Vector profile and gauge invariant observables of string field theory solutions for constant magnetic field background

Journal of High Energy Physics, May 2018

Abstract We study profiles and gauge invariant observables of classical solutions corresponding to a constant magnetic field on a torus in open string field theory. We numerically find that the profile is not discontinuous on the torus, although the solution describes topologically nontrivial configurations in the context of low energy effective theory. From the gauge invariant observables, we show that the solution provide correct couplings of closed strings to a D-brane with constant magnetic field.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2FJHEP05%282018%29144.pdf

Vector profile and gauge invariant observables of string field theory solutions for constant magnetic field background

Accepted: May Vector pro le and gauge invariant observables of string eld theory solutions for constant magnetic Nobuyuki Ishibashi 1 5 Isao Kishimoto 1 3 Toru Masuda 0 1 4 Tomohiko Takahashi 1 2 0 CORE of STEM, Nara Women's University 1 University of Tsukuba , Tsukuba, Ibaraki 305-8571 , Japan 2 Department of Physics, Nara Women's University 3 Faculty of Education, Niigata University 4 CEICO, Institute of Physics of the Czech Academy of Sciences 5 Tomonaga Center for the History of the Universe We study pro les and gauge invariant observables of classical solutions corresponding to a constant magnetic eld on a torus in open string eld theory. We numerically nd that the pro le is not discontinuous on the torus, although the solution describes topologically nontrivial con gurations in the context of low energy e ective theory. From the gauge invariant observables, we show that the solution provide correct couplings of closed strings to a D-brane with constant magnetic eld. Bosonic Strings; String Field Theory; D-branes; Tachyon Condensation - 1 Introduction 2 Classical solutions for constant magnetic eld background 3 Pro les of the classical solution 3.1 3.2 3.3 3.4 4.1 4.2 4.3 Pro les and dual states Quasi-periodicity of the solution Tachyon pro le Vector pro le Calculation of gauge invariant observables Comparison with Dirac-Born-Infeld action 4 Gauge invariant observables for the classical solution 5 Concluding remarks A Conformal transformation of the vector vertex operator B Calculation of correlation functions in the vector pro le moduli space of string theory. In bosonic open string eld theory, Erler and Maccaferri [1] proposed a way to construct classical solutions representing any time-independent open string background by use of boundary condition changing (BCC) operators. Following their method, a solution corresponding to constant magnetic eld background has been constructed by some of the present authors in [ 2 ]. It was found that the classical action of the solution calculated from the operator product expansions (OPEs) of BCC operators agrees with the Dirac-Born-Infeld action. This magnetic solution has several new features compared with the solutions discovered de ned as follows. Using the Fock space expression, a solution j i of the open string eld theory can be expanded as j i = X tp jTpi + X Ap Vp p p + ; where the lower mass states, corresponding to the tachyon and the massless vector eld are expressed as jTpi = c1 jpi = ceip X (0) j0i ; Vp = c1 1 jpi = i r 2 0 The position representation of the component elds is given by the Fourier transform of tp; Ap; : to ask whether the magnetic solution needs multiple patches in string eld theory. This question may be examined by evaluating the vector pro le A (x) corresponding to the U(1) gauge eld. If A (x) has discontinuities, we need to divide the torus by coordinate patches to represent the con guration by smooth gauge elds. Secondly, we will evaluate gauge invariant observables for the magnetic solution. In conventional eld theory, topologically non-trivial con gurations are characterized by some gauge invariant quantities which take discrete values. Such quantities have not been found in string eld theory. Instead, we have the gauge invariant observable which is associated with on-shell closed string vertex operators [3]: OV ( ) hIj c(i)c( i)V (i; i) j i ; (1.5) where I is the identity string eld, V (z; z) denotes an on-shell closed string