Loops, surfaces and Grassmann representation in two- and three-dimensionalIsing models

We construct an algebraic representation of the geometrical objects (loop a nd surface variables) dual to the spins in 2 and 3D Ising models. This alge braic calculus is simpler than dealing with the geometrical objects, in par ticular when analyzing geometry factors and counting problems. For the 2D c ase we give the corrected loop expansion of the free energy and the radius of convergence for this series. For the 3D case we give a simple derivation of the geometry factor which prevents overcounting of surfaces in the intr insic geometry representation of the partition function, and find a classif ication of the surfaces to be summed over. For 2 and 3D we derive a compact formula for an-point functions in loop (surface) representation.