text:
\section{Introduction}

The study of various aspects of classical and quantum field theory
in higher dimensions attracts attention basically due to connections
with the low-energy limit of superstring theory and miraculous
cancelations of some divergences in supersymmetric field models. One
of such aspects is a description of the massless representations of
the Poincar\'e group in multi-dimensional spaces (see e.g. the
recent works \cite{W,KUZ,BFIP,BFI-21})\footnote{For earlier activity in this direction see
e.g. \cite{BB1,BB2} and the references therein.}. In this
paper, we continue our study of field irreducible massless
representations of the six-dimensional Poincar\'e group \cite{BFIP,BFI-21} focusing on the infinite spin representations and
their Lagrangian formulation.

The study of the infinite spin representations of the Poincar\'e
group \cite{Wigner39,Wigner47,BargWigner}, their field realizations and dynamical description aroused
considerable interest, which led to the formation of a certain
research branch mainly in the context of the theory of higher spin
fields (see e.g. the review \cite{BekSk} and earlier references
therein, and recent papers \cite{BekMou,Bekaert:2017xin,Najafizadeh:2017tin,HabZin,AlkGr,Metsaev18,BFIR,
BuchKrTak,BuchIFKr,ACG18,Metsaev18a,BFI,Metsaev19,BKSZ,MN20,MN22,BuchIFKr22}) where
aspects of interactions and supersymmetry of infinite spin fields
have been examined). Since field realizations of the Poincar\'e
group representations in each concrete dimension have specific
features, infinite spin fields in higher dimensions deserve a separate
study.

To construct infinite spin fields in six-dimensional Minkowski
space, we should describe a possible spectrum of states corresponding
to these fields and, first of all, to clarify the spin structure. It can be
achieved by considering massless representations of the $6D$
Poincar\'e group in state space in terms of the canonically
conjugate position and momentum operators, as well as the canonically
conjugate pair of spinor operators. Irreducible representations are
formulated in terms of second-, fourth- and sixth-order Casimir
operators, respectively. The corresponding eigenvalues for these
Casimir operators in the irreducible infinite spin representation are
$0\,,-\mu^2\,,-\mu^{2}s(s+1)$ respectively, where $\mu$ is a
nonzero real parameter and $s$ is a non-negative integer or
half-integer number (see the details in \cite{BFIP}).

To describe the spin structure of the $6D$ infinite spin fields, it is
natural to refer to the light-cone frame for massless fields,
where the eigenvalues of the energy-momentum operator are
$p^0=p^5=k$, $p^{\hat a}=0$, ${\hat a}=1,2,3,4\,$ with some nonzero
real parameter $k$. Here a remarkable result was unexpectedly discovered that any infinite spin field in this frame is
necessarily a function on bi-harmonic space with the harmonics
$u^{\pm}\,,v^{\pm}$ which were earlier essentially used to construct
the unconstrained superfield formulation of $4D\,, {\cal N}=2$
supersymmetric field theories \cite{GIKOS,GIOS}.  Taking into
account this result, it is natural then to go to the light-front
coordinate system $x^{\pm},x^{\hat a}\,, a=1,2,3,4\,$, which
inherits the properties of the light-cone frame \cite{Dirac49}. Thus,
we arrive at the function of both $x^{\pm},x^{\hat a}$ and harmonics
$u^{\pm}\,,v^{\pm}$ which is considered as the infinite spin field
in the light-front coordinate system. The field dynamics
in the light-front coordinates can be constructed following the generic
scheme \cite{Dirac49} (see also
\cite{Bengtsson2Brink,Bengtsson2Linden,Sieg88,Sieg99,Metsaev05,PoSk}).

The paper is organized as follows. In Sect.\,2, we discuss the
description of irreducible infinite spin representations of the $6D$
Poincar\'e group in state space formulated in terms of the
position and momentum operators and spin operators. These operators
are $6D$ vectors and a pair of $\mathrm{SU}(2)$ Majorana-Weyl
spinors. We find expressions of the fourth- and sixth-orders Casimir
operators for the system under consideration and discuss the
conditions leading to fixing the eigenvalues of these operators on
physical states. In Sect.\,3, we derive infinite spin fields in
the light-cone frame. Here we show that these fields are the function on
bi-harmonic space with two sets of $\mathrm{SU}(2)$
harmonics $v_{i}^\pm$ and $v_{\underline{i}}^\pm$. The harmonics
obtained here describe the coset space
$[\mathrm{SU}(2)\,{\otimes}\,\mathrm{SU}(2)]/\mathrm{U}(1)$. Such a
harmonic field possesses a harmonic charge which is determined by
the eigenvalue of the sixth-order Casimir operator. We describe the
general structure of the harmonic field in the light-cone frame and show
that it is given by an infinite expansion in the harmonics. Using these
results, in Sect.\,4, we develop the light-front dynamical formulation
of an infinite spin field.  We find the generators of the $6D$
Poincar\'e group for the fields under consideration and propose the
corresponding action. An important point in this approach is the use
of harmonics as additional coordinates, which greatly simplifies the
field analysis. In Sect.\,5, we summarize the results obtained.
Appendix\,A  is devoted to the calculation of the sixth-order Casimir
operator for the system considered. In Appendix\,B, we find the spinor
part of the $6D$ Lorentz algebra generators.

\setcounter{equation}0
\section{Irreducible massless representation of the $D6$ Poincar\'e group}

In this section, we discuss the construction of a massless
irreducible representation of the six-dimensional Poincar\'e group
emphasizing the specific use of spinor operators.

We consider the representations in the space of states described by vectors
$|\Psi\rangle \,$. The basic operators acting in this space are
\begin{equation}
\label{op}
x^{a}\,,\quad p_{a}\,;\qquad \xi_\alpha^I\,,\quad \rho^{\alpha I}\,.
\end{equation}
Here the Hermitian coordinate $x^{a}=(x^{a})^\dagger$ and momentum $p_{a}=(p_{a})^\dagger$ operators are
components of the six-vectors, $a=0,1,\ldots,5$ and they obey the standard commutation relations
\begin{equation}
\label{x-p}
[x^{a},p_{b}]= i\delta^{a}_{b}\,.
\end{equation}
The operators $\xi_\alpha^I$, $\rho^{\alpha I}$ are the
$\mathrm{SU}(2)$ Majorana-Weyl spinors\footnote {We use the spinor
conventions of the works \cite{BFIP,BFI-21}.}, where
$\alpha=1,2,3,4$ and $I=1,2$ are, respectively, the spinorial
$\mathrm{SU}^*(4)$ and internal $\mathrm{SU}(2)$ indices. The
Hermitian conjugation for these operators is defined as follows:
\begin{equation}
\label{xi}
(\xi_\alpha^I)^\dagger =\epsilon_{IJ}B_{\dot\alpha}{}^\beta \xi_\beta^J\,,\qquad
(\rho^{\alpha I})^\dagger = \epsilon_{IJ}\rho^{\beta J}(B^{-1})_\beta{}^{\dot\alpha}\,,
\end{equation}
where $B_{\dot\alpha}{}^\beta$ is the matrix related to complex
conjugation, and the antisymmetric tensors $\epsilon_{IJ}$,
$\epsilon^{IJ}$ have the components $\epsilon_{12}=\epsilon^{21}=1$ (see
\cite{BFI-21} for details). Nonzero commutation relations for the
operators $\xi_\alpha^I$, $\rho^{\alpha I}$ have the form
\begin{equation}
\label{com-sp}
\left[ \xi_\alpha^I\,, \rho^\beta_J\right] = i\delta_\alpha^\beta \delta^I{}_J \,.
\end{equation}
The operators $\xi_\alpha^I$, $\rho^{\alpha I}$ are to describe the
spin degrees of freedom. The state space will be specified in the
next section.

We assume that the operators $p_a$ generate space-time translations.
In this case, the generators $\{P_a, M_{ab} \}$ of the algebra $\mathfrak{iso}(1,5)$ of the Poincar\'e group are realized as
\begin{eqnarray}
\label{P-gen}
P_a&=&  p_a \,, \\ [7pt]
\label{M-gen}
M_{ab}&=&
p_{a}x_{b}-p_{b}x_{a}+S_{ab} \,,
\end{eqnarray}
where the spin part of the Lorentz group generators looks like
\begin{equation}
\label{M-1}
S_{ab} = \xi_\alpha^I (\tilde\sigma_{ab})^{\alpha}{}_{\beta}\rho^\beta_I
= -\rho^\alpha_I (\sigma_{ab})_{\alpha}{}^{\beta}  \xi_\beta^I\,.
\end{equation}

We consider the massless representations where the quadratic Casimir operator $C_2 = P^2 = P^{a}P_{a}$
of the algebra $\mathfrak{iso}(1,5)$ has zero eigenvalues
\begin{equation}
\label{P2-0}
p^{a}p_{a}|\Psi\rangle =  0\,.
\end{equation}
In this case, the projection of the fourth-order Casimir operator in subspace \p{P2-0} has the form \cite{BFIP}
\begin{equation}
\label{C4-def}
C_4= \Pi^{a} \Pi_{a}  \, ,
\end{equation}
where
\begin{equation}
\label{Pi-def}
\Pi_a = P^{b}  M_{ba}  \, .
\end{equation}
As a result, we can see that in the representation \p{P-gen}, \p{M-gen}
and under the condition \p{P2-0} the operator \p{C4-def} takes the
following form:
\begin{equation}
\label{C4-expr}
C_4= - \,\tilde\ell\,\ell  \,,
\end{equation}
where the scalar operators $\ell$, $\tilde\ell$ are defined by the relations
\begin{equation}
\label{l-expr}
\ell:=\frac12\, \rho^\alpha_I(p_a\sigma^{a})_{\alpha\beta}\rho^\beta{}^I\,,\qquad
\tilde\ell:=\frac12\,  \xi_\alpha^I(p_a\tilde\sigma^{a})^{\alpha\beta}\xi_\beta{}_I\,.
\end{equation}
When deriving expression \p{C4-expr}, we used the relation
\p{sigma-id} for the $6D$ $\sigma$-matrices. The algebra of
operators \p{l-expr} is written in the form
\begin{equation}
\label{l-com}
[\tilde\ell,\ell]=N\, p^a p_a\,,
\end{equation}
where the operator $N$ is defined by the anticommutator
\begin{equation}
\label{N-def}
N:=\frac{i}{2}\, \{\xi_\alpha^I,\rho^\alpha_I\} \,.
\end{equation}
Besides, the operators \p{l-expr} are the Poincar\'e group
invariants and hence they commute with the generators \p{P-gen},
\p{M-gen}
\begin{equation}
\label{l-PM}
[P_a,\ell]=[P_a,\tilde\ell]=0\,\qquad [M_{ab},\ell]=[M_{ab},\tilde\ell]=0\,.
\end{equation}

The infinite spin representation is characterized by the condition that the fourth-order Casimir
operator has nonzero negative eigenvalue
\begin{equation}
\label{C4-const}
C_4\,|\Psi\rangle \ = \ -\,\mu^2\,|\Psi\rangle \,,
\end{equation}
where $\mu \neq 0$ is the dimensional real parameter which can be taken
positive $\mu \in \mathbb{R}_{> 0}$ without loss of generality. Using relations \p{C4-const} and \p{C4-expr}, we can see that
it is sufficient to define infinite spin states by the constraints
\begin{equation}
\label{C4-constr}
\ell\,|\Psi\rangle \ = \ \mu\,|\Psi\rangle \,,\qquad \tilde\ell\,|\Psi\rangle \ = \ \mu\,|\Psi\rangle \, .
\end{equation}

For massless representations \p{P2-0}, the sixth-order Casimir operator has the form \cite{BFIP}
\begin{equation}
\label{C6-Casimir}
C_6=
-\, \Pi^b M_{ba}\, \Pi_c M^{ca}
\ + \ \frac12\,\Big(M^{ab}M_{ab}-8\Big)\, \Pi^{a} \Pi_{a} \,,
\end{equation}
where the operator $\Pi_a$ is defined in \p{Pi-def}.
In the representation \p{P-gen}, \p{M-gen} and under the conditions \p{P2-0}, \p{C4-const}, we obtain\footnote{See the details in Appendix A.}
\begin{equation}
\label{C6-Cas}
C_6\,|\Psi\rangle \ = \ -\, \mu^2\, J_{\mathrm{i}}J_{\mathrm{i}}\,|\Psi\rangle\,,
\end{equation}
where the operators $J_{\mathrm{i}}$ $(\mathrm{i}=1,2,3)$ are defined as follows:
\begin{equation}\label{Ta-def}
J_{\mathrm{i}} \ := \ \frac{i}{2}\,\xi_\alpha^I(\sigma_{\mathrm{i}})_I{}^J \rho^\alpha_J\,.
\end{equation}
Here $\sigma_{\mathrm{i}}$ are the Pauli matrices. The operators $J_{\mathrm{i}}$ form the $\mathfrak{su}(2)$ algebra
\begin{equation}\label{Ta-alg}
[J_{\mathrm{i}},J_{\mathrm{j}}]= i\epsilon_{{\mathrm{i}}{\mathrm{j}}{\mathrm{k}}}J_{\mathrm{k}}\,.
\end{equation}
Expression \p{C6-Cas} for the operator $C_6$ is the same as in
\cite{BFIP} but the realization of the generators $J_{\mathrm{i}}\in
\mathfrak{su}(2)$ in \cite{BFIP} is different.

