Structure of Gauge-Invariant Lagrangians

TL;DR

This paper proves the existence of a finite generating set of gauge-invariant Lagrangians for gauge theories, enabling any such Lagrangian to be expressed as a smooth function of these generators.

Contribution

It establishes the finite generation of gauge-invariant Lagrangians and provides explicit examples, advancing the mathematical understanding of gauge field theories.

Findings

Existence of finite generating set of gauge-invariant Lagrangians

Any gauge-invariant Lagrangian can be expressed as a smooth function of generators

Explicit examples illustrating the theory

Abstract

The theory of gauge fields in Theoretical Physics poses several mathematical problems of interest in Differential Geometry and in Field Theory. Below we tackle one of these problems: The existence of a finite system of generators of gauge-invariant Lagrangians and how to compute them. More precisely, if $p\colon C\to M$ is the bundle of connections on a principal $G$-bundle $\pi\colon P\to M$, then a finite number $L_1,\dotsc,L_{N^\prime }$ of gauge-invariant Lagrangians defined on $J^1C$ is proved to exist such that for any other gauge-invariant Lagrangian $L\in C^\infty (J^1C)$ there exists a function $F\in C^\infty (\mathbb{R}^{N^\prime })$ such that $L=F(L_1,\dotsc,L_{N^\prime})$. Several examples are dealt with explicitly.

Equations

$$\begin{array}[c]{ll}Y=v^{\alpha}\frac{\partial}{\partial y^{\alpha}},&v^{\alpha}\in C^{\infty}(E),\\ Y^{(1)}=v^{\alpha}\frac{\partial}{\partial y^{\alpha}}+v_{i}^{\alpha}\frac{\partial}{\partial y_{i}^{\alpha}},&v_{i}^{\alpha}=\frac{\partial v^{\alpha}}{\partial x^{i}}+\frac{\partial v^{\alpha}}{\partial y^{\beta}}y_{i}^{\beta},\end{array}$$

$$0\to \mathrm{ad}P\to T(P)/G\stackrel{{\pi }_{\ast }}{⟶}TM\to 0.$$

$$\left[(u,A{)}_{\mathrm{ad}},(u,B{)}_{\mathrm{ad}}\right]=(u,-\left[A,B\right]{)}_{\mathrm{ad}},\forall u\in {\pi }^{-1}\left(x\right),\forall A,B\in \mathfrak{g},$$

$$s_{\Gamma}\Bigl{(}\frac{\partial}{\partial x^{i}}\Bigr{)}=\Bigl{(}\frac{\partial}{\partial x^{i}}\Bigr{)}^{h_{\Gamma}}=\frac{\partial}{\partial x^{i}}-\left(A_{i}^{\alpha}\circ s_{\Gamma}\right)\tilde{B}_{\alpha},\quad 1\leq i\leq n.$$

$$X=g^{\alpha }{\tilde{B}}_{\alpha },g^{\alpha }\in C^{\infty }(U),$$

$$X_C=-\left(\frac{\partial g^{\alpha }}{\partial x^i}-c_{\beta \gamma }^{\alpha }{g}^{\beta }{A}_i^{\gamma }\right)\frac{\partial }{\partial A_i^{\alpha }},$$

$${\mathcal{L}}_{X_C^{(1)}}(Lv)=X_C^{(1)}(L)v+L({\mathcal{L}}_{X_C^{(1)}}v)=0,\forall X\in \mathrm{aut}P.$$

$$\begin{array}[c]{l}\Omega\colon J^{1}C\to\bigwedge\nolimits^{2}T^{\ast}M\otimes\mathrm{ad}P\\ \Omega(j_{x}^{1}\sigma_{\Gamma})=\left(\Omega_{\Gamma}\right)_{x}\end{array}$$

$${\eta }_2=\sum _{j<k}{R}_{jk}^{\alpha }({\eta }_2){\left(d{x}^j\wedge d{x}^k\otimes {\tilde{B}}_{\alpha }\right)}_x,\forall {\eta }_2\in {\bigwedge }^2{T}_x^{\ast }M\otimes (\mathrm{ad}P{)}_x,\forall x\in U.$$

$$\begin{array}[c]{rl}R_{jk}^{\alpha}\circ\Omega=&A_{j,k}^{\alpha}-A_{k,j}^{\alpha}-\sum_{\beta<\gamma}c_{\beta\gamma}^{\alpha}\left(A_{j}^{\beta}A_{k}^{\gamma}-A_{j}^{\gamma}A_{k}^{\beta}\right),\vskip 3.0pt plus 1.0pt minus 1.0pt\\ &1\leq j<k\leq n,\quad 1\leq\alpha\leq m.\end{array}$$

$$\tilde{L}:{\bigwedge }^2{T}^{\ast }M\otimes \mathrm{ad}P\to \mathbb{R}$$

$$\begin{array}[c]{rll}\chi_{\alpha}(\tilde{L})=&0,&1\leq\alpha\leq m.\\ \chi_{\alpha}=&{\displaystyle\sum\nolimits_{i<j}}c_{\gamma\alpha}^{\beta}R_{ij}^{\gamma}\dfrac{\partial}{\partial R_{ij}^{\beta}}.&\end{array}$$

$$\begin{array}[c]{rll}\Lambda(\eta_{2})=&\left(\lambda_{\beta,ij}^{\alpha}(\eta_{2})\right)_{1\leq\beta\leq m,1\leq i<j\leq n}^{1\leq\alpha\leq m},&\lambda_{\beta,ij}^{\alpha}(\eta_{2})=c_{\gamma\alpha}^{\beta}R_{ij}^{\gamma}(\eta_{2}).\end{array}$$

$$\begin{array}[c]{rlrlll}c_{ij}^{k}=&\!\!0,&c_{ij}^{\alpha}=&\!\!0,&i,j=1,\dotsc,l,&\!\!\alpha\in\Delta_{\mathfrak{g}},\\ c_{i\alpha}^{k}=&\!\!0,&c_{i\alpha}^{\beta}=&\!\!\delta_{\alpha}^{\beta}\left\langle\alpha|\alpha_{i}\right\rangle,&1\leq i\leq l,&\!\!\alpha,\beta\in\Delta_{\mathfrak{g}},\\ c_{\alpha,-\alpha}^{i}=&\!\!e^{i},&c^{\beta}_{\alpha,-\alpha}=&\!\!0,&1\leq i\leq l,&\!\!\alpha,\beta\in\Delta_{\mathfrak{g}},\\ c_{\alpha\beta}^{i}=&\!\!0,&c_{\alpha\beta}^{\gamma}=&\!\!0&\alpha,\beta,\gamma\in\Delta_{\mathfrak{g}},&\!\!\alpha+\beta\neq 0,\alpha+\beta\notin\Delta_{\mathfrak{g}},\\ c_{\alpha\beta}^{i}=&\!\!0,&c_{\alpha\beta}^{\gamma}=&\!\!\pm\delta_{\alpha+\beta}^{\gamma}(p+1),&1\leq i\leq l,&\!\!\alpha+\beta\in\Delta_{\mathfrak{g}},\gamma\in\Delta_{\mathfrak{g}}.\end{array}$$

$${\eta }_2=\sum _{j<k}\sum _{i=1}^l{\left(R_{jk}^i({\eta }_2)d{x}^j\wedge d{x}^k\otimes {\tilde{h}}_i\right)}_x+\sum _{j<k}\sum _{a=1}^{m-l}{\left(R_{jk}^{{\alpha }_a}({\eta }_2)d{x}^j\wedge d{x}^k\otimes {\tilde{e}}_{{\alpha }_a}\right)}_x.$$

$$\begin{array}[c]{rll}\lambda_{\beta_{b},jk}^{t}(\eta_{2})=&-\left\langle\beta_{b}|\beta_{t}\right\rangle R_{jk}^{\beta_{b}}(\eta_{2}),&1\leq t\leq l,1\leq b\leq m-l,\end{array}$$

$$\begin{array}[c]{rll}\lambda_{t,jk}^{\alpha_{a}}(\eta_{2})=&-e^{t}R_{jk}^{-\alpha_{a}}(\eta_{2}),&1\leq t\leq l,1\leq a\leq m-l,\end{array}$$

