We relate the quantum cohomology of minuscule flag manifolds to the tt*-Toda equations, a special case of the topological–antitopological fusion equations which were introduced by Cecotti and Vafa in their study of supersymmetric quantum field theories. To do this, we combine the Lie-theoretic treatment of the tt*-Toda equations of Guest–Ho with the Lie-theoretic description of the quantum cohomology of minuscule flag manifolds from Chaput–Manivel–Perrin and Golyshev–Manivel.

We construct and study some vertex theoretic invariants associated with Poisson varieties, specializing in the conformal weight $0$ case to the familiar package of Poisson homology and cohomology. In order to do this conceptually, we sketch a version of the calculus, in the sense of [12], adapted to the context of vertex algebras. We obtain the standard theorems of Poisson (co)homology in this chiral context. This is part of a larger project related to promoting noncommutative geometric structures to chiral versions of such.

We obtain simple quadratic recurrence formulas counting bipartite maps on surfaces with prescribed degrees (in particular, $2k$-angulations) and constellations. These formulas are the fastest known way of computing these numbers.

Our work is a natural extension of previous works on integrable hierarchies (2-Toda and KP), namely, the Pandharipande recursion for Hurwitz numbers (proved by Okounkov and simplified by Dubrovin–Yang–Zagier), as well as formulas for several models of maps (Goulden–Jackson, Carrell–Chapuy, Kazarian–Zograf). As for those formulas, a bijective interpretation is still to be found. We also include a formula for monotone simple Hurwitz numbers derived in the same fashion.

These formulas also play a key role in subsequent work of the author with T. Budzinski establishing the hyperbolic local limit of random bipartite maps of large genus.

For almost any compact connected Lie group $G$ and any field $\mathbb{F}_{p}$, we compute the Batalin–Vilkovisky algebra $H^{\star +\text{dim}\,G}(\text{LBG};\mathbb{F}_{p})$ on the loop cohomology of the classifying space introduced by Chataur and the second author. In particular, if $p$ is odd or $p=0$, this Batalin–Vilkovisky algebra is isomorphic to the Hochschild cohomology $HH^{\star }(H_{\star }(G),H_{\star }(G))$. Over $\mathbb{F}_{2}$ , such an isomorphism of Batalin–Vilkovisky algebras does not hold when $G=\text{SO}(3)$ or $G=G_{2}$. Our elaborate considerations on the signs in string topology of the classifying spaces give rise to a general theorem on graded homological conformal field theory.

Conformal loop ensembles (CLEs) are random collections of loops in a simply connected domain, whose laws are characterized by a natural conformal invariance property. The set of points not surrounded by any loop is a canonical random connected fractal set — a random and conformally invariant analog of the Sierpinski carpet or gasket.

In the present paper, we derive a direct relationship between the CLEs with simple loops ($\text{CLE}_{\unicode[STIX]{x1D705}}$ for $\unicode[STIX]{x1D705}\in (8/3,4)$, whose loops are Schramm’s $\text{SLE}_{\unicode[STIX]{x1D705}}$-type curves) and the corresponding CLEs with nonsimple loops ($\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ with $\unicode[STIX]{x1D705}^{\prime }:=16/\unicode[STIX]{x1D705}\in (4,6)$, whose loops are $\text{SLE}_{\unicode[STIX]{x1D705}^{\prime }}$-type curves). This correspondence is the continuum analog of the Edwards–Sokal coupling between the $q$-state Potts model and the associated FK random cluster model, and its generalization to noninteger $q$.

Like its discrete analog, our continuum correspondence has two directions. First, we show that for each $\unicode[STIX]{x1D705}\in (8/3,4)$, one can construct a variant of $\text{CLE}_{\unicode[STIX]{x1D705}}$ as follows: start with an instance of $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$, then use a biased coin to independently color each $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ loop in one of two colors, and then consider the outer boundaries of the clusters of loops of a given color. Second, we show how to interpret $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ loops as interfaces of a continuum analog of critical Bernoulli percolation within $\text{CLE}_{\unicode[STIX]{x1D705}}$ carpets — this is the first construction of continuum percolation on a fractal planar domain. It extends and generalizes the continuum percolation on open domains defined by $\text{SLE}_{6}$ and $\text{CLE}_{6}$.

