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

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Kalb-Ramond Solitons in Bosonic String Theory by Benjamin Jacob Sussman B.Sc , 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 OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E 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 depart-ment 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. Un-der appropriate configurations the cosmological constant becomes a periodic function of the Kalb-Ramond field. This vacuum degeneracy permits the formation of sine-Gordon solitons. The energy and length scale of the soliton is inversely proportional to the string coupling constant gs. The stability of the the solitons is discussed and it is shown that these objects are stable. 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 History and Motivation 3 2.2 From Particles to Strings 4 2.2.1 Particles 4 2.2.2 Strings 4 2.3 Symmetries 6 Chapter 3: The Covariant Path Integral and Modular Invariance 12 3.1 The Covariant Path Integral 12 3.2 Modular Invariance and the Torus Amplitude 13 Chapter 4: Strings in a Kalb-Ramond Field 16 4.1 The Massless States of The Closed Bosonic String 16 4.2 Mass-Shell Condition and Level Matching 17 Chapter 5: The Vacuum Energy 21 5.1 Vacuum energy of the uncompactified string 21 5.2 Vacuum Energy with one compact direction 24 5.3 Vacuum Energy with many compact directions 25 5.4 Vacuum Energy with 2 compact directions and a B field 27 Chapter 6: Elimination of the Tachyon Contribution 32 6.1 Spectrum Truncation 32 6.2 Relative Subtraction 33 Chapter 7: String Theory Effective Action 34 7.1 The Effective Action as the Spacetime Action 35 7.2 Kaluza-Klein Theory 37 7.3 Compactification of the Effective Action 39 7.4 Adding the Cosmological Term 40 Table of Contents iv Chapter 8: Equat ions of M o t i o n 42 8.1 Equivalence of String Frame and Einstein Frame 42 8.2 Variation off the Action 43 Chapter 9: S ine-Gordon Solitons 47 9.1 Two Dimensional Sine-Gordon Solutions 47 9.2 B-field Solitons 49 Chapter 10: Topological Indices and V a c u u m M a n i f o l d 51 10.1 Topological Charge 51 10.2 Modularity of B 52 10.3 Unstable Topological Configurations 54 Chapter 11: S tabi l i ty 56 11.1 The Bounce 56 11.2 Modelling the Surface Tension 62 Chapter 12: Conclus ion 64 References 66 A p p e n d i x A : Dimens iona l Reduc t ion 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. Chap-ter 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 ex-pect 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 imme-diately applicable to the discussion B-field solitons. Chapter 2. Review of Bosonic String Theory 4 2.2 From Particles to Strings 2.2.1 Particles In special relativity, the action of a particle is given by the integral of the invariant length ds2 — glll,dxtldxu times an arbitrary constant m: The connection with non-relativistic physics is obtained by parameterizing r = X and taking the following limit: 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 {ca\a = 1, 2} and metric hob- The element of length for any Riemannian (or pseudo-Riemmanian) surface is given by ds2 = habdcadcb. Thus the length of a small coordinate displacement is given by ||dca|| = \/haadca (no sum). Therefore the area of a particular (parallelogram) element of the grid is given (2.1) S = -m J dx°y/i-x* = j dx°{^ by: Chapter 2. Review of Bosonic String Theory 5 AA = | |dc° x dc 1!! (2.3) = ||cic01| Hcic11| sin(6>) (2.4) = | |dc°| | Hdc1!! (1 -cos 2 (0)) (2.5) dc° 2 dc 1 2 - dc° • dc 1 2 (2.6) = v^oofcn - (hQl)2dc°dcl (2.7) = v ^ t i c ^ c 1 (2.8) where h — det(hab). Writing r = c°,o = c1 we then write the so-called Nambu-Goto action as the total area of the surface times arbitrary constant T of dimension (length)'2: S = -T I dodr\f^h J do r\f:-h (2.9) To determine the induced metric hab in terms of the spacetime metric note that the line element as calculate in both frames should be the same: ds2 = habdcadcb = 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^X1)2 = 1, -§^X° = 0. Substituting these values into (2.11) and subsequently into (2.9) we obtain: S = -TJ d r d o ^ l - ( ^ ) 2 + (^-X^Xr (2.12) Chapter 2. Review of Bosonic String Theory 6 Writing v2 = (-^X1)2 — (^Xl-^X1)2 and expanding in small v, we obtain 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 drda^l^hd-Xil^X^ (2-14) The equation of motion for 7 a h 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 ^ : Tab = - 4 . 