This paper is a review of monopoles, lowest Landau level, fuzzy spheres, and their mutual relations. The Hopf maps of division algebras provide a prototype relation between monopoles and fuzzy spheres. Generalization of complex numbers to Clifford algebra is exactly analogous to generalization of fuzzy two-spheres to higher dimensional fuzzy spheres. Higher dimensional fuzzy spheres have an interesting hierarchical structure made of "compounds" of lower dimensional spheres. We give a physical interpretation for such particular structure of fuzzy spheres by utilizing Landau models in generic even dimensions. With Grassmann algebra, we also introduce a graded version of the Hopf map, and discuss its relation to fuzzy supersphere in context of supersymmetric Landau model.
We apply the modern Batalin-Vilkovisky quantization techniques of Costello and Gwilliam to noncommutative field theories in the finite-dimensional case of fuzzy spaces. We further develop a generalization of this framework to theories that are equivariant under a triangular Hopf algebra symmetry, which in particular leads to quantizations of finite-dimensional analogs of the field theories proposed recently through the notion of `braided $L_\infty$-algebras'. The techniques are illustrated by computing perturbative correlation functions for scalar and Chern-Simons theories on the fuzzy $2$-sph…
One of the most celebrated works of Professor Madore is the introduction of fuzzy sphere. I briefly review how the fuzzy two-sphere and its higher dimensional cousins are realized in the (spherical) Landau models in non-Abelian monopole backgrounds. For extracting quantum geometry from the Landau models, we evaluate the matrix elements of the coordinates of spheres in the lowest and higher Landau levels. For the lowest Landau level, the matrix geometry is identified as the geometry of fuzzy sphere. Meanwhile for the higher Landau levels, the obtained quantum geometry turns out to be a nested m…
We consider a set of physical degrees of freedom coupled to a finite-dimensional Hilbert space, which may be taken as modeling a fuzzy space or as the lowest Landau level of a Landau-Hall problem. These may be viewed as matter fields on a fuzzy space. Sequentially generalizing to arbitrary backgrounds, we argue that the effective action is given by the Chern-Simons form associated with the Dirac index density (with gauge and gravitational fields), with an abelian gauge field shifted by the Poincaré-Cartan form for matter dynamics. The result is an action for matter fields where the Lagrangian …
The fuzzy S^2 structure of M2-M5 systems in ABJM membrane theories
我们分析的基态/漏斗解决方案的波动提出来描述M2-M5系统的水平k质量变形/纯陈-西蒙斯物质ABJM理论的多膜。我们表明，在大N限制的波动接近2球，而不是天真地期望3球上的函数空间。这是一个新的实现的模糊2球的背景下，矩阵理论，它使用bifundamental而不是伴随标量。从多重M2-膜作用量出发，在大N下恢复了R^{2，1} x S^2上的U（1）杨-米尔斯理论，这与IIA型弦论中的单一D4-膜解释是一致的。这是在大k时所预期的，其中半经典分析是有效的。几个方面的波动分析，基态/漏斗的解决方案和质量变形/纯ABJM方程可以理解的离散非交换实现的霍普夫纤维化。我们讨论了在M2-M5系统的经典S^3几何中找到一个M2膜世界体积导子的可能性。使用重写的SO（4）-协变模糊3球结构的方程，我们也直接比较这个模糊3球对ABJM基态/漏斗解决方案，并显示它们是不同的。
3d Conformal Field Theories via Fuzzy Sphere Algebra
Walter Burke Institute for Theoretical Physics and Department of Physics, California Institute of Technology（加州理工学院沃尔特·伯克理论物理研究所和物理系）；Department of Mathematics, University of California, Santa Barbara（加州大学圣塔芭芭拉分校数学系）
We investigate the $SO(5)$ Landau problem in the $SO(4)$ monopole gauge field background by applying the techniques of the non-linear realization of quantum field theory. The $SO(4)$ monopole carries two topological invariants, the second Chern number and a generalized Euler number, specified by the $SU(2)$ monopole and anti-monopole indices, $I_+$ and $I_-$. The energy levels of the $SO(5)$ Landau problem are grouped into $\text{Min}(I_+, I_-) +1$ sectors, each of which holds Landau levels. In the $n$-sector, $N$th Landau level eigenstates constitute the $SO(5)$ irreducible representation wit…
Conformal Data for the $O(2)$ Wilson-Fisher CFT in $(2+1)$-Dimensional Spacetime from Exact Diagonalization and Matrix Product States on the Fuzzy Sphere
Laboratory for Theoretical and Computational Physics, PSI Center for Scientific Computing, Theory and Data（理论计算物理实验室）；Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院物理研究所）；Univ. Grenoble Alpes, CNRS, LPMMC（格勒诺布尔阿尔卑斯大学）；Institut des Hautes Études Scientifiques（高等科学研究所）
The $\mathrm{SO}(5)$ Deconfined Phase Transition under the Fuzzy Sphere Microscope: Approximate Conformal Symmetry, Pseudo-Criticality, and Operator Spectrum