As it was shown in \cite{BFIP}, the space $V$ of irreducible infinite
spin representation is induced from the space of finite dimensional representation of \p{Ta-def} and the operator $C_6$ acts as follows:
\begin{equation}
\label{C6-ir}
C_6\,|\Psi\rangle \ = \ -\,\mu^2\, s(s+1)\,|\Psi\rangle\,,
\end{equation}
where $s$ is a nonzero integer or half-integer number, $s \in \mathbb{Z}_{\geq 0}/2$. Therefore,
the states corresponding to the infinite spin irreducible representation obey the constraints
\begin{equation}\label{eq-3}
J_{\mathrm{i}}J_{\mathrm{i}}\,|\Psi\rangle \ = \ s(s+1)\,|\Psi\rangle \,,
\end{equation}
where the operators  $J_{\mathrm{i}}$ are defined in \p{Ta-def}.

Note that the $\mathfrak{su}(2)$ algebra generators \p{Ta-def} commute with the $\mathrm{SU}(2)$ scalar
operators \p{l-expr}:
\begin{equation}
\label{su2-l} [J_{\mathrm{i}},\ell]=[J_{\mathrm{i}},\tilde\ell]=0
\,.
\end{equation}
Besides, the operators \p{Ta-def} commute with generators of six-dimensional translations \p{P-gen} and
with the Lorentz algebra $\mathfrak{so}(1,5)$ generators \p{M-gen}, \p{M-1}:
\begin{equation}
\label{M-J}
[P_{a},J_{\mathrm{i}}]=0\,,\qquad [M_{ab},J_{\mathrm{i}}]= [S_{ab},J_{\mathrm{i}}]=0\,.
\end{equation}

It is worth noting that the algebra
$\mathfrak{so}(1,5)=\mathfrak{su}^*(4)$ generated by \p{M-1} is dual
to the algebra $\mathfrak{su}(2)$ with the generators \p{Ta-def} in
the sense of Howe duality \cite{Howe}.

It was shown earlier \cite{BekMou,BB1,BB2} that in the vector
approach an infinite spin representation of $\mathfrak{so}(1,5)$
requires the use of the 6-dimensional Heisenberg algebra \p{x-p}
generated by the operators of position $x^a$ and momentum $p_a$ and two
additional 6-dimensional Heisenberg algebras with the coordinate
operators $y_1^a$, $y_2^a$ and their momentum operators
$p^{(y)}_{1a}$, $p^{(y)}_{2a}$. On the other hand, we proved in
\cite{BFI-21} that, in the twistor formulation, the
$\mathfrak{iso}(1,5)$ representations of infinite spin are
necessarily described in the bi-twistor space, which is defined by
two pairs of canonically-conjugated $\mathrm{SU}(2)$ Majorana–Weyl
spinors of type \p{com-sp} (but having nonzero mass dimensions).
Here we have shown that $6D$ infinite spin representations can be
described in the space defined by only one 6-dimensional Heisenberg
algebra \p{x-p} and one pair of canonically conjugated
$\mathrm{SU}(2)$ Majorana–Weyl spinors \p{com-sp}, as it was
indicated in  \p{op}.

In the next section, we will describe the infinite spin vectors $|\Psi\rangle$ in terms of appropriate fields.

\setcounter{equation}0
\section{Infinite spin fields in the light-cone frame}

We consider a structure of $D6$ infinite spin fields in the light-cone frame. This frame is defined
by the following conditions on eigenvalues of the energy-momentum operator:
\begin{equation}
\label{P-st}
p^0=p^5=k\,,\qquad p^{\hat a}=0\,,\qquad {\hat a}=1,2,3,4\,,
\end{equation}
where $k$ is a nonzero real parameter of mass dimension.
Using the light-cone coordinates
$p^\pm=\left(p^0\pm p^5\right)/{\sqrt{2}}$, one gets for the same frame
\begin{equation}
\label{P-st-lc}
p^+=\sqrt{2}k\,,\qquad p^-=p^{\hat a}=0\,.
\end{equation}

Consider the $\mathrm{SU}^*(4)$ spinors $\xi_\alpha^I$, $\rho^{\alpha I}$ with a four-component spinor index
$\alpha$ and present them as objects with two-component indices as follows:
\begin{equation}
\label{pi-2-2}
\xi_\alpha^I=(\xi_i^I,\xi_{\underline{i}}^I)\,,\qquad\quad \rho^\alpha_I=(\rho^i_I,\rho^{\underline{i}}_I)\,,
\end{equation}
where the two-component indices take the values
$i=1,2$ and $\underline{i}=1,2$, i.e. $i=\alpha$ for $\alpha=1,2$, and
 $\underline{i}=\alpha-2$ for $\alpha=3,4$.

In the light-cone frame \p{P-st} the operators  \p{l-expr} take the form
\begin{equation}
\label{l-tl}
\tilde\ell \ = \  k\,\epsilon^{ij}\epsilon_{IJ}\,\xi_i^I \xi_j^J\,,\qquad\quad
\ell \ = \ k\,\epsilon_{\underline{i}\underline{j}}\epsilon^{IJ}\,\rho^{\underline{i}}_I \rho^{\underline{j}}_J\,,
\end{equation}
where the matrices $\tilde\sigma^-$ \p{t-sigma+-} and $\sigma^-$ \p{sigma+-} were used.
Then in this frame the constraints
\begin{equation}
\label{C4-constr-1}
\tilde\ell= \mu \, , \qquad \ell= \mu
\end{equation}
from \p{C4-constr} are written in the form
\begin{equation}
\label{harm-cond-1}
\epsilon_{IJ}u_i^I u_j^J=\epsilon_{ij}\,,\qquad\quad \epsilon_{IJ}v_{\underline{i}}^I v_{\underline{j}}^J=\epsilon_{\underline{i}\underline{j}}\,,
\end{equation}
where we
have used the spinor variables
\begin{equation}
\label{harm-1}
u_i^I:= \sqrt{2k/\mu}\; \xi_i^I\,;\qquad\quad v_{\underline{i}}^I:= \sqrt{2k/\mu}\; \rho_{\underline{i}}^I\,,\qquad \rho_{\underline{i}}^I
=\epsilon_{\underline{i}\underline{j}}\epsilon^{IJ}\rho^{\underline{j}}_J\,.
\end{equation}
The conditions \p{xi} in terms of the $\mathrm{SU}(2)$ spinors
\p{harm-1} look like
\begin{equation}
\label{u-real}
(u_i^I)^* =-\epsilon_{IJ}\epsilon^{ij}u_j^J\,, \qquad\quad (v_{\underline{i}}^I)^* =-\epsilon_{IJ}\epsilon^{\underline{i}\underline{j}}v_{\underline{j}}^J\,.
\end{equation}
Conditions \p{harm-cond-1}, \p{u-real} are nothing but the ones of unimodularity, $\det u=1$, $\det v=1$, and
unitarity, $u^\dagger u=1$, $v^\dagger v=1$, of the $2{\times}2$ matrices
\begin{equation}
\label{u-matrix}
u \ := \  \parallel\! u_i{}^I \!\parallel\,, \qquad v \ := \  \parallel\! v_{\underline{i}}{}^I \!\parallel\,.
\end{equation}
As a result, in the light-cone frame the variables $u_i^I$ and $v_{\underline{i}}^I$ \p{harm-1} are the elements of the $\mathrm{SU}(2)$ groups
and parameterize the compact space. Further, analogously to \cite{GIKOS,GIOS}, we will use the following notation:
\begin{equation}
\label{harm}
u_i^1=u_i^+\,,\quad  u_i^2=u_i^-\,,\qquad\quad v_{\underline{i}}^1=v_{\underline{i}}^+\,,
\quad v_{\underline{i}}^2=v_{\underline{i}}^-\,.
\end{equation}
In this notation, relations \p{harm-cond-1} are rewritten in the form \footnote{Note that the
$\mathrm{U}(1)$ charges $\pm$ in the variables $u^{\pm}$ and
$v^{\pm}$ have a different meaning than the light-cone indices
$\pm$ in quantities $p^{\pm}$. The later are the $SO(1,1)$ vector
indices.}
\begin{equation}
\label{harm-cond-1a}
u_i^+ u_j^- - u_j^+ u_i^-=\epsilon_{ij}\,,\qquad\quad
v_{\underline{i}}^+ v_{\underline{j}}^- - v_{\underline{j}}^+ v_{\underline{i}}^-=\epsilon_{\underline{i}\underline{j}}
\end{equation}
or, in the equivalent form
\begin{equation}
\label{harm-cond-2b}
u^i{}^+ u_i^- =1\,,\qquad\quad
v^{\underline{i}}{}^+ v_{\underline{i}}^- =1\,,
\end{equation}
where $u^i{}^\pm=\epsilon^{ij}u_j^\pm$, $u^{\underline{i}}{}^\pm=\epsilon^{\underline{i}\underline{j}}u_{\underline{j}}^\pm$.
Note that both $u^{\pm}$ and $v^{\pm}$ have the same indices $\pm$,
since they are obtained from the common $\mathrm{SU}(2)$-index $I$
for the $\mathrm{SU}(2)$ Majorana-Weyl spinors \p{pi-2-2}.

We emphasize that the variables $u^{\pm}$ and $v^{\pm}$ introduced in
\p{harm-1} completely determine the operators $\ell$, $\tilde\ell$ in \p{l-tl} and represent half of the different
canonical pairs in the algebra \p{com-sp}. We treat the second half
of the operators in \p{com-sp} as differential operators. Thus, one
considers a representation
where the operators
$\rho^i_I$ and $\xi_{\underline{i}}^I$ in the algebra \p{com-sp} are
realized as differential operators
\begin{equation}
\label{vw-dif}
\rho^i_I=-i\frac{\partial}{\partial \xi_i^I}=-i\sqrt{2k/\mu}\,\frac{\partial}{\partial u_i^I} \,,\qquad
\xi_{\underline{i}}^I=i\frac{\partial}{\partial \rho^{\underline{i}}_I}=
i\sqrt{2k/\mu}\,\epsilon_{ij}\epsilon^{IJ}\frac{\partial}{\partial v_j^J} \,.
\end{equation}
In the representation chosen, the operators $\ell$ and $\tilde\ell$
are realized by operators of multiplication by the functions of
$u^\pm$, $v^\pm$.

In such a representation, the $\mathfrak{su}(2)$-generators
$J_\pm:=J_1\pm i J_2$ and $J_3$, given by \p{Ta-def}, are written as
follows
\begin{equation}
\label{Ja-ab3}
J_\pm = D^{\pm\pm}_u +D^{\pm\pm}_v\,, \qquad
J_3 =\frac{1}{2}\left(D^{0}_u + D^{0}_v\right)\,,
\end{equation}
where
\begin{eqnarray}
\label{D-u}
D^{\pm\pm}_u:= u_i^\pm \frac{\partial}{\partial u_i^\mp}\,,&&
D^{0}_u:=u_i^+ \frac{\partial}{\partial u_i^+} - u_i^- \frac{\partial}{\partial u_i^-}\,, \\ [6pt]
\label{D-w}
D^{\pm\pm}_v:= v_{\underline{i}}^\pm \frac{\partial}{\partial v_{\underline{i}}^\mp}\,,&&
D^{0}_v:=v_{\underline{i}}^+ \frac{\partial}{\partial v_{\underline{i}}^+} - v_{\underline{i}}^- \frac{\partial}{\partial v_{\underline{i}}^-}
\end{eqnarray}
coincide with the harmonic derivatives in the notation \cite{GIKOS,GIOS}.

Relations \p{Ta-alg} are written in the form $[J_+,J_-]=2J_3$,
$[J_3,J_\pm]=\pm J_\pm$, and the Casimir operator of the
$\mathfrak{su}(2)$ algebra has the standard expression
$$
J_{\mathrm{i}}J_{\mathrm{i}} \ = \ J_3(J_3+1)+J_-J_+  \ = \  -J_3(-J_3+1)+J_+J_-\,.
$$
As a solution to the irreducibility condition \p{C6-Cas}, \p{C6-ir} for $6D$ representations
we take the highest weight vector $|\Psi^{(2s)}\rangle$
which is defined by the equations
\begin{eqnarray}
\label{constr-J-a}
J_+ |\Psi^{(2s)}\rangle &=& 0\,, \\ [6pt]
\label{constr-J3a}
(J_3-s) |\Psi^{(2s)}\rangle &=& 0\,,
\end{eqnarray}
where the operators $J_+$ and $J_3$
are expressed via harmonic derivatives \p{D-u}, \p{D-w} in \p{Ja-ab3}.
Recall that the vector $|\Psi^{(2s)}\rangle$ also obeys the conditions \p{C4-constr}:
\begin{equation}
\label{C4-constr1}
\ell\,|\Psi^{(2s)}\rangle \ = \  \mu\,|\Psi^{(2s)}\rangle \,,\qquad \tilde\ell\,|\Psi^{(2s)}\rangle \ = \ \mu\,|\Psi^{(2s)}\rangle \, .
\end{equation}

Now, we show that the vectors of the states $|\Psi^{(2s)}\rangle$ are
realized as fields. In the representation \p{vw-dif} the
corresponding fields in the light-cone frame are the functions
$\Psi^{(2s)}(u^\pm, v^\pm)$ of four $\mathrm{SU}(2)$ spinors
$u_i^\pm$, $v_{\underline{i}}^\pm$.
Then, it is natural to present the solution of equations \p{C4-constr1} by using $\delta$-functions
\begin{equation}
\label{sol-0}
\Psi^{(2s)}(u^\pm, v^\pm)=\delta(\ell- \mu) \delta(\tilde\ell-\mu)\Phi^{(2s)}(u^\pm, v^\pm) \, ,
\end{equation}
where the arguments of the field $\Phi^{(2s)}(u^\pm, v^\pm)$ satisfy \p{harm-cond-1a}, \p{harm-cond-2b},
and it means that the field $\Psi^{(2s)}(u^\pm, v^\pm)$ is a function on the $\mathrm{SU}(2)\,{\otimes}\,\mathrm{SU}(2)$ group.

