Pontryagin Index Research Articles

Three-dimensional PT -symmetric topological phases with a Pontryagin index

We report on a certain class of three-dimensional topological insulators and semimetals protected by spinless PT symmetry, hosting an integer-valued bulk invariant. We show using homotopy arguments that these phases host multigap topology, providing a realization of a single Z invariant in three spatial dimensions that is distinct from the Hopf index. We identify this invariant with the Pontryagin index, which describes Belavin-Polyakov-Schwartz-Tyupkin (BPST) instantons in particle physics contexts and corresponds to a three-sphere winding number. We study naturally arising multigap linked nodal rings, topologically characterized by split-biquaternion charges, which can be removed by non-Abelian braiding of nodal rings, even without closing a gap. We additionally recast the describing winding number in terms of gauge-invariant combinations of non-Abelian Berry connection elements, indicating relations to Pontryagin characteristic class in four dimensions. These topological configurations are furthermore related to fully nondegenerate multigap phases that are characterized by a pair of winding numbers relating to two isoclinic rotations in the case of four bands and can be generalized to an arbitrary number of bands. From a physical perspective, we also analyze the edge states corresponding to this Pontryagin index as well as their dissolution subject to the gap-closing disorder. Finally, we elaborate on the realization of these novel non-Abelian phases, their edge states, and linked nodal structures in acoustic metamaterials and trapped-ion experiments. Published by the American Physical Society 2024

Black holes with topological charges in Chern-Simons AdS5 supergravity

We study static black hole solutions with locally spherical horizons coupled to non-Abelian field in mathcal{N} = 4 Chern-Simons AdS5 supergravity. They are governed by three parameters associated to the mass, axial torsion and amplitude of the internal soliton, and two ones to the gravitational hair. They describe geometries that can be a global AdS space, naked singularity or a (non-)extremal black hole. We analyze physical properties of two inequivalent asymptotically AdS solutions when the spatial section at radial infinity is either a 3-sphere or a projective 3-space. An important feature of these 3-parametric solutions is that they possess a topological structure including two SU(2) solitons that wind nontrivially around the black hole horizon, as characterized by the Pontryagin index. In the extremal black hole limit, the solitons’ strengths match and a soliton-antisoliton system unwinds. That limit admits both non-BPS and BPS configurations. For the latter, the pure gauge and non-pure gauge solutions preserve 1/2 and 1/16 of the original supersymmetries, respectively. In a general case, we compute conserved charges in Hamiltonian formalism, finding many similarities with standard supergravity black holes.

A comment on instantons and their fermion zero modes in adjoint QCD_2

The adjoint 2-dimensional QCD with the gauge group SU(N)/Z_NSU(N)/ZN admits topologically nontrivial gauge field configurations associated with nontrivial \pi_1[SU(N)/Z_N] = Z_Nπ1[SU(N)/ZN]=ZN. The topological sectors are labelled by an integer k=0,\ldots, N-1k=0,…,N−1. However, in contrast to QED_2QED2 and QCD_4QCD4, this topology is not associated with an integral invariant like the magnetic flux or Pontryagin index. These instantons may admit fermion zero modes, but there is always an equal number of left-handed and right-handed modes, so that the Atiyah-Singer theorem, which determines in other cases the number of the modes, does not apply. The mod. 2 argument [1] suggests that, for a generic gauge field configuration, there is either a single doublet of such zero modes or no modes whatsoever. However, the known solution of the Dirac problem for a wide class of gauge field configurations indicates the presence of k(N-k) zero mode doublets in the topological sector k. In this note, we demonstrate in an explicit way that these modes are not robust under a generic enough deformation of the gauge background and confirm thereby the conjecture of Ref. [1]. The implications for the physics of this theory (screening vs. confinement issue) are briefly discussed.

Chiral asymmetries in near-horizon region of charged black hole and teleparallelism

The issue of encoding physical information into metric structure of physical theories has been discussed recently by the author in the case of black hole teleparallelism. In this paper, one obtains a teleparallel chiral currents from quantum anomalies and topological torsional invariants of Nieh-Yan type. The Pontryagin index is also obtained in the case of rotating Kerr spacetime metric of non-static black holes. Magnetic monopoles which appears in this approach can be eliminated by a torsion constraint. These ideas are applied to Kerr and Kerr–Newmann charged black holes.

Weyl Node and Spin Texture in Trigonal Tellurium and Selenium.

We study Weyl nodes in materials with broken inversion symmetry. We find based on first-principles calculations that trigonal Te and Se have multiple Weyl nodes near the Fermi level. The conduction bands have a spin splitting similar to the Rashba splitting around the H points, but unlike the Rashba splitting the spin directions are radial, forming a hedgehog spin texture around the H points, with a nonzero Pontryagin index for each spin-split conduction band. The Weyl semimetal phase, which has never been observed in real materials without inversion symmetry, is realized under pressure. The evolution of the spin texture by varying the pressure can be explained by the evolution of the Weyl nodes in k space.

