In this post, the mathematics applies to both, Randall-Sundrum-1and-2 models, hence I will not distinguish between them here. One of the most powerful aspects of M-theory's braneworld scenarios is that the bosonic and fermionic fields of the Standard Model of physics can be interpreted as low-lying Kaluza-Klein excitations of Randall-Sundrum bulk fields, after extra dimensional modulus stabilization, and recalling that Randall-Sundrum bulk/brane interactions yield a very deep solution to the EW hierarchy problem. Start with the theory defined by the following action:

\[S = \frac{1}{2}\int {{d^4}} x\int\limits_{ - \pi }^\pi {d\phi \sqrt G } \left( {{G^{AB}}{\partial _A}\Phi {\partial _B}\Phi - {m^2}{\phi ^2}} \right)\]

with the bulk field given by:

\[\Phi \left( {x,\phi } \right) = \sum\limits_n {\frac{{{\Upsilon _n}\left( \phi \right)}}{{\sqrt {{\tau _c}} }}} \]

where generally, the bulk action, with worldsheet-uplift, is given by:

\[\begin{array}{l}{S_B} = - \frac{1}{{2{k^2}}}\int {{d^D}} X\left\{ {\frac{1}{2}} \right.\left[ {{\eta ^{\mu \rho }}} \right.{\eta ^{\nu \sigma }} + {\eta ^{\mu \sigma }}{\eta ^{\nu \rho }}\\ - \frac{2}{{D - 2}}{\eta ^{\mu \nu }}\left. {{\eta ^{\rho \sigma }}} \right]{h_{\mu \nu }}{\partial ^2}{h_{\rho \sigma }}\left. { + \frac{4}{{D - 2}}\bar \Phi {\partial ^2}\bar \Phi } \right\}\end{array}\]

and \({\Upsilon _n}\left( \phi \right)\) satisfying:

\[\int\limits_{ - \pi }^\pi {d\phi {e^{ - \sigma \left( \phi \right)}}} {\Upsilon _n}\left( \phi \right){\Upsilon _m}\left( \phi \right) = {\delta _{nm}}\]

with a Dirac-Born-Infeld brane interaction term:

\[{S_{BI}} = - {\tau _\rho }\int {{d^{p + 1}}} \xi \left( {\left( {\frac{{2p - D + 4}}{{D - 2}}} \right)\bar \Phi - \frac{1}{2}{h_{aa}}} \right)\]

which, after integration by parts and upon substituting \({e^{ - \sigma \left( \phi \right)}}\) in our action, we get the Horava-Witten action variant:

\[\begin{array}{c}S = \frac{1}{2}\int {{d^4}} x\int\limits_{ - \pi }^\pi {{r_c}d\phi } \left( {{e^{ - 2\sigma \left( \phi \right)}}} \right.\\{\eta _{\mu \nu }}{\partial _\mu }\Phi {\partial _\nu }\Phi + \frac{1}{{r_c^2}}\left( {{e^{ - 4\sigma \left( \phi \right)}}\partial \Phi } \right)\\ - {m^2}{e^{ - 4\sigma \left( \phi \right)}}\left. {{\Phi ^2}} \right)\end{array}\]

Now, the bulk fields manifest themselves to 4-D 'observers' as infinite towers of scalars \({\psi _n}\left( x \right)\) with masses \({m_n}\). After change of variables to:

\[\left\{ {\begin{array}{*{20}{c}}{{z_n} = {m_n}{e^{\sigma \left( \phi \right)}}/k}\\{{f_n} = {e^{ - 2\sigma \left( \phi \right)}}{\Upsilon _n}}\end{array}} \right.\]

our actions reduce to two interaction terms:

\[S_{{\mathop{\rm int}} }^G = \int {{d^4}} x\int\limits_{ - \pi }^\pi {d\phi \sqrt G } \frac{\lambda }{{{M^{5m - 5}}}}{\left( {{G^{AB}}{\partial _A}\Phi {\partial _B}\Phi } \right)^m}\]

and:

\[\begin{array}{l}S_{{\mathop{\rm int}} }^\Upsilon = \int {{d^4}} x\int\limits_{ - \pi }^\pi {{r_c}} d\phi {e^{ - 4\sigma \left( \phi \right)}}\frac{\lambda }{{{M^{5m - 5}}}} \cdot \\\psi _n^{2m}\left( {\frac{{{{\left( {{\partial _\phi }{\Upsilon _n}} \right)}^2}}}{{r_c^3}}} \right)\end{array}\]

where we have

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with the bulk field given by:

where generally, the bulk action, with worldsheet-uplift, is given by:

and satisfying:

with a Dirac-Born-Infeld brane interaction term:

which, after integration by parts and upon substituting in our action, we get the Horava-Witten action variant:

Now, the bulk fields manifest themselves to 4-D ‘observers’ as infinite towers of scalars with masses . After change of variables to:

our actions reduce to two interaction terms:

and:

where we have:

and the Bessel functions of order:

yield the standard Bertotti-Robinson-solutions. Hence, we have:

with a normalization factor. That the differential operator on the LHS of:

is self-adjoint means that the derivative of is continuous at the orbifold fixed points, giving us:

Four-dimensionally, these induce couplings between the Kaluza-Klein modes and so the exponential factor in:

where are Lorentz coordinates on the four-dimensional surfaces of constant thus plays an essential role in determining the effective scale of the couplings. If the Planck scale sets the scale of the five-dimensional couplings, the low-lying Kaluza-Klein modes will have TeV-range self-interactions.

Now, a Klebanov-Strassler geometry naturally arises by considering string theory compactification on where is the Einstein manifold in five dimensions, with the interaction-Lagrangian of the massless Klebanov-Strassler field and the brane fields fermions is:

which, after integrating over the extra dimensional part, the effective 4-D Lagrangian reduces to:

with the fundamental Planck scale and the 4-D Planck scale related as

Hence, in light of the Klebanov-Strassler/Randall-Sundrum throat-bulk isomorphism, this defines a background geometry given by:

with and the induced metric on the hidden and visible brane-sectors, the 5-D metric, with the 5-D Planck scale, the cosmological ‘constant’, the scalar field and the corresponding potential.

