h_1 + 1\,, \qquad h_1 = |h_2|>1/2\,, \hspace{2.5cm} \hbox{ self-dual massive fields} \\ \label{13102014-16} && \hspace{-0.8cm} E_0 = h_1 + 2\,, \qquad h_1 > |h_2|>1/2\,, \hspace{2.5cm} \hbox{ mixed-symmetry massless fields} \\ \label{13102014-17} && \hspace{-0.8cm} E_0 > h_1 + 2\,, \qquad h_1 > |h_2|>1/2\,, \hspace{2.5cm} \hbox{ mixed-symmetry massive fields} \\ \label{13102014-18} && \hspace{-0.8cm} E_0 = h_1 + 2\,, \qquad h_1 > |h_2|,\quad h_2 = 0, \pm 1/2\,, \qquad \hbox{ totally symmetric massless fields}\qquad \\ \label{13102014-19} && \hspace{-0.8cm} E_0 > h_1 + 2\,, \qquad h_1 > |h_2|, \quad h_2=0,\pm 1/2\,, \qquad \hbox{ totally symmetric massive fields}\qquad \eeq Often, in place of the labels $E_0$, $h_1$, $h_2$, we use the respective labels $\kappa$, $j_1$, $j_2$ defined by the relations \be \label{13102014-18-x1} \kappa \equiv E_0 - 2\,, \qquad j_1 \equiv \half(h_1 + h_2)\,, \qquad j_2 \equiv \half (h_1-h_2)\,. \ee For fields in~\eqref{13102014-18},~\eqref{13102014-19}, light-cone gauge formulation was developed in refs.~\cite{Metsaev-ml-1999ui,Metsaev-ml-2003cu}.\footnote{In refs.~\cite{Metsaev-ml-1999ui,Metsaev-ml-2003cu}, light-cone gauge formulation was developed for totally symmetric fields in $AdS_{d+1}$ with arbitrary $d$, $d\geq 3$. Light-cone gauge formulation for fields in $AdS_3$ was investigated in ref.~\cite{Metsaev-ml-2000qb}.} Light-cone gauge actions in refs.~\cite{Metsaev-ml-1999ui,Metsaev-ml-2003cu} lead to \emph{coupled} equations of motion for fields in~\eqref{13102014-18},~\eqref{13102014-19}. Light-cone gauge actions leading to \emph{decoupled} equations of motion for fields in~\eqref{13102014-18},~\eqref{13102014-19} were obtained recently in ref.~\cite{Metsaev-ml-2013kaa}. For fields in~\eqref{13102014-15},~\eqref{13102014-16}, light-cone gauge formulation was developed in ref.~\cite{Metsaev-ml-2002vr}, while, for fields in~\eqref{13102014-17}, light-cone gauge formulation was studied in ref.~\cite{Metsaev-ml-2004ee}.\footnote{\looseness=-1 We note that it is light-cone gauge approach in refs.~\cite{Metsaev-ml-2002vr,Metsaev-ml-2004ee} that made it possible to develop a Lagrangian formulation of AdS fields in~\eqref{13102014-15}--\eqref{13102014-17} for the first time. Also, for the reader's convenience, we recall that, in Lorentz covariant approaches, the mixed-symmetry fields in~\eqref{13102014-15}--\eqref{13102014-17} are described by tensor (or tensor-spinor) fields whose $so(4,1)$ space-time tensor indices have the structure of the Young tableaux with two rows. Namely, mixed-symmetry bosonic fields are associated with Young tableaux having row lengths $h_1$, $|h_2|$, while mixed-symmetry fermionic fields are associated with Young tableaux having row lengths $h_1-\half$, $|h_2|- \half$. } For fields in~\eqref{13102014-15},~\eqref{13102014-16}, light-cone gauge actions in ref.