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Open strings in a background guage field Laidlaw, Mark 2000

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O P E N STRINGS IN A B A C K G R O U N D G U A G E FIELD  By Mark Laidiaw B. Sc. University of Victoria, 1-998  A THESIS S U B M I T T E D THE  IN P A R T I A L  REQUIREMENTS MASTER  FULFILLMENT  FOR T H E DEGREE OF  OF  SCIENCE  in THE  FACULTY OF GRADUATE  DEPARTMENT  STUDIES  O F PHYSICS A N D A S T R O N O M Y  We accept this thesis as conforming to the required standard  THE  UNIVERSITY O F BRITISH  COLUMBIA  August 2000 © Mark Laidiaw, 2000  OF  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives.  It is understood that copying or publication of this thesis for  financial gain shall not be allowed without my written permission.  Department of Physics and Astronomy The University of British Columbia 6224 Agricultural Road Vancouver, B.C., Canada , V 6 T 1Z1  Date:  2 5 / W 10QO  Abstract  The vacuum energy for open strings in a constant gauge field background is calculated for the disk and the annulus in a path integral formalism. This same formalism is then used to reproduce and generalize previous results for D-brane non-commutative geometry at string disk level. The contribution to non-commutative geometry from string annulus diagrams is calculated, and shown to be proportional to the divergence of the open string tachyon.  11  Table of Contents  Abstract  ii  Table of Contents  iii  Acknowledgments  vi  1  2  I n t r o d u c t i o n to the Free S t r i n g  1  1.1  Introduction  1  1.2  Boundary Conditions and Mode Expansion  2  1.3  Commutation Relations  6  1.4  Virasoro Generators and Algebra  7  1.5  Ghosts and Vacuum Energy  10  V a c u u m Energy by P a t h Integral Calculations  13  2.1  Introduction  13  2.2  The Annulus  15  2.2.1  The Bosonic Propagator  15  2.2.2  Bosonic Vacuum Energy  16  2.2.3  The Fermionic Propagator  20  2.2.4  Fermionic Vacuum Energy  24  2.2.5  Superstring Vacuum Energy  28  2.3  The Disk  31  2.3.1  31  Boson Vacuum Energy  iii  2.3.2 2.4 3  Fermion Vacuum Energy  33  The Bosonic Mobius Strip  34  Non-Commutative Geometry  36  3.1  Introduction  36  3.2  Disk Calculations  38  3.3  One Loop Correction to Non-Commutative Geometry  40  3.3.1  The Charged String  43  3.3.2  The Neutral String  44  3.3.3  Distinct Branes  46  3.4  Non-Commutative Geometry from the Mobius Strip  47  4  Conclusion  49  A  Alternate Calculation of World-sheet Propagators  51  A.l  The Bosonic Propagator  51  A. 1.1  52  Equivalence to Published Expressions  A.2 The Fermion Propagator  ,  55  B  A Lemma  58  C  C a n o n i c a l Q u a n t i z a t i o n o f t h e O p e n S t r i n g i n a (7(1) B a c k g r o u n d  61  Cl  61  Motivation  C.2 Bosonic Degrees of Freedom  62  C.2.1  Boundary Conditions and Mode Expansion  62  C.2.2  Commutation Relations and the Virasoro Algebra  64  C.3 Fermionic Degrees of Freedom C.3.1  68  Boundary Conditions and Mode Expansion  iv  68  C.3.2  Commutation Relations and the Virasoro Algebra  71 74  Bibliography  v  Acknowledgments  I gratefully acknowledge the encouragement, assistance and constructive criticism from Dr. G . Semenoff, and the patient ears and ready minds of Kirk Buckley, Joel Erickson, Todd Fugleberg, Matthew Horton and Michel Olivier who helped clarify the problem for me many times. I would like to thank my wife for her support and encouragement. This work is supported by N S E R C .  vi  Chapter 1 Introduction to the Free String  1.1  Introduction  The purpose of this text is to calculate in a well motivated manner the vacuum energy of a superstring in a background gauge field, and to show how one can recover noncommutative geometry from the study of stringy degrees of freedom. This is a problem that is of historical interest [1] [2] [3] [4], as the effective action for this background field, which can be obtained by integrating out the string degrees of freedom, is the BornInfeld action, an non-linear generalization of Maxwell's equation. This problem is also of interest in current research as open string which end on D-branes are used in such diverse areas as the A d S / C F T correspondence and non-commutative geometry. A non-trivial question to ask is whether the question is well posed, since the photons that make up a background F^ field are in principle described by stringy excitations u  themselves, would it not be more useful to fully understand the interaction of two open strings and leave it at that? Two considerations make the approach taken reasonable, the first is that string phenomenology is not yet fully understood, so that the determination of what excitations constitute a photon is a difficult task in and of itself. The second is that in the limit of low energies the spectrum of the string is dominated by the massless excitations anyways, and it could be appropriate to regard the background field as some condensate of strings. To the end of better understanding of the subject, this chapter is intended to review  1  2  Chapter 1. Introduction to the Free String  the development of the open string. While certainly not exhaustive, as the subject is covered in detail in the literature [5] [6] this has two motivations. The first is to make this text as self contained as possible, and the second is to highlight some of the differences between a string that interacts with a background gauge field and the free case.  1.2  B o u n d a r y Conditions and M o d e  Expansion  When discussing any field theory, a useful starting place is the action, and string theory can be characterized as a free field theory on a two dimensional manifold with some arbitrary topology, with interactions being described by computations on higher genus surfaces. In the case of open string theory interactions only take place at the edge of the strings, in the sense that interactions involve a string breaking off and propagating freely, and this changes the topology of the surface. The action for a free Neveu-Schwarz-Ramond open string is given [5] as 5  = 7 ^ 7 / dadr (d X»d X, a  a  + V (-i-fd ) a  Vv)  (1-1)  where i i is the index on the target space, a is an index on the world-sheet, which is parameterized by o 6 [0,7r] and r and the target space metric is taken to be minkowskian, which eliminates the terms such as \f—g as they become trivial. In addition, the two (0 i\ (0 -i\ dimensional 7 matrices are given by 7 = ,^ = which satisfv the T  Clifford algebra  J 7  A  , 7  6  }  =  a  2r) , where r] is the world sheet metric. ab  ab  The X fields  represent bosonic degrees of freedom on the world-sheet, and commute, while the ip fields are world-sheet fermions and hence anticommuting, but they transform under target space Lorentz transforms as vectors. In a sense they can be regarded as auxiliary degrees of freedom, since they do not have the attractive physical interpretation of target space position that the X fields do, however, they contribute the necessary number of degrees of freedom to have a critical theory in 10 dimensional target space.  Chapter 1. Introduction to the Free String  3  A classical method for calculations in string theory is to expand the fields in terms of orthogonal modes, and then to use the canonical commutation relations of the theory to deduce the commutators of the mode coefficients which are then interpreted as raising and lowering operators. In addition, the constraints on the theory which are given by conditions on the energy momentum tensor are conveniently expressed in terms of these operators, and can be calculated. The first step in finding the orthogonal modes is to find boundary conditions that the fields must satisfy, and this is best accomplished by using variational arguments. Using the above and letting the X field vary by 5X it is clear that SS  =  —!- /  =  4 ^  /  dodr2d 5X^d X a  a  d  a  d  T  [9<l  (  tl  « *)  25X>ld  X  ~ ^X"d d X^} a  a  .  (1.2)  This gives the equation of motion for the boson, d d X^ a  a  = 0.  (1.3)  Imposing that the X fields either vanish at T — ±oo or are periodic in compact r, depending on the topology of the string, this leaves the boundary condition j drSX^X^l  = 0.  (1.4)  There are numerous boundary conditions that satisfy this, however, the literature singles out two cases, Neumann boundary conditions, where the normal derivative vanishes at the ends of the string, and Dirichlet boundary conditions, where the position of the string end is held fixed. Dirichlet boundary conditions, which give rise to D-branes, explicitly break Lorentz invariance, while Neumann boundary conditions are chosen because they preserve the maximum possible symmetry for open strings. The boundary conditions and  4  Chapter 1. Introduction to the Free String  demands for orthogonality only apply to the o dependence of X and the r dependence is undetermined. As given in [5] the mode expansion satisfies X» = of + p^r + iJ2  ~<e-  cos (rur).  inT  (1.5)  so the modes are (f) = — -  cos (ma).  imT  m  m  e  (1.6)  These modes satisfy the orthogonality relation [2] / — \<j>mid <l>n - c/>„Z(9 ^ ] = — S T  T  m  (1.7)  mn  and redefining the a oscillators to be the conventional raising and lowering operators eliminates the factor of —. m  In a similar manner it is necessarv to determine mode expansions for the world sheet fermions. Writing the fermion as a spinor in two dimensions, ip = I  , and varying  11  the above action results in the change in the action SS  =  ^ Ana  1  J dadr  J  + S^T  L  r  "  (d + d ) c % _ +  ~'  v  T  a  '  +  (d + d ) T  • ^  a  (d T  {d ~ d„) T  S^  ---^+  ^+] •  (1.8)  Imposing that the fermion fields are majorana gives, using the fact that the '0s anticommute, ^  j dadr [2S^  (d + d ) T  + 25^  a  + (Or + d ) (^^-)  + (d - d )  a  T  a  (d  T  - d) a  .  ^  (1.9)  This gives the standard equations of motion, (d + d )ib.  =  0  (d -d )^  =  o  T  T  a  a  +  (I.IO)  Chapter 1. Introduction to the Free String  5  for the fermion fields, which can be summarized as i^ d ip = 0 using the matrices listed a  a  above. The boundary terms are of interest, and assuming, as in the case of the bosons, either a periodicity in r or that the fields vanish at r = ± o o the conditions are  ~ r ^ )Zl  J dr Naively, an ansatz  = 0-  = M^ip  + +  u+  (1-11)  at o = 0, ir will solve this for any M that satisfies  M^M = 1, and M may be different at o = 0 and o = n. This causes the surface terms to vanish at both ends of the string independently. However, there is another consideration which restricts the choice of boundary conditions. The theory ought to be invariant under Lorentz transforms on the indices /J, since the J ' are regarded as target space 1  coordinates. The only boundary conditions that preserve this symmetry are  = ±<5^.  Due to the fact that there is ambiguity in the choice of boundary conditions for the fermions, there are two inequivalent choices of boundary conditions. Since the overall sign of •0_ is a normalization convention one can unambiguously set  =  at o = 0.  The most general mode expansion of ip that satisfies the condition that (d — d ) tp+ = 0 +  is ip = Y, d e ^ \ im  +  m  m  T+a  T  a  and similarly 0_ can only depend on r — a. In the case where  VV- = Vv+ at a = 7r, the Ramond sector of the theory, the expansion for tp is  (  -im(T-<j)  e  where m is an integer, whereas for the case  \  = — ip^ at o = ir the expression in the +  Neveu-Schwarz sector becomes  m  (  -im(T-o)  \  -irn(T+o)  J  e  e  where m is half-integral. The Ramond sector gives rise to target space fermions because of the non-commuting zero modes, while the Neveu-Schwarz sector gives rise to target-space bosons.  Chapter 1. Introduction  to the Free  String  6  The fermion modes are also orthogonal in the following sense,  JO  JO  7T  7T  '  V  • do  -in(r+a)  e  7r  JOo  1.3  Commutation  i (  7T  Relations  The commutation relations for the modes must be determined, and the easiest way is to express canonically conjugate variables in terms of raising and lowering operators, impose commutation relations, and fourier analyze the results. For the bosonic variables X and d X^ the commutation relation is 11  T  [drX^a),  X {a')\  = -i5{a - a')rf\  v  (1.15)  and substitution of the expansion 1.5 gives -i5(a  - a')jf"  =  =  P^ + Y. <  [P">  e m T  cos(ncr), x" + fr  +  ~ rn+n 5  +i £  —a e~  COs(flo) COs(mcr') [ < ,  v  imT  m  <  cos(mr)  J  (1.16)  using the fact that the commutator must be independent of r . Fourier analysis of the above immediately gives the standard commutation relationships for the modes, [x^p"]  K,c4]  =  irf  v  = ncWT-  (1-17)  For the fermions the momentum canonically conjugate to ip± is ib±, and so the anticommutation relationship reads  {^(CT),  A V ) } = ™5(a - ^rTSat  (1.18)  Chapter 1. Introduction  7  to the Free String  where a, b = ± . Applying the expansion for the fermions given above gives for the case of a = b = —  = {E merf  JM(R  - ,E< " <T)  im  E  M(T  "' ' } J  n  )  »  (1-19)  )  which again is forced to give the standard relationship since there must be no r dependence in the anticommutator, giving  {^,<} = »T*m+n  (1-20)  with an identical result for a = b — +, while the cross terms vanish, giving the proper orthogonality.  