Mapping class group representations from non-semisimple TQFTs
Copyright policy )Mapping class group representations from non-semisimple TQFTs
In [M. De Renzi, A. Gainutdinov, N. Geer, B. Patureau-Mirand and I. Runkel, 3-dimensional TQFTs from non-semisimple modular categories, preprint (2019), arXiv:1912.02063[math.GT]], we constructed 3-dimensional topological quantum field theories (TQFTs) using not necessarily semisimple modular categories. Here, we study projective representations of mapping class groups of surfaces defined by these TQFTs, and we express the action of a set of generators through the algebraic data of the underlying modular category [Formula: see text]. This allows us to prove that the projective representations induced from the non-semisimple TQFTs of the above reference are equivalent to those obtained by Lyubashenko via generators and relations in [V. Lyubashenko, Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity, Comm. Math. Phys. 172(3) (1995) 467–516, arXiv:hep-th/9405167]. Finally, we show that, when [Formula: see text] is the category of finite-dimensional representations of the small quantum group of [Formula: see text], the action of all Dehn twists for surfaces without marked points has infinite order.
- French National Centre for Scientific Research France
- University of Zurich Switzerland
- Universität Hamburg Germany
- University of Southern Brittany France
- Utah State University United States
High Energy Physics - Theory, FOS: Physical sciences, Quantum groups (quantized enveloping algebras) and related deformations, mapping class groups, 510, Mathematics - Geometric Topology, 510 Mathematics, 2604 Applied Mathematics, Braided monoidal categories and ribbon categories, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], 57K16, Mathematics - Quantum Algebra, FOS: Mathematics, Quantum Algebra (math.QA), 2600 General Mathematics, [MATH.MATH-GT] Mathematics [math]/Geometric Topology [math.GT], topological quantum field theories (TQFTs), [MATH.MATH-QA] Mathematics [math]/Quantum Algebra [math.QA], quantum groups, [PHYS.HTHE]Physics [physics]/High Energy Physics - Theory [hep-th], 500, Geometric Topology (math.GT), 17B37, ribbon categories, Topological quantum field theories (aspects of differential topology), 10123 Institute of Mathematics, MSC : 57K20, High Energy Physics - Theory (hep-th), Finite-type and quantum invariants, topological quantum field theories (TQFT), [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], [PHYS.HTHE] Physics [physics]/High Energy Physics - Theory [hep-th], 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.), 18M15
High Energy Physics - Theory, FOS: Physical sciences, Quantum groups (quantized enveloping algebras) and related deformations, mapping class groups, 510, Mathematics - Geometric Topology, 510 Mathematics, 2604 Applied Mathematics, Braided monoidal categories and ribbon categories, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], 57K16, Mathematics - Quantum Algebra, FOS: Mathematics, Quantum Algebra (math.QA), 2600 General Mathematics, [MATH.MATH-GT] Mathematics [math]/Geometric Topology [math.GT], topological quantum field theories (TQFTs), [MATH.MATH-QA] Mathematics [math]/Quantum Algebra [math.QA], quantum groups, [PHYS.HTHE]Physics [physics]/High Energy Physics - Theory [hep-th], 500, Geometric Topology (math.GT), 17B37, ribbon categories, Topological quantum field theories (aspects of differential topology), 10123 Institute of Mathematics, MSC : 57K20, High Energy Physics - Theory (hep-th), Finite-type and quantum invariants, topological quantum field theories (TQFT), [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], [PHYS.HTHE] Physics [physics]/High Energy Physics - Theory [hep-th], 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.), 18M15
selected citations These citations are derived from selected sources.This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).6 popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.Top 10% influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).Average impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.Top 10%
Citations provided by BIP!Popularity provided by BIP!