  Using a perturbative renormalization group approach, we show that the
extended ($J_1$-$J_2$-$J_d$) Heisenberg model on the kagome lattice with a
staggered chiral interaction ($J_\chi$) can exhibit a gapless chiral quantum
spin liquid phase. Within a coupled-chains construction, this phase can be
understood as a chiral sliding Luttinger liquid with algebraic decay of spin
correlations along the chain directions. We calculate the low-energy properties
of this gapless chiral spin liquid using the effective field theory and show
that they are compatible with the predictions from parton mean-field theories
with symmetry-protected line Fermi surfaces. These results may be relevant to
the state observed in the kapellasite material.