1.4  V i r a s o r o Generators a n d A l g e b r a  From the free string action, it is possible to determine the two dimensional energymomentum tensor by varying the action with respect to that metric. As given above the action is S  =  4 ^ / drdo^jg^X^X,  -  -^ ^^d ^  l  - ^v^ W0°</v  b  (1-21)  a  g  where g = det (#„(,). Taking the variation of this action with respect to g  ab  is simple with  the identity that det(g ) — exp(Tr In g ). Then for a change in the metric of Sg ab  ab  the  ab  change in the action is 6  S  4^c7> I °^g5g d X»d X^  =  dTd  a  -  b  ab  -  \ ^ g  a  V l  b  b  d  a  ^  -^5g r d tP  l  ab  a  b  1  -  -8g S.  ll  (1.22)  l  ab  Then, because the energy-momentum tensor is given by varying with respect to the two dimensional metric, T  ab  SS = —  = d X»d X» a  b  i -  + -rj d ^ a  b  ?-  + ~rj"d ^ a  1  - -g  ab  /  (d X»d X, c  c  ?-  \  + -^7 0cVv) c  (1.23)  Chapter 1. Introduction to the Free String  8  as in [5]. Since the left and right moving degrees of freedom on the string are mixed, the independent fourier components of this are defined as L  = ^J*  m  do (e T  + e- T__)  ima  .  ima  ++  (1.24)  This expression can be explicitly evaluated for the modes as developed above, and it is found that L  m  =  ^Yl -n  •a  a  + (n + ^rnj cL„  m+n  -dm+n.  (1.25)  This is unambiguous because of the diagonal commutation relationships except for the case of m = 0 in which case the operators are defined to be normal ordered, and there is some constant addition to LQ which reflects this. In the sum above the sum is implicitly over half-odd integers for the case of anti-periodic fermions. Now, the condition which arises as an equation of motion, that J^E = 0 arises as a constraint, which implies that all components of T b vanish. This condition is very a  restrictive, and is relaxed so only the positive fourier components of this will annihilate any physical states, so that L \<f>>=0,  (1.26)  n>0  n  and L -a\(t>>=0  (1.27)  o  for some value of a. It is also possible to calculate the commutators of the Ls, and explicitly one finds that [L , Lb a  —  ^  a  -n  • &a+n +  ^ Y tt-m • a  (n + ^aj d-  b+m  mn  n  • d ,  + (m + ^bj d_  a+n  m  d{ *• b+m a  Chapter 1. Introduction to the Free String  9  + \ n + -a) \ m + -bj [d_ • d , d _ • d ] n  =  (a - b) L  a+n  m  b+m  + A(a)6 .  a+b  (1.28)  a+b  The term A{d) is the central extension of the algebra, present because of the normal ordering ambiguity in L - To determine the normal ordering constant, one takes the 0  expectation value of commutators [5], between two states in the perturbative vacuum which have zero momentum. It can be shown that because of the Jacobi identity for commutators the central extension can be determined by only evaluating the expectation values of  and [ L , . L _ ] . 2  2  For the Ramond sector the calculation is (OKLi.L-xllO)  =  <0|i [d  do] |0)  =  (0| i d ^ d - i X |0)  =  (0| j ^ d o V 10) = IW  0  = \D.  (1.29)  This makes use of the facts that OLQ |0) = 0 on a zero momentum state and the normal ordering of the terms in the fermionic part of L\. The expectation for L gives 2  (0|[L ,L_ ]|0> 2  2  =  ^(0|[a,  =  ^ (0| 2 a ^  =  \(2^rr  - a - i j + ^do -d ,2d_ - d ] |0) 2  l  V  ^ + Ad»d d_ „dl 2ll  + l^rr)  2  2  0  |0)  =D  (1-30)  which confirms that for the Ramond sector the central extension is A(m) = lm D 8 3  where D is the dimension of the target space.  (1.31)  Chapter  1. Introduction  to the Free  10  String  In the Neveu-Schwarz sector where the fermionic excitations are half-integrally moded a similar calculation can be performed and one obtains (0|[Li,L_i]|0) = 0  (1.32)  because the bosonic part gives zero, and the fermionic part has normal ordered creation and annihilation operators, which will annihilate the vacuum. However, (0| [L ,L- ] 2  2  |0)  =  ^ (0| [ai • a i . a - i • a_i] + [d  =  \D  1/2  •d  3 / 2  ,  rf_ ci_ ] 3/2  1/2  |0)  (1.33)  which shows that the central extension for the NS sector is  A(m) 1.5  — ^ (m - m) D. 8 ^ ' 3  (1.34)  Ghosts and Vacuum Energy  For the free string in a Minkowski target space there is an additional consideration, which is that some of the states created by naive application of the raising and lowering operators for world-sheet bosons and fermions have negative norm. These negative norm states are known as ghosts, and they do not correspond to physical excitations. There are several approaches [5] [6] that will eliminate them, such as quantizing in light cone coordinates or proving that they decouple in critical dimensions. It is shown in the literature that string theory is critical, that is that the negative norm states decouple from the physical spectrum, in the case of D = 10 and a = 1, where a is as defined above in 1.27. The approach that generalizes best to the case of an interacting string is to introduce additional non-physical degrees of freedom to the system which have statistics that will cancel the undesired excitations. This is analogous to the field-theory formalism  Chapter 1. Introduction to the Free String  11  of adding 'Fadeev-Popov ghosts' in Q E D to cancel non-physical excitations. In addition, the perturbative spectrum must be truncated by the GSO projection. The reason for this is that there are naively too many degrees of freedom for the preservation of target-space supersymmetry, which would require that the number of space-time bosons and fermions should be equal. This condition is such that for fermions the trace becomes the operator 1 + ( —l)^ where / is the fermion number operator. It only remains to calculate the vacuum energy for the free string. Since the general form of the partition function for any field theory is Z = [ [Df] exp J  / dxf  / „ = constfdetpropagator)  propagator vronanator J  J  ^.  l  2  (1.35)  This is in general infinite, with the constant proportional to the volume of the space that the calculation is done in. On the critical dimensions the propagator for the string is Lo — 1. [5] Using an identity for determinants we find that Z = V exp  (1.36)  ~Trln(L -l) 0  This implies that E  v  =  ^Trln(L -l) 0  •\ f -Tre- < °- \ l  To evaluate J V c - M ^ o - i ) ^ no  o  a  =  e  j-jQ^h £  L  (1.37)  l  J t  2J  _ l 2 ^ Y^  q  n>1  a  a_ a n  n  and the convention that  P - This, together with the mass shell condition that a'M = y j 11  2  gives the vacuum energy as  \  '  pays  In critical theory with a' = \ this becomes -D '  v  /  phys  n > ]  a _ „ a „ — 1,  12  Chapter 1. Introduction to the Free String  The sum over physical states is nothing but the product of geometric series, since excitations at a given level have a unique polarization tensor. The sum over physical states then becomes  =  2~2 hys ~^ e  p  Mphys  Yi >i  ent  n  (1  —  e~ )~ . ntn  8  This then gives the contribution  from the bosonic components to be  agreeing with [4]. However, the contribution from the fermions is multiplicative, since the fermion and boson oscillators commute, and it vanishes by virtue of the G S O projection, as is appropriate for a theory which has space-time supersymmetry.  Chapter 2  V a c u u m Energy by P a t h Integral Calculations  2.1  Introduction  The previous development was exclusively for free strings that do not interact with any form of background. The generalization that is now being considered is that of a string that interacts at its endpoints with a background antisymmetric field, and the goal is to compute the vacuum energy of these strings. The original motivation for this type of calculation was to produce a 'stringy' nonlinear generalization of Maxwell's equations, since the procedure can equally well be thought of as integrating out all string degrees of freedom, and interpreting the resulting vacuum energy as an action for the background fields. This chapter presents a method which stands in contrast to the canonical quantization method of the previous chapter, and that of the treatment for the background dependent string vacuum energy presented in the appendix. The philosophy which is embraced is to take the fields that live on the string world-sheet and calculate a free Greens function for them for a given topology, which is in this case the annulus or the disk. Once the propagator has been determined the interaction with the background field is treated as a perturbation, and the vacuum energy of the string is calculated by simply summing all the possible world-sheet vacuum diagrams for a particular field. This method, unlike canonical quantization can be seen to generalize much more readily. In the case of a non-constant field the interaction at the end of the string gives  13  Chapter  2.  Vacuum Energy by Path Integral Calculations  14  rise to higher order vertices on the world-sheet [2]. These, in turn will give higher loop corrections to the theory that exists on the world-sheet. In general these vertices need renormalization, and that provides a check on the effective action for the external fields. In addition, the method obviously extends to interactions between the world-sheet fields and the background in the bulk of the world-sheet. In the case of canonical quantization, this would present a conceptual challenge, whereas in this method, it adds computational rather than conceptual difficulties. Another strength of particular interest for current research is that the coupling to antisymmetric fields which themselves are confined to D-branes is immediately accounted for in this methodology. This is because there is no reliance on a particular form of the background than for convenience in the presentation of a compact final answer. The point of this exercise is to compute the vacuum energy of a superstring in a background gauge field, and the calculation is performed explicitly for both the fermion and boson degrees of freedom. The starting place for the calculation is the string action as given in [4] [7] for the string coupled to a background 5  =  ^  1  field,  with the world-sheet topology of the annulus  dadr (y/=dg & X»d X l  ab  + ^ l  2  tl  i$»j d %) a  a  j dTF,  (x»d X»  -  Ul>  j drF^  (x d X»  - J f f f )  l=a-  u  - -e  -  b  T  v  T  (2.1)  The generalization to an arbitrary topology coupled to the background is accomplished by the addition of a Wilson line along each boundary [8]. The addition of a bulk field is trivial, since it is a total derivative, and is treated explicitly in the next chapter.  Chapter 2.  2.2  Vacuum Energy by Path Integral  Calculations  15  The Annulus  2.2.1  T h e Bosonic Propagator  The first task is to calculate the Greens function for the Bose degrees of freedom. While this calculation can be performed in several different ways (see Appendix A ) , the most transparent is to perform a conformal transformation of the Euclidean annulus of inner radius a and outer radius 1 to a cylinder of length In a. If the annulus is parameterized by the complex coordinate z = re  10  then this transformation is accomplished by the  identification of the cylinder coordinates p, <f> as p = l n r , cp = 6. The equation that must be solved to determine the Greens function is  w) ' ^ ''  +  G(p  /9  0,)=27ro/J(<A  ~  ~  {  2  '  2  where 5 is a modified 5-function with the constant mode subtracted so that 6 = 5—  ) TT-\—,  2TT In a '  and the 5-function in ct> understood to have a periodicity 2-K. Obviously G will have to be periodic in cp, and Neumann boundary conditions are imposed. The correct boundary conditions are then d  -G^,p-^J)  = 0  d  p=0  l-G{<t>, -<l>', ') P  P  =  0.  (2.3)  p=\n a  Similar conditions apply for p' also. A properly normalized eigenfunction of the Laplacian is 1  ^  x  / 777T  \  (p, cfy = - = = c o s (T—P), v 2 7 r In lna a Vina / y/2ir  (2.4)  with eigenvalue A = m + ( ^ ) , and so using the relation 2  G  M  , ' P  A  '  )  =  -^^dmMi  .  (2 5)  Chapter 2. Vacuum Energy by Path Integral  Calculations  16  and taking into account the 2-KCX multiplying the 5 function, we obtain the Greens function  In a  , m m ^ O , m=—oo 2a' ~ ~ nrr nit , + -— > > e * * cos-—pcos-—p 2  n  ,(  ,m  1  o-(2.6) \\naj  This can be re-written to obtain G{pA, 'A')  =  P  E l  •  n  f  • •  l  E  m ^ = - « > ™ =  •  ^  e ^ ) ™ c o s ^ p c o s ^ oo 717T  In a »7T  In  . / 111  + 27ra'Vcos-—pcos-—p'I ~^ In a m a \mr  j  a ,  .  (2.7)  This expression turns out to be a useful form of the bosonic propagator, and in the appendices, this is shown to be equivalent to other similarly derived expressions which are found in the literature. 2.2.2  Bosonic V a c u u m Energy  The bosonic part of the action given in equation 2.1 is 5  =  4 ^ /  j drF^X'drX^o-^  ^99a d X»d X,+^  d(7dT  a  b  b  ei  J  drF^X"d X%= . r  (2.8) This means that the interaction Lagrangian takes the form L  = ^e F X d X>'6(a) ,/  M  1  llv  T  - ^F^X'drX^ia  - In a),  (2.9)  which is identical to the statement that the X field can interact at either of the strings ends with the background field. To determine the vacuum energy in field theory one procedure is to take all the vacuum Feynman diagrams, sum them, and exponentiate the result. Now the possible elements in these diagrams can be enumerated. Due to the  a  Chapter  2.  