- Email: [email protected]
Bosonic D-branes at finite temperature
10 August 2000
Physics Letters B 487 Ž2000. 175–179 www.elsevier.nlrlocaternpe
Bosonic D-branes at finite temperature I.V. Vancea 1 Instituto de Fısica Teorica, UniÕersidade Estadual Paulista (UNESP), Rua Pamplona 145, 01405-900 Sao ´ ´ ˜ Paulo-SP, Brazil Received 31 May 2000; accepted 27 June 2000 Editor: L. Alvarez-Gaume´
Abstract We derive the finite temperature description of bosonic D-branes in the thermo field approach. The results might be relevant to the study of thermical properties of D-brane systems. q 2000 Published by Elsevier Science B.V. PACS: 11.25.-w; 11.10.Wx; 11.25.Db
During the past several years, researches on the properties of D-branes, viewed either as solitons of low energy field theories w1x or as states in perturbative spectra of strings w2x, has continued to grow unabated. Theoretical studies predicted perturbative w3x as well as non-perturbative dualities w4x among string theories and M-theory. D-branes offered new approaches to gauge theories w5x, a first microscopic description of the Beckenstein–Hawking entropy of black-holes w6x and new insights in string cosmology w7x. More recently, D-branes have been used to conjecture a relationship between gravity and quantum field theory w8x and to study the stable non-BPS states of string spectra w9x. In lights of these achievements, its is worthwhile to understand deeper the physical properties of D-branes. In this Letter we aim at constructing bosonic D-brane at finite temperature as boundary states of perturbative closed strings w10x. Although the con-
1
E-mail address: [email protected] ŽI.V. Vancea.. On leave from Babes-Bolyai University of Cluj.
struction of a single D-brane works in principle for any temperature, in considering statistical ensembles one should limit the discussion to low temperatures mainly because of the following reason. The bosonic string theories contain tachyons and, at zero mass level, dilatons and gravitons. The presence of these fields make the ensemble of strings to suffer a first order phase transition at Tc - TH , where TH is the Hagedorn temperature. As was argued in w11x, the latent heat involved in the transition is large enough to break down the notion of temperature shortly above Tc Žfor a review see w12x.. The thermodynamics of a generic Dp-brane in the imaginary time formalism was studied for the first time in w13x while the D-brane free energy in external electro-magnetic field has been computed in w14x and in w15x by using a boundary state formalism. In what follows, we are going to apply the thermo field theory w16x to construct the equations that define the D-branes and the boundary states of closed strings that satisfy these equations. We note that the quasi-particle picture in this theory no longer holds at sufficient large temperatures. This might be another reason for studying the ensembles of branes at
0370-2693r00r$ - see front matter q 2000 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 7 9 9 - 1
I.V. Vancear Physics Letters B 487 (2000) 175–179
176
low temperatures. However, it is not clear yet what is the validity of the quasi-particle interpretation in the case of strings and to what extent it may affect the ensembles of branes at high temperature. We hope to clarify these aspects in a forthcoming paper w17x. Consider a free bosonic closed string in the Minkowski spacetime and in the conformal gauge at zero temperature T s 0. We denote by a nm and bnm, m s 0,1, . . . ,25, n g Z ) , the right- and left-moving modes, respectively. In order to apply the thermo field method we pass to the oscillator description of string modes. This is given in terms of creation and annihilation operators for each mode n and in each sector m A nm† s ayn ,
A nm s a nm
m Bnm† s byn ,
Bnm s bnm ,
Ž 1.
where n g N ) . In units such that " s 1 the above operators describe independent harmonic oscillators of frequencies v n s n. The vacuum states of rightand left-moving sectors are denoted by <0:a and <0:b , respectively, and the fundamental state of string is given by the following product <0: s <0:a <0:b < p s 0: ,
Ž 2.
where < p : is the eigenstate of the linear momentum of the center-of-mass corresponding to the eigenvalues p m. A D p-brane is obtained by imposing the Neumann boundary conditions along a s 0,1, . . . , p directions and the Dirichlet boundary conditions along i s p q 1, . . . ,25 directions of the open string at the endpoint s s 0. These are equivalent with the following operatorial equations on the Hilbert space of the closed string w10x
Ž Aan q Bna† . < Bmat : s Ž Aa†n q Bna . < Bmat : s 0 Ž Ain y Bni† . < Bmat : s Ž Ai†n y Bni . < Bmat : s 0, pˆ Bmat : s Ž Xˆ y y . < Bmat : s 0, i
i
A nm , Anm† s A˜nm , A˜mm † s dn , mh mn A nm , A˜nm s A nm , A˜nm† s A nm , B˜mn s PPP s 0.
Ž 5. The extended Hilbert space of the total system is given by the direct product of the two Hilbert spaces of closed strings H0 s H = H˜
Ž 6.
and we denoted a state from H0 by < ::. The vacuum state of the total system is given by <0:: s <0::a <0::bs Ž <0:a <0˜ :a . Ž <0:b <0˜ :b . s Ž <0:a <0:b . <0˜ :a <0˜ :b ,
ž
Ž 7.