Using the relations \p{Ja-ab3}, one rewrites the remaining conditions \p{constr-J-a} and
\p{constr-J3a} in the form
\begin{eqnarray}
\label{constr-J-b}
\left( D^{++}_u +D^{++}_v\right) \Phi^{(2s)}(u^\pm, v^\pm) &=& 0\,, \\ [6pt]
\label{constr-J3b}
\left( D^{0}_u +D^{0}_v-2s\right) \Phi^{(2s)}(u^\pm, v^\pm) &=& 0\,.
\end{eqnarray}

Equation \p{constr-J3b} means the $\mathrm{U}(1)$ covariance of the
field $\Phi^{(2s)}(u^\pm,v^\pm)$:
\begin{equation}
\label{u1-field}
\Phi^{(2s)}(e^{\pm i\varphi}u^\pm, e^{\pm i\alpha}v^\pm)=e^{2si\alpha}\Phi^{(2s)}(u^\pm, v^\pm) \,.
\end{equation}
The charge $(2s)$ in the notation of the field $\Phi^{(2s)}$ reflects the property \p{u1-field} of this field.
Transformations of the arguments of the field $\Phi^{(2s)}(u^\pm,v^\pm)$ in \p{u1-field} are obtained from
the right action on the matrices \p{u-matrix} by the diagonal unitary matrix $h$:
\begin{equation}
\label{uv-u1}
u_i{}^J\,\to \,u_i{}^K h_K{}^J\,,\qquad v_{\underline{i}}{}^J\,\to \,v_{\underline{i}}{}^K h_K{}^J\,,\qquad\quad
h=\parallel\! h_K{}^J \!\parallel:=
\left(\!
\begin{array}{cc}
e^{ \,i\alpha} & 0 \\
0 & e^{- i\alpha} \\
\end{array}
\!\right)
,
\end{equation}
where $K,J=(+,-)$.
In the standard stereographic parametrization of the $SU(2)$ matrices
\begin{equation}
\label{su2-par}
\begin{array}{rcl}
\left(\!
\begin{array}{cc}
u_1^+ & u_1^- \\
u_2^+ & u_2^- \\
\end{array}
\!\right)&=&\displaystyle{\frac{1}{\sqrt{1+t_1\bar t_1}}}
\left(\!
\begin{array}{cc}
1 & -\bar t_1 \\
t_1 & 1 \\
\end{array}
\!\right)
\left(\!
\begin{array}{cc}
e^{ \,i(\psi+\varphi)} & 0 \\
0 & e^{- i(\psi+\varphi)} \\
\end{array}
\!\right)
, \\ [10pt]
\left(\!
\begin{array}{cc}
v_1^+ & v_1^- \\
v_2^+ & v_2^- \\
\end{array}
\!\right)&=&\displaystyle{\frac{1}{\sqrt{1+t_2\bar t_2}}}
\left(\!
\begin{array}{cc}
1 & -\bar t_2 \\
t_2 & 1 \\
\end{array}
\!\right)
\left(\!
\begin{array}{cc}
e^{ \,i(\psi-\varphi)} & 0 \\
0 & e^{- i(\psi-\varphi)} \\
\end{array}
\!\right)
,
\end{array}
\end{equation}
the transformation \p{uv-u1} is represented by the phase shift $\psi\to\psi+\alpha$.
Moreover, due to the fact that the field $\Phi^{(2s)}(u^\pm, v^\pm)$ has a fixed $\mathrm{U}(1)$-charge equal to $2s$,
its dependence on the phase variable $\psi$ is factorized:
\begin{equation}
\label{field-fact}
\Phi^{(2s)}(u^\pm, v^\pm)=e^{2si\psi}\hat\Phi(t_1,t_2,\bar t_1,\bar t_2,\varphi) \,.
\end{equation}
The field $\hat\Phi(t_1,t_2,\bar t_1,\bar t_2,\varphi)$ on the
right-hand side of equality \p{field-fact} is the function on the
coset $[\mathrm{SU}(2)\,{\otimes}\,\mathrm{SU}(2)]/\mathrm{U}(1)$
where the variable $\psi$ is the coordinate of the stability
subgroup $\mathrm{U}(1)$. Thus, the field $\Phi^{(2s)}(u^\pm,
v^\pm)$ having a fixed $\mathrm{U}(1)$-charge is in a one-to-one
correspondence with the function on the coset space
$[\mathrm{SU}(2)\,{\otimes}\,\mathrm{SU}(2)]/\mathrm{U}(1)$
\cite{GIKOS,GIOS}\footnote{Various aspects of functions on such a
coset are discussed in \cite{Kuz}.}. For this reason, we may refer
to the variables $u_i^\pm$, $v_{\underline{i}}^\pm$ used here as the
$[\mathrm{SU}(2)\,{\otimes}\,\mathrm{SU}(2)]/\mathrm{U}(1)$
harmonics. Since the variables $u_i^\pm$, $v_{\underline{i}}^\pm$
consist of twice the number of harmonics used in \cite{GIKOS,GIOS},
the space parameterized by these four $\mathrm{SU}(2)$ spinors can
be called the bi-harmonic space.

Note that a slightly different type of the bi-harmonic space was
previously used in the study of various supersymmetric models. For
example, two types of harmonics were employed in \cite{IvSut} for
constructing an off-shell superfield formulation of the $2D, (4,4)$
sigma-model. In those papers, the harmonics were used to parameterize
the coset space $\mathrm{SU}_L(2)/\mathrm{U}_L(1) \otimes
\mathrm{SU}(2)_R/\mathrm{U}_R(1)$, where the spaces
$\mathrm{SU}_L(2)/\mathrm{U}_L(1)$ and
$\mathrm{SU}(2)_R/\mathrm{U}_R( 1) $ were associated with the harmonics
$u^{\pm,0}$ and $v^{0,\pm}$, respectively, having the charges of
different $\mathrm{U}(1)$ groups. As a result, the fields on this
harmonic coset have two $\mathrm{U}(1)$ charges, which are defined
as eigenvalues of the operators $D^{0}_u$ and $D^{0}_v$. On the
other hand, in the case considered here, the field is defined by
equation \p{constr-J3b}, where the only $\mathrm{U}(1)$ charge $2s$
of the field $\Phi^{(2s)}(u^\pm, v^\pm)$ is given as the eigenvalue
of the $\mathrm{U}(1)$ generator $D^{0}=D^{0}_u +D^{0}_v$. Besides,
the $\mathrm{U}(1)$ charges $(\pm)$ of the two pairs of harmonics
$u^\pm$, $v^\pm$ coincide unlike the harmonics in \cite{IvSut}. As
discussed above, this means that the field $\Phi^{(2s)}(u^\pm,
v^\pm)$ of a special type is defined on bi-harmonic space where the
coordinates $u^\pm$ and $v^\pm$ parameterize the coset
$[\mathrm{SU}(2)\,{\otimes}\,\mathrm{SU}(2)]/\mathrm{U}(1)$, as
shown in \p{u1-field}. Another type of bi-harmonics was used, e.g. in
\cite{BII}, to describe the effective actions ${\cal N}=4$ SYM theory
(see the details in \cite{BII} and the references therein).

Now we describe the general solution to equations
\p{constr-J-b} and \p{constr-J3b}.

First, we note that any function of the variables
\begin{equation}
\label{y-+}
y_{i\underline{j}}:=u_{i}^+ v_{\underline{j}}^- -  u_{i}^- v_{\underline{j}}^+
\end{equation}
satisfies eqs. \p{constr-J-b}, \p{constr-J3b} for $s=0$.

Then, in general, the field $\Phi^{(2s)}(u^\pm, v^\pm)$,
obeying equations \p{constr-J-b}, \p{constr-J3b} for $2s\in
\mathbb{Z}{\geq}0$, is written in the form
\begin{equation}
\label{field-gen++}
\Phi^{(2s)}(u^\pm, v^\pm) \ = \ \sum_{r=0}^\infty
\Phi^{(2s)}_{k(r)\,\underline{l}(r)}(u^+, v^+) \,y^{k(r)\,\underline{l}(r)} \,,
\end{equation}
where
\begin{equation}
\label{field-gen++a}
\Phi^{(2s)}_{k(r)\,\underline{l}(r)}(u^+, v^+) \ = \ \sum_{\substack{p, q=0, \\ p+q=2s }}^{2s}
\phi_{k(r)\,\underline{l}(r)}^{i(p)\, \underline{j}(q)} u_{i(p)}^+ v_{\underline{j}(q)}^+  \,.
\end{equation}
Expressions \p{field-gen++} and \p{field-gen++a} use the following concise notation for the monomials:
\begin{equation}
\label{y-n}
u_{i(r)}^+:=u_{i_1}^+\ldots u_{i_r}^+ \,,\qquad
v_{\underline{i}(r)}^+:=v_{\underline{i}{}_1}^+\ldots v_{\underline{i}{}_r}^+ \,,\qquad
y^{i(r)\underline{j}(r)}:= y^{i_1\underline{j}{}_1}\ldots y^{i_r\underline{j}{}_r}\,,
\end{equation}
and we use the standard convention $y^{i\underline{j}}=\epsilon^{ik}\epsilon^{\underline{j}\underline{l}}y_{k\underline{l}}$ for raising and lowering the $\mathrm{SU}(2)$ indices.

The field $\Phi^{(2s)}(u^\pm,v^\pm)$ in \p{field-gen++}
that satisfies  \p{constr-J-b} and \p{constr-J3b}
is a linear combination with the constant coefficients $\phi_{k(r)\,\underline{l}(r)}^{i(p)\, \underline{j}(q)}$
of an infinite number of basis states
$u_{i(p)}^+ v_{\underline{j}(q)}^+y_{k(r)\,\underline{l}(r)}$.
The corresponding combinations of these basis vectors allow us to define the space for the irreducible infinite spin $\mathfrak{iso}(1,5)$ representation in the light-cone frame.

As a solution of the irreducibility condition \p{C6-Cas},
we chose one of the $2s{+}1$ possible vectors in the space of the $\mathfrak{su}(2)$ irreps with spin $s$,
namely, we took the higher weight vector $|\Psi^{(2s)}\rangle$ defined
by conditions \p{constr-J-a}, \p{constr-J3a}.
This choice does not lead to loss of generality.
The remaining $2s$ vectors are obtained from this vector $|\Psi^{(2s)}\rangle$
by acting of the operator $(J_-)^k$ at $k=1,\ldots,2s$.
In the representation \p{Ja-ab3}, \p{constr-J-b} and \p{constr-J3b} the fields $\Phi^{(2s-2k)}$
are obtained by the action of the operator $(D^{--})^k$ on the field $\Phi^{(2s)}$:
$\Phi^{(2s-2k)}=(D^{--})^k\Phi^{(2s)}$.
Note that the action of $(D^{--})^k$ decreases the degree of the polynomial $\Phi^{(2s)}_{k(r)\,\underline{l}(r)}(u^+, v^+)$ \p{field-gen++a}
in the variables $(u^+,v^+)$ and increases it in the variables $(u^-,v^-)$.
Choosing any other fields $\Phi^{(2s-2k)}$, $k=1,\ldots,2s$ leads to the equivalent
infinite spin representations of $\mathfrak{iso}(1,5)$.

\setcounter{equation}0
\section{Field theory in the light-front coordinates}

In the previous section, we have developed a description of the irreducible $6D$ infinite spin
representations in the light-cone frame and shown that this description is formulated in terms
of fields in bi-harmonic space. Now we extend this analysis to the light-front coordinate system and construct the
corresponding field theory.

The formulation of the field theory on the light-front was proposed
by Dirac \cite{Dirac49}, its further development and applications
were considered by many authors (see e.g.
\cite{Bengtsson2Brink,Bengtsson2Linden,Metsaev05,PoSk} and the
references therein). The light-front is defined as the surface
$x^+=const$ in the six-dimensional Minkowski space $\mathbb{R}^{1,5}$.
It means that the coordinate $x^+$ is interpreted as a "time"
evolution parameter. Therefore, the role of the Hamiltonian in the case
under consideration is played by the operator
\begin{equation}
\label{P-H}
H=P^-\,.
\end{equation}

To define an infinite spin field in the light-front coordinates, we
will use the results of the previous section, where the
corresponding field is given by \p{field-gen++}, \p{field-gen++a}.
Note that the light-cone coordinate system is obtained from the
light-front coordinate system by vanishing the coordinates $x^{\hat
a}$ and fixing the coordinates $x^\pm$. Therefore, it is natural to
assume that the infinite spin field in the light-front coordinates
should have the form \p{field-gen++}, \p{field-gen++a}, where,
however, the coefficients $\phi_{k(r)\,\underline{l}(r)}^{i(p)
\underline{i}(q)}$ are functions of $x^\pm$ and $x^{\hat a}$. As a
result, the irreducible infinite spin field depending on the light-front
coordinates is defined as
\begin{equation}
\label{field-gen++1}
\Phi^{(2s)}(x^\pm,x^{\hat a},u^\pm, v^\pm)\ = \ \sum_{\substack{p, q=0, \\ p+q=2s }}^{2s} \sum_{r=0}^\infty
\phi_{k(r)\,\underline{l}(r)}^{i(p) \underline{i}(q)}(x^\pm,x^{\hat a}) \, u_{i(p)}^+ v_{\underline{j}(q)}^+ \,y^{k(r)\,\underline{l}(r)}\,.
\end{equation}
Taking into account a general principle of the light-cone dynamics
\cite{Dirac49}, one concludes that the equation of motion for the
field \p{field-gen++1} is the Schr\"{o}dinger-type equation
\begin{equation}
\label{Schr-eq}
\left(-i\frac{\partial}{\partial x^+}-H \right)\Phi^{(2s)}(x^\pm,x^{\hat a},u^\pm, v^\pm)=0 \,,
\end{equation}
where the coordinate $x^+$ plays the role of time.

