Appears in collection : Combinatorics and Arithmetic for Physics

Steinberg symbol is a multiplicative bicharacter of a ring satisfying additional (Steinberg) relation. Beilinson in 1982 suggested an explicit expression for a canonical symbol for the ring functions on a circle. This symbol can be interpreted as a multiplicative analogue of a residue and also as a cocycle defining the Heisenberg group. We will show how one can define the boson-fermion correspondence using the symbol and show that the finite-gap tau-function can be interpreted as an automorphic form.