Instantons in Large Order of the Perturbative Series

Behavior of the Euclidean path integral at large orders of the perturbation series is studied. When the model allows tunneling, the path-integral functional in the zero instanton sector is known to be dominated by bounce-like configurations at large order of the perturbative series, which causes non-convergence of the series. We find that in addition to this bounce the perturbative functional has a subleading peak at the instanton and anti-instanton pair, and its sum reproduces the non-perturbative valley.Comment: 9 pages (without figures), KUCP-6

The Canonical Lattice Isomorphism between Topologies Compatible with a Linear Space

We consider all compatible topologies of an arbitrary finite-dimensional vector space over a non-trivial valuation field whose metric completion is a locally compact space. We construct the canonical lattice isomorphism between the lattice of all compatible topologies on the vector space and the lattice of all subspaces of the vector space whose coefficient field is extended to the complete valuation field. Moreover, in this situation, we use this isomorphism to characterize the continuity of linear maps between finite-dimensional vector spaces endowed with given compatible topologies, and also, we characterize all Hausdorff compatible topologies.Comment: 17 pages, no figures, references and a new proposition (Proposition 5.2) are adde

The Berkovits Method for Conformally Invariant Non-linear Sigma-models on G/H

We discuss 2-dimmensional non-linear sigma-models on the Kaehler manifold G/H in the first order formalisim. Using the Berkovits method we explicitly construct the G-symmetry currents and primaries, when G/H are irreducible. It is a variant of the Wakimoto realization of the affine Lie algebra using a particular reducible Kaehler manifold G/U(1)^r with r the rank of G.Comment: 13 page

The Fuzzy S^4 by Quantum Deformation

The fuzzy algebra of S^4 is discussed by quantum deformation. To this end we embed the classical S^4 in the Kaehler coset space SO(5)/U(2). The harmonic functions of S^4 are constructed in terms of the complex coordinates of SO(5)/U(2). Being endowed with the symplectic structure they can be deformed by the Fedosov formalism. We show that they generate the fuzzy algebra \hat A_\infty (S^4) under the * product defined therein, by using the Darboux coordinate system. The fuzzy spheres of higher even dimensions can be discussed similarly. We give basic arguments for the generalization as well.Comment: 20 pages, LaTex, no figur

Valley Views: Instantons, Large Order Behaviors, and Supersymmetry

The elucidation of the properties of the instantons in the topologically trivial sector has been a long-standing puzzle. Here we claim that the properties can be summarized in terms of the geometrical structure in the configuration space, the valley. The evidence for this claim is presented in various ways. The conventional perturbation theory and the non-perturbative calculation are unified, and the ambiguity of the Borel transform of the perturbation series is removed. A `proof' of Bogomolny's ``trick'' is presented, which enables us to go beyond the dilute-gas approximation. The prediction of the large order behavior of the perturbation theory is confirmed by explicit calculations, in some cases to the 478-th order. A new type of supersymmetry is found as a by-product, and our result is shown to be consistent with the non-renormalization theorem. The prediction of the energy levels is confirmed with numerical solutions of the Schr\"{o}dinger equation.Comment: 78 pages, Latex, 22 eps figure