scholarly journals ON A CLASS OF TOPOLOGICAL QUANTUM FIELD THEORIES IN THREE DIMENSIONS

1996 ◽  
Vol 11 (25) ◽  
pp. 4577-4596
Author(s):  
MASAKO ASANO
Keyword(s):  
Field Theory ◽  
Three Dimensional ◽  
Space Structure ◽  
Three Dimensions ◽  
Topological Field Theory ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Manifolds With Boundary ◽  
Topological Quantum ◽  
Topological Field

We investigate the Chung–Fukuma–Shapere theory, or Kuperberg theory, of three-dimensional lattice topological field theory. We construct a functor which satisfies Atiyah’s axioms of topological quantum field theory by reformulating the theory as a Turaev–Viro type state sum theory on a triangulated manifold. This corresponds to giving the Hilbert space structure to the original theory. The theory can be extended to give a topological invariant of manifolds with boundary.

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.

scholarly journals State sum construction of two-dimensional topological quantum field theories on spin surfaces

2015 ◽  
Vol 24 (05) ◽  
pp. 1550028 ◽  
Author(s):  
S. Novak ◽  
I. Runkel
Keyword(s):  
Spin Structure ◽  
Topological Field Theory ◽  
Quantum Field Theories ◽  
Extra Condition ◽  
Admissibility Condition ◽  
Topological Quantum ◽  
Combinatorial Model ◽  
Topological Field ◽  
Pachner Moves ◽  
Combinatorial Data

We provide a combinatorial model for spin surfaces. Given a triangulation of an oriented surface, a spin structure is encoded by assigning to each triangle a preferred edge, and to each edge an orientation and a sign, subject to certain admissibility conditions. The behavior of this data under Pachner moves is then used to define a state sum topological field theory on spin surfaces. The algebraic data is a Δ-separable Frobenius algebra whose Nakayama automorphism is an involution. We find that a simple extra condition on the algebra guarantees that the amplitude is zero unless the combinatorial data satisfies the admissibility condition required for the reconstruction of the spin structure.

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 Quantum Field Theory of Open Spin Networks and New Spin Foam Models

2003 ◽  
Vol 18 (supp02) ◽  
pp. 83-96 ◽  
Author(s):  
A. Miković
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Vector Bundles ◽  
Tensor Category ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Spin Networks ◽  
Spin Foam Models ◽  
Topological Quantum ◽  
Spin Foam

We describe how a spin-foam state sum model can be reformulated as a quantum field theory of spin networks, such that the Feynman diagrams of that field theory are the spin-foam amplitudes. In the case of open spin networks, we obtain a new type of state-sum models, which we call the matter spin foam models. In this type of state-sum models, one labels both the faces and the edges of the dual two-complex for a manifold triangulation with the simple objects from a tensor category. In the case of Lie groups, such a model corresponds to a quantization of a theory whose fields are the principal bundle connection and the sections of the associated vector bundles. We briefly discuss the relevance of the matter spin foam models for quantum gravity and for topological quantum field theories.

TOPOLOGICAL QUANTUM FIELD THEORY AND INVARIANTS OF SPATIAL GRAPHS

1992 ◽  
Vol 06 (11n12) ◽  
pp. 1825-1846
Author(s):  
KENNETH C. MILLETT
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Three Dimensional ◽  
Topological Quantum Field Theory ◽  
Dimensional Sphere ◽  
Quantum Field ◽  
Topological Quantum ◽  
Spatial Graphs ◽  
Jones Polynomials ◽  
Knots And Links

According to Sir Michael Atiyah [At], the study of topological quantum field theory is equivalent to the study of invariant quantities associated to three-dimensional manifolds. Although one has long considered the classical homology and cohomology structures and their extremely successful generalizations, the real subject of the Atiyah assertion is the new invariants proposed by Witten associated to the Jones polynomials of classical knots and links in the three-dimensional sphere. There have been many manifestations described by Reshetikhin & Turaev [Re1&2], Turaev & Viro [TV], Lickorish [Li 11– 15]. Kirby & Melvin [KM1&2], and Blanchet, Habegger, Mausbaum & Vogel [BHMV]. In these notes I describe some of the fundamental aspects of this theory, discuss the interest in these invariants and their extensions to the class of spatial graphs by Jonish & Millett [JonM], Kauffman & Vogel [KauV], Yamada [Ya2], Millett [Mi1&2], Kuperberg [Ku1&2], and Jaeger, Vertigan and Welsh [JaVW].

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.

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