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 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 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.

SYMMETRY BREAKING IN TOPOLOGICAL QUANTUM GRAVITY

2013 ◽  
Vol 22 (05) ◽  
pp. 1330009 ◽  
Author(s):  
ECKEHARD W. MIELKE
Keyword(s):  
Field Theory ◽  
Symmetry Breaking ◽  
High Energy ◽  
Topological Field Theory ◽  
Energy Limit ◽  
Gauge Invariant ◽  
High Energy Limit ◽  
Four Dimensions ◽  
Topological Quantum ◽  
Topological Field

A SL (5, ℝ) gauge-invariant topological field theory of gravity and possible gauge unifications are considered in four-dimensions (4D). The problem of quantization is evaluated in the asymptotic safety scenario. "Minimal" BF type models for the high energy limit are physically not quite realistic, a tiny symmetry breaking is needed to recover standard Einsteinian gravity for the macroscopic metrical background with induced cosmological constant.

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.

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 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.

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.

scholarly journals Domain walls in 4d $$ \mathcal{N} $$ = 1 SYM

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Diego Delmastro ◽  
Jaume Gomis
Keyword(s):  
Domain Wall ◽  
Domain Walls ◽  
Simons Theory ◽  
Partition Functions ◽  
Quantum Field Theories ◽  
Yang Mills ◽  
Stable Domain ◽  
Coxeter Number ◽  
Topological Quantum ◽  
Simply Connected

Abstract 4d$$ \mathcal{N} $$ N = 1 super Yang-Mills (SYM) with simply connected gauge group G has h gapped vacua arising from the spontaneously broken discrete R-symmetry, where h is the dual Coxeter number of G. Therefore, the theory admits stable domain walls interpolating between any two vacua, but it is a nonperturbative problem to determine the low energy theory on the domain wall. We put forward an explicit answer to this question for all the domain walls for G = SU(N), Sp(N), Spin(N) and G2, and for the minimal domain wall connecting neighboring vacua for arbitrary G. We propose that the domain wall theories support specific nontrivial topological quantum field theories (TQFTs), which include the Chern-Simons theory proposed long ago by Acharya-Vafa for SU(N). We provide nontrivial evidence for our proposals by exactly matching renormalization group invariant partition functions twisted by global symmetries of SYM computed in the ultraviolet with those computed in our proposed infrared TQFTs. A crucial element in this matching is constructing the Hilbert space of spin TQFTs, that is, theories that depend on the spin structure of spacetime and admit fermionic states — a subject we delve into in some detail.

A COMMENT ON TOPOLOGICAL FIELD THEORY QUANTIZATION AND STOCHASTIC PROCESSES

1990 ◽  
Vol 05 (19) ◽  
pp. 3777-3786 ◽  
Author(s):  
L.F. CUGLIANDOLO ◽  
G. LOZANO ◽  
H. MONTANI ◽  
F.A. SCHAPOSNIK
Keyword(s):  
Field Theory ◽  
Equilibrium State ◽  
Stochastic Processes ◽  
Topological Field Theories ◽  
Topological Charge ◽  
Field Theories ◽  
Langevin Equations ◽  
Topological Field Theory ◽  
Topological Field

We discuss the relation between different quantization approaches to topological field theories by deriving a connection between Bogomol’nyi and Langevin equations for stochastic processes which evolve towards an equilibrium state governed by the topological charge.

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