7 T -1 6 -7 8fab a The equation of motion for Xth is SP = - l ( d a A ^ d % - l^daX^d'Xj - 0. (2.15) da(V=llabdbX») - ^d2X» = 0 (2.16) 2.3 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'c do'd ^Ta^'cd(r',a') = lab(r,o). (2.18) An additional symmetry satisfied by Sp is that of D-dimensional Poincare invari-ance: X'"{T', a') = KX"{r, a) + a" (2.19) 7ab(T,o-)=LAB(T,o-). (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) (2.21) iab(r,o) = e2^hab(r,a). (2.22) An 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: Jab = e2^r,ab(T,o) (2.23) Chapter 2. Review of Bosonic String Theory 8 where nab is the flat Lorentzian two-dimensional metric. Subsequently, Weyl invari-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 trans-formation. These compound transformations are called conformal transformations. Introducing o± = r ± cr we note that in the conformal gauge — det(7a&) = e 4 w and = e~2uJr]ab. The action becomes S = 4 ^ / d r d C T v / = 7 7 a 6 9 t t X " ^ X A 4 (2.24) Thus, the equation of motion is the simple wave equation: d2 d2 1 A ^ = 0. (2.25) dr2 do2 we may then factorize the differential operator yielding 9 d ' A ^ = 0. (2.26) s do+ do The general solution is then the sum of a holomorphic and anti-holomorphic (right-and left-moving, respectively) part: X»{o, T) = XL(o+) + XR(o-) (2.27) X^(a~) = \x» + o y e r - + £ ^e~2ina X£(o+) = \x» + a 'p"<7 + + YJ fe~2ina+ (2.28) We may now quantize in the usual way. In the conformal gauge we have S = | J d2odaX»daXt, (2.29) Chapter 2. Review of Bosonic String Theory 9 and thus P " = — — L = TIC1. (2.30) To quantize the system, we introduce the equal time canonical commutators [X"(c, T), Pv{o', T)] = z5(a - o')rf\ (2-31) We then arrive at [xi*,tf\ = irr (2.32) [ < , < ] = (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 ^ m > 0 (2.36) a £ m = Vma^ m < 0. (2.37) One then constructs the various string states by using the raising operators on the vacuum state |0;pM) . It will turn out that the higher states are increasingly 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 ' T - ^ = s b [ ( 2 - 3 9 ) After substituting in the mode expansion we find that for the right modes 1 0 0 L m = 2 ^  a m ~ n ' a n = ° (2-41) —oo however, since [a_ n, a n] 7^  0 , after quantization there is an operator ordering ambiguity in the m = 0 case ^ 00 L 0 = - c / p 2 + - a-n • Oin = a. (2.42) 1 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 m2 = —p^p41 then the Virasoro generators give the important 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 11 N-N = 0 m2 = —{N + N-2). a' (2.43) (2.44) Chapter 3. The Covariant Path Integral and Modular Invariance 12 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 ana-lytically continuing the matrix ^ to the Euclidean signature metric gab- The path integral we are interested is given by J DXDge~s (3.1) where S is the sum of the action and the topologically invariant Euler character S = ^bJ d'a^daX^X, + d2a^gR. (3.2) 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 d2o^R = 2 - 2 7 . (3.3) In addition, however, we also need to add a counter term of the form /j,2 f d2o\fG 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 ~VG~C~ vw This integral was first evaluated by operator methods [11] but has also been de-termined via the path integral method [12] for the torus amplitude. For the torus amplitude it was determined to be Zt0rus = T13V26 J ^ e 4 ^ ( 2 7 r r 2 ) - 1 2 | / ( e 2 ^ ) | - 4 8 (3.5) where r = Ti + ir2, d2r = dridr2, f(q) = TI^°=i(l — T IS the string tension, and V2e is the volume of space time. The one loop cosmological constant is just zly™ • Of critical importance here is the fact that the integral is performed over the fundamental domain F defined as \ < TI < ^ , r 2 > 0, \T\ > 1. (3.6) 3.2 Modular Invariance and the Torus Amplitude The restriction to the fundamental domain can be understood by recalling the defini-tion of a torus. Consider a parallelogram in the complex plane with sides represented by the complex parameters Ai and A 2 (that behave as vectors in the complex plane). 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 14 z ~ z + n iAi + n 2 A 2 for m = 0 , ± 1 , ± 2 , - - - . There is, however, a certain amount of redundancy here. Consider (3.7) n iAi + n 2 A 2 = n'-^X^ + n 2A' 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 A' = M A (3.9) M = V a b (3.10) J where M is an SL(2,Z) matrix are equivalent tori. The world sheet action is confor-mally 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 (3.11) cr + d 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) Chapter 3. The Covariant Path Integral and Modular Invariance 15 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<\> r 2 > 0 , 1-7-1 > 1. (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 ( 3 ' 1 5 ) These regions are all equivalent and can be mapped to each other via (3.11). 