~\cite{Metsaev-ml-2002vr} lead to \emph{decoupled} equations of motion, while, for fields in~\eqref{13102014-17}, light-cone gauge actions in ref.~\cite{Metsaev-ml-2004ee} lead to \emph{coupled} equations of motion. Two main aims of this paper are as follows. The first aim is to develop light-cone gauge formulation which leads to \emph{decoupled} equations of motion for mixed-symmetry massive fields~\eqref{13102014-17}. The second aim is to apply such light-cone formulation to study of AdS/CFT correspondence. In this paper, our primary interest are the massive mixed-symmetry fields. The massless and self-dual massive fields are realized as appropriate limits of the massive fields. This allows us to study a Lagrangian formulation and the AdS/CFT correspondence for the mixed-symmetry massive, mixed-symmetry massless, and self-dual massive fields on an equal footing. Before proceeding to main theme of this paper let us briefly review various Lorentz covariant approaches in the literature which have been developed for the description of mixed-symmetry fields in AdS space. At the level of equations of motion given in Lorentz gauge, massless and massive mixed-symmetry fields in $AdS_{d+1}$, with arbitrary $d$ were studied in refs.~\cite{Metsaev-ml-1995re,Metsaev-ml-2003cu,Metsaev-ml-1998xg}. On-shell BRST invariant formulation of massless and massive mixed-symmetry fields may be found in refs.~\cite{Alkalaev-ml-2009vm,Alkalaev-ml-2009vm.m001}. On-shell formulation of massless mixed-symmetry fields in terms of generalized connections of AdS symmetry algebras was studied in refs.~\cite{Boulanger-ml-2008up,Boulanger-ml-2008up.m001,Skvortsov-ml-2009zu,Skvortsov-ml-2009zu.m001}. Lagrangian frame-like description of massless and massive mixed-symmetry fields may be found in the respective refs.~\cite{Alkalaev-ml-2005kw,Alkalaev-ml-2005kw.m001} and refs.~\cite{Zinoviev-ml-2009gh,Zinoviev-ml-2009gh.m001}. Lagrangian metric-like formulation for some particular mixed-symmetry massless field is considered in ref.~\cite{Brink-ml-2000ag}. Maxwell-like Lagrangians for massless mixed-symmetry fields are considered in ref.~\cite{Campoleoni-ml-2012th}. Discussion of Lagrangian formulation of massive fields corresponding to two-column Young tableaux may be found in ref.~\cite{deMedeiros-ml-2003px}. BFV-BRST Lagrangian formulation of mixed-symmetry massive fields is discussed in refs.~\cite{Reshetnyak-ml-2010ga,Reshetnyak-ml-2010ga.m001}. In the framework of world-line spinning particle approach, the self-dual massive fields are studied in ref.~\cite{Bastianelli-ml-2014lia}. Interesting recent discussion of mixed-symmetry fields may be found in ref.~\cite{Vasiliev-ml-2012tv}. Interacting mixed-symmetry fields were studied in refs.~\cite{Alkalaev-ml-2010af,Alkalaev-ml-2010af.m001,Alkalaev-ml-2010af.m002,Alkalaev-ml-2010af.m003}.