Gauge fields with vanishing scalar invariants

Let us briefly summarize a few basic notions of algebraic classification based on null alignment. See also definitions 2.1, 2.2 and 2.6 of [2].

2.1.1. Boost weight.

2.1.2. Boost order.

2.1.3. Boost order decomposition.

At this point, one can perform the boost order decomposition of any tensor T

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle T= \sum_b T_{(b)}, \nonumber \end{align} \tag{ 2.2 }$$

defining the boost weight b part T_(b) of T as the sum of all $T_{a \dots b} e^{(a)} \dots e^{(b)}$ with $b(T_{a \dots b}) = b$. The index b then ranges from $-b_{\rm max}$ to $b_{\rm max}$, where the value of $b_{\rm max}$ depends on rank and symmetries of T.

2.1.4. Algebraic types.

Let G denote a real non-Abelian Lie group of a finite dimension and $\mathfrak{g}$ the associated Lie algebra spanned by generators {t_a}, $a = 1, \dots, \dim \mathfrak{g}$⁴. The generators of G satisfy the commutation relations

where are real structure constants of $\mathfrak{g}$.

2.2.1. Gauge field and field strength.

Introduction of a $\mathfrak{g}$-valued gauge field $A_{\mu} = A^a_{\mu} t_a$ gives rise to the gauge covariant derivative $\newcommand{\DE}{{\mathscr{D}}} \DE_\mu$ associated with $A_\mu$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\DE}{{\mathscr{D}}} \displaystyle \DE_\mu \equiv \nabla_\mu - {\rm i}[A_\mu,\cdot], \nonumber \end{align} \tag{ 2.4 }$$

and the field strength (or curvature) $\newcommand{\e}{{\rm e}} F_{\mu \nu} \equiv \nabla_{\mu} A_\nu - \nabla_\nu A_\mu - {\rm i} [A_\mu,A_\nu]$. The field strength of $A_\mu$ is then subject to the Bianchi identity

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\DE}{{\mathscr{D}}} \displaystyle \label{bianchi} \DE_\mu F_{\nu \rho} + \DE_{\nu} F_{\rho \mu} + \DE_\rho F_{\mu \nu} = 0. \nonumber \end{align} \tag{ 2.5 }$$

Under a gauge transformation $U \in G$, the gauge field and its field strength undergo the following change

$$\begin{align} \renewcommand{\dag}{\dagger} \newcommand{\e}{{\rm e}} \newcommand{\re}{{\rm Re}} \displaystyle A_\mu \mapsto UA_\mu U^\dagger +{\rm i} U \partial_\mu U^\dagger, \label{Atrans} \nonumber \end{align} \tag{ 2.6 }$$

$$\begin{align} \renewcommand{\dag}{\dagger} \newcommand{\e}{{\rm e}} \newcommand{\re}{{\rm Re}} \displaystyle F_{\mu \nu} \mapsto U F_{\mu \nu} U^\dagger.\label{Ftrans} \nonumber \end{align} \tag{ 2.7 }$$

2.2.2. Gauge covariant fields.

We will say that a field $T_{\mu \dots \nu}$ is gauge covariant if it transforms under any $U\in G$ as

$$\begin{align} \renewcommand{\dag}{\dagger} \newcommand{\e}{{\rm e}} \newcommand{\re}{{\rm Re}} \displaystyle \label{g-invariant} T_{\mu \dots \nu} \mapsto U T_{\mu \dots \nu} U^\dagger. \nonumber \end{align} \tag{ 2.8 }$$

Gauge covariant derivative of a gauge covariant quantity $T_{\mu \dots \nu}$ is also gauge covariant, i.e.

$$\begin{align} \renewcommand{\dag}{\dagger} \newcommand{\e}{{\rm e}} \newcommand{\re}{{\rm Re}} \newcommand{\DE}{{\mathscr{D}}} \displaystyle \label{Dtransform} \DE_\rho T_{\mu \dots \nu} \mapsto U \DE_\rho T_{\mu \dots \nu} U^\dagger. \nonumber \end{align} \tag{ 2.9 }$$

Another useful relation is the commutator relation for gauge covariant derivatives

for any $\mathfrak{g}$-valued tensor $T_{\mu \dots \nu}$. If F is a YM field, i.e. a solution to the source-free YM equation $\newcommand{\DE}{{\mathscr{D}}} \DE^\mu F_{\mu \nu} = 0$, the Bianchi identity (2.5) together with (2.10) give us the following relation that will be useful later

There is a number of approaches to classification of YM fields or more general gauge fields on four-dimensional Lorentzian manifolds proposed in the literature, see e.g. [32] and references therein. These are based either on classification of polynomial Lorentz/gauge invariants or on the eigenvalue problem for $F_{\mu \nu}$. Our approach in this section is based on null alignment and hence is not restricted only to gauge fields and four dimensions. Of course, when applied to $F_{\mu \nu}$, there is an overlap of our classification with some of those already proposed for gauge fields in the literature. A detailed comparison is out of the scope of the paper, but some aspects of the overlap will be apparent at the end of this section and in section 2.5.

