Singular metrics and associated conformal groups underlying ...

10 K. R. Payne
x2∂1X 1 − x2∂2X 2 − m
2 X2 = 0 (4.13)
x m 2 ∂1X 2 + ∂2X 1 = 0 (4.14)
where we have used also divgX = ∂1X 1 + ∂2X 2 − mX 2 /(2x2). Hence the
system is equivalent to the corresponding (ξ, η) = (X 1 , X 2 ) part of the system
(2.18) and (2.20) describing the symmetry group. These two equations in two
unknowns have an infinite number of independent solutions including those
corresponding to the translation, dilation and inversion; namely
(X 1 , X 2 ) (1) = (1, 0), (4.15)
(X 1 , X 2 ) (2) = ((m + 2)x1, 2x2), (4.16)
(X 1 , X 2 ) (3) = ((m + 2) 2 x 2 1 − 4x m+2
2 , 4(m + 2)x1x2) (4.17)
as well as a remarkable sequence of independent solutions defined inductively
by (X 1 , X 2 ) (k) = (ξ (k), η (k)) with
ξ (k+1) = ξ 2 (k) − xm 2 η 2 (k) , and η (k+1) = 2ξ (k)η (k), k ≥ 2 (4.18)
as a simple calculation shows. Hence the conformal algebras have infinite
dimension.
All of the solutions above extend smoothly across the singular line x2 = 0
where the component X 2 tends to zero as x2 → 0; hence in this sense each X
is tangential to the singular line. Moreover, the equation (4.13) shows that
any C 1 continuation of X 2 must vanish as well.
As in the higher dimensional cases, we can now define the conformal algebra
Kg(R1+1 ) as the Lie algebra of smooth extensions across Σ of Kg(R 1+1
± )
and the conformal group associated to the singular metric g as the group of
local diffeomorphisms generated by Kg(R1+1 ) which is infinite dimensional.
On the other hand, the non-trivial part of the symmetry group associated to
the operator (1.1) is only three dimensional and corresponds to the generators
(4.15) − (4.17). The point being that the remainder of the system describing
the symmetry group (2.15) − (2.17) forces the pair (ξ, η) = (X1 , X2 ) to be
a linear combination of the generators (4.15) − (4.17) as has been shown in
[15]. On the other hand, if one considers the operator (3.2) one recovers the
complete conformal group as the non-trivial part of its symmetry group.
Theorem 4.5. Let K be a pure power type change function on R 1+1 . Then
the conformal algebra Kg(R 1+1 ) agrees with the natural projection onto T R 1+1
of the Lie algebra A(R 1+1 ) of generators of a one parameter symmetry group
for the operator
Lg = K(y)∂ 2 x + ∂ 2 y − m
2y ∂y. (4.19)
Proof: As a first step, we need to find the system analogous to (2.15) −
(2.20) which characterizes the generators v = ξ(x, y, u)∂/∂x+η(x, y, u)∂/∂y+
ϕ(x, y, u)∂/∂u of one parameter symmetry groups for the operator (4.19). In