LAGRANGIANS FOR CHIRAL BOSONS AND THE HETEROTIC STRING. J. M. F. LABASTIDA and M. PERNICI The Institute for Advanced Study Princeton, NJ 08540, USA

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1 October, 1987 IASSNS-HEP-87/57 LAGRANGIANS FOR CHIRAL BOSONS AND THE HETEROTIC STRING J. M. F. LABASTIDA and M. PERNICI The Institute for Advanced Study Princeton, NJ 08540, USA ABSTRACT We study the symmetries of the Siegel lagrangian for chiral bosons and of its supersymmetric extensions, which we obtain in chiral superspace. Only the N = 2 Siegel lagrangian can be coupled consistently to (super)gravity at the quantum level. We present the superfield formulation for the heterotic string lagrangian in the bosonic formulation, in which the internal chiral bosons are described at the lagrangian level by N = 2 supersymmetric Siegel models. Research supported by U. S. DOE contract DE-AC02-76ER02220 On leave from Instituto de Estructura de la Materia, CSIC, Serrano 119, Madrid, Spain

2 1. INTRODUCTION Superstring theories are the most successful attempt so far to obtain a consistent quantum theory unifying gravity with the other fundamental forces. Among these theories, the heterotic string [1] is the one which is phenomenologically the most promising. The heterotic string is essentially chiral in nature, being formed by a right-moving NRS superstring and a left-moving bosonic string. The internal 16 degrees of freedom of the bosonic string can be represented either by 16 chiral bosons or by 32 chiral fermions. The lagrangian for the heterotic string in the fermionic formulation [1] has been studied in detail. A lagrangian for the heterotic string in the bosonic formulation has been proposed recently [2]. The idea of this construction is based on a paper by Siegel [3], in which Lagrange multiplier gauge fields are introduced to implement the self-duality condition on the chiral bosons at the lagrangian level. In ref. [4] it has been shown that the bosonic Siegel lagrangian describes at the quantum level two chiral bosons for each Lagrange multiplier gauge field. To consider only one scalar field leads to inconsistencies, and for more than two scalar fields the theory does not describe chiral bosons. The bosonic Siegel lagrangian contains a local counterterm introduced to cancel a gauge anomaly [5,4], which is an obstruction for coupling it to gravity, and therefore it seems to be of little use in string theory. Supersymmetric extensions of the bosonic Siegel lagrangian were proposed for N = 1 and N = 2 by Siegel [3] and for N = 4 in ref. [6] by truncating the corresponding N =1 [7], N =2 [8] and N =4 [9] spinning string lagrangians. In all these models the left-moving fermions are decoupled from the rest and they can be consistently discarded. After doing this for N =1, 2 one obtains theories with bosonic physical states. The case N = 4 is inconsistent at the 1

3 quantum level. All these models are better described using chiral superspaces. The resulting N =(1, 0) Siegel lagrangian consists of the N =(1, 0) scalar multiplet proposed in ref. [10] and further studied in ref. [11]. Its formulation is known in components and in N =(1, 0) superfields. The resulting N =(2, 0) Siegel lagrangian was presented in components in ref. [6]. In this paper we give its formulation in N =(1, 0) superspace. For the corresponding N =(4, 0) Siegel lagrangian only the component formulation is known [6]. In ref. [2] it has been shown that the N =(2, 0) Siegel model describes consistently at the quantum level four chiral bosons, following a reasoning similar to that in ref. [4]. Unlike the N = 0 Siegel model, or the N =(1, 0) Siegel model, the N =(2, 0) Siegel lagrangian does not have a local counterterm to cancel a gauge anomaly, and therefore a coupling to gravity or to N =(1, 0) supergravity is possible. Using this model a lagrangian for the heterotic string in the bosonic formulation was constructed in ref. [2]. In this paper we review the Siegel lagrangian for the chiral boson, making an analysis of the symmetries of the theory and of the physical states. We find the superspace formulation for the N =(2, 0) Siegel lagrangian and its coupling to N =(1, 0) supergravity. We construct an N =(1, 0) superspace lagrangian for the heterotic string in the bosonic formulation, in which the chiral bosons are described in terms of N =(1, 0) scalar multiplets. We study the quantization of the resulting model showing that the physical states are those corresponding to the heterotic string. The paper is organized as follows. In sect. 2 we review the bosonic Siegel lagrangian, we discuss its coupling to gravity, and we give the transformation laws and the algebra of its symmetries. In sect. 3 we show that the N =(1, 0) Siegel lagrangian describes two left-moving states at the quantum level when the number of N =(1, 0) scalar superfields 2

4 is 2; otherwise the model is inconsistent or it contains right-moving states. In sect. 4 we give the superspace formulation of the N =(2, 0) Siegel lagrangian in N =(1, 0) superspace and we project it to components to make contact with previous formulations. In sect. 5 we couple the N =(2, 0) Siegel model to N =(1, 0) supergravity in N =(1, 0) superspace, and we give the corresponding component expansion. In sect. 6 the lagrangian for the heterotic string in the bosonic formulation is presented. In sect. 7 its quantization is studied, showing that the physical states of the heterotic string in this formulation are the same as in ref. [1]. Finally in sect. 8 we state our conclusions. 3

5 2. Coupling of the bosonic Siegel lagrangian to gravity Siegel [3] proposed a lagrangian for the chiral boson, L = ϕ + ϕ λ ( ϕ) 2. (2.1) This lagrangian is invariant under the gauge transformations, δϕ =η ϕ, δλ =2 + η + η λ λ η. (2.2) At the classical level the equation of motion of λ is ( ϕ) 2 = 0, which implies ϕ = 0. However this is not true at the quantum level. One could guess that there might be problems because in the classical hamiltonian system corresponding to (2.1) there is a first class constraint, but no allowed gauge-fixing function. At the quantum level the BRST lagrangian is [2] L = ϕ + ϕ + ib ++ + c, (2.3) where c and b ++ are the ghost and the antighost respectively. This theory does not describe a chiral boson as a physical state; in fact a chiral boson would give a determinant contribution in the partition function, det 1 2, while the theory described by (2.3) gives (det )det +, where the first factor comes from the scalar, and the second factor from the ghosts. The det + factors do not cancel. This argument suggests that the Siegel lagrangian could describe consistently two chiral bosons, in which case the det + factors cancel. 4