Sphaleron solutions of the Skyrme model from Yang–Mills holonomy

We discuss how an approximation to the axially symmetric sphalerons in the Skyrme model can be constructed from the holonomy of a non-BPS Yang–Mills calorons. These configurations, both in the Skyrme model and in the Euclidean Yang–Mills theory, are characterized by two integers n and m, where ±n are the winding numbers of the constituents and the second integer m defines type of the solution, it has zero topological charge for even m and for odd values of m the corresponding chain has topological charge n. It is found numerically that the holonomy of the chains of interpolating calorons–anticalorons provides a reasonably good approximation to the corresponding Skyrmion–antiSkyrmion chains when the topological charge of the Skyrmion constituents is two times more than the Chern–Pontryagin index of the caloron.

Gauge-invariant magnetic monopole dominance in quark confinement

We show that a gauge-invariant magnetic monopole can be defined in Yang-Mills theory without matter fields, using a non-Abelian Stokes theorem and change of field variables a la Cho-Faddeev-Niemi, instead of using the Abelian projection. In fact, we give a first exact solution representing a magnetic monopole loop due to a meron pair with the non-trivial Pontryagin index. In addition, we summarize remarkable results of numerical simulations based on a new formulation of Yang-Mills theory: “Abelian” dominance and magnetic monopole dominance in the string tension. Finally, we discuss a novel feature in the case of SU ( 3 ) gauge group.

The Chow ring of the moduli space of bundles on $\mathbb{P}^2$ with charge 1

For an algebraically closed field $K$ with $\mathrm{ch}(K)\neq 2$, let $\mathcal{O}M(1, SO(n,K))$ denote the moduli space of holomorphic bundles on $\mathbb{P}^{2}$ with the structure group $SO(n,K)$ and half the first Pontryagin index being equal to 1, each of which is trivial on a fixed line $l_{\infty}$ and has a fixed holomorphic trivialization there. In this paper we determine the Chow ring of $\mathcal{O}M(1, SO(n,K))$.

UnconstrainedSU(2)Yang-Mills theory with a topological term in the long-wavelength approximation

The Hamiltonian reduction of SU(2) Yang-Mills theory for an arbitrary \theta angle to an unconstrained nonlocal theory of a self-interacting positive definite symmetric 3 \times 3 matrix field S(x) is performed. It is shown that, after exact projection to a reduced phase space, the density of the Pontryagin index remains a pure divergence, proving the \theta independence of the unconstrained theory obtained. An expansion of the nonlocal kinetic part of the Hamiltonian in powers of the inverse coupling constant and truncation to lowest order, however, lead to violation of the \theta independence of the theory. In order to maintain this property on the level of the local approximate theory, a modified expansion in the inverse coupling constant is suggested, which for a vanishing \theta angle coincides with the original expansion. The corresponding approximate Lagrangian up to second order in derivatives is obtained, and the explicit form of the unconstrained analogue of the Chern-Simons current linear in derivatives is given. Finally, for the case of degenerate field configurations S(x) with rank|S| = 1, a nonlinear \sigma-type model is obtained, with the Pontryagin topological term reducing to the Hopf invariant of the mapping from the three-sphere S^3 to the unit two-sphere S^2 in the Whitehead form.

Quark zero modes in intersecting center vortex gauge fields

The zero modes of the Dirac operator in the background of center vortex gauge field configurations in $\R^2$ and $\R^4$ are examined. If the net flux in D=2 is larger than 1 we obtain normalizable zero modes which are mainly localized at the vortices. In D=4 quasi-normalizable zero modes exist for intersecting flat vortex sheets with the Pontryagin index equal to 2. These zero modes are mainly localized at the vortex intersection points, which carry a topological charge of $\pm 1/2$. To circumvent the problem of normalizability the space-time manifold is chosen to be the (compact) torus $\T^2$ and $\T^4$, respectively. According to the index theorem there are normalizable zero modes on $\T^2$ if the net flux is non-zero. These zero modes are localized at the vortices. On $\T^4$ zero modes exist for a non-vanishing Pontryagin index. As in $\R^4$ these zero modes are localized at the vortex intersection points.

Center vortex model for the infrared sector of Yang–Mills theory — topological susceptibility

A definition of the Pontryagin index for SU(2) center vortex world-surfaces composed of plaquettes on a hypercubic lattice is constructed. It is used to evaluate the topological susceptibility in a previously defined random surface model for vortices, the parameters of which have been fixed such as to reproduce the confinement properties of SU(2) Yang-Mills theory. A prediction for the topological susceptibility is obtained which is compatible with measurements of this quantity in lattice Yang-Mills theory.

Center projection vortices in continuum Yang–Mills theory

The maximal center gauge, combined with center projection, is a means to associate Yang–Mills lattice gauge configurations with closed center vortex world-surfaces. This technique allows to study center vortex physics in lattice gauge experiments. In the present work, the continuum analogue of the maximal center gauge is constructed. This sheds new light on the meaning of the procedure on the lattice and leads to a sketch of an effective vortex theory in the continuum. Furthermore, the manner in which center vortex configurations generate the Pontryagin index is investigated. The Pontryagin index is built up from self-intersections of the vortex world-surfaces, where it is crucial that the surfaces be globally non-oriented.