Vacuum Energy  by Path Integral  17  Calculations  fact that this is a vacuum diagram there can be no external X propagators, and since the only interactions take place at p = 0, In a the possibilities are that the field either experiences two subsequent interactions on the same edge of the cylinder, or on opposite edges, for a total of four possibilities. It is shown in Appendix B that the appropriate thing to do in this situation is to generalize the propagator from a number to a matrix, sacrificing the p dependence, which has the following structure: (2.10)  In this ,4(0,0') = G(0, 0,0,0'). and 5(0,0') = G(ln a, 0, 0, 0') with A' and B' being the obvious generalizations, and in the appendix this is shown to be a special case of a more general result. In this language, the vertex can be re-written as [1] (2.11) The rational for this is in direct analogy with the re-expression of the propagator, and the reason that this matrix is diagonal is that the interaction does not mix the ends of the string. A n interesting feature of this is that since the interactions of the strings bosonic fields are only at the boundary this has converted a 1 + 1 dimensional scalar theory into a 0 + 1 dimensional theory described by matrices. The calculation of C7(0, 0') must be done carefully. The starting point is the propagator calculated above, a  oo  oo  Im  cos  In a  pcos  1 In a  (2.12) and so it is immediately clear that  .4(0,0')  C7(0,0,0,0')  Chapter  2.  Vacuum Energy  "  by Path Integral  /  In a  oo ^  oo ^ i(4>-<t>')m_  E  E< 0  m^0,m=—oo n.=—oo 0 0  + 2 W  lna\  /  ^  71=1 \  a' In a  ,  ~ 2^  In a  e  i(*-*')"»  n  ,ln a 2  T r a ' — -  2  . v 7T.7T  „  =  18  Calculations  i com(ra In a) + 7ra —— 2  n  m  a  „ , v - , cos(m(d) — 0 ' ) ) . . + 2a' ] T —T-LL oth{m In a).  . (2.13)  C  m>0  3  m  Clearly the same result holds for A'((J),<j)') = C7(lna, 0 , In a, 0 ' ) , since the only change in the calculation would be that the sum over n would now be multiplied by  which  cos (n7r) 2  is identically 1. Similarly, for B(4>, 4>') the sum is multiplied by cos(rar) = (—1)", and the identity J2  n  m  immediately gives  I + 2 = ^csch(mir) n  ,-./ , ,»\ ^ / / , ,/\ /In i cos(m(d> — d>')) , . B((f>, (f)') = B'(<f>, (j)') = T r a ' — - + 2a' V — ^ c s c h i m In a). 2  a  n  (2.14)  Thus the Greens function can be compactly expressed as G  (  A  ,  )  =  2 R F E  £5!M*Z*2) m>o  G m  ,  c  .  HI" ""' 1 1  \csch(m  m  In a)  - M - l - A  (  2  I  5  )  coth(rn In a) /  It is also possible to obtain similar expressions for arbitrary p and p', however, that is unnecessary in this analysis. Now, armed with both a propagator and an interaction vertex it is possible to calculate any Feynman diagram that would arise in the theory. Because the coupling to the external field has only two X legs, there is a unique world-sheet diagram at each order in n, and it can be explicitly calculated as  J  ™ , ™, r> mi...m >0  777*1  n  d,,  "- TrlG ,a...G ,n].  C0S{mM  M  m  772  m  (2.16)  Chapter 2.  Vacuum Energy by Path Integral  19  Calculations  The factor of ^ is due to the order of the diagram, (n — 1)! comes from the inequivalent orderings of the vertices, 2 is a result of the fact that there are two propagators on each n  vertex that can be connected either way, and Tr(F )  together with ^ come from the  n  interaction Lagrangian, and the factors of 2 cancel. The reason that the derivatives n  are shown operating on distinct propagators is that if two operate on the same one, the expression can be converted into the above form by integration by parts and using the antisymmetry of F^, noting that the surface terms vanish because the world-sheet is compact in the (j) direction. The (j) integrals enforce the equality of the m,, so the final expression for the diagram is n  1  =  th  2ira') Tr(F ) n  n  The matrix G fL m  E  n  Tr [G Q]  n  m>0  m  .  (2.17)  can be diagonalized, and will generically have two eigenvalues, A  m +  and A _ which can be determined explicitly. This results in the simplification m  V2W)"Tr(F") £  ( A " + A^_)  (2.18)  +  m>0  n  Now that there is an expression for the n diagram in the set of all vacuum Feynman th  diagrams, it is possible to sum explicitly, {-2iva')  n  Y, connected  =  E  ^—^—Tr(F )  n>l  =  E  n  - E  T  r  t  ln  (<W + 2 W F „ A M  m > l  = - E  (Ki+ +  Ki-)  m > l  7 1  Tr In  (V + 2na'A ( m  m +  ) + In - e )F  +  ei  2  lll/  +  2Tra'F \m-)] fll/  (2W) i^,(-e )) , 2  ie2  m>\  (2.19)  where the last manipulation is obtained by evaluating the secular equation for G VL and m  having set A  m  e  Econnected  = =  • Thus, the exponential of the connected diagrams is jj m > l  e  x  p  _  T  r l  n  ^  +  2  W A  m  (  e  i  - e )F 2  + (2W) F%,{-e e )) 2  llv  x  2  Chapter 2. Vacuum Energy by Path Integral  Calculations  II (det [8^ + 2-KdA (e m  20  - e )F , + ( 2 W ) F 2  x  2  fU  2 I /  (-e e )J) 1  2  1  , (2.20)  m>l  which is the expression for the vacuum energy of the bosonic degrees of freedom of the string, and agrees with [1] [4] [9]. Note that in the case of the neutral string (e = e ) x  2  this is exactly the square of the Born Infeld action for the background field F^. This is because of the contribution of the different ends with opposite charges, each giving rise to a copy of the Born Infeld action.  2.2.3  T h e Fermionic  Propagator  We wish to analogously determine the fermion propagator, and use an identical technique to determine the contribution to the partition function from fermions. Since the topology under consideration is a cylinder with a Euclidean metric, and following the convention [5] for the two dimensional 7 matrices we have  *  C  =  0 ) ' ' (! 7  ~l\  =  ( 2  '  2 1 )  Then the eigenvalue problem for the Dirac operator j d (f) = \(p is expanded as a  a  L+^A  0  0>\  J ^ r . A  I .  (2.22)  Taking as an ansatz for ip E^M  IpJ  n  .  \tb„  I  (2.23)  and noting that e ^ is a complete set on cp 6 [0, 27r] then for each mode the eigenvalue m  problem becomes  l~n  0  JUJ  which is equivalent to '  d  2  \  UJ  Chapter  2.  Vacuum Energy by Path Integral  —2-  Now,  21  Calculations  -n  tin = An(2.25) with the definition k = A — n it is clear that the most general solution to the 2  2  2  first of those equations is i)  = Ae  + Be- .  ikp  n  (2.26)  lkp  This implies in turn that  ^  =  zi!LlJl ^P Ae  i±ZJl  +  A  (2.27)  -*P.  B e  A  Now it is necessary to impose the boundary conditions, and since these are free fermions either Ramond or Neveu-Schwarz boundary conditions will suffice. The condition that ibi = tp at p = 0 immediately implies that B = — w h i c h can be simplified using r  the definition of k to obtain B = — A ^.  This means that the condition  11  = ±ip at  p = In a is converted to gifclna _  n  +  -ik\na  _  c  — TI ^  _j_ f ~ i k  i  k  l  n  a  ^  ^-iklna\  (2  28)  which has the effect of quantizing k as nm  k =  ma  .  2.29  where m is either an integer or half integer depending on whether the + or — sign was chosen in the boundary conditions. Choosing the appropriate normalization constant A  =  \J \ A%  Da  the final expression for the eigenstate of the Dirac operator is ,  ^  = ^ 7 =  /  ikp  e  — n+ik -ikp  .. . /  V 4 7 r l n a y — n+ik^kp +  \  c  .,  +  e  -ik  P  j  •  (2-30)  Noting that the sign of A is not chosen by the definition means that, in addition to the value of k and n, the sign of A is also a choice. In analogy with the approach taken by  Chapter 2. Vacuum Energy by Path Integral  Calculations  22  [5] in quantizing the Fermionic modes of the open string, note that the above expression for tp implies that ipmn±{pA)  =1pmn±(-p,(f>)-  (2-31)  This means that the ip and ib fields on the domain 0 < p < In a can be combined into a single field which lives on the interval — In a < p < In a and which has either periodic (Ramond) or anti-periodic (Neveu-Schwarz) boundary conditions ib(<b,p + 2\na) = ±V>(0,p),  (2.32)  and the eigenvalue equation for the Dirac operator is trivially rewritten in the form (  d .3 dp ' d~4> '  ^  %  P  (2.33)  =  which explains the mixing of left- and right-movers in the expression for ib, which stands in contrast to the free string case. The inner product of the eigenfunctions is defined as /•in  (i>mn±i>m'n'±')  a  =  r£K  Jo  (2.34)  d(ptpl 1p,m'n'±' •  dp  nn±  Jo  This is in direct analogy with the orthogonality condition of the previous chapter. This can calculated explicitly as follows, r2ir  fin a  /  Jo  dp  Jo  d4>A 'ib , mn±  mln ±l  =  dp  I Jo  d(f> i)* ib > > +  Jo  mn  L  m  n  VCmVWn'  -i(n-n')4> ( ^kp  e  ik'p  e  +e  _ '' + n  -i(n-n')4>  x /"' i <W In a Jo 1  na  % k  '  +  %  e  x  -ik'p"  A  1  ™ A ikp |  c  (_n '  _  e  Jl -ik  ikp ] (  l  e  n-ikn' —j;  P  +  &  + ik'\  n>  A'  + '^ p tk  , —Uos(k-k')p  +  -ik'p"  £  Chapter 2. Vacuum Energy by Path Integral  Calculations  'n — ik  n' + ik'  s  A _  cos(£; + k')p  A'  /  23  n — ikn' + ik'\  —  Onn'°k-k'  =  5mm'finn'8±±',  IH  (n — ik I — Onn'Ok+k' I  ^  ^  n' + ik'\ 1  ^  (2.35)  upon imposition of the condition that k is exclusively positive. This condition can also be deduced from the observation that  m  = (—l) -^ '4>mn which means that those 11  ip(- )  n  1  degrees of freedom are not linearly independent, and is reminiscent of the condition for the bosons. Now the Greens function can be calculated, and it is simply G(p,1p,p',lp')  =  E  m>0,n,±  (2-36)  ^-^mn±(p,tp)lpLn±, ^  which is a 2 x 2 matrix, with the factor of 2na' accounting for the normalization of the Dirac operator in the action (2.1). Also, the sum is restricted to being over positive m by the observation above that the terms for negative m are not linearly independent. The elements of this matrix can be explicitly evaluated, and are 2l\a' Gn = m >E0 , n , ±  a  V V , m n ± (P, (t>Wr, n±  <t>')  m  E  }i in(4>-4>') (jkp e  _  71  + -ikp\  A  m>0,n,±  lk c  e  A"  "  f -ikp'  _ ™  l^Jkp .(2.37) 1  A  The sum over the sign of A has the effect of eliminating the odd power terms, and what is left can be written as a.  1 {4>-4>') ^ J_ -2n(e ik(p+p') A e m>0,n pin  i— E — 2  -ik{p+p>)^  e  k( ( ^ lk  +  2l  p+  e  - e-  )\.  lk{p+pl)  (2.38)  Similar In acalculations for the other components reveal that the structure of the Greens function is G(p^,p',^')  T L X  V  _L„in(0-*')  m>0,n  +  I  -ncos k(p + p')  — ncos k(p — p')  -ncosk(p-p')  —ncosk(p + p')  -k sin k(p + p')  +k sin k(p — p')  -k sin k(p-p')  +k sin k(p + p')  (2.39)  Chapter 2. Vacuum Energy by Path Integral  2.2.4  Calculations  24  Fermionic Vacuum Energy  The fermionic part of the action (2.1) is  (2.40) The interaction Lagrangian can be written as L  int  = \eiF^rW8(p)  -  -e F^~rYV5{\na  l  2  - p),  (2.41)  which means just as in the case of the bosons that the fermions only interact at the ends of the string. Exactly as in the case of the bosons the fermion propagator can be restricted to the edges and made into a larger matrix of the form G(p,<t>,p'A')^G((f>,(f>')=[  ,  VB'OM')  (2-42)  AM, <j>'))  with the same definitions for A and B as in the bosonic case. Now it is necessary to calculate the explicit form of A, A', B, B' for both Ramond and Neveu-Schwarz boundary conditions. The case of periodic fermions is straightforward, and explicitly An'  A  R  = 0(0,4,0,4')  =  E l  n  m>o,n  a  4a' l  n  a  v  1  / ~  ~  n  n  T W ^ ^ \ -n  A  _  m>o,n "  1 2  -n 1  -n —n  \-n  -n  in(<t>—4 ')  -e  +  (  j  cothin ln a)  cothin ln a) \  coth(n\na)  coth(n\na) J  •  (2-43)  The calculation of A' is identical, because m is quantized as an integer, so cos(27rm) = R  cos(0) = 1 while sin(m7r) is vanishing, and because there should be no physical difference between the two ends of the string, the two values are expected to be equal. In particular,  Chapter 2. Vacuum Energy by Path Integral  Calculations  25  the vacuum energy is not changed by the exchange of e\ and — e . Similarly, B 2  R  can be  calculated as W l  n  1 t £ n  a m  A  (-n(-l)  -n(-l)  m  V-^(-l)  2  (  -n(-l)  m  cschin In a)  cschin In a) \  csch(n In a)  csch(n In a) /  and the calculation is identical for .B^. For the Neveu-Schwarz sector the calculation of A^s and B^s requires more care, due to the fact that m is half integer instead of integer. The calculation for A^s can be done as Am' A  N  S  = GMM')  =  E  p -  (—n  1  (  —n\  tanh(nlna)  tanh(nlna)\  tanh(n\na)  tanh(nIn  a) /  • (2-45)  The calculation for A ^ - is analogous, resulting in 5  A;  s  = + W ^  » ( H ' )  e  > / —tanh(nIn a)  n  \ tanh(n\na)  tanh(nlna)  \  —tanh(n\na)  .  (2  .46)  /  The subtlety comes from the evaluation of / n cos k(+ In a) — n cos — In a) \ E i e - ^ M m>o,n \ — n cosA:( — In a) — ncos/c(+lna) /  An'  B  NS  = G(0,A\na,A)  = m  a  A  zW l n a  -  1  —A; sin&(+ In a)  +/c sin k(— In a)  -A: sin k(— In a)  +£; sin k(+ In a)  i  E  /—^-smlmn) re  m>o,n '  — P^sin^/wr)  I FsinfTriTr)  A 2  (  \  lna  v  sechtn In a)  —sech(n In a)  '  ^sin(mTr) In a  sechin In a) \  — sech(n In a) /  '  ,  (2.47)  Chapter 2.  