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:

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

F-Theory, the D-Term Equation and Representation Theory

Let us see how the Yukawa couplings among 4-D fermionic fields can be derived from the F-theory superpotential and relate them to the tree-level superpotential. This is of utmost importance since D7/D3-brane-phenomenology of 4-D F-theory can be promoted to M-theory in light of the F/M-theory duality and the compactness of Calabi-Yau 4-folds. Start with a Kähler coordinate expansion of \gamma which gives us, after inserting it in:

    \[{W^0} = \int_{\tilde S} {S{\rm{Tr}}} \left\{ {{\rm{P}}\left[ {{e^{i\lambda \iota \Phi \iota \Phi }}} \right] \wedge {e^{\lambda F}}} \right\}\]

the following:

    \[\begin{array}{l}{W^0} = {\lambda ^2}\int_{\tilde S} {{\rm{Tr}}} \left\{ {\phi dx \wedge dy \wedge F} \right\} = \\{\lambda ^2}\int_{\tilde S} {{\rm{Tr}}} \left\{ {\iota \Phi \Omega \wedge F} \right\}\end{array}\]

which is the exact 7-brane superpotential for F-theory and the integrand is independent of \lambda, entailing that the F-term conditions are purely topological and in no need for \alpha '-corrections.

However, the D-term in:

    \[{D^K} = \int_{\tilde S} {S\left\{ {{\rm{P}}\left[ {{e^{i\lambda \iota \Phi \iota \Phi }}{\mathop{\rm Im}\nolimits} {e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}}} \right\}} \]

is in need of \alpha '-corrections, since it is evaluable as:

    \[\begin{array}{l}D = \int_{\tilde S} {S\left\{ {\lambda P\left[ J \right]} \right.} \wedge F - \frac{{i\lambda }}{6}{\iota _\Phi }{\iota _\Phi }{J^3} + \\\frac{{i{\lambda ^3}}}{2}{\iota _\Phi }{\iota _\Phi }J \wedge F \wedge F - {\rm{P}}\left[ {J \wedge B} \right] \wedge F\\\left. { + i{\lambda ^2}{\iota _\Phi }{\iota _\Phi }\left( {J \wedge B} \right) \wedge \frac{{i\lambda }}{2}{\iota _\Phi }{\iota _\Phi }\left( {J \wedge {B^2}} \right)} \right\}\end{array}\]

and the non-Abelian D-term has the form:

    \[D = \int_{\tilde S} {\rm{P}} \left[ {{\rm{Im}}{e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}} \wedge \sqrt {\tilde A\left( {\tilde T} \right)/\tilde A\left( {\tilde N} \right)} \]

With {Y_4} our target Calabi-Yau 4-fold and Lie algebra G, for:

    \[\left[ {{{D'}_i}} \right] \in {H^2}\left( {{Y_4}} \right)\]

we have:

    \[\int_{{Y_4}} {\left[ {{{D'}_i}} \right]} \wedge \left[ {{{D'}_j}} \right] \wedge \tilde \omega = - {C_{ij}}\int_S {\tilde \omega } \]