Benoit Charbonneau, Anuk Dayaprema, C. J. Lang, Ákos Nagy, Haoyang Yu
We study solutions to Nahm's equations with continuous symmetries and, under certain (mild) hypotheses, we classify the corresponding Ansätze. Using our classification, we construct novel Nahm data, and prescribe methods for generating further solutions. Finally, we use these results to construct new BPS monopoles with spherical symmetry.
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We study supersymmetric deformations of N = 4 quantum mechanics with a Kahler target space admitting a holomorphic isometry. We show that the twisted mass deformation generalises to a deformation constructed from matrix-valued functions of the moment map, which obey the Nahm equations. We also explain how N = 4 supersymmetry implies that the Berry connection on the vacuum bundle for this theory satisfies the BPS monopole equations. In the case where the target space is a Riemann sphere, our analysis reduces to the standard Nahm construction of monopoles. This generalises an earlier result by S…
We study the relationship between ADHM/Nahm construction and ``solution generating technique'' of BPS solitons in noncommutative gauge theories. ADHM/Nahm construction and ``solution generating technique'' are the most strong ways to construct exact BPS solitons. Localized solitons are the solitons which are generated by the ``solution generating technique.'' The shift operators which play crucial roles in ``solution generating technique'' naturally appear in ADHM/Nahm construction and we can construct various exact localized solitons including new solitons: localized periodic instantons (=loc…
In a paper of Braam and Austin, $\text{SU}(2)$ magnetic monopoles in hyperbolic space $H^{3}$ were shown to be the same as solutions to matrix-valued difference equations called the discrete Nahm equations. Here, I discover the $(N-1)$-interval discrete Nahm equations and show that their solutions are equivalent to $\text{SU}(N)$ hyperbolic monopoles. These discrete time evolution equations on an interval feature a jump in matrix dimensions at certain points in the evolution, which are given by the mass data of the corresponding monopole. I prove the correspondence with higher rank hyperbolic …
The discrete Nahm equations, a system of matrix valued difference equations, arose in the work of Braam and Austin on half-integral mass hyperbolic monopoles. We show that the discrete Nahm equations are completely integrable in a natural sense: to any solution we can associate a spectral curve and a holomorphic line-bundle over the spectral curve, such that the discrete-time DN evolution corresponds to walking in the Jacobian of the spectral curve in a straight line through the line-bundle with steps of a fixed size. Some of the implications for hyperbolic monopoles are also discussed.
We show that any cyclically symmetric monopole is gauge equivalent to Nahm data given by Sutcliffe's ansatz, and so obtained from the affine Toda equations. Further the direction (the Ercolani-Sinha vector) and base point of the linearising flow in the Jacobian of the spectral curve associated to the Nahm equations arise as pull-backs of Toda data. A theorem of Accola and Fay then means that the theta-functions arising in the solution of the monopole problem reduce to the theta-functions of Toda.
We elaborate on a novel model of N=4 supersymmetric mechanics with extra spin variables. A dynamical linear (1,4,3) multiplet is coupled to a "semi-dynamical" linear (3,4,1) multiplet representing spin degrees of freedom in a Wess-Zumino action. The unique coupling of these two multiplets relates the dynamical bosonic variable to an arbitrary harmonic function of the SU(2) triplet of spin variables. As we prove at the classical and quantum level, N=4 supersymmetry is equivalent to the Nahm equations for the spin variables, with the dynamical boson as evolution parameter. We treat in detail the…
On the construction of monopoles with arbitrary symmetry breaking
本文提出从一类新的Nahm数据出发，成功构造出具有任意预设对称性破缺的有限能量BPS单极子。
BPS monopoles on $\mathbb{R}^2\times S^1$ correspond, via the generalized Nahm transform, to certain solutions of the Hitchin equations on the cylinder $\mathbb{R}\times S^1$. The moduli space M of two monopoles with their centre-of-mass fixed is a 4-dimensional manifold with a natural hyperkähler metric, and its geodesics correspond to slow-motion monopole scattering. The purpose of this paper is to study the geometry of M in terms of the Nahm/Hitchin data, i.e. in terms of structures on $\mathbb{R}\times S^1$. In particular, we identify the moduli, derive the asymptotic metric on M, and disc…
Analytic Nahm data is re-examined for SU(2) calorons, or periodic instantons, of instanton charge 2. The Nahm equations are solved analytically in terms of Jacobi elliptic functions and the possible matching conditions are classified. The dimensions of framed moduli space for charge 2 caloron is enumerated and the maximal parameter, which is 16 for charge 2, case is identified. The monopole and instanton limits are also considered. It is found that the Nahm data which does not correspond to the standard monopoles plays significant role for calorons.
