"Science, Faculty of"@en . "Physics and Astronomy, Department of"@en . "DSpace"@en . "UBCV"@en . "Gardezi, Jaffer"@en . "2009-11-18T19:19:27Z"@en . "2004"@en . "Master of Science - MSc"@en . "University of British Columbia"@en . "An electric version of the well-known magnetic Melvin solution of closed string theory is derived. By analogy with the Kaluza-Klein Melvin solution, which is flat space with points identified under a simultaneous rotation and translation in a compact dimension, an orbifold of Minkowski space involving identifications under a Lorentz boost and a translation is introduced. When dimensional reduction to 9 dimensions is performed, the resulting background involves an electric Kaluza-Klein gauge field, giving rise to the electric Melvin interpretation. As was done by other authors for the magnetic Melvin background, a curved generalization of this orbifold is derived using a series of T-duality transformations. The closed string is quantized on the resulting space, and the string spectrum and partition function are calculated."@en . "https://circle.library.ubc.ca/rest/handle/2429/15248?expand=metadata"@en . "2262356 bytes"@en . "application/pdf"@en . "T H E E L E C T R I C M E L V I N S O L U T I O N I N S T R I N G T H E O R Y By Jaffer Gardezi B. Math. Carleton University, 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 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 M A S T E R OF S C I E N C E in T H E F A C U L T Y OF G R A D U A T E STUDIES D E P A R T M E N T OF PHYSICS A N D A S T R O N O M Y We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y OF BRITISH C O L U M B I A January 2004 \u00C2\u00A9Jaffer Gardezi, 2004 Library Authorization 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. Name of Author (please print) Date (dd/mm/yyyy) Title of Thesis: Jfo^ fUAr)^. tlAvtV) 6o\iAiniA /V? $ - / y ; jUftnry _____ Degree: f\, \u00C2\u00A3 C t Year: 200^ Department of pf)y^j^C, grA A^MyiO/rt) The University of British Columbia Vancouver, BC Canada Abstract A b s t r a c t A n electric version of the well-known magnetic Me lv in solution of closed string theory is derived. B y analogy with the Ka luza -Kle in Me lv in solution, which is flat space with points identified under a simultaneous rotation and translation in a compact dimension, an orbifold of Minkowski space involving identifications under a Lorentz boost and a translation is introduced. When dimensional reduction to 9 dimensions is performed, the resulting background involves an electric Ka luza -Kle in gauge field, giving rise to the electric Melv in interpretation. As was done by other authors for the magnetic Me lv in back-ground, a curved generalization of this orbifold is derived using a series of T-duali ty transformations. The closed string is quantized on the resulting space, and the string spectrum and parti t ion function are calculated. Table of Contents T a b l e o f C o n t e n t s Abstract Table of Contents List of Figures Acknowledgements 1 Introduction 2 The Magnetic Melvin Solution 2.1 The Closed Bosonic String 2.1.1 Solution of the Equations of Motion 2.1.2 Boundary Conditions 2.1.3 Hamiltonian 2.1.4 Light-Cone Gauge Quantization 2.2 The Type II Superstring 3 The Electric Melvin Solution 3.1 The Electric Melvin Background 3.2 Quantization of the Closed String 3.3 Partition Function 4 Conclusion A Dimensional Reduction B T-Duality '.. Bibliography List of Figures iv L i s t o f F i g u r e s 3.1 Regions I and II of the X - T plane 33 3.2 The coordinates t and x 34 3.3 Regions with CTC's in the electric KK-Melv in space 37 3.4 The 2-parameter electric Melvin background 38 Acknowledgements A c k n o w l e d g e m e n t s I would like to thank my supervisor, Dr. Gordon Semenoff, as well as Dr. Dominic Brecher and Dr. Don Witt for their help with this thesis. Chapter 1: Introduction 1 C h a p t e r 1 I n t r o d u c t i o n Before string theory can become a complete description of nature on the most fundamental level, the behaviour of strings in background electromag-netic fields must be understood. One way to introduce electromagnetism into string theory is to add boundary terms to the open string worldsheet action which couple charges at the string endpoints to a background gauge field. This is only an approximation, valid for weak fields, however, since it does not take into account the effect of the energy of the gauge field on the curvature of the background spacetime. Moreover, since it does not change the bulk string worldsheet action, it does not include the effect of such a field on closed strings. It is difficult, however, to include these effects, since this involves finding a conformally invariant string theory background by solving the equations of string theory to all orders in perturbation theory, and few solutions are known. One such solution is the Melvin background. Classically, there exists a class of static, cylindrically symmetric solutions of the Einstein-Maxwell system of equations involving electric and magnetic fields, known as Melvin solutions. An example of such a solution has a line element and electromag-netic field strength given by [6]: ds2 = + (-dt2 + dz2 + dr2) + * V (1.1) V 4 ) V J 0 + Brdr A d

(p + bdx^. By completing the square, the metric, (1.2), may be rewritten in the form: ds210 = -dt2 + dx] + dr2 + ( l + b2r2) (dx9 + ^^^A (L3) r2 1 + b2r2 In this form, dimensional reduction may be carried out along the x 9 direction (see appendix A), giving the 9-dimensional background: r 2 dsl = -dt2 + dx2 + dr2 + - z^-zdip2 (1.4) 1 + bzrz br2 A\u00C2\u00AB = e2\u00C2\u00B0 = l + b2r2 v \u00E2\u0080\u00A2 i + b2r2 where Av is the Kaluza-Klein one-form gauge field, and e2a is the Brans-Dicke scalar. The field strength, dA, generated by Av is that of a magnetic flux tube perpendicular to the r \u00E2\u0080\u0094

