## LATTICE GAUGE FIELD THEORY (2001)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Akyar01latticegauge,

author = {Bedia Akyar and Johan L. Dupont},

title = {LATTICE GAUGE FIELD THEORY},

year = {2001}

}

### OpenURL

### Abstract

The inspiration for this thesis comes from mathematical physics, especially path integrals and the Chern-Simons action. Path integrals were introduced by Feynman in late 1940’s and they have recently been applied to purely geometric problems. The work [33] of Edward Witten on the topological quantum field theory has been found very attractive by many enthusiastic

### Citations

612 |
Quantum field theory and the Jones polynomial
- Witten
- 1989
(Show Context)
Citation Context ...cal physics, especially path integrals and the Chern-Simons action. Path integrals were introduced by Feynman in late 1940’s and they have recently been applied to purely geometric problems. The work =-=[33]-=- of Edward Witten on the topological quantum field theory has been found very attractive by many enthusiastic mathematicians. Although the ideas of quantum field theory are far from mathematically und... |

212 |
Topological Vector Spaces, Distributions and Kernels, volume 25 of Pure and Applied Mathematics
- Treves
- 1967
(Show Context)
Citation Context ...ex Ω∗(X) of compactly supported currents on X. One needs to equip the space A ∗ �X� with a natural Frechét topology in order to give a definition of the simplicial currents. One can see the monograph =-=[31]-=- for the theory of Frechét spaces. (p. 85). Definition 13.5: The simplicial n-forms have been defined in definition 1.9 as the space A n �X� = {φ ∈ ΠkA n (∆ k × Xk) | (ε i × id) ∗ φ (k) = (id × εi)φ (... |

122 |
Differential characters and geometric invariants
- Cheeger, Simons
- 1985
(Show Context)
Citation Context ...hern-Simons form and character have been introduced by S.S. Chern-J. Simons [7] and by taking this article as a base J. Cheeger and J. Simons have studied on differential characterssINTRODUCTION 3 in =-=[6]-=-. Lattice gauge fields were introduced by K. Wilson in [32] to represent classical field configurations in Monte Carlo evaluations of path integral solutions of quantum field theories. L.g.f.’s have b... |

116 | Topological gauge theories and group cohomology
- Dijkgraaf, Witten
- 1990
(Show Context)
Citation Context ...space of all connections on the 3-manifold. In 1989 Witten proposed a formulation of a class of 3-manifold invariants as generalized Feynman integrals taking the form above (see also Dijkgraaf-Witten =-=[8]-=-, Rabin[29]). In Chern-Simons theory for which we follow Chern-Simons [7], the parameter which defines the “path” varies the connection. In our case, the path corresponds to a connection and the subdi... |

101 |
Confinement of quarks,” Phys
- Wilson
- 1974
(Show Context)
Citation Context .... Chern-J. Simons [7] and by taking this article as a base J. Cheeger and J. Simons have studied on differential characterssINTRODUCTION 3 in [6]. Lattice gauge fields were introduced by K. Wilson in =-=[32]-=- to represent classical field configurations in Monte Carlo evaluations of path integral solutions of quantum field theories. L.g.f.’s have been used to find a new way of computing the second Chern nu... |

97 |
Characteristic forms and geometric invariants
- Chern, Simons
- 1974
(Show Context)
Citation Context ...mulation of a class of 3-manifold invariants as generalized Feynman integrals taking the form above (see also Dijkgraaf-Witten [8], Rabin[29]). In Chern-Simons theory for which we follow Chern-Simons =-=[7]-=-, the parameter which defines the “path” varies the connection. In our case, the path corresponds to a connection and the subdivision in the path corresponds to the subdivision in order to compute the... |

83 | Classifying spaces and spectral sequences - Segal - 1968 |

78 |
On the groups of H(Π, n
- Eilenberg, Lane
- 1953
(Show Context)
Citation Context ...struction of a pseudo section. They use the pseudo section to integrate the pulled-back Chern-Simons form from the universal bundle. By using the bar construction established by Eilenberg and MacLane =-=[14]-=-, they compute the Chern-Simons character from each 3-simplex σ ∈ Λ. They compare the Chern-Simons class S(ω) with the Chern-Simons class for the canonical connection for SU(2) and the second one is a... |

45 |
Simplicial de Rham cohomology and characteristic classes of flat bundles
- Dupont
- 1976
(Show Context)
Citation Context ... ij > of Λ, subject to the condition uji = uij −1 . They construct from u a principal SU(2)-bundle ξ over M and a piecewise smooth connection ω. They define a canonical connection follows from Dupont =-=[9]-=- in Milnor’s universal G-bundle ξ∆G = (π∆ : E∆G → B∆G) and a corresponding canonical Chern-Simons form. They extend l.g.f. to a G-valued parallel transport function (p.t.f.) over Λ which consists of a... |

