CECS-PHY-2205 Nonlinear automorphism of the conformal algebra in 2D and continuousp

2025-04-27 0 0 398.26KB 14 页 10玖币

侵权投诉

CECS-PHY-22/05

Nonlinear automorphism of the conformal algebra in

2D and continuous √T

Tdeformations

David Tempo,aRicardo Troncosob,c

aDepartamento de Ciencias Matem´aticas y F´ısicas, Universidad Cat´olica de Temuco, Montt 56,

Casilla 15-D, Temuco, Chile.

bCentro de Estudios Cient´ıﬁcos (CECs), Av. Arturo Prat 514, Valdivia, Chile.

cFacultad de Ingenier´ıa, Arquitectura y Dise˜no, Universidad San Sebasti´an, sede Valdivia, General

Lagos 1163, Valdivia 5110693, Chile

E-mail: jtempo@uct.cl,troncoso@cecs.cl,ricardo.troncoso@uss.cl

Abstract: The conformal algebra in 2D (Diﬀ(S1)⊕Diﬀ(S1)) is shown to be preserved un-

der a nonlinear map that mixes both chiral (holomorphic) generators Tand ¯

T. It depends

on a single real parameter and it can be regarded as a “nonlinear SO(1,1) automorphism.”

The map preserves the form of the momentum density and naturally induces a ﬂow on

the energy density by a marginal √T¯

Tdeformation. In turn, the general solution of the

corresponding ﬂow equation of the deformed action can be analytically solved in closed

form, recovering the nonlinear automorphism. The deformed CFT2can also be described

through the original theory on a ﬁeld-dependent curved metric whose lapse and shift func-

tions are given by the variation of the deformed Hamiltonian with respect to the energy

and momentum densities, respectively. The conformal symmetries of the deformed the-

ories can then also be seen to arise from diﬀeomorphisms that fulﬁll suitably deformed

conformal Killing equations. Besides, Cardy formula is shown to map to itseft under the

nonlinear automorphism. As a simple example, the deformation of Nfree bosons is brieﬂy

addressed, making contact with recent related results and the dimensional reduction of the

ModMax theory. Furthermore, the nonlinear map between the conformal algebra in 2D

and its ultra/non-relativistic versions (BMS3≈CCA2≈GCA2), including the corresponding

ﬁnite √T¯

Tdeformation, are recovered from a limiting case of the nonlinear automorphism.

The extension to a three-parameter nonlinear ISO(1,1) automorphism of the conformal

algebra, and a discrete nonlinear automorphism of BMS3are also brieﬂy discussed.

arXiv:2210.00059v1 [hep-th] 30 Sep 2022

Contents

1 Introduction 2

2 Nonlinear SO(1,1) automorphism 3

3 Continuous √T¯

Tdeformations 3

4 Geometric realization 5

5 Overview and ending remarks 7

–1–

1 Introduction

Symmetries spanned by inﬁnite-dimensional Lie algebras tightly constrain the possibilities

to devise theories or computing amplitudes, among others. Along these lines, consistent

nonlinear maps between the generators of inﬁnite-dimensional linear algebras can also be

very useful, but since they are just sporadically found, the systematic study of this kind

of mappings turns out to be uncharted territory either in physics or mathematics (for

a recent related discussion see [1]). A time-honored example of this sort is the Sugawara

construction [2], which is known to play a very relevant role in the context of conformal ﬁeld

theories (CFT’s) in two-dimensional spacetimes. In this class of mapping, the generators

of the Virasoro algebra are obtained from a precise quadratic combination of those of an

aﬃne Kac-Moody algebra (see e.g. [3,4]). Similar quadratic Sugawara-like relationships,

realized in a variety of setups, are also known to exist for the ultra/non-relativistic limit

of the conformal algebra in 2D (CCA2≈GCA2≈BMS3)) [5–9], and its supersymmetric

extensions [10–15] whose generators also emerge from diﬀerent quadratic combinations of

current algebras.

Another type of nonlinear mappings of the class under discussion is naturally formu-

lated in the context of integrable systems. Indeed, since the Virasoro algebra describes

one of the Poisson structures of the KdV hierarchy (see e.g., [16–18]), their generators are

also nonlinearly related to the inﬁnite-dimensional Abelian algebra of conserved charges,

spanned by the subset of commuting generators of the enveloping algebra. A similar non-

linear relationship of this kind is also known to hold for the generators of the BMS3algebra

[19].

A diﬀerent kind of nonlinear map that requires going beyond the enveloping algebra has

been recently introduced in [20], which relates the generators of the classical (nonanoma-

lous) conformal algebra in 2D (Diﬀ(S1)⊕Diﬀ(S1)) with those of the BMS3algebra. Under

this mapping, the Hamiltonian of a generic CFT2, with chiral (holomorphic) conformal gen-

erators given by Tand ¯

T, acquires a ﬁnite marginal non-analytic deformation determined

by √T¯

T, so that the deformed theory it is no longer a CFT2, but a conformal Carrollian

ﬁeld theory instead, because its energy and momentum densities fulﬁll the BMS3algebra1.

Remarkably, no limiting process is involved in the nonlinear map between the conformal

algebra and its ultra/non-relativistic version. This map, and its corresponding deformation

were then shown to be recovered through a class of “inﬁnite boosts” spanned by certain

degenerate (non-invertible) linear transformations acting on the coordinates [31]. Further

recent interesting results about continuous √T¯

Tdeformations have also been addressed for

ﬁeld theories in 2D along diﬀerent approaches and points of view in [32], [33], [34] (see also

[35], [36]), and for the deformation of two free harmonic oscillators in classical mechanics

[37].

One of the main purposes of our work is showing that the conformal algebra in 2D

admits a “nonlinear SO(1,1) automorphism” that maps the algebra to itself, which once

1These marginal deformations are clearly diﬀerent from the well-known irrelevant T¯

Tones [21–23], whose

diverse properties have been studied in e.g., [24–29] (for a review and further references see also [30]).

–2–

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

CECS-PHY-22/05Nonlinearautomorphismoftheconformalalgebrain2DandcontinuouspTTdeformationsDavidTempo,aRicardoTroncosob;caDepartamentodeCienciasMatematicasyFsicas,UniversidadCatolicadeTemuco,Montt56,Casilla15-D,Temuco,Chile.bCentrodeEstudiosCientcos(CECs),Av.ArturoPrat514,Valdivia,Chile.cFacult...

展开>> 收起<<

CECS-PHY-2205 Nonlinear automorphism of the conformal algebra in 2D and continuousp.pdf

共14页,预览3页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

CECS-PHY-2205 Nonlinear automorphism of the conformal algebra in 2D and continuousp

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: