Diseases of triangulated random surface models, and possible cures

A proposal for strings at D > 62; 1

“…The first lattice formulations of string theories in arbitrary dimensions appeared in the mid-eigthies [4,5,6,7], and there was a good deal of activity in solving some two-dimensional models on random triangulated surfaces [10]. A breakthrough took place with the work in ref.…”

Quantum Gravity via Causal Dynamical Triangulations

“…One can characterize this set of geometries as being constructed from gluing together equilateral simplicial building blocks in all possible ways, compatible with certain constraints (typically, a fixed topology and fixed boundary components). Consequently, the variation in geometry (the way in which the geometric degrees of freedom are encoded) is linked to the mutual connectivity of the building blocks created by the gluing and not to variations in the link lengths, giving rise to the name Dynamical Triangulations (DT) [5,6,7]. From a path-integral perspective this approach has the advantage that distinct triangulations correspond to physically distinct geometries.…”

“…The set of causal dynamical triangulations (CDT) -which can be defined in any dimension (not just d = 2) -obeys a strong version of causality of this kind, which is implemented by requiring each triangulation to be the product of a one-dimensional "triangulation" (a line with equidistant points), representing discrete proper time, and other triangulated degrees of freedom, representing the spatial directions of the geometry, which may be thought of as triangulated fibres over a one-dimensional base space. 5 As an added bonus, each triangulation in the class of CDT can be analytically continued to Euclidean signature, and the associated gravitational Regge actions satisfy the standard relation between actions defined in spacetimes of Lorentzian and Euclidean signatures, namely,…”

The Kosterlitz-Thouless phenomenon on a fluid random surface

“…The scaling law follows from (27) by requiring the 2nd term to be of the same order as the rst h 2 = h 2 R 3 log M : (28) This agrees with KPZ scaling for theñ = 1 operator at r = 1 = 2 but there is a logarithmic scaling violation; such violations are well-known at c = 1, occurring also for the cosmological constant [28]. The latter is determined in terms of M by expanding (24) and (25) (29) which, omitting + : : :now, determines A = 2 =4 and…”

“…As a representation of the uctuating lattice itself, it is convenient to use a simplicial complex [9,27] ( g.1), say triangulation, with the usual identi cation of curvature and area per vertex, which in the present case is embedded in the line parametrized by X. The Feynman diagrams generated by the NxN Hermitian matrix quantum mechanics with action…”