) + ir gg) (5.12) 2  2  2  2  2  J  29  Chapter 5. Total Number of Particles Emitted where A = A\ — A . Reference [1] also tells us that after evaluating (5.5) and factoring out the ZDD contribution from the Y boson, 2  Z b b  =  =  C  q  l  A _ q  ~  1/12  /  \neZ  i  ( 5 1 3 )  n=l  We have attempted many different ways to bring our result back to Mikowski space and all have failed. There is only method which gives a clear indication as to why it fails. This method begins by checking that we get the expected result for the total number i n the Neumann limit of the boundary state (5.2). Substituting our partition functions into (5.1) we get 24  V  P +  25-p'  i (5.14)  which can be split into two parts = -Trim V i~ P+  /  dX  """Vo  \>  0A  + /  ^ )  dX—-Lj-i  (5.15)  rj"(=±  Jo  with m)  = ( / f e ^ ) .  (5.16)  We now observe that in the first integral A is analytic in the upper half plane which allows us to rotate this contour to the positive imaginary axis. Similarly, i n the second integral A is analytic in the lower half plane which allows us to rotate this contour to the negative imaginary axis. After rotating these contours we see that these two integrals cancel one another giving N = 0  (5.17)  exactly as expected for a static D-brane. It was thought that perhaps a similar rotation could be used for the unstable D-brane. One can see from (5.13),however, that it is impossible to consistently perform an analogous Wick rotation. The reason is that the n in q'r is the discretized version of the square of momentum in the time direction, so as we rotate back to 2  Minkowski space n —> —n . This makes e~ ~ non-analytic in the upper half plane, so that the first integral is neither analytic in the upper half 2  2  lX  30  Chapter 5. Total Number of Particles Emitted plane nor in the lower half plane. The same argument can be applied to the second integral. This lack of analyticity prevents us from rotating these contours at all. The exception of course is n = 0 which is just the same as the Neumann case.  31  Chapter 6  Summary and Conclusion We have calculated the total number of massless particles emitted by a decaying Dp-brane in the covariant gauge using the method of fermionization  V  -  p  For the full brane \Ij(E)\  2  p  J  (2rr)25-P  2E  '  [  j  can be found by taking g = g in (4.13). For the  half brane | / / ( i ? ) | was given by (4.22). We have written ^ - in this form to 2  compare with the massless contribution to the total number obtained in [2]  l^/E^l/,^)!'.  (6.2)  The sum s includes both the sum over level n as well as over the momenta transverse to the brane and they found that only left-right symmetric states contribute to their sum. Also, we are taking their result \If(E )\ as integrated along the real contour for both the full brane and half brane, so as to be able to make a comparison. In their expression E — \fk + M so at the massless level E = E. So we see that their result agrees with ours, after picking up a factor of 24 from the 24 left-right symmetric states at this level. We have also attempted to verify (6.2) using the fermionic technique but we have been unsuccessful due to the ambiguity in rotating our result back to Minkowski space. 