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Victor G. Kac

Publications and source records attributed to Victor G. Kac.

At least 19 recordsLinked to original sources
arXiv2026

Spectral flow and application to unitarity of representations of minimal $W$-algebras

Using spectral flow, we provide a proof of [11, Theorem 9.17] on unitarity of Ramond twisted non-extremal representations of unitary minimal $W$-algebras that does not rely on the still conjectural exactness of the twisted quantum reduction functor (see Conjecture 9.11 of [11]). When $\mathfrak g = spo(2|2n)$, $F (4$), $D(2, 1; \frac{m}{n})$, it is also proven that the unitarity of extremal (=massless) representations of the unitary minimal $W$-algebra $W^k_{\min}(\mathfrak g)$ in the Ramond sector is equivalent to the unitarity of extremal representations in the Neveu-Schwarz sector.

math.RT↗
arXiv2026

Extremal unitary representations of big $N=4$ superconformal algebra

In this paper we give a detailed proof of the classification of extremal (=massless) unitary highest weight representations in the Neveu Schwarz and Ramond sectors of the big $N=4$ superconformal algebra which can be found in [5]. Our results agree with the general conjectures about classification of unitary highest weight representation of minimal $W$-algebras attached to basic Lie superalgebras formulated in [10], [11], and complete their proof for the big $N=4$ superconformal algebra.

math.RT↗
arXiv2025

Unitarity of minimal $W$-algebras and their representations II: Ramond sector

In this paper we study unitary Ramond twisted representations of minimal $W$-algebras. We classify all such irreducible highest weight representations with a non-Ramond extremal highest weight (unitarity in the Ramond extremal case, as well as in the untwisted extremal case, remains open). We compute the characters of these representations and deduce from them the denominator identities for all superconformal algebras in the Neveu-Schwarz and Ramond sector. Some of the results rely on conjectures about the properties of the quantum Hamiltonian reduction functor in the Ramond sector.

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arXiv2025

On modular invariance of quantum affine $W$-algebras

We find modular transformations of normalized characters for the following $W$-algebras: (a) $W^{min}_k(\frak{g})$, where $\frak{g}=D_n \, (n \geq 4)$, or $E_6$, $E_7$, $E_8$, and $k$ is a negative integer $\geq -2$, or $\geq -\frac{h^{\vee}}{6}-1$, respectively; (b) quantum Hamiltonian reduction of the $\hat{\frak{g}}$-module $L(kΛ_0)$, where $\frak{g}$ is a simple Lie algebra, $f$ is its non-zero nilpotent element, and $k$ is a principal admissible level with the denominator $u > θ(x)$, where $2x$ is the Dynkin characteristic of $f$ and $θ$ is the highest root of $\frak{g}$. We prove that these vertex algebras are modular invariant. A conformal vertex algebra is called modular invariant if its character $tr_V q^{L_0-c/24}$ converges to a holomorphic modular function in the complex upper half-plane on a congruence subgroup. We find explicit formulas for their characters. Modular invariance of $V$ is important since, in particular, conjecturally it implies that $V$ is simple, and that $V$ is rational, provided that it is lisse.

math.RT↗
arXiv2024

An application of collapsing levels to the representation theory of affine vertex algebras

We discover a large class of simple affine vertex algebras $V_{k} (\mathfrak g)$, associated to basic Lie superalgebras $\mathfrak g$ at non-admissible collapsing levels $k$, having exactly one irreducible $\mathfrak g$-locally finite module in the category ${\mathcal O}$. In the case when $\mathfrak g$ is a Lie algebra, we prove a complete reducibility result for $V_k(\mathfrak g)$-modules at an arbitrary collapsing level. We also determine the generators of the maximal ideal in the universal affine vertex algebra $V^k (\mathfrak g)$ at certain negative integer levels. Considering some conformal embeddings in the simple affine vertex algebras $V_{-1/2} (C_n)$ and $V_{-4}(E_7)$, we surprisingly obtain the realization of non-simple affine vertex algebras of types $B$ and $D$ having exactly one non-trivial ideal.

