ADESubalgebras of the Triplet Vertex Algebra đť’˛(p):E6,E7

10.1093/imrn/rnu147 â—˝  
2014 â—˝  
Vol 2015 (15) â—˝  
pp. 6752-6792 â—˝  
Author(s):  
Xianzu Lin
Keyword(s):  
Vertex Algebra

scholarly journals A Vertex Algebra Attached to the Flag Manifold and Lie Algebra Cohomology

10.1063/1.3460161 â—˝  
2010 â—˝  
Author(s):  
T. Arakawa â—˝  
F. Malikov â—˝  
Vladimir Dobrev
Keyword(s):  
Lie Algebra â—˝  
Vertex Algebra â—˝  
Flag Manifold â—˝  
Lie Algebra Cohomology

scholarly journals A note on the zeroth products of Frenkel–Jing operators

2017 â—˝  
Vol 16 (03) â—˝  
pp. 1750053 â—˝  
Author(s):  
Slaven Kožić
Keyword(s):  
Lie Algebra â—˝  
Vertex Operator â—˝  
Vertex Operator Algebra â—˝  
Vertex Algebra â—˝  
Infinite Dimensional â—˝  
Enveloping Algebra â—˝  
Highest Weight â—˝  
Finite Dimensional â—˝  
Quantum Vertex â—˝  
Algebra Theory

Let [Formula: see text] be an untwisted affine Kac–Moody Lie algebra. The top of every irreducible highest weight integrable [Formula: see text]-module is the finite-dimensional irreducible [Formula: see text]-module, where the action of the simple Lie algebra [Formula: see text] is given by zeroth products arising from the underlying vertex operator algebra theory. Motivated by this fact, we consider zeroth products of level [Formula: see text] Frenkel–Jing operators corresponding to Drinfeld realization of the quantum affine algebra [Formula: see text]. By applying these products, which originate from the quantum vertex algebra theory developed by Li, on the extension of Koyama vertex operator [Formula: see text], we obtain an infinite-dimensional vector space [Formula: see text]. Next, we introduce an associative algebra [Formula: see text], a certain quantum analogue of the universal enveloping algebra [Formula: see text], and construct some infinite-dimensional [Formula: see text]-modules [Formula: see text] corresponding to the finite-dimensional irreducible [Formula: see text]-modules [Formula: see text]. We show that the space [Formula: see text] carries a structure of an [Formula: see text]-module and, furthermore, we prove that the [Formula: see text]-module [Formula: see text] is isomorphic to the [Formula: see text]-module [Formula: see text].

Vertex algebra bundles

10.1090/surv/088/06 â—˝  
2004 â—˝  
pp. 99-120
Author(s):  
Edward Frenkel â—˝  
David Ben-Zvi
Keyword(s):  
Vertex Algebra

scholarly journals On fusion rules and intertwining operators for the Weyl vertex algebra

10.1063/1.5098128 â—˝  
2019 â—˝  
Vol 60 (8) â—˝  
pp. 081701 â—˝  
Author(s):  
Dražen Adamović â—˝  
Veronika Pedić
Keyword(s):  
Vertex Algebra â—˝  
Intertwining Operators â—˝  
Fusion Rules

Chiral Hodge Cohomology and Mathieu Moonshine

10.1093/imrn/rnz298 â—˝  
2019 â—˝  
Author(s):  
Bailin Song
Keyword(s):  
Finite Order â—˝  
Unitary Representation â—˝  
Central Charge â—˝  
Vertex Algebra â—˝  
K3 Surface â—˝  
Hodge Cohomology

Abstract We construct a filtration of chiral Hodge cohomolgy of a K3 surface $X$, such that its associated graded object is a unitary representation of the $\mathcal{N}=4$ superconformal vertex algebra with central charge $c=6$ and its subspace of primitive vectors has the property; its equivariant character for a symplectic automorphism $g$ of finite order acting on $X$ agrees with the McKay–Thompson series for $g$ in Mathieu moonshine.

scholarly journals Vertex Lie algebras and cyclotomic coinvariants

2017 â—˝  
Vol 19 (02) â—˝  
pp. 1650015 â—˝  
Author(s):  
Benoît Vicedo â—˝  
Charles Young
Keyword(s):  
Lie Algebra â—˝  
Lie Algebras â—˝  
Bethe Ansatz â—˝  
Riemann Sphere â—˝  
Marked Point â—˝  
Vertex Algebra â—˝  
The Other â—˝  
Gaudin Models â—˝  
Equivariant Functions â—˝  
Marked Points

