迈向${\mathcal W}_3$代数分类的研究

通过维度约化实现的$\mathcal{N}=2$双重分阶超对称量子力学

$\mathcal{N}=2$ Double graded supersymmetric quantum mechanics via dimensional reduction

通过维度约化与量子化，提出首个实现$\mathcal{N}=2$ $\mathbb{Z}_2^2$-超对称代数八维不可约表示的二维$\mathbb{Z}_2^2$-SQM系统，并研究其基态。

Confinement and Graded Partition Functions for $\mathcal{N}=4$ SYM

Gauge theories with confining phases at low temperatures tend to deconfine at high temperatures. In some cases, for example in supersymmetric theories, confinement can persist for all temperatures provided the partition function includes a grading by $(-1)^F$. When it is possible to define partition functions which smoothly interpolate between no grading and $(-1)^F$ grading, it is natural to ask if there are other choices of grading that have the same effect as $(-1)^F$ on confinement. We explore how this works for $\mathcal{N}=4$ SYM on $S^1\times S^3$ in the large $N$ limit at both small an…

M（k+1，2k+3）模型的新路径描述和对偶Z_k分次仿费米子

New path description for the M(k+1,2k+3) models and the dual Z_k graded parafermions

本文给出了非么正M（k+1，2k+3）模型态的一种新的路径描述.这种描述不同于由Forrester-Baxter解决方案引起的，在配置和方面，他们的限制固体上固体模型。所提出的路径表示实际上非常类似于酉极小模型M（k+1，k+2）的基本表示，具有类似的费米气体解释。这种解释导致有限化M（k+1，2k+3）字符的费米子表达式，其无限长的限制代表新的费米子字符的不可约模。M（k+1，2k+3）模型也通过一个（q到1/q）对偶变换与Z_k分次仿费米子相联系。

Graded quivers and B-branes at Calabi-Yau singularities

A graded quiver with superpotential is a quiver whose arrows are assigned degrees $c\in \{0, 1, \cdots, m\}$, for some integer $m \geq 0$, with relations generated by a superpotential of degree $m-1$. Ordinary quivers ($m=1)$ often describe the open string sector of D-brane systems; in particular, they capture the physics of D3-branes at local Calabi-Yau (CY) 3-fold singularities in type IIB string theory, in the guise of 4d $\mathcal{N}=1$ supersymmetric quiver gauge theories. It was pointed out recently that graded quivers with $m=2$ and $m=3$ similarly describe systems of D-branes at CY 4-f…

The graded Lie algebra of general relativity

We construct a graded Lie algebra $\mathcal{E}$ in which the Maurer-Cartan equation is equivalent to the vacuum Einstein equations. The gauge groupoid is the groupoid of rank 4 real vector bundles with a conformal inner product, over a 4-dimensional base manifold, and the graded Lie algebra construction is a functor out of this groupoid. As usual, each Maurer-Cartan element in $\mathcal{E}^1$ yields a differential on $\mathcal{E}$. Its first homology is linearized gravity about that element. We introduce a gauge-fixing algorithm that generates, for each gauge object $G$, a contraction to a muc…

Hamiltonian systems, Toda lattices, Solitons, Lax Pairs on weighted Z-graded graphs

We consider discrete one dimensional nonlinear equations and present the procedure of lifting them to Z-graded graphs. We identify conditions which allow one to lift one dimensional solutions to solutions on graphs. In particular, we prove the existence of solitons {for static potentials} on graded fractal graphs. We also show that even for a simple example of a topologically interesting graph the corresponding non-trivial Lax pairs and associated unitary transformations do not lift to a Lax pair on the Z-graded graph.

Non-unitary TQFTs from 3D $\mathcal{N}=4$ rank 0 SCFTs

We propose a novel procedure of assigning a pair of non-unitary topological quantum field theories (TQFTs), TFT$_\pm [\mathcal{T}_{\rm rank \;0}]$, to a (2+1)D interacting $\mathcal{N}=4$ superconformal field theory (SCFT) $\mathcal{T}_{\rm rank \;0}$ of rank 0, i.e. having no Coulomb and Higgs branches. The topological theories arise from particular degenerate limits of the SCFT. Modular data of the non-unitary TQFTs are extracted from the supersymmetric partition functions in the degenerate limits. As a non-trivial dictionary, we propose that $F = \max_α\left(- \log |S^{(+)}_{0α}| \right) = …

Building 1D lattice models with $G$-graded fusion category

We construct a family of one-dimensional (1D) quantum lattice models based on $G$-graded unitary fusion category $\mathcal{C}_G$. This family realize an interpolation between the anyon-chain models and edge models of 2D symmetry-protected topological states, and can be thought of as edge models of 2D symmetry-enriched topological states. The models display a set of unconventional global symmetries that are characterized by the input category $\mathcal{C}_G$. While spontaneous symmetry breaking is also possible, our numerical evidence shows that the category symmetry constrains the the models t…

霍普夫代数的Weil代数-Ⅰ- A非交换框架

The Weil Algebra of a Hopf Algebra - I - A noncommutative framework

推广了Henri Cartan提出的分次微分代数$Ω$中李代数$\mathfrak g$的运算的概念.定义了阶化微分代数PDMATHAMN中的霍普夫代数$\mathcal H$的运算的概念，称之为$\mathcal H$ -运算.然后，我们推广这样一个操作的概念代数连接。最后讨论了相应的非交换形式的Weil代数：霍普夫代数$\mathcal H$的Weil代数$W(\mathcal H)$是带联络的$\mathcal H$ -运算范畴的泛始对象.

关于酉2-群对称性

On Unitary 2-Group Symmetries

本文将量子系统全局对称性的酉作用推广到线算符，提出有限2-群对称性以酉2-表示作用于线算符，并给出酉等价分类，恢复了已知分类并揭示了新的反射异常数据。

The Graded Extension of Thomas-Whitehead Gravity

Thomas-Whitehead (TW) gravity was recently introduced as a projective gauge theory of gravity over a d-dimensional manifold that embeds reparameterization invariance into the action functional for gravitation through the use of the Thomas-Whitehead connection. The projective invariance in this d-dimensional theory enjoys an intimate relationship with the Virasoro coadjoint elements found in string theory as one of the components of the connection, $\mathcal{D}_{ab}$, is directly related to the coadjoint elements of the Virasoro algebra. TW Gravity exploits projective Gauss-Bonnet terms in the …

分次Hopf映射与Fuzzy超球

Graded Hopf Maps and Fuzzy Superspheres

我们讨论了基于酉正交辛代数$uosp(N|2)$和$uosp(N|4)$的模糊二球面和模糊四球面的超对称推广。利用超对称的Schwinger构造导出了模糊超球的分次全对称表示。作为模糊超球的经典对应，引入了第一和第二阶Hopf映射的分次形式，并研究了它们的基本几何结构.结果表明，模糊超球表示为一个“叠加”的模糊超球具有较低的超对称性。我们还研究了模糊二超球和模糊四超球的代数结构，以确定$su(2|N)$和$su(4|N)$作为其增强的代数结构，分别。对关联函数的计算显示了模糊超球的量子涨落等增强结构。