Oct 2022 Three-point functions of conserved currents in 3D CFT general formalism for arbitrary spins

2025-05-02 0 0 2.45MB 56 页 10玖币

侵权投诉

Oct, 2022

Three-point functions of conserved currents in 3D

CFT: general formalism for arbitrary spins

Evgeny I. Buchbinder and Benjamin J. Stone

Department of Physics M013, The University of Western Australia

35 Stirling Highway, Crawley W.A. 6009, Australia

Email: evgeny.buchbinder@uwa.edu.au,

benjamin.stone@research.uwa.edu.au

Abstract

We analyse the general structure of the three-point functions involving conserved

bosonic and fermionic higher-spin currents in three-dimensional conformal ﬁeld the-

ory. Using the constraints of conformal symmetry and conservation equations, we

use a computational formalism to analyse the general structure of hJs1J0

s2J00

s3i, where

Js1,J0

s2and J00

s3are conserved currents with spins s1,s2and s3respectively (integer

or half-integer). The calculations are completely automated for any chosen spins

and are limited only by computer power. We ﬁnd that the correlation function is

in general ﬁxed up to two independent “even” structures, and one “odd” structure,

subject to a set of triangle inequalities. We also analyse the structure of three-point

functions involving higher-spin currents and fundamental scalars and spinors.

arXiv:2210.13135v2 [hep-th] 26 Oct 2022

Contents

1 Introduction 2

2 Conformal building blocks 5

2.1 Two-point building blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Three-point building blocks . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 General formalism for correlation functions of primary operators 8

3.1 Two-pointfunctions .............................. 8

3.2 Three-pointfunctions.............................. 9

3.2.1 Conserved currents . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2.2 Comments on diﬀerential constraints . . . . . . . . . . . . . . . . . 10

3.2.3 Auxiliary spinor formalism . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.4 Generating function method . . . . . . . . . . . . . . . . . . . . . . 15

4 Correlation functions involving bosonic currents 20

4.1 Energy-momentum tensor and vector current correlators . . . . . . . . . . 22

4.2 Higher-spin correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Mixed correlators involving fermionic currents 28

5.1 Spin - 3/2 current correlators . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.2 Higher-spin correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6 Correlators involving scalars and spinors 37

6.1 Low-spincorrelators .............................. 38

6.2 Higher-spin correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

7 Discussion 43

A 3D conventions and notation 44

B More examples of higher-spin correlators 46

References 50

1 Introduction

It is widely understood that in any conformal ﬁeld theory, the general structure of

three-point correlation functions is determined up to ﬁnitely many parameters by confor-

mal symmetry. However, it remains a non-trivial problem to construct explicit solutions

for three-point functions for various classes of primary operators. Among the most im-

portant primary operators are conserved currents, whose scale dimension saturates the

unitarity bound. The fundamental examples of conserved currents in any conformal ﬁeld

theory are the energy-momentum tensor and vector currents; the three-point functions of

these currents were analysed in [1, 2], where a systematic approach to study correlation

functions of primary operators was introduced (see also refs. [3–12] for earlier works). The

analysis was performed in general dimensions, however, it did not consider higher-spin

conserved currents, which can exist in more general conformal ﬁeld theories. It also did

not account for the possibility of parity-violating structures, which appear in the three-

point functions of the energy-momentum tensor and vector currents in three-dimensions.

These structures were found in [13], where correlation functions of higher-spin conserved

currents were considered, and were also found to contain parity-violating structures. Soon

after, it was proven in [14] that under certain assumptions (which are, however, violated

in the presence of fermionic higher-spin currents) that all correlation functions involving

the energy-momentum tensor and higher-spin currents are equal to those of free theories.

This is an extension of the Coleman-Mandula theorem [15] to conformal ﬁeld theories; it

was originally proven in three dimensions and was later generalised to four- and higher-

dimensional cases in [16–20]. There are also approaches to the construction of correlation

functions of conserved currents which make use of embedding formalisms [21–26] (see also

[27, 28] for supersymmetric extensions), while others carry out the calculations in momen-

tum space [29–38]. Results have also been obtained within the framework of the AdS/CFT

correspondence (see e.g. [39–43]). The study of correlation functions of conserved currents

has also been extended to superconformal ﬁeld theories in diverse dimensions [44–59].

The general structure of the three-point functions of conserved higher-spin, bosonic,

vector currents was proposed by Giombi, Prakash and Yin [13] in three dimensions, and

further analysis was undertaken by Stanev [17, 18, 60] (see also [61, 62]) in the four

dimensional case, and by Zhiboedov [16] in general dimensions. Despite the obvious

success, the analysis in [13, 16, 17] appears to have some limitations. First, the results

only apply to conserved currents of integer spin. Second, it is unclear how the results

comprise all linearly independent structures for a given choice of spins. In particular,

in [16, 17], the conserved three-point functions are presented in the form of generating

functions which are proposed (to best of our understanding) without proof of the latter.

