## Homotopy field theory in dimension 2 and group-algebras (1999)

Citations: | 33 - 5 self |

### BibTeX

@MISC{Turaev99homotopyfield,

author = {Vladimir Turaev},

title = {Homotopy field theory in dimension 2 and group-algebras},

year = {1999}

}

### OpenURL

### Abstract

### Citations

217 |
Geometry of 2D topological field theories, Integrable Systems and Quantum Groups, Montecatini Terme
- Dubrovin
- 1993
(Show Context)
Citation Context ...in low dimensions d = 1 and d = 2. Deep algebraic theories come up in both cases. The (1 + 1)-dimensional TQFT’s bijectively correspond to finite-dimensional commutative Frobenius algebras (see [Di], =-=[Du]-=-). The (2+1)-dimensional TQFT’s are closely related to quantum groups and braided categories (see [Tu]). In this paper we apply the basic ideas of a TQFT to maps from manifolds into topological spaces... |

183 |
State sum invariants of 3-manifolds and quantum 6j-symbols, Topology 31
- Turaev, Viro
- 1992
(Show Context)
Citation Context ... which can be computed in terms of partition functions (or state sums) on triangulations or cellular decompositions of manifolds. Lattice TQFT’s are well known in dimensions 2 and 3, see [BP], [FHK], =-=[TV]-=-. In this section we introduce a lattice HQFT in dimension 2. 7.1. Biangular π-algebras. Let L = ⊕α∈πLα be a π-algebra. Given ℓ ∈ L, denote by µℓ the left multiplication by ℓ sending any a ∈ L into ℓa... |

123 |
Topological field theories
- Atiyah
- 1989
(Show Context)
Citation Context ...imensional X-cobordism W = (W, g : W → X), the homomorphism τ(W) is preserved under any homotopy of g relative to ∂W. Axioms (1.2.1) - (1.2.7) form a version of the standard definition of a TQFT, cf. =-=[At]-=-, [Tu, Chapter III]. It is sometimes convenient to consider the homomorphism τ(W) associated to an X-cobordism (W, M0, M1) as a vector τ(W) ∈ HomK(AM0, AM1) = A ∗ M0 ⊗ AM1. In this language, axiom (1.... |

94 |
A presentation for the mapping class group of a closed orientable surface, Topology
- Hatcher, Thurston
- 1980
(Show Context)
Citation Context ... class (relative to the base points on ∂W). We claim that the homomorphism τ(W) does not depend on the choice of a splitting system of loops on W. The crucial argument is provided by the fact (see 36=-=[HT]-=-) that any two splitting systems of loops on W are related by the following transformations: (i) isotopy in W; (ii) adding to a splitting system of loops {α1, ..., αN } a simple loop α ⊂ W \ ∪i αi; (i... |

65 | Associative algebras, Graduate Text - Pierce - 1982 |

54 |
Chern-Simons theory with finite gauge group
- Freed, Quinn
- 1993
(Show Context)
Citation Context ...define a (d + 1)-dimensional HQFT (A, τ) with target X called the primitive cohomological HQFT associated with θ and denoted (A θ , τ θ ). This construction is inspired by the work of Freed and Quinn =-=[FQ]-=- on TQFT’s associated with finite groups. Choose a singular (d + 1)-dimensional cocycle on X with values in K ∗ representing θ. By abuse of notation we denote this cocycle by the same symbol θ. Let M ... |

37 |
Lattice topological field theory in two-dimensions
- Fukuma, Hosono, et al.
- 1994
(Show Context)
Citation Context ...t K(π, 1). We introduce two lattice models derived from biangular (resp. non-degenerate) π-algebras. The first model generalizes the well known lattice model for (1 + 1)-dimensional TQFT’s, see [BP], =-=[FHK]-=-. We first present a map from a surface to K(π, 1) by a π-system, i.e., a system of elements of π associated with 1-cells of a CW-decomposition of the surface. We fix a biangular π-algebra and use it ... |

28 |
A geometric approach to two dimensional conformal field theory
- Dijkgraaf
- 1989
(Show Context)
Citation Context ...ssful in low dimensions d = 1 and d = 2. Deep algebraic theories come up in both cases. The (1 + 1)-dimensional TQFT’s bijectively correspond to finite-dimensional commutative Frobenius algebras (see =-=[Di]-=-, [Du]). The (2+1)-dimensional TQFT’s are closely related to quantum groups and braided categories (see [Tu]). In this paper we apply the basic ideas of a TQFT to maps from manifolds into topological ... |

28 |
Quantum Invariants of Knots and 3-Manifolds
- Turaev
- 1994
(Show Context)
Citation Context ...nal TQFT’s bijectively correspond to finite-dimensional commutative Frobenius algebras (see [Di], [Du]). The (2+1)-dimensional TQFT’s are closely related to quantum groups and braided categories (see =-=[Tu]-=-). In this paper we apply the basic ideas of a TQFT to maps from manifolds into topological spaces. This suggests a notion of a (d + 1)-dimensional homotopy quantum field theory (HQFT) which may be br... |

8 |
Topological Models on the Lattice and a Remark on String Theory
- Bachas, Petropoulos
- 1993
(Show Context)
Citation Context ... target K(π, 1). We introduce two lattice models derived from biangular (resp. non-degenerate) π-algebras. The first model generalizes the well known lattice model for (1 + 1)-dimensional TQFT’s, see =-=[BP]-=-, [FHK]. We first present a map from a surface to K(π, 1) by a π-system, i.e., a system of elements of π associated with 1-cells of a CW-decomposition of the surface. We fix a biangular π-algebra and ... |