Let $T_{\mu \dots \nu}=T_{\mu \dots \nu}^a t_a$ be a gauge covariant quantity. Given a null frame, one can assign boost order ${\rm bo}(T)$ to T as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm bo}(T_{\mu \dots \nu}) \equiv \max_{a} \{{\rm bo}(T_{\mu \dots \nu}^a) \}, \nonumber \end{align} \tag{ 2.12 }$$

where ${\rm bo}(T^a)$ is boost order of a tensor T^a. If $T_{(b)}^a$ denotes the boost weight b part of a tensor T^a, then $\newcommand{\e}{{\rm e}} T_{(b)} \equiv T_{(b)}^a t_{a}$ corresponds to the (gauge covariant) boost weight b part T_(b) of T.

This enables one to classify any gauge covariant field T exactly as in section 2.1. It is easy to see that such definition is equivalent to the following, more practical one.

Definition 2.1. We will say that a gauge covariant tensor $T_{\mu \dots \nu}$ is of algebraic type X if all its Lie algebra components $T_{\mu \dots \nu}^a$ are aligned and of type X.

Remark 2.2. The notion of algebraic type of gauge covariant quantities is gauge invariant. Indeed, given a frame and b, T_(b) vanishes if and only if $T_{(b)}^\prime$ of a gauge transformed field $\newcommand{\re}{{\rm Re}} \renewcommand{\dag}{\dagger}T^\prime = U T U^\dagger$ vanishes as well.

Hence, algebraic type of a gauge covariant quantity is well defined in this sense. Note that e.g. algebraic type of $A_\mu$ is not defined, since it transforms as (2.6), and hence, in general, the alignment type would not be preserved.

In the paper, we are focused mainly on null field strengths, i.e. those of type N. Note that this definition of null field strength is consistent with the one of [28, 29]. In four dimensions, null solutions of the YM equation have already been studied extensively, see e.g. [29, 33, 34] and references therein.

One of the key ingredients in studying invariants, universality and proving various related results for tensor fields on Lorentzian manifolds is the notion of balanced tensors [3–6] which, in appropriate background spacetimes (the so called degenerate Kundt spacetimes, see e.g. [35]), guarantees that all covariant derivatives of the field are well behaved. Therefore, let us first provide appropriate extension of the notion of k-balancedness to gauge covariant fields.

Definition 2.3. Let $k \in \mathbb{N}_0$. In a frame parallely propagated along a null geodetic affinely parameterized vector field $\newcommand{\e}{{\rm e}} \ell$, a gauge covariant quantity T is said to be k-balanced if only its boost weight b  <  −k parts T_(b) are possibly non-vanishing and satisfy $D_A^{-b-k} T_{(b)} = 0$, where $\newcommand{\DE}{{\mathscr{D}}} \newcommand{\e}{{\rm e}} D_A \equiv \ell^\mu \DE_{\mu}$. If T is 0-balanced, we will simply say it is balanced.

Remark 2.4. From (2.8) and (2.9), it can be seen that the notion of k-balancedness is again gauge invariant.

In the appendix A, we have proven a number of results on balanced gauge covariant fields. Here, we present only the following three statements that will be of useful in the following sections.

Lemma 2.5. Let g be a degenerate Kundt spacetime. If a field strength $F_{\mu \nu}$ and its derivative $\newcommand{\DE}{{\mathscr{D}}} \DE_\rho F_{\mu \nu}$ are both aligned and of type II, then gauge covariant derivative of k-balanced gauge covariant quantity $T_{\mu \dots \nu}$ is again k-balanced.

By induction, one has that gauge covariant derivatives of T of arbitrary order are all aligned with $\newcommand{\e}{{\rm e}} \ell$ and of boost order b  <  k. Note that, in the case of $F_{\mu \nu}$ being null, the Bianchi identity (2.5) implies $D_A F_{\mu \nu} = 0$. Thus, for an aligned null $F_{\mu \nu}$ in a degenerate Kundt spacetime, conditions of 2.5 are automatically satisfied and we conclude

Lemma 2.6. Let g be a degenerate Kundt spacetime. Then, aligned field strength F is balanced if and only if it is null.

In view of lemma 2.5, one immediately arrives at

Corollary 2.7. Let $F_{\mu \nu}$ be an aligned null field strength in a degenerate Kundt spacetime. Then, its gauge covariant derivatives of arbitrary order are aligned with $F_{\mu \nu}$ and of type III.

And since Hodge duality does not alter alignment, algebraic type nor behavior with respect to D_A, the conclusions hold also for the Hodge dual $\star F_{\mu \dots \nu}$ and its gauge covariant derivatives.

The notion of $VSI$ and CSI spacetimes has been around for some time, see e.g. [4, 8] and has proven useful in a number of applications. Lately, also $VSI$ and CSI p-forms gained interest, mainly in applications in the context of universal Maxwell fields, see [6, 10, 16]. Here, we consider a natural extension of the notion to gauge covariant fields.

Definition 2.8. We will say that $T_{\mu \dots \nu}$ is CSI_k if all scalar polynomial gauge invariants constructed⁵ from $T_{\mu \dots \nu}$ and its gauge covariant derivatives up to order k are constant. If $T_{\mu \dots \nu}$ is CSI_k for all k, we will say it is CSI. Analogically, we will also define $VSI_k$ and $VSI$ fields as a subset of CSI_k and CSI fields, respectively, for which the corresponding invariants vanish.