2  s  2  2  s  s  32  Bibliography [1] M . Hassefield, Taejin Lee, G . W . Semenoff, and P . C . E . Stamp. Critical boundary sine-gordon revisited. arXiv:hep-th/0512219v2, 2006. [2] Neil Lambert, Hong L i u , and Juan Maldacena. decaying D-branes. arXiv:hep-th/0303139, 2003.  Closed strings from  [3] T . Lee and G . W . Semenoff. Fermion representation of the rolling tachyon boundary conformal field theory. arXiv:hep-th/0502236vl, 2005. [4] J . Polchinski. String Theory, volume 1 of Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge; New York, 2004, c2001. [5] A . Sen. Non-BPS states and branes in string theory. th/9904207vl, 1999.  arXiv:hep-  [6] A . Sen. Rolling tachyon. arXiv:hep-th/0203211, 2002. [7] A . Sen. Tachyon dynamics in open string theory. th/0410103v2, 2004.  arXivrhep-  [8] Ashoke Sen. Rolling tachyon boundary state, conserved charges and two dimensional string theory. JHEP, 0405:076, 2004. [9] S. Weinberg. Photons and gravitons in perturbation theory: Derivation of Maxwell's and Einstein's equations. Physical Review, 138(4B), 1965.  33  Appendix A  Calculation of  (n n \aiai\BD) L  R  A.l  (ir  7T \aiai\BD)  L  R  NS  We begin by writing out some of the equations and results from section 3 for the convenience of the reader. The first of these is the boundary state (2.38) written in fermion variables in the NS sector \BD)NS = 2 4 _  -  f[ exp UlrU-Ha^-r  4>l i<T UiP- ] |0) l  r  r  (A.l)  and we will also require the dual form of the momentum states  *L,*R > 0  KL,-KR>0  <7T  L )  (lT ,7T \ L  R  ^|  l  l  *X-2"  "'H-2  ]J ( V 4 r l M I I1 (Ml)  = (0|  = (0|  rj (4 i> ) r  lr  1 -•"R-2  f J ^ U  (-) A  2  ( -) A  3  -7T L  -n ,7T >0 L  R  {n ,7r \ L  R  = (0\  (^Uar) I I (^ir)  1  ( -) A 4  1 -7TL-2 ir ,iTR<0 L  (n ,ir \ L  R  = (0\  J]  (^U*)  II  ( M i )  (A.5)  34  Appendix  A. Calculation of (TTL  -K \a\a.\\BD) R  Placed between these two states we have the a a terms of fermion variables (3.20) x  t  operator written in  x  (  ^Vi-  (A.6)  n=Z+±  A.l.l  7r >0,7r L  F  L  >0  After acting with (A.6) to the left on the bra momentum state only two terms in each of the sums survive (ni  TT[i\aiai\BD)NS  L  x  ^r(^L _ )^ ^ i/2 + ^ CT ^_ _i)) 3  3  X{TT  (7ri/  1/2  +1/2  i+  (^+i/ ^ ^-(^-i/2) + ^ l ^ _ 3  2  (  (7ri  ^ ^ i/2) 3  1 / 2 )  (A.7)  \BD) .  i +  NS  Noticing that the right hand side of (A.7) can be broken down into the product of two inner products 2  l/4  S(TT  -7r )(7r  L  x(0|(^  C T  R  i2^ ^t ) (  ( c r  -  L  1,7T  i2 t^ )  1\BD)  -  L  ) ( V )  t  NS  i r C 7  3  V ) r + i  ^  +  r + i 0  -^_  r )  x ( $ a < / i _ + -<pL o- jr+i)\BD)  (A.8)  3  3  +1  r  r  NS  where we have set r = irr, — 1/2 in order to unclutter our equations. The first inner product we have already calculated i n (3.12) (TTL - 1,TT - l\BD) L  = 2174 ( - i t / j a ) ^ " ) . 