Dirac Structure Research Articles

Courant algebroids and generalized complex geometry

In this survey, we recall the background and major developments in Courant algebroids and their applications. We mainly recall the origin and motivation of the definition of Courant algebroids. A brief introduction to the theories of Dirac structures, Lie bialgebroids, the Clifford construction, spinor representation, generalized complex geometry and regular Courant algebroids are given.

On dual pairs in Dirac geometry

In this note we discuss dual pairs in Dirac geometry. We show that this notion appears naturally when studying the problem of pushing forward a Dirac structure along a surjective submersion, and we prove a Dirac-theoretic version of Libermann's theorem from Poisson geometry. Our main result is an explicit construction of strong self-dual pairs for Dirac structures. This theorem not only recovers the global construction of symplectic realizations from [Crainic-Marcut 2011], but allows for a more conceptual understanding of it, yielding a simpler and more natural proof. As an application of the main theorem, we present a different approach to the recent normal form theorem around Dirac transversals from [Bursztyn-Lima-Meinrenken 2016].

Exploring interacting Floquet states in black phosphorus: Anisotropy and bandgap laser tuning

Non-adiabatic interacting Floquet (or dressed) states arising from the intrinsic coupling between electrons and holes with off-resonant electromagnetic radiation have been investigated for recently synthesized ultrathin gapped and anisotropic black phosphorus. Our analytical calculations were carried out for the low-energy electronic subbands near the Γ point. Optical dressed states for both linear and circular polarizations of the incoming radiation have been obtained and analyzed. We focus our attention on linearly polarized light with arbitrary polarization direction since this case has not been considered for dressing fields imposed on initially anisotropic structures. We have examined and compared various cases for one- and few-layer phosphorus, including massless Dirac fermions with laser tunable in-plane anisotropy. The electronic properties, such as bandgaps, Fermi velocities, and effective masses, are renormalized in a very different way compared to those for the previously studied gapped Dirac structures. More importantly, material anisotropy and angular dependence of energy dispersions could be tuned by incident polarized light in the individual direction. The interaction of electrons with photons brings out a product term for two wave-vector components in energy dispersions, which leads to non-vanishing off-diagonal elements in inverse effective-mass and momentum-relaxation-time tensors and is expected to have a significant effect on the conductivities of electrons and holes. Such unique features in the band structure can be applied to new designs and fabrications of optical transistors and logic devices used in optical computers and communications.

On Higher Dirac Structures

We study higher-order analogues of Dirac structures, extending the multisymplectic structures that arise in field theory. We define higher Dirac structures as involutive subbundles of $TM+\wedge^k TM^*$ satisfying a weak version of the usual lagrangian condition (which agrees with it only when $k=1$). Higher Dirac structures transversal to $TM$ recover the higher Poisson structures introduced in [8] as the infinitesimal counterparts of multisymplectic groupoids. We describe the leafwise geometry underlying an involutive isotropic subbundle in terms of a distinguished 1-cocycle in a natural differential complex, generalizing the presymplectic foliation of a Dirac structure. We also identify the global objects integrating higher Dirac structures.

RK and RK* beyond the standard model

Measurements of the ratio of $B \to K^* \mu \mu $ to $B \to K^* e e $ branching fractions, $R_{K^*}$, by the LHCb collaboration strengthen the hints from previous studies with pseudoscalar kaons, $R_K$, for the breakdown of lepton universality, and therefore the Standard Model (SM), to $\sim 3.5 \sigma$. Complementarity between $R_K$ and $R_{K^*}$ allows to pin down the Dirac structure of the new contributions to be predominantly SM-like chiral, with possible admixture of chirality-flipped contributions of up to ${\cal{O}}( \mbox{few} 10 \%)$. Scalar and vector leptoquark representations $(S_3,V_1,V_3)$ plus possible ($\tilde S_2,V_2$) admixture can explain $R_{K,K^*}$ via tree level exchange. Flavor models naturally predict leptoquark masses not exceeding a few TeV, with couplings to third generation quarks at $O(0.1)$, implying that this scenario can be directly tested at the LHC.

