\documentclass[10pt]{article} \usepackage{amsmath} \usepackage{amssymb} % \pdfpagewidth 8.5in % \pdfpageheight 11in \setlength{\topmargin}{-.25in} \setlength{\headheight}{0in} \setlength{\headsep}{0in} \setlength{\topskip}{0in} \setlength{\textheight}{9.50in} \setlength{\footskip}{0.375in} \setlength{\oddsidemargin}{0in} \setlength{\evensidemargin}{0in} \setlength{\textwidth}{6.5in} % \newcommand\deb{\mbox{\boldmath$\delta$}} \def\noss{\noalign{\smallskip}} \def\pa{\partial} \def\al{\alpha} \def\be{\beta} \def\ga{\gamma} \def\Ga{\Gamma} \def\de{\delta} \def\De{\Delta} \def\ep{\epsilon} \def\ka{\kappa} \def\la{\lambda} \def\La{\Lambda} \def\si{\sigma} \def\Si{\Sigma} \def\om{\omega} \def\Om{\Omega} \def\albe{{\al\be}} \def\munu{{\mu\nu}} \def\nurho{{\nu\;\rho}} \def\munurho{{\mu\nu\;\rho}} \def\nurhosi{{\nurho\.\si}} \def\rhosi{{\rho\.\si}} \def\epbar{{\overline\epsilon}} \def\labar{{\overline\lambda}} \def\chibar{{\overline\chi}} \def\psibar{{\overline\psi}} \def\Gtilde{{\widetilde G}} \def\Xtilde{{\widetilde X}} \def\Ytilde{{\widetilde Y}} \def\bAA{{\mathbb{A}}} \def\bRR{{\mathbb{R}}} \def\bCC{{\mathbb{C}}} \def\bHH{{\mathbb{H}}} \def\bOO{{\mathbb{O}}} \def\SO{{\rm SO}} \def\SU{{\rm SU}} \def\SL{{\rm SL}} \def\USp{{\rm USp}} \def\A{{\rm A}} \def\B{{\rm B}} \def\C{{\rm C}} \def\D{{\rm D}} \def\E{{\rm E}} \def\F{{\rm F}} \def\N{{\rm N}} \def\U{{\rm U}} \def\Half{\textstyle \frac12} \def\Third{\textstyle \frac13} \def\Quarter{\textstyle \frac14} \def\sss{\scriptscriptstyle} \def\ssr{\scriptscriptstyle\rm} \def\.{{\mkern 1.5mu}} \def\;{{\mkern -1mu}} \def\myst{\rule[-1.5mm]{0mm}{5mm}} \def\mysta{\rule[-2mm]{0mm}{5.5mm}} \def\mystb{\rule[-1.5mm]{0mm}{5.5mm}} \def\mystab{\rule[-2mm]{0mm}{6mm}} \newcommand\1[1]{\frac{1}{#1}} \newcommand\fraq[2]{\frac{\mystb\displaystyle #1}{\mysta\displaystyle #2}} \newcommand\atoq[2]{{\mystb\displaystyle #1}\atop{\mysta\displaystyle #2}} \newcommand\bu[1]{{\underline{#1}}} \newcommand\bz[2]{{\bu{#1}}^{\scriptscriptstyle\rm(#2)}} \newcommand\bzz[2]{{#1}^{\scriptscriptstyle\rm(#2)}} \def\myss{\\[4pt]} \pagestyle{plain} \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title{{\Large\bf Six-dimensional $\N=2$, $\F_4/\SO(9)$ Supergravity\\ and the roof of the Magic Square}} \author{{\large L. J. Romans}} \date{{\normalsize\tt Larry.J.Romans@gmail.com}} \maketitle \begin{abstract} Motivated by the construction by G\"unaydin, Sierra and Townsend of Maxwell-Einstein N = 2 supergravity theories in 3, 4 and 5 dimensions realizing symmetries appearing in the lower three rows of the $4\times4$ Freudenthal-Tits Magic Square, we present a six-dimensional theory with $F_4/\SO(9)$ structure, corresponding to the maximal ``octonionic" item in the top row of the square. The construction involves detailed interplay between the scalar coset structure and the spin-1 sector comprised of vectors, self-dual and anti-self-dual antisymmetric