Instantons in curvilinear coordinates

The multi-instanton solutions by ’tHooft and Jackiw, Nohl & Rebbi are generalized to curvilinear coordinates. Expressions can be notably simplified by the appropriate gauge transformation. This generates the compensating addition to the gauge potential of pseudoparticles. Singularities of the compensating connection are irrelevant for physics but affect gauge dependent quantities. The third connection The years that passed since the discovery of instantons, [1], did not bring answer to the question about the role of instantons in QCD, [2, 3]. As far as confinement remains a puzzle all references to instantons at long scales are ambiguous. Indications may come from studies of instanton effects in phenomenological models. These could tell whether confinement may seriously affect pseudoparticles and v. v. Common confinement models look most natural in non-Cartesian coordinate frames. The obvious choice for bags are 3+1-cylindrical, i. e. 3-spherical+time, coordinates while strings would prefer 2+2-cylindrical (2+1-cylindrical+time) geometry. Nevertheless instantons were usually discussed in the Cartesian frame (that was ideal in vacuum). The purpose of the present work is to draw attention to the problem and to develop the adequate technique. We shall generalize to curvilinear coordinates the multi-instanton solutions by ’tHooft and Jackiw, Nohl & Rebbi, [4], and simplify the formulae by the gauge transformation. Presently I don’t know whether the procedure is good for other topological configurations† but I would expect that it makes sense for the AHDM‡, [5], solution. We start from the basics of curvilinear coordinates in Sect. 1.1 and introduce the first two connections, namely the Levi-Civita connection and the spin connection. In Sect. 1.2 we describe the multi-instanton solutions. In Sect. 2 we shall rewrite instantons in nonCartesian coordinates and propose the gauge transform that makes formulae compact. The price will be the appearance of the third, so called compensating, gauge connection. The example of the O(4)-spherical coordinates is sketched in Sect. 3. Singularities of the gauged solution are discussed in Sect. 4. The last part summarizes the results. ∗The work is done with partial support of the RFBR grant 97-02-16131. I’m grateful to L. Lipatov† and S. Moch‡ for the questions.