Abstract
Motivated by applications to soft supersymmetry breaking, we revisit the expansion of the Seiberg-Witten solution around the multi-monopole point on the Coulomb branch of pure SU(N) mathcal{N} = 2 gauge theory in four dimensions. At this point N − 1 mutually local magnetic monopoles become massless simultaneously, and in a suitable duality frame the gauge couplings logarithmically run to zero. We explicitly calculate the leading threshold corrections to this logarithmic running from the Seiberg-Witten solution by adapting a method previously introduced by D’Hoker and Phong. We compare our computation to existing results in the literature; this includes results specific to SU(2) and SU(3) gauge theories, the large-N results of Douglas and Shenker, as well as results obtained by appealing to integrable systems or topological strings. We find broad agreement, while also clarifying some lingering inconsistencies. Finally, we explicitly extend the results of Douglas and Shenker to finite N , finding exact agreement with our first calculation.
Highlights
Motivated by applications to soft supersymmetry breaking, we revisit the expansion of the Seiberg-Witten solution around the multi-monopole point on the Coulomb branch of pure SU(N ) N = 2 gauge theory in four dimensions
Upon softly breaking N = 2 → N = 1 (as reviewed below (1.4)) the threshold matrix is needed to determine the spectrum of light particles, as well as the confining string tensions. This is due to the fact that tk is the matrix of gauge-kinetic terms in the low-energy U(1)ND−1 gauge theory that couples to the N − 1 massless monopole hypermultiplets at the multi-monopole point
In upcoming work [18] we extend this to G = SU(N ) for all N, where the soft supersymmetry-breaking mass deformation leads to a rich structure of phases and phase transitions that can be analyzed by focusing on the multi-dyon points
Summary
The computations described in this paper were motivated by applications of Seiberg-Witten theory that require more detailed information about the multi-monopole point than the leading logarithmic running of the couplings in (1.4) or the value of the ak-periods in (1.5). (Two such applications are mentioned below.) Our primary interest will be the leading regular terms in (1.4), which we parametrize as follows,. Upon softly breaking N = 2 → N = 1 (as reviewed below (1.4)) the threshold matrix is needed to determine the spectrum of light particles, as well as the confining string tensions Speaking, this is due to the fact that tk is the matrix of gauge-kinetic terms in the low-energy U(1)ND−1 gauge theory that couples to the N − 1 massless monopole hypermultiplets at the multi-monopole point. In upcoming work [18] we extend this to G = SU(N ) for all N , where the soft supersymmetry-breaking mass deformation leads to a rich structure of phases and phase transitions that can be analyzed by focusing on the multi-dyon points This analysis crucially depends on the detailed properties of the threshold matrix tk in (1.6).
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