Atsushi Nakamula

One-parameter family of selfdual solutions in classical Yang-Mills theory

Abstract The ADHM construction, which yields (anti-)selfdual configurations in classical Yang-Mills theories, is applied to an infinite dimensional l2 vector space, and as a consequence, a family of (anti-)selfdual configurations with a parameter q is obtained for SU(2) Yang-Mills theory. This l2 formulation can be seen as a q-analog of Nahms monopole construction, so that the configuration approaches the BPS monopole at q→1 limit.

Riemann–Liouville integrals of fractional order and extended KP hierarchy

An attempt to formulate the extensions of the KP hierarchy by introducing fractional-order pseudo-differential operators is given. In the case of the extension with the half-order pseudo-differential operators, a system analogous to the supersymmetric extensions of the KP hierarchy is obtained. Unlike the supersymmetric extensions, no Grassmannian variable appears in the hierarchy considered here. More general hierarchies constructed by the 1/Nth-order pseudo-differential operators, their integrability and the reduction procedure are also investigated. In addition to finding the new extensions of the KP hierarchy, a brief introduction to the Riemann–Liouville integral is provided to yield a candidate for the fractional-order pseudo-differential operators.

Multi-Calorons Revisited

Analytic Nahm data is re-examined for SU(2) calorons, or periodic instantons, of instanton charge 2. The Nahm equations are solved analytically in terms of Jacobi elliptic functions and the possible matching conditions are classified. The dimensions of framed moduli space for charge 2 caloron is enumerated and the maximal parameter, which is 16 for charge 2, case is identified. The monopole and instanton limits are also considered. It is found that the Nahm data which does not correspond to the standard monopoles plays significant role for calorons.

Magnetic solutions to 2 + 1 dimensional gravity with dilaton field

Abstract We show a general method to solve 2 + 1 dimensional dilatonic Maxwell-Einstein equation with a positive or negative cosmological constant. All the physical solutions are listed with assumptions that they are static, rotationally symmetric, and has a nonzero magnetic field and a nonzero dilaton field. On the contrary to the magnetic solution without a dilaton field, some of the present solutions with a dilaton field possess a horizon.

Atiyah-Manton construction of Skyrmions in eight dimensions

A bstractWe show that the eight-dimensional instanton solution, which satisfies the self-duality equation F ∧ F = ∗8F ∧ F, realizes the static Skyrmion configuration in eight dimensions through the Atiyah-Manton construction. The relevant energy functional of the Skyrme field is obtained by the formalism developed by Sutcliffe. By comparing the Skyrmion solution associated with the extreme of the energy, with the Atiyah-Manton solution constructed by the instantons, we find that they agree with high accuracy. This is a higher-dimensional analogue of the Atiyah-Manton construction of Skyrmions in four dimensions. Our result indicates that the instanton/Skyrmion correspondence seems to be an universal property in 4k (k = 1, 2,…) dimensions.

Numerical Nahm transform for 2-caloron solutions

Abstract A new numerical method for performing the Nahm transform for charge k = 2 caloron is presented. The Weyl equations with boundary impurities are solved directly and the determination of the appropriate basis to the linear system is established. The action densities of the 2-calorons with 10 moduli parameters are shown.

A q-analogue of the ADHMN construction and the axisymmetric multi-instantons

In the preceding paper, the present authors presented a q-analogue of the ADHMN construction and obtained a family of anti-selfdual configurations with a parameter q for classical SU(2) Yang-Mills theory in four-dimensional Euclidean space. The family of solutions can be seen as a q-analogue of the single BPS monopole preserving (anti-)selfduality. Further discussion is given of the relation to the axisymmetric ansatz on the anti-selfdual equation given by Witten in the late 1970s. It is found that the q-exponential functions familiar in q-analysis appear as analytic functions categorizing the anti-selfdual configurations yielded by the axisymmetric ansatz.

Aspects of C3-symmetric calorons from numerical Nahm transform

Calorons are finite action solutions to the anti-selfdual Yang-Mills equations on R3 × S1. They are generally constructed by the so called Nahm construction. We perform the numerical Nahm transform for the Nahm data of 3-calorons with C3-symmetry, which do not have the monopole limits. Dissimilar to the cases of having monopole limits, we can trace the zero-circumference limit of S1. It is found that the action density of the calorons tends to fade away as S1 shrinks.

Variant of ADHMN construction associated with q-analysis

A q-deformation of the ADHMN caloron construction is considered, under which the anti-self-dual (ASD) conditions of the gauge fields are preserved. It is shown that the q-dependent Nahm data with certain constraints are crucial to determine the ASD gauge fields, as in the case of ordinary caloron construction. As an application of the q-deformed ADHMN construction, we give a q-deformed caloron of Harrington-Shepard type. Some limits of the parameters are also considered.

Selfdual solution of classical yang—mills fields through a q-analog of ADHM construction

Abstract The ADHMN construction of selfdual Yang—Mills gauge fields is reformulated by introducing a functional space defined on a discrete point set. Using this formalism, a q -analogous object of the BPS monopole is obtained.