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 u can be decomposed as: 1 1 1 1 [-(eij + en) - -^z^^ijtre] + [-(e^ - eji)] + [jy^S^tre] (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 Chapter 4. Strings in a Kalb-Ramond Field 17 S=l~bf d2°V99a\udaX»dbX\ (4.4) 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) + ieabB^{X))daX»dbX» + 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 d2(T^9ahGAX) + e^B^X^daX^X"} (4.6) If B^u is constant, the second term can be written as a total divergence da(tabBlluX^dbXu). This term's contribution to the canonical momentum is zero except when X^ is non-periodic. Therefore, to determine the allowed momenta we consider only the non-periodic parts of X11 and write X* = ^ ( T ) + w^Ra + oscillators (4.7) where is an integer that represents the number of times the string wraps around the X11 direction. The action then becomes (4.8) Chapter 4. Strings in a Kalb-Ramond Field 18 The Lagrangian L for the system is just the integrand. We may form the canonical momenta P» = ^{Glu,xv + Btu,w"R). (4.9) 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 M = We may now write the mode expansion as X" = x» + loRvf + 2 r ( G " " V - ^ - B"uwuR) + oscillators (4.10) R Since we have X^(o,r)=XL(o+)+XR(o-) (4.11) with X£(o+) = \x» + oltfLo+ + £ fe~2in°+ We may determine the right and left momenta as and (4.12) 77 olpl = w^R + (G^a'-zj - B^w'R) (4.13) R Tl ay = -w^R + (G^a'-Z - B^uTR) (4.14) R For illustrative purposes, we consider a special case for non-zero B — B 1 2 with X1 toroidally compactified. The momentum conjugate to X1* for flat spacetime (G^u = Chapter 4. Strings in a Kalb-Ramond Field 19 n^) is P " = - ^ r L = /"W" + BSdaX"]da. (4.15) We write P " = p41 +P%- The first term here is the usual kinetic momentum p1 that multiplies the term linear in r in the mode expansion of X11. In the X1 direction the left and right moving parts are modified to: XR{o~) = \xl + {-wR + a'| - BwR)o~ + £ s^e-*™-n * ° (4.16) XlL(o+) = \xl + {wR + oi\ - BwR)o+ + £ °*-e-2in(J+ The second term in (4.15) is the field momentum pB. This term only contributes when X1 is compactified on a torus with radius R then we have X1(T,O + n) = X1(T,O) + 2'KRW. In this case P7T / daXldo = 2irRw (4.17) Jo and hence the field momentum is non-zero. The zero-mode Virasoro constraints are implemented as usual: T r • 2 U = y A-R dcr = 1 (4.18) L0 = - j XL do = 1. (4.19) 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 X1 direction: P1 — ^. Also, we may replace the kinetic momentum with the canonical one which changes the mass-shell condition to [13]: ( P 2 - 2*BRWf + £ p"p„ = (If + (^ )2 + -V + N - 2) (4.22) ^—' R a a ^1,2 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 ampli-tude in the euclidean formalism. Z= (0|e — HT |0> = e -E0T (5.1) For the scalar field <j> in D non-compact dimensions we have Z = j D<t>e~s = j D<f>e-ydDx ^+™2)* = e - | r n „ ( - ^ W ) ( 5 2 ) We may evaluate the trace as the sum of the eigenfunctions and obtain the vacuum Chapter 5. The Vacuum Energy 22 energy £ ° r = W ( 3 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 m2 = £(N + N - 2). We may enforce the level matching condition (4.21) with a delta function. ;l/2 $N-N - -1/2 /•1/2 / d n e 2 ^ N - N ^ (5.7) J-1and then trace over N and N: A = ~ \ (^ v) / d r i d r 2 r 2 - 1 4 e 4 7 r T 2 i r ( 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 2 = X/ct'ir r = Ti + zr 2. For the evaluation of the trace we recall the relations (2.36) and (2.37) and define q = e 2 7 U T. Now we note / oo \ 24 t r ( g E - = 1 - U m ) = T " [ _ ^ _ . ( 5 . 9 ) \ m = l ^ / The factor of 24 comes from performing the trace over the transverse states. We now define the Dedekind eta function as oo T}(q) = q1,U]l(l-qm)- (5-10) m = l In particular, then, we have y y ™ » * = f'2 dn ^ ( e ^ p 4 8 (5.11) and finally: The integral here ranges from —1/2 < T\ < 1/2 and 0 < r 2 < oo. However as we have learned from the path integral formalism we should in fact truncate the integral to the fundamental region —1/2 < TJ < 1/2, r 2 > 0 and abs(r) > 1. The physical 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. Chapter 5. The Vacuum Energy 24 5.2 Vacuum Energy with one compact direction In a configuration with one toroidally compact direction X1 we make the identification X1~X1 + 2nR (5.13) Because the momentum is the generator of spacetime translation, the momentum Pl must be discrete P 1 = - n = 0 , ± l , ± 2 , - - - . (5.14) R 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 X1 = xl + 2oRw + 2 a ' + iJjJ2~ + oie~2in^) . (5.15) Evaluating the zero mode Virasoro generators L0 = L0 = 1 we obtain the mass-shell (for the 25- and 26-dimensional mass - see section 7.2 for their definition) and level matching conditions m. a ' / „ r »~r _ \ n2 R2w2 = _ ( A . + A r _ 2 ) + _ + _ _ (5.16) a',„ ,-r ^ R2w ™26 f ( i V + i V - 2 ) + w (5'17) N-N = nw (5.18) Chapter 5. The Vacuum Energy 25 with oo N = Y1 (a-nann + Oi\nanl) (5.19) 71 = 1 00 N = J2 ( a - n Q V + c t n a n l ) (5.20) 71=1 where n here excludes the compactified dimensions X1. 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 ^ J2n ana-also include the winding modes in the mass by replacing m with m26- The vacuum energy is then modified to ^ ' w,n F (5.21) where D = 26. 