RelationTrγ5=0and the index theorem in lattice gauge theory

The relation $\mathrm{Tr}{\ensuremath{\gamma}}_{5}=0$ implies the contribution to the trace from unphysical (would-be) species doublers in lattice gauge theory. This statement is also true for Pauli-Villars regularization in continuum theory. If one insists on $\mathrm{Tr}{\ensuremath{\gamma}}_{5}=0,$ one thus inevitably includes unphysical states in Hilbert space. If one truncates the trace to the contribution from physical species only, one obtains $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{T}\mathrm{r}{\ensuremath{\gamma}}_{5}{=n}_{+}\ensuremath{-}{n}_{\ensuremath{-}}$ which is equal to the Pontryagin index. A smooth continuum limit of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{T}\mathrm{r}{\ensuremath{\gamma}}_{5}=\mathrm{Tr}{\ensuremath{\gamma}}_{5}[1\ensuremath{-}(a/2)D]{=n}_{+}\ensuremath{-}{n}_{\ensuremath{-}}$ for the Dirac operator D satisfying the Ginsparg-Wilson relation leads to a natural treatment of a chiral anomaly in the continuum path integral. In contrast, the continuum limit of $\mathrm{Tr}{\ensuremath{\gamma}}_{5}=0$ is not defined consistently. It is shown that the nondecoupling of heavy fermions in the anomaly calculation is crucial to understand the consistency of the customary lattice calculation of the anomaly where $\mathrm{Tr}{\ensuremath{\gamma}}_{5}=0$ is used. We also comment on a closely related phenomenon in the analysis of the photon phase operator where the notion of index and the modification of index by a finite cutoff play a crucial role.

Magnetic monopoles and topology of Yang-Mills theory in Polyakov gauge

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.

Resolution of Gauss' law in Yang-Mills theory by gauge-invariant projection: Topology and magnetic monopoles

An efficient way of resolving Gauss' law in Yang-Mills theory is presented by starting from the projected gauge-invariant partition function and integrating out one spatial field variable. In this way one obtains immediately the description in terms of unconstrained gauge-invariant variables which was previously obtained by explicitly resolving Gauss' law in a modified axial gauge. In this gauge, which is a variant of 't Hooft's Abelian gauges, magnetic monopoles occur. It is shown how the Pontryagin index of the gauge field is related to the magnetic charges. It turns out that the magnetic monopoles are sufficient to account for the non-trivial topological structure of the theory.

Multiple Instantons Representing Higher-Order Chern-Pontryagin Classes

It has been shown in the work of Chakrabarti, Sherry and Tchrakian that the chiral SO±(4 p) Yang–Mills theory in the Euclidean 4 p (p≥ 2) dimensions allows an axially symmetric self-dual system of equations similar to Witten's instanton equations in the classical 4-dimensional SU(2)∼SO±(4) theory and the solutions represent a new class of instantons. However the rigorous existence of these higher-dimensional instanton solutions has remained open except for the solution of unit charge representing a single instanton. In this paper we establish an existence and uniqueness theorem for multi-instantons of arbitrary charges in the case p≥ 2. These solutions are the first known instantons, with the Chern–Pontryagin index greater than one, of the Yang–Mills model in higher dimensions. Our approach is a study of a nonlinear variational equation defined on the Poincare half plane.

Existence of a new instanton in constrained Yang-Mills-Higgs theory

Our goal is to discover possible new four-dimensional euclidean solutions (instantons) in fundamental SU(2) Yang-Mills-Higgs theory, with a constraint added to prevent collapse of the scale. We show that, most likely, there exists one particular new constrained instanton (I∗) with vanishing Pontryagin index. This is based on a topological argument that involves the construction of a non-contractible loop of four-dimensional configurations with a certain upper bound on the action, which we establish numerically. We expect I∗ to be the lowest action non-trivial solution in the vacuum sector of the theory. There also exists a related static, but unstable, solution, the new sphaleron S∗. Possible applications of I∗ to the electroweak interactions include the asymptotics of perturbation theory and the high-energy behaviour of the total cross section.

Radial excitation of hadronic states with null pontryagin index

Radial excitations of hadrons carrying zero instanton number are investigated. The mass relations among relevant non-strange and strange mesons and baryons have been derived consistently with the observed data.

Mass Formula for Hadrons with Zero Pontryagin index

Mass Formula for Hadrons with Zero Pontryagin index

Mass Formula for Hadrons with Zero Pontryagin index

We assume that the hadron carries zero instanton number. The diquark of baryon is associated with an instanton bag (I) and quark with the anti-instanton one (Ī), and their all possible permutations, and q and q' of a meson are associated with I and Ī and vice versa. The masses of the baryon and meson are given by the sum of the constituent quark masses and the parameters of q-q'(q') exchange potential which operates solely attractively at spin-zero state for them in I and Ī, with additional binding energies for mesons due to the structural difference between the baryon and meson