6 Let us write down the Siegel lagrangian for M scalars ϕ n, n =1,..., M, L = ϕ n + ϕ n λ ( ϕ n ) 2. (2.4) One arrives to the same conclusion as above if one counts the number of physical states containing oscillators in the right-moving sector. If the theory describes left-moving scalars, there will be only one state in the right-moving sector, the ground state. The partition function is Trx H = n (1 x n ) M (1 x n ) 2 = n p n x n, (2.5) where H is the hamiltonian. There are p n states in the right-moving sector at level n. The first term comes from M scalars; the second term comes from the ghosts. If there is only one scalar (M = 1) the theory is inconsistent because the on-shell condition is never satisfied as was shown in ref. [4]. If there are two scalars (M = 2), there is only one state, the ground state. For 2 <M 26 there are infinitely many states in the right-moving sector, and the theory does not describe chiral bosons. A more rigorous proof of this fact has been given studying the BRST cohomology [4]. Let us remark that the Siegel lagrangian for M scalars, 2 <M 26, is consistent quantum mechanically, even if does not describe chiral bosons as physical states, provided one adds the local counterterm, (26 M) L = λ ϕ 1. (2.6) 48π This counterterm was found for M = 1 in ref. [5]. The case M = 26, in which there is no local counterterm, has been investigated in ref. [12]. The anomaly term is present also when the theory describes two chiral bosons [4]. In ref. [4] it was shown that it is unlikely 5

7 that the lagrangian of this theory can be coupled to gravity, due to the presence of the local counterterm (2.6). In the following sections we will discuss the N = 1 and N = 2 supersymmetric extensions of the bosonic Siegel lagrangian. In the N = 2 model the local counterterm is absent, and the coupling of it to (super)gravity is possible. To exhibit a few features of this coupling, let us study the simpler coupling of the bosonic Siegel lagrangian to gravity at the classical level (i.e., in absence of the local counterterm). Add to the bosonic Siegel lagrangian a spectator field X, and couple this system to gravity: e 1 L = D + XD X D + ϕ n D ϕ n λ (D ϕ n ) 2. (2.7) The gauge transformations are δϕ n =η D ϕ n, δλ =2D + η + η D λ λ D η, δx =0, (2.8) δe a m =0. In (2.7) and (2.8), D a = e a m m + ω a M are the Lorentz covariant derivatives and ω a is the spin connection (see (B.8) and (B.20)). The reparametrizations are and δϕ n =ξ a D a ϕ n, δλ =ξ a D a λ, δd a =(D a ξ b )D b ξ b [D b, D a ], δe m a =(D a ξ b )e m b. (2.9) (2.10) The algebra of these symmetries is [δ ξ1,δ ξ2 ]=δ ξ3, (2.11a) 6

8 [δ ξ,δ η ]=δ η, (2.11b) [δ η1,δ η2 ]=δ η3, (2.11c) where ξ a 3 =ξ b 2D b ξ a 1 ξ b 1D b ξ a 2, η =η D ξ ξ a D a η, (2.12) η3 =η2 D η1 η1 D η2. The algebra of Siegel transformations (2.11c) is similar to that of reparametrizations (2.11a). The Siegel symmetry does not commute with reparametrizations. To understand this fact, let us consider the generators of these symmetries. One can use reparametrizations and the Siegel symmetry to gauge-fix e m a to the conformal gauge and λ to zero. The equations of motion of e m a and λ give, in this gauge, the constraints, T ++ =( + X) 2 +( + ϕ n ) 2, T =( X) 2 +( ϕ n ) 2, (2.13) T =( ϕ n ) 2. The generator T of the Siegel symmetry is part of the energy-momentum tensor T. In the commutator between T and T, only the T term of T contributes; since T has the form of an energy-momentum tensor, the algebra (2.11) is explained. At the quantum level the local counterterm (2.6) induces a modification in the generator of the Siegel symmetry, which for M =2is T =( ϕ n ) π ϕ 1. (2.14) For consistency (closure of the algebra of the constraints) the same extra term must be added to T ; this creates an asymmetry in the scalar sector of the energy-momentum tensor, since 7

9 T ++ has no such a term; the extra term in T could only come from a term grφ 1 in the lagrangian, which however would give the same contribution to T ++. This shows that the modified energy-momentum tensor cannot be obtained from the field equation of an auxiliary gravity field, and that the bosonic Siegel lagrangian cannot be coupled consistently to gravity at the quantum level. It is therefore unlikely that this model can be used in string theory to describe chiral bosons at lagrangian level. 8

10 3. N =1Siegel theory In this section we will discuss the N = 1 Siegel theory [3] which is obtained upon truncation of the N = 1 spinning string. First we will start describing it in N =(1, 1) superspace. Consider the N =(1, 1) matter multiplet Φ and the Siegel gauge superfield ˆΞ +. The action of the theory is [3], S = [ ] d 2 xd 2 θ (D + Φ)(D Φ) + ˆΞ + (D Φ)(i = Φ). (3.1) This action is invariant under the following gauge transformations, δφ =Ŝ i = Φ+ 1 2 (D Ŝ )(D Φ), δˆξ + = D + Ŝ + Ŝ i = ˆΞ + ˆΞ + i = Ŝ (D Ŝ )(D ˆΞ + ). (3.2a) (3.2b) Next, we will project the theory to N =(1, 0) superspace. Let us choose (τ,σ,θ )to be the coordinates of this superspace and let us define therefore the following N =(1, 0) superfields: φ =Φ, Ψ + = D + Φ, Ξ + = ˆΞ +, (3.3) Ξ = D + ˆΞ +, where the slash denotes θ + -independent part. Similarly, we define the N =(1, 0) components of the gauge parameters as S = Ŝ, S + = D + Ŝ. (3.4) From now on we will use and = for the flat vector indices, reserving + and for the spinor indices. The notation is summarized in appendix A. 9

