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Kalb-Ramond solitons in basonic string theory Sussman, Benjamin Jacob 2002-12-31

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Kalb-Ramond Solitons in Bosonic String Theory by  Benjamin Jacob Sussman B . S c , University of Waterloo, 2000 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 T H E REQUIREMENTS FOR THE DEGREE OF  Master of Science in  The Faculty of Graduate Studies Department of Physics and Astronomy We accept this thesis as conforming to the required standard  The University of British Columbia January 2002 © Benjamin Jacob Sussman, 2002  In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of Physics and Astronomy The University of British Columbia Vancouver, Canada Date  Abstract  ii  Abstract A soliton of the Kalb-Ramond field in closed bosonic string theory is introduced. Under appropriate configurations the cosmological constant becomes a periodic function of the Kalb-Ramond field. This vacuum degeneracy permits the formation of sineGordon solitons. The energy and length scale of the soliton is inversely proportional to the string coupling constant g . The stability of the the solitons is discussed and it is shown that these objects are stable. s  Table of Contents  iii  Table of Contents  Abstract  ii  Table of Contents  iii  List of Figures  v  Acknowledgements  vi  Chapter 1:  Introduction  1  Chapter 2:  Review of Bosonic String Theory  3  2.1 2.2  2.3  History and Motivation From Particles to Strings 2.2.1 Particles 2.2.2 Strings Symmetries  3 4 4 4 6  Chapter 3: The Covariant Path Integral and Modular Invariance 3.1 The Covariant Path Integral 3.2 Modular Invariance and the Torus Amplitude  12 12 13  Chapter 4: Strings in a Kalb-Ramond Field 4.1 The Massless States of The Closed Bosonic String 4.2 Mass-Shell Condition and Level Matching  16 16 17  Chapter 5: The Vacuum Energy 5.1 Vacuum energy of the uncompactified string 5.2 Vacuum Energy with one compact direction 5.3 Vacuum Energy with many compact directions 5.4 Vacuum Energy with 2 compact directions and a B  21 21 24 25 27  field  Chapter 6: Elimination of the Tachyon Contribution 6.1 Spectrum Truncation 6.2 Relative Subtraction  32 32 33  Chapter 7: String Theory Effective Action 7.1 The Effective Action as the Spacetime Action 7.2 Kaluza-Klein Theory 7.3 Compactification of the Effective Action 7.4 Adding the Cosmological Term  34 35 37 39 40  Table of Contents  iv  C h a p t e r 8: E q u a t i o n s of M o t i o n 8.1 Equivalence of String Frame and Einstein Frame 8.2 Variation off the Action  42 42 43  C h a p t e r 9: S i n e - G o r d o n Solitons 9.1 Two Dimensional Sine-Gordon Solutions 9.2 B-field Solitons  47 47 49  C h a p t e r 10: T o p o l o g i c a l Indices and V a c u u m M a n i f o l d 10.1 Topological Charge 10.2 Modularity of B 10.3 Unstable Topological Configurations  51 51 52 54  C h a p t e r 11: S t a b i l i t y 11.1 The Bounce 11.2 Modelling the Surface Tension  56 56 62  C h a p t e r 12:  64  Conclusion  References  66  Appendix A : Dimensional Reduction  69  List of Figures  v  List of Figures 7.1  Zero loop string diagram  35  7.2  One loop string diagram  36  10.1 Homotopy classes  55  11.1 Bubble forming in a domain wall  60  Acknowledgements  vi  Acknowledgements I wish to gratefully acknowledge Prof. Gordon Semenoff for his patient supervision and support. Tom Davis and Mark Laidlaw could always be counted on for friendly and engaging discussions. I would also like to thank my friends and family, without whom, this thesis would not have been possible. This work was supported in part by the Natural Sciences and Engineering Council of Canada.  Chapter 1.  Introduction  1  1: Introduction  String theory purports to be a 'theory of everything' and it is currently the lone serious candidate for the unification of gravity and quantum mechanics. Decades of intense theoretical work have explored only a few avenues of the large landscape that is string theory, but to determine if string theory really is a 'theory of everything', we must continue to investigate its various aspects. A particularly interesting area of research involves looking at certain topological features of string theory. Herein we discuss one topological feature that occurs in closed bosonic string theory: the Kalb-Ramond soliton. One of the massless states of closed bosonic string theory gives rise to a field known as the Kalb-Ramond field, which is often called simply the B-field. Under appropriate compactifications, the B-field can couple to closed strings. In this situation, the B-field shifts the momentum, much like the shift in momentum caused by the application of an electromagnetic field to an electron. The shift in momentum drastically changes the value of the vacuum energy density. In fact, the cosmological constant becomes a function that is periodic in the B-field. This degeneracy of the vacuum can produce domain walls. To an approximation, the system is equivalent to the well known Sine-Gordon system. The energy, size, and structure of the domain wall depends on the compactification radius and may be observable in certain configurations. We begin what follows with a brief review of bosonic string theory.  Chapter 2  provides the necessary prerequisites for a discussion of Kalb-Ramond solitons. Chapter 3 contains a discussion of the covariant path integral formalism. The key point here is the idea of modular invariance. The influence of the Kalb-Ramond field on  Chapter 1.  Introduction  2  closed strings is analyzed in chapter 4. It is shown that the Kalb-Ramond field can shift the canonical momenta string modes. Chapter 5 details the calculation of the vacuum energy with and without toroidal compactification. The calculation is then consider for the configuration of a constant Kalb-Ramond field. The finding is that that vacuum energy is a periodic function of the Kalb-Ramond field. After a brief discussion of the tachyon divergence in chapter 6, the string theory effective action is introduced in chapter 7. The cosmological constant is added to the effective action and dimensional reduction is considered. Chapter 8 shows that under appropriate circumstances the Kalb-Ramond field can have sine-Gordon solitons, the details of which are discussed in chapter 9. The remainder of the thesis discuss the stability of these solitons. Chapter 10 begins with an explanation of why one might at first expect the solitons to be stable and concludes by explaining that the soliton may decay to the true vacuum through its coupling to massive modes. Chapter 11 estimates the rate of decay of the soliton through nucleation. Chapter 12 concludes the thesis with a summary.  Chapter 2. Review of Bosonic String Theory  3  2: Review of Bosonic String Theory  2.1  History  and  Motivation  Einstein's general theory of relativity is the currently accepted theory of gravitation. However, it is a classical theory. In order to describe gravitational effects in extreme circumstances a quantum version of general relativity is required. After many years of effort, though, gravity has prooven impossible to quantize. The efforts to do so have persisted since the early days of quantum mechanics, but all conventional attempts have failed [1]. During the 1960s and 1970s, the so called dual resonance model was explored as a possible theory of the hadrons. Ultimately it failed and was replaced by quantum chromodynamics.  One of its failures was that it predicted a spin-2 particle that  was not observed in hadronic reactions. This failure was, however, was somewhat fortuitous.  For as the dual resonance model was abandoned, some realized that  the spin-2 particle may in fact be a candidate for the graviton - the quanta of the gravitational field. That is, the dual resonance model should not be taken as a theory of hadronic interactions, but as a quantum theory of everything. In modern times, string theory has become the leading candidate for a consistent theory of quantum gravity. Thorough monographs have been written on string theory [2, 3, 4, 5] as well as excellent review articles [6, 7, 8, 9]. Herein we shall give a brief review of the elements of string theory that are immediately applicable to the discussion B-field solitons.  Chapter 2. Review of Bosonic String Theory 2.2  From Particles  to  4  Strings  2.2.1 Particles In special relativity, the action of a particle is given by the integral of the invariant length ds — g ,dx dx 2  tl  lll  u  times an arbitrary constant m:  (2.1) The connection with non-relativistic physics is obtained by parameterizing r = X and taking the following limit:  S = - m J dx°y/i-x*  = j  dx°{^  We therefore conclude that m is the particle mass.  2.2.2 Strings To move to string theory, we note that a one dimensional string sweeps out a two dimensional sheet as it moves through spacetime.  