5 We are near to the end of our journey. We still have to compute the critical exponent and the critical behaviour of the correlation of q ∗ $_{h}$( x ). Techniques introduced in [29] could be used to achieve this goal. (14) has many solutions, the solution relevant for the dy- namics is uniquely identified as q ∗ $_{ω}$( x ) = lim s $_{→∞}$q$_{ω}$ ( x, s ) . (20) [5] S. Franz and G. Parisi, J. Phys. I (France) 5 , 1401 (1995); Phys. Rev. Lett. 79 , 2486 (1997); Physica A 261 , 317 (1998). [6] L.F. Cugliandolo and J. Kurchan, Phys. Rev. Lett. 71 , 173 (1993); J. Phys. A: Math. Gen. 27 , 5749 (1994). [7] S. Franz and M. M´ ezard, Europhys. Lett. 26 , 209 (1994). [8] C. Dasgupta, A. Indrani, S. Ramaswamy and M. Phani, Europhys. Lett. 15 , 307 (1991). [9] C. Donati, J.F. Douglas, W. Kob, S.J. Plimpton, P.H. Poole and S.C. Glotzer, Phys. Rev. Lett. 80 , 2338 (1998). [10] G. Parisi, J. Phys. Chem. B 103 , 4128 (1999). [11] S. Franz and G. Parisi, Eur. Phys. J. B 8 , 417 (1999). [12] M. 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We remark that the fictitious time s in the equation (19) should not be identified with the real time: the de- pendence of q ( s ) if we solve equation (19) is quite different from the behaviour of q ( t ) in mode-coupling theory and there should be no confusion among the two variables. However one may make en passant the conjecture that, if we consider only reparametrization invariant quantities, the mode-coupling equations and eq. (19) stay in the same universality class: in other words, if in the mode- coupling theory we eliminate the real time t in favour of q and we write everything as function of q and in eq. (19) we eliminate the fictitious time s in favour of q , the dependence of the physical quantities on q should be the same near the critical point for the models. This con- jecture is trivially valid if there is only one solution of equation (14). The point is to understand its correctness beyond perturbation theory This conjecture may be generalized by introducing a more general evolution equation that can be used to com- pute the properties of reparametrization invariant quan- tities when activated processes are present below T$_{c}$ : where η ( x, s ) is a thermal noise. The discussions of the consequences of these conjectures cannot be done here, however it has not escaped to how attention that eq. (21) can be used to explain the experimental results of [30]. ∂q ( x, s ) ∂s = − Δ q ( x, s )+ W $^{′}$( q ( x, s )) − ω ( x )+ η ( x, s ) (21) [21] T. Temesvari, C. De Dominicis and I.R. Pimentel, Eur. Phys. J. B 25 , 361 (2002). [22] R. Monasson, Phys. Rev. Lett. 75 , 2847 (1995). [23] S. Franz, G. Parisi, F. Ricci-Tersenghi and T. Rizzo (in preparation). [24] G. Parisi and N. Sourlas, Phys. Rev. Lett. 43 , 744 (1979). [25] G. Parisi and N. Sourlas, Phys. Rev. Lett. 46 , 871 (1981). [26] G. Parisi, Les Houches Session XXXIX on Recent Ad- vances in Field Theory and Statistical Mechanics (North- Holland, 1984). [27] T. Sarlat, A. Billoire, G. Biroli and J.-P. Bouchaud, J. Stat. Mech., P08014 (2009). [28] D. Lancaster, E. Marinari and G. Parisi, J. Phys. A 28 , 3959 (1995). [29] G. Parisi and L. Pietronero, Europhys. Lett. 16 , 321 (1991). [30] R. Candelier, A. Widmer-Cooper, J.K. Kummerfeld, O. Dauchot, G. Biroli, P. Harrowell and D.R. Reichman, Avalanches and Dynamical Correlations in supercooled liquids , preprint arXiv:0912.0193. [31] Typically the ratio of a with the inter-particle distance is taken to be smaller that 1 and not far from the Linde- mann constant. [32] We implicitly assume that the maximum χ ∗ $_{4}$( ϵ ) happens in the critical region C ≈ C$_{P}$ . [33] The value 8 for the the upper critical dimension was first suggested in [12]. 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