Masatoshi Sato

N fold supersymmetry in quantum mechanics: General formalism

Abstract We report general properties of N -fold supersymmetry in one-dimensional quantum mechanics. N -fold supersymmetry is characterized by supercharges which are N th polynomials of momentum. Relations between the anti-commutator of the supercharges and the Hamiltonian, the spectra, the Witten index, the non-renormalization theorems and the quasi-solvability are examined. We also present further investigation about a particular class of N -fold supersymmetric models which we dubbed typexa0A. Algebraic equations which determine a part of spectra of typexa0A models are presented, and the non-renormalization theorem are generalized. Finally, we present a possible generalization of N -fold supersymmetry in multi-dimensional quantum mechanics.

Topological superconductors: a review

This review elaborates pedagogically on the fundamental concept, basic theory, expected properties, and materials realizations of topological superconductors. The relation between topological superconductivity and Majorana fermions are explained, and the difference between dispersive Majorana fermions and a localized Majorana zero mode is emphasized. A variety of routes to topological superconductivity are explained with an emphasis on the roles of spin-orbit coupling. Present experimental situations and possible signatures of topological superconductivity are summarized with an emphasis on intrinsic topological superconductors.

General Forms of a N-fold Supersymmetric Family

Abstract We report general forms of one family of the N -fold supersymmetry in one-dimensional quantum mechanics. The N -fold supersymmetry is characterized by the supercharges which are N th order in differential operators. The family reported here is defined as a particular form of the supercharges and is referred to as “type A”. We show that a quartic and a periodic potentials, which were previously found to be N -fold supersymmetric by the authors, are realized as special cases of this type A family.

Valley views: instantons, large order behaviors, and supersymmetry

Abstract 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. Evidence for this claim is presented in various ways. The conventional perturbation theory and the non-perturbatioe calculation are unified, and the ambiguity of the Borel transform of the perturbation series is removed. A ‘proof’ of Bogomolnys “trick” is presented, which enables us to go beyond the dilute-gas approximation. The prediction of the large order behavior of perturbation theory is confirmed by explicit calculations, in some cases to the 478th 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 Schrodinger equation.

Multi-instanton calculus in N = 2 supersymmetric QCD

Microscopic tests of the exact results are performed in N = 2 SU(2) supersymmetric QCD. We construct the multi-instanton solution in N = 2 supersymmetric QCD and calculate the two-instanton contribution F2 to the prepotential F explicitly. For Nf = 1,2, instanton calculus agrees with the prediction of the exact results, however, for Nf = 3, we find a discrepancy between them.

“Topological” formulation of effective vortex strings

We present a “topological” formulation of arbitrarily shaped vortex strings in four dimensional field theory. By using a large Higgs mass expansion, we then evaluate the effective action of the closed Abrikosov-Nielsen-Olesen vortex string. It is shown that the effective action contains the Nambu-Goto term and an extrinsic curvature squared term with negative sign. We next evaluate the topological FμνFμν term in the case where the vortex string manifold extends out to the boundary of the time direction of space-time and find that it becomes the sum of an ordinary self-intersection number and Polyakovs self-intersection number of the world sheet swept by the vortex string. These self-intersection numbers are related to the self-linking number and the total twist number, respectively. Furthermore, the FμνFμν term turns out to be the difference between the sum of the writhing numbers and the linking numbers of the vortex strings at the initial time and the one at the final time. When the vortex string is coupled to fermions, the chiral fermion number of the vortex string becomes the writhing number (modulo Z) through the chiral anomaly. Our formulation is also applied to “global” vortex strings in a model with a broken global U(1) symmetry.

N fold supersymmetry for a periodic potential

Abstract We report a new type of supersymmetry, “ N -fold supersymmetry”, in one-dimensional quantum mechanics. Its supercharges are N th order polynomials of momentum: it reduces to ordinary supersymmetry for N =1 , but for other values of N the anticommutator of the supercharges is not the ordinary Hamiltonian, but is a polynomial of the Hamiltonian. (For this reason, the original Hamiltonian is referred to as the “Mother Hamiltonian”.) This supersymmetry shares some features with the ordinary variety, the most notable of which is the non-renormalization theorem. An N -fold supersymmetry was earlier found for a quartic potential whose supersymmetry is spontaneously broken. Here we report that it also holds for a periodic potential, albeit with somewhat different supercharges, whose supersymmetry is not broken.

sl(2) construction of type A N-fold supersymmetry

N-fold supersymmetry is an extension of the ordinary supersymmetry in one-dimensional quantum mechanics. One of its major property is quasi-solvability, which means that energy eigenvalues can be obtained for a portion of the spectra. We show that recently found type A N-fold supersymmetry can be constructed by using sl(2) algebra, which provides a basis for the quasi-solvability. By this construction we find a condition for the type A N-fold supersymmetry which is less restrictive than the condition known previously. Several explicitly known models are also examined in the light of this construction.

MULTI-INSTANTON CALCULUS VERSUS EXACT RESULTS IN N=2 SUPERSYMMETRIC QCD

Microscopic tests of the exact results are performed in N = 2 supersymmetric SU(2) QCD. We present the complete construction of the multi-instanton in N = 2 supersymmetric QCD. All the defining equations of the super instanton are reduced to the algebraic equations. Using this result, we calculate the two-instanton contribution F2 to the prepotential F for the arbitrary Nƒ theories. For Nƒ = 0, 1, 2, instanton calculus agrees with the prediction of the exact results, however, for Nƒ = 3, 4, we find discrepancies between them. We propose improved curves of the exact results for the massive Nƒ = 3 and massless Nƒ = 4 theories.

Recent Developments of the Theory of Tunneling

Path-integral approach in imaginary and complex time has been proven successful in treating the tunneling phenomena in quantum mechanics and quantum field theories. Latest developments in this field, the proper valley method in imaginary time, its application to various quantum systems, complex time formalism, asympton theory for the large order analysis of the perturbation theory, are reviewed in a self-contained manner.