Cover the surface with a two  dimensional grid parameterized by {c \a = 1, 2} and metric hob- The element of length a  for any Riemannian (or pseudo-Riemmanian) surface is given by ds = h dc dc . 2  a  b  ab  Thus the length of a small coordinate displacement is given by ||dc || = \/h dc (no a  a  aa  sum). Therefore the area of a particular (parallelogram) element of the grid is given by:  5  Chapter 2. Review of Bosonic String Theory  AA  = ||dc° x dc !!  (2.3)  =  ||cic 1| Hcic 1| sin(6>)  (2.4)  =  ||dc°|| Hdc !! (1 -cos (0))  1  0  1  1  dc°  where h — det(h ). ab  dc  2  (2.5)  2  1  -  2  dc° • d c  1  (2.6)  2  =  v^oofcn - (h ) dc°dc  (2.7)  =  v^tic^c  (2.8)  2  l  Ql  1  Writing r = c°,o = c we then write the so-called Nambu1  Goto action as the total area of the surface times arbitrary constant T of dimension (length)' : 2  -h S = -T JI dodr\f dodr\f^h  (2.9)  :  To determine the induced metric h  in terms of the spacetime metric note that the  ab  line element as calculate in both frames should be the same:  ds = h dc dc 2  a  ab  b  = G^dX^dX"  (2.10)  dX»dX"  To make the connection with non-relativistic physics, we parameterize r = X° . We then parametrize o as the proper length perpendicular to r: (-J^X ) = 1, -§^X° = 0. 1  2  Substituting these values into (2.11) and subsequently into (2.9) we obtain:  S = -TJ  drdo^l-(^)  2 +  (^-X^Xr  (2.12)  Chapter 2. Review of Bosonic String Theory Writing v = (-^X ) 2  1  2  — (^X -^X ) l  1  6  and expanding in small v, we obtain  2  5 = T | d r d o Q,  2  - 1+C V ) ) .  (2.13)  Thus we may interpret v as the transverse velocity of the string and the second term as a potential energy term. I.e., T is the classical tension of the string. T is often written as T =  TT—F  We will find that an alternative, but equivalent version of the string action (2.9) will be easier to work with. Introduce a Lorentzian (—, +) metric 7  a h  for the so-called  Polyakov action:  S p  =  4^J  The equation of motion for 7  a h  ^l^ - ^ ^  drda  hd  Xil  (- )  X  2  14  following from this action can be used to eliminate  itself from this equation and recover the Nambu-Goto action (2.9). We may also define the energy-momentum tensor via the equation of motion for ^ :  T  ab  = -4.7T-  1 6 S = - l ( d A ^ d % - l^daX^d'Xj -7 8f a a  ab  P  - 0.  (2.15)  The equation of motion for X is th  d (V=ll d X») a  2.3  ab  b  - ^d X» 2  = 0  (2.16)  Symmetries  The concept of symmetry plays an important role in string theory. Demanding certain symmetries puts extreme constraints on string theory. First we note that the action 5^ should be independent of intrinsic properties of  Chapter 2. Review of Bosonic String Theory  7  the world-sheet. That is, physics should be independent of parameterization of the world-sheet. This is called diffeomorphism invariance under:  X'^T^O')  = X»{T,O)  (2.17)  do' do' c  d  ^T ^'c (r',a') a  = (r,o).  d  lab  (2.18)  A n additional symmetry satisfied by Sp is that of D-dimensional Poincare invariance:  X'"{T', a') = KX"{r, a) + a"  7ab(T,o-)= (T,o-). LAB  (2.19)  (2.20)  A third symmetry is unique to that fact that the world-sheet is two dimensional and is called Weyl Symmetry:  A " " ( T V ) =X"(T,<7)  i (r,o)  = e ^ha (r,a). 2  ab  b  (2.21)  (2.22)  A n arbitrary symmetric matrix has D(D + l ) / 2 free components. Diffeomorphism invariance implies that we can fix D components. Therefore there is D(D — l ) / 2 degrees of freedom. In particular, in D = 2 we have 1 degree of freedom which means that we can write the world-sheet metric as:  J  ab  =  e ^r, (T,o) 2  ab  (2.23)  8  Chapter 2. Review of Bosonic String Theory  where n b is the flat Lorentzian two-dimensional metric. Subsequently, Weyl invaria  ance implies that that the action is independent of the final degree of freedom. Of particular importance are conformal transformations:  There are coordinate  changes that scale the metric which can subsequently be undone via a Weyl transformation. These compound transformations are called conformal transformations. Introducing o  ±  = r ± cr we note that in the conformal gauge — det(7 &) = e a  4w  and  = e~ r] . The action becomes 2uJ  ab  S  4 ^ /  =  d r d C T  v 77 9 X"^X / =  (2.24)  a 6  t t  A 4  Thus, the equation of motion is the simple wave equation: d  d  2  2  dr  1  do  2  2  A ^ = 0.  (2.25)  we may then factorize the differential operator yielding  9 +  s  ' A ^ = 0.  d  do  do  (2.26)  The general solution is then the sum of a holomorphic and anti-holomorphic (rightand left-moving, respectively) part:  X»{o, T) = X (o )  + X (o-)  +  L  X^(a~) = \x» + o y e r - + £ X£(o ) +  (2.27)  R  = \x» + a ' p " < 7 + +  ^e~  2ina  (2.28)  YJ fe~  2ina+  We may now quantize in the usual way. In the conformal gauge we have  S = | J d od X»d X , 2  a  a  t  (2.29)  Chapter 2. Review of Bosonic String Theory  9  and thus  P " = — — L = TIC . 1  (2.30)  To quantize the system, we introduce the equal time canonical commutators  [X"(c, T), P {o', T)] = z5(a - o')rf\  (2-31)  [ i*,tf\ = irr  (2.32)  v  We then arrive at  x  [<,<] =  (2-33)  [<, <] = Sm+nVr  (2.34)  K , a @ = 0.  (2.35)  We recognize these as the harmonic oscillator raising and lower operators if we write  04 = v ^ a ^  a£  m  = Vma^  m>0  (2.36)  m < 0.  (2.37)  One then constructs the various string states by using the raising operators on the vacuum state |0;p ) . It will turn out that the higher states are increasingly M  more massive. However, not all states constructed in this fashion are physical. The  Chapter 2. Review of Bosonic String Theory  10  equation of motion for 7 in (2.15 ) yielded the energy-momentum tensor which may be written as  T  ±  - d o ^ d o ^ - °  ±  ( 2  '  3 8 )  In order to facilitate the application of this constraint we take it's Fourier modes. For example:  L  - =i  h  [ ^  ,  m  '  - ^ = s b  T  [  ( 2  -  3 9 )  After substituting in the mode expansion we find that for the right modes 1 L  m  0 0  = 2^  a m  ~ ' n  a  °  n =  (- ) 2  41  —oo  however, since [a_ , a ] 7^ 0 , after quantization there is an operator ordering n  n  ambiguity in the m = 0 case ^ 00  L = -c/p + -  -n • Oi = a.  2  a  0  1  n  (2.42)  n=l  That is, this normal ordered expression is ambiguous up to a constant a. The constant must be determined. For closed strings we find a — I [4]. Similar relations hold for the left modes. If we define mass as m = —p^p then the Virasoro generators give the important 2  41  level matching and mass-shell condition. Taking the sum and the difference of the zero-mode Virasoro constraints, we find  Chapter 2. Review of Bosonic String Theory  N-N  2  (2.43)  =0  m = —{N + a'  11  N-2).  (2.44)  12  Chapter 3. The Covariant Path Integral and Modular Invariance  3: The Covariant Path Integral and Modular Invariance  3.1  The Covariant  Path  Integral  The Polyakov path integral formalism is essential for a proper understanding of string theory [10]. Here we investigate the the closed bosonic string path integral by analytically continuing the matrix ^  to the Euclidean signature metric g b- The path a  integral we are interested is given by  J DXDge~  (3.1)  s  where S is the sum of the action and the topologically invariant Euler character  S  =  ^bJ  d'a^daX^X,  +  d a^gR.  (3.2)  2  This is the only local functional consistent with Weyl invariance, two-dimensional general covariance and Poincare invariance. The Euler character is often denoted by X and and is related to the number of handles (genus) of the surface in question:  X= ^  J d o^R 2  = 2- 2 .  (3.3)  7  In addition, however, we also need to add a counter term of the form /j, f 2  d o\fG 2  to (3.2). This is required to compensate the regulator that is used to define path integrals of products of X^ at the same point. In D = 26 all anomalies cancel and the action is invariant under two-dimensional  Chapter 3. The Covariant Path Integral and Modular Invariance  13  coordinate transformations and Weyl transformations. We must therefore divide out the group volumes for the general coordinate transforms (diff-invariance) and Weyl invariance: V = VQCVW- This is in fact an oversimplified notation since the volumes are not quite independent. The proper path integral is then given by  r DX*. J  vw ~VG~C~  This integral was first evaluated by operator methods [11] but has also been determined via the path integral method [12] for the torus amplitude. For the torus amplitude it was determined to be  Z rus = T V26 J  ^e ^(27rr )-  13  4  t0  where r = Ti + ir , d r 2  2  1 2  2  = dridr , f(q) = TI^°=i(l  |/(e ^)|2  T  —  2  IS  4 8  (3.5)  the string tension,  and V e is the volume of space time. The one loop cosmological constant is just 2  ly™ • Of critical importance here is the fact that the integral is performed over the  z  fundamental domain F defined as  \  3.2  Modular  Invariance  < TI  < ^,  r > 0, 2  and the Torus  \T\ >  1.  (3.6)  Amplitude  The restriction to the fundamental domain can be understood by recalling the definition of a torus. Consider a parallelogram in the complex plane with sides represented by the complex parameters Ai and A (that behave as vectors in the complex plane). 2  A torus is obtained by identifying opposite edges of the parallelogram. If the plane is tiled with these parallelograms then we may identify equivalent points on the tori:  Chapter 3. The Covariant Path Integral and Modular Invariance  z ~ z + niAi + n A 2  14  (3.7)  2  for m = 0 , ± 1 , ± 2 , - - - . There is, however, a certain amount of redundancy here. Consider  n i A i + n A = n'-^X^ + n A' . 2  2  2  2  (3.8)  If, for a given A and for arbitrary n we can find always find a A' that satisfies this equation for a particular choice of n', then A and A' describes the same torus. To be more precise all A that are related by  (3.9)  A' = M A  M =  V  a  J  b  (3.10)  where M is an SL(2,Z) matrix are equivalent tori. The world sheet action is conformally invariant so, in fact, the absolute size of the tori is irrelevant, only the ratio of its sides is important. We thus define r = ^ Equation (3.9) then implies that all the relationships between all equivalent tori are given by  CLT + b  cr + d  (3.11)  A l l of these transformations can be generated by the two simple transformations  S:  T—y—  r  (3.12)  and T : T ^ - r + 1.  (3.13)  15  Chapter 3. The Covariant Path Integral and Modular Invariance  If we wish to parametrize the path integral in terms of unique tori then we must restrict r to a region where it is unique. One such region is the fundamental domain  -\<TI<\>  1-7-1 > 1.  2>0,  r  (3.14)  The remaining infinite number of unique regions lie within the semi-circle in the strip of the upper half plane  ~ \  <  T  x  <  h  *  r  >  °'  |  r  |  <  L  These regions are all equivalent and can be mapped to each other via (3.11).  (  3  '  1  5  )  Chapter 4. Strings in a Kalb-Ramond Field  16  4: Strings in a Kalb-Ramond Field  4.1  The Massless  States of The Closed  Bosonic  String  The massless states of the closed bosonic string are the graviton G^„, the dilaton <fi, and the Kalb-Ramond field (or antisymmetric field) B^.  To be explicit, the massless  states of momentum p in the light-cone gauge are given as  etj-aLioiilOsp)  (4.1)  where j = 1, 2 , . . . (D — 2) and e^- is the transverse polarization tensor. The states transform as a two index tensor under SO(D-2). We may reduce this representation into a symmetric traceless tensor, an antisymmetric tensor, and a scalar that do not mix under SO(D-2) transformations. Any tensor e  1  1  u  can be decomposed as:  1  1  [-(eij + en) - -^z^^ijtre]  + [-(e^ - )] + [jy^S^tre] eji  (4.2)  The three terms in brackets correspond, respectively, as: M  +  + [8v*]  (4.3)  which represent the graviton, the Kalb-Ramond field and the dilaton, respectively. Herein, we shall be mostly interested in B. The world sheet action for strings is the Polyakov action  17  Chapter 4. Strings in a Kalb-Ramond Field  l~bf  S=  °V99 \ud X»d X\  d2  (4.4)  a  a  b  In the presence of background fields corresponding to coherent superpositions of the massless states, the action is modified to  S  =  I^b  / d^VdWG^iX)  + ie B^{X))d X»d X» ab  a  b  + a'R(j){X)].  (4.5)  This equation will be justified in 7.1  4.2  Mass-Shell  Condition  and Level  Matching  We wish to consider the allowed momenta for a string in only a background B field where space time is toroidally compactified on a 26-torus of radius R. In Euclidean space we have  S  ^bJ  =  ^9 GAX)  d2(T  ah  + e^B^X^daX^X"}  If B^ is constant, the second term can be written as a total divergence u  (4.6) d (t B X^d X ). ab  a  u  llu  This term's contribution to the canonical momentum is zero except when X^ is nonperiodic. Therefore, to determine the allowed momenta we consider only the nonperiodic parts of X  11  and write  X* = ^ ( T ) + w^Ra + oscillators where the X  11  (4.7)  is an integer that represents the number of times the string wraps around direction. The action then becomes  (4.8)  b  18  Chapter 4. Strings in a Kalb-Ramond Field  The Lagrangian L for the system is just the integrand. We may form the canonical momenta  P» = ^{G ,x  + B ,w"R).  v  lu  (4.9)  tu  We recognize this momentum as a sum of the conventional momentum and and a field momentum contributed from B. Under quantization it is the canonical momentum that must be quantized and we thus require P =  We may now write the mode  M  expansion as  X" = x» + loRvf  + 2 r ( G " " V - ^ - B" w R) R  + oscillators  u  u  (4.10)  Since we have  X^(o,r)=X (o )+X (o-)  (4.11)  +  L  R  with  (4.12) X£(o ) +  = \x» + oltf o+ + £  fe~ ° 2in  L  +  We may determine the right and left momenta as  77  olpl = w^R + (G^a'-zj - B^w'R) R  (4.13)  and y  a  Tl  = -w^R + (G^a'-Z - B^uTR) R  For illustrative purposes, we consider a special case for non-zero B — B  (4.14) with X  1  1 2  toroidally compactified. The momentum conjugate to X * for flat spacetime (G^ = 1  u  Chapter 4. Strings in a Kalb-Ramond Field  19  n^) is  P" = -^r  /"W"  =  L  + BSd X"]da.  (4.15)  a  We write P " = p +P%- The first term here is the usual kinetic momentum p that 41  1  multiplies the term linear in r in the mode expansion of X . 11  In the X direction the left and right moving parts are modified to: 1  X {o~) R  = \x + {-wR + a'| - BwR)o~ + £ l  n  X (o ) l  +  L  = \x + {wR + oi\ - BwR)o l  +  +£  s^e-*™-  *°  (4.16) °*- -  2in(J+  e  The second term in (4.15) is the field momentum p . This term only contributes B  when X  1  X (T,O) 1  is compactified on a torus with radius R then we have X (T,O 1  + n) =  + 2'KRW. In this case  P7T  / d X do Jo  = 2irRw  l  a  (4.17)  and hence the field momentum is non-zero. The zero-mode Virasoro constraints are implemented as usual:  r  T  U =  y  L =- j 0  • 2 A-R dcr = 1  (4.18)  X  (4.19)  L  do = 1.  Inserting (5.26) and (4.16) into these constraints we obtain the mass-shell and level matching rules:  (4.20)  Chapter 4. Strings in a Kalb-Ramond Field  20  and  N-N  = wn.  (4.21)  Since the canonical momentum is the generator of translations, it must be discrete in the X  1  direction: P — ^. Also, we may replace the kinetic momentum with the 1  canonical one which changes the mass-shell condition to [13]:  ( P - 2*BR f 2  W  + £ p"p„ = (If ^—' R  ^1,2  + (^) a 2  + -V a  + N - 2)  (4.22)  This contrasts the the free-field string case in that there is now a shift in the mass.  Chapter 5. The Vacuum Energy  21  5: The Vacuum Energy  The vacuum energy can be calculated in essentially two different ways: via the operator formalism, or via the path integral formalism. Application of either of these methods will reveal that for an appropriate configuration the vacuum energy is a periodic function of B. The configuration we shall choose to investigate is two toroidally compactified dimensions with a nonzero B field in the component of those directions. The operator formalism is instructive, but the more correct method in this case is the path integral formalism. However, as discussed above, the operator formalism can be corrected by performing the restriction to the fundamental domain. We will proceed in this manner.  5.1  Vacuum  energy of the uncompactified  string  We may determine the vacuum energy by evaluating the vacuum persistence amplitude in the euclidean formalism.  Z=  (0|e — HT |0> = e -E T  (5.1)  0  For the scalar field <j> in D non-compact dimensions we have  Z = j D<t>e~ = j D<f>e-y s  dDx  ^+™ )* 2  = -|rn„(-^W) e  (  5  2  We may evaluate the trace as the sum of the eigenfunctions and obtain the vacuum  )  22  Chapter 5. The Vacuum Energy energy £  ° W(3 r =  -  j i n ( p 2 + m 2 )  (5  3)  where V is the spatial volume of D — 1 dimensions and T is the temporal length or time. A convenient representation of the logarithm for an arbitrary operator O is  Hd) This can be verified by repeated differentiation and by observing that ln(l) = 0.We see that the divergent part is independent of O and can neglected.  Defining the  vacuum energy density as A — ^ we find that  This is the vacuum energy for scalar field on an uncompactified flat space.  For  bosonic strings, we should sum over the masses of the spectrum while satisfying the level matching condition (4.21). That is, the vacuum energy becomes  with m = £(N + N - 2). We may enforce the level matching condition (4.21) with 2  a delta function. /•1/2 ;l/2  / d e ^ - ^ J-1/2 - -1/2  $N-N  2  N  (5.7)  N  n  and then trace over N and N:  A  =  ~ \  (^v)  /  d r  i 2r - e 1 4  d r  2  4 7 r T 2  ir(e  2 7 r i T A r  -  2 7 r i f  ^)  (5.8)  Chapter 5. The Vacuum Energy  23  where we have defined r = X/ct'ir r = Ti + zr . 