Vacuum Energy by Path Integral  Calculations  and the subtle point is that when evaluating B'  26  there is a difference compared with the  NS  above calculation. The signs of the off-diagonal elements change, because they depend on p — p', while p and p' are interchanged. The result is that  (  sechin In d)  —sechin In a) \  sech(n In a)  — sech(n\i\d)  (2.48) )  From the interaction Lagrangian it is clear that we can re-write it as /^(-ry*)  L  0 -  \  = i / d^Ffu,  mt  .  2.49  Given this interaction term, it is possible to write down the contribution from the unique diagram at order n. This takes the form n  =  th  ^ ( n - l ! ) T r ( F ; j a r l  -  £  m  ^  e  ™i(*2-*i)  mi...m„  • • • e »(*-*»> x Tr [G Q...  G ft],  iro  mi  (2.50)  m B  where Q is the matrix component of the interaction term, and G  m  is the matrix portion  of the fermion propagator. The term ^ comes from the order of the diagram, and (n —1!) counts the number of ways of ordering the vertices. The integration over the (ps is trivial and the result is n  th  1  S  Tr(Fp  (a'n)  n  £  Tr (G tt)  .  n  m  (2.51)  Now, using the fact that the matrix can be diagonalized, and denoting the four eigenvalues of G fl m  by A i • • • ^4m this can be re-expressed as m  1 * Tr(Fp n  n  th  n  (oV)" £  (A? + X  n  m  2m  + A? + A ? J . m  (2.52)  To obtain the vacuum energy it is necessary to sum all the diagrams so £  connected  =  £  \ Tr{F^)  (a'-n) A? n  mnj  E  mj  l o  S  T  r  (<W + (<*'*) F^Xjm),  (2.53)  Chapter 2.  Vacuum Energy by Path Integral  Calculations  27  so _,y] connected  (2.54)  and the sum in the determinant depends on the eigenvalues of  G fl. m  The task is then to determine the eigenvalues of G Q for both the case of Ramond, m  and Neveu-Schwarz boundary conditions. In the periodic sector the matrix ( A  R  G R$l m  —  B'  R  B\  /e *  R  0  l 7  A'J{  0  f coth(m\na)  \  -e V 27  coth(m\na)  csch(mlna)  coth(m\na)  coth(m\n a)  csch(m\na)  csch(m\na)  csch(m\na)  csch(m\na)  coth(m\na)  coth(m\na)  csch[m\na)  coth[m\na)  \csch(mlna) ( 0  e  x  0  0  ei  0  0  0  0  0  0  -e  V0  0  -e  csch(m\na)\  coth(mlna))  \ (2.55) 2  0  2  /  had two non-zero eigenvalues Ai#, A # that satisfy 2  Ai/j + A /j = 2  AiflA K 2  =  2(ei — e )cot/i(m In a) 2  (2.56)  -4eie . 2  In the antiperiodic sector the result is different, but analogous. Evaluating ftanh(m\na)  G NS^ m  tanh(m\na)  sech(m\na)  sech(mlna)  tanh(m\na)  tanh(m\na)  —sech(mlna)  —sech(m\na)  sech(m\na)  —sech(m\na)  —tanh(m\na)  tanh(m\na)  —sech(m\na)  tanh(m\na)  \  X  \sech(m\na)  —tanh(m\na)J  Chapter  2.  Vacuum Energy by Path Integral  / 0  ei  0  0  ei  0  0  0  0  0  0  -e  VO  0  -e  Calculations  28  \ (2.57)  0  2  2  /  A g a i n , two non-zero eigenvalues are found, which satisfy ^INS + A JVS 2  ^INS^2NS  =  2(ei — e )tanh(m  =  —4eie .  2  In a) (2.58)  2  W i t h this information the contribution to the vacuum energy in both the R a m o n d and Neveu Schwarz sectors can be written down explicitly. T h e result is ^  connected  JJ  _  D E T  +  2 ' ( e - e )coth(m In a ) + a  7 r  1  2  (2aV) (-eie )F^) 2  2  m  X>"" ' ec  e  erf  =  JJdet  + 2af 7r(e - e )tanh(m /  2  1  In a)F^  + (2a'7r) (-eie )F ,,) , 2  2  2  m  (2.59) in agreement w i t h the results of [4].  2.2.5  Superstring V a c u u m Energy  For the bosons, the energy associated w i t h an annulus of inner radius a and charges e\ and — e at the ends was determined to be 2  E  = II m>\  with A  m  (  det  [V + 2 W i  m  (  e i  -  e )F , + (2Txa'fF%{-e e )\) 2  tu  1  2  1  ,  (2.60)  defined as above. Since the external field can be block diagonalized, it is only  necessary to determine the effect of one of those blocks on the energy, and the rest w i l l be identical calculations.  If the field strength is / in this pair of directions, w i t h the  identifications /?! = 2 7 r a ' e i / ,  /? = -2ira'e f 2  2  (2.61)  Chapter 2.  Vacuum Energy by Path Integral  29  Calculations  then for the two dimensions in question, the energy is -i  JJ det m>l  AmPl  +  1  fo  -  P1P2  -1 ( l - A / ? 2 )  II  \m>\  n  t  \m>\  +  2  1-  (1 - a ™ ) 2  \m>\  2  + 1  n  ( ^ m ( / ? l + / ? ) )  +  1+ 2a 1 - 2a  +a  2 m  4 m  +  2 m  n ( l + /? )(l + /5 ) x 2  2  2  1  n>l  2  -I  + p\p\ n>!V 1-2 1 - p\ (-l +PIj S-? )AP,p (l+#) 2  m  4m  2 m  (2.62)  Now, making the identification _1_  _j I-PI2n °  V  PI - 4Aft + P\Pl  (1+/? )(1 2  C S  +  (2.63)  /?I)  the energy can be re-written in terms of a Jacobi 9 function as Z  / _  _  ,  na+A )(i+^ ) 2  2  /  1  1  T-)  \  f ^ n ^a  \  m>l  U  '  1  3  —OiU—)  ] 2sin7r^  2m  2  TI>1  _  V  ml  -1  (2.64)  which can be re-cast with a Dedekind r\ function as  i ^M^)\ n(i+/? )(i+/? ) 2sin7r^ rfl I Ina \ 2  X  2  2  \ iir J  in>l  ((l+/? )(l+/? ))2  2  1 / 2  ^ {Pl  +  p2?  (Ina \ \ in J  (2.65)  2(Pi+P ) 2  The second step makes use of the fact that sin(|cos /c) = \f^r _1  and also £ function  regularization. This gives the effect of one independent block of energy of the bosons.  on the vacuum  '  Chapter  2.  Vacuum Energy by Path Integral  Calculations  30  The above must be integrated over the Teichmuller parameter, and the obvious generalization is that the vacuum energy in 10 dimensions is  *=/irr 2<A.+A»>rJMr n d  J  where as before, /  1 ;  U  .../  5  n=l  Vl \ n \ ^ - ) u  (i - « - * * ) ' .  (*•«»>  k>l  are the independent components of the background F^  field,  equation 2.61 is extended to read  p  in  =  2Tra'eif , n  fan  = -27ro:'e / 2  (2.67)  2j  and the generalization of 2.63 is  T ""  S °  =  C  _,!-/??„ - pi - 4 f t „ f e „ + m o i (! + «.)(!+«.)  S  •  < 2  -  6 8 )  The contribution from the ghosts is simply the vacuum energy in the limit of zero field for a pair of the coordinates, but in the numerator, as is appropriate for the degrees of freedom which reduce the number of bosons. The integration has an imaginary part for a background electric field, which indicates, as discussed in [4] that there is pair production of strings from the vacuum. The contribution from the fermions is more subtle.  They are multiplicative with  the bosonic contribution, but must be summed over four spin structures [10] [11] which is equivalent to performing the GSO projection on the string. These spin structures correspond to periodicity or antiperiodicity in both the p and <j> directions on the annulus, and have their origin in the fact that the annulus is also a tree level closed string diagram. The contribution of the two terms that have periodic (j> e a r  1 ^ NS,P  J  -  2(A+A)  #i (H—)  '  rf(4)  (2 69)  -w.- " n ( i + «  2(1-Aft) „(!M)  1  2  i  .  (2.70)  Chapter 2.  Vacuum Energy by Path Integral  31  Calculations  while the terms antiperiodic in (j) are ZR,A  =  , ] ^f^a-W V ( l + /5i )(l + /5 ) /  2  II ( l - ^ -  ->o  2  2  ZNS,A  = ,  ^ H^) -l/12  }  3  a  V  JJ (  1  +  a  1  )"  (2.71)  2  J  2 m - ! N -  2  (  2  7  2  )  As in [11] the sum over spin structures is given as ZR,A +  Z s,p N  Zp  —  Rj  —  (2.73)  Z S,A N  and as noted in [4] the zero field limit has this exactly vanishing, as expected for a model with unbroken supersymmetry.  2.3  The Disk  2.3.1  Boson Vacuum Energy  The amplitude for the disk is a simpler calculation than that for the annulus, and can be obtained as a limit of the annulus calculation. The calculation starts in the same place as the others, with the action for a set of bosonic fields on the string that are only coupled at the boundary to the external field. This action is [1] 1 l' S = - J dadrdaXTX^  +  i2ixn'p  c J dTFp,X"d X», T  (2.74)  where the X fields have been rescaled by a factor of y j ^ y compared with the previous section, for ease of computation. Reading off from this the term associated with each vertex is -eira'F^J  drX^rX",  (2.75)  and the analogy with the annulus is that because there is only one boundary il, = 1 here. A convenient way to do calculations on the disk is to parameterize it as the unit  Chapter 2. Vacuum Energy by Path Integral  Calculations  32  disk in the complex plane, because the Greens function satisfying Neumann boundary conditions is well known there and is [12] G{z, z') = -^- In \\z-z'\  z - z'- 1 .  (2.76)  1  2TT  Using the parameterization z = re , z' = r'e ' and denoting £ = r — r', because there is lT  lT  only one boundary in analogy with the calculations done for the annulus it is necessary to calculate the propagator restricted to the disk's edge. With r = r' = 1 the propagator immediately becomes G(r,r') = -^ln(2-2cosC),  (2.77)  In ( l + b - 26 cos C) = - 2 E — cos m(,  (2.78)  and using the identity 2  n=l  m  then G(T,T')  /x  1^  cos mC  = - J 2 — - ± .  (2.79)  This expression also has an analogy in terms of the annulus, where the matrix A  m  =1  in this case. This considerable simplification makes the closed form for the disk simpler than that for the annulus. Now it is necessary to calculate an arbitrary Feynman diagram, and exactly as in the case of the annulus, the n  th  n  th  = 1 ( - e W ) " Tr(F ) n  diagram is (n - 1)!2" f dn . . . dr ^-G(r n  u  r )Ct... 2  ^-G(r , n  (2.80) The term 2 comes from the observation that each vertex has two equivalent bosonic n  legs, and no matter how they are connected the diagram can be restored to the form above by and integration by parts. Using the above expression for the propagator this  Chapter 2.  Vacuum Energy  by Path Integral  33  Calculations  can be re-expressed as =  - ( - 2 e W ) " Tr(F )  \  ]T  n  j dr ...  dr sin m (r  x  n  x  - r) x  x  2  m\...m  n  . . .sin m„(r„ - T )A Vt... X  1  (-2ena')  Tr(F )  n  A il  mi  £  n  mn  (A Q)  .  n  m  (2.81)  m>0  n  So the sum of the connected vacuum diagrams is Y, connected  =  ^  £  ^ (-2 ro/) Tr(F") (^ ft) n  e7  ?i>l m > l  =  ~  n  m  ^ T rl n  (V + 2e-Ka'F^A Q)  .  m  (2.82)  m > l  So that  e  £  connect  =  J J  exp ( - T r In (5^ +  2eTia'F^A Q)) m  m>\  det (5^ + 2e-na'F A Q)' ilv  But since the factors A  m  and  ( 2  m  '  8 3 )  are trivial, this becomes = yjdet (5^ + 2ena'F )  (2.84)  llv  by £ function regularization, and this is exactly the Born-Infeld action. It is worth noting that the expression for free energy of the annulus exactly reproduces this in the limit where a —> 0 and e  2.3.2  Fermion V a c u u m  2  0, which corresponds to the disk amplitude.  Energy  It is more difficult to place fermions conventionally on the disk, because while the topology had a Greens function that satisfied the two dimensional Laplacian, the mapping to appropriate boundary conditions is non-trivial for the fermions.  However, since the  question has already been solved for the annulus, and since for the bosons one can get  Chapter 2.  Vacuum Energy by Path Integral  34  Calculations  the disk amplitude from the annulus amplitude by taking the limits a —> 0 and e —> 0, 2  it is reasonable to suppose that the same procedure will work for the fermions. In this limit the energy for the fermions from the Ramond sector (2.59) is = II det (V + 2 a V e F ) ,  e£ ° c  nnected  1  (2.85)  / u /  m  and equally the sum in the Neveu-Schwarz sector is e  E  connected  =  JJ  ^  (  ^  +  j  (  2  gg)  m  so the contribution from either sector is equal, which confirms the naive expectation that there should only be one way to include fermions on such a simple topology. After C function regularization the fermion contribution becomes 1  x/det (5^ + 2e7ra'F ,)  (2.87)  tll  This means that for superstrings in a background U(l) field the disk amplitude becomes trivial, and the tachyonic divergences are cancelled.  2.4  The Bosonic M o b i u s Strip  The mobius strip is very similar to the annulus, and the propagator and vacuum energy can be calculated similarly. The fields must obey the same Neumann boundary conditions as on the annulus, with the only significant difference coming from the topological boundary condition G(p, <j>, p', </>') = G(\na - p, <f> + 2vr, p\ (/>').  (2.88)  This has the effect of suppressing the odd modes in p and p', which leaves the Greens function in the form G{pA,p'A')  = f-  E  ^ - . e ^ ' ^ c o s ^ p c o s ^ p ' ,  (2.89)  Chapter  Vacuum  2.  Energy  by Path  Integral  Calculations  35  which is exactly the same form as the propagator for the annulus, with the multiplicative change in the constant multiplying k. When restricted to the edge of the strip, of which there is only one, we find that C7(0, 0,0,0')  = C7(lna,0,O,0') = C7(0, 0, Ina, 0') = G(lna, 0, Ina, 0') ,  =  COS 777.(0 — 0')  ,^  Y,  4a'  , 777 In a  (2.90)  — — — c o t h — — ,  2  m  m>0  .  and so, following the exact same steps as in the case of the disk we find that the partition function, given by z  =  e  Econn  =  jj  e  x  p  (  Tr  l  n  (  ^  +  2  .  , _, J] Idet o^u + 2 • 27ra m>0  27  ra'eF^cothH^)  \  m>0  V  . r n In a  eF^coth—-—  2  X  - l  (2.91)  Several things can be observed. In the limit a —> 0 this reproduces the form of the Born-Infeld action for the field F^. The additional 2 multiplying the field strength can be thought of as encoding the information that the mobius strip has an edge twice as long as that of the disk, and the lack of a F term indicates that there is one and only 2  one edge, in contrast to the annulus where the "squaring" of the Born-Infeld action is precisely because of the existence of two edges.  