Bose-Fermi equivalence of the two-dimensional torus for simply-laced groups

Abstract

Bosonic and fermionic partition functions on the two-dimensional torus are explicitly calculated to establish a Bose-Fermi equivalence for each fundamental representation of every simple-laced group G. A one-to-one correspondence exists for bosons with momenta generated by the root lattice of G shifted by the element w k, where w k is a minimal weight of G, i.e. and element of the center of the covering group of G, and fermions with generalized boundary conditions ψ → (exp iθ k ) ψ (under rotation around the torus) where exp( iθ k ) is the same element of Z as is w k . The non-orthogonal weight lattices of G are converted into orthogonal lattices with constraints. It is shown that only for spin(2 N) can these constraints be written as projection operators, which correspond to sums over spin structure in fermion language. For SU( N), E 6 and E 7 the appropriate constraints are imposed by coupling the fermions to background U(1) world-sheet gauge fields which act as Lagrange multipliers, i.e. the equivalence is to constrained fermions. Explicit modular invariant partition functions are presented for closed strings compactified ti simply-laced groups associated to Kac-Moody algebras with central charge k = 1.