  This is the first part of a two-part work on a unified mathematical theory of
gapped and gapless edges of 2d topological orders. We analyze all the possible
observables on the 1+1D world sheet of a chiral gapless edge of a 2d
topological order, and show that these observables form an enriched unitary
fusion category, the Drinfeld center of which is precisely the unitary modular
tensor category associated to the bulk. This mathematical description of a
chiral gapless edge automatically includes that of a gapped edge (i.e. a
unitary fusion category) as a special case. Therefore, we obtain a unified
mathematical description and a classification of both gapped and chiral gapless
edges of a given 2d topological order. In the process of our analysis, we
encounter an interesting and reoccurring phenomenon: spatial fusion anomaly,
which leads us to propose the Principle of Universality at RG fixed points for
all quantum field theories. Our theory also implies that all chiral gapless
edges can be obtained from a so-called topological Wick rotations. This fact
leads us to propose, at the end of this work, a surprising correspondence
between gapped and gapless phases in all dimensions.
