Tensor networks, $p$-adic fields, and algebraic curves: arithmetic and the AdS$_3$/CFT$_2$ correspondence

Understand

One of the many remarkable properties of conformal field theory in two dimensions is its connection to algebraic geometry.

• Since every compact Riemann surface is a projective algebraic curve, many constructions of interest in physics (which a priori depend on the analytic structure of the spacetime) can be formulated in purely algebraic language.
• This opens the door to interesting generalizations, obtained by taking another choice of field: for instance, the $p$-adics.
• We generalize the AdS/CFT correspondence according to this principle; the result is a formulation of holography in which the bulk geometry is discrete---the Bruhat--Tits tree for $\mathrm{PGL}(2,\mathbb{Q}_p)$---but the group of bulk isometries nonetheless agrees with that of boundary conformal transformations and is not broken by discretization.