A TQFT of Turaev–Viro Type on Shaped Triangulations

Kashaev, Rinat; Vartanov, Grigory; Luo, Feng

doi:10.1007/s00023-015-0427-8

Journal Article

PUBDB-2016-03856

A TQFT of Turaev–Viro Type on Shaped Triangulations

Kashaev, R. ; Luo, F. ; Vartanov, G. (Corresponding author)DESY*

2016
Birkhäuser Basel

Annales Henri Poincaré 17(5), 1109 - 1143 (2016) [10.1007/s00023-015-0427-8]

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Please use a persistent id in citations: doi:10.1007/s00023-015-0427-8 doi:10.3204/PUBDB-2016-03856

Report No.: DESY-12-195; arXiv:1210.8393

Abstract: A shaped triangulation is a finite triangulation of an oriented pseudo-three-manifold where each tetrahedron carries dihedral angles of an ideal hyperbolic tetrahedron. To each shaped triangulation, we associate a quantum partition function in the form of an absolutely convergent state integral which is invariant under shaped 3–2 Pachner moves and invariant with respect to shape gauge transformations generated by total dihedral angles around internal edges through the Neumann–Zagier Poisson bracket. Similarly to Turaev–Viro theory, the state variables live on edges of the triangulation but take their values on the whole real axis. The tetrahedral weight functions are composed of three hyperbolic gamma functions in a way that they enjoy a manifest tetrahedral symmetry. We conjecture that for shaped triangulations of closed three-manifolds, our partition function is twice the absolute value squared of the partition function of Techmüller TQFT defined by Andersen and Kashaev. This is similar to the known relationship between the Turaev–Viro and the Witten–Reshetikhin–Turaev invariants of three-manifolds. We also discuss interpretations of our construction in terms of three-dimensional supersymmetric field theories related to triangulated three-dimensional manifolds.

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