\footnote{\looseness=-1 Interacting mixed-symmetry fields in flat space were studied by using light-cone gauge approach in refs.~\cite{Metsaev-ml-1993mj}--\cite{Metsaev-ml-2007rn} and by BRST method in refs.~\cite{Koh-ml-1986vg}--\cite{Metsaev-ml-2012uy} (see also refs.~\cite{Henneaux-ml-2012wg,Henneaux-ml-2012wg.m001}). Taking into account successful applications of these two approaches in string theory (see, e.g., refs.~\cite{Green-ml-1983hw,Green-ml-1983hw.m001}), the light-cone gauge and BRST approaches seem to be promising for developing Lagrangian formulation of interacting mixed-symmetry AdS fields.} \looseness=-1 This paper is organized as follows. In section~\ref{mix-lagr}, we develop light-cone gauge Lagrangian formulation which leads to decoupled equations of motion for mixed-symmetry massive fields in $AdS_5$. Also we describe light-cone gauge realization of relativistic symmetries of actions for such fields. In sections~\ref{masless},~\ref{self-dual}, mixed-symmetry massless and self-dual massive fields are considered. We demonstrate how light-cone gauge Lagrangian for such fields can be obtained from the Lagrangian for the mixed-symmetry massive field. In section~\ref{mix-cur-sh}, light-cone gauge formulation for mixed-symmetry anomalous conformal currents and shadows is developed. Two-point vertices for such currents and shadows are discussed. Section~\ref{ads-cft} is devoted to study of the AdS/CFT correspondence for mixed-symmetry AdS fields and boundary mixed-symmetry conformal currents and shadows. Mixed-symmetry conformal fields are considered in section~\ref{conf-field}. We demonstrate how the action for such fields can be obtained from 2-point vertex of mixed-symmetry shadows. Our conventions and notation are explained in appendix~A. ]]>

h_1 + 2$ (see~\eqref{13102014-17}). To realize the limit of the mixed-symmetry massless field, we take \be \label{13102014-04} E_0 \rightarrow h_1 +2\,,\qquad h_1 > |h_2|\,. \ee In terms of labels $\kappa$, $j_1$, $j_2$,~\eqref{13102014-18-x1}, limit in~\eqref{13102014-04} can equivalently be represented as \be \label{13102014-05} \kappa \rightarrow j_1 + j_2\,,\qquad j_1 \ne 0 \,, \quad j_2 \ne 0\,. \ee Below, we will demonstrate, that limit~\eqref{13102014-04} leads to appearance of invariant subspace in $\phik$~\eqref{12102014-02}. Namely we state that, in the limit~\eqref{13102014-04}, the following set of fields \beq && \phi_{-j_1,j_2}\,, \nonumber\\ && \phi_{m,j_2}\,, \qquad \ \ m = -j_1+1,-j_1+2,\ldots , j_1\,, \nonumber\\ \label{13102014-06} && \phi_{-j_1,m}\,, \qquad m=-j_2,-j_2+1,\ldots, j_2-1\,, \eeq are invariant under action of $so(4,2)$ algebra generators given in~\eqref{12102014-18}--\eqref{12102014-27}. In other words, in limit~\eqref{13102014-04}, fields~\eqref{13102014-06} transform into themselves under the action of the generators given in~\eqref{12102014-18}--\eqref{12102014-27}. Before proving this statement let us present our result for Lagrangian of mixed-symmetry massless field. \paragraph{Lagrangian for mixed-symmetry massless field.