34 | The Geometric Realization of a Semi-Simplicial Complex - Milnor - 1957 |

22 | Homotopy coherent category theory and A∞-structures in monoidal categories - Batanin - 1998 |

16 | operations on higher K-theory, K-theory 6 - Grayson, Adams - 1992 |

9 |
Extended moduli spaces, the Kan construction, and lattice gauge theory. Topology 38
- Huebschmann
- 1999
(Show Context)
Citation Context ...ath corresponds to a connection and the subdivision in the path corresponds to the subdivision in order to compute the variation of the Chern-Simons class for a connection. Now, we recall Huebschmann =-=[17]-=- for the Chern-Simons function, let M be a smooth closed, oriented manifold and consider a G-bundle on I × M with a connection ω on M, parametrized by I = [0, 1] or equivalently, as a path of connecti... |

8 |
Coherent categories with respect to monads and coherent prohomotopy theory. Cahiers Topologie Géom. Différentielle Catégoriques 34
- Batanin
- 1993
(Show Context)
Citation Context ...a connection. Now, we recall Huebschmann [17] for the Chern-Simons function, let M be a smooth closed, oriented manifold and consider a G-bundle on I × M with a connection ω on M, parametrized by I = =-=[0, 1]-=- or equivalently, as a path of connections. The Chern-Weil construction assigns to an invariant homogeneous degree k polynomial P on the Lie algebra g the characteristic form P (Fω) on I × M of degree... |

8 | A solution of Deligne’s conjecture
- McClure, Smith
(Show Context)
Citation Context ... SU(2)-bundle ξ over a triangulated 4-manifold M by Phillips-Stone [25], if ξ has a connection. The prismatic subdivision plays an important role in the thesis. It has also been used by McClure-Smith =-=[21]-=- to give a solution of Deligne’s Conjecture. An affirmative answer to the Deligne’s conjecture has been given by Kontsevich and Voronov. The prismatic subdivision was discovered independently and vari... |

6 |
Curvature and Characteristic
- Dupont
- 1978
(Show Context)
Citation Context ... brief review of classifying spaces of lie group G. Much more information of the ingredient in this chapter can be found in the mathematical literature, for example in Dijkgraaf-Witten [[8]], Dupont [=-=[10]-=-], Milnor [], Segal []. Definition 1.1 ( Standard n-Simplex ) : Let us consider ∆ n in R n+1 , the convex hull of the set of canonical basis vectors ei = (0, ..., 1, ..., 0) with 1 on the i-th place, ... |

6 |
Topology of lattice gauge
- Lüscher
- 1982
(Show Context)
Citation Context ...et a section. Therefore we would have to follow another way to calculate the related formula for the Chern-Simons character. For further information, one can see Freed [15], Huebschmann [17], Lüscher =-=[19]-=-, Phillips-Stone [23],[24],[26],[28]. The Chern-Simons form and character have been introduced by S.S. Chern-J. Simons [7] and by taking this article as a base J. Cheeger and J. Simons have studied on... |

6 |
Lattice gauge fields, principal bundles and the calculation of topological charge
- Phillips, Stone
- 1986
(Show Context)
Citation Context ...ield theories. L.g.f.’s have been used to find a new way of computing the second Chern number (the topological charge) of a principal SU(2)-bundle ξ over a triangulated 4-manifold M by Phillips-Stone =-=[25]-=-, if ξ has a connection. The prismatic subdivision plays an important role in the thesis. It has also been used by McClure-Smith [21] to give a solution of Deligne’s Conjecture. An affirmative answer ... |

5 | F.: On a generalization of Cheeger-Chern-Simons classes
- Dupont, Kamber
- 1990
(Show Context)
Citation Context ...acteristic classes, Classical ChernWeil theory and Chern-Simons theory. We also give the Chern-Simons form as a differential character due to Chegeer-Simons [6]. In corollary 15.11 (see Dupont-Kamber =-=[12]-=-), we give the Chern-Simons class SP,u(ω) ∈ H2k−1 (M, R/Z), where P ∈ Ik (G) and u ∈ H2k (BG, Λ). At last, we give the difference of the evaluation of the Chern-Simons classes for two different connec... |

4 | Characteristic numbers and generalized path integrals
- Freed
(Show Context)
Citation Context ...n and such a cycle property to get a section. Therefore we would have to follow another way to calculate the related formula for the Chern-Simons character. For further information, one can see Freed =-=[15]-=-, Huebschmann [17], Lüscher [19], Phillips-Stone [23],[24],[26],[28]. The Chern-Simons form and character have been introduced by S.S. Chern-J. Simons [7] and by taking this article as a base J. Cheeg... |

