We develop techniques to construct isomorphisms between simple affine W-algebras and affine vertex algebras at admissible levels. We then apply these techniques to obtain many new, and conjecturally all, admissible collapsing levels for affine W-algebras. In short, if a simple affine W-algebra at a given level is equal to its affine vertex subalgebra generated by the centraliser of an ${\mathfrak {sl}}_2$-triple associated with the underlying nilpotent orbit, then that level is said to be collapsing. Collapsing levels are important both in representation theory and in theoretical physics. Our approach relies on two fundamental invariants of vertex algebras. The first one is the associated variety, which, in the context of admissible level simple affine W-algebras, leads to the Poisson varieties known as nilpotent Slodowy slices. We exploit the singularities of these varieties to detect possible collapsing levels. The second invariant is the asymptotic datum. We prove a general result asserting that, under appropriate hypotheses, equality of asymptotic data implies isomorphism at the level of vertex algebras. Then we use this to give a sufficient criterion, of combinatorial nature, for an admissible level to be collapsing. Our methods also allow us to study isomorphisms between quotients of W-algebras and extensions of simple affine vertex algebras at admissible levels. Based on such examples, we are led to formulate a general conjecture: for any finite extension of vertex algebras, the induced morphism between associated Poisson varieties is dominant.

We obtain a new interpretation of the cohomological Hall algebra $\mathcal {H}_Q$ of a symmetric quiver Q in the context of the theory of vertex algebras. Namely, we show that the graded dual of $\mathcal {H}_Q$ is naturally identified with the underlying vector space of the principal free vertex algebra associated to the Euler form of Q. Properties of that vertex algebra are shown to account for the key results about $\mathcal {H}_Q$. In particular, it has a natural structure of a vertex bialgebra, leading to a new interpretation of the product of $\mathcal {H}_Q$. Moreover, it is isomorphic to the universal enveloping vertex algebra of a certain vertex Lie algebra, which leads to a new interpretation of Donaldson–Thomas invariants of Q (and, in particular, re-proves their positivity). Finally, it is possible to use that vertex algebra to give a new interpretation of CoHA modules made of cohomologies of non-commutative Hilbert schemes.

In this paper, we classify simple smooth modules over the mirror Heisenberg–Virasoro algebra ${\mathfrak {D}}$, and simple smooth modules over the twisted Heisenberg–Virasoro algebra $\bar {\mathfrak {D}}$ with non-zero level. To this end we generalize Sugawara operators to smooth modules over the Heisenberg algebra, and develop new techniques. As applications, we characterize simple Whittaker modules and simple highest weight modules over ${\mathfrak {D}}$. A vertex-algebraic interpretation of our result is the classification of simple weak twisted and untwisted modules over the Heisenberg–Virasoro vertex algebras. We also present a few examples of simple smooth ${\mathfrak {D}}$-modules and $\bar {\mathfrak {D}}$-modules induced from simple modules over finite dimensional solvable Lie algebras, that are not tensor product modules of Virasoro modules and Heisenberg modules. This is very different from the case of simple highest weight modules over $\mathfrak {D}$ and $\bar {\mathfrak {D}}$ which are always tensor products of simple Virasoro modules and simple Heisenberg modules.

We give a lattice theoretical interpretation of generalized deep holes of the Leech lattice VOA $V_\Lambda $. We show that a generalized deep hole defines a ‘true’ automorphism invariant deep hole of the Leech lattice. We also show that there is a correspondence between the set of isomorphism classes of holomorphic VOA V of central charge $24$ having non-abelian $V_1$ and the set of equivalence classes of pairs $(\tau , \tilde {\beta })$ satisfying certain conditions, where $\tau \in Co.0$ and $\tilde {\beta }$ is a $\tau $-invariant deep hole of squared length $2$. It provides a new combinatorial approach towards the classification of holomorphic VOAs of central charge $24$. In particular, we give an explanation for an observation of G. Höhn, which relates the weight one Lie algebras of holomorphic VOAs of central charge $24$ to certain codewords associated with the glue codes of Niemeier lattices.

In this work, we introduce Urod algebras associated to simply laced Lie algebras as well as the concept of translation of W-algebras.

Both results are achieved by showing that the quantum Hamiltonian reduction commutes with tensoring with integrable representations; that is, for V and L an affine vertex algebra and an integrable affine vertex algebra associated with $\mathfrak {g}$, we have the vertex algebra isomorphism $H_{DS,f}^{0}(V\otimes L)\cong H_{DS,f}^{0}(V)\otimes L$, where in the left-hand-side the Drinfeld–Sokolov reduction is taken with respect to the diagonal action of $\widehat {\mathfrak {g}}$ on $V{\otimes } L$.

The proof is based on some new construction of automorphisms of vertex algebras, which may be of independent interest. As corollaries, we get fusion categories of modules of many exceptional W-algebras, and we can construct various corner vertex algebras.