$$\begin{array}[c]{rl}\lambda_{\alpha_{v},jk}^{\alpha_{u}}(\eta_{2})=&\sum_{t=1}^{l}\delta_{\alpha_{v}}^{\alpha_{u}}\left\langle\alpha_{v}|\alpha_{t}\right\rangle R_{jk}^{t}(\eta_{2})\\ &\lx@intercol\hfil\mp\sum_{r=1}^{m-l}(p_{r}+1)\delta_{\alpha_{r}+\alpha_{v}}^{\alpha_{u}}R_{jk}^{\alpha_{r}}(\eta_{2}),\vskip 3.0pt plus 1.0pt minus 1.0pt\lx@intercol\\ &\lx@intercol\hfil u,v=1,\dotsc,m-l.\lx@intercol\end{array}$$

$$\begin{array}[c]{ll}\lambda_{\alpha_{v},jk}^{\alpha_{u}}(\eta_{2})=&\sum_{t=1}^{l}\delta_{\alpha_{v}}^{\alpha_{u}}\left\langle\alpha_{v}|\alpha_{t}\right\rangle R_{jk}^{t}(\eta_{2})\\ \lx@intercol\hfil\lx@intercol&\lx@intercol\hfil\mp(p_{r_{0}}+1)R_{jk}^{\alpha_{r_{0}}}(\eta_{2}),\vskip 3.0pt plus 1.0pt minus 1.0pt\lx@intercol\\ \lx@intercol\hfil\lx@intercol&\lx@intercol\hfil u,v=1,\dotsc,m-l.\lx@intercol\end{array}$$

$$\begin{array}[c]{ll}\lambda_{\alpha_{v},jk}^{\alpha_{u}}(\eta_{2})=&\sum_{t=1}^{l}\delta_{\alpha_{v}}^{\alpha_{u}}\left\langle\alpha_{v}|\alpha_{t}\right\rangle R_{jk}^{t}(\eta_{2}),\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \lx@intercol\hfil\lx@intercol&\lx@intercol\hfil u,v=1,\dotsc,m-l.\lx@intercol\end{array}$$

$$\Lambda_{jk}(\eta_{2})=\left(\begin{array}[c]{cc}O&A_{jk}(\eta_{2})\\ B_{jk}(\eta_{2})&C_{jk}(\eta_{2})\end{array}\right),$$

$$A_{jk}({\eta }_2)={\left({\lambda }_{{\alpha }_a,jk}^t({\eta }_2)\right)}_{1\le a\le m-l}^{1\le t\le l},B_{jk}({\eta }_2)={\left({\lambda }_{t,jk}^{{\alpha }_a}({\eta }_2)\right)}_{1\le t\le l}^{1\le a\le m-l},$$

$$C_{jk}({\eta }_2)={\left({\lambda }_{a_b,jk}^{{\alpha }_a}({\eta }_2)\right)}_{1\le b\le m-l}^{1\le a\le m-l}.$$

$$A_{jk}(\eta_{2})=-\left(\begin{array}[c]{ccc}\left\langle\beta_{1}|\beta_{1}\right\rangle R_{jk}^{\beta_{1}}(\eta_{2})&\ldots&\left\langle\beta_{m-l}|\beta_{1}\right\rangle R_{jk}^{\beta_{m-l}}(\eta_{2})\\ \vdots&\ddots&\vdots\\ \left\langle\beta_{1}|\beta_{l}\right\rangle R_{jk}^{\beta_{1}}(\eta_{2})&\ldots&\left\langle\beta_{m-l}|\beta_{l}\right\rangle R_{jk}^{\beta_{m-l}}(\eta_{2})\end{array}\right),$$

$$B_{jk}(\eta_{2})=-\left(\begin{array}[c]{ccc}e^{1}R_{jk}^{-\alpha_{1}}(\eta_{2})&\ldots&e^{l}R_{jk}^{-\alpha_{1}}(\eta_{2})\\ \vdots&\ddots&\vdots\\ e^{1}R_{jk}^{-\alpha_{m-l}}(\eta_{2})&\ldots&e^{l}R_{jk}^{-\alpha_{m-l}}(\eta_{2})\end{array}\right),$$

$$\begin{array}[c]{rlll}\mu_{u,ij}=\lambda_{\alpha_{u},ij}^{\alpha_{u}}(\eta_{2})=&\sum_{t=1}^{l}\left\langle\alpha_{u}|\alpha_{t}\right\rangle R_{ij}^{t}(\eta_{2}^{0}),&1\leq u\leq m-l,&1\leq i<j\leq n.\end{array}$$

$$\det A_{jk}^{\prime }({\eta }_2^0)=R_{jk}^{{\alpha }_1}({\eta }_2^0)\cdots R_{jk}^{{\alpha }_l}({\eta }_2^0)\det (⟨{\alpha }_u|{\alpha }_v⟩{)}_{u,v=1}^l.$$

$$\Lambda^{\prime\prime}(\eta_{2}^{0})=\left(\begin{array}[c]{cc}A_{12}(\eta_{2}^{0})&A_{13}^{\prime}(\eta_{2}^{0})\\ C_{12}(\eta_{2}^{0})&C_{13}^{\prime}(\eta_{2}^{0})\end{array}\right),$$

$$C_{13}^{\prime}(\eta_{2}^{0})=\left(\begin{array}[c]{cccc}\mu_{1,13}&0&\ldots&0\\ 0&\mu_{2,13}&0&0\\ \vdots&0&\ddots&\vdots\\ 0&0&\ldots&\mu_{l,13}\end{array}\right).$$

$$\left(\begin{array}[c]{c}A_{13}^{\prime}(\eta_{2}^{0})\\ C_{13}^{\prime}(\eta_{2}^{0})\end{array}\right)$$

Full text

PACS numbers: 2.20.Sv Lie algebras of Lie groups; 02.30.Ik Integrable systems; 02.40.-k Geometry, differential geometry, and topology; 03.50. z Classical field theories; 11.10.Ef Lagrangian and Hamiltonian approach; 11.15.-q Gauge field theories

Mathematics Subject Classification 2000: Primary 35F20; Secondary 53C05, 58A20, 58D19, 58E15, 58E30, 81T13.

Key words and phrases: Bundle of connections, gauge invariance, jet bundles, curvature mapping, functionally independent gauge-invariant Lagrangians, structure of Lie algebras

Acknowledgments: ¡¡¡ Incluir proyecto quien lo tenga !!!

1 Introduction

The notion of gauge invariance—defined as invariance under the group of base-preserving automorphisms of a principal fibre $G$-bundle $\pi\colon P\rightarrow M$ over an oriented space-time—is fundamental in the theory of gauge fields and their associated fields, such as Yang-Mills-Higgs fields; for example, see the classical expositions [13] or [14].

Below we are concerned with gauge invariance of the variational problems determined by a free (i.e., without any interaction term with a particle field) gauge invariant Lagrangian function defined on the fibre bundle $p\colon C\rightarrow M$ of connections on $\pi\colon P\rightarrow M$.

The fundamental step in determining such Lagrangians, is the so-called geometric formulation of Utiyama’s theorem (see [4, 10.2.15 Theorem]), according to which a Lagrangian $L$ of first order on $C$ is gauge invariant if and only if $L$ factors through the curvature map by means of a zero-order Lagrangian on the vector bundle of differential $2$-forms on $M$ with values in the adjoint bundle of $P$ (also called “the curvature bundle”), which, in turn, must be invariant under the natural representation of the gauge group on that bundle. See [3] for the generalization of Utiyama’s theorem to Lagrangians for gauge-particle field interaction.

This reduces the problem of determining gauge-invariant Lagrangians to the problem of determining the zero-order gauge invariant Lagrangians defined on the curvature bundle. If $G$ is connected, then the second problem can infinitesimally be solved by proving that zero-order gauge invariant Lagrangians on the curvature bundle are the first integrals of an involutive distribution $\mathcal{D}$.

In this paper, we prove that $\mathcal{D}$ is of constant rank on a dense open subset and we compute this rank. If $\mathfrak{g}$ is the Lie algebra of $G$, and $n=\dim M$, $m=\dim\mathfrak{g}$, $l=\operatorname*{rank}\mathfrak{g}$, then we obtain the following results (see §3.2 below):

1st) If $n=2$, then the generic rank of $\mathcal{D}$ equals $m-l$, and

2nd) If $n\geq 3$, then the generic rank is $m$.

