A pair of homotopy-theoretic version of TQFT’s induced by a Brown functor. (English) Zbl 1493.57025
The paper under review applies the author’s earlier work on integrals of bimonoid homomorphisms [M. Kim, Appl. Categ. Struct. 29, No. 4, 577–627 (2021; Zbl 1472.18015)] to construct generalizations of the Dijkgraaf-Witten TQFTs and the Turaev-Viro TQFTs.
A spanical and cospanical path integral are constructed, from a category of Brown functors on the cospan category of finite pointed CW-spaces (such functors were defined and studied by the author in a previous paper [M. Kim, “An extension of Brown functor to cospans of spaces”, Preprint, arXiv:2005.10621]) to a space of homotopy-theoretic projective TQFTs. Several properties of these path integrals are proven, including them being inverse to one another, and their tensor product yielding the Hopf-algebra dimension.
The construction may be interpreted as a generalization of the relationship between the Turaev-Viro TQFTs and the Kitaev lattice Hamiltonian model (toric code) to an arbitrary ground field and pointed finite CW-complex.
A spanical and cospanical path integral are constructed, from a category of Brown functors on the cospan category of finite pointed CW-spaces (such functors were defined and studied by the author in a previous paper [M. Kim, “An extension of Brown functor to cospans of spaces”, Preprint, arXiv:2005.10621]) to a space of homotopy-theoretic projective TQFTs. Several properties of these path integrals are proven, including them being inverse to one another, and their tensor product yielding the Hopf-algebra dimension.
The construction may be interpreted as a generalization of the relationship between the Turaev-Viro TQFTs and the Kitaev lattice Hamiltonian model (toric code) to an arbitrary ground field and pointed finite CW-complex.
Reviewer: Daniel Moskovich (Beer-Sheva)
MSC:
| 57R56 | Topological quantum field theories (aspects of differential topology) |
Keywords:
TQFT; Brown functor; bicommutative Hopf algebra; Dijkgraaf-Witten theory; Turaev-Viro theory; finite path-integral; cospan categoryCitations:
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