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Strings at the tachyonic vacuum

Strings at the tachyonic vacuum JHEP03(2001)014 Received: February 2, 2001, Accepted: March 8, 2001 HYPER VERSION Revised: February 20, 2001 a ab Ulf Lindstr¨om and Maxim Zabzine Institute of Theoretical Physics, University of Stockholm Box 6730, S-113 85 Stockholm Sweden Institute Henri Poincare 11, rue P. et M. Curie, 75231 Paris Cedex 05, France E-mail: [email protected], [email protected] Abstract: We study the world-volume e ective action of Dp-brane at the tachy- onic vacuum which is equivalent to the zero tension limit. Using the Hamiltonian formalism we discuss the algebra of constraints and show that there is a non-trivial ideal of the algebra which corresponds to Virasoro like constraints. The Lagrangian treatment of the model is also considered. For the gauge xed theory we construct the important subset of classical solutions which is equivalent to the string theory solutions in conformal gauge. We speculate on a possible quantization of the system. At the end a brief discussion of di erent background elds and fluctuations around the tachyonic vacuum is presented. Keywords: Bosonic Strings, D-branes. JHEP03(2001)014 Contents 1. Introduction and motivation 1 2. Hamiltonian treatment of BI theory 2 3. Lagrangian analysis 7 4. Background elds 10 5. Discussion 11 A. Basic de nitions 12 1. Introduction and motivation The problem of tachyon condensation is an old subject in the context of the string theory[1]. Recently ithasbeenconjectured thatthetachyonic vacuuminopenstring theory on the D-brane describes the closed string vacuum without D-branes and that various soliton solutions of the theory describe D-branes of lower dimension [2]. This conjecture has been supported by number calculations within the rst and second quantized string theory. Onecanget someinsight intotheproblem byconsidering theworld-volume e ec- tive action [3] which describes the D-brane around the tachyonic vacuum. Recently there has been some e ort directed towards identifying of string-like classical solu- tions whose tension matches that of fundamental string [4]{[8]. In fact the D-brane action at the tachyonic vacuum is equivalent to the zero tension limit of the D-brane action. The zero tension limit was with a di erent motivation studied previously in [9]{[11] and a string picture was obtained there as well. In this short note we would like to revise and clarify certain arguments from [9]{ [11] in light of the new motivation. Unlike [7, 8] we do not start from the gauge xed model. The ultimate goal is to relate two di erent theories (zero tension D- brane theory and string theory) to each other. These two theories have di erent gauge symmetries and are unrelated a priori. We study the algebra of constraints of D-brane theory at the tachyonic vacuum in detail. It turns out that fundamental string-like Hamiltonian constraints generate a subalgebra (in fact an ideal) of the full 1 JHEP03(2001)014 algebra of the model at the tachyonic vacuum and that there is natural embedding of this subalgebra provided by the \electric flux". We hope that the present analysis will clarify the general situation and as well as explore the relation between the static gauge results [7, 8] and the general situation. Our formalism is Poincarecovariant, i.e. we avoid the gauge xing which breaks the Poincare invariance (i.e static gauge). In addition we study the lagrangian which describes the dynamics of D-brane at the tachyonic vacuum. In the set of solutions of the classical equations of the gauge xed model we identify the subset of solutions that correspond to string theory solu- tions in conformal gauge. We argue that the model might be consistently quantized if we regard the electric and magnetic elds as a type of background elds. At the tachyonic vacuum the RR background elds decouple completely from the dynamics since the Wess-Zumino couplings are proportional to the derivative of the tachyon eld. This is natural since one expects that the tachyonic vacuum is equivalent to the closed string vacuum without D-branes. However the fluctuations around the vacuum should describe the D-branes of lower dimension. Therefore it is natural to study these fluctuation within the present framework. At the end of the paper we thus briefly discuss small fluctuation around the tachyonic vacuum. Thepaper isorganizedasfollows: insection2westudy thealgebraofconstraints within the hamiltonian formalism. Section 3 is devoted to the lagrangian treatment of the model and to the classical solutions of the gauge xed theory. In the section 4 the role of the background antisymmetric tensor elds and small fluctuations around the vacuum are discussed. In the last section 5 we summarize the results and propose some future research. 2. Hamiltonian treatment of BI theory In this section we study the algebra of constraints in detail and nd that the string- like constraints generate an ideal inside the full algebra. Let us start by considering the e ective D-braneaction [3] with constant tachyon eld T p+1 S = T V(T) d x −det(γ +2 F ); (2.1) where γ = @ X @ X G is the pullback of the space-time metric, F = @ A [ ] is the eld strength for the U(1) gauge eld A and V(T) is a tachyon potential. For the moment we ignore the antisymmetric background tensor elds (NS two-form and RR forms). In what follows we assume that the D-brane is closed (i.e. that the appropriate periodicity conditions on the elds are imposed) or that there is appropriate fall o of the eld at the spatial in nity. This assumption is needed to avoid possible boundary terms. 2 JHEP03(2001)014 Since (2.1) is a generally covariant system the naive hamiltonian vanishes. The constraints can be straightforwardly derived from the action (2.1) [9] H = P @ X + F ; (2.2) a  a ab a b 2 2 0 H = P G P +  γ  +T (V(T)) det(γ +2 F ); (2.3) ab ab ab 0 2 (2 ) G = @ ; =0; (2.4) a 0 where we use lower case latin letters for the spatial indices. There are (p +3) constraints as there should be corresponding to the (p + 1) di eomorphisms and U(1) symmetries. It is convenient to smear the constraints with test functions Z Z a p a p H [N ]= d xN (x)H (x); H[M]= d xM(x)H(x); a a G[] = d x(x)G(x) (2.5) where N is a p-dimensional vector,  is scalar and M is scalar density of weight minus one. M has this weight because the constraint (2.3) transforms as a scalar density of weight two and we would like to write the Hamiltonian as a sum of the constraints smeared by test functions. In analogy with general relativity we call N and M shift vector and lapse function, respectively. As usual for gauge theories we canidentifywiththezerocomponentofthegaugevectorpotential. Theconstraints obey an algebra whose non-zero brackets are a b a a b fH [N ];H [M ]g = H [L M ]+G[N M F ]; (2.6) a b a ~ ba a a b fH [N ];H[M]g = H[L M]+G[2M γ N ]; (2.7) a ~ ab a b (ab) fH[N];H[M]g = H [4(  +A )(N@ M −M@ N)]; (2.8) a b b with the following notation ab 2 2 aa :::a bb :::b 1 p−1 1 p−1 A = T V(T)   (γ +F ):::(γ +F ) (2.9) a b a b a b a b p 1 1 1 1 p−1 