Related papers: W-algebras, Gaussian Free Fields and $\mathfrak{g}…

In this thesis steps are taken in the direction of formulating non-critical strings in the framework of the $G/G$ approach. A major part of the thesis is concerned with conformal field theory based on affine $SL(2)$ current algebra, in…

In these notes we consider integrable structure of the conformal field theory with the algebra of symmetries $\mathcal{A}=W_{n}\otimes H$, where $W_{n}$ is $W-$algebra and $H$ is Heisenberg algebra. We found the system of commuting…

We develop a general theory of $W$-algebras in the context of supersymmetric vertex algebras. We describe the structure of $W$-algebras associated with odd nilpotent elements of Lie superalgebras in terms of their free generating sets. As…

Using the invariant theory of arc spaces, we find minimal strong generating sets for certain cosets of affine vertex algebras inside free field algebras that are related to classical Howe duality. These results have several applications.…

In Section 1 we review various equivalent definitions of a vertex algebra V. The main novelty here is the definition in terms of an indefinite integral of the lambda-bracket. In Section 2 we construct, in the most general framework, the Zhu…

Let $\mathfrak{g}$ be a simple finite-dimensional Lie superalgebra with a non-degenerate supersymmetric even invariant bilinear form, $f$ a nilpotent element in the even part of $\mathfrak{g}$, $\Gamma$ a good grading of $\mathfrak{g}$ for…

Feigin-Frenkel duality is the isomorphism between the principal $\mathcal{W}$-algebras of a simple Lie algebra $\mathfrak{g}$ and its Langlands dual Lie algebra ${}^L\mathfrak{g}$. A generalization of this duality to a larger family of…

We develop a method of quantization for free field theories on manifolds with boundary where the bulk theory is topological in the direction normal to the boundary and a local boundary condition is imposed. Our approach is within the…

We continue the study of the Hrushovski-Kazhdan integration theory and consider exponential integrals. The Grothendieck ring is enlarged via a tautological additive character and hence can receive such integrals. We then define the Fourier…

Consider a doubly-infinite array of iid centered variables with moment conditions, from which one can extract a finite number of rectangular, overlapping submatrices, and form the corresponding Wishart matrices. We show that under basic…

We review the recent development in the representation theory of the $W_{1+\infty}$ algebra. The topics that we concern are, Quasifinite representation, Free field realizations, (Super) Matrix Generalization, Structure of subalgebras such…

We formulate a $q$-Schur algebra associated to an arbitrary $W$-invariant finite set $X_{\texttt f}$ of integral weights for a complex simple Lie algebra with Weyl group $W$. We establish a $q$-Schur duality between the $q$-Schur algebra…

FL$_\mathrm{ew}$-algebras form the algebraic semantics of the full Lambek calculus with exchange and weakening. We investigate two relations, called satisfiability and positive satisfiability, between FL$_\mathrm{ew}$-terms and…

Certain integrable hierarchies appearing in random matrix theory, enumerative geometry, and conformal field theory are governed by Virasoro/$W$-algebra constraints and their $W$-representations.Motivated by the Gaussian Hermitian…

We prove that when Kontsevich's deformation quantization is applied on weight homogeneous Poisson structures, the operators in the $\ast-$ product formula are weight homogeneous. We then consider the linear Poisson case…

We show that the Zamolodchikov's and Polyakov-Bershadsky nonlinear algebras $W_3$ and $W_3^{(2)}$ can be embedded as subalgebras into some {\em linear} algebras with finite set of currents. Using these linear algebras we find new field…

The symmetric algebra g (denoted S(\g)) over a Lie algebra \g (frak g) has the structure of a Poisson algebra. Assume \g is complex semi-simple. Then results of Fomenko- Mischenko (translation of invariants) and A.Tarasev construct a…

A new parafermionic algebra associated with the homogeneous space $A^{(2)}_2/U(1)$ and its corresponding $Z$-algebra have been recently proposed. In this paper, we give a free boson representation of the $A^{(2)}_2$ parafermion algebra in…

A class of classical affine W-algebras are shown to be isomorphic as differential algebras to the coordinate rings of double coset spaces of certain prounipotent proalgebraic groups. As an application, integrable Hamiltonian hierarchies…

We survey a number of results regarding the representation theory of $W$-algebras and their connection with the resent development of the four dimensional $N=2$ superconformal field theories in physics.