The result for $n=2$ explains why the theory of Yang-Mills fields on a surface presents special features. In fact, according to a classical theorem by Chevalley (e.g., see [15, Theorem 4.9.3]) there exist $l$ homogeneous algebraically independent polynomials $p_{1},\ldots,p_{l}$ such that the algebra of polynomial functions on $\mathfrak{g}$ that are invariant with respect to the adjoint representation of $G$ on $\mathfrak{g}$, is isomorphic to the algebra of polynomials in $p_{1},\ldots,p_{l}$, thus providing a basis with geometric meaning for the algebra of first integrals of $\mathcal{D}$, as stated in Remark 4.1.

Next, for $n\geq 3$ we also obtain a basis of first integrals spanning differentiably the ring of zero-order Lagrangians. Assuming $G$ connected and semisimple, in the local case such a basis is deduced from Hilbert-Nagata theorem (see Theorem 3.2).

Finally, we include several worked examples in low dimensions illustrating the previous general results.

2 Preliminaries

2.1 Jet prolongation

Let $p\colon E\to M$ be a fibred manifold over an orientable connected smooth manifold oriented by a volume form $\mathbf{v}$. A pair of diffeomorphisms $\phi\in\mathrm{Diff}M$, $\Phi\in\mathrm{Diff}E$, such that $p\circ\Phi=\phi\circ p$, is said to be an automorphism of $p$; the group of all automorphisms is denoted by $\mathrm{Aut}E$.

If $Y\in\mathfrak{X}(E)$ then there exists a unique vector field $Y^{(1)}$ in $\mathfrak{X}(J^{1}E)$—the $1$-jet prolongation of $Y$ to the first-order jet bundle $J^{1}E$—projectable onto $Y$ such that $\mathcal{L}_{Y^{(1)}}$ keeps invariant the module of contact $1$-forms spanned by the following forms $\theta^{\alpha}=dy^{\alpha}-y_{i}^{\alpha}dx^{i}$, $1\leq\alpha\leq m$ on $\Omega^{1}(J^{1}E)$, where $n=\dim M$, $m+n=\dim E$, and $(x^{i},y^{\alpha})$, $1\leq\alpha\leq m$, $1\leq i\leq n$, is a fibred coordinate system for $p\colon E\to M$ and $(x^{i},y^{\alpha},y_{i}^{\alpha})$ is the induced coordinate system on $J^{1}E$.

The Lie algebra of $\mathrm{Aut}E$ is the Lie subalgebra of $p$-projectable vector fields $\mathfrak{X}_{p}(E)\subset\mathfrak{X}(E)$, namely, if $\Phi_{t}$ is the local flow of $Y\in\mathfrak{X}(E)$, then $\Phi_{t}\in\mathrm{Aut}E$, $\forall t$, if and only if $Y\in\mathfrak{X}_{p}(E)$; in this case, $\Phi_{t}^{(1)}$ is the flow of $Y^{(1)}$. If $Y$ is a $p$-vertical vector field, then the formulas of $1$-jet prolongation are as follows:

[TABLE]

2.2 $\mathrm{aut}P$ and $\mathrm{gau}P$

Let $G$ be a Lie group. An automorphism of a principal $G$-bundle $\pi\colon P\to M$ is a $G$-equivariant diffeomorphism $\Phi\colon P\to P$. The group of all automorphisms of $P$ is denoted by $\mathrm{Aut}P$. Every $\Phi\in\mathrm{Aut}P$ determines a unique diffeomorphism $\phi\colon M\to M$, such that $\pi\circ\Phi=\phi\circ\pi$. If $\phi$ is the identity map on $M$, then $\Phi$ is said to be a gauge transformation (cf. [4, 3.2.1]); the subgroup of all gauge transformations is denoted by $\mathrm{Gau}P\subset\mathrm{Aut}P$.

A vector field $X\in\mathfrak{X}\left(P\right)$ is said to be $G$-invariant if $R_{g}\cdot X=X$, $\forall g\in G$; if $\Phi_{t}$ is the flow of $X$, then $X$ is $G$-invariant if and only if $\Phi_{t}\in\mathrm{Aut}P$, $\forall t\in\mathbb{R}$. The Lie subalgebra of $G$-invariant vector fields on $P$ is denoted by $\mathrm{aut}P\subset\mathfrak{X}\left(P\right)$. Each $G$-invariant vector field on $P$ is $\pi$-projectable.

Similarly, a $\pi$-vertical vector field $X\in\mathfrak{X}\left(P\right)$ is $G$-invariant if and only if $\Phi_{t}\in\mathrm{Gau}P$, $\forall t\in\mathbb{R}$. Let $\mathrm{gau}P\subset\mathrm{aut}P$ be the ideal of all $\pi$-vertical $G$-invariant vector fields on $P$, which is usually called the gauge algebra of $P$.

The quotient $T(P)/G$ exists as a differentiable manifold and it is endowed with a vector bundle structure over $M$ (see [1]), whose global sections can naturally be identified to $\mathrm{aut}P$; i.e., $\mathrm{aut}P\cong\Gamma\left(M,T(P)/G\right)$.

If $G$ acts on the left on a manifold $F$ via a map $G\times F\to F$, $(g,y)\mapsto g\star y$, then $G$ acts on the right on the product $P\times F$ by setting $(u,y)\cdot g=(u\cdot g,g^{-1}\star y)$, $\forall g\in G$, $\forall u\in P$, $\forall y\in F$. The quotient manifold $(P\times F)/G$ of this action exists and it defines a fibre bundle $\pi_{F}\colon P\times^{G}F\to M$, $\pi_{F}((u,y)\operatorname{mod}G)=\pi(u)$, called the bundle associated to $P$ by the action on $F$; e.g., see [4, §3.1], [10, §35], [11, p.54].

Every $\phi\in\mathrm{Aut}P$ induces a diffeomorphism $\Phi_{F}\colon P\times^{G}F\to P\times^{G}F$ by setting $\Phi_{F}((u,y)\operatorname{mod}G)=(\Phi(u),y)\operatorname{mod}G$, $\forall u\in P$, $\forall y\in F$. If $\phi\in\mathrm{Diff}M$ is the diffeomorphism determined by $\Phi$, then $\pi_{F}\circ\Phi_{F}=\phi\circ\pi_{F}$.

Smooth sections $s\colon M\to P\times^{G}F$ of $\pi_{F}\colon P\times^{G}F\to M$ are in one-to-one correspondence with $G$-equivariant smooth maps $\hat{s}\colon P\to F$, which means in this case that $\hat{s}(u\cdot g)=g^{-1}\star\hat{s}(u)$, $\forall u\in P$, $\forall g\in G$, see [10, Proposition 35.1].

If $F$ is a Lie group and every left translation $L_{g}\colon F\to F$ of the action of $G$ on $F$ is an automorphism, then the fibres of $\pi_{F}\colon P\times^{G}F\to M$ are endowed with a structure of Lie group isomorphic to $F$. In fact, we can represent two given points $f,f^{\prime}\in\pi_{F}^{-1}(x)$ as $f=(u,y)\operatorname{mod}G$, $f^{\prime}=(u,y^{\prime})\operatorname{mod}G$, with the same $u\in P$, as $G$ acts transitively on $\pi^{-1}(x)$, and thus the operation given by $f\cdot f^{\prime}=(u,y\cdot y^{\prime})\operatorname{mod}G$, makes sense. Hence $\pi_{F}$ is Lie-group fibre bundle.

In particular, if $G$ acts on itself by conjugation, $G\times G\to G$, $(g,x)\to gxg^{-1}$, $\forall g,x\in G$, then the associated bundle $\pi_{G}\colon\mathrm{Ad}P=P\times^{G}G\to M$ is a Lie-group fibre bundle. Consequently, smooth sections $\Gamma(M,\mathrm{Ad}P)$ of $\pi_{G}$ admit a group structure given by $(s\cdot s^{\prime})(x)=s(x)\cdot s^{\prime}(x)$, for all $s,s^{\prime}\in\Gamma(M,\mathrm{Ad}P)$, $x\in M$, and the group $\mathrm{Gau}P$ is isomorphic to $\Gamma(M,\mathrm{Ad}P)$, see [10, Proposition 35.2].