p−1 p−1 p−1 (p−1)! (ab) ab ba 0 and A =(1=2)(A +A ). For the time being we drop a factor (2 )toavoid cluttering the formulae. This factor can be easily restored in the nal expressions. The full Hamiltonian is given by H = H [N ]+H[M]+G[]; (2.10) and provides the time evolution of the elds. The algebra (2.6){(2.8) is not closed and has eld dependent structure constants. As far as the authors are aware the classical BRST charge for this system has not been constructed. By analogy to the 3 JHEP03(2001)014 p-brane case [12] one may expect that the BRST charge will have quite a high rank (maybe p as for p-branes [12]). Therefore it seems dicult to quantize this model from rst principles. If the tachyon eld is frozen at its minimum T = T (V(T ) = 0) the constraints 0 0 are reduced to the following set H = P @ X + F ; (2.11) a  a ab and a b H = P G P +  γ  ; (2.12) ab 0 2 (2 ) together with Gauss' law @  =0 and  = 0. The constraints (2.11), (2.12) obey a 0 the following algebra a b a a b fH [N ];H [M ]g = H [L M ]+G[N M F ]; (2.13) a b a ~ ba a a b fH [N ];H[M]g = H[L M]+G[2M γ N ]; (2.14) a ~ ab a b fH[N];H[M]g = H [4  (N@ M −M@ N)]: (2.15) a b b The algebra (2.13){(2.15) is similar to that of the full theory (2.6){(2.8). In fact at the tachyonic vacuum the system has the gauge symmetries of the full theory. For the usual p-branes (i.e. with the gauge eld in (2.1) equal to zero) this is not the case. The zero tension algebra for a p-brane has a completely di erent form from that of the full algebra. There is one important di erence in the eld dependent structure constants of (2.15) and (2.8) which plays a key role in nding the string like subalgebra (2.21){ (2.23). Explicitly the right hand side of (2.15) is a b p a b  c H [4  (N@ M −M@ N)] = d x4  (N@ M −M@ N)(P @ X + F ); a b b b b  a ac (2.16) where the last term vanishes identically because of the antisymmetry of F .Thus ab a  a the right hand side of (2.16) leads to P  @ X =0 showing that  @ X =0 is a a a \preferred" direction on the world-volume. As a direct result of (2.11), (2.12) there is not much dynamics for the U(1) degrees of freedom. For instance, the momenta  satisfy the following equations a a a @  =0; _ =L  ; (2.17) a ~ Even though we call the BI theory the e ective theory it is still an unsettled question what is more fundamental, strings or D-branes. Thus a possible consistent quantization of the BI theory is an important issue. 2 0 This is not the case in the !1 limit, where the gauge algebra has a di erent form. 4 JHEP03(2001)014 and thus  completely decouples from the other elds, except from the lagrangian multiplier, N, (the shift vector). Because of (2.16) one may decompose the con- straints (2.11), (2.12) in the following fashion a   a b H = P  @ X ;H = P G P +  γ  ; (2.18) a   ab 0 2 (2 ) the other (p−1) generators being ? a p a  b ~ ~ H [N ]= d xN (P @ X + F ); (2.19) a ab which generate the general coordinate transformations with parameter N (see ap- pendix A). The decomposition (2.18)-(2.19) may be thought of as a decomposition a a of the shift vector N along directions \parallel" and \orthogonal" to a a a N =N +N ; (2.20) where N transforms as density of weight minus one. To really implement the concept \parallel" and \orthogonal" involves a metric, e.g., (the spatial part of) the induced metric, and leads to unwanted additional constraints. All we need is for N and N to be linearly independent. The details of this decomposition are not essential can in what follows, however. The appropriate decomposition of the constraint H always be done locally. At the global level (2.20) may imply restrictions on the topology of the D-brane world-volume. In the following discussion we will disregard this potential complication. For the given  we decompose the full set constrains as in (2.18), (2.19). The algebra of (2.18) is given by fH [N];H [M]g = H [ (N@ M −M@ N)]; (2.21) a a a a fH [N];H[M]g = H[4 (N@ M −M@ N)+NM(@  )]; (2.22) a a a [4 (N@ M −M@ N)]: (2.23) fH[N];H[M]g = H a a This subalgebra closely resembles the algebra of constraints of the Nambu-Goto string. We have thus found a non trivial embedding of one algebra into another with eld dependent structure constants. Guided by this we introduce the following constraints p a   b [N]=H[N]2H [N]= d xN(P G  @ X )G (P G  @ X ); a   b (2.24) which are analogs of Virasoro constraints. In terms of the new constraints the alge- bra (2.13){(2.15) becomes + + + a fQ [N];Q [M]g = Q [8 (N@ M −M@ N)]; (2.25) a a 5 JHEP03(2001)014 − − − a fQ [N];Q [M]g = Q [8 (N@ M −M@ N)]; (2.26) a a + − + a − a fQ [N];Q [M]g = Q [NM@  ]+Q [NM@  ]; (2.27) a a ? a   a b ~ ~ [N ];Q [M]g = Q [L M]+G[2N  Mγ ]; (2.28) fH ~ ab a ~ ? a ? b ? a a b ~ ~ ~ ~ ~ fH [N ];H [M ]g = H [L M ]+G[N M F ]: (2.29) ba a b a This algebra is thus exactly the same as (2.13){(2.15) but written in a di erent form. The relation (2.27) can also be written as + −  a b fQ [N];Q [M]g=G[2NM(P G P + γ  )]: (2.30) ab The algebra (2.25)-(2.29) follows straightforwardly from the previous calculations. The only relation which needs checking is fH [N ];H [M]g = H [L M]; (2.31) a   ~ where there are no restrictions on N . The algebra (2.25){(2.29) contains the Vira- soro like generators Q which together with Gauss law G generate an ideal of the full algebra. We know of no other nontrivial ideal of a gravity algebra with eld de- pendent structure constants. It is unclear how this ideal is manifested in the BRST charge and other gauge theory quantities, but the existence of a nontrivial ideal may perhaps throw some light on the relation between theories with di erent gauge symmetries. We hope to return to this question elsewhere. The algebra (2.25){(2.29) is not closed and has eld dependent structure con- stants. However since  decouples (see (2.17)) it is tempting to assume that Gauss' law holds strongly; @  = 0. Thus one may regard it as an \background eld". a a Thus, considering a de nite eld con guration  with @  = 0 we may study the behavior of the system with this given \electric flux"  . Introducing a mode a  ? expansion for  and a mode expansion of the constraints Q , H ~ ~ ~ a a −iN~x   −iM~x ? −iN~x =  e ;L =Q e ;H = H [e ] (2.32) ~ a ~ a;N we get the following classical algebra + + a + fL ;L g = i  (N −M )L ; (2.33) a a ~ ~ ~ ~ ~ N M N+M+S − − a − fL ;L g = i  (N −M )L ; (2.34) a a ~ ~ ~ ~ ~ N M N+M+S + − fL ;L g=0; (2.35) ~ ~ N M fH ;L g = i(N −M )L ; (2.36) ~ a a a;N ~ ~ ~ M N+M fH ;H g = iN H −iM H : (2.37) ~ ~ b ~ ~ a ~ ~ a;N b;M a;N+M b;N+M + − and L is an ideal of the whole gauge We see that the subalgebra generated by L ~ ~ N N algebra. 