Analysis of the strong coupling form factors of ΣbNB and ΣcND in QCD sum rules**Supported by Fundamental Research Funds for the Central Universities (2016MS133)

In this article, we study the strong interaction of the vertices ΣbNB and ΣcND using the three-point QCD sum rules under two different Dirac structures. Considering the contributions of the vacuum condensates up to dimension 5 in the operation product expansion, the form factors of these vertices are calculated. Then, we fit the form factors into analytical functions and extrapolate them into time-like regions, which gives the coupling constants. Our analysis indicates that the coupling constants for these two vertices are GΣbNB = 0.43±0.01 GeV−1 and GΣcND = 3.76±0.05 GeV−1.

Dirac Geometry of the Holonomy Fibration

In this paper, we solve the problem of giving a gauge-theoretic description of the natural Dirac structure on a Lie Group which plays a prominent role in the theory of D- branes for the Wess-Zumino-Witten model as well as the theory of quasi-Hamiltonian spaces. We describe the structure as an infinite-dimensional reduction of the space of connections over the circle. Our insight is that the formal Poisson structure on the space of connections is not an actual Poisson structure, but is itself a Dirac structure, due to the fact that it is defined by an unbounded operator. We also develop general tools for reducing Courant algebroids and morphisms between them, allowing us to give a precise correspondence between Hamiltonian loop group spaces and quasi- Hamiltonian spaces.

Transport coefficients of two-flavor quark matter from the Kubo formalism

The transport coefficients of quark matter at non-zero chemical potential and temperature are computed within the two-flavor Nambu--Jona-Lasinio model. We apply the Kubo formalism to obtain the thermal ($\kappa$) and electrical ($\sigma$) conductivities as well as an update of the shear viscosity ($\eta$) by evaluating the corresponding equilibrium two-point correlation functions to leading order in the $1/N_c$ expansion. The Dirac structure of the self-energies and spectral functions is taken into account as these are evaluated from the meson-exchange Fock diagrams for on-mass-shell quarks. We find that the thermal and electrical conductivities are decreasing functions of temperature and density above the Mott temperature $T_{\rm M}$ of dissolution of mesons into quarks, the main contributions being generated by the temporal and vector components of the spectral functions. The coefficients show a universal dependence on the ratio $T/T_{\rm M}$ for different densities, i.e., the results differ by a chemical-potential dependent constant. We also show that the Wiedemann-Franz law for the ratio $\sigma/\kappa$ does not hold. The ratio $\eta/s $, where $s$ is the entropy density, is of order of unity (or larger) close to the Mott temperature and, as the temperature increases, approaches the AdS/CFT bound $1/4\pi$. It is also conjectured that the ratio $\kappa T/c_V $, with $c_V$ being the specific heat, is bounded from below by $1/18$.

G-Invariant Deformations of Almost-Coupling Poisson Structures

On a foliated manifold equipped with an action of a compact Lie group $G$, we study a class of almost-coupling Poisson and Dirac structures, in the context of deformation theory and the method of averaging.

Splitting theorems for Poisson and related structures

Abstract According to the Weinstein splitting theorem, any Poisson manifold is locally, near any given point, a product of a symplectic manifold with another Poisson manifold whose Poisson structure vanishes at the point. Similar splitting results are known, e.g., for Lie algebroids, Dirac structures and generalized complex structures. In this paper, we develop a novel approach towards these results that leads to various generalizations, including their equivariant versions as well as their formulations in new contexts.

Removable Presymplectic Singularities and the Local Splitting of Dirac Structures

We call a singularity of a presymplectic form $\omega$ removable in its graph if its graph extends to a smooth Dirac structure over the singularity. An example for this is the symplectic form of a magnetic monopole. A criterion for the removability of singularities is given in terms of regularizing functions for pure spinors. All removable singularities are poles in the sense that the norm of $\omega$ is not locally bounded. The points at which removable singularities occur are the non-regular points of the Dirac structure for which we prove a general splitting theorem: Locally, every Dirac structure is the gauge transform of the product of a tangent bundle and the graph of a Poisson structure. This implies that in a neighborhood of a removable singularity $\omega$ can be split into a non-singular presymplectic form and a singular presymplectic form which is the partial inverse of a Poisson bivector that vanishes at the singularity. An interesting class of examples is given by log-Dirac structures which generalize log-symplectic structures. The analogous notion of removable singularities of Poisson structures is also studied.