Let us start with some simple lemmas on arbitrary gauge covariant fields, which will be particularly useful in obtaining characterization of $VSI$ field strengths.

Lemma 2.9. If a gauge-covariant field $T_{\mu \dots \nu}$ is $VSI_k$, so is every tensor U constructed as a trace of polynomial in $T_{\mu \dots \nu}$. In particular, U and its covariant derivatives up to order k are aligned and of type III.

Proof. Let $H_{\mu \dots \nu}$ be any polynomial in $T_{\mu \dots \nu}$. Obviously, $\newcommand{\e}{{\rm e}} U \equiv {\rm Tr} H$ has to be $VSI_0$, since any scalar polynomial constructed from U is gauge invariant and therefore vanishes by assumption. It remains to notice that ${\rm Tr} [A_\rho,H_{\mu \dots \nu}]$ vanishes due to the cyclic property of ${\rm Tr}$, and hence

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\DE}{{\mathscr{D}}} \displaystyle \nabla_\rho U_{\mu \dots \nu} = {\rm Tr} \DE_\rho H_{\mu \dots \nu} \nonumber \end{align} \tag{ 2.13 }$$

is also gauge invariant. It is thus clear that also any scalar polynomial invariant constructed from U and its covariant derivatives of arbitrary order is a gauge invariant and, in particular, U has to be $VSI_k$. The algebraic VSI theorem [9] then guarantees that $\nabla^{(\,j)} U$ for all $j=0,1, \dots, k$ are aligned and of type III. □

From lemma 2.9 and lemma B.5 of [10], it immediately follows

Lemma 2.10. Let $T_{\mu \dots \nu}$ be a gauge covariant $VSI_2$ field. If there exists a non-vanishing rank-2 tensor constructed as a trace of polynomial in $T_{\mu \dots \nu}$, then $T_{\mu \dots \nu}$ is aligned with a Kundt vector.

In this section, we discuss explicit form of gauge fields with null curvature in a degenerate Kundt spacetime. These are all necessarily $VSI$ (theorem 3.1) and exhaust all $VSI$ gauge fields in the case of a compact and semi-simple gauge group G (theorem 3.3).

Local form of a general d-dimensional degenerate Kundt metric in adapted Kundt coordinates $(r,u,x^\alpha)$ is given by [35]:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\de}{{{\rm d}}} \displaystyle \label{degKundt} \de s^2 = 2H(r,u,x) {\rm \de} u^2 + 2 {\rm \de} u \de r + 2W_\alpha(r,u,x){\rm \de} u {\rm \de} x^\alpha + g_{\alpha \beta}(u,x) {\rm \de} x^\alpha {\rm \de} x^\beta, \nonumber \end{align} \tag{ 3.5 }$$

with $\alpha,\beta = 2, \dots , d-1$ and functions $W_\alpha$ and H at most linear and at most quadratic in r, respectively

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{Wform} W_\alpha (r,u,x)= W^{(1)}_\alpha(u,x) r + W^{(0)}_\alpha(u,x), \nonumber \end{align} \tag{ 3.6 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{Hform} H(r,u,x)=H^{(2)}(u,x) r^2 + H^{(1)}(u,x) r + H^{(0)}(u,x). \nonumber \end{align} \tag{ 3.7 }$$

Here, r denotes affine parameter of the corresponding Kundt vector $\newcommand{\e}{{\rm e}} \ell = \partial_r$.

For any null field strength $F_{\mu \nu}$ aligned with $\newcommand{\e}{{\rm e}} \ell$ in a degenerate Kundt spacetime (3.5), the only non-vanishing coordinate components are $F_{u \alpha}^a$:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\de}{{{\rm d}}} \displaystyle F_{\mu \nu}^a {\rm \de} x^\mu {\rm \de} x^\nu = F_{u \alpha}^a (r,u,x) {\rm \de} u \wedge {\rm \de} x^\alpha. \nonumber \end{align} \tag{ 3.8 }$$

Exploiting gauge freedom, the form of the gauge potential $A_\mu$ can be then simplified considerably. Indeed, we can fix the light-cone gauge first to obtain $A_r^a=0$. Then, $F_{r \alpha}^a = 0$ reduces to $\partial_r A_\alpha^a = 0$ for all $\alpha$, which means that the remaining gauge freedom can be used to transform away also some of the $A_\alpha^a$. In fact, $F_{\alpha \beta} = 0$ guarantees that this can be done for all $A_\alpha^a$. Finally, F_ru  =  0 implies $\partial_r A_u = 0$ and we arrive at

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\de}{{{\rm d}}} \displaystyle \label{localgauge} A_\mu^a {\rm \de} x^\mu = A^a(u,x) {\rm \de} u. \nonumber \end{align} \tag{ 3.9 }$$

Any $VSI$ gauge field in the background spacetime (3.5) can be then obtained by a gauge transformation of (3.9). Particular examples of $VSI$ gauge fields already given in the literature will be discussed in section 4.4 in the context of universal YM fields.