/
where the last equality is a consequence of the fact that the original string and the ; string are independent. The first line in Ž7. shows explicitly the doubling of each oscillator while the second one shows the string-tilde string structure of the vacuum state. In order to obtain the fundamental state of the enlarged system we have to multiply Ž7. by < p :< p˜ :. The thermal description of the system can be obtained from the above one by acting with Bogoliubov operators Gna and Gnb on each sector of the Hilbert space and on the creation and annihilation operators. The G-operators are defined as follows Gna s yi u Ž b T . A n P A˜n y A†n P A˜†n
ž . ž B P B˜ y B
/ P B˜ / .
Ž 3.
Gnb s yi u Ž b T
Ž 4.
Here, unŽ b T . is real and depends on the statistics of the nth oscillator w16x
for oscillator modes n ) 0, and a<
this end, firstly we have to double the system. We denote by ; the quantities that correspond to an identical copy of the original bosonic string. The corresponding operators obey the usual commutation rules, the right- and left-sector operators commute and they also commute with the operators corresponding to the original string. For example
for the momentum and position operators of the center-of-mass of string. Here, y i 4 represent the coordinates of the D p-brane in the transverse space. Let us construct the counterpart of Eqs. Ž3. and Ž4. at T / 0 using the thermo field approach w16x. To
n
n
† n
† n
Ž 8.
1
cosh un Ž b T . s Ž 1 y ey b T n .
y 2
.
Ž 9.
Therefore, u is the same in both right- and left-sectors for a given mode n. The dot in Ž8. means the usual scalar product in the Minkowski space A n P A˜n
I.V. Vancear Physics Letters B 487 (2000) 175–179
s A nm P A˜n m . Since the oscillators are independent, the vacuum state at T / 0 has the following structure
Ł eyi G
<0 Ž b T . :: s
a n
<0::a
n)0
Ł eyi G
b m
<0::b .
Ž 10 .
m)0
The creation and annihilation operators at T / 0 corresponding to Ž10. are obtained by acting on the zero temperature operators A† , A, A˜† , A˜4 and B †, B, B˜†, B˜4 with the G-operators given in Ž8.. Taking into account the algebraic properties of Gna and Gnb it is easy to show that the finite temperature annihilation operators can be cast into the following form a
a
A nm Ž b T . s eyi G n A nm e iG n
s u n Ž b T . A nm y Õn Ž b T . A˜nm† yiG na
˜
s u n Ž b T . A˜nm y Õn Ž b T . A nm †
Ž 11 .
in the right-moving sector and Bnm
yi G nb
Ž bT . s e
Bnm e
functions.. Therefore, we may construct the D p-brane states at T / 0 by imposing the Neumann and Dirichlet boundary conditions on the appropriate spacetime directions
Et X a Ž t , s . Ž b T . < ts0 s 0 , X i Ž t , s . Ž b T . < ts0 s y i ,
Ž 14 .
and similarly for X˜ m Žt , s .Ž b T .. Here, a s 0,1, . . . , p and i s p q 1, . . . ,25. When imposed on the extended Hilbert space, these relations define two sets ˜ p-boundary of equations that determine D p- and D states at T / 0. We denote these states by < Bmat Ž b T .:: and < B˜mat Ž b T .::, respectively. If we introduce the following matrices w10x S mn s Ž h a b ,y d i j . ,
a A nm e iG n
˜ Ž bT . s e
A nm
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S˜mn s Ž h˜ a b ,y d˜ i j . ,
we can see that the equations defining D p- and ˜ p-branes have the following form D u n Ž b T . Ž A nm q Snm Bnn † . q Õn Ž b T . B˜nm q S˜nm A˜nn †
ž
iG nb
< Bm Ž b T . :: s 0
s u n Ž b T . Bnm y Õn Ž b T . B˜nm†
ž
Ž 12 .
< Bm Ž b T . :: s 0
/
Ž 17 .
and
in the left-moving sector. Here,
u n Ž b T . A˜nm q S˜nm B˜nn † q Õn Ž b T . Ž Bnm q Snm Ann † .
u n Ž b T . s cosh un Ž b T . , Õn Ž b T . s sinh un Ž b T . .
/
Ž 16 .
u n Ž b T . Ž A nm† q Snm Bnn . q Õn Ž b T . B˜nm † q S˜nm A˜nn
b b B˜nm Ž b T . s eyiG n B˜nm e iG n
s u n Ž b T . B˜nm y Õn Ž b T . Bnm †
Ž 15 .
ž
Ž 13 .
ˆ p˜ˆ and X˜ˆ commute with all Since the operators p, ˆ X, oscillator operators, they are not affected by the transformations of the type Ž11. and Ž12.. If we construct zero mode G-operators as we did for the oscillators we see that the momenta commute with them. Therefore, we take the position and momenta operators of both the string and the tilde string to be invariant under the transformations above. The corresponding eigenstates of the momenta operators are taken to be invariant, too. Then it is not difficult to see that the string coordinate operators X m Žt , s .Ž b T . and X˜ m Žt , s .Ž b T . constructed from Ž11. and Ž12. are solutions of the string equations of motion Ž Gna and Gnb do not act on the two dimensional wave
/
< B˜m Ž b T . :: s 0
Ž 18 .
u n Ž b T . A˜nm† q S˜nm B˜nn q Õn Ž b T . Ž Bnm † q Snm Ann † .