2  2  For the evaluation of the trace we recall the relations (2.36) and (2.37) and define q= e  27UT  . Now we note / t  r  (  g  E -  =  1  - U  m  )  oo  \  24  T"[_^_  =  \m=l  .  ^  ( 5  .  9 )  /  The factor of 24 comes from performing the trace over the transverse states. We now define the Dedekind eta function as oo  T}(q) = q ]l(l-q )1,U  (5-10)  m  m=l  In particular, then, we have  y y ™ » *  f'  2  =  dn  ^ ( e ^ p  4  (5.11)  8  and finally:  The integral here ranges from —1/2 < T\ < 1/2 and 0 < r < oo. However as we 2  have learned from the path integral formalism we should in fact truncate the integral to the fundamental region —1/2 < T J < 1/2, r > 0 and abs(r) > 1. The physical 2  reason for this is that the above is effectively a quantum field theory calculation and that we are over counting states. From the path integral perspective, we are calculating the torus amplitude. The Tj correspond to the Teichmuller parameters that characterize an arbitrary torus. However, some different Tj correspond to the same torus and we must only count each unique torus once. Hence we only integrate over the fundamental domain.  24  Chapter 5. The Vacuum Energy 5.2  Vacuum  Energy  with one compact  direction  In a configuration with one toroidally compact direction X we make the identification 1  X ~X 1  + 2nR  1  (5.13)  Because the momentum is the generator of spacetime translation, the momentum P  l  must be discrete  P  = R  1  n = 0,±l,±2,--- .  (5.14)  Additionally, there exists the winding modes, so we must include modes that wrap around the compact direction. Denoting the number of wraps by w, we write the mode expansion as  X = x + 2oRw + 2 a ' + iJjJ2~ 1  + oie~ ^)  l  2in  .  (5.15)  Evaluating the zero mode Virasoro generators L = L = 1 we obtain the mass-shell 0  0  (for the 25- and 26-dimensional mass - see section 7.2 for their definition) and level matching conditions  m.  _a ' / „.r  =  r»~r _ _ \_ n _ _ R w 2  +  ( A  A  ™  a',„  26  f  2  ) +  ,-  ( i V + i V  N-N  r  2  +  ^  -  Rw 2  2 ) +  = nw  w  2  (5.16)  '  (5 17)  (5.18)  25  Chapter 5. The Vacuum Energy with oo N  = Y1 ( -n nn a  (5.19)  + Oi\ a )  a  n  nl  1  71 =  00  N =  J2 ( a - n Q V + c t  n  a  n l  (5.20)  )  71=1  where n here excludes the compactified dimensions  X. 1  To convert (5.12) into the correct form 26-dimensional vacuum energy for a single toroidal compactification, we replace the integration f dpi with a sum ^ J2  -  ana n  also include the winding modes in the mass by replacing m with m 6- The vacuum 2  energy is then modified to  ^  w,n  '  F  (5.21) where D = 26.  5.3  Vacuum  Energy  with many compact  We now consider the case of M xT . d  k  directions  In this case use label index m for the compact  directions and write the mode expansion as  X  m  =x  m  + 2oL  m  +  2a' T m  P  «^  +  2  i  <  T  ~  a  )  + o^e~ ^ ). 2in  a)  (5.22)  It turns out that the are various ways one can consistently choose compactified momenta. To make this manifest, we use the left and right momenta.  a'p™ = L  m  + a'p'  (5.23)  26  Chapter 5. The Vacuum Energy  a'p% = L  - a'p  (5.24)  m  m  with  X»(a, T) = X {o )  + X (a~)  +  L  (5.25)  R  (5.26) -2in<r  +  n#0  The zero-mode Virasoro constraints now give a' N - N = - (p| -  p  y .  (5.27)  where the boldface indicates a vector over the indices m. The left hand side of this equation is an integer and therefore we must demand that the right hand side of this equation be integers. The modification here contrasts the T compactification 1  because the periodic coordinate identifications we make need not be at right angles. For example, we may identify points as  X = X + 27rv o7E /  Where the basis vectors E  m  (5.28)  are complete over T . Thus we may write k  m  L = Vo7w E . m  m  (5.29)  In order to make sure that the momenta are the generators of translations we write  p = Va7n e m  m  (5.30)  Chapter 5. The Vacuum Energy  27  where we have formed the reciprocal lattice vectors ei  e • E = 6pq. p  (5.31)  9  In this notation we may write the mass shell and level matching condition as  m 2  D  d  = j(N  + N - 2) + p + ^  (5.32)  2  N - N = n •w  (5.33)  We will typically be interested in the case where the basis vectors Ej are all orthogonal and equal to R in length. In this case we have simply  ^ -  d  = y(iV + iV-2) + ^  + ^  .  (5.34)  Now the cosmological constant is adjusted to  l( ')(*-£)/2 a  w,n  dndn -( + -*)/ -°'' *«*> (*>')- * """ D  r  5.4  Vacuum  Energy  !  2  !  1+  2  i  \ (q)\-  1  K  e  with 2 compact  directions  v  and a B field  In the configuration where two spatial dimensions are compactified (say X  and X )  1  and a non-zero (and constant) B-field is permitted only in the B  i2  2  = —B \ compo2  nent, an unusual feature appears: the vacuum energy becomes a periodic function of B. This configuration will be the central focus of what follows, as it can give rise to a unique topological structure.  Chapter 5. The Vacuum Energy  28  In the free field case, two compactified dimensions have a cosmological constant of  ^  '  w,n  (5.36) where D = 26. However, if we turn on B = B\ = —B \ we must now incorporate the 2  2  momenta shifts as discussed in section 4.2. The mode expansions for the directions that are modified display the momenta shift (the coefficient of 2ra'):  X  = x + 20-Rw + 2r(a'— - Bw R) R  1  l  1  2  n = x + 2oRw + 2r(a'— + Bw R) XL  + oscillators  (5.37)  + oscillators  (5.38)  2  X  2  2  2  l  Thus the cosmological constant is modified to  '  K  .  w,n  - Q ' r 7 r ( ^ ( r - * « ) 2 ) + ^-(n2+*u;i) +(^) )-27riwnT ^^-48 2  e  2  2  2  n  1  ^  We have defined the flux as $ = 2-K^-B. Note that the level matching condition is left unchanged after modifying the right a left momenta. While it is not immediately apparent, (5.39) is periodic in <5. To make the periodicity manifest we must perform some rearrangement. Consider the component  Q =  -Q T27r(^(ni-$ ;2) + ^(n2+$wi) -K^f ) )-27riw-nri _ ,  e  2  tl  We shall now perform a Poisson re-sum:  2  2  (5.40)  Chapter 5. The Vacuum Energy  29  OO  dP  e  2  /  •oo  2 T i k  ' e ' p  _ a  T 2 W  (^  ( p i  ~*  W 2 ) 2 +  ^  ( p 2 +  *  W l ) 2 + (  ^ ) ^ ' ) 2  _ 2  w  p T l  .  (5-41)  k  Define the dual of w as w = (—w,Wi) and shift p —» p + w $ and perform the 2  integration over p:  R  2  ^  -^(fci-^iri) -^(fc2-^2T ) - 'r 7r(^f)^ 2  c  2  1  Q  2  2 7 r i k  .w^  a'T a'r2 2  k  and insert this back into (5.39) and we obtain  9  , = ^ - ^ [ ( ' . - " ^ ^ - w l - ™ ^ ^ . ^  ( 5  4 4 )  w,k  Immediately the periodicity in $ is manifest. Further, if gi is a rapidly decreasing function for large negative and positive I we find that the most important contribution to the periodic function is the first harmonic. We thus write the cosmological constant as  (5.45)  AICOS(2TT—B).  a'  In summary, we have found that the cosmological constant for T x M 2  2  4  immersed  in a constant B-field is periodic in B. Obtaining the value of A i is not trivial. On dimensional grounds A i must scale as (a')~ l . D 2  We also expect A i to tend to zero  as R —> 0 (this is the no compactification limit and there should be no periodic term). Further, since A i arises from the torus amplitude, we expect there to be no g dependence. A dimensional estimation then yields s  30  Chapter 5. The Vacuum Energy  (5.46) A more careful analysis involving the method of steepest descent modifies this by adding an exponential suppression. Consider the term in the exponent found in the cosmological constant  -7r^-C/(ri,r ) = cr 2  Uki - Win)  + (k  2  a r  2  w Ti) } 2  -  2  -  T  7 T +  a  2  2  2nik • w $ . (5.47)  By using 'Polchinski's Trick' [14] we may eliminate w in favour of integrating over 2  the full strip, and not just the fundamental domain. Then we may trade integrating over Ti = [—1/2,1/2] to T\ — [—oo, oo] but summing only over ki < w\. Minimizing U with respect to T\ and r we obtain the extrema, respectively 2  (5.48) Wl  (5.49)  U = The exponent at the extremum now becomes  R |& u>i| + a' 2  -2-7T—  2  2irik wi$.  (5.50)  2  Now we may proceed to write  U{n,T ) 2  = U(ti,t ) 2  + ]-Ui i ( n - h)  2  t  + \u  2 a  (r  - t)  2  2  2  + U  h2  ( n - h) (r  2  -1 ) 2  +  (5.51) and perform the integration over the upper-half plane. Noting  Chapter 5. The Vacuum Energy  oo poo I - {x-xtf-b(y- ) -c{x-x ){y-y ) / ooi-oo 2  e  a  yo  0  0  dxdy  =  31  n _ _ ^ VAab-c =  =  (553)  We find that the dominant periodic term behaves as  R  2  A COS(2TT—) 1  a' r\i'  where  (5.53)  32  Chapter 6. Elimination of the Tachyon Contribution  6: Elimination of the Tachyon Contribution  While the discovery of a periodic cosmological constant is rather interesting, it is mired by the fact that the constant which multiplies the periodic term is in fact divergent. We must eliminate this divergence to make a physically reasonable interpretation of the vacuum energies. We may proceed in two ways.  6.1  Spectrum  Truncation  The tachyon contribution comes from the N = N = 0 modes.  