2  NS  (A.9)  1  To evaluate the second inner product we first act the very last term in brackets on \BD)NS followed by acting (i/>l <7 i/v + ipl a ip- ) on everything to the right of it so as to eliminate the r+1 modes 3  3  +1  r  x (-V>L(r+1)(^V)cT ^_ 3  x {ipl <j {iU- a )a 4>„ 3  r  l  l  3  r  ^_ o- {ic- U)i>_ )\BD) 3  r  r  r  l  3  R  l  {R+L)  - ^_ a {ia U)a ^- )\BD) . 3  r  +l  NS  NS  (A.10) 35  Appendix A. Calculation of (TTL  ir \a\a\\BD) R  Next we anti-commute away the r modes in the final line of (A. 10) to get -(0|(V4a Vv)($a 12  12t  Vv)  x ( ^ . ^ ( i t f - V y =  (0\i>y  3  1  - &o  3  3  r  NS  a </v  3  [a {ia U)a }  X2X  =  1  1  12t  [^(iU- * )* ]  2  4-y(^ U)<J iP- )\BD)  ^ -  l  * H \BD)  3  L  r  NS  -(t)\iplcj (iU- o- )o- \ia U)il>l2  l  l  l2  l  r  + 1  ^ a \ia U)a (iU- a )^- \BD) l2  1  12  l  1  r  r  Tr(o- {iU- a )a \ia U)) l2  L  '21/4  2  l  l  12  NS  12t/„-Tr{a l j \ia U)a {iU- a ))  +  l  l2  l  l2  l  1  Tr{a {iU- a )a \ia U)) l2  "21/4  l  l  l2  l  .^(ilZ-V )!!^ !/^ 1  2  1  (-it/ ) . 2  (A.ll)  12  Substituting (A.9) and ( A . l l ) into (A.8) we complete the desired calculation fa TT \a\ai\BD) R  A.1.2  = -^(-iUn) S(Tv  (A.12)  - ir )  27rL  NS  L  R  7r >0,7r <0 L  f l  Following the same lines of reasoning as in the previous subsection we can arrive at the equivalent of equation (A.8) by taking (717, 2  that  -K \aiai\BD) R  NS  l/4  —  (-XL +  n )(n -l,-(ir -l)\BD) s  x  R  X (0|(^a Vr)(^O1  x{^l a' ^ i  +1  1  L  L  N  ^r)(^L a ^ 3  2  P  r  +  ^  i +  +  ( 7  1  + 4_ a i>r+i)\BD) s. r  ^ _  r  )  (A.13)  :i  r  3  N  The first inner product was given in (3.15) by fa - 1, - f a -  = 7 ^ {U22? * {  L  l)  (A.14)  and we can find the second inner product by taking the third last line of ( A . l l ) and also remembering to pick up an extra minus sign 36  Appendix A.  Calculation  of (TTL  K \a\a\\BD) R  between the second and third line, giving  NS  =  ^f  [Tr(a^(iU-W)a^U))]  4  =  -^(iU-'a^ia'U)^ -7^(t/22) .  =  (A.15)  2  Substituting (A.14) and (A.15) into (A.13) we get the result fa Tr \a a \BD) R  1  1  =  NS  (U ) ^  «5fa + rr ).  2  22  R  (A.16)  A.1.3 vr<0,7r >0 L  FL  This time we take a  12  —> cr \ l2  n \aiai\BD)Ns  fa  2 / 1  so that (A.8) becomes  R  4  — < ^ f a + ^ f l ) f a + 1 , - f a + 1)\BD)  NS  (0|(^a  1 2  V )(^o-  1 2 t  r  ^)(^L^ ^  +  3  + 1  NS-  (A.17) Once again we can evaluate the first inner product using (3.15) fa + 1, - f a + l)\BD)  = ^  NS  and taking cr  12  {U r^  L+l)  n  (A.18)  —> <r ^ in the third last line of ( A . l l ) we get 12  (0|(v4aVv)(V;VVv)(V< V V v + i 12t  2t  +  NS  2 2 / 1  = =  4  Tr-^  1 2  ^^- ^ )^ 1  1  1 2  ^^ ^)) 1  -^(^-VM^C/^ - ^ n )  2  .  (A.19) 37  Appendix A. Calculation of (717,  -n^aiai\BD)  for the second inner product. Substituting (A.19) and (A.18) in (A.17) we find {n ir \a a \BD) L  A.1.4  R  1  1  =  NS  (U )~^  S(n + n ).  L  n  L  R  (A.20)  7r <0,7r <0 L  R  For this case we need to take a (A.8) becomes (717, 2  —> a ^ and a ^ —> a  1 2  12  12  12  so that equation  -K \a\a\\BD) R  NS  l/4 .  -j-S(n  +  L  n )(n +l,Tr +l\BD) R  x(4>l a ^ 3  +1  L  L  NS  + ^l a ^ )\BD) .  (A.21)  3  r  r  r+l  NS  Using (3.15) we can find the first inner product ( T T L +l,n +  1\BD)  L  and taking a get  -> <r  12  and a  12t  x^l a ^ +l  3  =^  NS  1 2 t  -> a  2{  (A.22)  L+l)  in the third last line of (A.11), we  12  +  r  {-iU^)- ^  ^_ a A )\BD) 3  r  +1  NS  2 Tr(a \iU- a )(j {ia U)) "2V L 12]  l  l  l2  l  4  =  4h [~  i U 2 l ) 2  -  ( A  -  2 3 )  for the second inner product. Substituting (A.21) and (A.22) into (A.23) we find (TT n \a a \BD) l  R  1  1  NS  =^  (-W^ 2  S(n -n ) L  R  (A.24)  38  Appendix A. Calculation of (TTL irR\a\a\\BD) A.1.5  7 r = 0,7T L  = 0  fi  We see that by acting (3.20) to the left on the bra (717, = 0 TTR = 0| only one term in each sum survives (ir = Qir L  =  R  0\a ai\BD) ' 1  NS  =  - \ (  =  -i(0|^ a [«r ^ r [»C/- cr ]^_  Q  \ (^1/2^1/2) ( ^ 3  1  / 2  3  / 2  ^i/ )  1  NS  1  t  1  \BD)  2  1 / 2  |S£)) s A f  Tr[o- (cr C/)cT ((7-V )] 2 • 2V4 1 (U U + U U ) 3  2i/4  3  21  n  1  22  1 2^(l-2f/nI/22).  = A.2  12  1  (A.25)  (TTL 7Tij|Q!iQ:i|BD)i2  Again we will rewrite the fermion boundary state (2.39) in the Ramond sector 1  0  0  = 2 - 4 JT exp j ^ t / " ^ ^ - ^ L ^ C / V - n ]  \BD)  1  n=l  R  1  x exp 4U- ia ^ ] I  | - + -+)  l  0  (A.26)  as well as the momentum states from [1] for this sector  1 7r ,TTR>0 L  (n ,7r \ L  = (0U J J  n=1  R  1 ( ^ m ) ] J  L  f l  >0  (7r ,rr | L  f l  = (0| J J 2  (  1  L  R  (Tr  L j  TT  f l  |  = (0|  1 -Ti,~2  ir ,ir <0 L  R  (TT ,-KR\ = L  (0|  ^  m  II  )  * " R -  4  JT  (&"$n)  )  (A.28)  1 2  I]  (V-L^n)  3  2 7  1  -TL—2  -Tr ,Tr >0  A  n=1  1  Tr ,-Tr  ( "  (^L)-(A.29)  1 ~ R—2 n  (VL^n)  J J  (&»$n)  (A.30) 39  Appendix A.  Calculation  of"(ITL -K \a\a\\B  D)  R  where the degenerate ground states are given by |0)i = | + - - + >  (A.31)  |0> = *! + + - - >  (A.32)  2  |0> = -%\ - - + +)  -  3  (A.33)  |0) = | - + + - ) .  (A.34)  4  In the Ramond sector the fermion modes are integers so that a a x  \ (j2  ti^l-n)  Vn=Z  /  (£  tl^l-rr)  "  \m=Z  x  is  ( -35) A  /  7r >0 7r >0  A.2.1  I  L  f i  Acting (A.35) to the left on the momentum state gives the same result as in the NS sector, only two terms in each sum survive, (717,  -K \aiai\BD) R  R  (TT TT| (^L ^_ ^ ^ +I/ + ^ ^ V'-( -i)) x (^ i/2 ^-(^-i/2) + ^L(^-i )^ ^ i/2) \BD) 3  L  l  (  1/2)  3  L  2  +1/2  7rL  ff3  3  i+  /2  i+  R  (A.36)  and for wr, > 1/2 neither of these terms contains a zero mode. In this case the calculation is the same as in the NS sector, except that we still have to calculate the remaining zero mode parts (ir  -K \aia \BD)  L  R  1  =  2U4 (- i2) - (+  =  ^(-iU ) S(ir -ir )  iU  2nL  ~ - + I exp[4u- ia^ }\-  1  l  0  2KL  12  R  R  L  + - +> (A.37)  40  Appendix  A.  Calculation of (TTL  -K \a\a\\BD) R  where we have used the fact that if c is a Grassman variable then c = 0. A special case occurs when TTL = 1/2, equation (A.36) becomes 2  (1/2 1 / 2 1 0 ^  =  -\(+ - ~ + I ( 4 ^ i + 4^o)  =  -\(+--  + \4 ^x4x^  +  0  2~f^  (  1  + 4 ^i>i)  R  4i^4y^i\BD)  R  R  < + ~ + -lexP^^" ^ ^)! - +  +  \BD)  0  < + - + -\Mix)\BD)  •= ~ \ ((- + - + 1^1,1^1,1+ =  (4^0  1  1  •  -+>)  -iUi 2V4  (A.38)  2  where in the second last line we have used the fact that {U- ia ) {U- ia ) l  x  l  l  lx  - (U- ia ) (U- ia ) i  = 1.  (A.39)  = -L(-iU ) ^6(7r  - n)  (A.40)  1  22  1  1  12  1  2  Thus we conclude that (rr 7r \a a \BD) L  A.2.2  R  1  l  2  R  12  R  L  rr >0,7r <0 L  R  Following the same lines of reasoning as in the previous subsection we can immediately write down the solution for %]_, > 1/2 n \aiai\BD)  (TT  l  R  -^i(U ) ^- (+ 2  1  22  -^(U ) ^6(7T 2  22  R  R  + - - | exp[4u~ ia M\l  + n) L  l  + - +) (A.41)  41  Calculation of (-Kj,  Appendix A.  itn\a.\a.\\BD)  so that the only thing left to calculate is (1/2 1  -l/2\a a \BD) l  1  R  = -(+ + - - I (fo^i 2  + 1>Wik)  =  -\{+ + - - l^yMl^o  =  - \ ( < - + - + 1^1,1^,1 + { + - + 2 2 -U 22 V 4-(1 1  A.2.3  TT < 0,n L  +  (#<r$o + V ^ i ) 3  ^\a Myfa\BD) 3  R  \ )R BD  +<+-+- I exp(^C/- Ja ^ )| - + - +)) 1  1  0  > 0  R  Once again we can immediately write down the solution for nr. < 1/2 (ir  ir \aiai\BD)  L  R  =  ^i(U )-  =  -^(Un)- « S(«R  R  - ( - - + +\^Pbi'lU- ia^o}\-  2 7 r L  1  n  2  +  1  + irL)-  L  -+) (A.43)  Let's now calculate the special case TTL = —1/2 (1/2  -l/2\a cl \BD} , 1  1  R  ;(-- + + ! (Vi^Vi + V^Vo) =  ^(--  ++\ 4 ^ ^ \ ^ o +  =  ~\ ( ( - + - + W2,i#,i + ( + - + -  ($\<r j>o'+ 3  \BD)  R  4(r ^la fa\BD) 3  3  R  |^I,i^2,i)  \BD)  R  2 • 2/  1 4  42  \BD)  R  Appendix A. Calculation of (TTL 7 T R | a i o : i \ B D ) 7r <0,7r <0  A.2 A  L  i ?  Once again we find  (717,  ir \aia\\BD) R  R  =  -^(U ir - (-  =  ^(U2i)- 5(n -n )  2nL  l  + + - I e x p ^ f / " ^ ^ ] ! - + - +) 1  2  1  (A.45)  27TL  R  L  for TVI < —1/2. Finally we consider the special case when TTI. = —1/2 (-1/2 =  -l/2\a a \BD) 1  1  - \ ( - + + - I (Vo^i  R  + 4^o)  = . -^(- + +-\4 a fa^a ^ 3  0  =  -\((-  + - +  % U % 1  ( $ ^ 0 + 4° 4>i)  +  3  0  R  .  4a iJo4a fa\BD) 3  3  R  <+ - +  -  \  (1 + ( + - + - | e x p ^ t / - ^ ^ ) ! 1  2 • 2V4 -iU \  \BD)  3  1  B  D  )  R  - + - +» (A.46)  2  so that (n n \a a \BD) L  R  1  1  = -^(-iU )- "H(7r 2  R  21  R  - -K ) L  (A.47)  43  Appendix B  BRST invariance of the M = 0 state 2  We begin by writing down the most general closed string state with M in D = 26 flat space-time dimensions  2  +Bc6-i5-i + B c - i 6 - i + B6-16_ + C _ 5_ |0 0;/c)| ||) c  1  c  1  1  I  = 0  (B.l)  and as was shown in [4] B R S T invariance of the ground state demands k = M = 0 and that the other degenerate ground states are not allowed. B R S T invariance demands that the charges 2  2  OO  OO  Q=  /  .  (CnL™ )+  B  9  n  n=—oo  m , n =  and  (B.2)  oCmCnb-rn-nl  —oo  oo  QB = ^2 (ajL™ ) + n  oo  n=—  ^2  -"^-m-nS  -—  (B.3)  m , n = — o o  annihilate any physical state of the closed string spectrum. The operators L™ , L™ are the left and right moving matter Virasoro modes and % I indicates creation-annihilation normal ordering. Acting the holomorphic B R S T charge on | ^ ) gives n  n  c  QB|* ) C  +( ^  1  y ( c - i f c ^ a ^ + cik^ati)  +  CQ  ) ( ^ i ) + B c ^ a ^ 5 - i + B~b- (k^_ )}\0,0;k)\ 1  which demands that k^e^, = k^p^ = k^j^  1  1  l * c )  ||)  (B.4)  = Be = /3 = B = 0 for physical M  44  Appendix B. BRST invariance of the M  = 0 state  2  states. Acting the antiholomorphic B R S T charge on \$> ) gives c  CO  |*c>  = y-2-(*'V<£ c_i + ( ^ ) c - i b - i - ( V H c - i c - i . 