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 l 2 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 l 2 formulation can be seen as a q-analog of Nahm's monopole construction, so that the configuration approaches the BPS monopole at q → 1 limit.

Variant of ADHMN construction associated with q-analysis

“…and the * conjugation of this quaternion valued vector v * (z; q) = v * + (z; q) ⊕ v * − (z; q) being defined by the hermite conjugation in addition to the "twist" of q, v Under this definition of the ℓ 2 inner product, we can confirm the "hermiticity" of q-difference operator iD z [14]…”

“…The procedure to fix φ −1/2 is in the similar manner to the q-deformation of BPS monopole case [14]. We finally find,…”

“…The anti-selfduality of the gauge field is obvious from (25) as in the q-deformed monopole case [14]. The entire information on the gauge connection and the curvature is included in the vector V q , so that they can be obtained by the canonical way but are very complex form.…”

“…In [14], we have made such a deformation of the BPS monopole by introducing an ℓ 2 functional space in place of L 2 functional space. We apply the same approach to the caloron construction described above.…”

On the Second-Order Quantum (p,q)-Difference Equations with Separated Boundary Conditions

“…The -Coulomb problem and -hydrogen atom were studied by [18][19][20][21]. In addition, Yang-Mills theories and also -Yang-Mills equation were developed by [22][23][24]. The theory of quantum group applied to vibration and rotation molecules with -algebra and -Heisemberg algebra technique was established in [25][26][27].…”