scholarly journals Vertex Algebras Associated with Hypertoric Varieties

2020 ◽  
Author(s):  
Toshiro Kuwabara
Keyword(s):  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Vertex Algebra ◽  
Hamiltonian Reduction ◽  
Vertex Algebras ◽  
Torus Action ◽  
Level 1 ◽  
Hypertoric Varieties ◽  
Construction Works

Abstract We construct a family of vertex algebras associated with a family of symplectic singularity/resolution, called hypertoric varieties. While the hypertoric varieties are constructed by a certain Hamiltonian reduction associated with a torus action, our vertex algebras are constructed by (semi-infinite) BRST reduction. The construction works algebro-geometrically, and we construct sheaves of $\hbar $-adic vertex algebras over hypertoric varieties, which localize the vertex algebras. We determine when it is a vertex operator algebra by giving an explicit conformal vector. We also discuss the Zhu algebra of the vertex algebra and its relation with a quantization of the hypertoric variety. In certain cases, we obtain the affine ${\mathcal{W}}$-algebra associated with the subregular nilpotent orbit in $\mathfrak{s}\mathfrak{l}_N$ at level $N-1$ and simple affine vertex operator algebra for $\mathfrak{s}\mathfrak{l}_N$ at level $-1$.

Mathieu–Zhao subspaces of vertex algebras

2019 ◽  
Vol 18 (12) ◽  
pp. 1950225
Author(s):  
Gaywalee Yamskulna
Keyword(s):  
Associative Algebra ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Quotient Space ◽  
Vertex Algebra ◽  
Jacobian Conjecture ◽  
Vertex Algebras ◽  
Text Type ◽  
Commutative Associative Algebra

We introduce a notion of Mathieu–Zhao subspaces of vertex algebras. Among other things, we show that for a vertex algebra [Formula: see text] and its subspace [Formula: see text] that contains [Formula: see text], [Formula: see text] is a Mathieu–Zhao subspace of [Formula: see text] if and only if the quotient space [Formula: see text] is a Mathieu–Zhao subspace of a commutative associative algebra [Formula: see text]. As a result, one can study the famous Jacobian conjecture in terms of Mathieu–Zhao subspaces of vertex algebras. In addition, for a [Formula: see text]-type vertex operator algebra [Formula: see text] that satisfies the [Formula: see text]-cofiniteness condition, we classify all Mathieu–Zhao subspaces [Formula: see text] that contain [Formula: see text].

scholarly journals CONSTRUCTING QUANTUM VERTEX ALGEBRAS

2006 ◽  
Vol 17 (04) ◽  
pp. 441-476 ◽  
Author(s):  
HAISHENG LI
Keyword(s):  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Operator Algebras ◽  
Vertex Algebra ◽  
Vertex Operator Algebras ◽  
Vertex Algebras ◽  
Quantum Vertex

This is a sequel to [23]. In this paper, we focus on the construction of quantum vertex algebras over ℂ, whose notion was formulated in [23] with Etingof and Kazhdan's notion of quantum vertex operator algebra (over ℂ[[h]]) as one of the main motivations. As one of the main steps in constructing quantum vertex algebras, we prove that every countable-dimensional nonlocal (namely, noncommutative) vertex algebra over ℂ, which either is irreducible or has a basis of PBW type, is nondegenerate in the sense of Etingof and Kazhdan. Using this result, we establish the nondegeneracy of better known vertex operator algebras and some newly constructed nonlocal vertex algebras. We construct a family of quantum vertex algebras closely related to Zamolodchikov–Faddeev algebras.

scholarly journals A LOGARITHMIC GENERALIZATION OF TENSOR PRODUCT THEORY FOR MODULES FOR A VERTEX OPERATOR ALGEBRA

2006 ◽  
Vol 17 (08) ◽  
pp. 975-1012 ◽  
Author(s):  
YI-ZHI HUANG ◽  
JAMES LEPOWSKY ◽  
LIN ZHANG
Keyword(s):  
Tensor Product ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Vertex Algebra ◽  
Intertwining Operators

We describe a logarithmic tensor product theory for certain module categories for a "conformal vertex algebra". In this theory, which is a natural, although intricate, generalization of earlier work of Huang and Lepowsky, we do not require the module categories to be semisimple, and we accommodate modules with generalized weight spaces. The corresponding intertwining operators contain logarithms of the variables.

scholarly journals SEMI-INFINITE FORMS AND TOPOLOGICAL VERTEX OPERATOR ALGEBRAS

2000 ◽  
Vol 02 (02) ◽  
pp. 191-241 ◽  
Author(s):  
YI-ZHI HUANG ◽  
WENHUA ZHAO
Keyword(s):  
Moduli Spaces ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Riemann Surfaces ◽  
Vertex Operator Algebra ◽  
Operator Algebras ◽  
Vertex Algebra ◽  
Vertex Operator Algebras ◽  
Topological Vertex ◽  
Type K

Semi-infinite forms on the moduli spaces of genus-zero Riemann surfaces with punctures and local coordinates are introduced. A partial operad of semi-infinite forms is constructed. Using semi-infinite forms and motivated by a partial suboperad of the partial operad of semi-infinite forms, topological vertex partial operads of type k<0 and strong topological vertex partial operads of type k<0 are constructed. It is proved that the category of (locally-)grading-restricted (strong) topological vertex operator algebras of type k<0 and the category of (weakly) meromorphic ℤ×ℤ-graded algebras over the (strong) topological vertex partial operad of type k are isomorphic. As an application of this isomorphism theorem, the following conjecture of Lian-Zuckerman and Kimura-Voronov-Zuckerman is proved: A strong topological vertex operator algebra gives a (weak) homotopy Gerstenhaber algebra. These results hold in particular for the tensor product of the moonshine module vertex operator algebra, the vertex algebra constructed from a rank 2 Lorentz lattice and the ghost vertex operator algebra, studied in detail first by Lian and Zuckerman.

Extension of Vertex Operator Algebra $V_{\widehat{H}_{4}}(\ell,0)$

2014 ◽  
Vol 21 (03) ◽  
pp. 361-380 ◽  
Author(s):  
Cuipo Jiang ◽  
Song Wang
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Vertex Algebra ◽  
Irreducible Representations ◽  
Natural Conditions ◽  
Irreducible Modules ◽  
Heisenberg Algebras

We classify the irreducible restricted modules for the affine Nappi-Witten Lie algebra [Formula: see text] with some natural conditions. It turns out that the representation theory of [Formula: see text] is quite different from the theory of representations of Heisenberg algebras. We also study the extension of the vertex operator algebra [Formula: see text] by the even lattice L. We give the structure of the extension [Formula: see text] and its irreducible modules via irreducible representations of [Formula: see text] viewed as a vertex algebra.

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