coupled to a Chern-Simons Wess-Zumino term:

the worldvolume pullback, and the corresponding p-orientifold action is:

with the string-coupling given by:

and the pullbacks SUGRA fields to the -brane worldvolume. Thus, we can derive the action:

with:

which yields a 10-D SYM action:

with:

and the non-abelian field strength of the gauge field :

where the gauge covariant derivative is defined by:

giving us the -worldvolume SYM action:

with:

with the potential:

that will figure in the Kähler form that will characterize -brane cosmology. -branes are central not simply because of F-theory compactification due to backreaction and Type-IIB axio-dilaton, but also because the Kähler moduli space induced by the -brane F-theoretic backreaction has a modulus that can naturally be identified with the ΛCDM-inflaton. Thus, inflationary cosmology can be derived, up to isomorphism, from 6D F-theory compactification using the Type-IIB string-theory/11D SUGRA duality upon dimensional reduction to 4D, where the Type-IIB SUGRA action in the string frame is given by:

with:

In light of flux moduli stabilization, this is a natural framework since the Kähler moduli potential apriori allows a Type-IIB-modulus/ΛCDM-inflaton identification. In such a SYM -brane scenario, we have an -flaton model in the Large Volume Scenario that evades the -problem, and periodicity of the modulus is achieved by defluxing the pullback of the -term on the superstring worldsheet. This can be readily seen since the total supersymmetric worldsheet action in conformal gauge:

with:

has modular-invariance. The skeleton of Type-IIB flux compactification on a Calabi-Yau threefold is constructed from a superpotential of the form:

where:

is the Gukov-Vafa-Witten superpotential stabilization complex term, as well as the axio-dilaton field:

Given the presence of -brane instantons, are of Kähler moduli Type-IIB-orbifold class:

with being the volume of the divisor and the 4-form Ramond-Ramond axion field corresponding to:

and:

where is the Kähler form:

and:

an integral-form basis and the associated intersection coefficients. Hence, the Kähler potential is given by:

with the Calabi-Yau volume, and in the Einstein frame, is given by:

The -term is given by:

and in the large volume scenario (LVS) is:

Thus, the LVS -term is given by:

with:

and the Fayet-Illopoulos terms being:

where are the -brane charge-vectors. The Kähler LVS potential is now derivable, and takes the form:

One can now choose any as the Kähler-modulus-inflaton, and inflation takes place in:

We can build our model now from the Kähler form and the superpotential. For a Witten-deformed -brane, the Kähler potential is thus derivable in the weak string-coupling limit of the F-theory Kähler potential arising from the elliptic fibration base Calabi-Yau four-fold:

Expanded in the weak string coupling limit gives us:

with the periods of integrals of the holomorphic (3,0)-form over the symplectic basis of 3-cycles with intersection matrix and is a function of symplectic and brane-moduli that figures in the flux-compactification moduli-stabilization but is independent of the axio-dilaton. Using mirror symmetry, the Kähler potential:

can be identified with the Kähler-moduli-potential of the mirror fourfold, which in the LVS-limit, involves the volume moduli of the elliptic fourfold, but not the corresponding axions …

The post Type-IIB String-Theory, 4D N=1 SUGRA, D7-Branes and Kähler Inflation appeared first on George Shiber.

]]>coupled to a Chern-Simons Wess-Zumino term:

the worldvolume pullback, and the corresponding p-orientifold action is:

with the string-coupling given by:

and the pullbacks SUGRA fields to the -brane worldvolume. Thus, we can derive the action:

with:

which yields a 10-D SYM action:

with:

and the non-abelian field strength of the gauge field :

where the gauge covariant derivative is defined by:

giving us the -worldvolume SYM action:

with:

with the potential:

that will figure in the Kähler form that will characterize -brane cosmology. -branes are central not simply because of F-theory compactification due to backreaction and Type-IIB axio-dilaton, but also because the Kähler moduli space induced by the -brane F-theoretic backreaction has a modulus that can naturally be identified with the ΛCDM-inflaton. Thus, inflationary cosmology can be derived, up to isomorphism, from 6D F-theory compactification using the Type-IIB string-theory/11D SUGRA duality upon dimensional reduction to 4D, where the Type-IIB SUGRA action in the string frame is given by:

with:

