Anyons, Monopole and Coulomb Problem

Spin 1 particle in the magnetic monopole potential, nonrelativistic approximation. Minkowski and Lobachevski spaces

The spin 1 particle is treated in the presence of the Dirac magnetic monopole in the Minkowski and Lobachevsky spaces. Separating the variables in the frame of the matrix 10-component Duffin-Kemer-Petiau approach (wave equation) and making a nonrelativistic approximation in the corresponding radial equations, a system of three coupled second order linear differential equations is derived for each type of geometry. For the Minkowski space, the nonrelativistic equations are disconnected using a linear transformation, which makes the mixing matrix diagonal. The resultant three unconnected equatio…

3D Oscillator and Coulomb Systems reduced from Kahler spaces

We define the oscillator and Coulomb systems on four-dimensional spaces with U(2)-invariant Kahler metric and perform their Hamiltonian reduction to the three-dimensional oscillator and Coulomb systems specified by the presence of Dirac monopoles. We find the Kahler spaces with conic singularity, where the oscillator and Coulomb systems on three-dimensional sphere and two-sheet hyperboloid are originated. Then we construct the superintegrable oscillator system on three-dimensional sphere and hyperboloid, coupled to monopole, and find their four-dimensional origins. In the latter case the metri…

分离余切型与正交辛箭图的量子库仑分支

Quantized Coulomb Branches of Separated Cotangent Type and Orthosymplectic Quivers

该研究提出分离余切型量子库仑分支的定义，证明其经典构造可恢复非余切型库仑分支，推导拟极小单极算子公式，计算正交辛箭图的单极算子并构造了相关代数同态。

Fundamental monopole operators and embeddings of Kac-Moody affine Grassmannian slices

Braverman, Finkelberg, and Nakajima define Kac-Moody affine Grassmannian slices as Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories and prove that their Coulomb branch construction agrees with the usual loop group definition in finite ADE types. The Coulomb branch construction has good algebraic properties, but its geometry is hard to understand in general. In finite types, an essential geometric feature is that slices embed into one another. We show that these embeddings are compatible with the fundamental monopole operators (FMOs), remarkable regular functions arising from the …

Magnetic Monopoles, Electric Neutrality and the Static Maxwell-Dirac Equations

We study the full Maxwell-Dirac equations: Dirac field with minimally coupled electromagnetic field and Maxwell field with Dirac current as source. Our particular interest is the static case in which the Dirac current is purely time-like -- the "electron" is at rest in some Lorentz frame. In this case we prove two theorems under rather general assumptions. Firstly, that if the system is also stationary (time independent in some gauge) then the system as a whole must have vanishing total charge, i.e. it must be electrically neutral. In fact, the theorem only requires that the system be {\em asy…

Monopole Chern-Simons Term: Charge-Monopole System as a Particle with Spin

The topological nature of Chern-Simons term describing the interaction of a charge with magnetic monopole is manifested in two ways: it changes the plane dynamical geometry of a free particle for the cone dynamical geometry without distorting the free (geodesic) character of the motion, and in the limit of zero charge's mass it describes a spin system. This observation allows us to interpret the charge-monopole system alternatively as a free particle of fixed spin with translational and spin degrees of freedom interacting via the helicity constraint, or as a symmetric spinning top with dynamic…

类点全局单极子时空与吴-杨磁单极子中的Klein-Gordon粒子：不变性与等谱性

Klein-Gordon particles in a quasi-pointlike global monopole spacetime and a Wu-Yang magnetic monopole: invariance and isospectrality

本研究通过关联度规函数将点状全局单极子时空形变为类点版本，证明其中Klein-Gordon粒子与原时空粒子等谱且不变，还分析了两种单极子对两类KG模型光谱的影响，兼具基础与教学意义。

Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal N=4$ gauge theories, I

Consider the $3$-dimensional $\mathcal N=4$ supersymmetric gauge theory associated with a compact Lie group $G$ and its quaternionic representation $\mathbf M$. Physicists study its Coulomb branch, which is a noncompact hyper-Kähler manifold, such as instanton moduli spaces on $\mathbb R^4$, $SU(2)$-monopole moduli spaces on $\mathbb R^3$, etc. In this paper and its sequel, we propose a mathematical definition of the coordinate ring of the Coulomb branch, using the vanishing cycle cohomology group of a certain moduli space for a gauged $σ$-model on the $2$-sphere associated with $(G,\mathbf M)…