} Equating to zero all component fields $\phi_{m_1,m_2}$ in~\eqref{12102014-02} with the exception of ones appearing in~\eqref{13102014-06}, we note that ket-vector which describes massless mixed-symmetry field is given by \beq \label{13102014-07} \phik & = & |\phi_0\rangle + |\phi_1\rangle + |\phi_2\rangle\,, \\ \label{13102014-07-a1} |\phi_0\rangle & = & \frac{v_1^{2j_1} u_2^{2j_2} }{\sqrt{(2j_1)!(2j_2)!}}\, \phi_{-j_1,j_2}|0\rangle\,, \nonumber\\ |\phi_1\rangle &= & \sum_{m=-j_1+1}^{j_1} \frac{u_1^{j_1+m} v_1^{j_1-m} u_2^{2j_2} }{\sqrt{(j_1+m)!(j_1-m)!(2j_2)!}}\, \phi_{m,j_2}|0\rangle\,, \nonumber\\ |\phi_2\rangle & = & \sum_{m=-j_2}^{j_2-1} \frac{ v_1^{2j_1} u_2^{j_2+m} v_2^{j_2-m} }{\sqrt{(2j_1)!(j_2+m)!(j_2-m)!}}\, \phi_{-j_1,m}|0\rangle\,. \eeq Plugging~\eqref{13102014-07} into Lagrangian~\eqref{12102014-05} and using $\kappa$ given in~\eqref{13102014-05}, we find the following Lagrangian: \beq \label{13102014-08-c1} \LL &=& \LL_0 + \LL_1 + \LL_2\,, \\ \LL_0 & = & \langle \phi_0| (\Box + \partial_z^2 + \frac{1}{4z^2}) |\phi_0\rangle\,, \nonumber\\ \LL_1 &= & \langle\phi_1| \biggl( \Box + \partial_z^2 - \frac{1}{z^2}\biggl((j_1+S_1)^2 -\frac{1}{4}\bigg)\biggr)|\phi_1\rangle\,, \nonumber\\ \label{13102014-08} \LL_2 &= & \langle\phi_2| \biggl( \Box + \partial_z^2 - \frac{1}{z^2}\biggl((j_2 - S_2)^2 -\frac{1}{4}\bigg)\biggr)|\phi_2\rangle\,, \eeq where $S_1$, $S_2$ are given in~\eqref{12102014-09}. In terms of component fields~\eqref{13102014-06}, expressions for $\LL_0$, $\LL_1$, $\LL_2$~\eqref{13102014-08} can be represented as \beq \LL_0 & = & \phi_{-j_1,j_2}^\dagger \bigg(\Box + \partial_z^2 + \frac{1}{4z^2}\bigg)\phi_{-j_1,j_2}\,, \nonumber\\ \LL_1 &= & \sum_{m=-j_1+1}^{j_1} \phi_{m,j_2}^\dagger \biggl( \Box + \partial_z^2 - \frac{1}{z^2}\biggl((j_1+m)^2 -\frac{1}{4}\bigg)\biggr) \phi_{m,j_2}\,, \nonumber\\ \label{13102014-08-c2} \LL_2 &= & \sum_{m=-j_2+1}^{j_2} \phi_{-j_1,-m}^\dagger \biggl( \Box + \partial_z^2 - \frac{1}{z^2}\biggl((j_2+m)^2 -\frac{1}{4}\bigg)\biggr) \phi_{-j_1,-m}\,. \eeq \looseness=-1 According to the analysis in ref.~\cite{Metsaev-ml-1995re}, for $h_1 > |h_2|$, there are two solutions for $E_0$ corresponding to the massless fields: $E_{0(1)}=h_1+2$ and $E_{0(2)} = |h_2| + 1$. Also, in ref.~\cite{Metsaev-ml-1995re} it was demonstrated that only $E_{0(1)}$ leads to unitary massless representation. In ref.~\cite{Brink-ml-2000ag}, it was pointed out that the $E_{0(2)}$ leads to wrong (negative sign) kinetic terms for some fields in the Lorentz covariant Lagrangian of massless field. In light-cone gauge approach, from the very beginning, we start with Lagrangian for massive field having the correct (positive sign) kinetic terms for all fields entering the Lagrangian. We expect however that, if we relax the restriction on signs of kinetic terms, then the massless limit $E_0\rightarrow E_{0(2)}$ will lead to negative signs of kinetic terms for some fields in light-cone gauge Lagrangian of massless field. \paragraph{Transformations of mixed-symmetry massless fields.