vertex operator and z = +i; z = i in the complex plane correspond to the midpoint = =2 of the open string. If is a classical solution, the observable represents a coupling of the closed string vertex operator V to the D-brane to which the solution corresponds. Accordingly, we expect that the observable for a massless anti-symmetric tensor vertex has a non-trivial value since the corresponding D-brane has constant background magnetic eld. In order to con rm the existence of background magnetic eld and nd a clue for topological invariants in string eld theory, we calculate the gauge invariant observables with massless and zeromomentum closed string vertex operators. This paper is organized as follows. In section 2, we brie y review the constant magnetic solution on torus. In section 3, we study tachyon and vector pro les of the solution. As { 2 { (1.1) (1.2) (1.3) (1.4) a by-product, we show periodic and quasi-periodic properties of the solutions. In section 4, we evaluate the gauge invariant observables of the solution. By comparing the resulting observables with the Dirac-Born-Infeld (DBI) action, we nd that the solution indeed corresponds to constant magnetic eld background. In section 5, we will give concluding remarks. In the appendices, details of calculations are exhibited. 2 Classical solutions for constant magnetic eld background We would like to consider the con guration with a constant background F . We concentrate on the spatial directions X1 and X2, since a real antisymmetric tensor F can be transformed into a block diagonal form with 2 2 blocks. Let us consider the bosonic open string eld theory in which these spatial directions are toroidally compacti ed with radii R1 and R2, and the Neumann boundary conditions are imposed on the variables X1; X2. The time direction X0 is required to be noncompact in order to construct the solution following Erler-Maccaferri's method [1], but other directions are unspeci ed here. To nd the classical solution corresponding to a constant F12 background, we need to prepare the BCC operators which changes the open string boundary conditions for X1; X2 from the Neumann boundary condition to the one with F12 and vice versa. Such operators correspond to the open strings with one edge with the free boundary conditions and the other coupled to the constant magnetic eld. The zero mode coordinates x1; x2 of these open strings become noncommutative and we need to introduce the following operators jN j pairs of BCC operators: conformal weight The zero mode algebra has a jN j dimensional representation. Correspondingly, we can nd k (k = 1; ; jN j) [ 2 ], which are primary elds with h = ; ( tan = 2 0F12 ): Following Erler-Maccaferri's method, we multiply k ; k by the vertex operators e iphX0 k ; k. They are primary elds with conformal weight zero and satisfy the OPEs and appropriate normalization factors and construct jN j pairs of modi ed BCC operators k ; (1 2 x 1 R1 ) { 3 { k(s) l(0) k;l; l(s) k(0) l;k j cos j = l;k for small positive s. Having these BCC operators, the classical solution corresponding to the constant magnetic eld background [ 2 ] are given as follows: where tv is the Erler-Schnabl solution for the tachyon vacuum [4], : 1 + K ; (2.8) (2.9) It should be noted that 1k;l and 2k;l independently satisfy the string eld reality condition,1 X Ak;l 1 k;l k;l z = X Ak;l 1k;l; k;l X Ak;lQB 2 k;l k;l z = X Ak;lQB 2k;l: k;l (2.10) These are useful in checking the correctness of the pro le calculation presented later. A = diag( 1; ; 1; 0; ; 0 ); | {z } | jN{jzM } M then the solution can be regarded as describing M D-branes with magnetic eld condensation. Using the OPEs (2.3), it is easy to show that the energy of the solution is given by M times that of a single D-brane with magnetic eld condensation. For later discussion, it is convenient to rewrite the solution (2.4) in the following way. By using the algebra of K, B, c and k , l, the solution k;l can be decomposed to three parts: and Ak;l is a hermitian jN j jN j matrix satisfying A2 = A. The second term is a solution to the equations of motion around the tachyon vacuum. Suppose that A is given by where 1k;l and 2k;l are de ned as Pro les of the classical solution Pro les and dual states Now let us study the pro les of the magnetic solution 0. In order to extract