Torus algebra and logical operators at low energy

We can study all universal properties of 2+1D topologically ordered states under the unitary monoidal tensor category (UMTC) descriptions. In this work, we investigate the torus algebra, originally introduced by Ma etc. [Phys. Rev. B 107, 085123 (2023)], and show that it is exactly the algebra of all local operators acting on the low energy subspaces of a topologically ordered state on a torus. The simple modules over torus algebra (irreducible blocks after decomposition) enumerates all possible low-energy subspaces. We discovered the underlying vector spaces of modules corresponding to the ground state subspaces of a punctured torus. We also discuss an example of idempotent decomposition of a chiral ising model using our algorithm.