ž
/
< B˜m Ž b T . :: s 0
Ž 19 .
If for some mode n and some critical temperature Tc the equation u nŽ b Tc . s ÕnŽ b Tc . holds, then the corresponding Eqs. Ž16.,Ž17. and Ž18.,Ž19. are identical and give the same nth oscillator contribution to < Bmat Ž b T .:: and < B˜mat Ž b T .::. This condition is actually equivalent to the limit Tc ` for any finite n, case in which Ž16.,Ž17. and Ž18.,Ž19. are identical for all n. The equations above are consistent with the zero temperature description of the D p-branes as
™
I.V. Vancear Physics Letters B 487 (2000) 175–179
178
™
boundary states since in the limit T 0 they reduce to two copies of the Eqs. Ž3.. Let us look for the solutions of the equations above. We put the state < Bmat Ž b T .:: under the form < Bmat Ž b T . :: s Bˆmat <0 Ž b T . :: s Bˆmat Ž b T . <0:: .
Ž 20 . Then, by analogy with the zero temperature limit, we can write the operator Bˆmat Ž b T . in the following way `
Bˆmat Ž b T . ; d 25yp Ž Xˆ i y y i . Oˆmat Ł e iG k .
Ž 21 .
ks1
After some simple algebra one can see that the Eqs. Ž16.,Ž17. have solution of the form Ž20.. Moreover, this solution is not unique since it can be constructed with the following different operators ` 1 Oˆmat s
` †
˜ † ˜ ˜†
†
Ł eyA n S B n Ł eyB r S A r
ns1
rs1
` 2,3 Oˆmat s
` † n
Ł eyA S B
ns1
† n
=1"1=
Ł eyB˜ S˜ A˜ . † n
† n
Ž 22 .
ns1
Thus, the D p-brane states at T / 0 from the extended Hilbert space are degenerate in the sense of the equations above. The same considerations apply to < B˜mat Ž b T .:: states. In this case we obtain the first solutions from Ž22. as a consequence of the fact that the right- and left-modes commute for either the original string as well as for its copy. Each of D p- or ˜ p-branes depend on the oscillator operators of the D original as well as the tilde strings. However, according to the thermofield theory, the physical quantities in a thermal D p-brane state may depend only on the vev of the operators without tilde w16x. In this sense, the solutions derived above in Ž22. are equivalent. Note that one has to include in < Bmat Ž b T .:: the eigenvectors of momenta operators at p s 0. However, since these operators and their eigenvalues do not transform under G-transformations, their contribution is the same as in T s 0 case. The normalization constant that enter Ž20. is related, at T s 0, to the tension of the brane w2x. To determine it at finite
temperature, one has to perform calculations in both open and closed string sectors w17x. The above boundary states make sense only if they belong to the physical sector of the theory. At T s 0, this sector is defined using the conformal invariance of the string theory. The same definition can be applied here since the conformal invariance is not broken. Indeed, using the fact that G-operators commute, it is easy to show that the string oscillation modes satisfy the same algebra at finite temperature. Then, since X m Žt , s .Ž b T . and X˜ m Žt , s .Ž b T . satisfy the string equations of motion the operators L nŽ b T . and L˜ nŽ b T . have formally the same expansion in terms of modes a mŽ b T . as the Virasoro operators at zero temperature and satisfy the same algebra in both right- and left-sectors and for both the original string and the tilde string. The algebras in all of these sectors are independent of each other. Using the conformal symmetry, one can impose the physical state conditions on the Fock space of the extended system as for T s 0 w17x. Also, the problem of maintaining the conformal invariance is related to the possibility of mapping the open string sector to the closed one and to constructing the BRST invariant states. Note that the conformal invariance at finite temperature is due to the specific form of G-operators taken in Ž8.. This is not the most general form allowed in thermo field approach w16x. Nevertheless, the conformal symmetry is maintained by other transformations, too. The point here is that G-operators do not mix the various oscillation mode which is a reasonable assumption at low temperature. At high temperature we might expect that G-operators involve interacting terms between various oscillators, case in which the conformal invariance is not automatically satisfied. If we stick on the definition of strings as conformal invariant field theories in two dimensions, the conformal invariance imposes some constraints on G-operators w17x. On the other hand, breaking the conformal invariance by G-operators might be a consequence of the fact that the temperature is not well defined for string systems at T ) Tc as was showed in w11x. In the case of super D-branes one has to repeat the same construction in both NS-NS and R-R sectors of the closed string. It is quite easy to construct the corresponding Bogoliubov transformations for NS and R spinors and the corresponding fermionic
I.V. Vancear Physics Letters B 487 (2000) 175–179
modes at finite temperature. However, it is not clear yet what is the natural extension of the supersymmetry at T / 0 of the theory that is supersymmetric at T s 0 and whether the superconformal invariance is broken or not w17x. The supersymmetry in right- and left-modes splits into susy
a X p m q Ý u n Ž b T . E. X nm ; n
Ý u t Ž bT . c.m Ž t . t
susy
Ý Õn Ž bT . E. Y˜nm ; y Ý Õt Ž bT . x˜ .m Ž t . , n
Ž 23 .
