Comments on real tachyon vacuum solution without square roots

Journal of High Energy Physics, Jan 2018

We analyze the consistency of a recently proposed real tachyon vacuum solution without square roots in open bosonic string field theory. We show that the equation of motion contracted with the solution itself is satisfied. Additionally, by expanding the solution in the basis of the curly ℒ0 and the traditional L0 eigenstates, we evaluate numerically the vacuum energy and obtain a result in agreement with Sen’s conjecture.

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Comments on real tachyon vacuum solution without square roots

HJE Comments on real tachyon vacuum solution without square roots E. Aldo Arroyo 0 1 0 Santo Andre , 09210-170 S~ao Paulo, SP , Brazil 1 Centro de Ci 2 encias Naturais e Humanas, Universidade Federal do ABC We analyze the consistency of a recently proposed real tachyon vacuum solution without square roots in open bosonic string eld theory. We show that the equation of motion contracted with the solution itself is satis ed. Additionally, by expanding the solution in the basis of the curly L0 and the traditional L0 eigenstates, we evaluate numerically the vacuum energy and obtain a result in agreement with Sen's conjecture. String Field Theory; Tachyon Condensation - 1 Introduction 2 3 4 5 6 Conservation laws and the two point vertex in the sliver frame Curly L0 level expansion analysis of the real solution L0 level expansion analysis of the real solution Summary and discussion Introduction Er-Sch = c(1 + K)Bc 1 1 + K ; { 1 { In open string eld theory [1], we say that a string eld is real if obeys the following reality condition z = ; where the double dagger denotes a composition of Hermitian and BPZ conjugation introduced in Gaberdiel and Zwiebach's seminal work [2]. Analytic tachyon vacuum solutions that satisfy the above reality condition (1.1) exist in the literature [3, 4], however they carry some technical complications. For instance, Schnabl's original solution is real, but has some subtleties, the solution contains a singular, projector-like state known as the phantom term [5]. Solutions without the phantom term, known as simple solutions or Erler-Schnabl's type solutions have been proposed [6{10], but they often fail to satisfy the reality condition. By performing a gauge transformation over a non-real simple solution, a real phantom-less solution has been constructed in reference [6]. However, as noted in reference [11], the cost of having this real solution is the introduction of somewhat awkward square roots. It would be desirable to have a solution that is both real and simple, namely without square roots and phantom terms. This is precisely the issue that has been studied in a recent paper [11], where the author has presented an alternative prescription to obtain a real solution from a non-real one which does not make use of a similarity transformation. Basically, it has been shown that given a tachyon vacuum solution together with its corresponding homotopy operator A [12{14], the string eld de ned by = Re( ) + Im( ) A Im( ) is a real solution for the tachyon vacuum. Applying this prescription for the case of the non-real Erler-Schnabl's tachyon vacuum solution [6] the corresponding real solution [11] has been constructed where the QB-exact terms are given by = 1 For this real solution the corresponding energy has been computed and shown that the value is in agreement with the value predicted by Sen's conjecture [15, 16]. HJEP01(28)6 Nevertheless, for the evaluation of the energy, the equation of motion contracted with the solution itself was simply assumed to be satis ed. In this paper, we compute the cubic term of the action for the real solution (1.3) and discuss the validity of the previous assumption. Additionally, by expanding the solution in the basis of curly L0 eigenstates, we evaluate the energy numerically and obtain a