momentum space pro les from the solution expanded as (1.1), we de ne the states dual to the tachyon 1We follow the de nition of the conjugate z given in [1]. K, B and c are self-conjugate and z = . In (2.4) (2.5) (2.6) (2.7) and massless vector states, jTpi and Vp , by ~ Tp = ~ V p = 1 1 ; 1 i and they are orthogonal to other higher massive states. The momentum space pro le is derived from the inner product of the dual state and 0 : HJEP05(218)4 tp = T~p; 0 ; Ap = V~p ; 0 : Using (2.7), we can see that the pro les (3.4) are decomposed to three parts. The calculations are simpli ed by using the following identities: QB T~p QB V~p Consequently, for example, we only need to deal with the tachyon pro le. 3.2 Quasi-periodicity of the solution Before starting the calculation of the pro les, we would like to point out that the pro les satisfy quasi-periodic relations. Here we deal with the solution corresponding to a single D-brane with constant magnetic eld F12 6= 0, namely M = 1 in (2.6). We can construct jN j independent solutions 0k = tv + k;k (k = 1; ; jN j): To derive the tachyon pro le from 0k, we have to calculate the inner products T~p; tv and T~p; 1k;k as seen from (3.5). The former has been known to be a constant [4].2 The latter can be calculated by rewriting in terms of the correlation function including the tachyon vertex and the BCC operators, which has been derived in [ 2 ]: 2 D e ip X ( ) k( 1) l ( 2) E = ( ( 1 Cl;kp = ! n12n2 n2l 2) 0p2 Cl;kp; k l; n1 (modN) 0p2 2 ; T~p; tv = Z 1 0 dt Z 1 0 ds e s t 4 2pts (s + t + 1)2 1 + cos s + t + 1 (t s) = 0:509038 : { 5 { tv and 1k;l for the calculation of (3.1) (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) (3.9) where ! is an N -th root of unity, ! = exp(2 i=N ), and ni are momentum quantum numbers, pi = ni=Ri (i = 1; 2), and ln ( ) (x). Here, it is important to notice that the correlation function depends on the parameters k and l only through the normalization factor, Ck;pl. Given this, we nd that the inner product T~p; 1k;k depends on k as follows, T~p; 1k;k = Ck;pk = e i 2 RN2 k p2 : Therefore, it turns out that the tachyon pro le tk(x) of 0k satis es the quasi-periodic Moreover, since the factor Ck;pk includes 0; n1 (modN) we nd t k x1 + 2 R1 ; x2 N = tk x1; x2 : Similarly to the tachyon case, other pro les can be calculated also by using a 3-point functions of the BCC operators. The matter vertex operators are of the form @nX eipX . Since the operators U; V are not included in the derivatives of X1; X2 [ 2 ], the 3-point function depends on the parameters k and l only through Ck;pl. Consequently, we nd that all pro les of the solution satisfy the quasi-periodic relations similar to (3.11) and (3.12), namely, the space representation of the solution satis es k 0 k 0 x1 + x1; x2 + 2 R1 ; x2 N Notice that the quasi-periodicity relations of these forms arise because we have taken the BCC operators corresponding to the eigenstates of V = exp(ix2=R2) [ 2 ]. If we take the BCC operators corresponding to the eigenstates of U = exp(ix1=R1), we have a set of classical solutions which is periodic in the x 1 direction and quasi-periodic in the x 2 direction. In addition, we should comment that, due to the translational symmetry of the they correspond to degenerate states in magnetic elds, so-called Landau level.3 theory, we can generate other set of solutions from torus. In our case, we simply just choose 0k by arbitrary displacement of the 0k as jN j independent solutions in the sense that 3In a naive expectation, the jN j independent solutions 0k (k = 1; ; jN j) are physically equivalent as a result of the translational symmetry and the relation (3.14). However, it is di cult to connect the translation is represented as a gauge transformation in open string eld theory. { 6 { (3.10) (3.11) (3.12) (3.13) (3.14) Now let us calculate the tachyon pro le. The inner product T~p; 1k;l is rewritten in terms of correlation functions with the help of KBc algebra: T~p; 1k;l = dt2 L L 0 Z 1 dt1 e t1 t2 t3 1 2 1 2 iL denotes a correlation function on the in nite cylinder of circumference L. The correlation functions which appear in (3.15) can be obtained by conformally transforming (3.8) and the ghost correlation functions which are de ned on the complex plane. We eventually get the tachyon pro le for the jN j-th