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:

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

with T({x^\mu })\mu = 0,1,...,p + 1, the metaplectic D-field whose potential achieves its maximum at T = 0 and asymptotes to zero (closed string vacuum) at large T. Note now, the action above gives the known exponentially super-decreasing pressure at late-times while being consistent with the string-theory calculation, where V(T) is interpreted as an exponential function of T.

Since the energy:

    \[\epsilon = V(T)/\sqrt {1 - {T^2}} \]

is conserved, one gets the homogeneous solution {T_{cl}}\left( {{x^0}} \right)

    \[{x^0} = \int_0^{{T_{cl}}} {\frac{{dT}}{{\sqrt {1 - V{{(T)}^2}/{\epsilon ^2}} }}} \]

When D-fields approach their minimum, V(T) \to 0, their time-dependence simplifies to T \sim {x^0}. Hence, the location of a static domain wall is determined by the equation {T_{cl}}\left( {{x^\mu }} \right) = 0 where {T_{cl}} is the semi-classical solution of the domain wall, so time-dependent D-field solutions are analogously characterized by T = 0 and the S-brane is found wherever T = 0. So, from

    \[{x^0} = \int_0^{{T_{cl}}} {\frac{{dT}}{{\sqrt {1 - V{{(T)}^2}/{\epsilon ^2}} }}} \]

it follows that we must choose the Sp-brane field solution to be the space-like p+1-dimensional space {x^0} = 0. So now, we are in a position to deform the S-brane worldvolume as given by analyzing Heisenberg fluctuations of D-fields around semi-classical solutions given above,

    \[T = {T_{cl}}\left( {{x^0}} \right) + t\left( {{x^\mu }} \right)\]

Substituting this into

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

while keeping terms quadratic in t, one gets the Heisenberg fluctuation action

    \[{S_f} = \frac{{ - \epsilon }}{2}\int {d{x^0}} {d^{p + 1}}x{\mkern 1mu} \hat \mu \left( {\frac{{ - {\epsilon^2}}}{{{V^2}}}{{\left( {\dot t} \right)}^2} + {{\left( {{\partial _{\hat \mu }}t} \right)}^2} + {M^2}\left( {{x^0}} \right){t^2}} \right)\]

with

    \[\frac{{ - {\epsilon ^2}}}{{{V^2}}}{\left( {\dot t} \right)^2} + {\left( {{\partial _{\hat \mu }}t} \right)^2} + {M^2}\left( {{x^0}} \right){t^2}\]

being the key to time-like transformation, with \widehat \mu = 1,2,...,p + 1 and the time-dependent mass is

    \[{M^2}\left( {{x^0}} \right) = {\left[ {\frac{{V''}}{V} - \frac{{{{\left( {V'} \right)}^2}}}{{{V^2}}}} \right]_{T = {T_{cl}}}}\]

The factor in front of {\left( {\dot t} \right)^2} in the Heisenberg fluctuation action diverges at late time {x^0} \to \infty hence the Heisenberg fluctuation is governed by the Carrollian bulk-metric and ceases to propagate, which is what we expect. Now, since

    \[{x^0} = \int_0^{{T_{cl}}} {\frac{{dT}}{{\sqrt {1 - V{{(T)}^2}/{\epsilon ^2}} }}} \]

breaks translation invariance along the time direction, there is a zero mode on the defect S-brane, which gives us

    \[t\left( {{x^\mu }} \right) = \, - {X^0}\left( {{X^{\widehat \mu }}} \right){\dot T_{cl}}\left( {{x^{^0}}} \right)\]

with {X^0} depending only on the coordinates along the Sp-brane. By substituting into the fluctuation action, the mass term in

    \[{M^2}\left( {{x^0}} \right) = {\left[ {\frac{{V''}}{V} - \frac{{{{\left( {V'} \right)}^2}}}{{{V^2}}}} \right]_{T = {T_{cl}}}}\]

cancels with the contribution from the term {\left( {\dot t} \right)^2}. Hence, the effective action for a massless displacement field {X^0}\left( {{x^{\widehat \mu }}} \right) is

    \[{S^\dagger } = {\mkern 1mu} - \Im (\epsilon )\int {{d^{p + 1}}} {x^{\hat \mu }}\frac{1}{2}{\left( {{\partial _{\hat \nu }}{X^0}} \right)^2}\]

with the constant \Im depending only on the energy \epsilon, and hence, the S-brane effective action for a Euclidean world-volume to lowest order has been determined. Now, one naturally expects gauge fields on the S-branes, just like on D-branes. So, to proceed, first note that the constant gauge field strength appears in the S-field action only through the overall Born-Infeld factor

    \[\sqrt { - {\rm{det}}{{\left( {\eta + F} \right)}_{\mu \nu }}} \]

and the open string metric

    \[{G^{\mu \nu }} = {\left( {\left( {\eta + F} \right)_{Sym}^{ - 1}} \right)^{\mu \nu }}\]

used for contracting the indices of the derivatives. Since the equations of motion for the gauge fields are also …