Related papers: Virasoro Algebra, Dedekind $\eta$-function and Spe…

We describe a categorification of the Double Affine Hecke Algebra (${\mathcal{H}\kern -.4em\mathcal{H}}$) associated with an affine Lie algebra $\widehat{\mathfrak{g}}$, including a categorification of the polynomial representation and…

Using certain results for the vertex operator algebras associated with affine Lie algebras we obtain recurrence relations for the characters of integrable highest weight irreducible modules for an affine Lie algebra. As an application we…

Suppose that $\ell \geq 5$ is prime. For a positive integer $N$ with $4 \mid N$, previous works studied properties of half-integral weight modular forms on $\Gamma_0(N)$ which are supported on finitely many square classes modulo $\ell$, in…

The famous Rogers-Ramanujan and Andrews--Gordon identities are embedded in a doubly-infinite family of Rogers-Ramanujan-type identities labelled by positive integers m and n. For fixed m and n the product side corresponds to a specialised…

We prove the unbounded denominators conjecture in the theory of noncongruence modular forms for finite index subgroups of SL_2(Z). Our result includes also Mason's generalization of the original conjecture to the setting of vector-valued…

Whenever the group $\R^n$ acts on an algebra $\calA$, there is a method to twist $\cal A$ to a new algebra $\calA_\theta$ which depends on an antisymmetric matrix $\theta$ ($\theta^{\mu \nu}=-\theta^{\nu \mu}=\mathrm{constant}$). The…

In this thesis, we obtain the formula for the Kac determinant of the algebra arising from the level $N$ representation of the Ding-Iohara-Miki algebra. This formula can be proved by decomposing the level $N$ representation into the deformed…

The paper is devoted to the derivation of the expansion formula for the powers of the Euler Product in terms of partition hook lengths, discovered by Nekrasov and Okounkov in their study of the Seiberg-Witten Theory. We provide a refinement…

We show that modular forms of fractional weights on principal congruence subgroups of odd levels, which are found by T. Ibukiyama, naturally appear as characters being multiplied $\eta^{c_{\text{eff}}}$ of the so-called minimal models of…

The Macdonald operator is known to coincide with a certain element of the quantum toroidal $\mathfrak{gl}(1)$ algebra in the Fock representation of levels $(1,0)$. A generalization of this operator to higher levels $(r,0)$ can be built…

We associate certain curves over function fields to given algebraic power series and show that bounds on the rank of Kodaira-Spencer map of this curves imply bounds on the exponents of the power series, with more generic curves giving lower…

We show that the so called $\lambda$ deformed $\sigma$-model as well as the $\eta$ deformed one belong to a class of the ${\cal E}$-models introduced in the context of the Poisson-Lie-T-duality. The $\lambda$ and $\eta$ theories differ…

A $(q,t)$-deformation of the 2d Toda integrable hierarchy is introduced by enhancing the underlying symmetry algebra $\mathfrak{gl}(\infty)\simeq \text{q-W}_{1+\infty}$ to the quantum toroidal $\mathfrak{gl}(1)$ algebra. The…

The wonderful formulas by I.Dumitriu and A.Edelman rewrite $\beta$-ensemble, with eigenvalue integrals containing Vandermonde factors in the power $2\beta$, through integrals over tridiagonal matrices, where $\beta$-dependent are the powers…

We give an explicit formula for an operator that sends a wreath Macdonald polynomial to the delta function at a character associated to its partition. This allows us to prove many new results for wreath Macdonald polynomials, especially…

We consider the algebra $\mathcal O(\mathsf M)$ of observables and the (formally) versal morphism $\eta: A \to \mathcal O(\mathsf M)$ defined by the noncommutative deformation functor $\mathsf{Def}_{\mathsf M}$ of a family $\mathsf M = \{…

We interpret aspects of the Schur indices, that were identified with characters of highest weight modules in Virasoro $(p,p')=(2,2k+3)$ minimal models for $k=1,2,\dots$, in terms of paths that first appeared in exact solutions in…

We extend the recently developed theory of Roehrig and Zwegers on indefinite theta functions to prove certain power series are modular forms. As a consequence, we obtain several power series identities for powers of the generating function…

We study zeta functions enumerating submodules invariant under a given endomorphism of a finitely generated module over the ring of ($S$-)integers of a number field. In particular, we compute explicit formulae involving Dedekind zeta…

We continue our study of the AGT correspondence between instanton counting on C^2/Z_p and Conformal field theories with the symmetry algebra A(r,p). In the cases r=1, p=2 and r=2, p=2 this algebra specialized to: A(1,2)=H+sl(2)_1 and…