Gravity and Strings

284 Gravitational pp-waves
The Heisenberg algebra H(2n + 2) is the semidirect sum of H(2n + 1) and the Lie algebra
generated by the automorphism U whose action is determined by the new non-vanishing
Lie brackets
[U, qi] = pi, [U, pi] =−qi. (10.10)
In the complex basis
the Lie brackets take the form
[αi,α †
αi = 1
√ 2 (qi + ipi), I = iV, N =−iU, (10.11)
j ] = δijI, [N,αi] =−αi, [N,α †
i ] =+α† i , (10.12)
in which we recognize N as the number operator.
All the Heisenberg algebras are solvable and have a singular Killing metric. 1 V (I )is
always central.
The Heisenberg algebras can be deformed as follows: let us denote by xr, r = 1,...,2n
the column vector formed by the qis and pis. The Lie brackets can be written in this form:
[xr, xs] = ηrsV, [U, xr] = ηrsxs, (ηrs) =
Now, we can define a new (solvable) Lie algebra with brackets
0 In×n
−In×n 0
. (10.13)
[xr, xs] = MrsV, [U, xr] = Nrsxs, MN T − NM T = 0. (10.14)
In some cases, but not always, this algebra is equivalent to the original Heisenberg algebra
up to a GL(2n) transformation.
The (n + 2)-dimensional Hpp-wave spacetimes are constructed starting from a (2n + 2)-
dimensional algebra of the above form with
(Mrs) =
0 −2A
2A 0
, (Nrs) =
0 In×n
, Aij = A
2A 0
ji, (10.15)
which is inequivalent to the original Heisenberg algebra H(2n + 2). Inthe coset construction
h will be the Abelian subalgebra generated by the pi ≡ Mis and its orthogonal complement
k is generated by qi ≡ Pi, V ≡ Pv, and U ≡ Pu. h and k are a symmetric pair.
Using the coset representative
we obtain the 1-forms
u = e v Pv e uPu e x i Pi , (10.16)
eu =−du, ei =−dxi ,
ev =−(dv + Aijx i x jdu), ϑi =−xidu. 1 Actually the algebras H(2n + 1) are nilpotent, which implies an identically vanishing Killing metric.
(10.17)