} Now we demonstrate that, in limit~\eqref{13102014-04}, fields~\eqref{13102014-06} transform into themselves under the action of the generators given in~\eqref{12102014-18}--\eqref{12102014-27}. From~\eqref{12102014-18}--\eqref{12102014-27}, we see that we should verify that fields~\eqref{13102014-06} transform into themselves under the action of the operators $A$, $B$, $M^{\Rsm\Lsm}$, $M^{zi}$, $M^{\oplussm i}$. Using expressions for operators $A$~\eqref{12102014-07}, $B$~\eqref{12102014-34-a}, and $M^{\Rsm\Lsm}$~\eqref{12102014-30}, we see that these operators are diagonal on space of the fields $\phi_{m_1,m_2}$. Taking this into account and using~\eqref{12102014-28-a}, we see that if fields~\eqref{13102014-06} are invariant under the action of the operator $M^{zi}$ then they also are invariant under the action of the operator $M^{\oplussm i}$. Thus all that remains is to check that, in limit~\eqref{13102014-04}, fields~\eqref{13102014-06} transform into themselves under the action of the operator $M^{zi}$. To this end we use~\eqref{13102014-01d},~\eqref{13102014-02d} and note that, under the action of the operators $M^{z\Rsm}$, $M^{z\Lsm}$, fields~\eqref{13102014-06} transform as \beq \label{11102014-01} M^{z\Rsm}\phi_{_{m_1,j_2}} & = & \biggl(\frac{\kappa+j_2+m_1}{ \kappa-j_2+m_1}\biggr)^{1/2} r_{j_1,m_1} \phi_{_{m_1-1,j_2}} \nonumber\\ & &+ \biggl(\frac{(\kappa+1+j_1-j_2)(\kappa-j_1-j_2)}{ (\kappa+1-j_2+ m_1)(\kappa-j_2+m_1)}\biggr)^{1/2} r_{j_2,j_2} \phi_{_{m_1,j_2-1}}\,, \nonumber\\ m_1 &=& -j_1+1,-j_1+2,\ldots, j_1\,, \\ M^{z\Rsm}\phi_{_{-j_1,m_2}} & = & \biggl(\frac{\kappa+1+j_1-m_2}{ \kappa + 1 -j_1-m_2}\biggr)^{1/2} r_{j_2,m_2} \phi_{_{-j_1,m_2-1}}\,, \nonumber\\ m_2 &=& -j_2,-j_2+1,\ldots, j_2\,, \\[10pt] \label{11102014-03} -M^{z\Lsm}\phi_{_{-j_1,m_2}} & = & \biggl(\frac{(\kappa+1+j_2-j_1)(\kappa-j_1-j_2)}{ (\kappa - j_1 - m_2)(\kappa+1 - j_1-m_2)}\biggr)^{1/2} r_{j_1,-j_1+1} \phi_{_{-j_1+1,m_2}} \nonumber\\ & & +\biggl(\frac{\kappa+ j_1-m_2}{ \kappa - j_1-m_2}\biggr)^{1/2} r_{j_2,m_2+1} \phi_{_{-j_1,m_2+1}}\,, \nonumber\\ m_2 &=& -j_2,-j_2+1,\ldots, j_2-1\,, \\[10pt] \label{11102014-04} -M^{z\Lsm}\phi_{_{m_1,j_2}} & = & \biggl(\frac{\kappa+1+j_2+m_1}{\kappa+1-j_2+m_1}\biggr)^{1/2} r_{j_1,m_1+1} \phi_{_{m_1+1,j_2}}\,, \nonumber\\ m_1 &=& -j_1,-j_1+1,\ldots, j_1\,. \eeq From~\eqref{11102014-01},~\eqref{11102014-03}, we see that, for arbitrary $\kappa$, fields in~\eqref{13102014-06} are not invariant under the action of the operators $M^{z\Rsm}$, $M^{z\Lsm}$. However, we note that, in