arXiv 0901.22802009-01-15math.RTmath.AP

The minimal representation of the conformal group and classic solutions to the wave equation

Markus Hunziker, Mark R. Sepanski, Ronald J. Stanke

备注：: 13 pages, 1 figure

英文摘要：

We give a uniform realization of the minimal representation of a double cover of the conformal group SO(2,n+1)_0 in the kernel of the wave operator on flat Minkowski space as a positive energy representation H^+ for n even and odd. Using this realization, we obtain an explicit orthonormal basis for H^+ that is well behaved with respect to energy and angular momentum. Of special note, for n odd, all functions in our basis are rational functions. Finally, using Fourier analysis with respect to this basis, we prove that every classical real-valued solution to the wave equation is the real part of a unique continuous element in the representation H^+.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Minimal representations and reductive dual pairs in conformal field theory

A minimal representation of a simple non-compact Lie group is obtained by ``quantizing'' the minimal nilpotent coadjoint orbit of its Lie algebra. It provides context for Roger Howe's notion of a reductive dual pair encountered recently in the description of global gauge symmetry of a (4-dimensional) conformal observable algebra. We give a pedagogical introduction to these notions and point out that physicists have been using both minimal representations and dual pairs without naming them and hence stand a chance to understand their theory and to profit from it.

2010-06-10hep-thmath-phmath.MP

Minimal unitary representation of SO*(8) = SO(6,2) and its SU(2) deformations as massless 6D conformal fields and their supersymmetric extensions

We study the minimal unitary representation (minrep) of SO(6,2) over an Hilbert space of functions of five variables, obtained by quantizing its quasiconformal realization. The minrep of SO(6,2), which coincides with the minrep of SO*(8) similarly constructed, corresponds to a massless conformal scalar field in six spacetime dimensions. There exists a family of "deformations" of the minrep of SO*(8) labeled by the spin t of an SU(2)_T subgroup of the little group SO(4) of lightlike vectors. These deformations labeled by t are positive energy unitary irreducible representations of SO*(8) that d…

2010-06-14hep-thmath-phmath.MP

Conformal embeddings of affine vertex algebras in minimal $W$-algebras I: structural results

We find all values of $k\in \mathbb C$, for which the embedding of the maximal affine vertex algebra in a simple minimal W-algebra $W_k(\mathfrak g,θ)$ is conformal, where $\mathfrak g$ is a basic simple Lie superalgebra and $-θ$ its minimal root. In particular, it turns out that if $W_k(\mathfrak g,θ)$ does not collapse to its affine part, then the possible values of these $k$ are either $-\frac{2}{3} h^\vee$ or $-\frac{h^\vee-1}{2}$, where $h^\vee$ is the dual Coxeter number of $\mathfrak g$ for the normalization $(θ,θ)=2$. As an application of our results, we present a realization of simple…

2016-04-17math.RT

Analysis on the minimal representation of O(p,q) -- I. Realization via conformal geometry

This is the first in a series of papers devoted to an analogue of the metaplectic representation, namely, the minimal unitary representation of an indefinite orthogonal group; this representation corresponds to the minimal nilpotent coadjoint orbit in the philosophy of Kirillov-Kostant. We begin by applying methods from conformal geometry of pseudo-Riemannian manifolds to a general construction of an infinite-dimensional representation of the conformal group on the solution space of the Yamabe equation. By functoriality of the constructions, we obtain different models of the unitary representa…

2001-11-07math-phmath.DGmath.MP

SU(2) deformations of the minimal unitary representation of OSp(8*|2N) as massless 6D conformal supermultiplets

Minimal unitary representation of SO*(8) = SO(6,2) realized over the Hilbert space of functions of five variables and its deformations labeled by the spin t of an SU(2) subgroup correspond to massless conformal fields in six dimensions as was shown in arXiv:1005.3580. In this paper we study the minimal unitary supermultiplet of OSp(8*|2N) with the even subgroup SO*(8) x USp(2N) and its deformations using quasiconformal methods. We show that the minimal unitary supermultiplet of OSp(8*|2N) admits deformations labeled uniquely by the spin t of an SU(2) subgroup of the little group SO(4) of light…

2010-08-04hep-thmath-phmath.MP

Minimal representations of conformal groups and generalized Laguerre functions

We give a unified construction of the minimal representation of a finite cover $G$ of the conformal group of a (non necessarily euclidean) Jordan algebra $V$. This representation is realized on the $L^2$-space of the minimal orbit $\mathcal{O}$ of the structure group $L$ of $V$. We construct its corresponding $(\mathfrak{g},\mathfrak{k})$-module and show that it can be integrated to a unitary irreducible representation of $G$ on $L^2(\mathcal{O})$. In particular, we obtain a unified approach to the two most prominent minimal representations, namely the Segal--Shale--Weil representation of the …

2012-08-27math.CAmath.RT

更多相关论文收起更多论文

The inversion formula and holomorphic extension of the minimal representation of the conformal group

The minimal representation $π$ of the indefinite orthogonal group $O(m+1,2)$ is realized on the Hilbert space of square integrable functions on $\mathbb R^m$ with respect to the measure $|x|^{-1} dx_1... dx_m$. This article gives an explicit integral formula for the holomorphic extension of $π$ to a holomorphic semigroup of $O(m+3, \mathbb C)$ by means of the Bessel function. Taking its `boundary value', we also find the integral kernel of the `inversion operator' corresponding to the inversion element on the Minkowski space $\mathbb R^{m,1}$.

2006-06-30math.FAmath.RT

Conformal classical Yang-Baxter equation, $S$-equation and $\mathcal{O}$-operators

Conformal classical Yang-Baxter equation and $S$-equation naturally appear in the study of Lie conformal bialgebras and left-symmetric conformal bialgebras. In this paper, they are interpreted in terms of a kind of operators, namely, $\mathcal O$-operators in the conformal sense. Explicitly, the skew-symmetric part of a conformal linear map $T$ where $T_0=T_λ\mid_{λ=0}$ is an $\mathcal O$-operator in the conformal sense is a skew-symmetric solution of conformal classical Yang-Baxter equation, whereas the symmetric part is a symmetric solution of conformal $S$-equation. One byproduct is that a …

2018-08-18math-phmath.MPmath.RA

Conformal embeddings of affine vertex algebras in minimal $W$-algebras II: decompositions

We present methods for computing the explicit decomposition of the minimal simple affine $W$-algebra $W_k(\mathfrak g, θ)$ at a conformal level $k$ as a module for its maximal affine subalgebra $\mathcal V_k(\mathfrak g^{\natural})$. A particular emphasis is given on the application of affine fusion rules to the determination of branching rules. In almost all cases when $\mathfrak g^{\natural}$ is a semisimple Lie algebra, we show that, for a suitable conformal level $k$, $W_k(\mathfrak g, θ)$ is isomorphic to an extension of $\mathcal V_k(\mathfrak g^{\natural})$ by its simple module. We are …

2017-04-12math-phmath.MPmath.QA