\u00E2\u0080\u00A2 tp + bx9. This parameter is analogous to the field strength parameter b in the T-dualized space. Performing this shift, the background (1.5) becomes ds\0 = -dt2 + dx2 + dxl + 1 + b2r2 K> +(b + l>) dx9] [dtp+ (1.6) (b \u00E2\u0080\u0094 dx9 _ br2 2 ( * - * 0 ) 1 l + b2r2 , l + b2r2 Another T-duality transformation along x9 interchanges b and its T-dual b, giving r2 dsl0 = -dt2 + dx] + dx\ + 1 p 2 dip + (b + fc) dx9 [dtp+ (1.7) (b \u00E2\u0080\u0094 dxg^ hr2 U - \u00E2\u0080\u0094 p 2 ( * - * o ) 1 + 6 2r 2 1 + b2r2 This is a generalization of the KK Melvin background, since the choice b = 0 yields (1.2). Dimensional reduction of (1.7) to nine dimensions gives: r 2 dsl = -dt2 + dx2 + dr2 H -, -\u00E2\u0080\u0094^dtp2 (1.8) (1 + b2r2) (l + b2r2) Chapter 1: Introduction 4 A = ^ B ~br\" c2(*-*\u00C2\u00BB] = 1 \"*> l + b 2 r 2 D * 1 + ~b2r2 1 + ~b2r2 e2\u00C2\u00B0 1 + b2r2 1 + ~b2r2 where Av is a Kaluza-Klein gauge field and Bv is a gauge field arising from the dimensional reduction of the NS-NS 2-form. This 9-dimensional back-ground is a generalization of the background (1.4) involving two magnetic gauge fields, with field strength parameters b and b. Thus the string theory background (1.7) is a Melvin background which generalizes the K K Melvin model. It can be shown that (1.7), like (1.2), is also a solution of string the-ory to all orders in a', and that it admits an exact solution of closed string theory [2] [9]. In addition to magnetic backgrounds such as (1.1), there also exist similar solutions of the classical Einstein-Maxwell equations involving electric fields, and it would be useful to find electric Melvin solutions in string theory as well. The study of an electric Melvin background may improve our understanding of aspects of string theory in electric fields, such as Schwinger pair creation of strings. Moreover, it has been suggested that such backgrounds may have implications for string cosmology [11] [12]. The purpose of this thesis is to construct an electric Melvin background analogous to (1.7), and to study closed string theory on this background. The background studied is a generalization of the one described in [11] to a curved space involving two electric field parameters. In chapter 2, the quantization of closed string theory on (1.7) is reviewed. This will illustrate the method that will be used to quantize the string on the electric Melvin background, and will also allow a comparison of the magnetic and electric cases. In chapter 3, the electric Melvin background is derived, and the geometry of the resulting space is described. The closed string is then quantized on this space, and the partition function is calculated. Chapter 2: The Magnetic Melvin Solution 5 C h a p t e r 2 T h e M a g n e t i c M e l v i n S o l u t i o n The pupose of this chapter is to review the quantization of the closed string on the magnetic Melvin background (1.7), as given in [2] and [3], in order to provide the necessary background for an understanding of the electric Melvin solution described in the following chapter. In section 2.1, the closed bosonic string is quantized on this background, and its quantum Hamiltonian is derived in terms of free string oscillator modes. The following section will extend these results to the type II superstring. 2.1 T h e C l o s e d B o s o n i c S t r i n g 2.1.1 Solu t ion of the Equat ions of M o t i o n The bosonic string worldsheet Lagrangian for the background (1.7) can be obtained from the usual worldsheet action [5]: Choosing the conformal gauge, the worldsheet metric hap becomes the flat Minkowski metric and the worldsheet Ricci scalar R is zero, giving Substituting the spacetime metric and antisymmetric tensor from (1.7) into (2.2) gives, for the Lagrangian, S = -S = j\u00E2\u0080\u0094 \d+

1 , x 2% \ x'*, This gives d\u00C2\u00B1tp' 2i -d\u00C2\u00B1x' :d\u00C2\u00B1x\" Substituting (2.15) into (2.13), L = -1 1 + b2x'x'* 8+u - \u00E2\u0080\u0094 {x'*d+x' - x'd+x'*) (2.14) (2.15) (2.16) d-v + - (x'*d-x' - x'd-x'*) 2i + d+x'd.x' The equations of motion of the Lagrangian in the form (2.16) can be solved in terms of free fields. Taking F(x') = , r , 1 , and \u00C2\u00B0 x ' 1+b^x'x'* 2i (x'*d\u00C2\u00B1x' \u00E2\u0080\u0094 x'd\u00C2\u00B1x'*), the equations of motion for u and v are A d-[F(x')(d+u-bA+)] = 0 (2.17) d+[F(x'){d-v + bAJ)] = 0 (2.18) Integrating (2.17) and (2.18) once with respect to a_ and a+, respectively, gives F(x') d+u-bA+ = h+{a+) F(x') d-v + bA-(2.19) (2.20) where h+ and h- are arbitrary functions of a+ and cr_, respectively. From (2.16), the equation of motion for x' is: dL n dL dL d. d+ b + d(d+x') b d(d_x') dx1 = 0 2i x'*h- + d-x\" x'*hA ~-~b2x'*h+h-+ \u00E2\u0080\u0094d+x'*h_ d_x'*h+ 2i + 2i + 0 d+d-x'* + ibh\u00E2\u0080\u009Ed+x'* - ibh+d.x'* + b2h+h_x'* = 0 d+d-x' \u00E2\u0080\u0094 ibh-d+x' + ibh+d-x' + b2h+h-.x' = 0 (2.21) Chapter 2: The Magnetic Melvin Solution 9 It can be checked by substitution that the solution to (2.21) is x = e JbV- -i ibU+X . (2.22)' where U+ and V_ are arbitrary functions of a+ and cr_, respectively, such that h+ \u00E2\u0080\u0094 d+U+ and h- = 9_V_, and X satisfies the free string equation of motion d+d-X = 0. X can be expressed as X = X+ + X_ X+ = e 2 ^ X + (2-23) x_ = e-2i^-x_ where x+ a n d X- a r e single-valued free fields with oscillator expansions f~cV 00 X+ = J2 anexp{-2ina+) ' (2.24) ' n=\u00E2\u0080\u0094oo [o7 \u00C2\u00B0\u00C2\u00B0 X- = \u00C2\u00AB 1 I T E o\u00E2\u0080\u009Eeajp(-2mo-_) (2.25) ' ra=\u00E2\u0080\u0094oo and 7 is chosen so that the physical coordinate, x = rel,p, is single-valued: x(a + Tr, r) = x(cr, r) (2.26) The light-cone coordinates can also be solved for in terms of the free field X. Substituting (2.22) into (2.19) and simplifying gives d+u - (-ibd+UXX* + X*d+X - ibd+UXX* - Xd+X*) F(x') = d+U d+u = d+U{l + b2XX*) - \l2d+uxx* - \l2d+uxx* + \u00C2\u00A3 Zi \il(x*d+x - xe+x*) Z d+u = d+U + \ib{Xd+X* - X*8+X) (2.27) Z Similarly, d-v = d-V- - \ib (Xd\u00E2\u0080\u009EX* - X*d\u00E2\u0080\u009EX) (2.28) z Chapter 2: The Magnetic Melvin Solution 10 These equations can be integrated to give u = U+ + U- - b(p (2.29) v = V+ + V- - b(p where U- and V+ are arbitrary functions of <7_ and a+, respectively, and

n - ( - 2 m ) a > n ] + - E < a \u00C2\u00AB 4a' 4 ~ n = - o o n=\u00E2\u0080\u0094oo 1 \u00C2\u00B0\u00C2\u00B0 + - E ( 2m) ( -2m)a*a\u00E2\u0080\u009E oo 1 0 0 -i oo _ _ _ . + 7 \u00C2\u00A3 na> n + i 7 2 E E \" 2< a\" 4a' Z n = - o o n = - o o n=\u00E2\u0080\u0094oo oo ____ 1 , ^ 2 * (2.52) In the derivation of (2.52), the contribution of the V term vanished by periodicity: fW d-V'_do = - T((9+ - d-)V'_da = - T daVLda = -V% = 0 (2.53) yo Jo Jo Similarly, it can be shown that L \u00C2\u00B0 = ^ f r + \ E (n - 7) 2 K\u00C2\u00B0+(6-6>_x 9 ] ^ ( 6 - 6 ) [d+ip + (6 + b)d+x9] + dTx9 (2.61) The first two terms in (2.61) can be integrated using a relation between

= -r + r 2 [d+r + ir (d+