the massless limit, \be \kappa \rightarrow j_1 + j_2\,, \ee relations~\eqref{11102014-01}--\eqref{11102014-04} take the form \beq \label{11102014-05} M^{z\Rsm}\phi_{_{m_1,j_2}} & = & \biggl(\frac{j_1+2j_2+m_1}{ j_1+m_1}\biggr)^{1/2} r_{j_1,m_1} \phi_{_{m_1-1,j_2}}\,, \nonumber\\ m_1 &=& -j_1+1,-j_1+2,\ldots, j_1\,, \\ M^{z\Rsm}\phi_{_{-j_1,m_2}} & = & \biggl(\frac{2j_1 +j_2+1-m_2}{ j_2 + 1 - m_2}\biggr)^{1/2} r_{j_2,m_2} \phi_{_{-j_1,m_2-1}}\,, \nonumber\\ m_2 &=& -j_2,-j_2+1,\ldots, j_2\,, \\[10pt] -M^{z\Lsm}\phi_{_{-j_1,m_2}} & =& \biggl(\frac{2j_1 + j_2 -m_2}{ j_2-m_2}\biggr)^{1/2} r_{j_2,m_2+1} \phi_{_{-j_1,m_2+1}}\,, \nonumber\\ m_2 &=& -j_2,-j_2+1,\ldots, j_2-1\,, \\[10pt] \label{11102014-08} -M^{z\Lsm}\phi_{_{m_1,j_2}} & = & \biggl(\frac{j_1 + 2j_2+1 + m_1}{j_1+1+m_1}\biggr)^{1/2} r_{j_1,m_1+1} \phi_{_{m_1+1,j_2}} \nonumber\\ m_1 &=& -j_1,-j_1+1,\ldots, j_1\,. \eeq From~\eqref{11102014-05}--\eqref{11102014-08}, we see that, in massless limit, fields in~\eqref{13102014-06} are indeed invariant under the action of the operators $M^{z\Rsm}$, $M^{z\Lsm}$. \paragraph{Lagrangian for totally symmetric massless fields.} For totally symmetric bosonic fields, we have the condition $j_1 = j_2$. Using the notation $j\equiv j_1$, we can impose the reality condition \beq \label{20102014-10} \phi_{-j,j}^\dagger &=& \phi_{-j,j}\,, \\ \label{20102014-11} \phi_{m,j}^\dagger &=& \phi_{-j,-m}\,, \qquad m= -j+1, -j+2, \ldots , j\,. \eeq Introducing then the real-valued field $\phi_0$, $\phi_0^\dagger=\phi_0$, and a set of the complex-valued fields $\phi_m$ by the relations \be \label{20102014-01} \phi_0 \equiv \sqrt{2} \phi_{-j,j}, \qquad \phi_{j+m} \equiv \sqrt{2} \phi_{m,j}\,, \qquad m =-j+1,-j+2,\ldots j\,, \ee we see that Lagrangian~\eqref{13102014-08-c1},~\eqref{13102014-08-c2} takes the form \be \label{20102014-02} \LL = \half \phi_0 \biggl( \Box + \partial_z^2 + \frac{1}{4z^2} \biggr) \phi_0 + \sum_{m=1}^{2j} \phi_m^\dagger \biggl( \Box + \partial_z^2 - \frac{1}{z^2}\biggl(m^2 -\frac{1}{4}\bigg)\biggr) \phi_m\,. \ee Number of propagating real-valued D.o.F. entering Lagrangian~\eqref{20102014-02} is equal to $2s+1$, where $s\equiv 2j$. Lagrangian~\eqref{20102014-02} describes light-cone gauge dynamics of spin-$s$ totally symmetric massless field in $AdS_5$. \paragraph{Fermionic mixed-symmetry and totally symmetric massless fields.} To get Lagrangian for mixed-symmetry massless fermionic field we use a ket-vector $\psik$ which is obtained from~\eqref{13102014-07} by substituting anticommuting fields $\psi_{-j_1,j_2}$, $\psi_{m,j_2}$, $\psi_{-j_1,m}$ in the respective ket-vectors in~\eqref{13102014-07-a1}. Plugging then such ket-vector $\psik$ into~\eqref{12102014-45} and setting $\kappa=j_1+j_2$, we get Lagrangian for mixed-symmetry massless fermionic field. Setting $j_1=\half (h_1+\half)$, $j_2=\half (h_1-\half)$ in such Lagrangian for mixed-symmetry fermionic field, we get Lagrangian for totally symmetric massless fermionic field. ]]>

