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arXiv:hep-th/9802079v1 11 Feb 1998

February 1, 2008LBNL-41142, UCB-PTH-98/01

hep-th/9802079

Conformal Field Theories: From Old to New1

J. de Boer and M. B. Halpern

Department of Physics, University of California at Berkeley

366 LeConte Hall, Berkeley, CA 94720-7300, U.S.A.

and

Theoretical Physics Group, Mail Stop 50A–5101

Ernest Orlando Lawrence Berkeley National Laboratory

Berkeley, CA 94720, U.S.A.

Abstract

In a short review of recent work, we discuss the general problem of con-

structing the actions of new conformal field theories from old conformal field

theories. Such a construction follows when the old conformal field theory ad-

mits new conformal stress tensors in its chiral algebra, and it turns out that

the new conformal field theory is generically a new spin-two gauge theory. As

an example we discuss the new spin-two gauged sigma models which arise in

this fashion from the general conformal non-linear sigma model.

1To appear in a memorial issue of Theoretical and Mathematical Physics in memory of F.A. Lunev.

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1 Introduction

We are saddened by the death of Dr. F.A. Lunev and offer this contribution in his

honor.

The problem of constructing new conformal field theories from old conformal field

theories dates back to K-conjugation covariance [1–5], the coset constructions [1–3] and

the general affine-Virasoro construction [5–7]. Such a construction follows when the old

conformal field theory admits new conformal stress tensors in its chiral algebra. The

simplest examples of this construction are the coset constructions, whose (new) spin-

one gauged WZW actions [8–11] are obtained from the (old) WZW actions. The coset

constructions are however a special case of higher symmetry, and the problem of finding

the actions of the generic affine-Virasoro constructions was solved in [12–14], where the

(new) spin-two gauged WZW actions of these theories were obtained from the (old) WZW

action.

Recently, we have extended [15–17] this program to the new conformal field theories

which can be obtained in this way from the (old) general non-linear sigma model, and

this memorial issue provides an opportunity to review the program here in general terms.

The discussion of the second and third sections is based on material originally discussed

in [12,18], and the discussion of Section 4 is based on material originally discussed in

[15–17].

2 Action Formulation of a CFT

Consider a conformal field theory (CFT) C∗with chiral/antichiral stress tensors T∗,

¯T∗and associated chiral/antichiral algebras A∗,¯ A∗

T∗∈ A∗,

¯T∗∈¯ A∗

(2.1)

where A∗,¯ A∗are defined to include all mutually local holomorphic/antiholomorphic ob-

jects in the theory. The naive Hamiltonian of C∗is

H∗0=

?2π

0

dσ H∗0,H∗0= T∗+¯T∗

(2.2)

but C∗is a gauge theory if the centralizers A′

∗,¯ A′

∗of H∗0in A∗,¯ A∗,

A′

∗= {X(z) ∈ A∗|[X(z),T∗] = 0},

¯ A′

∗= {X(¯ z) ∈¯ A∗|[X(¯ z),¯T∗] = 0} (2.3)

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are non-trivial1. We assume for simplicity that all elements of the centralizer can be ex-

pressed as differential polynomials in terms of a finite local set of basis elements, which

we also denote by A′

bras, and the corresponding positive frequency modes of the centralizers will be denoted

by A′

the Hamiltonian H∗0acting on a physical Hilbert space defined by

∗,¯ A′

∗. The centralizers are in fact infinite dimensional Z-graded alge-

∗(+),¯ A′

∗(+). Following Gupta and Bleuler, the theory C∗can then be described by

A′

∗(+)|phys? =¯ A′

∗(+)|phys? = 0. (2.4)

This means that the physical states are primary under the algebras A′

∗,¯ A′

∗.

When C∗has a smooth classical limit, we expect that the theory has an action de-

scription, To find the action one must first find the classical limit or Poisson bracket

description of the objects and algebras described above. We assume for simplicity that

the classical limit of the centralizer contains no central terms, that is, the basis elements

of the centralizer form a set of first-class constraints in the language of Dirac. Then the

full classical Hamiltonian of the the CFT C∗can be written as

H∗=

?

dσ H∗,H∗= H∗0+ v · A′

∗+ ¯ v ·¯ A′

∗

(2.5)

where v, ¯ v are Lagrange multipliers, and the action S∗ of C∗ is obtained by the usual

canonical method. The multipliers v, ¯ v form world-sheet gauge fields whose spins are

those of the corresponding elements of the centralizers A′

∗,¯ A′

∗of C∗.

