GRANGIAN QUANTIZATION OF THE HETEROTIC STRING IN THE BOSONIC FORMULAT

Transcription

1 September, 1987 IASSNS-HEP-87/draft GRANGIAN QUANTIZATION OF THE HETEROTIC STRING IN THE BOSONIC FORMULAT J. M. F. LABASTIDA and M. PERNICI The Institute for Advanced Study Princeton, NJ 08540, USA ABSTRACT We consider the BRSTquantization of the N=2 supersymmetric Siegel lagrangian, and we show that it describes four chiral bosons. This lagrangian can be consistently coupled to gravity. We apply this method to the heterotic string, obtaining a lagrangian formulation for the heterotic string in the bosonic formulation. 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 Chiral bosons are a basic ingredient in the heterotic string []. A covariant lagrangian quantization of chiral bosons in a gravity background is not known. A classical bosonic lagrangian formulation for chiral bosons has been presented by Siegel []. This theory has anomalies [3,4], which can be eliminated introducing a Liouville term. In ref. [] it has been shown that the theory is consistent only when it describes two chiral bosons. A consistent formulation of this model in a gravity background is lacking; the difficulty in obtaining it is due to the presence of the Liouville term [4]. To solve this problem one can try to modify this model in such a way to get rid of the Liouville term. In ref. [5] a Siegel lagrangian describing 26 bosons is considered; this theory can be consistently coupled to gravity, since the Liouville term vanishes, however it does not describe chiral bosons as physical states. In ref. [2] N = 1 and N = 2 supersymmetric extensions of the model describing chiral bosons are also considered at the classical level. In ref. [6] the N = 4 supersymmetric extension of this model is found. In this letter we show that the N = 2 model describes four chiral bosons at the quantum level; the Liouville term is absent, and it is possible to couple this model to background gravity. We show that this method can be applied to find a covariant lagrangian quantization for the heterotic string in the bosonic formulation. The N = 2 supersymmetric Siegel lagrangian describing four chiral bosons in a gravity background is the following e 1 L = + X a X a + Y a Y a + i 2 ψ ia ρ + D + ψ ia λ ( X a X a + Y a Y a ) i 4 λ ψia ρ + D ψ ia +2 χ i ψ ia X a +2 χ i Ω ij ψ ja Y a + i 2 ψ ia ρ + Ω ij ψ ja A, (1) In our conventions: ± = 1 2 ( σ± τ ), ρ ± = 1 2 (ρ 1 ±ρ 0 ), Ω 01 =1, Ω ij = Ω ji. 1

3 where D ± = e α ±D α, D α are covariant derivatives, and the ± indices are tangent space indices. The matter multiplet is formed by the real scalar bosonic fields X a and Y a, and by the Majorana-Weyl fermions ψ ia, a =1, 2, i =1, 2, and gauge multiplet by the doubly self-dual gauge field λ, the Majorana-Weyl gravitini χ i, and the self-dual U(1) gauge field A. The gauge and super-gauge transformations are the following [6] : δλ =2D + ξ + ξ D λ λ D ξ, δχ i = ξ D χ i 1 2 χ D ξ +ΛΩ ij χ j, δa = ξ D A λ Λ + Λ, δψ ia = ξ D ψ ia ψia D ξ +ΛΩ ij ψ ja, δx a = ξ X a, δy a = ξ Y a, δλ = 4i ɛ i ρ χ i, (2) δχ i = D + ɛ i ɛi D λ 1 2 λ D ɛ i + A Ω ij ɛ j, δa = i ɛ i Ω ij ρ D χ j i(d ɛ i )Ω ij ρ χ j, δψ ia = iρ ɛ i X a + iρ Ω ij ɛ j Y a, δx a = ɛ i ψ ia, δy a = ɛ i Ω ij ψ ja, 2