5.3 Vacuum Energy with many compact directions We now consider the case of Md xTk. In this case use label index m for the compact directions and write the mode expansion as Xm = xm + 2oLm + 2a'PmT + « ^ 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™ = Lm + a'p' (5.23) Chapter 5. The Vacuum Energy 26 a'p% = L m - a'pm (5.24) with X»(a, T) = XL{o+) + XR(a~) (5.25) n#0 -2in<r+ (5.26) 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 1 compactification because the periodic coordinate identifications we make need not be at right angles. For example, we may identify points as X = X + 27rv /o7Em (5.28) Where the basis vectors E m are complete over Tk. Thus we may write L = Vo7wmEm. (5.29) In order to make sure that the momenta are the generators of translations we write p = V a 7 n m e m (5.30) Chapter 5. The Vacuum Energy 27 where we have formed the reciprocal lattice vectors ei ep • E 9 = 6pq. (5.31) In this notation we may write the mass shell and level matching condition as m2D-d = j(N + N - 2) + p 2 + ^ (5.32) N - N = n • w (5.33) We will typically be interested in the case where the basis vectors Ej are all orthog-onal and equal to R in length. In this case we have simply ^ - d = y ( i V + i V - 2 ) + ^ + ^ . (5.34) Now the cosmological constant is adjusted to l ( a ' ) ( * - £ ) / 2 w,n dndn r-(D+!-*)/2e-°''!*«*>1+(*>')-2*i"""1 \v(q)\-K 5.4 Vacuum Energy with 2 compact directions and a B field In the configuration where two spatial dimensions are compactified (say X1 and X2) and a non-zero (and constant) B-field is permitted only in the Bi2 = —B2\ compo-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\2 = —B2\ we must now incorporate the 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'): X1 = xl + 20-Rw1 + 2r(a'— - Bw2R) + oscillators (5.37) R n2 X2 = x2 + 2oRw2 + 2r(a'— + BwlR) + oscillators (5.38) XL Thus the cosmological constant is modified to K ' w,n . e - Q ' r 2 7 r ( ^ ( r n - * « ) 2 ) 2 + ^ -(n2+*u;i)2+(^)2)-27riwnT1 ^^-48 ^ 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 = e-Q ,T27r(^(ni-$ t l;2)2 + ^ (n2+$wi)2-K^f )2)-27riw-nri _ (5.40) We shall now perform a Poisson re-sum: Chapter 5. The Vacuum Energy 29 /OO d2P e 2 T i k ' p e _ a ' T 2 W ( ^ ( p i ~ * W 2 ) 2 + ^ ( p 2 + * W l ) 2 + ( ^ ) 2 ) _ 2 ^ w ' p T l . (5-41) •oo k Define the dual of w as w = (—w2,Wi) and shift p —» p + w $ and perform the integration over p: ^ a'r2 k R 2 c - ^ ( f c i - ^ i r i ) 2 - ^ ( f c 2 - ^ 2 T 1 ) 2 - Q ' r 2 7 r ( ^ f ) ^ 2 7 r i k . w ^ a'T2 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 A I C O S ( 2 T T — B ) . (5.45) a' In summary, we have found that the cosmological constant for T 2 x M 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')~Dl2. 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 gs dependence. A dimensional estimation then yields Chapter 5. The Vacuum Energy 30 (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/(r i , r 2 ) = Uki - Win)2 + (k2 - w2Ti)2} - T 2 7 T + 2nik • w $ . cr a r2 a (5.47) By using 'Polchinski's Trick' [14] we may eliminate w2 in favour of integrating over 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 2 we obtain the extrema, respectively Wl (5.48) U = The exponent at the extremum now becomes (5.49) R2 -2-7T— |&2u>i| + 2irik2wi$. a' (5.50) Now we may proceed to write U{n,T2) = U(ti,t2) + ]-Uiti ( n - h)2 + \ u 2 a (r 2 - t2)2 + Uh2 ( n - h) (r 2 -1 2) + (5.51) and perform the integration over the upper-half plane. Noting Chapter 5. The Vacuum Energy 31 /oo poo n I e-a{x-xtf-b(y-yo)2-c{x-x0){y-y0)dxdy = _ _ ^ = = ( 5 5 3 ) o o i - o o VAab-c We find that the dominant periodic term behaves as R2 A 1 COS(2TT— ) (5.53) r\i' a' where Chapter 6. Elimination of the Tachyon Contribution 32 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 inter-pretation 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 e-2niTN+2irirN = | / ( g ) | - 4 8 . (6.1) 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)p4 8 \v(Q)r48 - e 4 ? r T 2 (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 r = A{B) - A(B = 0). (6.4) We should note, that there is some ambiguity in the choice of regularization. In re-ality, 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 gs (the string Feynman diagrams are weighted such that each additional loop provides a weight of gs): 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 oo (7.1) genus h=Q Chapter 7. String Theory Effective Action 35 constant that fixes their relative weights of the two diagrams. More precisely we see that $ —> $ • const, sends gs -» econstgs. However, while the relative weighting of two 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 gs —> econstgs, — 2const.. We see that the constant part of the dilaton shifts gs and in a sense, gs is set by the vacuum 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»u{daX»daX» - l-5aadcX»dcXn) ~ (1 - |) = 0 (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%„ + 2G/V^V„<S> - -H^Hj* + 0(a'2) (7.3) 0 = P% = ~ VAi/A„„ + a\Vx<S>)Hx,u + 0{a'2) (7.4) 0 = £* = - ^ V 2 $ + a'(V$)2 - ±-a'H2 + 0(a'2) (7.5) Chapter 7. String Theory Effective Action 37 where is the covariant derivative. These three equations are linear combinations of the equations of motion which can be derived from the action s = h I dDx^e~2* \2{D3J6) +R- ^H^H^X+2d^d^+°^') (7.6) Specifically, we have SS = [ dDXv^Ge-2* \8G^ (Bg^ - l-G^{B^ - 40*)) 2K,0a J i \ z j [1-1) + 5B^ (8B^) + 5$ (2$p - 80*)] = 0. 