Multidimensional analogues of the geometrics⇌t duality

Abstract

The usual propetry of s ↔ t duality for scattering amplitudes, e.g. for Veneziano amplitude, is deeply connected with the 2-dimensional geometry. In particular, a simple geometric construction of such amplitudes was proposed in a joint work by this author and S. Saito (solv-int/9812016). Here we propose analogs of one of those amplitudes associated with multidimensional euclidean spaces, paying most attention to the 3-dimensional case. Our results can be regarded as a variant of "Regge calculus" intimately connected with ideas of the theory of integrable models.

Invariants of PL Manifolds from Metrized Simplicial Complexes. Three-Dimensional Case

“…The work [5] suggests that invariants of the same type as in this Letter can be constructed for higher-dimensional manifolds as well. This may be combined with other ways of generalization such as the use of noncommutative "lengths".…”

“…Note that ∂ω new /∂l new has been calculated in Section 1, see formula (5). With this taken into account, formula (18) shows that the following theorem is valid.…”

“…This simple fact can be proved in different ways. We prefer to link it to the ideas of paper [5]. Let us begin with the case where there is only one vertex D in the complex besides A, B and C. The already mentioned trigonometry (see (31) and (32)) shows that with fixed A, B and C…”

“…Then, l AB and l DE satisfy one constraint (Cayley-Menger equation) which we can, using arguments like those in [5], represent in the following differential form:…”

Path integral measure and triangulation independence in discrete gravity

“…To do so one has to compute terms of the form ∂ω ∂l for which we will present a general strategy, similar to [46,47]. We need to extend the ideas in [46,47] in order to also obtain the matrix elements of the Hessian indexed by edges in the boundary. First we will derive two auxiliary formulas, which is the subject of the next section.…”