Mikhail Shifman, Gianni Tallarita, Alexei Yung
We extend the Georgi-Glashow model of the t'Hooft-Polyakov monopoles to include additional collective coordinates "orientational isospin moduli". The low-energy theory of these solitonic solutions can be interpreted as dyons with isospin.
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We discuss a core instability of 't Hooft Polyakov monopoles in Alice electrodynamics type of models in which charge conjugation symmetry is gauged. The monopole may deform into a toroidal defect which carries an Alice flux and a (non-localizable) magnetic Cheshire charge.
We present explicit BPS field configurations representing one nonabelian monopole with one minimal weight 't Hooft operator insertion. We explore the SO(3) and SU(2) gauge groups. In the case of SU(2) gauge group the minimal 't Hooft operator can be completely screened by the monopole. If the gauge group is SO(3), however, such screening is impossible. In the latter case we observe a different effect of the gauge symmetry enhancement in the vicinity of the 't Hooft operator.
We show how to generalize our method, based on projective modules and matrix models, which enabled us to derive noncommutative monopoles on a fuzzy sphere, to the non-abelian case, recovering known results in literature. We then discuss a possible candidate for deforming the commutative Chern class to the non-commutative case.
We exploit the 't Hooft-Polyakov monopole to define closed algebra of the quantum field operators and the BRST charge $Q_{BRST}$. In the first-class configuration of the Dirac quantization, by including the $Q_{BRST}$-exact gauge fixing term and the Faddeev-Popov ghost term, we find the BRST invariant Hamiltonian to investigate the de Rham-type cohomology group structure for the monopole system. The Bogomol'nyi bound is also discussed in terms of the first-class topological charge defined on the extended internal 2-sphere.
In this paper, we extend the analysis of the Lorentz-violating Quantum Eletrodynamics to the non-Abelian case: an SO(3) Yang-Mills Lagrangian with the addition of the non-Abelian Chern-Simons-type term. We consider the spontaneous symmetry breaking of the model and inspect its spectrum in order to check if unitarity and causality are respected. An analysis of the topological structure is also carried out and we show that a 't Hooft-Polyakov solution for monopoles is still present.
We recall the quaternionic fomulation, which can simplify the computation of the linearized Yang-Mills-Higgs equation in the background of a 't Hooft-Polyakov monopole. We then study the solutions in the cases $j=0$, $j=1$ and $j\geq 2$ separately. In particular, we investigate the spectral properties of the monopoles. We focus on some of the bound states and show that as the generalized momentum increases, the $k-$th eigenvalue tends to 1. We show the existence of Feshbach resonance for $\om <1$ in the coupled system and calculated the partial cross section when $\om >1$.
Singularities of Magnetic Monopoles for Dirac and 't Hooft-Polyakov Theories by Pre-potential Method
本研究采用预势方法，探究Dirac与't Hooft-Polyakov磁单极子理论中的奇点问题，断言两类规范理论下的磁单极子均存在磁奇点，且通过正则化方法验证了该结论。
The dependence of the energies of axially symmetric monopoles of magnetic charges 2 and 3, on the Higgs self-interaction coupling constant, is studied numerically. Comparing the energy per unit topological charge of the charge-2 monopole with the energy of the spherically symmetric charge-1 monopole, we confirm that there is only a repulsive phase in the interaction energy between like monopoles
In this paper, making use of the 't Hooft-Polyakov-Julia-Zee ansatz for the SU(2) Yang-Mills-Higgs gauge field theory, we present the straightforward generalization of the Bogomoln'yi equations and its several consequences. Particularly, this is shown that this idea is able to generate new types of non-abelian both dyons and magnetic monopoles and, moreover, that within the new model the scalar field can be described through the Coulomb potential, whereas, up to a constant, the non-abelian gauge field becomes the Wu-Yang monopole.
The 'tHooft-Polyakov monopole is treated as constrained system using the Hamilton-Jacobi method. The set of the Hamilton-Jacobi partial differential equations and the equations of motion are obtained. The quantization of the system is also discussed.