solution 0 jNj as where t0 = T~p; tv . The function Gt(u) is de ned by g( 0u 1) exp ( ) (1 ) 2 log 4) ; (3.17) Z 1 0 ds Z 1 s+t+1 dL sin (s+t+1) L L 2L sin (s+1=2) sin (t+1=2) L !z : The tachyon pro le for the k-th solution can be derived by using (3.11). Now, we carry out numerical evaluation of the tachyon pro le. In order to do so, we need to evaluate the triple integration on the right hand side of (3.18). However, this expression is inappropriate for numerical integration, because the rate of convergence is very slow due to the in nite integration region and it gives an inaccurate value even by use of Monte Carlo method. An expression of g (z) which is better suited than (3.18) for numerical integration can be obtained by a change of variables: g(z) = of the pro le depends on the behavior of g (z) when z is very large. 4 z The Fourier coe cients of (3.16) are evaluated by using the numerical results of g(z) and the tachyon pro le can be obtained numerically by summing up the Fourier series. We nd that the Fourier coe cient approaches zero fast enough as n; m ! 1 so that we can approximate the series by a nite summation over lower modes. The plot of a result is depicted in gure 2.4 3.4 Vector pro le Next, we consider the vector pro le of the magnetic solution. Since the vector pro le of tv vanishes, the momentum space vector pro le of 0k is given by Ap = V~p ; 1k;k QBV~p ; 2k;k : (3.20) As in the case of the tachyon pro le, each inner product can be rewritten in terms of correlation functions on in nite cylinder of circumference L = t1 + t2 + t3 + 1: V~p ; 1k;l = 2 0p2 1 4The reason why the x1 and x2 dependence are di erent from each other is that the solution is constructed by using the BCC operators corresponding to the eigenstates of V = exp(ix2=R2). { 8 { D L 2 0p2 (2 )2R1R2 1 2 1 2 1 Z 1 dt3 Z 1 c(t1) E L c(t1 + t2)B L E L ; (3.21) Bc(t1) ; (3.22) 2 0 where we have used the expressions for the dual vector vertex operator, (3.2) and (3.6). The computation of the correlation functions on the right hand side of (3.21) is not so simple because the vector vertex operator in (3.21) is not a primary eld. As derived in appendix A, under the conformal transformation z = f ( ), the vector vertex transforms as QBV~p ; 2k;l follows; E L 9 > = > ; d dz 0p2+1 >< 8 > : i 0p 2 d 2 d dz 2 dz2 e ip X ( ) : (3.23) Thus, we need correlation functions for vector and tachyon vertex operators to derive the vector pro le. A detailed derivation of these correlation functions is presented in appendix B. Using the results, it turns out that only the term (3.21) contributes and we nd: Ap = the position space representation of the vector pro le is obtained as follows: A1(x) = A2(x) = 1 p where the function G(u) is de ned in terms of g (z) in (3.18) as G(u) = ( 0u + 1) g( 0u) exp 2p3. Here, we take 0 = 1. 2p3. Here, we take 0 = 1. As in the tachyon pro le, we can numerically calculate the vector pro le by using the numerical results of g(z) and summing up the Fourier series. The di erence of the two pro les is in the fact that the Fourier coe cients of the vector pro le include momentum factors, N m=R1 or n=R2. Hence the asymptotic behavior of the pro le may change in the ultraviolet region and discontinuities could be found for the vector pro le. However, we observe that the Fourier coe cients for the vector pro le rapidly converges to zero as n; m ! 1 and the position space representation of the pro le seems to be absolutely convergent. Consequently, we get the plot shown in gures 3 and 4, and we conclude that the vector pro le has no discontinuities.5 Here, we would like to comment on pro les for other massive modes. The reason why the tachyon and vector pro les are not discontinuous is because the momentum space pro le becomes zero rapidly for large momenta. This behavior is due to the exponential factor 5From the vector pro le, we can calculate the quantity F~12 not as a constant but as a smooth function, and the space average of F~12 over the torus becomes zero. Since F~12 is not invariant under gauge transformations in string eld theory, it is no wonder that F~12 does not correspond to the constant magnetic eld. 0p2=2 in the normalization factor of the 3 point function (3.9).6 Since this exponential factor always appears in the expression of other pro les, we expect that