Torus algebra and logical operators at low energy

Given a modular tensor category $\mathscr{C}$, we construct an associative algebra $\mathrm{Tor({\mathscr{C}}})$, which we call the torus algebra. We prove that the torus algebra is semisimple by explicitly constructing all the simple modules. Suppose that a topological ordered phase described by $\mathscr{C}$ is put on a torus. Physically, each simple module over $\mathrm{Tor({\mathscr{C}}})$ consists of the low energy states on the torus with one anyon excitation, or equivalently, the ground states on a punctured torus where the anyon is enclosed by the puncture. Elements in $\mathrm{Tor({\mathscr{C}}})$ can be physically interpreted as anyon hopping processes on the torus. We give the precise formula how an arbitrary logical operator on the low energy states on a torus can be realized by moving anyons on the torus. Our work thus provides a theoretical proposal that the low energy states on a torus can serve as topological qudits and one can arbitrarily manipulate them by moving anyons around.