  The problem of constructing twisted modules for a vertex operator algebra and
an automorphism has been solved in particular in two contexts. One of these two
constructions is that initiated by the third author in the case of a lattice
vertex operator algebra and an automorphism arising from an arbitrary lattice
isometry. This construction, from a physical point of view, is related to the
space-time geometry associated with the lattice in the sense of string theory.
The other construction is due to the first author, jointly with C. Dong and G.
Mason, in the case of a multi-fold tensor product of a given vertex operator
algebra with itself and a permutation automorphism of the tensor factors. The
latter construction is based on a certain change of variables in the worldsheet
geometry in the sense of string theory. In the case of a lattice that is the
orthogonal direct sum of copies of a given lattice, these two very different
constructions can both be carried out, and must produce isomorphic twisted
modules, by a theorem of the first author jointly with Dong and Mason. In this
paper, we explicitly construct an isomorphism, thereby providing, from both
mathematical and physical points of view, a direct link between space-time
geometry and worldsheet geometry in this setting.