other pro les also do not have any discontinuities. Gauge invariant observables for the classical solution Calculation of gauge invariant observables Let us consider the gauge invariant observable (1.5) for the k-th solution (3.7) with the following closed string vertex operators at zero momentum: HJEP05(218)4 B(z; z) G(z; z) where X = (X1 + iX2)=p2 and X~ = (X1 iX2)=p2. These correspond to the antisymmetric tensor eld B12 and the sum of graviton eld G11 + G22 in the spatial directions X1 and X2. Substituting the k-th solution (3.7) into (1.5) and expressing it in terms of the correlation functions on the in nite cylinder, the observables can be rewritten as OV ( tv) 1 j cos Z 1 dt1 Z 1 0 c(s + t1 + i1)c(s + t1 i1)c(s) E CL ; E CL where L = s + t1 + t2 . For B(z; z) and G(z; z), OV ( tv) has been calculated as7 OB( tv) = 0; OG( tv) = 0 2 i (2 )2R1R2: Here we normalize the observable by dividing it by the volume of the directions other than X1 and X2. be derived from the correlator For the vertex operators (4.1) and (4.2), the matter correlation function in (4.3) can 0 2 z 1 w + (1 (z w)2 ) wz k l ; 6It gives exponential factor in (3.17) and (3.27), where 2 (1) ( ) (1 ) 2 log 4 0 for 0 7As in [ 7 ], if V (z; z) is decomposed by the matter primary eld Vn(z) as V (z; z) = P m;n mnVm(z)Vn(z), and the OPE of the primary elds is Vm(z)Vn(z0) vmn=(z z0)2, the observable for the tachyon vacuum solutions is given by (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) < 1. The matter 4-point function in (4.3) on the in nite cylinder can be obtained from (4.6) by a conformal transformation. To make our calculation well-de ned, we regularize the correlators by replacing i1 by iM and taking the limit M ! 1. As a result, we nd iM ) k(x1) k(x2) CL ! 0 2 L 2 ( 4)e L 4M n ei 2L (1 )(x1 x2) + (1 )e i 2L (x1 x2)o Multiplying the ghost correlator, the 4-point function in (4.3) is given as c(s + t1 + i1)c(s + t1 i1)c(s) CL where L = s + t1 + t2. Similarly its conjugate is given by c(s + t1 + i1)c(s + t1 i1)c(s) CL which is also obtained by acting the X2 parity transformation $ 1 on (4.9) [ 2 ]. Having calculated the correlation functions, the observables can be obtained by intek l = j cos ) ne i 2L s ei 2L (1 )so ) ne i 2L (1 )s ei 2L so 2 0F12 j cos j cos j j ; ; j cos j and its conjugate. Here gration of (4.3): (2 )2R1R2 Comparison with Dirac-Born-Infeld action Let us check if the gauge invariant observables (4.11) and (4.12) are consistent with what is known about the magnetic background. Here we consider a D2-brane extended in the X1 and X2 directions for simplicity. The coupling of the D2-brane to massless modes of closed strings is described by the Dirac-Born-Infeld (DBI) action:8 S = T2e (2 )2R1R2pdet(Gab + Bab + 2 0Fab); (4.13) 8Hereafter, we adopt the static gauge and normalize the action by dividing it by the volume of the time direction. where T2 is the D2-brane tension, and , Gab and Bab denote the induced elds on the brane. Fixing to the static gauge, we nd the variation of the DBI action under an in nitesimal variation of B12 around the at background, namely hGabi = ab and hBabi = 0: The observable (4.11) can be expressed by using the DBI action as T2e ( S B12 F126=0 B12 F12=0 0F12)2 S B12: ) nd the variation of the DBI action under an in nitesimal variation G12 does not appear in this result and the graviton included implicitly in the dilaton eld as = 24 + (1=4) ln det Gab when the space-time is of the form M 24 T 2 [ 8 ]. From (4.16) we get the relation T2e ( S G11 + S G22 These results (4.15) and (4.17) show that the observables correctly re ect the coupling of the D2-brane with the constant magnetic eld to the closed string modes. Consequently, it is explicitly found that (3.7) can be regarded as the classical solution corresponding to magnetic eld condensation. 