Integrality of Gopakumar–vafa Invariants of Toric Calabi–yau Threefolds
نویسنده
چکیده
The Gopakumar–Vafa invariant is a number defined as certain linear combination of the Gromov–Witten invariants. We prove that the GV invariants of the toric Calabi–Yau threefold are integers and that the invariants at high genera vanish. The proof of the integrality is the elementary number theory and that of the vanishing uses the operator formalism and the exponential formula.
منابع مشابه
Topological Strings, Instantons and Asymptotic Forms of Gopakumar–Vafa Invariants
We calculate the topological string amplitudes of Calabi–Yau toric threefolds corresponding to 4D, N = 2 SU(2) gauge theory with Nf = 0, 1, 2, 3, 4 fundamental hypermultiplets by using the method of the geometric transition and show that they reproduce Nekrasov’s formulas for instanton counting. We also determine the asymptotic forms of the Gopakumar–Vafa invariants of the Calabi–Yau threefolds...
متن کاملAsymptotic Form of Gopakumar–Vafa Invariants from Instanton Counting
We study the asymptotic form of the Gopakumar–Vafa invariants at all genus for Calabi–Yau toric threefolds which have the structure of fibration of the An singularity over P. We claim that the asymptotic form is the inverse Laplace transform of the corresponding instanton amplitude in the prepotential of N = 2 SU(n + 1) gauge theory coupled to external graviphoton fields, which is given by the ...
متن کاملNew Exact Quantization Condition for Toric Calabi-Yau Geometries.
We propose a new exact quantization condition for a class of quantum mechanical systems derived from local toric Calabi-Yau threefolds. Our proposal includes all contributions to the energy spectrum which are nonperturbative in the Planck constant, and is much simpler than the available quantization condition in the literature. We check that our proposal is consistent with previous works and im...
متن کاملA Simple Proof of Gopakumar–vafa Conjecture for Local Toric Calabi-yau Manifolds
We prove Gopakumar-Vafa conjecture for local toric CalabiYau manifolds. It is also proved that the local Goparkumar-Vafa invariants of a given class at large genus vanish.
متن کاملA Mathematical Theory of the Topological Vertex
We have developed a mathematical theory of the topological vertex— a theory that was originally proposed by M. Aganagic, A. Klemm, M. Mariño, and C. Vafa on effectively computing Gromov-Witten invariants of smooth toric Calabi-Yau threefolds derived from duality between open string theory of smooth Calabi-Yau threefolds and Chern-Simons theory on three manifolds.
متن کامل