LARGE CONFORMAL DEFORMATIONS AND SPACETIME ...

arXiv:hep-th/9501082v1 18 Jan 1995

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RU95-1-B

LARGE CONFORMAL DEFORMATIONS AND SPACETIME STRACTURE Gregory Pelts ∗ Department of Physics The Rockefeller University 1230 York Avenue New York, NY 10021-6399

Abstract We demonstrate in detail how the space of two-dimensional quantum field theories can be parametrized by off-shell closed string states. The dynamic equation corresponding to the condition of conformal invariance includes an infinite number of higher order terms, and we give an explicit procedure for their calculation. The symmetries corresponding to equivalence relations of CFT are described. In this framework we show how to perform nonperturbative analysis in the low-energy limit and prove that it corresponds to the Brans-Dicke theory of gravity interacting with a skew symmetric tensor field. Talk presented at the G¨ ursey memorial conference, Istanbul, Turkey, June, 1994.

∗

Supported by the DOE grant DOE-91ER4651 Task B

1

Introduction

Classical closed string states are believed to be associated with quantum conformal field theories in two dimensions (CFT), which are usually defined as theories of the single string moving in some nontrivial spacetime background. The condition of anomaly cancellation leads to the so-called βfunction equation on the background fields. The main advantage of this approach is its more or less explicit connection to spacetime geometry, and the main drawback is that it usually focuses only on massless fields. Treatment of massive fields is problematic and, therefore, characterization of dynamical degrees of freedom is obscure. Symmetries also are not explicit because classically equivalent CFT may correspond to inequivalent quantum theories. Approaches [1-5] based on the operator formalism [6] encounter problems dealing with ambiguity of the vertex operator commutator due to contact singularity of their T -product. As we will see, this ambiguity is principal as it actually makes the string theory nonlinear.

2

Vertex operators

We will consider CFT as a family of amplitudes h0iΣ assigned to Riemann surfaces Σ and obeying sewing property (see details in [7]). It is similar to the Segal’s definition [8]. In this formalism vertex operators Ψ(z0 ) inserted at the point z0 can be defined as a family of states hΨ(z0 )iD ∈ H∂D associated with disc like environments D of z0 obeying the condition hΨ(z0 )iD2 = Sp∂D1 h0iD2 \D1 ⊗ hΨ(z0 )iD1

(D1 ⊂ D2 ).

Here SpΓ denotes contraction of amplitudes corresponding to sewing of Riemann surfaces along the contour Γ. The T -product of such vertex operators can be defines as hΨ0 (z0 ) · · · Ψn (zn )iΣ = Sp∂Σext

n O i=0

hΨi (zi )iDi ⊗ h0iΣext .

(2.1)

Here Di are nonintersecting environments of zi and Σext is their complement in Σ. The Virasoro algebra does not have a bounded natural representation in H∂D . The conformal transformations deform the boundary of the disk and, 1

therefore, corresponding to them linear operators are not automorphisms. However, we can define such a representation in the space of vertex operators, which is independent of the position of the boundary.

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Infinitesimal Deformation of CFT

Infinitesimal deformations of amplitudes can be parametrized by (1,1)-primary fields 1Z h0iΣ −→ h0iΣ + δh0iΣ , δh0iΣ = hΨ(z)iΣ d2z. π Σ Formally we can parametrize vertex operators of the deformed theory by vertex operators of the initial theory as hΥiD −→ hΥiD + δhΥiD , δhΥ(z0 )iD =

1 π

Z

D

hΨ(z)Υ(z0 )iD d2z.

However, in general, the integral here may be divergent because of the contact singularity of the T -product. The simple cutoff regularization δR hΥ(z0 )iΣ =

1 π

Z

Σ\Dz0 ,R

hΨ(z)Υ(z0 )iΣ d2z, Dz0 ,R = {z ∈ Σ, |z − z0 | ≤ R}

will violate sewing properties for small disks. Instead we will make an overage of such regularizations over infinitely small cutoff radiuses δhΥ(z0 )iΣ =

Z

0

∞

δr hΥ(z0 )iΣ dµ(r).

(3.2)

Here dµ is a generalized measure in R+ having support in 0 and integrable in a product with all the functions having a finite degree singularity at r = 0. Such a measure exists and is fully described by the function Λ(α) =

Z

∞

0

r 2α µ(r) dr (α ∈ R),

satisfying Λ(0) = 1,

Λ(α) = 0 (α > A)

for some positive A.

2

4

Residue-like operations

We will call local linear operators from the space of functions on ΣN +1 having diagonal singularities to the space of functions on Σ residue-like operators of rank N and denote them as G(z0 ) = RzN =···=z0 F (z1 , . . . , z0 ) or G(z0 ) = Rz¯N =···=¯z0 F (z1 , . . . , z0 ). Using T -product (2.1) we can define representation of residue-like operations by multilinear products in the space of vertex operator functions: {Υi }N i=1 −→ Υ = RzN =···=z0 Υ0 . . . ΥN , def

hΥ(z0 )iΣ = RzN =···=z0 hΥ0 . . . ΥN iΣ .