11 From (3.2b) it is simple to observe that the N =(1, 0) gauge superfield Ξ + transforms as δξ + = S (3.5) and so it can be gauged away algebraically using the gauge parameter S +. In this gauge the transformations of the N =(1, 0) superfields turn out to be, δφ =S i = φ (D S )(D φ), δψ + =S i = Ψ (D S )(D Ψ + ), δξ = i S + S i = Ξ Ξ i = S (D S )(D Ξ ), (3.6a) (3.6b) (3.6c) while the action becomes, S = ] d 2 xdθ [(i φ)(d φ)+ψ + (D Ψ + )+Ξ (D φ)(i = φ). (3.7) From (3.6-7) we observe that the Siegel N = 1 theory is made out of the N = (1, 0) multiplet first discussed in ref. [10] and further analyzed in ref. [11] plus an N =(1, 0) spinor multiplet, which contains a component spinor and an auxiliary field. These two multiplets are decoupled. After the projection, the N =(0, 1) global supersymmetry of the theory is not manifest. However, the corresponding supersymmetry transformations of the N =(1, 0) superfields are easily computed using the fact that in N =(1, 1) superspace any superfield A transforms as δa =[ ˆω D + ˆω + D,A], (3.8) where ˆω ± are the supersymmetry parameters. In N =(1, 0) superspace the manifest supersymmetry is parametrized by ˆω +, while the non-manifest one uses ω ˆω. The transformations of the N =(1, 0) superfields corresponding to the latter are easily obtained 10

12 from (3.8). One finds, δφ = ω Ψ +, (3.9a) δψ + = ω i φ ω Ξ i = φ 1 2 ω (D Ξ )(D φ), (3.9b) δξ =0, (3.9c) where we have used the fact that we work in the gauge where Ξ + vanishes. We define the following component fields, X =φ, ψ =D φ, ψ + =Ψ +, σ =D Ψ +, (3.10) λ = 2Ξ, χ + = 1 2 D Ξ, and the following component gauge parameters: η = is, ɛ + = i 2 D S, (3.11) where the slash now denotes the θ -independent part. The action (3.7), after integrating out the auxiliary field σ becomes, S = [ d 2 x ( X)( = X) ψ + i = ψ + ψ i ψ + 1 ( ) ] 2 λ ( = X)( = X)+ψ i = ψ 2iχ + ψ = X, (3.12) 11

13 and the gauge transformations, δx = η = X iɛ + ψ, δψ = η = ψ ψ = η + ɛ + = X, δ λ =2 η + η = λ λ = η 4iɛ + χ +, δχ + = η = χ + + ɛ χ + = η (3.13) 1 2 λ = ɛ ɛ + = λ, δψ + =0. The global supersymmetry transformations are easily obtained from (3.9): δx = ω ψ +, δψ + = ω i X λ ω i = X + ω (χ + ψ ), (3.14) δ λ =δχ + = δψ =0. The other global supersymmetry that one should obtain under projection from N =(1, 0) superspace to x-space is just the one in (3.13) when one considers ɛ + constant. As it was mentioned before, the two N =(1, 0) multiplets entering the action (3.7) are decoupled. Since we are interested in extensions of the bosonic Siegel action (2.1) which describe only bosonic physical states, we discard the spinor superfield Ψ + and keep only the scalar multiplet φ, i.e., the model consists of one N =(1, 0) scalar multiplet of the type considered in ref. [10,11]. Let us analyze the classical field equations of this reduced system, requiring that the symmetries of the theory be maintained. From (3.12) we obtain, in the gauge where the fields λ and χ + vanish, = X =0, ψ =0, (3.15a) (3.15b) 12

14 ( = X)( = X)+ψ i = ψ =0, (3.15c) ψ i = X =0. (3.15d) Equation (3.15d) implies that either X is chiral, or ψ is zero. In the latter case the system of equations clearly reduces to the bosonic one so let us consider the first case. From (3.15c) it follows that ψ is either chiral or proportional to = ψ. The second case violates Lorentz invariance, and it can be ruled out. The first case, together with (3.15b) implies that ψ is constant, and therefore zero by Lorentz invariance. This shows that at the classical level the system is made out of a chiral boson and no fermion [10]. As it is the case in the previous paragraph, this is not true at the quantum level. If the system were made out of two N =(1, 0) scalar multiplets instead of one, then it would describe two chiral bosons as physical states. This can be easily seen from the partition function argument. Let us assume that we have M copies of N =(1, 0) scalar multiplets φ n, n =1,..., M. In the right-moving sector there would be M scalars, M fermions, the ghosts b and c corresponding to the gauge field λ and the supergauge ghosts β and γ corresponding to χ +. The partition function for the states formed by the right-moving oscillators is Trx H = n (1 x n ) M (1 + x n ) M (1 x n ) 2 (1 + x n ) 2 = n p n x n (3.16) For M = 2 there is only the ground state in the right-moving sector, and the theory describes two left-moving chiral bosons. As in the bosonic chiral lagrangian, also in this case one must We are indebted to L. Mezincescu for explaining to us this argument as well as a similar one for the N=2 Siegel theory. 13

15 introduce a local counterterm, (10 M) L = i d 2 xdθ Ξ D = φ 1 8π (10 M) = i d 2 x(2χ + = ψ 1 + i 8π 2 λ = = φ 1 ), (3.17) to the action. This term is an obstruction for coupling the lagrangian to gravity and therefore the N = 1 Siegel model cannot be used to describe chiral bosons in a gravity background. However, as we will show in the next three sections, the N = 2 Siegel model can be used for this type of description if one considers the chiral bosons in groups of four. Furthermore, in this case one is able to couple the system to background N =(1, 0) supergravity. 14

16 4. Flat N =2Siegel theory In this section we will consider the N = 2 Siegel theory, which is obtained by truncating of the N = 2 spinning string [13]. This truncation was carried out in ref. [3] in N = 2 superspace, and in ref. [6] this model was obtained in component fields using the Noether method. In the latter formulation the action contains at most terms linear in the gauge fields, while in the N = 2 superspace formulation of ref. [3] the action contains terms with arbitrary powers of the gauge fields. This N = 2 superspace formulation contains a constrained gauge parameter. The constraint is originated by the truncation procedure. This, together with the fact that the action and the gauge transformations have an infinite power series in the gauge fields, makes the formulation rather cumbersome. On the other hand, the existence of the simple formulation of ref. [6] at the component level indicates that it may be possible to formulate the theory in superspace in a manner much simpler than in ref. [3]. In this section we present a simple formulation of the theory in N = 1 superspace. Whether or not this formulation can be obtained from the one in ref. [3] after projection from N = 2 superspace in the corresponding Wess-Zumino gauge is an open question which we will not address in this paper. However, we expect that our formulation is equivalent to the one in ref. [3]. Consider two N =(1, 1) scalar superfields Φ i, i =1, 2, which constitute the matter multiplets of the theory. Additionally, let us introduce the N =(1, 1) gauge superfields ˆΛ + and ˆΣ. The N =(1, 1) superspace action of the theory is S = [ ] d 2 xd 2 θ (D + Φ i )(D Φ i )+ˆΛ + (D Φ i )(i = Φ i )+ˆΣ Ω ij (D Φ i )(D Φ j ) (4.1) where Ω ij = Ω ji,ω 12 = 1. This action is invariant under the following gauge transforma- 15