Chapter 3  Non-Commutative  3.1  Geometry  Introduction  Non-commutative geometry is a subject that has received a great deal of attention in the past few years [13] [14] [15] [16] [17] [18] [19]. It is interesting for two particular reasons. The first is that field theory on non-commutative spaces are currently being studied because of interesting properties such as U V / I R mixing, and the second is that if the X^ fields on the string world-sheet are regarded as the coordinates of the string embedded into a larger space it can be shown that they are themselves non-commutative. The general form of the non-commutativity among the coordinates is encoded in the non-commutativity parameter 9  pi/  which is defined by the relation [X*,X ] V  =  itf"",  (3.1)  and it is taken to be a constant. The algebra of the functions on the non-commutative space is governed by the Moyal product, defined as f*g(x)  =  e ^ ^f(x L  d  + ci)g(x + Ok=c=o,  (3.2)  which is associative and non-commutative to order 9. The way that non-commutative geometry appears is by considering the end-points of open strings, for which it is possible to derive an exact propagator in the presence of a antisymmetric background. Noncommutativity appears in the guise of coordinate dependent coefficients in the O P E .  36  Chapter 3. Non-Commutative Geometry  37  The purpose of this section is to derive the non-commutativity of space time coordinates in the perturbative framework developed previously for the calculation of the partition function for strings in an electro-magnetic field background. The methodology is as follows, given the action for a string ending on a Dp-brane with a constant gauge field confined to the brane and an antisymmetric B^ field in the bulk of target space, the two fields appear in a gauge invariant combination, which is treated as a perturbation which gives rise to an interaction term in the two dimensional field theory of the bosons on the string world sheet exactly as in the previous section. The modification to the X field propagator due to an arbitrary number of such interactions is calculated, and using a technique due to [14] the non-commutative nature of the Dpbrane is revealed both at the disk and annulus levels. This is generalized for the case of a string ending on two separate branes. It is not surprising that there should be some modification to the standard commutation relations given an antisymmetric background. The well known problem of the motion of a charged particle in a constant magnetic field background gives rise to a nontrivial commutation relation with the anticommutativity parameter depending on the magnetic field strength.  Indeed, it is noted in [14] the a' —> 0 limit of string theory,  which is equivalent to that of a large (T 3> 1) background field, has an action which is exactly that describing the motion of electrons in a large background field. It is shown [14] [20] that the dynamics of the X fields is that of electrons in the lowest Landau level, and the commutation relations, derived later, reduce to (3.3) in the large field limit. Correspondingly, the noncommutativity parameter, (3.4)  Chapter 3. Non-Commutative Geometry  38  goes in the limit of small fields (F -C 1) to 9 — T which illustrates the fact that commutativity should be restored in the case with no background field. 3.2  Disk Calculations  It is well known that the action for a fundamental string ending on a Dp-brane in the presence of a B-field, with a gauge field confined to the brane is [21] S  =  4 ^  /  [^G^X^dpX"  + J drA^X",  + ncx'e B d X' dpX' ] a0  i  lll/  [22]  /  a  (3.5)  and since the term involving the B field is a total derivative, and in the approximation of constant B and F fields the last two terms can be combined in the form S  =  tta!  I  9 G^d X»dpX  d2(J  a0  + i | drT^drX^  v  a  (3.6)  with •Faw  Fp^  (3.7)  v  as the gauge invariant field strength that appears in the Born-Infeld action. Using the gauge transformations A -> A  A ^ A  + X,  + dA,  B ->  B ^ B  B  (3.8)  + dX  it is possible to leave only components of T which are on the brane. It is equally well know that for the disk amplitude the world-sheet of the string can be conformally transformed into a unit disk on the complex plane, and using the propagator from the previous section  G(z, z') = -^-ln\z-  z'\ z- z'- . 1  (3.9)  Chapter 3. Non-Commutative  Geometry  39  When this is restricted to the boundaries \z\ = \z'\ = 1 it becomes ^ e ' ^ l f  C  O  ^ - ^ ,  S  (3.10)  so using the standard Feynman rules for the now interacting field theory the correction to the propagator from N interactions with the external field is G (pe ",p'e 1 ) i<  i  /  N  ^...d^pe'V^  =  j  =  (-2ita'ef  (T")^  J d9 ...d9 -^ l  N  J2  l  _j_  p'm  N  x sinmi(^i — 9 ) sinm (9 2  „  . V  2  . „,. 1 *  1  m  m  N  —9) ...  2  n  J-m  smm (9N  3  im  n  1  fJ-m  (  COS Vflid)  —  6')  [ in m(0-0')  2  rn  — </>')  N  S  NeveTl Nodd  (3.11) Then summing the contributions from all the values of N gives the propagator as r  i r J *  G (pe*,pe*) int  I  n'^'\  =  1  [_ l  {2nale)  f  1  v  (PPT  + (P/PT  2^)  , .  /  V  -cosm^-^Neven  (3.12) The only thing that remains is to calculate the equal time commutator of the X fields, and to do that using the technique due to Witten's paper, [X"(T),  X"{T)\  = limT(X»(T)X»(T  - e) - X»{T)X»{T  + e)),  (3.13)  where T represents time ordering. In the case p = p' = 1 the sine series above is nothing but a step function in the limit e —> 0, while for either p or p' not on the boundary this converges to something proportional to t a n (f^^iyf)> which vanishes in the limit - 1  Chapter 3. Non-Commutative Geometry-  AO  e —> 0. This shows that for the end-points of the open string at the disk amplitude there is a non-trivial commutation relation, which is the signature of non-commutative geometry on the brane, and this relation is, after rescaling the Xs by \J2KOL' to restore the fields to a form comparable to those used in analyzing the annulus,  ^ •  J  f  ' W '  =  w  ( i - " ( w ^ ) " -  ( 3  -  1 4 )  This reproduces the non-commutative geometry on the brane [14] [21], and it also demonstrates that the non-commutative geometry is in fact a perturbative effect, and not due to instantons, or other non-perturbative effects.  3.3  One L o o p Correction to N o n - C o m m u t a t i v e G e o m e t r y  Since the foundation of the non-commutative geometry is given in tree level diagrams, it is interesting to consider the corrections that come from one loop string diagrams. In this section one loop diagrams are discussed with two main thrusts, the first being to examine the effect of an open string stretched between two branes, and the second, to give the one loop correction to the commutator found at the disk level for the X^ fields. As is developed elsewhere, the free propagator for the X fields on the annulus with Teichmuller parameter Ina, which has been conformally transformed to a cylinder is (2.7)  + 2 W T cos ^ nTo  l  cos ^ n  a  l  (—\ n  a  \  n  7  r  | , /  (3.15)  /  and as discussed in the calculation of the partition function of the string, this propagator can be re-cast as a matrix valued propagator of only one variable when the interaction happens only at the edges. The Lagrangian for the string coupled to the two branes is the same as that given before, with the Ti being the gauge invariant field strength  Chapter 3. Non-Commutative  on the i  th  Geometry  41  brane. In this language, the N  diagram for the propagator on the annulus  th  <X'( ,T)XV,T')> ^ P  G (P,T, ',T') N  P  ^N\jd9 ...d9 (G(p,T,O,0 ),G(p,T,\na,9 ))  =  N  1  u  -e^rt)  0  1  u  \  n P  N  l  )de M(9 9 )x...x  G(0,9 , ',T')  xdg \  1  ~  1  2  lkN  V  l  0  -e Tl2k 2  ,  \G(lna,9 ,p',T')  N  (3.16)  N  where , , T  n  ^  M(9,9')  = Y,  cosm(9-9')  m>o  fcothmlna  cschm\na\  \ cschm ln a  171  cothm ln a )  •  (3.17)  The integration is trivial and gives G (P,T,P',T>) n  =  ( sin m(r — T') Nodd , m>o [ cos m(r — r ) Neven  i  (2aT (-*) A-Y,{ +1  N  m  ^  x >  i  n  a  rnrp  cos l  n  a  1  y, cos  ^777, + t e ) 2  nixp l  n  (-1)™  ik\  1 f i^i e  y  °™ +fe)  J V  2  0  / cothm ln a  cschm ln a \  / cothm ln a  cschm ln a  cschm ln a  cothm ln a /  \ cschm ln a  cothm ln a  ' e i ^ . ,  0  0  \ _  /cos  l  n  l  n  -e Tt 2  a  - e ^ J »' Vcos^^g, £  u  a  ™  2  2  + ( i n i )  .  (  V In a /  Note that this expression reproduces and generalizes the non-commutativity relation found by Chu [22] for strings between two branes, namely that 1 -  (27ra'eiJ )i r  <  7ra'27ra'e T 2  2  Vl-(27ra'e ^)iy  p = p' = ln a  2  0 otherwise.  (3.19)  3  .  Chapter 3. Non-Commutative  Geometry  42  The generalization is that there is no requirement that the fields on the two branes are aligned in any way with each other, which is no surprise, but reassuring to discover. The limit in which the above expression reproduces Chu's result is in the limit a —> 0. The way to understand this is to note that this limit corresponds to the disk limit, since a is the Teichmuller parameter for the annulus. Chu's results rely upon a time average to calculate commutators, and this average is taken over non-compact time (on the string world-sheet) which corresponds to an infinite string world-sheet extending between the branes in question. This is nothing but a disk amplitude. The last thing to note is that in the most general case, with two branes with independent T fields on them the commutator of the X fields will generally depend on both the Fs in a non-trivial way. This means that in a Randall-Sundrum type scenario in which there are two branes, with us living on one, subtle deviations in the non-commutativity parameter would enable the measurement of the T field on the other brane, which is an exciting possibility. Now, consider a string beginning and ending on the same D-brane. This brane has a gauge field with field strength F on it, and there is a B field in the background. The commutation relation for the X fields will be determined exclusively by the values of the string endpoint charges.  The dependence on the background fields occurs in the  combination T = B — F which is gauge invariant. The action is re-written so that the X and T fields have a derivative interaction at the edge of the string world-sheet. This process in itself suggests that the string coupling to the T should be the same strength at each end because the bulk interaction with the B field is a total derivative, and appears with equal coefficients when rewritten as a surface term on the world-sheet. This is in direct analogy with the procedure used previously to determine the string partition function.  Chapter  3.3.1  3. Non-Commutative  43  Geometry  T h e Charged String  For a charged string with equal coupling at each end, we have e\ = —e = e and the 2  above expression (3.18) can be modified. With equal charges and p = p' = 0 analysis identical to that for the partition function that we've discussed before gives that (X*(0,0)^(0, cj>'))  N  =  i  f smm(6 — d>') N odd 1  2c/(2W)"(e"F") £U  m  / (1 + 2a  + a )-  V  0  m  2m  N  [ cos m((f> — (j)') N even J  171  0  1  (1 ~2a  \  + a  m  2 m  ) - ) N  1  1 \  (I  1 (1 + a ,2a ),  x I  2m  (3.20)  m  and so i  . j,m  ((1 + 2 a + a ) m  2m  r  f sinm((f) - (j)') N odd ] i / [ cos m(<p — (f>) N even J  n  + (1 - 2 a + a  N+1  m  2 m  )  / v + 1  z  V  i "  i  ) .  N  /  (3.21)  The same calculation for p — p' = l n a gives an identical result. The results for p = 0, p' = In a and p = In a, (X (0,cf>)X^lna, f>')) l  (  N  = 0 are equal to each other with =  2a'(2na')  (e T )  N  x- (  _  N  a 2 r n  N  3  )  £ m  i  f sinm(c!> — d>) N odd  -  { cos m((f) — (f)') N even  m  ((1 + 2a  m  +a  2 m  )  N + l  - (1 - 2 a + a ) ) m  2m  N+1  .  (3.22) Thus, the sum of these four contributions, which must be taken to evaluate the commutator gives . i,-  <A^)A-(0')>* =  i  2a'(2W)-( ^) E^ i J  e  m  m  f  sinm(0 —  <t>')  N odd 1  „  [cosm(0 — (j)) N even J  N+I  /  2  ,  1  m 7v+i x  ( T T T ^ ) V  i  a  '  Chapter 3. Non-Commutative  44  Geometry  (3. Since in the end, a commutator is going to be evaluated, it is obvious that the terms which have even N will cancel, while the odd N terms add. Summing over all odd iV and integrating over the Teichmuller parameter one obtains \X\X>] = lim / — V - s i n m ( e ) x 4a' * L J e^o+J a ^ m 1- fk / f c  w  where / = 2-Ka'eT, and k =  2  2  ,  v  (3.24) '  . The sum of sine terms is the fourier series for a  sawtooth wave, and so the limit e — > 0 can be evaluated as  This integral has two poles, one as a —>• 0, and one near a — 1. The second is a simple pole, and the contributions of the integral on either side of it cancel, and the pole at a = 0 can be regulated by introducing a small cutoff, and then up to constant terms "i da 1  J  Je  ,  a  fk Pk 2  1  2  f_ -27ra' -^- lne, 7  I-/  2  7i  (3.26)  Thus the contribution at one loop level from a charged string to the non-commutative geometry is a renormalization effect. The divergence is the same encountered in the free energy for the free annulus, which is interpreted in [6] as originating from the open string tachyon.  