h_1+1\,, \qquad h_1 = |h_2|>1/2\,. \ee Using labels $\kappa$, $j_1$, $j_2$~\eqref{13102014-18-x1} and considering $h_2>1/2$, we represent~\eqref{13102014-20} as \be \label{13102014-21} \kappa > j_1 - 1\,, \qquad j_2 = 0\,, \quad \hbox{ for } \quad h_2 > 1/2\,. \ee Using the relation $j_2=0$ in~\eqref{12102014-02}, we see that ket-vector of the bosonic field~\eqref{12102014-02} simplifies as \be \label{13102014-22} \phik = \sum_{m=-j_1}^{j_1} \frac{u_1^{j_1 + m} v_1^{j_1 - m}}{\sqrt{(j_1+m)!(j_1-m)!}} \phi_{m,0}|0\rangle\,. \ee Ket-vector~\eqref{13102014-22} describes the massive self-dual field. Plugging this ket-vector into~\eqref{12102014-05}, we obtain Lagrangian of the massive self-dual field with the operator $A$ given by \be \label{13102014-23} A = \nu^2 - \frac{1}{4}\,, \qquad \nu = \kappa + S_1 \,, \qquad \kappa \equiv E_0-2\,, \ee where $S_1$ is given in~\eqref{12102014-09}. In terms of the component fields $\phi_{m,0}$, Lagrangian of the massive self-dual field takes the form \be \LL= \sum_{m = -j_1}^{j_1} \phi_{m,0}^\dagger \biggl( \Box + \partial_z^2 - \frac{1}{z^2}\biggl((\kappa + m)^2 -\frac{1}{4}\bigg)\biggr) \phi_{m,0}\,. \ee Note that the relation $j_2=0$ implies the relations $S_2\phik=0$, $S_2^{\Rsm,\Lsm}\phik=0$. Using these relations in~\eqref{12102014-30}--\eqref{12102014-34-a}, we get operators entering generators of the $so(4,2)$ algebra~\eqref{12102014-18}--\eqref{12102014-27} corresponding to the self-dual massive field, \begin{align} \label{13102014-24} M^{\Rsm\Lsm} &= S_1\,, & M^{z\Rsm} &= S_1^\Rsm\,, & M^{z\Lsm} &= - S_1^\Lsm\,, \\ \label{13102014-25} M^{\oplussm \Rsm} &= - \TT_{-\nu + \half} S_1^\Rsm \,, & M^{\oplussm \Lsm} &= S_1^\Lsm \TT_{\nu - \half}\,,& \TT_\nu &\equiv \partial_z + \frac{\nu}{z}\,, \\ \label{13102014-26} B &= \kappa S_1 + S_1^2 - j_1(j_1+1)\,. \end{align} \paragraph{Fermionic self-dual massive fields.} Our result for bosonic fields above presented can be extended to the case of fermionic fields is a rather straightforward way. For this case, we use the fermionic ket-vector given by \be \label{14102014-01} \psik = \sum_{m=-j_1}^{j_1} \frac{u_1^{j_1 + m} v_1^{j_1 - m}}{\sqrt{(j_1+m)!(j_1-m)!}} \psi_{m,0}|0\rangle\,, \ee while the operator $A$ takes the form given in~\eqref{13102014-23}. Plugging~\eqref{14102014-01} and $A$~\eqref{13102014-23} into~\eqref{12102014-45}, we get Lagrangian of fermionic self-dual massive field \be \label{14102014-02} \LL= \sum_{m = -j_1}^{j_1} \psi_{m,0}^\dagger\, \frac{\irm }{\partial^+}\biggl( \Box + \partial_z^2 - \frac{1}{z^2}\biggl((\kappa + m)^2 -\frac{1}{4}\bigg)\biggr) \psi_{m,0}\,. \ee \looseness=-1 Operators for bosonic fields in~\eqref{13102014-24}--\eqref{13102014-26} take the same form for fermionic self-dual massive fields. ]]>