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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 E1(J1,J2) plotted in Figure 3View Image 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 R × S3. As here J2 = 0 = S1 = S2 the relevant operators will be built from the two complex scalars Z := f1 + if2 and W := f3 + i f4 with a total number of J1 Z-fields and J2 W-fields, i.e.

J1,J2 J1 J2 O a = Tr[Z W ] + ..., (44)
where the dots denote suitable permutations of the Z and W to be discussed. An operator of the form (44View Equation) may be pictured as a ring of black (“Z”) and red (“W”) beads - or equivalently as a configuration of an s = 1/2 quantum spin chain, where W corresponds to the state | |^ > and Z to | |, >.
2 4 Tr[ZW ZW ] <==> <==> | |, |^ |^ |, |^ |^ |^ |^ >cyclic

How does one compute the associated scaling dimensions at (say) one loop order for J1,J2 --> oo? Clearly one is facing a huge operator mixing problem as all J,J O 1i 2 with arbitrary permutations of Z’s and W’s are degenerate at tree level where OJi1,J2 D 0 = J1 + J2.

A very efficient tool to deal with this problem is the dilatation operator D, which was introduced in [2021]. It acts on the trace operators OJ1,J2 a at a fixed space-time point x and its eigenvalues are the scaling dimensions D

J1,J2 sum J1,J2 D o O a (x) = Dab O b (x). (45) b
The dilatation operator is constructed in such a fashion as to attach the relevant diagrams to the open legs of the
View Image

Figure 4: The action of the dilatation operator on a trace operator.
“incoming” trace operators (as depicted in Figure 4View Image) and may be computed in perturbation theory
sum oo (n) D = D , (46) n=0
where D(n) is of order g2n YM. For the explicit computation of the one-loop piece D(1) 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 Z and W it takes the rather simple form
(0) (1) g2YM- --d- D = Tr(ZZ + W W ), D = - 8p2 Tr [Z, W ][Z, W ], where Zij := dZ . (47) ji
Note that the tree-level piece D(0) simply measures the length of the incident operator (or spin chain) J1 + J2. The eigenvalues of the dilatation operator then yield the scaling dimensions we are looking for - diagonalization of D solves the mixing problem.

We shall be exclusively interested in the planar contribution to D, as this sector of the gauge theory corresponds to the “free” (in the sense of gs = 0) AdS5 × S5 string. For this, it is important to realize that the planar piece of D(1) only acts on two neighboring fields in the chain of Z’s and W’s. This may be seen by evaluating explicitly the action of D(1) on two fields Z and W separated by arbitrary matrices A and B

Tr [Z,W ][Z,W ] o Tr(Z A W B) = - Tr(A) Tr([Z,W ]B) + Tr(B) Tr([Z, W ]A) . (48)
Clearly, there is an enhanced contribution when A = 1 or B = 1, i.e. Z and W are nearest neighbors on the spin chain. From the above computation we learn that
L (1) -c-- sum D planar = 8p2 (1i,i+1 - Pi,i+1) (49) i=1
where Pi,j permutes the fields (or spins) at sites i and j and periodicity PL,L+1 = P1,L is understood. Remarkably, as noticed by Minahan and Zarembo [92], this spin chain operator is the Heisenberg XXX1/2 quantum spin chain Hamiltonian, which is the prototype of an integrable spin chain. Written in terms of the Pauli matrices si acting on the spin at site i one finds
sum L D(1p)lanar = -c--HXXX1/2 = -c-- (1-- si .si+1). (50) 8p2 4p2 i=1 4
Due to the positive sign of the sum, the spin chain is ferromagnetic and its ground state is | |, |, ..., | > <==> Tr(ZL) cyclic: 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 | |, ..., | ^ | , | ... |, >cyclic has vanishing energy due to the cyclic property of the trace, it corresponds to a zero mode plane-wave string excitation a †0|0>. Two-magnon excitations are the first stringy excitations which are dual to the a†a † |0> n - n state of the plane-wave string in the BMN limit.

The integrability of the spin chain amounts to the existence of L - 1 higher charges Qk which commute with the Hamiltonian (alias dilatation operator) and amongst themselves, i.e. [Qk, Ql] = 0. Explicitly the first two charges of the Heisenberg chain are given by

sum L Q2 := HXXX1/2 Q3 = (si × si+1) .si+2 . (51) i=1
The explicit form of all the higher Qk may be found in [67]. Note that Qk will involve up to k neighboring spin interactions.

 4.1 The coordinate Bethe ansatz
 4.2 The thermodynamic limit of the spin chain
 4.3 Higher Loops in the SU(2) sector and discrepancies
 4.4 Further developments

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