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Classical formulation Consider the (untwisted) Dijkgraaf-Witten invariant, defined for an oriented, connected, closed manifold $M$ and a finite group $G$: $$DW_G(M) := \lvert \operatorname{Hom}(\pi_ …

In the quantum harmonic oscillator, there exists a family of states called coherent states which form an overcomplete set of states. They are regarded as "the states most resembling classical states", …

A state sum model is a smooth invariant defined on smooth triangulated, or PL manifolds, by summing a local partition function over labels attached to the elements of the triangulation. Typical exampl …

Often, TQFTs are defined in families, parametrised by some algebraic data. For example, the Turaev-Viro-Barrett-Westbury TQFTs are parametrised by spherical fusion categories, the Crane-Yetter TQFTs a …

Mathematical physicists in solid state physics and topological insulators talk a lot about Walker-Wang models, which are a family of Hamiltonians defined on a 3d lattice. Unfortunately, the original p …

Geometric realisation of simplicial sets can be roughly thought of like this: In some category $\mathcal{C}$, we choose an object for every abstract $n$-simplex. In topological spaces, we would choo …

Chris Schommer-Pries has classified 2d extended TQFTs (topological quantum field theories) in his PhD thesis. The result is a (not necessarily abelian) separable symmetric Frobenius algebra (possibly …

I've heard that Reshetikhin-Turaev (RT) is stronger than homotopy, and it can distinguish certain homotopy-equivalent, but non-homeomorphic Lens spaces (I think $L(7,1)$ and $L(7,2)$). Now the Turaev- …

This is a follow up to this question. Imagine there is a finitely presented crossed module $\mathcal{G} = (G,H, -\triangleright-\colon G \to \operatorname{Aut}(H), \delta\colon H \to G)$ which I do …

The bordism hypothesis says that the $(\infty, n)$-category of smooth, framed $n$-bordisms, $(n-1)$-dimensional boundaries, and corners down to points, is freely generated symmetric monoidal with dual …

Introduction Axiomatic TQFTs An axiomatic $n$-dimensional TQFT is a symmetric monoidal functor $\mathcal{Z}\colon \operatorname{Bord}_n \to \operatorname{Hilb}$ from $n$-dimensional oriented bordism …

In general, one extracts a manifold invariant from a TQFT by interpreting the closed manifold as a bordism from the empty set to the empty set. The TQFT sends this bordism to a homomorphism of the gro …

In Walker and Wang's article about (3+1)-TQFTs from premodular categories, they say on page 14 that you can take a quotient of a premodular category $\mathcal{C}$ by its symmetric fusion subcategory …