The Non-Compact Weyl Equation

for three n × n anti-hermitian matrices T i (the so-called Nahm data) of complex-valued functions of the variable s, where n is the magnetic charge of the BPS monopole configuration.

The tensor ε ijk is the totally antisymmetric tensor.

In the ADHMN approach, the construction of SU(n + 1) monopole solutions of the Bogomolny equation with topological charge n is translated to the following problem which is known as the inverse Nahm transform [1]. Given the Nahm data for a n-monopole the one-dimensional Weyl equation

for the complex 2n-vector v(x, s), must be solved. I n denotes the n × n identity matrix,

x = (x 1 , x 2 , x 3 ) is the position in space at which the monopole fields are to be calculated. In the minimal symmetry breaking case, the Nahm data T i ’s can be cast as (see Reference [2],

for a more detailed discussion)

where τ i ’s form the n-dimensional representation of SU(2) and satisfy:

Let us choose an orthonormal basis for these solutions, satisfying v † v ds = I.

(5)

Given v(x, s), the normalized vector computed from ( 2) and ( 5), the Higgs field Φ and the gauge potential A i are given by

In [3,4], we applied the ADHMN construction to obtain the SU(n+1) (for generic values of n) BPS monopoles with minimal symmetry breaking, by solving the Weyl equation. In this paper, we present a non-compact approach of the ADHMN transform by introducing an infinite dimensional spin zero representation of the sl 2 algebra for the Nahm data. The aforementioned representation is expressed in terms of appropriate differential operators;

hence, the Weyl equation is also written in terms of the aforementioned differential operators, and not in terms of n × n matrices as in its conventional form (see, for example, Ref. [3,4]).

In the Appendix, we present the equivalence between the two approaches, i.e. matrix versus differential operator description of the Weyl equation, which leads us to conjecture that the results of the present investigation should by construction satisfy the Bogomolny equation. This is mainly due to the structural similarity between the equations arising in the present case, and the ones emerging in the finite dimensional case described in the Appendix and in Ref. [3]. Nevertheless, this is an intriguing issue, which merits further investigation, in particular when azimuthal dependence is also implemented along the lines described in [4].

In order to construct the non-compact BPS monopole solutions of the Weyl equation, let us consider the sl 2 algebra, and focus on the non-trivial spin zero representation.

Consider the general case: i.e. the spin S ∈ R representation of sl 2 of the form

Also take the inner product, in the basis of polynomials of ξ on the unit circle (ξ = e iθ ), to be of the form:

and immediately obtain the formula

Next consider the generic state

where

Notice that using the representation ( 8), for S being an integer or half integer; together with the inner product ( 9) of an appropriate orthonormal basis {v 1 , . . . , vn+1 } where n = 2S + 1 being the dimension of the representation (see also Appendix for more details):

one may recover the Higgs field obtained in [3] from the formula

Next, we focus on the the spin zero representation of sl 2 , associated to the Möbius transformation and also relevant in high energy QCD (see for example, Ref. [5,6]). Again we consider the spherically symmetric case (that is, x i = rδ i3 ) where the Nahm data are given by (3) for f i = f = -1 s . Substituting the Nahm data (3) where τ i ’s are defined by (8) for S = 0 to the Weyl equation ( 2) and expressing σ i in terms of the spin 1 2 representation; that is equation ( 8) for S = 1 2 :

one gets

Next, by setting

Here, ẇk and uk are the total derivatives of the functions w k and u k with respect to the argument s. Note that our results are analogous to the ones obtained in [3].

Let us now solve these equations. The coupled equations for u k+1 and w k are equivalent by expressing u k+1 in terms of w k :

to the single second-order equation

Then, the solution of ( 18) is given in a closed form, in terms of the Kummer functions as

where c i (r) for i = 1, 2 are constants. M (-k, 2, 2rs) is the regular confluent hypergeometric Kummer function and U (-k, 2, 2rs) is the Tricomi confluent hypergeometric function defined in Table 1 1 . These functions are widely known as the Kummer functions of first and second kind, respectively, and are linearly independent solutions of the Kummer equation [7]. Finally, the corresponding function u k+1 given by (17) takes the simple form

The next step is to choose an orthogonal basis of the infinite dimensional space. Consider the following functions:

which are orthogonal by construction. Then the norm of such a function is given by

As it can be observed from Table 1 the arbitrary constant c 2 (r) at ( 19) and (20) should be set equal to zero in order t

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