4 The algebra of these gauge and supergauge transformations is [7] [δ(ɛ 1 ),δ(ɛ 2 )] =δ(ξ )+δ(λ), [δ(epsilon),δ(ξ )] = [δ(epsilon),δ(λ)] = [δ(ξ 1 ),δ(ξ 2 )] = (3) [δ(ξ ),δ(λ)] = where, [δ(λ 1 ),δ(λ 2 )] = ξ = 2i ɛ i 2ρ ɛ i 1, Λ=iΩ kl [( ɛ k 2)ρ ɛ l 1 ɛ k 2ρ ( ɛ l 1)]. (4) Let us prove that this model describes four chiral bosons. The physical states are given by Q Φ >= 0 (5) where Q is the BRSTcharge, given by [fb] Q = c i G i + f k ijc i c j c k (6) where G i are the generators of the algebra (3), c i and c j are the ghosts and the antighosts respectively and f k ij the structure constants. Following ref. [8,4], let us expand Q in the zero modes of the bosonic reparametrization ghosts: Q = c o H + Q B + b 0 M (7) where H = 1 2 (p2 X + a p2 Y a)+n a (8) 3

5 is the hamiltonian, p X a and p Y a are the left sector momenta of the scalar fields X a and Y a, N is the number operator of all the matter fields, ghosts and antighosts and a the intercept. Taking (6), it turns out that Q 2 = 0 and that, =0, (9) where the contributions are given in order by the scalars, the fermions, the reparametrization ghosts b and c, the superconformal ghosts β i and γ i, and the ghosts of the U(1) symmetry [9]. Since the physical states satisfy the condition b 0 Φ >= 0 (10) it follows that H Φ >= 0 (11) Since H is positive semi-definite, and it vanishes only on the states built by oscillators and momenta belonging to the right-moving sector, it follows that the model describes four rightmoving scalars only. This analysis is similar to the one done in ref. [4] for the bosonic Siegel lagrangian. As in that case, it is crucial that the intercept be zero. As in ref. [4], one could prove the no-ghost theorem also in the Minkowski case, that is when the scalars X 1 and Y 1 have the wrong sign in the kinetic term while the other two have the right one. This case is analogous to the N = 2 spinning string [ade], in which it was proved that the only physical state is the ground state, a massless scalar. To describe the chiral bosons in the Minkowski case one must put the momentum of thi physical scalar to zero (such an artificial process is absent in the Euclidean case ). This model can be used to obtain a lagrangian for the heterotic string in the bosonic formulation. There are the 16 chiral bosons X,I =1,..., 16, which can be grouped four by 4

6 four as in (1). The bosons X have the Fourier expansion X = X p(τ + σ)+ p(τ σ)+... (12) 2π 2π where p and p are the left and right momenta respectively. The physical state condition (5) implies that p = 0. The p belong to a self-dual lattice []. Using the lagrangian (1) it is simple to write the covariant lagrangian formulation of the heterotic string without the need of fermionazing the 16 internal bosonic coordinates X. We organize these 16 bosonic coordinates in groups of 4 so we label them as X a, Y a, a =1, 2, J =1,..., 4, and we use the condensed index A, A = {µ, i, a, J}, µ =1,..., 10, i = 1 for the X a and i = 2 for the Y a to denote all the 26 bosonic coordinates X. Similarly for the 26 left-handed Majorana spinors ψ = ψ µ,ψ ia. The lagrangian is [], e 1 L hs = 1 2 (eµ α µ X A ) 2 + i 2 ψρ D ψ i 2 ξ αρ β ρ α ψ µ β X µ λ ( X a X aj +( Y a Y aj ) i 4 λ ψia ρ + D ψ ia (13) +2 χ i ψ ia X a +2 χ i Ω ij ψ ja Y a + i 2 ψ ia ρ + Ω ij ψ ja A where e µ α and ξ µ form the N = 1 supergravity multiplet in two space-time dimensions. This lagrangian is invariant under N = 1 local supersymmetry and under the transformations (2) Let us finally comment that the bosonic Siegel lagrangian quantization in ref. [] has drawbacks also in flat space when there is an internal symmetry among the scalars, for instance on a group manifold, due to the Liouville term, breaking such symmetries. This drawback is absent in the present formulation, which however holds only for n = 4k chiral bosons. 5

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