, r), x(a', r)] = -i6(a - a1) (2.75) where Px = |(Pi + iP2) and P* = \{Pl-iP2). The commutation relations of the Fourier modes can be derived from (2.75). The calculation can be simplified using a relation between the fields x and x* and.the free fields X and X*. In [2] it is shown that the action associated with x and x* is related to a free string action involving X and X* via the duality that relates (1.2) to (1.7). This relation implies the equivalence of the canonical commutation relations (2.75) to those of the free fields, [Px(a,r),X*(a',r)} = [P*x(a,r), X(o>, r)] = -iS(a - a') (2.76) where, since the action of X is the free string one, Px = 4^9rX is the usual canonical momentum from free string theory. Substituting the mode expansions (2.24) and (2.25) in the left-hand side of (2.76) gives 11[drX{a,T)tX' (cr',r)] 47rcy 1 47T oo e 2 i 7 ( r + a ) (ij - in) ane-2in{T+a) + e - 2 i ^ T ^ -]T ( - i 7 - m ) a n e - 2 i n ( T - C T ' , e - 2 i 7 ^ \u00C2\u00A3 a*ne2in^ + 3 2\u00C2\u00ABT(T-(T) E < * 2m(r\u00E2\u0080\u0094 an+k, a* \u00E2\u0080\u0094> a*n+k, an \u00E2\u0080\u0094>\u00E2\u0080\u00A2 a\u00E2\u0080\u009E_fc, and 5* \u00E2\u0080\u0094> a*n_k are made. The Hamiltonian is invariant under these redefinitions, since they leave the infinite sums over oscillators unchanged. The theory is therefore invariant under shifting 7 by an integer, and 7 may be restricted to the range 0 < 7 < 1. For 7 in this range, the normalized creation and annihilation operators corresponding to the Fourier modes in (2.78) and (2.79) are given by: bl n+ bL ui\u00C2\u00B1 (2.80) where the b operators satisfy bo,bl = b~n\u00C2\u00B1 > b\n\u00C2\u00B1 (2.81) 1, MSI = 1 Using these quantized oscillator modes, an operator expression can be derived for the quantum Hamiltonian corresponding to the classical Hamil-tonian (2.56). The quantized version of (2.52) and (2.54) is obtained by Chapter 2: The Magnetic Melvin Solution 21 symmetrizing the sums over modes, giving \ __ in + T ) 2 a*nan = j E (n + 7? \u00C2\u00AB a n + ana*n) (2.82) ^ _ _ ( \u00C2\u00AB - 7 ) 2 ^ a n = J E (n - 7) 2 (S;a\u00E2\u0080\u009E + ana*n) (2.83) In terms of the normalized operators (2.81), (2.82) is 1 4 7 E (n + 7 ) 2 \u00C2\u00AB \u00C2\u00AB n + ana*n) \u00E2\u0080\u00A2 oo oo = o E (\u00C2\u00AB + 7 ) (bLbn- + &\u00E2\u0080\u009E-&],-) + - E in - 7) \u00E2\u0080\u00A2 Z n = l Z n = l (bn+bl_, + bl+bn+) + i 7 (blb0 + b0bl) (2.84) The infinite sums arising in the normal-ordering of this expression can be evaluated using a generalized zeta function regularization: 00 2 n E ( \" + C) = - T 7 5 + 9 C ( 1 - C ) ( 2 - 8 5 ) n = l i Z Z where c is an arbitrary constant. This gives, for (2.84), \ E ( n + 7 ) 2 (o>\u00E2\u0080\u009E + a\u00E2\u0080\u009Ea*) n 00 1 1 \u00C2\u00B0\u00C2\u00B0 = E ( n + 7 ) ^ - 6 n - - ^ T + T 7 ( l - 7 ) + E ( n - 7 ) -n = l / 4 4 n = l - 24 - 47 (1 + 7) + 7 & 0 & 0 + ^ 7 00 00 = E ( \u00E2\u0084\u00A2 + 7 ) & t A - + E ( \u00E2\u0084\u00A2 - 7 ) O n + + 7 ^ o ( 2- 8 6) n = l n = l h -7(1 ~ 7) 12 2 ' V '/ Similarly, for the left-movers, \ E ( n ~ 7 ) 2 ( 5 n 5 n + 00 = E ( \" - 7 ) ^ - & n - + E ( ^ + 7 ) ^ A + + 7 ^ o (2.87) n=l n\u00E2\u0080\u00941 1 1 , - 7 7 2 + 2 - 7 ( 1 - 7 ) Chapter 2: The Magnetic Melvin Solution 22 Substituting the expressions (2.86) and (2.87) for the quantized version of the infinite sums in (2.56) gives the following expression for the quantum Hamiltonian: -I CO H = \u00E2\u0080\u0094(4w2R2 + S 2 - P 2 ) + Y / ( n + y)bn-bn- + n = l \u00C2\u00A3 (n - 7) h{+bn+ + + \u00C2\u00A3 (n - 7) b L k - + (2.88) n = l oo n = l \u00C2\u00A3 (n + 7) ~b{+bn+ + jbl~b0 - bwRJ - 2 + 71 = 1 7 ( 1 - 7 ) + ^ ^ The normal-ordering constant in (2.88) was obtained by adding the normal-ordering constants in (2.86) and (2.87) to the usual contributions from the remaining 22 trivial coordinates. The infinite sums in (2.88) can be identified with level number and angular momentum operators. Right- and left-moving level number operators can be defined by oo NR = \u00C2\u00A3 n (&],_&\u00E2\u0080\u009E_ + b]n+bn+) n=l oo NL = J2n{bn-~bn-+~bi+~bn+) (2.89) 71 = 1 Similar expansions can be obtained for the angular momentum operators JL and JR, using (2.31) and (2.23). For example, the classical expression for JR is JR = T do-(X-d-X*_-X*_d-X-) Ana' Jo v ' % - j* da_ [X- (2ijX- + d-X-) ~ X- (-2i7X- + d-X-) \u00C2\u00A3 (2.7 + 2m) a*nan - \u00C2\u00A3 (-2.7 - 2m) a > \u00E2\u0080\u009E .71 = \u00E2\u0080\u0094OO 7l= \u00E2\u0080\u0094 OO ^ OO -- \u00C2\u00A3 (n + j)a*nan (2.90) Chapter 2: The Magnetic Melvin Solution 23 Symmetrizing to obtain the quantum version, JR = - 7 E (n + j) (a*nan + ana*n) (2.91) n=\u00E2\u0080\u0094oo where a hat is used to distinguish this operator from its normal-ordered form. Similarly, oo h = - T E (\" - 7) ( & X + a na;) (2.92) n=\u00E2\u0080\u0094oo Using (2.80) and normal ordering, -j^ oo oo JR = ~o E ( f e i - & n - + & \u00E2\u0080\u009E _ & ] , _ ) + - E (bn+b]n+ + bll+bn+) n=\ n=l -\(b0b0 + b0bl) oo oo -I = - E bLbn- - C (0) + E O n + + C (0) - &J&\u00E2\u0080\u009E - r = - E bl-bn- + - + E - 2 ~ 6\u00C2\u00AB^o - 2 n = l n = l oo ^ = E ( O n + - bi_bn_) - blbQ - - (2.93) n-1 where zeta function regularization was used to compute the normal ordering constants. Similarly, it can be shown that oo JL = E {bn+bn+ ~ bnX-) + b0b0 + - (2.94) n=l 1 The normal ordering constants of JL and JR cancel in the angular momentum operator J = JL + JR, giving a normal ordering constant of zero for J. The first and second terms in (2.93) are the spin, SR, and the orbital angular momentum, or Landau level, IR, respectively: Similarly, JR = -IR + Sr-^ (2.95) JL = IL + Sl + 1 (2.96) Chapter 2: The Magnetic Melvin Solution 24 Using (2.60), (2.72), (2.89), (2.93), and (2.94), the Hamiltonian, (2.88), can be expressed in terms of N and J : H = J -8a' 4w2R2 + 4a'2{^-bj) \u00E2\u0080\u0094 4a'2 E2 R \u00E2\u0080\u00A2+ [NR - lJR, + [NL + 7JL) - bwRJ + \a'b2J2 - 2 + 7 (1 - 7) Z = -\a'E2^NL + NR + \a'(^-bj) + (2.97) 1 ( wR GbJ) - 7 ( ^ - ^ L ) - 2 + 7 ( 1 - 7 ) This operator expression for the Hamiltonian must be supplemented by the level-matching condition, L 0 \u00E2\u0080\u0094 L0 = 0. From (2.52) and (2.54), P-P- ^1X-( __ ^ * P X 1 4a' = 4_ [\" H \" + P \" l [\" H \" + P 1 - 4cV - a'&j) + pu\ [{wR - a'bJ) + pv] + {NR - yJR) - (NL + 1JL) = ~ ^ i { w R - a'~bJ) (PU + PV) + NR-NL-JJ = \u00E2\u0080\u0094^-l(wR-a'bj)s + NR-NL-'yJ (wR - a'bJ) m R bJ)+NR-NL-1J T a mb . ~ 0 mw \u00E2\u0080\u0094 wRbJ \u00E2\u0080\u0094\u00E2\u0080\u0094 J + a bbJ R = \u00E2\u0080\u0094mw + 7 J + NR \u00E2\u0080\u0094 NL \u00E2\u0080\u0094 \u00E2\u0080\u00A2yJ \u00E2\u0080\u0094 NR \u00E2\u0080\u0094 NL \u00E2\u0080\u0094 mw Therefore, the level matching condition is: NR \u00E2\u0080\u0094 NL \u00E2\u0080\u0094 mw N R - N L - 7 J (2.98) (2.99) The Hamiltonian (2.97) is a periodic function of the parameter 7, which also implies periodicity in the magnetic field strength parameters b and b. Chapter 2: The Magnetic Melvin Solution 25 The Hamiltonian was derived for 7 in the range 0 < 7 < 1. For 7 outside this range, because of a change in normal ordering constants, an integer must be added to 7 so that it satisfies this condition. From the form (2.73) of 7, it can be seen that this implies invariance of the spectrum under the shift b-+b+^ (2.100) XL for any integer n, and that, in (2.97), b should be restricted to the interval . 