Similarly $\mathrm{gau}P\cong\Gamma\left(M,\mathrm{ad}P\right)$, where $\pi_{\mathfrak{g}}\colon\mathrm{ad}P\to M$ denotes the adjoint bundle, i.e., the bundle associated to $P$ by the adjoint representation of $G$ on its Lie algebra $\mathfrak{g}$; namely $\mathrm{ad}P=\left(P\times\mathfrak{g}\right)/G$, the action of $G$ on $P\times\mathfrak{g}$ being defined by $\left(u,A\right)\cdot g=(u\cdot g,\mathrm{Ad}_{g^{-1}}\left(A\right))$, $\forall u\in P$, $\forall A\in\mathfrak{g}$, $\forall g\in G$. The $G$-orbit of $\left(u,A\right)\in P\times\mathfrak{g}$ in $\mathrm{ad}P$ is denoted by $(u,A)_{\mathrm{ad}}$. We thus obtain an exact sequence of vector bundles over $M$ (the so-called Atiyah sequence, see [1, Th. 1]):

[TABLE]

The fibres $\left(\mathrm{ad}P\right)_{x}$ are endowed with a Lie algebra structure determined by

[TABLE]

where $\left[\cdot,\cdot\right]$ denotes the bracket in $\mathfrak{g}$, but this is no longer true for the fibres of $T(P)/G$. The sign of the bracket in (3) is needed in order to ensure that the natural identification $\mathrm{gau}P\simeq\Gamma\left(M,\mathrm{ad}P\right)$ is a Lie algebra isomorphism, when $\mathrm{gau}P$ is considered as a Lie subalgebra of $\mathfrak{X}(P)$.

3 Bundle of connections

Let $X^{h_{\Gamma}}\in\mathfrak{X}\left(P\right)$ be the horizontal lift of $X\in\mathfrak{X}\left(M\right)$ with respect to a connection $\Gamma$ on $\pi\colon P\to M$. The vector field $X^{h_{\Gamma}}$ is $G$-invariant and projects onto $X$ (cf. [11, II. Proposition 1.2]). Hence we have a splitting of (2), $s_{\Gamma}\colon TM\to T(P)/G$, $s_{\Gamma}\left(X\right)=X^{h_{\Gamma}}$. Conversely, any splitting $\sigma\colon TM\to T(P)/G$ of that sequence comes from a unique connection on $P$. Therefore there is a natural bijection between connections on $P$ and splittings of the sequence above. Connections on $P$ can be identified to the global sections of a bundle $p\colon C\to M$; the section of $p$ induced by $\Gamma$ is denoted by $s_{\Gamma}\colon M\to C$. Moreover, $C$ is an affine bundle modelled over $\mathrm{Hom}\left(TM,\mathrm{ad}P\right)\cong T^{\ast}M\otimes\mathrm{ad}P$. For more details we refer the reader to [6].

Let $\left(U;x^{i}\right)$ be a coordinate system on an open domain $U\subset M$ over which $\pi$ admits a section $s\colon U\to P$, so that $\pi^{-1}(U)\cong U\times G$. For every $B\in\mathfrak{g}$ let $\tilde{B}$ be the infinitesimal generator of the flow of gauge transformations over $U$ defined by $\varphi_{t}^{B}\left(x,g\right)=\left(x,\exp\left(tB\right)\cdot g\right)$, $x\in U$. As $\pi\circ\varphi_{t}^{B}=\pi$ the vector field $\tilde{B}\in\mathfrak{X}(\pi^{-1}(U))$ is $\pi$-vertical. If $(B_{1},\dotsc,B_{m})$ is a basis of $\mathfrak{g}$, then $\tilde{B}_{1},\dotsc,\tilde{B}_{m}$ is a basis of $\Gamma(U,\mathrm{ad}P)$. The horizontal lift with respect to $\Gamma$ of the basic vector field $\partial/\partial x^{i}$ is given as follows:

[TABLE]

The functions $(x^{i},A_{j}^{\alpha})$, $i,j=1,\dotsc,n=\dim M$, $1\leq\alpha\leq m=\dim G$, induce a coordinate system on $p^{-1}\left(U\right)=C\left(\pi^{-1}U\right)$ (cf. [6]); hence $\dim C=n(m+1)$.

Each automorphism $\phi\in\mathrm{Aut}P$ acts on connections by pulling back connection forms; i.e., $\Gamma^{\prime}=\Phi\left(\Gamma\right)$ where $\omega_{\Gamma^{\prime}}=(\Phi^{-1})^{\ast}\omega_{\Gamma}$ (cf. [11, II. Proposition 6.2-(b)]). For each $\phi\in\mathrm{Aut}P$ there exists a unique diffeomorphism $\Phi_{C}\colon C\to C$ such that $p\circ\Phi_{C}=\phi\circ p$, where $\phi\colon M\to M$ is the diffeomorphism induced by $\Phi$ on the ground manifold. We thus obtain a group homomorphism $\mathrm{Aut}P\to\mathrm{Diff}C$ and for every connection $\Gamma$ on $P$ we have $\Phi_{C}\circ s_{\Gamma}=s_{\Phi\left(\Gamma\right)}$. If $\Phi_{t}$ is the flow of a $G$-invariant vector field $X\in\mathrm{aut}P$, then $\left(\Phi_{t}\right)_{C}$ is a one-parameter group in $\mathrm{Diff}C$ with infinitesimal generator denoted by $X_{C}$, and the map $\mathrm{aut}P\to\mathfrak{X}\left(C\right)$, $X\mapsto X_{C}$ is a Lie-algebra homomorphism.

By using a coordinate domain $\left(U;x^{i}\right)_{i=1}^{n}$ in $M$ and the basis $(\tilde{B}_{\alpha})_{\alpha=1}^{m}$ of $\mathrm{ad}\pi^{-1}(U)$ introduced above, it follows that each $X\in\mathrm{gau}\pi^{-1}(U)$ can be written as

[TABLE]

and, as a computation shows (e.g., see [6]), we have

[TABLE]

where $c_{\beta\gamma}^{\alpha}$ are the structure constants: $[B_{\beta},B_{\gamma}]=c_{\beta\gamma}^{\alpha}B_{\alpha}$.

4 Gauge invariance

4.1 $\mathrm{gau}P$ and $\mathrm{aut}P$ invariance

A Lagrangian density $\Lambda=Lv$, $L\in C^{\infty}(J^{1}C)$, on the bundle of connections is said to be gauge invariant if $X_{C}^{(1)}(L)=0$, $\forall X\in\mathrm{gau}P$, where $X_{C}^{(1)}$ denotes the $1$-jet prolongation of the natural representation $\mathrm{aut}P\to\mathfrak{X}(C)$. Similarly, a Lagrangian density is said to be $\mathrm{aut}P$-invariant if

[TABLE]

The vector field $X_{C}^{(1)}$ is $p_{1}$-projectable onto $X^{\prime}=\pi_{\ast}X$, where $p_{1}\colon J^{1}C\to M$ is the canonical projection. We thus have $\mathcal{L}_{X_{C}^{(1)}}(Lv)=(X_{C}^{(1)}(L)+L\mathrm{div}X^{\prime})v$ and the condition of $\mathrm{aut}P$-invariance yields $X_{C}^{(1)}(L)+L\mathrm{div}X^{\prime}=0$, $\forall X\in\mathrm{aut}P$. It turns out, every $\mathrm{aut}P$-invariant Lagrangian density is variationally trivial, as it is proved in [7, Corollary 1]. Thus, the notion of $\mathrm{aut}P$-invariance is too restrictive to be useful in Field Theory.

If $X\in\mathrm{gau}P$, then $X^{\prime}=0$ and the definition of gauge invariance is recovered. As every $\Phi\in\mathrm{gau}P$ induces the identity map on $M$, the function $L$ is gauge invariant if and only if the gauge group is a group of symmetries of the Lagrangian density $\Lambda=L\mathbf{v}$, where $\mathbf{v}$ is the volume form on the ground manifold. For more details we refer the reader to [6], [7], and [8].

