Related papers: Almost harmonic Maass forms and Kac-Wakimoto chara…
We give a new interpretation and proof of the "quasi-particle" type character formulas for integrable representations of the simply-laced affine Kac-Moody algebras through a new "semi-infinite" construction of such representations. We…
We consider several vertex operator (super)algebras closely related to $V_{-1}(\frak{sl} (n) )$, $n \ge 3$ : (a) the parafermionic subalgebra $K(\frak{sl}(n),-1)$ for which we completely describe its inner structure, (b) the vacuum algebra…
The relationship between the quasi-exactly solvable problems and W-algebras is revealed. This relationship enabled one to formulate a new general method for building multi-dimensional and multi-channel exactly and quasi-exactly solvable…
We interpolate the Gauss-Manin connection in p-adic families of nearly overconvergent modular forms. This gives a family of Maass-Shimura type differential operators from the space of nearly overconvergent modular forms of type r to the…
We discuss polyharmonic Maass forms of even integer weight on PSL(2, Z)\H, which are a generalization of classical Maass forms. We explain the role of real-analytic Eisenstein series E_k(z, s) and the differential operator…
Modules over affine Lie superalgebras ${\cal G}$ are studied, in particular, for ${\cal G}=\hat{OSP(1,2)}$. It is shown that on studying Verma modules, much of the results in Kac-Moody algebra can be generalized to the super case. Of most…
We revisit the study of the multiplets of the conformal algebra in any dimension. The theory of highest weight representations is reviewed in the context of the Bernstein-Gelfand-Gelfand category of modules. The Kazhdan-Lusztig polynomials…
We generalize the results of [KMST] concerning equivariant quantization by means of Verma modules $M(\lambda)$ for generic weight $\lambda$ to the case of general $\lambda$. We consider the relationship between the Shapovalov form on an…
Rational conformal field theories in 2d have partition functions built from holomorphic characters, whose classification can be addressed via the holomorphic modular bootstrap. This is facilitated by a special basis of ``quasi-characters''…
We determine the size of spaces of higher order Maass forms of even weight for cofinite discrete subgroups of PSL(2,R) with cusps. If exponential growth at the cusps is allowed, the spaces of Maass forms of a given order are as large as…
Let $M$ be a compact orientable 3-manifold. The set of characters of $SL_2({\mathbb C})$ representations of the fundamental group of $M$ forms a closed affine algebraic set. We show that its coordinate ring is isomorphic to a specialization…
In this paper we review some connections between harmonic analysis and the modern theory of automorphic forms. We indicate in some examples how the study of problems of harmonic analysis brings us to the important objects of the theory of…
This is the second of a series of articles devoted to the study of relaxed highest-weight modules over affine vertex algebras and W-algebras. The first studied the simple "rank-$1$" affine vertex superalgebras $L_k(\mathfrak{sl}_2)$ and…
We extend our previous work arXiv:1012.5721 [hep-th] on the non-compact N=2 SCFT_2 defined as the supersymmetric SL(2,R)/U(1)-gauged WZW model. Starting from path-integral calculations of torus partition functions of both the axial-type…
Inspired by a series of conjectures related to higher coinvariant algebras, we present two families of relations involving harmonic differential forms of the symmetric group. Our relations, together with a novel bijection, are sufficient to…
Let $w$ be an unbounded radial weight on the complex plane. We study the following approximation problem: find a proper holomorphic map $f: \mathbb{C}\to\mathbb{C}^n$ such that $|f|$ is equivalent to $w$. We give several characterizations…
We present the Wakimoto construction of the super OSp(1,2) and SL(2,1) Kac-Moody algebras and the free field representation of the corresponding WZW models. After imposing suitable constraints, we can lead the Feigin-Fuchs representation of…
In a recent paper, Bruinier and Ono prove that the coefficients of certain weight -1/2 harmonic Maass forms are traces of singular moduli for weak Maass forms. In particular, for the partition function $p(n)$, they prove that…
We introduce a class of modules over Kac-Moody superalgebras; we call these modules snowflake. These modules are characterized by invariance property of their characters with respect to a certain subgroup of the Weyl group. Examples of…
We prove homological stability for both general linear groups of modules over a ring with finite stable rank and unitary groups of quadratic modules over a ring with finite unitary stable rank. In particular, we do not assume the modules…