The 't Hooft-Polyakov monopole solution in Yang-Mills theory is given new physical interpretation in the geometric theory of defects. It describes solids with continuous distribution of dislocations and disclinations. The corresponding densities of Burgers and Frank vectors are computed. It means that the 't Hooft-Polyakov monopole can be seen, probably, in solids.
We examine whether the analysis of quantum corrections for the kink soliton carries over to the 't Hooft-Polyakov monopole. For the kink, it is central that the quantum fluctuations are eigenmodes of a Hermitian operator. For the monopole, we show the analogous operator is not Hermitian. This property raises some questions about the quantization procedure for a 't Hooft-Polyakov monopole.
The Kalb-Ramond monopole, as discussed by Nepomechie, is identical with the (singular) Dirac monopole in d=3 dimensions. The latter can be described by the (regular) 't Hooft-Polyakov monopole, via the 't Hooft tensor construction. This construction is extended to arbitrary odd dimensions by performing the d=5 case explicitly, exploiting the (regular) `monopoles' of generalised Georgi-Glashow models and identifying their 't Hooft tensors as the Kalb-Ramond fields. The relevant `magnetic charges' are expressed as topological invariants.
At classical level, dynamical derivation of the properties and conservation laws for topologically non-trivial systems from Noether theorem versus the derivation of the system's properties on topological grounds are considered as distinct. We do celebrate any agreements in results derived from these two distinct approaches: i.e. the dynamical versus the topological approach. Here we consider the Corrigan-Olive-Fairlie-Nuyts solution based on which we study the stability of the 't Hooft- Polyakov outer field, known as its Higgs vacuum, and derive its stability, dynamically, from the equations o…
A high precision numerical analysis of the static, spherically symmetric SU(2) magnetic monopole equations is carried out. Using multi-shooting and multi-domain spectral methods, the mass of the monopole is obtained rather precisely as a function of $β=M_H/M_W$ for a large $β$-interval ($M_H$ and $M_W$ denote the mass of the Higgs and gauge field respectively). The numerical results necessitated the reexamination and subsequent correction of a previous asymptotic analysis of the monopole mass in the literature for $β\ll1$.
We express the Pontryagin index in Polyakov gauge completely in terms of magnetically charged gauge fixing defects, namely magnetic monopoles, lines, and domain walls. Open lines and domain walls are topologically equivalent to monopoles, which are the genuine defects. The emergence of non-genuine magnetically charged closed domain walls can be avoided by choosing the temporal gauge field smoothly. The Pontryagin index is then exclusively determined by the magnetic monopoles.
Fermion Zero Modes and Fermion number $1/2$ of the 't Hooft-Polyakov Monopole
研究电弱SU(2)单极背景下的费米束缚态，分析不同耦合常数对零模的影响，发现零模局域性随g和λ增加而增强，右手法成分随g增加而减小，但随y_q增加而增加，揭示系统非线性和非微扰特性。
Lorentz invariance is broken for the non-Abelian monopoles. Here we will consider the case of 't Hooft-Polyakov monopole and show that the Lorentz invariance of its field will be restored using Dirac quantization.
We construct a compactification of Polyakov model on $T^2 \times \mathbb R $ down to quantum mechanics which remembers non-perturbative aspects of field theory even at an arbitrarily small area. Standard compactification on small $T^2 \times \mathbb R $ possesses a unique perturbative vacuum (zero magnetic flux state), separated parametrically from higher flux states, and the instanton effects do not survive in the Born-Oppenheimer approximation. By turning on a background magnetic GNO flux in co-weight lattice corresponding to a non-zero 't Hooft flux, we show that $N$-degenerate vacua appear…
The mass and interactions of a quantum 't Hooft-Polyakov monopole are measured nonperturbatively using correlation functions in lattice Monte Carlo simulations. A method of measuring the form factors for interactions between the monopole and fundamental particles, such as the photon, is demonstrated. These quantities are potentially of experimental relevance in searches for magnetic monopoles.
In this article, we carry out the Hamiltonization in the axial gauge, of the t'Hooft-Polyakov monopole field outside the localized region, which represents the monopole's core. One feature of the treatment here, is using the Higgs vacuum condition as both strong and weak equation instead of using it in the degree of freedom reduction.