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 4 x Sl with 0 < y < 2nR where R is the compactification radius. The 5-D Einstein action is S* = 7T2 [ d5x^f5R5 (7.8) ZKb 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*VnvlT (7-10) n=—oo and similar expansions for the other fields. If we substitute the Fourier expansions and the parametrization of g5fll/ into (7.8) and integrate over y only keeping the n = 0 terms (<f>0 = </>, etc.) we obtain (after some algebra, the details of which are discussed in the appendix) S=hj d ' x ^ R - \d^<t> - \e-^F,uFn (7.11) where the field strength is defined as F^ = <9MA, — duA^ and K2 = KI/2TVR. We have discovered that gravity on M4 x S1 is equivalent to gravity on M 4 plus electromag-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 R4 x Sl: d2J(x,y) = 0. (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>n = 0 (7.13) Chapter 7. String Theory Effective Action 39 where m 2 = j^. To a four dimensional observer (who defines mass as m\ = p2,— p\ — 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\ = p2a — p\ — p\ — p\— p\. 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 2 4 x T2. We do this by way of Kaluza-Klein compactification. Parametrize the metric as ds2 = G^NdxMdxN = G^dafdx" + Gmn{dxm + A™dx»){dxn + A^dx11) (7.14) Then, the effective action can be dimensionally reduced to S0,k = / ddxy/=G~de-2** [Rd + 4 c ^ d " $ d 2K 0 - \GmnG™ (d.G^Gn, + d,BmpduBnq) (715) - \ G m n F $ m F ^ - \GmnHmiwH^ - ^H^H^] where the following definitions have been employed: $ d = $ _ I _ det(Gmn) (7.16) F$m = d,A^m - dvA^™ (7.17) H = F^ —B F^n (7 18) Chapter 7. String Theory Effective Action 40 FuJm ~ ^ 4 1 ^ ^ L m (7-19) 4 m = B,m + BmnA^> (7.20) H^x = d,BuX - \ (4 1 )™ F%m + 4 M ) m ) + cyclic perms. (7.21) B,v = B,v + \A^mA^l - \A^mA^l - A^mBmnA^n (7.22) the details of which are explained in the appendix. 7.4 Adding the Cosmological Term The general form of the partition function is: OO p Z= J2 92sh~2 / DxDge-Sp^3\ (7.23) genus h=0 Tree level amplitudes correspond h = 0 and the torus amplitude is h = 1. Alterna-tively, we take the effective action as the spacetime action. The effective action will then generate the string theory S-matrix. The effective action is invariant under c 9s -> 9se2c for constant c. Hence we can always absorb the string constant into the dilaton. We expect the same of the effective action (which must reproduce the same S-matrix). Again, the torus amplitude goes as g®, so we expect the following form for the effective Chapter 7. String Theory Effective Action 41 action with the inclusion of the vacuum energy: S = S0tk + j dDxv^GA(B) (7.24) where we may replace 2k2 = (a')l2g2s in So,k- Any uncertainty in the relative size of A term from the rest has been absorbed in gs We should note that the vacuum energy calculation was performed for a constant B. However, here, we assume B is not constant. We must be in a regime where treating B as non-constant is applicable. Chapter 8. Equations of Motion 42 8: Equations of Motion 8.1 Equivalence of String Frame and Einstein Frame To touch basis with conventional gravity we shall now demonstrate the equivalence of (8.5) and the Einstein-Hilbert action [2]. Consider the action s=h I d°xV=Ue~ 2$ (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- K2 = 8TTGN- We shall show that this is equivalent to the string effective action and vice-versa. Under Weyl transformations of the metric G>„ -> e 2 w ^GV (8.2) the Ricci Scalar Transforms as [18] R _> e - M « ) (R _ 2(D - 1) - (D - 2){D - 1)8^8^) (8.3) and G - ) • e2D^X)G. (8.4) $ 0 — $ If, for constant $o> w e define K — K0e 0 and choose w = 2 ——— , this transforma-Chapter 8. Equations of Motion 43 tion turns the action () into the string action S0 = -L [ dDxv^Ge-2* [R + Ad^d^}. (8.5) 2«o J This form of the action not as conveinient for determining the equations of motion as (8.1). We have learnt that string theory predicts general relativity coupled to a scalar field. 