MATHPH In the literature: open

Rotational and conformal invariance of the critical 3D Ising scaling limit

In plain words

At its critical point the 3D Ising model is believed to look the same under rotations and angle-preserving maps (conformal symmetry), which physicists use to compute exponents. In 2D this was proven; in 3D even rotation invariance is unproven.

Precise statement

Show that the scaling limit of critical spin correlations on $Z^3$ exists, is rotation invariant, and transforms covariantly under conformal maps of $R^3$ with scaling dimension $\Delta_{\sigma}\ \text{approximately}\ 0.51815$. Answer: a proof.

What would settle it

A proof of rotation invariance of the 3D scaling limit, then of conformal covariance.

Status in the literature

Conformal invariance in 2D was proven by Smirnov (2010) and by Chelkak, Hongler and Izyurov (2015).

See also