• M-Theory, Kaluza-Klein Splitting, U-Duality and F-Theory

    There is a deep connection between the U-duality groups of M-theory and the embedding of the 11-dimensions in the extended superspace which under the gauge and diffeomorphism group actions, induces a continuous {E_{d(d)}} symmetry. Here, I will relate the F-theory action to that of M-theory in the context of the F-theory/M-theory duality with an {\rm{SL}}\left( N \right) \times {\mathbb{R}^ + } representation. Recall that F-theory is a one-time theory, so let us start with how to make a space-like brane time-like in M-theory. Keeping in mind that the total action of M-theory is given by:

        \[\begin{array}{*{20}{l}}{{S_M} = \frac{1}{{{k^9}}}\int\limits_{world - vol} {{d^{11}}} \sqrt {\frac{{ - {g_{\mu \nu }}}}{{ - \gamma }}} T_p^{10}d\Omega {{\left( {{\phi _{Inst}}} \right)}^{26}}\left( {{R_{icci}} - A_\mu ^H\frac{1}{{48}}G_4^2} \right)}\\{ + \sum\limits_{Dp} {D_\mu ^S} {e^{ - H_3^b}}/S_{Dp}^{WV} + \sum\limits_{Dp} {D_\mu ^S} {e^{ - H_3^b}}/{S^{Total}}}\end{array}\]

    as I showed here, with {T_p} \sim {\alpha ^\dagger }\frac{{p + 1}}{2} the D-p-brane world-volume tension, and the Yang-Mills field strength being:

        \[{F_{\mu \nu }} = {\partial _\mu }A_\mu ^H - {\partial _\nu }\bar A_\mu ^H + \left[ {A_\mu ^H,\Upsilon _{2\kappa }^i(\cos \varphi )} \right]\]

    and by a Paton-Chern-Simons factor, we get:

        \[\left[ {A_\mu ^H,A_\nu ^H} \right] = \sum\limits_{k = 1}^N {A_\mu ^{H,ac}} A_\nu ^{H,cb} - A_\nu ^{H,ac}A_\mu ^{H,cb}\]

    {\phi _{Inst}} the instanton field, with:

        \[{e^{ - {\phi _{Inst}}{g_{\mu \nu }}}} = {e^{ - 2{\phi _{Inst}}\left( {{g_{\mu \nu }} - 1} \right)}}\]

    and {g_{\mu \nu }} = {e^{{{\left( {{\phi _{Inst}}} \right)}^2}}}.

    Space-like branes are a class of time-dependent solutions of M-theory with topological defects localized in (P + 1)-dimensional space-like surfaces and exist at a moment in time, and are time-like super-tachyonic kink solutions of unstable D(P + 1)-branes in string theory and provide the topology of the throat-bulk. Let us start with a Dp-Dp pair Lagrangian, fixing the boundary of the string field theory superspace, so that the action is:

        \[S = {\mkern 1mu} - 2{T_{D9}}\int {{d^{10}}} x{e^{ - \pi {{\left| T \right|}^2}}}F\left( {X + \sqrt Y } \right)F\left( {X - \sqrt Y } \right)\]

    with

        \[\left\{ {\begin{array}{*{20}{c}}{X \equiv {\partial _\mu }T{\partial ^\mu }\bar T}\\{Y \equiv {{\left( {{\partial _\mu }T} \right)}^2}{{\left( {{\partial ^\nu }\bar T} \right)}^2}}\end{array}} \right.\quad p = 9\]

    and

        \[T = {T_{cl(st)}}(x) = x + \sum\limits_{cl{{(st)}_x}} {\int_{cl{{(st)}_x}} {{e^{\tilde T(x)}}} } \gg 0\]

    A Teichmuller BPS D(P+1)-brane 2-D reduction gives us the throat action:

        \[S = - \int {{d^{p + 2}}} xV(T)\sqrt {1 + {{\left( {{\partial _\mu }T} \right)}^2}} \]