These constructions allow us to prove several conjectures made by the second author and provide new and perhaps surprising interpretations of the relationship between CLEs and the Gaussian free field. Along the way, we obtain new results about generalized $\text{SLE}_{\unicode[STIX]{x1D705}}(\unicode[STIX]{x1D70C})$ curves for $\unicode[STIX]{x1D70C}<-2$, such as their decomposition into collections of $\text{SLE}_{\unicode[STIX]{x1D705}}$-type ‘loops’ hanging off of $\text{SLE}_{\unicode[STIX]{x1D705}^{\prime }}$-type ‘trunks’, and vice versa (exchanging $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D705}^{\prime }$). We also define a continuous family of natural $\text{CLE}$ variants called boundary conformal loop ensembles (BCLEs) that share some (but not all) of the conformal symmetries that characterize $\text{CLE}$s, and that should be scaling limits of critical models with special boundary conditions. We extend the $\text{CLE}_{\unicode[STIX]{x1D705}}$/$\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ correspondence to a $\text{BCLE}_{\unicode[STIX]{x1D705}}$/$\text{BCLE}_{\unicode[STIX]{x1D705}^{\prime }}$ correspondence that makes sense for the wider range $\unicode[STIX]{x1D705}\in (2,4]$ and $\unicode[STIX]{x1D705}^{\prime }\in [4,8)$.

We show that the massless particle spectrum in a four-dimensional conformal Haag–Kastler net is generated by a free field subnet. If the massless particle spectrum is scalar, then the free field subnet decouples as a tensor product component.

We present a procedure for constructing families of local, massive and interacting Haag–Kastler nets on the two-dimensional spacetime through an operator-algebraic method. A proof of existence of local observables is given without relying on modular nuclearity. By a similar technique, another family of wedge-local nets is constructed using certain endomorphisms of conformal nets recently studied by Longo and Witten.

We study a family of semi-ample divisors on the moduli space of n-pointed genus 0 curves given by higher-level conformal blocks. We derive formulae for their intersections with a basis of 1-cycles, show that they form a basis for the Sn-invariant Picard group, and generate a full-dimensional subcone of the Sn-invariant nef cone. We find their position in the nef cone and study their associated morphisms.

The most basic structure of chiral conformal field theory (CFT) is the Verlinde ring. Freed-Hopkins-Teleman have expressed the Verlinde ring for the CFT's associated to loop groups, as twisted equivariant K-theory. We build on their work to express K-theoretically the structures of full CFT. In particular, the modular invariant partition functions (which essentially parametrise the possible full CFTs) have a rich interpretation within von Neumann algebras (subfactors), which has led to the developments of structures of full CFT such as the full system (fusion ring of defect lines), nimrep (cylindrical partition function), alpha-induction etc. Modular categorical interpretations for these have followed. For the generic families of modular invariants (i.e. those associated to Dynkin diagram symmetries), we provide a K-theoretic framework for these other CFT structures, and show how they relate to D-brane charges and charge-groups. We also study conformal embeddings and the modular invariant of SU(2), as well as some families of finite group doubles. This new K-theoretic framework allows us to simplify and extend the less transparent, more ad hoc descriptions of these structures obtained previously within CFT.

Twenty-five years ago, Conway and Norton published in this journal their remarkable paper ‘Monstrous Moonshine’, proposing a completely unexpected relationship between finite simple groups and modular functions. We review the progress made in broadening and understanding that relationship.

We begin by reviewing Monstrous Moonshine. The impact of Moonshine on algebra has been profound, but so far it has had little to teach number theory. We introduce (using ‘postcards’) a much larger context in which Monstrous Moonshine naturally sits. This context suggests Moonshine should indeed have consequences for number theory. We provide some humble examples of this: new generalisations of Gauss sums and quadratic reciprocity.

We give some higher dimensional analogues of the Durfee square formula and point out their relation to dissections of multipartitions. We apply the results to write certain affine Lie algebra characters in terms of Universal Chiral Partition Functions.

The ‘1-loop partition function’ of a rational conformal field theory is a sesquilinear combination of characters, invariant under a natural action of $\text{S}{{\text{L}}_{2}}(\mathbb{Z})$, and obeying an integrality condition. Classifying these is a clearly defined mathematical problem, and at least for the affine Kac-Moody algebras tends to have interesting solutions. This paper finds for each affine algebra $B_{r}^{\left( 1 \right)}$ and $D_{r}^{(1)}$ all of these at level $k\le 3$. Previously, only those at level 1 were classified. An extraordinary number of exceptionals appear at level 2—the $B_{r}^{(1)},D_{r}^{(1)}$ level 2 classification is easily the most anomalous one known and this uniqueness is the primary motivation for this paper. The only level 3 exceptionals occur for $B_{2}^{(1)}\cong C_{2}^{(1)}$ and $D_{7}^{(1)}$. The ${{B}_{2,3}}$ and ${{D}_{7,3}}$ exceptionals are cousins of the ${{\varepsilon }_{6}}$-exceptional and ${{\varepsilon }_{8}}$-exceptional, respectively, in the $\text{A-D-E}$ classification for $A_{1}^{(1)}$, while the level 2 exceptionals are related to the lattice invariants of affine $u(1)$.