Coset construction of logarithmic minimal models: branching rules and branching functions

Working in the Virasoro picture, it is argued that the logarithmic minimal models LM(p,p’)=LM(p,p’;1) can be extended to an infinite hierarchy of logarithmic conformal field theories LM(p,p’;n) at higher fusion levels n=1,2,3,… From the lattice, these theories are constructed by fusing together n x n elementary faces of the appropriate LM(p,p’) models. It is further argued that all of these logarithmic theories are realized as diagonal cosets (A_1^{(1)})_k \oplus (A_1^{(1)})n / (A_1^{(1)}){k+n} where n is the integer fusion level and k=np/(p’-p)-2 is a fractional level. These cosets mirror the cosets of the higher fusion level minimal models of the form M(m,m’;n), but are associated with certain reducible representations. We present explicit branching rules for characters in the form of multiplication formulas arising in the logarithmic limit of the usual Goddard-Kent-Olive coset construction of the non-unitary minimal models M(m,m’;n). The limiting branching functions play the role of Kac characters for the LM(p,p’;n) theories.