4.2 The number of gauge-invariant Lagrangians

Let

[TABLE]

be the curvature mapping. The curvature form $\Omega_{\Gamma}$ of the connection $\Gamma$ corresponding to a section $s_{\Gamma}$ of $p$ is seen to be a two form on $M$ with values in the adjoint bundle $\mathrm{ad}P$. On the vector bundle $\bigwedge\nolimits^{2}T^{\ast}M\otimes\mathrm{ad}P$ we consider the coordinate systems $(x^{i};R_{jk}^{\alpha})$, $j<k$, induced by a coordinate system $(U;x^{i})_{i=1}^{n}$ on $M$, and a basis $(B_{\alpha})_{\alpha=1}^{m}$ of $\mathfrak{g}$, as follows:

[TABLE]

With respect to the coordinate systems $(x^{i},A_{j}^{\alpha},A_{j,k}^{\alpha})$ and $(x^{i};R_{jk}^{\alpha})$, $j<k$, on $J^{1}C$ and $\bigwedge\nolimits^{2}T^{\ast}M\otimes\mathrm{ad}P$, respectively, the equations of the curvature mapping are as follows:

[TABLE]

The geometric formulation of Utiyama’s Theorem (e.g., see [4]) states that a Lagrangian $L\colon J^{1}C\to\mathbb{R}$ is gauge invariant if and only if $L$ factors through $\Omega$ as $L=\tilde{L}\circ\Omega$, where

[TABLE]

is a $C^{\infty}$ function that is invariant under the adjoint representation of $G$ on the curvature bundle. As the curvature map (7) is surjective, the function $\tilde{L}$ is unique.

If the group $G$ is connected, then according to the formulas (6) and (1), and taking the equations of the curvature mapping (9) into account, the function $\tilde{L}$ in the formula (10) is invariant under the adjoint representation of $G$ on the curvature bundle if and only if

[TABLE]

Alternatively, this equivalence can also be deduced from the formula for $X_{C}^{(1)}$ in [7, (2.10)].

Theorem 4.1**.**

Assume the group $G$ is connected.

The distribution $\mathcal{D}$ on $\bigwedge\nolimits^{2}T^{\ast}M\otimes\mathrm{ad}P$ generated by the vector fields $\chi_{\alpha}$, $1\leq\alpha\leq m$, given in (11), is involutive.

1. (i)

The Lie algebra $\mathfrak{g}$ is Abelian if and only if $\mathcal{D}=\{0\}$. 2. (ii)

If $\mathfrak{g}$ is not Abelian, then the rank of $\mathcal{D}$ on a dense open subset is constant. 3. (iii)

If $\dim M=n=2$ and $\dim\mathfrak{g}=m$, $\operatorname*{rank}\mathfrak{g}=l$, then the generic rank of $\mathcal{D}$ is equal to $m-l$. 4. (iv)

If $\dim M=n\geq 3$, $\dim\mathfrak{g}=m$, and $\mathfrak{g}$ is semisimple, then the generic rank of $\mathcal{D}$ is equal to $m$.

Proof.

As a computation shows, we have $[\chi_{\rho},\chi_{\sigma}]=c_{\rho\sigma}^{\gamma}\chi_{\gamma}$ for $1\leq\rho<\sigma\leq m$; hence $\mathcal{D}$ is involutive.

(i) The vector fields $\chi_{\alpha}$, $1\leq\alpha\leq m$, vanish if and only if $c_{\gamma\alpha}^{\beta}R_{ij}^{\gamma}=0$ for all $\alpha,\beta=1,\dotsc,m$, $1\leq i<j\leq n$, and these equations are obviously equivalent to saying that $c_{\gamma\alpha}^{\beta}=0\,$for all $\alpha,\beta,\gamma=1,\dotsc,m$.

(ii) The rank of $\mathcal{D}$ at a point $\eta_{2}\in\bigwedge\nolimits^{2}T_{x_{0}}^{\ast}M\otimes(\mathrm{ad}P)_{x_{0}}$ equals the rank of the $m\times m\tbinom{n}{2}$ matrix

[TABLE]

As the entries of $\Lambda$ are polynomial functions in the coordinates $R_{ij}^{\alpha}$, $1\leq\alpha\leq m$, $1\leq i<j\leq n$, it follows that the rank of $\mathcal{D}$ takes its maximum value on a dense open subset, and we have $\max_{\eta_{2}}\operatorname*{rank}\Lambda(\eta_{2})\leq m$, $\forall\eta_{2}\in\bigwedge\nolimits^{2}T^{\ast}M\otimes\mathrm{ad}P$.

(iii) If $n=2$, then $\Lambda(\eta_{2})$ is a square matrix of size $m$. If $B=R_{12}^{\gamma}(\eta_{2})B_{\gamma}$, then the matrix of the linear map $\mathrm{ad}_{B}\colon\mathfrak{g}\to\mathfrak{g}$ in the basis $(B_{\gamma})_{\gamma=1}^{m}$ coincides with $\Lambda(\eta_{2})$; in fact, we have $\mathrm{ad}_{B}(B_{\alpha})=[B,B_{\alpha}]=c_{\gamma\alpha}^{\beta}R_{12}^{\gamma}(\eta_{2})B_{\beta}$, $1\leq\alpha\leq m$. Hence, if $B\in\mathfrak{g}$ is a regular element, then the rank of $\Lambda(\eta_{2})$ is $m-l$ exactly.

(iv) For every pair of indices $1\leq j<k\leq n$, let $\Lambda_{jk}(\eta_{2})$ be the $m\times m$ matrix $\Lambda_{jk}(\eta_{2})=\left(\lambda_{\beta,jk}^{\alpha}(\eta_{2})\right)_{\alpha,\beta=1}^{m}$. Then the $m\times m\tbinom{n}{2}$ matrix $\Lambda(\eta_{2})$ can be written in blocks as follows: $\Lambda(\eta_{2})=\left(\Lambda_{12}(\eta_{2}),\dotsc,\Lambda_{1n}(\eta_{2}),\dotsc,\Lambda_{n-1,n}(\eta_{2})\right)$.

In order to prove this case, we can use a Chevalley basis (e.g., see [9, Chapter 3, Theorem 1.19]); more precisely: Let $\{\alpha_{1},\dotsc,\alpha_{l}\}$ be a system of simple roots in the set $\Delta_{\mathfrak{g}}=\{\alpha_{1},\dotsc,\alpha_{l},\alpha_{l+1},\dotsc,\alpha_{m-l}\}$, and let $h_{i}=h_{\alpha_{i}}$ for $1\leq i\leq l$. The basis $h_{i}$, $1\leq i\leq l$, $e_{\alpha}$, $\alpha\in\Delta_{\mathfrak{g}}$, of $\mathfrak{g}$ satisfies the following properties: $[h_{i},h_{j}]=0$ for $i,j=1,\dotsc,l$, $[h_{i},e_{\alpha}]=\left\langle\alpha|\alpha_{i}\right\rangle e_{\alpha}$, $\alpha\in\Delta_{\mathfrak{g}}$, $1\leq i\leq l$, and $[e_{\alpha},e_{-\alpha}]=h_{\alpha}=e^{i}h_{i}$, $e^{i}\in\mathbb{Z}$, $1\leq i\leq l$, and if $\alpha,\beta\in\Delta_{\mathfrak{g}}$, $\alpha+\beta\neq 0$, and $\beta-p\alpha,\dotsc,\beta+q\alpha$ is the $\alpha$-string of roots containing $\beta$, then $[e_{\alpha},e_{\beta}]=0$ if $q=0$, $[e_{\alpha},e_{\beta}]=\pm(p+1)e_{\alpha+\beta}$ if $\alpha+\beta\in\Delta_{\mathfrak{g}}$; or equivalently

[TABLE]

According to the general notations (8) in this case we can write

[TABLE]

With these notations, for $1\leq i<j\leq n$ we have $\lambda_{u,jk}^{t}(\eta_{2})=0$ for all $t,u=1,\dotsc,l$, and

[TABLE]

[TABLE]

[TABLE]

Distinguishing cases, (14) is readily seen to be equivalent to the following two formulas:

If $\alpha_{u}-\alpha_{v}=\alpha_{r_{0}}\in\Delta_{\mathfrak{g}}$, for some $1\leq r_{0}\leq m-l$, then

[TABLE]

If $\alpha_{u}-\alpha_{v}\notin\Delta_{\mathfrak{g}}$, then

[TABLE]

Hence the $m\times m$ matrix $\Lambda_{jk}(\eta_{2})$ is given by

[TABLE]

where $O$ denotes the $l\times l$ zero matrix, and $A_{jk}(\eta_{2})$, $B_{jk}(\eta_{2})$, $C_{jk}(\eta_{2})$ are the matrices with sizes $l\times(m-l)$, $(m-l)\times l$, $(m-l)\times(m-l)$, respectively, given by

[TABLE]

[TABLE]

According to (12) we have

[TABLE]

and according to (13) we have

[TABLE]

and $C_{jk}(\eta_{2})=(\lambda_{\alpha_{v},jk}^{\alpha_{u}}(\eta_{2}))_{1\leq v\leq m-l}^{1\leq u\leq m-l}$ is given by the formulas (15) and (16).