We explain a new construction of self-dual string solutions to the non-abelian two-form self-duality equation proposed in \cite{CK}. This class of self-dual strings is determined by the BPS monopoles in four-dimensions and the self dual string charge is given by the charge of the monopole. Our construction covers the SO(4) invariant self-dual string solutions found previously. We have also constructed, based on the 't Hooft-Polyakov monopole, a singular solution that describes two finitely separated M5-branes meeting midway in between. We comment that as BPS monopoles are generally given by th…
I survey the theory of Bogomolny monopoles and the various approaches to their study. These include the spectral curve, Nahm correspondence and rational maps.
We discuss the explicit formulation of the transcendental constraints defining spectral curves of SU(2) BPS monopoles in the twistor approach of Hitchin, following Ercolani and Sinha. We obtain an improved version of the Ercolani-Sinha constraints, and show that the Corrigan-Goddard conditions for constructing monopoles of arbitrary charge can be regarded as a special case of these. As an application, we study the spectral curve of the tetrahedrally symmetric 3-monopole, an example where the Corrigan-Goddard conditions need to be modified. A particular 1-cycle on the spectral curve plays an im…
We develop a method for finding the zero modes of the Dirac operator in the presence of BPS monopoles. We use it to find the zero modes in the case of Abelian BPS monopoles in $\mathbb R^3$.
Periodic instantons, also called calorons, are the BPS solutions to the pure Yang-Mills theories on $\mathbb{R}^3\times S^1$. It is known that the calorons interconnect with the instantons and the BPS monopoles as the ratio of their size to the period of $S^1$ varies. We give, in this paper, the action density configurations of the $SU(2)$ calorons of higher instanton charges with several platonic symmetries through the numerical Nahm transform, after the construction of the analytic Nahm data. The calorons considered are 5-caloron with octahedral symmetry, 7-caloron with icosahedral symmetry,…
We show that the Higgs and gauge fields for a BPS monopole may be constructed directly from the spectral curve without having to solve the gauge constraint needed to obtain the Nahm data. The result is the analogue of the instanton result: given ADHM data one can reconstruct the gauge fields algebraically together with differentiation. Here, given the spectral curve, one can similarly reconstruct the Higgs and gauge fields. This answers a problem that has remained open since the discovery of monopoles.
We apply the ADHMN construction to obtain the SU(n+1)(for generic values of n) spherically symmetric BPS monopoles with minimal symmetry breaking. In particular, the problem simplifies by solving the Weyl equation, leading to a set of coupled equations, whose solutions are expressed in terms of the Whittaker functions. Next, this construction is generalized for non-commutative SU(n+1) BPS monopoles, where the corresponding solutions are given in terms of the Heun B functions.
Non-BPS Monopoles and Dyons via Resurgent Transseries
该研究通过 resurgence 转换级数构造了非BPS单极子和双极子，揭示了其在BPS极限下的行为特征。
We construct SU(n+1) BPS spherically symmetric monopoles with minimal symmetry breaking by solving the full Weyl equation. In this context, we explore and discuss the existence of open spin chain-like part within the Weyl equation. For instance, in the SU(3) case the relevant spin chain is the 2-site spin 1/2 XXX chain with open boundary conditions. We exploit the existence of such a spin chain part in order to solve the full Weyl equation.