As usual, the generators $P_{a}$ and $M_{ab}$ of $\mathfrak{iso}(1,5)$ in
the the light-front formulation are divided into kinematic and dynamic generators. One can show that
these divisions in the field realization \p{field-gen++1} have the form
\begin{itemize}
\item
Kinematic generators
\begin{equation}
\label{P-kin}
P^+=p^+\,,\qquad P^{\hat a}=p^{\hat a} \,,
\end{equation}
\begin{equation}
\label{M-kin}
M^{\hat a \hat b}=x^{\hat b}p^{\hat a} -x^{\hat a}p^{\hat b}+ S^{\hat a \hat b}\,,\qquad
M^{+\hat a}=x^{\hat a}p^+ + S^{+\hat a}\,,\qquad
M^{+-}=x^{-}p^+ + S^{+-} \,;
\end{equation}
\item
Dynamic generators
\begin{equation}
\label{P-dyn}
P^-=\frac{p^{\hat a}p^{\hat a}}{2p^+}=H\,,
\end{equation}
\begin{equation}
\label{M-dyn}
M^{-\hat a}=x^{\hat a}H -x^{-}p^{\hat a} + S^{-\hat a}\,,
\end{equation}
\end{itemize}
where
\begin{equation}
\label{P-dif}
p^{\hat a}=i\frac{\partial}{\partial x^{\hat a}}\,,\qquad p^{+}=-i\frac{\partial}{\partial x^{-}}
\end{equation}
and all spin parts of the Lorentz rotation generators  $S^{ab}=(S^{\hat a\hat b},S^{\pm\hat a},S^{+-})$
depend on the spinors  $u_i^\pm$ and $v_{\underline{i}}^\pm$ in the same way as the operators
\p{Mpm-eta-summ}, \p{Si-pm-ch}, \p{Si-pm-ch-1}, \p{Si-pm-ch-2}
depend on the spinors  $\xi_i^I$ and $\rho_{\underline{i}}^I$
\begin{eqnarray}
\label{Mab-lc}
S^{\hat a\hat b}&=& \frac{1}{2}\,\eta^{\mathrm{i}}_{\hat a\hat b}
\left[\frac{\partial}{\partial u_i^+} (\tau_{\mathrm{\,i}})_i{}^j u_j^+
+ \frac{\partial}{\partial u_i^-} (\tau_{\mathrm{\,i}})_i{}^j u_j^- \right]
+\frac{1}{2}\,\bar\eta^{\mathrm{i}^\prime}_{\hat a\hat b}
\left[\frac{\partial}{\partial v_{\underline{i}}^+} (\tau_{\mathrm{\,i}^\prime})_{\underline{i}}{}^{\underline{j}}v_{\underline{j}}^+
+\frac{\partial}{\partial v_{\underline{i}}^-} (\tau_{\mathrm{\,i}^\prime})_{\underline{i}}{}^{\underline{j}}v_{\underline{j}}^-\right] ,
\\ [5pt]
\label{M+a-lc}
S^{+\hat a}&=&
\frac{i}{\sqrt{2}}\left[
\frac{\partial}{\partial u_i^+}(\tau_{\hat a})_{i}{}^{\underline{j}}\epsilon_{\underline{j}\underline{k}}\frac{\partial}{\partial v_{\underline{k}}^-} -\frac{\partial}{\partial u_i^-}(\tau_{\hat a})_{i}{}^{\underline{j}}\epsilon_{\underline{j}\underline{k}}\frac{\partial}{\partial v_{\underline{k}}^+}\right],\ \ \ \mbox{at} \ \ \hat a=1,2,3\,,
\\ [5pt]
\label{M-a-lc}
S^{-\hat a}&=&
\frac{i}{\sqrt{2}}\left[v_{\underline{k}}^-\epsilon^{\underline{k}\underline{i}}(\tau_{\hat a})_{\underline{i}}{}^{j}u_{j}^+
-v_{\underline{k}}^+\epsilon^{\underline{k}\underline{i}}(\tau_{\hat a})_{\underline{i}}{}^{j}u_{j}^-\right]
,\ \ \ \mbox{at} \ \ \hat a=1,2,3\,,
\\ [5pt]
\label{M+4-lc}
S^{+4}&=&
\frac{1}{\sqrt{2}}\left[
\frac{\partial}{\partial u_i^+}\epsilon_{i\underline{k}}\frac{\partial}{\partial v_{\underline{k}}^-} -\frac{\partial}{\partial u_i^-}\epsilon_{i\underline{k}}\frac{\partial}{\partial v_{\underline{k}}^+}\right],
\\ [5pt]
\label{M-4-lc}
S^{-4}&=&
\frac{1}{\sqrt{2}}\left[v_{\underline{k}}^-\epsilon^{\underline{k}i}u_{i}^+
-v_{\underline{k}}^+\epsilon^{\underline{k}i}u_{i}^-\right],\qquad
\\ [5pt]
\label{M+--lc}
S^{+-}&=&
\frac{1}{2}
\left[u_i^+ \frac{\partial}{\partial u_i^+}
+ u_i^- \frac{\partial}{\partial u_i^-} +
v_{\underline{i}}^+ \frac{\partial}{\partial v_{\underline{i}}^+}
+v_{\underline{i}}^- \frac{\partial}{\partial v_{\underline{i}}^-} +4\right].
\end{eqnarray}
Turn attention that all the generators $S^{ab}=(S^{\hat a\hat b},S^{\pm\hat a},S^{+-})$
defined in \p{Mab-lc}-\p{M+--lc} have zero  $\mathrm{U}(1)$-charge.

After acting by the operator $p^+$ on equation \p{Schr-eq}, this equation takes the form\footnote{For a discussion of the operator $p^+$ invertibility,  see e.g. \cite{Sieg88,Sieg99}.}
\begin{equation}
\label{Dal-eq}
\Box\,\Phi^{(2s)}(x^\pm,x^{\hat a},u^\pm, v^\pm)=0 \,,
\end{equation}
where $\Box$ is the d'Alambertian operator in the six-dimensional Minkowski space in the light-front coordinates
\begin{equation}
\label{Dal-op}
\Box \ := \ 2\frac{\partial}{\partial x^+}\frac{\partial}{\partial x^-}-\frac{\partial}{\partial x^{\hat a}}\frac{\partial}{\partial x^{\hat a}} \,.
\end{equation}
Equation \p{Dal-eq} is the equation of motion corresponding to the action
\begin{equation}
\label{act-lc}
S \ = \ \int d^{\,6}x\, du\, dv\ \bar\Phi^{(-2s)}\Box\,\Phi^{(2s)} \,,
\end{equation}
where $d^{\,6}x = dx^+ dx^- d^{\,4}x$ is the $6D$ Minkowski space measure and  $du dv$
is the bi-harmonic space measure \cite{GIKOS,GIOS}. The function $\bar\Phi^{(-2s)}$ is obtained by complex conjugation of
the function $\Phi^{(2s)}$:
\begin{equation}
\label{cc-field}
\bar\Phi^{(-2s)} \ = \ (\Phi^{(2s)})^* \,.
\end{equation}
Integration over harmonics is defined by simple rules (see \cite{GIKOS,GIOS} for details).
The integral is is a linear operation and it does not vanish only for $\mathrm{SU}(2)$-scalars with
the following normalization condition:
\begin{equation}
\label{int-harm-0}
\int du = 1\,,\qquad  \int dv = 1\,.
\end{equation}
For all other harmonic monomials, the harmonic integral is equal to zero:
\begin{equation}
\label{int-harm-1}
\int du \, u^+_{(i_1}\ldots u^+_{i_m}u^-_{j_1}\ldots u^-_{j_n)}= 0\,,\qquad
\int dv \, v^+_{(\underline{i}_1}\ldots v^+_{\underline{i}_m}v^-_{\underline{j}_1}\ldots v^-_{\underline{j}_n)}= 0\,,
\end{equation}
at arbitrary integers $m$ and $n$ which are not equal to zero simultaneously.

Reality conditions \p{u-real} are now written as follows:
\begin{equation}
\label{harm-conj}
(u^{i\pm})^*=\pm\, u_i^\mp\,,\qquad  (v^{i\pm})^*=\pm\, v_i^\mp\,.
\end{equation}
Therefore, at complex conjugation the charge $2s$ of the harmonic field $\Phi^{(2s)}$ changes to
$-2s$ in accordance with \p{cc-field}. As a result, the integrand in \p{act-lc} has a zero harmonic charge
as it should be for the non-vanishing harmonic integral.\footnote{Note that in the harmonic superspace approach to
${\cal N}=2$ supersymmetric field theories \cite{GIKOS,GIOS} another rule of conjugation was used that
combines complex conjugation with an antipodal map. However, in the case under consideration, the
ordinary complex conjugation is totally appropriate.}

In expansion of the harmonic field \p{field-gen++1} the indices $i$ and $\underline{i}$ of the
component fields
$\phi^{(i(m)j(n))(\underline{i}(k)\underline{j}(n))}(x)$ are half of the $6D$
$\mathrm{SU}^*(4)$-indices $\alpha$. It means, for half-integer $s$, the harmonic field
$\Phi^{(2s)}(x,u^\pm,v^\pm)$ is an odd order polynomial in $u^\pm$, $v^\pm$  and describes half-integer spin fields with an odd number of indices.
Therefore, the fermionic fields should be endowed by the corresponding odd statistics. Besides, in the
fermionic case, the natural Lagrangian is the one of the first order in space derivatives. This type of Lagrangian in
the light-front formalism is obtained from the Lagrangian \p{act-lc} by replacement
\begin{equation}
\label{Psi-Phi}
\Psi^{(2s)} \ = \ \sqrt{p^+}\,\Phi^{(2s)} \,.
\end{equation}
Then, for the field $\Psi^{(2s)}$ expression \p{act-lc} leads to the following Lagrangian:\footnote{For discussion of the light-front describing the fields with different statistics see e.g.
\cite{Sieg88,Sieg99}.}
\begin{equation}
\label{La-lc}
\bar{\Psi}^{(-2s)}\left(p^- - H \right) \Psi^{(2s)} \,.
\end{equation}

Thus, the action \p{act-lc} determines the field dynamics of
infinite spin fields on the light front. A specific feature of the
obtained theory is its formulation in terms of harmonic variables.

\setcounter{equation}0
\section{Summary}

We have developed the $6D$ Minkowski space infinite spin free
Lagrangian field theory in the light-cone formalism. First, we have
studied this theory in the light-cone frame and unexpectedly found
that the corresponding infinite spin field is a function on a
special bi-harmonic space associated with the coset
$[\mathrm{SU}(2)\,{\otimes}\,\mathrm{SU}(2)]/\mathrm{U}(1)$. Second,
the result obtained was generalized to the light-front coordinate
system, where the infinite spin field is described by the function
$\Phi^{(2s)}(x^\pm,x^{\hat a},u^\pm, v^\pm)$ \p{field-gen++1}
depending on the light-front coordinates and harmonics. Representations
of all the $6D$ Poincar\'e group generators in this coordinate
system are constructed. The field equation of motion in the light-front
coordinate system has the form of Schr\"{o}dinger-type equation
\p{Schr-eq} with the Hamiltonian \p{P-H}. The corresponding action
is given by \p{act-lc}.

The harmonic light-front approach formulated in this paper opens a
possibility to construct an interacting theory for $6D$ infinite
spin fields. One can expect that introducing an interaction will
lead to a modification of the dynamic generators \p{P-dyn}, \p{M-dyn}
by the interaction terms (see the description of interactions in
the light-front formalism, e.g., in \cite{PoSk} and the references
therein). In particular, the Hamiltonian \p{P-dyn} should go to
\begin{equation}
\label{P-P-int}
H\quad \rightarrow \quad H+H_{\mathrm{int}}\,.
\end{equation}
The harmonic formalism allows one from the very beginning to make some
simple predictions on the structure of the interacting Hamiltonian
$H_{\mathrm{int}}$. To preserve zero harmonic charge of the action,
this Hamiltonian should have zero harmonic charge as well. It
immediately means that an arbitrary order self-interaction of the same harmonic fields
$\Phi^{(2s)}$ is possible only for $s=0$ if other charged
harmonic quantities in the action are absent.
Self-interaction of charged fields $\Phi^{(2s)}$, $s\neq0$ can only be of an even order, such as
$\,\sim \bar{\Phi}^{(-2s)}\bar{\Phi}^{(-2s)}{\Phi}^{(2s)}{\Phi}^{(2s)}$.
Although for fields with different charges there is an additional choice in the structure of the interaction Lagrangian.
For example, the following interacting terms
$\,\sim \bar{\Phi}_1^{(-2s)}\left({\Phi}_2^{(0)}+\bar{\Phi}_2^{(0)}\right){\Phi}_1^{(2s)}$
or
$\,\sim\!\left({\Phi}_1^{(q_1)}{\Phi}_2^{(q_2)}{\Phi}_3^{(q_3)}+c.c.\right)$ at $q_1+q_2+q_3=0$
are
allowed in the action. In general, the requirement of zero charge of
interacting contributions to the action controls both charges of
interacting fields and their number. We plan to construct
interacting infinite spin $6D$ theories in the forthcoming works.