0 < b < 4 (2.101) R This periodicity in b is to be expected, since, from the form of the K K Melvin metric (1.2), the background space is such that a translation in the \u00C2\u00A39 direction is accompanied by a rotation in a plane by an angle 2irbR, where R is the radius of the compact direction. The shift (2.100) thus adds 2im to this angle, leaving the theory unchanged. Similarly, the spectrum is periodic under the shift (2.102) a Consequently, b in (2.97) should be restricted to the interval 0 < b < \u00E2\u0080\u0094 (2.103) a' This periodicity also follows from the periodicity in b, since b is the equivalent of b in the T-dualized space, for which the radius of the compact dimension is i. 2.2 Type II Superstring The results of section 2.1 for the closed bosonic string can be extended to the type II superstring. This can be done by constructing a worldsheet supersymmetric version of the action (2.2) [3]. In conformal gauge, the RNS superstring Lagrangian has the form L = 1 + b2x'x'* d+u- y. (x'*d+x' - x'd+x'* - 2i\'*L\'L) 2% [d-v + - (x'*d-x' - x'd-x'* - 2iX'RX'R) 2i +iX'*Ld-X'L + d+x'd-x'* + iX'*Rd+X'R (2.104) Chapter 2: The Magnetic Melvin Solution 26 d+-^^ = 0 As in (2.16), the coordinate x is given by x = relv> = xi + ix2, where x\ and x2 are cartesian coordinates in the r \u00E2\u0080\u0094 (p plane. XL and XR are the left- and right-moving components of the spinor, A = Ax + iX2, corresponding to the x coordinate. The primes indicate the same rotation of the physical fields as in (2.9): x' = e^bX9+h)x (2.105) y = ei(bxg+bt) ^ The solution of this theory is similar to that of the bosonic theory. The equations of motion (2.19) and (2.20), for u and v become, dL d{d+u) d+ [F(x') (d-v + bA- - bX'RX'R)] = 0 F{x') (d-v + lA. - bX'RX'R) = h- (2.106) and d~dltv) = \u00C2\u00B0 c9_ [F(x') (d+u - bA+ + bX'lA _)] = 0 F(x') (d+u - bA_ + bX'*LX'L) = h+ (2.107) The equation of motion, (2.21), for x' and its solution, (2.22), in the bosonic string case are unchanged in the superstring theory, except for the new form oih+ and h_, (2.106) and (2.107). Thus the solution of (2.106) and (2.107) for d-V and d+u parallels (2.27) and (2.28), except for the addition of the fermionic term. Therefore, d+u = d+U+ + \ib (Xd+X* - X*d+X) - bX'*LX'L (2.108) d.v = d-V- - \ib {Xd-X* - X*d-X) + bX'RX'R (2.109) Integrating (2.108) and (2.109) gives r<7+ -u = ubosonic - / bX'*LX'Lda+ + Ku(o-) \u00E2\u0080\u00A2 (2.110) . v = vbosonic + / bX'*RX'Rda- + Kv(a+) (2-111) Jo Chapter 2: The Magnetic Melvin Solution 27 where ubosonic and Vbosonic are the corresponding solutions for the bosonic string, (2.29), and Ku and Kv are arbitrary functions. Taking Ku = f0a~ b\'RX'Rda- and Kv = \u00E2\u0080\u0094 /0CT+ bX'*LX'Lda+, the solution in the super-string case becomes the same as that in the bosonic string case, except with fermionic contributions to the angular momentum currents: J\u00C2\u00B1{a\u00C2\u00B1) = 7 ^ 7 7 d\u00C2\u00B0\u00C2\u00B1 (X\u00C2\u00B1d\u00C2\u00B1X*\u00C2\u00B1 - X*\u00C2\u00B1d\u00C2\u00B1X\u00C2\u00B1 +2i\'l\'L R R For the fermions, the equations of motion are given by dL dL (2.112) d+ and d. d{d+x'R) ^ dX'R d+X'R + ibh+X'n dL _ dL d(d-\'L) ~ WL d-X'L - ibh-\'L 0 0 0 0 (2.113) (2.114) As can be checked by substitution, the solution is (c.f. (2.22) and (2.23)): e-ilu++ilv. A r j e-iiu++ibv.AL t X'R = X'L = AR = A L = e2i^rj+ d+AR = 0 c L A L = 0 (2.115) where r]+ and 77_ are free fermionic fields with expansions U E -2ira- (2.116) / 2a 7 E dne~ \u00E2\u0080\u00A22ina-and similarly for r]+NS^ and rj+R\ Since the boundary conditions (2.32) are unchanged, the form of U\u00C2\u00B1 and V\u00C2\u00B1 is still (2.33), so that 7 is the same as in Chapter 2: The Magnetic Melvin Solution 28 (2.39). Moreover, it can be shown using (2.104) that the form of E and pxg, (2.60) and (2.71), is unchanged, so that 7 is again given by (2.73). As in the bosonic string theory, the energy-momentum tensor reduces to the free string theory form: T++ = d+U+d+V+ + d+Xd+X* + iA*Ld+KL (2.117) T__ = d-U-d-V- + d.Xd-X* + iA*Rd-AR The bosonic parts of the Virasoro operators L0 and L0 are given by (2.52) and (2.54). Substituting (2.115) and (2.116) into (2.117) gives the fermionic parts, I/Q and L(, of L0 and L0: a = ^j\ARd.ARda j E r e z + | (r + 7) c*cT , NS sector 1 T,nGZ (n + 7) d*ndn , R sector (2.118) If = i ^rez+\iT ~l)Kcr , NS sector \u00C2\u00B0 I T,nez {n - 7) d*ndn , R sector The classical Hamiltonian is given by (2.56) with (2.118) and (2.119) added. Quantization of the theory in the light-cone gauge leads to the same Hamiltonian, (2.97), except with fermionic contributions to N and J. From the Lagrangian (2.104), the anticommutation relations for the right-moving fermion fields are, {\R,{a,T),\R(a',T)} = 2ira'5{o--o-') (2.120) From (2.115), this anticommutator is the same as that of the free fields, (2.116), since the phase factors cancel. Therefore, the oscillator commutation relations are the same as those of free string theory, {c*r,Cl} = 5rl (2.121) {dn,dm} = 5\u00E2\u0080\u009E Jnm The quantum version of L{ and L{ is obtained from (2.118) and (2.119) by antisymmetrizing the oscillator products. For LQ in the Ramond sector, this Chapter 2: The Magnetic Melvin Solution 29 gives: I oo 4 = 2 5_ (n + 7) (d*ndn - dnd*n) n=\u00E2\u0080\u0094oo 1 OO 1 oo = 7 5 \u00C2\u00A3 ( \u00E2\u0084\u00A2 + 7 ) \u00C2\u00AB ^ - \u00C2\u00AB ) - x \u00C2\u00A3 ( n - 7 ) - (2-122) Z n = l Z n = l (d-nd-n - d-nd*_n) + ^7 [dj, d0] This expression can be normal-ordered using the expression for generalized zeta function regularization, (2.85). The result is OO OO I L{ = J2(n + ~,)d*ndn + J2(n-l)d-nd*_n + -j[d*0,d0} (2.123) 71 = 1 71=1 Z 1 1 2 _i y 12 2 ' A similar calculation can be used to derive the normal-ordered expressions for the left movers and for the NS sector. The normal-ordered level number operators are _ f Nto, + EZi r (c;cr + c_PcI r) , iVS sector ^ * ~ j iV 6 o , + E~=i n \d*ndn + d_ n d*\u00E2\u0080\u009E) , R sector and the same for NL with tildes over the oscillators, where Nbos is the bosonic part, (2.89). The classical angular momentum operators are: if7 JR - (JR)bos + ~.