Let $A\subset\Delta_{\mathfrak{g}}$ be the set of elements $\alpha_{r_{0}}\in\Delta_{\mathfrak{g}}$ for which there exist $\alpha_{u},\alpha_{v}\in\Delta_{\mathfrak{g}}$ such that $\alpha_{u}-\alpha_{v}=\alpha_{r_{0}}$, and let $E_{x_{0}}\subset\bigwedge\nolimits^{2}T_{x_{0}}^{\ast}M\otimes(\mathrm{ad}P)_{x_{0}}$ be the closed subset of $(\mathrm{ad}P)_{x_{0}}$-valued $2$-covectors $\eta_{2}^{0}$ such that $R_{ij}^{\alpha_{r_{0}}}(\eta_{2}^{0})=0$ for all indices $1\leq i<j\leq n$, as long as $\alpha_{r_{0}}\in A$. If $\eta_{2}^{0}\in E_{x_{0}}$, then $C_{ij}(\eta_{2}^{0})$ is a diagonal square matrix of order $m-l$, whose non-vanishing entries are given by

[TABLE]

Hence, by taking the values $R_{ij}^{t}(\eta_{2}^{0})$ in a suitable dense open subset in $E_{x_{0}}$, it follows that $\det C_{ij}(\eta_{2}^{0})\neq 0$.

We have $m>2l$ (see Remark 5.2), as $\mathfrak{g}$ is semisimple. We can thus decompose the matrix $A_{jk}(\eta_{2})$ into two blocks as follows: $A_{jk}(\eta_{2})=(A_{jk}^{\prime}(\eta_{2}),A_{jk}^{\prime\prime}(\eta_{2}))$, where $A_{jk}^{\prime}(\eta_{2})$ denotes the $l\times l$ submatrix of $A_{jk}(\eta_{2})$ determined by its $l$ rows and its first $l$ columns of $A_{jk}(\eta_{2})$, whereas $A_{jk}^{\prime\prime}(\eta_{2})$ denotes the $l\times(m-2l)$ submatrix of $A_{jk}(\eta_{2})$ determined by its $l$ rows and its $m-2l$ columns. As a computation shows, we have

[TABLE]

Hence $A_{jk}^{\prime}(\eta_{2}^{0})$ is non-singular on a dense open subset in $E_{x_{0}}$.

Moreover, since $n\geq 3$, we can consider the $m\times 2m$ submatrix of $\Lambda(\eta_{2}^{0})$ given by $\Lambda^{\prime}(\eta_{2}^{0})=\left(\Lambda_{12}(\eta_{2}^{0}),\Lambda_{13}(\eta_{2}^{0})\right)$, and also the $m\times m$ submatrix $\Lambda^{\prime\prime}(\eta_{2}^{0})$ of $\Lambda^{\prime}(\eta_{2}^{0})$ defined by

[TABLE]

where $C_{13}^{\prime}(\eta_{2}^{0})$ is the $l\times l$ matrix

[TABLE]

Next, the determinant of $\Lambda^{\prime\prime}(\eta_{2}^{0})$ is evaluated by the Laplacian expansion along the $l\times l$ minors of its $l$ last columns; e.g., see [5, III, §8, formula (21)].

Let $S_{l}$ be the set of $l$-element subsets of $\{1,2,...,m\}$ and for every $I=\{i_{1}<\ldots<i_{l}\}\in S_{l}$ let us denote by $\Delta_{I}$ the $l\times l$ submatrix of

[TABLE]

determined by the rows $i_{1},\dotsc,i_{l}$; for example

[TABLE]

If $\Delta_{I}^{\prime}$ denotes the complement of $\Delta_{I}$ in $\Lambda^{\prime\prime}(\eta_{2}^{0})$, then by setting $|I|=i_{1}+\ldots+i_{l}$, we have $\det\Lambda^{\prime\prime}(\eta_{2}^{0})=\sum_{I\in S_{l}}(-1)^{\frac{(m-l+1)+m}{2}l+|I|}(\det\Delta_{I})(\det\Delta_{I}^{\prime})$.

In this formula the functions $\det\Delta_{I}$ depend on the values $R_{13}^{t}(\eta_{2}^{0})$ only, while the functions $\det\Delta_{I}^{\prime}$ depend on the values $R_{12}^{t}(\eta_{2}^{0})$ only. Hence $\det\Lambda^{\prime\prime}(\eta_{2}^{0})$ is written as a sum of double products of functions depending on disjoint values. Furthermore, by separating the first summand from the right-hand side of the previous formula, it follows:

[TABLE]

with which, one concludes the proof. ∎

4.3 Generators for gauge-invariant Lagrangians

Given a vector $\alpha\in\oplus^{N}(\mathrm{ad}P)_{x}$ and an element $u\in\pi^{-1}(x)$, there exist unique elements $A_{1},\dotsc,A_{N}\in\mathfrak{g}$ such that $\alpha=\left((u,A_{1})_{\mathrm{ad}},\dotsc,(u,A_{N})_{\mathrm{ad}}\right)$. Actually, $\alpha$ can be written as $\alpha=\left((u_{1}^{\prime},A_{1}^{\prime})_{\mathrm{ad}},\dotsc,(u_{N}^{\prime},A_{N}^{\prime})_{\mathrm{ad}}\right)$, with $u_{h}^{\prime}\in\pi^{-1}(x)$, $A_{h}^{\prime}\in\mathfrak{g}$, $1\leq h\leq N$. As $G$ operates freely and transitively on the fibre $\pi^{-1}(x)$, there exist unique elements $g_{1},\dotsc,g_{N}\in G$ such that $u_{h}^{\prime}=u\cdot g_{h}$, $1\leq h\leq N$. Hence

[TABLE]

If $I\colon\oplus^{N}\mathfrak{g}\to\mathbb{R}$ is a polynomial function invariant under the diagonal action induced by the adjoint representation of $G$ on its Lie algebra $\mathfrak{g}$, then a function $\tilde{I}\in C^{\infty}(\oplus^{N}\mathrm{ad}P)$ can be associated by setting

[TABLE]

as the formula above makes sense because it does not depend on the representative chosen. In fact, any other representative of the element

[TABLE]

is of the form $\alpha=\left((u\cdot g^{-1},\mathrm{Ad}_{g}A_{1})_{\mathrm{ad}},\dotsc,(u\cdot g^{-1},\mathrm{Ad}_{g}A_{N})_{\mathrm{ad}}\right)$, $\forall g\in G$. As $I$ is invariant under the diagonal action we have

[TABLE]

Theorem 4.2**.**

Assume the group $G$ is connected and semisimple. If $(U;x^{i})_{i=1}^{n}$ is a coordinate system such that $P$ is trivial over $U$, then a $\mathrm{Gau}P|_{U}$-equivariant vector-bundle isomorphism $\Psi\colon\bigwedge^{2}T^{\ast}U\otimes\mathrm{ad}P|_{U}\to\oplus^{N}\mathrm{ad}P|_{U}$, $N=\frac{1}{2}n(n-1)$, is defined by

[TABLE]

There exists a finite system of generators $I^{i}$, $1\leq i\leq\nu$, of the algebra of polynomial functions $\mathcal{P}(\oplus^{N}\mathfrak{g})^{G}$ invariant under the diagonal action induced by the adjoint representation of $G$ on $\mathfrak{g}$ and the functions $\tilde{I}^{i}\circ\Psi$, $1\leq i\leq\nu$, generate the algebra $C^{\infty}(\bigwedge^{2}T^{\ast}U\otimes\mathrm{ad}P|_{U})^{\mathrm{Gau}P|_{U}}$ over $C^{\infty}(U)$ differentiably.

