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.
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“incoming” trace operators (as depicted in Figure 4) and may be computed in perturbation theory , 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
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, 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
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. Note that will involve up to neighboring spin interactions.
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