  We analyze the nature of quantum phase transition to a superfluid state of
flexible chains in a gas of polar bosonic molecules confined in a stack of $N$
identical 1d ("cigar" type) optical lattice layers and polarized
perpendicularly to the layers. Monte Carlo simulations within the $J$-current
model show that, in the absence of the inter-tube tunneling, the transition to
the $N$-layered superfluid is in the Berezinskii-Kosterlitz-Thouless
universality class in the one-particle density matrix channel. The inter-layer
tunneling changes it to the $q=N$ 2d Potts universality. The low energy field
descriptions of the transition are discussed in terms of conformal field
theories.