Let G be a semi-simple Lie group. Consider a theory with gauge group G and Lagrangian $\mathcal{L}$ being a function of a finite set of gauge invariant scalars {I_k} constructed as traces of polynomials in $F_{\mu \nu}$, its Hodge dual $\star F_{\mu \dots \nu}$ and their gauge covariant derivatives (one of the scalars being $I={\rm Tr} F_{\mu \nu} F^{\mu \nu}$). Moreover, assume that $\mathcal{L}$ is analytic at zero and its power series expansion takes the form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{theory} \mathcal{L} = \mathcal{L}_{\rm YM} + \mathcal{L}_{\rm HC}, \nonumber \end{align} \tag{ 4.1 }$$

where $\mathcal{L}_{\rm YM}$ is the Yang–Mills Lagrangian

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\kap}{{\kappa}} \displaystyle \mathcal{L}_{\rm YM} = -\frac{1}{2 \kappa} {\rm Tr} F_{\mu \nu} F^{\mu \nu}, \nonumber \end{align} \tag{ 4.2 }$$

with $\newcommand{\kap}{{\kappa}} \kappa$ being a coupling constant and $\mathcal{L}_{\rm HC}$ (higher-order corrections) consists strictly of monomials at least cubic in $F_{\mu \nu}$ or containing its gauge covariant derivatives. Since $\mathcal{L}$ and $\mathcal{L}_{\rm YM}$ are analytic function of scalar polynomials I_k, so is $\mathcal{L}_{\rm HC}$.

4.1.1. Reduction of EOM for CSI fields.

It turns out that the form of equations of motion corresponding to (4.1) reduce significantly for CSI fields $F_{\mu \nu}$. This fact will be of great importance in the next section.

Assume that the invariants {I_m} involved in the Lagrangian $\mathcal{L}$ are constructed from $F_{\mu \nu}$, $\star F_{\mu\dots \nu}$ and their derivatives up to order k, i.e. schematically $\newcommand{\DE}{{\mathscr{D}}} I_m = {\rm Tr} \mathcal{P}_m (F, \DE F, \dots , \DE^k F)$ with $\mathcal{P}_m$ denoting the corresponding polynomial. Then, assuming the boundary terms vanish, variation of the action $S_{\rm HC}$ corresponding to $\mathcal{L}_{\rm HC}$ yields

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\de}{{{\rm d}}} \newcommand{\del}{{\delta}} \displaystyle \delta S_{\rm HC} = \int \star 1 \sum_n \frac{\partial \mathcal{L}_{\rm HC}}{\partial I_n} \delta I_n, \nonumber \end{align} \tag{ 4.3 }$$

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\de}{{{\rm d}}} \newcommand{\DE}{{\mathscr{D}}} \newcommand{\del}{{\delta}} \displaystyle = \int \star 1 \sum_n \frac{\partial \mathcal{L}_{\rm HC}}{\partial I_n} {\rm Tr} \bigg\{\frac{\partial \mathcal{P}_n}{\partial F^{\mu \nu}} \delta F^{\mu \nu} + \frac{\partial \mathcal{P}_n}{\partial \DE^\rho F^{\mu \nu}} \delta \DE^\rho F^{\mu \nu} + \dots \bigg\}, \nonumber \end{align} \tag{ 4.4 }$$

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\de}{{{\rm d}}} \newcommand{\del}{{\delta}} \displaystyle = \int \star 1 {\rm Tr} \bigg\{\sum_n \frac{\partial \mathcal{L}_{\rm HC}}{\partial I_n} \frac{\delta \mathcal{P}_n}{\delta A^{\nu}} \delta A^{\nu} \bigg\} + {\rm terms~ involving} \nabla_\mu \frac{\partial \mathcal{L}_{\rm HC}}{\partial I_n}. \nonumber \end{align} \tag{ 4.5 }$$

But terms involving covariant derivatives of $\partial \mathcal{L}_{\rm HC} / \partial I_n$ will vanish for any CSI_k field $F_{\mu \nu}$. Moreover, variation of any of the individual polynomials $\mathcal{P}_{n}$ takes the form $\newcommand{\DE}{{\mathscr{D}}} \DE^\mu {H}_{\mu \nu} + H_\nu$, where $H_{\mu \nu}$ and $H_\nu$ is some 2-form and 1-form, respectively, constructed polynomially from $F_{\mu \nu}$, $\star F_{\mu \dots \nu}$ and their gauge covariant derivatives (not necessarily only up to order k). Thus, the full field equations reduce for a CSI_k field $F_{\mu \nu}$ (possessing invariants {I_m} within the domain of convergence of the Taylor series) to

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\DE}{{\mathscr{D}}} \displaystyle \label{CSIEOM} \DE^\mu F_{\mu \nu} = \DE^\mu \tilde{F}_{\mu \nu} + F_\nu, \nonumber \end{align} \tag{ 4.6 }$$

where $\tilde{F}_{\mu \nu}$ and $F_\nu$ are some polynomial gauge covariant $\mathfrak{g}$-valued forms with $F_\nu$ in general not expressible as a divergence of a 2-form. Let us also note that the term $F_\nu$ has origin in variation of the non-Abelian contribution [A,D^mF] to the gauge covariant derivative D^m+1F and is thus present only in the non-Abelian case with k  >  0, i.e. when invariants {I_n} involve not only $F_{\mu \nu}$ (or $\star F_{\mu \dots \nu}$), but also its gauge covariant derivatives. In Abelian theories or non-Abelian theories involving only algebraic invariants {I_n}, the field equations take the form (4.6) with $F_\nu = 0$.

