Discretized superstring in three-dimensional superspace.

The partition function of the discretized superstring in a target superspace of three (Euclidean) bosonic dimensions, is shown, for a fixed triangulation of the random world sheet, to be derived from the partition function of a discretized bosonic string with an external field present in the action in the form of a specific constant matrix, using first order forms of the actions. This latter partition function appears more amenable to an exact analytical treatment. The theory of strings in spacetimes of dimension lower than the critical dimension stands far from complete. Most attempts are stymied by the so-called cm ≤ 1 barrier, [1,2] where cm is the central charge of the matter two dimensional conformal field theory (CFT) on the (random) string world sheet. In the continuum approach, the conformal mode of the world sheet metric plays the role of an additional target space dimension, [3] so that D = cm + 1 for the bosonic string. It follows that, for the susceptibility of the bosonic non-critical string to have real critical exponents, D ≤ 2. This pathology manifests even more clearly in the discretized version of the theory (by dynamical triangulations of the random 2d surface) [4–6], in that the lattice string tension does not scale to zero at the critical point [7,8]. This implies that the discrete random surface does not admit a continuum limit for embeddings in (bosonic) spacetimes of D > 2, possibly degenerating into an ensemble of branched polymers (see, e.g., ref. [9]). In this letter, we consider an extension of the Polyakov string with target space supersymmetry, by embedding the 2d random surface in a superspace with three bosonic (Euclidean) directions. If a continuum limit exists for this theory, the Liouville field will supply the