M-Theory And A Chern-Simons Derivation Of Open String Action Via Kappa-Clifford Analysis

By George Shiber Wednesday, August 19, 2015Saturday, August 17, 2019 In Grand Unification Physics, M Theory, Mathematical Physics, Philosophy Of Physics, Super String Theory, Supersymmetry chern, Clifford algebra

INDEED STEPHEN, AND THE ONLY ‘GAME IN TOWN’ AS WELL!

Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere”~ W. S. Anglin

At the end of this post, marvel at the beauty of M-Theory’s action and find out why it really is called ‘M’-theory. In this post, unlike the last one, I will derive the string world-sheet action for open strings, also in curved space-time, and there are deep differences between closed and open strings. For one, as we saw, the second quantized integral of the D-p-brane partition function for closed strings is:

$P_{{\rm{int}}}^{Dp} \equiv \not Z = \sum\limits_{\gamma = 0}^\infty {\underbrace {\int {{{\not D}^{SuSy}}\gamma {{\not D'}^{SuSy}}X{e^{S_{cld}^s}}} }_{{\rm{Topologies}}}}$

where ${\not D^{SuSy}}$ is the supersymmetry group covariant derivative.

and, since:

$\begin{array}{c}\delta S_{cld}^s = - \frac{1}{{2\pi \alpha '}}\int_{\partial E_S^5} {{d^2}} \sigma d\,\Omega {\left( {{\phi _{INST}}} \right)^2}{\varepsilon ^{\alpha \beta }}{{\not \partial }_\alpha }{X^\mu }{{\not \partial }_\mu }{\lambda _\nu }\\ = - \frac{1}{{2\pi \alpha '}}\int_{\partial E_S^5} {{d^2}} d\,\Omega {\left( {{\phi _{INST}}} \right)^{ - 1/2}}\sigma \,{{\not \partial }_\mu }X\nu {\left( {{\varepsilon ^{\alpha \beta }}{{\not \partial }_\beta }{X^{^\nu }}{\lambda _\mu }} \right)^{{e^{ - H_3^b}}}}\end{array}$

it follows that no topology in the sum is degenerate, and thus the closed string has a solvable action in curved space-time that needs no renormalization. Matters are not that simple for the open string. The main reason being, open strings have their ends attached to D-branes of lower and odd dimensionality: max = 9, and that is key to deriving the open string partition function:

$P_{{\mathop{\rm int}} }^{Dp}(op) \equiv \not Z = \sum\limits_\gamma ^\infty {\underbrace {\int {{{\not D}^{SuSy}}\gamma {{\not D}^{SuSy}}X{e^{S_{open}^s}}} }_{{\rm{Brane - worldvolumes}}}}$

where $S_{open}^s$ is given by:

$S_{open}^s = {\mkern 1mu} - \frac{1}{{4\pi {\alpha ^\dagger }}}\int_{\partial E_S^5} {{d^2}} \sigma d{\mkern 1mu} \Omega {\left( {{\phi _{INST}}} \right)^2}\sigma \sqrt { - \gamma } \left( {\phi \left( {{{\bar X}^\mu }} \right)} \right.{R_{icci}} + {\gamma ^{\alpha \beta }}{\not \partial _\alpha }{X^\mu }{g_{\mu \nu }}\left( {{{\overline X }^\nu }} \right) + \frac{1}{{\sqrt { - \gamma } }}{\varepsilon ^{ - {c_{2n}}/{Y_k}\left( {{{\cos }^2}\varphi } \right)}}{\not \partial _\beta }{\overline X ^\nu }{b_{\mu \nu }}{\left( {\overline X } \right)^2}$

with ${c_{2n}}$ and ${\Upsilon _\kappa }\left( {{{\cos }^2}\varphi } \right)$ the spherical harmonics on $E_S^5$ . Now, realizing that for open strings:

$P_{{\mathop{\rm int}} }^{Dp}(op) \equiv \not Z = \sum\limits_\gamma ^\infty {\underbrace {\int {{{\not D}^{SuSy}}\gamma {{\not D}^{SuSy}}X{e^{S_{open}^s}}} }_{{\rm{Brane - worldvolumes}}}}$

has a genus $g$ suppressed by the factor:

${e^{ - {\phi _{INST}}{g_{\mu \nu }}}} = {e^{ - 2{\phi _{INST}}\left( {{g_{\mu \nu }} - 1} \right)}}\quad \quad \quad {\rm{here: }}\quad \quad \quad {g_{\mu \nu }} = {e^{{{\left( {{\phi _{INST}}} \right)}^2}}}$

it follows that one can derive, given that D-p-branes have conformal invariance, the n-loop level equation of motion with conditions:

$\left\{ {\begin{array}{*{20}{c}}{{R_{\mu \nu }} = \, - \frac{1}{4}H_{3,\mu \lambda l}^bH_{3,\nu }^{b,\lambda \rho } + 2{{\not D}_{_\mu }}^{SuSy}{{\left( {{\phi _{INST}}} \right)}^2}{{\not D}_\nu }^{SuSy}{{\left( {{\phi _{INST}}} \right)}^{ - 1/2}}}\\{{{\not D}_\lambda }^{SuSy}{\phi _{INST}}H_{3,\lambda \mu \nu }^b - 4{{\not D}^{SuSy}}{{\left( {{\phi _{INST}}} \right)}^2}}\\{4{{\not D}^{SuSy}}{\phi _{INST}} - 4\left( {{{\not D}^{SuSy}}{{\left( {{\phi _{INST}}} \right)}^2}} \right) = {R_{icci}} + \frac{1}{{12}}{{\left( {H_3^b} \right)}^2}}\end{array}} \right.$

with:

$H_{3,\mu \nu \lambda }^b = 3\,{\not \partial _{\left[ {\mu {b_\nu }\lambda } \right]}}$

This all describes a string propagating in curved space-time that must be interpreted as a background derived from non-trivial massless open string modes, and thus pick up an infinite series of ${\alpha ^\dagger }$ correction terms and an infinite genus correction series – so it follows that the space-time effective action is:

${S_{effct}} = {\mkern 1mu} - \frac{1}{{2{\alpha ^\dagger }12}}\int {{d^{26}}} x{\mkern 1mu} d{\mkern 1mu} \Omega {\left( {{\phi _{INST}}} \right)^2}\sqrt { - \gamma } {e^{ - {\phi _{INST}}}}\left( {{R_{icci}}} \right. - 4{\left( {{\phi _{INST}}} \right)^{ - 1/2}} + \frac{1}{{12}}H_{3,\mu \nu \lambda }^bH_3^{b,\mu \nu \lambda } + ...$

Now, since open strings can geometrically interact to form closed strings, and since closed strings give rise to the graviton and their gravitational quantum modes, and additionally, since everything gravitates, we get a highly non-trivial warping of space-time – hence, to go further in the analysis, one must consider D-p-branes in closed string theory, since a D-p-brane is a p+1 dimensional world-volume in space-time on which open strings end. The boundary conditions are:

$\delta {X^i} = 0\quad \quad \quad {\rm{at}}\quad \quad \quad \sigma = (0,\pi )$

Let a D-p-brane parallel to the ${x^0},{x^1},...,{x^p}$ dimension, and so lives at a specific location in ${x^{p + 1}},...,{x^{p + 26}}$ dimension, we hence get the mode expansion of their fields:

${X^\mu } = {x^\mu } + 2{\alpha ^\dagger }p\,\tau d\,\Omega {\left( {{\phi _{INST}}} \right)^2} + \sqrt {2{\alpha ^\dagger }i} \sum\limits_{n \ne 0} {\frac{{a_n^\mu }}{n}} \,{e^{ - {c_{2n}}/{\Upsilon _\kappa }(\cos \varphi )}}\cos \left( {{c_{2n}}\sigma } \right)$

and on the transverse coordinates to the D-p-brane with boundary conditions:

