# Counting BPS States via Holomorphic Anomaly Equations

@article{Hosono2002CountingBS,
title={Counting BPS States via Holomorphic Anomaly Equations},
author={Shinobu Hosono},
journal={arXiv: High Energy Physics - Theory},
year={2002}
}
• S. Hosono
• Published 2002
• Physics, Mathematics
• arXiv: High Energy Physics - Theory
We study Gromov-Witten invariants of a rational elliptic surface using holomorphic anomaly equation in [HST1](hep-th/9901151). Formulating invariance under the affine $E_8$ Weyl group symmetry, we determine conjectured invariants, the number of BPS states, from Gromov-Witten invariants. We also connect our holomorphic anomaly equation to that found by Bershadsky,Cecotti,Ooguri and Vafa [BCOV1](hep-th/9302103).

#### Tables from this paper

Quantum geometry of elliptic Calabi-Yau manifolds
• Physics, Mathematics
• 2012
We study the quantum geometry of the class of Calabi-Yau threefolds, which are elliptic brations over a two-dimensional toric base. A holomorphic anomaly equation for the topological string freeExpand
Topological Strings on Elliptic Fibrations
• Physics, Mathematics
• 2012
We study topological string theory on elliptically fibered Calabi-Yau manifolds using mirror symmetry. We compute higher genus topological string amplitudes and express these in terms of polynomialsExpand
On topological approach to local theory of surfaces in Calabi-Yau threefolds
• Mathematics, Physics
• 2016
We study the web of dualities relating various enumerative invariants, notably Gromov-Witten invariants and invariants that arise in topological gauge theory. In particular, we study Donaldson-ThomasExpand
Modularity from monodromy
A bstractIn this note we describe a method to calculate the action of a particular Fourier-Mukai transformation on a basis of brane charges on elliptically fibered Calabi-Yau threefolds with andExpand
Refined stable pair invariants for E-, M- and [p, q]-strings
• Physics
• 2013
A bstractWe use mirror symmetry, the refined holomorphic anomaly equation and modularity properties of elliptic singularities to calculate the refined BPS invariants of stable pairs on non-compactExpand
Wall-Crossing Holomorphic Anomaly and Mock Modularity of Multiple M5-Branes
• Physics, Mathematics
• 2010
Using wall-crossing formulae and the theory of mock modular forms we derive a holomorphic anomaly equation for the modified elliptic genus of two M5-branes wrapping a rigid divisor inside aExpand
E-, M- and (p;q)-strings
• Mathematics
• 2013
We use mirror symmetry, the rened holomorphic anomaly equation and modularity properties of elliptic singularities to calculate the rened BPS invariants of stable pairs on non-compact Calabi-YauExpand
BCOV ring and holomorphic anomaly equation
We study certain differential rings over the moduli space of Calabi-Yau manifolds. In the case of an elliptic curve, we observe a close relation to the differential ring of quasi-modular forms due toExpand
The Topological Vertex
• Mathematics, Physics
• 2005
We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact toric Calabi-Yau threefolds. The topology of a given Feynman diagram encodes theExpand
Topological string amplitudes for the local $\frac{1}{2}$K3 surface
We study topological string amplitudes for the local half K3 surface. We develop a method of computing higher-genus amplitudes along the lines of the direct integration formalism, making full use ofExpand

#### References

SHOWING 1-10 OF 71 REFERENCES
Holomorphic Anomaly Equation and BPS State Counting of Rational Elliptic Surface
• Physics, Mathematics
• 1999
We consider the generating function (prepotential) for Gromov-Witten invariants of rational elliptic surface. We apply the local mirror principle to calculate the prepotential and prove a certainExpand
Holomorphic anomalies in topological field theories
• Physics
• 1993
We study the stringy genus-one partition function of N = 2 SCFTs. It is shown how to compute this using an anomaly in decoupling of BRST trivial states from the partition function. A particular limitExpand
Euler Characteristics of SU(2) Instanton Moduli Spaces on Rational Elliptic Surfaces
Abstract:Recently, Minahan, Nemeschansky, Vafa and Warner computed partition functions for N = 4 topological Yang–Mills theory on rational elliptic surfaces. In particular they computed generatingExpand
Relative Lefschetz action and BPS state counting
• Mathematics, Physics
• 2001
In this paper, we propose a mathematical definition of a new numerical invariants" of Calabi--Yau 3-folds from stable sheaves of dimension one, which is motivated by the Gopakumar-Vafa conjectureExpand
Topological field theory and rational curves
• Mathematics, Physics
• 1993
We analyze the quantum field theory corresponding to a string propagating on a Calabi-Yau threefold. This theory naturally leads to the consideration of Witten's topological non-linear σ-model andExpand
BPS states of curves in Calabi-Yau 3-folds
• Mathematics, Physics
• 2001
The Gopakumar-Vafa conjecture is defined and studied for the local geometry of a curve in a Calabi-Yau 3-fold. The integrality predicted in Gromov-Witten theory by the Gopakumar-Vafa BPS count isExpand
Gromov-Witten classes, quantum cohomology, and enumerative geometry
• Physics, Mathematics
• 1994
The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomaticExpand
Localization of virtual classes
• Mathematics
• 1997
We prove a localization formula for virtual fundamental classes in the context of torus equivariant perfect obstruction theories. As an application, the higher genus Gromov-Witten invariants ofExpand
Knot Invariants and Topological Strings
• Physics
• 1999
We find further evidence for the conjecture relating large N Chern-Simons theory on S^3 with topological string on the resolved conifold geometry by showing that the Wilson loop observable of aExpand
EXCEPTIONAL STRING: INSTANTON EXPANSIONS AND SEIBERG–WITTEN CURVE
We investigate instanton expansions of partition functions of several toric E-string models using local mirror symmetry and elliptic modular forms. We also develop a method to determine theExpand