Direct integration of the topological string

Abstract

We present a new method to solve the holomorphic anomaly equations governing the free energies of type B topological strings. The method is based on direct integration with respect to the non-holomorphic dependence of the amplitudes, and relies on the interplay between non-holomorphicity and modularity properties of the topological string amplitudes. We develop a formalism valid for any Calabi-Yau manifold and we study in detail two examples, providing closed expressions for the amplitudes at low genus, as well as a discussion of the boundary conditions that fix the holomorphic ambiguity. The first example is the non-compact Calabi-Yau underlying Seiberg-Witten theory and its gravitational corrections. The second example is the Enriques Calabi-Yau, which we solve in full generality up to genus six. We discuss various aspects of this model: we obtain a new method to generate holomorphic automorphic forms on the Enriques moduli space, we write down a new product formula for the fiber amplitudes at all genus, and we analyze in detail the field theory limit. This allows us to uncover the modularity properties of SU(2), N = 2 super Yang-Mills theory with four massless hypermultiplets.

Residue formulas for prepotentials, instanton expansions and conformal blocks

“…We derive a generalization of the Zamolodchikov renormalization formula [19][20][21] (see also [8,[22][23][24][25]) for this class of constrained quiver theories, which includes the first-order differential equations for the effective couplings and their implicit solution via the Thomae formulas. Finally, we study another kind of non-linear differential equations for the ex-…”

Enhanced supersymmetry from vanishing Euler number

“…It is self-mirror and thereby does not have experience any worldsheet instanton corrections at the twoderivative level [33]. There is however a non-trivial pattern of higher-derivative corrections for this background (graviphoton-curvature couplings), some of which have been computed in [35,36]. We begin by considering the harmonic forms on the Enriques Calabi-Yau Y = (K3 × T 2 )/Z τ 2 .…”

Exact Results for Topological Strings on Resolved Y p,q Singularities

“…Turning now to the case of g > 1 closed amplitudes, let us first of all recall the main statements put forward in [7,56] for the computation of higher genus free energies. As mentioned in section 6.3.2, modular symmetry is an amazingly stringent constraint.…”

“…c (g) 0 (τ ). In the 1-parameter cases analyzed in [7,56], this is systematically done by plugging into (6.52) an ansatz for c (g) 0 (τ ) which is then determined from extra boundary data. More in detail, this works as follows: at fixed genus g, c (g) 0 (τ ) is a weight w = 6g − 3 modular form 8 .…”

“…At a computational level, the formalism of [8] is somewhat lengthier than the one of [56] for computing higher genus F g . On the other hand, the recursion of [8] has the great advantage of providing unambiguous results, with the holomorphic ambiguity c (g) 0 (τ ) automatically fixed.…”