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Orbifolds of $n$–dimensional defect TQFTs

Nils Carqueville, Ingo Runkel and Gregor Schaumann

Geometry & Topology 23 (2019) 781–864
Abstract

We introduce the notion of n–dimensional topological quantum field theory (TQFT) with defects as a symmetric monoidal functor on decorated stratified bordisms of dimension n. The familiar closed or open–closed TQFTs are special cases of defect TQFTs, and for n = 2 and n = 3 our general definition recovers what had previously been studied in the literature.

Our main construction is that of “generalised orbifolds” for any n–dimensional defect TQFT: Given a defect TQFT Z, one obtains a new TQFT ZA by decorating the Poincaré duals of triangulated bordisms with certain algebraic data A and then evaluating with Z. The orbifold datum A is constrained by demanding invariance under n–dimensional Pachner moves. This procedure generalises both state sum models and gauging of finite symmetry groups for any n. After developing the general theory, we focus on the case n = 3.

Keywords
TQFT, orbifold, triangulation-invariance, stratified bordism
Mathematical Subject Classification 2010
Primary: 57R56
References
Publication
Received: 20 August 2017
Revised: 3 November 2017
Accepted: 12 May 2018
Published: 8 April 2019
Proposed: Peter Teichner
Seconded: Stefan Schwede, Ralph Cohen
Authors
Nils Carqueville
Fakultät für Mathematik
Universität Wien
Wien
Austria
Ingo Runkel
Fachbereich Mathematik
Universität Hamburg
Hamburg
Germany
Gregor Schaumann
Fakultät für Mathematik
Universität Wien
Wien
Austria