tensors. \end{abstract} \bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{{\large 1. Introduction}} The celebrated Magic Square of Freudenthal and Tits % magic square \begin{equation*} \begin{array}{|r|c|c|c|c|} \hline & \mystab \bAA=\bOO & \bAA=\bHH & \bAA=\bCC & \bAA=\bRR \\ \hline d=6 & \mystab \F_4 & \C_3 & \A_2 & \A_1 \\ \hline d=5 & \mystab \E_6 & \A_5 & \A_2 \times \A_2 & \A_2 \\ \hline d=4 & \mystab \E_7 & \D_6 & \A_5 & \C_3 \\ \hline d=3 & \mystab \E_8 & \E_7 & \E_6 & \F_4 \\ \hline \end{array} \end{equation*} % magic square sugras \begin{equation*} \begin{array}{|r|c|c|c|c|} \hline & \mystab \bAA=\bOO & \bAA=\bHH & \bAA=\bCC & \bAA=\bRR \\ \hline d=6 & \fraq{\F_{4(-20)}}{\SO(9)} & \fraq{\USp(4,2)}{\USp(4)\times\SU(2)} & \fraq{\SU(2,1)}{\SU(2)\times\U(1)} & \fraq{\SO(2,1)}{\SO(2)} \\ \hline d=5 & \fraq{\E_{6(-26)}}{\F_4} & \fraq{\SU^*(6)}{\USp(6)} & \fraq{\SL(3,\bCC)}{\SU(3)} & \fraq{\SL(3,\bRR)}{\SO(3)} \\ \hline d=4 & \fraq{\E_{7(-25)}}{\E_6\times\U(1)} & \fraq{\SO^*(12)}{\SU(6)\times\U(1)} & \fraq{\SU(3,3)}{\SU(3)\times\SU(3)\times\U(1)} & \fraq{\USp(6)}{\SU(3)\times\U(1)} \\ \hline d=3 & \fraq{\E_{8(-24)}}{\E_7\times\SU(2)} & \fraq{\E_{7(-5)}}{\SO(12)\times\SU(2)} & \fraq{\E_{6(2)}}{\SU(6)\times\SU(2)} & \fraq{\F_{4(4)}}{\USp(6)\times\SU(2)} \\ \hline \end{array} \end{equation*} % magic square sugras \begin{equation*} \begin{array}{|r|c|c|c|c|} \hline d=6 & \atoq{\bu{26} = \bz{1}{t+} + \bz{16}{t-} + \bz{9}{v}}{52 - 36 = \bzz{16}{0}} & \atoq{\bu{14} = \bz{1}{t+} + \bz{8}{t-} + \bz{5}{v}}{21 - (10+3) = \bzz{8}{0}} & \atoq{\bu{8} = \bz{1}{t+} + \bz{4}{t-} + \bz{3}{v}}{8 - (3+1) = \bzz{4}{0}} & \atoq{\bu{5} = \bz{1}{t+} + \bz{2}{t-} + \bz{2}{v}}{3 - 1 = \bzz{2}{0}} \\ \hline d=5 & \atoq{\bu{27} = \bz{1}{v} + \bz{26}{v}}{78 - 52 = \bzz{26}{0}} & \atoq{\bu{15} = \bz{1}{v} + \bz{14}{v}}{35 - 21 = \bzz{14}{0}} & \atoq{\bu{9} = \bz{1}{v} + \bz{8}{v}}{16 - 8 = \bzz{8}{0}} & \atoq{\bu{6} = \bz{1}{v} + \bz{5}{v}}{8 - 3 = \bzz{5}{0}} \\ \hline d=4 & \atoq{\bu{28} = \bz{1}{v} + \bz{27}{v}}{133 - (78+1) = \bzz{54}{0}} & \atoq{\bu{16} = \bz{1}{v} + \bz{15}{v}}{66 - (35+1) = \bzz{30}{0}} & \atoq{\bu{10} = \bz{1}{v} + \bz{9}{v}}{35 - (8+8+1) = \bzz{18}{0}} & \atoq{\bu{7} = \bz{1}{v} + \bz{6}{v}}{21 - (8+1) = \bzz{12}{0}} \\ \hline d=3 & \mystab 248 - (133+3) = \bzz{112}{0} & 133 - (66+3) = \bzz{64}{0} & 78 - (35+3) = \bzz{40}{0} & 52 - (21+3) = \bzz{28}{0} \\ \hline \end{array} \end{equation*} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \smallskip \section*{{\large 2. $\F_{4(-20)}$ description in an $\SO(9)$ basis}} % F4 -> so9 \begin{equation*} \left( \begin{array}{cc} Z_0 & {\underline{1}} \\[4pt] Z_i & {\underline{9}} \\[4pt] Z_a & {\underline{16}} \end{array} \right) \end{equation*} % quadratic form \begin{eqnarray} {\cal Q} &\equiv& \eta^{IJ}\,Z_I\.Z_J \nonumber \\[4pt] &=& Z_0^2\ +\ Z_i\.Z_i\ \mp\ Z_a\.Z_a \end{eqnarray} % cubic form \begin{eqnarray} {\cal