3.3.2  The Neutral String  For the neutral string, a very similar process can be followed. In particular, taking the initial form of the N  th  interaction, and substituting e\ = e = e, which is a neutral string, 2  Chapter 3. Non-Commutative  Geometry  45  one obtains  i f sinm(0 — d>') N odd (X\PA)X>{ 'A'))N  2a'(2W ) (jfE w  =  P  i  e  m  fl  x (/ (p,0),/ (p,lna)) m  m  0  [ cos m((j) — (f)') N even 0 \ / 1 o!  m  \0  -1 / \ - a  ( - l ^ l - a  1  \ ~ N  X  /  ,l-a "V  ™ ) " -  1  ; \-a  m  -J  V/ (lna,p') m  1  Vl-a  2  2  -1  m  2 m  (3.27)  ,  Since the properties which determine whether a particular TV survives the difference in the commutator depends only on / and sin, and from previous experience we know that only the odd N survive, in the following we only consider odd N. For the case of p = p' = 0 the expression simplifies to A-(O,0)A^'(O,0'))iv  =  (F YY-^m{ct>-—A 1N  2a\2na'e)  N  i  x fl + a  2 m  i  0  i  - ( 1 -a ) 2m  j \  sin m{<j> -</>')(—-  N  m  m  ) -4a 2  3  I I  2a'27ta'e)  2 m  n  \ / 11 + a„ 2 m '  m  V  x ((l+ a  n 0  2m  2 m  — II.  1  ,2a ) I  V  =  m l-a  /  2 m  \1  —  a  ,2m  2a 2  m  )  ).  (3.28)  Following similar logic, we find that for p = p' = ln a, (^(lna^X^lna,^))^  =  2a\2na'e)  (T y  N  (-(l +a  Y-smm(<f> ui- in  N  2 m  )  2  + 4a  2 m  ),  - <f>') \1  Cl  J  771  (3.29)  and for p = 0, p' = ln a, the result is  = 2a'(2We) (^ ) ^lsinm(0-0')f^-^) yv  (X\0A)X^\naA'))  N  Ar  tJ  2  Chapter 3. Non-Commutative  46  Geometry  ( l + a,2m  x ( ( l + a ) 2o -2a 2m  =  m  m  ))  0.  (3.30)  So since the four contributions must be summed to obtain the annulus correction we see explicitly that the contribution at each order in perturbation theory from the neutral string to the commutation relations vanishes.  3.3.3  Distinct Branes  As noted above for an open string stretched between two branes, in the limit a —> 0 the non-commutativity parameter is 9 =  j ^ where T is the gauge invariant field strength  1  2  appropriate to the brane. This will, of course, be modified when not in the disk amplitude limit. Since the field strengths on the two branes are not a priori related to each other, it is unreasonable to expect that they have similar decomposition or block diagonal forms. Here, we examine the limit T  <C T\, which gives leading order corrections to the non-  2  commutativity parameter. With T <C T\ we find that the contribution of the N interactions with either back2  ground field gives  +csch m 2  ln a (T T^  +  1  2  (3.31)  T T T[ X  2  Since the expression multiplying csch m ln a is manifestly odd for odd N, and even for 2  even TV it is clear that following the same procedure as above [X",X [ V  N  =  ira.•! {2na'f  coth - m  +csch m\na  (T T"-  2  N  l  1  2  ln a [coth m ln a ( j f ) 2  +  T T T[ X  2  Tt F Y] l  2  .  (3.32)  Chapter 3. Non-Commutative  Geometry  47  Now, making the assumption that T\ and J- have the same block diagonal form we 2  obtain. 2na''  T\Coth mma 2  (2ira'J i) coth m r  2  2  fl + (27ra'J i) coth m In a , + 7 - —. — csch VI — (2i:a'j iYcoth mma r  2  In a  2  2  2  7  2  m\naT  2  (3.33)  l  Using the same techniques as before we can see that upon integration over the Teichmuller parameter, there will be the same divergent structure for the full commutator as in the case of the charged string.  3.4  N o n - C o m m u t a t i v e Geometry from the M o b i u s Strip  The mobius strip is the simplest non-orientable surface that can be incorporated in string theory, and as such it is interesting to study its effects in the context of non commutative geometry, since it will give the contribution of non-planar diagrams. As was derived previously, the propagator for bosons on the annulus is GMJA')  = ^~ E l n a ^ m  o  m  2  +  2 ^ - ^ c o s ^ p c o s ^ p ' , ( M ) lna lna  (3.34)  and following the procedure given for the annulus, it can be shown that the correction to this due to N interactions with the background field can be written as / i x 2 f s'mmU - <b')N odd ) / m]nn\ ~ V (-M \ hcoth ^) k^ [cosm(<f>-<l>')N even) \ J N  (X*Xi)  N  =  2ct{-2*otef  (F ) N  V  cosf^p xm  - +(s) 2  J  1  7  '  l n a J  2  m k  cosf^-p  T^T2>  -+m 2  and similarly to the case for the disk, for p, p' ^ 0, In a this will identically vanish for  \ 3.35  Chapter 3. Non-Commutative  Geometry  48  Now, calculating the commutator of the fields on the endpoint, we find that after summing the contributions of all possible interactions with the background field for a particular value of a, (3.36) where k = 2coth mlna and / = 2-Ko!eT. This is clearly of the same form as the case 2 of the annulus, and so when integrating over the parameter a, we see a logarithmic divergence near a = 0 and the pole for the critical value of a near a = 1 does not contribute any infinite pieces to the integral, because it is a simple pole. Thus, we see that the contribution to non-commutativity from the mobius strip is the same as that of the charged annulus.  Chapter 4  Conclusion  There are two major themes that are presented in this thesis. The first is the verification of previous work on the vacuum energy of strings. The contributions from both the disk and the annulus for the bosonic degrees of freedom were analyzed and found to be in exact agreement with the results presented by Fradkin and Tseytlin [1]. The fermions on the world-sheet were similarly analyzed and their contribution to the vacuum energy were also in agreement with those of Bachas and Porrati [4] for the disk and annulus, however the method used to obtain these results are distinct from those used in the original paper and thus add confidence in the final results. Finally, the bosonic vacuum energy was found for the Mobius strip, and while it had the same form as that for the disk, namely the Born-Infeld action for the background field, the coefficient multiplying the field-strength encoded the fact that the boundary of the nonorientable surface is twice the length of the boundary of the disk. Using the mechanism and propagator developed for the calculation of vacuum energies several results pertaining to the non-commutativity of D-brane world-sheets were obtained. It was possible to reproduce the classic disk contribution to non-commutative geometry, and also more recent calculation for non-commutativity from strings stretched between two separate branes. The new results obtained from this analysis were the determination that the uncharged string makes no contribution to the non-commutativity at the annulus level, while for the charged string there is a logarithmic divergence which has as its coefficient the standard non-commutativity parameter, and the origin relations for  49  Chapter  4.  Conclusion  50  the Mobius strip are shown to have the same structure as those for the annulus. Perhaps the most significant result is the determination of corrections to non-commutativity on one brane due to a weak independent antisymmetric field on another brane. There are several opportunities for further research presented by these results. One is the calculation of tachyon and vector particle scattering amplitudes in the presence of an antisymmetric background, using the techniques applied above. Another is the exciting possibility of combining the results for non-commutativity within a Randall-Sundrum framework, which could provide a cosmological explanation for non-commutativity in our world.  Appendix A  Alternate Calculation of World-sheet Propagators  A.l  T h e Bosonic Propagator  Another method of calculating the bosons green function can be obtained from a path integral formalism on the string world sheet. The action for the boson is written Sboson  =  4^7/  =  ^  d ad X^X 2  a  ll  j dpd^X^X^  + dpX^dpX^)  (A.l)  in Euclidean space. Since the boson satisfies Neumann boundary conditions an ansatz is E  /£„e  i r a  *cos^.  n>0.m  l  Q  (A.2)  a  Applying this it is clear that S  =  ^ 7 J dpd(f> E  [- m'F^ F ,p e ^ c o s ^ e ^ c o s ^ m  n  mnm'n'  ma  mln  \  In a In a  In a  " T vE- mnF™F. p^ 4 a' l l n a  p  n  mn  In a ,  /U__2 +, ( ^ f) \ * ) . n7T  r  Now the fact that A' is a real valued field implies that F _ —llna^-,  lna  .  (  S= ^rE™»KJ (™ T  l2  2  9  (A.S)  = F,^ so  m n  n  / mr \ \ 2  2  +  (^) j-  ,,  .  (A.4)  Now consider the path integral / [dF}e~  s  =  J  J>„  [dReF][dImF}e  W  51  | £ | (™+ (j^)')) , 2  F  B  2  (A.5)  Appendix  A. Alternate  Calculation  of World-sheet  Propagators  52  with the assumption that the target space metric is Euclidean  n>on>o^ n l n a [m  -2d  2a'n 2  (A.6)  +  (^)  The power of 2 in for the result of the Gaussian integral is due to the degrees of freedom of the real and imaginary parts of F and the d is from the number of free bosons, the dimension of the target space. Now consider the integral J[dF}e- X^ j ,p)X^^,p')  =  s  ( )  J[dF)  1 ln a exp  [ —4 —a'  . ,_,„ ,n i J2  m  n  \ mn\  o ™  F  x E ^ c o s lna ^E^cos^. ao  mr +  ln a (A.7)  cd  The only non-vanishing contributions to the integral come from c = —a, b = d and p = u, so this becomes \ p im{4>-<t>') e  cos  mrp  ln a  cos  mrp' 2a'  1  ln a ln a ™ 2  (  nir  \na)  \2  (A.8)  '  which is equivalent to the propagator calculated previously.  A . 1.1  Equivalence to P u b l i s h e d Expressions  The bosonic Greens function was determined above in two different ways, resulting in identical expressions.  However, both of these methods are for the cylinder, and it is  instructive to verify that they correspond to the the Greens function on the annulus, which can be obtained from complex analysis. The free propagator on the annulus with Neumann boundary conditions is [1] [2] In \z — z'\ z — z ' - 1  + 5>  ,2n  \l-a ^\\l-a ^\\l-a "zz'\\l- 2  2  2  a  n>l  (A.9)  Appendix  A.  Alternate  Calculation  of World-sheet  53  Propagators  and it is desired to compare this with the previous result for the Greens function, which, appropriately scaled, is (2.7) G{ A,P'A')  =  P  TT4— 27T In a  E  Lcfzf— „  E e^^cos^cos™ '  777^7^0,771= — O O 7 7 , =  ^  In a  OO  mr  n7r  . / ln a ,  ln a  TR  1  ™ +2  I  2  + E i—/° i—P\— • cos  A.IO  cos  In a  ma  \mr  I  J  This expression has to be converted into a form that facilitates comparison. The first thing to note is that the second term is formally divergent, and further is eliminated from consideration by the fact that the  fields  have a derivative coupling to the external  field. The above expression, without the m = 0 modes, is even in n, and can be rewritten  C7(p,0,p',0') =  ^E / d n e ^ ^  x L _ * c o t 7 r n . ^2 j_ ( HJL) 2  cos ^-pcos^-pf-^—  1^) as a contour integral:  lna  lna  mr  mr  In a , 1 7T  Vina,/  1  —i  m  2  \ in a /  (A.H) In the preceding the factor ^ came from using the evenness in n to expand the sum overall integers, and the factor of ^ is to ensure that the coefficient of the residue at each integer is 1, and the contour is an infinitesimal circle around each element in the sum. The contour can then be continuously deformed to a pair of semi-circles in the upper and lower half plane respectively, that do not overlap the real line. This integral can then be evaluated at the poles n = ± " , imX  and the objection to the procedure that the  a  cosine terms diverge at n — ±ioo is taken care of by the fact that the cotangent in the expression gives an additional sign change, so the curves at infinity cancel. This implies ^/ , / ,i\ s.—* /, ,-A h mp cosh mp' , ., . G(p,<p,p,<f>) = > cos m(<p — phi) cothm l n a (A. 12) c o s  rT^o  mTrlna  is an equivalent expression. Now explicitly expand the above in terms of 4> and p to obtain e  ~2>o  im{4>-<j>') _|_ g-im(0-«4') grip _j_ -mp £  m7rlna  2  £  mp'  _|_ ~mp' e  2  ra  fl  _j_  a  -m  a -a- ' m  m  (A.13)  Appendix  A. Alternate  Calculation  of World-sheet  and using the identification z = e  54  Propagators  ^ this becomes  p+l<  1  = E mirS In a  2  m>0 +  (  £  r  +  ^  (  2  m>0,n>0  (  i  (  +  (zz')  (  r+  _L  ,z' + (-) z  + (-) z'  m  £  a" + a  r  22:  m7r8 In a r  22  1  i  (  2 i  E  +  r +  2  m  m  +  {zzj  2m\ alm)a„ 2 m n  (1 +  r  + (—) z'z  m  (A-14)  Now, doing the sum carefully this becomes  E  1  (22') + (^) m  m7r8 In a  m  + (|)  + {^z)  m  + (ZZ'Y  m  m>0  z'  z  1  E  2  zz'  m>0,n>0  (22')" + ( - ) 2 1  m7r8 In a  + (-) 2  m  + (— ) 22  m  m  + (22')  +(4r + (-r + (-) 2'  1 ,r2 m n 7  2  T  (A.15)  ZZ'  This expression simplifies to 1 rrnr8 In a  In M l - 22I 1  . V  + 2Eln 71>0  1  r  1  2n  1 — a 22  /, 2-2' 2 V  2 In  + 2Eln n>0  -I  1  2  1  2'  1-a "2  1 z'  2  1  1  2'2  1 - a2 n '  1-a " — 2  - 21n Ul  1 - a "22' 1 - a2n 2  2'  1-a "2 2  2n  1- a  (A.16)  Now, this can be seen to be equal to the expression from [1] apart from the In \z\ term, which is clearly harmonic in the region. Due to a general property of differential equations any function / that satisfies V / = 0 and does not affect the boundary conditions can  Appendix  A.  Alternate  Calculation  of World-sheet  Propagators  55  be added to the Greens function and it is still a Greens function. This demonstrates the equivalence of the two different representations of the Greens function.  