Working in the -warp-factor metric:

the corresponding 5-D Einstein and scalar field equations are:

and

with the index over the branes and our boundary-conditions of and are given by:

To analytically solve in the backreacted Randall-Sundrum model-type, we use the quadratic/quartic bulk/brane dualized potential:

with:

Now we can derive solutions:

where is the scalar field on the Planck brane. Hence, and are given by:

and

We can now address the modulus stability of the braneworld. Substituting into:

gives us the 4-D potential for the radion:

One then achieves inter-brane stabilization by minimizing the above potential with respect to the radion:

Hence, for the modulus field , the stabilization condition is:

Note now, in a backreacted RS model,

has no minima that is consistent with inflationary coupling-running. Thus, a quartic term of the bulk stabilising field potential must be coupled to the action factoring the mass of the scalar field:

Consider brane-fluctuations localized at stable inter-brane separation as a function of brane-coordinates. The metric is hence:

The KK-modified brane warping is given by:

for the 5-D modulus brane angular coordinates. The Einstein-Hilbert action now is given by:

and by integrating over the 5th dimensional scalar field, we get

with kinetic sub-part:

where is the normalised radion field:

Thus, coupling the mass to the effective potential term and the inter brane measure gives us:

and:

hence yielding the mass term:

Standard Model physics naturally arises now. One first derives the scalar radion field via interaction terms in the Standard Model, since the metric:

implies that the visible RS brane:

and:

couple to the Higgs sector of the SM via the action:

where is the Higgs field. Normalizing via:

thus reduces our action to:

where:

By an Euler-Lagrange derivation, we get the Higgs-field energy-momentum tensor:

thus coupling the radion to the Higgs via:

It is straightforward now to generalize to all fields of the Standard Model. Let be any field

By the above Higgs method, the corresponding RS-SM coupling term is:

Hence, yielding the coupled action:

Going back to the Klebanov-Strassler/Randall-Sundrum throat-bulk isomorphism, a stack of N D3-branes placed at the singularity backreacts on the KS-geometry, creating a warped background with the following ten dimensional line element

with the metric

and the warp factor is

and

the deep part is that this AdS background is an explicit realization of the Randall-Sundrum scenario in string theory

that I discussed here and here. So in line with the AdS/CFT duality, the geometry

has a dual gauge theory interpretation

namely, an gauge theory coupled to bifundamental chiral superfields, and adding D5-branes wrapped over the inside , then the gauge group becomes

giving a cascading gauge theory. The three-form flux induced by the wrapped D5-branes – fractional D3-branes – satisfies

and the Klebanov-Strassler warp-throat factor is

with

Thus from:

one derives the wave-function and the superposition-principle for every SM field from Kirchhoff ’s integral theorem. That gets very philosophically deep in the context of the Wheeler–DeWitt equation and the Hartle–Hawking wave function , since as one might expect, boundary conditions and the potential condition corresponding to the path-integral of:

are highly problematic, to say the least, though in upcoming posts, I will show how M-theory successfully deals with both via geometric surgery/quantum engineering methods in homological mirror symmetry.

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]]>T-branes are supersymmetric intersecting brane configurations such that the non-Abelian Higgs field \(\Phi \) that describes D-brane deformations is not diagonalisable and satisfies nilpotency conditions where the worldvolume flux has non-commuting expectation values and their worldvolume adjoint Higgs field is given a VEV that cannot be captured by its characteristic polynomial, and thus derive their importance from the fact that heterotic string compactifications are dual to T-branes in F-theory. Let's probe their dynamics. Starting with the D-term potential:

\[\begin{array}{c}{{\hat V}_D} = \frac{1}{{2{\mathop{\rm Re}\nolimits} \left( {{f_h}} \right)}}{\left( {\sum\limits_j {{q_{{\phi _j}}}{\phi _j}\frac{{\partial K}}{{\partial {\phi _j}}} + M_P^2\sum\limits_j {{q_{hj}}} } } \right)^2}\\ = \frac{\pi }{{{\mathop{\rm Re}\nolimits} \left( {{T_h}} \right)}}{\left( {\sum\limits_j {{q_{{\phi _j}}}\frac{{{{\left| {{\phi _j}} \right|}^2}}}{s} - {\xi _h}} } \right)^2}\end{array}\]

with the \(U\left( 1 \right)\)-charge:

\[{q_{hj}} = \frac{1}{{l_s^4}}\int_{{D_h}} {{{\hat D}_j}} \wedge {F^G}\]

and \({F^G}\) the gauge flux that yields the Fayet-Iliopoulos term:

\[\begin{array}{l}\frac{{{\xi _h}}}{{M_P^2}} = \frac{{{e^{ - \phi /2}}}}{{4\pi \mathcal{V}}}\frac{1}{{l_s^4}}\int_{{D_h}} {J \wedge {F^G}} = \frac{1}{{2\pi }}\sum\limits_j {\frac{{{q_{hj}}}}{\mathcal{V}}} \\ = - \sum\limits_j {{q_{hj}}} \frac{{\partial K}}{{\partial {T_j}}}\end{array}\]

where the D-brane partition function for closed strings is given by:

\[P_{{\rm{int}}}^{Dp} \equiv Z = \sum\limits_{\gamma = 0}^\infty {\underbrace {\int {{D^K}\gamma {{D'}^K}X{e^{S_{cld}^s}}} }_{{\rm{Topologies}}}} \]

with a non-Abelian D-term:

\[D_{\hat A}^K = \int_{\tilde S} {\rm{P}} \left[ {{\rm{Im}}{e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}} \wedge \sqrt {{{\rm{A}}^\prime }(\Gamma )/{{\overline {\rm{A}} }^\prime }({\rm{N}})} {\rm{ }}\]

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with the -charge:

and the gauge flux that yields the Fayet-Iliopoulos term:

where the D-brane partition function for closed strings is given by:

with a non-Abelian D-term:

and

is the first Pontryagin class-term, and is the flat space Kähler form:

where is given by:

Then the non-Abelian profiles for and must satisfy the 7-brane functional equations of motion. Non-Abelian generalisation of:

are built up as follows. Write locally:

and localize the integral in:

as:

thus,

the non-Abelian generalisation of and have both the form of the D7-brane Chern-Simons action and hence satisfy the T-brane equation of motion

So effectively, we have a Kähler-equivalence of the derivatives in the pull-back with gauge-covariant ones, yielding:

with the inclusion of the complex Higgs field , and represents the symmetrization over gauge indices.