6 JHEP03(2001)014 The subalgebra (2.33) (as well as (2.34)) can be thought of as generalizations of the Virasoro algebra. To illustrate this let us choose  to be constant and thus the subalgebra (2.33) to be + + a + fL ;L g= i (N −M )L : (2.38) a a ~ ~ ~ ~ N M N+M Generically this algebra contains p copies of the standard Virasoro algebra + + 1 + fL ;L g = i (n−m)L ; (n;0;0;:::;0) (m;0;0;:::;0) (n+m;0;0;:::;0) + + 2 + fL ;L g = i (n−m)L ; (0;n;0;:::;0) (0;m;0;:::;0) (0;n+m;0;:::;0) ::: (2.39) can be absorbed into a rede nition of the generators to bring these to the standard Virasoro algebra form. The embedding of the Virasoro algebras depends on the a p relative orientation of  in R . Thus at the level of the classical gauge algebra we see that there is a string sector of the D-brane at the tachyonic vacuum. In the quantum theory the algebra (2.33){(2.37) would have a central extension which should be related to the \electric flux"  . Thus a consistent quantization of the system may impose restrictions on the allowed \electric fluxes". Since (2.33){ (2.37) is a standard Lie algebra the classical BRST charge can be constructed and it will have rank one. Therefore in principle one may quantize the system. However, the relation of the BRST charge for (2.33){(2.37) to the full BRST charge of the system (2.25){(2.29) remains to be determined. In the next section we propose another way of quantizing the system where an algebra similar to (2.33){(2.37) appears as the algebra of residual symmetries of the model, after a partial gauge- xing. 3. Lagrangian analysis In this section we would like to review the problem fromthe lagrangianpoint of view. We learn that all, as is usually the case, results can be obtained from the lagrangian approach without any direct reference to Hamiltonian analysis. The lagrangian (2.1) is not suited for freezing the tachyon eld to its minimum (or taking the zero tension limit). Thus the natural approach is to rewrite the lagrangian (2.1) in a di erent but classically equivalent form which is appropriate for the limit in. Following calculation in [9] one constructs the following equivalent action for the model p 0 2 2 S = d x (E E −E E )γ +2 E E F − T (V(T)) 1 1 2 2 [2 p 1] 0 (2E ) det(γ +2 F ) : (3.1) ab ab 7 JHEP03(2001)014 Eliminating E andE gives backthe\classical"BIaction(2.1). TostudyaD-brane 1 2 at the tachyonic vacuum we drop the last term in (3.1) p 0 S = d x[(E E −E E )γ +4 @ (E E )A ]: (3.2) 1 1 2 2 [1 2] This action corresponds to the Hamiltonian constraints (2.11), (2.12) (and may be derived from them). The action (3.2) gives rise the following equations of motion @ (E E )=0; (3.3) [1 2] γ E −2 F E =0; (3.4) 1 2 γ E −2 F E =0; (3.5) 2 1 @ [(E E −E E )@ X ]=0; (3.6) 1 1 2 2 where forthesake ofsimplicity we use a flat space-time metric G =  (The gener- alization to a general metric is straightforward). In the gaugeE =  equation (3.3) 1 0 reduces to a a @ E =0;@ E =0: (3.7) a 0 2 2 a 0 a From the action (3.2), the canonical momentum  conjugated to A is −2 E and therefore there is a constraint E = 0. In the present gauge the 2p equa- tions (3.4), (3.5) reduce to p independent equations (constraints) 0 b a b γ −2 F E =0;γ +E γ E =0: (3.8) 0a ab 00 ab 2 2 2 These constraints correspond to residual symmetries left after gauge xing. Also 0 a from the action (3.2) the canonical momentum conjugated to A is −2 E (i.e. ) and that the canonical momentum conjugated to X is X (i.e. P ). Thus the constraints (3.8) coincide with those we have discussed previously (up to factor a 2 ). The equation (3.6) becomes 2  a b @ X −E @ E @ X =0: (3.9) a b 0 2 2 Now we can analyze the solutions of the equations of motion in the given gauge. We see that there are no dynamical equations of motion for the \electric" E and magnetic F elds. There are only a Gauss' law for E and Bianchi identities ab for F . As a rst example, let us take E =(E;0;:::;0) with E constant and ab F = 0 and make the following ansatz ¨ for the solution X (x ;x ;x ;:::;x )= ab 0 1 2 p Y (x ;x )f(x ;:::;x ). Because of (3.8) and (3.9) we see that Y 's satisfy the 0 1 2 p following equations 0  2 0 0 _ _ _ Y Y =0; Y Y +E Y Y =0; (@ −E@ )(@ +E@ )Y =0; (3.10) 0 1 0 1 Modulo the linear dependent equations and F . 0a 8 JHEP03(2001)014 where Y  @ Y and Y  @ Y . The function f should satisfy the following 0 1 (p−1) equations Y )(f@ f)=0;a=2;:::;p: (3.11) (Y As a result of (3.10) Y can be interpreted as a string solution in the conformal gauge, with tension jEj. The equations (3.11) are solved by requiring f = const. Thus the string solutions are completely delocalized in the world-volume coordinates ;:::;x ). In other words, the solution corresponds to a set of strings distributed (x 2 p uniformly in the \transverse directions" (x ;:::;x ). 2 p If we now take the same \electric" eld E as before but a magnetic eld F ab di erent from zero, then the equation (3.10) stays the same while equation (3.11) gets modi ed to (Y Y )(f@ f)=2 EF ;a=2;:::;p: (3.12) a a1 This equation is equivalent to 4 E @ (f )= dx F ; (3.13) a 1 a1 rv 2 1 where we assume that x 2 [0;]andr  dx Y Y , v  r_. Writing F = @ A 1 1  a1 [a 1] one may solve this equation explicitly. The solution f of this equation will de ne how the string world-sheets are distributed in the directions (x ;:::;x ). 2 p So far we have discussed solutions which have an interpretation as a collection of strings lling the world-volume of the brane. One can also construct solutions which correspond to a single string world-sheet which is completely localized in the trans- verse directions (e.g. f = (x ):::(x ) in the previous example). However these 2 p solutions are singular and require highly singular electric or magnetic con gurations which may be problematic from a computational point of view. These solutions are limiting cases of regular solutions of the theory (in contemporary parlance, they correspond to the boundary points of the moduli space of solutions.). The analysis can be extended along similar lines for other con gurations of E and F . As result one sees that any classical solution of string theory (Polyakov's ab action) in conformal gauge can be naturally embedded into the present theory in the given gauge. The details of the embedding are governed by E and F .In ab a di erent setup (static gauge) similar results were obtained by Sen [8]. However not all solutions of the classical equations of motions are string-like excitations. For instance, assuming that X is independent of x (for the case E =(E;0;:::;0) and F = 0) we nd a gauge xed tensionless (p−1)-brane solution, other ansatz ¨ es give ab point particle solutions etc. In quantizing the system one may adopt the same approach as in the previous classical consideration, i.e. treat the U(1) degrees of freedom as background elds. 9 JHEP03(2001)014 Thus one may choose speci c con gurations of E and F satisfying Gauss' law ab and the Bianchi identities and then quantize the system considering only the X as quantum excitations. Within this semiclassical treatment the positive modes of the (p+1) constraints (3.8) should be imposed on the physical states. These constraints generate a closed algebra similar to (2.33){(2.37) but not the same (special care is needed for (2.36)). The algebra is di erent because E and A are not regarded as a quantum canonical pair of operators, they are just xed classical background elds. It is not to be expected that this way of quantizing is reliable in all regimes of the theory. However it might give a rst insight into the theory and technically it is straightforward to carry out since the algebra of constraints is a Lie algebra. The case of E =0, F = 0 corresponds to a tensionless p-brane and using the BRST ab approach this has been quantized previously [13] (no restrictions such as critical dimensions were found). The next natural generalization is the case of a non-zero electric eld E and zero magnetic eld F . This case should be non-trivial since ab the Virasoro algebra is contained in the full algebra of constraints. 