Abstract
The topological string captures certain superstring amplitudes which are also encoded in the underlying string effective action. However, unlike the topological string free energy, the effective action that comprises higher-order derivative couplings is not defined in terms of duality covariant variables. This puzzle is resolved in the context of real special geometry by introducing the so-called Hesse potential, which is defined in terms of duality covariant variables and is related by a Legendre transformation to the function that encodes the effective action. It is demonstrated that the Hesse potential contains a unique subsector that possesses all the characteristic properties of a topological string free energy. Genus $g\leq3$ contributions are constructed explicitly for a general class of effective actions associated with a special-K\"ahler target space and are shown to satisfy the holomorphic anomaly equation of perturbative type-II topological string theory. This identification of a topological string free energy from an effective action is primarily based on conceptual arguments and does not involve any of its more specific properties. It is fully consistent with known results. A general theorem is presented that captures some characteristic features of the equivalence, which demonstrates at the same time that non-holomorphic deformations of special geometry can be dealt with consistently.
Highlights
Couplings.1 In case these higher-order derivative couplings are absent, we will denote the function by F (0), which is always holomorphic and homogeneous and encodes an action that is at most quadratic in space-time derivatives
In the previous section we introduced holomorphic functions that encode either the Wilsonian action or the topological string free energy, as well as a real function known as the Hesse potential
Based on the observation that the duality transformations act differently on the function that encodes the effective action than on the topological free energy, we have proposed a conceptual framework based on the Hesse potential of real special geometry to understand the relation between the two
Summary
In the previous section we introduced holomorphic functions that encode either the Wilsonian action or the topological string free energy, as well as a real function known as the Hesse potential. The topological string free energy is a function of the Calabi-Yau moduli which are subject to dualities related to the homology group of the underlying holomorphic threeform As explained, these moduli are associated with a holomorphic function F (0)(X), which encodes a corresponding vector multiplet Lagrangian with at most two space-time derivatives coupled to supergravity. Note that the replacement of the original variables XI by Y I is relevant, as it would not make sense to consider linear combinations of the original variables XI and their complex conjugates in view of the fact that they are projectively defined.5 As it turns out, nonholomorphic corrections can be encoded in a real function Ω(Y, Y ), which is incorporated into the function F in the following way [32],. The reason for this convention is that we will often not have the situation where holomorphic and anti-holomorphic indices are contracted consistently
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