PUBLIC POLICY EDUCATION- WHOM DO WE TEACH?

Public Economics,

TRANSPORTATION POLICY WORKSHOP

Public Economics,

LIMITS OF PUBLIC POLICY EDUCATION

Teaching/Communication/Extension/Profession,

PUBLIC POLICY EDUCATION FOR ENVIRONMENTAL AND ECONOMIC DEVELOPMENT ISSUES

Environmental Economics and Policy,

Factorization of formal exponentials and uniformization

Let $\mathfrak{g}$ be a Lie algebra in characteristic zero equipped with a vector space decomposition $\mathfrak{g}=\mathfrak{g}^-\oplus \mathfrak{g}^+$, and let $s$ and $t$ be commuting formal variables. We prove that the Campbell-Baker-Hausdorff map $C:s\mathfrak{g}^- [[s,t]]\times t\mathfrak{g}^+[[s,t]]\to s\mathfrak{g}^-[[s,t]]\oplus t\mathfrak{g}^+[[s,t]]$ given by $e^{sg^-}e^{tg^+}=e^{C(sg^-,tg^+)}$ for $g^\pm\in\mathfrak{g}^\pm[[s,t]]$ is a bijection, as is well known when $\mathfrak{g}$ is finite-dimensional over $\mathbb{R}$ or $\mathbb{C}$, by geometry. It follows that there exist unique $\Psi^\pm\in\mathfrak{g}^\pm[[s,t]]$ such that $e^{tg^+}e^{sg^-}= e^{s\Psi^-}e^{t\Psi^+}$ (also well known in the finite-dimensional geometric setting). We apply this to $\mathfrak{g}$ consisting of certain formal infinite series with coefficients in a Lie algebra $\mathfrak{p}$. For $\mathfrak{p}$ the Virasoro algebra (resp., a Grassmann envelope of the Neveu-Schwarz superalgebra), the result was first proved by Huang (resp., Barron) as a step in the construction of a (super)geometric formulation of the notion of vertex operator (super)algebra. For the Virasoro (resp., N=1 Neveu-Schwarz) algebra with zero central charge the result gives the precise expansion of the uniformizing function for a sphere (resp., supersphere) with tubes resulting from the sewing of two spheres (resp., superspheres) with tubes in two-dimensional genus-zero holomorphic conformal (resp., N = 1 superconformal) field theory. The general result places such uniformization problems into a broad formal algebraic context.Comment: LaTex file, 31 page

An equivalence of two constructions of permutation-twisted modules for lattice vertex operator algebras

The problem of constructing twisted modules for a vertex operator algebra and an automorphism has been solved in particular in two contexts. One of these two constructions is that initiated by the third author in the case of a lattice vertex operator algebra and an automorphism arising from an arbitrary lattice isometry. This construction, from a physical point of view, is related to the space-time geometry associated with the lattice in the sense of string theory. The other construction is due to the first author, jointly with C. Dong and G. Mason, in the case of a multi-fold tensor product of a given vertex operator algebra with itself and a permutation automorphism of the tensor factors. The latter construction is based on a certain change of variables in the worldsheet geometry in the sense of string theory. In the case of a lattice that is the orthogonal direct sum of copies of a given lattice, these two very different constructions can both be carried out, and must produce isomorphic twisted modules, by a theorem of the first author jointly with Dong and Mason. In this paper, we explicitly construct an isomorphism, thereby providing, from both mathematical and physical points of view, a direct link between space-time geometry and worldsheet geometry in this setting.Comment: 35 pages. Further exposition added, with the referee's helpful comments taken into account. Final version to appear in Journal of Pure and Applied Algebr