Physics Letters B 759 (2016) 154–158
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Studying critical string emerging from non-Abelian vortex in four
dimensions
P. Koroteev
a,b
, M. Shifman
c,∗
, A. Yung
c,d,e
a
Perimeter Institute for Theoretical Physics, Waterloo, ON N2L2Y5, Canada
b
Department of Mathematics, University of California, Davis, CA 95616, USA
c
William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, USA
d
National Research Center “Kurchatov Institute”, Petersburg Nuclear Physics Institute, Gatchina, St. Petersburg 188300, Russia
e
St. Petersburg State University, Universitetskaya nab., St. Petersburg 199034, Russia
a r t i c l e i n f o a b s t r a c t
Article history:
Received
13 May 2016
Accepted
23 May 2016
Available
online 26 May 2016
Editor:
M. Cveti
ˇ
c
Recently a special vortex string was found [5] in a class of soliton vortices supported in four-dimensional
Yang–Mills theories that under certain conditions can become infinitely thin and can be interpreted as
a critical ten-dimensional string. The appropriate bulk Yang–Mills theory has the U(2) gauge group and
the Fayet–Iliopoulos term. It supports semilocal non-Abelian vortices with the world-sheet theory for
orientational and size moduli described by the weighted CP(2, 2) model. The full target space is R
4
×Y
6
where Y
6
is a non-compact Calabi–Yau space.
We
study the above vortex string from the standpoint of string theory, focusing on the massless states
in four dimensions. In the generic case all massless modes are non-normalizable, hence, no massless
gravitons or vector fields are predicted in the physical spectrum. However, at the selfdual point (at strong
coupling) weighted CP(2, 2) admits deformation of the complex structure, resulting in a single massless
hypermultiplet in the bulk. We interpret it as a composite “baryon.”
© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Studies of non-Abelian vortex solitons
1
[1–4] supported by
some four-dimensional Yang–Mills theories with N = 2 supersym-
metry,
resulted in identification of a bulk theory which under sev-
eral
conditions gives rise to a critical ten-dimensional string [5].
The appropriate four-dimensional Yang–Mills theory has the U (2)
gauge group, the Fayet–Iliopoulos term ξ and four matter hyper-
multiplets.
Its non-Abelian sector would be conformal if it were
not for the parameter ξ . It supports semilocal non-Abelian vortices
with the world-sheet theory for orientational and size moduli de-
scribed
by the weighted CP(2, 2) model.
2
The target space is a
six-dimensional non-compact Calabi–Yau manifold Y
6
, namely, the
resolved conifold. Including the translational moduli with the R
4
*
Corresponding author.
E-mail
address: shifman@umn.edu (M. Shifman).
1
The non-Abelian vortex strings are understood as strings carrying non-Abelian
moduli on their world sheet, in addition to translational moduli.
2
A more accurate mathematical term is the two-dimensional sigma model with
the target space O(−1)
⊕(2)
CP
1
. For brevity we will use WCP(2, 2).
target space one obtains a bona fide critical string, a seemingly
promising discovery of [5].
In
this paper we explore the spectrum of the massless modes
of the above critical closed string theory. In [5] a hypothesis was
formulated regarding parameters of the bulk theory and the cor-
responding
world sheet model necessary to make the thickness of
the vortex string vanish at strong coupling, which is required in
order to neglect higher derivative corrections in the world sheet
theory on the vortex. Here, using duality we derive an exact for-
mula
for the relation between the two-dimensional (2D) coupling
β and four-dimensional (4D) gauge coupling g
2
of N = 2SQCD.
Moreover, we identify the critical point β
∗
= 0at which the vortex
string can become infinitely thin. At this point the resolved coni-
fold
mentioned above becomes a singular conifold. As the only 4D
massless mode of the string which emerges at β
∗
= 0we identify
a single matter hypermultiplet associated with the deformation of
the complex structure of the conifold. Other states arising from the
ten-dimensional graviton are not dynamical in four dimensions. In
particular 4D graviton and unwanted vector multiplets are absent.
This is due to non-compactness of the Calabi–Yau manifold we
deal with and non-normalizability of the corresponding modes.
Then
we discuss the question of how the states seen in the
bulk theory at weak coupling are related to what we obtain from
http://dx.doi.org/10.1016/j.physletb.2016.05.075
0370-2693/
© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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