Hessian geometry and the holomorphic anomaly
HJE
Hessian geometry and the holomorphic anomaly
G.L. Cardoso 0 1 3 4 5
T. Mohaupt 0 1 2 4 5
Instituto Superior Tecnico 0 1 4 5
Universidade de Lisboa 0 1 4 5
0 Peach Street, Liverpool L69 7ZL, U.K
1 Av. Rovisco Pais , 1049-001 Lisboa , Portugal
2 Department of Mathematical Sciences, University of Liverpool
3 Center for Mathematical Analysis , Geometry and Dynamical Systems
4 multiple derivatives @
5 the di erence @
We present a geometrical framework which incorporates higher derivative corrections to the action of N = 2 vector multiplets in terms of an enlarged scalar manifold which includes a complex deformation parameter. This enlarged space carries a deformed version of special Kahler geometry which we characterise. equation arises in this framework from the integrability condition for the existence of a
Di erential and Algebraic Geometry; Supergravity Models; Topological
-
Strings
ArXiv ePrint: 1511.06658
1 Introduction
2
3
2.1
2.2
3.1
3.2
3.3
3.4
3.5
3.6
4.1
4.2
4.3
4.4
4.5
Review of special Kahler geometry
A ne special Kahler manifolds
Conical a ne special Kahler manifolds
The holomorphic deformation
Deformation of the immersion
Real coordinates and the Hesse potential
Deformed special Kahler geometry
Stringy complex coordinates
The symplectic covariant derivative
Anomaly equation from the Hessian structure
Non-holomorphic deformation of the prepotential
Real coordinates and the Hesse potential
The symplectic covariant derivative
The holomorphic anomaly equation
From Hessian structure to the full anomaly equation
4
The non-holomorphic deformation 5
Concluding remarks
A Connections on vector bundles
B Special coordinates C Symplectic transformations and functions 1 5
are required for consistency with electric-magnetic duality and are essential for
incorporating higher derivative corrections to black hole entropy. While supergravity provides a
powerful tool to organise an e ective action for quantum gravity, the actual computation
of couplings requires a speci c theory. String theory is the natural candidate, and in
particular the higher derivative corrections to N = 2 vector multiplets are captured by the
topological string. However, the relation between supergravity and the topological string is
subtle, and non-holomorphic corrections are incorporated di erently in the respective
formalisms. In this paper we develop a new geometrical description of the higher derivative
corrections on the supergravity side, by showing that they can be understood in terms of
an extended scalar manifold which carries a deformed version of special geometry. We also
derive various exact relations between the variables used in supergravity and in the
topological string. The most interesting result we obtain is that the holomorphic anomaly equation
which controls the non-holomorphic corrections in both the supergravity and topological
string formalism can be derived from the integrability condition for the existence of a Hesse
potential on the extended scalar manifold.
Let us next introduce our topic in more technical terms. In four dimensions, the
complex scalar elds residing in N = 2 vector multiplets parametrize a scalar manifold which
is the target space of the non-linear sigma-model that enters in the Wilsonian Lagrangian
describing the couplings of N = 2 vector multiplets at the two-derivative level. The scalar
manifold is an a ne special Kahler manifold in global supersymmetry, and a projective
special Kahler manifold in local supersymmetry; both types of target space geometry are
referred to as special geometry [1{7]. Special geometry, when formulated in terms of
complex variables Y I , is encoded in a holomorphic function F (0)(Y ), called the prepotential.
When formulated in terms of special real coordinates, it is the Hesse potential that plays
a central role. For a ne special Kahler manifolds, the Hesse potential is related to the
prepotential by a Legendre transform [8].
When coupling the N = 2 vector multiplets to the square of the Weyl multiplet, the
resulting Wilsonian Lagrangian, which now contains higher derivative terms proportional
to the square of the Weyl tensor, in encoded in a generalized prepotential F (Y; ), where
denotes a complex scalar eld residing in the lowest component of the square of the Weyl
multiplet. The complex scalar
elds (Y I ; ) will be called supergravity variables in the
following. The prepotential F (0)(Y ) is obtained from F (Y; ) by setting
= 0.
Electricmagnetic duality, a central feature of N = 2 systems based on vector multiplets, then acts
by symplectic transformations of the vector (Y I ; FI ), where FI = @F=@Y I . While F (Y; )
itself does not transform as a function under symplectic transformations, F
does [9]. The associated Hesse potential, obtained by a Legendre transform of Im F (Y; ),
is also a symplectic function.
Away from the Wilsonian limit, the coupling functions encoded in F
receive
nonholomorphic corrections, in general. In supergr (...truncated)