Minimal models and modular invariance

For central charge below one, positivity of Virasoro descendant Gram matrices severely restricts unitary highest-weight representations. The nontrivial unitary minimal series is labeled by an integer m at least three. Its allowed weights can be placed in a rectangular Kac table with an identification pairing entries. Counting the rectangle without that identification doubles the number of distinct families. Other nonunitary minimal theories exist, but they obey different positivity premises and must not be presented as additional members of this unitary series.

The finite label set means finitely many primary representations of the chosen chiral algebra, not finitely many local operators or Hilbert states. Each representation includes descendants, with null descendants removed. A full theory must choose how left and right representations are paired, and may have additional algebraic symmetry. Even the same central charge can support distinct consistent full spectra, so a central charge alone is not a complete identification of an experimental universality class.

A character traces the radial energy grading over an irreducible chiral module, including its descendants and the central vacuum shift. Null states have already been quotiented, so replacing every minimal-model character by an unrestricted Verma-module count gives the wrong partition function. For a full left-right spectrum, nonnegative integer multiplicities specify how chiral modules occur together. A unique ordinary vacuum gives a unit coefficient for the vacuum pair in the simplest connected theory.

The torus is specified by a complex shape parameter in the upper half-plane. Changing the primitive cycles by an integer matrix of determinant one describes the same unmarked torus, so the bosonic partition function must transform consistently. The generators are a unit shift and inversion. The exponential variables are dimensionless; they encode geometry and Euclidean evolution, rather than a new coupling in a flat-space OPE. A fixed spin-structure fermionic partition function need not be invariant by itself.

Characters of a rational theory transform among themselves under finite modular matrices. In a convention writing the partition function as a character-vector sesquilinear form, modular invariance requires its multiplicity matrix to commute appropriately with those transformations. The diagonal identity pairing is invariant when the character matrices are unitary, but not every abstract nonnegative matrix produces consistent local correlation functions. Sphere crossing and the sewing of more general surfaces impose further requirements on OPE data.

For the diagonal Ising theory, the three weights are zero, one half and one sixteenth, with left and right matched. Its spin field has full dimension one eighth, illustrating the factor of two between a holomorphic weight and a scalar dimension. A pure-qubit overlap plot cannot calculate a character, modular transformation or Kac determinant. The lab only supplies a limited example of normalized weights; the finite number of visible outcomes is not evidence for a minimal CFT.