4 The Dual Gauge Theory Side
Let us now turn to the identification of the folded and circular string solutions in the dual gauge
theory.
Our aim is to reproduce the obtained energy functions
plotted in Figure 3 from a dual
gauge theory computation at one-loop. For this, we need to identify the gauge theory operators, which are
dual to the spinning strings on
. As here
the relevant operators will be built
from the two complex scalars
and
with a total number of
-fields
and
-fields, i.e.
where the dots denote suitable permutations of the
and
to be discussed. An operator of the form
(44) may be pictured as a ring of black (“
”) and red (“
”) beads - or equivalently as a configuration
of an
quantum spin chain, where
corresponds to the state
and
to
.
How does one compute the associated scaling dimensions at (say) one loop order for
?
Clearly one is facing a huge operator mixing problem as all
with arbitrary permutations of
’s
and
’s are degenerate at tree level where
.
A very efficient tool to deal with this problem is the dilatation operator
, which was introduced in
[20, 21]. It acts on the trace operators
at a fixed space-time point
and its eigenvalues are the
scaling dimensions
The dilatation operator is constructed in such a fashion as to attach the relevant diagrams to the open legs
of the
“incoming” trace operators (as depicted in Figure 4) and may be computed in perturbation theory
where
is of order
. For the explicit computation of the one-loop piece
see
e.g. the review [97], where the concrete relation to two-point functions is also explained. In our
“minimal” SU(2) sector of complex scalar fields
and
it takes the rather simple form
Note that the tree-level piece
simply measures the length of the incident operator (or spin chain)
. The eigenvalues of the dilatation operator then yield the scaling dimensions we are looking for -
diagonalization of
solves the mixing problem.
We shall be exclusively interested in the planar contribution to
, as this sector of the gauge theory
corresponds to the “free” (in the sense of
)
string. For this, it is important to realize
that the planar piece of
only acts on two neighboring fields in the chain of
’s and
’s. This
may be seen by evaluating explicitly the action of
on two fields
and
separated by arbitrary
matrices
and 
Clearly, there is an enhanced contribution when
or
, i.e.
and
are nearest
neighbors on the spin chain. From the above computation we learn that
where
permutes the fields (or spins) at sites
and
and periodicity
is
understood. Remarkably, as noticed by Minahan and Zarembo [92], this spin chain operator is the
Heisenberg
quantum spin chain Hamiltonian, which is the prototype of an integrable
spin chain. Written in terms of the Pauli matrices
acting on the spin at site
one finds
Due to the positive sign of the sum, the spin chain is ferromagnetic and its ground state is
: the gauge dual of the rotating point particle of Section 2.1. Excitations of
the ground state are given by spin flips or “magnons”. Note that a one-magnon excitation
has vanishing energy due to the cyclic property of the trace, it corresponds
to a zero mode plane-wave string excitation
. Two-magnon excitations are the first
stringy excitations which are dual to the
state of the plane-wave string in the BMN
limit.
The integrability of the spin chain amounts to the existence of
higher charges
which
commute with the Hamiltonian (alias dilatation operator) and amongst themselves, i.e.
.
Explicitly the first two charges of the Heisenberg chain are given by
The explicit form of all the higher
may be found in [67]. Note that
will involve up to
neighboring spin interactions.