Part 13 of 15. The pivot. Everything in the series so far has been classical applications of information geometry to classical statistics. The last three parts point at an unclaimed territory, the Moonshine ecosystem of modular forms, Monster-group symmetries, and conformal field theories, where the machinery of Acts I and II is conspicuously missing from the standard toolkit and where it has the potential to say something new.
Explore: Review the correspondence table below. It maps familiar information-geometry concepts to their Moonshine equivalents. This is the Rosetta Stone for the final three parts of this series.
| Information geometry | Moonshine ecosystem |
|---|---|
| Parametric family of distributions | Family of conformal field theories at varying moduli τ |
| Parameter space / manifold | Moduli space of the CFT (e.g., complex structure of a K3 surface) |
| Fisher information metric | Weil-Petersson or Zamolodchikov metric on the moduli space |
| Probability density p(x | θ) | Normalised character χ_i(τ) / Z(τ) of a primary field |
| Log-partition function ψ(θ) | log Z(τ), i.e. the free energy of the CFT |
| Kullback-Leibler divergence | Thermodynamic length / relative entropy between modular invariants |
| Natural gradient descent | Renormalisation group flow as information-geometric gradient descent |
| Asymmetry of KL | c-theorem: information loss under RG flow is unidirectional |
| Exponential family / dually flat | Characters of a rational CFT with their primal/dual structure |
| Koszul-Souriau moment map | j-function as entropic potential on the Lie algebra of the Monster |
A conformal field theory (CFT) has a partition function that depends on a complex modular parameter . Physically, this is a trace over the Hilbert space of the theory. But mathematically, once normalized, becomes a generating function for a probability distribution over the quantum states of the theory:
is the probability of a state with energy . Because it's a parameterized probability distribution, it has a parameter space (the moduli space), a notion of distance between nearby distributions, and a KL divergence. It has every ingredient needed for Information Geometry.
Yet, the fields studying Moonshine (string theory, algebraic geometry, number theory) rarely use the statistical toolkit. By applying Information Geometry to these objects, we open the door to unifying disparate proofs and potentially discovering new theorems.
Part 14 makes one row concrete: the instanton moduli of a 2D CFT. The Fisher metric, Weil-Petersson metric, and Euclidean anti-de Sitter metric all coincide as 3D hyperbolic space.
Part 15 takes the most speculative row and runs with it: the -function as a Koszul-Souriau potential on the Lie algebra of the Monster.