\u00E2\u0080\u0094: / (2iX*RXR)da-4.TTOC Jo 1 fn = {JR)*. ~ ^ Jq V-V-da-= { ( J * L ~ C ^ ' NS sector ^ ^ and similarly for J L . Antisymmetrizing and normal-ordering to obtain the quantum version, j = ) {Jx)bos + Zrez+\u00C2\u00B1{c;cr-C-rc*_r) , NS sector R (jR)bos + En=i {d*ndn - d.nd*_n) + \ [d*Q, d0] , R sector Chapter 2: The Magnetic Melvin Solution 30 When the fermionic parts of L0 and L0 are added to the bosonic contribu-tions, the superstring Hamiltonian becomes 1 1 f m \ 2 H = --a'E2 + NL + NR + -a'(--bJ) + (2.127) tfbj) - J { J R - J l ) + C + J where c is the normal-ordering constant of free type II superstring theory. The range of 7 in the superstring Hamitonian (2.127) is \u00E2\u0080\u0094 1 < 7 < 1, rather than 0 < 7 < 1 as in the bosonic theory. This is because super-string theory is only invariant under rotations by an even multiple of 27r, since fermions are multiplied by -1 when rotated by an odd multiple of 2TY. Consequently, the shifts (2.100) and (2.102) under which b and b are periodic become 2n . b -> b+\u00E2\u0080\u0094 (2.128) ix b -> b-\ From the form (2.73) of 7, this implies that 7 is periodic only under even integer shifts. Chapter 3: The Electric Melvin Solution 31 C h a p t e r 3 T h e E l e c t r i c M e l v i n S o l u t i o n 3.1 T h e E l e c t r i c M e l v i n B a c k g r o u n d The KK-Melvin metric ds210 = -dt2 + dx] + dx\ + dr2 + r2 (dip + bdx9)2 (3.1) is equivalent to an orbifold of Minkowski space which identifies points under a combination of a translation in the x 9 direction and a rotation in the r \u00E2\u0080\u0094 ip plane [10]. This can be shown by introducing the coordinate ip' =

1 E (3.20) For spacetime regions satisfying (3.20), any finite translation in x9 with its accompanying boost becomes everywhere a timelike translation in spacetime, so that (3.11) identifies points separated by a timelike interval, and there exist CTC's connecting these points. Thus, these regions should be excluded when discussing string theory on the electric Melvin space. The regions are shown in Figure 3.3. The electric KK-Melv in background can be generalized to a curved 2-parameter background in the same way as was done for the corresponding magnetic background in chapter 1. Starting with (3.14), a T-duality trans-formation along xd yields B. ds = dxs =p dt + dxl ^ Et2 t2 XXg = \u00C2\u00B1 1 \u00C2\u00B1 EH2 2 ( * - * o ) _ _ (dx + Edx9) (dx - Edx9) (3.21) 1 \u00C2\u00B1 E2t2 1 \u00C2\u00B1 E2t2 The shift x \u00E2\u0080\u0094> x + Edx9 introduces a second electric field parameter T-dual to E. This gives. t2 (3.22) ds' = dx2s =F dt2 4- dxl \u00C2\u00B1 1 \u00C2\u00B1 E2t2 dx+(E + E) dxg] \u00E2\u0080\u00A2 (3.23) B. dx + - E) dxc, Et2 XXg \u00C2\u00B1 - 2 ( * - * 0 ) l\u00C2\u00B1E2t2 Another T-duality along XQ gives 1 \u00C2\u00B1 E2t2 ds2 BT dx2s T dt2 + dxl ^ dx + (E - E) dx9 Et2 1 \u00C2\u00B1 E2t2 dx+ (E + E) dx9] \u00E2\u0080\u00A2 (3.24) \u00C2\u00B1 -1 \u00C2\u00B1E2t2 2 ( * - # o ) _ 1 \u00C2\u00B1 E2t2 When this background is dimensionally reduced, the resulting background has two electric gauge fields similar to that in (3.16) with parameters E and Chapter 3: The Electric Melvin Solution 37 X Figure 3.3: Regions with CTC's in the electric KK-Melvin space. E. (3.24) can therefore be considered an electric Melvin background, like the background (3.14) of which it is a generalization. The properties of the spacetime (3.24) are similar to those of (3.14), but it has some additional pathologies. Timelike singularities occur in region II at \t\ = A. This results from the fact that, at these points, the x9 direction along which T-duality was carried out to produce (3.24) becomes null in (3.14). Also, the shift (3.22) results in an identification of points in the T-dualized space: / a' ~a'\ {X9,X')EE \X9 + 2?T-,X'+ 2TTE\u00E2\u0080\u0094\ (3.25) where x' = x + Ex9. This produces an additional region, t > A, where CTC's occur. These properties are illustrated in Figure 3.4. Chapter 3: The Electric Melvin Solution 38 3.2 Quantization of the Closed String Like its magnetic version, the electric Melvin background (3.24) admits an exact solution of closed string theory. Since the steps involved in the quan-tization are very similar to those in the magnetic case, only an outline will be given in this section. In a suitably-chosen coordinate system, the Lagrangian reduces to a form similar to (2.16), for which the equations of motion may be solved in terms of free fields. The bosonic string Lagrangian for (3.24) is, omitting the flat space dimensions, xs, L = ^d+td-t + d+xgd-x$ \u00C2\u00B1 d.x + (E~E) 8-XQ Introducing coordinates t2 1 \u00C2\u00B1 EH2 d+x+ (E + E)d+x9] -(3.26) x' = x 4- Ex9 Chapter 3: The Electric Melvin Solution 39 X+l = tex' = eEx9X+ X~' =\u00E2\u0080\u00A2 Tte (3.27) -x = e - E x 9 X -(3.26) reduces to E2X+X-d+x9 + ^ {x-'d+x+l - x+'d+x-') (3.28) d-x9 - | (x-'d_x+l - x+'d_x-') d+X+'d-X~' T h e l igh t -cone coord ina tes u a n d v t ha t were used i n the magne t i c case canno t be used i n (3.28), because of the n o n - t r i v i a l t i m e dependence. D e f i n i n g F(X-',X+I) = - 1 EX+X-A \u00C2\u00B1 = - (x-'d\u00C2\u00B1x+l - x+,d\u00C2\u00B1x-') the equat ions of m o t i o n of x9, X+l, a n d X ' become F (d+x9 + EA+) = h+ (=n-^ \" Z n=-oo z n=-oo oo oo * 7 E n a n Q = n + 2 E n 2 ( l n a - n n=\u00E2\u0080\u0094oo n=\u00E2\u0080\u0094oo = + \u00C2\u00A3 t ay_n + l E (n-^7)2a>:\u00E2\u0080\u009E ^ n=-oo z n=-oo (3.46) The second term, which is the oscillator contribution from Xg, was not present in the magnetic case because these oscillators were eliminated by the choice of light-cone gauge. The light-cone gauge cannot be used in the electric case, because of the non-trivial time dependence of the Lagrangian. The expression for LQ is Z 1 o u i o u (3.47) Therefore, the Hamiltonian is: H = LQ + Lo J _ 4a 7 ^ 0 0 - (wR - a'Ej) + -p + 4a' L (wR - j) + ^p + oo 2 E a V - n + 2 E + n=\u00E2\u0080\u0094oo oo oo 2 E ( n - n ) 2 ^ a : n + 2 E (\u00C2\u00AB + i 7 ) 2 ^ s : \u00E2\u0080\u009E n=\u00E2\u0080\u0094oo n=\u00E2\u0080\u0094oo' 