where is the orbifold delta function:

with the dilaton and are terms derived from the D5-brane backreaction and such that varying the orbifold function with respect to the Type-IIB action induces orbifold-compactifications that locally inject 4-D gauge actions written as:

Hence, the Ramond-Ramond coupling is given by:

and since for Type-IIB, is odd, the potential for the Type-IIB theory compactified on a Calabi-Yau threefold takes the form:

where the translational, rotational, and Chern-Simons 3-form of gauge-class:

are respectively:

and

which are derived by varying the Lagrangian density:

with respect to and . This yields us the crucial NS-NS field equations:

Noting that the Einstein-Hilbert terms in the metaplectic Riemann-Cartan formalism constitute systolic algebraic 1-forms as well as a super-Lie-algebraic dual of the Lorentz connection:

more precisely:

it follows that the corresponding dual-field strength is the 2-form Kähler torsion:

with curvature form:

The Poisson-Lie duality allows us to add Chern-Simons forms, and by gauging the super-Poincaré group, we get the desired Mielke-Baekler theory that solves for the Einstein-Cartan Lagrangian:

Now, combining the Chern-Simons VEV equations:

and:

by modularity, we get the torsion and Riemann-Cartan curvature, respectively:

where is the Picard constant:

and:

is the CS-Witten term. Now, coupling to matter fields, we get the torsion condition:

and the Riemann-Cartan form reduces to:

which yields the 4-D action for :

In order to show the CS-H Yang-Mills GUT construction modulo a Teichmüller orbifold, note that by the F/M-theory duality, flux-compactification yields moduli-stabilization via double-Higgsing and solving the Yukawa coupling integral-equation:

Now, taking the Hodge dual gives us the Hodge-Fukaya form:

and realizing that the elliptic fibration induces a Calabi-Yau potential as a polynomial in that yields via Kaluza-Klein reduction and -backreaction PBS conditions on the string spectrum to derive the SYM Green’s function and then coupling the Hodge-Fukaya form to the Heterotic action in the Einstein-frame:

where is the 3-flux form:

and is the Chern-Simons 3-form given by:

gives an action that is isomorphic for the class of Lorentzian manifolds to the path integral of F-theory, and visually, it gives rise to the following picture:

where the Type-IIB action is:

with the Neveu-Schwarz, Ramond-Ramond, and Chern-Simons actions are, respectively:

and

giving us the CS-H Yang-Mills D3-brane GUT model:

The key is the topological coupling of solutions to the Yukawa coupling integral-equation:

on the orbifolded fibration to:

and using the variation-principle with respect to the CS action term:

The GUT is hence achieved by solving the monodromy action-equation on the elliptic-curve line-bundle by -cusp D7-brane intersections on the base with divisors defining the elliptic singularities, where is the number of degeneracies of the torus-modulus as a varying function of the RR-C-form and the dilaton, and integrating the exact holomorphic 2-form :

giving us a K3-class model over an integral basis of 2-cycles and where the corresponding period integrals elliptic over 1-cycles:

factoring in the base modulus cusp -terms and summing over divisor points over resolutions of D-7-branes localized on the line degeneracies. Pictorially, we get our desired F/M-GUT as:

Note that the Hauptmodul-function maps the fundamental region to the 7-plane geometry via D7-brane backreaction as a function of the Type-IIB Jacobi-term of the -plane for:

guaranteeing moduli stabilization, and by the Heterotic/F-theory duality, topological mirror …

The post Chern–Simons Theory, the Orbifold Delta Function, and GUT-Models appeared first on George Shiber.

]]>where is the orbifold delta function:

with the dilaton and are terms derived from the D5-brane backreaction and such that varying the orbifold function with respect to the Type-IIB action induces orbifold-compactifications that locally inject 4-D gauge actions written as:

Hence, the Ramond-Ramond coupling is given by:

and since for Type-IIB, is odd, the potential for the Type-IIB theory compactified on a Calabi-Yau threefold takes the form:

where the translational, rotational, and Chern-Simons 3-form of gauge-class:

are respectively:

and

which are derived by varying the Lagrangian density:

with respect to and . This yields us the crucial NS-NS field equations:

Noting that the Einstein-Hilbert terms in the metaplectic Riemann-Cartan formalism constitute systolic algebraic 1-forms as well as a super-Lie-algebraic dual of the Lorentz connection:

more precisely:

it follows that the corresponding dual-field strength is the 2-form Kähler torsion:

with curvature form:

The Poisson-Lie duality allows us to add Chern-Simons forms, and by gauging the super-Poincaré group, we get the desired Mielke-Baekler theory that solves for the Einstein-Cartan Lagrangian:

Now, combining the Chern-Simons VEV equations:

and:

by modularity, we get the torsion and Riemann-Cartan curvature, respectively:

where is the Picard constant:

and:

is the CS-Witten term. Now, coupling to matter fields, we get the torsion condition:

and the Riemann-Cartan form reduces to:

which yields the 4-D action for :

In order to show the CS-H Yang-Mills GUT construction modulo a Teichmüller orbifold, note that by the F/M-theory duality, flux-compactification yields moduli-stabilization via double-Higgsing and solving the Yukawa coupling integral-equation:

Now, taking the Hodge dual gives us the Hodge-Fukaya form:

and realizing that the elliptic fibration induces a Calabi-Yau potential as a polynomial in that yields via Kaluza-Klein reduction and -backreaction PBS conditions on the string spectrum to derive the SYM Green’s function and then coupling the Hodge-Fukaya form to the Heterotic action in the Einstein-frame:

where is the 3-flux form:

and is the Chern-Simons 3-form given by:

gives an action that is isomorphic for the class of Lorentzian manifolds to the path integral of F-theory, and visually, it gives rise to the following picture:

where the Type-IIB action is:

with the Neveu-Schwarz, Ramond-Ramond, and Chern-Simons actions are, respectively:

and

giving us the CS-H Yang-Mills D3-brane GUT model:

The key is the topological coupling of solutions to the Yukawa coupling integral-equation:

on the orbifolded fibration to:

and using the variation-principle with respect to the CS action term:

The GUT is hence achieved by solving the monodromy action-equation on the elliptic-curve line-bundle by -cusp D7-brane intersections on the base with divisors defining the elliptic singularities, where is the number of degeneracies of the torus-modulus as a varying function of the RR-C-form and the dilaton, and integrating the exact holomorphic 2-form :

giving us a K3-class model over an integral basis of 2-cycles and where the corresponding period integrals elliptic over 1-cycles:

factoring in the base modulus cusp -terms and summing over divisor points over resolutions of D-7-branes localized on the line degeneracies. Pictorially, we get our desired F/M-GUT as:

Note that the Hauptmodul-function maps the fundamental region to the 7-plane geometry via D7-brane backreaction as a function of the Type-IIB Jacobi-term of the -plane for:

guaranteeing moduli stabilization, and by the Heterotic/F-theory duality, topological mirror …

The post Chern–Simons Theory, the Orbifold Delta Function, and GUT-Models appeared first on George Shiber.

]]>

in Type IIB SUGRA background satisfies the Gaillard-Zumino duality condition and exhibits exact self-duality. Dp-branes are p + 1 dimensional Ramond-Ramond charged dynamical hypersurfaces that open strings end on and admit perturbative worldsheet description in terms of open strings satisfying Dirichlet boundary conditions in p + 1 dimensions. Naturally, for 4-D spacetime physics, D3 branes are especially important for string-phenomenology due to mirror symmetry on Calabi-Yau 3-folds where they holomorphically wrap Fukaya super-Lagrangians. The D3-brane effective action in the NS5-brane geometry, given that it satisfies D3-brane self-duality and Poincaré invariance, is given by:

with

being the D3-brane tension, and , are the RR-4 and RR-2 exterior forms, and generally, the DBI action is:

and the D-brane WZ action is given by:

In order for the effective action to be integrable with fields in 2nd-quantized form, one must work under the Gaillard-Zumino Condition: that is – using 8-loop counterterms with superspace torsion:

where is the superfield torsion. One starts with a Lagrangian:

in D = 4, with a dependence on a gauge field strength , metric , and matter field . So, we now have:

and the Hodge dual components for the tensor are given by:

The Gaillard-Zumino condition is an infinitesimal duality transformation of and and fermionic transformation given by:

Now, the Lagrangian must transform as:

and one has an transformation given by , , and so the Lagrangian is given by:

and by D3-brane self-duality, it follows that:

Now we are in a position to analyse the D3-brane action, and for simplicity but with no loss of generality, without scalar supergravity backgrounds. Let be a bosonic brane coordinatization in D = 10 flat target bulk space and its fermionic partner given by the Majorana-Weyl spinor index with N = 2 SUSY index . The D3 action for the brane coordinates and worldvolume gauge field must have Kappa symmetry and we require N = 2 SUSY. Hence:

where

with being the Pauli matrices and act on the N = 2 SUSY indices. The 1-form defined by:

and

By use of exterior differential forms on the bulk, with an RR pull-back 2-form and 4-form , we get:

and

where is given by:

with . To check whether the Gaillard-Zumino condition is met, we must calculate the first 2 terms of the condition:

so, is given by:

where I have made an explicit use of the determinant formula for the four-by-four matrix:

and by Hodge duality, can be derived as:

and by the GKP-Witten relation for the D3-brane action:

one gets

and by conjugation, one derives the essential identity:

A few remarks are in order now on the bosonic truncation of the D3-brane action. Note in the above equation, the right-hand-side vanishes completely, and so the Lagrangian transforms accordingly as:

The supersymmetry situation under the Gaillard-Zumino condition can now be considered for the matter field contributions. For a D3-brane, the matter fields transform as:

and hence we get:

while …

The post SO(2) Duality, Type IIB SuperGravity and the Super-D3-Brane Action appeared first on George Shiber.

]]>

in Type IIB SUGRA background satisfies the Gaillard-Zumino duality condition and exhibits exact self-duality. Dp-branes are p + 1 dimensional Ramond-Ramond charged dynamical hypersurfaces that open strings end on and admit perturbative worldsheet description in terms of open strings satisfying Dirichlet boundary conditions in p + 1 dimensions. Naturally, for 4-D spacetime physics, D3 branes are especially important for string-phenomenology due to mirror symmetry on Calabi-Yau 3-folds where they holomorphically wrap Fukaya super-Lagrangians. The D3-brane effective action in the NS5-brane geometry, given that it satisfies D3-brane self-duality and Poincaré invariance, is given by:

with

being the D3-brane tension, and , are the RR-4 and RR-2 exterior forms, and generally, the DBI action is:

and the D-brane WZ action is given by:

In order for the effective action to be integrable with fields in 2nd-quantized form, one must work under the Gaillard-Zumino Condition: that is – using 8-loop counterterms with superspace torsion:

where is the superfield torsion. One starts with a Lagrangian:

in D = 4, with a dependence on a gauge field strength , metric , and matter field . So, we now have:

and the Hodge dual components for the tensor are given by:

The Gaillard-Zumino condition is an infinitesimal duality transformation of and and fermionic transformation given by:

Now, the Lagrangian must transform as:

and one has an transformation given by , , and so the Lagrangian is given by:

and by D3-brane self-duality, it follows that:

Now we are in a position to analyse the D3-brane action, and for simplicity but with no loss of generality, without scalar supergravity backgrounds. Let be a bosonic brane coordinatization in D = 10 flat target bulk space and its fermionic partner given by the Majorana-Weyl spinor index with N = 2 SUSY index . The D3 action for the brane coordinates and worldvolume gauge field must have Kappa symmetry and we require N = 2 SUSY. Hence:

where

with being the Pauli matrices and act on the N = 2 SUSY indices. The 1-form defined by:

and

By use of exterior differential forms on the bulk, with an RR pull-back 2-form and 4-form , we get:

and

where is given by:

with . To check whether the Gaillard-Zumino condition is met, we must calculate the first 2 terms of the condition:

so, is given by:

where I have made an explicit use of the determinant formula for the four-by-four matrix:

and by Hodge duality, can be derived as:

and by the GKP-Witten relation for the D3-brane action:

one gets

and by conjugation, one derives the essential identity:

A few remarks are in order now on the bosonic truncation of the D3-brane action. Note in the above equation, the right-hand-side vanishes completely, and so the Lagrangian transforms accordingly as:

The supersymmetry situation under the Gaillard-Zumino condition can now be considered for the matter field contributions. For a D3-brane, the matter fields transform as:

and hence we get:

while …

The post SO(2) Duality, Type IIB SuperGravity and the Super-D3-Brane Action appeared first on George Shiber.

]]>

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 …

The post Randall-Sundrum Braneworld Scenario, Klebanov-Strassler Geometry and the Standard Model appeared first on George Shiber.

]]>

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 …

The post Randall-Sundrum Braneworld Scenario, Klebanov-Strassler Geometry and the Standard Model appeared first on George Shiber.

]]>

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 …

The post T-Branes, the Chern-Simons Action and the Kähler Pull-Back appeared first on George Shiber.

]]>

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 …

The post T-Branes, the Chern-Simons Action and the Kähler Pull-Back appeared first on George Shiber.

]]>

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 …

The post Randall-Sundrum Cosmology and Dp-Brane Dynamics appeared first on George Shiber.

]]>

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 …

The post Randall-Sundrum Cosmology and Dp-Brane Dynamics appeared first on George Shiber.

]]>

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, and

nota 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 of

M-theory

In the **closed** membrane case:

the gauge fields are defined in terms of

11-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:

…

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

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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:

…

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

]]>