1  + Bck^c-!  + (0,*of )(k a _ ) v  1  v  1  + Bb-i(k^W,0;  *)| | | )  '(B.5)  which demands that We^v = k^j = B = (3^ = 0 for physical states. Eliminating unphysical states ( B . l ) reduces to c  e^a^iO^i  +  7^o; l c_ /  1  1  +  T^Q^C-I  + Cc_ic_i |0, 0;  ||),  (B.6)  with the restrictions k^e^ = k^e^ = k^y^ = = 0 and k — 0. Some of these remaining states, however, are B R S T exact and have zero norm. B y inspection eliminating the zero norm states sets 7 = 7 = C = 0. One can also check that polarizations of the form e „ = k^e and = e^k^ have zero norm and can be eliminated. In conclusion the 24x24 states that are B R S T invariant and have positive norm are given by 2  M  |*B>  =  e^i«_i|0,0;A:>|  with the restrictions e fc'*. = e^k" = 0, /u/  v  (B.7)  ||)  ^ e^k^ and  ^ k^e,  45  Appendix C  Polarization sums We are interested in evaluating  = X>£;(*)<£(*)-  (c.i)  A  As discussed by Polchinski [4], the B R S T invariant states (B.7) derived in the previous appendix transform as a 2-tensor under SO(D — 2). Since this is a reducible representation any tensor gij can be decomposed as  9ij  -  \  [913  + 9ji  ~  D  _  iJ9kkj  S 2  +  2 (Sij  ~ 9ji)  +  D  _  ij9kk-  (C.2)  x 2  Under this decomposition reference [9] gives a method for computing ( C . l ) for the graviton. We can also use this method for the dilaton and antisymmetric tensor so that  J" \ (n n + n n^ w  ^fit/pa —  \  w  MCT  ^n^Il^)  • -5 ( n ^ I I ^ — n^n^)  Graviton Antisymmetric Tensor  •^z^U^Upcr  Dilaton  (C.3) where  is the sum over polarizations of the gauge boson  I V = $ » 2 . A  (C.4) _  Note that there is no actual gauge boson, this is just a mathematical trick. Since (C.3) is rotationally invariant we choose a choice of coordinates such that = (A: , 0,0, ...,fc ), k° = k which gives linear polarizations e\ = (0,1,0,...); el = (0, 0,1, 0,...); e = (0, 0 , 1 , 0 ) . This choice of coordinates gives non-zero as 0  25  25  24  n  ii  = l;* = l , . . , 2 4  •  (C.5)  with no sum on i . 46  Appendix  C. Polarization  sums  Following equation (4.15) we see that we are interested in calculating the special case n i , „ with no sum on \x or v. Substituting (C.5) into (C.3) we see that the non-zero components of are W  V  Graviton Antisymmetric Tensor Dilaton  D-2  U  0  n,  l D-2  (C.6)  with i,j = 1...24 and again no summation of the i's and j ' s . As advertised these sums are independent of k^. We can now simplify (4.15) to give 25  S  =  Sq + SB + 5<j> =  eoo •  E - ~  E  e  ^33  J=P+1 24  E  E  e  i  i  E  _  p  24  p  24  (C.7) i = l J=p-|-1  and using equation (C.6) we can fully evaluate this sum for each of SG,SB and 5$ independently Sr i,j=p-\-\  i,j = l  S  B  =  j=p-\-l  1(24-p)  (C.8)  0  (C.9)  ,  /  P  pp  24  E \i,3 = l  =  i=l  i>3=P+\  24  1 - 2C EE E  1  (C.10)  24-^(24-p).  Adding these together gives S = 24.  (C.ll)  47  

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