Geometric quantization of Dirac manifolds

We extend a prequantization procedure to Dirac manifolds by using singular distributions obtained from 2-cocycles associated with Dirac structures. Given a Dirac manifold (M, D), we describe a prequantization formula in terms of a Lie algebroid connection and show that it is a representation of a Poisson algebra consisting of admissible functions on (M, D) on the space of global sections of a hermitian line bundle over M if and only if the curvature derived from the Lie algebroid connection is represented by a skew-symmetric operation which is naturally defined for (M, D). Moreover, we describe a necessary and sufficient condition for the prequantization formula to be the representation in terms of a Lie algebroid cohomology. We introduce the notion of a polarization for (M, D) and construct a representation of a subalgebra of admissible functions. Lastly, we discuss procedures for quantization in two cases: where M is compact and where M is not compact.

Revised variational approach to QCD in Coulomb gauge

The variational approach to QCD in Coulomb gauge is revisited. By assuming the non-Abelian Coulomb potential to be given by the sum of its infrared and ultraviolet parts, i.e.~by a linearly rising potential and an ordinary Coulomb potential, and by using a Slater determinant ansatz for the quark wave functional, which contains the coupling of the quarks and the gluons with two different Dirac structures, we obtain variational equations for the kernels of the fermionic vacuum wave functional, which are free of ultraviolet divergences. Thereby, a Gaussian type wave functional is assumed for the gluonic part of the vacuum. By using the results of the pure Yang--Mills sector for the gluon propagator as input, we solve the equations for the fermionic kernels numerically and calculate the quark condensate and the effective quark mass in leading order. Assuming a value of $\sigma_{\mathrm{C}} = 2.5 \sigma$ for the Coulomb string tension (where $\sigma$ is the usual Wilsonian string tension) the phenomenological value of the quark condensate $\langle \bar{\psi} \psi \rangle \simeq (-235 \, \mathrm{MeV})^3$ is reproduced with a value of $g \simeq 2.1$ for the strong coupling constant of the quark-gluon vertex.

Beyond the standard gauging: gauge symmetries of Dirac sigma models

In this paper we study the general conditions that have to be met for a gauged extension of a two-dimensional bosonic sigma-model to exist. In an inversion of the usual approach of identifying a global symmetry and then promoting it to a local one, we focus directly on the gauge symmetries of the theory. This allows for action functionals which are gauge invariant for rather general background fields in the sense that their invariance conditions are milder than the usual case. In particular, the vector fields that control the gauging need not be Killing. The relaxation of isometry for the background fields is controlled by two connections on a Lie algebroid L in which the gauge fields take values, in a generalization of the common Lie-algebraic picture. Here we show that these connections can always be determined when L is a Dirac structure in the H-twisted Courant algebroid. This also leads us to a derivation of the general form for the gauge symmetries of a wide class of two-dimensional topological field theories called Dirac sigma-models, which interpolate between the G/G Wess-Zumino-Witten model and the (Wess-Zumino-term twisted) Poisson sigma model.

The Dirac form factor predicts the Pauli form factor in the Endpoint Model

We compute the momentum-transfer dependence of the proton Pauli form factor $F_{2}$ in the endpoint overlap model. We find the model correctly reproduces the scaling of the ratio of $F_{2}$ with the Dirac Form factor $F_{1}$ observed at the Jefferson Laboratory. The calculation uses the leading-power, leading twist Dirac structure of the quark light-cone wave function, and the same endpoint dependence previously determined from the Dirac form factor $F_{1}$. There are no parameters and no adjustable functions in the endpoint model's prediction for $F_{2}$. The model's predicted ratio $F_{2}(Q^{2})/F_{1}(Q^{2})$ is quite insensitive to the endpoint wave function, which explains why the observed ratio scales like $1/Q$ down to rather low momentum transfers. The endpoint model appears to be the only comprehensive model consistent with all form factor information as well as reproducing fixed-angle proton-proton scattering at large momentum transfer. Any one of the processes is capable of predicting the others.

