Physics Letters B 784 (2018) 307–311
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
D-branes (or not) in the non-Abelian T-dual of the SU(2) WZW model
Benjo Fraser
CUniverse Group, Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
a r t i c l e i n f o a b s t r a c t
Article history:
Received
8 July 2018
Received
in revised form 5 August 2018
Accepted
8 August 2018
Available
online 13 August 2018
Editor: N.
Lambert
Following on from a previous article [1], we consider open strings in the non-Abelian T-dual of the
SU(2)
k
WZW model, with respect to the vector SU(2) isometry. Since in this case the dual theory has
an exact CFT description, we look at the chiral algebra-preserving D-branes. The general conclusion is
that a large set of branes, while consistent boundary conditions on the worldsheet, are sent to infinity in
the target space. This suggests inconsistency of the strict non-Abelian duality with open strings, in this
case.
© 2018 The Author(s). 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
WZW models and their cosets [2]are fruitful models to con-
sider
in string theory. This is because they enjoy an extension of
the Virasoro algebra to larger chiral algebras, with respect to which
they are rational. This allows the spectrum to be worked out ex-
actly
as a function of the level k, which typically parametrizes the
curvature of the target manifold. It also provides, via arguments
originally due to Cardy [3], an exact CFT description of certain open
string sectors, i.e. D-brane configurations. D-branes possess charges
conjectured [4]to be valued in the twisted K-theory of the tar-
get
space, and in the case of the SU(N)
k
WZW model the ‘Cardy
branes’, with respect to appropriate subalgebras, were shown to
furnish a complete set of representatives of this space of charges.
Given
this sort of solubility, is natural to explore these mod-
els
as fully as possible. One avenue is Abelian and non-Abelian
T-duality [5]. It was shown in [6] that the classical action for the
non-Abelian T-dual of the SU(2)
k
WZW model with respect to the
vector SU(2) isometry can be obtained as the limit of the coset
model
ˆ
su(
2)
k
⊕
ˆ
su(2)
ˆ
su(
2)
k+
(1)
in which →∞, and we must also zoom in close to the identity in
the corresponding coset geometry, parametrizing the SU(2)
group
element as g
2
=1 +i v/. The recent paper [1]developed a corre-
sponding
modular invariant truncation of the exact coset CFT. The
torus partition (namely, the closed string sector) was calculated,
and the fusion rules were discussed.
E-mail address: benjojazz@gmail.com.
The goal of the present paper is to consider open strings in
the truncation, in the process fleshing out certain aspects of the
picture. We fail to find a consistent picture unless the whole coset
geometry is included.
2. D-branes in rational CFT
Cardy branes in rational CFT are constructed in the following
way. We are interested in boundary states which preserve a given
chiral algebra A. In the closed string picture these correspond to
boundary states |B satisfying
J
n
+ (
¯
J
−n
)
|B=0(2)
where J
n
,
¯
J
n
are the (anti-)holomorphic modes of the algebra gen-
erators,
and is some outer automorphism. For us will always
be trivial. An orthogonal basis of solutions to (2)is given by the
Ishibashi states |i, one for each representation i of A appearing
in the spectrum (we consider here the diagonal modular invari-
ant).
Consistency with modular transformations picks specific lin-
ear
combinations of the |i to be good boundary states for the
CFT. These are the ‘Cardy states’
1
|i
C
≡
S
ij
√
S
0i
|j . (3)
The embedding of the D-brane in the target space may be deduced
from identifying the zero-mode distributions of the Ishibashi states
with the solutions of the scalar Laplacian – this yields functions
which reduce at large level to delta functions supported on the
D-brane worldvolume.
1
These also satisfy certain ‘sewing’ constraints [7].
https://doi.org/10.1016/j.physletb.2018.08.010
0370-2693/
© 2018 The Author(s). 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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