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.

TOPOLOGICAL FIELD THEORIES ON THE COMPACT KÄHLER COMPLEX SURFACES

1994 ◽  
Vol 09 (10) ◽  
pp. 903-911
Author(s):  
HYUK-JAE LEE
Keyword(s):  
Topological Field Theories ◽  
Chern Class ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Complex Surfaces ◽  
Topological Quantum ◽  
Topological Field ◽  
Universal Bundle ◽  
Compact Kähler Manifold

The structure of topological quantum field theories on the compact Kähler manifold is interpreted. The BRST transformations of fields are derived from universal bundle and the observables are found from the second Chern class of universal bundle.

TWO-DIMENSIONAL TOPOLOGICAL QUANTUM FIELD THEORIES AND FROBENIUS ALGEBRAS

1996 ◽  
Vol 05 (05) ◽  
pp. 569-587 ◽  
Author(s):  
LOWELL ABRAMS
Keyword(s):  
Field Theories ◽  
Algebra Structure ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Two Dimensional ◽  
Direct Sums ◽  
Frobenius Algebras ◽  
Topological Quantum ◽  
Simple Demonstration ◽  
Topological Quantum Field Theories

We characterize Frobenius algebras A as algebras having a comultiplication which is a map of A-modules. This characterization allows a simple demonstration of the compatibility of Frobenius algebra structure with direct sums. We then classify the indecomposable Frobenius algebras as being either “annihilator algebras” — algebras whose socle is a principal ideal — or field extensions. The relationship between two-dimensional topological quantum field theories and Frobenius algebras is then formulated as an equivalence of categories. The proof hinges on our new characterization of Frobenius algebras. These results together provide a classification of the indecomposable two-dimensional topological quantum field theories.

scholarly journals THE RELATIONS BETWEEN GENERALIZED FIELDS AND SUPERFIELDS FORMALISMS OF THE BATALIN–VILKOVISKY METHOD OF QUANTIZATION

2004 ◽  
Vol 19 (14) ◽  
pp. 2339-2353 ◽  
Author(s):  
ÖMER F. DAYI
Keyword(s):  
General Solution ◽  
Master Equation ◽  
Gauge Theories ◽  
Relativistic Particle ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Yang Mills ◽  
Topological Quantum ◽  
Topological Quantum Field Theories

A general solution of the Batalin–Vilkovisky master equation was formulated in terms of generalized fields. Recently, a superfields approach of obtaining solutions of the Batalin–Vilkovisky master equation is also established. Superfields formalism is usually applied to topological quantum field theories. However, generalized fields method is suitable to find solutions of the Batalin–Vilkovisky master equation either for topological quantum field theories or the usual gauge theories like Yang–Mills theory. We show that by truncating some components of superfields with appropriate actions, generalized fields formalism of the usual gauge theories result. We demonstrate that for some topological quantum field theories and the relativistic particle both of the methods possess the same field contents and yield similar results. Inspired by the observed relations, we give the solution of the BV master equation for on-shell N=1 supersymmetric Yang–Mills theory utilizing superfields.

scholarly journals STATE SUM CONSTRUCTION OF TWO-DIMENSIONAL OPEN-CLOSED TOPOLOGICAL QUANTUM FIELD THEORIES

2007 ◽  
Vol 16 (09) ◽  
pp. 1121-1163 ◽  
Author(s):  
AARON D. LAUDA ◽  
HENDRYK PFEIFFER
Keyword(s):  
The State ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Two Dimensional ◽  
Topological Quantum ◽  
Finite Set ◽  
Manifolds With Corners ◽  
The Relationship ◽  
Topological Quantum Field Theories

We present a state sum construction of two-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from triangulations of conventional two-dimensional cobordisms to those of open-closed cobordisms, i.e. smooth compact oriented 2-manifolds with corners that have a particular global structure. This construction reveals the topological interpretation of the associative algebra on which the state sum is based, as the vector space that the TQFT assigns to the unit interval. Extending the notion of a two-dimensional TQFT from cobordisms to suitable manifolds with corners therefore makes the relationship between the global description of the TQFT in terms of a functor into the category of vector spaces and the local description in terms of a state sum fully transparent. We also illustrate the state sum construction of an open-closed TQFT with a finite set of D-branes using the example of the groupoid algebra of a finite groupoid.

scholarly journals QUASI-TOPOLOGICAL QUANTUM FIELD THEORIES AND ℤ2 LATTICE GAUGE THEORIES

2012 ◽  
Vol 27 (23) ◽  
pp. 1250132 ◽  
Author(s):  
MIGUEL J. B. FERREIRA ◽  
VICTOR A. PEREIRA ◽  
PAULO TEOTONIO-SOBRINHO
Keyword(s):  
Partition Function ◽  
Wilson Loop ◽  
Gauge Theories ◽  
Temperature Limit ◽  
Field Theories ◽  
Discrete Version ◽  
Quantum Field Theories ◽  
Gauge Model ◽  
Topological Quantum ◽  
Topological Field