In this local description, the Higgs field is given by:

where is a matrix in the complexified adjoint representation of and its Hermitian conjugate. Thus, locally, we have:

with:

a Kähler coordinate expansion of and gives us, after inserting it in:

the following:

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

Fixing our induced Dp-brane worldvolume metric:

we can write the Dirac-Born-Infeld action as:

which is a Higgsed gauge theory in dimensions with scalar fields. Thus, by dimensional reduction, this action is equivalent to a Yang-Mills gauge theory in 10-spacetime-dimensions with action:

with:

and the action is invariant under the supersymmetric transformations:

with the infinitesimal Majorana-Weyl spinor. By double-gauging, we get our desired Dp-brane action:

Crucially, note that the theory contains intersecting D2-D4-branes, since in the Casimir representation, the open string worldsheet boundary is a vertex vacuum connection coupled to a closed string state. This is the worldsheet-state correspondence in F-theory. Hence, the n-th loop open string Casimir force is equivalent to the n-th tree-level closed string charge exchange between two D-branes. It follows that the complete action of the Ramond-Ramond D-brane is an integral over the full space :

Hence, the gauged supergravity action is derivable as:

with:

and is the Ramond-Ramond potential, thus yielding the Chern-Simons action:

The non-Abelian D-term thus takes the form:

In the local patch on the C-manifold, we take the flat-space-Kähler-form:

and decompose the Kähler-background B-field as:

with:

thus giving us:

with the Abelian pull-back to given by:

Hence we have:

Now: realize that is a zero-form and does not have transverse-legs to , and thus the pull-back has a trivial action. So, after solving:

the D-term equations amount to with:

and with the -field vanishing on the sheave of the C-manifold, one gets a reduction to:

which yields a non-Abelian -corrected Chern-Simons action for a stack of D7-branes with both terms at leading order in

entailing that for matrix algebras:

they are the matrix products in the fundamental representation of

and so the -corrections on D-terms with the gauge flux F diagonalization yield the D-term equations:

Hence, the -corrections are given entirely by the abelian pull-back of the Kähler-form to

establishing a deep Wess–Zumino-based connection between T-branes and Abelian gauge field theory. This has vast implications for building realistic N/M-stack intersecting Dp-brane theories embedding the Standard Model of particle physics as well as that of cosmology.

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]]>Let me work in the large-warped Randall-Sundrum D-brane scenario and consider how an \({S^1}/{Z_2}\) orbifolding leads to a non-singular cosmic bounce on the brane thus alleviating the rip of the big-bang singularity. The Randall-Sundrum 5-D braneworld action is:

\[\begin{array}{*{20}{c}}{S = \int {{d^5}} x\sqrt { - {g_{\left( 5 \right)}}} \left[ {{M^3}{R_{\left( 5 \right)}} - \Lambda } \right]}\\{ + \int {{d^4}x} \sqrt { - {g_{\left( 4 \right)}}} {\rm{ \tilde L}}_{Brane}^{{\rm{large}}}}\end{array}\]

\(M\) the 5-D Planck mass, \({M^3} = 1/\left( {16\pi {G_5}} \right)\), and \(\Lambda \) the cosmological constant in the bulk, yielding the metric on the brane:

\[\begin{array}{c}d{s^2} = {e^{ - 2\left| y \right|/L}}\left[ {{\eta _{\mu \nu }} + {h_{\mu \nu }}\left( {x,y} \right)} \right] \cdot \\d{x^\mu }d{x^\nu } + d{y^2}\end{array}\]

with \(L\) being the radius of AdS, defined by:

\[R_{MNPQ}^{\left( 5 \right)} = - \frac{1}{{{L^2}}}\left( {g_{MP}^{\left( 5 \right)}g_{NQ}^{\left( 5 \right)} - g_{MQ}^{\left( 5 \right)}g_{NP}^{\left( 5 \right)}} \right)\]

thus allowing us to derive the CFT-brane relation:

\[\begin{array}{l}{h_{\mu \nu }}\left( p \right) = - \frac{2}{{L{M^3}}}{\Delta _3}\left[ {{T_{\mu \nu }}\left( p \right) - \frac{1}{2}{\eta _{\mu \nu }}{T^\alpha }_\alpha \left( p \right)} \right]\\ - \frac{1}{{{M^3}}}{\Delta _{KK}}\left( p \right)\left[ {{T_{\mu \nu }}\left( p \right) - \frac{1}{3}{\eta _{\mu \nu }}{T^\alpha }_\alpha \left( p \right)} \right]\end{array}\]

where the dynamics of N-parallel topologically intersecting Dp-branes with gauge group \(U\left( N \right)\) is:

\[\begin{array}{l}S = - \frac{{{T_p}{g_s}{{\left( {2\pi \alpha '} \right)}^2}}}{4}\int {{{\rm{d}}^{p + 1}}} \xi {\rm{tr}}\left( {{F_{ab}}{F^{ab}} + } \right.\\2{D_a}{\Phi ^m}{D^a}{\Phi _m} + \sum\limits_{m \ne n} {{{\left[ {{\Phi ^m},{\Phi ^n}} \right]}^2} + \left. {{\rm{fermions}}} \right)} \end{array}\]

with the Yang-Mills potential

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the 5-D Planck mass, , and the cosmological constant in the bulk, yielding the metric on the brane:

with being the radius of AdS, defined by:

thus allowing us to derive the CFT-brane relation:

where the dynamics of N-parallel topologically intersecting Dp-branes with gauge group is:

with the Yang-Mills potential being:

and we have:

The Einstein field equations in the RS theory are derived for both, the 2-stack 3-branes as well as inter-brane separation. Our 5-D ADS spacetime geometry is an

orbifolding and our branes are localized at orbifolded fixed points:

with the Planck brane. The action is hence given by:

with the metric being:

with

the spacetime warped brane factor along the extra dimensions. From this, one gets the Einstein field equations as:

with the constraint:

We also impose the condition that the brane curvature radius is much larger than the bulk curvature :

Visually, we get the following model:

First, let us recall the remarkable way in which the Randall-Sundrum scenario solves the hierarchy problem. I will make it easy by simply quoting Graham D. Kribs:

Randall and Sundrum (RS) proposed a fascinating solution to the hierarchy problem. The setup involves two 4D surfaces (“branes”) bounding a slice of 5D compact AdS space taken to be on an S1/Z2 orbifold. Gravity is effectively localized one brane, while the Standard Model (SM) fields are assumed to be localized on the other. The wavefunction overlap of the graviton with the SM brane is exponentially suppressed, causing the masses of all fields localized on the SM brane to be exponentially rescaled. The hierarchy problem can be solved by assuming all fields initially have masses near the 4D Planck scale, and arranging that the exponential suppression rescales the Planck mass to a TeV on the SM brane. This requires stabilizing the size of the extra dimension to be about thirty-five times larger than the AdS radius. Goldberger and Wise proposed adding a massive bulk scalar field with suitable brane potentials causing it to acquire a vev with a nontrivial x5-dependent profile. The desired exponential suppression could be obtained without any large fine-tuning of parameters. Fluctuations about the stabilized RS model include both tensor and scalar modes

Now, integrating over the dimensions yields the effective 4-D action:

with:

and since:

is the induced RS visible-sector brane Ricci scalar, we have a Brans-Dicke type theory. Our metric equation:

splits the hidden and the visible RS sectors along a path as such:

and is our 4-D modulus field and is identical to the field occurring in:

From the effective action, one can derive the scalar and gravitational equations of motion:

where is the Einstein tensor and the corresponding covariant derivatives emerge on the visible-sector brane metric .

Visually:

Now take the Friedmann–Robertson–Walker brane-metric:

with the cosmic scale factor and we switched to polar coordinates. Given the ansatz defined above by our metric, our field equations reduce to:

with overdot being and is the Hubble parameter and is the 4-D RS modulus. We introduce conformal time defined via:

Hence, we get the reduction to:

and after integrating, we get a solution:

If we define via:

then:

reduces to:

Hence, expressing the Friedmann–Robertson–Walker-equations in terms of , we obtain solutions to the 4-D RS modulus and cosmic scale function in various ways. Here are four.

One, by integrating:

giving us the solution:

Another solution can be obtained by dividing both sides of:

we get the integral of the 4-D RS modulus:

thus giving us:

where and are our integration constants.

Yet a third solution is gotten by substituting and with respect to conformal time in:

to obtain:

and the crucial one involves:

with respect to cosmic time:

Since has a minimum that is non-zero for , it follows that the presence of warped extra dimension allows a non-singular bounce of the scale factor with respect to cosmic time in the Einstein-Friedmann–Robertson–Walker 4-D universe. And from:

we have:

which sets the constraints on the stability of the 4-D RS modulus. One introduces a time dependent scalar brane field in the bulk with action:

The hidden and visible bulk-brane interaction terms are thus:

and:

with:

The above action leads to:

and it gives us, in the large delta limits, the boundary conditions:

and:

Hence, the stabilizing scalar field solution is given by:

with:

Hence, our boundary conditions give us:

and given the time-dependence of the scalar solution:

we get:

inserting back into solutions of and , we get the bulk-field equation:

where is given by:

Thus, the 4-D RS modulus stabilization can be carried in this model via the field . Minimizing the 5-D RS modulus potential, we get:

Thus, the branes are maximally stabilized by the bulk massive scalar field . So the solution above to yields the asymptoticity of as such:

hence, we get the following two conditions:

and:

thus, the 4-D RS stabilized modulus (RSSM) condition is:

and we have:

which yields a solution to the gauge hierarchy problem. Moreover, all D-brane theories share the property that a large warped brane leads to a solution of the initial big-bang singularity problem, basically, as Edward Witten keeps stressing, in a structurally similar way in which the topology of string worldsheets solves the singularity problem — UV-divergences — in particle interactions — as represented by Feynman diagrams — which are isomorphic to 1-D manifolds with a 1-D general relativity living on them. In our situation, one can see that by substituting into the Freidmann equation:

Now since the Ramond-Ramond RS-warped brane action:

is a smoothing of the Hubble parameter and since the Einstein field equations in the RS theory are derived for both, the 2-stack 3-branes as well as the inter-brane separation wall, then the Dp-brane/Ramond-Ramond field-coupling action:

implies that the bounce is guaranteed since it is a universal property of solutions to the above -Freidmann equation. This is just a generalization of Witten’s point above about the string-worldsheet topology/Feynman diagrammatic geometry to Dp-braneworld scenarios applied to the RS theory.

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]]>Continuing from my last post where I discussed the triangular interplay between string-string duality, string field theory, and the action of Dp/M5-branes, here I shall discuss Stueckelberg string fields and derive the BRST invariance of the Landau-Stueckelberg action. Recalling that the action of M-theory in the Witten gauge is:

\[\begin{array}{l}{S_M} = \frac{1}{{{k^9}}}\int\limits_{{\rm{world - volumes}}} {{d^{11}}} \sqrt {\frac{{ - {g_{\mu \nu }}}}{{ - \gamma }}} {T_p}^{10}d\Omega {\left( {{\phi _{INST}}} \right)^{26}}\left( {{R_{icci}} - A_\mu ^H\frac{1}{{48}}G_4^2} \right) + \\\sum\limits_{Dp} {D_\mu ^{SuSy}} {e^{ - H_3^b}}/S_{Dp}^{WV} + \sum\limits_{Dp} {D_\nu ^{SuSy}} {e^{H_3^b}}/S_{Dp}^{SV}\end{array}\]

with \(k\) the kappa symmetry term, \({g_{mn}}\) the metric on \({M^{11}}\), and \({x^m}\) the corresponding coordinates with \({B_{mnp}}\) an antisymmetric 3-tensor. Hence, the worldvolume \({M^3}\) is:

\[R \times {S^1} \times {S^1}/{Z_2}\]

and the worldsheet action:

\[{S_{het}} = {S_{st}} + {S_{KK}} + {S_{\bmod }}\]

being the sum of three terms:

\[{S_{st}} = \int {{d^2}} \sigma \frac{1}{2}\left( {{g_{mn}}{\eta ^{ij}} + {b_{mn}}{\varepsilon ^{ij}}} \right){\partial _i}{x^m}{\partial _j}{x^n}\]