We also think that the appearance of bi-harmonic space in the
infinite spin representations of $\mathfrak{iso}(1,5)$ can indicate
the existence of manifest $\mathcal{N}{=}\,(1,0)$
supersymmetrization of the theory \p{act-lc},
\p{P-P-int}.\footnote{See, e.g., harmonic superfield formulation of the
six-dimensional $\mathcal{N}{=}\,(1,0)$ and $\mathcal{N}{=}\,(1,1)$
supersymmetric theories in \cite{BIS} and references therein.}

\section*{Acknowledgements}
The authors are grateful to E.A.\,Ivanov for discussing the aspects
of the harmonic formalism. The work of ILB and SAF is supported by
the Russian Science Foundation, project No 21-12-00129. The work of API was
partially supported by the Ministry of Education of the Russian
Federation, project FEWF-2020-0003.

\section*{Appendix A. Calculation of the Casimir operator $C_6$}
\def\theequation{A.\arabic{equation}}
\setcounter{equation}0

\quad\,

The six-order Casimir operator in the $6D$ massless theory is given by \p{C6-Casimir}. The derivation of this
operator in \cite{BFIP} was based on the relations:
\begin{equation}
\label{C6-Casimir-def}
C_6=\frac{1}{64}\, \Upsilon^a\Upsilon_a\,,\qquad
\Upsilon_a=\varepsilon_{abcdef}P^{b}  M^{cd} M^{ef}
\end{equation}
These expressions show that only the spin part $S_{ab}$ \p{M-1} of the Lorentz group generators
$M_{ab}$ \p{M-gen} contributes to the Casimir operator \p{C6-Casimir-def}.
However, if we substitute $S_{ab}$ for $M_{ab}$ into \p{C6-Casimir},
an incorrect result is obtained, since when passing from \p{C6-Casimir-def} to \p{C6-Casimir},
one has to rearrange the operators $ P_{c}$ and $M_{ab} $ using commutators.
At the replacement $M_{ab}\to S_{ab}$, a correct expression $C_6$ can be obtained only after preliminary "untangling" of the generators
$P_{c}$ и $M_{ab}$ in expression \p{C6-Casimir}.

We will act in the following way. First, we rearrange with the help
of commutation relations all the operators $P_{c}$ to the right on all the operators $M_{ab}$ in
expression \p{C6-Casimir}. Second, after such an ordering is done, we replace the operator $M_{ab}$ by
the operator $S_{ab}$ in the obtained expression.

Using the commutator $[M_{ab},\Pi_{c}]=i\left(\eta_{ac}\Pi_{b}-\eta_{bc}\Pi_{a}\right)$, we rearrange
the operators $\Pi_{a}$ to the right in the first term of expression \p{C6-Casimir}. All the terms
proportional to $2\Pi_{[a} \Pi_{b]}=[\Pi_a, \Pi_b] = -i \, M_{ab} \, P^2 $ can be omitted for a
massless representation where $P^2=0$. Besides, since we consider irreducible infinite spin
for which the condition \p{C4-const} holds,
we replace the operator $C_4$ by its eigenvalue $-\mu^2$ in the second term of \p{C6-Casimir}.
Now all operators $M_{ab}$ to the left of the operators $\Pi_a$ are replaced by the operators $S_{ab}$.
As a result, one obtains
\begin{equation}
\label{C6-Casimir-1}
C_6=
-\, S^{(b}{}_{a}\, S^{c)a}\Pi_b \Pi_c
\ - \ \frac12\,\mu^2\,\Big(S^{bc}S_{bc}-8\Big) \,.
\end{equation}

Using the relations for the $\sigma$-matrices from \cite{BFI-21}, the identity
\begin{equation}
\label{sigma-id}
(\tilde\sigma^b{}_a)^\alpha{}_\beta (\tilde\sigma^{ca})^\gamma{}_\delta
= \frac14\,\eta^{bc}\left(\delta^\alpha_\beta \delta^\gamma_\delta -2 \delta^\alpha_\delta \delta^\gamma_\beta\right)
+\frac12 (\tilde\sigma^{(b})^{\alpha\gamma} (\sigma^{c)})_{\beta\delta}
+\frac12\,\delta^\alpha_\delta (\tilde\sigma^{bc})^\gamma{}_\beta
- \frac12\,\delta^\gamma_\beta (\tilde\sigma^{bc})^\alpha{}_\delta\,,
\end{equation}
the commutation relations \p{com-sp} and the realization \p{M-1} for the operators $S_{ab}$, one gets the equality
\begin{equation}
\label{S2-exp1}
S^{(b}{}_aS^{\,c)a}
= \eta^{bc}\left[\frac12\,(\xi_{(I}\rho_{K)})(\xi^{I}\rho^{K})
+i (\xi^{I}\rho_I)\right]
+\frac12\,\xi_\alpha^I \rho^\beta_I \xi_\gamma^J \rho^\delta_J
(\tilde\sigma^{(b})^{\alpha\gamma}(\sigma^{c)})_{\beta\delta}\,,
\end{equation}
which leads to
\begin{equation}
\label{S2-exp2}
S^{bc}S_{bc}
= - \frac12\,(\xi^{I}\rho_I)^2
+4i (\xi^{I}\rho_I)
+2 (\xi_{(I}\rho_{K)})(\xi^{I}\rho^{K})\,,
\end{equation}
where the notation $(\xi^{I}\rho_{K}):=\xi_\alpha^{I}\rho^\alpha_{K}$ has been used.
After substituting  \p{S2-exp1} and  \p{S2-exp2} into \p{C6-Casimir-1}, one gets
\begin{equation}
\label{C6-Casimir-2}
C_6=
\mu^2\left[\frac14\,(\xi^{I}\rho_I)^2-\frac12\,(\xi_{(I}\rho_{J)})(\xi^{I}\rho^{J})
-i (\xi^{I}\rho_I)+4\right]
-\frac12\,\xi_\alpha^I \rho^\beta_I \xi_\gamma^J \rho^\delta_J
(\tilde\sigma^{b})^{\alpha\gamma}(\sigma^{c})_{\beta\delta}\Pi_b \Pi_c \,.
\end{equation}
The last term in this expression is represented in the following form:
\begin{equation}
\label{2-C6-Casimir-2}
-\frac12\,\xi_\alpha^I \rho^\beta_I \xi_\gamma^J \rho^\delta_J
(\tilde\sigma^{b})^{\alpha\gamma}(\sigma^{c})_{\beta\delta}\Pi_b \Pi_c =
-\frac{i}{2}\,\mu^2 (\xi^{I}\rho_I)
+\frac14\, (\xi^I\tilde\sigma^b\xi_I)(\rho_J\sigma^c\rho^J)\Pi_b \Pi_c\,,
\end{equation}
where
$(\xi^I\tilde\sigma^b\xi_I):=\xi_\alpha^I(\tilde\sigma^b)^{\alpha\beta}\xi_{\beta I}$,
$(\rho_J\sigma^c\rho^J):=\rho^\alpha_J(\sigma^c)_{\alpha\beta}\rho^{\beta J}$.

The last step in deriving the expression for $C_6$ is to move the operators $P_m$ to the right in expression \p{2-C6-Casimir-2}.
Using the equality $\Pi_a = M_{ba}\, P^{b} - 5 i \, P_a$, we write the expression  $\Pi_b \Pi_c$ in the form:
\begin{equation}
\label{Pi-2}
\Pi_b \Pi_c=M_{eb}M_{fc}P^eP^f-6iM_{eb}P^eP_c -5iM_{ec}P^eP_b-30P_bP_c\,.
\end{equation}
Now that all operators $M_{ab}$ on the right side of \p{Pi-2} are to the left of all operators $P_c$, we replace the operators $M_{ab}$ with their spin parts $S_{ab}$.

After such a replacement $M_{ab}\to S_{ab}$, where $S_{ab}$ are defined in \p{M-1},  the substitution \p{Pi-2} into \p{2-C6-Casimir-2} and \p{C6-Casimir-2} and using the equalities
\begin{eqnarray}
\label{c-a}
\frac{1}{4}\,(\xi^I\tilde\sigma^b\xi_I)(\rho_J\sigma^c\rho^J)S_{eb}S_{fc}P^e P^f &=&
\mu^2\left[-\frac14\,(\xi^{I}\rho_I)^2+2i (\xi^{I}\rho_I)-6\right], \\ [6pt]
\label{c-b}
-\frac{3i}{2}\,(\xi^I\tilde\sigma^b\xi_I)(\rho_J\sigma^c\rho^J)S_{eb}P^e P_c &=&
\mu^2\left[-3i (\xi^{I}\rho_I)+12\right], \\ [6pt]
\label{c-c}
-\frac{5i}{4}\,(\xi^I\tilde\sigma^b\xi_I)(\rho_J\sigma^c\rho^J)S_{ec}P^e P_b &=&
\mu^2\left[\frac{5i}{2} (\xi^{I}\rho_I)+20\right], \\ [6pt]
\label{c-d}
-\frac{15}{2}\,(\xi^I\tilde\sigma^b\xi_I)(\rho_J\sigma^c\rho^J) P_b P_c &=&  -30\mu^2\,,
\end{eqnarray}
which are valid at $P^2=0$ and $(\xi^I\tilde\sigma^b\xi_I)(\rho_J\sigma^c\rho^J)P_b P_c=4\mu^2$
(second equality is the condition \p{C4-const} for the fourth-order Casimir operator), one obtains
\begin{equation}
\label{C6-Casimir-fin}
C_6= -\frac12\,\mu^2(\xi_{(I}\rho_{J)})(\xi^{I}\rho^{J}) \,.
\end{equation}
When using the operators \p{Ta-def}, this final expression \p{C6-Casimir-fin} is represented as
\begin{equation}
\label{C6-Casimir-fin1}
C_6= -\mu^2J_{\mathrm{i}}J_{\mathrm{i}}\,,
\end{equation}
which is the same as \p{C6-Cas}.

\section*{Appendix B. Spinor part of the $\mathfrak{so}(1,5)$-generators}
\def\theequation{B.\arabic{equation}}
\setcounter{equation}0

