The Adhm Construction of Instantons on Noncommutative Spaces

Abstract

Abstract. We present an account of the ADHM construction of instantons on Euclidean space-time R 4 from the point of view of noncommutative geometry. We recall the main ingredients of the classical construction in a coordinate algebra format, which we then deform using a cocycle twisting procedure to obtain a method for constructing families of instantons on noncommutative space-time, parameterised by solutions to an appropriate set of ADHM equations. We illustrate the noncommutative construction in two special cases: the Moyal-Groenewold plane R 4 and the Connes-Landi plane R 4 θ .

Moduli spaces of instantons on toric noncommutative manifolds

“…An index argument expresses the dimension of the moduli space in terms of Chern classes of the underlying vector bundles, in parallel with the classical analysis [2] although now in noncommutative parlance. In the special example of the toric noncommutative four-sphere S 4 θ , we find that the moduli space of instantons on a vector bundle with fixed topological charge k ∈ Z has dimension 8k − 3, in agreement with the value suggested by the results of [6,18].…”

“…As in previous works [5,6], the crucial tool in the present paper will be a functorial deformation procedure to derive the noncommutative geometry of M θ from the classical geometry of M in a systematic way, by deforming along an action of the N -torus T N . This "quantisation functor" constitutes the foundation upon which our construction is built, in the sense that it explains very precisely which aspects of the classical geometry are preserved by the deformation.…”

Higher abelian gauge theory associated to gerbes on noncommutative deformed M5-branes and S-duality

“…The nondegeneracy is explained in [4] for example. 7 Either of the two expressions, (2.15) or (2.17), can be taken to be the kinetic term of the H-field.…”

“…The equations of motion (EOM) and Bianchi identity 3 of the H-field can be derived as in the case of noncommutative Yang-Mills [23] to be 4) where d † θ is the adjoint of d with respect to the deformed inner product…”

“…Therefore, for p = 1, the noncommutative deformation removes this subtlety or difficulty that has plagued the study of the M5-brane. 4 3. The third goal is to calculate the partition function of the model.…”