The perturbative expansion of Chern-Simons gauge theory leads to invariants of knots and links, the finite type invariants or Vassiliev invariants. It has been proven that at any order in perturbation theory the resulting expression is an invariant of that order. Bott-Taubes integrals on configuration spaces are introduced in the present context to write Feynman diagrams at a given order in perturbation theory in a geometrical and topological setting. One of the consequences of the configuration space formalism is that the resulting amplitudes are given in cohomological terms. This cohomological structure can be used to translate Bott-Taubes integrals into Chern-Simons perturbative amplitudes and vice versa. In this article this program is performed up to third order in the coupling constant. This expands some work previously worked out by Thurston. Finally we take advantage of these results to incorporate in the formalism a smooth and divergenceless vector field on the $3$-manifold. The Bott-Taubes integrals obtained are used for constructing higher-order asymptotic Vassiliev invariants extending the work of Komendarczyk and Volić.
Algebraic structures of Vassiliev invariants for knot families
Institute for Theoretical and Experimental Physics（理论与实验物理研究所）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Institute for Information Transmission Problems（信息传输问题研究所）；NRC ”Kurchatov Institute”（库尔恰托夫国家研究中心）
It has been a long-standing problem how to relate Chern-Simons theory to the quantum groups. In this paper we recover the classical $r$-matrix directly from a 3-dimensional Chern-Simons theory with boundary conditions, thus creating a direct link to the quantum groups. It is known that the Jones polynomials can be constructed using an $R$-matrix. We show how these constructions can be seen to arise directly from 3-dimensional Chern-Simons theory.
We find a Chern-Simons propagator on the ball with the chiral boundary condition. We use it to study perturbatively Chern-Simons boundary conditions related to 2-dim $σ$-models and to Poisson-Lie T-duality. In particular, we find their renormalization group flow, given by the generalized Ricci tensor. Finally we briefly discuss what happens when the Chern-Simons theory is replaced by a Courant $σ$-model or possibly by a more general AKSZ model.
We perform a resurgence analysis of the $SU(2)$ Chern-Simons partition function on a Brieksorn homology sphere $Σ(2,5,7)$. Starting from an exact Chern-Simons partition function, we study the Borel resummation of its perturbative expansion.
A note on universality in refined Chern-Simons theory
Lebedev Physics Institute（列别捷夫物理研究所）；NRC ”Kurchatov Institute”（库尔恰托夫国家研究中心）；Institute for Information Transmission Problems（信息传输问题研究所）；Alikhanyan National Science Laboratory（阿里赫纳扬国家科学实验室）
An existence theorem is established for the solutions to the non-Abelian relativistic Chern--Simons--Higgs vortex equations over a doubly periodic domain when the gauge group $G$ assumes the most general and important prototype form, $G=SU(N)$
In this article it is studied the 3-brane world in the context of five-dimensional Einstein-Chern-Simons gravity. We started by considering Israel's junction condition for AdS-Chern-Simons gravity. Using the S-expansion procedure, we mapped the AdS-Chern-Simons junction conditions to Einstein-Chern-Simons gravity, allowing us to derive effective four-dimensional Einstein-Chern-Simons field equations.
Chern-Simons gravities and gravitational Chern-Simons densities are constructed using the non-Abelian Yang-Mills Chern-Simons densities. As such, they are defined only in odd dimensions. We propose instead an analogous construction employing what we term Higgs--Chern-Simons (HCS) densities, which are defined in $all\ dimensions$. This enables the definition of extended versions of Chern-Simons gravities in all dimensions. Employing the same prescription, the definition of gravitational Chern-Simons densities is extended to all even dimensions, but only to $4p-1$ $odd$ dimensions. All our consi…
