## 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 [2021]. 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.