In the following sections, we will show that for certain types of $VSI_k$ solutions of the original Yang–Mill equations and certain background spacetimes, $F_{\mu \nu}$ is universal. In particular, above corrections necessarily vanish, and hence the field equations of any higher-order theory (4.1) are automatically satisfied. Necessary conditions for universality of YM fields will be also studied.

In the particular case of all invariants {I_k} being only algebraic (i.e. constructed solely from $F_{\mu \nu}$ and $\star F_{\mu \dots \nu}$ with no derivatives of these involved), $\mathcal{L}_{\rm HC}$ represents algebraic corrections to the YM Lagrangian and proposition 2.4 of [6] can be extended to the non-Abelian case.

Theorem 4.2 (Sufficient conditions for 0-universality). Null Yang–Mills fields are 0-universal and hence solve any theory (4.1) with algebraic corrections $\mathcal{L}_{\rm HC}$.

Proof. Since the YM field $F_{\mu \nu}$ is null, all polynomials at least quadratic in F are of boost order at most $(-2)$ and hence any rank-1 contraction of such polynomial trivially vanishes, making F 0-universal.

By theorem 3.1, F is $VSI_0$. The same holds also for $\star F_{\mu \dots \nu}$ and mixed invariants of both fields vanish as well. Consequently, as we saw in the previous section, the field equations of theory (4.1) reduce to (4.6) with $F_\nu = 0$ and with $\tilde{F}_{\mu \nu}$ constructed polynomially from $F_{\mu \nu}$ and its Hodge dual. But since $\tilde{F}_{\mu \nu}$ has to be at least quadratic in $F_{\mu \nu}$ (recall that power series expansion of $\mathcal{L}_{\rm HC}$ at zero consists of monomials at least cubic in $F_{\mu \nu}$), it vanishes due to 0-universality of F. □

In this section, we consider the general case of higher-order corrections constructed employing also gauge covariant derivatives of $F_{\mu \nu}$. Before proceeding further, let us first prove the following technical lemma on possible forms of skew-symmetric rank-2 contractions of gauge covariant derivatives $\newcommand{\DE}{{\mathscr{D}}} \DE_\mu \dots \DE_\nu F_{\alpha \beta}$.

Lemma 4.3. Let $F_{\mu \nu}$ be an aligned null YM field in a degenerate Kundt spacetime of Weyl and traceless Ricci type III. Then, any skew-symmetric rank-2 contraction of k-th gauge covariant derivative $\newcommand{\DE}{{\mathscr{D}}} \DE^k F_{\mu \nu}$ of $F_{\mu \nu}$ reduces to $F_{\mu \nu}$ multiplied by some polynomial in the Ricci scalar.

Proof. Let $H_{\mu \nu}$ denote an arbitrary 2-form constructed by contractions (and antisymmetrization, if needed) of $\newcommand{\DE}{{\mathscr{D}}} \DE^k F_{\mu \nu}$. Clearly, to construct H, k/2 contractions of tensorial indices will be required and hence k has to be even. Consequently, there are two possible types of terms in H—either there is at least one contraction of some derivative index with a field strength index or all derivative indices are contracted.

It turns out that, by commuting derivatives, these two types of terms in $\newcommand{\DE}{{\mathscr{D}}} \DE^k F_{\mu \nu}$ can be, cast in the form

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\DE}{{\mathscr{D}}} \displaystyle \label{forms} \DE^{k-1}\DE^\mu F_{\mu \nu}, \qquad \DE^{k-2} \DE^\alpha \DE_\alpha F_{\mu \nu}, \nonumber \end{align} \tag{ 4.7 }$$

respectively, for the price of producing new (but lower order) terms linear in $\newcommand{\DE}{{\mathscr{D}}} \DE^{k-2} F_{\mu \nu}$. Indeed, the formula (2.10) tells us that, schematically, $\newcommand{\DE}{{\mathscr{D}}} \DE^l [\DE,\DE] \DE^{m}F$ is given by $\newcommand{\DE}{{\mathscr{D}}} \DE^l [F,\DE^{m} F]$ and terms of type $\newcommand{\DE}{{\mathscr{D}}} \DE^l ({\rm Riem} \ast \DE^m F)$. However, for a balanced F, we have that $\newcommand{\DE}{{\mathscr{D}}} [F,\DE^{m}F]$ is of boost order $(-2)$ and hence will not contribute to H and only boost weight (0) part of the Riemann tensor (given by the Ricci scalar R and the metric) can survive in contribution from $\newcommand{\DE}{{\mathscr{D}}} {\rm Riem} \ast \DE^m F$. Thus, this term is simply proportional to $\newcommand{\DE}{{\mathscr{D}}} R \DE^m F$ and, since in a degenerate Kundt spacetime of Weyl and traceless Ricci type III, $\newcommand{\e}{{\rm e}} \nabla_\mu R \propto\ell_\mu$ is of boost order $(-1)$ (see the proof of proposition A.8 in [6]), $\newcommand{\DE}{{\mathscr{D}}} \DE^l (R \DE^m F)$ reduces to $\newcommand{\DE}{{\mathscr{D}}} R \DE^{k-2} F$.