The deformation (3.2) induce the deformation of this representation . =···=z Υ0 (z0 ) · · · Υn (zn )Ψ(z). δRzn =···=z0 Υ0 (z0 ) · · · Υn (zn ) = Rz =z n 0

(4.3)

. =···=z is a next rank residue-like operation defined as Here Rz =z n 0 . =···=z F = Rzn =···=z0 Rz =z n 0

1 π

Z

Σ

F d2z −

1 π

Z

Σ

Rzn =···=z0 F d2z.

(4.4)

It is, indeed, a residue-like operation, because the right part of (4.4) does not depend on the area of integration as far as it includes z0 . We will call this operation a successor of Rzzn =···=z0 . A successor of antiholomorphic derivative can be shown to be . z F = −Resz=z0 F. ∂z¯=¯ 0 Here Resz=zi is a generalized residue operation defined for nonholomorphic functions as Resz=z0

5

(z − z0 )k Λ(k − α) = δk,−1 2α |z − z0 | k!(k + 1)!

(α ∈ R, k ∈ Z).

Finite Deformations

Let deformed amplitudes be defined by the formula h0iΨ Σ



1 = exp π 3

Z

2

Ψd z Σ



. Σ

Here Ψ is some vertex operator function (not necessarily primary), and the contact divergences are regularized by the method (3.2). Then the sewing property is automatically satisfied, and only the condition of conformal invariance remains to be implemented. Hereafter we mark all the deformed objects with the superscript symbol of the vertex operator function parameterizing the deformation. For parametrization of deformed vertex operators we will use the formula 

hΥ(z0 )iΨ Σ = Υ(z0 ) exp

1Z Ψ(z) d2z π Σ



. Σ

The energy-momentum tensors for the family of deformed theories parametrize by scaled vertex operator function τ Ψ (τ ∈ R) can be shown to obey the differential equation ∂ τΨ τΨ T (z) = Bzτ1Ψ=z Ψ(z1 )Tzz (z) + Aτz¯1Ψ=¯z Ψ(z1 )Tzτz¯Ψ (z) ∂τ zz ∂ τΨ τΨ T (z) = Aτz1Ψ=z Ψ(z1 )Tzz (z) + Bzτ¯1Ψ=¯z Ψ(z1 )Tzτz¯Ψ (z) + Ψ(z). (5.5) ∂τ z z¯ Here Az1 =z , Bz1 =z are residue-like operations satisfying ∂z Az1 =z + ∂z¯Bz1 =z = Resz1 =z + Resz=z1 . Differentiating (5.5) with respect to τ and applying (4.3) we can recurrently calculate all the higher derivatives of the energy-momentum tensor and then substitute them to the Taylor expansion. Thus we have a perturbative formula for T Ψ with higher order terms expressed through residue-like operations Resz=z0 , Az=z0 , Bz=z0 and their successors (4.4). Details on calculation of these operations can be found in [7]. The transformation of energymomentum tensor Ψ,Φ Ψ Tzz = Tzz − ∂zΨ Φz , Ψ Ψ Tz¯Ψ,Φ z¯ = Tz¯z¯ − ∂z¯ Φz¯,

Ψ Ψ TzΨ,Φ z¯ = Tz z¯ + ∂z¯ Φz , Tz¯Ψ,Φ = Tz¯Ψz + ∂zΨ Φz¯ z

does not affect translation operators L−1 , L−1 . If the theory is conformally symmetrical, there exists Φz , Φz¯ trivializing the contradiagonal components of T Ψ,Φ . Then the diagonal components of T Ψ,Φ , will be (anti)holomorphic. They can be used for calculation of the deformed Virasoro representation. 4

6

Symmetries

The transformations of Ψ δξ Ψ(z) = ∂z¯ξz (z) + Resz1 =z ξz (z1 )Ψ(z ) + z ↔ z¯ + O(Ψ2)

(6.6)

| {z } asym

can be shown not to affect equivalence classes of theories. They are parametrized by the pair of vertex operator functions ξ = (ξz , ξz¯). The corresponding covariant transformation of vertex operators are ˆ 0 ) = Resz¯=¯z ξz (z)Υ(z0 ) + 1 Res . ¯ + O(Ψ2 ). ξΥ(z 0 z2 =z1 =z0 Ψ(z2 )ξ(z1 )Υ(z0 ) + z ↔ z 2 The symmetry transformation of Φ is defined by the requirement for the energy-momentum tensor T Ψ,Φ to transform covariantly. Note that the commutator of such symmetries depend on Ψ and regularization parameters. Some symmetries related to global spacetime transformations where also described in [9,10]

7

Linear approximation

In the linear approximations the energy-momentum tensor and symmetries can be shown to be TzΨ,Φ Tz¯Ψ,Φ = OΨ − L−1 Φz , z¯ = OΨ − L−1 Φz , z Ψ,Φ Ψ,Φ Tzz = Tzz + L−1 Φz , Tz¯z¯ = Tz¯z¯ + L−1 Φz¯, ¯ 1 ξz − L1 ξz¯, δξ Φz = Oξz , δξ Φz¯ = Oξz¯. δξ Ψ = −L Here O =1+

∞ X

(L−1 )j Lj , j=0 (j + 1)!

