Aq-analogue of the ADHMN construction and the axisymmetric multi-instantons

Abstract

In the preceding paper (1999 Phys.Lett. B463 257), the authors presented a q-analog 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-analog of the single BPS monopole preserving (anti-)selfduality. Further discussion is made on the relation to axisymmetric ansatz on anti-selfdual equation given by Witten in the late seventies. It is found that the q-exponential functions familiar in q-analysis appear as analytic functions categorizing the anti-selfdual configurations yielded by axisymmetric ansatz.

On a q-Analog of ADHMN Construction for Self-Dual Yang-Mills

“…Another example is given in ref. [8] in which a function space defined on a multiplicatively located discrete point set such as ±1/2, ±q/2, ±q 2 /2, ±q 3 /2, . .…”

“…Here we consider the q-analog formulation in ref. [8] and its generalization, in which we need appropriate inner products < w, v > q compatible with SD condition on the qderivative operator in new ∆'s. As we will see the q-analog formulations approach the Nahm formulation for monopoles when q → 1, since then q-derivative tends to ordinary derivative.…”

“…Hence, to understand more about the SDYM equations, and/or the reduced systems, we need to investigate further concrete examples of exact analytic solutions. In this context the author and the collaborator obtained a new family of solutions to SDYM through constructing a q-analog of the ADHM method [8].…”