Dihedrally Symmetric Monopoles and Affine Toda Equations
本文证明了具有二面体对称性的$SU(2)$ BPS单极子在不同电荷下规范等价于特定类型的仿射Toda方程导出的Nahm数据。
Recently a Nahm transform has been discovered for magnetic bags, which are conjectured to arise in the large n limit of magnetic monopoles with charge n. We interpret these ideas using string theory and present some partial proofs of this conjecture. We then extend the notion of bags and their Nahm transform to higher gauge theories and arbitrary domains. Bags in four dimensions conjecturally describe the large n limit of n self-dual strings. We show that the corresponding Basu-Harvey equation is the large n limit of an equation describing n M2-branes, and that it has a natural interpretation …
We study SU(2) BPS monopoles with spectral curves of the form $η^3+χ(ζ^6+b ζ^3-1)=0$. Previous work has has established a countable family of solutions to Hitchin's constraint that $L^2$ was trivial on such a curve. Here we establish that the only curves of this family that yield BPS monopoles correspond to tetrahedrally symmetric monopoles. We introduce several new techniques making use of a factorization theorem of Fay and Accola for theta functions, together with properties of the Humbert variety. The geometry leads us to a formulation purely in terms of elliptic functions. A more general c…
We present a semi-classical description of BPS monopoles interacting with Wilson lines. The Wilson lines are represented as non-Abelian spin impurities. These spins interact with the monopole degrees of freedom through a natural connection on the moduli space. We employ this technology in N = 2 SU(2) gauge theory to count the number of framed BPS states of a single monopole bound to Wilson lines in different representations.
The space of all possible boundary conditions that respect self-adjointness of Hamiltonian operator is known to be given by the group manifold $U(2)$ in one-dimensional quantum mechanics. In this paper we study non-Abelian Berry's connections in the space of boundary conditions in a simple quantum mechanical system. We consider a system for a free spinless particle on a circle with two point-like interactions described by the $U(2) \times U(2)$ family of boundary conditions. We show that, for a certain $SU(2) \subset U(2) \times U(2)$ subfamily of boundary conditions, all the energy levels bec…
The ADHMN construction of magnetic monopoles is given in terms of the (normalizable) solutions of an associated Weyl equation. We focus here on solving this equation directly by algebro-geometric means. The (adjoint) Weyl equation is solved using an ansätz of Nahm in terms of Baker-Akhiezer functions. The solution of Nahm's equation is not directly used in our development.
We use the well-known Bogomolny's equations, in general coordinate system, for BPS monopoles and dyons in the $SU(2)$ Yang-Mills-Higgs model to obtain an explicit form of BPS Lagrangian density under the BPS Lagrangian method. We then generalize this BPS Lagrangian density and use it to derive several possible generalized Bogomolny's equations, with(out) additional constraint equations, for BPS monopoles and dyons in the generalized $SU(2)$ Yang-Mills-Higgs model. We also compute the stress-energy-momentum tensor of the generalized model, and argue that the BPS monopole and dyon solutions are …
Simulations of Magnetic Monopole Collisions
本文通过数值模拟研究BPS磁单极子散射，提出多种多单极子构型 ansatz，覆盖平面与非平面过程，验证模空间近似预测，并扩展至相对论速度及超出BPS极限的参数。
On BPS Equations of Generalized $SU(2)$ Yang-Mills-Higgs Model with Scalars-Dependent Coupling $θ$-term
本研究通过 BPS 拉格朗日量方法，为含标量依赖耦合 $θ$ 项的广义 $SU(2)$ Yang-Mills-Higgs 模型推导出更一般的 Bogomolnyi 方程，揭示 BPS 磁单极子解由参数 $γ$ 刻画，而 $θ$ 项耦合仅影响双子电荷，与 Witten 结果一致。
We describe in detail a general scheme for embedding several BPS monopoles into a theory with a larger gauge group, which generalizes embeddings of SU(2) monopoles discussed by several authors. This construction is applied to obtain explicit fields for monopoles with several charge configurations in the SU(5)\to [SU(3)\times SU(2)\times U(1)]/Z_6 model.
We explicitly construct all kink solitons arising in the recent study of Auzzi, Bolognesi, and Shifman of a monopole confinement problem in ${\cal N}=2$ supersymmetric QCD. In particular, we show that all finite-energy kink solitons must be BPS.
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…
The equations of motion of multiple M0{brane (multiple M-wave or mM0) system in an arbitrary D = 11 supergravity superspace, which generalize the Matrix model equations for the case of inter- action with a generic 11D supergravity background, are obtained in the frame of superembedding approach. We also derive the BPS equations for supersymmetric bosonic solutions of these mM0 equations and show that the set of 1/2 BPS solutions contain a fuzzy sphere modeling M2 brane as well as that the Nahm equation appears as a particular case of the 1/4 BPS equations.
We construct, numerically, a solution of the SU(2) Bogomolny equations corresponding to a sheet of BPS monopoles. It represents a domain wall between a vacuum region and a region of constant energy density, and it is the smoothed-out version of the planar sheet of Dirac monopoles obtained by linear superposition.
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…
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