Now, each of these new terms proportional to $\newcommand{\DE}{{\mathscr{D}}} \DE^{k-2} F$ can be again, by the commutation procedure above, cast in the form (4.7) introducing new lower-order terms proportional to $\newcommand{\DE}{{\mathscr{D}}} \DE^{k-4}F$. Therefore, the procedure above can be used iteratively until one ends up with all terms in H of the form (4.7) for some k even. The first one obviously vanishes for any YM field, it thus remains to examine the second one. Employing the formula (2.11), we obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\DE}{{\mathscr{D}}} \displaystyle \DE^\alpha \DE_\alpha F_{\mu \nu} \propto R F_{\mu \nu}, \nonumber \end{align} \tag{ 4.8 }$$

where R denotes the Ricci scalar. Recalling that $\nabla_\mu R$ is of boost order $(-1)$, the second term in (4.7) reduces to $\newcommand{\DE}{{\mathscr{D}}} R \DE^{k-2} F_{\mu \nu}$ with all derivative indices contracted. As before, we can repeat the procedure above until we end up with all terms in the form $R^n F_{\mu \nu}$ for some n up to a multiplication constant. Consequently, we obtain that $\tilde{F}$ can be actually always put in the form of F multiplied by some polynomial in R. □

Remark 4.4. From the proof of lemma 4.3, it is easy to see that, under assumption of the lemma, any symmetric rank-2 contraction $H_{\mu \nu}$ of $\newcommand{\DE}{{\mathscr{D}}} \DE^k F$ can only consist of boost order (−2) terms of type $\newcommand{\DE}{{\mathscr{D}}} \DE^m F \otimes \DE^l F$, $m,l \geqslant 0$. Such terms would appear while commuting the gauge covariant derivatives in the procedure described above.

Theorem 4.5 (Sufficient conditions for universality). An aligned null Yang–Mills field in a degenerate Kundt spacetime of Weyl and traceless Ricci type III is universal and hence solves any theory (4.1).

Proof. Recall that, according to corollary 2.7, $F_{\mu \nu}$ and its gauge covariant derivatives of arbitrary order are aligned and of boost order at most (−1). Therefore, any nonvanishing 1-form constructed polynomially from these fields can be at most linear in them. Consequently, it is sufficient to prove that all rank-1 contractions $H_\mu$ of $\newcommand{\DE}{{\mathscr{D}}} \DE^k F$ vanish for k  >  0.

There are two possible types of $H_\mu$—either the index $\mu$ in $\newcommand{\DE}{{\mathscr{D}}} \DE_\mu \dots \DE_\nu F_{\rho \sigma}$ is left uncontracted, in which case $\newcommand{\DE}{{\mathscr{D}}} H_\mu = \DE_\mu I$, where I is some $\mathfrak{g}$-valued tensorial invariant of F, or $\mu$ gets contracted, which means $\newcommand{\DE}{{\mathscr{D}}} H_\mu = \DE^\nu H_{\mu \nu}$ for some $H_{\mu \nu}$ constructed from D^k−1F. Obviously, as a boost order (0) object, I has to vanish and consequently also the corresponding $H_\mu$. It remains to discuss the second possibility. We can decompose $H_{\mu \nu}$ into symmetric and skew-symmetric part and discuss their contributions to $H_\mu$ separately.

According to remark 4.4, the symmetric part of $H_{\mu \nu}$ consists of boost order $(-2)$ terms of type $\newcommand{\DE}{{\mathscr{D}}} \DE^m F \otimes \DE^l F$ and hence cannot contribute to $H_\mu$, since such contribution can again only be of boost order $(-2)$. On the other hand, lemma 4.3 tells us that the skew-symmetric part of $H_{\mu \nu}$ takes the form $\mathcal{P}(R) F_{\mu \nu}$ for some polynomial $\mathcal{P}$ of the Ricci scalar R. Thus, $\newcommand{\DE}{{\mathscr{D}}} H_\mu = F_{\mu \nu} \nabla^\nu \mathcal{P}(R) + \mathcal{P}(R) \DE^\nu F_{\mu \nu}$. But $\newcommand{\e}{{\rm e}} \nabla_\mu R \propto \ell_\mu$ in our background spacetime, and hence both terms in $H_\mu$ necessarily vanish for a null YM field. □

Theorem 4.6 (Necessary conditions for universality). Let $F_{\mu \nu}$ be a non-vanishing null Yang–Mills field solving any theory (4.1). Then, $F_{\mu \nu}$ is either CSI or it is aligned with gradient $\nabla_\mu I$ of all of its invariants I, these are all aligned and generate a twist-free geodetic null congruence.