\quad\,

We consider a representation where the $(4{\times}4)$ $\sigma$-matrices \cite{KuTow,IR,BFI-21}
$\sigma^a=\| (\sigma^a)_{\alpha\dot\beta}\|$,
$\tilde\sigma^a=\| (\tilde\sigma^a)^{\dot\alpha\beta}\|$ with the $6D$ vector index $a=0,1,\ldots,5$
and spinor indices  $\alpha,\dot\alpha=1\dots,4$ are realized in the form of the following matrices:
\begin{equation}
\label{sigma-a}
\sigma^a=(\sigma^0,\sigma^{\hat a},\sigma^5)\,,\qquad \tilde\sigma^a=(\sigma^0,-\sigma^{\hat a},-\sigma^5)\,,\qquad \hat a=1,\ldots,4\,,
\end{equation}
where
\begin{equation}
\label{sigma-a-r}
\sigma^0=1_4\,, \qquad \sigma^{\hat a}=\tau_2\otimes \tau_{\hat a} \ \ \mbox{at} \ \ \hat a=1,2,3\,,
\qquad \sigma^4=-\tau_1\otimes 1_2\,,\qquad
\sigma^5=\tau_3\otimes 1_2
\end{equation}
and $\tau_{1,2,3}$ are the Pauli matrices.\footnote{ In \cite{KuTow,BFI-21} the
representation $\sigma^1=\tau_1\otimes 1_2$, $\sigma^{\hat a}=
\tau_2\otimes \tau_{\hat a-1}$ at $\hat a=2,3,4$ and $\sigma^0$, $\sigma^5$ was used as in \p{sigma-a-r}.
That is, distinction between the representations \cite{KuTow,BFI-21} and  \p{sigma-a-r} lies in the difference
in the notation of the four space coordinates labeled by $\hat a$. However, the representation
\p{sigma-a-r} is more convenient when using the standard realizations \p{eta} for the 't\,Hooft symbols.}
The antisymmetric $\sigma$-matrices with non-dotted spinor subscripts and superscripts are defined as follows:
\begin{equation}
\label{sigma-m}
(\sigma^{a})_{\alpha\beta}=(\sigma^{a})_{\alpha\dot\gamma}(B^{-1})_\beta{}^{\dot\gamma}\,,\qquad
(\tilde\sigma^{a})^{\alpha\beta}=B_{\dot\gamma}{}^\alpha(\tilde\sigma^{a})^{\dot\gamma\beta}\,,
\end{equation}
where
$B=\| B_{\dot\alpha}{}^{\beta}\|=1_2\otimes i\tau_{2}=
\left(
\begin{array}{cc}
i\tau_2 & 0 \\
0 & i\tau_2 \\
\end{array}
\right)
$
is the matrix defining complex conjugation of the $6D$ Weyl spinors \cite{KuTow,IR,BFI-21}. In particular,  the
matrices $\sigma^\pm=(\sigma^0\pm\sigma^5)/\sqrt{2}$,
$\tilde\sigma^\pm=(\tilde\sigma^0\pm\tilde\sigma^5)/\sqrt{2}$ with undotted indices  have the form:
\begin{equation}
\label{sigma+-}
(\sigma^{+})_{\alpha\beta}=
\sqrt{2}\left(\!
\begin{array}{cc}
i\tau_2 & 0 \\
0 & 0 \\
\end{array}
\!\right)=
\left(\!
\begin{array}{cc}
\sqrt{2}\epsilon_{ij} & 0 \\
0 & 0 \\
\end{array}
\!\right),\quad
(\sigma^{-})_{\alpha\beta}=
\sqrt{2}\left(\!
\begin{array}{cc}
0 & 0 \\
0 & i\tau_2 \\
\end{array}
\!\right)=
\left(\!
\begin{array}{cc}
0 & 0 \\
0 & \sqrt{2}\epsilon_{\underline{i}\underline{j}} \\
\end{array}
\!\right),
\end{equation}
\begin{equation}
\label{t-sigma+-}
(\tilde\sigma^{+})^{\alpha\beta}=
\sqrt{2}\left(\!
\begin{array}{cc}
0 & 0 \\
0 & -i\tau_2 \\
\end{array}
\!\right)=
\left(\!
\begin{array}{cc}
0 & 0 \\
0 & \sqrt{2}\epsilon^{\underline{i}\underline{j}} \\
\end{array}
\!\right),\quad
(\tilde\sigma^{-})^{\alpha\beta}=
\sqrt{2}\left(\!
\begin{array}{cc}
-i\tau_2 & 0 \\
0 & 0 \\
\end{array}
\!\right)=
\left(\!
\begin{array}{cc}
\sqrt{2}\epsilon^{ij} & 0 \\
0 & 0 \\
\end{array}
\!\right).
\end{equation}
Also we use the standard representation for the 't\,Hooft symbols
$\eta^{\mathrm{i}}_{\hat a\hat b}=-\eta^{\mathrm{i}}_{\hat b\hat a}$,
$\mathrm{i}=1,2,3$ and $\bar\eta^{\mathrm{i}^\prime}_{\hat a\hat b}=
-\bar\eta^{\mathrm{i}^\prime}_{\hat b\hat a}$, $\mathrm{i}^\prime=1,2,3$
(see e.g. \cite{tHooft,IR1,BFIP})
\begin{equation}
\label{eta}
\eta^{\mathrm{i}}_{\hat a\hat b}=
\left\{
\begin{array}{l}
\epsilon_{\mathrm{i}\hat a\hat b} \qquad \hat a,\hat b=1,2,3,\\
\,\delta_{\mathrm{i}\hat a} \qquad \,\hat b=4,
\end{array}
\right.
\qquad\quad
\bar\eta^{\mathrm{i}^\prime}_{\hat a\hat b}=
\left\{
\begin{array}{l}
\,\,\epsilon_{\mathrm{i}^\prime \hat a\hat b} \qquad \hat a,\hat b=1,2,3,\\
-\delta_{\mathrm{i}^\prime \hat a} \qquad \hat b=4.
\end{array}
\right.
\end{equation}

First, we will consider the $\mathfrak{so}(4)$-part of the generators \p{M-1}, i.e. the operators
\begin{equation}
\label{M-1-a}
S_{\hat a\hat b} = \xi_\alpha^I (\tilde\sigma_{\hat a\hat b})^{\alpha}{}_{\beta}\rho^\beta_I\,,\ \ \ {\hat a}=1,2,3,4\,.
\end{equation}
These six generators  $S_{\hat a\hat b}$ are written as the sum
\begin{equation}
\label{M-Mpm}
S_{\hat a\hat b}=S_{\hat a\hat b}^{(+)} + S_{\hat a\hat b}^{(-)}\,,
\end{equation}
where the $\mathrm{SO}(4)$-(anti-)self-dual parts
$
S^{(\pm)}_{\hat a\hat b} = \pm{\displaystyle \frac12}\,\epsilon_{\hat a\hat b\hat c\hat d}S^{(\pm)}_{\hat c\hat d}
$
are expressed in terms of the $\mathrm{SO}(3)$-vectors $S^{(+)}_{\mathrm{\,i}}$, $S^{(-)}_{\mathrm{\,i}^\prime}$:
\begin{equation}
\label{Mpm-eta}
S^{(+)}_{\hat a\hat b} = -\eta^{\mathrm{i}}_{\hat a\hat b}S^{(+)}_{\mathrm{\,i}}\,,\qquad
S^{(-)}_{\hat a\hat b} = -\bar\eta^{\mathrm{i}^\prime}_{\hat a\hat b}S^{(-)}_{\mathrm{\,i}^\prime}\,,
\end{equation}
if we use the 't\,Hooft symbols \p{eta}. Thus, the generator \p{M-1-a} has the expansion
\begin{equation}
\label{Mpm-eta-summ}
S_{\hat a\hat b}= -\eta^{\mathrm{i}}_{\hat a\hat b}S^{(+)}_{\mathrm{\,i}}
-\bar\eta^{\mathrm{i}^\prime}_{\hat a\hat b}S^{(-)}_{\mathrm{\,i}^\prime}\,,
\end{equation}
where the operators $S^{(+)}_{\mathrm{\,i}}$ и $S^{(-)}_{\mathrm{\,i}^\prime}$ form two $\mathfrak{su}(2)$ algebras:
\begin{equation}
\label{M-M-2}
[S^{(+)}_{\mathrm{\,i}},S^{(+)}_{\mathrm{\,j}}]=i\epsilon_{\mathrm{i}\mathrm{j}\mathrm{k}}S^{(+)}_{\mathrm{\,k}}\,,\qquad
[S^{(-)}_{\mathrm{\,i}^\prime},S^{(-)}_{\mathrm{\,j}^\prime}]=
i\epsilon_{\mathrm{i}^\prime\mathrm{j}^\prime\mathrm{k}^\prime}S^{(-)}_{\mathrm{\,k}^\prime}\,,\qquad
[S^{(+)}_{\mathrm{\,i}},S^{(-)}_{\mathrm{\,j}^\prime}]=0\,.
\end{equation}

Using the equalities
$\eta^{\mathrm{i}}_{ab}\eta^{\mathrm{j}}_{ab}=4\delta^{\mathrm{i}\mathrm{j}}$,
$\bar\eta^{\mathrm{i}^\prime}_{ab}\bar\eta^{\mathrm{j}^\prime}_{ab}=
4\delta^{\mathrm{i}^\prime\mathrm{j}^\prime}$ and
$\eta^{\mathrm{i}}_{ab}\bar\eta^{\mathrm{j}^\prime}_{ab}=0$,
one finds the inverse to \p{Mpm-eta} relations
\begin{equation}
\label{Mpm-eta-inv}
S^{(+)}_{\mathrm{\,i}} = -\frac14\,\eta^{\mathrm{i}}_{\hat a\hat b}S_{\hat a\hat b}\,,\qquad
S^{(-)}_{\mathrm{\,i}^\prime} = -\frac14\,\bar\eta^{\mathrm{i}^\prime}_{\hat a\hat b}S_{\hat a\hat b}\,.
\end{equation}
However, using \p{sigma-a}, \p{sigma-a-r}, \p{sigma-m} and \p{eta} we obtain that the matrices present in the definition of the generators \p{Mpm-eta-inv} have only one diagonal $2{\times}2$ matrix block:
\begin{equation}
\label{eta-sigma1}
-\frac14\,\eta^{\mathrm{i}}_{\hat a\hat b}(\tilde\sigma_{\hat a\hat b})^{\alpha}{}_{\beta}=
\frac{i}{2}\left(
\begin{array}{cc}
-\tau^T_{\mathrm{\,i}} & 0 \\
0 & 0 \\
\end{array}
\right)
\,,\qquad
-\frac14\,\bar\eta^{\mathrm{i}^\prime}_{\hat a\hat b}(\tilde\sigma_{\hat a\hat b})^{\alpha}{}_{\beta}=
\frac{i}{2}\left(
\begin{array}{cc}
0 & 0 \\
0 & -\tau^T_{\mathrm{\,i}^\prime} \\
\end{array}
\right)\,.
\end{equation}
Substituting \p{M-1-a} and \p{eta-sigma1} into \p{Mpm-eta-inv}, one finds
\begin{equation}
\label{Si-pm-ch}
S^{(+)}_{\mathrm{\,i}} = -\frac{i}{2}\,\rho^i_I (\tau_{\mathrm{\,i}})_i{}^j\xi_j^I\,,\qquad
S^{(-)}_{\mathrm{\,i}^\prime} = -\frac{i}{2}\,\rho^{\underline{i}}_I (\tau_{\mathrm{\,i}^\prime})_{\underline{i}}{}^{\underline{j}}\xi_{\underline{j}}^I\,.
\end{equation}
Thus, the generators $S^{(+)}_{\mathrm{\,i}}$ are built using the canonical pairs $(\xi_i^I,\rho^j_J)$ from \p{pi-2-2},
while the generators $S^{(-)}_{\mathrm{\,i}^\prime}$ are built using the other canonical pairs $(\xi_{\underline{i}}^I,\rho^{\underline{j}}_J)$.

Now using the matrix expressions
\begin{equation}
\label{eta-sigma2}
(\tilde\sigma^{+\hat a})^{\alpha}{}_{\beta}=
\frac{i}{\sqrt{2}}\,
\left(
\begin{array}{cc}
0 & 0 \\
-\tau_{\hat a}^T & 0 \\
\end{array}
\right)
\,,\quad
(\tilde\sigma^{-\hat a})^{\alpha}{}_{\beta}=
\frac{i}{\sqrt{2}}\,
\left(
\begin{array}{cc}
0 & \tau_{\hat a}^T \\
0 & 0 \\
\end{array}
\right)\,,\ \ \mbox{at} \ \ \hat a=1,2,3\,,
\end{equation}
\begin{equation}
\label{eta-sigma3}
(\tilde\sigma^{+4})^{\alpha}{}_{\beta}=
\frac{-1}{\sqrt{2}}\,
\left(
\begin{array}{cc}
0 & 0 \\
1_2 & 0 \\
\end{array}
\right)
\,,\quad
(\tilde\sigma^{-4})^{\alpha}{}_{\beta}=
\frac{-1}{\sqrt{2}}\,
\left(
\begin{array}{cc}
0 & 1_2 \\
0 & 0 \\
\end{array}
\right)\,,\quad
(\tilde\sigma^{+-})^{\alpha}{}_{\beta}=
\frac{1}{2}\,
\left(
\begin{array}{cc}
-1_2 & 0 \\
0 & 1_2 \\
\end{array}
\right)
\end{equation}
obtained from \p{sigma-a}, \p{sigma-a-r}, \p{sigma-m}, and expansion \p{pi-2-2}, we find the spin part of the remaining
Lorentz group generators \p{M-1}:
\begin{equation}
\label{Si-pm-ch-1}
S^{+\hat a}=
-\frac{i}{\sqrt{2}}\,\rho^i_I(\tau_{\hat a})_{i}{}^{\underline{j}}\xi_{\underline{j}}^I
\,,\quad
S^{-\hat a}=
\frac{i}{\sqrt{2}}\,\rho^{\underline{i}}_I(\tau_{\hat a})_{\underline{i}}{}^{j}\xi_{j}^I
\,,\ \ \mbox{at} \ \ \hat a=1,2,3\,,
\end{equation}
\begin{equation}
\label{Si-pm-ch-2}
S^{+4}=
-\frac{1}{\sqrt{2}}\,
\rho^i_I \delta_{i}{}^{\underline{j}}\xi_{\underline{j}}^I
\,,\qquad
S^{-4}=
-\frac{1}{\sqrt{2}}\,
\rho^{\underline{i}}_I \delta_{\underline{i}}{}^{j}\xi_{j}^I\,,\qquad
S^{+-}=
-\frac{1}{2}\left(\xi_{i}^I \rho^{i}_I -  \xi_{\underline{i}}^I \rho^{\underline{i}}_I\right)\,.
\end{equation}

The found expressions \p{Mpm-eta-summ}, \p{Si-pm-ch}, \p{Si-pm-ch-1}, \p{Si-pm-ch-2} are used in Sect.\,4 to construct
the spin part of the Lorentz algebra generators in the bi-harmonic space.

