G. Landi, F. Lizzi, R.J. Szabo
String Geometry and the Noncommutative Torus
(122K, LATeX 2e)
ABSTRACT. We describe an intimate relationship between the noncommutative geometry
associated with a lattice vertex operator algebra A and the noncommutative
torus. We show that the tachyon subalgebra of A is naturally isomorphic to
a class of twisted modules representing quantum deformations of the algebra of
functions on the torus. We construct the corresponding even real spectral
triples and determine their Morita equivalence classes using string duality
arguments. These constructions yield simple proofs of the O(d,d;Z) Morita
equivalences between d-dimensional noncommutative tori and give a natural
physical interpretation of them in terms of the target space duality group of
toroidally compactified string theory. We classify the automorphisms of the
twisted modules and construct the most general gauge theory which is invariant
under the automorphism group. We compute bosonic and fermionic actions
associated with these gauge theories and show that they are explicitly
duality-symmetric. The duality-invariant gauge theory is manifestly covariant
but contains highly non-local interactions. We show that it also admits a new
sort of particle-antiparticle duality which enables the construction of
instanton field configurations in any dimension. The duality non-symmetric
on-shell projection of the field theory is shown to coincide with the standard
non-abelian Yang-Mills gauge theory minimally coupled to massive Dirac
fermions.