In evaluating the  vacuum energy we traced over  -2niTN+2irirN  e  |/( )|- . 4 8  =  g  ( .1) 6  This causes the integral to diverge when N = N = 0. One way to remove the tachyon contribution, is to simply subtract off the N = N = 0 contribution, which amounts to replacing  i/(9)r  4 8  -H/(</)r  4 8  -i  (6-2)  or equivalently  b7(g)p  48  \v(Q)r 48  e  4?rT2  (6-3)  Note however, that the modular invariance of the vacuum energy is now spoiled since r —> r + 1 and r —>• — ^ is no longer a symmetry. The value of the cosmological  Chapter 6. Elimination of the Tachyon Contribution  33  constant has been carried out numerically for this scheme [15].  6.2  Relative  Subtraction  Perhaps a more reasonable (i.e., modular invariant) way is to look at the change in vacuum energy with respect to the non-compactified, field free vacuum energy.We define the renormalized cosmological constant as  A = A{B) - A(B = 0). r  (6.4)  We should note, that there is some ambiguity in the choice of regularization. In reality, any contribution to the vacuum energy should be observable due to its coupling to gravity. Bosonic string theory is pathological in this sense and cannot be remedied. However, for exploratory purposes we must choose a method of regularization so that we may have tenable results. Further, the contribution to A i is finite, so we may proceed by considering this term only.  Chapter 7. String Theory Effective Action  34  7: String Theory Effective Action  The path integral formalism is a general prescription for quantizing theories with classical actions. The key element of this method is the vacuum amplitude Z. The vacuum amplitude is formed by taking the weighted sum over all possible classical configurations. For strings, the different configurations are the different worldsheets. The weight is given as e~  s  where S is the action for the worldsheet.  S is the  area of the surface multiplied by the dimensional parameter T which produces the dimensionless action plus some additional terms. One of the additional terms (as discussed in section 3.1) is the Euler character. The value of this term is set by the number of handles in the world sheet and thus becomes the coupling constant g  s  (the string Feynman diagrams are weighted such that each additional loop provides a weight of g ): s  oo  (7.1) genus h=Q  Consider the worldsheet depicted in figure 7.1. Here we have a closed string that propagates to another closed string. Consider, now, the more complicated worldsheet shown in figure 7.2. Here we have a closed string that propagates, splits into two strings, and recombines in to one string. This diagram is analogous to a one-loop Feynman diagram. The relative weighting of these diagrams is set by the expectation value of the dilaton though the Euler character. There is no way to guess at a value for the coupling constant and we must therefore introduce it as an arbitrary string coupling  Chapter 7. String Theory Effective Action  35  constant that fixes their relative weights of the two diagrams. More precisely we see that $ — > $ • const, sends g -» e g .  However, while the relative weighting of two  const  s  s  diagrams is arbitrary, once we choose a value, the relative weight with respect to a third diagram is fixed. We shall be interested in topological features of the massless Kalb-Ramond field. The most convenient way to deal with these features is through the use of the effective action. The effective action method transforms the world-sheet perspective to the spacetime perspective. The effective action is the spacetime action for string theory. By this, we mean that is should generate the spacetime S-matrix elements of interest. The ambiguity of the relative weighting of string diagrams is also apparent in thee effective action. It is invariant under g —> e g ,  — 2const.. We see that  const  s  s  the constant part of the dilaton shifts g and in a sense, g is set by the vacuum s  s  expectation value of the dilaton.  7.1  The Effective  Action  as the Spacetime  Action  The Polyakov action (4.5) is the worldsheet action for describing a string moving in a gravitational, Kalb-Ramond, and dilaton field. There is, however, an alternative way to examine strings propagating in background fields: the string effective action. The effective action formalism in string theory provides and spacetime perspective for  Chapter 7. String Theory Effective Action  36  Figure 7.2: One loop string diagram  strings in background fields. The complete derivation of the string spacetime action is straightforward, but somewhat involved [16]. However, the motivation is brief. The equations of motion for the internal metric (2.15) is traceless when the world sheet is two-dimensional:  T% = ^G» {d X»d X»  -  a  u  a  -5 d X»d X )  l  a  c  ~ (1 - |) = 0  n  a  c  (7.2)  where q = 2 is the dimension of the worldsheet. This relationship is anomalous. That is, after quantization this condition only holds in special circumstances. The vanishing of the trace corresponds to maintaining conformal invariance in the quantized theory. It turns out that in order to guarantee the disappearance of the anomalies three differential equations must be satisfied:  0 = 0%, = a%„  0 = P% = ~  + 2G/V^V„<S> - -H^Hj*  V i/ „„ + a\V <S>)H , x  A  x  A  2  2  2  2  + 0{a' ) 2  u  0 = £* = - ^ V $ + a'(V$) - ±-a'H  + 0(a' )  + 0(a' ) 2  (7.3)  (7.4)  (7.5)  37  Chapter 7. String Theory Effective Action where  is the covariant derivative. These three equations are linear combinations  of the equations of motion which can be derived from the action  s  =h I ^~* dDx  e  2  \ J 2{D  ^ ^ ^ + ^ ^+°^')  6) +R  H  3  H  X  2d  d  (7.6) Specifically, we have  SS =  \8G^ (B ^  [ d Xv^Ge- * D  2K, a 0  2  -  g  J  i  + 5B^ (8 ^)  \  z  + 5$ (2$  B  -G^{B^  l  - 40*)) j  [1-1)  - 80*)] = 0.  p  The action (8.5) is the effective action for the low energy spacetime fields. It can be shown that using this action to generate S-matrix elements is equivalent to generating S-matrix elements from the standard operator formalism (for one-loop closed strings coupling to massless fields). Equation (8.5) will be of central focus in what follows.  7.2  Kaluza-Klein  Theory  Kaluza-Klein theory is a classical attempt to unify electromagnetism with gravity [17]. The original idea was to consider gravity on a five dimensional manifold where one of the directions is periodic. In this section we let the compact direction be specified by y and the non-compact directions by x^. The manifold is M x S 4  l  with  0 < y < 2nR where R is the compactification radius. The 5-D Einstein action is  * = 7T2 [ d x^f R  S  5  5  ZK  b  5  (7.8)  J  We may parametrize the metric any way we wish and we choose the following:  Chapter 7. String Theory Effective Action  38  (7.9)  Here, 0 is a scalar, A^ is a vector, and  is to be the usual 4-D spacetime metric.  Owing to the periodicity of the fifth dimension we may expand <f> as a Fourier series oo  <Kx,y)= YI  U*V  (7-10)  nvlT  n=—oo  and similar expansions for the other fields. If we substitute the Fourier expansions and the parametrization of g  5fll/  into (7.8) and integrate over y only keeping the n = 0  terms (<f> = </>, etc.) we obtain (after some algebra, the details of which are discussed 0  in the appendix)  hj  S=  d  '  x  ^  R  - \ ^<t>  - \e-^F, Fn  d  (7.11)  u  where the field strength is defined as F^ = <9 A, — d A^ and K = KI/2TVR. We have 2  M  u  discovered that gravity on M x S is equivalent to gravity on M plus electromag4  1  4  netism plus a massless scalar field. The presence of the dilaton 4> was troublesome at first, but in modern times it has become an important theoretical element. Kaluza-Klein theory also has a mechanism for constructing massive fields out of massless ones. Consider the free massless scalar propagating on R x S : 4  d J(x,y) 2  = 0.  l  (7.12)  If we plug the Fourier expansion for <j> in we obtain an infinite number of uncoupled equations of the form  (d^  - ml)<f> = 0 n  (7.13)  Chapter 7. String Theory Effective Action  39  where m = j^. To a four dimensional observer (who defines mass as m\ = p ,— p\ — 2  2  p\ — pi) these modes appear as massive scalar fields. In regards to these modes, one speaks of the Kaluza-Klein tower of states. Of course, the five dimensional observer still defines mass as m\ = p — p\ — p\ — p\— p\. 