Holographic Complex Conformal Field Theories
Antón F. Faedo ,
1
Carlos Hoyos ,
2,3
David Mateos ,
1,4
and Javier G. Subils
1
1
Departament de Física Qu `antica i Astrofísica & Institut de Ci`encies del Cosmos (ICC), Universitat de Barcelona,
Martí i Franqu`es 1, 08028 Barcelona , Spain
2
Deptartment of Physics, Universidad de Oviedo, Federico García Lorca 18, 33007 Oviedo, Spain
3
Instituto Universitario de Ciencias y Tecnologías Espaciales de Asturias (ICTEA), Universidad de Oviedo,
Calle de la Independencia, 13, 33004 Oviedo, Spain
4
Institució Catalana de Recerca i Estu dis Avançats (ICREA), Lluís Companys 23, 08010 Barcelona, Spain
(Received 8 November 2019; revised manuscript received 3 February 2020; accepted 24 March 2020; published 20 April 2020)
The loss of criticality in the form of weak first-order transitions or the end of the conformal window in
gauge theories can be described as the merging of two fixed points that move to complex values of the
couplings. When the complex fixed points are close to the real axis, the system typically exhibits walking
behavior with Miransky (or Berezinsky-Kosterlitz-Thouless) scaling. We present a novel realization of
these phenomena at strong coupling by means of the gauge/gravity duality, and give evidence for the
conjectured existence of complex conformal field theories at the fixed points.
DOI: 10.1103/PhysRevLett.124.161601
Introduction.—Fixed-point annihilation (FPA) is an inter-
esting phenomenon in which two fixed points (FPs) of the
renormalization group (RG) flow merge and disappear as
some parameter is varied. In the context of phase transitions
and critical phenomena it is associated to a change from
continuous to weak first-order transitions. Examples include
the superconducting transition in the Abelian Higgs model
[1,2], the related N´eel-valence bond-solid transition in
antiferromagnets [3–6], the ferromagnetic transition in the
Potts model [7–9], metal-Mott insulator transitions [10] and
six-dimensional OðNÞ models [11,12]. FPA has also been
associated to the boundaries of the conformal window in
gauge theories with flavors, both in (2 þ 1)-dimensional
quantum electrodynamics [13–16] and in non-Abelian
gauge theories in 3 þ 1 dimensions [17–21].
More generally, FPA has been proposed as a natural
mechanism to produce “walking behavior” in gauge theories
[22]. The idea is that, just after the merging, the critical
points leave a footprint in the form of approximate scale
invariance over a large range of scales. Typically the range
of the walking region increases exponentially as parameters
are tuned to the merging point, following Miransky (or
Berezinsky-Kosterlitz-Thouless) scaling [23–25]. This
behavior can be explained by continuing the theory to
complex values of the couplings, so that the annihilation
is understood as a migration of the FPs to the complex plane
after the merger [19]. Their effect on the RG flow is
noticeable as long as they remain close to the real axis. It
has been recently conjectured [22] that a nonunitary,
complex conformal field theory (CCFT) exists at each of
the two complex fixed points (CFPs), so that the properties
of the theory in the walking region can be derived from
perturbations of the CCFTs. Each CCFT has a complex
spectrum of operators that is the conjugate of its compan-
ion’s, implying that CFPs should always come in pairs.
Although FPA and CFPs are expected to exist generically,
their study has been mostly limited to weakly coupled
theories (see, e.g., [26–29] for recent examples), as their
identification requires computing the beta functions for the
different couplings in the theory, a task that often can only be
donevia perturbation theory. In this paper we will construct a
simple holographic model that realizes FPA and CFPs, thus
showing that these phenomena can also occur at strong
coupling. In addition, our analysis provides nonperturbative
evidence that CFPs have the conjectured properties of
CCFTs regarding the spectrum of local operators.
Fixed-point annihilation and complex CFTs.—Consider
a system with a dimensionless coupling g whose β function
depends on an external parameter α in such a way that, for
α ≃ α
,
βðgÞ ≃ ðα − α
Þ − ðg − g
Þ
2
: ð1Þ
We will see an explicit example in the next section. If
α > α
, discarding higher-order terms, the β function
vanishes at two values
g
¼ g
ffiffiffiffiffiffiffiffiffiffiffiffiffi
α − α
p
: ð2Þ
To make sure that the theory is well defined in the far
ultraviolet (UV) we may imagine that β has another zero at
Published by the American Physical Societ y under the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation,
and DOI . Funded by SCOAP
3
.
PHYSICAL REVIEW LETTERS 124, 161601 (2020)
0031-9007=20=124(16)=161601(6) 161601-1 Published by the American Physical Society
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