A major motivation for this work is that Urod algebras of type A provide a representation theoretic interpretation of the celebrated Nakajima–Yoshioka blowup equations for the moduli space of framed torsion free sheaves on $\mathbb {CP}^{2}$ of an arbitrary rank.

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 observe that every self-dual ternary code determines a holomorphic $\mathcal N=1$ superconformal field theory. This provides ternary constructions of some well-known holomorphic $\mathcal N=1$ superconformal field theories (SCFTs), including Duncan’s “supermoonshine” model and the fermionic “beauty and the beast” model of Dixon, Ginsparg, and Harvey. Along the way, we clarify some issues related to orbifolds of fermionic holomorphic CFTs. We give a simple coding-theoretic description of the supersymmetric index and conjecture that for every self-dual ternary code this index is divisible by $24$ ; we are able to prove this conjecture except in the case when the code has length $12$ mod $24$ . Lastly, we discuss a conjecture of Stolz and Teichner relating $\mathcal N=1$ SCFTs with Topological Modular Forms. This conjecture implies constraints on the supersymmetric indexes of arbitrary holomorphic SCFTs, and suggests (but does not require) that there should be, for each k, a holomorphic $\mathcal N=1$ SCFT of central charge $12k$ and index $24/\gcd (k,24)$ . We give ternary code constructions of SCFTs realizing this suggestion for $k\leq 5$ .

We prove the longstanding physics conjecture that there exists a unique two-parameter ${\mathcal {W}}_{\infty }$-algebra which is freely generated of type ${\mathcal {W}}(2,3,\ldots )$, and generated by the weights $2$ and $3$ fields. Subject to some mild constraints, all vertex algebras of type ${\mathcal {W}}(2,3,\ldots , N)$ for some $N$ can be obtained as quotients of this universal algebra. As an application, we show that for $n\geq 3$, the structure constants for the principal ${\mathcal {W}}$-algebras ${\mathcal {W}}^k({\mathfrak s}{\mathfrak l}_n, f_{\text {prin}})$ are rational functions of $k$ and $n$, and we classify all coincidences among the simple quotients ${\mathcal {W}}_k({\mathfrak s}{\mathfrak l}_n, f_{\text {prin}})$ for $n\geq 2$. We also obtain many new coincidences between ${\mathcal {W}}_k({\mathfrak s}{\mathfrak l}_n, f_{\text {prin}})$ and other vertex algebras of type ${\mathcal {W}}(2,3,\ldots , N)$ which arise as cosets of affine vertex algebras or nonprincipal ${\mathcal {W}}$-algebras.

For ${\mathcal{C}}$ a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show:

1. (1) ${\mathcal{C}}$ always contains a simple projective object;

2. (2) if ${\mathcal{C}}$ is in addition ribbon, the internal characters of projective modules span a submodule for the projective $\text{SL}(2,\mathbb{Z})$-action;

3. (3) the action of the Grothendieck ring of ${\mathcal{C}}$ on the span of internal characters of projective objects can be diagonalised;

4. (4) the linearised Grothendieck ring of ${\mathcal{C}}$ is semisimple if and only if ${\mathcal{C}}$ is semisimple.

Results (1)–(3) remain true in positive characteristic under an extra assumption. Result (1) implies that the tensor ideal of projective objects in ${\mathcal{C}}$ carries a unique-up-to-scalars modified trace function. We express the modified trace of open Hopf links coloured by projectives in terms of $S$-matrix elements. Furthermore, we give a Verlinde-like formula for the decomposition of tensor products of projective objects which uses only the modular $S$-transformation restricted to internal characters of projective objects. We compute the modified trace in the example of symplectic fermion categories, and we illustrate how the Verlinde-like formula for projective objects can be applied there.

We prove duality isomorphisms of certain representations of ${\mathcal{W}}$-algebras which play an essential role in the quantum geometric Langlands program and some related results.

Under the local Langlands correspondence, the conductor of an irreducible representation of $\text{G}{{\text{l}}_{n}}\left( F \right)$ is greater than the Swan conductor of the corresponding Galois representation. In this paper, we establish the geometric analogue of this statement by showing that the conductor of a categorical representation of the loop group is greater than the irregularity of the corresponding meromorphic connection.

We consider a lifting of Joseph ideals for the minimal nilpotent orbit closure to the setting of affine Kac–Moody algebras and find new examples of affine vertex algebras whose associated varieties are minimal nilpotent orbit closures. As an application we obtain a new family of lisse ($C_{2}$-cofinite) $W$-algebras that are not coming from admissible representations of affine Kac–Moody algebras.

