scholarly journals Construction of two-dimensional topological field theories with non-invertible symmetries

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Tzu-Chen Huang ◽  
Ying-Hsuan Lin ◽  
Sahand Seifnashri
Keyword(s):  
Topological Field Theories ◽  
Hilbert Spaces ◽  
Topological Defect ◽  
Field Theories ◽  
Boundary Crossing ◽  
Two Dimensional ◽  
Fusion Categories ◽  
Explicit Constructions ◽  
Crossing Symmetry ◽  
Topological Field

Abstract We construct the defining data of two-dimensional topological field theories (TFTs) enriched by non-invertible symmetries/topological defect lines. Simple formulae for the three-point functions and the lasso two-point functions are derived, and crossing symmetry is proven. The key ingredients are open-to-closed maps and a boundary crossing relation, by which we show that a diagonal basis exists in the defect Hilbert spaces. We then introduce regular TFTs, provide their explicit constructions for the Fibonacci, Ising and Haagerup ℋ3 fusion categories, and match our formulae with previous bootstrap results. We end by explaining how non-regular TFTs are obtained from regular TFTs via generalized gauging.

scholarly journals DIFF(Σ) AND METRICS FROM HAMILTONIAN-TQFT'S IN 2+1 DIMENSIONS

1993 ◽  
Vol 08 (24) ◽  
pp. 2277-2283 ◽  
Author(s):  
ROGER BROOKS
Keyword(s):  
Topological Field Theories ◽  
Gauge Theories ◽  
Dimensional Space ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Two Dimensional ◽  
Canonical Gravity ◽  
Topological Quantum ◽  
Topological Field

The constraints of BF topological gauge theories are used to construct Hamiltonians which are anti-commutators of the BRST and anti-BRST operators. Such Hamiltonians are a signature of topological quantum field theories (TQFTs). By construction, both classes of topological field theories share the same phase spaces and constraints. We find that, for (2+1)- and (1+1)-dimensional space-times foliated as M=Σ × ℝ, a homomorphism exists between the constraint algebras of our TQFT and those of canonical gravity. The metrics on the two-dimensional hypersurfaces are also obtained.

scholarly journals Solitons in two-dimensional topological field theories

1993 ◽  
Vol 309 (3-4) ◽  
pp. 304-311 ◽  
Author(s):  
S. Penati ◽  
M. Pernici ◽  
D. Zanon
Keyword(s):  
Topological Field Theories ◽  
Field Theories ◽  
Two Dimensional ◽  
Topological Field

scholarly journals Topological field theories and symmetry protected topological phases with fusion category symmetries

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Kansei Inamura
Keyword(s):  
Topological Field Theories ◽  
Time Reversal ◽  
Equivalence Classes ◽  
Field Theories ◽  
Fusion Category ◽  
Topological Phases ◽  
Fusion Categories ◽  
Self Duality ◽  
Topological Field ◽  
Symmetry Protected

Abstract Fusion category symmetries are finite symmetries in 1+1 dimensions described by unitary fusion categories. We classify 1+1d time-reversal invariant bosonic symmetry protected topological (SPT) phases with fusion category symmetry by using topological field theories. We first formulate two-dimensional unoriented topological field theories whose symmetry splits into time-reversal symmetry and fusion category symmetry. We then solve them to show that SPT phases are classified by equivalence classes of quintuples (Z, M, i, s, ϕ) where (Z, M, i) is a fiber functor, s is a sign, and ϕ is the action of orientation- reversing symmetry that is compatible with the fiber functor (Z, M, i). We apply this classification to SPT phases with Kramers-Wannier-like self-duality.

scholarly journals POISSON STRUCTURE INDUCED (TOPOLOGICAL) FIELD THEORIES

1994 ◽  
Vol 09 (33) ◽  
pp. 3129-3136 ◽  
Author(s):  
PETER SCHALLER ◽  
THOMAS STROBL
Keyword(s):  
Topological Field Theories ◽  
Poisson Structure ◽  
General Scheme ◽  
Hamiltonian Formulation ◽  
Field Theories ◽  
Two Dimensional ◽  
Poisson Structures ◽  
Yang Mills ◽  
Topological Field ◽  
Equations Of Motions

A class of two-dimensional field theories, based on (generically degenerate) Poisson structures and generalizing gravity-Yang–Mills systems, is presented. Locally, the solutions of the classical equations of motions are given. A general scheme for the quantization of the models in a Hamiltonian formulation is found. A BRS-formulation is outlined briefly.

scholarly journals On dualizability of braided tensor categories

2021 ◽  
Vol 157 (3) ◽  
pp. 435-483
Author(s):  
Adrien Brochier ◽  
David Jordan ◽  
Noah Snyder
Keyword(s):  
Field Theory ◽  
Quantum Group ◽  
Topological Field Theories ◽  
Characteristic Zero ◽  
Three Dimensional ◽  
Field Theories ◽  
Topological Field Theory ◽  
Tensor Categories ◽  
Fusion Categories ◽  
Topological Field

We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided multi-fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively two-, three- and four-dimensional framed local topological field theories. In particular, we produce a framed three-dimensional local topological field theory attached to the category of representations of a quantum group at any value of $q$ .

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