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Physics Letters B 768 (2017) 305–308

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Discussion of the duality in three dimensional quantum ﬁeld theory

Chen-Te Ma

Department of Physics and Center for Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan, ROC

a r t i c l e i n f o a b s t r a c t

Article history:

Received

6 January 2017

Received

in revised form 22 February 2017

Accepted

3 March 2017

Available

online 9 March 2017

Editor: N.

Lambert

Keywords:

Renormalization

group

Chern–Simons

theories

We discuss the duality in three dimensional quantum ﬁeld theory at infrared limit. The starting point

is to use a conjecture of a duality between the free fermion and the interacting scalar ﬁeld theories at

the Wilson–Fisher ﬁxed point. The conjecture is useful for deriving various dualities in three dimensions

to obtain a duality web. The study is also interesting for understanding the dualities, or equivalence

of different theories from the perspective of the renormalization group ﬂow. We ﬁrst discuss the

“derivation” without losing the holonomy. Furthermore, we also derive these dualities from the mean-

ﬁeld

study, and consider the extension of the conjecture or dualities at ﬁnite temperature.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

Duality is an equivalence of two theories from some non-trivial

transformations. The duality is a useful tool to study theoretical

physics because we could use different mathematical ways to see

one physical phenomenon. Especially, we are particularly inter-

ested

in dualities between the weakly coupled and strongly cou-

pled

ﬁeld theories to study the strongly coupled ﬁeld theories from

the weakly coupled ﬁeld theories.

Three

dimensional quantum ﬁeld theory at infrared limit is par-

ticularly

interesting because of time reversal symmetry anomaly in

free Dirac fermion. The massless Dirac fermion at quantum level

is not a well-deﬁned theory and the massive Dirac fermion the-

ory

could be related to the Abelian Chern–Simons term so one

way of ﬁnding a well-deﬁned theory at infrared limit is to com-

bine

the massless Dirac fermion and the induced Abelian Chern–

Simons

term, which comes from the one-loop effective potential

of the Dirac fermion [1,2]. When one considers ﬁnite tempera-

ture,

the Dirac fermion is not enough to be combined only the

lowest non-trivial order term, the induced Abelian Chern–Simons

term, to obtain the gauge symmetry in the effective theory. Thus,

the resummation is necessary for restoring the gauge symmetry of

the effective theory at the Wilson–Fisher ﬁxed point. The theory

at zero temperature is also conjectured to dual to the interact-

ing

scalar ﬁeld theory at the Wilson–Fisher ﬁxed point coupled to

the BF and Chern–Simons theories in the manifold with a chosen

spin structure. The conjecture at zero temperature could exhibit

the known particle-vortex duality [3] so it should be a concrete

E-mail address: yefgst@gmail.com.

evidence. An interesting fact is that the conjecture also gives a

duality web to ﬁnd the relation between the free theory and in-

teracting

theory [3–13].

this paper, we discuss the “derivation” (We use the “deriva-

tion”

because one derives these dualities from a conjecture.) of the

dualities in three dimensional quantum ﬁeld theory. The “deriva-

tion”

starts from one conjecture to derive other dualities. When

one “derives” the dualities, one needs to integrate out gauge ﬁeld

to solve da = db, where a and b are gauge ﬁelds. The general so-

lution

of da = db is a = b + df with a globally deﬁned f if a

and

b satisfy the Dirac quantization condition. Roughly speaking,

we could set f = 0or zero holonomy due to the gauge trans-

formation.

When a theory only has dynamical gauge ﬁelds, then

the case is not problematic. If one theory has background gauge

ﬁelds, then setting zero homonomy is not always doable because

the gauge transformation could be performed only on the dynam-

ical

ﬁelds. However, we ﬁnd that the non-trivial holonomy could

be absorbed by the scalar ﬁeld, and the non-trivial holonomy does

not affect the dualities when a boundary term is absent, and all

ﬁelds are globally deﬁned. The mean-ﬁeld study is a powerful

tool to study the phase diagram so we consider the order pa-

rameter

ψψ

in the fermion side. The above studies only focus on

the zero temperature. The phase diagram is particularly interest-

ing

when one considers the phase transition with respect to ﬁnite

temperature. The diﬃculty of the conjecture at ﬁnite temperature

is that it is necessary to include higher order terms beyond the

induced Abelian Chern–Simons term. The inclusion of the higher

order terms possibly affects the validity of the conjecture because

of the kinetic term of a gauge ﬁeld. Thus, only the induced Abelian

Chern–Simons term with a temperature dependent coeﬃcient is

considered in our conjecture. Although the effective theory loses

http://dx.doi.org/10.1016/j.physletb.2017.03.003

0370-2693/

SCOAP

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