Spherical Twist

A spherical twist is an autoequivalence associated with a spherical object in an appropriate derived category. For an -spherical object , its self-extension space is and the Serre functor takes to . The twist is defined by the evaluation cone

For a smooth quintic threefold, the structure sheaf is 3-spherical. Its twist, composed with tensoring by , gives the autoequivalence in the quintic Gepner stability problem.