M. Katanaev, B. Volkov
We use the Chern-Simons action for a SO(3)-connection for the description of point disclinations in the geometric theory of defects. The most general spherically symmetric SO(3)-connection with zero curvature is found. The corresponding orthogonal spherically symmetric SO(3) matrix and n-field are computed. Two examples of point disclinations are described.
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The Chern--Simons term is used in the geometric theory of defects. The equilibrium equations with $δ$-function source are explicitly solved with respect to the $SO(3)$ connection. This solution describes one straight linear disclination and corresponds to the new kind of geometrical defect: it is the defect in the connection but not the metric which is the flat Euclidean metric. This is the first example of a disclination described within the geometric theory of defects. The corresponding angular rotation field is computed.
Some properties of Chern-Simons terms are presented and their physical utility is surveyed.
We use some BRS techniques to construct Chern-Simons forms generalizing the Chern character of K_1 groups in the Cuntz-Quillen description of cyclic homology.
Chern-Simons Theories and Integrability
本文系统研究陈-西蒙斯理论与可积场论在不同维度间的对应关系，推广了四维及扭子空间上的已有结果，并新发现五维全纯2-陈-西蒙斯理论与三维可积场论的对应。
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 derive discrete and oscillatory Chern-Simons matrix models. The method is based on fundamental properties of the associated orthogonal polynomials. As an application, we show that the discrete model allows to prove and extend the recently found relationship between Chern-Simons theory and q-deformed 2dYM. In addition, the equivalence of the Chern-Simons matrix models gives a complementary view on the equivalence of effective superpotentials in N=1 gauge theories.
The dimensional reduction of the three-dimensional fermion-Chern-Simons model (related to Hopf maps) of Adam et el. is shown to be equivalent to (i) either the static, fixed--chirality sector of our non-relativistic spinor-Chern-Simons model in 2+1 dimensions, (ii) or a particular Heisenberg ferromagnet in the plane.
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.
In the geometric theory of defects, media with a spin structure, for example, ferromagnet, is considered as a manifold with given Riemann--Cartan geometry. We consider the case with the Euclidean metric corresponding to the absence of elastic deformations but with nontrivial ${\mathbb S}{\mathbb O}(3)$-connection which produces nontrivial curvature and torsion tensors. We show that the 't Hooft--Polyakov monopole has physical interpretation in solid state physics describing media with continuous distribution of dislocations and disclinations. The Chern--Simons action is used for the descriptio…
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.
Being gauge non-invariant, a Chern-Simons (2k-1)-form seen as a Lagrangian of gauge theory on a (2k-1)-dimensional manifold leads to the gauge conservation law of a modified Noether current.
A note on universality in refined Chern-Simons theory
本文探讨了陈-西蒙斯理论中普遍性的精炼形式，指出原始普遍性适用于任意简单李群，而精炼理论中则限制于单列李群。
Though a Chern-Simons (2k-1)-form is not gauge-invariant and it depends on a background connection, this form seen as a Lagrangian of gauge theory on a (2k-1)-dimensional manifold leads to the energy-momentum conservation law.
Higher Chern-Simons based on (2-)crossed modules
该研究利用（2-）交叉模和广义微分演算构造高阶陈–西蒙斯理论，建立广义联络、陈–西蒙斯作用量及高阶第二陈形式与陈–韦伊定理。
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, $G=SU(N)$
We show that the Chern-Simons-CP(1) model can support a singular soliton solution in which the magnetic field is a Dirac delta.
We study self-dual multi-vortex solutions of Chern-Simons Higgs theory in a background curved spacetime. The existence and decaying property of a solution are demonstrated.
The Chern-Simons-Higgs and the Chern-Simons-Dirac systems in Lorenz gauge are locally well-posed in suitable Fourier-Lebesgue spaces $\hat{H}^{s,r}$. Our aim is to minimize $s=s(r)$ in the range $1<r \le 2$ . If $r \to 1$ we show that we almost reach the critical regularity dictated by scaling. In the classical case $r=2$ the results are due to Huh and Oh. Crucial is the fact that the decisive quadratic nonlinearities fulfill a null condition.