1 1 00 -| oo = -L( 4 \u00C2\u00AB, a # + \u00C2\u00A3 2 \u00C2\u00ABV-\u00E2\u0080\u009E + \u00C2\u00A3 E \u00C2\u00AB n \u00C2\u00AB 9 - n + (3-48) \u00C2\u00B0 \" ^ n=-oo L n=-oo 2 E (n - ^ )2 On a l n + \u00C2\u00AB ^ (n + ^ )2 o+a:\u00E2\u0080\u009E -n=\u00E2\u0080\u0094oo n=\u00E2\u0080\u0094oo EwitV + \a'E2J2 The usual contribution from the 23 flat space coordinates should be added to this Hamiltonian. Chapter 3: The Electric Melvin Solution 43 Quantization of this theory is similar to that of the magnetic Melvin the-ory, except that the light-cone gauge cannot be used because of the non-linear time dependence. B R S T quantization can be used instead. The canonical commutation relation is 1 1dTZ+ (a,r),Z~ (cr',r) 4na = -i5 (a - a') -dTZ~ (a,r))Z+ (a',r) (3.49) The left-hand side is i [ ^ + ( ^ ) , r K r ) ] = _ L J e 2 7 ( r + * ) ( 7 - m ) a + e - 2 ^ + ^ + e - 2 ^ - ^ -47T \u00E2\u0080\u009E_ ^ 00 00 \u00C2\u00A3 ( - 7 - m ) a + e - 2 i \" ( r - ^ , e - 2 ^ T + f f ) \u00C2\u00A3 a ~ e - 2 i n ^ + ; 2 7 ( r - a ) ^ a ~ e _ 2 i n ( r _ 0 ' ) = i^'^W \u00C2\u00A3 e-^-^[a:,~aZn](n + ll) I n=-oo 1 00 1 i E e 2 i \"(\u00E2\u0080\u0094')[a+ a : J ( n - i 7 ) n=\u00E2\u0080\u009400 J This leads to the commutators an > fl-m] = 2 (n - v y ) - 1 5 n + m fln > \u00C2\u00B0 - m ] = 2 (n + 17 ) _ 1 5 \u00E2\u0080\u009E + m The normalized creation and annihilation operators are: (3.50) (3.51) (3.52) fe.. n+ t n\u00E2\u0080\u0094 t v+aln bn+ = uj+an 6\u00E2\u0080\u009E_ = w _ a + w _ a l n bn + (3.53) Chapter 3: The Electric Melvin Solution 44 b~l = y^7flo b0 = J^i^ u\u00C2\u00B1 = y^(n\u00C2\u00B1z7), n = l,2,... The b operators satisfy the commutation relations (2.81). The commutation relations (3.51) and (3.52) can be used to normal-order the expression (3.48) to produce a quantum Hamiltonian. By symmetrizing and normal-ordering, it can be shown that 2 oo - E (n ~ an a - n = E (n ~ h) bi-bn- + (3.54) 1 n n=l oo ^ ^ E ( n + fy) 6\u00E2\u0080\u009E+&\u00E2\u0080\u009E+ + hblb0 - \u00E2\u0080\u0094 + -17 (1 - 47) n=l i Z ^ 00 oE(\" + *7)2 5 n a - n = E (n + *7) & n - & n - + (3.55) ^ n n=l oo _ ^ 1 E (\u00E2\u0084\u00A2 ~ \u00C2\u00AB7) &n+&r.+ + *7&0&0 - ^ + -17 (1 - 17) n=l The quantum Hamiltonian is therefore 1 oo oo H = 7 ^ ( 4 ^ 2 + p 2 ) + E \u00C2\u00AB + E \u00C2\u00AB + 8 a n= l n = l 00 oo 53 (n - 17) 6\u00E2\u0080\u009E_6n_ + E (n + *7) &n+&n-r + 17&J&0 + (3.56) n=l n=l 00 00 53 ( n + *7) bl-K- + 53 ( n - ?'7) &n+&n+ + f-y6j60 -n=l n=l \u00C2\u00A3w.RJ - 2 + 17 (1 - 17) + -a'E2J2 2 This can also be expressed in terms of level number operators and boost operators. The level operators are given by 00 NR = Y,[n{hn-bn-+bUbn+)+a\al] (3.57) n=l oo N L = E [n (bl-bn- + b ]n+bn+) + a9_na9n] n=l Chapter 3: The Electric Melvin Solution 45 From (3.38), the boost generators have the form CO o o o o n - 1 oo JR \u00E2\u0080\u0094 i 5_ bl_bn- - i \u00C2\u00A3 b]l+bn+ - iblb0 (3.58) (3.59) n = l n-1 Substituting (3.57), (3.58), and (3.59) into (3.56), 1 /m \ 2 H = NL + NR + -a1 - EJ) + (3.60) /oJEjj - 7 {JR - JL) - 2 + _ 7 (1 - .7) 1 ( w R 2 IT rV The level-matching condition, LQ \u00E2\u0080\u0094 L0 = 0, is ]VR \u00E2\u0080\u0094 iVr, = m \u00E2\u0080\u009E (3.61) These results can be extended to the type II superstring. The RNS su-perstring Lagrangian is L = 1 1 - E2X+X~ d+X9 + j (x-'d+x+l - x+,d+x-'- (3.62) E 2iAZ'A^)] d.x9 - - (x-'d-X+l - X+'d-X~' - 2iXR'XR + d+X+'d-X-' + iX^'d+XR1 + iXl'd-Xf where A^ and A^ are the left- and right-moving components of the spinors X\u00C2\u00B1 = Ax \u00C2\u00B1 Ay, which correspond to the coordinates X\u00C2\u00B1, and (3.63) Quantization of this theory leads to a Hamiltonian with the same form as (3.60), except with the operators N and J replaced with their fermionic ver-sions, and the normal-ordering constant changed to c + ry, where c is the normal-ordering constant of free superstring theory. Chapter 3: The Electric Melvin Solution 46 3.3 Partition Function The partition function for the bosonic string in the electric Melvin back-ground can be calculated from the usual expression in terms of a trace over states of the string: . J2 .23 co Z = j-rjl[dPa T r e x P [ 2\u00E2\u0084\u00A2 ( r L o - flo)] (3.64) a = l vn.w\u00E2\u0080\u0094\u00E2\u0080\u0094oo where r = T\ + ir2, T\ and r 2 are the modular parameters of the torus, pa are the momenta in the 23 free directions, and m and w are the momentum and winding modes in the x 9 direction. The domain of integration for T_ and r 2 is the fundamental region, = { T _ , T 2 | \T\ > 1, \u00E2\u0080\u0094 ^ < T_ < y 0 < r 2 < ooj (3.64) can be expressed as __ T2 ^ 2 3 ^ 0 - ^ 3 (3.65) (3.66) where Z23 and _ j are the usual contributions of 23 free bosons and of the B R S T ghosts, respectively, and Z3 is the contribution of the three non-trivial coordinates, X+, X~, and x$. Z3 is given by oo Z-i = Trexp fain ( _ 0 - L0) - 27rr 2F (3.67) m,w=\u00E2\u0080\u0094oo where H is given by (3.60) and L0 \u00E2\u0080\u0094 L0 = NR \u00E2\u0080\u0094 NJJ \u00E2\u0080\u0094 mw. The calculation of Z% is facilitated by expressing H in the form 1 2 D 2 - - a' (m H (E + E)jR+(E- E) (3.68) JL}}2 - wR 7 2 + ry - 2 Substituting (3.68) into (3.67) gives OO r \u00C2\u00A3 Trexp < \u00E2\u0080\u0094 2 (E + E)JR-(E- E) JL] + 2a'E2JRJL + iTTlTi mw + N - N \u00E2\u0080\u0094 27rr2 m,w=\u00E2\u0080\u0094oo NR + NL + J { ^ - ((E + E)JR+(E- E) JL))2 - (3.69) wR i[(E + E)JR-(E- E) JL) + 2a'E2JRJL + j2 + ij - 2]} Chapter 3: The Electric Melvin Solution This expression contains terms quadratic in JL and JR, which must be lin-earized before the trace can be calculated. The j2 term can be linearized using the identity e~2^2 = ^ 2Y2 p due \u00E2\u0080\u0094 2 7 T T 2 I / \u00E2\u0080\u0094 4 T T I T _ 7 I ' (3.70) (3.70) can be verified by completing the square in the exponent and evaluat-ing the resulting Gaussian integral. From the form (3.60) of H , it can be seen that the remaining terms linear in 7 in (3.69) and (3.70) can be absorbed into a redefinition of JR and JL'-R J'r J R - i v - -i J L + iv + ~-i (3.71) The contribution to (3.69) of the fourth term in (3.68) becomes linear in JL and JR after Poisson resummation over m is carried out. After completing the square, the m-dependent part of (3.69) becomes J2 exp-Ur2a - + 1\u00E2\u0080\u0094\u00E2\u0080\u0094 - (E + E) J R - (3.72) m,w=\u00E2\u0080\u009400 \u00E2\u0080\u00A2 ( E ~ E ) J L ] 2 + 2mRwriT2 [(E + E ) J R + ( E - E) JL] + TTR2W2T2 \ a'r2 J The Poisson resummation formula