3 New CFT’s

We focus now on a case of particular interest, when the chiral/antichiral algebras

A∗,¯ A∗of C∗contain two new chiral/antichiral spin-two objects T and¯T which satisfy

commuting Virasoro algebras.In this case, one expects the existence of two further

chiral/antichiral spin-two tensors˜T,˜¯T, so that all four stess tensors

T,˜T,¯T,˜¯T; T,˜T ∈ A∗,

¯T,˜¯T ∈¯ A∗

(3.1)

are commuting Virasoro operators. The four new stress tensors sum in pairs to the stress

tensors T∗,¯T∗of C∗,

T∗= T +˜T,

¯T∗=¯T +˜¯T (3.2)

1Usually one defines the chiral algebra of a conformal field theory such that these centralizers contain

only the unit operator, the gauge degrees of freedom having already been modded out. However, we first

allow for a more general situation where the conformal field theory is embedded in a larger gauge-covariant

system, modding out later by the gauge degrees of freedom.

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and this is called K-conjugation covariance [1–5]. This phenomenon is most familiar in

the coset constructions (where T∗= Tg, T = Tg/h,˜T = Th) and is explicit in the general

affine-Virasoro construction, but the phenomenon of K-conjugation covariance was argued

quite generally in [4].

In the presence of K-conjugation covariance we see that C∗is a tensor product CFT

composed of the K-conjugate pair of CFTs C and˜C

C∗= C ⊗˜C (3.3a)

C : T,¯T,

˜C :˜T,˜¯T (3.3b)

whose stress tensors are shown in (3.3b). In what follows we focus on the C theory,

but the corresponding description of the˜C theory can be obtained at any stage of the

discussion by K-conjugation.

The naive Hamiltonian of the C theory is

H0=

?

dσH0,H0= T +¯T (3.4)

and we see that C is generically1a spin-two gauge theory because the Virasoro operators

˜T and˜ ¯T are in the centralizer A′,¯ A′of H0. In the classical limit, the full Hamiltonian of

the generic theory C is therefore

H =

?

dσH,H = T +¯T + v˜T + ¯ v˜¯T (3.5)

where v and ¯ v form a spin-two gauge field on the world-sheet.

Generically, the centralizer of H0is nothing but˜T and˜ ¯T so the Hamiltonian (3.5) is

the proper description of the generic new CFT C. In special cases of higher symmetry

however, we must gauge the new theory C by the full centralizer of H0,

H = T +¯T + v · A′+ ¯ v ·¯ A′

(3.6)

where A′, ¯ A′may satisfy Virasoro algebras, W3 algebras, etc. Adding to (3.6) a term

proportional to v¯ v, one can also include local spin-one symmetries, which are associated

to affine Lie algebras. The actions of the new CFTs then follow by the usual canonical

prescription.

1The case of the g/h coset constructions is a special case of higher symmetry: These are spin-one gauge

theories because the centralizer of Tg/his generated by the h-currents, with Tha composite operator.

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4New Spin-Two Gauged Sigma Models

We have recently studied this program [16,17] starting with the action of the general

conformal non-linear sigma model

S∗= SG=

?

d2ξ LG,LG= (Gij+ Bij)∂+xi∂−xj

(4.1)

with conformal stress tensors TG,¯TG. Taking the sigma model as a background, we looked

for all four new chiral/antichiral stress tensors T,˜T,¯T,˜¯T in the generic form

T ∼ Lij∂+xi∂+xj

(4.2)

and found that they exist in K-conjugate pairs

T∗= TG= T +˜T,

¯T∗=¯TG=¯T +¯TG

(4.3)

under certain conditions on the coefficients Lij, given below (see (4.5a) and (4.5b)). In-

cluding the dilaton, the quantum extension of (4.2) and (4.3) has been verified at the

one-loop level in [15,16].

Following the program outlined above, we found that the generic new conformal field

theories C = T,¯T are described by the following set of new spin-two gauged sigma models

S =

?

d2ξ L(4.4a)

L = LG+

1

4πα′[α˜LijBiBj+ ¯ α˜ ¯Lij¯Bi¯Bj

−(Bi− ∂+xi)Gij(¯Bj− ∂−xj)](4.4b)

ˆ∇+

i˜Ljk=ˆ∇−

i˜ ¯Ljk= 0

˜ ¯Lij= 2˜ ¯Lik˜ ¯Lkj.

(4.5a)

˜Lij= 2˜Lik˜Lkj, (4.5b)

Here α, ¯ α form a spin-two gauge field, closely related to the multipliers v, ¯ v, and B,

¯B are auxiliary fields. One new C = T,¯T conformal field theory is obtained for each

solution of the conditions in (4.5a), (4.5b). The gradientsˆ∇±in (4.5a) are generalized

covariant derivatives with torsion. See [17] for further details, including the spin-two gauge

invariance of these actions and their non-linear form after integrating out the auxiliary

fields.

The relations (4.5a), (4.5b) are nothing but the conditions that the classical chiral

algebras are closed, and necessary and sufficient conditions for their solution are also

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