Graded-fusion 2-categories and quantum homotopy invariants of 4-manifolds
by
B/2nd floor-B.203 - Seminar room
Marc de Hemptinne (chemin du Cyclotron, 2, Louvain-la-Neuve)
A fundamental notion in quantum topology is that of topological quantum field theory (TQFT) formulated by Witten and Atiyah. This notion originates in ideas from quantum physics and constitutes a framework that organizes certain topological invariants of manifolds, called quantum invariants, which are defined by means of quantum groups.
Homotopy quantum field theories (HQFTs) are a generalization of TQFTs. The idea is to use TQFT techniques to study principal bundles over manifolds and, more generally, homotopy classes of maps from manifolds to a (fixed) topological space called the target. The resulting invariants are called quantum homotopy invariants.
Turaev and Virelizier have constructed quantum homotopy invariants of 3-manifolds (by state-sum) when the target space is aspherical (i.e. its n-th homotopy groups are trivial for n>1) and Sözer and Virelizier have recently constructed quantum homotopy invariants of 3-manifolds when the target space is a 2-type (i.e. its n-th homotopy groups are trivial for n>2). Using state sum techniques, Douglas and Reutter have constructed quantum invariants of 4-manifolds using fusion 2-categories. In this talk, we combine both these approaches: we construct quantum homotopy invariants of 4-manifolds with a 3-type target from fusion 2-categories graded by a 3-group.