We investigate several properties of Ginsparg-Wilson fermion on fuzzy 2-sphere. We first examine chiral anomaly up to the second order of the gauge field and show that it is indeed reduced to the correct form of the Chern character in the commutative limit. Next we study topologically non-trivial gauge configurations and their topological charges. We investigate 't Hooft-Polyakov monopole type configuration on fuzzy 2-sphere and show that it has the correct commutative limit. We also consider more general configurations in our formulation.
The quantum mechanical mass of 't Hooft-Polyakov monopoles in the four-dimensional Georgi-Glashow is calculated non-perturbatively using lattice Monte Carlo simulations. This is done by imposing twisted boundary conditions that ensure there is one unit of magnetic charge on the lattice, and measuring the free energy difference between this ensemble and the vacuum. In the weak-coupling limit, the results can be used to determine the quantum correction to the classical mass, once renormalisation of couplings is taken properly into account. The methods can also be used to study the masses at stro…
We present regular solutions for a brane world scenario in the form of a 't Hooft-Polyakov monopole living in the three-dimensional spherical symmetric transverse space of a seven-dimensional spacetime. In contrast to the cases of a domain-wall in five dimensions and a string in six dimensions, there exist gravity-localizing solutions for both signs of the bulk cosmological constant. A detailed discussion of the parameter space that leads to localization of gravity is given. A point-like monopole limit is discussed.
A transport equation of the representative quasi-particles is derived, by solving Dirac (or Dyson) equations under the semi-classical approximation, for describing the entangled motions of the red and blue chiral quarks in the vicinity of a SU(2) t'Hooft-Polyakov monopole. The representative quasi-particles have two states characterized by different dispersion relations. A ground state quasi-particle has a zero pole mass, a zero longitudinal inertial mass and a finite transverse inertial mass, while an excited quasi-particle has a finite pole mass and finite inertial masses. At equilibrium, th…
We present the results of an explicit numerical computation of a novel instanton in Georgi-Glashow SU(2) theory. The instanton is physically relevant as a mediator of Schwinger production of 't Hooft-Polyakov magnetic monopoles from strong magnetic fields. In weak fields, the pair production rate has previously been computed using the worldline approximation, which breaks down in strong fields due to the effects of finite monopole size. Using lattice field theory we have overcome this limit, including finite monopole size effects to all orders. We demonstrate that a full consideration of the i…
We show that in the SU(2) Georgi-Glashow model, 't Hooft-Polyakov monopoles are produced by a classical instability in magnetic fields above the Ambjorn-Olesen critical field, which coincides approximately with the field at which Schwinger pair production becomes unsuppressed. Below it, monopoles can be produced thermally, and we show that the rate is higher than for pointlike monopoles by calculating the sphaleron energy as a function of the magnetic field. The results can be applied to production of monopoles in heavy-ion collisions or in the early Universe.
The time evolution of strongly exited SU(2) Bogomolny-Prasad-Sommerfield (BPS) magnetic monopoles in Minkowski spacetime is investigated by means of numerical simulations based on the technique of conformal compactification and on the use of hyperboloidal initial value problem. It is found that an initially static monopole does not radiate the entire energy of the exciting pulse toward future null infinity. Rather, a long-lasting quasi-stable `breathing state' develops in the central region and certain expanding shell structures -- built up by very high frequency oscillations -- are formed in …
Internal force distributions in 't Hooft-Polyakov monopole and Julia-Zee dyon
本文研究 't Hooft-Polyakov 磁单极子和 Julia-Zee 双子的能量-动量张量，提出用修正的 't Hooft 定义分解长程与短程部分，得到新平衡方程并计算质量、D 项和均方半径。