Proof. Consider a correction Lagrangian of the form $\newcommand{\e}{{\rm e}} \mathcal{L}_{\rm HC} \equiv I \mathcal{L}_{\rm YM}$, where I is arbitrary scalar qauge invariant of $F_{\mu \nu}$. Since the invariant $\mathcal{L}_{\rm YM}$ vanishes for a null $F_{\mu \nu}$, varying $\mathcal{L}_{\rm HC}$ and employing the YM equation, the requirement of universality reduces to

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{univpodm} F_{\mu \nu} \nabla^\mu I = 0, \nonumber \end{align} \tag{ 4.9 }$$

for all invariants I. Since $F_{\mu \nu}$ is null and non-vanishing, decomposing the covariant derivative into null frame, $\newcommand{\del}{{\delta}} \newcommand{\de}{{{\rm d}}} \newcommand{\e}{{\rm e}} \nabla_\mu = \ell_\mu \Delta + n_\mu D + m_\mu^{(i)} \delta_i$, (4.9) implies that both DI and $\newcommand{\del}{{\delta}} \newcommand{\de}{{{\rm d}}} \delta_i I$ vanish, hence $\newcommand{\e}{{\rm e}} \nabla_\mu I = \ell_\mu \Delta I$. Therefore, gradients of all invariants I are aligned. Moreover, since the null vector $\newcommand{\e}{{\rm e}} \ell$ is proportional to gradient of a function, it generates a twistfree geodetic congruence. □

Remark 4.7. Note that, while in four dimensions, any null YM field is necessarily aligned with a shear-free geodetic null congruence [29], in higher dimensions, this is not the case even for Maxwell fields, see appendix B of [36] and lemmas 3, 4 in [37]. In fact, the congruence has necessarily non-vanishing shear in higher-dimensions, unless it is non-expanding. Employing the YM equation and Bianchi identity in the generalized GHP formalism given by (B.6)–(B.9) in the appendix, we provide an appropriate extension of lemma 4 in [37] to non-vanishing null YM fields, see lemma B.1.

However, under additional conditions on curvature of the background spacetime, the admissible geometries of the corresponding null congruence further reduce due to Ricci and Bianchi equations. In particular, in a spacetime of Weyl and traceless Ricci type N with constant Ricci scalar, any null YM field is necessarily aligned with shear-free, twist-free and non-expanding null geodetic (i.e. Kundt) congruence, as we prove in the appendix (theorem B.2).

We first give explicit form of all universal solutions satisfying the hypothesis of theorem 4.5 in adapted coordinates. Then, discussion of special cases already known in the literature follows.

4.4.1. The metric.

Recall that local form of a general degenerate Kundt metric in Kundt coordinates $(r,u,x^\alpha)$ is given by (3.5). Requirement that g is of Weyl and traceless Ricci type III then puts further restrictions on functions in (3.5), see discussion in section 4 of [38]. In particular, $\newcommand{\de}{{{\rm d}}} \newcommand{\e}{{\rm e}} g^T \equiv g_{\alpha \beta}(u,x) {\rm \de} x^\alpha {\rm \de} x^\beta$ has to be a Riemannian metric on a $(d-2)$-dimensional transverse space spanned by coordinates $\{x^\alpha \}$, which is, for any fixed u, of constant sectional curvature $K(u)$ depending on the Ricci scalar $R=R(u)$ of (3.5) and the dimension d as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{sectcurv} K \equiv \frac{R}{d(d-1)}, \nonumber \end{align} \tag{ 4.10 }$$

and functions H and $W_{\alpha}$ are constrained by the following equations, where $\nabla^T$ denotes the covariant derivative associated with the transverse space metric g^T

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{constraint1} 2H^{(2)} = \frac{1}{4}W_\alpha^{(1)}W^{(1)\alpha} + K, \nonumber \end{align} \tag{ 4.11 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{constraint2} \nabla_{(\beta}^T W_{\alpha)}^{(1)} - \frac{1}{2} W_\alpha^{(1)} W_\beta^{(1)} = 2K g_{\alpha \beta}, \qquad \partial_{[\beta} W_{\alpha]}^{(1)}=0. \nonumber \end{align} \tag{ 4.12 }$$

4.4.2. The gauge field.

Exploiting gauge freedom, we can without loss of generality start with gauge potential of the form (3.9). The function A^a is then subject to the YM equation, which in our case reduces to the wave equation in the $(d-2)$-dimensional transverse space of constant sectional curvature (4.10):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{YMtransverse} \Box^T A^a = 0, \nonumber \end{align} \tag{ 4.13 }$$

where $\newcommand{\e}{{\rm e}} \Box^T \equiv g^{T \alpha \beta} \nabla_\alpha^T \nabla_\beta^T$ is the Laplace–Beltrami operator. Any null YM field in the background spacetime (3.5) can be then obtained by a gauge transformation of (3.9) and is necessarily $VSI$ (recall theorem 3.1).

Note that the YM equation (4.13) depends only on the transverse metric g^T, while the functions H and $W_\alpha$ in (3.5) can be arbitrary, as long as the constraints (4.11) and (4.12) hold.

4.4.3. Examples in the literature.

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\vphi}{\varphi} \displaystyle \eth_{[i}F_{j k]}^a = A_{[i}^b F_{jk]}^c f_{cb}^a + 2 (\vphi_{[i} \rho_{j k]}^{\prime a} + \vphi_{[i}^{\prime a} \rho_{j k]}), \label{max:4} \nonumber \end{align} \tag{ B.9 }$$

together with the primed equations (B.6)$'$–(B.8)$'$. Beware of the fact that, under priming operation, the scalars f ^a transform as $f^{\prime a} = -f^a$.