\[{S_{KK}}\int {{d^2}} \sigma {\varepsilon ^{ij}}{\partial _i}{x^I}{\partial _j}{x^m}A_m^I\]

\[{S_{\,\bmod \,}} = \int {{d^2}} \sigma \frac{1}{2}\left( {{g_{IJ}}{\eta ^{ij}} + {b_{IJ}}{\varepsilon ^{ij}}} \right){\partial _i}{x^J}{\partial _j}{x^I}\]

and the index I = 1, ... , 22 labels 22 gauge fields: 16 coming from the internal dimensions of the heterotic string, and the other 6 gauge fields are the KK modes of the metric and antisymmetric tensor. The action \({S_{\bmod }}\) has a massless spectrum given by moduli fields corresponding to deformations of the Narain lattice and thus take values in the group manifold:

\[\frac{{SO\left( {19,3} \right)}}{{SO\left( {19} \right) \times SO\left( 3 \right)}}\]

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with the kappa symmetry term, the metric on , and the corresponding coordinates with an antisymmetric 3-tensor. Hence, the worldvolume is:

and the worldsheet action:

being the sum of three terms:

and the index I = 1, … , 22 labels 22 gauge fields: 16 coming from the internal dimensions of the heterotic string, and the other 6 gauge fields are the KK modes of the metric and antisymmetric tensor. The action has a massless spectrum given by moduli fields corresponding to deformations of the Narain lattice and thus take values in the group manifold:

Something deep has occurred: all the gauge fields of the action have appeared within a two-dimensional theory, andnota three-dimensional theory

which is precisely the long wavelength limit behavior of the **open** membrane:

the gauge fields are defined in terms of fields that live on 10-dimensional boundaries ofM-theory

In the **closed** membrane case:

the gauge fields are defined in terms of11-dimensional fields

which brought us to the connection between string field theory and Dp-branes. Recall that one derives the string propagator by an evaluation of the Witten super-symmetric quantum path integral on a fiber-strip with the Polyakov string action:

with:

for and the Regge parameter clear from context. In the proper-time gauge and the normal modes of the lapse and shift function in 2-D, the Polyakov metric has the following property:

allowing us to derive the open string field Polyakov propagator on the Dp-branes:

with:

and the momentum operators are given by:

Since open string end-points are topologically glued to Dp-branes, open strings must have inequivalent quantum states and thus, the string field has to carry the gauge group indices of :

where are the generators of the SU(N) group, with . Hence, the string propagator on multi-Dp-branes takes the following form, with contraction and indices ordering:

which yields the field theory action:

BRST-invariantly as:

Hence, the above field theory action implies that the string-string duality associates to every Dp–Brane a solution corresponding to the d–dimensional string–frame Lagrangian:

with the dilaton, the curvature of a (p + 1)–form gauge field:

where the two–index NS/NS tensor and the dual six-index heterotic five–brane tensor are given by:

and

Now we have the general form of a 10-D p-brane solution:

with:

and:

with

The general form of 11-D Mp–branes solutions, noting the absence of the dilaton field, with the following Lagrangian:

is:

Hence, the M2-brane solution is:

squaring the field strength gives the following M5-brane solution:

In the string-frame Ramond-Ramond gauge field Lagrangian:

Dp-brane solutions have the following form:

From the string-string duality above and , we can derive the kinetic term of Dp–branes in terms of the Born–Infeld action with the following form:

with the embedding metric and the gauge field world-volume curvature manifest, entailing the existence of a WZ/RR term that couples to Dp-branes:

and where the heterotic 5–brane, the IIA five–brane and the D5–brane dual potentials are given by:

Parallels for the M5-brane are formally similar. We have the quadratic kinetic term:

with the WZ term:

and the dual 6–form potential:

By the field-property of the Polyakov propagator on the Dp-branes:

combined with the string-string duality, we can prove that all Dp-and-Mn–brane solutions preserve half of the SUSY. With the SUSY rules for the gravitino and dilatino in the string-frame given by:

Let us consider the gauge covariantization of the proper-time gauge and the Ramond-Ramond gauge discussed above. The action for the covariant bosonic open string field theory is implicitly defined by the BRST operator :

with respect to the BPZ conjugation-derived inner product, where the string field has the following Fock space expansion:

where the following holds:

for the bosonic case, and:

are the associated space-time fields. We can now write the action as:

and is invariant under the gauge transformation:

with the gauge parameter being a Grassmann string field of

given as:

In terms of and , the gauge transformation is expressible as:

It follows then that:

is gauge invariant. Hence, in terms of , the action:

becomes:

Note that the gauge invariance of each of:

and

entails that the string field, by conformal gauge theory, has the following form:

and we also have:

Thus, our action becomes a sum of two gauge invariant terms:

and

Now, crucially:

is equivalent to a gauge invariant action of massless vector field

and by the metaplecticity of:

it follows that gauge transformation up to level N = 1 is expandible in terms of a gauge parameter as:

We perform now a Virasoro reparametrization of the evolving string surface as a transformation:

with

the wave-function of the string, which in string field theory, must be interpreted as a functional , giving us the functional action:

where the inner product is defined in terms of integrals over the whole string configuration space and the string field kinetic energy operator. By reparametrization invariance, we can derive the following:

and the following relations can be easily checked:

Now, what makes unique contrastively to is its invariance under an large group of extra symmetries in addition to reparametrization implicitly expressed by:

specifically, given by shifts:

and is a type of meta-gauge symmetry acting directly on the metaplectic phase space. Let us study some properties of this metaplectic gauge group as well as .

We expand in terms of the eigenstates of the mass operator:

where the state is annihilated by all . Then the string field functional can be written as:

The kinetic gauge of is given by the action of on new string functionals. At first order we get the following equation of motion:

and since satisfies:

it follows that:

has the property of linearized Yang-Mills gauge invariance and the following transformation laws can be derived:

where is the Chan-Paton field term.

With a 0-level state and an eigenstate of with eigenvalue , we have the following definition for the contravariant form

Now we must define:

satisfying:

with the 0-th projection operator:

Combining, we have at n-th-mass level a Klein-Gordon equation:

which is gauge-invariant, thus the existence of .

