Results 1  10
of
36
Topological strings and (almost) modular forms
, 2007
"... The Bmodel topological string theory on a CalabiYau threefold X has a symmetry group Γ, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural way, governed by the quantum mechanics of the phase space H 3 (X). We show that, depending on the cho ..."
Abstract

Cited by 93 (10 self)
 Add to MetaCart
The Bmodel topological string theory on a CalabiYau threefold X has a symmetry group Γ, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural way, governed by the quantum mechanics of the phase space H 3 (X). We show that, depending on the choice of polarization, the genus g topological string amplitude is either a holomorphic quasimodular form or an almost holomorphic modular form of weight 0 under Γ. Moreover, at each genus, certain combinations of genus g amplitudes are both modular and holomorphic. We illustrate this for the local CalabiYau manifolds giving rise to SeibergWitten gauge theories in four dimensions and local IP2 and IP1×IP1. As a byproduct, we also obtain a simple way of relating the topological string amplitudes near different points in the moduli space, which we use to give predictions for GromovWitten invariants of the orbifold C 3 / Z3.
Topological string theory on compact CalabiYau: Modularity and boundary conditions
, 2006
"... The topological string partition function Z(λ,t, ¯t) = exp(λ 2g−2 Fg(t, ¯t)) is calculated on a compact CalabiYau M. The Fg(t, ¯t) fulfill the holomorphic anomaly equations, which imply that Ψ = Z transforms as a wave function on the symplectic space H 3 (M, Z). This defines it everywhere in the m ..."
Abstract

Cited by 83 (11 self)
 Add to MetaCart
(Show Context)
The topological string partition function Z(λ,t, ¯t) = exp(λ 2g−2 Fg(t, ¯t)) is calculated on a compact CalabiYau M. The Fg(t, ¯t) fulfill the holomorphic anomaly equations, which imply that Ψ = Z transforms as a wave function on the symplectic space H 3 (M, Z). This defines it everywhere in the moduli space M(M) along with preferred local coordinates. Modular properties of the sections Fg as well as local constraints from the 4d effective action allow us to fix Z to a large extent. Currently with a newly found gap condition at the conifold, regularity at the orbifold and the most naive bounds from Castelnuovo’s theory, we can provide the boundary data, which specify Z, e.g. up to genus 51 for the quintic.
Holomorphic Anomaly and Matrix Models
, 2007
"... The genus g free energies of matrix models can be promoted to modular invariant, nonholomorphic amplitudes which only depend on the geometry of the classical spectral curve. We show that these nonholomorphic amplitudes satisfy the holomorphic anomaly equations of Bershadsky, Cecotti, Ooguri and Vaf ..."
Abstract

Cited by 29 (10 self)
 Add to MetaCart
The genus g free energies of matrix models can be promoted to modular invariant, nonholomorphic amplitudes which only depend on the geometry of the classical spectral curve. We show that these nonholomorphic amplitudes satisfy the holomorphic anomaly equations of Bershadsky, Cecotti, Ooguri and Vafa. We derive as well holomorphic anomaly equations for the open string sector. These results provide evidence at all genera for the Dijkgraaf–Vafa conjecture relating matrix models to type B topological strings on certain local Calabi–Yau threefolds.
Direct Integration of the Topological String
, 2007
"... 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 modular ..."
Abstract

Cited by 23 (5 self)
 Add to MetaCart
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 noncompact 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.
The Real Topological String on a local CalabiYau
, 2009
"... We study the topological string on local P2 with Oplane and Dbrane at its real locus, using three complementary techniques. In the Amodel, we refine localization on the moduli space of maps with respect to the torus action preserved by the antiholomorphic involution. This leads to a computation o ..."
Abstract

Cited by 13 (4 self)
 Add to MetaCart
We study the topological string on local P2 with Oplane and Dbrane at its real locus, using three complementary techniques. In the Amodel, we refine localization on the moduli space of maps with respect to the torus action preserved by the antiholomorphic involution. This leads to a computation of open and unoriented GromovWitten invariants that can be applied to any toric CalabiYau with involution. We then show that the full topological string amplitudes can be reproduced within the topological vertex formalism. We obtain the real topological vertex with trivial fixed leg. Finally, we verify that the same results derive in the Bmodel from the extended holomorphic anomaly equation, together with appropriate boundary conditions. The expansion at the conifold exhibits a gap structure that belongs to a so far unidentified