Given a vertex Lie algebra [Formula: see text] equipped with an action by automorphisms of a cyclic group [Formula: see text], we define spaces of cyclotomic coinvariants over the Riemann sphere. These are quotients of tensor products of smooth modules over “local” Lie algebras [Formula: see text] assigned to marked points [Formula: see text], by the action of a “global” Lie algebra [Formula: see text] of [Formula: see text]-equivariant functions. On the other hand, the universal enveloping vertex algebra [Formula: see text] of [Formula: see text] is itself a vertex Lie algebra with an induced action of [Formula: see text]. This gives “big” analogs of the Lie algebras above. From these we construct the space of “big” cyclotomic coinvariants, i.e. coinvariants with respect to [Formula: see text]. We prove that these two definitions of cyclotomic coinvariants in fact coincide, provided the origin is included as a marked point. As a corollary, we prove a result on the functoriality of cyclotomic coinvariants which we require for the solution of cyclotomic Gaudin models in [B. Vicedo and C. Young, Cyclotomic Gaudin models: Construction and Bethe ansatz, preprint (2014); arXiv:1409.6937]. At the origin, which is fixed by [Formula: see text], one must assign a module over the stable subalgebra [Formula: see text] of [Formula: see text]. This module becomes a [Formula: see text]-quasi-module in the sense of Li. As a bi-product we obtain an iterate formula for such quasi-modules.

Moduli of wild Higgs bundles on with -actions

2021 â—˝  
pp. 1-34
Author(s):  
LAURA FREDRICKSON â—˝  
ANDREW NEITZKE
Keyword(s):  
Fixed Points â—˝  
Moduli Spaces â—˝  
Vector Bundles â—˝  
Central Charge â—˝  
Higgs Field â—˝  
Vertex Algebra â—˝  
Higgs Bundles â—˝  
Irregular Singularity â—˝  
Isomorphism Classes â—˝  
Type Decomposition

Abstract We study a set $\mathcal{M}_{K,N}$ parameterising filtered SL(K)-Higgs bundles over $\mathbb{C}P^1$ with an irregular singularity at $z = \infty$ , such that the eigenvalues of the Higgs field grow like $\vert \lambda \vert \sim \vert z^{N/K} \mathrm{d}z \vert$ , where K and N are coprime. $\mathcal{M}_{K,N}$ carries a $\mathbb{C}^\times$ -action analogous to the famous $\mathbb{C}^\times$ -action introduced by Hitchin on the moduli spaces of Higgs bundles over compact curves. The construction of this $\mathbb{C}^\times$ -action on $\mathcal{M}_{K,N}$ involves the rotation automorphism of the base $\mathbb{C}P^1$ . We classify the fixed points of this $\mathbb{C}^\times$ -action, and exhibit a curious 1-1 correspondence between these fixed points and certain representations of the vertex algebra $\mathcal{W}_K$ ; in particular we have the relation $\mu = {k-1-c_{\mathrm{eff}}}/{12}$ , where $\mu$ is a regulated version of the L 2 norm of the Higgs field, and $c_{\mathrm{eff}}$ is the effective Virasoro central charge of the corresponding W-algebra representation. We also discuss a Białynicki–Birula-type decomposition of $\mathcal{M}_{K,N}$ , where the strata are labeled by isomorphism classes of the underlying filtered vector bundles.

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

2006 â—˝  
Vol 17 (08) â—˝  
pp. 975-1012 â—˝  
Author(s):  
YI-ZHI HUANG â—˝  
JAMES LEPOWSKY â—˝  
LIN ZHANG
Keyword(s):  
Tensor Product â—˝  
Vertex Operator â—˝  
Operator Algebra â—˝  
Vertex Operator Algebra â—˝  
Vertex Algebra â—˝  
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 -constraints for the total descendant potential of a simple singularity

2013 â—˝  
Vol 149 (5) â—˝  
pp. 840-888 â—˝  
Author(s):  
Bojko Bakalov â—˝  
Todor Milanov
Keyword(s):  
Analytic Continuation â—˝  
Vertex Algebra â—˝  
Weight Vector â—˝  
Simple Singularity â—˝  
Moody Algebra â—˝  
Basic Representation â—˝  
Highest Weight â—˝  
Finite Dimensional â—˝  
Dynkin Diagrams â—˝  
Twisted Representation

AbstractSimple, or Kleinian, singularities are classified by Dynkin diagrams of type $ADE$. Let $\mathfrak {g}$ be the corresponding finite-dimensional Lie algebra, and $W$ its Weyl group. The set of $\mathfrak {g}$-invariants in the basic representation of the affine Kac–Moody algebra $\hat {\mathfrak {g}}$ is known as a $\mathcal {W}$-algebra and is a subalgebra of the Heisenberg vertex algebra $\mathcal {F}$. Using period integrals, we construct an analytic continuation of the twisted representation of $\mathcal {F}$. Our construction yields a global object, which may be called a $W$-twisted representation of $\mathcal {F}$. Our main result is that the total descendant potential of the singularity, introduced by Givental, is a highest-weight vector for the $\mathcal {W}$-algebra.

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