A.2  T h e Fermion Propagator  In close analogy with the case of the boson, a path integral calculation of the fermion propagator can also be done. Again, the starting point for the calculation is the free fermion action, which can be written as  S  =  4 ^ /  ° (-W-fM*)  S  •  (A-17)  In this case the mode expansion for the fermions is  VKp,0) = £ V w  .  '  e" *.  (A.18)  im  mn  This expansion clearly satisfies Neveu-Schwarz or Ramond boundary conditions depending on whether n is integer or half integer. Using the convention for the 7 matrices  o)'  7  " 0  o ) "  / In this case the Dirac operator —ij d a  a  (  0  A  "  1  9  )  d^-dA  becomes  . Making the as-  sumption that the target space metric is Euclidean, and recalling that the Dirac conjugate of a spinor is i/> = 7/^7^ the action can be rewritten as .1  I e  im(i>  r  47ra J  MN  0  ^  n  u  L  J  ,,  M N  dt-d \ p  mn  'n'  m  (  e '^ m  \  1  /0  %  Appendix  A.  Alternate  le  ln  ° , e  Calculation  l n a  of World-sheet  d * - d  0  I  d<t, + d  m'n' "n<P ~im ie  ^Ct  '  mn 'n  p  \ ( e  e  cp  &  M  )  p  56  Propagators  \e- '& in  x  m  mnm'n'  V  na/  for each of the d fermion fields. Now consider the free vacuum energy / [oV] e* = / [dtf] M  exp  s  (_  £  m  +  .  (A.21)  This exponential can be Taylor expanded very simply since the coefficients ip  are  mn  Grassman parameters, so / m  [d^t]  (i+  n  mn  =n — ( -  Ana  .+  / +  ^  *  m +  n7r  'mn I  In  a  d-  (A  -  22)  Now, to calculate the Greens function for the fermions, it is necessary to similarly consider the quantity  / [dV>KVKp,^V>')5  (A.23)  Now, using the expression for tp from above it is possible to directly substitute into the above expression and find =  / m  [dtf] n (l + ^ L t f V n n (-m + —))  E^e-^ ab  ( e ^  , -*rfr). c  1  e l b l n  -i  \e  W  °  7Tp  lnc  (A.24)  Appendix  A.  Alternate  Calculation  of World-sheet  57  Propagators  It is immediately obvious that the Berezin integral forces the conditions a = c, b = d, and noting that the mode coefficients anticommute,  M n fi + ^ L ^ m n  = /m  mn \  J  j  e  \  e  i b  (  - ^  p + p  p  -  '^ p  a  e ^- '^Tta ib  ' ^  e  ( - m + ( - i )  v  ma//  £  ^  e  w  -  v  X  a b  \  p  - ^ + P ' ) i ^ )  )  = EH)e"l'-«  n  i l ^ _ iHL m+  .(A.25)  From the above result it is immediately apparent that for the fermions the propagator can be written as  .  G(p, <f>, p', <j>') = T  1  .  (  (  in e  i — - " - ^ - * )  '^  e ( ~ '^  p+p  in  p  . \  p  . ( .26) A  which can be trivially re-expressed as =  m+  JUL [ n o  \ in a /  <  ( e -  a j ^>_^ | n  c  m(  in{  ^  p+pl)  e  -  v  |  (A 27)  x  which can be seen to be of the same form as the propagator calculated from the eigenfunctions of the Dirac operator, with the observation that because this calculation was performed with a Minkowski world-sheet metric it is necessary to let n —> in for the purposed of comparison.  Appendix B  A Lemma  The purpose of this lemma is to offer a proof of the validity of the technique used above to promote the bosonic and fermionic propagators to matrices. Consider a field X(x\ .. .x ) n  which is defined on a manifold of dimension n with p  boundaries all of dimension n — 1, and interacts with some external potential F only on these boundaries. Further, assume that there exists some choice of coordinates such that each of the p boundaries corresponds to a unique value of x . n  A Greens function  can be determined for the field on the manifold, and in general it will have the form G(xi,...  x , x^,... n  x ). n  The statement to be proved is: 'The vacuum Feynman diagrams of a field which only interacts at the boundaries of a manifold can be evaluated by integrating over all the coordinates transverse to the boundary and replacing the propagator by the matrix constructed in the following way (A  xl  G(x-[,  ^ G(X\  • %n)  2  n  ...  A  A2  •••  A? i2p  l2  A\ • . • X —\, X^ . . .  A  2  l  p  \  , (B.l)  % _i) n  V Ai  A2  p  p  A  /  where Ab a  G(x\ , . . . , X —\ , X , Xy,...  with x^ as the coordinate of the a  n  th  n  x_, n  x  x ), n  (B-2)  boundary. The interaction vertex is given by a  58  Appendix  B. A  59  Lemma  diagonal matrix whose entries are the coupling to the field at each boundary with ti = diag(g g , u  2  (B-3)  • ••g )F P  and taking the trace of the result.' The proof is by induction, and the case of 0 interactions with the external potential the statement is trivial. In the case of one interaction it must be shown that the two methods of calculating the vacuum diagrams are the same. In the case of p boundaries, there are p integrals of the form  J ctai...  dx G(xi,...,  x , xi,...  n  x )g F5(x  n  n  a  (B.4)  - x ) a  n  n  and the sum of them is r  P  J dxi. ..dx -i n  E (G(xi,..  (B.5)  .,i -i,<)Ffl )).  .,x -i,x%,xi,..  B  n  a=l  a  Now, for the matrix method one evaluates  I  /An  A  A  A 22  12  21  dx\...  dx -\Tr  A  \  (9x  0  A 2p  0  £2  0  V0  0  9J  l p  F  n  VAi  A  p  p2  A  J PP '  (B.6)  P  with x = x' which can be trivially seen to be identical to the expression above. Now, assume that the expression above is valid for the interactions of order m, the task is now to show that it is valid for order m + 1. Consider a general Feynman diagram for the field X interacting with the potential F. There is a propagator for X between the m  th  point (which is on boundary a ) and the I ' point (which is on boundary a,i). To s  m  construct a diagram of order m + 1 that propagator will be replaced by two propagators and a vertex. There are a total of p possibilities for which boundary the inserted vertex  Appendix  B. A Lemma  60  is on, and the modification to the Feynman diagrams will be as follows Prior to the insertion the additional vertex, the Feynman diagram would be evaluated as j dx ... l  where x is the i l  th  dx G{x\x^,  x  / dx\..dx dx G{x\x \x\x^)g  6=1  J  m+l  ax  x\ x^)g  (B.7)  am  Summing the p possible insertions gives  ..  a  n  ... G(x , m  ai  set of coordinates x\ ... x\_ .  E  m  x\ x%)g  m  .G(x ,x*~,x \xl)g G(x \xlx ,x% m  m+  m+  1  am  (B.8) The analogous operation with the matrices would be to replace G{x™,x ) l  ^ jdx G{x ,x^ )m{x \x ) m+l  m  +l  m+  t  which is equivalent to the above expression. This justifies the approach used in the calculations using the propagators.  (B.9)  Appendix C  Canonical Quantization of the O p e n String in a U(l)  C l  Background  Motivation  The interacting open string is different in several respects from the free string. The most notable difference is that the world-sheet fields, which were free apart from boundary conditions, have those boundary conditions changed. In a sense the external gauge field acts as a source on the boundary for the degree of freedom on the string. Another difference is that spacetime supersymmetry is broken by the presence of background field which chooses particular directions in space by its components. The consequence of this breaking of supersymmetry is the absence of certain zero modes that are present in the spectrum of the open string. The method used throughout this chapter can be summarized as follows: First, using variational arguments the world-sheet equations of motion are reproduced, and coincide with those of the free string. Second, the surface terms are analyzed and appropriate boundary conditions which involve the external  are determined. The boundary con-  ditions and the equations of motion determine the mode expansion, and care is taken to produce modes that are orthogonal with respect to the inner product that is natural given the action. Using canonical commutation relationships between the fields, commutation relations for the coefficients of the modes are determined, and those coefficients are promoted to operators in the usual way. The Virasoro generators and central extension are calculated.  61  Appendix  C. Canonical Quantization  of the Open String in a U(l)  Background  62  This method is useful, since it follows very closely from the development of the free string which is well understood. It has limitations however. In particular, it relies on the fact that the external field is constant. If it were not, there would be higher derivative interactions at the edge of the string which would complicate the boundary conditions. In addition, a non-constant external field could not necessarily be block diagonalized consistently at both ends, that is to say different rotations would be necessary to obtain an identical block diagonal form, which would mean that there would be a non-trivial interaction between the different fields on the string world-sheet, which would complicate quantization. This same objection makes this an inappropriate vehicle for the discussion of open string coupled to an antisymmetric field living on a stack of D-branes.  C.2  B o s o n i c Degrees of F r e e d o m  C.2.1  Boundary Conditions and M o d e Expansion  To use the operator formalism, it is necessary to construct orthogonal modes for use in the expansion of the fields, which account for the background F „ field. As in the case ((  of free strings, the first step is to determine appropriate boundary conditions from a variational method. Re-writing the bosonic part of the superstring action gives S =  -1  Ana' (Cl)  so varying the 5S  =  field gives  -1 (7 = 0  Arra'  (C.2) Imposing that the string world-sheet metric is Euclidean, this becomes SS  =  -1  2ira'  j dadr [SX  P  (-d  2  - a ) X " + d (SXpdrX*) 2  T  + d  a  (SX&X")  Appendix  C.  Canonical  Quantization  of the Open String in aU(l)  + -e J drFpU (d (X 5X»)  + 28X»d X»)  \  a = 0  j drFpU (d (X»5X»)  + 28X»d X»)  \  a = a  X  U  x  - -e l  2  T  T  T  T  Background  63  (C.3)  .  Integrating over the total derivatives gives, after imposing the periodicity of X on the annulus,  -J + I (2^ l  2-Kd  dodrSXp  dT  (-dl-  d) X 2  p  a  " SX» d° Xli  +  + jdr  (^-SX^X*  l^F^SX'drX^  \  a = 0  - ±e F ,26X"d X'^ 2  fU  .  \  T  (CA)  This gives the equations of motion ( - % - % ) X »  =  (C.5)  0,  which is the same as those for the free string, and the boundary conditions d X«  =  dX  =  a  p  a  -2va'e F»d X \ =a 2  T  v a  (C.6)  -2ixde F»d X \ ^. x  T  v a  The mode expansion for the bosons must satisfy both the boundary conditions and the wave equation, and also be orthogonal. One of the principal differences between the case of the free string and the charged string in a (7(1) gauge field is that while the boundary conditions for the free string are diagonal in the target space indices, the boundary conditions for the charged string mix them. The boundary conditions can be made diagonal however by use of the fact that F  pil  can be re-cast in a block diagonal  form by a rotation of target space coordinates, so /  F,  0  /1  -/1  0  •••  0 -h  h  •••  0  ...  \  (C7)  f f = a  Appendix  C. Canonical Quantization  of the Open String in a U(l) Background  64  Now, this can be diagonalized for any individual block by a change of basis 1  1  X+n = ^ = (X2n-1 + iX2n) j X-  n  = —j= ( A ^ n - l ~ iX2 )  ,  n  (C8)  as noted in [2] this has the effect of changing the boundary conditions to dX a  =  +n  d<jX  n  n  n  =  +n  where a = 2ira'e±f  ia d X \ o T  +n a=  i(3 d X \ , n  T  (C.9)  +n a=a  and (3 = 27ra'e2f , and since X_ n  n  = X\_ the sign of the boundary  n  n  conditions changes for X _ . n  Restricting attention for a moment to one of the n independent components of the field strength and, for instance the + component, the normalized mode that satisfies the boundary conditions and the wave equation is X  = '  m  t  1 [ 1 / 2  \n - e\ '  cos [(m - e) a + ] e"^""^  (CIO)  7  with e =  - ( 7 ' - 7 ) ,  7  =  tan  a,  - 1  7'  =  tan  /?,  - 1  (Cll)  7T  where a and /3 are the appropriate constants as defined above for the mode in question. The particularly interesting thing about this mode expansion is that the right and left moving terms that were in the free mode expansion are mixed by the boundary conditions. The orthogonality condition for the modes can be expressed as fir  . _  /  _  — X id X JO . 7T n  T  m  - X id X m  T  _  n  -  - (35(a - 7r)X X  + a6(a)X X  n  m  n  L  C.2.2  = 5 sign(n  m  mn  - e).  J  (C.12)  C o m m u t a t i o n Relations and the Virasoro Algebra  In terms of the modes given above, the expansion for X  +  is [2] [4]  00  X  +  =x  +  - iblX  0  +% Y 71 = 1  [a X n  n  - biX^ ] n  •  (C.13)  Appendix  C. Canonical Quantization  of the Open String in a U(l) Background  65  and similarly for X _ oo  + ib X 0  +iE  0  bX n  n  -  a X.—n  (C.14)  n  n=l  An interesting point to note is that unlike the case of the open string in this mode expansion there is no term linear in r ± o. This is simply due to the fact that there can be no such term that satisfies the boundary conditions. Using the orthogonality relation above, the oscillators can be obtained by taking the inner product defined above with X  n  in the following manner, (C.15)  where Xd Y T  = Xd Y  — Yd X  T  T  given that X and Y commute. Similar expressions hold  for a*, b , and b^. This can be written in an easier way for the purpose of calculating n  commutation relations. The momentum conjugate to X  +  satisfies (C.16)  and the conjugate of this for P_, since X  +  and X_ are complex conjugates of each other.  