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2 The Setup

With the embedding coordinates Xm(t, s) and Y m(t,s) the Polyakov action of the AdS5 × S5 string in conformal gauge (jab = diag(- 1,1)) takes the form (m, n = 1,2,3,4,5)
V~ -- integral [ ] I = - --c- dt ds G(AmdnS5)@aXm @aXn + G(Sm5n) @aY m @aY n + fermions , (4) 4p
where we have suppressed the fermionic terms in the action, as they will not be relevant in our discussion of classical solutions (the full fermionic action is stated in [89100]). A natural choice of coordinates for the 5 AdS5 × S space (“global coordinates”) is3
2 2 2 2 2 2 2 2 2 2 dsAdS5 = dr - cosh rdt + sinh r (dy + cos ydf1 + sin y df2 ) ds2S5 = dg2 + cos2 g df32 + sin2g (dy2 + cos2y df12 + sin2 y df22) . (5)
Moreover, we have directly written the string action with the help of the effective string tension V~ -- 2 ' c = R /a of (2View Equation). It is helpful to picture the AdS5 space-time as a bulk cylinder with a four-dimensional boundary of the form R × S3 (see Figure 2).
View Image

Figure 1: Sketch of the AdS5 (bulk cylinder with boundary R × S3) space-time and the S5 (sphere) space.
A consequence of the conformal gauge choice are the Virasoro constraints, which take the form
! m ' p ' ! m p m' ' p' ' 0 = X X m + Y Yp 0 = X Xm + Y Yp + X X m + Y Y p , (6)
where the dot refers to @ t and the prime to @ s derivatives. Of course the contractions in the above are to be performed with the metrics of (5View Equation).

As mentioned in the introduction, it is presently unknown how to perform an exact quantization of this model. It is, however, possible to perform a quantum fluctuation expansion in V~ -- 1/ c. For this one expands around a classical solution of (4View Equation) and integrates out the fluctuations in the path-integral loop order by order. This is the route we will follow. Of course, in doing so, we will only have a patch-wise access to the full spectrum of the theory, with each patch given by the solution expanded around.

 2.1 The rotating point-particle
 2.2 N = 4 Super Yang-Mills

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