17 tions: δφ i = ˆK i = Φ i (D ˆK )(D Φ i )+ˆL + Ω ij (D Φ j ), (4.2a) δ ˆΛ + = D + ˆK + ˆK i = ˆΛ + ˆΛ + i = ˆK (D ˆK )(D ˆΛ + )+2ˆL + ˆΣ δ ˆΣ =D + ˆL [ 2 ˆK i = ˆΣ ˆΣ (i ] = ˆK )+(D ˆK )(D ˆΣ ) (4.2b) + 1 ] (i = ˆΛ + )+2ˆΛ + (i = ˆL+ ) (D ˆL+ )(D ˆΛ + ). (4.2c) 2[ˆL+ To show that this formulation is equivalent to the one in ref. [6] we will project it into components. We will do this in two steps: first we will project it to chiral superspace, which in the next section is considered in the presence of background N =(1, 0) supergravity, and, second, from it we will project it to x-space component fields. The coordinates of our chiral superspace are taken to be (τ,σ,θ ). Therefore, we define the following N =(1, 0) superspace components for our N =(1, 1) superfields: φ i =Φ i, Ψ i + =D + Φ i, Λ + =ˆΛ +, Λ =D + ˆΛ +, Σ =ˆΣ, Σ + =D + ˆΣ, (4.3a) (4.3b) (4.3c) (4.3d) (4.3e) (4.3f) where the slash denotes θ + -independent part. Defining the following N =(1, 0) superspace components for the gauge parameters, K = ˆK, (4.4a) 16

18 K + =D + ˆK, L =ˆL +, L =D + ˆL+, (4.4b) (4.4c) (4.4d) it is simple to find the gauge transformations of the N =(1, 0) superfields. It is important first to observe that according to (4.2b) and (4.2c), δλ + = K , δσ =L +..., (4.5a) (4.5b) and therefore these fields can be gauged away algebraically using the gauge parameters K + and L. In this gauge the transformations of the rest of the N =(1, 0) superfields are: δφ i =K i = φ i (D K )(D φ i )+L + Ω ij D φ j, δψ i + =K i = Ψ i (D K )(D Ψ i +)+L + Ω ij D Ψ j +, (4.6a) (4.6b) δλ = i K + K i = Λ Λ i = K (D K )(D Λ ) 2L + Σ +, (4.6c) δσ + =i L [ 2 [ ] 2K i = Σ + Σ + i = K +(D K )(D Σ + ) ] L + i = Λ +2Λ i = L + +(D L + )(D Λ ), (4.6d) and the action (4.1) becomes, S = d 2 xdθ [(i φ i )(D φ i )+Ψ i +(D Ψ i +) ] +Λ (D φ i )(i = φ i )+Σ + Ω ij (D φ i )(D φ j ). (4.7) We observe from (4.6) and (4.7) that the spinor multiplets Ψ i + are decoupled from the rest although their presence is essential to keep the second or N =(0, 1) supersymmetry. 17

19 Each of these two spinor multiplets contains a component spinor and an auxiliary field. The scalar multiplets φ i are of the type considered in ref. [10,11]. However, in (4.7) these scalar superfields are coupled via the gauge field Σ +. The N =(0, 1) supersymmetry of the theory is not manifest. The transformations of the fields under this supersymmetry can be easily computed from the fact that in N =(1, 1) superspace any superfield A transform in the following way: δa =[ ˆω 1 D + ˆω 1 +D,A], (4.8) where ˆω ± 1 are the supersymmetry parameters. In N =(1, 0) superspace the manifest supersymmetry is parametrized by ˆω +, 1 while the non-manifest one uses ω 1 ˆω. 1 The transformations of the N =(1, 0) superfields corresponding to the latter are easily obtained from (4.8). One finds, δφ i = ω 1 Ψ i +, (4.9a) δψ i + = ω 1 i φ i ω 1 Λ i = φ i 1 2 ω1 (D Λ )(D φ i )+ω 1 Σ + Ω ij (D φ j ), δλ =δσ + =0, (4.9b) (4.9c) where we have used the fact that we work in the gauge where Λ + and Σ vanish. The action (4.7) contains another global supersymmetry which one can deduce simply analyzing the way cancellations occur in the variation of (4.7) under (4.9a-c). The transformations of the N =(1, 0) superfields under this supersymmetry are, δφ i = ω 2 Ω ij Ψ j +, (4.9d) δψ i + =ω 2 Ω ij i φ j + ω 2 Λ Ω ij i = φ j 18

20 + 1 2 ω2 (D Λ )Ω ij (D φ j )+ω 2 Σ + (D φ i ), (4.9e) δλ =δσ + =0, (4.9f) where ω 2 is a global parameter. In the next section we will use the action (4.7) to couple the system to N =(1, 0) supergravity. In the rest of this section we obtain the x-space component formalism of the theory. We define the following component fields: X =φ 1, Y =φ 2, ψ i =D φ i, ψ+ i =Ψ i +, σ i =D Ψ i +, (4.10) λ = 2Λ, χ 1 + = 1 2 D Λ, χ 2 + =Σ +, A =D Σ +, and the following component gauge parameters: η = ik, ɛ 1 + = i 1 2 D K, ɛ 2 + = ik +, (4.11) Λ= id K +, where now the slash denotes θ -independent part. Using these definitions and (4.6) one finds 19

