馬克斯偉-陳-西蒙斯： 陳數與晶格規范場論

Abstract

在本論文中，我們介紹了陳-西蒙斯（Chern-Simons, CS）理論與麥克斯韋-陳-西蒙斯（Maxwell-Chern-Simons, MCS）理論，並展示了它們所具有的一些有趣特性，例如能級量子化、分数量子統計等。我們接著在平面和環面上對 MCS 理論進行量子化。同時，我們也簡要介紹了陳數（Chern number）。特別地，計算定義在空間環面上的U(1) MCS 理論的平直（零）模的陳數，是本論文的主要目標。我們發現，為了實現這一目標，有必要引入扭曲邊界條件。最後，我們採用了修改後的 Villain 表達式與轉移矩陣方法推導了晶格 MCS 理論的哈密頓量。我們發現該晶格哈密頓量在結構上與連續情況相似，但由於引入了杯積（cup product）而產生了一些有趣的修正。最終，在晶格上計算陳數的過程，與連續情況頗為相似。

In this thesis, we introduce Chern-Simons (CS) theory and Maxwell-Chern-Simons (MCS) and demonstrate they exhibits some interesting features, such as level quanitzation, fractional statistics, etc. We proceed by quantizing the MCS on a plane and torus. We also give a brief introduction to the Chern number. In particular, the calculation of the Chern number of the flat (zero) modes of $U(1)$ MCS on a spatial torus comprises our primary goal. We find that, to achieve this aim, it is necessary to employ the twisted boundary conditions. Finally, we use the modified Villain formulation and transfer matrix methods to derive the Hamiltonian of lattice MCS theory. We see that the lattice Hamiltonian has similar structure as in the continuum, except some interesting modifications coming from the use of cup product. Finally, the computation of the Chern number on the lattice turns out to be reminiscent of the case in the continuum.