BPS states of curves in Calabi–Yau 3–folds

Abstract

The Gopakumar-Vafa integrality conjecture is defined and studied for the local geometry of a super-rigid curve in a Calabi-Yau 3-fold. The integrality predicted in Gromov-Witten theory by the Gopakumar-Vafa BPS count is verified in a natural series of cases in this local geometry. The method involves Gromov-Witten computations, Möbius inversion, and a combinatorial analysis of the numbers ofétale covers of a curve.

Exceptional String: Instanton Expansions and Seiberg–witten Curve

“…The ambient toric variety in this case is a P 3 bundle over P 1 . This space admits the quotient realization ( 4 , µx 5 , µx 6 ), and the bad point set…”

“…(y) 2 = 4(x) 3 − f 0 (τ )u 4 x − (g 0 (τ )u 6 + u 5 ), (9.6) which is the Seiberg-Witten curve of the E 8 model found in [12].…”

Lectures on the Topological Vertex

“…where q is given in (12). By using that s R is homogeneous of degree (R) in the variables x i we finally obtain…”

Integrality of relative BPS state counts of toric del Pezzo surfaces

“…1 They can nonetheless be defined via Equation (1.1), or alternatively, via generating functions as follows. …”