Stephen L. Adler
I describe how my involvement with monopoles related to the multimonopole existence proof of Taubes, and how my later work on quaternionic quantum mechanics led to the classification theorem for generalized projective group representations of Tao and Millard.
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We use the SU(2) 't Hooft-Polyakov monopole configuration, and its BPS version, to test the integral equations of the Yang-Mills theory. Those integral equations involve two (complex) parameters which do not appear in the differential Yang-Mills equations, and if they are considered to be arbitrary it then implies that non-abelian gauge theories (but not abelian ones) possess an infinity of integral equations. For static monopole configurations only one of those parameters is relevant. We expand the integral Yang-Mills equation in a power series of that parameter and show that the 't Hooft-Pol…
The well-known Bogomol'nyi-Prasad-Sommerfeld (BPS) monopole is considered in the limit of the infinite mass of the Higgs field as a basis for constructing the Yang-Mills vacuum with the finite energy density. In this limit the Higgs field disappears at the spatial infinity, but it leaves, nevertheless, its trace as vacuum Yang-Mills BPS monopoles transformed into Wu-Yang monopoles obtained in the pure Yang-Mills theory by a spontaneous scale symmetry breaking in the class of functions with zero topological charges. The topological degeneration of a vacuum BPS monopole manifests itself via Grib…
We show that the closed-form monotone solution linking two different vacuum states, for the exterior Yang--Mills wave equation over the extremal Reissner--Nordström spacetime found in a recent work of Bizoń and Kahl, is the unique solution to an associated domain-wall type energy minimization problem. We also derive the accompanying interior Yang--Mills wave equation within the same formalism. We obtain a few more closed-form solutions, regular and singular, among other solutions of oscillatory behavior, and discuss several interesting features of the solutions based on some energy considerati…
In this expository review we discuss various aspects of gauge theory. While the focus is on mathematics, wherever possible we make contact with theoretical high energy physics. Particular emphasis is placed on instantons and monopoles, which admit physical interpretation, and yield interesting and nontrivial mathematics. We give a clear and essentially self-contained exposition of the mathematical structure of the Seiberg-Witten monopole equations. Other topics include Donaldson's theorem on moduli spaces of monopoles, compactification of spaces of ASD connections, The Abelian monopole equatio…
We show that, in some limit, gravitational instantons correspond to exotic smooth geometry on R^4. The geometry of this exotic R^4 represents magnetic monopoles of Polyakov-'t Hooft type and the BPS condition allows for the generation of the charges from the gravitational sources of exotic R^4. Higgs field is, as usual, present in the monopole configurations. We indicate some possible scenarios in cosmology, particle physics and condensed matter where pure SU(2) Yang-Mills theory on the exotic R^4 aquires non-zero mass for its gauge boson when the smoothness is changed to the standard R^4 and,…
We establish that the Wu-Yang monopole needs the introduction of a magnetic point source at the origin in order for it to be a solution of the differential and integral equations for the Yang-Mills theory. That result is corroborated by the analysis through distribution theory, of the two types of magnetic fields relevant for the local and global properties of the Wu-Yang solution. The subtlety lies on the fact that with the non-vanishing magnetic point source required by the Yang-Mills integral equations, the Wu-Yang monopole configuration does not violate, in the sense of distribution theory…
We present a systematic approach to the linearised Yang-Mills-Higgs equations in the background of a 't Hooft-Polyakov monopole and use it to unify and extend previous studies of their spectral properties. We show that a quaternionic formulation allows for a compact and efficient treatment of the linearised equations in the BPS limit of vanishing Higgs self-coupling, and use it to study both scattering and bound states. We focus on the sector of vanishing generalised angular momentum and analyse it numerically, putting zero-energy bound states, Coulomb bound states and infinitely many Feshbach…
In this paper we give a mathematical proof of the existence of the time independent and spherically symmetric solution to the 't Hooft-Polyakov model of magnetic monopole by using 2D-shooting method.
For any locally defined $G_{2}-$structure and Hermitian-Yang-Mills connection on $S^{6}$, the $G_{2}-$monopole equation always admits a local solution that is asymptotic to the HYM-connection.
In this paper we study spherically symmetric monopoles, which are critical points for the Yang-Mills-Higgs functional over a disk in 3 dimensions, with prescribed degree and covariant constant at the boundary. This is a 3-dimensional gauge-theory generalization of the Ginzburg-Landau model in 2 dimensions.
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