result in agreement with Sen's conjecture. Since the numerical evaluation of the energy by means of the curly L0 level expansion of the solution is not a trivial task, in order to automate the computations of relevant correlation functions de ned on the sliver frame, we have developed conservation laws. This paper is organized as follows. In section 2, we evaluate the cubic term of the action for the real solution and test the validity of the equation of motion when contracted with the solution itself. In section 3, in order to automate the computations involved in the numerical evaluation of the energy associated with the solution, we developed conservation laws for operators de ned on the sliver frame. In sections 4, and 5, we compute the energy by means of the curly L0 and the standard Virasoro L0 level expansion of the solution and after using Pade approximants we show that the numerical results obtained for the energy are in agreement with Sen's conjecture. In section 6, a summary and further directions of exploration are given. 2 Computing the cubic term for the real solution In reference [11], a new real solution for the tachyon vacuum has been proposed. This solution in the KBc subalgebra [17, 18] takes the form = 1 1 real solution (2.1). Recall that these terms were not necessary in the evaluation of the kinetic term. The QB-exact terms in (2.1) are given by 1 2 ], after a lengthy algebraic manipulations, we arrive to experience with other solutions [9, 18{20] that this assumption is not a trivial one. In general, a priori there is no justi cation for assuming the validity of without an explicit calculation. Therefore the cubic term of the action must be evaluated. The computation of the kinetic term has been already done in reference [11] given as a result tr[ QB + Thus, for equation (2.3) to be valid, we must show that HJEP01(28)6 by means of the following basic correlators tr ce t1K ce t2K ce t3K t2 (t1 + t2 + t3) 3 sin t1+t2+t3 t3 sin t1+t2+t3 tr Be t1K ce t2K ce t3K ce t4K c s2(t2 + t3 + t4) sin s2(t3 + t4) sin 3 3 + 2 s t34 sin t3 s t4 s sin t2 s where s = t1 + t2 + t3 + t4. 1 16 1 = = (2.3) (2.4) (2.5) (2.6) (2.8) 3 16 cKc c To compute correlators containing the B string eld, we proceed in the same manner. As an illustration, let us explicitly evaluate the correlator trhB (1+1K)2 cKc (1+1K)2 cKci. The integral representation of this correlator is given by Z 1 0 dt1dt2 t1t2e t1 t2 @s1;s2 trhBe t1K ce s1K ce t2K ce s2K ci s1=s2=0 Using the correlator (2.9), from equation (2.12) we obtain Performing the change of variables t1 ! uv, t2 ! u the above double integral (2.13), we get uv, R01 dt1dt2 ! R01 du R01 dv u into For instance, employing the correlator (2.8), let us explicitly compute the correlator tr cKc 1 c 1 To evaluate the above double integral, we perform the change of variables t1 ! uv, t2 ! Therefore, we have just shown that tr B 1 1 + K cKc c cKc = cKc = 1 1 + K c = cKc = cKc 2 6 3 2 2 15 4 ; ; ; 15 4 4 2 ; 2 2 Employing these results (2.16){(2.23) into equation (2.7) and adding up all terms, we obtain the value for the cubic term tr[ ] = 0, it is guaranteed that the energy associated with the solution (2.1) is directly proportional to the kinetic term term by means of the curly L0 level expansion of the solution. As we are going to show, when we insert the curly L0 level expansion of the solution into the kinetic term, we are required to evaluate two point vertices for string elds containing the operators L^ , B^ and c~p. These two point vertices can be evaluated by means of the so-called conservation laws which will be studied in the next section. 