A conformal restriction system is a commutative, associative, unital algebra
equipped with a representation of the groupoid of univalent conformal maps on
connected open sets of the Riemann sphere, and a family of linear functionals
on subalgebras, satisfying a set of properties including conformal invariance
and a type of restriction. This embodies some expected properties of
expectation values in conformal loop ensembles CLE. In the context of conformal
restriction systems, we study certain algebra elements associated with
hypotrochoid simple curves (including the ellipse). These have the CLE
interpretation of being "renormalized random variables" that are nonzero only
if there is at least one loop of hypotrochoid shape. Each curve has a center w,
a scale \epsilon\ and a rotation angle \theta, and we analyze the renormalized
random variable as a function of u=\epsilon e^{i\theta} and w. We find that it
has an expansion in positive powers of u and u*, and that the coefficients of
pure u (u*) powers are holomorphic in w (w*). We identify these coefficients
(the "hypotrochoid fields") with certain Virasoro descendants of the identity
field in conformal field theory, thereby showing that they form part of a
vertex operator algebraic structure. This largely generalizes works by the
author (in CLE), and the author with his collaborators V. Riva and J. Cardy (in
SLE 8/3 and other restriction measures), where the case of the ellipse, at the
order u^2, led to the stress-energy tensor of CFT. The derivation uses in an
essential way the Virasoro vertex operator algebra structure of conformal
derivatives established recently by the author. The results suggest in
particular the exact evaluation of CLE expectations of products of hypotrochoid
fields as well as non-trivial relations amongst them through the vertex
operator algebra, and further shed light onto the relationship between CLE and
CFT.Comment: 1 figure, 39 page