Conformal field theory
The Hilbert space of physical states is a unitary module of the Virasoro algebra corresponding to a fixed value of "c". Stability requires that the energy spectrum of the Hamiltonian be nonnegative. The modules of interest are the highest weight modules of the Virasoro algebra.
Hilbert series and Hilbert polynomial
Thus formula_62 is an Artinian ring, which is a -vector space of dimension , and Jordan–Hölder theorem may be used for proving that formula_63 is the degree of the algebraic set . In fact, the multiplicity of a point is the number of occurrences of the corresponding maximal ideal in a composition series.