 Heisenberg current algebra of a fermion on the circle

ID: heisenberg-current-algebra-of-a-fermion-on-the-circle

For integer modes filled at indices below zero, set

E_{rs} = c_{r}^{†} c_{s} - δ_{rs} 1_{r < 0}

and

J_{n} = \sum_{r} E_{r, r + n}

on the finite-energy domain. The canonical anticommutation relations give a central term

δ_{s t} δ_{u r} (1_{r < 0} - 1_{s < 0})

[E_{rs}, E_{t u}]

. Summing across the filled-empty boundary gives

[J_{n}, J_{m}] = n δ_{n + m, 0} I

, with

J_{n}^{*} = J_{- n}

. The current's level is the coefficient of this central term.

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