M-Theory, Space-like Branes, and their Dirac-Born-Infeld Action

By George Shiber Saturday, December 12, 2015Friday, October 26, 2018 In Cosmology, Grand Unification Physics, M Theory, Mathematical Physics, Philosophy Of Physics, Physics, Super String Theory, Supersymmetry compactification, D-Branes

Anyone can count the seeds in an apple, but no one can count the apples in a seed ~ Anonymous

How to make a space-like brane time-like, and why it matters. In this post, I will derive the Dirac-Born-Infeld S-brane action for Euclidean D-world-volumes in the S-brane context of super-condensation of non-BPS branes. Space-like branes are a class of time-dependent solutions of string/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. They are also deeply explanatory in quantum cosmology. To understand what a tachyonic S-brane is, I will start with a Lagrangian of a Dp-Dp pair, choosing the Lagrangian of the boundary string field theory, so the action is

$S = \, - 2{T_{D9}}\int {{d^{10}}} x{e^{ - \pi {{\left| T \right|}^2}}}\not F\left( {X + \sqrt Y } \right)\not F\left( {X - \sqrt Y } \right)$

with

$\left\{ {\begin{array}{*{20}{c}}{X \equiv {{\not \partial }_\mu }T{{\not \partial }^\mu }\overline T }\\{Y \equiv {{\left( {{{\not \partial }_\mu }T} \right)}^2}{{\left( {{{\not \partial }^\nu }\overline T } \right)}^2}}\end{array}} \right.\quad p = 9$

and

$T = {T_{cl(st)}}(x) = x + \left[ {{\rm{exponentially small terms for }}x} \right]$

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

$S = \, - \int {{d^{p + 2}}} xV(T)\sqrt {1 + {{\left( {{{\not \partial }_{\mu T}}} \right)}^2}}$

with $T({x^\mu })$ , $\mu = 0,1,...,p + 1$ ,the metaplectic tachyonic 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

$\varepsilon = V(T)/\sqrt {1 - {T^2}}$

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

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

When tachyonic fields approache their minimum, $V(T) \to 0$ , the time-dependence of the tachyon 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 tachyonic field solutions are analogously characterized by $T = 0$ and the S-brane is found wherever $T = 0$ . So, from

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

it follows that we must choose the Sp-brane tachyonic field solution to be the spacelike 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 the tachyon field around its semi-classical solution given above,

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

Substituting this into

$S = \, - 2{T_{D9}}\int {{d^{10}}} x{e^{ - \pi {{\left| T \right|}^2}}}\not F\left( {X + \sqrt Y } \right)\not F\left( {X - \sqrt Y } \right)$

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

${S_f} = \frac{{ - \varepsilon }}{2}\int {d{x^0}} {d^{p + 1}}x\,\widehat \mu \left( {\frac{{ - {\varepsilon ^2}}}{{{V^2}}}{{\left( {\dot t} \right)}^2} + {{\left( {{{\not \partial }_{\widehat \mu }}t} \right)}^2} + {M^2}\left( {{x^0}} \right){t^2}} \right)$

with

${\frac{{ - {\varepsilon ^2}}}{{{V^2}}}{{\left( {\dot t} \right)}^2} + {{\left( {{{\not \partial }_{\widehat \mu }}t} \right)}^2} + {M^2}\left( {{x^0}} \right){t^2}}$

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: the true vacuum of the tachyon theory open string degrees of freedom disappear and we therefore concentrate on the fluctuations around S-branes. Now, since

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

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 substitution 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

${S^\dagger } = \, - \Im (\varepsilon )\int {{d^{p + 1}}} {x^{\widehat \mu }}\frac{1}{2}{\left( {{{\not \partial }_{\widehat \nu }}{X^0}} \right)^2}$

with the constant $\Im$ depending only on the energy $\varepsilon$ , 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 tachyonic field action only through the overall Born-Infeld factor

$\sqrt { - {\rm{det}}{{\left( {\eta + \not F} \right)}_{\mu \nu }}}$

and the open string metric

${G^{\mu \nu }} = {\left( {\left( {\eta + \not F} \right)_{Sym}^{ - 1}} \right)^{\mu \nu }}$

used for contracting the indices of the derivatives. Since the equations of motion for the gauge fields are also satisfied in the time-dependent homogeneous tachyon background, the open string metric satisfies

$\left\{ {\begin{array}{*{20}{c}}{{G^{00}} = \, - 1}\\{{G^{D\widehat \mu }} = 0}\end{array}} \right.$

So, by allowing such an introduction of dynamical gauge fields, while also preserving the tachyon equations of motion, essentially entails that we can not turn on electric fields on a Euclidean worldvolume and the dependence on the zero mode ${X^0}$ in the tachyon action ought to then be