C} &\equiv& c^{IJK}\,Z_I\.Z_J\.Z_K \nonumber \\[4pt] &=& Z_0^3\ -\ 3\.Z_0\.Z_i\.Z_i\ \mp\ {\frac32}\.Z_0\.Z_a\.Z_a \ \pm\ {\frac{3\sqrt3}2}\left(\Ga_i^{ab}\right)\.Z_i\.Z_a\.Z_b \end{eqnarray} % F4 trans \begin{equation} \left( \begin{array}{ccc} \mystab 0 & 0 & {\sqrt3}Y^b \\[4pt] \mystab 0 & X^{ij} & \left(\Ga^{bc}_i\right)Y^c \\[4pt] \mystab {\sqrt3}\.Y^a & \left(\Ga^{ac}_j\right)Y^c & \Quarter\left(\Ga^{ab}_{k\ell}\right)X^{k\ell} \end{array} \right) \end{equation} \smallskip \section*{{\large 2. The scalar sector}} % [v u] = [1] \begin{equation} \ \left( \begin{array}{ccc} v^I{}_0 & v^I{}_i & v^I{}_a \end{array} \right) \left( \begin{array}{c} u^0{}_J \\[4pt] u^i{}_J \\[4pt] u^a{}_J \end{array} \right) \ =\ \left( \de^I{}_J \right) \end{equation} % [u v] = [1] \begin{equation} \left( \begin{array}{c} u^0{}_I \\[4pt] u^i{}_I \\[4pt] u^a{}_I \end{array} \right) \ \left( \begin{array}{ccc} v^I{}_0 & v^I{}_j & v^I{}_b \end{array} \right) \ =\ \left( \begin{array}{ccc} 1 & 0 & 0 \\[4pt] 0 & \de^i{}_j & 0 \\[4pt] 0 & 0 & \de^a{}_b \end{array} \right) \end{equation} % [u dv] = [Q+R] \begin{equation} \left( \begin{array}{c} u^0{}_I \\[4pt] u^i{}_I \\[4pt] u^a{}_I \end{array} \right) \ \left( \begin{array}{ccc} \pa_\mu\.v^I{}_0 & \pa_\mu\.v^I{}_j & \pa_\mu\.v^I{}_b \end{array} \right) \ =\ \left( \begin{array}{ccc} \mystab 0 & 0 & {\sqrt3}R^b_\mu \\[4pt] \mystab 0 & Q^{ij}_\mu & \left(\Ga^{bc}_i\right)R^c_\mu \\[4pt] \mystab {\sqrt3}\.R^a_\mu & \left(\Ga^{ac}_j\right)R^c_\mu & \Quarter\left(\Ga^{ab}_{k\ell}\right)Q^{k\ell}_\mu \end{array} \right) \end{equation} % [u Dv] = [R] \begin{equation} \left( \begin{array}{c} u^0{}_I \\[4pt] u^i{}_I \\[4pt] u^a{}_I \end{array} \right) \ \left( \begin{array}{ccc} {\cal D}_\mu\.v^I{}_0 & {\cal D}_\mu\.v^I{}_j & {\cal D}_\mu\.v^I{}_b \end{array} \right) \ =\ \left( \begin{array}{ccc} \mystab 0 & 0 & {\sqrt3}R^b_\mu \\[4pt] \mystab 0 & 0 & \left(\Ga^{bc}_i\right)R^c_\mu \\[4pt] \mystab {\sqrt3}\.R^a_\mu & \left(\Ga^{ac}_j\right)R^c_\mu & 0 \end{array} \right) \end{equation} % [Dv] = \begin{eqnarray} {\cal D}_\mu\.v^I{}_0 &\equiv& \pa_\mu\.v^I{}_0 \\ {\cal D}_\mu\.v^I{}_i &\equiv& \pa_\mu\.v^I{}_i \ -\ Q^{ij}_\mu\.v^I_j \\ {\cal D}_\mu\.v^I{}_a &\equiv& \pa_\mu\.v^I{}_a \ -\ \Quarter\.Q^{ij}_\mu\left(\Ga^{ab}_{ij}\right)v^I_b \end{eqnarray} The $\SO(9)$ Fierz identities % so9 fierz \begin{eqnarray} \Ga^{a(b}_i\,\Ga^{cd)}_i &=& \de^{a(b}\,\de^{cd)} \\ 3\.\de^{a[c}\,\de^{d]b} + \Ga^{a[c}_i\,\Ga^{d]b}_i &=& \Quarter\,\Ga^{ab}_{ij}\,\Ga^{cd}_{ij} \end{eqnarray} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \smallskip \section*{{\large 3. The spin-1 sector}} % F,G defs \begin{eqnarray} F^I_{\mu\nu} & = & 2\,\pa^{}_{[\mu}\.A^I_{\nu]} \\ G^I_{\mu\nurho} & = & 3\,\pa^{}_{[\mu}\.B^I_{\nurho]}\ +\ 3\,c^I_{JK}\,F^J_{[\mu\nu}\,A^K_{\rho]} \end{eqnarray} % gauge trans \begin{eqnarray} \de A^I_\mu & = & \pa^{}_\mu\.