t
where by Y˜nm and x˜ nm we denoted the bosonic and fermionic modes of the tilde string present in the equations of the original string. ŽSimilar expressions should be considered for the tilde string, too.. The problem is due to the presence of u nŽ b T . and ÕnŽ b T . in the bosonic mode expansion and to similar terms u t Ž b T . and Õt Ž b T . in the fermionic mode expansion, where t g Zqq 1r2 in NS sector and t g Zq in R sector, respectively. We note that the second set of coefficients is different from the first one since it is related to Fermi–Dirac distributions of the fermionic modes. In summary, we have used the thermo field theory to find out the equations that define the bosonic D p-branes at finite temperature. We found that the corresponding boundary states from the extended Hilbert space are degenerate, but equivalent at thermical equilibrium. They do not come in pairs since the solutions are the same for the equations of the original string and for the tilde string. In this respect ˜ p-branes are the states that describe D p-branes and D degenerate, opposite to the quasi-particle states in field theory.
Acknowledgements I have greatly benefited from discussion with M.C.B. Abdalla, N. Berkovits, A.L. Gadelha, B.M. Pimentel, H.Q. Placido, D.L. Nedel and B.C. Vallillo. I also acknowledge to M.A. De Andrade and J.A. Helayel-Neto for hospitality at GFT-UCP where part of this work was done and to A. Sen for correspon-
179
dence. The work was financially supported by a FAPESP postdoc fellowship.
References w1x M.J. Duff, R.R. Khuri, J.X. Lu, Phys. Rep. 259 Ž1995. 213. w2x J. Polchinski, Phys. Rev. Lett. 75 Ž1995. 4724. w3x A. Giveon, M. Porrati, E. Rabinovici, Phys. Rep. 244 Ž1994. 77. w4x C.M. Hull, P.K. Townsend, Nucl. Phys. B 439 Ž1995. 109; E. Witten, Nucl. Phys. B 443 Ž1995. 85. w5x For a review see A. Giveon, D. Kutasov, Rev. of Mod. Phys. 71 Ž1999. 983. w6x A. Strominger, C. Vafa, Phys. Lett. B 379 Ž1996. 99; J. Maldacena, Black Holes in String Theory, PhD Thesis, Princeton University Ž1996. hep-thr9607235. w7x P. Horava, E. Witten, Nucl. Phys. B 460 Ž1996. 506; Nucl. Phys. B 475 Ž1996. 94; E. Witten, Nucl. Phys. B 471 Ž1996. 135; L. Randall, R. Sundrum, Phys. Rev. Lett. 83 Ž1999. 3370. w8x J. Maldacena, Adv. Theory. Math. Phys. 2 Ž1998. 231; S.S. Gubser, I. Klebanov, A. Polyakov, Phys. Lett. B 428 Ž1998. 105; E. Witten, Adv. Theory. Math. Phys. 2 Ž1998. 253. w9x A. Sen, JHEP 06 Ž1998. 007; JHEP 08 Ž1998. 010; JHEP 08 Ž1998. 012; JHEP 09 Ž1998. 023; JHEP 10 Ž1998. 021; JHEP 12 Ž1998. 021; M. Gaberdiel, A. Sen, hep-thr9908060. w10x C.G. Callan, C. Lovelace, C.R. Nappi, S.A. Yost, Nuc. Phys. B 288 Ž1987. 525; Nuc. Phys. B 293 Ž1987. 83; Nucl. Phys. B 308 Ž1988. 221; J. Polchinski, Y. Cai, Nucl. Phys. B 296 Ž1988. 91; M.B. Green, P. Wai, Nucl. Phys. B 431 Ž1994. 131; M.B. Green, Nucl. Phys. B 381 Ž1992. 201; M.B. Green, M. Gutperle, Nucl. Phys. B 476 Ž1996. 484; M. Li, Nucl. Phys. B 460 Ž1996. 351; C. Callan Jr., I. Klebanov, Nucl. Phys. B 465 Ž1996. 473; P. Di Vecchia, A. Liccardo, hep-thr9912161, hep-thr9912275; P. Di Vecchia, M. Frau, A. Lerda, A. Liccardo, hep-thr9906214, to be published. w11x J.J. Atick, E. Witten, Nucl. Phys. B 310 Ž1988. 291. w12x L. Alvarez-Gaume, in: B. Julia, J. ´ M.A. Vazquez-Mozo, ´ Zinn-Justin ŽEds.., Topics in String Theory and Quantum Gravity, Les Houches, Session LVII, 1992; S. Odintsov, Rivista Nuovo Cim. 15 Ž1992. 1. w13x M.V. Mozo, Phys. Lett. B 388 Ž1996. 494; J.L.F. Barbon, M.V. Mozo, Nucl. Phys. B 497 Ž1997. 236. w14x A. Bytsenko, S. Odintsov, L. Granada, Mod. Phys. Lett. A 11 Ž1996. 2525. w15x J. Ambjorn, Yu. Makeenko, G.W. Semenoff, R. Szabo, Phys. Rev. D 60 Ž1999. 106009. w16x Y. Takahashi, H. Umezawa, Collaboration Phenom. 2 Ž1975. 55. w17x M.C.B. Abdalla, I.V. Vancea, Strings at Finite Temperature in Thermo Field Approach, in preparation.