$S = \int {d{x^0}} {d^{p + 1}}x\,\not L\left( {{T_{cl}}\left( {\frac{{{x^0} - {X^0}\left( {{x^{\widehat \mu }}} \right)}}{{\beta \left( {{X^0}} \right)}}} \right)} \right)$

with

${{T_{cl}}\left( {\frac{{{x^0} - {X^0}\left( {{x^{\widehat \mu }}} \right)}}{{\beta \left( {{X^0}} \right)}}} \right)}$

giving us world-volume spacetime continuity and $\beta \left( {{X^0}} \right)$ can be fixed by the global Lorentz invariance in the world-volume spacetime. The condition that the Lorentz super-boost preserves the open string symplectic metric is

$\Lambda _\mu ^\nu {G_{\nu \rho }}{\left( {{\Lambda ^t}} \right)^\rho }_\sigma = {G_{\mu \nu }}$

which allows us to define the Lorentz boost as

$\Lambda _\mu ^\nu = \left( {\begin{array}{*{20}{c}}{1/\beta }&{ - \,{{\not \partial }_{\widehat \mu }}{X^0}/\beta }\\ * & * \end{array}} \right)$

hence deriving

$\beta = \sqrt {1 - {G^{\widehat \mu \widehat \nu }}{{\not \partial }_{\widehat \mu }}{X^0}{{\not \partial }_{\widehat \nu }}{X^0}}$

Now, integration over ${x^0}$ in

$S = \int {d{x^0}} {d^{p + 1}}x\,\not L\left( {{T_{cl}}\left( {\frac{{{x^0} - {X^0}\left( {{x^{\widehat \mu }}} \right)}}{{\beta \left( {{X^0}} \right)}}} \right)} \right)$

and including the $\not F$ dependence, we obtain the Dirac-Born-Infeld S-brane action

$\begin{array}{l}S = {S_0}(\varepsilon )\int {{d^{p + 1}}} x\beta \left( {{X^0}} \right)\sqrt {{\rm{det}}{{\left( {\delta + \not F} \right)}_{\widehat \mu \widehat \nu }}} \\ = {S_0}(\varepsilon )\int {{d^{p + 1}}} x\sqrt {{\rm{det}}\left( {{\delta _{\widehat \mu \widehat \nu }} - {{\not \partial }_{\widehat \mu }}{X^0}{{\not \partial }_{\widehat \nu }}{X^0} + {{\not F}_{\widehat \mu \widehat \nu }}} \right)} \end{array}$

Note however the above Dirac-Born-Infeld S-brane action differs from the usual D-brane action in two deep respects: first, the action is defined on a Euclidean world-volume, and second the kinetic term of the transverse scalar field ${X^0}$ has a wrong sign since it represents time translation. Covariantizing the Dirac-Born-Infeld S-brane action reduces the lagrangian to $\sqrt {{\rm{det}}\left( {g + \not F} \right)}$

with $g$ the induced metric on the brane. It differs from the usual DBI lagrangian only by a factor of $i$ , and therefore has the same equations of motion. Finally, I must show that this transversality has no D-brane charge at future infinity. Take the Ramond-Ramond coupling for an S-brane to be the same as that for a D-brane. So, the coupling of RR fields to the particular S-brane above is

$\mu \int {A = \mu \int {{A^{p + 1r\,\Omega }}} } {r^{p - 1}}dr\,d{x^{p + 1}}d{\Omega _{p - 1}}$

Transforming $r$ into the embedding time ${X^0}$ , it follows that

$\mu \int {{A^{p + 10\,\Omega }}} {\left( {\frac{{{X^0}}}{{{c_p}}}} \right)^{ - \frac{{p - 1}}{{p - 2}}}}d{X^0}d{x^{p + 1}}d{\Omega _{p - 1}}$

hence the D-brane charge of this solution shrinks to zero at future infinity due to

$\int {{A^p}} xH = \frac{n}{{2\pi \alpha '}}\int {d{x^{p + 1}}} d\,{\Omega _{p - 1}}$

Deep point is that in the T-dual picture by compactifying ${x^{p + 1}}$ , ${A_{p + 1}}$ becomes a spatial coordinate, and the S-brane solution

${X^0} = {A_{p + 1}} = \frac{{{c_p}}}{{{r^{p - 2}}}}$

implies that although by definition S-branes are spacelike objects, they are however constructed using the open string tachyon and hence governed by the open string metric and have time-like holographic embedding on the brane-bulk, thus deriving a duality between brane-world cosmology and quantum cosmology with dimensional reduction where branes dynamically drive and dominate solutions to 4-D quantum cosmological equations.