Obviously, the GHP equations (B.6)–(B.9) differ from the GHP Maxwell equations in [37] only by presence of the additional terms emanating from the commutator $[A_\mu,F_{\nu \rho}]$ in $\newcommand{\DE}{{\mathscr{D}}} \DE_{\mu} F_{\nu \rho}$. Therefore, it can be expected that some of the results obtained for Maxwell fields from the GHP Maxwell equations can be appropriately extended also to YM fields.

Indeed, since for a null field strength $F_{\mu \nu}$, null frame $\newcommand{\e}{{\rm e}} \{\ell, n, m_{(i)} \}$ can be chosen such that only scalars $\newcommand{\vphi}{\varphi} \vphi_i^{\prime a}$ are non-vanishing, equations (B.6)– (B.9) reduce to

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\de}{{{\rm d}}} \newcommand{\del}{{\delta}} \newcommand{\kap}{{\kappa}} \newcommand{\vphi}{\varphi} \displaystyle \kap_i \vphi_{i}^{\prime a} = 0, \qquad \kap_{[i} \vphi_{j]}^{\prime a} = 0, \qquad (\rho_{i j} + \rho_{j i} -\rho\del_{i j})\vphi_{j}^{\prime a}=0, \qquad \vphi_{[i}^{\prime a} \rho_{j k]} = 0. \nonumber \end{align} \tag{ B.10 }$$

Therefore, lemma 4 of [37] can be extended to YM fields in the following way

Lemma B.1. If a non-vanishing null Yang–Mills field $F_{\mu \nu}$ is aligned with a null vector $\newcommand{\e}{{\rm e}} \ell$, then $\newcommand{\e}{{\rm e}} \ell$ is geodetic. Moreover, for its optical matrix $\rho_{ij}$ and the corresponding GHP scalars $\newcommand{\vphi}{\varphi} \vphi_i^{\prime a}$ associated with $F_{\mu \nu}$, there exists $\omega_i^a$ such that

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\vphi}{\varphi} \displaystyle \label{optical} \rho_{(ij)}\vphi_j^{\prime a} = (\rho/2)\vphi_i^{\prime a}, \qquad \rho_{[ij]} = \omega_{[i}^a \vphi_{j]}^{\prime a}. \nonumber \end{align} \tag{ B.11 }$$

Note that, in four dimensions, the lemma reduces to the well-known result that any non-vanishing null YM field is aligned with shear-free geodetic congruence [29, 44].

In a similar way, more results on YM fields (both the test fields in a fixed background and the ones coupled to gravity) can be obtained. Here, we provide an elementary proof of the following result stated originally for Einstein–Maxwell solutions in arbitrary dimension [38].

Theorem B.2. If a spacetime of Weyl and traceless Ricci type N with constant Ricci scalar admits a non-vanishing null Yang–Mills field, then the spacetime is Kundt.

Proof. Due to proposition 3.1 of [45], the Weyl and the Ricci tensors (and consequently also $F_{\mu \nu}$) are aligned with the same null vector, say $\newcommand{\e}{{\rm e}} \ell$. Moreover, according to proposition 5.4 of [45], null frame be chosen such that the optical matrix $\rho_{ij}$ takes the form such that only the following components are non-vanishing

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \rho_{22}=s, \qquad \rho_{33}= s b, \qquad \rho_{23}= s a, \qquad \rho_{32}= -s a \nonumber \end{align} \tag{ B.12 }$$

with $b\neq 1$, otherwise $F_{\mu \nu}$ would have to be zero. Thus, the expansion of $\newcommand{\e}{{\rm e}} \ell$ reads $\rho = s (1+b)$ and the first equation of (B.11) tells us that of the eigenvectors is $\rho/2$. This means that either $\rho = 0$ or b  =  1 and, since b  =  1 is forbidden, we have $\rho =0$. Consequently, either s  =  0 (in which case the spacetime is Kundt and we are done) or b  =  −1. Also, if a  =  0, the spacetime is again Kundt due to the Sachs equation [46]. Assume thus that $s,a \neq 0$ and b  =  −1. Then, the first equation of (B.11) implies $\newcommand{\vphi}{\varphi} \vphi_2^{\prime a},\vphi_3^{\prime a} = 0$. The second of (B.11) then gives us $\newcommand{\vphi}{\varphi} \vphi_i^{\prime a} = 0$ for all i  >  3. Hence $F_{\mu \nu}$ vanishes completely, which is a contradiction and the spacetime is indeed Kundt. □

Consider now a solution $(g_{\mu \nu},F_{\mu \nu})$ of the Einstein–YM equations with a non-vanishing null field strength $F_{\mu \nu}$. Then, $g_{\mu \nu}$ is of traceless Ricci type N. Therefore, in view of theorem B.2, we immediately obtain

Corollary B.3. All Weyl type N solutions of the Einstein–Yang–Mills equations with a non-vanishing null field strength are aligned and Kundt.

Thus, although null YM test fields are not necessarily aligned with a shear-free null congruence in higher dimensions, it is still geodetic and, once additional assumptions on structure of curvature tensors of the background spacetime are taken into account, further restrictions on geometry of the corresponding null congruence follow.