Physics Letters B 487 Ž2000. 175–179 www.elsevier.nlrlocaternpe
Bosonic D-branes at finite temperature I.V. Vancea 1 Instituto de Fısica Teorica, UniÕersidade Estadual Paulista (UNESP), Rua Pamplona 145, 01405-900 Sao ´ ´ ˜ Paulo-SP, Brazil Received 31 May 2000; accepted 27 June 2000 Editor: L. Alvarez-Gaume´
Abstract We derive the finite temperature description of bosonic D-branes in the thermo field approach. The results might be relevant to the study of thermical properties of D-brane systems. q 2000 Published by Elsevier Science B.V. PACS: 11.25.-w; 11.10.Wx; 11.25.Db
During the past several years, researches on the properties of D-branes, viewed either as solitons of low energy field theories w1x or as states in perturbative spectra of strings w2x, has continued to grow unabated. Theoretical studies predicted perturbative w3x as well as non-perturbative dualities w4x among string theories and M-theory. D-branes offered new approaches to gauge theories w5x, a first microscopic description of the Beckenstein–Hawking entropy of black-holes w6x and new insights in string cosmology w7x. More recently, D-branes have been used to conjecture a relationship between gravity and quantum field theory w8x and to study the stable non-BPS states of string spectra w9x. In lights of these achievements, its is worthwhile to understand deeper the physical properties of D-branes. In this Letter we aim at constructing bosonic D-brane at finite temperature as boundary states of perturbative closed strings w10x. Although the con-
1
E-mail address: [email protected] ŽI.V. Vancea.. On leave from Babes-Bolyai University of Cluj.
struction of a single D-brane works in principle for any temperature, in considering statistical ensembles one should limit the discussion to low temperatures mainly because of the following reason. The bosonic string theories contain tachyons and, at zero mass level, dilatons and gravitons. The presence of these fields make the ensemble of strings to suffer a first order phase transition at Tc - TH , where TH is the Hagedorn temperature. As was argued in w11x, the latent heat involved in the transition is large enough to break down the notion of temperature shortly above Tc Žfor a review see w12x.. The thermodynamics of a generic Dp-brane in the imaginary time formalism was studied for the first time in w13x while the D-brane free energy in external electro-magnetic field has been computed in w14x and in w15x by using a boundary state formalism. In what follows, we are going to apply the thermo field theory w16x to construct the equations that define the D-branes and the boundary states of closed strings that satisfy these equations. We note that the quasi-particle picture in this theory no longer holds at sufficient large temperatures. This might be another reason for studying the ensembles of branes at
0370-2693r00r$ - see front matter q 2000 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 7 9 9 - 1
I.V. Vancear Physics Letters B 487 (2000) 175–179
176
low temperatures. However, it is not clear yet what is the validity of the quasi-particle interpretation in the case of strings and to what extent it may affect the ensembles of branes at high temperature. We hope to clarify these aspects in a forthcoming paper w17x. Consider a free bosonic closed string in the Minkowski spacetime and in the conformal gauge at zero temperature T s 0. We denote by a nm and bnm, m s 0,1, . . . ,25, n g Z ) , the right- and left-moving modes, respectively. In order to apply the thermo field method we pass to the oscillator description of string modes. This is given in terms of creation and annihilation operators for each mode n and in each sector m A nm† s ayn ,
A nm s a nm
m Bnm† s byn ,
Bnm s bnm ,
Ž 1.
where n g N ) . In units such that " s 1 the above operators describe independent harmonic oscillators of frequencies v n s n. The vacuum states of rightand left-moving sectors are denoted by <0:a and <0:b , respectively, and the fundamental state of string is given by the following product <0: s <0:a <0:b < p s 0: ,
Ž 2.
where < p : is the eigenstate of the linear momentum of the center-of-mass corresponding to the eigenvalues p m. A D p-brane is obtained by imposing the Neumann boundary conditions along a s 0,1, . . . , p directions and the Dirichlet boundary conditions along i s p q 1, . . . ,25 directions of the open string at the endpoint s s 0. These are equivalent with the following operatorial equations on the Hilbert space of the closed string w10x
Ž Aan q Bna† . < Bmat : s Ž Aa†n q Bna . < Bmat : s 0 Ž Ain y Bni† . < Bmat : s Ž Ai†n y Bni . < Bmat : s 0, pˆ Bmat : s Ž Xˆ y y . < Bmat : s 0, i
i
A nm , Anm† s A˜nm , A˜mm † s dn , mh mn A nm , A˜nm s A nm , A˜nm† s A nm , B˜mn s PPP s 0.
Ž 5. The extended Hilbert space of the total system is given by the direct product of the two Hilbert spaces of closed strings H0 s H = H˜
Ž 6.
and we denoted a state from H0 by < ::. The vacuum state of the total system is given by <0:: s <0::a <0::bs Ž <0:a <0˜ :a . Ž <0:b <0˜ :b . s Ž <0:a <0:b . <0˜ :a <0˜ :b ,
ž
Ž 7.