8.2 Variation off the Action We wish to find the equations of motion for the doubly compactified action 5o,2 = f ddxV^G~de-2^ [Rd + Ad^ddu^d - \GmnG™ {d,GmpduGnq + d^B^Bng) (8-6) + J* dPx VGA. Write S = SGD + SBFG + S\, where SGD refers to the terms on the first line, SBFG refers to the terms on the second and third line, and SA refers to the final line. The cosmological constant term is constructed from A = A i ( l -cos(AB)). (8.7) We now perform the variation of the action to obtain the equations of motion for 9nu-SSQD = V=G~de-2** 5G^ {o.')l2g2 Rd^ + 2D„A, - ^G^(Rd + AD^$ - Ad^d^) (8.8) Chapter 8. Equations of Motion 44 SSBFG _ SG^ 1 V^G~d 2(c*')12<?2< G p,u GmnGpq ( D P G M P D P G N G + dpBmpdPBnq) 1 /~i rpm rpnpa ^ rvmn TJ TJ pa ^ TJ TjpaX ~~^Gmnr p(Jt — -G h m p a H n H - — llpa\tlH 2* r f 1 2 QmnQpq (d^GmpduGnq + d^Bjnpd^Bng) rpn a \/-vmn TJ TTX — T-f TJ PA ' -^mn-T p,ar v r, ^  •nmfj.\rlni' A n ^ p a n v (8.9) (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^v is then For Gr 6&' ;{SGD + SGBF + 5A) = 0. (8.11) 5St GD 5Gr< (8.12) SS BFG 8Gr (aiy2g2~ [ -^Gpq (dpGmpdpGnq +dpBrnpdpBnq) d^aG^d^Gn, - (GolGpid»Gnq)^ GlpG,t -L.-C ,Q F° FLPA - -JJ JJP° > ^Jnl^Jom1 pa1 ^ 1 1 m p a 1 1 n (8.13) JQ^ - -^V-G~dV-GD-dGmn^- (8.14) Chapter 8. Equations of Motion 45 The equation of motion for Gmn is then 5 5Gr (SGD + SGBF + SA) = 0. (8.15) For $ d = "C?i2~~r [ - 2 ^ + Sd&d*** - SD^d] (8.16) SSBFG = V^G~de-2^ 6$d (a') 1 20? 1 ' [-\GmnG™ (d,GmpduGnq + dliBmpduBnq) (8-17) - iGmnF$mFWn»" _ \GmnHm(U,H^ - ±H^XH^} t = ° <8-18> The equation of motion for $^ is then (S G j D + S G B F + 5 A ) = 0. (8.19) 5®d For B 5SGp _ Q 5Bmn ~ J E ^ = i f f ? ^ (e-^G-G^Boq) GlmG 0 { m , n } ^ {1,2} SBr AA0 sm(BA) y/^G~d x / G ^ {m, n) = {1,2} The equation of motion for Bmn is then Chapter 8. Equations of Motion 46 -(SGD + OGBF + SA) = 0 (8.20) 5Bmn We are interested in static solutions about flat space with all fields flat except for B = Bi2(X3) and therefore the additional equations of motion do not couple in this regime. The equation of motion for Bmn then leads us to the equation of motion for the celebrated sine-Gordon equation: 1 dlB{Xz) - A A i sm(B(X3)A) = 0 (8.21) W29. 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 gs Equation (8.21) admits a first integral. We multiply by 83B and integrate to find 1 [B'f + A x cos(£,4) = c (8.22) 2(a'Y292 The constant is determined by requiring B'—O and BA — 2-KU at spatial infinity: c = A i . And thus (B')2 + 2(a') 1 2^ 2Ai(cos(i3A) - 1) = 0. (8.23) This equation can be solved (as will be done in what follows), however, the most important feature is that (B1)2 ~ g2. Since equations of motion for the effective action satisfy the flat solution, we may consider the sine-Gordon solution as pertubative solution in gs. Since (B1)2 ~ g2, to lowest order, we may ignore the coupling of B in (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 2<^+^=sin(—0) = 0. (9.3) y/X rn Y J The naming of the constants m and A is clear if we expand out the cosine in the Lagrangian to reveal C=1-(d<l>r-1-m^ + ^ + ... (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 equa-tion of motion 9.3 by ^ 0 and integrating. Choosing boundary conditions such that at spacetime infinity dx(f)=0 and ^<j) = 2irn for integer n we obtain the so called soliton solution Am i <f>{x) = ^ a r c t a n ( e ± m x ) . (9.5) v A 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 = pdt<t> - U (9.6) V = ^ (9-7) d0(p which yields E = J dxU = jdx^(d(f>)2 - ^ - ^ o s ( ^ % ) - l j . (9.8) Applying this formula to our solution (9.5) we obtain the energy of the soliton: E = 8^-. (9.9) A 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 — v2. There are also various superposition solutions. For example, the so-called soliton-antisoliton: Chapter 9. Sine-Gordon Solitons 49 4>{x) = -= arctan( ) / v ; ) . (9.10) VA v cosh(mx/y/1 — v 9.2 B-field Solitons We are considering the configuration where we have two compactified directions A" 1 , X2 and a non zero B field in that various only in a third direction B = B^ix3) = —B2i(X3). To relevant order in gs we may then write the action as W292s / d26X { ~(d3B)2 - Y(l - cos(AB)) } (9.11) where we have defined Y = (a!)l2g2Ai. We my proceed as in the previous section to obtain the energy an size of the domain wall: B = j avctan(e±AVVx) (9.12) e = —^= (9.13) AVY y^2W r-E - / . A S . V A 1 j d22X (9.14) (a')5gs where the integral is over d22X = dX4dX5 • • -dX25. Using the estimation for A i from (5.54) we obtain a' /Ri\ e{^> (9.15) 2nRg. V2l6n2 _r«i :—; e 2^"' (d)uRg. Since we have assumed ^ > 1 we find that £ is large and E is small. The depen Chapter 9. Sine-Gordon Solitons 50 dance of E on gs is inverse and that is somewhat peculiar. Note the similarity to the physical D-brane tension, which has the same gs dependance [2] ^ (27rv/o7)11-p (9.17) I6a'bgs for a p dimensional brane. 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 ) 2 - u{4>) (10.1) where the potential is given by m 4 / , v A . . U= — \ cos(—(£) - 1 . 10.2 A \ m Classically there are many ground states corresponding to the to minima of the po-tential 4>o = 2nn-^=. (10.3) v A However, the equation of motion also admits another solution A 'YY) (j)(x) = _ a rc tan(e ± m ( l - X o ) ) . (10.4) v A which has finite energy. 