On the composition of two spherical twists

TL;DR

This paper presents a construction to compose two spherical twists into a single twist around a combined spherical functor, clarifying their structure and cotwist behavior, especially for spherical objects and P-objects.

Contribution

It introduces a method to realize the composition of two spherical twists as a single twist around a semiorthogonally decomposed spherical functor, expanding understanding of autoequivalence compositions.

Findings

The composition of two spherical twists can be realized as a single twist around a specific spherical functor.

The cotwist of this combined spherical functor is described, and in special cases, it equals the Serre functor up to a shift.

Explicit treatment provided for the case of P-objects.

Abstract

E. Segal proved that any autoequivalence of an enhanced triangulated category can be realised as a spherical twist. However, when exhibiting an autoequivalence as a spherical twist one has various choices for the source category of the spherical functor. We describe a construction that realises the composition of two spherical twists as the twist around a single spherical functor whose source category semiorthogonally decomposes into the source categories for the spherical functors we started with. We give a description of the cotwist for this spherical functor and prove, in the special case when our starting twists are around spherical objects, that the cotwist is the Serre functor (up to a shift). We finish with an explicit treatment for the case of P-objects.