Numerical Nahm transform for 2-caloron solutions

Abstract

A new numerical method for performing the Nahm transform for charge $k=2$ caloron is presented. The Weyl equations with boundary impurities are solved directly and the determination of the appropriate basis to the linear system is established. The action densities of the 2-calorons with 10 moduli parameters are shown.Comment: 8 pages, 4 figures, 1 table, v2: coincides with Phys.Lett.B versio

Symmetric calorons and the rotation map

“…for G ∈ GL(k, C). We denote by M j k (ρ 2 ) the set of matrices in M k satisfying (29)- (32), and denote by M j k (ρ) the set of matrices in M j k (ρ 2 ) satisfying (23)- (28). The purpose behind the second consideration is due to the inclusion M j k (ρ) ⊂ M j k (ρ 2 ), which is seen by setting…”

“…• Finally we fix the matrix g appearing in (23)- (28) in such a way which preserves these gauge choices, and this allows us to solve the equations completely.…”

“…The parameters α p , u, v, y, z ∈ C * , and β p ∈ C are ultimately constrained by the monad equation (9), full-rank-conditions (10), and the symmetry equations (23)- (28). We would like to fix a gauge such that the symmetry equations are more manageable.…”

“…In this case, the monad equations tell us nothing useful about the other parameters. For this, we rely on the symmetry equations (23)- (28), for which we need a gauge transformation g ∈ GL(k, C). A convenient way to write such a matrix is in terms of the 'basis' {S p } k p=1 :…”

“…and (25)- (28) imply that the only non-zero entry in columns j and j/2, and rows k − j/2 − 1 and k − 1 are given by γ x k− j 2 (for the corresponding value of x). In other words, for a = k − j 2 γ p a = 0, for all p ∈ a + j, a +…”