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 —= x = —oo <f>{x) = { . (10.5) •y/X Consider the following integral: 1 \/X f°° d m 27rn2—F= x = +oo / • O O Q I —<f>dx = rii — n 2 = A n (10.6) ./-oo dx 2n m 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 j0dx = A n . (10.8) 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 2 . For example, since B^V is a gauge field -» + d,j.xv - d„Xn- (10.9) Chapter 10. Topological Indices and Vacuum Manifold 53 We may change gauge with Xv = | (^X 1 — S^X2). This shifts by SB^^ciSlSl-SlSl). (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 Sb=4^G7>J d2°[zabBAX))daX»dbXv}. (10.11) 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 ASF 2ira' i 2i\ot i - J <Po-[eabdliXvdaX"dbX1'} (10.12) 7 J d2a[dlxXu{dlX^d2Xv - diX^X")} (10-13) = 2^ J d2°ldMdi(x"d2X") - d2(X»d1X»))} (10.14) since d^Xv is a constant we may use Green's theorem to turn this integral into a line integral ASF = 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 54 B ~ (10.16) 10.3 Unstable Topological 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 corre-spond 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 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] (11.1) Chapter 11. Stability 57 r v Ae-B^{l + 0{h)). (11.2) where (11.3) B = SE(<j>) - SE(j>+) (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 We wish to investigate the system when the energy difference between the two vacuo 4>+ and 4>- is small (11.5) The equation of motion is 2<jf+ r<j>" - r —f/ = 0. (11.6) 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. Stability 58 U(<i>) = Uo(<f>) + pU1(<i>) + --- . (11.8) where Uo(<f>+) = Uo(4>-) and J^oU± = 0. An example for Uo is Uo = ^2-x)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—UQ = 0. (11.10) Multiply by 4-(f) and integrate using the boundary condition (/>(oo) = <j)+: | : 0 = V / 2(f /o(0)-f /o(0 + ) ) 1 / 2 . (11.11) Letting r be the radius at which 4> is midway between and </>_ we find r<t> Q r = r0+ —rd<f> (11.12) ./(</>++</,_ )/2 0(P For (11.9) we find 0 = - ^ t a n h ( ^ 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. Stability 59 poo 1 B = 4 T T d r rX-m2 - (0'+) 2) + U0(J>) - W + ) ] (11.14) This integral may separated over three significant regions: the region inside r 0 , the region about r 0 , and the region outside r0. B = Bin + Bon + Bout (11.15) /R ° 47T dr r2[UM - U0(<i>+)] = - — r i p (11.16) this is the volume of a 3-sphere times the density p. Bon = 47rr02 /" d r [ ^ ' ) 2 + U0(<f>) - U0(<j>+)] = ^T\O (11.17) This is the the surface area of a 3-sphere times the surface density o — fon dr[\((j)')2+ U0{(j>) - C/o(0+)] = / /_ + # (2U0{<t>) ~ 2U0{<f>+))1/2• The right hand sides follows from the equation of motion. For (11.9) we find (11,8) Outside the bubble cf> = 4>+ so we find Bout = 0. (11.19) In total the bounce is given by 47T B = -—r30p + 4irr2o. (11.20) The extremum of B occurs when r§ = 2j which gives 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. _ 1 U / I u . . B = — - 2 . (11.21) If B > 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 = -Sld-p + ndrd-1a (11.22) where p and a are the d-dimensional volume and surface density respectively. The Chapter 11. Stability 61 extremum of B with respect to r gives the critical radius r 0 cr r 0 = ( d - l ) - . (11.23) P Then, the d-dimensional bounce is given as (11-24) d pd For large d we may use Sterling's formula for T(x) F(x) ~ V2^e~xxx-1/2 (11.25) and expand ( d - l ) ^ 1 - e " 1 ^ " 1 . (11.26) For large d we find the asymptotic bounce: B « 2d/Vd-1)/Vd-2)/Vd-3>/24LT (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 ) 2 2 a 2 3 (11.30) Chapter 11. Stability 62 where we have written g's to absorb the unnecessary pure number. 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. 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 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')~ 2 3 / 2 In the free field limit, coupling to the massive modes is not present and the sine-Gordon soliton must be stable. Therefore we dimensionally estimate w = A 2 1 ^ 5 2 ^ - , where n is some positive integer. The gs dependence must be inverse because we must have B = oo for gs = 0. The power n can not be obtained easily but is likely some small integer power. Thus the surface tension is U0 / 4 ) 2 (11.31) (11.32) 6 6 9\ ,n s(11.33) Chapter 11. Stability 63 and the bounce is B = c{ct')-*m9? " 2 3 " A - 8 9 / 5 2 (11.34) where c is a constant that can be absorbed into gs. Note that B scales inversely as gs since n is a positive integer. We may also insert our estimation for A i from (5.54): / p 2 \ 89/52 B = ( ~ \ g 2 2 - 2 3 " e i ^ (11.35) where we have dropped c (which is equivalent to absorbing it into gs). Since we require ^ >^ 1 and we expect gs to be small, we find that the decay is exponentially 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 Kalb-Ramond 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 where the integral is over the spatial directions not including the compactified direc-tions 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 gs. 