2  a  7.3  Compactitication  of the Effective  Action  The effective action discussed in the previous section was for D = 26. We wish to analyze the effective action on the space M  x T . We do this by way of Kaluza-Klein  2 4  2  compactification. Parametrize the metric as  ds = G^ dx dx 2  M  = G^dafdx"  N  N  + G {dx  + A™dx»){dx  m  n  mn  + A^dx ) 11  (7.14)  Then, the effective action can be dimensionally reduced to  S, = 0 k  / d xy/=G~ e- ** [R + 4 c ^ d " $ d  2K mn  m  d  d  0  - \G G™ - \ G  2  d  n  (d.G^Gn, F $  + d,B d B )  715  nq  F ^ - \G H H^  m  ( )  u  mp  -  mn  miw  ^H^H^]  where the following definitions have been employed:  $  d  = $ _ I _ det(G ) mn  F$  = d,A^  H  = F^  m  m  - d A^™ v  —B  F^  n  (7.16)  (7.17)  (7 18)  40  Chapter 7. String Theory Effective Action  FuJm ~ ^ 4 1  4 m = B,  m  H^x = d,B  )  m  m  = B, + \A^ A^l  - \A^ A^l m  v  (7.20)  mn  - \ ( 4 ) ™ F% + 4 M  m  v  (7-19)  + B A^>  1  uX  B,  ^^Lm  ) + cyclic perms.  - A^ B A^ m  n  mn  (7.21)  (7.22)  the details of which are explained in the appendix.  7.4  Adding  the Cosmological  Term  The general form of the partition function is: p  OO  Z=  J2 9 s ~ / DxDge- ^ \ genus h=0 2  h  2  Sp  3  (7.23)  Tree level amplitudes correspond h = 0 and the torus amplitude is h = 1. Alternatively, we take the effective action as the spacetime action. The effective action will then generate the string theory S-matrix. The effective action is invariant under  c 9s -> 9 e  2c  s  for constant c. Hence we can always absorb the string constant into the dilaton. We expect the same of the effective action (which must reproduce the same S-matrix). Again, the torus amplitude goes as g®, so we expect the following form for the effective  Chapter 7. String Theory Effective Action  41  action with the inclusion of the vacuum energy:  S =S  where we may replace 2k = (a') g 2  l2  + j d xv^GA(B) D  0tk  2 s  (7.24)  in So,k- Any uncertainty in the relative size of  A term from the rest has been absorbed in g  s  We should note that the vacuum energy calculation was performed for a constant B.  However, here, we assume B is not constant. We must be in a regime where  treating B as non-constant is applicable.  42  Chapter 8. Equations of Motion  8: Equations of Motion  8.1  Equivalence  of String  Frame  and Einstein  Frame  To touch basis with conventional gravity we shall now demonstrate the equivalence of (8.5) and the Einstein-Hilbert action [2]. Consider the action  =h I °  s  d  ~  xV=Ue2$  (8.1)  This is the Einstein-Hilbert action coupled to a scalar field $. In D = 4 we identify K with Newton's gravitational constant GN- K = 8TTGN- We shall show that this is 2  equivalent to the string effective action and vice-versa. Under Weyl transformations of the metric  G>„ -> e ^ G V  (8.2)  2w  the Ricci Scalar Transforms as [18]  R _> - M « ) (R _ (D - 1) - (D - 2){D - 1)8^8^) e  2  (8.3)  and  G - ) • e ^ G. 2D  (8.4)  X)  $  If, for constant $o> e define K — K e w  0  0  0  — $  and choose w = 2 — — — , this transforma-  Chapter 8. Equations of Motion  43  tion turns the action () into the string action  S = -L [ d xv^Ge- * 2«o J D  [R + Ad^d^}.  2  0  (8.5)  This form of the action not as conveinient for determining the equations of motion as (8.1). We have learnt that string theory predicts general relativity coupled to a scalar field.  8.2  Variation  off the  Action  We wish to find the equations of motion for the doubly compactified action  5o, =  f d xV^G~ e- ^ d  2  [R +  2  d  - \G G™ mn  {d,G d G u  mp  nq  d  +  Ad^ d ^ u  d  d  d^B^Bng) (8-6)  + J* dPx  VGA.  Write S = SGD + SBFG + S\, where SGD refers to the terms on the first line, SBFG refers to the terms on the second and third line, and SA refers to the final line. The cosmological constant term is constructed from  A = A i ( l -cos(AB)).  (8.7)  We now perform the variation of the action to obtain the equations of motion for 9nu-  SSQD  5G^  V=G~ e- ** {o.') 2g 2  =  d  l  2  Rd^ + 2 D „ A , - ^G^(R  d  + AD^$  -  Ad^d^)  (8.8)  44  Chapter 8. Equations of Motion  SSBFG  _  SG^ 1 V^G~ 2(c*') <? 1 /~i  2<  rpm  ~~^G r mn  rpnpa  t  p(J  2*  1 p,a  r  G  G  h  m  (  q  ^ rvmn TJ  — -G  D  P  TJH pa  p  a  H  n  G  M  P  D  - —  P  ^  G  N  +  G  TJ  dB dB ) P  p  TjpaX llpa\tl  mp  nq  (8.9)  H  QmnQpq (d^GmpduGnq + d^Bjnpd^Bng)  rf  rpn a  '2 -^mn-T  mn p  G p,u  d  12  v  \/-vmn  TJ  r, ^  • mfj.\ ni'  — T-f  TTX  n  TJ P  A  A ^pa v  rl  n  n  (8.10) The first of these three terms is obtained most easily from the Einstein-Hilbert form of the action (8.1). In this case the variation of the Ricci scalar can be carried out in the usual way [19]. The equation of motion for G^ is then v  (8.11)  ;{SGD + SGBF + 5A) = 0.  6&' For G  r  5SGD t  (8.12)  5G < r  SS BFG  8G  r  ( iy2g2~  [ -^G (dpG d G pq  a  +d B d B )  p  mp  p  nq  p  rnp  nq  (8.13) d^aG^d^Gn, -L.-C ,Q > ^Jnl^Jom  -  F° F P pa L  1  1  JQ^  -  A  -  (G G id»G )^ ol  p  nq  G G, lp  t  -JJ JJP° ^ mpa 1 1  1 1  n  -^V-G~dV- D-dGmn^G  (8.14)  Chapter 8. Equations of Motion The equation of motion for G  is then  mn  5  (S  +S  GD  5G  r  For $  45  + S ) = 0.  GBF  (8.15)  A  d  = "C?i2~~r SS  [ - 2 ^ + Sd&d***  - SD^ ] d  (8.16)  V^G~ e- ^ 2  BFG  =  d  6$  (a') 0? 12  d  [-\G G™  '  1  (d,G d G  mn  + dB dB )  u  mp  - iG F$ FW »" m  li  mp  8  nq  _ \G H ,H^  n  -  mn  mn  (- )  u  nq  m(U  17  ±H^ H^} X  t =°  <-> 88 1  The equation of motion for $^ is then  (S  5®  GjD  +S  G B F  + 5 ) = 0.  (8.19)  A  d  For B  5S p _ 5B ~ G  Q  mn  J E ^  = i f f ? ^  (e-^G-G^B ) oq  0 SB  GG lm  { m , n } ^ {1,2}  r  AA  sm(BA) y/^G~ x / G ^  0  The equation of motion for B  d  mn  is then  {m, n) = {1,2}  Chapter 8. Equations of Motion  -(S  +  GD  5B  mn  OGBF  46  + S) = 0  (8.20)  A  We are interested in static solutions about flat space with all fields flat except for B = Bi2(X )  and therefore the additional equations of motion do not couple in this  3  regime. The equation of motion for B  then leads us to the equation of motion for the  mn  celebrated sine-Gordon equation:  dlB{X )  1  - A A i sm(B(X )A)  z  W 9.  3  =0  (8.21)  2  Unfortunately, allowing non-zero B means we must consider the other equations of motion that B couples to. We will resolve this shortly, but to do this we must see how (8.21) scales with g  s  Equation (8.21) admits a first integral. We multiply by 83B and integrate to find  [B'f + A cos(£,4) = c  1  x  2(a'Y 9 2  2  (8.22)  The constant is determined by requiring B'—O and BA — 2-KU at spatial infinity: c = A i . And thus  (B') + 2(a') ^ Ai(cos(i3A) - 1) = 0. 2  12  2  (8.23)  This equation can be solved (as will be done in what follows), however, the most important feature is that (B ) 1  2  ~ g . Since equations of motion for the effective action 2  satisfy the flat solution, we may consider the sine-Gordon solution as pertubative solution in g . Since (B ) 1  s  2  ~ g , to lowest order, we may ignore the coupling of B in 2  (8.11, 8.20) and consider (8.21) alone.  Chapter 9. Sine-Gordon Solitons  47  9: Sine-Gordon Solitons  9.1  Two Dimensional  Sine-Gordon  Solutions  The sine-Gordon model is well known from field theory and many other fields [20]. It is one of few important example where various analytic results can be obtained. The sine-Gordon model exhibits various interesting solitonic solutions. We examine the model here in (1 + 1) dimensions with a (-1—) metric. The Lagrangian density is constructed from a kinetic term and a periodic potential  The action is obtained by integrating over the two spacetime dimensions:  S = j dtdxC  (9.2)  The equation of motion for this Lagrangian follow as the extremum of the action:  a <^+^=sin(—0) = 0.  (9.3)  2  rn  y/X  Y J  The naming of the constants m and A is clear if we expand out the cosine in the Lagrangian to reveal  C= -(d<l>r- -m^ 1  1  +  ^  + ...  (9.4)  Chapter 9. Sine-Gordon Solitons  48  We recognize m as a mass term (of dimension one in D=2) and A as a coupling constant (of dimension 2 in D=2). We may obtain a solution by multiplying the time-independent version of the equation of motion 9.3 by ^ 0 and integrating. Choosing boundary conditions such that at spacetime infinity d (f)=0 and ^<j) = 2irn for integer n we obtain the so called x  soliton solution Am <f>{x) = ^ a r c t a n ( e vA i  ± m x  ).  (9.5)  the length scale of the soliton is thus given by £ = 1/ra. To determine the energy of this solution we appeal to the canonical formalism by forming the the Hamiltonian density  U = pd <t> - U t  V = ^ d (p  (9.6)  (9-7)  0  which yields  E = J dxU = jdx^(d(f>) - ^ - ^ o s ( ^ % ) - l j . 2  (9.8)  Applying this formula to our solution (9.5) we obtain the energy of the soliton:  E = 8^-. A  (9.9)  The sine-Gordon model also admits a time-dependent version of (9.5) which may be obtained by performing a Lorentz boost on x. Likewise, its energy is increased by the Lorentz factor 1 / V l — v . There are also various superposition solutions. For 2  example, the so-called soliton-antisoliton:  Chapter 9. Sine-Gordon Solitons  4>{x) = -= arctan( ) ). VA v cosh(mx/y/1 — v /  9.2  B-field  v  ;  49  (9.10)  Solitons  We are considering the configuration where we have two compactified directions A" , X 1  and a non zero B field in that various only in a third direction B = B^ix )  2  3  —B2i(X ). 