/
where the last equality is a consequence of the fact that the original string and the ; string are independent. The first line in Ž7. shows explicitly the doubling of each oscillator while the second one shows the string-tilde string structure of the vacuum state. In order to obtain the fundamental state of the enlarged system we have to multiply Ž7. by < p :< p˜ :. The thermal description of the system can be obtained from the above one by acting with Bogoliubov operators Gna and Gnb on each sector of the Hilbert space and on the creation and annihilation operators. The G-operators are defined as follows Gna s yi u Ž b T . A n P A˜n y A†n P A˜†n
ž . ž B P B˜ y B
/ P B˜ / .
Ž 3.
Gnb s yi u Ž b T
Ž 4.
Here, unŽ b T . is real and depends on the statistics of the nth oscillator w16x
for oscillator modes n ) 0, and a<
this end, firstly we have to double the system. We denote by ; the quantities that correspond to an identical copy of the original bosonic string. The corresponding operators obey the usual commutation rules, the right- and left-sector operators commute and they also commute with the operators corresponding to the original string. For example
for the momentum and position operators of the center-of-mass of string. Here, y i 4 represent the coordinates of the D p-brane in the transverse space. Let us construct the counterpart of Eqs. Ž3. and Ž4. at T / 0 using the thermo field approach w16x. To
n
n
† n
† n
Ž 8.
1
cosh un Ž b T . s Ž 1 y ey b T n .
y 2
.
Ž 9.
Therefore, u is the same in both right- and left-sectors for a given mode n. The dot in Ž8. means the usual scalar product in the Minkowski space A n P A˜n
I.V. Vancear Physics Letters B 487 (2000) 175–179
s A nm P A˜n m . Since the oscillators are independent, the vacuum state at T / 0 has the following structure
Ł eyi G
<0 Ž b T . :: s
a n
<0::a
n)0
Ł eyi G
b m
<0::b .
Ž 10 .
m)0
The creation and annihilation operators at T / 0 corresponding to Ž10. are obtained by acting on the zero temperature operators A† , A, A˜† , A˜4 and B †, B, B˜†, B˜4 with the G-operators given in Ž8.. Taking into account the algebraic properties of Gna and Gnb it is easy to show that the finite temperature annihilation operators can be cast into the following form a
a
A nm Ž b T . s eyi G n A nm e iG n
s u n Ž b T . A nm y Õn Ž b T . A˜nm† yiG na
˜
s u n Ž b T . A˜nm y Õn Ž b T . A nm †
Ž 11 .
in the right-moving sector and Bnm
yi G nb
Ž bT . s e
Bnm e
functions.. Therefore, we may construct the D p-brane states at T / 0 by imposing the Neumann and Dirichlet boundary conditions on the appropriate spacetime directions
Et X a Ž t , s . Ž b T . < ts0 s 0 , X i Ž t , s . Ž b T . < ts0 s y i ,
Ž 14 .
and similarly for X˜ m Žt , s .Ž b T .. Here, a s 0,1, . . . , p and i s p q 1, . . . ,25. When imposed on the extended Hilbert space, these relations define two sets ˜ p-boundary of equations that determine D p- and D states at T / 0. We denote these states by < Bmat Ž b T .:: and < B˜mat Ž b T .::, respectively. If we introduce the following matrices w10x S mn s Ž h a b ,y d i j . ,
a A nm e iG n
˜ Ž bT . s e
A nm
177
S˜mn s Ž h˜ a b ,y d˜ i j . ,
we can see that the equations defining D p- and ˜ p-branes have the following form D u n Ž b T . Ž A nm q Snm Bnn † . q Õn Ž b T . B˜nm q S˜nm A˜nn †
ž
iG nb
< Bm Ž b T . :: s 0
s u n Ž b T . Bnm y Õn Ž b T . B˜nm†
ž
Ž 12 .
< Bm Ž b T . :: s 0
/
Ž 17 .
and
in the left-moving sector. Here,
u n Ž b T . A˜nm q S˜nm B˜nn † q Õn Ž b T . Ž Bnm q Snm Ann † .
u n Ž b T . s cosh un Ž b T . , Õn Ž b T . s sinh un Ž b T . .
/
Ž 16 .
u n Ž b T . Ž A nm† q Snm Bnn . q Õn Ž b T . B˜nm † q S˜nm A˜nn
b b B˜nm Ž b T . s eyiG n B˜nm e iG n
s u n Ž b T . B˜nm y Õn Ž b T . Bnm †
Ž 15 .
ž
Ž 13 .
ˆ p˜ˆ and X˜ˆ commute with all Since the operators p, ˆ X, oscillator operators, they are not affected by the transformations of the type Ž11. and Ž12.. If we construct zero mode G-operators as we did for the oscillators we see that the momenta commute with them. Therefore, we take the position and momenta operators of both the string and the tilde string to be invariant under the transformations above. The corresponding eigenstates of the momenta operators are taken to be invariant, too. Then it is not difficult to see that the string coordinate operators X m Žt , s .Ž b T . and X˜ m Žt , s .Ž b T . constructed from Ž11. and Ž12. are solutions of the string equations of motion Ž Gna and Gnb do not act on the two dimensional wave
/
< B˜m Ž b T . :: s 0
Ž 18 .
u n Ž b T . A˜nm† q S˜nm B˜nn q Õn Ž b T . Ž Bnm † q Snm Ann † .