Adding the Stueckelberg string fields to the fundamental string field , we have the local Stueckelberg action:

We now define N-th-projection operators:

along with:

with

Now, since we have:

it follows that the Stueckelberg action:

is equivalent to the kinetic metaplectic gauge field action:

Now we introduce a new gauge:

At N=1 gauge field level, we thus have:

which is a hybrid Landau-Stueckelberg gauge. So for

this fixes the gauge invariance of the Stueckelberg action above.

Note that under this condition, since the following holds:

any string field:

satisfies the Landau-Stueckelberg gauge condition given:

with:

This in turn entails that:

holds.

The action for this gauge condition is then:

with:

with the odd/even Grassmann string fields and the projection operator defined implicitly by:

Putting all together, it follows that the action is BRST invariant.

The post String Field Theory, Gauge Theory and the Landau-Stueckelberg action appeared first on George Shiber.

]]>The D=6 string-string duality, crucial for allowing the interchanging of the roles of 4-D spacetime and string-world-sheet loop expansion, entails that there is an equivalence between the K-3 membrane action and the \({T^3} \times {S^1}/{Z^2}\) orbifold action. Here are some thoughts and reflections.

In the bosonic sector, the membrane action is:

\[\begin{array}{*{20}{l}}{S = {S_M} + \int_{\partial {M^3}} {\left\{ {\frac{1}{2}} \right.} \left( {{g_{mn}}{\eta ^{ij}} + {b_{mn}}{\varepsilon ^{ij}}} \right)}\\{{\partial _i}{x^m}{\partial _j}{x^n} + \frac{1}{2}\left( {{g_{IJ}}{\eta ^{ij}} + {b_{IJ}}{\varepsilon ^{ij}}} \right)}\\{{\partial _i}{x^I}{\partial _j}{x^J} + {\varepsilon ^{ij}}{\partial _i}{x^J}{\partial _j}{x^m}\left. {A_m^J(x)} \right\}}\end{array}\]

where:

\[\begin{array}{*{20}{l}}{{S_M} = \int_{{M^3}} {\left( {\sqrt { - {g_{mn}}{\partial _i}{x^m}{\partial _j}{x^n}} } \right.} + }\\{\frac{1}{6}{\varepsilon ^{ijk}}{\partial _i}{x^m}{\partial _j}{x^n}{\partial _k}{x^p}\left. {{B_{mnp}}} \right)}\end{array}\]

Recall I derived the total action:

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

which is highly non-trivial since Clifford algebras are a quantization of exterior algebras. Applying to the Einstein-Minkowski fibre-bundle, we get via Gaussian matrix elimination, an expansion of \({D^{SuSy}}\) via Green's-functions, yielding the on-shell action of M-theory in the Witten gauge:

\[\begin{array}{l}{S_M} = \frac{1}{{{k^9}}}\int\limits_{{\rm{world - volumes}}} {{d^{11}}} \sqrt {\frac{{ - {g_{\mu \nu }}}}{{ - \gamma }}} {T_p}^{10}d\Omega {\left( {{\phi _{INST}}} \right)^{26}}\left( {{R_{icci}} - A_\mu ^H\frac{1}{{48}}G_4^2} \right) + \\\sum\limits_{Dp} {D_\mu ^{SuSy}} {e^{ - H_3^b}}/S_{Dp}^{WV} + \sum\limits_{Dp} {D_\nu ^{SuSy}} {e^{H_3^b}}/S_{Dp}^{SV}\end{array}\]

with \(k\) the kappa symmetry term. With \({g_{mn}}\) the metric on \({M^{11}}\), and \({x^m}\) the corresponding coordinates with \({B_{mnp}}\) an antisymmetric 3-tensor. Hence, the worldvolume \({M^3}\) is:

\[R \times {S^1} \times {S^1}/{Z_2}\]

The bosonic sector lives on the boundary of the open membrane: two copies of \(R \times {S^1}\), which naturally couple to the U(1) connections \({A^J}\).

Now, double dimensional reduction of the twisted supermembrane on:

\[{M^{10}} \times {S^1}/{Z_2}\]

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]]>In the bosonic sector, the membrane action is:

where:

Recall I derived the total action:

which is highly non-trivial since Clifford algebras are a quantization of exterior algebras. Applying to the Einstein-Minkowski fibre-bundle, we get via Gaussian matrix elimination, an expansion of via Green’s-functions, yielding the on-shell action of M-theory in the Witten gauge:

with the kappa symmetry term. With the metric on , and the corresponding coordinates with an antisymmetric 3-tensor. Hence, the worldvolume is:

The bosonic sector lives on the boundary of the open membrane: two copies of , which naturally couple to the U(1) connections .

Now, double dimensional reduction of the twisted supermembrane on:

of

entails that the bosonic sector is that of the heterotic string:

with gauge group indices I = 1, … , 16.

It gets interesting when we consider:

with dimension:

since the worldsheet action:

is now just a sum of three terms:

and the index I = 1, … , 22 labels 22 gauge fields: 16 coming from the internal dimensions of the heterotic string, and the other 6 gauge fields are the KK modes of the metric and antisymmetric tensor. The action has a massless spectrum given by moduli fields corresponding to deformations of the Narain lattice and thus take values in the group manifold:

Now, something fundamentally deep has occurred: all the gauge fields of the action have appeared within a two-dimensional theory, andnota three-dimensional theory

This is precisely the long wavelength limit behavior of the **open** membrane:

the gauge fields are defined in terms of fields which live on 10-dimensional boundaries ofM-theory

In the **closed** membrane case:

the gauge fields are defined in terms of11-dimensional fields

Hence, the gauge fields of the closed membrane must be defined over M3 and not over its boundary, unlike the closed membrane, whose action on is:

where is with the spacetime being .

Hence, the closed membrane action on reduces to:

with:

and

and since surfaces have no one-cycles, it follows that the three-form potential that appears in of the action:

can be expanded in terms of the cocycles of .

For the 22 2-cocycles of , one can decompose in a similar way for the two-form potential:

with I = 1, …, 22 labeling the two-cycles of . So after insertion into , we can derive:

Applying reparametrization invariance, one can set:

where is a worldvolume coordinate, and now one performs a dimensional reduction of:

Here are the key propositions relevant to the** membrane/string duality** of the low energy theory in D=7.