4. Background elds In this section we would like to consider two related questions: the vacuum fluctua- tions and the e ects of antisymmetric background elds. Let us begin with a comment on the role of the B- eld in the tachyonic vacuum. The e ect of the B- eld comes from the replacement of F by F = @ A + [ ] @ X B @ X in (2.1). In the Hamiltonian formalism this results in a rede nition of the momenta P P ! P −B  @ X : (4.1) With this replacement all expressions in Section 2 are still correct. The new string- like constraints Q correspond to string constraints in a non-trivial B- eld back- ground. Thus the e ect of the B- eld is rather trivial. Now let us turn to the RR elds. When the tachyon is frozen at the vacuum all RR background eld decouple completely from the theory. In general the tachyon T is a world-volume degree of freedom and has a corresponding kinetic term. Proposals forthee ectiveactionincludingatachyonkinetictermhavebeenputforwardin[14]{ [16]. However in the present discussion (and as is often the practice) we ignore the dynamics of the tachyon eld itself and consider T as a background eld. Hence the action has the form Z Z p+1 2 F 2 S = T d xV(T) −det(γ +2 F )+ C^dT ^e +O((@T) ); p p (4.2) 10 JHEP03(2001)014 where C is the sum of RR forms. Assuming the expansion of the tachyon potential around the vacuum 00 2 3 V(T)=V(T )+ V (T )(T −T ) +O(T ) (4.3) 0 0 0 we rewrite the action as Z Z p 0 2 F 2 2 S= d x[(E E −E E )γ +2 E E F ]+ C^dT^e +O((@T) ;T ): 1 1 2 2 [2 1] (4.4) We may regard T asa static con gurationinterpolating fromone vacuum to another, like a kink or a vertex. The point is that the energy density of these con gurations is localizedinsomeofthespatialworld-volume coordinates. Inanextreme situation dT is a -function along those directions. Thus in the approximation used the e ect of the Wess-Zumino term is to insert  sources on the right hand side of (3.3) and (3.6). InthegaugeE =  theequation(3.8)stays thesame (thustheVirasorosubalgebra 2 0 is still present). Apart from those sources all the discussion in the previous section goes through. The presence of -source on the right hand side of (3.7) will allow string-like solutions to end where the -source sits (since it is a source for the flux @ E = j ). There are thus open strings which can end on \planes" localized in some a a of the spatial world-volume coordinates, i.e. on lower dimensional D-branes. 2 2 It is far from obvious that one can drop terms of order O((@T) ;T )whendis- cussing fluctuations around the vacuum. Most likely that one cannot freely do so. Nevertheless the above qualitative picture is quite reasonable and agrees with expec- tations. To study the fluctuations around the vacuum more carefully the tachyon eld should be treated as dynamical. 5. Discussion In the present note we have analyzed the classical e ective theory of D-branes at the tachyonic vacuum. We have established the following two facts: the string gauge algebra (the Virasoro algebra) is a subalgebra of the D-brane theory at the tachy- onic vacuum and the classical string solutions is a subset of the D-brane solutions in this regime. Thus the Nambu-Goto strings can be embedded into D-branes at the tachyonic vacuum. However, it is not clear to what extent the string picture suces to describe the D-branes in this regime. In the similar situation when !1 in the fundamental string the quantum theory describes (collection of) massless par- ticles [17]. By analogy one might expect the quantum theory here to describe (a collection of) strings. In fact, the analogy goes even further than indicated, since constraint algebra of tensionless string also contains an ideal (P G P =0). 4 −T The expansion (4.3) is not always valid, for instance not for V(T)=(T+1)e around T = 1. However in this case the argument still goes through, since as a result of (3.1) the correction to the 2 2 expansion (4.4) is O((@T) ;(V(T)) ) (i.e. exponentially small around T = 1). 11 JHEP03(2001)014 Wehave emphasized thatthereareproblemswithquantizing (3.2), butthestudy of the gauge algebra leads to a suggestion for quantizing the system treating the U(1) degrees of freedom as background degrees of freedom. Acknowledgments MZ is grateful to Gary Gibbons for valuable discussion. We thank Inegemar Bengts- son for the comments on the manuscript. The work of UL was supported in part by NFR grant 650-1998368 and by EU contract HPRN-CT-2000-0122. A. Basic de nitions The Lie derivative of arbitrary tensor density of weight n in the direction of the eld N , is de ned as b b :::b c b b :::b c b b :::b 1 2 l 1 2 l 1 2 l L T = N @ T +@ N T +:::− c a N a a :::a a a :::a 1 ca :::a 1 2 k 1 2 k 2 k j cb :::b c b b :::b 1 2 l 1 2 l −@ N T −+n@ N T : (A.1) c c a a :::a a a :::a 1 2 k 1 2 k We are using the following basic Poisson brackets b b (p)   (p) fA (x); (y)g=  (x−y); fX (x);P (y)g=  (x−y): (A.2) There is the following action of momentum constraint on the elds a  a fX ;H [N ]g = L X ; fP ;H [N ]g = L P ; a ~  a ~ N N (A.3) c a c a a c fF ;H [N ]g = L F ; f ;H [N ]g = L  −N @  : ab c ~ ab c ~ c N N References [1] K. Bardakci, Dual Models and Spontaneous Symmetry Breaking, Nucl. Phys. B68 (1974) 331; K. Bardakci and M.B. Halpern, Explicit spontaneous breakdown in a dual model, Phys. Rev. D10 (1974) 4230; K. Bardakci and M.B. Halpern, Explicit spontaneous breakdown in a dual model II: N point functions, Nucl. Phys. B96 (1975) 285; K. Bardakci, Spontaneous symmetry breakdown in the standard dual string model, Nucl. Phys. B 133 (1978) 297. [2] A. Sen, Descent relations among bosonic D-branes,Int.J.Mod.Phys. A14 (1999) 4061 [hep-th/9902105]. [3] A. Sen, Supersymmetric world-volume action for non-BPS D-branes, J. High Energy Phys. 10 (1999) 008 [hep-th/9909062]. 