ž
/
< B˜m Ž b T . :: s 0
Ž 19 .
If for some mode n and some critical temperature Tc the equation u nŽ b Tc . s ÕnŽ b Tc . holds, then the corresponding Eqs. Ž16.,Ž17. and Ž18.,Ž19. are identical and give the same nth oscillator contribution to < Bmat Ž b T .:: and < B˜mat Ž b T .::. This condition is actually equivalent to the limit Tc ` for any finite n, case in which Ž16.,Ž17. and Ž18.,Ž19. are identical for all n. The equations above are consistent with the zero temperature description of the D p-branes as
™
I.V. Vancear Physics Letters B 487 (2000) 175–179
178
™
boundary states since in the limit T 0 they reduce to two copies of the Eqs. Ž3.. Let us look for the solutions of the equations above. We put the state < Bmat Ž b T .:: under the form < Bmat Ž b T . :: s Bˆmat <0 Ž b T . :: s Bˆmat Ž b T . <0:: .
Ž 20 . Then, by analogy with the zero temperature limit, we can write the operator Bˆmat Ž b T . in the following way `
Bˆmat Ž b T . ; d 25yp Ž Xˆ i y y i . Oˆmat Ł e iG k .
Ž 21 .
ks1
After some simple algebra one can see that the Eqs. Ž16.,Ž17. have solution of the form Ž20.. Moreover, this solution is not unique since it can be constructed with the following different operators ` 1 Oˆmat s
` †
˜ † ˜ ˜†
†
Ł eyA n S B n Ł eyB r S A r
ns1
rs1
` 2,3 Oˆmat s
` † n
Ł eyA S B
ns1
† n
=1"1=
Ł eyB˜ S˜ A˜ . † n
† n
Ž 22 .
ns1
Thus, the D p-brane states at T / 0 from the extended Hilbert space are degenerate in the sense of the equations above. The same considerations apply to < B˜mat Ž b T .:: states. In this case we obtain the first solutions from Ž22. as a consequence of the fact that the right- and left-modes commute for either the original string as well as for its copy. Each of D p- or ˜ p-branes depend on the oscillator operators of the D original as well as the tilde strings. However, according to the thermofield theory, the physical quantities in a thermal D p-brane state may depend only on the vev of the operators without tilde w16x. In this sense, the solutions derived above in Ž22. are equivalent. Note that one has to include in < Bmat Ž b T .:: the eigenvectors of momenta operators at p s 0. However, since these operators and their eigenvalues do not transform under G-transformations, their contribution is the same as in T s 0 case. The normalization constant that enter Ž20. is related, at T s 0, to the tension of the brane w2x. To determine it at finite
temperature, one has to perform calculations in both open and closed string sectors w17x. The above boundary states make sense only if they belong to the physical sector of the theory. At T s 0, this sector is defined using the conformal invariance of the string theory. The same definition can be applied here since the conformal invariance is not broken. Indeed, using the fact that G-operators commute, it is easy to show that the string oscillation modes satisfy the same algebra at finite temperature. Then, since X m Žt , s .Ž b T . and X˜ m Žt , s .Ž b T . satisfy the string equations of motion the operators L nŽ b T . and L˜ nŽ b T . have formally the same expansion in terms of modes a mŽ b T . as the Virasoro operators at zero temperature and satisfy the same algebra in both right- and left-sectors and for both the original string and the tilde string. The algebras in all of these sectors are independent of each other. Using the conformal symmetry, one can impose the physical state conditions on the Fock space of the extended system as for T s 0 w17x. Also, the problem of maintaining the conformal invariance is related to the possibility of mapping the open string sector to the closed one and to constructing the BRST invariant states. Note that the conformal invariance at finite temperature is due to the specific form of G-operators taken in Ž8.. This is not the most general form allowed in thermo field approach w16x. Nevertheless, the conformal symmetry is maintained by other transformations, too. The point here is that G-operators do not mix the various oscillation mode which is a reasonable assumption at low temperature. At high temperature we might expect that G-operators involve interacting terms between various oscillators, case in which the conformal invariance is not automatically satisfied. If we stick on the definition of strings as conformal invariant field theories in two dimensions, the conformal invariance imposes some constraints on G-operators w17x. On the other hand, breaking the conformal invariance by G-operators might be a consequence of the fact that the temperature is not well defined for string systems at T ) Tc as was showed in w11x. In the case of super D-branes one has to repeat the same construction in both NS-NS and R-R sectors of the closed string. It is quite easy to construct the corresponding Bogoliubov transformations for NS and R spinors and the corresponding fermionic
I.V. Vancear Physics Letters B 487 (2000) 175–179
modes at finite temperature. However, it is not clear yet what is the natural extension of the supersymmetry at T / 0 of the theory that is supersymmetric at T s 0 and whether the superconformal invariance is broken or not w17x. The supersymmetry in right- and left-modes splits into susy
a X p m q Ý u n Ž b T . E. X nm ; n
Ý u t Ž bT . c.m Ž t . t
susy
Ý Õn Ž bT . E. Y˜nm ; y Ý Õt Ž bT . x˜ .m Ž t . , n
Ž 23 .