3  =  To relevant order in g we may then write the action as s  dX  { ~(d B)  26  W9 2  2 s  /  where we have defined Y = (a!) g Ai. l2  - Y(l - cos(AB)) }  2  3  (9.11)  We my proceed as in the previous section  2  to obtain the energy an size of the domain wall:  B = j avctan(e  )  (9.12)  ±AVVx  e = —^=  (9.13)  AVY  y^2W E -  where the integral is over d X 22  r-  / . A S (a') g. V 1 5  A  j dX 22  (9.14)  s  = dX dX 4  • • -dX .  5  25  Using the estimation for A i  from (5.54) we obtain  a' /Ri\ e ^> 2nRg. {  (9.15)  V2l6n _ «i 2  r  :—; e ^2"' (d) Rg. Since we have assumed ^ > 1 we find that £ is large and E is small. The depen u  Chapter 9. Sine-Gordon Solitons  50  dance of E on g is inverse and that is somewhat peculiar. Note the similarity to the s  physical D-brane tension, which has the same g dependance [2] s  ^  (27rv o7) /  I6a' g b  s  for a p dimensional brane.  11 p  (9.17)  Chapter 10. Topological Indices and Vacuum Manifold  51  10: Topological Indices and Vacuum Manifold  10.1  Topological  Charge  We now take a brief digression into topology. Topological defects have become of considerable interest in recent times in field theory, in string theory, and in cosmology [20] [21] [22] Consider a (1+1) scalar field theory with Lagrangian  £ = i ( d , 0 ) - u{4>) 2  (10.1)  where the potential is given by m / ,vA . U= — \ cos(—(£) - 1 . A \ m 4  .  10.2  Classically there are many ground states corresponding to the to minima of the potential  4>o = 2nn-^=. vA  (10.3)  However, the equation of motion also admits another solution  A 'YY)  (j)( ) = _ a r c t a n ( e vA x  which has finite energy.  ±m(l  -  Xo)  ).  (10.4)  For all finite energy solutions we expect the asymptotic  Chapter 10. Topological Indices and Vacuum Manifold  52  values of <> / to take on the values m 27rni —=  <f>{x) = {  x =  —oo  x =  +oo  m  27rn —F= •y/X 2  .  (10.5)  Consider the following integral: /•OO Q 1 \/X f°° d I —<f>dx = rii — n = A n 2n m ./-oo dx  (10.6)  2  where A n = 0 corresponds to the true vacuum and A n = ± 1 corresponds to the soliton and anti-soliton. We see what appears to be a charge that characterizes the solutions. This motivates us to define the topological charge  Jn - ^-—e^r*  (10.7)  This current is conserved identically (i.e., independent of the field equations)  = 0  and its zeroth component is the topological charge  Q  j j dx = A n .  (10.8)  0  We should note that Q does not generate a continuous symmetry transformation, i.e., it is not a Noether charge. Regardless, it is a charge that is conserved and this indicates that the soliton is stable.  10.2  Modularity  of B  Note that if (j> is a periodic variable then only Q is relevant, not ni, n . For example, 2  since B^ is a gauge field V  -»  + d,j.x  v  -  d„Xn-  (10.9)  Chapter 10. Topological Indices and Vacuum Manifold We may change gauge with Xv = | ( ^ X  — S^X ). This shifts  1  2  SB^^ciSlSl-SlSl).  53  by  (10.10)  That is we can shift the field in the 1-2 direction and leave the other directions unchanged and it is the same physical state. This will have important consequences for the B-field solitons. Consider the term in the Polyakov action associated with the B-field  4^G7>J  °[z AX))daX»d X }.  Sb=  d2  abB  (10.11)  v  b  For the path integral to be invariant under gauge transformations, this term must change by 2nni, for n 6 Z . We must be careful about the fact that we are on a compactified manifold. Under the gauge transformation (10.9) SB changes by  - J <Po-[e d XvdaX"d X '} ab  AS  F  2ira'  i  =  b  J d a[d u{d X^d X 2i\ot i 2^ J °l Mdi( "d2X") 2  l  lxX  d2  - diX^X")}  v  7  d  (10.12)  1  li  2  - d (X»d X»))}  x  2  1  (10-13) (10.14)  since d^Xv is a constant we may use Green's theorem to turn this integral into a line integral  AS  F  =  2^a7 j  X  '  d  X  2  ~ * X  d  X  l  ( 1 0 - 1 5 )  Since the string must wrap around the compact directions a discrete number of times and the momentum modes are also discrete, we find the equivalence between physical states  Chapter 10. Topological Indices and Vacuum Manifold  B ~  10.3  Unstable  Topological  54  (10.16)  Configurations  Ordinarily one would expect the sine-Gordon soliton to be stable: It has a topological charge that commutes with the Hamiltonian [20]. However, we have been dealing with an effective field theory where other modes have been 'integrated out'. These modes can change the topology of the configuration space and permit a decay [23, 24, 25]. Figure illustrates two different homotopy classes corresponding to the vacuum (Q = 0) and the soliton (Q = 1). The vacuum cannot be continuously transformed into the soliton (and vice versa) without passing through the forbidden region. In reality, though, the problem is not two dimensional. Through coupling to other modes the problem introduces a third dimension perpendicular to the plane. In this case, the forbidden region becomes a hump barrier. Loops around the hump can be deformed to Q = 0 loops by dragging them over the hump. This of course has some energy cost. The hump will be modeled in the next chapter.  Chapter 10. Topological Indices and Vacuum Manifold  55  Figure 10.1: Homotopy classes. The black region is forbidden. The two loops correspond to the vacuum (Q = 0) and the soliton (Q = 1).  Chapter 11.  Stability  56  11: Stability  11.1  The  Bounce  The decay of false vacuum states has been discussed extensively in the context of homogeneous states [26, 27, 28] as well as in the application to domain walls [24, 25]. Consider a (2+1) scalar field theory with action  (11.1)  where U(<f>) is a potential with two uniform (constant field) extrema: one local minima at (j) and a global minima at </>_. We wish to consider the probability for +  decay from (f> to +  The relevance of false vacuum decay to cosmology is important. In the early universe the energy density was very high and was not nearly a vacuum. The universe may have settled into a false vacuum </> and may decay toward the true vacuum </>_. +  There are also more terrestrial issues where false vacuum decay is important. For example, the nucleation process in a supercooled fluid as occurs in cold clouds. Physically the process is facilitated by quantum fluctuations. A fluctuation bubble of 4>- is formed in the (f> state. If the energy decrease from the volume of the bubble +  exceeds the energy increase due to the wall of the bubble, the bubble will rapidly grow, consuming the  solution.  In the semi-classical approximation, the decay rate per unit volume per unit time for a tunneling particle process is given as [27]  Chapter 11.  Stability  r  + 0{h)).  v  Ae- ^{l B  57  (11.2)  where (11.3)  B = S (<j>) E  S (j> ) E  (11.4)  +  and 4> here is the bounce solution. The bounce solution is so called because after forming the euclidean action, the sign of the potential is reversed. Hence the solution corresponding to tunneling through a barrier is actually a solution that slips down the inverted barrier, stops at the other end, and returns to its starting point. If the bounce has 0(3) symmetry then we may convert the action integral into a radial integral. We introduce the D dimensional solid angle QD = rm/l)  ^  an<  W  T  ^  e  (11.5)  The equation of motion is  2<jf+ r<j>" - r —f/ = 0.  (11.6)  We wish to investigate the system when the energy difference between the two vacuo 4> and 4>- is small +  p = U{<j> )-U{^) +  (11.7)  and think of p as a perturbative parameter that turns on the asymmetry of the energy levels in in U . We thus write  Chapter 11.  58  Stability  U(<i>) = Uo(<f>) + pU (<i>) + --- .  (11.8)  1  where Uo(<f>+) = Uo(4>-) and  Uo  J^oU± = 0. A n example for Uo is =  ^ -x 2  )2  ( 1 L 9 )  We are looking for bubble solutions to the equation of motion: the bubble has <f>in its interior and at some large r = ro it quickly grows to 4> . We can therefore +  ignore the j^cf) in the equation of motion as it is either small, or small with respect to r. Our approximate equation of motion is then  r<f>" - r—U  Q  = 0.  (11.10)  Multiply by 4-(f) and integrate using the boundary condition (/>(oo) = <j) : +  | 0 = V 2(f/o(0)-f/o(0 )) . /  (11.11)  1/2  :  +  Letting r be the radius at which 4> is midway between  r = r+  r<t>  0  ./(</>++</,_ )/2  Q  —rd<f>  and </>_ we find  (11.12)  P  0(  For (11.9) we find  0=-^tanh(^i(r-r )) o  (11.13)  Now that we have explored the nature of the bubble solutions, we my evaluate B by treating the bubble as having a thin wall. B is given to lowest order by an integral over all space  Chapter 11.  