O =1+

¯ −1 )j L ¯j (L . j=0 (j + 1)! ∞ X

Then the equations Ψ,Φ TzΨ,Φ =0 z¯ = Tz¯z

are satisfied if Φ is trivial and Ψ is a primary field. It corresponds to the deformations of the Virasoro operators LΨ k = Lk +

1 2πi

I

Γ

¯Ψ = L ¯k + 1 L k 2πi

Ψ(z − z0 )k d¯ z, 5

I

Γ

Ψ(¯ z − z¯0 )k dz,

which is equivalent to the deformations proposed in [1]. Some deformations corresponding to nonprimary fields were first found in [11,12] in the lowenergy limit. This relaxation of the equations of motion is compensated by the symmetries and does not create additional physical degrees of freedom.

8

Low-energy limit

Let us consider deformations corresponding to the vertex operator function ¯ µ (z ′ ), where Hνµ is a Hermitian matrix with slowly-varying Ψ = Hνµ ∂X ν ∂X coefficients. Then the coordinate vertex operators can be shown to have the obey νµ Ψ ′ hX ν (z ′ )X µ (z)iΨ Σ ≈ −2h: g (z) :iΣ ln |z − z|, ∂¯Ψ∂ Ψ : ψ : ≈ : (ψ;νµ + iψ ;η dωηνµ ) ∂ ΨX ν ∂¯ΨX µ : .

The contravariant metric g νµ and skew symmetric tensor ωνµ here are equal to ! Z 1 −1 T ∂ T −1 (U ) g = U(1)U (1), ω = ℑ U dτ , ∂τ 0 where 

√

U(τ ) = cosh τ HH T Local spacetime symmetries δgνµ = εν;µ + εµ;ν ,



  √ T HH sinh τ √ + HT . HH T

δωνµ = εη ωνµ;η + εν;η ωηµ + εµ;η ωνη + dςνµ

are a particular case of more general symmetries (6.6) with the following choice of parameters: ¯ µ: . ξz¯ =: (εµ − iςµ )∂X

ξz =: (εν + iςν )∂X ν :,

The contradiagonal components of the energy-momentum tensor can be shown to be 1 TzΨz¯ ≈ Tz¯Ψz ≈ − : (Rνµ − dωνσρ dωµσρ + idωηνµ;η ) ∂ ΨX ν ∂¯ΨX µ : . 2

6

Putting here Φz = ∂zΨ : φ :, Φz¯ = ∂z¯Ψ : φ : we will come to the following equations of motion: Rνµ = dωνσρ dωµσρ + 2φ;νµ ,

dωηνµ;η = 2φ;η dωηνµ .

As a consequence of this equations it can be shown that 2 22φ + 4φ;ν φ;ν = m2 − dω νσρ dωνσρ . 3 The parameter m here is the topological constant of the theory which can be interpreted as a dilaton mass It is responsible for deformation of central charge 1 c = D + m2 . 2

Acknowledgement I am very grateful to Mark Evans for inspiring me to develop the deformation approach, for discussions and for many important suggestions. It is also a pleasure to thank Toni Weil for significant editing help.

7

References [1] M. Evans and B. Ovrut, Phys. Rev. D 41, 3149 (1990). [2] A. Sen, Nucl. Phys. B 345, 551 (1990). [3] M. Campbell, P. Nelson and E. Wong, Int. J. Mod. Phys. A 6, 4909 (1991). [4] B. Zwiebach, Nucl. Phys. B 390, 33 (1993), hep-th/9206084. [5] K. Ranganathan, H. Sonoda and B. Zwiebach, Report FERMILAB-PUB-MIT-CTP-2193, April 1993, hepth/9304053. [6] Alvarez-Gaume, C. Gomez, G. Moore and C.Vafa, Nucl. Phys B303, 445 (1988). [7] G. Pelts, Phys. Rev. D 51, No 2, 30 pp., hep-th/9406022. [8] G.B. Segal, in Proceedings of the XVI International Conference on Differential Geometrical Methods in Theoretical Physics, Como, Italy, 1987, edited by K. Bleuler and M. Werner, (Univ. di Como, Como, Italy, 1987), pp. 165-171. [9] M. Evans and I. Giannakis , Symmetry Principles for String Theory CTP-TAMU-64/94, In Proceedings of G¨ ursey memorial conference, Istanbul, Turkey, June, 1994, ACT-22/94, RU-94B-8. 5pp (to be published by Spring Verlag), hep-th/9411165 [10] M. Evans and I. Giannakis, Phys. Rev. D 44, 2467 (1991). [11] Mark Evans, Ioannis Giannakis and D.V. Nanopoulos, Report RU93-8, CTP-TAMU-2/94, CERN-TH.7022/93, hepth/9401075. [12] M. Evans and I. Giannakis, in Proceedings of the XX International Conference on Differential Geometric Methods in Theoretical Physics, edited by S. Catto and A. Rocha, New York, 1991 (City College, New York, 1991), pp. 759-768, hepth/9109055. 8