Chiral Hodge Cohomology and Mathieu Moonshine

2019 ◽  
Author(s):  
Bailin Song
Keyword(s):  
Finite Order ◽  
Unitary Representation ◽  
Central Charge ◽  
Vertex Algebra ◽  
K3 Surface ◽  
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.

Moduli of wild Higgs bundles on with -actions

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

ADE SUBALGEBRAS OF THE TRIPLET VERTEX ALGEBRA $\mathcal{W}(p)$: A-SERIES

2013 ◽  
Vol 15 (06) ◽  
pp. 1350028 ◽  
Author(s):  
DRAŽEN ADAMOVIĆ ◽  
XIANZU LIN ◽  
ANTUN MILAS
Keyword(s):  
Central Charge ◽  
Strong Support ◽  
Vertex Algebra ◽  
Finite Subgroup ◽  
Irreducible Representations ◽  
Finite Subgroups ◽  
Complete Set ◽  
Ade Classification ◽  
Charge Formula

Motivated by [On the triplet vertex algebra [Formula: see text], Adv. Math.217 (2008) 2664–2699], for every finite subgroup Γ ⊂ PSL(2, ℂ) we investigate the fixed point subalgebra [Formula: see text] of the triplet vertex [Formula: see text], of central charge [Formula: see text], p ≥ 2. This part deals with the A-series in the ADE classification of finite subgroups of PSL(2, ℂ). First, we prove the C2-cofiniteness of the Am-fixed subalgebra [Formula: see text]. Then we construct a family of [Formula: see text]-modules, which are expected to form a complete set of irreducible representations. As a strong support to our conjecture, we prove modular invariance of (generalized) characters of the relevant (logarithmic) modules. Further evidence is provided by calculations in Zhu's algebra for m = 2. We also present a rigorous proof of the fact that the full automorphism group of [Formula: see text] is PSL(2, ℂ).

On parafermion vertex algebras of 𝔰𝔩(2) and 𝔰𝔩(3) at level −3 2

2020 ◽  
pp. 2050086
Author(s):  
Dražen Adamović ◽  
Antun Milas ◽  
Qing Wang
Keyword(s):  
Lie Algebra ◽  
Central Charge ◽  
Vertex Algebra ◽  
Vertex Algebras ◽  
Algebraic Proof ◽  
Direct Sums ◽  
Direct Sum ◽  
Irreducible Modules ◽  
Level 3 ◽  
Charge Formula

We study parafermion vertex algebras [Formula: see text] and [Formula: see text]. Using the isomorphism between [Formula: see text] and the logarithmic vertex algebra [Formula: see text] from [D. Adamović, A realization of certain modules for the [Formula: see text] superconformal algebra and the affine Lie algebra [Formula: see text], Transform. Groups 21(2) (2016) 299–327], we show that these parafermion vertex algebras are infinite direct sums of irreducible modules for the Zamolodchikov algebra [Formula: see text] of central charge [Formula: see text], and that [Formula: see text] is a direct sum of irreducible [Formula: see text]-modules. As a byproduct, we prove certain conjectures about the vertex algebra [Formula: see text]. We also obtain a vertex-algebraic proof of the irreducibility of a family of [Formula: see text] modules at [Formula: see text].

Automorphisms of finite order of nilpotent groups II

2014 ◽  
Vol 51 (4) ◽  
pp. 547-555 ◽  
Author(s):  
B. Wehrfritz
Keyword(s):  
Nilpotent Group ◽  
Finite Order ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

Finite-order Integration Weights Can be Dangerous

2007 ◽  
Vol 7 (3) ◽  
pp. 239-254 ◽  
Author(s):  
I.H. Sloan
Keyword(s):  
Function Space ◽  
Finite Order ◽  
Small Subset ◽  
Worst Case ◽  
Integration Rule ◽  
Dimensional Lattice ◽  
3 Dimensional ◽  
Sum Of Functions ◽  
Integration Rules

Abstract Finite-order weights have been introduced in recent years to describe the often occurring situation that multivariate integrands can be approximated by a sum of functions each depending only on a small subset of the variables. The aim of this paper is to demonstrate the danger of relying on this structure when designing lattice integration rules, if the true integrand has components lying outside the assumed finiteorder function space. It does this by proving, for weights of order two, the existence of 3-dimensional lattice integration rules for which the worst case error is of order O(N¯½), where N is the number of points, yet for which there exists a smooth 3- dimensional integrand for which the integration rule does not converge.

On Commutation Properties of the Composition Relation of Convergent and Divergent Permutations (Part I)

2014 ◽  
Vol 58 (1) ◽  
pp. 13-22
Author(s):  
Roman Wituła ◽  
Edyta Hetmaniok ◽  
Damian Słota
Keyword(s):  
Finite Order ◽  
Convergent Series ◽  
The Other ◽  
Other Hand ◽  
The Family ◽  
Composition Relation ◽  
The Continuum

Abstract In the paper we present the selected properties of composition relation of the convergent and divergent permutations connected with commutation. We note that a permutation on ℕ is called the convergent permutation if for each convergent series ∑an of real terms, the p-rearranged series ∑ap(n) is also convergent. All the other permutations on ℕ are called the divergent permutations. We have proven, among others, that, for many permutations p on ℕ, the family of divergent permutations q on ℕ commuting with p possesses cardinality of the continuum. For example, the permutations p on ℕ having finite order possess this property. On the other hand, an example of a convergent permutation which commutes only with some convergent permutations is also presented.

Sign in / Sign up

Export Citation Format

Share Document