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A.S.\,Galperin, E.A.\,Ivanov, V.I.\,Ogievetsky, E.S.\,Sokatchev,
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\end{document}

redpajama_set_name: RedPajamaArXiv


Miss Portharcourt City 2017 heritage edition was a huge success as 14 contestants competed for the coveted crown on the 1st of December 2017 at the L.A Kings Dome Stadium Road (inside l.a kings).
In attendance were high profiled personalities as the Q.L.O.P.H.A (Quintessential Ladies Of Portharcourt City Awards) also took place alongside the beauty pageant. Some of the guest were Hon Ibim Semenitari, Mrs Loveth Ogedegbe, Mrs Mercy Abu, Mrs Izobe Spiff, Muma Gee, Arinze Baba, K.O Baba and a host of other distinguished Portharcourt Personalities.
Miss Oroma Shedrack, a 22 years old undergraduate of Business Management at the University of Portharcourt defeated 13 other contestants to emerge Miss Portharcourt City 2017/2018. Other crowned Queens were Cherrie Miller Face of Portharcourt City, Desire Suku Diboye Top Model Portharcourt City, Beauty Stanley Miss Portharcourt City (Tourism) Wariboko Otokini Miss Portharcourt City (humanity).


text:
Submitted by oscarluke on November 15, 2012 - 12:30am
Challenges to National Policy: Who Will Be in Charge of Carbon Trading? Webinar 10.22.2008 12:00am
The introduction of proposed cap and trade regulation has been premised on the addition of CO2 costs to the decision-making processes of emitters, so that improved control and innovation will result. This is under scrutiny in the EU and the US. New models are likely to emerge by 2009 built around cap & trade modified, cap & dividends, technology and R&D, and carbon fees.
The "cap and trade" methodology embraces use of existing commercial tools, and is expected to be implemented through a network of commercial and financial trading transactions. For it to succeed politically and in the marketplace, it is important that the regulation of its operations be transparent and legally comprehensible to lawyers for execution, implementation and enforcement purposes.
The commercial and regulatory issues are made more complex by several emerging developments: new oversight institutions have been proposed with regulatory roles in the cap & trade arena and new environmental regulatory roles for existing agencies i.e. EPA, FERC; existing institutions with jurisdiction over aspects of commodities, securities, energy trading, and finance have yet to weigh-in i.e. SEC, CFTC, Federal Reserve and Treasury; current national financial and lending difficulties may lead to new financial regulatory mechanisms relating to carbon trade, either domestically or in connection with international trade. Permeating all of these developments is a concern that there not be unintended consequences of the new regulatory regime, as experienced under electric deregulation (e.g. "Enron"), the specter of concurrent jurisdiction among federal agencies and departments and overlapping jurisdiction with the states.
Other Federal agencies as well as Congressional and state regulatory bodies are reviewing the issue, holding hearings, and suggesting actions focused on the provision of consumer protection, transparency and adequacy of disclosure, and enforcement protections. Options may broaden to include carbon fees, innovative R&D and renewable portfolio standards.
To analyze these emerging regulatory issues systematically, and provide a foundation for guidance as to the implications of both the shaping of GHG regulation and the sorting out of other commercial responsibilities and their impacts on renewables, this October Webinar scoped matters through three basic perspectives: provision of overall regulatory /legislative policy with respect to national cap and trade; administration of regulatory enforcement; and practical operation of trading operations from a commercial standpoint.
Moderated by
Williams Mullen attorney, Robert F. Riley
Please visit the ACORE website or contact abaquestions@acore.org
Robert F. Riley
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redpajama_set_name: RedPajamaCommonCrawl


Tomlin allowed one earned run on four hits and no walks while striking out five across 4.2 innings Thursday against the Royals. He did not factor into the decision.
Tomlin pitched on three days rest and was likely to be limited, so it came as little surprise that he was allowed to throw only 74 pitches despite the effective outing. The lone run he surrendered came on a solo home run by Adalberto Mondesi in the third inning. His regular season is likely over with a 6.14 ERA and 1.48 WHIP across 70.1 innings.


As the father of a growing girl I am interested in the world of child safety seats for the first time ever. I actually already own one but I can't work out how to make the blooming thing work.
No one told me that this could be a problem. I kind of assumed that you strapped the wriggling little lady into the seat, handed her a plastic dinosaur and drove about without any hassles. Now it turns out that I can't seem to connect the seat to the car. And I don't even have a plastic dinosaur to give her.
It seems I am not alone though. I had kind of hoped that other people had suffered from the same kind of problem but my tentative attempts to bring up the subject never led me anywhere.
Then I got a stroke of good luck yesterday when I found out the type of news which always makes me feel a bit better; it is someone else's fault. I love it when I realise that my own failings and incompetence are beaten by another person's and here we have a classic case of someone being sillier than me.
A study by those nice people at Which? has revealed that a lot of High Street shops give out misleading and even dangerous advice to parents. Mystery shoppers were out and about asking about child seats in Mothercare, John Lewis, Mamas & Papas, Babies R Us and Halfords.
The results make for pretty bad reading, as the shop staff missed out important questions like what model of car the seat is to be used in. In fact, none of the different chains managed to give the right advice in more than 2 of their stores.
I might have the odd rant about other things but this is a serious business. The big name shops need to give parents the right advice to keep their kids safe. Now, if they want to make it up to me and keep me happy then they can send me through a free plastic dinosaur if they like.
This entry was posted in News and Views and tagged car, ford, High Street, It Wasn, John Lewis, LS, Mamas Papas, MoT, seat by Robert Bell. Bookmark the permalink.


text:
Places to pump, park and change a diaper are scarce at Purdue Northwest, student moms say
By Meredith Colias-Pete
Oct 21, 2018 | 5:00 AM
Ayana Clark laughs as she speaks with other mothers about the importance of college campuses providing amenities to parents on Wednesday, Oct. 10, 2018. (Kyle Telechan/Post Tribune)
Halfway into her pregnancy, Julia Cook-Jones remembers walking into a bathroom at Purdue Northwest during summer school and discovered it had no changing stations.
Becoming pregnant at 21 was unexpected, Cook-Jones, now 22, said. As the youngest of five children, only she had attended college more than a few weeks.
"It was a lot of pressure, because everybody was living through me," said Cook-Jones, of Hammond, a communications major and Morton High School graduate.
As she began to look around the Hammond campus, amenities that could easily help pregnant students or new parents seemed to be lacking, she said.
After she had her son, Jace, in November, she began to raise the issue with other parents. She wrote suggestions on comments forms to a Dean of Students, she said.
Few bathrooms appeared to have a changing stations. There were no maternity parking spots that could help heavily pregnant women get to class, she said.
Purdue had set aside a lactation room on the third floor of the library/student union. Kept locked, it was across the hall from a theater and also used at times as a dressing room — raising privacy concerns for women wanting to pump breast milk, she said.
The Riley Child Care Center's closure on-campus in June 2017 also had an impact, Cook-Jones said. While open to the general public, Purdue announced its closure, citing dropping enrollment and stating it had been losing money for years.
"I felt like I was running out of (options)," she said.
Purdue Northwest in Hammond has expanded in recent years to include resident halls, a growing number of international students and a new $40 million Bioscience building set to open in 2020.
Yet, Cook-Jones and other mothers said the university can do more to provide simple fixes to help young parents rooted in the Region working to get an education.
Purdue Northwest student Julia Cook-Jones smiles as her son, Jace, takes a bite of a donut on Wednesday, Oct. 10, 2018. (Kyle Telechan/Post-Tribune)
College addressing concerns
Purdue Northwest spokesman Doug Clark said the university takes steps to provide services to pregnant students over the minimum federal Title IX requirements. Parking requests are usually determined on an individual basis, he said.
"Even though our enrollment has been trending toward traditional-aged students, we continue to be sensitive and supportive of students who come to PNW with a variety of life circumstances to achieve their academic potential," he said via email.
"While Title IX is more explicit in protecting the rights of pregnant students," he said, "Purdue Northwest has gone beyond the minimal requirements in providing advocacy and education for parenting students as well.
"Through the Dean of Students Office, Purdue Northwest established a Student Support Advocacy Program in 2015. Two full-time staff positions, one at each campus location, assist students with a variety of personal and life situations, including pregnancy and parenting."
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Parking problems
When Ayana Clark's first pregnancy progressed in 2015, parking was the issue. As a broadcast major, she also was determined to finish her degree.
"I feel like a lot of moms fall into that trap to where once you have a kid, your life is over," Clark, now 24, said.
As her belly grew, her classes fell in the afternoon when parking was nearly full. During the winter, pregnant women can slip on ice, she said.
Her solution was to park in the campus' 15-minute parking spaces, then fight the tickets.
"When I would go contest them, I was like, 'Look, I'm pregnant, I'm not going to walk all the way (from the back lot) and they would throw them out," she said.
"We shouldn't have to go through that," Clark said. "There's motorcycle parking."
Her son, now 3, often comes with her when she works as a teaching assistant. She is now a media criticism graduate student and expects to finish in May.
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Pregnant again, she is also concerned about how she will be able to have guaranteed privacy in the library's third floor room if she needs to pump breast milk.
Occasionally, theater students are there "taking a nap and doing their makeup," Clark said.
"That's the closest thing to pumping rooms that we have," she said. "You would have to go to a classroom and you have to hope that nobody comes in there."
Aliya Adams, left, sits with her daughter Ameah, 3, as she and other mothers gather to speak about the importance of college campuses providing amenities to parents on Wednesday, Oct. 10, 2018. (Kyle Telechan/Post-Tribune)
Changing child on dorm bed
Aliya Adams found out she was pregnant at 17, while still a student at Morton High School.
Worried about how to manage a baby and college, Adams, of Hammond, had a full scholarship and still signed up for classes. Her daughter, Ameah, was 4 months old when she started, she said.
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"At that time, she was still waking up in the middle of the night," Adams, now 21, said. "I'd have her in my arms the entire time I was studying."
Initially enthusiastic about bringing Ameah while she got involved in campus life, it became less practical as she couldn't find a place to change her diapers.
Adams said she would sometimes go to one of the dorms and would change her on a friend's bed.
Cook-Jones noted Purdue Northwest's Westville campus, with two-thirds as many students, continues to have a on-campus day care.
The Lions Cubs Child Care facility is only available to children of Westville students and faculty. It is funded through campus fees.
Hoping to help parents like her, she intends to start an informational support services group on the Hammond campus, she said.
An Indiana University Northwest spokeswoman said via email that campus once had an on-site day care that was closed several years ago.


Series: Foundation Stones, Dr. Bob Clanton finishes up the series, Foundation Stones.
Series: Foundation Stones, Dr. Bob Clanton continues he Foundation Stones series with the second message entitled, "The River."
Series: Foundation Stones, Dr. Bob Clanton begins the new series, Foundation Stones. The first in a series of three messages is. The Rock" and discusses the importance and benefit of regularly reading the Bible.


The instructional program serves as an integral and vital component of health physical education co-curricular activities in the Arlington Public Schools. The program teaches safety awareness, develops basic swimming skills, and enhances proficiency levels of all aquatic skills. It also improves personal and community safety skills and helps students develop a commitment to lifetime fitness.
Students in third and fourth grade receive five hours of swimming and water safety instruction during the school year. Ninth and tenth graders spend three to four week of their physical education time at the swimming pool, and participate in swimming, water safety, small craft safety, fitness, and aquatic recreation. All students learn swimming and water safety skills that are appropriate to their proficiency level and that enhance their ability to participate fully and safely in aquatic activities. The program includes fitness swimming, small craft and water recreation, and competition. The American Red Cross Learn-to-Swim program provides the framework for our swimming and water safety curriculum, our instructors are certified and authorized by the program co-sponsor, Arlington County Chapter of the American Red Cross.
Arlington Secondary School students may also participate in interscholastic swimming and diving. Middle school students receive instruction and coaching in swimming and diving and engage in athletic competition among the five APS schools. High school students participate in regional athletic competition as part of the Virginia High School League (VHSL). Yorktown High School Girls Swim Team won the title of District Champions in 2005-06 and 2006-07. The Athletic Program provides opportunities for students to demonstrate their swimming and diving proficiency through competitive events as well as an opportunity for students to excel and experience success, while promoting and applying concepts of team work, sportsmanship, respect for others, and high standards of conduct and honor.
For more information, please contact the Aquatics Management Office at 703-228-6263.


