Discrete Hamitonian models for 3+1 D topological phases derived from higher gauge theory

Joao Faria Martins (Leeds)

Abstract. Higher gauge theory is a higher order version of gauge theory that makes possible the definition of 2-dimensional holonomy along surfaces embedded in a manifold where a gauge 2-connection is present. I will review the recent construction of discrete Hamiltonian models for 3+1D topological phases derived from discrete higher gauge theory defined on a lattice decomposition of a manifold. The construction is similar to Kitaev quantum double model, but replacing a gauge connection discretised on a lattice with a gauge 2-group 2-connection discretised on a lattice. This means that we enrich the local variables of lattice gauge theory (holonomies along edges) to include non-abelian 2-dimensional holonomies along the faces of the lattice. I will prove that the ground state degeneracy is a topological invariant.

The construction will be mostly combinatorial and should be accessible to someone who has not been exposed to higher gauge theory.

--Alex Bullivant, Marcos Calcada, Zoltán Kádár, João Faria Martins, Paul Martin:
-Higher lattices, discrete two-dimensional holonomy and topological phases in (3+1) D with higher gauge symmetry. arXiv:1702.00868 [math-ph]
-Topological phases from higher gauge symmetry in 3+1 dimensions. PHYSICAL REVIEW B 95, 155118 (2017).

--A Yu Kitaev. Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1):2–30, 2003.

This visit is funded by the Leverhulme trust grant: “RPG-2018-029: Emergent Physics From
Lattice Models of Higher Gauge Theory”