1\chapter{SUSY2: Super Symmetry} 2\label{SUSY2} 3\typeout{{SUSY2: Super Symmetry}} 4 5{\footnotesize 6\begin{center} 7Ziemowit Popowicz \\ 8Institute of Theoretical Physics, University of Wroclaw\\ 9pl. M. Borna 9 50-205 Wroclaw, Poland \\ 10e-mail: ziemek@ift.uni.wroc.pl 11\end{center} 12} 13\ttindex{SUSY2} 14 15 16This package deals with supersymmetric functions and with algebra 17of supersymmetric operators in the extended N=2 as well as in the 18nonextended N=1 supersymmetry. It allows us 19to make the realization of SuSy algebra of differential operators, 20compute the gradients of given SuSy Hamiltonians and to obtain 21SuSy version of soliton equations using the SuSy Lax approach. There 22are also many additional procedures encountered in the SuSy soliton 23approach, as for example: conjugation of a given SuSy operator, computation 24of general form of SuSy Hamiltonians (up to SuSy-divergence equivalence), 25checking of the validity of the Jacobi identity for some SuSy 26Hamiltonian operators. 27 28To load the package, type \quad {\tt load susy2;} \\ 29\\ 30For full explanation and further examples, please refer to the 31detailed documentation and the susy2.tst which comes with this package. 32 33\section{Operators} 34 35\subsection{Operators for constructing Objects} 36 37The superfunctions are represented in this package by \f{BOS}(f,n,m) for superbosons 38and \f{FER}(f,n,m) for superfermions. The first index denotes the name of the given 39superobject, the second denotes the value of SuSy derivatives, and the last gives the 40value of usual derivative. \\ 41In addition to the definitions of the superfunctions, also the inverse and the exponential 42of superbosons can be defined (where the inverse is defined as \f{BOS}(f,n,m,-1) 43with the property {\it bos(f,n,m,-1)*bos(f,n,m,1)=1}). The exponential of the superboson 44function is \f{AXP}(\f{BOS}(f,0,0)). \\ 45The operator \f{FUN} and \f{GRAS} denote the classical and the Grassmann function. \\ 46Three different realizations of supersymmetric derivatives are implemented. To select 47traditional realization declare \f{LET TRAD}. In order to select chiral or chiral1 algebra 48declare \f{LET CHIRAL} or \f{LET CHIRAL1}. For usual differentiation the operator 49\f{D}(1) stands for right and \f{D}(2) for left differentiation. SuSy derivatives are 50denoted as {\it der} and {\it del}. \f{DER} and \f{DEL} are one component argument operations 51and represent the left and right operators. The action of these operators on the 52superfunctions depends on the choice of the supersymmetry algebra. 53 54\flushleft 55{\small\begin{center} 56\begin{tabular}{ l l l l l l} 57 \f{BOS}(f,n,m)\ttindex{BOS} & \f{BOS}(f,n,m,k)\ttindex{BOS} & 58 \f{FER}(f,n,m)\ttindex{FER} & \f{AXP}(f)\ttindex{AXP} & 59 \f{FUN}(f,n)\ttindex{FUN} & \f{FUN}(f,n,m)\ttindex{FUN} \cr 60 \f{GRAS}(f,n)\ttindex{GRAS} & \f{AXX}(f)\ttindex{AXX} & 61 \f{D}(1)\ttindex{D} & \f{D}(2)\ttindex{D} & 62 \f{D}(3)\ttindex{D} & \f{D}(-1)\ttindex{D} \cr 63 \f{D}(-2)\ttindex{D} & \f{D}(-3)\ttindex{D} & 64 \f{D}(-4)\ttindex{D} & \f{DR}(-n)\ttindex{DR} & 65 \f{DER}(1)\ttindex{DER} & \f{DER}(2)\ttindex{DER} \cr 66 \f{DEL}(1)\ttindex{DEL} & \f{DEL}(2)\ttindex{DEL} 67\end{tabular} 68\end{center} } 69\vspace{1cm} 70 71{\bf Example}: 72\begin{verbatim} 731: load susy2; 74 752: bos(f,0,2,-2)*axp(fer(k,1,2))*del(1); %first susy derivative 76 772*fer(f,1,2)*bos(f,0,2,-3)*axp(fer(k,1,2)) 78 79 - bos(k,0,3)*bos(f,0,2,-2)*axp(fer(k,1,2)) 80 81 + del(1)*bos(f,0,2,-2)*axp(fer(k,1,2)) 82 833: sub(del=der,ws); 84 85bos(f,0,2,-2)*axp(fer(k,1,2))*der(1) 86 87\end{verbatim} 88 89\subsection{Commands} 90 91There are plenty of operators on superfunction objects. Some of them are introduced 92here briefly. 93\begin{itemize} 94\item By using the operators \f{FPART}, \f{BPART}, \f{BFPART} and \f{BF\_PART} 95 it is possible to compute the coordinates of the arbitrary SuSy expressions. 96\item With \f{W\_COMB}, \f{FCOMB} and \f{PSE\_ELE} there are three operators to be able to 97 construct different possible combinations of superfunctions and 98 super-pseudo-differential elements with the given conformal dimensions . 