The broad class of U(N) and SU(N) Polyakov loop models on the lattice are solved exactly in the combined large N, Nf limit, where N is a number of colors and Nf is a number of quark flavors, and in any dimension. In this 't Hooft-Veneziano limit the ratio N/Nf is kept fixed. We calculate both the free energy and various correlation functions. The critical behavior of the models is described in details at finite temperatures and non-zero baryon chemical potential. Furthermore, we prove that the calculation of the N-point (baryon) correlation function reduces to the geometric median problem in t…
The well known topological monopoles originally investigated by 't Hooft and Polyakov are known to arise in classical Yang-Mills-Higgs theory. With a pure gauge theory it is known that the classical Yang-Mills field equation do not have such finite energy configurations. Here we argue that such configurations may arise in a semi-quantized Yang-Mills theory, where the original gauge group, SU(3), is reduced to a smaller gauge group, SU(2), and with some combination of the coset fields of the SU(3) to SU(2) reduction acting as effective scalar fields. The procedure is called semi-quantized since…
We revisit the localization computation of the expectation values of 't Hooft operators in $\mathcal{N}=2^*$ SU(N) theory on $\mathbb{R}^3 \times S^1$. We show that the part of the answer arising from "monopole bubbling" on $\mathbb{R}^3$ can be understood as an equivariant integral over a Kronheimer-Nakajima moduli space of instantons on an orbifold of $\mathbb{C}^2$. It can also be described as a Witten index of a certain supersymmetric quiver quantum mechanics with $\mathcal{N}=(4,4)$ supersymmetry. The map between the defect data and the quiver quantum mechanics is worked out for all value…
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…
We study monopoles and corresponding 't Hooft tensor in a generic gauge theory. This issue is relevant to the understanding of the color confinement in terms of dual symmetry.
Universal global analytic expansion for the 't Hooft-Polyakov monopole profiles
基于近期重聚合法论文 arXiv:2602.14620，本文针对任意 λ/e^2>0 的 t Hooft-Polyakov 单极子轮廓方程，提出了一种全局解析展开方案，通过构建围绕非微扰背景的均匀收敛泛函微扰级数，完美匹配了全解在零和无穷远处的局部行为，并给出了局部不可达数值参数的简单解析公式。
In our previous paper (Fukui et al., 2010), we have shown that the Jackiw-Nohl-Rebbi two-instanton generates a circular loop of magnetic monopole in the four-dimensional Euclidean SU(2) Yang-Mills theory. On the other hand, it is claimed in Brower, Orginos and Tan (1997) that the 't Hooft two-instanton does not generate magnetic monopole loop. It seems that two results are inconsistent with each other, since the JNR two-instanton converges to the 't Hooft two-instanton in a certain limit. In this paper, we clarify that two results are compatible with each other by demonstrating how the magneti…
We study intersecting brane systems that realize a class of singular monopole configurations in four-dimensional Yang-Mills-Higgs theory. Singular monopoles are solutions to the Bogomolny equation on R^3 with a prescribed number of singularities corresponding to the insertion of 't Hooft defects. We use the brane construction to motivate a recent conjecture on the conditions for which the moduli space of solutions is non-empty. We also show how branes provide physical intuition for various aspects of the dimension formula derived in {arXiv:1404.5616}, including the contribution to the dimensio…
We study indices for 5d gauge theories on S^1 \times S^4/Z_n. In the large orbifold limit, n \rightarrow \infty, we find evidence that the indices become 4d indices in the presence of a 't Hooft line operator. The non-perturbative part of the index poses some subtleties when being compared to the 4d monopole bubbling which happens in the presence of 't Hooft line operators. We study such monopole bubbling indices and find an interesting connection to the Hilbert series of the moduli space of instantons on an auxiliary ALE space.
We study monopoles and corresponding 't Hooft tensor in QCD with a generic compact gauge group. This issue is relevant to the understanding of color confinement in terms of dual symmetry.
We consider pure Yang Mills theory on the four torus. A set of non-Abelian transition functions is presented which encompass all instanton sectors. It is argued that these transition functions are a convenient starting point for gauge fixing. In particular, we give an extended Abelian projection with respect to the Polyakov loop, where $A_0$ is independent of time and in the Cartan subalgebra. In the non-perturbative sectors such gauge fixings are necessarily singular. These singularities can be restricted to Dirac strings joining monopole and anti-monopole like ``defects''.