4.3 T-dual of a D2-brane with F12 6= 0 We would like to examine if the couplings (4.11) and (4.12) are consistent with T-duality. It is well known that the D2-brane is transformed into a D-string tilted in the dual torus, which is extended along the line [ 8 ] where X01 denotes the coordinate dual to X1. From the Dirac quantization condition (2.1), it follows that the D-string winds N times around the X2 direction. The DBI action for the D-string is given by (4.14) (4.15) (4.16) eld is (4.17) (4.18) (4.19) on the D-string to coincide with X2. Then, the In this gauge, the induced metric is calculated as 0F12)2G011 + 4 0F12G012 + G022: Substituting this into the DBI action (4.19), we can calculate the variation of S under an HJEP05(218)4 in nitesimal variation of G0ab: S = T1e 0 + S can be rewritten as T2e where we have used the relation 0 = 0 24 (1=4) ln det G0ab as in the D2-brane case. By using the relations between the parameters [ 8 ] 0 = p 0 R1 e ; T1 = 2 p 0T2; Combining with the results (4.11) and (4.12), we nd that G012 F126=0 G012 F12=0 G012 0F12)2 + 0F12)2 + S S G022 ) ; These relations implies that the gauge invariant observables reproduce correct couplings of the D-sting to the closed string modes in the dual space, because, in the dual space, the vertex operators (4:1) and (4.2) correspond to G012 and a G011 + G022 respectively. 1 1 2 p1 + (2 2 p1 + (2 1 1 0F12)2 0F12)2 ! ! ( G011) ) G022 : S ( G011) + S G022 F12=0 (4.20) (4.21) (4.23) (4.24) (4.25) ) : (4.26) Concluding remarks In this paper, we have studied and explicitly calculated the tachyon and vector pro les for the constant magnetic eld solution on a torus constructed by following Erler-Maccaferri's method. In addition, we have calculated gauge invariant observables for the solution and found that the solution reproduces correct couplings of the D-brane with a constant magnetic eld to the closed string modes. A remarkable feature of the resulting pro les is that they have no discontinuity on the torus. This result does not seem to be consistent with the fact that the solution corresponds to a topologically nontrivial con guration. In low energy eld theory, the gauge eld will have discontinuities if one tries to describe the con guration without dividing the torus into patches. We expect that the same thing happens in string eld theory, but we have found that pro les for any states do not have discontinuities. One possibility is that pro les for normalized states are not the right quantities to be chosen in observing such phenomena. Instead, we might have to consider a sum of in nitely many pro les or even a nonlinear functional of the string eld. Another possibility is that our results could be an indication of nonlocality of string eld theory. Since the star product is a nonlocal operation from the target space viewpoint, even if a string eld is de ned in a coordinate patch, it can spread beyond the boundary after a gauge transformation. Therefore, the non-existence of discontinuity itself may be a natural consequence of the nonlocality of string eld theory. Another important question about the solution is how we can de ne the topological invariant characterizing the constant magnetic eld solution in the framework of string eld theory. The magnetic eld is proportional to an integer due to the Dirac quantization condition, which is derived based on low energy theory. This quantization condition should be derived from the string eld theory itself. Unfortunately, the gauge invariant observables calculated in this paper do not provide any clue about such a quantization condition, although it is interesting to notice that the observable with the antisymmetric tensor mode becomes non-zero as a result of the magnetic eld background. In order to capture the \topological" nature, we may need new insights from noncommutative geometry, matrix theory and so on [9{12]. There are many interesting possibilities for future exploration related to the constant magnetic eld solution. Suppose that we consider the con guration which is T-dual to the magnetic eld solution along both X1 and X2 directions. The resulting con guration may correspond to a con guration of multiple D-particles on the torus. Such a system was studied in the case of a non-compact space [13]. In this case, the coordinates of the D-particles become noncommutative and the observable (4.11) represents the coupling of the D-particle to a symplectic form characterizing the noncommutativity. Therefore, D-particles with noncommutative coordinates may be described in terms of the string eld theory in this dual background. Such a string eld theory may be considered as another version of Matrix theories [14, 15] although it has no supersymmetry. It is also possible to make the relation between noncommutative geometry and constant magnetic