99\item The three operators \f{S\_PART}, \f{D\_PART} and \f{SD\_PART} are implemented to 100 obtain the components of the (pseudo)-SuSy element. 101\item \f{RZUT} is used to obtain the projection onto the invariant subspace (with respect 102 to commutator) of algebra of pseudo-SuSy-differential algebra. 103\item To obtain the list of the same combinations of some superfunctions and (SuSy) 104 derivatives from some given operator-valued expression, the operators 105 \f{LYST}, \f{LYST1} and \f{LYST2} are constructed. 106\end{itemize} 107 108 109\begin{center} 110\begin{tabular}{ l l} 111 \f{FPART}(expression)\ttindex{FPART} & 112 \f{BPART}(expression)\ttindex{BPART} \cr 113 \f{BF\_PART}(expression,n)\ttindex{BF\_PART} & 114 \f{B\_PART}(expression,n)\ttindex{B\_PART} \cr 115 \f{PR}(n,expression)\ttindex{PR} & 116 \f{PG}(n,expression)\ttindex{PG} \cr 117 \f{W\_COMB}(\{\{f,n,x\},...\},m,z,y)\ttindex{W\_COMB} & 118 \f{FCOMB}(\{\{f,n,x\},...\},m,z,y)\ttindex{FCOMB} \cr 119 \f{PSE\_ELE}(n,\{\{f,n\},...\},z)\ttindex{PSE\_ELE} \cr 120 \f{S\_PART}(expression,n)\ttindex{S\_PART} & 121 \f{D\_PART}(expression,n)\ttindex{D\_PART} \cr 122 \f{SD\_PART}(expression,n,m)\ttindex{SD\_PART} & 123 \f{CP}(expression)\ttindex{CP} \cr 124 \f{RZUT}(expression,n)\ttindex{RZUT} & 125 \f{LYST}(expression)\ttindex{LYST} \cr 126 \f{LYST1}(expression)\ttindex{LYST1} & 127 \f{LYST2}(expression)\ttindex{LYST2} \cr 128 \f{CHAN}(expression)\ttindex{CHAN} & 129 \f{ODWA}(expression)\ttindex{ODWA} \cr 130 \f{GRA}(expression,f)\ttindex{GRA} & 131 \f{DYW}(expression,f)\ttindex{DYW} \cr 132 \f{WAR}(expression,f)\ttindex{WAR} & 133 \f{DOT\_HAM}(equations,expression)\ttindex{DOT\_HAM} \cr 134 \f{N\_GAT}(operator,list)\ttindex{N\_GAT} & 135 \f{FJACOB}(operator,list)\ttindex{FJACOB} \cr 136 \f{JACOB}(operator,list,\{$\alpha,\beta,\gamma$\})\ttindex{JACOB} & 137 \f{MACIERZ}(expression,x,y)\ttindex{MACIERZ} \cr 138 \f{S\_INT}(number,expression,list)\ttindex{S\_INT} 139\end{tabular} 140\end{center} 141\vspace{1cm} 142 143{\bf Example}: 144\begin{verbatim} 1454: xxx:=fer(f,2,3); 146 147xxx := fer(f,2,3) 148 1495: fpart(xxx); % all components 150 151 - fun(f0,4) + 2*fun(f1,3) gras(ff2,4) 152{gras(ff2,3), ----------------------------,0, -------------} 153 2 2 1546: bpart(xxx); % bosonic sector 155 156 - fun(f0,4) + 2*fun(f1,3) 157{0,----------------------------,0,0} 158 2 159 1609: b_part(xxx,1); %the given component in the bosonic sector 161 162 - fun(f0,4) + 2*fun(f1,3) 163---------------------------- 164 2 165\end{verbatim} 166 167\section{Options} 168The are several options defined in this package. Please note that they are 169activated by typing \f{let $<$option$>$}. See also above. \\ 170The \f{TRAD}, \f{CHIRAL} and \f{CHIRAL1} select the different realizations of the 171supersymmetric derivatives. By default traditional algebra is selected. \\ 172If the command {\tt LET INVERSE} is used, then three indices {\it bos} objects are 173transformed onto four indices objects. 174\begin{center} 175\begin{tabular}{ l l l l l l } 176\f{TRAD}\ttindex{TRAD} & \f{CHIRAL}\ttindex{CHIRAL} & 177\f{CHIRAL1}\ttindex{CHIRAL1} & \f{INVERSE}\ttindex{INVERSE} & 178\f{DRR}\ttindex{DRR} & \f{NODRR}\ttindex{NODRR} 179\end{tabular} 180\end{center} 181\vspace{1cm} 182 183{\bf Example}: 184\begin{verbatim} 18510: let inverse; 186 18711: bos(f,0,3)**3*bos(k,3,1)**40*bos(f,0,3,-2); 188 189bos(k,3,1,40)*bos(f,0,3,1); 190 19112: clearrules inverse; 192 19313: xxx:=fer(f,1,2)*bos(k,0,2,-2); 194 195xxx := fer(f,1,2)*bos(k,0,2,-2) 196 19714: pr(1,xxx); % first susy derivative 198 199- 2*fer(k,1,2)*fer(f,1,2)*bos(k,0,2,-3) + bos(k,0,2,-2)*bos(f,0,3) 200 20115: pr(2,xxx); %second susy derivative 202 203- 2*fer(k,2,2)*fer(f,1,2)*bos(k,0,2,-3) - bos(k,0,2,-2)*bos(f,3,2) 204 20516: clearrules trad; 206 20717: let chiral; % changing to chiral algebra 208 20918: pr(1,xxx); 210 211- 2*fer(k,1,2)*fer(f,1,2)*bos(k,0,2,-3) 212\end{verbatim} 213 214