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The definition of an action functional for the Jacobi sigma models, known for Jacobi brackets of functions, is generalized to Jacobi bundles, i.e. Lie brackets on sections of (possibly non-trivial) line bundles, with the particular case of contact manifolds. Different approaches are proposed, but all of them share a common feature: the presence of a homogeneity structure appearing as a principal action of the Lie group ${ℝ}^{\times }=\mathrm{\text{GL}}(1;ℝ)$. Consequently, solutions of the equations of motions are morphisms of certain Jacobi algebroids, i.e. principal ${ℝ}^{\times }$-bundles equipped additionally with a compatible Lie algebroid structure. Despite the different approaches we propose, there is a one-to-one correspondence between the space of solutions of the different models. The definition can be immediately extended to almost Poisson and almost Jacobi brackets, i.e. to brackets that do not satisfy the Jacobi identity. Our sigma models are geometric and fully covariant.

There are two groups which act in a natural way on the bundle $TP$ tangent to the total space $P$ of a principal $G$-bundle $P(M,G)$: the group $Au{t}_{0}TP$ of automorphisms of $TP$ covering the identity map of $P$ and the group $TG$ tangent to the structural group $G$. Let $Au{t}_{TG}TP\subset Au{t}_{0}TP$ be the subgroup of those automorphisms which commute with the action of $TG$. In the paper, we investigate $G$-invariant symplectic structures on the cotangent bundle $T^{\ast }P$ which are in a one-to-one correspondence with elements of $Au{t}_{TG}TP$. Since, as it is shown here, the connections on $P(M,G)$ are in a one-to-one correspondence with elements of the normal subgroup $Au{t}_{N}TP$ of $Au{t}_{0}TP$, so the symplectic structures related to them are also investigated. The Marsden–Weinstein reduction procedure for these symplectic structures is discussed.

The moduli space of G-bundles on an elliptic curve with additional flag structure admits a Poisson structure. The bivector can be defined using double loop group, loop group and sheaf cohomology constructions. We investigate the links between these methods and for the case SL₂ perform explicit computations, describing the bracket and its leaves in detail.

Let f : M → A be a smooth surjective algebraic morphism, where M is a connected complex projective manifold and A a complex abelian variety, such that all the fibers of f are rationally connected. We show that an algebraic principal G-bundle E_G over M admits a flat holomorphic connection if E_G admits a holomorphic connection; here G is any connected reductive linear algebraic group defined over ℂ. We also show that E_G admits a holomorphic connection if and only if any of the following three statements holds.

(1) The principal G-bundle E_G is semistable, c₂(ad(E_G)) = 0, and all the line bundles associated to E_G for the characters of G have vanishing rational first Chern class.

(2) There is an algebraic principal G-bundle E'_G on A such that f*E'_G = E_G, and all the translations of E'_G by elements of A are isomorphic to E'_G itself.

(3) There is a finite étale Galois cover and a reduction of structure group to a Borel subgroup B ⊂ G such that all the line bundles associated to Ê_B for the characters of B have vanishing rational first Chern class.

In particular, the above three statements are equivalent.

We consider principal bundles over G/P, where P is a parabolic subgroup of a semi-simple and simply connected linear algebraic group G defined over ℂ. We prove that a holomorphic principal H-bundle E_H → G/P, where H is a complex reductive group, and is homogeneous if the adjoint vector bundle ad(E_H) is homogeneous. Fix a faithful H-module V. We also show that E_H is homogeneous if the vector bundle E_H ×^H V associated to it for the H-module V is homogeneous.

Let G be a connected reductive linear algebraic group defined over an algebraically closed field k of positive characteristic. Let Z(G) ⊂ G be the center, and , where each G_i is simple with trivial center. For i ∈ [1, m], let ρ_i : G → G_i be the natural projection. Fix a proper parabolic subgroup P of G such that for each i ∈ [1, m], the image ρ_i(G) ⊂ G_i is a proper parabolic subgroup. Fix a strictly anti-dominant character χ of P such that χ is trivial on Z(G). Let M be a smooth projective variety, defined over k, equipped with a very ample line bundle ξ. Let E_G → M be a principal G-bundle. We prove that the following six statements are equivalent:

(1) The line bundle E_G(χ) → E_G/P associated to the principal P-bundle E_G → E_G/P for the character χ is numerically effective.

(2) The sequence of principal G-bundles is bounded, where F_M is the absolute Frobenius morphism of M.

(3) The principal G-bundle E_G is strongly semistable with respect to ξ, and c₂(ad(E_G)) is numerically equivalent to zero.

(4) The principal G-bundle E_G is strongly semistable with respect to ξ, and [c₂(ad(E_G))c₁(ξ)^d-2] = 0.

(5) The adjoint vector bundle ad(E_G) is numerically effective.

(6) For every pair of the form (Y,ψ), where Y is an irreducible smooth projective curve and ψ : Y → M is a morphism, the principal G-bundle ψ*E_G → Y is semistable.

