AI 中文总结
研究求解规范李代数含秩\(K \geq 1\)阿贝尔直和项的四维规范理论局部反常消除条件的阿贝尔部分，通过与代数/算术几何问题等价转化，求解了秩为2的例子，如\(U(1)^2\)和六个费米子的情况，描述了相关几何结构及解的性质。
AI 中文摘要
我们证明，对于规范李代数具有秩\(K \geq 1\)的阿贝尔直和项的四维规范理论，求解局部反常消除条件的阿贝尔部分，等同于在代数/算术几何中寻找有理数上三次超曲面的\((K - 1)\)维射影线性子空间的问题，其中三次曲面由半单直和项的数据及其外尔费米子携带的表示确定。然后我们用这种重新表述来求解各种秩为2的例子。最简单的非平凡物理例子，即规范李群\(U(1)^2\)和六个费米子，具有丰富（且已充分研究）的几何结构：它对应于塞格雷三次原三重曲面中的线的法诺簇。这是一个具有15个不可约分量（为平面，对应于塞格雷三次原曲面中的15个平面中的线，产生非手征费米子）和6个分量（为5度分裂德尔佩佐曲面，产生手征费米子）的曲面。这些分量都是有理簇，使得反常消除条件的所有解都能被参数化，并描述它们的集体性质（如拓扑和渐近分布）。
英文摘要
We show that solving the abelian part of the local anomaly cancellation conditions for a 4-d gauge theory whose gauge Lie algebra has an abelian summand with rank $K \geq 1$ is equivalent to the problem in algebraic/arithmetic geometry of finding $(K-1)$-dimensional projective linear subspaces of a cubic hypersurface over the rational numbers, where the cubic is determined by the data of the semisimple summand and the representation thereof carried by the Weyl fermions. We then use this reformulation to solve a variety of examples with rank 2. The simplest non-trivial example from physics, namely gauge Lie group $U(1)^2$ and six fermions, nevertheless has a rich (and well-studied) geometry: it corresponds to the Fano variety of lines in the Segre cubic primal threefold. This is a surface with 15 irreducible components that are planes (which correspond to lines lying in the 15 planes in the Segre cubic primal and which give rise to non-chiral fermions) and 6 components that are split del Pezzo surfaces of degree 5 (which give rise to chiral fermions). These components are all rational varieties, enabling all solutions to the anomaly cancellation conditions to be parametrized and their collective properties (e.g. their topology and asymptotic distribution) to be described.
Comments39 pp