t
where by Y˜nm and x˜ nm we denoted the bosonic and fermionic modes of the tilde string present in the equations of the original string. ŽSimilar expressions should be considered for the tilde string, too.. The problem is due to the presence of u nŽ b T . and ÕnŽ b T . in the bosonic mode expansion and to similar terms u t Ž b T . and Õt Ž b T . in the fermionic mode expansion, where t g Zqq 1r2 in NS sector and t g Zq in R sector, respectively. We note that the second set of coefficients is different from the first one since it is related to Fermi–Dirac distributions of the fermionic modes. In summary, we have used the thermo field theory to find out the equations that define the bosonic D p-branes at finite temperature. We found that the corresponding boundary states from the extended Hilbert space are degenerate, but equivalent at thermical equilibrium. They do not come in pairs since the solutions are the same for the equations of the original string and for the tilde string. In this respect ˜ p-branes are the states that describe D p-branes and D degenerate, opposite to the quasi-particle states in field theory.
Acknowledgements I have greatly benefited from discussion with M.C.B. Abdalla, N. Berkovits, A.L. Gadelha, B.M. Pimentel, H.Q. Placido, D.L. Nedel and B.C. Vallillo. I also acknowledge to M.A. De Andrade and J.A. Helayel-Neto for hospitality at GFT-UCP where part of this work was done and to A. Sen for correspon-
179
dence. The work was financially supported by a FAPESP postdoc fellowship.
References w1x M.J. Duff, R.R. Khuri, J.X. Lu, Phys. Rep. 259 Ž1995. 213. w2x J. Polchinski, Phys. Rev. Lett. 75 Ž1995. 4724. w3x A. Giveon, M. Porrati, E. Rabinovici, Phys. Rep. 244 Ž1994. 77. w4x C.M. Hull, P.K. Townsend, Nucl. Phys. B 439 Ž1995. 109; E. Witten, Nucl. Phys. B 443 Ž1995. 85. w5x For a review see A. Giveon, D. Kutasov, Rev. of Mod. Phys. 71 Ž1999. 983. w6x A. Strominger, C. Vafa, Phys. Lett. B 379 Ž1996. 99; J. Maldacena, Black Holes in String Theory, PhD Thesis, Princeton University Ž1996. hep-thr9607235. w7x P. Horava, E. Witten, Nucl. Phys. B 460 Ž1996. 506; Nucl. Phys. B 475 Ž1996. 94; E. Witten, Nucl. Phys. B 471 Ž1996. 135; L. Randall, R. Sundrum, Phys. Rev. Lett. 83 Ž1999. 3370. w8x J. Maldacena, Adv. Theory. Math. Phys. 2 Ž1998. 231; S.S. Gubser, I. Klebanov, A. Polyakov, Phys. Lett. B 428 Ž1998. 105; E. Witten, Adv. Theory. Math. Phys. 2 Ž1998. 253. w9x A. Sen, JHEP 06 Ž1998. 007; JHEP 08 Ž1998. 010; JHEP 08 Ž1998. 012; JHEP 09 Ž1998. 023; JHEP 10 Ž1998. 021; JHEP 12 Ž1998. 021; M. Gaberdiel, A. Sen, hep-thr9908060. w10x C.G. Callan, C. Lovelace, C.R. Nappi, S.A. Yost, Nuc. Phys. B 288 Ž1987. 525; Nuc. Phys. B 293 Ž1987. 83; Nucl. Phys. B 308 Ž1988. 221; J. Polchinski, Y. Cai, Nucl. Phys. B 296 Ž1988. 91; M.B. Green, P. Wai, Nucl. Phys. B 431 Ž1994. 131; M.B. Green, Nucl. Phys. B 381 Ž1992. 201; M.B. Green, M. Gutperle, Nucl. Phys. B 476 Ž1996. 484; M. Li, Nucl. Phys. B 460 Ž1996. 351; C. Callan Jr., I. Klebanov, Nucl. Phys. B 465 Ž1996. 473; P. Di Vecchia, A. Liccardo, hep-thr9912161, hep-thr9912275; P. Di Vecchia, M. Frau, A. Lerda, A. Liccardo, hep-thr9906214, to be published. w11x J.J. Atick, E. Witten, Nucl. Phys. B 310 Ž1988. 291. w12x L. Alvarez-Gaume, in: B. Julia, J. ´ M.A. Vazquez-Mozo, ´ Zinn-Justin ŽEds.., Topics in String Theory and Quantum Gravity, Les Houches, Session LVII, 1992; S. Odintsov, Rivista Nuovo Cim. 15 Ž1992. 1. w13x M.V. Mozo, Phys. Lett. B 388 Ž1996. 494; J.L.F. Barbon, M.V. Mozo, Nucl. Phys. B 497 Ž1997. 236. w14x A. Bytsenko, S. Odintsov, L. Granada, Mod. Phys. Lett. A 11 Ž1996. 2525. w15x J. Ambjorn, Yu. Makeenko, G.W. Semenoff, R. Szabo, Phys. Rev. D 60 Ž1999. 106009. w16x Y. Takahashi, H. Umezawa, Collaboration Phenom. 2 Ž1975. 55. w17x M.C.B. Abdalla, I.V. Vancea, Strings at Finite Temperature in Thermo Field Approach, in preparation.