${X^i} = {a^i}\quad \quad \quad {\rm{at}}\quad \quad \quad \sigma = (0,\pi )$

we get:

$\begin{array}{c}{\overline X ^i} = {x^i} + {w^i}\sigma + {\alpha ^\dagger }{p^i}\tau d\,\Omega \,{\left( {{\phi _{INST}}} \right)^i} + \sqrt {\frac{{{\alpha ^\dagger }}}{{2\pi i}}} \sum\limits_{n \ne 0} {\left( {\frac{{a_n^i}}{n}\,} \right.} {e^{ - {c_{2n}}/{\Upsilon _\kappa }(\cos \varphi )}} + \\\frac{{\overline a _n^i}}{n}{e^{ - \,{c_{2n/{\Upsilon _\kappa }}}(\cos \varphi )}}{e^{ - \,{{\left( {{\phi _{INST}}} \right)}^2}}}\left. {\left( {\tau - \sigma } \right)} \right)\end{array}$

And so, one gets the philosophical importance of D-p-branes … they are metaplectic solitons in closed string theory, and hence, by the von Neumann boundary condition, there is a natural coupling of the super-Higgs field $A_\mu ^H$ to the world-sheet of a string through its boundary:

${S_{open}} = {S_{cld}} + \int\limits_{{\rm{end - points}}} {d\tau } A_\mu ^H{\overline X ^\mu }{e^{ - H_3^b}}d\,\Omega {\left( {{\phi _{INST}}} \right)^{\exp \,({c_{2n}}/{\Upsilon _\kappa }(\cos \varphi )}}$

One finds then that $A_\mu ^H$ lives on a p+1 dimensional subspace with a ${\Upsilon _\kappa }(\cos \varphi )$ contribution. Thus, the world-volume action is:

$S_{Dp}^{WV} = {S_{cld}} + \int\limits_{{\rm{end - points}}} {d\tau } A_\mu ^H{\overline X ^\mu }d\,\Omega {\left( {{\phi _{INST}}} \right)^2}{e^{ - \left( {H_3^b} \right)/{\Upsilon _\kappa }(\cos \varphi )}} + {e^{{c_{2n}}/{\Upsilon _\kappa }(\cos \varphi )}}/H_3^b$

and by solving, one gets:

$\begin{array}{c}S_{Dp}^{WV} = \, - {T_p}\int {{d^{p + 1}}} x\,{e^{ - {\phi _{INST}}}}{\rm{Tr}}\left( {1 + \frac{1}{4}\left( {{F_{\mu \nu }} + d\,\Omega {{\left( {{\phi _{INST}}} \right)}^2}} \right) + {{\left( {2\pi {\alpha ^\dagger }} \right)}^{ - 1}}{b_{\mu \nu }}} \right) \cdot \\\left( {A_\mu ^H + {{\left( {2\pi {\alpha ^\dagger }} \right)}^{ - 1}}{b_{\mu \nu }}} \right) + \frac{1}{2}{{\not D}^{SuSy}}{\Upsilon _\kappa }(\cos \varphi ) - \frac{1}{4}\sum\limits_{i \times j} {\left[ {\Upsilon _{2\kappa }^i(\cos \varphi ,\Upsilon _{2\kappa }^j(\cos \varphi } \right]} \cdot \\\left[ {\Upsilon _{2\kappa }^j(\cos \varphi ),\Upsilon _{2\kappa }^i(\cos \varphi )} \right]\end{array}$

with ${T_p} \sim {\alpha ^\dagger }\frac{{p + 1}}{2}$ being the D-p-brane world-volume tension, and:

${F_{\mu \nu }} = {\not \partial _\mu }A_\mu ^H - {\not \partial _\nu }\overline A _\mu ^H + \left[ {A_\mu ^H,\Upsilon _{2\kappa }^i(\cos \varphi } \right]$

being the Yang-Mills field strength, and by a Paton-Chern-Simons factor, we get:

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

and since D-p-branes warp space-time, we can deduce:

$\begin{array}{c}\delta {S_{open}} = \, - \frac{1}{{2\pi {\alpha ^\dagger }}}\int\limits_{{\rm{end - points}}} {{d^2}\sigma } {\varepsilon ^{\alpha \beta }}{e^{ - {c_{2n}}/{\Upsilon _\kappa }(\cos \varphi )}}{{\not D}_\alpha }^{SuSy}{\overline X ^\mu }{{\not D}_\beta }^{SuSy}{X^\nu }{{\not \partial }_{\mu \nu }}{\lambda _{\mu \nu }}\\ = \, - \frac{1}{{2\pi {\alpha ^\dagger }}}\int\limits_{{\rm{world - sheets}}} {{d^2}} \sigma \,{{\not \partial }_\alpha }\left( {{\varepsilon ^{\alpha \beta }}d\,\Omega {{\left( {{\phi _{INST}}} \right)}^2}} \right){{\not D}^{SuSy}}{\overline X ^\mu }{\lambda _\nu } = \\ - \frac{1}{{2\pi {\alpha ^\dagger }}}\int\limits_{{\rm{world - volumes}}} {d\tau {{\not \partial }_0}} {X^\nu }{\lambda _\nu }\,{e^{ - H_3^b/S_{cld}^s}}\end{array}$

and finally, for the mathematical geeks ‘here’, whenever 2 or more D-branes coincide, there is a Clifford algebraic symmetry whose generators allow us to derive the total action:

$\begin{array}{l}{S^{Total}} = \frac{1}{{2\pi {\alpha ^\dagger }12}}\int\limits_{{\rm{world - volumes}}} {{d^{26}}} x\,d\,\Omega {\left( {{\phi _{INST}}} \right)^2}\sqrt {\frac{{ - {g_{\mu \nu }}}}{{ - \gamma }}} \,{e^{ - {c_{2n}}/{\Upsilon _\kappa }(\cos \varphi )}} \cdot \\\left( {{R_{icci}} - 4{{\left( {{{\not D}^{SuSy}}\left( {{\phi _{INST}}} \right)} \right)}^2}} \right) + \frac{1}{{12}}H_{3,\mu \nu \lambda }^bH_3^{b,\mu \nu \lambda }/A_\mu ^H + \sum\limits_{D - p - branes} {S_{Dp}^{WV}} \end{array}$

Finally one can see why M-Theory is called ‘M-Theory’, and appreciate the deep philosophical connection of the coupling with the super-Higgs field: because Clifford algebras are a quantization of exterior algebras: applying to Einsteinian space-time’s tangent bundle, their ‘M’-atrices, via Gaussian elimination, allow us to expand ${\not D^{SuSy}}$ , via Green-functions, and by SuSy, the M-Theory action is:

${S_M} = \frac{1}{{{k^9}}}\int\limits_{{\rm{world - volumes}}} {{d^{11}}} \sqrt {\frac{{ - {g_{\mu \nu }}}}{{ - \gamma }}} {T_p}^{10}{\mkern 1mu} d{\mkern 1mu} \Omega {\left( {{\phi _{INST}}} \right)^{26}}\left( {{R_{icci}} - A_\mu ^H\frac{1}{{48}}G_4^2} \right) + \sum\limits_{Dp} {\not D_\mu ^{SuSy}} {e^{ - H_3^b}}/S_{Dp}^{WV} + \sum\limits_{Dp} {\not D_\nu ^{SuSy}} {e^{H_3^b}}/S_{Dp}^{SV}$

with $k$ the kappa symmetry gravitonic term. That is where the ‘M’ comes from, and why it unifies all string theories and is the only complete, categorical ‘theory-of-everything’: you have the bosons, fermions and a Higgs-field coupling in ONE action-principle that includes the graviton, and hence describes quantum gravity: NO other Grand Unification Theory/Theory Of Everything can derive such an action with such ‘completeness’: in fact, any theory that does is provably model-theoretically isomorphic to M-Theory!