We consider 't Hooft-Polyakov monopole (TPM) as a topological soliton in the 5-dimensional (5D) theory of gauge-Higgs unification. This scenario provides a very natural framework to incorporate the TPM, since the adjoint scalar is builtin as the extra-space component of higher dimensional gauge field. In the process of the analysis, we realize that the condition to be satisfied by the Bogomolny-Prasad-Sommerfield (BPS) state of TPM, the BPS monopole, is equivalent to an (anti-)self-dual condition for the higher dimensional gauge field. This observation, in turn, suggests the presence of instan…
Confinement by Monopole Loops in Inhomogeneous Magnetic Field
研究通过非平凡磁场中单极环的约束机制，揭示了弱耦合下约束的产生方式，将单极与反单极转化为去约束的单极环片段。
We propose a new order parameter of a $Z_N$ symmetry in SU(N) gauge theories in $4$ dimensional Minkowski space-time, assuming spatial periodic boundary conditions. It is given by $Tr(P\exp(i\int_c A_μdx^μ))$ where the spatial path $c$ is taken, for example, along $x_1$ axis. The parameter vanishes when the $Z_N$ symmetry is preserved. We calculate the contribution of QCD monopoles to the order parameter and show that when the monopoles condense $\langleΦ\rangle\neq 0$, it vanishes, while it does not vanish when they do not condense. These calculations are performed using a monopole field $Φ$ …
Using the embedded defect method, we classify the possible embeddings of a 't Hooft-Polyakov monopole in a general gauge theory. We then discuss some similarities with embedded vortices and relate our results to fundamental monopoles.
The spherically symmetric magnetic monopole in an SU(2) gauge theory coupled to a massless Higgs field is shown to possess an infinite number of resonances or quasinormal modes. These modes are eigenfunctions of the isospin 1 perturbation equations with complex eigenvalues, $E_n=ω_n-iγ_n$, satisfying the outgoing radiation condition. For $n\to\infty$, their frequencies $ω_n$ approach the mass of the vector boson, $M_W$, while their lifetimes $1/γ_n$ tend to infinity. The response of the monopole to an arbitrary initial perturbation is largely determined by these resonant modes, whose collectiv…
We discuss the effect of the choice of an Abelian projection on the dual superconductor mechanism of confinement in SU(2) gluodynamics. Using qualitative arguments we show that the dual superconductor Lagrangian corresponding to the Abelian Polyakov gauge has a different structure compared to the dual Lagrangian in the Maximal Abelian gauge. A difference between these Lagrangians reflects the fact that in continuum limit the monopoles should be static in the Abelian Polyakov gauge and therefore these monopoles can not give rise to the confinement of static quarks. Using the Abelian Polyakov ga…
We construct exact t Hooft-Polyakov monopole solutions in a non-Hermitian field theory with local non-Abelian SU(2) gauge symmetry and a modified antilinear CPT symmetry. The solutions are obtained in a fourfold Bogomolny-Prasad-Sommerfield scaling limit giving rise to two different types of monopole masses that saturate the lower energy bound. These two masses only coincide in the Hermitian limit and in the limit in which the symmetry breaking vacuum tends to the trivial symmetry preserving vacuum. In the two theories corresponding to the two known Dyson maps these two masses are exchanged, u…
Resurgent structure of the 't Hooft-Polyakov monopole
本文从 resurgence 理论的角度对‘t Hooft-Polyakov单极的空间轮廓所受的微分方程进行了全面分析，揭示了其渐近行为的普遍性以及 Borel 变换和 Volterra 方程核的相对简单性，从而得到一个具有良好控制的 perturbative 展开，能够有效控制 Borel 平面奇点的繁衍，并提供关于相关对数不连续性的完整信息。此外，其部分 resummation 显示出简单的普遍解析非 perturbative 背景轮廓，围绕这些轮廓可以发展出一致收敛的全局 perturbative 展开，用于任何 λ/e² > 0 的精确解。这还提供了对 gauge 和 scalar 轮廓函数在原点和无穷远处的展开参数的解析控制。
We investigate the Higgs mechanism for gravity, which has been recently put forward by 't Hooft, when the Polyakov-type action for scalar fields is added to the original action. We find that from the Polyakov-type action, it is very natural to derive an 'alternative' metric tensor composed of the scalar fields. The positivity condition on the determinant can be also derived easily by requiring that this term does not change the dynamics at all and becomes a topological number, that is, the wrapping number. It turns out that the gauge conditions adopted by 't Hooft are nothing but the restricti…
The motion of a one-dimensional kink and its energy losses are considered as a model of interaction of nontrivial topological field configurations with external fields. The approach is based on the calculation of the zero modes excitation probability in the external field. We study in the same way the interaction of the t'Hooft-Polyakov monopole with weak external fields. The basic idea is to treat the excitation of a monopole zero mode as the monopole displacement. The excitation is found perturbatively. As an example we consider the interaction of the t'Hooft-Polyakov monopole with an extern…
Generalisations of the 't Hooft-Polyakov monopole which can exhibit repulsion only, attraction only, and both attraction and repulsion, between like monopoles, are studied numerically. The models supporting these solitons are SO(3) gauged Higgs models featuring Skyrme-like terms.