12 JHEP03(2001)014 [4] P. Yi, Membranes from ve-branes and fundamental strings from Dp-branes, Nucl. Phys. B 550 (1999) 214 [hep-th/9901159]. [5] O. Bergman, K. Hori and P. Yi, Con nement on the brane, Nucl. Phys. B 580 (2000) 289 [hep-th/0002223]. [6] J.A. Harvey, P. Kraus, F. Larsen and E.J. Martinec, D-branes and strings as non- commutative solitons, J. High Energy Phys. 07 (2000) 042 [hep-th/0005031]. [7] G.Gibbons,K.HoriandP.Yi, String fluid from unstable D-branes, Nucl. Phys. B 596 (2001) 136 [hep-th/0009061]. [8] A. Sen, Fundamental strings in open string theory at the tachyonic vacuum, hep-th/0010240. [9] U. Lindstrom and R. von Unge, A picture of D-branes at strong coupling, Phys. Lett. B 403 (1997) 233 [hep-th/9704051]. [10] H. Gustafsson and U. Lindstrom, A picture of D-branes at strong coupling II: spin- ning partons, Phys. Lett. B 440 (1998) 43 [hep-th/9807064]. [11] U. Lindstrom, M. Zabzine and A. Zheltukhin, Limits of the D-brane action, J. High Energy Phys. 12 (1999) 016 [hep-th/9910159]. [12] M. Henneaux, Transition amplitude in the quantum theory of the relativistic mem- brane, Phys. Lett. B 120 (1983) 179. [13] P. Saltsidis, Tensionless p-branes with manifest conformal invariance, Phys. Lett. B 401 (1997) 21 [hep-th/9702081]. [14] M.R. Garousi, Tachyon couplings on non-BPS D-branes and Dirac-Born-Infeld ac- tion, Nucl. Phys. B 584 (2000) 284 [hep-th/0003122]. [15] E.A. Bergshoe , M. de Roo, T.C. de Wit, E. Eyras and S. Panda, T-duality and ac- tions for non-BPS D-branes, J. High Energy Phys. 05(2000) 009[hep-th/0003221]. [16] J. Kluson, Proposal for non-BPS D-brane action, Phys. Rev. D62 (2000) 126003 [hep-th/0004106]. [17] A. Karlhede and U. Lindstr¨om, The Classical Bosonic String In The Zero Tension Limit, Class. and Quant. Grav. 3 (1986) 73; J. Isberg, U. Lindstrom and B. Sundborg, Space-time symmetries of quantized ten- sionless strings, Phys. Lett. B 293 (1992) 321 [hep-th/9207005]. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of High Energy Physics IOP Publishing

Strings at the tachyonic vacuum

Lindström, Ulf; Zabzine, Maxim
Journal of High Energy Physics , Volume 2001 (03) – Mar 19, 2001
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JHEP03(2001)014 Received: February 2, 2001, Accepted: March 8, 2001 HYPER VERSION Revised: February 20, 2001 a ab Ulf Lindstr¨om and Maxim Zabzine Institute of Theoretical Physics, University of Stockholm Box 6730, S-113 85 Stockholm Sweden Institute Henri Poincare 11, rue P. et M. Curie, 75231 Paris Cedex 05, France E-mail: [email protected], [email protected] Abstract: We study the world-volume e ective action of Dp-brane at the tachy- onic vacuum which is equivalent to the zero tension limit. Using the Hamiltonian formalism we discuss the algebra of constraints and show that there is a non-trivial ideal of the algebra which corresponds to Virasoro like constraints. The Lagrangian treatment of the model is also considered. For the gauge xed theory we construct the important subset of classical solutions which is equivalent to the string theory solutions in conformal gauge. We speculate on a possible quantization of the system. At the end a brief discussion of di erent background elds and fluctuations around the tachyonic vacuum is presented. Keywords: Bosonic Strings, D-branes. JHEP03(2001)014 Contents 1. Introduction and motivation 1 2. Hamiltonian treatment of BI theory 2 3. Lagrangian analysis 7 4. Background elds 10 5. Discussion 11 A. Basic de nitions 12 1. Introduction and motivation The problem of tachyon condensation is an old subject in the context of the string theory[1]. Recently ithasbeenconjectured thatthetachyonic vacuuminopenstring theory on the D-brane describes the closed string vacuum without D-branes and that various soliton solutions of the theory describe D-branes of lower dimension [2]. This conjecture has been supported by number calculations within the rst and second quantized string theory. Onecanget someinsight intotheproblem byconsidering theworld-volume e ec- tive action [3] which describes the D-brane around the tachyonic vacuum. Recently there has been some e ort directed towards identifying of string-like classical solu- tions whose tension matches that of fundamental string [4]{[8]. In fact the D-brane action at the tachyonic vacuum is equivalent to the zero tension limit of the D-brane action. The zero tension limit was with a di erent motivation studied previously in [9]{[11] and a string picture was obtained there as well. In this short note we would like to revise and clarify certain arguments from [9]{ [11] in light of the new motivation. Unlike [7, 8] we do not start from the gauge xed model. The ultimate goal is to relate two di erent theories (zero tension D- brane theory and string theory) to each other. These two theories have di erent gauge symmetries and are unrelated a priori. We study the algebra of constraints of D-brane theory at the tachyonic vacuum in detail. It turns out that fundamental string-like Hamiltonian constraints generate a subalgebra (in fact an ideal) of the full 1 JHEP03(2001)014 algebra of the model at the tachyonic vacuum and that there is natural embedding of this subalgebra provided by the \electric flux". We hope that the present analysis will clarify the general situation and as well as explore the relation between the static gauge results [7, 8] and the general situation. Our formalism is Poincarecovariant, i.e. we avoid the gauge xing which breaks the Poincare invariance (i.e static gauge). In addition we study the lagrangian which describes the dynamics of D-brane at the tachyonic vacuum. In the set of solutions of the classical equations of the gauge xed model we identify the subset of solutions that correspond to string theory solu- tions in conformal gauge. We argue that the model might be consistently quantized if we regard the electric and magnetic elds as a type of background elds. At the tachyonic vacuum the RR background elds decouple completely from the dynamics since the Wess-Zumino couplings are proportional to the derivative of the tachyon eld. This is natural since one expects that the tachyonic vacuum is equivalent to the closed string vacuum without D-branes. However the fluctuations around the vacuum should describe the D-branes of lower dimension. Therefore it is natural to study these fluctuation within the present framework. At the end of the paper we thus briefly discuss small fluctuation around the tachyonic vacuum. Thepaper isorganizedasfollows: insection2westudy thealgebraofconstraints within the hamiltonian formalism. Section 3 is devoted to the lagrangian treatment of the model and to the classical solutions of the gauge xed theory. In the section 4 the role of the background antisymmetric tensor elds and small fluctuations around the vacuum are discussed. In the last section 5 we summarize the results and propose some future research. 2. Hamiltonian treatment of BI theory In this section we study the algebra of constraints in detail and nd that the string- like constraints generate an ideal inside the full algebra. Let us start by considering the e ective D-braneaction [3] with constant tachyon eld T p+1 S = T V(T) d x −det(γ +2 F ); (2.1) where γ = @ X @ X G is the pullback of the space-time metric, F = @ A [ ] is the eld strength for the U(1) gauge eld A and V(T) is a tachyon potential. For the moment we ignore the antisymmetric background tensor elds (NS two-form and RR forms). In what follows we assume that the D-brane is closed (i.e. that the appropriate periodicity conditions on the elds are imposed) or that there is appropriate fall o of the eld at the spatial in nity. This assumption is needed to avoid possible boundary terms. 