We consider a two-parameter family of ℤ2 gauge theories on a lattice discretization [Formula: see text] of a three-manifold [Formula: see text] and its relation to topological field theories. Familiar models such as the spin-gauge model are curves on a parameter space Γ. We show that there is a region Γ0 ⊂ Γ where the partition function and the expectation value 〈WR(γ)〉 of the Wilson loop can be exactly computed. Depending on the point of Γ0, the model behaves as topological or quasi-topological. The partition function is, up to a scaling factor, a topological number of [Formula: see text]. The Wilson loop on the other hand, does not depend on the topology of γ. However, for a subset of Γ0, 〈WR(γ)〉 depends on the size of γ and follows a discrete version of an area law. At the zero temperature limit, the spin-gauge model approaches the topological and the quasi-topological regions depending on the sign of the coupling constant.

scholarly journals Branes and integrable lattice models

2017 ◽  
Vol 32 (03) ◽  
pp. 1730003 ◽  
Author(s):  
Junya Yagi
Keyword(s):  
Lattice Models ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Two Dimensional ◽  
Supersymmetric Field Theories ◽  
Topological Quantum ◽  
Integrable Lattice ◽  
Integrable Lattice Models ◽  
Topological Quantum Field Theories

This is a brief review of my work on the correspondence between four-dimensional [Formula: see text] supersymmetric field theories realized by brane tilings and two-dimensional integrable lattice models. I explain how to construct integrable lattice models from extended operators in partially topological quantum field theories, and elucidate the correspondence as an application of this construction.

TOPOLOGICAL QUANTUM FIELD THEORIES ASSOCIATED TO FINITE GROUPS AND CROSSED G-SETS

1992 ◽  
Vol 01 (01) ◽  
pp. 1-20 ◽  
Author(s):  
DAVID N. YETTER
Keyword(s):  
Gauge Group ◽  
Quantum Groups ◽  
Finite Groups ◽  
Gauge Theories ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Link Invariants ◽  
Topological Quantum ◽  
Topological Quantum Field Theories

Using methods suggested by the work of Turaev and Viro [11, 12], we provide a detailed construction of topological quantum field theories associated to finite crossed G-sets. Our construction of theories associated to finite groups fills in some details implicit in Dijkgraaf and Witten's [3] discussion of topological gauge theories with finite gauge group, while the theories associated to finite crossed G-sets simultaneously extend Dijkgraaf and Witten's [3] results to 3-manifolds equipped with links and Freyd and Yetter's [5] construction of link invariants from crossed G-sets from links in the 3-sphere to links in arbitrary 3-manifolds. Topological interpretations of the manifold and link invariants associated to these TQFT's are provided. We conclude discussion of our results as a toy model for QFT and of their relation to quantum groups.

scholarly journals a×b=c in 2+1D TQFT

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 468
Author(s):  
Matthew Buican ◽  
Linfeng Li ◽  
Rajath Radhakrishnan
Keyword(s):  
Gauge Theories ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Mathieu Group ◽  
Gauge Groups ◽  
Global Properties ◽  
Topological Quantum ◽  
The One ◽  
Chern Simons

We study the implications of the anyon fusion equation a×b=c on global properties of 2+1D topological quantum field theories (TQFTs). Here a and b are anyons that fuse together to give a unique anyon, c. As is well known, when at least one of a and b is abelian, such equations describe aspects of the one-form symmetry of the theory. When a and b are non-abelian, the most obvious way such fusions arise is when a TQFT can be resolved into a product of TQFTs with trivial mutual braiding, and a and b lie in separate factors. More generally, we argue that the appearance of such fusions for non-abelian a and b can also be an indication of zero-form symmetries in a TQFT, of what we term "quasi-zero-form symmetries" (as in the case of discrete gauge theories based on the largest Mathieu group, M24), or of the existence of non-modular fusion subcategories. We study these ideas in a variety of TQFT settings from (twisted and untwisted) discrete gauge theories to Chern-Simons theories based on continuous gauge groups and related cosets. Along the way, we prove various useful theorems.

scholarly journals A general state-sum construction of 2-dimensional topological quantum field theories with defects

2017 ◽  
Vol 26 (04) ◽  
pp. 1750014
Author(s):  
Gathoni Kamau-Devers ◽  
Gail Jardine ◽  
David Yetter
Keyword(s):  
Knot Theory ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Two Dimensional ◽  
General State ◽  
Topological Quantum ◽  
Witten Theory ◽  
Pachner Moves ◽  
Topological Quantum Field Theories

We derive a general state sum construction for 2D topological quantum field theories (TQFTs) with source defects on oriented curves, extending the state-sum construction from special symmetric Frobenius algebra for 2D TQFTs without defects (cf. Lauda and Pfeiffer [State-sum construction of two-dimensional open-closed topological quantum field theories, J. Knot Theory Ramifications 16 (2007) 1121–1163, doi: 10.1142/S0218216507005725]). From the extended Pachner moves (Crane and Yetter [Moves on filtered PL manifolds and stratified PL spaces, arXiv:1404.3142 ]), we derive equations that we subsequently translate into string diagrams so that we can easily observe their properties. As in Dougherty, Park and Yetter [On 2-dimensional Dijkgraaf–Witten theory with defects, to appear in J. Knots Theory Ramifications], we require that triangulations be flaglike, meaning that each simplex of the triangulation is either disjoint from the defect curve, or intersects it in a closed face, and that the extended Pachner moves preserve flaglikeness.

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