We will review our joint work with Omar Foda, Masato Okado and Trevor Welsh, on a class of (multi-)partitions arising from the combinatorial description of branching functions of affine sl(n).

In this talk, I discuss the torus partition function of an integrable relativistic theory characterised by a purely elastic two-particle scattering matrix. The simplest example of such field theory is the sinh-Gordon model. As far as the partition function is concerned, the original field theory is reformulated as an effective field theory involving a pair of bosonic fields associated with the two cycles of the torus and a pair of fermionic fields which generate the Gaudin measure. This effective field theory generalizes the Thermodynamical Bethe Ansatz which applies to the limit when one of the periods of the torus becomes asymptotically large.

Omar was very fond of combinatorial constructions. Here, I recall one of our most successful, which used lattice paths to analyse the particle structure of the conformal minimal models $M(p,p')$, and also construct generalisations of the Rogers--Ramanujan identities. These lattice paths, whose generating functions are the characters $\chi^{p,p'}_{r,s}$ of the Virasoro Lie algebra, were based on those arising in the celebrated statistical models of Andrews, Baxter and Forrester, and Forrester and Baxter. Our alternative weighting, using a novel banding structure, enabled a recursive combinatorial derivation of the particle structure, giving rise to fermionic expressions for $\chi^{p,p'}_{r,s}$.

If time allows, I will also describe our attempts to extend these constructions to the much trickier $W_n$ conformal algebras, and how these have inspired more recent developments.

In an early work together with Omar and Mathew Zuparic, we computed the partition function of a certain free-fermionic vertex model with domain wall boundary conditions. This partition function is fully factorized. I will present a higher-rank analogue of this result, and talk about its role in proving the Cauchy identity for Lascoux--Leclerc--Thibon (LLT) polynomials.

I review fermionic character formulas of various affine Lie algebra modules and list up problems yet to be settled.

In this talk, I will go over my collaboration and relevant works with Omar. We worked on two slightly different directions that both connect higher-dimensional gauge theories and two-dimensional conformal field theories. The common key idea behind our works is the powerful tool of vertex operator. At the end I will also briefly mention the recent progress made with a new type of vertex operator, called O-vertex.

I plan to talk about fermionic formulas --- old things and if it will be possible about relatively new. I mean formulas for the characters of triplet-like algebras and maybe relations with knot invariants.

Boundary states in integrable spin chains feature remarkable overlap formulas that generalize the famous Gaudin norm. In virtue of the AdS/CFT integrability, spin chain overlaps naturally describe defects and D-branes in 4D gauge theories. As examples I will discuss domain walls in the N=4 super-Yang--Mills theory and ABJM model.