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We consider discrete random fractal surfaces with negative Hurst exponent $H, Comment: 35 pages. Replacement takes into account the comments of the two SciPost referees, which can be found at https://scipost.org/submissions/2005.11830v1/#report_1

We perform Monte-Carlo computations of four-point cluster connectivities in the critical 2d Potts model, for numbers of states $Q\in (0,4)$ that are not necessarily integer. We compare these connectivities to four-point functions in a CFT that interpolates between D-series minimal models. We find that 3 combinations of the 4 independent connectivities agree with CFT four-point functions, down to the $2$ to $4$ significant digits of our Monte-Carlo computations. However, we argue that the agreement is exact only in the special cases $Q=0, 3, 4$. We conjecture that the Potts model can be analytically continued to a double cover of the half-plane $\{\Re c, 53 pages, v2: small improvements and clarifications

It has been recently shown that the deformation of an arbitrary two-dimensional conformal field theory by the composite irrelevant operator $T \bar T$, built from the components of the stress tensor, is solvable; in particular, the finite-size spectrum of the deformed theory can be obtained from that of the original CFT through a universal formula. We study a similarly universal, Lorentz-breaking deformation of two-dimensional CFTs that posess a conserved $U(1)$ current, $J$. The deformation takes the schematic form $J \bar T$ and is interesting because it preserves an $SL(2,\mathbb{R}) \times U(1)$ subgroup of the original global conformal symmetries. For the case of a purely (anti)chiral current, we find the finite-size spectrum of the deformed theory and study its thermodynamic properties. We test our predictions in a simple example involving deformed free fermions., Comment: v3: institutional and grant information added