Anyon is collective excitation of two dimensional electron gas subjected to strong magnetic field, carrying fractional charges and exotic statistical character beyond fermion and boson. So far, anyons with serial fractional charges only exist in fractional quantum Hall effect. It is still a challenge to find new serial of fractional charges in other physical system and develop an unified mathematical physics theory based on the same root. Here a topological path fusion theory of propagating electrons in magnetic flux lattice is proposed to explore the physical origin of fractional charges based on a generalization of Feynman's path integral theory and Thurston's train track theory. This mathematical physics theory generated the existed serial of fractional charges in fractional quantum Hall effect and predicted new serial of fractional charges. A serial of irrational charges are predicted in one dimensional lattice of magnetic fluxes. Fractionally charged anyons are also generated in two dimensional and three dimensional lattice of train tracks of electric currents, revealing an exact correspondence between knot lattice model and train track model. A new explanation for the modular symmetry of complex Hall conductance and composite fermion in fractional quantum Hall effect is also derived from this topological path fusion theory. Experimental observation of anyon in three dimensions can be realized by constructing three dimensional interlocking magnetic fluxes or mapping magnetic fluxes into forbidden zones in multi-connected space filled by solid state material. A photonic crystal with porous nano-structures is a promising system for detecting fractional charges and paves a new way for topological quantum computation.

任意子是二维电子气在强磁场下的集体激发，携带分数电荷和超越费米子和玻色子的奇异统计特性。到目前为止，具有串联分数电荷的任何子只存在于分数量子霍尔效应中。在其他物理系统中寻找新的分数电荷序列并发展基于同一根的统一数学物理理论仍然是一个挑战。在这里，基于费曼路径积分理论和瑟斯顿火车轨道理论的推广，提出了一种在磁通量晶格中传播电子的拓扑路径融合理论，以探索分数电荷的物理起源。该数学物理理论产生了分数量子霍尔效应中存在的分数电荷序列，并预测了新的分数电荷序列。在磁通量的一维晶格中预测了一系列无理电荷。在电流的火车轨道的二维和三维格子中也产生了分数带电任意子，揭示了结点阵模型和火车轨道模型之间的精确对应关系。对于分数量子霍尔效应中复霍尔电导和复合费米子的模对称性的新解释也源于这种拓扑路径融合理论。通过构建三维互锁磁通量或将磁通量映射到由固态材料填充的多连通空间中的禁区，可以实现对任意子的三维实验观察。具有多孔纳米结构的光子晶体是一种很有前途的检测分数电荷的系统，并为拓扑量子计算开辟了一条新途径。