eld background manifest by a similarity transformation for string elds [16, 17]. It will be interesting to nd such a similarity transformation in the background of the magnetic eld solution. Acknowledgments The authors would like to thank T. Erler, M. Kudrna, C. Maccaferri, M. Schnabl, Y. Kaneko and S. Watamura for helpful comments. We also thank the organizers of \SFT@HIT" at Holon, especially M. Kroyter, for hospitality. T. M. also would like to thank the organizers of \String Field Theory and String Phenomenology" at Harish-Chandra Research Institute, Allahabad, for their hospitality and for providing him a stimulating environment. This work was supported in part by JSPS Grant-in-Aid for Scienti c Research (C) (JP25400242), JSPS Grant-in-Aid for Young Scientists (B) (JP25800134), and JSPS Grant-in-Aid for Scienti c Research (C) (JP15K05056), and Nara Women's University Intramural Grant for Project Research. The research of T.M. was supported by the Grant Agency of the Czech Republic under the grant 17-22899S. The numerical computation in this work was partly carried out at the Yukawa Institute Computer Facility. A Conformal transformation of the vector vertex operator In this appendix, we would like to derive the conformal transformation (3.23) of the vector vertex operator. The OPE of the vertex operator with the energy momentum tensor is ( i 0p )3 eip X ( ) + ( 0p2 + 1 )2 1 The right hand side includes the tachyon vertex operator which is a primary eld of weight given by transformed as T ( ) eip X ( ) ( 0p2 )2 1 eip X ( ) + Let us consider a conformal transformation ! z = f ( ), under which a eld ( ) is ( ) ! f ( ) = Uf ( )Uf 1: Here, the operator Uf is given in terms of the energy-momentum tensor as follows: Since the tachyon vertex is primary, it is transformed as I d 2 i Uf = eT (v); T (v) = v( )T ( ); Uf eip X ( ) Uf 1 = dz d 0p2 eip X (z): From the OPEs (A.1) and (A.2), we can derive the commutation relations of the vertex operators with T (v): i 2 (A.2) (A.3) (A.4) (A.5) (A.7) Let us consider the transformation of the vector vertex operator under the following one parameter family of conformal transformations: By using the commutation relations (A.6) and (A.7), we can see that the result should be expressed as Yt( ) @X eip X ( ) + Zt ( ) eip X ( ); dYt( ) dZt ( ) dt dt = ( 0p2 + 1)v0 ft( ) Yt( ); = ( 0p2)v0 ft( ) Zt ( ) Yt( ) = ft0( ) where Yt( ) and Zt( ) are some functions of t and . Di erentiating these with respect to t and using (A.6) and (A.7), we obtain the equations where ft( ) is de ned by ft( ) = exp(t v( )@) . Integrating these with the initial conditions Yt=0 = 1 and Zt=0 = 0, we nd 2 i 0p v00 ft( ) Yt( ); i 2 p ft00( ) ft0( ) ft0( ) e ip X ( ) k( 1) l ( 2) + regular terms; ( E ! By setting t = 1 and p p , we obtain the formula (3.23). B Calculation of correlation functions in the vector pro le In this appendix, we will show how to calculate the correlation functions which appear in (3.21) and (3.22). Here we begin with the calculation of the following 4-point functions on the complex The rst two are derived directly from boundary conditions of X. Moreover, in the limit , we nd that the 4-point function behaves as plane: behaves as D E Here X = (X1 + iX2)=p2, and , 1 and 2 are taken to be real and satisfy 2 < 1 < , so that the operator e ip X is on the boundary with the Neumann boundary condition. The correlation function is a 1-form on the complex plane with respect to the variable and it E E which xes the normalization of the four point function. These conditions determine the 4-point function as E = i 0p ( where the 3-point function on the right hand side is given by (3.8) [ 2 ]. By taking the limit in (B.4), we obtain a correlation function involving the vector vertex operator on the complex plane: D e ip X ( ) k( 1) l ( 2) ; E (B.4) + (1 ( 1)( ) 1 + 2) ( Taking the normalization of the modi ed BCC operators [ 2 ] into account, we obtain = i 0p Cl;kp (2 )2R1R2 f j cos j + (1 ) 1 + 2g ( ( 1 The correlation function for the conjugate vertex operator @X~ e ip X (X~ = (X1 iX2)=p2) can be calculated in a similar way and we nd E = i 0p~Cl;kp (2 )2R1R2 f j cos j + 1 + (1 ) 2g ( ( 1 Eq. (B.7) can also be obtained by using the fact that the X2 parity transformation X2 X2 corresponds to the following transformations of the parameters:[ 2 ] Having found the 3-point functions on the complex plane, we can nd the one on the in nite cylinder with circumference L by the conformal transformation: By using the transformation law (3.23), we obtain the following 3-point functions on the cylinder from (B.6) and (B.7): D = L h Cl;kp (2 )2R1R2 i 0p 2 1 2 j cos 0 sin j p sin (z1 L E L (z2 L z2) cos where h = 0p2 + 1 and 12 = 21 = 1. p ! p~; ! 