Noncommutative instantons and solitons

Abstract

I explain how to construct noncommutative BPS configurations in four and lower dimensions by solving linear matrix equations. Examples are instantons in D=4 Yang-Mills, monopoles in D=3 Yang-Mills-Higgs, and (moving) solitons in D=2+1 YangMills-Higgs. Some emphasis is on the latter as a showcase for the dressing method. Self-duality and BPS equationsIn this talk I shall present a powerful method for and results of constructing classical field configurations with finite action or energy in four-dimensional noncommutative gauge theory and its lower-dimensional descendants:I am setting up the formalism in such a way that it is completely transparent to the (Moyal-type) noncommutative deformation. In other words, the noncommutative equations below differ from the commutative ones merely in the interpretation of the symbols or their product (stars are suppressed). This will be briefly explained in Section 6. The Yang-Mills field equations are implied by first-order (self-duality or BPS) equations:where F and φ are u(n) valued and Greek indices run from 1 to 4 while Latin ones stop at 3. In complex coordinates (note the signs!) y = x 1 +ix 2 and z = x 3 ∓ix 4 the self-duality equationwhere the upper and lower signs belong to the signatures (4,0) and (2,2), respectively. Dimensional reduction to D=3 is accomplished via

Integrable noncommutative sine-Gordon model

“…This property has been shown to extend also to the noncommutative case 2 where integrable models in three and in two dimensions have been constructed [6,7,8,9,10,11]. In fact, almost all integrable noncommutative models in less than four dimensions 3 can be obtained in this fashion.…”

“…In the ordinary commutative case it is well known 1 that dimensional reduction of four dimensional SDYM gives rise to many integrable systems in three and two dimensions. This property has been shown to extend also to the noncommutative case 2 where integrable models in three and in two dimensions have been constructed [6,7,8,9,10,11]. In fact, almost all integrable noncommutative models in less than four dimensions 3 can be obtained in this fashion.…”

“…Summing all the contributions, for the ϕϕ → ϕϕ amplitude we arrive at 11) which perfectly describes a causal amplitude.…”

Noncommutative Instantons Revisited

“…The instanton number is computed to be −k. For reviews on noncommutative instantons, see [14,15,24,26,31,37].…”

Exact construction of noncommutative instantons

“…The instanton number is computed to be k. We will focus mainly on the case of (N, k) = (2, 1) in the subsequent discussions. For further reviews on noncommutative instantons, see [16,20,31,33,40,47].…”