poo  1  59  - (0' ) ) + U (J>) - W + ) ]  rX-m  B = 4TTdr  Stability  2  (11.14)  2  +  0  This integral may separated over three significant regions: the region inside r , the 0  region about r , and the region outside r . 0  0  B =B  + B  in  R  /  °  + B  on  (11.15)  out  47T  dr r [UM  - U (<i> )] = - — r i p  2  0  (11.16)  +  this is the volume of a 3-sphere times the density p.  B  on  = 47rr  2  0  /" d r [ ^ ' )  + U (<f>) - U (<j>+)] =  2  0  0  ^T\O  This is the the surface area of a 3-sphere times the surface density o — f  (11.17)  dr[\((j)') + 2  on  U {(j>) - C/ (0+)] = //_ # (2U {<t>) ~ 2U {<f>+)) • The right hand sides follows from +  0  1/2  o  0  0  the equation of motion. For (11.9) we find  (11,8) Outside the bubble cf> = 4> so we find +  B  = 0.  out  (11.19)  In total the bounce is given by  47T  B = -—r p 3  0  + 4irr o. 2  The extremum of B occurs when r§ = j which gives 2  (11.20)  Chapter 11.  Stability  60  Figure 11.1: The lines represent the value of the Kalb-Ramond flux $. The light line passes through the bubble and travels from $ = 0—> 0 as z = — oo —> oo. The dark line avoids the bubble and travels from $ = 0 -4 2n as z = —oo —> oo.  _  1U/I  B = — If B >  u 2  .  .  .  (11.21)  1 we find the decay process is strongly suppressed via the exponential in  (11.2). So far, we have been discussing a bubble of 0_ forming in a uniform c6+ world. However, this can also be applied to the case of a domain wall that separates two physically equivalent vacuua in arbitrary dimension. For example, the above (2+1) theory is equivalent to a (3+1) theory where a 3-dimensional bubble forms in the wall. In this case we write <f> for the domain wall solution and 0_ for the empty +  vacuum. Figure 11.1 illustrates a bubble forming in a domain wall. For bubbles of dimensions d we have  B = -Sl -p d  + nr-a d  d  1  (11.22)  where p and a are the d-dimensional volume and surface density respectively. The  Chapter 11.  61  Stability  extremum of B with respect to r gives the critical radius r  0  cr  r = (d-l)-. 0  (11.23)  P  Then, the d-dimensional bounce is given as  d  (11-24)  p  d  For large d we may use Sterling's formula for T(x)  F(x) ~ V2^e~ x -  (11.25)  (d-l)^ -e" ^" .  (11.26)  x  x  1/2  and expand  1  1  1  For large d we find the asymptotic bounce:  B «  2 /V - )/V - )/V - >/ 4 T d  d 1  d 2  d 3  2  L  (11-27)  For our domain walls the energy is given by (9.14)  E = ^ y  /  % f d  2  2  X  (11.28)  and hence  Evaluating the bounce for our domain wall requires d = 23. In this case we find  B  =  (Q,)  ^y  )22  a  23  (11.30)  Chapter 11.  Stability  62  where we have written g' to absorb the unnecessary pure number. s  11.2  Modelling  the Surface  Tension  Our final task then is to estimate a. Unfortunately we know very little about the detailed structure Uo((f>) for this case. We must proceed cautiously with an estimation. Qualitatively we know that (f> and </>_ must correspond to the minima of UQ and there +  must be some barrier between them. We shall model UQ with (11.9), but to analyze the problem it is more convenient to write it in terms of its width between minima w and its barrier height h.  U  /4)  0  (11.31)  2  where the momentum dimensions of the constants are: [h] = 23 and [w] = 21/2. We obtain these dimensions by modeling the bubble in the wall (excluding the X  3  direction) as a 25-dimensional scalar field theory. That is, we choose an action like  (11.32) This is consistent with obtaining the bounce (11.30). We should expect that the height of the barrier goes as the string energy scale scale.  Therefore we dimensionally estimate h = (a')~ / 23  In the free field limit,  2  coupling to the massive modes is not present and the sine-Gordon soliton must be stable. Therefore we dimensionally estimate w = A ^ ^ - , where n is some positive 2 1  5 2  integer. The g dependence must be inverse because we must have B = oo for g = 0. s  s  The power n can not be obtained easily but is likely some small integer power. Thus the surface tension is  6  6  ,n  9\s  (11.33)  Chapter 11.  Stability  63  and the bounce is B = c{ct')-* 9?  "  m  2 3  "A-  (11.34)  8 9 / 5 2  where c is a constant that can be absorbed into g . Note that B scales inversely as s  g since n is a positive integer. We may also insert our estimation for A i from (5.54): s  /  p2 \  B=(~\  89/52  g  2 2  -  2 3  "ei^  (11.35)  where we have dropped c (which is equivalent to absorbing it into g ). s  Since we  require ^ ^> 1 and we expect g to be small, we find that the decay is exponentially s  suppressed: the Kalb-Ramond soliton is stable.  Chapter 12.  Conclusion  64  12: Conclusion After a brief introduction to the relevant parts of string theory, a soliton of the KalbRamond field in closed bosonic string theory was introduced. The soliton discussed forms under a double toroidal compactification. With a constant Kalb-Ramond field permitted in a components of the compact directions, the vacuum energy is found to be a periodic function of the field. This vacuum degeneracy permits the formation of domain walls, as in ferromagnets. We examined the configuration where two halves of the universe are in separate domains. The region in between is approximated by a sine-Gordon soliton whose length and energy are given as  1=  (12.1)  2nRg  s  (12.2) where the integral is over the spatial directions not including the compactified directions and the direction in which the field varies. The stability of this structure is threatened by decay through nucleation. However, the decay process is exponentially suppressed by inverse powers of g . s  The are many other structures and features that may be investigated. Alternative compacitifactions may form other topological structures.  As well, instanton gasses  may occur. It is also important to investigate how particles would scatter of these objects and permit their observational detection. Bosonic string theory is pathological and is not the correct model for nature. However, type II string theory is more realistic and should have the same solitonic struc-  Chapter 12. 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Kibble, G . Lazarides, and Q. Shafi. Walls bounded by strings. Phys. Rev., D26:435, 1982. [26] Jr. Callan, Curtis G . and Sidney R. Coleman. The fate of the false vacuum. 2. first quantum corrections. Phys. Rev., D16:1762-1768, 1977. [27] Sidney R. Coleman. The fate of the false vacuum. 1. semiclassical theory. Phys. Rev., D15:2929-2936, 1977. [28] Sidney R. Coleman and Frank De Luccia. Gravitational effects on and of vacuum decay. Phys. Rev., D21:3305, 1980. [29] Jnanadeva Maharana and John H . Schwarz. Noncompact symmetries in string theory. Nucl. Phys., B390:3-32, 1993.  69  Appendix A. Dimensional Reduction  A: Dimensional Reduction  Here we summarize the procedure of dimensional reduction [29]. We choose to parametrize the D-dimensional spacetime metric G^  AS N  ds = G dx dx 2  M  = G d^dx  N  where M,N  + G (dx  v  MN  + A dx»){dx  m  ilv  m  mn  + A^dx ).  n  11  (A.l)  = 0,1, • • - 25 and fj, and u index the non-periodic dimensions and m  and n index the periodic dimensions. In matrix form, we may write the metric and its inverse as  (  (~i  _i_  n  fi  Am An  Am  GMN — ri  r  -A"  G» + u  G MN  J  G,  mn- -v  KJ  \  _ Hn  Qmn  A  pm  +  A  (A.2)  (A.3) A  n  This parametrization has a convenient property:  \/—\GMN\  —  \J  —  (A.4)  \G^\\/— \G  T  Using the above relations we find that  ^  [d xyf-\G \e- * D  2  MN  [R + D  2d $d $] M  M  (A.5)  Appendix A. Dimensional  70  Reduction  is equivalent to {2nR)  k  /d xV=G~ e-™«  [R +  d  2 Kn  d  l mn G  (  GPq  \ G  m  F $  m  u  d  d  (A.6)  dG ) U  diiGmp  n  4d^ d $  d  nq  F ^ \  with  Fj»  = d^  m  $  -  m  d  d„AV  (A.7)  m  = $ - -det(G/ )  (A.8)  mn  for A; periodic dimensions with d = D — k. Typically, we will write Au^  n  = A/j, . n  The (1) is make a distinction between another tensor that will be defined shortly. Now we consider the action associated with Kalb-Ramond field  f J  TT  d xV^Ge~ * D  2  T T M N L  ~ 12  M  N  (A.9)  L  The metric parametrization allows us to separate the terms and write  - ^ T "  f d xx/^G~ e- ^H H d  2  d  + ^ H^  mnl  1  mnl  Hfinl  + ^H H^  1  (lul  +  ^H, H^ ] x  uX  (A.10) where the following definitions have been employed:  (ATI)  ff  —  P(2)  _  D  p(l)n  (A.12)  Appendix A. Dimensional  Ffflm — ^li-^vm  AW=B„  - \ (4)™  D  _  D  i I  (A. 13)  (A.14)  mn  F% + A^lFT)  1  vX  dvA$n  + B A^  m  H^x = d„B  + cyclic perms.  m  A / t ( 2 )  71  Reduction  _  1 4(l)m/t(2)  _  -j^mrj  ^(l)n  (A.15)  /*  1 fi\  Adding all the components together, we find that the effective action can be dimensionally reduced to  S,  0 k  ! d x^rGde-^ d  = - \G G™ mn  - \ G  m  n  (d G d G tt  lt  F $  m  mp  F ^  [R +  4d^ d^  d  d  d  + dB dB )  •  u  nq  li  mp  - \G™H ,H<? mtu  nq  -  ^H^H^}  ^- ) A  17  


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