As noted long ago by Atiyah and Bott, the classical Yang-Mills action on a Riemann surface admits a beautiful symplectic interpretation as the norm-square of a moment map associated to the Hamiltonian action by gauge transformations on the affine space of connections. Here I explain how certain Wilson loop observables in Chern-Simons gauge theory on a Seifert three-manifold can be given an analogous symplectic description. Among other results, this symplectic description implies that the stationary-phase approximation to the Wilson loop path integral is exact for torus knots, an empirical observation made previously by Lawrence and Rozansky. This article reviews selected material from the larger work "Localization for Wilson Loops in Chern-Simons Theory," arXiv:0911.2687.
The path integral for the partition function of Chern-Simons gauge theory with a compact gauge group is evaluated on a general Seifert 3-manifold. This extends previous results and relies on abelianisation, a background field method and local application of the Kawasaki Index theorem.
Chern-Simons (CS) theories with rank $N$ and level $k$ on Seifert manifold are discussed. The partition functions of such theories can be written as a function of modular transformation matrices summed over different integrable representations of affine Lie algebra $u(N)_k$ associated with boundary Wess-Zumino-Witten (WZW) model. Using properties of modular transform matrices we express the partition functions of these theories as a unitary matrix model. We show that, the eigenvalues of unitary matrices are discrete and proportional to hook lengths of the corresponding integrable Young diagram…
We calculate homological blocks for a knot in Seifert manifolds when the gauge group is $SU(N)$. We obtain the homological blocks with a given representation of the gauge group from the expectation value of the Wilson loop operator by analytically continuing the Chern-Simons level. We also obtain homological blocks with the analytically continued level and representation for a knot in the Seifert integer homology spheres.
We give a pedagogical introduction to the study of supersymmetric partition functions of 3D $\mathcal{N}{=}2$ supersymmetric Chern-Simons-matter theories (with an $R$-symmetry) on half-BPS closed three-manifolds---including $S^3$, $S^2 \times S^1$, and any Seifert three-manifold. Three-dimensional gauge theories can flow to non-trivial fixed points in the infrared. In the presence of 3D $\mathcal{N}{=}2$ supersymmetry, many exact results are known about the strongly-coupled infrared, due in good part to powerful localization techniques. We review some of these techniques and emphasize some mor…
Level-rank duality of the U(N) WZW model, Chern-Simons theory, and 2d qYM theory
我们研究了WZW、Chern-Simons和2dqYM规范群理论。U（N）WZW模型只在奇能级K下才有定义，并且该模型表现出比SU（N）更简单的能级-秩对偶性.塞弗特流形上的U（N）陈-西蒙斯理论也表现出类似的对偶性，与S^3上SU（N）陈-西蒙斯理论的能级-秩对偶性不同。当q = e^{2 pi i/（N+K）}时，2d U（N）qYM理论的观测量可以表示为U（N）表示的有限子集上的和。当N和K是奇数时，qYM理论表现出N<-->-K对偶性，条件是q = e^{2 pi i/（N+K）}和θ = 0 mod 2 pi /（N+K）。
$\hat{Z}$-invariants, which can reconstruct the analytic continuation of the SU(2) Chern-Simons partition functions via Borel resummation, were discovered by GPV and have been conjectured to be a new homological invariant of 3-manifolds which can shed light onto the superconformal and topologically twisted index of 3d $\mathcal{N}=2$ theories proposed by GPPV. In particular, the resurgent analysis of $\hat{Z}$ has been fruitful in discovering analytic properties of the WRT invariants. The resurgent analysis of these $\hat{Z}$-invariants has been performed for the cases of $Σ(2,3,5),\ Σ(2,3,7)$…
We consider $U(N)_k$ Chern-Simons theory on $S^3$ in Seifert framing and write down the partition function as a unitary matrix model. In the large $k$ and large $N$ limit the eigenvalue density satisfies an upper bound $\frac{1}{2πλ}$ where $λ=N/(k+N)$. We study the partition function under saddle point approximation and find that the saddle point equation admits a gapped solution for the eigenvalue density. The on-shell partition function on this solution matches with the partition function in the canonical framing up to a phase. However the eigenvalue density saturates the upper cap at a cri…
It is shown that Chern-Simons gravities in $2n+1$ dimensions admit solutions described by the same lapse function which describes the BTZ black hole in the $n=1$ case. This has been carried out explicitly for $n=1,2,3$. Moreover, it is seen that these solutions are unique.
Einstein-Cartan formulation of Chern-Simons Lorentz-violating Gravity
我们考虑四维标准爱因斯坦理论的一个修正，替代R。Jackiw和S.- Y. Pi，Phys.Rev.D68，104012（2003），因为它是基于广义相对论的一阶（爱因斯坦-嘉当）方法，其规范结构是显而易见的。这是通过在作用量中引入一个额外的拓扑项来实现的，由于耦合对时空点的依赖性，该拓扑项成为洛伦兹破坏项。我们得到了一个条件的解决方案的爱因斯坦方程，使他们坚持在变形理论，并表明解决方案显着对应的经典解决方案的一个集合的独立2+1-d（拓扑）Chern-Simons引力。最后，我们研究了它与标准二阶近似的关系，并证明了它们在Lorentz破坏向量场的模中都是首阶的。
Based on the gauge-gravity duality, we study the three-dimensional QCD ($\mathrm{QCD}_{3}$) and Chern-Simons theory by constructing the anisotropic black D3-brane solution in IIB supergravity. The deformed bulk geometry is obtained by performing a double Wick rotation and dimension reduction which becomes an anisotropic bubble configuration exhibiting confinement in the dual theory. And its anisotropy also reduces to a Chern-Simons term due to the presence of the dissolved D7-branes or the axion field in bulk. Using the bubble geometry, we investigate the the ground-state energy density, quark…
