Simplicial vs. Continuum String Theory and Loop Equations

We derive loop equations in a scalar matrix field theory. We discuss their solutions in terms of simplicial string theory -- the theory describing embeddings of two--dimensional simplicial complexes into the space--time of the matrix field theory. This relation between the loop equations and the simplicial string theory gives further arguments that favor one of the statements of the paper hep-th/0407018. The statement is that there is an equivalence between the partition function of the simplicial string theory and the functional integral in a continuum string theory -- the theory describing embeddings of smooth two--dimensional world--sheets into the space--time of the matrix field theory in question.Comment: 6 page

Towards the Theory of Non--Abelian Tensor Fields I

We present a triangulation--independent area--ordering prescription which naturally generalizes the well known path ordering one. For such a prescription it is natural that the two--form ``connection'' should carry three ``color'' indices rather than two as it is in the case of the ordinary one--form gauge connection. To define the prescription in question we have to define how to {\it exponentiate} a matrix with three indices. The definition uses the fusion rule structure constants.Comment: 22 pages, 18 figure

Expansion in Feynman Graphs as Simplicial String Theory

We show that the series expansion of quantum field theory in the Feynman diagrams can be explicitly mapped on the partition function of the simplicial string theory -- the theory describing embeddings of the two--dimensional simplicial complexes into the space--time of the field theory. The summation over two--dimensional geometries in this theory is obtained from the summation over the Feynman diagrams and the integration over the Schwinger parameters of the propagators. We discuss the meaning of the obtained relation and derive the one--dimensional analog of the simplicial theory on the example of the free relativistic particle.Comment: Latex, 11pp, Minor mintakes are correcte