Magic bases, metric ansätze and generalized graph theories in the Virasoro master equation

Abstract

We define a class of “magic” Lie group bases in which the Virasoro master equation admits a class of simple metric ansätze { g metric}, whose structure is visible in the high-level expansion. When a magic basis is real on compact g, the corresponding g metric is a large system of unitary, generically irrational conformal field theories. Examples in this class include the graph-theory ansatz SO( n) diag in the Cartesian basis of SO( n) and the ansatz SU( n) metric in the Pauli-like basis of SU( n). A new phenomenon is observed in the high-level expansion of SU( n) metric: Due to the trigonometric structure constants of the Pauli-like basis, irrational central charge is clearly visible at finite order of the expansion. We also define the “sine-area graphs” of SU( n), which label the conformal field theories of SU( n) metric and note that, in a similar fashion, each magic basis of g defines a generalized graph theory on g which labels the conformal field theories of g metric.