Direct integration of the topological string

By Thomas W. Grimm, A. Klemm, Marcos Mariño, Marlene Weiss. 2007

158 citations recorded by the source corpus.

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.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/62eaf5e4008e65ce12178e423e80a5494fa007f9

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Astronomy and Astrophysics

In Nuclear and High Energy Physics

In Statistical and Nonlinear Physics

Citations in the corpus

The Ω Deformed B-model for Rigid N = 2 Theories (2011, methodology): “The relation between the holomorphic anomaly and target space modularity was developed in [10, 42] and led, upon imposing suitable boundary conditions, to the direct integration method [11, 12, 14, 9].”
Topological String Theory on Compact Calabi–Yau: Modularity and Boundary Conditions (2006, background): “Moreover these cases have N = 2 field theory limits, which contain further information, which might be sufficient to solve these models [ 66 ].”
Remodeling the B-Model (2007, background): “Recently, boundary conditions have been found in the closed sector [32] (the so-called conifold gap condition and regularity at orbifold point) which fully fix these data in many local geometries (like the Seiberg–Witten geometry or local P(2)) [32, 28].”
From Weak to Strong Coupling in ABJM Theory (2010, methodology): “The resulting non-holomorphic objects satisfy the holomorphic anomaly equations of [30], as shown in [28,63], and these can be in turn solved with the technique of direct integration developed in [28,29,31,58] for local CY manifolds and matrix models.”
Penner type matrix model and Seiberg-Witten theory (2009, result): “comparison of the M-theory curve and the standard Seiberg-Witten curve leads to a relation between the UV and IR gauge coupling constants [17, 18, 1, 4].”
Liouville Correlation Functions from Four-Dimensional Gauge Theories (2009, background): “3, the cross ratio is actually given by the exponential of the UV coupling, q = exp(2πiτUV); this precise relation was first noticed by [9].”