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. How-ever, type II string theory is more realistic and should have the same solitonic struc-1 = 2nRgs (12.1) (12.2) Chapter 12. Conclusion 65 tures. This is worth investigating in future work. References 66 References [1] S. Carlip. Quantum gravity: A progress report. Rept. Prog. Phys., 64:885, 2001. [2] J. Polchinski. String theory, vol. 1: An introduction to the bosonic string. Cam-bridge, U K : Univ. Pr. (1998) 402 p. [3] J. Polchinski. String theory, vol. 2: Superstring theory and beyond. Cambridge, U K : Univ. Pr. (1998) 531 p. [4] M . B. Green, J. H. Schwarz, and Edward Witten. Superstring theory, vol. 1: Introduction. Cambridge, Uk: Univ. Pr. ( 1987) 469 P. ( Cambridge Monographs On Mathematical Physics). [5] B. Hatfield. Quantum field theory of point particles and strings. Redwood City, USA: Addison-Wesley (1992) 734 p. (Frontiers in physics, 75). [6] Hirosi Ooguri and Zheng Yin . Tasi lectures on perturbative string theories. 1996. http: //arXiv.org/abs/hep-th/9612254/. [7] U . Danielsson. Introduction to string theory. Rept. Prog. Phys., 64:51-96, 2001. [8] Ashoke Sen. An introduction to non-perturbative string theory. 1998. http: //arXiv.org/abs/hep-th/9802051. [9] Stefan Forste. Strings, branes and extra dimensions. 2001. http://arXiv.org/abs/hep-th/0110055/. References 67 [10] Steven Weinberg. Covariant path integral approach to string theory. Lectures given at 3rd Jerusalem Winter School of Theoretical Physics, Jerusalem, Israel, Dec 30, 1986 - Jan 8, 1987. [11] J. A. Shapiro. Loop graph in the dual-tube model. Phys. Rev., D5:1945-1948, 1972. [12] Joseph Polchinski. Evaluation of the one loop string path integral. Commun. Math. Phys., 104:37, 1986. [13] Gianluca Grignani, Marta Orselli, and Gordon W. Semenoff. Matrix strings in a b-field. JHEP, 07:004, 2001. [14] Gordon W. Semenoff. Dlcq strings and branched covers of torii. 2001. http: //arXiv.org/abs/hep-th/0112043/. [15] N . Sakai and I. Senda. Vacuum energies of string compactified on torus. Prog. Theor. Phys., 75:692, 1986. [16] Jr. Callan, Curtis G. and L. Thorlacius. Sigma models and string theory. In *Providence 1988, Proceedings, Particles, strings and supernovae, vol. 2* 795-878. [17] M . J. Duff. Kaluza-klein theory in perspective. 1994. http://arXiv.org/abs/hep-th/9410046/. [18] R. M . Wald. General relativity. Chicago, Usa: Univ. Pr. (1984) 491p. [19] Steven Weinberg. General relativity. Toronto, Canada: Wiley (1984) 657p. References 68 [20] R. Rajaraman. Solitons and instantons. an introduction to solitons and instan-tons in quantum field theory. Amsterdam, Netherlands: North-holland ( 1982) 409p. [21] Alejandro Gangui. Topological defects in cosmology. 2001. http://arXiv.org/abs/astro-ph/0110285/. [22] Mirjam Cvetic and Harald H. Soleng. Supergravity domain walls. Phys. Rept., 282:159-223, 1997. [23] John Preskill and Alexander Vilenkin. Decay of metastable topological defects. Phys. Rev., D47:2324-2342, 1993. [24] Michael McNeil Forbes and Ariel R. Zhitnitsky. Domain walls in qcd. JEEP, 10:013, 2001. [25] T. W. B. 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. Appendix A. Dimensional Reduction 69 A: Dimensional Reduction Here we summarize the procedure of dimensional reduction [29]. We choose to parametrize the D-dimensional spacetime metric G^N AS ds2 = GMNdxMdxN = Gilvd^dxv + Gmn(dxm + Amdx»){dxn + A^dx11). (A. l) where M,N = 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 GMN — ( (~i _i_ n Am An fi Am \ KJmn-ri-v Gr, J (A.2) G MN G»u+ -A" _AHn Qmn + Apm A n This parametrization has a convenient property: \/—\GMN\ — \J — \G^\\/— \GT Using the above relations we find that (A.3) (A.4) ^ [dDxyf-\GMN\e-2* [RD + 2dM$dM$] (A.5) Appendix A. Dimensional Reduction 70 is equivalent to {2nR)k 2 K n /ddxV=G~de-™« [Rd + 4d^ddu$d lGmnGPq (diiGmpdUGnq) \ G m n F $ m F ^ \ (A.6) with Fj»m = d^m - d„AVm (A.7) $ d = $ - -det (G/ m n ) (A.8) 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 dDxV^Ge~2* J TT T T M N L ~ 12 M N L (A.9) The metric parametrization allows us to separate the terms and write - ^ T " f ddxx/^G~de-2^HmnlHmnl + ^HfinlH^1 + ^H(lulH^1 + ^H,uXH^x] (A.10) where the following definitions have been employed: (ATI) ff — P (2 ) _ D p ( l ) n (A.12) Appendix A. Dimensional Reduction 71 Ffflm — ^ li-^vm dvA$n (A. 13) AW=B„m + BmnA^ (A.14) H^x = d„BvX - \ (41)™ F%m + A^lFT) + cyclic perms. (A.15) D _ D i I A / t ( 2 ) _ 1 4 ( l )m / t (2) _ - j ^ m r j ^( l )n / * 1 fi\ Adding all the components together, we find that the effective action can be dimen-sionally reduced to S0,k = ! ddx^rGde-^ [Rd + 4d^dd^d - \GmnG™ (dltGmpdttGnq + dliBmpduBnq) • ^A-17) - \ G m n F $ m F ^ - \G™Hmtu,H<? - ^H^H^} 


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