21 the following gauge transformations for the component fields: δx =η = X iɛ i +ψ, i δy =η = Y iω ij ɛ i +ψ, j δψ i =η = ψ i ψi = η + ɛ i + = X Ω ij ɛ j + = Y iλω ij ψ, j δλ =2 η + η = λ λ = η 4iɛ i +χ i +, (4.12) δχ i + =η = χ i + + ɛ i χi + = η 1 2 λ = ɛ i ɛi + = λ iω ij ɛ j A iλω ij χ j +, δa =η = A +Ω ij (ɛ i + = χ j + + χj + =ɛ i +)+ Λ, δψ i + =0, and the action (4.7) becomes, [ S = d 2 x ( X)( = X) ( Y )( = Y ) ψ+i i = ψ+ i ψ i ψ i i + 1 ( ) 2 λ ( = X)( = X)+( = Y )( = Y )+ψ i i = ψ i 2χ i +ψ i i = X ] 2χ i +Ω ij ψ i j = Y + A Ω ij ψ ψ i j, (4.13) where the auxiliary fields σ i have been integrated out. The global supersymmetry transformations of these component fields are obtained from (4.9): δx = ω i ψ i +, δy = Ω ij ω i ψ j +, δψ i + = ω i i X +Ω ij ω j i Y λ ω i i = X (4.14) 1 2 λ Ω ij ω j i = Y + ω i (χ j + ψj )+Ω ij ω j (Ω kl χ k +ψ l ), δλ =δχ i + = δa = δψ i =0. 20

22 The other global supersymmetry that one should obtain under projection from N =(1, 0) superspace to x-space is just the one in (4.13) when one considers ɛ 1 + constant. In fact, there is one more of this type corresponding to the transformations where ɛ 2 + is constant. This one was already present in the formulation of the theory in N =(1, 1) superspace. 21

23 5. N =(2, 0) Siegel theory coupled to N =(1, 0) supergravity In this section we couple the action (4.7) without the spinor superfields Ψ i + to background N =(1, 0) supergravity. The coupling of spinor superfields is well known [14,11] and it is of no interest for our purposes since we are interested in supersymmetric extensions of the bosonic Siegel action which describe only bosonic states. The formulation of N =(1, 0) supergravity is reviewed in appendix Band we will use the results stated there through all this section. The coupling of this theory to background N =(1, 0) supergravity is described by the following action: S = [ d 2 xdθ E (i φ i )( φ i )+Λ ( φ i )(i = φ i ) ] +Σ + Ω ij ( φ i )( φ j ), (5.1) where the covariant derivatives are defined in (B.1) and satisfy the constraints (B.2), and E = sdet(e M A ). This action is invariant under the following gauge transformations, δφ i =K i = φ i ( K )( φ i )+L + Ω ij φ j, (5.2a) δλ = i K + K i = Λ Λ i = K ( K )( Λ ) 2L + Σ +, (5.2b) δσ + =i L [ 2 [ ] 2K i = Σ + Σ + i = K +( K )( Σ + ) ] L + i = Λ +2Λ i = L + +( L + )( Λ ), (5.2c) while all the pure supergravity fields are inert. These transformations are simply the covariantization of the transformations (4.6). The fact that (5.1) is invariant under these transformations is not obvious because the covariant derivatives satisfy different commutation relations than in the flat case. However, the constraints (B.2) are such that (5.2) constitute a set of gauge symmetries of the action (5.1). 22

24 N =(1, 0) supergravity possesses the superweyl symmetry described in (B.5). Our theory also has this symmetry. The superweyl transformations of the matter and gauge fields are, δφ i =δλ =0, δσ + = 1 2 W Σ +, (5.3) where W is the N =(1, 0) scalar superfield in (B.5). The projection of the theory to component fields is carried out in the superconformal- Wess-Zumino gauge described in appendix B. We define our component fields in the following way, X =φ 1, Y =φ 2, ψ i = φ i, ψ+ i =Ψ i +, σ i = Ψ i +, λ = 2Λ, χ 1 + = 1 2 Λ, χ 2 + =Σ +, A = Σ +, (5.4) and the gauge parameters as, η = ik, ɛ 1 + = i 2 K, ɛ 2 + = ik +, (5.5) Λ= i K +. 23

25 The gauge transformations of the component fields are, δx =η D = X iɛ i +ψ, i δy =η D = Y iω ij ɛ i +ψ, j δψ i =η D = ψ i ψi D = η + ɛ i +D = X Ω ij ɛ j +D = Y iλω ij ψ, j δλ =2D η + η D = λ λ D = η 4iɛ i +χ i + 4iξ ɛ 1 +, δχ 1 + =η D = χ D ɛ χ1 +D = η 1 2 λ D = ɛ ɛ1 +D = λ iω ij ɛ j +A iλω ij χ j + (5.6) η D = ξ ξ D = η, δχ 2 + =η D = χ D ɛ χ2 +D = η 1 2 λ D = ɛ ɛ2 +D = λ iω ij ɛ j +A iλω ij χ j + iξ Λ, δa =η D = A +Ω ij (ɛ i +D = χ j + + χj + D =ɛ i +)+D Λ + ɛ 2 +D = ξ + ξ D = ɛ 2 +, while all the supergravity fields are inert. The action (4.1) becomes, S = [ d 2 xe (D X)(D = X) (D Y )(D = Y ) ψ id ψ i i + 1 ( ) 2 λ (D = X)(D = X)+(D = Y )(D = Y )+ψ id i = ψ i 2iχ i +ψ D i = X ] 2iχ i +Ω ij ψ D j = Y + A Ω ij ψ ψ i j +2iξ (ψ D 1 = X + ψ D 2 = Y ). (5.7) The local supersymmetry transformations of the matter and gauge fields are easily com- 24

26 puted using the methods described in appendix B. We find, δx = ω ψ, 1 δy = ω ψ, 2 δψ 1 = ω id = X, δψ 2 = ω id = Y, δλ = 4ω χ 1 +, (5.8) δχ 1 + = 1 4 ω id = λ λ id = ω, δχ 2 + = ω A, δa = ω id = χ 2 + χ 2 +id = ω, where use has been made of the fact that the superconformal- Wess-Zumino gauge has been taken (see (B.15) and (B.16)). The transformations of the pure N =(1, 0) supergravity objects are given in (B.18). The Weyl transformations of these fields are simply computed from (5.3): δx =δy = δλ =0, δψ i = 1 2 wψi, δχ i + = 1 2 wχi +, (5.9) δa =wa. The corresponding transformations for the supergravity objects are shown in (B.19). From the equations of motion in the superconformal gauge, and requiring that the solutions of the equations of motion maintain the symmetries of the theory, it follows that the N =(2, 0) chiral model describes two left-moving chiral bosons; the analysis is analogous to the one made in the N =(1, 0) model. To describe chiral bosons at the quantum level one 25

