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 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 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:

as I showed here, with the D-p-brane world-volume tension, and the Yang-Mills field strength being:

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

the instanton field, with:

and .

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:

with

and

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

with , , the metaplectic D-field whose potential achieves its maximum at and asymptotes to zero (closed string vacuum) at large . Note now, the action above gives the known exponentially super-decreasing pressure at late-times while being consistent with the string-theory calculation, where is interpreted as an exponential function of .

Since the energy:

is conserved, one gets the homogeneous solution

When D-fields approach their minimum, , their time-dependence simplifies to . Hence, the location of a static domain wall is determined by the equation where is the semi-classical solution of the domain wall, so time-dependent D-field solutions are analogously characterized by and the S-brane is found wherever . So, from

it follows that we must choose the Sp-brane field solution to be the space-like p+1-dimensional space . 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,

Substituting this into

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

with

being the key to time-like transformation, with and the time-dependent mass is

The factor in front of in the Heisenberg fluctuation action diverges at late time hence the Heisenberg fluctuation is governed by the Carrollian bulk-metric and ceases to propagate, which is what we expect. Now, since

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

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

cancels with the contribution from the term . Hence, the effective action for a massless displacement field is

with the constant depending only on the energy , 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

and the open string metric

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

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