Gravity duals of \( \mathcal{N} \) = 2 SCFTs and asymptotic emergence of the electrostatic description
P. Marios Petropoulos
2
Konstadinos Sfetsos
0
Konstadinos Siampos
1
0
Department of Nuclear and Particle Physics, Faculty of Physics, University of Athens
,
Athens 15784, Greece
1
Mecanique et Gravitation,
Universite de Mons Hainaut
, 7000 Mons, Belgique
2
Centre de Physique Theorique
, Ecole Polytechnique, CNRS UMR 7644, 91128 Palaiseau Cedex,
France
We built the first eleven-dimensional supergravity solutions with SO(2, 4) SO(3) U(1)R symmetry that exhibit the asymptotic emergence of an extra U(1) isometry. This enables us to make the connection with the usual electrostatics-quiver description. The solution is obtained via the Toda frame of Kahler surfaces with vanishing scalar curvature and SU(2) action.
Contents
1 Introduction
Scalar-flat four-dimensional Kahler spaces
Toda from Pedersen-Poon
3.1 Boundary conditions and general equations
3.2 Enhanced SU(2) U(1) symmetry and electrostatics
3.3 Strict SU(2) symmetry and new solutions 4 Conclusion and outlook
Introduction
Finding explicit solutions of eleven-dimensional supergravity admitting dual N = 2 field
theories is a challenging, though well-owed problem. The first example was presented in [1],
while general features and properties have been developed since in [26], making contact
in particular with N = 2 quiver gauge theories.
Assuming a specific form for the metric and the antisymmetric fields, the problem boils
down to finding solutions of the continual Toda equation, subject to appropriate boundary
conditions. The solution of Toda equation can exhibit a symmetry, which translates at the
level of the geometry into an extra U(1) isometry. When this happens, the Toda problem
is equivalent to solving a Laplace equation [7] and addresses the cylindrically symmetric
electrostatic problem of a perfectly conducting plane with a line charge distribution normal
to it [3].
The electrostatic picture is useful for unravelling the quiver interpretation of the dual
field theory. It is however a stringent limitation and it is desirable to understand more
general situations without electrostatic analogue. A first step in that direction was taken
in [8], where an explicit two-parameter family of solutions of the Toda equation without
extra symmetry was exhibited. The idea underlying the construction was to borrow
solutions from other systems, where Toda equation governs the dynamics. Four-dimensional
gravitational configurations are among those, and in particular self-dual gravitational
instantons of the Boyer-Finley type [911]. Assuming that these are furthermore Bianchi
IX foliations, Toda solutions are obtained by solving other integrable systems such as
Darboux-Halphen [12], which are well understood irrespective of the symmetry, and using
the mapping provided in [13, 14].
The analysis performed in [8] is a real tour de force in terms of finding
elevendimensional supergravity solutions. The solutions obtained in this way have no smearing
and thus no extra U(1) symmetry, even asymptotically. This good feature in terms of
novelty is altogether a caveat because it does not provide any handle for the interpretation
of the dual field theory.
In the present note, we propose another set of supergravity solutions, for which the
absent U(1) is restored in some asymptotic corner of the geometry. These are technically
less involved than that in [8]. They are based on solutions of Toda equation as they
appear in another class of remarkable four-dimensional geometries, namely metrics with
a symmetry, vanishing scalar curvature and Kahler structure. The specific metrics we
consider here belong to the more general class of LeBrun metrics [15], and combine again
the Bianchi IX feature as it emerges in a class known as Pedersen-Poon Kahler surfaces
with zero scalar curvature [16].
Scalar-flat four-dimensional Kahler spaces
The purview of this section is to set-up the contact with Toda equation via the so-called
Kahler-plus-symmetry LeBrun metrics [15] for the Pedersen-Poon class [16].
The LeBrun geometries possess a U(1) isometry, are Kahler and have vanishing scalar
curvature. The presence of the U(1) isometry, realised with the Killing vector , enables
the metric to be set in the form
dA = xU dy dz + yU dz dx + z U e dx dy ,
with integrability condition
also known as linearised Toda equation. Imposing in addition the vanishing of the scalar
curvature R gives the differential equation
which is precisely the continual Toda.1
One should stress that according to LeBrun [15], every Kahler-plus-symmetry metric
with vanishing R is locally of the form (2.1) and (2.2), with A, U, satisfying (2.3)(2.5),
1Notice that the left-hand side of the Toda equation can be recast as e23, where 3 refers to the
three-dimensional metric (2.2).
= 0 ,
and conversely every metric in the class (2.1)(2.5) is Kahler-plus-symmetry with vanishing
R. The Kahler form reads:
J = (d + A) dz U edx dy,
and satisfies dJ = 0.
Let us for completeness and later use remind that a four- (...truncated)