2 JHEP03(2001)014 Since (2.1) is a generally covariant system the naive hamiltonian vanishes. The constraints can be straightforwardly derived from the action (2.1) [9] H = P @ X + F ; (2.2) a  a ab a b 2 2 0 H = P G P +  γ  +T (V(T)) det(γ +2 F ); (2.3) ab ab ab 0 2 (2 ) G = @ ; =0; (2.4) a 0 where we use lower case latin letters for the spatial indices. There are (p +3) constraints as there should be corresponding to the (p + 1) di eomorphisms and U(1) symmetries. It is convenient to smear the constraints with test functions Z Z a p a p H [N ]= d xN (x)H (x); H[M]= d xM(x)H(x); a a G[] = d x(x)G(x) (2.5) where N is a p-dimensional vector,  is scalar and M is scalar density of weight minus one. M has this weight because the constraint (2.3) transforms as a scalar density of weight two and we would like to write the Hamiltonian as a sum of the constraints smeared by test functions. In analogy with general relativity we call N and M shift vector and lapse function, respectively. As usual for gauge theories we canidentifywiththezerocomponentofthegaugevectorpotential. Theconstraints obey an algebra whose non-zero brackets are a b a a b fH [N ];H [M ]g = H [L M ]+G[N M F ]; (2.6) a b a ~ ba a a b fH [N ];H[M]g = H[L M]+G[2M γ N ]; (2.7) a ~ ab a b (ab) fH[N];H[M]g = H [4(  +A )(N@ M −M@ N)]; (2.8) a b b with the following notation ab 2 2 aa :::a bb :::b 1 p−1 1 p−1 A = T V(T)   (γ +F ):::(γ +F ) (2.9) a b a b a b a b p 1 1 1 1 p−1 p−1 p−1 p−1 (p−1)! (ab) ab ba 0 and A =(1=2)(A +A ). For the time being we drop a factor (2 )toavoid cluttering the formulae. This factor can be easily restored in the nal expressions. The full Hamiltonian is given by H = H [N ]+H[M]+G[]; (2.10) and provides the time evolution of the elds. The algebra (2.6){(2.8) is not closed and has eld dependent structure constants. As far as the authors are aware the classical BRST charge for this system has not been constructed. By analogy to the 3 JHEP03(2001)014 p-brane case [12] one may expect that the BRST charge will have quite a high rank (maybe p as for p-branes [12]). Therefore it seems dicult to quantize this model from rst principles. If the tachyon eld is frozen at its minimum T = T (V(T ) = 0) the constraints 0 0 are reduced to the following set H = P @ X + F ; (2.11) a  a ab and a b H = P G P +  γ  ; (2.12) ab 0 2 (2 ) together with Gauss' law @  =0 and  = 0. The constraints (2.11), (2.12) obey a 0 the following algebra a b a a b fH [N ];H [M ]g = H [L M ]+G[N M F ]; (2.13) a b a ~ ba a a b fH [N ];H[M]g = H[L M]+G[2M γ N ]; (2.14) a ~ ab a b fH[N];H[M]g = H [4  (N@ M −M@ N)]: (2.15) a b b The algebra (2.13){(2.15) is similar to that of the full theory (2.6){(2.8). In fact at the tachyonic vacuum the system has the gauge symmetries of the full theory. For the usual p-branes (i.e. with the gauge eld in (2.1) equal to zero) this is not the case. The zero tension algebra for a p-brane has a completely di erent form from that of the full algebra. There is one important di erence in the eld dependent structure constants of (2.15) and (2.8) which plays a key role in nding the string like subalgebra (2.21){ (2.23). Explicitly the right hand side of (2.15) is a b p a b  c H [4  (N@ M −M@ N)] = d x4  (N@ M −M@ N)(P @ X + F ); a b b b b  a ac (2.16) where the last term vanishes identically because of the antisymmetry of F .Thus ab a  a the right hand side of (2.16) leads to P  @ X =0 showing that  @ X =0 is a a a \preferred" direction on the world-volume. As a direct result of (2.11), (2.12) there is not much dynamics for the U(1) degrees of freedom. For instance, the momenta  satisfy the following equations a a a @  =0; _ =L  ; (2.17) a ~ Even though we call the BI theory the e ective theory it is still an unsettled question what is more fundamental, strings or D-branes. Thus a possible consistent quantization of the BI theory is an important issue. 2 0 This is not the case in the !1 limit, where the gauge algebra has a di erent form. 4 JHEP03(2001)014 and thus  completely decouples from the other elds, except from the lagrangian multiplier, N, (the shift vector). Because of (2.16) one may decompose the con- straints (2.11), (2.12) in the following fashion a   a b H = P  @ X ;H = P G P +  γ  ; (2.18) a   ab 0 2 (2 ) the other (p−1) generators being ? a p a  b ~ ~ H [N ]= d xN (P @ X + F ); (2.19) a ab which generate the general coordinate transformations with parameter N (see ap- pendix A). The decomposition (2.18)-(2.19) may be thought of as a decomposition a a of the shift vector N along directions \parallel" and \orthogonal" to a a a N =N +N ; (2.20) where N transforms as density of weight minus one. To really implement the concept \parallel" and \orthogonal" involves a metric, e.g., (the spatial part of) the induced metric, and leads to unwanted additional constraints. All we need is for N and N to be linearly independent. The details of this decomposition are not essential can in what follows, however. The appropriate decomposition of the constraint H always be done locally. At the global level (2.20) may imply restrictions on the topology of the D-brane world-volume. In the following discussion we will disregard this potential complication. For the given  we decompose the full set constrains as in (2.18), (2.19). The algebra of (2.18) is given by fH [N];H [M]g = H [ (N@ M −M@ N)]; (2.21) a a a a fH [N];H[M]g = H[4 (N@ M −M@ N)+NM(@  )]; (2.22) a a a [4 (N@ M −M@ N)]: (2.23) fH[N];H[M]g = H a a This subalgebra closely resembles the algebra of constraints of the Nambu-Goto string. We have thus found a non trivial embedding of one algebra into another with eld dependent structure constants. Guided by this we introduce the following constraints p a   b [N]=H[N]2H [N]= d xN(P G  @ X )G (P G  @ X ); a   b (2.24) which are analogs of Virasoro constraints. In terms of the new constraints the alge- bra (2.13){(2.15) becomes + + + a fQ [N];Q [M]g = Q [8 (N@ M −M@ N)]; (2.25) a a 5 JHEP03(2001)014 − − − a fQ [N];Q [M]g = Q [8 (N@ M −M@ N)]; (2.26) a a + − + a − a fQ [N];Q [M]g = Q [NM@  ]+Q [NM@  ]; (2.27) a a ? a   a b ~ ~ [N ];Q [M]g = Q [L M]+G[2N  Mγ ]; (2.28) fH ~ ab a ~ ? a ? b ? a a b ~ ~ ~ ~ ~ fH [N ];H [M ]g = H [L M ]+G[N M F ]: (2.29) ba a b a This algebra is thus exactly the same as (2.13){(2.15) but written in a di erent form. The relation (2.27) can also be written as + −  a b fQ [N];Q [M]g=G[2NM(P G P + γ  )]: (2.30) ab The algebra (2.25)-(2.29) follows straightforwardly from the previous calculations. The only relation which needs checking is fH [N ];H [M]g = H [L M]; (2.31) a   ~ where there are no restrictions on N . The algebra (2.25){(2.29) contains the Vira- soro like generators Q which together with Gauss law G generate an ideal of the full algebra. We know of no other nontrivial ideal of a gravity algebra with eld de- pendent structure constants. It is unclear how this ideal is manifested in the BRST charge and other gauge theory quantities, but the existence of a nontrivial ideal may perhaps throw some light on the relation between theories with di erent gauge symmetries. We hope to return to this question elsewhere. The algebra (2.25){(2.29) is not closed and has eld dependent structure con- stants. However since  decouples (see (2.17)) it is tempting to assume that Gauss' law holds strongly; @  = 0. Thus one may regard it as an \background eld". a a Thus, considering a de nite eld con guration  with @  = 0 we may study the behavior of the system with this given \electric flux"  . Introducing a mode a  ? expansion for  and a mode expansion of the constraints Q , H ~ ~ ~ a a −iN~x   −iM~x ? −iN~x =  e ;L =Q e ;H = H [e ] (2.32) ~ a ~ a;N we get the following classical algebra + + a + fL ;L g = i  (N −M )L ; (2.33) a a ~ ~ ~ ~ ~ N M N+M+S − − a − fL ;L g = i  (N −M )L ; (2.34) a a ~ ~ ~ ~ ~ N M N+M+S + − fL ;L g=0; (2.35) ~ ~ N M fH ;L g = i(N −M )L ; (2.36) ~ a a a;N ~ ~ ~ M N+M fH ;H g = iN H −iM H : (2.37) ~ ~ b ~ ~ a ~ ~ a;N b;M a;N+M b;N+M + − and L is an ideal of the whole gauge We see that the subalgebra generated by L ~ ~ N N algebra. 