Finally, if $(U_{\alpha};x_{\alpha}^{i})$, $1\leq i\leq n$, $\alpha\in A$, is a coordinate system such that $\cup_{\alpha\in A}U_{\alpha}=M$, $P$ is trivial over every $U_{\alpha}$, and $(\eta_{\alpha})$, $\alpha\in A$, is a partition of unity subordinate to $(U_{\alpha})$, $\alpha\in A$, then the functions $J^{i}=\sum_{\alpha\in A}\eta_{\alpha}(\tilde{I}_{\alpha}^{i}\circ\Psi)$, $1\leq i\leq\nu$, generate the algebra $C^{\infty}(\bigwedge^{2}T^{\ast}U\otimes\mathrm{ad}P)^{\mathrm{Gau}P}$ over $C^{\infty}(M)$ differentiably.

Proof.

From the very definition of $\Psi$, it follows that the map $\Psi_{x}$ induced on every fibre $\bigwedge^{2}T_{x}^{\ast}U\otimes(\mathrm{ad}P)_{x}$, $x\in U$, is $\mathbb{R}$-linear.

If $\eta_{2}=\sum_{1\leq i<j\leq n}(dx^{i})_{x}\wedge(dx^{j})_{x}\otimes A_{ij}$, $A_{ij}\in(\mathrm{ad}P)_{x}$, belongs to $\ker\Psi_{x}$, then $A_{ij}=0$ for $1\leq i<j\leq n$. Hence $\Psi_{x}$ is injective and since the vector bundles $\bigwedge^{2}T^{\ast}M\otimes\mathrm{ad}P$ and $\oplus^{N}\mathrm{ad}P$ have te same rank, we conclude that $\Psi$ is an isomorphism. Moreover, every $\phi\in\mathrm{Gau}P$ acts on $\mathrm{ad}P$ as explained in §2.2, namely $\Phi_{\mathfrak{g}}((u,v)\operatorname{mod}G)=(\Phi(u),v)\operatorname{mod}G$, $\forall u\in P$, $\forall v\in\mathfrak{g}$, and acts trivially on $\bigwedge^{2}T^{\ast}M$, thus proving that $\Psi$ is $\mathrm{Gau}P|_{U}$-equivariant.

As $G$ is assumed to be connected and semisimple, according to Weyl’s Theorem every (finite dimensional) linear representation of $G$ is completely reducible. Hence by virtue of Hilbert-Nagata Theorem there exists a finite system of generators $I^{i}$, $1\leq i\leq\nu$, of the algebra of polynomial functions $\mathcal{P}(\oplus^{N}\mathfrak{g})^{G}$ invariant under the diagonal action induced by the adjoint representation of $G$ on $\mathfrak{g}$, as said in the statement. If $I\colon\oplus^{N}\mathfrak{g}\to\mathbb{R}^{\nu}$ is the map whose components are $I^{1},\dotsc,I^{\nu}$, then by virtue of the main result in [12] it follows that for every $f\in C^{\infty}(\oplus^{N}\mathfrak{g})^{G}$ there exists $g\in C^{\infty}(\mathbb{R}^{\nu})$ such that $f=g\circ I$. Moreover, as $P|_{U}$ is trivial by virtue of the hypothesis we can choose a trivialization $\tau\colon P|_{U}\to U\times G$, i.e., $\tau$ is an isomorphism of principal $G$-bundles such that $\mathrm{pr}_{1}\circ\tau=\pi$, which induces an isomorphism of vector bundles over $U$, $\tau_{\mathrm{ad}}\colon\oplus^{N}\mathrm{ad}P|_{U}\to U\times\oplus^{N}\mathfrak{g}$. Hence every $F\in C^{\infty}(\bigwedge^{2}T^{\ast}U\otimes\mathrm{ad}P|_{U})^{\mathrm{Gau}P}$ can be written as $F=(g\circ I)\circ\mathrm{pr}_{2}\circ\tau_{\mathrm{ad}}\circ\Psi$, where $\mathrm{pr}_{2}\colon U\times\oplus^{N}\mathfrak{g}\to\oplus^{N}\mathfrak{g}$ denotes the canonical projection onto the second factor.

Finally, as $\operatorname*{support}\eta_{\alpha}\subset U_{\alpha}$ and $\eta_{\alpha}\in C^{\infty}(M)$, it follows that $\eta_{\alpha}(\tilde{I}_{\alpha}^{i}\circ\Psi)$ is globally defined for $1\leq i\leq\nu$, and taking account of the fact that the vector fields $\chi_{\alpha}$, $1\leq\alpha\leq m$, in (11) spanning the distribution $\mathcal{D}$, are vertical with respect to the natural projection $\bigwedge^{2}T^{\ast}M\otimes\mathrm{ad}P\to M$, we can conclude that $X(J^{i})=0$ for every $1\leq i\leq\nu$ and every $X\in\Gamma(M,\mathcal{D})$, thus finishing the proof. ∎

5 Remarks and Examples

Remark 5.1*.*

Let $G$ be a connected Lie group. As $M$ is also assumed to be connected and oriented, if $n=2$, then $\bigwedge\nolimits^{2}T^{\ast}M$ is a trivial bundle of rank $1$; hence $\bigwedge\nolimits^{2}T^{\ast}M\otimes\mathrm{ad}P$ is isomorphic to the adjoint bundle by means of the map $\bigwedge\nolimits^{2}T^{\ast}M\otimes\mathrm{ad}P\to\mathrm{ad}P$, $\mathbf{v}_{x}\otimes X\mapsto X$, $\forall X\in(\mathrm{ad}P)_{x}$, $\mathbf{v}$ being the volume form on $M$. As $G$ is connected, the first integrals of $\mathcal{D}$ coincide with the $C^{\infty}$ functions on $\mathrm{ad}P$ invariant under the adjoint representation of $G$ on $\mathfrak{g}$, namely  $C^{\infty}(\mathrm{ad}P)^{G}=\left\{f\in C^{\infty}(\mathrm{ad}P):X(f)=0,\forall X\in\mathcal{D}\right\}$. If $p\in S^{\bullet}(\mathfrak{g}^{\ast})$ is a polynomial invariant under the adjoint representation of $G$ on its Lie algebra and if $G$ is a semisimple complex Lie group, then we can consider the function $\tilde{p}\in C^{\infty}(\mathrm{ad}P)$ defined as above and Chevalley’s theorem (e.g., see [15, Theorem 4.9.3]) ensures the existence of $l$ homogeneous algebraically independent polynomials $p_{1},\dotsc,p_{l}$ such that the algebra of polynomial functions on $\mathfrak{g}$ that are invariant with respect to the adjoint representation of $G$ on $\mathfrak{g}$, is isomorphic to $\mathbb{C}[p_{1},\dotsc,p_{l}]$, i.e., $S^{\bullet}(\mathfrak{g}^{\ast})^{G}=\mathbb{C}[p_{1},\dotsc,p_{l}]$. If $\varphi\colon\mathrm{ad}P\to M\times\mathbb{C}^{l}$ is the map given by

[TABLE]

then according to [12, Theorem. 3)] we have $C^{\infty}(\mathrm{ad}P)^{G}=\varphi^{\ast}C^{\infty}(M\times\mathbb{C}^{l})$ and by applying Frobenius theorem we conclude that $C^{\infty}(\bigwedge\nolimits^{2}T^{\ast}M\otimes\mathrm{ad}P)^{G}$ is generated by the $l$ functions $\tilde{p}_{1},\dotsc,\tilde{p}_{l}$.

This fact explains why equations of Yang-Mills type on a surface admit a geometric treatment; see [2].

