Gopakumar–Vafa invariants and wall-crossing
Yukinobu Toda
Abstract
In this paper, we give a generalization of a mathematical definition of Gopakumar–Vafa (GV) invariants on Calabi–Yau 3-folds introduced by Maulik and the author, using an analogue of BPS sheaves introduced by Davison–Meinhardt on the coarse moduli spaces of one dimensional twisted semistable sheaves with arbitrary holomorphic Euler characteristics. We show that our generalized GV invariants are independent of twisted stability conditions, and conjecture that they are also independent of holomorphic Euler characteristics, so that they define the same GV invariants. As an application, we will show the flop transformation formula of GV invariants.
Topics & Concepts
MathematicsHolomorphic functionConjecturePure mathematicsGeneralizationEuler's formulaModuli spaceModuliStability (learning theory)Algebra over a fieldMathematical analysisComputer scienceMachine learningQuantum mechanicsPhysicsAlgebraic Geometry and Number TheoryNonlinear Waves and SolitonsGeometry and complex manifolds