27 can consider two matter superfields φ in, n =1, 2 in (5.1) and the following action, [ S = d 2 xdθ E (i φ in )( φ in )+Λ ( φ in )(i = φ in ) ] +Σ + Ω ij ( φ in )( φ jn ), (5.10) A partition function argument similar to the one in (3.16) shows that this theory describes four left-moving chiral bosons. 26

28 6. Heterotic string lagrangian in the bosonic formulation In sect. 2 we reviewed the essential features of the analysis of the quantization of the bosonic Siegel action (2.1). The gauge symmetry (2.2) has an anomaly which, however, can be cancelled by adding the local counterterm (2.6) to the Siegel action. This cancellation can be achieved for M bosonic fields (M 26) in the theory and the coefficient of this local counterterm vanishes for 26 bosonic fields. However, it was shown in ref. [4] and reviewed in sect. 2, that if one wants to have a Fock space with only left-moving bosonic states one is forced to introduce a different Siegel gauge field for each pair of bosonic fields. Therefore, in a lagrangian of this type, describing only left-moving bosonic states, the counterterm is always present. The presence of the local counterterm makes the coupling of this system to background gravity difficult. This was pointed out in ref. [4] and reviewed in sect. 2. In order to achieve this coupling one may think of making use of supersymmetric extensions of the bosonic Siegel action (2.1). For the case of N = 1, it was shown in sect. 3 that the local counterterm will be always present for M < 10 since the critical dimension of the corresponding spinning string is 10. Also in this case one needs a super Siegel gauge multiplet for each pair of scalar multiplets if one wants to have only left-moving states. Therefore, in the case N = 1 one faces the same problems as in the bosonic case. The features of the N = 2 extension are rather different. The corresponding spinning string has complex critical dimension 2 [13,15,16] and so if one is willing to describe 4 leftmoving bosons there is no need for a counterterm because there is no anomaly. Furthermore, the physical states of the theory are all left-moving when the number of scalar multiplets is 27

29 exactly 4. This is in correspondence with the fact that in the N = 2 spinning string there is only one physical state. Therefore, it seems that the problems of the previous cases are not present for the N = 2 extension of the Siegel action and so one should be able to achieve the coupling to gravity. In fact, one is able to go further and to couple the system to N =(1, 0) supergravity. The fact that this coupling can be carried out maintaining the Siegel gauge symmetries is not trivial. However, the constraints of N =(1, 0) supergravity are such that these symmetries are maintained after the coupling [2]. This N = 2 extension of the Siegel action, in which one is able to describe left-moving bosons in groups of 4 can be used to describe the 16 left-moving bosons of the heterotic string. Let us carry out the construction of this formulation. To describe the 16 left-moving bosons of the heterotic string one needs 4 sets of N =(1, 0) scalar superfields φ in, I =1,..., 4 and the corresponding gauge fields for each set: Λ and Σ +. They are described by four copies of the chiral lagrangian (5.10). Additionally one needs to introduce the 10 scalar N =(1, 0) superfields φ µ, µ =0,..., 9. The action of the heterotic string in the bosonic formulation in N =(1, 0) superspace is S = [ d 2 xdθ E (i φ µ )( φ µ )+ 4 (i φ in )( φ in )+ =1 4 ] +Ω ij Σ + ( φ in )( φ jn ). =1 4 Λ ( φ in )(i = φ in ) =1 (6.1) This action is invariant under the following gauge transformations: δφ in =K i = φ in ( K )( φ in )+L + Ω ij φ jn, (6.2a) δλ = i K + K i = Λ Λ i = K ( K )( Λ ) 2L + Σ +, (6.2b) 28

30 δσ + =i L [ 2 [ ] 2K i = Σ + Σ + i = K +( K )( Σ + ) ] L + i = Λ +2Λ i = L + +( L + )( Λ ), (6.2c) δφ µ =0, (6.2d) while all the pure supergravity fields are inert. Notice that in (6.1) and (6.2) sums over the index I are specified explicitly. The convention of summing over repeated indices does not apply to this index. The action (6.1) is manifestly invariant under N =(1, 0) local supersymmetry. Finally, to connect with the component expressions given in ref. [2] we reproduce here the projection to component fields of the action (6.1) S = [ d 2 xe (D X µ )(D = X µ ) ψ id ψ µ µ =1 4 =1 ( ) (D X n )(D = X n )+(D Y n )(D = Y n )+ψ in id ψ in 1 ( ) 2 λ (D = X n )(D = X n )+(D = Y n )(D = Y n )+ψ in id = ψ in 4 =1 +Ω ij 4 =1 χ i + ( ψ in id = X n +Ω ij ψ jn id = Y n ) A ψ in ψ jn +2ξ 4 ] (ψ 1n id = X n + ψ 2n id = Y n ), =1 (6.3) where the component fields are defined as in (5.4) and X µ = φ µ, ψ µ = φ µ. (6.5) The component gauge parameters are defined as in (5.5). The gauge transformations are 29

31 obtained from (6.2). One finds, δx n =η D = X n iɛ i +ψ in, δy n =η D = Y n iω ij ɛ i +ψ jn, δψ in =η D = ψ in ψin D = η + ɛ i +D = X n Ω ij ɛ j +D = Y n iλω ij ψ jn, δλ =2D η + η D = λ λ D = η 4iɛ i +χ i + 4iξ ɛ 1 +, δχ 1 + =η D = χ D ɛ χ1 +D = η 1 2 λ D = ɛ ɛ1 +D = λ iω ij ɛ j +A iλω ij χ j + (6.6) η D = ξ ξ D = η, δχ 2 + =η D = χ D ɛ χ2 +D = η 1 2 λ D = ɛ ɛ2 +D = λ iω ij ɛ j +A iλω ij χ j + iξ Λ, δa =η D = A +Ω ij (ɛ i +D = χ j + + χj + D =ɛ i +)+D Λ + ɛ 2 +D = ξ + ξ D = ɛ 2 +, δx µ =δψ µ =0. 30

