论文标题
从2D液滴到2D Yang-Mills
From 2d Droplets to 2d Yang-Mills
论文作者
论文摘要
我们在\ emph {generalized} \ emph {q} {Q}赋予Yang-Mills Theories的时间演变与分区功能之间建立了连接。 $（0+1）的经典阶段$尺寸统一矩阵模型可以以二维为单位的Fermi液滴来表征。我们对这些液滴进行量化，并发现模式满足Abelian Kac-Moody代数。 The Hilbert spaces $\mathcal{H}_+$ and $\mathcal{H}_-$ associated with the upper and lower free Fermi surfaces of a droplet admit a Young diagram basis in which the phase space Hamiltonian is diagonal with eigenvalue, in the large $N$ limit, equal to the quadratic Casimir of $u(N)$.我们在$ \ Mathcal {H} _ \ pm $和液滴的几何形状中建立了一个确切的状态映射。特别是，$ \ Mathcal {H} _ \ pm $中的相干状态对应于上和下部费米表面的经典变形。我们证明，在$ \ Mathcal {h} _ \ pm $中，两个连贯状态之间的相关性等于$ 2D $ YANG-MILLS理论的手性和反性手续分区功能。使用完整的Hilbert Space $ \ Mathcal {H} _+ \ otimes \ Mathcal {H} _- $允许A \ Emph {Composite}基础的事实，我们表明，两个经典的液滴几何形状之间的相关性等于完整的$ u（n）$ u（n）$ u（n）$ yang-mills pition cylinder cylinder cylinder function。我们进一步建立了$ \ Mathcal {h} _ \ pm $中的高点相关器与Riemann Surface上$ 2D $ YANG-MILLS中的高点相关器之间的连接。 $ \ Mathcal {h} _ \ pm $中有特殊状态的过渡幅度等于$ 2D $ \ emph {q}的分区函数，并赋予了Yang-mills的Yang-mills，并且在一般角色中扩展了反派动作。我们强调，Yang-Mills侧中的\ Emph {Q}形式与液滴几何形状的特殊变形有关，而无需变形与矩阵模型相关的量规组。
We establish a connection between time evolution of free Fermi droplets and partition function of \emph{generalised} \emph{q}-deformed Yang-Mills theories on Riemann surfaces. Classical phases of $(0+1)$ dimensional unitary matrix models can be characterised by free Fermi droplets in two dimensions. We quantise these droplets and find that the modes satisfy an abelian Kac-Moody algebra. The Hilbert spaces $\mathcal{H}_+$ and $\mathcal{H}_-$ associated with the upper and lower free Fermi surfaces of a droplet admit a Young diagram basis in which the phase space Hamiltonian is diagonal with eigenvalue, in the large $N$ limit, equal to the quadratic Casimir of $u(N)$. We establish an exact mapping between states in $\mathcal{H}_\pm$ and geometries of droplets. In particular, coherent states in $\mathcal{H}_\pm$ correspond to classical deformation of upper and lower Fermi surfaces. We prove that correlation between two coherent states in $\mathcal{H}_\pm$ is equal to the chiral and anti-chiral partition function of $2d$ Yang-Mills theory on a cylinder. Using the fact that the full Hilbert space $\mathcal{H}_+ \otimes \mathcal{H}_-$ admits a \emph{composite} basis, we show that correlation between two classical droplet geometries is equal to the full $U(N)$ Yang-Mills partition function on cylinder. We further establish a connection between higher point correlators in $\mathcal{H}_\pm$ and higher point correlators in $2d$ Yang-Mills on Riemann surface. There are special states in $\mathcal{H}_\pm$ whose transition amplitudes are equal to the partition function of $2d$ \emph{q}-deformed Yang-Mills and in general character expansion of Villain action. We emphasise that the \emph{q}-deformation in the Yang-Mills side is related to special deformation of droplet geometries without deforming the gauge group associated with the matrix model.
