## BPS Z(N) String Tensions, Sine Law and Casimir Scaling and Integrable Field Theories

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Kneipp, Marco A. C.

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We consider a Yang-Mills-Higgs theory with spontaneous symmetry breaking of
the gauge group G -> U(1)^r -> C(G), with C(G) being the center of G. We study
two vacua solutions of the theory which produce this symmetry breaking. We show
that for one of these vacua, the theory in the Coulomb phase has the mass
spectrum of particles and monopoles which is exactly the same as the mass
spectrum of particles and solitons of two dimensional affine Toda field theory.
That result holds also for N=4 Super Yang-Mills theories. On the other hand, in
the Higgs phase, we show that for each of the two vacua the ratio of the
tensions of the BPS Z(N) strings satisfy either the Casimir scaling or the sine
law scaling for G=SU(N). These results are extended to other gauge groups: for
the Casimir scaling, the ratios of the tensions are equal to the ratios of the
quadratic Casimir constant of specific representations; for the sine law
scaling, the tensions are proportional to the components of the left
Perron-Frobenius eigenvector of Cartan matrix and the ratios of tensions are
equal to the ratios of the soliton masses of affine Toda field theories.

Comment: 21 pages, 10 figures. A correction on the title page

Comment: 21 pages, 10 figures. A correction on the title page

##### Keywords

High Energy Physics - Theory, High Energy Physics - Lattice, High Energy Physics - Phenomenology