Pohlmeyer reduction of string theory for flat and AdS spaces

Question

The definition of Pohlmeyer invariants in flat-space (as per eq-2.16 in Urs Schreiber's DDF and Pohlmeyer invariants of (super)string) is the following:

$ Z^{\mu_1...\mu_N} (\mathcal{P}) = \frac{1}{N} \int\limits_0^{2\pi} d\sigma^1 \int\limits_{\sigma_1}^{\sigma_1+2\pi} d\sigma^2... \int\limits_{\sigma_{N-1}}^{\sigma_1} d\sigma^N \ \mathcal{P}^{\mu_1}(\sigma_1)\ \mathcal{P}^{\mu_2}(\sigma_2)...\mathcal{P}^{\mu_N}(\sigma_N) $

Another reference is eq-3.29 in Thomas Thiemann's The LQG -- String: Loop Quantum Gravity Quantization of String Theory I. Flat Target Space. These formulations of the Pohlmeyer invariants is used to show its equivalence to DDF invariants of light-cone gauge fixed strings in flat-space (cf. eq-2.53 in Urs Schreiber's paper and chap-2.3.2 in Green, Schwarz & Witten vol-I)

Now, what is the corresponding invariant for the Pohlmeyer reduction in AdS case (ref: Miramontes' "Pohlmeyer reduction revisited" arXiv:0808.3365)? I cant seem to find one that resembles the flat-space expression.

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