- the kinetic terms for the gauge fields in D=7 supergravity are:

derived by a split of the 4-4 field strength , of the 11-dimensional supergravity action:

from the following term:

- Membrane/string duality in D=7 requires the existence of a point in the moduli space of where all the 22 gauge fields are enhanced via U(1) gauging: this is key to preserving kappa symmetry. Thus, at the point in the moduli space when the 22 two-cycles vanish the following holds:

- Hence, dimensional reduction yields:

So, the S-duality map:

takes:

to:

and is equivalent to the term in:

So, the above map acts on the induced metric on the worldvolume. It follows then that the term in in:

yields, after a double dimensional reduction of , the following:

with:

which yields an equivalence between:

and

Thus, the S-duality map that takes to also takes to the dimensionally reduced .

To achieve the matching of gauge sectors of the closed and open membrane, we must generate the gauge fields of the closed membrane before dimensionally reducing the theory, as opposed to the gauge fields of the open membrane, which are always generated within the two-dimensional theory. This explains the origin of strong-weak duality in string theory. The strong coupling limit of type IIA string is 11-dimensional supergravity which is believed to arise as the long wavelength limit of supermembrane theory. So, gauge fields present in the 3-dimensional theory will be strongly interacting, and will continue to be strongly interacting after dimensional reduction to a two-dimensional theory. However, the open membrane has its gauge fields appearing in two dimensional theories, which are therefore weakly interacting.

So, we must consider the spacetime part of the action for the closed membrane:

The term:

can be dimensionally reduced to:

which is equivalent to the first term in:

and the term:

maps to:

with and members of

Now, since the term is topological, and S-duality of the seven dimensional space entails:

then one can reduce:

to:

with the Hodge dual and in turn, allows us to further reduce to:

Therefore the b-term in the spacetimestring actionis a direct consequence of thedualityof the seven dimensional duality between 3- and 4-forms, and so the dimensional reduction of yields the term , and this is tantamount to mapping the closed membrane action on to the open membrane action on , thus D=6 string-string duality follows and both theories will have the samespacetime supersymmetrysince they have the same massless spectra

This naturally brings us to the connection between string field theory and Dp-branes. Recall that one derives the string propagator by an evaluation of the Witten super-symmetric quantum path integral on a fiber-strip with the Polyakov string action:

with:

for and the Regge parameter clear from context. In the proper-time gauge and the normal modes of the lapse and shift function in 2-D, the Polyakov metric has the following property:

allowing us to derive the open string field Polyakov propagator on the Dp-branes:

with:

and the momentum operators are given by:

Since open string end-points are topologically glued to Dp-branes, open strings must have inequivalent quantum states and thus, the string field has to carry the gauge group indices of :

where are the generators of the SU(N) group, with . Hence, the string propagator on multi-Dp-branes takes the following form, with contraction and indices ordering:

which yields the field theory action:

BRST-invariantly as:

Hence, the above field theory action implies that the string-string duality associates to every Dp–Brane a solution corresponding to the d–dimensional string–frame Lagrangian:

with the dilaton, the curvature of a (p + 1)–form gauge field:

where the two–index NS/NS tensor and the dual six-index heterotic five–brane tensor are given by:

and

Now we have the general form of a 10-D p-brane solution:

with:

and:

with

The general form of 11-D Mp–branes solutions, noting the absence of the dilaton field, with the following Lagrangian:

is:

Hence, the M2-brane solution is:

squaring the field strength gives the following M5-brane solution:

In the string-frame Ramond-Ramond gauge field Lagrangian:

Dp-brane solutions have the following form:

From the string-string duality above and , we can derive the kinetic term of Dp–branes in terms of the Born–Infeld action with the following form:

with the embedding metric and the gauge field world-volume curvature manifest, entailing the existence of a WZ/RR term that couples to Dp-branes:

and where the heterotic 5–brane, the IIA five–brane and the D5–brane dual potentials are given by:

Parallels for the M5-brane are formally similar. We have the quadratic kinetic term:

with the WZ term:

and the dual 6–form potential:

By the field-property of the Polyakov propagator on the Dp-branes:

combined with the string-string duality, we can prove that all Dp-and-Mn–brane solutions preserve half of the SUSY. With the SUSY rules for the gravitino and dilatino in the string-frame given by:

for IIA:

and for IIB:

Since the Killing spinor is given by:

where is a constant spinor.

End of proof.

Hence, the triangular interplay between string-string duality, string-field theory, and the action of Dp/M5-branes establishes a duality between 4-D spacetime and string-world-sheet loop expansion, entailing the equivalence between the K-3 membrane action and the orbifold action. Here is a classic by Edward Witten et al. on why that is important.

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]]>String/M-[F]-theory remains by far the most promising - only? - theoretical paradigm for both, grand unification and quantization of general relativity. With the Dp-action given by:

\[S_p^D = - {T_p}\int\limits_{{\rm{worldvolumes}}} {{d^{p + 1}}} \xi \frac{{{D_{\mu \nu }}L}}{{{\partial _{{v_a}}}}}{e^{ - {\Phi _{bos}}}}{\rm{de}}{{\rm{t}}^{1/2}}G_{ab}^{\exp \left( {H_{p + 1}^{{\rm{array}}}} \right)}\]

for contextualization, note that a necessary condition for the world-sheet Dirac propagator \({\delta ^{\left( 2 \right)}}\left( {{\sigma _i} - {\sigma _j}} \right)\):

\[S = i\int {{d^2}} {\sigma _1}{d^2}{\sigma _2}\sum\limits_{i,j = + , - } {{\psi _i}\left( {{\sigma _1}} \right)} {A_{ij}}\left( {{\sigma _1},{\sigma _2}} \right)\psi \left( {{\sigma _2}} \right)\]

to be integrable, is that the Seiberg vacuum fluctuation of the string world-sheet:

\[{S_\eta } = \frac{1}{\beta }\sum\limits_{\frac{{i2\pi }}{\beta }} {{{\left( {i\frac{{2n + 1}}{\beta }} \right)}^\pi }} W + \alpha '{R_{\left( 2 \right)}}\Phi \]

with

\[W \equiv {h^{mn}}{\partial _m}{X^a}{\partial _n}{X^b}{g_{ab}}\left( X \right)\]

and \(\beta \) the bosonic frequency, be analytically summable. The string world-sheet is given by:

\[{S_{ws}} = \frac{1}{{4\pi \alpha '}}\int\limits_{c + o} {d\tilde \sigma } d\tau '\sqrt h \left( {W + \alpha '{R_{\left( 2 \right)}}\Phi } \right)\]