4 |
Characteristic numbers of U(1) valued lattice gauge
- Phillips
- 1985
(Show Context)
Citation Context ... would have to follow another way to calculate the related formula for the Chern-Simons character. For further information, one can see Freed [15], Huebschmann [17], Lüscher [19], Phillips-Stone [23],=-=[24]-=-,[26],[28]. The Chern-Simons form and character have been introduced by S.S. Chern-J. Simons [7] and by taking this article as a base J. Cheeger and J. Simons have studied on differential characterssI... |

3 |
Homology of O(n) and O 1 (1, n) made discrete: an application of edgewise subdivision
- Bökstedt, Brun, et al.
- 1998
(Show Context)
Citation Context ... of a simplex is a subdivision of the prismatic subdivision. A special case of the edgewise subdivision was given by Bökstedt-Brun-Dupont in an unpublished note [4] and applied by the same authors in =-=[3]-=-. It can be defined as a chain map from the singular chain complex of any topological space into itself. In the construction given in [4], one uses the Alexander-Whitney map and the definition is clos... |

3 |
Coherent prohomotopy and strong shape theory
- Lisica, Mardesi'c
- 1984
(Show Context)
Citation Context ...njecture. An affirmative answer to the Deligne’s conjecture has been given by Kontsevich and Voronov. The prismatic subdivision was discovered independently and various times by e.g., Lisica-Mardesic =-=[18]-=- and Batanin [1], [2]. There is a related edgewise subdivision discovered by Quillen, Segal, Bökstedt and Goodwille and it has also been studied by Hsiang and Madsen. The edgewise subdivision of a sim... |

3 |
Introduction to Quantum Field Theory for
- Rabin
- 1995
(Show Context)
Citation Context ...ll connections on the 3-manifold. In 1989 Witten proposed a formulation of a class of 3-manifold invariants as generalized Feynman integrals taking the form above (see also Dijkgraaf-Witten [8], Rabin=-=[29]-=-). In Chern-Simons theory for which we follow Chern-Simons [7], the parameter which defines the “path” varies the connection. In our case, the path corresponds to a connection and the subdivision in t... |

2 |
Lattice gauge field and Chern-Weil Theory, Geometry and Topology: Manifolds, varieties and knots
- Phillips, Stone
- 1985
(Show Context)
Citation Context ...re we would have to follow another way to calculate the related formula for the Chern-Simons character. For further information, one can see Freed [15], Huebschmann [17], Lüscher [19], Phillips-Stone =-=[23]-=-,[24],[26],[28]. The Chern-Simons form and character have been introduced by S.S. Chern-J. Simons [7] and by taking this article as a base J. Cheeger and J. Simons have studied on differential charact... |

2 |
The computation of characteristic classes of lattice gauge fields
- Phillips, Stone
- 1990
(Show Context)
Citation Context ...d have to follow another way to calculate the related formula for the Chern-Simons character. For further information, one can see Freed [15], Huebschmann [17], Lüscher [19], Phillips-Stone [23],[24],=-=[26]-=-,[28]. The Chern-Simons form and character have been introduced by S.S. Chern-J. Simons [7] and by taking this article as a base J. Cheeger and J. Simons have studied on differential characterssINTROD... |

2 |
The Chern-Simons character and a lattice gauge field, Quantum Topology
- Phillips, Stone
- 1993
(Show Context)
Citation Context ... for the variation of a ChernSimons class for a given bundle F → |S|, where S is a simplicial set, with a connection ω by using prismatic subdivision. I have been inspired by Phillips-Stone’s article =-=[27]-=- in which they compute the Chern-Simons character of a lattice gauge field. In their work, they take a generic SU(2)-valued lattice gauge field u on a triangulation Λ of a manifold M of dimension ≥ 3.... |

2 |
Topological Chern-Weil Theory
- Phillips, Stone
- 1993
(Show Context)
Citation Context ...e to follow another way to calculate the related formula for the Chern-Simons character. For further information, one can see Freed [15], Huebschmann [17], Lüscher [19], Phillips-Stone [23],[24],[26],=-=[28]-=-. The Chern-Simons form and character have been introduced by S.S. Chern-J. Simons [7] and by taking this article as a base J. Cheeger and J. Simons have studied on differential characterssINTRODUCTIO... |

1 |
A dual simplicial de Rham complex, Algebraic Topology-Rational Homotopy, proceedings Louvain-la-Neuve
- Dupont
- 1986
(Show Context)
Citation Context ...ed also some other tools in this thesis. One of the most important of them is the simplicial currents given by Dupont-Just [13]. The complex A ∗ �X� of simplicial differential forms defined in Dupont =-=[11]-=- plays the role of the differential forms on a manifold. In their article, the aim is to introduce a complex Ω∗�X� of simplicial currents on a simplicial manifold X, with properties similar to the com... |