is [5]: \u00E2\u0080\u0094TT (m \u00E2\u0080\u0094 6) 2 1 __ exp \u00E2\u0080\u0094 \[a \u00C2\u00A3 exP (^\u00E2\u0080\u0094^CLW'2 + 2iTibw') (3.73) where a and b are arbitrary constants. Using (3.73), (3.72) becomes R A f TTR2 __ exp w12 + ^ ^ w w 1 + 2niR [(E+ (3.74) E) J R + ( E - E) JL] W' - 2mRT! [(E + E) J r + (E-E)jL TTR2T? ' W- ; W OL'T2 Chapter 3: The Electric Melvin Solution 48 Substituting (3.70) and (3.74) into (3.69), and using (3.71), (3.69) reduces to Z* = V2R IR f\u00C2\u00B0\u00C2\u00B0 r / \ i =\u00E2\u0080\u00A2 / duTrexp \u00E2\u0080\u00942ITT2 (v2 \u00E2\u0080\u0094 2) exP \u00E2\u0080\u00A2KR2 OL'T2 (3.75) w,w \u00E2\u0080\u0094\u00E2\u0080\u0094oo (w' \u00E2\u0080\u0094 TW) (w' \u00E2\u0080\u0094 fw)} exp \2iri (TNR \u00E2\u0080\u0094 TNL)] exp [2m (\u00C2\u00AB/ \u00E2\u0080\u0094 TW) \u00E2\u0080\u00A2 R(E + E) J'r] exp [2m {w' - fw) R(E - E) J'L] \u00E2\u0080\u00A2 exp ( - ^ T 2 a ' E 2 J ' R J ' L ) The last factor in (3.75) can be made linear in the boost operators by using the identity 4_ r2 J d\i\2exp ir2\Ta'EJ'L] 47T T2 X R 2jd 7 [w' \u00E2\u0080\u0094 Tw) \u00E2\u0080\u0094 (3.76) R X H \u00E2\u0080\u0094j (w' - fw) - iT2Va'EJ'R 2Va where A = Ai 4- iX2. This identity can be proved by converting the right-hand side into a product of Gaussian integrals over Ai and A2. When (3.76) is inserted into (3.75), the first and last factors in (3.75) cancel out, giving Z* = AV2R \u00E2\u0080\u00947=\u00E2\u0080\u0094< ,47TT2 /OO . . duexp (\u00E2\u0080\u00942TXT2V2\ \u00E2\u0080\u00A2 -00 ^ ' (3.77) 0 0 r\u00C2\u00B0\u00C2\u00B0 ( 4TT E / dXidX2exp { \u00E2\u0080\u0094 \u00E2\u0080\u0094 w,w \u00E2\u0080\u0094\u00E2\u0080\u0094OO R + \u00E2\u0080\u00947= r2 XX-R 2 v V (\u00C2\u00AB/ \u00E2\u0080\u0094 TW) X (w' \u00E2\u0080\u0094 TW) X ) J J Trexp [2ni (TNR - XJ'R). Trexp [-2m (fNL + xJ'L). where X = -Va' R 2EX + E^= [w1 - TW) Vet' (3.78) The traces in (3.77) can be evaluated using (3.57), (3.58) and (3.59). For the right-movers, this gives Trexp [2m (TNR - xJ'R). Chapter 3: The Electric Melvin Solution 49 = Trexp2m' I r 53 [n (bli-bn- + b\ljrbn+ OO OO XI 53 b n - & n - + X* 53 + X\u00C2\u00AB&0&0+. n=l n=l ^x + ^ x j = e-2wx^+^Trexp\27rif^{nr-ix)bl_bn^ + I n=l oo oo 2ni 53 (nr + ix) &n A + - 27rx&J&0 + 2vrzr 53 a 9 . ^ n=l n=l -27rx(^+|) 1 - e - 2 ^ n I n=l X i g27ri(nr\u00E2\u0080\u0094i\) (3.79) g27ri(nr+ix) J \ \ g27rmr Similarly, Trexp [-2m (fNL + xJ'L)] eM'H) (-^\u00E2\u0080\u0094) f f ( - 1 -V i _ e 2 ^ ; 11 V l - e - 2 7 r 27ri(rjT+ix) (3.80) 1^ Q\u00E2\u0080\u00942ni(nT\u00E2\u0080\u0094ix) J \ \ g\u00E2\u0080\u009427rmr The integral over J/ in (3.77) can be evaluated using the ^-dependence of the traces given by (3.79) and (3.80). The result is: j duexp |-27rr 2^ 2 - 27rx (V + ^ + 27rx (v + ^ J V2r 2 Substituting (3.79), (3.80), and (3.81) into (3.77) gives (3.81) -^ye** 7 * 53 / d\xd\2exp -\u00E2\u0080\u0094 A A -it\" - R \u00E2\u0080\u0094 7 = (w' \u00E2\u0080\u0094 TW) A H T = ( \u00C2\u00AB / \u00E2\u0080\u0094 f IU) A\u00E2\u0080\u0094 2vo:' 2yQ!' (3.82) , - \ 2 g (x - x) 1 lOx (iv Oi ( ix,r) | 2 Chapter 3: The Electric Melvin Solution 50 where \u00C2\u00A9i is the Jacobi Oi-function given by Gi (u, T) = 2exp C^f) sinirv fj (l - e2lxinT) \u00E2\u0080\u00A2 (3.83) _ e2ni(nr+u) j ^ _ ^ i { n r - v ) j Unlike the partition functions of the magnetic Melvin solution and of free string theory, which are analytic everywhere in the interior of the fundamen-tal domain F0, the electric Melvin partition function has an infinite number of simple poles in Fo. This can easily be seen in the special cases E = 0 and E = 0. Choosing E = 0, and considering only the terms with w = 0, the 0i-function in (3.82) becomes 0i {ix, r) = 2exp C^f) sin {-inERw') fj (l \u00E2\u0080\u0094 e2ninT) \u00E2\u0080\u00A2 (3.84) ^ _ e2iri(nT-iERw')^ ^ _ e2ni(nT+iERw') j The second factor in the infinite product can become zero at values of T\ and r 2 satisfying 2?ri (TIT - _ i W ) = 2TTZA; (3.85) u for some integer k. Solving for r gives k ERw1 T = - + I (3.86 n n For values of k, n, and w' such that |\u00C2\u00A3 | < | , _\u00E2\u0080\u00A2' > 0, and k2 E2R2w'2 \u00E2\u0080\u0094 + ' 2 > 1 this zero lies in F0. Thus, the factor of [6i (ix, T ) ] - 1 in (3.82) has an infinite number of poles in the fundamental domain. The presence of poles can also be demonstrated in the case E \u00E2\u0080\u0094 0, E ^ 0. In this case, the @i-function is independent of w and w', and the sums over w and w' yield 5-functions, allowing the integrals over Xx and A 2 to be evaluated. The resulting expression contains a \u00C2\u00A9i-function with zeros at the points k a'Ew' T = - + I\u00E2\u0080\u0094\u00E2\u0080\u0094 3.87 n nR for arbitrary integers k and w', producing poles in the partition function at these points. Chapter 4'- Conclusion . 51 Chapter 4 Conclusion In this thesis, an electric Melvin solution of string theory analogous to the magnetic Melvin solution of [2] was presented. The electric KK-Melv in background described in [11] was generalized to a curved, 2-parameter back-ground using a T-duality transformation. The closed string was quantized on this space using techniques similar to those used in [2] and [3] for the mag-netic Melvin space. The partition function was calculated, and was shown to possess poles in the interior of the fundamental region for two special cases of the electric field parameters. There are many possibilities for further research on the electric Melvin solution. More insight into the nature of the solution and its implications for string theory would be provided by determining the physical significance of the poles in the partition function described in section 3.3. Also, the electric field in this background should result in pair creation of strings by the Schwinger mechanism. It may be possible to find an expression for the pair production rate by calculating the imaginary part of the partition function, as was done for open strings in a constant electric field in [14]. Another possibility is to find generalizations of the solution. Other string theories, such,as open string theories, could be quantized on the space. Combinations of the magnetic and electric Melvin solutions could also be constructed, and may be worth investigating. This could be done by quantizing the string on an orbifold which identifies points under both rotations and boosts. Appendix A: Dimensional Reduction 52 Appendix A Dimensional Reduction Dimensional reduction is a technique by