We test the 3d-3d correspondence for theories that are labelled by Lens spaces. We find a full agreement between the index of the 3d ${\cal N}=2$ "Lens space theory" $T[L(p,1)]$ and the partition function of complex Chern-Simons theory on $L(p,1)$. In particular, for $p=1$, we show how the familiar $S^3$ partition function of Chern-Simons theory arises from the index of a free theory. For large $p$, we find that the index of $T[L(p,1)]$ becomes a constant independent of $p$. In addition, we study $T[L(p,1)]$ on the squashed three-sphere $S^3_b$. This enables us to see clearly, at the level of partition function, to what extent $G_\mathbb{C}$ complex Chern-Simons theory can be thought of as two copies of Chern-Simons theory with compact gauge group $G$.
Chern-Simons Theory in Lens Spaces from 2d Yang-Mills on the Cylinder
利用二维Yang-Mills与布朗运动之间的关系，证明了在一类透镜空间中，柱体上的二维Yang-Mills与Chern-Simons理论之间的关系.或者，这可以被看作是计算共形场论中的某些标量的2dYM。我们发现，对于特定的YM耦合值但有限的k和N，2dYM的配分函数约化为Chern-Simons理论中的U=ST^pS型算子. U是一个算子，从中可以得到S^3/Z_p上的陈-西蒙斯配分函数，以及威尔逊圈的期望值。对应关系涉及杨-米尔斯耦合的虚部是一个有理数，可以看作是陈-西蒙斯/WZW理论和拓扑2dYM的Witten，和Blau ant Thompson之间的关系的推广。本重新制定的一些性质的2dYM的圆柱明确。特别是，我们表明，模变换性质的配分函数是密切相关的仿射字符。
We study a method to compute a topological phase factor of partition function for pure Chern-Simons theory incorporating the supersymmetric localization. We develop a regularization preserving supersymmetry and the topological phase appears as a result of the supersymmetric regularization. Applying this method to pure Chern-Simons theory on lens space we compute the background dependent phase factor coming from the Chern-Simons term. We confirm that the partition function computed in this method enjoys a couple of level rank dualities including the one recently proposed in arXiv:1607.07457 for…
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…
We compute the invariants for a class of knots and links in arbitrary representations in $S^3/\mathbb{Z}_p$ in the large $k$ (level), large $N$ (rank) limit, keeping $N/(k+N)=λ$ fixed, in $U(N)$ and $Sp(N)$ Chern-Simons theories. Using the relation between the saddle point description and collective field theory, we first find that the invariants for the Hopf link and unknot are given by the on shell collective field theory action. We next show that the results of these two invariants can be used to compute the invariants of other torus knots and links. We also discuss the large $N$ phase stru…
We discuss ensemble averages of two-dimensional conformal field theories associated with an arbitrary indefinite lattice with integral quadratic form $Q$. We provide evidence that the holographic dual after the ensemble average is the three-dimensional Abelian Chern-Simons theory with kinetic term determined by $Q$. The resulting partition function can be written as a modular form, expressed as a sum over the partition functions of Chern-Simons theories on lens spaces. For odd lattices, the dual bulk theory is a spin Chern-Simons theory, and we identify several novel phenomena in this case. We…
We obtain a charged black lens solution in the five-dimensional Einstein-Maxwell-Chern-Simons theory with a positive cosmological constant. It is shown that the solution obtained here describes the formation of a black hole with the spatial cross section of a sphere from that of the lens space of L(n,1) in five-dimensional de Sitter space.
The goal of this paper is to investigate the Theta invariant --- an invariant of framed 3-manifolds associated with the lowest order contribution to the Chern-Simons partition function --- in the context of the quantum BV-BFV formalism. Namely, we compute the state on the solid torus to low degree in $\hbar$, and apply the gluing procedure to compute the Theta invariant of lens spaces. We use a distributional propagator which does not extend to a compactified configuration space, so to compute loop diagrams we have to define a regularization of the product of the distributional propagators, wh…
We investigate physical implications of a gravitational analog of axion electrodynamics with a parity-violating gravitoelectromagnetic theta term. This is related to the Nieh-Yan topological invariant in gravity with torsion, in contrast to the well-studied gravitational Chern-Simons term quadratic in curvature, coupled via a dynamical axionlike scalar field. Axion gravitodynamics is the corresponding linearized theory. We find that potentially observable effects are over 80 orders of magnitude stronger than for its Chern-Simons counterpart and could be in reach for detection by experiments in…
The level-k U(1) Chern-Simons theory is a spin topological quantum field theory for k odd. Its dynamics is captured by the 2d CFT of a compact boson with a certain radius. Recently it was recognized that a dependence on the 2d spin structure can be given to the CFT by modifying it using the so-called Arf invariant. We demonstrate that one can reorganize the torus partition function of the modified CFT into a finite sum involving a finite number of conformal blocks. This allows us to reproduce the modular matrices of the spin theory. We use the modular matrices to calculate the partition functi…
Possible altitudinal, latitudinal and directional dependence of relativistic Sagnac effect in Chern-Simons modified gravity
为了检验引力理论中的宇称破缺，最近讨论了Chern-Simons引力对干涉仪的可能影响。在早期出版物[Okawara，Yamada和Asada，Phys. Rev. Lett. 109，231101（2012）]，我们研究了CS修正引力中相对论Sagnac效应可能的高度和方向依赖性。我们比较了Sagnac干涉仪的CS效应与广义相对论的Lense-Thirring（LT）效应。数值计算表明，东向Sagnac干涉仪可能是首选的测试CS单独，因为LT的影响，这个干涉仪抵消。在CS模型中引起的相移的大小可能也有振荡依赖于通过CS质量参数$m_{CS}$的干涉仪的高度。因此，对国际空间站选址以及进行地基实验也进行了讨论。
Patch-wise localization with Chern-Simons forms in five dimensional supergravity
Università di Torino（都灵大学）；INFN, Sezione di Torino（意大利国家核物理研究所都灵分部）；Università di Milano-Bicocca（米兰比可卡大学）；INFN, sezione di Milano-Bicocca（意大利国家核物理研究所米兰比可卡分部）
We compute the topological partition function (twisted index) of $\mathcal{N}=2$ $U(N)$ Chern-Simons theory with an adjoint chiral multiplet on $Σ_g \times S^1$. The localization technique shows that the underlying Frobenius algebra is the equivariant Verlinde algebra which is obtained from the canonical quantization of the complex Chern-Simons theory regularized by $U(1)$ equivariant parameter $t$. Our computation relies on a Bethe/Gauge correspondence which allows us to represent the equivariant Verlinde algebra in terms of the Hall-Littlewood polynomials $P_λ(x_B, t)$ with a specialization …
We study the expansion near roots of unity of the superconformal index of 4d $SU(N)$ $\mathcal{N}=4$ SYM. In such an expansion, middle-dimensional walls of non-analyticity are shown to emerge in the complex analytic extension of the integrand. These walls intersect the integration contour at infinitesimal vicinities and come from both, the vector and chiral multiplet contributions, and combinations thereof. We will call these intersections vector and chiral bits, and the complementary region bulk, and show that, in the corresponding limit, the integrals along the infinitesimal bits include, am…