3 Conservation laws and the two point vertex in the sliver frame The operators employed in the basis of curly L0 eigenstates are given in terms of the basic operators L^ , B^ and c~p. These operators are related to the worldsheet energy momentum tensor T (z), the b(z) and c(z) ghosts elds respectively. We are going to derive the conservation law for the L^ operator L^ = I dz 2 i (1 + z2)(arctan z + arccotz) T (z) : Using the conformal map z~ = 2 arctan z, we can write the expression of the L^ operator in the sliver frame L^ = I dz~ 2 i "(Rez~) T~(z~) ; where "(x) is the step function equal to 1 for positive or negative values of its argument respectively. { 5 { We need conservation laws such that the operator L^ acting on the two point vertex, which we denote as V2 , can be expressed in terms of non-negative Virasoro modes de ned HJEP01(28)6 on the sliver frame1 where an and bn are coe cients that will be determined below. v(2)(z~2) puncture, v(1)(z~1) To derive a conservation law of the form (3.5), we need a vector eld which behaves as "(Rez~2) + O(z~2) around puncture 2, and has the following behavior in the other O(z~1). A vector eld which does this job is given by The expression of the conservation law for Virasoro modes de ned on the sliver frame v(z) = (1 + z2)arccotz: V2 X2 I j=1 Cj 2 i 1 v(j)(z~j )T~(z~j )dz~j = 0 ; For vertex operators i de ned on the sliver frame, the two functions f1 and f2 which appear in the de nition of the two point vertex f1 1(0)f2 2(0) are given by f1(z~1) = tan (1 + z~1) ; f2(z~2) = tan 2 2 z~2 : (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) (3.9) (3.10) is given by2 j-puncture. where v(j)(z~j ) = (@z~j fj (z~j )) 1v(fj (z~j )), and Cj is a closed contour which encircles the Using equations (3.3), (3.4) and (3.6) into the de nition (@z~j fj (z~j )) 1v(fj (z~j )) of the vector elds v(1)(z~1) and v(2)(z~2), we nd that v(j)(z~j ) = v(1)(z~1) = z~1 v(2)(z~2) = "(Rez~2) z~2: Due to the presence of the step function we see that the vector eld v(2)(z~2) is discontinuous around puncture 2, since we are interested in the conservation law of the operator de ned in equation (3.2), this kind of discontinuity is what we want. Using (3.7) and noting that integration amounts to the replacement vn(i)z~in ! vn(i)L(ni) 1, we can immediately write the conservation law V2 L0 (1) + L^(2) (2) L0 = 0 : 1We are going to use the following notation O(i) to refer an operator O de ned around the i-th puncture. 2This formula can be derived using the general prescription for conservation laws shown in references [21, 22]. { 6 { We can write this conservation law (3.10) in the standard form as given in equation (3.5) By the symmetry property of the two vertex, the same identity (3.11) holds after replacing Regarding the conservation law for the B^ operator, since the b ghost is a conformal eld of dimension two, the conservation laws for operators involving this eld are identical HJEP01(28)6 to those for the Virasoro operators Employing these conservation laws for the operators L^ and B^, together with the commutator and anti-commutator relations [L(0i); L^(j)] = ij ^(j); L [B0(i); L^(j)] = ij ^(j); B [L(0i); B^(j)] = ij ^(j); B fB0(i); B^(j)g = 0; [L(0i); c~(pj)] = ij p c(pj) ; fB0(i); c~(pj)g = ij 0;p ; we can show that all two point correlation functions involving string elds constructed out of the operators L^, B^ and c~p can be reduced to the evaluation of the following basic correlators To evaluate explicitly the above correlators (3.17) and (3.18), the following formulas where the correlator c(x)c(y)c(z) in general is given by will be very useful Sa;b Ca;b 2 i 2 i I dz za sin(bz) = I dz za cos(bz) = b a 1 cos b a 1 sin ( a) ( a) { 7 { sin a 2 ; a For instance, let us compute correlator (3.17). Using (3.19) into equation (3.17), we have It is clear that the above equation (3.22) can be written in terms of the functions (3.20) and (3.21), so that we arrive to an explicit expression for the correlator (3.17) In the same way, we can also derive the explicit expression for the correlator (3.18) Note that in addition to the conservation laws, we will be required to know the action of the BRST charge QB on the operators L^, B^ and c~p [QB; L^(j)] = 0; fQB; B^(j)g = L^(j); fQB; c~(pj)g = k)c~(pj)kc~(kj): As an illustration of the use of conservation laws, we are going to compute a particular correlator involving the operators B^ and L^ string elds We choose, as an example, the following 1 X k= 1 (1 = B^L^c~0c~1j0i; = c~1j0i: { 8 { i i i (3.25) (3.26) (3.27) Using these string elds, let us evaluate the correlator Inserting equation (3.27) into equation (3.28) and using (3.26), we obtain Using the conservation law (3.14) and the anti-commutator relations (3.16), from equation (3.29) we get Employing the conservation laws (3.12), (3.14) and the commutator and anti-commutator relations (3.15), (3.16), from equation (3.30) we arrive to tr[ QB ] = V2 c~0 t1 c t2 Bc t3 = X (x1 p + y1 p)L^nc~pj0i 1 1 X where we have used equation (3.24). These kind of computations can be automated in a computer. Next, we are going to apply the results shown in this section to evaluate the kinetic term by means of the curly L0 level expansion of the real solution (2.1). 4 Curly L0 level expansion analysis of the real solution Since the kinetic term does not depend on the QB-exact terms, we are going to consider only the rst term of given in equation (2.1). Let us de ne this term as ^ = 1 1 { 9 { Using the integral representation of 1=(1 + K) we can write (4.1) as ^ = tc + c t + c tBc + dsdt e s t sc t : By writing the basic string elds K, B in terms of the operators L^, B^, and using the modes c~p of the ghost eld c(z) de ned in the z~-conformal frame z~ = 2 arctan z, we can show that 4 2 = 8 2 ; (3.31) (3.28) (3.29) (4.1) (4.2) (4.3) where = 1 2 1 2 where in this case double integral dsdt e s t x 1 p = (t1 + t2 + t3); x = (t3 t1 t2); y = Employing equation (4.4), it is possible to derive the curly L0 level expansion of the string eld de ned in equation (4.3). As a pedagogical illustration, let us explicitly compute the curly L0 level expansion of the last term appearing on the right hand side of equation (4.3) dsdt e s t sc t = dsdt e s t (x1 p + y1 p)L^nc~pj0i; As we can see from equations (4.6) and (4.7), we are required to evaluate the following dsdt e s t x 1 p = dsdt e s t 2 n+p 1( s t + 1)n(t s)1 p : (4.8) Performing the change of variables s ! uv, t ! u above integral (4.8), we obtain uv, R01 dsdt ! R01 du R01 dv u into the e u2 n+p 1(1 u)nu2 p(1 2v)1 p 1) 2 n+p 2 Z 1 du e u(1 u)nu2 p 1 2 1 X 1 X n! n 2n! 1 2 0 F (n; 2 p); M k 1 2 n! n! (4.6) (4.7) (4.9) (4.11) (4.12) (4.13) where we have de ned F (M; N ) = u)M uN = X( 1)M k (M + N k)! (4.10) Proceeding in the same way, we can also calculate the curly L0 level expansion of the rst terms appearing on the right hand side of equation (4.3). Adding up all the results, we show that the string eld (4.1) has the following curly L0 level expansion ^ = 1 X 1 X where the coe cients fn;p and fn;p;q are given by fn;p = fn;p;q = (1 ( 1)p) 2 n+p 4 3F (n; 1 p) + 2 1 p F (n; 2 (( 1)q ( 1)p) 2 n+p+q 6F (n; 2 p q) : p) ; To compute the kinetic term, we start by replacing the string eld ^ with zL0 ^ , so that states in the curly L0 level expansion will acquire di erent integer powers of z at di erent levels. As we are going to see, the parameter z is needed because we need to express the kinetic term as a formal power series expansion if we want to use Pade approximants. After doing our calculations, we will simply set z = 1. Let us start with the evaluation of the kinetic term as a formal power series expansion in z. By inserting the expansion (4.11) of the string eld ^ into the kinetic term, and using the conservation laws studied in section 3 to evaluate the corresponding two point vertices, we obtain tr[zL0 ^ QB zL0 ^ ] = + + 4 2z2 + 41 4 293 2 1 4 Considering terms up to order z6, and setting z = 1, from equation (4.14) we get 3328% of the expected result (2.4). In principle, we can compute the curly L0 level expansion of the kinetic term up to any desired order, however as we increase the order, the involved tasks demand a lot of computing