The n-Category Café

I’m happy to hear you’re re-interested in QFT! I get a little depressed every time you go off to work on climate change (even though that’s important) or pure math (even though that’s fascinating) but console myself that you’ll inevitably return, because your knowledge and skill in mathematical physics and its exposition are unparalleled and the subject is compelling.

I do have two directions in QFT of potential interest. One is the paper that’s (after a two month hold, perhaps a new record) on hep-th:

C, P, T, and Triality

and the second I don’t wish to talk about publicly but can privately if you’d like.

Your guess on planarity coming from n->oo in SU(n) is correct. They consider the planar stuff pretty much done at this point (the mathematicians sure don’t!) so a lot of the current work is on non-planar extensions.

I’ve been following this stuff for a long time and have discussed it a bunch with Nima. There’s a lot going on there.

The link labelled “pop science media” links to a preprint on the arXiv about scalar-scaffolded gluons. Presumably you had meant to actually link a pop science media article there.

One hub for references on the associahedra and their applications and related structures–stellahedra, polygons, hypertetrahedra/n-simplices, and more–is OEIS A133437; another, with extensive refs in the comments, “Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?”; and a third is my MO-Q “An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes”.

The refined Euler characteristic partition polynomials, a.k.a. the refined, signed, face- or f-partition polynomials, for the associahedra, $[A]$ of OEIS A133437 re-normalized by the factorials, are precisely the polynomials for the coefficients of the compositional inverse (CI) of a power series, or ordinary generating function (o.g.f.), the first few of which can be found in a letter dated 1676 from Newton to a Secretary of the Royal Society. They occur in explicit solutions to the inviscid Burgers-Hopf equation, as sketched in the MO-Q “Why is there a connection between enumerative geometry and nonlinear waves?”.

$[A]$ is related to the set of refined h-polynomials of the associahedra, $[N]$ of OEIS A134264, via the partition polynomials $[R]$ of A263633, the refined Pascal partition polynomials for the reciprocal, or multiplicative inverse (MI), of an o.g.f., by the substitutional identity denoted by $[A] = [N][R]$ (this forms the core of an infinite dihedral group under substitution/composition of indeterminates). $[N]$ is also the set of refined Narayana / Kreweras polynomials, related to Noncrossing partitions, primitive parking functions, forests of trees, Dyck lattice paths, and more. They are also known as the Voiculescu polynomials, giving the free moments from the free cumulants of free probability/random matrix theory (inverse is A350499), and are to be found in “Connecting Scalar Amplitudes using The Positive Tropical Grassmannian” by Freddy Cachazo and Bruno Giménez Umbert. See the MO-Q “Guises of the noncrossing partitions (NCPs)” for more notes on $[N]$.

Allen Knutson mentions the positroids. OEIS A046802 and A248727 give analytic relations among the coarse h- and f-polynomials of the associahedra and those of the stellahedra, whose coarse face polynomials enumerate the number of positroid cells of the totally nonnegative Grassmannians. Combinatorics of tropical Grassmannians and that of the associahedra are linked analytically by the simple scaling transformation between the CI of an o.g.f. and that of an exponential generating function (e.g.f.), or formal Taylor series. The counterpart to $[A]$ for the CI of an o.g.f. is the set of classic Lagrange partition polynomials $[L]$ of A134685 for the CI of an e.g.f.. The natural reduction A134991 of $[L]$ is related to tropical Grassmannians and phylogenetic trees. Both $[A]$ and $[L]$ are related to the flow equations characterized by the refined Euler partition polynomials $[E]$ of A145271. The counterpart to the identity $[A]=[N][R]$ for o.g.f.s is $[L]=[E][P]$ for e.g.f.s, where $[P]$ of A133314 is the set of refined Euler characteristic partition polynomials, a.k.a. the refined, signed f-polynomials, of the permutahedra, providing the MI of an e.g.f..

Given all these relationships, can you point me to the talks in the IAS 2024 summer school related to any of these sets of partition polynomials or their natural reductions?