# Compositional inversion and generating functions in algebraic geometry

The exponential generating function of the graded dimension of the cohomology ring of the moduli space of n-pointed curves of genus zero satisfying the associativity equations of physics (the WDVV equations) (cf. OEIS-A074060) is the compositional inverse of the generating function for the Betti numbers for $M_{0,n}$ (cf. OEIS-A049444 and A143491).

And, Brown and Bergstrom in "Inversion of series and the cohomology of the moduli spaces of $M_{0,n}^\delta$" state in the abstract:

For $n \geq 3$, let $M_{0,n}$ denote the moduli space of genus $0$ curves with $n$ marked points, and $\overline{M}_{0,n}$ its smooth compactification. A theorem due to Ginzburg, Kapranov and Getzler states that the inverse of the exponential generating series for the Poincare polynomial of $H^\bullet(M_{0,n})$ is given by the corresponding series for $H^\bullet(\overline{M}_{0,n})$. In this paper, we prove that the inverse of the ordinary generating series for the Poincare polynomial of $H^\bullet(M_{0,n})$ is given by the corresponding series for $H^\bullet(M^{\delta}_{0,n})$, where $\overline{M}_{0,n}\subset M^{\delta}_{0,n} \subset \overline{M}_{0,n}$ is a certain smooth affine scheme.

Is there an intuitive principle underlying this relationship of compositional inversion between the generating functions?

(Getzler makes use of a generalized Legendre transform-compositional inversion in disguise-to relate the generating functions.)

• There is an intriguing tale told by He and Jejalla in "Modular matrix models" interweaving the compositional inversion of generating series as in free probability theory, matrix models, gauge field theory, Calabi-Yau geometry, the Klein modular invariant j-function, and some monstrous moonshine. Jan 31, 2015 at 10:10
• To the degree that the inverse pair can be represented by certain types of trees, Drakes' thesis "An inversion theorem for labelled trees ..." people.brandeis.edu/~gessel/homepage/students/drakethesis.pdf has some insights. Dec 8, 2015 at 6:56
• See also "Brown's moduli spaces of curves and the gravity operad" by Dupont and Vallette. arxiv.org/abs/1509.08840 Apr 14, 2016 at 20:49
• Rather Jejjala in the first comment. For more on matrix integrals, see "Random matrices" by Eynard, Kimura, and Ribault and on matrix integrals and free probability, see "Three lectures on free probability" by Novak an LaCroix. Oct 16, 2020 at 9:32

I think you would enjoy reading Curt McMullen's paper "Moduli spaces in genus zero and inversion of power series". In some sense there is nothing there that isn't already in Getzler's paper, but everything is stated in a down-to-earth and combinatorial fashion.

Let me summarize the story, first for the spaces $\overline M_{0,n}$ and $M_{0,n}$. The space $\overline M_{0,n}$ has a stratification where a stratum corresponds to a tree with no vertices of valence two. The stratum itself is isomorphic to $\prod_v M_{0,\mathrm{val}(v)}$ where $v$ runs over interior vertices of the tree and $\mathrm{val}(v)$ denotes the number of incident edges. Since the virtual Poincaré polynomial is additive over stratifications, this shows that the virtual Poincaré polynomial of $\overline M_{0,n}$ is given by a sum over trees involving the virtual Poincaré polynomials of $M_{0,n'}$ for $n' \leq n$. Now using the relationship between compositional inversion and summing over trees, well-known to combinatorists, one can thus show that the exponential generating series of virtual Poincaré polynomials of $\overline M_{0,n}$ and $M_{0,n}$ are compositional inverses of each other. (If you don't know virtual Poincaré polynomials, think about any other invariant additive under stratification, e.g. Euler characteristic.)

Finally, both the spaces $\overline M_{0,n}$ and $M_{0,n}$ have pure cohomology in every degree: $H^k (\overline M_{0,n})$ is pure of weight $k$, and $H^k(M_{0,n})$ us pure of weight $2k$. Thus in both cases, the virtual Poincaré polynomial concides with the usual Poincaré polynomial (in the latter case up to a substitution $t \mapsto t^2$). This explains the second sentence in Bergström-Brown's abstract.

The story for $M_{0,n}^\delta$ and $M_{0,n}$ is completely similar, the only difference being that $M_{0,n}^\delta$ has a stratification indexed by trees without vertices of valence two and with a cyclic ordering of the edges incident to each vertex. In the same way as compositional inversion of exponential generating functions corresponds to sums over trees, compositional inversion of ordinary generating functions corresponds to sums over trees with such cyclic structure. McMullen touches upon something very similar at the very end of his paper. He doesn't consider $M_{0,n}$ and $M_{0,n}^\delta$, but instead considers the choice of a connected component of $M_{0,n}(\mathbf R)$ and its closure. Combinatorially this amounts to exactly the same thing: $M_{0,n}^\delta$ is defined by choosing a connected component of $M_{0,n}(\mathbf R)$ and taking the union of all strata meeting the closure of this component.

A final remark is that the duality between $H^\bullet(\overline M_{0,n})$ and $H^\bullet(M_{0,n})$ can be upgraded to a Koszul duality of two cyclic operads, the "Hypercommutative" and "Gravity" operads. This is a much stronger result than just that their generating series are compositional inverses, and this is what Getzler proves. On the other hand the cohomologies of $M_{0,n}^\delta$ and $M_{0,n}$ give rise to nonsymmetric cyclic operads (this notion is not defined in the literature, but it's not hard to give the definition). However, it turns out that they are not in any natural sense Koszul dual of each other, but it is still true that they are interchanged with each other under bar-cobar-duality, up to homotopy. (But first one needs to define a bar transform of nonsymmetric cyclic operads...) This is an operad-theoretic statement that improves on what Bergström-Brown proved. I worked this out with Johan Alm at one point but we never wrote it down properly.

• Can you give a bit more detail about how one goes from a Koszul duality between two operads to an observation about two generating functions being compositional inverses? Sep 19, 2014 at 21:03
• Thanks Dan for the great overview. Hadn't seen McMullen's paper. (Murri gives nice diagrams of ribbons and marked Riemann surfaces. For the interested, see the OEIS refs.). The relations of diff. ops to inversion inspired me to climb the Cayley trees for a clear view of the landscape years ago. I guess I must put my nose to the grindstone to familiarize myself with stratified spaces so that I can feel it's intuitive. Sep 20, 2014 at 3:19
• Brown and Bergstrom are inverting o.g.f.s, so the associahedra face vectors occur oeis.org/A133437, and McMullen is inverting e.g.f.s oeis.org/A134685. Sep 20, 2014 at 5:58
• Alm and Petersen, "Brown's dihedral moduli space and freedom of the gravity operad" arxiv.org/abs/1509.09274 Apr 14, 2016 at 20:54