time. We have determined the series (4.14) up to order z18, and setting z = 1, we obtain about 1:5036 1015% of the expected result. As we can see, if we naively set z = 1 and sum the series, we are left with a non-convergent result. Recall that in numerical curly L0 level truncation computations, a regularization technique based on Pade approximants provides desired results for gauge invariant quantities like the energy [6, 20, 23, 24]. Let us see if after applying Pade approximants, we can recover the expected result. term as follows To start with Pade approximants, rst let us de ne the normalized value of the kinetic Since the series for the kinetic term (4.14) is known up to order z18, we can write the series for E^(z) up to order z20, and after considering a numerical value for , we obtain (4.15) (4.16) In general, to construct a Pade approximant of order Pnn(z) for the normalized value of the kinetic term (4.15), we need to truncate the series (4.16) up to order z2n. 2:84231 3:28987z3 Expanding the right hand side of (4.17) around z = 0 up to order z4 and equating the coe cients of z0, z1, z2, z3, z4 with the expansion (4.16), we get a system of algebraic equations for the unknown coe cients a0, a1, a2, b1, and b2. Solving those equations we get HJEP01(28)6 a0 = Replacing the value of these coe cients inside the de nition of P22(z) (4.17), and evaluating this at z = 1, we get the following value (4.17) (4.19) (4.20) The results of our calculations are summarized in table 1. As we can see, the value of E^(z) at z = 1 by means of Pade approximants con rms the expected analytical result 1. Although the convergence to the expected answer gets irregE^(1) = 32 tr[ ^ QB ^ ] ! right value. ular at n = 4, by considering higher level contributions, we will eventually reach to the Using an alternative resummation technique, we would like to con rm the expected answer for the normalized value of the kinetic term. We have used a second method which is based on a combination of Pade and Borel resummation. We replace the Borel transform of E^(z), which is de ned as E^(z)Borel = P Ekzk=k!, by its Pade approximant Pnn(z)Borel and then evaluate the integral Penn(z) = dt e t Pnn(zt)Borel at z = 1. In the third column of table 1, we list the results obtained for E^(1) by means of Pade-Borel approximation. Note that starting at the value of n = 4, Pade-Borel does a little better than Pade. 5 L0 level expansion analysis of the real solution To expand the string eld (4.3) in the Virasoro basis of L0 eigenstates, we are going to use the following formulas e t1K ce t2K Bce t3K P22(z = 1) = 0:935125008: = r cos2 x r + r cos2 y r (r 2y) Pnn 1:3333333333 0:9351250080 0:7462344772 0:9803952323 0:9800827399 0:9997340118 Penn 1:3333333333 0:6792579899 0:9160629680 0:9938587065 1:0020031889 1:0017620332 1:3333333333 9:2413341787 3327:8214730 2:56791 9:62763 4:94676 107 109 1015 2 2 3 z tr[zL0 ^ QB zL0 ^ ] evaluated at z = 1. The second column shows the Pnn Pade last column, P02n represents a trivial approximation, a naively summed series. approximation. The third column shows the corresponding Penn Pade-Borel approximation. In the HJEP01(28)6 (5.1) (5.2) (5.3) (5.4) (5.5) where the operator Uer is de ned as To nd the coe cients un;r appearing in the exponentials, we use r 2 2 r tan arctan z f2;u2;r f4;u4;r f6;u6;r f8;u8;r f10;u10;r fN;uN;r (z) f2;u2;r (f4;u4;r (f6;u6;r (f8;u8;r (f10;u10;r ( (fN;uN;r (z)) : : : ))))) ; where the function fn;un;r (z) is given by fn;un;r (z) = (1 z un;rnzn)1=n : Employing the set of equations (5.1) (5.3) for the string eld (4.3), we obtain 0 + + dt ^ = Z 1 e tr sin2 2r where r = 1 + t. e s t(1 + s + t)2 cos2 (t s) 2(1+s+t) 8 0 2 tan (t s) 2(1+s+t) 1 + s + t A ; (5.6) Uer c 2r Uerb 2kc t 2 tan 2r r + c t 2 tan 2r r t 2 tan 2r r c t 2 tan 2r r 1 k=1 Uer = lim N!1 = lim N!1 By writing the c ghost in terms of its modes c(z) = Pm cm=zm 1 and employing equations (5.3) and (5.6), the