which the low-energy limit of string theory on a space with a compact dimension can be viewed as a gauge theory on a lower dimensional space [5]. For a d + 1-dimensional space with dimensions xM, M = 0, l , . . .d , with xd compact, the line element can be expressed as ds2 = gMNdxMdxN = g^dx^dx\" + 2gf,ddx\"dxd + gdd (dxdf (A. l ) where M and N range from 0 to d, and fj, and v range from 0 to d \u00E2\u0080\u0094 1. Completing the square, (A. l ) can be rewritten in the form ds2 = g'^dx^dx\" + gdd (dxd + A^dx\") for some vector A^. In this form, the original metric has been separated into a d-dimensional metric g' , a scalar gdd, called the Brans-Dicke scalar, and a vector field A^, called a Kaluza-Klein field. If the metric gMN of the d+ 1-dimensional space does not .depend on xd, this space can be viewed as a d-dimensional space with metric g' and gauge field Aft. The metric g' transforms as a d-dimensional metric for coordinate transformations of the form x^ \u00E2\u0080\u0094> x'^ (xu). Under the transformation x ' d = x d + A (x\") the metric (A.2) is invariant if the gauge transformation A'^A,- d,X (A.4) is made. Fields which are charged under the Kaluza-Klein gauge field can occur in the low-energy string spectrum. For example, because of the iden-tification xd = xd + 2TTR, a scalar field 4> c a n be expanded as 4>(xM)= \u00C2\u00A3 4>n(x*)exp^pJ (A.5) (A.2) (A.3) Appendix A: Dimensional Reduction 53 Under the translation (A.3), a given mode in this expansion is multiplied by a factor of e - 1* - . The modes are thus charged under A^, with charges equal to the quantized momentum ^ in the xd direction. This technique can be extended to backgrounds with an antisymmetric tensor field, B M N . The general string action (2.1) is invariant under the gauge transformation 5BMN = dM(N ~ 0n(M (A.6) where is an arbitrary vector. This transformation adds a total derivative to the action. For ( M independent of xd, (A.6) reduces to a gauge transfor-mation of an antisymmetric tensor field and of a vector: SB^ = d ^ - d ^ (A.7) bB^d = c^C xd Therefore, the dimensionally-reduced theory involves an antisymmetric ten-sor field, B^, and a one-form gauge field, B^ = B^xd. Appendix B: T-duality 54 Appendix B T-duality T-duality is a symmetry of 2-dimensional conformal field theory which relates different spacetime backgrounds in string theory. For a given back-ground with metric 5^ , antisymmetric tensor B^, and dilaton $, which are independent of some coordinate x\u00C2\u00B0, the general bosonic string worldsheet action (2.1) is equivalent on shell to the action [16] S = JdadT{VhhaP[gQ0VaVp + 2g0iVadpXi+ gijdaX%X>] + ea / 3 [BoiVadpX* + BijdaX'dpX*] + (B.l) eaf}X\u00C2\u00B0daVp + a'VhR\u00C2\u00AE (X)} where i and j run from 1 to 25. The field equation for X\u00C2\u00B0, ea0daVp = 0, has as a solution Va = daX\u00C2\u00B0, which when substituted into (B.l) gives back (2.1). Another action which is equivalent to (B.l) on shell can be derived by solving for Va through its equation of motion, and substituting the solution into (B.l). This gives an action of the form S = - \u00E2\u0080\u0094 * - / dadT\^daX^dpXvh^g.v^^daX^dpXvB.v{^,.2) 4.TJOL J L +a'VhR (1)] with goo 1 goo Boi , 9o% \u00E2\u0080\u0094 #00 9ij = 9ij 9oi9oj \u00E2\u0080\u0094 B0iB0j 9oo (B.3) Boi _ goi goo B^ = B^ 1 9oiBoj \u00E2\u0080\u0094 B0ig0j 9oo Appendix B: T-duality 55 In addition to preserving the classical properties of the background, the trans-formation (B.3) also preserves conformal invariance at one-loop order, pro-vided that it is accompanied by a shift in the dilaton: $ ->\u00E2\u0080\u00A2 $ - ^lng00 (BA) This can be shown by substituting (B.3) and (B.4) into the one-loop confor-mal invariance equations of string theory [15]. This transformation is called a T-duality transformation along the X\u00C2\u00B0 direction. When T-duality is done along a compact direction, a generalization is obtained of the R \u00E2\u0080\u0094> ^ duality of string theory on flat space with a compact dimension of radius R [17]. For a space with X\u00C2\u00B0 = X\u00C2\u00B0 + 2ir, if the identifi-cation ^/goo = R is made, the direction X\u00C2\u00B0 in the T-dual space is compact with radius -^ ==. In the case of flat space, g0i = B^ = 0 and (B.3) reduces to the R \u00E2\u0080\u0094> - 5 duality under which the string spectrum is invariant. Bibliography Bibliography 56 [I] M . B . Green, J .H. Schwarz, and E. Witten, Superstring Theory, Cambridge University Press, Cambridge U.K. , 1987. [2] J .G. Russo and A . A . Tseytlin, \"Exactly solvable string models of curved space-time backgrounds,\" Nucl. Phys. B 4 4 9 , 91 (1995). [3] J .G. Russo and A . A . Tseytlin, \"Magnetic flux tube models in superstring theory,\" Nucl. Phys. B 4 6 1 , 131 (1996). [4] J .G. Russo and A . A . Tseytlin, \"Magnetic backgrounds and tachyonic instabilities in closed superstring theory and M-theory,\" hep-th/0104238. [5] J. Polchinski, String Theory, Cambridge University Press, Cambridge U.K. , 1998. [6] M . Gutperle and A. Strominger, \"Fluxbranes in string theory,\" hep-th/0104136. [7] T. Suyama, \"Properties of string theory on Kaluza-Klein Melvin back-ground,\" hep-th/0110077. [8] T. Suyama, \"Melvin background in heterotic theories,\" hep-th/0107116. [9] G.T. Horowitz and A . A . Tseytlin, \"New class of exact solutions in string theory,\" Physical Review D 50, 2896 (1995). [10] T. Takayanagi and T. Uesugi, \"Orbifolds as Melvin geometry,\" hep-th/0110099. [II] L. Cornalba and M.S. Costa, \"A new cosmological scenario in string theory,\" hep-th/0203031. [12] B. Pioline and M . Berkooz, \"Strings in an electric field, and the Milne universe,\" hep-th/0307280. [13] T. Friedmann and H. Verlinde, \"Schwinger meets Kaluza-Klein,\" hep-th/0212163. Bibliography 57 [14] C. Bachas and M . Porrati, \"Pair creation of open strings in an electric field,\" hep-th/9209032. [15] T .H . Buscher, \"A symmetry of the string background field equations,\" Physics Letters B 194, 59 (1987). [16] T .H. Buscher, \"Path-integral derivation of quantum duality in nonlinear sigma-models,\" Physics Letters B 201, 466 (1987). [17] E. Kiritsis, \"Exact duality symmetries in C F T and string theory,\" Nu-clear Physics B 4 0 5 , 109 (1993). "@en . "Thesis/Dissertation"@en . "2004-05"@en . "10.14288/1.0091552"@en . "eng"@en . "Physics"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "The electric Melvin solution in string theory"@en . "Text"@en . "http://hdl.handle.net/2429/15248"@en .