6 JHEP03(2001)014 The subalgebra (2.33) (as well as (2.34)) can be thought of as generalizations of the Virasoro algebra. To illustrate this let us choose  to be constant and thus the subalgebra (2.33) to be + + a + fL ;L g= i (N −M )L : (2.38) a a ~ ~ ~ ~ N M N+M Generically this algebra contains p copies of the standard Virasoro algebra + + 1 + fL ;L g = i (n−m)L ; (n;0;0;:::;0) (m;0;0;:::;0) (n+m;0;0;:::;0) + + 2 + fL ;L g = i (n−m)L ; (0;n;0;:::;0) (0;m;0;:::;0) (0;n+m;0;:::;0) ::: (2.39) can be absorbed into a rede nition of the generators to bring these to the standard Virasoro algebra form. The embedding of the Virasoro algebras depends on the a p relative orientation of  in R . Thus at the level of the classical gauge algebra we see that there is a string sector of the D-brane at the tachyonic vacuum. In the quantum theory the algebra (2.33){(2.37) would have a central extension which should be related to the \electric flux"  . Thus a consistent quantization of the system may impose restrictions on the allowed \electric fluxes". Since (2.33){ (2.37) is a standard Lie algebra the classical BRST charge can be constructed and it will have rank one. Therefore in principle one may quantize the system. However, the relation of the BRST charge for (2.33){(2.37) to the full BRST charge of the system (2.25){(2.29) remains to be determined. In the next section we propose another way of quantizing the system where an algebra similar to (2.33){(2.37) appears as the algebra of residual symmetries of the model, after a partial gauge- xing. 3. Lagrangian analysis In this section we would like to review the problem fromthe lagrangianpoint of view. We learn that all, as is usually the case, results can be obtained from the lagrangian approach without any direct reference to Hamiltonian analysis. The lagrangian (2.1) is not suited for freezing the tachyon eld to its minimum (or taking the zero tension limit). Thus the natural approach is to rewrite the lagrangian (2.1) in a di erent but classically equivalent form which is appropriate for the limit in. Following calculation in [9] one constructs the following equivalent action for the model p 0 2 2 S = d x (E E −E E )γ +2 E E F − T (V(T)) 1 1 2 2 [2 p 1] 0 (2E ) det(γ +2 F ) : (3.1) ab ab 7 JHEP03(2001)014 Eliminating E andE gives backthe\classical"BIaction(2.1). TostudyaD-brane 1 2 at the tachyonic vacuum we drop the last term in (3.1) p 0 S = d x[(E E −E E )γ +4 @ (E E )A ]: (3.2) 1 1 2 2 [1 2] This action corresponds to the Hamiltonian constraints (2.11), (2.12) (and may be derived from them). The action (3.2) gives rise the following equations of motion @ (E E )=0; (3.3) [1 2] γ E −2 F E =0; (3.4) 1 2 γ E −2 F E =0; (3.5) 2 1 @ [(E E −E E )@ X ]=0; (3.6) 1 1 2 2 where forthesake ofsimplicity we use a flat space-time metric G =  (The gener- alization to a general metric is straightforward). In the gaugeE =  equation (3.3) 1 0 reduces to a a @ E =0;@ E =0: (3.7) a 0 2 2 a 0 a From the action (3.2), the canonical momentum  conjugated to A is −2 E and therefore there is a constraint E = 0. In the present gauge the 2p equa- tions (3.4), (3.5) reduce to p independent equations (constraints) 0 b a b γ −2 F E =0;γ +E γ E =0: (3.8) 0a ab 00 ab 2 2 2 These constraints correspond to residual symmetries left after gauge xing. Also 0 a from the action (3.2) the canonical momentum conjugated to A is −2 E (i.e. ) and that the canonical momentum conjugated to X is X (i.e. P ). Thus the constraints (3.8) coincide with those we have discussed previously (up to factor a 2 ). The equation (3.6) becomes 2  a b @ X −E @ E @ X =0: (3.9) a b 0 2 2 Now we can analyze the solutions of the equations of motion in the given gauge. We see that there are no dynamical equations of motion for the \electric" E and magnetic F elds. There are only a Gauss' law for E and Bianchi identities ab for F . As a rst example, let us take E =(E;0;:::;0) with E constant and ab F = 0 and make the following ansatz ¨ for the solution X (x ;x ;x ;:::;x )= ab 0 1 2 p Y (x ;x )f(x ;:::;x ). Because of (3.8) and (3.9) we see that Y 's satisfy the 0 1 2 p following equations 0  2 0 0 _ _ _ Y Y =0; Y Y +E Y Y =0; (@ −E@ )(@ +E@ )Y =0; (3.10) 0 1 0 1 Modulo the linear dependent equations and F . 0a 8 JHEP03(2001)014 where Y  @ Y and Y  @ Y . The function f should satisfy the following 0 1 (p−1) equations Y )(f@ f)=0;a=2;:::;p: (3.11) (Y As a result of (3.10) Y can be interpreted as a string solution in the conformal gauge, with tension jEj. The equations (3.11) are solved by requiring f = const. Thus the string solutions are completely delocalized in the world-volume coordinates ;:::;x ). In other words, the solution corresponds to a set of strings distributed (x 2 p uniformly in the \transverse directions" (x ;:::;x ). 2 p If we now take the same \electric" eld E as before but a magnetic eld F ab di erent from zero, then the equation (3.10) stays the same while equation (3.11) gets modi ed to (Y Y )(f@ f)=2 EF ;a=2;:::;p: (3.12) a a1 This equation is equivalent to 4 E @ (f )= dx F ; (3.13) a 1 a1 rv 2 1 where we assume that x 2 [0;]andr  dx Y Y , v  r_. Writing F = @ A 1 1  a1 [a 1] one may solve this equation explicitly. The solution f of this equation will de ne how the string world-sheets are distributed in the directions (x ;:::;x ). 