string eld ^ can be readily expanded and the individual coe cients can be numerically integrated. For instance, let us write the expansion of ^ up to level four states ^ = 0:45457753c1j0i + 0:17214438c 1j0i the resulting string eld zL0 ^ , we de ne, the analogue of equation (4.15) As in the case of the curly L0 level expansion analysis, to evaluate the normalized value of the vacuum energy, rst we perform the replacement ^ ! zL0 ^ and then using The normalized value of the vacuum energy is obtained just by setting z = 1. Since the kinetic term is diagonal in L0 eigenstates, the coe cients of the energy (5.8) at order z2L are exactly the contributions from elds at level L. We have expanded the string eld ^ given in equation (5.6) up to level twelve states, and hence the series of E~(z) can be determined up to the order z24 If we naively evaluate the truncated vacuum energy (5.9), i.e., setting z = 1 in the series before using Pade or Pade-Borel approximations, we obtain a non-convergent result. Note that the series (5.9) is less divergent than the series (4.16) that has been obtained in the case of the curly L0 level expansion analysis of the energy. Let us re-sum the divergent series (5.9). To obtain the Pade or Pade-Borel approximation of order Pnn for the energy, we will need to know the series expansion of E~(z) up to the order z2n. The results of these numerical calculations are summarized in table 2. 6 Summary and discussion We have analyzed the validity of the recently proposed real tachyon vacuum solution [11], in open bosonic string eld theory. We have found that the solution solves in a non trivial way the equation of motion when contracted with itself. Let us point out that a similar test of consistency was performed by Okawa [18], Fuchs, Kroyter [19] and Arroyo [20] for the case of the original Schnabl's solution [3]. As a second test of consistency, we have analyzed the solution from a numerical point of view. Using either the curly L0, or the Virasoro L0 level expansion of the solution, we have found that the expression representing the energy is given in terms of a divergent series, (5.7) HJEP01(28)6 (5.8) (5.9) z tr[zL0 ^ QB zL0 ^ ] evaluated at z = 1. The second column shows the Pnn Pade last column, P02n represents a trivial approximation, a naively summed series. approximation. The third column shows the corresponding Penn Pade-Borel approximation. In the which nevertheless can be re-summed, either by means of Pade technique or a combination of Pade-Borel resummation to bring the expected result in agreement with Sen's conjecture. It would be interesting to analyze other real solutions. For instance, the tachyon vacuum solution corresponding to the regularized identity based solution [8]. The real version of this solution, obtained by means of a similarity transformation, contains square roots and consequently the analytical and numerical computations of the energy become cumbersome [9, 23]. Employing the prescription studied in reference [11], it should be possible to nd an alternative real version for this regularized identity based solution. Finally, regarding to the modi ed cubic superstring eld theory [ 25 ] and Berkovits nonpolynomial open superstring eld theory [26], since these theories are based on Witten's associative star product, their mathematical setup shares the same algebraic structure of the open bosonic string eld theory, and thus the prescription developed in reference [11] and the results shown in this paper should be extended to construct and study new real solutions in the superstring context like the ones discussed in references [24, 27{32]. Acknowledgments I would like to thank Ted Erler and Max Jokel for useful discussions. Open Access. 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E. Aldo Arroyo. Comments on real tachyon vacuum solution without square roots, Journal of High Energy Physics, 2018, 6, DOI: 10.1007/JHEP01(2018)006