Quantization Commutes with Reduction of Chern-Simons Gauge Theory
本文在陈-西蒙斯规范理论中证明了规范化与约简的无穷维版本，利用复分析和Atiyah-Bott堆栈进行证明。
The essential role played by Chern--Simons terms in a variety of physical models provides yet another illustration of the unexpected but profound interactions between the two disciplines.
Our aim in this note is to clarify a relationship between covariant Chern-Simons 3-dimensional theory and Schwartz type topological field theory known also as BF theory.
Let $G=(V,E)$ be a connected finite graph. We study the relativistic non-Abelian Chern-Simons-Higgs vortex equations on the graph $G$. We establish an existence result to the relativistic non-Abelian Chern-Simons-Higgs vortex equations.
We introduce certain relative differential characters which we call Cheeger-Chern-Simons characters. These combine the well-known Cheeger-Simons characters with Chern-Simons forms. In the same way as the Cheeger-Simons characters generalize Chern-Simons invariants of oriented closed manifolds, the Cheeger-Chern-Simons characters generalize Chern-Simons invariants of oriented manifolds with boundary. Using Cheeger-Chern-Simons characters, we introduce the notion of differential trivializations of universal characteristic classes. Specializing to the class 1/2 $p_1 \in H^4(B\mathrm{Spin}_n;\math…
We compute the semiclassical formulas for the partition functions obtained using two different Lagrangians: the Chern-Simons functional and the symplectic action functional.
In this paper, we address the problem of local well-posedness of the Chern-Simons-Dirac (CSD) and the Chern-Simons-Higgs (CSH) equations in the Lorenz gauge for low regularity initial data. One of our main contributions is the uncovering of a null structure of (CSD). Combined with the standard machinery of $X^{s,b}$ spaces, we obtain local well-posedness of (CSD) for initial data $a_μ, ψ\in H^{1/4+ε}_{x}$. Moreover, it is observed that the same techniques applied to (CSH) lead to a quick proof of local well-posedness for initial data $a_μ \in H^{1/4+ε}_{x}$, $(ϕ, \partial_{t} ϕ) \in H^{3/4+ε}_…
We discuss a nexus among quantum topology, quantum physics and quantum computing.
We examine Chern-Simons theory as a deformation of a 3-dimensional BF theory that is partially holomorphic and partially topological. In particular, we introduce a novel gauge that leads naturally to a one-loop exact quantization of this BF theory and Chern-Simons theory. This approach illuminates several important features of Chern-Simons theory, notably the bulk-boundary correspondence of Chern-Simons theory with chiral WZW theory. In addition to rigorously constructing the theory, we also explain how it applies to a large class of closely related 3-dimensional theories and some of the conse…
In this paper, the decomposition of SU(2) gauge potential in terms of Pauli spinors is studied. Using this decomposition, the spinor strutures of the Chern-Simons form and the Chern density are obtained. Furthermore, by these spinor structures, the knot quantum number of non-Abelian gauge theory is discussed, and the second Chern number is characterized by the Hopf indices and the Brouwer degrees of $ϕ$-mapping.
Generalized differential forms are employed to construct generalized connections. Lorentzian four-metrics determined by certain of these connections satisfy Einstein's vacuum field equations when the connections are flat. Generalized Chern-Simons action principles with Einstein's equations as Euler-Lagrange equations are constructed by using these connections.
Chern-Simons theory on a closed contact three-manifold is studied when the Lie group for gauge transformations is compact, connected and abelian. A rigorous definition of an abelian Chern-Simons partition function is derived using the Faddeev-Popov gauge fixing method. A symplectic abelian Chern-Simons partition function is also derived using the technique of non-abelian localization. This physically identifies the symplectic abelian partition function with the abelian Chern-Simons partition function as rigorous topological three-manifold invariants. This study leads to a natural identificatio…
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