2 p So far we have discussed solutions which have an interpretation as a collection of strings lling the world-volume of the brane. One can also construct solutions which correspond to a single string world-sheet which is completely localized in the trans- verse directions (e.g. f = (x ):::(x ) in the previous example). However these 2 p solutions are singular and require highly singular electric or magnetic con gurations which may be problematic from a computational point of view. These solutions are limiting cases of regular solutions of the theory (in contemporary parlance, they correspond to the boundary points of the moduli space of solutions.). The analysis can be extended along similar lines for other con gurations of E and F . As result one sees that any classical solution of string theory (Polyakov's ab action) in conformal gauge can be naturally embedded into the present theory in the given gauge. The details of the embedding are governed by E and F .In ab a di erent setup (static gauge) similar results were obtained by Sen [8]. However not all solutions of the classical equations of motions are string-like excitations. For instance, assuming that X is independent of x (for the case E =(E;0;:::;0) and F = 0) we nd a gauge xed tensionless (p−1)-brane solution, other ansatz ¨ es give ab point particle solutions etc. In quantizing the system one may adopt the same approach as in the previous classical consideration, i.e. treat the U(1) degrees of freedom as background elds. 9 JHEP03(2001)014 Thus one may choose speci c con gurations of E and F satisfying Gauss' law ab and the Bianchi identities and then quantize the system considering only the X as quantum excitations. Within this semiclassical treatment the positive modes of the (p+1) constraints (3.8) should be imposed on the physical states. These constraints generate a closed algebra similar to (2.33){(2.37) but not the same (special care is needed for (2.36)). The algebra is di erent because E and A are not regarded as a quantum canonical pair of operators, they are just xed classical background elds. It is not to be expected that this way of quantizing is reliable in all regimes of the theory. However it might give a rst insight into the theory and technically it is straightforward to carry out since the algebra of constraints is a Lie algebra. The case of E =0, F = 0 corresponds to a tensionless p-brane and using the BRST ab approach this has been quantized previously [13] (no restrictions such as critical dimensions were found). The next natural generalization is the case of a non-zero electric eld E and zero magnetic eld F . This case should be non-trivial since ab the Virasoro algebra is contained in the full algebra of constraints. 4. Background elds In this section we would like to consider two related questions: the vacuum fluctua- tions and the e ects of antisymmetric background elds. Let us begin with a comment on the role of the B- eld in the tachyonic vacuum. The e ect of the B- eld comes from the replacement of F by F = @ A + [ ] @ X B @ X in (2.1). In the Hamiltonian formalism this results in a rede nition of the momenta P P ! P −B  @ X : (4.1) With this replacement all expressions in Section 2 are still correct. The new string- like constraints Q correspond to string constraints in a non-trivial B- eld back- ground. Thus the e ect of the B- eld is rather trivial. Now let us turn to the RR elds. When the tachyon is frozen at the vacuum all RR background eld decouple completely from the theory. In general the tachyon T is a world-volume degree of freedom and has a corresponding kinetic term. Proposals forthee ectiveactionincludingatachyonkinetictermhavebeenputforwardin[14]{ [16]. However in the present discussion (and as is often the practice) we ignore the dynamics of the tachyon eld itself and consider T as a background eld. Hence the action has the form Z Z p+1 2 F 2 S = T d xV(T) −det(γ +2 F )+ C^dT ^e +O((@T) ); p p (4.2) 10 JHEP03(2001)014 where C is the sum of RR forms. Assuming the expansion of the tachyon potential around the vacuum 00 2 3 V(T)=V(T )+ V (T )(T −T ) +O(T ) (4.3) 0 0 0 we rewrite the action as Z Z p 0 2 F 2 2 S= d x[(E E −E E )γ +2 E E F ]+ C^dT^e +O((@T) ;T ): 1 1 2 2 [2 1] (4.4) We may regard T asa static con gurationinterpolating fromone vacuum to another, like a kink or a vertex. The point is that the energy density of these con gurations is localizedinsomeofthespatialworld-volume coordinates. Inanextreme situation dT is a -function along those directions. Thus in the approximation used the e ect of the Wess-Zumino term is to insert  sources on the right hand side of (3.3) and (3.6). InthegaugeE =  theequation(3.8)stays thesame (thustheVirasorosubalgebra 2 0 is still present). Apart from those sources all the discussion in the previous section goes through. The presence of -source on the right hand side of (3.7) will allow string-like solutions to end where the -source sits (since it is a source for the flux @ E = j ). There are thus open strings which can end on \planes" localized in some a a of the spatial world-volume coordinates, i.e. on lower dimensional D-branes. 2 2 It is far from obvious that one can drop terms of order O((@T) ;T )whendis- cussing fluctuations around the vacuum. Most likely that one cannot freely do so. Nevertheless the above qualitative picture is quite reasonable and agrees with expec- tations. To study the fluctuations around the vacuum more carefully the tachyon eld should be treated as dynamical. 5. Discussion In the present note we have analyzed the classical e ective theory of D-branes at the tachyonic vacuum. We have established the following two facts: the string gauge algebra (the Virasoro algebra) is a subalgebra of the D-brane theory at the tachy- onic vacuum and the classical string solutions is a subset of the D-brane solutions in this regime. Thus the Nambu-Goto strings can be embedded into D-branes at the tachyonic vacuum. However, it is not clear to what extent the string picture suces to describe the D-branes in this regime. In the similar situation when !1 in the fundamental string the quantum theory describes (collection of) massless par- ticles [17]. By analogy one might expect the quantum theory here to describe (a collection of) strings. In fact, the analogy goes even further than indicated, since constraint algebra of tensionless string also contains an ideal (P G P =0). 4 −T The expansion (4.3) is not always valid, for instance not for V(T)=(T+1)e around T = 1. However in this case the argument still goes through, since as a result of (3.1) the correction to the 2 2 expansion (4.4) is O((@T) ;(V(T)) ) (i.e. exponentially small around T = 1). 11 JHEP03(2001)014 Wehave emphasized thatthereareproblemswithquantizing (3.2), butthestudy of the gauge algebra leads to a suggestion for quantizing the system treating the U(1) degrees of freedom as background degrees of freedom. Acknowledgments MZ is grateful to Gary Gibbons for valuable discussion. We thank Inegemar Bengts- son for the comments on the manuscript. The work of UL was supported in part by NFR grant 650-1998368 and by EU contract HPRN-CT-2000-0122. A. Basic de nitions The Lie derivative of arbitrary tensor density of weight n in the direction of the eld N , is de ned as b b :::b c b b :::b c b b :::b 1 2 l 1 2 l 1 2 l L T = N @ T +@ N T +:::− c a N a a :::a a a :::a 1 ca :::a 1 2 k 1 2 k 2 k j cb :::b c b b :::b 1 2 l 1 2 l −@ N T −+n@ N T : (A.1) c c a a :::a a a :::a 1 2 k 1 2 k We are using the following basic Poisson brackets b b (p)   (p) fA (x); (y)g=  (x−y); fX (x);P (y)g=  (x−y): (A.2) There is the following action of momentum constraint on the elds a  a fX ;H [N ]g = L X ; fP ;H [N ]g = L P ; a ~  a ~ N N (A.3) c a c a a c fF ;H [N ]g = L F ; f ;H [N ]g = L  −N @  : ab c ~ ab c ~ c N N References [1] K. Bardakci, Dual Models and Spontaneous Symmetry Breaking, Nucl. Phys. 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Journal of High Energy Physics – IOP Publishing

Published: Mar 19, 2001

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