Anomalous electronic structure and magnetoresistance in TaAs2.

The change in resistance of a material in a magnetic field reflects its electronic state. In metals with weakly- or non-interacting electrons, the resistance typically increases upon the application of a magnetic field. In contrast, negative magnetoresistance may appear under some circumstances, e.g., in metals with anisotropic Fermi surfaces or with spin-disorder scattering and semimetals with Dirac or Weyl electronic structures. Here we show that the non-magnetic semimetal TaAs2 possesses a very large negative magnetoresistance, with an unknown scattering mechanism. Density functional calculations find that TaAs2 is a new topological semimetal [ℤ2 invariant (0;111)] without Dirac dispersion, demonstrating that a negative magnetoresistance in non-magnetic semimetals cannot be attributed uniquely to the Adler-Bell-Jackiw chiral anomaly of bulk Dirac/Weyl fermions.

Implicit Lagrange–Routh equations and Dirac reduction

In this paper, we make a generalization of Routh’s reduction method for Lagrangian systems with symmetry to the case where not any regularity condition is imposed on the Lagrangian. First, we show how implicit Lagrange–Routh equations can be obtained from the Hamilton–Pontryagin principle, by making use of an anholonomic frame, and how these equations can be reduced. To do this, we keep the momentum constraint implicit throughout and we make use of a Routhian function defined on a certain submanifold of the Pontryagin bundle. Then, we show how the reduced implicit Lagrange–Routh equations can be described in the context of dynamical systems associated to Dirac structures, in which we fully utilize a symmetry reduction procedure for implicit Hamiltonian systems with symmetry.

Dynamical chiral symmetry with an infrared finite gluon propagator

In this work we study dynamical quark mass generation using an infrared finite gluon propagator obtained from quenched lattice simulations. The quark gap equation is solved using a purely non-Abelian Ansatz for the quark-gluon vertex, which displays a dependence on the ghost dressing function and the scalar component of quark-ghost scattering kernel. For the former quantity we use quenched lattice results, while for the latter we derive its own integral equation at the one-loop-dressed approximation. This latter quantity is then coupled to the system of equations governing the two Dirac structures of the quark propagator. It turns out that when a current quark mass of 5 MeV is introduced, the constituent quark mass generated from the gap equation is of the order of 310 MeV. In addition, the pion decay constant computed from the resulting quark propagator is in good agreement with the physical value.

A structured control model for the thermo-magneto-hydrodynamics of plasmas in tokamaks

ABSTRACTA thermo-magneto-hydrodynamics port-Hamiltonian model is derived for the plasmas in tokamaks. Electromagnetic field and material domain balance equations are expressed in covariant forms, together with the magneto-hydrodynamics interconnection structure connecting them together. The balance equations for the entropy, mass and momentum, as well as closure equations in the material domain, are derived from the Boltzmann equation (kinetic theory). The Gibbs–Duhem equation is used to compute the irreversible entropy source term and to define the interdomain ℜ-field of the model. All derived interdomain couplings in the material domain are represented using Dirac and Stokes–Dirac structures and the resistivity ℜ-field structure. The complete model is summarized in a Bond Graph.

Improved variational approach to QCD in Coulomb gauge

The variational approach to QCD in Coulomb gauge developed previously by the T\"ubingen group is improved by enlarging the space of the quark trial vacuum wave functionals through a new Dirac structure in the quark-gluon coupling. Our Ansatz for the quark vacuum wave functional ensures that all linear divergences cancel in the quark gap equation resulting from the minimization of the energy calculated to two-loop order. The logarithmic divergences are absorbed in a renormalized coupling which is adjusted to reproduce the phenomenological value of the quark condensate. We also unquench the gluon propagator and show that the unquenching effects are generally small and amount to a small reduction in the midmomentum regime.