scholarly journals FROM SUBFACTORS TO 3-DIMENSIONAL TOPOLOGICAL QUANTUM FIELD THEORIES AND BACK: A detailed account of Ocneanu’s theory

1995 ◽  
Vol 06 (04) ◽  
pp. 537-558 ◽  
Author(s):  
DAVID E. EVANS ◽  
YASUYUKI KAWAHIGASHI
Keyword(s):  
Field Theory ◽  
Finite Depth ◽  
Topological Quantum Field Theory ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Frobenius Reciprocity ◽  
3 Dimensional ◽  
Topological Quantum ◽  
Full Proof

A full proof of Ocneanu’s theorem is given that one can produce a rational unitary polyhedral 3-dimensional topological quantum field theory of Turaev-Viro type from a subfactor with finite index and finite depth, and vice versa. The key argument is an equivalence between flatness of a connection in paragroup theory and invariance of a state sum under one of the three local moves of tetrahedra. This was announced by A. Ocneanu and he gave a proof of Frobenius reciprocity and the pentagon relation, which produces a 3-dimensional TQFT via the Turaev-Viro machinery, but he has not published a proof of the converse direction of the equivalence. Details are given here along the lines suggested by him.

scholarly journals ON ALGEBRAIC STRUCTURES IMPLICIT IN TOPOLOGICAL QUANTUM FIELD THEORIES

1999 ◽  
Vol 08 (02) ◽  
pp. 125-163 ◽  
Author(s):  
Louis Crane ◽  
David Yetter
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hopf Algebra ◽  
Tensor Category ◽  
Topological Quantum Field Theory ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Topological Quantum ◽  
Natural Way

We show that any 3D topological quantum field theory satisfying physically reasonable factorizability conditions has associated to it in a natural way a Hopf algebra object in a suitable tensor category. We also show that all objects in the tensor category have the structure of left-left crossed bimodules over the Hopf algebra object. For 4D factorizable topological quantum filed theories, we provide by analogous methods a construction of a Hopf algebra category.

On 3-dimensional homotopy quantum field theory II: The surgery approach

2014 ◽  
Vol 25 (04) ◽  
pp. 1450027 ◽  
Author(s):  
Vladimir Turaev ◽  
Alexis Virelizier
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
3 Dimensional ◽  
Topological Quantum ◽  
Topological Quantum Field Theories

Homotopy Quantum Field Theories (HQFTs) generalize more familiar Topological Quantum Field Theories (TQFTs). In generalization of the surgery construction of 3-dimensional TQFTs from modular categories, we use surgery to derive 3-dimensional HQFTs from G-modular categories.

scholarly journals SPIN TOPOLOGICAL QUANTUM FIELD THEORIES

1998 ◽  
Vol 09 (02) ◽  
pp. 129-152 ◽  
Author(s):  
ANNA BELIAKOVA
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Spin Structure ◽  
Topological Quantum Field Theory ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Topological Quantum ◽  
The Relationship ◽  
Construct Operator

Starting from the quantum group [Formula: see text], we construct operator invariants of 3-cobordisms with spin structure, satisfying the requirements of a topological quantum field theory and refining the Reshetikhin–Turaev and Turaev–Viro models. We establish the relationship between these two refined theories.

ON THE SYMMETRIES OF TOPOLOGICAL QUANTUM FIELD THEORIES

1995 ◽  
Vol 10 (31) ◽  
pp. 4483-4499 ◽  
Author(s):  
LAURENT BAULIEU
Keyword(s):  
Field Theory ◽  
Gauge Symmetry ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Topological Quantum ◽  
Topological Quantum Field Theories

We display properties of the general formalism which associates with any given gauge symmetry a topological action and a system of topological BRST and anti-BRST equations. We emphasize the distinction between the antighosts of the geometrical BRST equations and the antighosts occurring in field theory. We propose a transmutation mechanism between these objects. We illustrate our general presentation by examples.