Remark 5.2*.*

The map $\pi^{(N)}\colon P^{(N)}=P\times_{M}\overset{(N}{\ldots}\times_{m}P\to M$ is a principal fibre bundle with structure group $G^{N}$, the adjoint bundle of which can be identified to $\mathrm{ad}P^{(N)}=\oplus^{N}\mathrm{ad}P$. If $G$ is complex and semisimple, then $G^{N}$ also is, and we have $\dim G^{N}=Nm$, $\operatorname*{rank}\mathfrak{g}^{N}=Nl$, $l=\operatorname*{rank}\mathfrak{g}$. Hence Chevalley’s theorem can be applied to the adjoint representation of $G^{N}$ on its Lie algebra, thus deducing that its ring of invariants admits a basis of $Nl$ algebraically independent homogeneous polynomials that can be constructed from a basis $p_{1},\dotsc,p_{l}$ of Chevalley for $\mathfrak{g}$ by defining the polynomial $p_{i}^{h}\colon\mathfrak{g}^{N}\to\mathbb{C}$ by $p_{i}^{h}(A_{1},\dotsc,A_{N})=p_{i}(A_{h})$, $1\leq i\leq l$, $1\leq h\leq N$. In particular, such polynomials are also invariant under the diagonal action induced by the adjoint representation of $G$ on $\mathfrak{g}^{N}$; but the polynomials $(p_{i}^{h})_{1\leq i\leq l}^{1\leq h\leq N}$ do not generate in general the ring of invariants for the diagonal action. In fact, if $n\geq 3$, then we know that the generic rank of the distribution $\mathcal{D}$ is $m$; hence, the maximum number of functionally independient gauge-invariant functions is $mN-m=m(N-1)$. As the number of polynomials $p_{i}^{h}$ is $Nl$, if such polynomials span the ring of gauge invariants it should be $Nl\geq m(N-1)$, i.e., $\frac{l}{m}\geq 1-\frac{1}{N}$, and this inequality never occurs in the semisimple case, as in this case we have $m>2l$. Actually, as dimension and rank are aditive, it suffice to prove the last inequality for simple algebras:

[TABLE]

as in the fourth case me must have $l\geq 2$; and for the exceptional algebras:

[TABLE]

Example 5.3*.*

If $\dim M=n=3$ y $\mathfrak{g}=\mathfrak{sl}(2,\mathbb{R})$, then $m=3$, $l=1$. The basic invariant is $p(X)=x_{12}x_{21}+(x_{11})^{2}=-\det(X)$, where

[TABLE]

The maximum number of functionally independent $G$-invariant functions over the curvature bundle $\bigwedge\nolimits^{2}T^{\ast}(M)\otimes\mathrm{ad}P\underset{\text{locally}}{\cong}\oplus^{3}\mathrm{ad}P$ is $3\tbinom{3}{2}-3=6$ in this case.

To the quadratic polynomial $p$ corresponds a unique symmetric bilinear form $s_{p}\colon\mathfrak{sl}(2,\mathbb{R})\times\mathfrak{sl}(2,\mathbb{R})\to\mathbb{R}$ obtained by polarization

[TABLE]

from which the following invariants on $\oplus^{3}\mathfrak{sl}(2,\mathbb{R})$ follow:

[TABLE]

As a calculation shows, these $6$ functions are functionally independent in the dense open subset defined by

[TABLE]

By using the local method in Theorem 4.2 by means of the map (17), it follows:

[TABLE]

Example 5.4*.*

If $(e_{\alpha\beta})_{\alpha,\beta=1}^{m}$ denotes the standard basis for $\mathfrak{gl}(m,\mathbb{R})$, then we have $[e_{\alpha\beta},e_{\rho\sigma}]=\delta_{\beta\rho}e_{\alpha\sigma}-\delta_{\alpha\sigma}e_{\rho\beta}$. If $\mathfrak{g}=\left\langle e_{12},e_{13},e_{23}\right\rangle$ denotes the Heisenberg Lie algebra and $B_{1}=e_{12}$, $B_{2}=e_{13}$, $B_{3}=e_{23}$, then $[B_{1},B_{2}]=0$, $[B_{1},B_{3}]=-B_{2}$, $[B_{2},B_{3}]=0$, or equivalently $c_{12}^{\alpha}=c_{23}^{\alpha}=0$, $c_{13}^{\alpha}=-\delta_{2}^{\alpha}$, $1\leq\alpha\leq 3$; it thus follows: $\chi_{1}={\textstyle\sum\nolimits_{i<j}}R_{ij}^{3}\tfrac{\partial}{\partial R_{ij}^{2}}$, $\chi_{2}=0$, $\chi_{3}=-{\textstyle\sum\nolimits_{i<j}}R_{ij}^{1}\tfrac{\partial}{\partial R_{ij}^{2}}$. We distinguish two cases: 1st) If $n=2$, then $\chi_{1}=R_{12}^{3}\tfrac{\partial}{\partial R_{12}^{2}}$, $\chi_{3}=-R_{12}^{1}\tfrac{\partial}{\partial R_{12}^{2}}$, and these two vectors are proportional on a dense open subset. Hence the generic rank of $\mathcal{D}$ is $1$ in this case; 2nd) If $n\geq 3$, then the matrix of $\chi_{1}$ and $\chi_{2}$ in the basis $\tfrac{\partial}{\partial R_{ij}^{2}}$, $1\leq i<j\leq n$, is as follows:

[TABLE]

and on the dense open subset on which at least one of the determinants

[TABLE]

does not vanish, the rank of $\mathcal{D}$ is $2$ in this case. Note that the algebra $\mathfrak{g}$ under consideration is not semisiple.

Example 5.5*.*

Let us consider $\mathfrak{g}=\mathfrak{sl}(2,\mathbb{R})$ with its standard basis $B_{1}=e_{21}$, $B_{2}=e_{11}-e_{22}$, $B_{3}=e_{12}$; $[B_{1},B_{2}]=2B_{1}$, $[B_{1},B_{3}]=-B_{2}$, $[B_{2},B_{3}]=2B_{3}$, i.e., $c_{12}^{\alpha}=2\delta_{1}^{\alpha}$, $c_{13}^{\alpha}=-\delta_{2}^{\alpha}$, $c_{23}^{\alpha}=2\delta_{3}^{\alpha}$, $1\leq\alpha\leq 3$. If $n=2$, then we have

[TABLE]

Consequently, the generic rank of $\mathcal{D}$ is $2$ in this case. If $n\geq 3$, then the components $\chi_{1}^{\prime}$, $\chi_{2}^{\prime}$, $\chi_{3}^{\prime}$ of the vector fields $\chi_{1}$, $\chi_{2}$, $\chi_{3}$, respectively, in the subspace generated by $\frac{\partial}{\partial R_{ij}^{\alpha}}$, $1\leq\alpha\leq 3$, $1\leq i<j\leq 3$, are

[TABLE]

or in matrix notation:

[TABLE]

and the determinant

[TABLE]

does not vanish identically. Accordingly, for $n\geq 3$ the generic rank of $\mathcal{D}$ is $3$.

Example 5.6*.*

If $n=3$, $\mathfrak{g}=\mathfrak{so}(3,\mathbb{R})$, then $m=3$, $l=1$, and by considering the standard basis $B_{1}=e_{12}-e_{21}$, $B_{2}=e_{13}-e_{31}$, $B_{3}=e_{23}-e_{32}$ we have $X=xB_{1}+yB_{2}+zB_{23}$, $\det(tI_{3}-X)=t^{3}+\left(x^{2}+y^{2}+z^{2}\right)t$. Thus, the basic invariant is $p(X)=x^{2}+y^{2}+z^{2}$. Hence

[TABLE]

and one can check that these invariants are functionally independent and therefore build a system of generators for the ring of (local) invariants.

Example 5.7*.*

If $n=3$, $\mathfrak{g}=\mathfrak{so}(4,\mathbb{R})$, then $m=6$, $l=2$ and the characteristic polynomial of $X=\sum_{1\leq i<j\leq 4}x_{ij}(e_{ij}-e_{ji})$ is

[TABLE]

Proceeding as above, by polarizing $p_{1}$ and $p_{2}$, we obtain $12$ generically independent invariant functions, which coincides with the corank of the ditribution $\mathcal{D}$ in this case.