Physics Letters B 779 (2018) 249–256
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Higher-n triangular dilatonic black holes
Anton Zadora
a
, Dmitri V. Gal’tsov
a,b,∗
, Chiang-Mei Chen
c
a
Department of Theoretical Physics, Faculty of Physics, Moscow State University, 119899, Moscow, Russia
b
Kazan Federal University, 420008 Kazan, Russia
c
Department of Physics, National Central University, Chungli 32001, Taiwan
a r t i c l e i n f o a b s t r a c t
Article history:
Received
28 December 2017
Received
in revised form 8 February 2018
Accepted
9 February 2018
Available
online 13 February 2018
Editor:
M. Cveti
ˇ
c
Dilaton gravity with the form fields is known to possess dyon solutions with two horizons for the
discrete “triangular” values of the dilaton coupling constant a =
√
n(n + 1)/2. This sequence first obtained
numerically and then explained analytically as consequence of the regularity of the dilaton, should have
some higher-dimensional and/or group theoretical origin. Meanwhile, this origin was explained earlier
only for n = 1, 2in which cases the solutions were known analytically. We extend this explanation to
n = 3, 5presenting analytical triangular solutions for the theory with different dilaton couplings a, b
in
electric and magnetic sectors in which case the quantization condition reads ab = n(n + 1)/2. The
solutions are derived via the Toda chains for B
2
and G
2
Lie algebras. They are found in the closed
form in general D space–time dimensions. Solutions satisfy the entropy product rules indicating on the
microscopic origin of their entropy and have negative binding energy in the extremal case.
© 2018 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
Einstein–Maxwell-dilaton (EMD) theory in four dimensions may
originate from different supersymmetric higher-dimensional theo-
ries
with various values of the dilaton coupling constant a in the
Maxwell term e
−2aφ
F
2
in the Lagrangian. For a = 0the dilaton
decouples and the theory reduces to Einstein–Maxwell (EM) sys-
tem,
which is the bosonic part of N = 2, D = 4supergravity. The
value a = 1 corresponds to N = 4, D = 4 supergravity [1,2]. The
value a =
√
3, corresponds to dimensionally reduced D = 5grav-
ity
[3,4], which have different supersymmetric extensions. In all
these cases analytical solutions are known for static black holes
possessing both electric and magnetic charges (dyons) with two
horizons between which the dilaton exhibits n oscillations. Such
solutions have non-singular extremal limits with AdS
2
× S
2
hori-
zons
contrary the one-charged dilatonic black hole which have
singular horizons in the extremal limit [1,2,5]for generic coupling
constant.
As it was first noticed by Poletti, Twamley and Wiltshire [6],
the values a =1,
√
3are just the two lowest members n = 1, 2of
the “triangular” sequence of dilaton couplings
*
Corresponding author.
E-mail
addresses: as .zadora @physics .msu .ru (A. Zadora), galtsov @phys .msu .ru
(D.V. Gal’tsov),
cmchen @phy.ncu .edu .tw (C.-M. Chen).
a
n
=
n(n +1)/2, (1)
for which dyons (known for higher n only numerically) exhibit
similar behavior of the dilaton. Analytically this triangle quantiza-
tion
rule was rederived in [7]as condition of regularity of the dila-
ton
in the case of coinciding horizons, i.e. for extremal dyons. The
same rule was shown to arise from the linearized dilaton equation
on the Reissner–Nordström dyonic background [8,9]as condition
of existence of dilaton bound states. However, exact higher-n dy-
onic
solutions were not known analytically. Here we give solutions
for n = 3, 5in the theory where the dilaton couples to electric
and magnetic sectors with different coupling constants a, b, in
which case the quantization rule generalizes to ab = n(n + 1)/2.
We also give generalization to arbitrary dimensions. These new so-
lutions
are derived via Toda chains for B
2
and G
2
algebras. The
solutions satisfy the entropy product rule: the product of the en-
tropies
of the internal and internal horizons are functions of their
charges only [10,11](see also [8]). This property is considered to
be an indication on possibility of statistical interpretation of the
entropy. Another interesting feature is that the extremal dyons
have negative binding energy like the SL(n, R) multi-scalar Toda
black holes [12].
Previous
work on Toda dilatonic black holes includes among
other [13–16]. Our results have some overlap with the recent
preprints [9,17].
https://doi.org/10.1016/j.physletb.2018.02.017
0370-2693/
© 2018 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
.