Correlation between instantons and QCD-monopoles is studied in the abelian-gauge-fixed QCD. From a simple topological consideration, instantons are expected to appear only around the QCD-monopole trajectory in the abelian-dominating system. The QCD-monopole in the multi-instanton solution is studied in the Polyakov-like gauge, where $A_4(x)$ is diagonalized. The world line of the QCD-monopole is found to be penetrate the center of each instanton. For the single-instanton solution, the QCD-monopole trajectory becomes a simple straight line. On the other hand, in the multi-instanton system, the …
We present a new type of spherically symmetric monopole and dyon solutions with the magnetic charge $ 4π/e$ in the standard Weinberg-Salam model. The monopole (and dyon) could be interpreted as a non-trivial hybrid between the abelian Dirac monopole and non-abelian 't Hooft-Polyakov monopole (with an electric charge). We discuss the possible physical implications of the electroweak dyon.
The 't Hooft-Veneziano limit of various U(N) and SU(N) Polyakov loop models is constructed at finite temperature and non-zero baryon chemical potential. In this paper we calculate the free energy, its derivatives, the Polyakov loop expectation values and the baryon density. The phase diagram is described in details.
We investigate monopole solutions for the Born-Infeld Higgs system. We analyze numerically these solutions and compare them with the standard 't Hooft-Polyakov monopoles. We also discuss the existence of a critical value of beta (the Born-Infeld "absolute field parameter") below which no regular solution exists.
Monopole bubbling contributions to supersymmetric 't Hooft loops in 4d $\mathcal{N}=2$ theories are computed by SQM indices. As recently argued, those indices are hard to compute due to the presence of Coulomb vacua that are not captured by standard localization techniques. We propose an algorithmic method to compute the full bubbling contributions that circumvent this issue, by considering SQM with more matter fields and isolating the bubbling terms as residues in flavor fugacities. The enlarged SQMs are read from brane configurations realizing the bubbling sector of a given 't Hooft loop. We…
In this paper, we propose a string theory description of generic 't Hooft defects in $\mathcal{N}=2$ $SU(N)$ supersymmetric gauge theories. We show that the space of supersymmetric ground states is given by the moduli space of singular monopoles and that in this setting, Kronheimer's correspondence is realized as T-duality. We conjecture that this brane configuration can be used to study the full dynamics of monopole bubbling.
Lattice calculations performed in Abelian gauges give strong evidence that confinement is realized as a dual Meissner effect, implying that the Yang--Mills vacuum consists of a condensate of magnetic monopoles. We show in Polyakov gauge how the Pontrjagin index of the gauge field is related to the magnetic monopole charges.
A relation between the total instanton number and the quantum-numbers of magnetic monopoles that arise in general Abelian gauges in SU(2) Yang-Mills theory is established. The instanton number is expressed as the sum of the `twists' of all monopoles, where the twist is related to a generalized Hopf invariant. The origin of a stronger relation between instantons and monopoles in the Polyakov gauge is discussed.
The interaction of a nontrivial topological field configuration with the external fields is considered. The approach is based on the calculation of the zero modes excitation probability. We consider the interaction of the t'Hooft-Polyakov monopole with an external weak uniform magnetic field and the field of another monopole.
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