text:
China says ex-Internet czar on trial over corruption charges
FILE - In this Sept. 23, 2015, file photo, Chinese President Xi Jinping, center, talks with Facebook Chief Executive Mark Zuckerberg, right, as Lu Wei, left, China's Internet czar, looks on during a gathering of CEOs and other executives at Microsoft's main campus in Redmond, Wash. Lu was standing trial Friday, Oct. 19, 2018 on corruption allegations, state media reported. (AP Photo/Ted S. Warren, Pool, File)
FILE - In this Sept. 23, 2015, file photo, China's Internet czar Lu Wei attends a gathering of CEOs and other executives at Microsoft's main campus in Redmond, Wash. Lu was standing trial Friday, Oct. 19, 2018 on corruption allegations, state media reported. (AP Photo/Ted S. Warren, Pool, File)
In this Sept. 13, 2015, photo, Lu Wei, then deputy head of the Propaganda Department of the Communist Party of China, speaks at a forum in Nanning in southern China's Guangxi Zhuang Autonomous Region. China's former internet censor, who once held high-profile meetings with industry leaders such as Apple chief executive Tim Cook and Facebook founder Mark Zuckerberg, was standing trial Friday, Oct. 19, 2018 on corruption allegations, state media reported. (Chinatopix via AP)
In this image taken from video, Lu Wei, former minister of China's Cyberspace Administration, speaks during his trial in the city of Ningbo in eastern China's Zhejiang province, Friday, Oct. 19, 2018. China's former internet censor, who once held high-profile meetings with industry leaders such as Apple chief executive Tim Cook and Facebook founder Mark Zuckerberg, stood trial on Friday on corruption allegations. Chinese headlines at the bottom reads: "Obey sentencing, Will not appeal." (CCTV via AP)
In this image made from video, Lu Wei, fourth from left, former minister of China's Cyberspace Administration, is escorted by police officers to his trial in the city of Ningbo in eastern China's Zhejiang province, Friday, Oct. 19, 2018. China's former internet censor, who once held high-profile meetings with industry leaders such as Apple chief executive Tim Cook and Facebook founder Mark Zuckerberg, stood trial on Friday on corruption allegations. (CCTV via AP)
BEIJING — China's former internet censor, who once held high-profile meetings with industry leaders such as Apple CEO Tim Cook and Facebook founder Mark Zuckerberg, was standing trial Friday on corruption allegations, state media reported.
Lu Wei is accused of accepting 32 million yuan ($4.6 million) in bribes and has admitted to his crimes and expressed remorse, according to state broadcaster CCTV. It said his trial began Friday in the eastern city of Ningbo.
China has 700 million people online, but it heavily censors content, especially that of a political nature, along with sites related to gambling, drugs and pornography.
Lu was responsible for leading the government's efforts to tighten control over domestic cyberspace and championing the party's position that governments have a right to filter and censor their countries' internet.
He took a hard line in demanding tough security checks on imported foreign tech products and keeping out foreign internet companies and social networks like Facebook in the name of preserving social stability.
Lu worked his way up the ranks of China's official Xinhua News Agency from a reporting job in the southern city of Guilin in the early 1990s to becoming the agency's vice president from 2004 to 2011. He was vice mayor of Beijing from 2011 to 2013.
While CCTV's report gave few details, previous accusations against Lu said he engaged in factionalism and "selectively enforced" the party's "strategic arrangements on internet work," in addition to being corrupt and abusing his powers.
Chinese President Xi Jinping has carried out a wide-ranging crackdown on corruption that observers say is also calibrated to bring down political opponents.
Among officials and military leaders recently ensnared is the former president of Interpol, Meng Hongwei, who vanished after traveling to China late last month from France, where the global anti-crime body is headquartered.
Days later, China said Meng was under investigation for graft and possibly other crimes, although there are heavy suspicions he had fallen out of political favor with Xi. His wife, Grace Meng, said she received a threatening phone call from a man speaking Chinese and is under police protection in France.
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text:
Adult Animation
Black Cinema
Cine en Espanol
Gay Cinema
Mystery / Thriller
Sports / Special Interest
Superheroes / Comics
24FRAMES
by Edythe Smith
by Aaron Sausedo
ARIZONA PHOSSILS
by Adam Payne
CASHIER DU CINEMA
by Robbie Ikegami
CINEMA SUICIDE
by Seamus Smith
CINEMANOME
by Chris Morris
COFFEE, CIGARETTES AND CINEMA
by Paul Losada
DICK LAURENT IS DEAD
by Travis King
EDDIE'S BLOG
by Eddie Vegas
ERIC'S BLOG
by Eric Branscum
FILM BORBOR
by Aiah Samba
FRANKLIN'S BLOG
by Franklin Lei
GIRLWONDER'S PICKS
by Grace Bartlett
ICONIC FRIGHTS
by Rob Galluzzo
JOSE'S BLOG
by Jose Palafox
KAMAN'S MYSTERIOUS JONES
by Jessica Kaman
KINOPTICON
by Joey Izzo
LAWS ON FILM
by Zach Laws
by Jonah Rust
by Mark Ayala
by Amoebite
ORSON AROUND
by Jed Leland
PHOTOPLAYS WE FANCY
by Eric Brightwell
by Joey Jenkins
READING THE SUB-TEXAN
by Charles Reece
SHAWN'S BLOG
by Shawn Scarborough
by Eric Kench
SUNDRY ARTICLES
by Gillian Horvat
SWEENEYRULES
by Sean Sweeney
TIFFANY'S TALES
by Tiffany Huang
WILL RECOMMENDS
by Will B
Archived Entries
BERNIE CASEY (1)
BIG DEAL ON MADONNA STREET (1)
BOOGIE NIGHTS (4)
CALIFORNIA SPLIT (1)
CYNTHIA GIBB (1)
EVENING SHADE (1)
HARLEY COKELISS (1)
HENRY GIBSON (1)
HIGH PLAINS DRIFTER (5)
IT'S A MAD MAD MAD MAD WORLD (3)
KEITH CARRADINE (3)
LILY TOMLIN (1)
MCCABE & MRS. MILLER (1)
MEAN STREETS (3)
MYSTERY / THRILLER (54)
MYSTERY/THRILLER (3)
NED BEATTY (6)
OCEAN'S ELEVEN (1)
OSCARS (160)
PALE RIDER (1)
POLICE WOMAN (1)
PRIMAL FEAR (1)
QUINTET (1)
RACHEL WARD (1)
RENT-A-COP (1)
REVENGE OF THE NERDS (2)
RONEE BLAKLEY (1)
THE LONGEST YARD (7)
THE NUDE BOMB! (1)
THE OUTLAW JOSEY WALES (2)
THE PARENT TRAP (2)
THE PLAYER (1)
THE RUSSIANS ARE COMING THE RUSSIANS ARE COMING (1)
THE THORN BIRDS (1)
TOOTSIE (2)
Dir: Paul Thomas Anderson, 1997. Starring: M. Wahlberg, J. Moore, B. Reynolds, D. Cheadle, J. C. Reilly, W. H. Macy. Drama.
From the opening sounds of sad circus music flowing into disco, you feel you are in for something unique. As the camera tracks across a street into a bustling nightclub, introducing us to a large array of characters in one take, you know you are in for one hell of a spectacle...
Boogie Nights is an epic tail about life in the swinging seventies through the lens of the porno industry of Southern California. It explores the transition of the business into the 1980s, where film was switched out for video, and the roof caved in for many. But it's not simply a story of the sex trade—it's about family. Although somewhat warped, the group of porn stars connect together as if they were brothers and sisters, mothers and fathers.
Seamus Smith
Dir: Harley Cokeliss, 1987. Starring: Burt Reynolds, Scott Wilson, Cynthia Gibb, Cliff Robertson. Action/Adventure.
At one time Burt Reynolds was a megastar. By 1987 the shine was thinning, as his hair piece was thickening. Just two years after his arch '70s box office king rival Clint Eastwood made his obligatory quasi-Shane remake Pale Rider (the least of the four Westerns Clint directed), Reynolds did his own less blatant variation of Shane, with an equally simple title: Malone. Updated, instead of an old-timey oater, Malone is more of an '80s muscle film. This could have been a by-the-numbers vehicle for any number of steroidy non-actors of the day; the ace up the sleeve here is the lovely British Columbia backdrop and Reynolds' considerable charm. Even when he seems to be barely trying he's much more likable then most of the action stars of the period. Back in his salad days (the '70s) Reynolds starred in two bona-fide classics, Deliverance and The Longest Yard, and had a massive box office hit with Smokey and The Bandit. But by the '80s--though Reynolds was still a very popular personality--none of the vehicles really matched his talent. Looking back years later, as the smoke has cleared, Malone is probably his most entertaining film of the decade.
Burt plays Malone, an ex-CIA hitman trying to escape his past (is there any other kind?) and make a break from his sexy handler (Lauren Hutton). After hitting the open road, his car ends up breaking down in a small mountain town where he befriends a clean cut gas station owning family, the Barlows. Father and his teenage daughter (Scott Wilson and Cynthia Gibb) take him in and luckily for them Malone also happens to be handy with a wrench for fixing engines. It turns out Malone happened to show up in the knick of time, as the town is being bought up by a nasty rich guy, Delany ( the alway dependable Cliff Robertson). The Barlows won't sell, even under pressure from the town's corrupt cops (lead by Kenneth McMillan) and Delany's own band of thugs, plus the usual suspects of '80s B-creeps--including many familiar faces such as Tracy Walter (Batman) and Dennis Burkley (the lovable big mute biker from Mask). The bad guys try to exert muscle and Burt kills a couple of them. Shane had that famous tree trunk digging scene, but Burt doesn't exert much sweat. Although he does get shot, giving Hutton a chance to come back to bandage him up and have a quickie romance before the creeps kill her. This leads to an all out war as Malone is forced to use his considerable killing skills to take out the security team and finally have it out with Delany. And then, very abruptly, it ends. But this was the '80s. Who wanted to sit through a movie much longer then ninety minutes? We had to get to the arcade.
Sean Sweeney
Dir: Robert Altman, 1975. Starring: Lily Tomlin, Keith Carradine, Ronee Blakley, Ned Beatty, Henry Gibson. Drama.
Even casual film historians know that the 1970s was the decade with the most creative freedom afforded to the director. Just as studios were beginning to become just pieces of larger corporate empires and the blockbuster became the only goal, filmmakers were given unprecedented access to seeing out their visions. No director took advantage of the era as unusually as Robert Altman managed to. After exploding as a brand name director with his huge hit MASH in '70 he spent the decade exploring a plethora of film quirks, with such notable titles as McCabe & Mrs. Miller, The Long Goodbye and California Split, as well as a number of oddities and misfires, ending the decade with the utterly unwatchable sci-fi bomb Quintet. But Altman's greatest masterpiece (with apologies to MASH and The Player) came in the middle of the decade: Nashville, a film that truly stands alone as one of those films that could never be repeated (and still proves very challenging to even write about) and, in the end, is the most Altman-y film Altman ever made.
Clocking in at 159 minutes, Nashville is a sorta satire, but also a real tribute to country music. The film takes place during a political rally for the Replacement Party presidential candidate that coincides with a number of musicians coming to town to record and play at the rally. With over twenty main characters coming and going, it's almost impossible to keep up with on a first viewing. The standout story lines start with Lily Tomlin as Linnea (outstanding in her first film), a gospel singer and mother to a pair of deaf kids, and her husband (Ned Beatty), a political operative for a campaign operator (Michael Murphy) who is putting together a fundraiser at Opryland. Meanwhile, country legend Haven Hamilton (the always entertaining Henry Gibson) is sought after by both the politicians, after he records a tribute to the bicentennial ("we must be doing something right, to last 200 years") and a fish-outta-water British journalist (Geraldine Chaplin) who has an affair with his son. Another country music star, the very damaged Barbara Jean (Ronee Sue Blakley, who then was known more as a singer, but proves herself as an actress wonderfully here) seems to be having a nervous breakdown and is followed by a lurking uniformed Vietnam vet (Scott Glenn). Up-and-coming singer Tom (Keith Carradine) has all the women chasing him, including a spaced out groupie (Shelly Duvall), but he appears to make a real connection with married mother Linnea. And that's just a taste of the story lines, which also includes a motley crew of characters giving fully lived-in performances, including Keenan Wynn, Gwen Welles, Barbara Baxley, Barbara Harris, Bert Remsen, Karen Black, Jeff Goldblum, Allen Garfield and cameos by Elliott Gould and Julie Christie as themselves. It's almost like a hee haw version of It's a Mad, Mad, Mad, Mad World.
Feb 3, 2017 1:03pm
Sharky's Machine
Dir: Burt Reynolds, 1981. Starring: Burt Reynolds, Rachel Ward, Bernie Casey, Charles Durning, Brian Keith. Mystery/Thriller.
The third and the best of the four movies Burt Reynolds directed and starred in, Sharky's Machine is often written off as Burt's attempt at a Dirty Harry like franchise starter since he and Clint Eastwood were often linked as rival '70s macho mega-stars. But where Clint would go on to reinvent himself as an awards bait elder statesman of economical directing, this would unfortunately be Burt's last memorable movie as a major leading man. (Of course, sixteen years later he would score his only Oscar nomination for his great supporting performance in Boogie Nights). Sharky's Machine now feels more reminiscent of '70s Italian crime flicks known as Poliziotteschi films than it does Dirty Harry, as these films often dealt with dirty and violent cops in the seedier side of politics, organized crime and prostitution. As Sharky, it's one of those rare, less winky performances from Reynolds. Though he can't help but ooze charm, he also creates a sometimes unlikable character as the film veers fairly effortlessly from rowdy Joseph Wambaugh type police station mayhem picture to a Rear Window inspired erotic thriller to a very gripping final confrontation.
Based on a book by the gritty novelist William Diehl (Primal Fear), Burt actually took over direction when his Deliverance helmer John Boorman got stuck finishing Excalibur. Atlanta narcotics cop Tom Sharky is one of those plays-by-his-own-rules badasses who has been bumped downstairs to vice when a sting goes wrong. He's surrounded by a motley crew of great character actors including Bernie Casey (Revenge of The Nerds), Richard Libertini (Fletch), John Fiedler (12 Angry Men and the voice of Piglet in Winnie the Pooh), Brian Keith (The Russians Are Coming, the Russians Are Coming, the original The Parent Trap) and the great Charles Durning (Dog Day Afternoon, Tootsie, at his most "Durningist"). They are stuck busting hookers to make the town look presentable to a candidate for governor, Donald Hotchkins (Earl Holliman of TV's Police Woman). It turns out the politician is tied to a high-priced call girl ring led by the Italian sleazoid, Victor D'Anton (Vittorio Gassman, a major Italian actor whose resume before Sharky's Machine spanned from Big Deal on Madonna Street to the Get Smart movie, The Nude Bomb!), who is killing off his own women with a hitman played by the always reliably creepy Henry Silva (one of the original Ocean's Eleven!)
Mar 29, 2018 2:31pm


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