1 Topological Quantum Field (TQFT) State on the Tetrahedron

Let us consider first a regular tetrahedron whose vertices are labeled by TQFT data associated with a quantum-field state. The vertices of the tetrahedron are located at

and will be labeled, respectively, as 1, 2, 3, 4.

Definition 1.1. A quantum field state ϕ provides a total order, denoted by ≤ _ϕ, on the vertices of the tetrahedron and thus assigns a direction to each edge of the tetrahedron, from the smaller to the larger vertex.

A quantum field state also labels each edge

by an element ϕ₁(e) of B_A, where B_A is a distinguished basis of a fusion algebra 𝒜; that is, a finite-dimensional, unital, involutive algebra over ℂ.

Moreover, the quantum field state assigns an element ϕ²(f), called an intertwiner, of a Hilbert space

to each face

of the tetrahedron such that

1.1 Remarks

A topological quantum field theory (TQFT) is a mathematical framework for quantum field theory in which topological information plays a central role. TQFTs are used to construct and compute topological invariants and have important applications in low-dimensional topology, condensed-matter physics, and related areas of mathematical physics.

The development of TQFT is associated with work by Michael Atiyah, Edward Witten, Maxim Kontsevich, Vaughan Jones, Simon Donaldson, and others. Related developments include knot invariants, topological invariants of manifolds, and homological mirror symmetry.

References