Gauge-natural Bundles Research Articles

Higgs fields induced by Yang–Mills type Lagrangians on gauge-natural prolongations of principal bundles

We address some new issues concerning spontaneous symmetry breaking. We define classical Higgs fields for gauge-natural invariant Yang–Mills type Lagrangian field theories through the requirement of the existence of canonical covariant gauge-natural conserved quantities. As an illustrative example, we consider the ‘gluon Lagrangian’, i.e. a Yang–Mills Lagrangian on the [Formula: see text]-order gauge-natural bundle of [Formula: see text]-principal connections, and canonically define a ‘gluon’ classical Higgs field through the split reductive structure induced by the kernel of the associated gauge-natural Jacobi morphism.

Classical Higgs fields on gauge gluon bundles

Classical Higgs fields and related canonical conserved quantities are defined by invariant variational problems on suitably defined gauge gluon bundles. We consider Lagrangian field theories which are assumed to be invariant with respect to the action of a gauge-natural group. As an illustrative example we exploit the ‘gluon Lagrangian’, i.e. a Yang-Mills Lagrangian on the (1, 1)-order gauge-natural bundle of SU (3)-principal connections. The kernel of the gauge-natural Jacobi morphism for such a Lagrangian, by inducing a reductive split structure, canonically defines a ‘gluon classical Higgs field’.

Higgs fields on spinor gauge-natural bundles

We show that the Lie derivative of spinor fields is parametrized by Higgs fields defined by the kernel of a gauge-natural Jacobi morphism associated with the Einstein–Cartan– Dirac Lagrangian. In particular, the generalized Kosmann lift to the total bundle of the theory is constrained by variational Higgs fields on gauge-natural bundles.

GAUGE-NATURAL NOETHER CURRENTS AND CONNECTION FIELDS

We study geometric aspects concerned with symmetries and conserved quantities in gauge-natural invariant variational problems and investigate implications of the existence of a reductive split structure associated with canonical Lagrangian conserved quantities on gauge-natural bundles. In particular, we characterize the existence of covariant conserved quantities in terms of principal Cartan connections on gauge-natural prolongations.

Reduction theorems for principal and classical connections

We prove general reduction theorems for gauge natural operators transforming principal connections and classical linear connections on the base manifold into sections of an arbitrary gauge natural bundle. Then we apply our results to the principal prolongation of connections. Finally we describe all such gauge natural operators for some special cases of a Lie group G.

Lagrangian reductive structures on gauge-natural bundles

A reductive structure is associated here with Lagrangian canonically defined conserved quantities on gauge-natural bundles. Parametrized transformations defined by the gauge-natural lift of infinitesimal principal automorphisms induce a variational sequence such that the generalized Jacobi morphism is naturally self-adjoint. As a consequence, its kernel defines a reductive split structure on the relevant underlying principal bundle.

The relation between the Jacobi morphism and the Hessian in gauge-natural field theories

We generalize a classic result, due to Goldschmidt and Sternberg, relating the Jacobi morphism and the Hessian for first-order field theories to higher-order gauge-natural field theories. In particular, we define a generalized gauge-natural Jacobi morphism where the variation vector fields are Lie derivatives of sections of the gauge-natural bundle with respect to gauge-natural lifts of infinitesimal principal automorphisms, and we relate it to the Hessian. The Hessian is also very simply related to the generalized Bergmann-Bianchi morphism, whose kernel provides necessary and sufficient conditions for the existence of global canonical superpotentials. We find that the Hamilton equations for the Hamiltonian connection associated with a suitably defined covariant strongly conserved current are satisfied identically and can be interpreted as generalized Bergmann-Bianchi identities and thus characterized in terms of the Hessian vanishing.

Higher-order utiyama invariant interaction

In this paper we prove higher order version of the Utiyama's invariant interaction of a particle and a gauge field (a principal connection on a principal bundle). To describe the Utiyama's interaction in order $r\ge 2$ we have to use an auxiliary gravitational field (a linear symmetric connection on the base manifold). We prove that any natural (invariant) operator of order $s$ in the gravitational field, of order $r$ in the gauge field and of order $k$ in the particle field, $s,r\ge k-1, \, s\ge r-2$, with values in a $(1,0)$-order $G$-gauge-natural bundle factorizes through the curvature tensors of both connections, the particle field and their covariant differentials up to sufficiently high orders. The covariant differentials are considered with respect to both connections.

Higher order Utiyama-like theorem

In this paper we prove higher order version of the Utiyama-like theorem. To prove the Utiyama-like theorem in order r ≥ 2 we have to use auxiliary classical connections on base manifolds. We prove that any natural (invariant) operator of order r for principal connections on principal G -bundles and for classical connections on base manifolds with values in a (1, 0)-order G -gauge-natural bundle factorizes through curvature tensors of both connections and their co-variant differentials, where the covariant differential of curvature tensors of principal connections is considered with respect to both connections.

A first-order purely frame-formulation of general relativity

In the gauge natural bundle framework, a new space is introduced and a first-order purely frame-formulation of general relativity is obtained.

Generalized Bianchi identities in gauge-natural field theories and the curvature of variational principles

By resorting to Noether's Second Theorem, we relate the generalized Bianchi identities for Lagrangian field theories on gauge-natural bundles with the kernel of the associated gauge-natural Jacobi morphism. A suitable definition of the curvature of gauge-natural variational principles can be consequently formulated in terms of the Hamiltonian connection canonically associated with a generalized Lagrangian obtained by contracting field equations.

Covariant gauge-natural conservation laws

When a gauge-natural invariant variational principle is assigned, to determine canonical covariant conservation laws, the vertical part of gauge-natural lifts of infinitesimal principal automorphisms—defining infinitesimal variations of sections of gauge-natural bundles—must satisfy generalized Jacobi equations for the gauge-natural invariant Lagrangian. Vice versa all vertical parts of gauge-natural lifts of infinitesimal principal automorphisms which are in the kernel of generalized Jacobi morphisms are generators of canonical covariant currents and superpotentials. In particular, only a few gauge-natural lifts can be considered as canonical generators of covariant gauge-natural physical charges.

Einstein-Dirac theory on gauge-natural bundles

We present a clear-cut example of the importance of the functorial approach of gauge-natural bundles and the general theory of Lie derivatives for classical field theory, where the sole correct geometrical formulation of Einstein (-Cartan) gravity coupled with Dirac fields gives rise to an unexpected indeterminacy in the concept of conserved quantities.

Metric-affine gravity and the Nester–Witten 2-form

In this paper we redefine the well-known metric-affine Hilbert Lagrangian in terms of a spin connection and a spin-tetrad. On applying the Poincaré–Cartan method and using the geometry of gauge-natural bundles, a global gravitational superpotential is derived. On specializing to the case of the Kosmann lift, we recover the result originally found by Kijowski [Gen. Rel. Gravity 9 (1978) (10) 857] for the metric (natural) Hilbert Lagrangian. On choosing a different, suitable lift, we can also recover the Nester–Witten 2-form, which plays an important role in the energy positivity proof and in many quasi-local definitions of mass.

Gauge-natural bundles and generalized gauge theories

Gauge-natural bundles and generalized gauge theories