In this paper, we develop a formalism to study the general structure of the three-point

correlation function

hJs1(x1)J0

s2(x2)J00

s3(x3)i,(1.1)

in three-dimensional conformal ﬁeld theory, assuming only the constraints imposed by

conformal symmetry and conservation equations. Here by Jswe denote a conserved cur-

rent of spin s. Our formalism is suitable for both integer and half-integer spin. Within

our approach we reproduce all known results concerning the structure of three-point func-

tions of bosonic conserved currents and also extend the results to three-point functions

involving currents of an arbitrary half-integer spin. We also apply it to correlation func-

tions of scalar/spinor operators thus covering essentially all possible three-point function

in three-dimensional conformal ﬁeld theory. Our method is exhaustive; ﬁrst we construct

all possible structures for the correlation function for a given set of spins s1, s2and s3,

consistent with its conformal properties. We then systematically extract the linearly inde-

pendent structures and then, ﬁnally, impose the conservation equations and symmetries

under permutations of spacetime points. As a result we obtain the three-point function

in a very explicit form which can be explicitly presented even for relatively high spins.1

Our method can be applied for arbitrary s1, s2and s3and is limited only by computer

power. Due to these limitations we were able to carry out computations up to si= 20,

however, with a suﬃciently powerful computer one could probably extend our results up

to si∼50 as in [17]. We demonstrate that in all cases with si≤20, including examples

involving conserved half-integer spin currents, that the correlation function is ﬁxed up to

the following form:

hJs1J0

s2J00

s3i=a1hJs1J0

s2J00

s3iE1+a2hJs1J0

s2J00

s3iE2+bhJs1J0

s2J00

s3iO.(1.2)

where hJs1J0

s2J00

s3iE1and hJs1J0

s2J00

s3iE2are parity-even solutions (in the bosonic case cor-

responding to free bosonic and fermionic theories respectively), while hJs1J0

s2J00

s3iOis a

1A similar analysis can also be done in the four-dimensional case and will appear elsewhere.

parity-violating (or parity-odd) solution. Parity-odd solutions are unique to three di-

mensions, and have been shown to correspond to Chern-Simons theories interacting with

parity-violating matter [63–73].2Further, the existence of the odd solution depends on a

set of triangle inequalities:

s1≤s2+s3, s2≤s1+s3, s3≤s1+s2.(1.3)

When the triangle inequalities are simultaneously satisﬁed there are two even solutions,

and one odd solution. However, when any one of the above relations is not satisﬁed there

are only two even solutions; the odd solution is incompatible with conservation equations.

The analysis quickly becomes cumbersome due to the proliferation of tensor indices; to

streamline the calculations we develop a hybrid, index-free formalism which combines the

approach of Osborn and Petkou [1] and a method based on contraction of tensor indices

with auxiliary spinors. This method is widely used throughout the literature to construct

correlation functions involving more complicated tensor operators. Our particular ap-

proach, however, describes the correlation function completely in terms of a polynomial

which is a function of a single conformally covariant three-point building block, X, and

the auxiliary spinor variables u,v, and w. Hence, one does not have to work with the

spacetime points explicitly when imposing conservation equations. To ﬁnd all solutions

for the polynomial, we construct a generating function which produces an exhaustive list

of all possible linearly dependent structures for a given set of spins using Mathematica.

With the use of pattern-matching functions, we then systematically apply linear depen-

dence relations to this set of structures to form a linearly-independent ansatz for the

correlation function. Once this ansatz is obtained, we impose conservation equations and

any symmetries due to permutation of spacetime points. The tensor structures (related to

the leading singular OPE coeﬃcient, as in [1]) may then be read oﬀ by acting on the poly-

nomials with appropriate partial derivatives in the auxiliary spinors. The computational

approach we have developed is essentially automatic and limited only by computer power;

one simply chooses the spins of the ﬁelds and the solution for the three-point function

consistent with conservation and point-switch symmetries is generated.

The results of this paper are organised as follows. In section 2 we review the essen-

tials of the group theoretic formalism used to construct correlation functions of primary

operators in three dimensions. In section 3 we develop the formalism necessary to impose

2The parity-odd terms in correlation functions involving scalars and spinors can also arise in theories

without a Chern-Simons term, for example in theories with fermions in three-dimensions, because ¯

ψψ is

a parity-odd pseudoscalar, see e.g. [74, 75]. We are grateful to S. Prakash for pointing this out.

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：

Oct,2022Three-pointfunctionsofconservedcurrentsin3DCFT:generalformalismforarbitraryspinsEvgenyI.BuchbinderandBenjaminJ.StoneDepartmentofPhysicsM013,TheUniversityofWesternAustralia35StirlingHighway,CrawleyW.A.6009,AustraliaEmail:evgeny.buchbinder@uwa.edu.au,benjamin.stone@research.uwa.edu.auAbstractW...

展开>> 收起<<

Oct 2022 Three-point functions of conserved currents in 3D CFT general formalism for arbitrary spins.pdf

共56页,预览5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！

Oct 2022 Three-point functions of conserved currents in 3D CFT general formalism for arbitrary spins

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: