Crane–Frenkel categorification conjecture for quantum mathfrak{sl}_2

Let $U_q(\mathfrak{sl}_2)$ be the quantum group at roots of unity. A categorification is a categorical structure whose decategorification recovers the quantum group, and a 4D TQFT is a four-dimensional topological quantum field theory.

Crane–Frenkel conjecture. There exists a categorification of the quantum group $U_q(\mathfrak{sl}_2)$ at roots of unity giving rise to a 4D TQFT.

The conjecture seeks a four-dimensional topological field-theoretic realization of the quantum-group invariants underlying Reshetikhin–Turaev theory. The source states that it remains open, although link homology theories and other developments in categorification constitute significant progress.