A major problem is that by the Heisenberg's uncertainty principle:

\[\left( {\Delta A/} \right)\left( {\left| {\frac{{d\left\langle A \right\rangle }}{{dt}}} \right|} \right)\left( {\Delta H} \right) \ge \hbar /2\]

the string time-parameter on the world sheet \({\sigma _t}\) with Feynman propagator in Euclidean signature being:

\[\begin{array}{c}G\left( {x,y} \right) = \int_0^\infty {d{\sigma _t}} G\left( {x,y;{\sigma _t}} \right)\\ = \int {\frac{{{d^D}p}}{{{{\left( {2\pi } \right)}^D}}}} \exp \left[ {ip \cdot \left( {y - x} \right)} \right]\frac{2}{{{p^2} + {m^2}}}\end{array}\]

violates

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for contextualization, note that a necessary condition for the world-sheet Dirac propagator :

to be integrable, is that the Seiberg vacuum fluctuation of the string world-sheet:

with

and the bosonic frequency, be analytically summable. The string world-sheet is given by:

A major problem is that by the Heisenberg’s uncertainty principle:

the string time-parameter on the world sheet with Feynman propagator in Euclidean signature being:

violates the integrability condition for the action:

and hence, in light of the principle of superposition, as a function of runs into the Riemann-Lebesgue Lemma problem, given that the Fourier transform of :

is non-convergent, with real, since the quantization of spacetime is an anti-smoothing dynamical breaking of the Ricci scalar . Hence we get,

which is incoherent. To see this, note that the anti-smoothing of spacetime implies that cannot be recovered from via

and that implies that the gravitonic wave-propagation travels in spacetime at infinite speed, given

and by wave-particle duality and the violation of special relativity, the graviton provably cannot exist. Or, by quantum tunnelling and the fact that gravitons self-gravitate, we have the instantaneous collapse of spacetime to a zero-dimensional singularity. Pick your poison.

A solution is to integrate over orbibolds and derive the Lagrangian of N=1 supergravity by orbifoidal D-11 and D-10 SUGRA-Barbero coupled actions. Let us see how this works. One must begin by giving a description of the field contents and the degrees of freedom, which will turn out to be a crucial number. At first, note that in D=11, SUGRA has a simple action: using exterior algebraic notation for the anti-symmetric tensor fields , with the field strength , it is surprisingly:

with the Newtonian constant in 11 dimensions. By dimensional reduction, the Type IIA can be derived from (1). Note that there are D=10 supergravity theories with only SUSY which couple to D=10 super-Yang-Mills theory. We still do not have a workable Type IIB theory since it involves an antisymmetric field with a self-dual field strength. Nonetheless, one may still derive an action that involves both dualities of . Then, by imposing the self-duality as a supplementary equation, we get:

with field strengths: , , , , , with the self-duality condition .

Note that the above action arises from the string low-energy limit and:

naturally yields the NS-NS sector of the theory, while:

is derivable from the RR sector of the theory. Now, Type IIB supergravity theory is invariant under the non-compact symmetry group and the key is that this symmetry is not manifest in:

To make it so, one must redefine fields, from the string metric in (2) to the Einstein metric , along with a complexification of the tensor fields:

Now, the action is easily seen to be:

the metric and fields are invariant under the symmetry of Type IIB supergravity. The axionic dilaton field varies with a Möbius super-transformation:

and , self-rotate under the Möbius super-transformation, and can most clearly be visualized as a complex 3-form field :

The SUSY transformation for Type IIB supergravity on the fermion fields are of the following form, via Seiberg–Witten analysis, without a need for bosonic transformation laws, with the dilaton and the gravitino :

It is crucial to realize that in the context, the SUSY transformation parameter essentially has charge, implying that has necessarily, given unitarity, and has .

The geometry of superstring theory in the Ramond-Neveu-Schwarz setting is given by the bosonic world-sheet fields and the fermionic world-sheet fields , with expressing chirality**,** and , must be functions of local world-sheet coordinates . Both and vectorially transform under the irreducible representation of the Lorentz group. By using Gliozzi-Scherk-Olive holographic projections, the spacetime supersymmetric derivative can act on (1) and (2) above. It is more informative to work with orientable strings. Type II and heterotic string theories are perfectly suited in this context. Field interactions in second quantization arise from the orbifoidal splitting and joining of the world-sheets, and causality is maintained. Moreover, the genus for orientable world-sheets equals the number of Witten-handles. The world-sheet bosonic field naturally gives rise to a non-linear sigma model:

with being the square root of the Planck length, being the world-sheet metric, and being the Gaussian curvature. The world-sheet fermionic field axially gives rise to a world-sheet supersymmetric completion of the sigma model. It suffices to give its form on a flat world-sheet metric with a vanishing world-sheet gravitino field:

and being the Riemann tensor for the metric . Now the all too important SUSY-covariant derivatives can be derived:

with the Levi-Civita connections for . Now one is in a position to solve the Seiberg-Lebesgue problem via the functional integral over all and by integrating over all world-sheet metrics and world-sheet gravitini fields via the amplitude:

This clearly solves the Feynman propagation problem. And the upshot is that the vacuum expectation value of the dilaton field is , and the string vacuum expectation value of the string amplitude is** **given by the Euler number of the world-sheet :

with the genus, is a number that counts the orbifoidal puntures in spacetime as the string world-sheet propagates under quantum fluctuations. Thus, a genus world-sheet with no boundary gets a multiplicative contribution: , thus deriving another truly remarkable identity, , representing the closed string coupling constant which lives on a Dp-brane, and hence by the supersymmetry action and the Witten Index:

where is the fermion number, and the trace is over all bound and continuum states of the supersymmetric-Hamiltonian, and PBC being the periodic boundary conditions on both the fermionic and bosonic fields**, **we get a finite, causal SUGRA action in D=11/ D=10, thus solving the Seiberg-Lebesgue problem.

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