\la^I \\ \de B^I_{\mu\nu} & = & 2\,\pa^{}_{[\mu}\.\La^I_{\nu]}\ +\ 3\,c^I_{JK}\,F^J_{\mu\nu}\,\la^K \end{eqnarray} % Bianchis \begin{eqnarray} \pa^{}_{[\mu}\.F^I_{\nurho]} & = & 0 \\ \pa^{}_{[\mu}\.G^I_{\nurhosi]} & = & 3\,c^I_{JK}\,F^J_{[\mu\nu}\,F^K_{\rhosi]} \end{eqnarray} % uF defs \begin{eqnarray} F^0_\munu & \equiv & u^0{}_I\,F^I_\munu \myss F^i_\munu & \equiv & u^i{}_I\,F^I_\munu \myss F^a_\munu & \equiv & u^a{}_I\,F^I_\munu \end{eqnarray} % F constraints \begin{eqnarray} F^0_\munu & = & 0 \myss F^a_\munu & = & 0 \end{eqnarray} % uG defs \begin{eqnarray} G^0_\munurho & \equiv & u^0{}_I\,G^I_\munurho \myss G^i_\munurho & \equiv & u^i{}_I\,G^I_\munurho \myss G^a_\munurho & \equiv & u^a{}_I\,G^I_\munurho \end{eqnarray} % G constraints \begin{eqnarray} G^0_\munurho & = & +\Gtilde^0_\munurho \myss G^a_\munurho & = & -\Gtilde^a_\munurho \end{eqnarray} \begin{eqnarray} G^i_\munurho & = & 0 \end{eqnarray} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \smallskip \section*{{\large 4. Supersymmetry transformations and field equations}} % susy trans \begin{eqnarray} \de\,e_\mu{}_\al & = & \psibar_\mu \ga^\al \ep \myss \de\,\psi_\mu & = & \nabla_\mu \ep + \1{4}\,G^0_\munurho\ga^\nurho \ep \myss \de\,A^I_\mu & = & v^I{}_i\,\labar^i \ga_\mu \ep \myss \de\,B^I_\munu & = & v^I{}_0\,\psibar^{}_{[\mu} \ga^{}_{\nu]} \ep - \1{2}\,v^I{}_a\,\chibar^a\,\ga_\munu \ep + 2 c^I_{JK}\,A^J_{[\mu}\de A^J_{\nu]} \myss \de\,\la^i & = & -\1{4}\,F^i_\munu\,\ga^\munu \ep \myss \de\,\chi^a & = & P^a_\mu\,\ga^\mu \ep + \1{12}\,G^a_\munurho\,\ga^\munurho \ep \end{eqnarray} % susy commutator \begin{eqnarray} [\ \de_{s.s.}(\ep_2)\,,\, \de_{s.s.}(\ep_1)\ ] & = & \de_{g.c.}(\xi^\mu) \ +\ \de_{l.l.}(\Si^\albe) \myss & & \ +\ \de_{gauge(v)}(\la^I) \ +\ \de_{gauge(t)}(\La^I_\mu) \myss & & \ +\ \de_{SO(9)}(L^{ij}) \end{eqnarray} % susy trans \begin{eqnarray} \xi^\mu & = & \epbar_2 \ga^\mu \ep_1 \myss \Si^\mu & = & \xi^\mu ( \om_\mu{}^\albe - G^0{}_\mu{}^\albe ) \myss \la^I & = & -\xi^\mu\,A^I_\mu \myss \La^I_\mu & = & \dots \myss L^{ij} & = & -\xi^\mu\,Q^{ij}_\mu \end{eqnarray} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \smallskip \section*{{\large 5. Discussion}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \smallskip \section*{{\large A. Spacetime conventions}} % tilde def \begin{eqnarray} \Xtilde_\munurho & = & \1{6} e_\munurho{}^{\si\tau\ka}\,X_{\si\tau\ka} \end{eqnarray} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \smallskip \renewcommand\refname{\large References} \begin{thebibliography}{9} \bibitem{ref1} M. G\"unaydin, G. Sierra and P. K. Townsend, Nucl. Phys. B242 (1984) 244. \bibitem{ref2} H. Freudenthal, Nederl. Akad. Wetensch. Proc. Ser. A, 57 (1954) 218; T. A. Springer, ibid. 65 (1962) 259; J. Tits, ibid. 65 (1962) 530; 69 (1966) 223. \bibitem{ref3} L. J. Romans, Nucl. Phys. B276 (1986) 71. \bibitem{ref4} J. H. Schwarz, Nucl. Phys. B226 (1983) 269. \end{thebibliography} \end{document}