Completely fixed point free isometry and cyclic orbifold of lattice vertex operator algebras (Q6639811)

Vertex operator algebras and their representation categories axiomatise 2-dimensional conformal field theories in physics.\N\NAn important problem is to study fixed-point vertex operator algebras \(V^G\) under finite groups of automorphisms \(G\leq\mathrm{Aut}(V)\). In particular, one can ask what the automorphism group \(\mathrm{Aut}(V^G)\) of \(V^G\) is. Many automorphisms of \(V^G\) will be induced from those of \(V\) that are in the normaliser of \(G\) in \(\mathrm{Aut}(V)\), but there may be \emph{extra automorphisms} that are not. It is a natural question to ask under what circumstances \(V^G\) has extra automorphisms.\N\NThis paper is concerned with the special case that \(V=V_L\) is a lattice vertex operator algebra (for a positive-definite, even lattice \(L\)) and \(G=\langle g\rangle\) is a cyclic group. We remark that \(V_L\) is strongly rational, and so is the fixed-point subalgebra \(V_L^g\).\N\NThe main result of the paper assumes that \(g\) is (a lift of) an isometry of \(L\) with the property that \(g^i\) only fixes \(0\in L\) for all \(1\leq i<|g|\) (i.e.\ \(\langle g\rangle\) has as few fixed points as possible). Moreover, \(L\) is assumed to possess no vectors of squared norm~\(2\) (meaning that \(L\) is small in some sense). Then, it is proved that \(V_L^g\) can have extra automorphisms only if either the order of \(g\) is prime or if \(L\) is the Leech lattice or a coinvariant sublattice of the Leech lattice.