Let M be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric g and a covariant constant volume form. Let G be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat principal G-bundle E_G over M admits a Hermitian–Einstein structure if and only if E_G is polystable. A polystable flat principal G-bundle over M admits a unique Hermitian–Einstein connection. We also prove the existence and uniqueness of a Harder–Narasimhan filtration for flat vector bundles over M. We prove a Bogomolov type inequality for semistable vector bundles under the assumption that the Gauduchon metric g is astheno-Kähler.

Let $X$ be a geometrically irreducible smooth projective curve, of genus at least three, defined over the field of real numbers. Let $G$ be a connected reductive affine algebraic group, defined over $ℝ$, such that $G$ is nonabelian and has one simple factor. We prove that the isomorphism class of the moduli space of principal $G$-bundles on $X$ determine uniquely the isomorphism class of $X$.

Let $E_G$ be a $\mathrm{\Gamma }$-equivariant algebraic principal $G$-bundle over a normal complex affine variety $X$ equipped with an action of $\mathrm{\Gamma }$, where $G$ and $\mathrm{\Gamma }$ are complex linear algebraic groups. Suppose $X$ is contractible as a topological $\mathrm{\Gamma }$-space with a dense orbit, and $x_0\in X$ is a $\mathrm{\Gamma }$-fixed point. We show that if $\mathrm{\Gamma }$ is reductive, then $E_G$ admits a $\mathrm{\Gamma }$-equivariant isomorphism with the product principal $G$-bundle $X{\times }_{\rho }{E}_G(x_0)$, where $\rho :\mathrm{\Gamma }\to G$ is a homomorphism between algebraic groups. As a consequence, any torus equivariant principal $G$-bundle over an affine toric variety is equivariantly trivial. This leads to a classification of torus equivariant principal $G$-bundles over any complex toric variety, generalizing the main result of [A classification of equivariant principal bundles over nonsingular toric varieties, Internat. J. Math.27(14) (2016)].

Let $G$ be a semisimple complex algebraic group with a simple Lie algebra $𝔤$, and let ${{ℳ}}_G^0$ denote the moduli stack of topologically trivial stable $G$-bundles on a smooth projective curve $C$. Fix a theta characteristic $\kappa$ on $C$ which is even in case $dim𝔤$ is odd. We show that there is a nonempty Zariski open substack ${{𝒰}}_{\kappa }$ of ${{ℳ}}_G^0$ such that $H^i(C,\mathrm{\text{ad}}(E_G)\otimes \kappa )=0$, $i=1,2$, for all $E_G\in {{𝒰}}_{\kappa }$. It is shown that any such $E_G$ has a canonical connection. It is also shown that the tangent bundle $T{U}_{\kappa }$ has a natural splitting, where $U_{\kappa }$ is the restriction of ${{𝒰}}_{\kappa }$ to the semi-stable locus. We also produce an isomorphism between two naturally occurring ${\mathrm{\Omega }}_{M_G^{rs}}^1$-torsors on the moduli space of regularly stable $M_G^{rs}$.

In gauge theory, Higgs fields are responsible for spontaneous symmetry breaking. In classical gauge theory on a principal bundle P, a symmetry breaking is defined as the reduction of a structure group of this principal bundle to a subgroup H of exact symmetries. This reduction takes place if and only if there exists a global section of the quotient bundle P/H. It is a classical Higgs field. A metric gravitational field exemplifies such a Higgs field. We summarize the basic facts on the reduction in principal bundles and geometry of Higgs fields. Our goal is the particular covariant differential in the presence of a Higgs field.

Let Λ ⊂ ℂ be the ℤ-module generated by 1 and , where τ is a positive real number. Let Z := ℂ/Λ be the corresponding complex torus of dimension one. Our aim here is to give a general construction of holomorphic principal Z-bundles over a complex manifold X.

Let θ₁ and θ₂ be two C^∞ real closed two-forms on X such that the Hodge type (0, 2) component of the form vanishes, and the elements in H²(X, ℂ) represented by θ₁ and θ₂ are contained in the image of H²(X, ℤ). For such a pair we construct a holomorphic principal Z-bundle over X. Conversely, given any holomorphic principal Z-bundle E_Z over X, we construct a pair of closed differential forms on X of the above type.

It is shown that, anti-BRST symmetry is the quantized counterpart of local axial symmetry in gauge theories. An extended form of descent equations is worked out, which yields a set of modified consistent anomalies.

Let $A$ be a complex abelian variety and $G$ a complex reductive affine algebraic group. We describe the connected component, containing the trivial bundle, of the moduli spaces of topologically trivial principal $G$-bundles and $G$-Higgs bundles on $A$. We also describe the moduli spaces of $G$-connections and the $G$-character variety for $A$.

We develop the concept of a double (more generally $n$-tuple) principal bundle departing from a compatibility condition for a principal action of a Lie group on a groupoid.