| A brief history of affine Lie algebra, the Virasoro algebra and its culmination in the Virasoro master equation is given. By studying ansatze of the master equation, we obtain exact solutions and gain insight in the structure of large slices of affine-Virasoro space.; We find an isomorphism between the constructions in the ansatz SO(n){dollar}sb{lcub}diag{rcub}{dollar}, which is a set of unitary, generically irrational affine-Virasoro constructions on SO(n), and the unlabelled graphs of order n. On the one hand, the conformal constructions, are classified by the graphs, while, conversely, a group-theoretic and conformal field-theoretic identification is obtained for every graph of graph theory.; We also define a class of "magic" Lie group bases in which the Virasoro master equation admits a simple metric ansatz {dollar}{lcub}{dollar}g{dollar}sb{lcub}metric{rcub}{rcub}{dollar}, whose structure is visible in the high-level expansion. When a magic basis is real on compact g, the corresponding g{dollar}sb{lcub}metric{rcub}{dollar} is a large system of unitary, generically irrational conformal field theories. Examples in this class include the graph-theory ansatz SO(n){dollar}sb{lcub}diag{rcub}{dollar} in the Cartesian basis of SO(n), and the ansatz SU(n){dollar}sb{lcub}metric{rcub}{dollar} in the Pauli-like basis of SU(n). Finally, we define the "sine-area graphs" of SU(n), which label the conformal field theories of SU(n){dollar}sb{lcub}metric{rcub}{dollar}, and we note that, in similar fashion, each magic basis of g defines a generalized graph theory on g which labels the conformal field theories of g{dollar}sb{lcub}metric{rcub}{dollar}. |