32 7. Quantization In this section we study the BRST quantization of the heterotic string in the bosonic formulation using N =(1, 0) superspace. Using the superconformal symmetry one can gauge-fix the supergravity fields to the superconformal gauge; using the Siegel symmetry one can gauge-fix Λ and Σ + to zero. The gauge fixed classical lagrangian is, L 0 = dθ [i( φ µ )D φ µ + 4 i( φ in )D φ in ]. (7.1) =1 The constraints coming from the field equations of the auxiliary gauge fields are the following: the supersymmetric extension of the energy-momentum tensor in the right-moving sector T (m) = = i(d φ µ )( = φ µ ) i 4 (D φ in )( = φ in ), (7.2) =1 where the label m indicates that this is the matter contribution to the full (matter plus ghost) T =. The matter part of energy-momentum tensor in the left-moving sector is, T == (m) = D T (m) = =iψ µ = ψ µ +( = X µ )( = X µ ) + 4 =1 [iψ in = ψ in +( = X n )( = X n )+( = Y n )( = Y n )]. (7.3) The constraints coming from the Siegel gauge symmetries are the following: T,(m) = = i( = φ in )(D φ in ), (7.4a) J,(m) = =Ω ij (D φ in )(D φ jn ). (7.4b) The ghost and antighost superfields associated to the N =(1, 0) superconformal constraint 31

33 T = are respectively c = =C =, γ =D C =, β + =B +, (7.5) b =D B +. The ghost and antighost associated to T are respectively c and b ==. The ghost and antighost superfields associated to the constraints (7.4a) are respectively c = C, γ 1+ =D C, β 1= = B =, (7.6) b== =D B=, where C is the ghost superfield corresponding to the gauge parameter superfield K. The ghost and antighost superfields associated to the constraints (7.4b) are γ 2+ = G +, g =D G+, f = = F =, (7.7) β 2= =D F=, where G + is the ghost superfield corresponding to the gauge parameter superfield L +. The ghost action is the following L gh = i dθ (B = C + 4 ( B = C + F = G+ )) + ib == = c. (7.8) =1 The quantum action is formed by the gauge-fixed classical action (7.1) and by the ghost 32

34 action (7.8). To determine the BRST charge Q one can use the following relations {Q, B = } =T = = T (m) = + T (B,C) = + {Q, B = } = T = = T,(m) = + T,(gh) =, 4 =1 T,(gh) =, (7.9) {Q, F = } = J = = J I(m) = + J,(gh) =. The BRST charge is Q = + dz [ 2πi dθ CT (m) CT(B,C) + C( 4 T gh ) 4 ( C T,(m) C T,(gh) + G J I,(m) ( G) )] 2 B +2 G(D F )(D C) =1 =1 (7.10) where m labels the matter part and gh the ghost part of the generators. In (7.9-10), T (B,C) = = ( = B + )C = i 2 (D B + )(D C = ) 3 2 B + = C =, T,(gh) = = ( = B= ) C i 2 (D B = )(D C ) 3 2 B = = C i 2 (D G + )(D F= ) 1 2 G + = F=, J =,(gh) = 2( B= G ) + (D F )(D C ). (7.11) J are the U(1) generators. One can choose either the Neveu-Schwarz or the Ramond boundary conditions. The analysis of the physical states for the right-moving sector is similar to the one for the N = 2 spinning string. With either boundary condition there is only one physical state, the ground state, as can be seen computing the partition function [17]. Consider the constraint H Φ >= (k 2 r + N r ) Φ >= 0 (7.12) The intercept is zero both in the NS and in the R sector. k 2 r = 4 [(k n X,r )2 +(k n Y,r )2 ], and k n X,r, kn Y,r and N r are respectively the internal momenta and the number operator in the 33

35 right-moving sector, and H is given by H = dz 2πiz T ==. (7.13) k 2 r is positive definite. N r is positive definite on the subspace of the states with the same ghost number as the vacuum (it is not positive definite in the NS sector on the states with ghost number different from the ghost number of the vacuum, because γ 1 2 does not annihilate the vacuum; however the physical states have the same ghost number as the vacuum). Therefore k 2 r and N r annihilate separately the physical states. From this it follows that there are no right-moving physical states apart from the ground state, in agreement with the partition function argument. Unlike the case of the N = 2 spinning string, in which the only physical state is a massless scalar, so that k 2 = 0 but k µ 0, here the physical state has k n X,r = kn Y,r =0. The vertex for a chiral boson is of the form e ik X,l.X l +ik Y,l.Y l, where the right-moving internal momenta k n X,r and kn Y,r are zero, while the left-moving momenta kn X,l and k n Y,l are different from zero, and they will belong to an even self-dual lattice [1]. X l and Y l are the left-moving parts of the scalars X and Y and the dot means sum over the indices n and I. It would seem natural to consider a N = 2 supersymmetric invariant vertex for a chiral boson of the form d 2 xdθ 4 (kx,rd n φ 2n ky,rd n φ 1n )e i(k X,r.φ 1 +k Y,r.φ 2 +k X,l.X l +k l,y.y l ). (7.14) =1 However this vertex is proportional to kx,r n and kn Y,r, and so it is zero. Therefore, we see that the physical vertex operators of this theory are the same as those of the heterotic string in the bosonic formulation [1]. 34

36 8. Conclusions In this paper we have analyzed the quantum properties of the Siegel lagrangians for chiral bosons and we have applied the results of this analysis to the construction of a lagrangian for the heterotic string in the bosonic formulation. The N = 0 Siegel model was studied in detail in ref. [4]. The N = 1 and N = 2 supersymmetric extensions of this model have been presented in this paper in their N =(1, 0) superspace and component forms. It turns out that the N = 0 and N = 1 models need a counterterm to obtain consistency at the quantum level and only left moving physical states. This counterterm makes difficult the coupling of these models to background (super)gravity and so these models do not seem useful to formulate chiral bosons in string theories. On the other hand, the N = 2 extension does not contain a counterterm when describing chiral bosons in groups of four. Additionally, it turns out that just for the description in groups of four the Fock space analysis indicates that there are only left moving physical states. These facts make the N = 2 extension suitable for applications to the heterotic string in the bosonic formulation. The N =(1, 0) superspace formulation of the heterotic string in the bosonic formulation using the N = 2 Siegel models, as well as the corresponding quantization has been presented. The analysis becomes very simple and it turns out that the physical states of the theory are those of the heterotic string as obtained in ref. [1] It is an interesting problem to study the heterotic string in this formulation at one loop. On the torus the Siegel gauge fields cannot be gauged away completely, and one remains with an integration over moduli [5]. Using this formalism it should be possible to compute 35

37 chiral boson determinants and to verify the chiral bosonization formulae. Another interesting problem would be to formulate the corresponding sigma model [20] and to make background field computations in the present formulation. 36

