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Article
Report number	hep-th/0702187 ; CERN-PH-TH-2007-039 ; MAD-TH-07-04 ; CERN-PH-TH-2007-039 ; MAD-TH-2007-04
Title	Direct Integration of the Topological String
Author(s)	Grimm, Thomas W. (Wisconsin U., Madison) ; Klemm, Albrecht (Wisconsin U., Madison) ; Marino, Marcos (CERN) ; Weiss, Marlene (CERN ; Zurich, ETH)
Affiliation	(CERN)
Publication	2007
Imprint	23 Feb 2007
Number of pages	75
In:	JHEP 08 (2007) 058
DOI	10.1088/1126-6708/2007/08/058
Subject category	Particle Physics - Theory
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.
Copyright/License	Open Access

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