Two Subfactors Arising from a Non-Degenerate Commuting Square II — Tensor Categories and TQFT's —

1997 ◽  
Vol 08 (03) ◽  
pp. 407-420 ◽  
Author(s):  
Nobuya Sato
Keyword(s):  
Field Theories ◽  
Complex Conjugate ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Tensor Categories ◽  
3 Dimensional ◽  
Topological Quantum ◽  
Topological Quantum Field Theories

We prove that two subfactors N ⊂ M and P ⊂ Q arising from a non-degenerate commuting square have 3-dimensional topological quantum field theories (based on triangulations) complex conjugate to each other. Applying the asymptotic inclusion construction to each subfactor, we have new subfactors M ∨ (M′ ∩ M∞) ⊂ M∞ and Q ∨ (Q′ ∩ Q∞) ⊂ Q∞, then we also prove that the tensor categories of the M∞-M∞ bimodules and the Q∞-Q∞ bimodules are isomorphic to each other, if the two fusion graphs are connected. These results are based on our previous work and give a finer answer to a question raised by V. F. R. Jones in June, 1995.

TOPOLOGICAL QUANTUM FIELD THEORIES FROM GENERALIZED 6J-SYMBOLS

1993 ◽  
Vol 05 (01) ◽  
pp. 1-67 ◽  
Author(s):  
BERGFINNUR DURHUUS ◽  
HANS PLESNER JAKOBSEN ◽  
RYSZARD NEST
Keyword(s):  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
3 Dimensional ◽  
Conformal Field ◽  
Topological Quantum ◽  
Finite Set ◽  
6J Symbols ◽  
Chiral Algebras ◽  
Natural Way

Given an associative algebra with a distinguished finite set of representations that is closed under a (deformed) tensor product, and satisfies some technical assumptions, we define generalized 6j-symbols, and show that they can be associated, in a natural way, with certain labeled tetrahedra. Given a 3-dimensional compact oriented manifold M with boundary ∂M = Σ we choose an arbitrary triangulation [Formula: see text] of M and exploit the above correspondence between 6j-symbols and labeled tetrahedra to construct a vectorspace UΣ and a vector Z(M) ∈ UΣ, independent of [Formula: see text], and fulfilling the axioms of a topological quantum field theory as formulated by Atiyah [11]. Examples covered by our approach are quantum groups corresponding to the classical simple Lie algebras as well as, expectedly, chiral algebras of 2-dimensional rational conformal field theories.

scholarly journals ON OCNEANU'S THEORY OF ASYMPTOTIC INCLUSIONS FOR SUBFACTORS, TOPOLOGICAL QUANTUM FIELD THEORIES AND QUANTUM DOUBLES

1995 ◽  
Vol 06 (02) ◽  
pp. 205-228 ◽  
Author(s):  
DAVID E. EVANS ◽  
YASUYUKI KAWAHIGASHI
Keyword(s):  
Field Theory ◽  
Hilbert Space ◽  
Finite Depth ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Natural Basis ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Topological Quantum

A fully detailed account of Ocneanu's theorem is given that the Hilbert space associated to the two-dimensional torus in a Turaev-Viro type (2+1)-dimensional topological quantum field theory arising from a finite depth subfactor N⊂M has a natural basis labeled by certain M∞- M∞ bimodules of the asymptotic inclusion M∨(M'∩M∞)⊂M∞, and moreover that all these bimodules are given by the basic construction from M∨(M'∩M∞)⊂M∞ if the fusion graph is connected. This Hilbert space is an analogue of the space of conformal blocks in conformal field theory. It is also shown that after passing to the asymptotic inclusions we have S- and T-matrices, analogues of the Verlinde identity and Vafa's result on roots of unity. It is explained that the asymptotic inclusions can be regarded as a subfactor analogue of the quantum double construction of Drinfel'd. These claims were announced by Λ. Ocneanu in several talks, but he has not published his proofs, so details are given here along the lines outlined in his talks.

QFT and unification of knot theories

2014 ◽  
Vol 29 (24) ◽  
pp. 1430025
Author(s):  
Alexey Sleptsov
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Knot Theory ◽  
Topological Quantum Field Theory ◽  
Quantum Field ◽  
Knot Invariants ◽  
Topological Quantum

We discuss relation between knot theory and topological quantum field theory. Also it is considered a theory of superpolynomial invariants of knots which generalizes all other known theories of knot invariants. We discuss a possible generalization of topological quantum field theory with the help of superpolynomial invariants.

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