1 : L z = arctan : L z3) sin (z2 z3) h 1 L sin (z1 z2) sin (z1 z3) h L ! (B.8) (z1 z3) L + sin (z1 L z3) cos (z1 z2) L ; (B.9) The matter correlation functions that appear in (3.21) are calculated by using (B.9) as (B.12) (B.13) 1) cos t2 sin 1) sin t2 cos p h sin h+1 sin 1) cos t2 cos t3+ 21 sin 2 t1+ 21 j cos j cos t2 sin sin h 2 t1+ 21 sin t2 h+1 p h sin h sin t1+ 21 sin t2 h+1 (h j cos cos t2 sin sin h 2 j t2 h sin h+1 +(h (h t1+ 21 sin t3+ 21 1) cos t2 sin 1) sin t2 cos t1+ 21 + sin t2 cos t1+ 21 t3+ 21 sin t3+ 21 sin t1+ 21 cos t1+ 21 cos t1+ 21 t1+ 21 D ip X and D ip X = L i 2 L i 2 1 2 C p sin h +(h (h 1 2 C p L h+1 0p L h+1 0p 1 2 1 2 L E 1 2 n 2 E n L 1 2 +h sin t2 sin t3+ 21 cos 2 t1+ 21 + sin t2 sin t3+ 21 sin 2 t1+ 21 o # ; (B.11) where s is de ned by s = s=L. The ghost correlation functions which appear in (3.21) are given as Combining these results, we nd that the integrand in (3.21) turns out to be D ip X D ip X k l (t1 + t2)@ (t1) L c(t1 + t2) 1 2 E E L L = = E D L L L 2 2 sin sin 2 2 1 2 L t3+ 21 ; t1+ 21 : 1 2 L h 1 Cl;kp (2 )2R1R2 " j cos j 2 cos t2 sin2 sinh + 1 2 t1+ 12 sinh t3+ 12 t1+ 12 + sin2 o 1) cos t2 sin t1+ 12 sin t3+ 12 sin t1+ 12 cos t1+ 12 The second term in the bracket on the right hand side proportional to p is antisymmetric under the interchange of t1 and t3. Since the integration measure in (3.21) is symmetric with respect to it, the second term does not contribute to the vector pro le. Next let us turn to (3.22). The correlation function which appears in the integrand in (3.22) is given as L j cos j sinh sinh 2 t2 t1+ 12 sinh t3+ 12 n sin t2 (1 + 2t1) cos t1+ 12 sin t3+ 12 (1 + 2t3) cos t3+ 12 sin t1+ 12 o 2 (t1 L t3) cos t2 sin t1+ 12 sin t3+ 12 2 sin t2 sin t1+ 12 sin t3+ 12 cos2 t1+ 12 cos2 t3+ 12 2 cos t2 cos t1+ 12 sin2 t1+ 12 sin t3+ 12 cos t3+ 12 sin2 t3+ 12 sin t1+ 12 : (B.15) Since (B.15) is antisymmetric under the interchange of t1 and t3, we nd that the right hand side of (3.22) vanishes. Open Access. Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [1] T. Erler and C. Maccaferri, String eld theory solution for any open string background, JHEP 10 (2014) 029 [arXiv:1406.3021] [INSPIRE]. constant background magnetic [INSPIRE]. [hep-th/9202015] [INSPIRE]. gauge elds, Nucl. Phys. B 280 (1987) 599 [INSPIRE]. Monographs on Mathematical Physics, Cambridge UNiversity Press, Cambridge U.K. (1998). compacti cation on tori, JHEP 02 (1998) 003 [hep-th/9711162] [INSPIRE]. [hep-th/0007175] [INSPIRE]. 2)-branes in the bosonic string theory, Nucl. Phys. B 539 [2] N. Ishibashi , I. Kishimoto and T. Takahashi , String eld theory solution corresponding to eld , PTEP 2017 ( 2017 ) 013B06 [arXiv: 1610 .05911] [3] B. Zwiebach , Interpolating string eld theories , Mod. Phys. Lett. A 7 ( 1992 ) 1079 [4] T. Erler and M. Schnabl , A simple analytic solution for tachyon condensation , JHEP 10 [5] A. Abouelsaood , C.G. Callan Jr. , C.R. Nappi and S.A. Yost , Open strings in background [6] G.T. Horowitz and A. Strominger , Translations as inner derivations and associativity anomalies in open string eld theory , Phys. Lett. B 185 ( 1987 ) 45 [INSPIRE]. [7] T. Kawano , I. Kishimoto and T. Takahashi , Gauge invariant overlaps for classical solutions in open string eld theory, Nucl . Phys. B 803 ( 2008 ) 135 [arXiv: 0804 .1541] [INSPIRE]. [8] J. Polchinski , String theory. Volume 1 : an introduction to the bosonic string , Cambridge [10] A. Connes , M.R. Douglas and A.S. Schwarz , Noncommutative geometry and matrix theory: [11] E. Witten , Overview of k-theory applied to strings , Int. J. Mod. Phys. A 16 ( 2001 ) 693


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP05%282018%29144.pdf

Nobuyuki Ishibashi, Isao Kishimoto, Toru Masuda, Tomohiko Takahashi. Vector profile and gauge invariant observables of string field theory solutions for constant magnetic field background, Journal of High Energy Physics, 2018, 144, DOI: 10.1007/JHEP05(2018)144