38 Appendix A In this appendix we summarize some facts about the notation used in the paper from sect. 3 to sect. 8 and in appendix B. The space-time indices are labeled in the light-cone form, = while spinor indices are labeled by +,. Flat spinor (vector) indices are denoted by letters from the beginning of the greek (latin) alphabet while world spinor (vector) indices are denoted by letters from the middle of the greek (latin) alphabets. The generator of the Lorentz transformations M is defined such that [M,A a ]=2ɛ ab A b, [M,ψ α ]=ɛ αβ ψ β, (A.1) (A.3) where ɛ ab = ɛ ba, ɛ = = 1 and ɛ αβ = ɛ βα, ɛ + =1. When working in N =(1, 1) superspace the superspace coordinates are the commuting ones τ, σ and the anticommuting ones θ +, θ. In this superspace the flat derivatives are =,, and D θ + iθ =, D + θ + + iθ+. (A.3) (A.4) which satisfy, {D,D } =2i =, (A.5) {D +,D + } =2i, (A.6) {D,D + } =0, (A.7) [D ±, = ]=0, (A.8) [D ±, ] =0, (A.9) 37

39 When working in N =(1, 0) superspace the superspace coordinates are the commuting ones τ, σ and the anticommuting one θ. In this superspace the flat derivatives are =,, and D as defined in (A.3). 38

40 Appendix B In this appendix we review the N =(1, 0) superspace geometry of N =(1, 0) supergravity and the corresponding projection in terms of component fields. The simplest way to obtain this formulation is by projection from N =(1, 1) supergravity in N =(1, 1) superspace [18] to N = (1, 0) superspace and then truncating the theory by setting to zero one of the Majorana-Weyl gravitini. The resulting theory has been discussed in ref. [14,19]. We introduce the N =(1, 0) supergravity supervielbein E M A and the Lorentz connection Γ A via the supercovariant derivatives A E A M D M +Γ A M, (B.1) where A = (, =, ), M = (, =, ), D = =, D, and M is the generator of local Lorentz transformations. These supercovariant derivatives satisfy the following constraints [14,19] : {, } =2i =, (B.2a) [,i = ]=0, (B.2b) [,i ] =R + M, (B.2c) [i =,i ] =R + +( R + )M, (B.2d) where R + is an N =(1, 0) superfield. All the N =(1, 0) objects transform in the following way δa =[ik, A], (B.3) where ik = ω A A λm, (B.4) 39

41 being ω A the parameters corresponding to superspace reparametrizations and λ the one to superspace local Lorentz transformations. Besides the manifest symmetry (B.3), the constraints on the supercovariant derivatives (B.2) are invariant under the following superweyl transformations: δ = 1 2 W ( W )M, δi = = Wi = +( W ) (i =W )M, δi = Wi 1 2 (i W )M, (B.5) δr + = 3 2 WR + ( i W ). In these transformations W is an arbitrary N =(1, 0) scalar superfield. The projection to component fields of this superspace formulation is carried out choosing a Wess-Zumino gauge. This gauge uses all but the θ -independent part of M. We will use the notation A to denote the θ -independent part of the superfield object A. From (B.3) follows that there exist a gauge where =. (B.6) In this gauge we have a = D a + ξ a, (B.7) where D a = e a m m + ω a M. (B.8) In (B.7) and (B.8) e a m are the component inverse vielbein, ξ a the component grvitini and ω a the component Lorentz connection. We define the components of the superfield R + in (B.2) as r + = R +, r = R +. (B.9) 40

42 Using (B.2), (B.7) and (B.8) one finds that in the Wess-Zumino gauge (B.6), the higher components of a are, = = ω = + iξ = D =, = ω + iξ D = + iξ ξ = ir + M. Using these results one finds from (B.2d) that in the Wess-Zumino gauge, (B.10) [D, D = ]= 2iξ ξ = D = +(r iξ = r + )M, r + = D ξ = D = ξ. (B.11a) (B.11b) Additionally, making use of the same relations and (B.3) one finds the local supersymmetry transformations of the component fields and covariant derivatives: δd = = 2iω ξ = D =, δd = 2iω ξ D = + iω r + M, δξ = =D = ω, δξ =D ω 2iω ξ ξ =, δe m a = 2iω ξ a e m =, δe =2ieω ξ =, (B.12a) (B.12b) (B.12c) (B.12d) (B.12e) (B.12f) where e is the determinant of the component vielbein e = det(e a m ). We define the components of the superweyl parameter W as w = W, (B.13) w = W. From (B.5) it is simple to derive the superweyl transformations of the component objects of the theory: δd = = wd = (D =w)m, (B.14a) 41

43 δd = wd 1 2 (D w +2ξ w )M, δξ = = 1 2 wξ = iw, δξ = 1 2 wξ, δe m a = we m a, δe = 2we. (B.14b) (B.14c) (B.14d) (B.14e) (B.14f) From these transformations we observe that we can use part of the superconformal invariance to gauge away algebraically one of the components of the gravitino. We go on and we choose a superconformal-wess-zumino gauge in which ξ = =0. (B.15) Equations (B.12c) and (B.14c) imply that in this gauge w = id = ω, (B.16) and so one must do gauge restoring transformations to stay in this gauge. This means that all the objects of the theory that contain w in their superweyl transformations get an extra term in their supersymmetry transformations upon substitution of the w by its value in (B.16). In this superconformal-wess-zumino gauge the commutation relation of the component covariant derivatives has the simple form, [D, D = ]=rm. (B.17) The local supersymmetry transformations are, δd = =0, (B.18a) 42

44 δd = 2iω ξ D = i(ω D = ξ ξ D = ω )M, δξ = D ω, δe m = =0, δe m = 2iω ξ e m =, δe =0, (B.18b) (B.18c) (B.18d) (B.18e) (B.18f) while the Weyl transformations become, δd = = wd = (D =w)m, δd = wd 1 2 (D w)m, δξ = 1 2 wξ, δe m a = we m a, δe = 2we. (B.19a) (B.19b) (B.19c) (B.19d) (B.19e) Finally, we give the expressions of the Lorentz connection ω a and the curvature r in terms of the vielbein, which are simply obtained from (B.17), ω a = 1 2 ɛab e 1 m (ee b m ), r = ɛ ab e a m m ω b + ω a ω a, (B.20) where ɛ ab = ɛ ba, ɛ = =1. 43

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