TQFT with corners and tilting functors in the Kac-Moody case

TL;DR

This paper classifies indecomposable projective functors in the Kac-Moody setting, linking them to tangle invariants and conjecturally to Khovanov homology, advancing understanding of categorical structures in representation theory.

Contribution

It provides a classification theorem for indecomposable projective functors in the Kac-Moody case and explores their connection to tangle invariants and Khovanov homology.

Findings

Complete classification of indecomposable projective functors in Kac-Moody case.

Establishment of a link between projective functors and tangle cobordism invariants.

Formulation of a conjectural connection to Khovanov homology.

Abstract

We study projective functors (i.e. direct summands of compositions of translations through walls) for parabolic versions of $\cO$ as well as for integral regular blocks outside the critical hyperplanes in the symmetrizable Kac-Moody case. It turns out that in both situations the functors are completely determined by their restriction to the additive category generated by (the limit of) a `full projective tilting' object. We describe how projective functors in the parabolic setup give rise to an invariant of tangle cobordisms and formulate a conjectural direct connection to Khovanov homology. Our main result, however, is the classification theorem for indecomposable projective functors in the Kac-Moody case verifying a conjecture of F. Malikov and I. Frenkel.