Explicit derivation of Yang-Mills self-dual solutions on non-commutative harmonic space

Abstract

We develop the noncommutative harmonic space (NHS) analysis to study the problem of solving the non-linear constraints eqs of noncommutative Yang-Mills selfduality in four-dimensions. We show that this space, denoted also as NHS(η, θ), has two SU(2) isovector deformations η (ij) and θ (ij) parametrising respectively two noncommutative harmonic subspaces NHS(η,0) and NHS(0,θ) used to study the selfdual and anti self-dual noncommutative Yang-Mills solutions. We reformulate the Yang-Mills self-dual constraint eqs on NHS(η,0) by extending the idea of harmonic analyticity to linearize them. Then we give a perturbative self dual solution recovering the ordinary one. Finally we present the explicit computation of an exact self-dual solution.

NC geometry and fractional branes

“…Moreover the solution of these eqs read, up to a normalization factor, as: 14) where x i are as in eqs(2.1), ̟ is the complex conjugate of ω and where 15) with α standing for ω k1 , ω k2 , ω k3 and their products. This solution shows clearly that Z 5 i , the product 4 i=1 Z i and their linear combination are all of them in the centre Z(Q nc ) of the NC algebra Q nc .…”

On noncommutative Calabi–Yau hypersurfaces

“…Since the original work of Connes et al on Matrix model compactification on non commutative (NC) torii [1], an increasing interest has been devoted to the study of non commutative spaces in connection with solitons in NC quantum field [2,3], and string field theories [4]. These NC solitons, which have been subject to an intensive interest during the last few years, are involved in the study of D(p − 4)/Dp brane systems (p > 3) of superstrings; in particular in the ADH construction of the D0/D4 system [5], in the determination of the vacuum field solutions of the Higgs branch of supersymmetric gauge theories with eight supercharges [6,7] and in tachyon condensation using the GMS approach [8].…”

Toric varieties with NC toric actions: NC type IIA geometry

“…Matrix model compactification of M theory on non commutative (NC) torii [1] has opened an increasing interest in the study of non commutative spaces, in relation with NC quantum field instantons [2], and open strings of the solitonic sector of type II string theories [3]- [5]. These NC structures have found remarkable applications in various areas of quantum physics such as in the analysis of D(p−4)/Dp brane systems (p > 3) [6,7] and in the study of tachyon condensation using the GMS method [8].…”