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arXiv2023

On simplicity of universal minimal W-algebras

Simplicity of universal minimal quantum affine W-algebras is studied. As an application, we find the values of the center, for which the vacuum module over a superconformal algebra is irreducible.

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arXiv2023

Poisson vertex algebras and Hamiltonian PDE

We discuss the theory of Poisson vertex algebras and their generalizations in relation to integrability of Hamiltonian PDE. In particular, we discuss the theory of affine classical W-algebras and apply it to construct a large class of integrable Hamiltonian PDE. We also discuss non commutative Hamiltonian PDE in the framework of double PVA, and differential-difference Hamiltonian equations in the framework of multiplicative PVA.

math-ph↗
arXiv2023

Unitarity of minimal $W$-algebras and their representations I

We begin a systematic study of unitary representations of minimal $W$-algebras. In particular, we classify unitary minimal $W$-algebras and make substantial progress in classification of their unitary irreducible highest weight modules. We also compute the characters of these modules.

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arXiv2022

Orbifolds of Lattice Vertex Algebras

To a positive-definite even lattice $Q$, one can associate the lattice vertex algebra $V_Q$, and any automorphism $σ$ of $Q$ lifts to an automorphism of $V_Q$. In this paper, we investigate the orbifold vertex algebra $V_Q^σ$, which consists of the elements of $V_Q$ fixed under $σ$, in the case when $σ$ has prime order. We describe explicitly the irreducible $V_Q^σ$-modules, compute their characters, and determine the modular transformations of characters. As an application, we find the asymptotic and quantum dimensions of all irreducible $V_Q^σ$-modules. We consider in detail the cases when the order of $σ$ is $2$ or $3$, as well as the case of permutation orbifolds.

math.QA↗
arXiv2022

Adler-Oevel-Ragnisco type operators and Poisson vertex algebras

The theory of triples of Poisson brackets and related integrable systems, based on a classical R-matrix R in End_F(g), where g is a finite dimensional associative algebra over a field F viewed as a Lie algebra, was developed by Oevel-Ragnisco and Li-Parmentier [OR89,LP89]. In the present paper we develop an "affine" analogue of this theory by introducing the notion of a continuous Poisson vertex algebra and constructing triples of Poisson lambda-brackets. We introduce the corresponding Adler type identities and apply them to integrability of hierarchies of Hamiltonian PDEs.

nlin.SI↗
arXiv2021

On Lax operators

We define a Lax operator as a monic pseudodifferential operator $L(\partial)$ of order $N\geq 1$, such that the Lax equations $\dfrac{\partial L(\partial)}{\partial t_k}=[(L^{\frac kN}(\partial))_+,L(\partial)]$ are consistent and non-zero for infinitely many positive integers $k$. Consistency of an equation means that its flow is defined by an evolutionary vector field. In the present paper we demonstrate that the traditional theory of the KP and the $N$-th KdV hierarchies holds for arbitrary scalar Lax operators.

nlin.SI↗
arXiv2021

Invariant Hermitian forms on vertex algebras

We study invariant Hermitian forms on a conformal vertex algebra and on their (twisted) modules. We establish existence of a non-zero invariant Hermitian form on an arbitrary $W$-algebra. We show that for a minimal simple $W$-algebra $W_k(\mathfrak g,θ/2)$ this form can be unitary only when its $\tfrac{1}{2}\mathbb Z$-grading is compatible with parity, unless $W_k(\mathfrak g,θ/2)$ "collapses" to its affine subalgebra.

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arXiv2021

Integrability of classical affine W-algebras

We prove that all classical affine W-algebras W(g,f), where g is a simple Lie algebra and f is its non-zero nilpotent element, admit an integrable hierarchy of bi-Hamiltonian PDEs, except possibly for one nilpotent conjugacy class in G_2, one in F_4, and five in E_8.

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