Given the expressions for P, the expression for a can be re-expressed as n  (C.17)  +  and so the calculation of commutators between the a operators becomes  =  j dodo'X (a)X (a') n  ( ( n - e) + \a8{o)  m  + ( i ( ( m - e) + \a5{o) =  / doX {o)X (o) n  mn  m  - ^B6(TT - a ) ) ) [X (a),  - ^ ( T T - a ) ) ) [P_(a),  +  P (o')} +  X.(o>)]  ((n - e) + (m - e) + aS(o) - B5(ir -  o)) (C.18)  Appendix  C. Canonical Quantization  of the Open String in a U{1) Background  66  after using the commutation relations between X and P and the orthogonality of the modes. Similar calculations for the other commutators give  [a ,a ]  =  [a ,a\\ = 0  [b ,b ]  =  [bl,bl} = 0  m  n  m  n  ]  m  (C.19) The conditions for x  +  and x_ appear unusual, but since there is no term linear in r it is  natural to expect that these will be conjugate to each other. Since the X fields are expressed in terms of raising and lowering operators, the V i rasoro operators can be calculated. Just as in the case of the free string, the bosonic contribution to the energy momentum tensor is T  = dcX'dpXp  aP  -  (C.20)  ^ga&XWXv  because there is no dependence on the metric in the surface interaction terms. The Virasoro generators will be determined in the same way as for the free string, by taking the fourier modes of T p, so that a  L  k  =  l l o  d  {  a  e l k < T d  + ^ ' X  T  where as for the free string x  +  )  a  d  +  X  a)d-Xp(r,  ^ °) + e-^d-X^r,  a))  ,  (C.21)  = r + a and x~ = r — a. Using the mode expansion for  X^ and noting that X^X^ — X X_ for the pair of coordinates under consideration, the +  expression becomes  L  k  =  n  C do (j2 J  °  E <*n<4  \n>0m>0  (e d X d X ikff  +  n  +  m  +  e- d.X d.X ) ikff  n  m  Appendix  C. Canonical  Quantization  -EE  of the Open String in aU(l)  (e^d X d X_ e-^d^X d.X^ )  nb  a  67  Background  m  +  n  +  m  n  m  n>Om>0  -EE  Wm  (e °d X_ d X e-* °d_X_ d„X )  b  ik  k  +  n  +  m  n  m  n>Om>0  +E E  n»m  b  n>Om>0  =  n  n  +  m  n  m  (C.22)  it can be verified that  n  +  k  +  /  and from the definition of X dX  J,  (e^d X_ d X„ e-* °d-X_ d-X_ )  -d X = +  .  n  7 1  ~ '  (_t) -'("-^ -'(("-^+7) e  e  |n — cy-i  1  d.X  =  n  -d_X  = , ~^ (-i)e- "|n — e l ' n  n  t (  6 ) T  / 0  1  e  t ( (  "-  £ ) < T + 7 )  .  (C.23)  2  Upon re-insertion of these terms into the above, the integration over a becomes trivial and the result for the Virasoro generator in the direction of the field is as in [4] L  n  =  E  \l{m  - e)(n + m - e)a a m  n+m  + E  m>0  \J(m + e)(n + m +  e)b b m  n+m  m>0 n-l  +E  V(  m  +  e  )(  n  m  _  (C.24)  £)b am-n, m  0  for n > 0 and L-  n  = Ll  (C.25)  as can be seen from direct evaluation of the above expression for L ^ . The L s m  obey the standard Virasoro algebra, as can be verified easily. The only  possible change comes from the central extension of the algebra, and this can be calculated in the same way as was done in the case of a free open string. Because the Virasoro algebra is the same up to the central extension, the central extension can be uniquely determined by taking the expectation value both  and [Z/ ,L_ ]. In the first case, taking 2  2  |0 > to be a state with zero momentum, <0|[L ,L_ ]|0>=<0| 1  1  |0>  (C.26)  Appendix  C. Canonical Quantization  of the Open String in a U(l)  68  Background  because of normal ordering (C.27)  < 0|e(l - e)6 &oaiai|0 >= e(l - e). 0  Similarly for L , 2  <0|[L ,L_ ]|0> 2  2  <0| / V  y/t(2 - e)b a + y/(l + e)(l - e ) M i , 0  e(2-e)6 a 0  2  + / ( l + e)(l-e)6 a t  2  A  e(2 - e) + (1 + e)(l - e),  1  t 1  |0 > (C.28)  which confirms that the central extension of the algebra for bosons degrees of freedom is (C.29)  C.3  F e r m i o n i c Degrees of F r e e d o m  C.3.1  Boundary Conditions and M o d e Expansion  To use an analogous technique to those in [5] it is necessary to find a mode expansion for the fermions that is orthogonal and that account for the background U(l) gauge field. The first task then is to determine the boundary conditions for the fermionic fields, which can be found by the usual variational arguments, in analogy with what is done for the bosons. The starting point is the fermionic part of the action on a Euclidean world sheet, S =  -1 (C.30)  world-sheet, (C.31)  Appendix  C. Canonical Quantization  of the Open String in a U(l)  Background  69  the action can be re-expressed as S  =  ^ | d a d r [ - # ? c 7 ^  j drF  - -e l  2  Now,  ,  +  av  (C.32)  Ulna-  varying the action according to Sip — I  the change in the action is, ignoring  the bulk term that gives the equation of motion, and ignoring the total derivative in r which is eliminated by the periodicity around the cylinder, 5S  =  4nb I + ^  e i  dadTda  j drF^  H*"^-  [$ _Sil)±  +  +  + S-4> _V- +  +  v  -\e drF^  V  2  Now,  9^u  +  ^+}  S  a  =  ty+i!>£\  0  a=lna  •  (C33)  impose that these are pure Majorana spinors, which as in the case of the free string  imposes constraints on the raising and lowering operators for the fermions. This also has the effect of simplifying the above to 8S  =  -1 ~Ja=\na  4ira'  J  l  +^  J drF^  (WV-  ei  + '^+)  a  =  0  ~ \e  L  2  ~^o•=0  J drF,  u  8^_ +  Imposing that this must vanish gives the boundary conditions  '^u^t5r--^Sr ) +  +  le F^(r-S^+r S^) 2  +  =  0, o = 0  =  0, a = l n a . ( C 3 5 )  These boundary conditions are solved first by rotating the F^ field into its canonical block diagonal form, and then going to coordinates that are an exact analog of the change  Appendix  C. Canonical Quantization  of the Open String in a U(l)  Background  70  of bosonic coordinates introduced above, that is ipa = ipi+ #2,  ij>b = il>i- 11P2,  (C.36)  and it is understood that similar definitions occur in every pair of coordinates, which correspond to the independent components of F. Writing for simplicity / i = 2-Ka'eiF where F is the component of F^ for the pair of fields in question, the the boundary v  conditions can be re-cast as Va-  = A/  1  _  .j.^iPa+\o=0,  Wa-  = \l _  (C.37)  .j,^a+\a=lna,  x  with the appropriate modification for ipb, which is the complex conjugate of tp . It is a  important to note that these boundary conditions are distinct from the ones found in the literature [4]. The difference is due to the fact that the variation of the fields must obey the same boundary conditions that the fields themselves obey. Now defining 7i =  o  2  l r  7^> T2 = ^ l n W — — 2 v i - ^  M/i  V 1 - */i  (C.38)  and e=-(72-7i),  (C39)  7T  the mode expansion for the fermions becomes 4>- = V+  J2 ne~* d  n  = Efi„ n  ~^  {n+l£)(T  j(n+ie)(r+  e  )+11  ^- , )  71  (C.40)  where as in the case of free fermions the index n can be either integer or half integer to correspond to the analog of Ramond or Neveu-Schwarz boundary conditions. The inner product that these are orthogonal with respect to is a generalization of the one for the the free case j doiPl (1 + 2 ™ '  e i  /f%) - 2 W e  2  / f ^ - Tr)) Vm = < W  (C.41)  Appendix  C.3.2  C. Canonical  Quantization  of the Open String in a 17(1) Background  71  Commutation Relations and the Virasoro Algebra  The action for the interacting string, given above (C.30) gives rise to the same cannonicaly conjugate momentum for the fermion fields as it does in the case of a free string, the reason being that the interaction at the string ends is not derivative with respect to the coordinate r. Since there is no change from the case of the free string and since the modes of the expansion are orthogonal, the commutation relationships are identical to those for the free string,  {d^,'d^} = rrS  (C.42)  m+nt  which is in agreement with the statement in [4] It is important to note that the coordinate system which diagonalized the boundary conditions has the same effect as the light cone coordinates of making the spacetime metric off diagonal. As in the case of the free string the Virasoro generators are given by Lk =  \Ll °\ {^ +^ ^ + e-^r-d^.,) d  d  lha  ,  (C.43)  and examining only the contribution from some particular coordinate, a, this becomes, since ip = iip , b  a  =  -?-/" da— £ £ dZd \ e In a J 2 b  n  m  + e  - ^I^r J  l k a  ^  ( e - ^ + ^ + ^ - i i  V  d  +  e  - ^ + - ) ( T + ^ ) - -  7  l  n  ( - (™+^)( - h^)+7i J  T  ff  e  Q_ -^n+ie)(T-a^)+ e  ll  (C.44)  _  It is important to note that since e and 7 are logarithms and the effect of conjugation is the same as inverting the external field, the field dependent parts of the exponential cancel, and what remains is =  ^  =  (  1  j do^  +  Y, £ -m n d  ^)Y, -rnd (rn d  m+k  d  (n + it) ( l + j ^ ) [e* -^°^ m+k  +  k +  t6).  +  -H™+k-n)a&  e  (C.45)  Appendix  C. Canonical  Quantization  of the Open String in a U(l)  Background  72  This can be converted into the standard form for the fermionic part of the Virasoro algebra except for the field dependent part, and that is what distinguishes the case of a non-zero external field. To complete the analogy with the free string, it remains to show that the Virasoro algebra is satisfied, and also to determine the central extensions. The commutator of two Ls, with the expression given above is  (C.46) after re-indexing. As for the bosons, expectation values for L commutators must be calculated to determine the central extension. In the sector where the fermions are integrally moded <0|[Li,L_i]|0>  (0| [d^d (l + ie)+d^d te,d _ d ie b  b  l  a  0  b  l  0  + d^d _ (-l b  1  + ie)] |0>  (0| (1 + z e ^ e d ^ d ^ d o + (-.1 + ^ e d ^ d ^ |0) (C.47)  (1 + ie)ie,  and (0|[L ,L_ ]|0> 2  2  (0| (2 + te)d d + (1 + ie)d\d\ + ied^d , (-2 + ie)<%d _ a  b  b  0  +(-1  b  0  + ie)d _ d _ + itd _ a% |0) a  b  1  a  1  2  2  Appendix  C. Canonical  Quantization  of the Open String in a U(l)  (0| (2 + ie)d^d iedt d b  +ied d (-2 l  2  + (1 + ie)d d (-l  b  2  2  a  Q  Background  73  + ie)d _ d _  b  a  1  b  l  l  + ie)d%d _ \0)  b  b  0  2  (0| (2 + ie)ied d a  + ie(~2 + ie)d d |0) + (1 + ie)(-l  b  0  b  0  (2 + ie)ie + (l + ie)(-l  + it)  a  0  0  + ie).  (C.48)  So for the Ramond sector the central extension is  2  (C.49)  A(m) = — (m  3  — mj + m ( l + ie)ie  which exactly analogous to the contribution in the bosonic sector. By contrast, in the case of Neveu-Schwarz boundary conditions one has  (OKLx, L^IO)  1  = (0| =  (0| Z  -l/2^-l/ (-^  rf  2  IP)  + «0  0)  ^1/2^1/2(2 + ^ ) ^ - l / 2 ^ - l / 2 ( 2 e  • W  1  (3  , 1  fc  ^1/2^/2(3 + "0>  + z e  —  ^  1  )^2  (C.50)  and <0|[L ,L_ ]|0) 2  2  =  + ie)d? 4 + ( - + ie)dS d;  (0|  /2  /2  + te)d _ d _ a  b  1/2  3/2  /2  + {~\+  /2>  i£)d _ d _ a  b  V2  l/2  10)  (0| (2 + e)(-2 + z6)< /2 4 2 d _ 3/2 fi _ 1/2 a  6  ?  (C.51)  (2 + ze)(-2 + ze). So for the Neveu-Schwarz sector the central extension is M  \  m  3  A(m) =  8 which is in agreement with the results of [4].  2  h me ,  (C.52)  Bibliography  [1] E.S. Fradkin A . A . Tseytlin. Non-linear Electrodynamics from Quantized Strings. Phys. Lett, 163B:123, 1985. [2] A . Abouelsaood C . G . Callan C R . Nappi S.A. Yost. Open Strings in Background Guage Fields. Nucl. Phys. B, 280:599, 1987. [3] C . G . Callan C. Lovelace C R . Nappi S.A. Yost. String Loop Corrections to Beta Functions. Nucl. Phys. B, 288:525, 1987. [4] C. Bachas M . Porrati. Pair Creation of Open Strings in an Electric Field, th/9209032. [5] M . B . Green J . H . Schwarz E . Witten. Superstring Press, 1987.  theory.  hep-  Cambridge University  [6] J. Polchinski. String Theory. Cambridge University Press, 1998. [7] P. Haggi-Mani U . Lindstrom M . Zabzine. Boundary Conditions, Supersymmetry, and A-field Coupling for an Open String in a B-field Background, hep-th/0004 061. [8] A . A . Tseytlin. th/9908105.  Born-Infeld action, supersymmetry  and string theory.  hep-  [9] A . A . Tseytlin. Vector Field Effective Action in the Open Superstring Theory. Nucl. Phys. B, 276:391, 1986. [10] A . A . Tseytlin. Open Superstring partition function in constant guage field background at finite temperature, hep-th/'9802133. [11] J.J. Atick E . Witten. The Hagedorn transition and the number of degrees of freedom of string theory. Nucl. Phys. B, 310:291, 1988. [12] C S . Hsue B. Sakita M . A . Virasoro. Formulation of Dual Theory in Terms of Functional Integrations. Phys. Rev. D, 2:2857, 1970. [13] F. Ardalan H. Arfaei M . M . Sheikh-Jabbari. Noncommutative Geometry from Strings and Branes. hep-th/'9810012. [14] N . Seiberg E . Witten. th/9908142.  String Theory and noncommutative Geometry,  74  hep-  Bibliography  75  [15] A . Connes. A Short Survey of Noncommutative Geometry,  hep-th/0003006.  [16] S. Lee Y . Kiem. U V / I R Mixing in Noncommutative Field Theory via Open String Loops, hep-th/0003145. [17] T. Lee. Canonical Quantization of Open String and Noncommutative Geometry. hep-th/9911140. [18] A . Bilal C.-S. Chu R. Russo. String Theory and Noncommutative Field Theories at One Loop, hep-th/'0003180. [19] P . - M . Ho Y . - T . Yeh. Noncommutative D-Brane in Non-Constant NS-NS B Field Background, hep-th/'0005159. [20] C. Zachos. A Survey of Star Product Geometry  hep-th/0008010  [21] C.-S. Chu P . - M . Ho. Noncommutative Open String and D-Brane.  hep-th/9812219.  [22] C.-S. Chu. Noncommutative Open String: Neutral and Charged, hep-th/0001144-  

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