### 2.1 The rotating point-particle

It is instructive to sketch this procedure by considering the perhaps
simplest solution to the equations of motion of (4): the rotating point-particle on , which is a
degenerated string configuration.
One easily shows that this configurations satisfies the equations of motion and the Virasoro
constraints.

A glance at Equations (4, 5) reveals that the cyclic coordinates of the action are
leading to the conserved charges , corresponding to the energy
and two spins on as well as the three angular momenta on the five
sphere respectively. The energy and the first angular momentum are the only non vanishing
conserved quantities of the above point-particle configuration and take the values (we also spell out for
later use)

Hence classically . One may now consider quantum fluctuations around this solution,
i.e. , and integrate out the quantum field in a perturbative
fashion. This will result in “quantum” corrections to the classical energy in terms of an expansion in
They key idea of Berenstein, Maldacena and Nastase [33] was to consider the limit with the
parameter held fixed. This limit of a large quantum number suppresses all the higher loop
contributions beyond one-loop, i.e.
Hence, the quadratic approximation becomes exact! This quadratic fluctuation action (including the
fermions) is nothing but the IIB superstring in a plane wave background [88], which arises from the
geometry through a so-called Penrose limit (see [38, 39] for this construction). The
quantization of this string model is straightforward in the light-cone gauge [88, 90] and leads to
a free, massive two-dimensional theory for the transverse degrees of freedom ()
with a compact expression for the spectrum
where is the excitation number operator for transverse string excitations with
. The Virasoro constraints (6) reduce to the level matching condition
known from string theory in flat Minkowski space-time. Hence, the first stringy excitation is
with . For a more detailed treatment of the plane wave superstring see
[94, 97, 104, 103].