We present the detailed derivation of the charge one periodic instantons -or calorons -with non-trivial holonomy for SU(2). We use a suitable combination of the Nahm transformation and ADHM techniques. Our results rely on our ability to compute explicitly the relevant Green's function in terms of which the solution can be conveniently expressed. We also discuss the properties of the moduli space, IR 3 × S 1 × Taub-NUT/Z 2 and its metric, relating the holonomy to the Taub-NUT mass parameter. We comment on the monopole constituent description of these calorons, how to retrieve topological charge in the context of abelian projection and possible applications to QCD. The Nahm transformationWe will consider a U(n) bundle E with self-dual gauge connection A µ on a four manifold M = IR 4 /H, with instanton number k. Here H is a subgroup of translation symmetries under which the physics is invariant. When H is a four dimensional lattice, M will be the four torus [17]. Other four manifolds are obtained by taking appropriate limits [18]. We demand the gauge potential to be invariant modulo gauge transformations under the action of H.An essential ingredient in Nahm's construction [15] is to add a curvature free abelian connection, −2πiz µ dx µ , to the gauge field and to study the Weyl operatorσ µ = (1 2 , i τ ) andσ µ = σ † µ = (1 2 , −i τ ) are unit quaternions. As compared to usual conventions [17,18], we replaced z by −z to facilitate matching with the ADHM construction. When A is without flat factors (WFF, meaning that the vector bundle E does not split in E ′ ⊕ L for any flat line bundle L), then D z (A) will have a trivial kernel [19]. For such gauge fields G z (x, y) = (D † z (A)D z (A)) −1 is well-defined. The index theorem [20,21] shows

We determine all SU(2) caloron solutions with topological charge one and arbitrary Polyakov loop at spatial infinity (with trace 2 cos(2πω)), using the Nahm duality transformation and ADHM. By explicit computations we show that the moduli space is given by a product of the base manifold R 3 × S 1 and a Taub-NUT space with mass M = 1/ 8ω(1 − 2ω), for ω ∈ [0, 1 2 ], in units where S 1 = R/Z. Implications for finite temperature field theory and string duality between Kaluza-Klein and H-monopoles are briefly discussed.

We present a simple result for the action density of the SU(n) charge one periodic instantons -or calorons -with arbitrary non-trivial Polyakov loop P ∞ at spatial infinity. It is shown explicitly that there are n lumps inside the caloron, each of which represents a BPS monopole, their masses being related to the eigenvalues of P ∞ . A suitable combination of the ADHM construction and the Nahm transformation is used to obtain this result.

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We describe in mathematical detail the Nahm transformation which maps anti-self dual connections on the four-torus ($1) 4 onto anti-self-dual connections on the dual torus. This transformation induces a map between the relevant instanton moduli spaces and we show that this map is a (hyperK/ihler) isometry.