We exhibit an action of Conway’s group – the automorphism group of the Leech lattice – on a distinguished super vertex operator algebra, and we prove that the associated graded trace functions are normalized principal moduli, all having vanishing constant terms in their Fourier expansion. Thus we construct the natural analogue of the Frenkel–Lepowsky–Meurman moonshine module for Conway’s group. The super vertex operator algebra we consider admits a natural characterization, in direct analogy with that conjectured to hold for the moonshine module vertex operator algebra. It also admits a unique canonically twisted module, and the action of the Conway group naturally extends. We prove a special case of generalized moonshine for the Conway group, by showing that the graded trace functions arising from its action on the canonically twisted module are constant in the case of Leech lattice automorphisms with fixed points, and are principal moduli for genus-zero groups otherwise.

A representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral ${{\mathbb{Z}}_{2}}$-lattice. The irreducible decomposition of the representation is explicitly computed and described. As a by-product, some fundamental representations of affine Kac–Moody Lie algebra of type $A_{n}^{\left( 2 \right)}$ are recovered by the new method.

Simple, or Kleinian, singularities are classified by Dynkin diagrams of type $ADE$. Let $\mathfrak {g}$ be the corresponding finite-dimensional Lie algebra, and $W$ its Weyl group. The set of $\mathfrak {g}$-invariants in the basic representation of the affine Kac–Moody algebra $\hat {\mathfrak {g}}$ is known as a $\mathcal {W}$-algebra and is a subalgebra of the Heisenberg vertex algebra $\mathcal {F}$. Using period integrals, we construct an analytic continuation of the twisted representation of $\mathcal {F}$. Our construction yields a global object, which may be called a $W$-twisted representation of $\mathcal {F}$. Our main result is that the total descendant potential of the singularity, introduced by Givental, is a highest-weight vector for the $\mathcal {W}$-algebra.

We present a superfield formulation of the chiral de Rham complex (CDR), as introduced by Malikov, Schechtman and Vaintrob in 1999, in the setting of a general smooth manifold, and use it to endow CDR with superconformal structures of geometric origin. Given a Riemannian metric, we construct an N=1 structure on CDR (action of the N=1 super-Virasoro, or Neveu–Schwarz, algebra). If the metric is Kähler, and the manifold Ricci-flat, this is augmented to an N=2 structure. Finally, if the manifold is hyperkähler, we obtain an N=4 structure. The superconformal structures are constructed directly from the Levi-Civita connection. These structures provide an analog for CDR of the extended supersymmetries of nonlinear σ-models.

We study the Fourier coefficients of generalized modular forms f(τ) of integral weight k on subgroups Γ of finite index in the modular group. We establish two Theorems asserting that f(τ) is constant if k = 0, f(τ) has empty divisor, and the Fourier coefficients have certain rationality properties. (The result is false if the rationality assumptions are dropped.) These results are applied to the case that f(τ) has a cuspidal divisor, k is arbitrary, and Γ = Γ0(N), where we show that f(τ) is modular, indeed an eta-quotient, under natural rationality assumptions on the Fourier coefficients. We also explain how these results apply to the theory of orbifold vertex operator algebras.

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.

The Jucys–Murphy elements for wreath products $\Gamma_n = \Gamma \wr S_n $ associated to any finite group $\Gamma$ are introduced and they play an important role in our study of the connections between class algebras of $\Gamma_n$ for all $n$ and vertex algebras. We construct an action of (a variant of) the $\mathcal W_{1 + \infty}$-algebra acting irreducibly on the direct sum ${\mathcal R}_\Gamma$ of the class algebras of $\Gamma_n$ for all $n$ in a group-theoretic manner. We establish various relations between convolution operators using JM elements and Heisenberg algebra operators acting on ${\mathcal R}_\Gamma$. As applications, we obtain two distinct sets of algebra generators for the class algebra of $\Gamma_n$ and establish various stability results concerning products of normalized conjugacy classes of $\Gamma_n$ and the power sums of Jucys--Murphy elements, etc. We introduce a stable algebra which encodes the class algebra structures of $\Gamma_n$ for all $n$, whose structure constants are shown to be non-negative integers. In the symmetric group case (that is, when $\Gamma$ is trivial), we recover and strengthen in a uniform approach various results of Lascoux and Thibon, Kerov and Olshanski, and Farahat and Higman, etc.

We classify the subalgebras of the general Lie conformal algebra $\text{g}{{\text{c}}_{N}}$ that act irreducibly on $\mathbb{C}\,{{\left[ \partial \right]}^{N}}$ and that are normalized by the $\text{s}{{\text{l}}_{2}}$-part of a Virasoro element. The problem turns out to be closely related to classical Jacobi polynomials $P_{n}^{\left( -\sigma ,\sigma \right)}$, $\sigma \,\in \,C$. The connection goes both ways—we use in our classification some classical properties of Jacobi polynomials, and we derive from the theory of conformal algebras some apparently new properties of Jacobi polynomials.