  We consider states of bosons in two dimensions that do not support anyons in
the bulk, but nevertheless have stable chiral edge modes that are protected
even without any symmetry. Such states must have edge modes with central charge
$c=8k$ for integer $k$. While there is a single such state with $c=8$, there
are, naively, two such states with $c=16$, corresponding to the two distinct
even unimodular lattices in 16 dimensions. However, we show that these two
phases are the same in the bulk, which is a consequence of the uniqueness of
signature $(8k +n, n)$ even unimodular lattices. The bulk phases are stably
equivalent, in a sense that we make precise. However, there are two different
phases of the edge corresponding to these two lattices, thereby realizing a
novel form of the bulk-edge correspondence. Two distinct fully chiral edge
phases are associated with the same bulk phase, which is consistent with the
uniqueness of the bulk since the transition between them, which is generically
first-order, can occur purely at the edge. Our construction is closely related
to $T$-duality of toroidally compactified heterotic strings. We discuss
generalizations of these results.