In light of flux moduli stabilization, this is a natural framework since the Kähler moduli potential apriori allows a Type-IIB-modulus/ΛCDM-inflaton identification. In such a SYM -brane scenario, we have an -flaton model in the Large Volume Scenario that evades the -problem, and periodicity of the modulus is achieved by defluxing the pullback of the -term on the superstring worldsheet. This can be readily seen since the total supersymmetric worldsheet action in conformal gauge:

with:

has modular-invariance. The skeleton of Type-IIB flux compactification on a Calabi-Yau threefold is constructed from a superpotential of the form:

where:

is the Gukov-Vafa-Witten superpotential stabilization complex term, as well as the axio-dilaton field:

Given the presence of -brane instantons, are of Kähler moduli Type-IIB-orbifold class:

with being the volume of the divisor and the 4-form Ramond-Ramond axion field corresponding to:

and:

where is the Kähler form:

and:

an integral-form basis and the associated intersection coefficients. Hence, the Kähler potential is given by:

with the Calabi-Yau volume, and in the Einstein frame, is given by:

The -term is given by:

and in the large volume scenario (LVS) is:

Thus, the LVS -term is given by:

with:

and the Fayet-Illopoulos terms being:

where are the -brane charge-vectors. The Kähler LVS potential is now derivable, and takes the form:

One can now choose any as the Kähler-modulus-inflaton, and inflation takes place in:

We can build our model now from the Kähler form and the superpotential. For a Witten-deformed -brane, the Kähler potential is thus derivable in the weak string-coupling limit of the F-theory Kähler potential arising from the elliptic fibration base Calabi-Yau four-fold:

Expanded in the weak string coupling limit gives us:

with the periods of integrals of the holomorphic (3,0)-form over the symplectic basis of 3-cycles with intersection matrix and is a function of symplectic and brane-moduli that figures in the flux-compactification moduli-stabilization but is independent of the axio-dilaton. Using mirror symmetry, the Kähler potential:

can be identified with the Kähler-moduli-potential of the mirror fourfold, which in the LVS-limit, involves the volume moduli of the elliptic fourfold, but not the corresponding axions …

The post Type-IIB String-Theory, 4D N=1 SUGRA, D7-Branes and Kähler Inflation appeared first on George Shiber.

]]>with:

and:

where the Chern-Simons-topological Lagrangian has covariant variational form:

with:

and the Yang-Mills field equation for the covariant field strength form is:

Thus, we can derive the Chern-Simons-type topological action:

with:

and:

and the covariant curvature form and holomorphic curvature form are, respectively:

and:

where the Ramond-Ramond gauge-coupling sector is given by the action:

and the Ramond-Ramond term being:

thus giving us the Type-IIB Calabi-Yau three-fold superpotential:

Before we can see the duality relations between M-theory and F-theory elliptic fibrational Standard-Model constructions, note that the topologically mixed Yang-Mills action:

where the corresponding Chern-Simons action is:

with the Ramond-Ramond coupling-term:

has variational action:

with:

Now, since 11-D SUGRA on a torus is equivalent to Type-IIB string-theory on a circle, the action of the modular group on the Type-IIB axio-dilaton allows us to take the zero limit of:

and by mirror symmetry, we get a Type-IIA dimensional uplift to M-theory, given that in the Einstein frame, the Type-IIB bosonic SUGRA action is:

with:

This is one aspect of appreciating, via the exceptional field theory (EFT) modular group action, the F-theory/M-theory duality. The essence of EFT is thus a deeper double duality relating M-theory/Type-IIA and F-theory/Type-IIB. Key is the role of U-duality in the modular holomorphic action on the Neveu-Schwarz sector of Type-IIB. Our generalized diffeomorphisms, generated by a vector , act fully locally on yielding the Lie derivative that differs from the classic Lie derivative by a Calabi-Yau induced -tensor and is implicitly defined by the transformation rules for a generalized vector:

The associated diffeomorphism algebra has an exceptional field bracket:

with closure condition:

The action diffeomorphism-symmetries are parametrized by vector bundles over the metaplectic space and take the form:

with:

where the gauge vector transforms as:

The corresponding generalized exceptional scalar metric has the following property:

which decomposes in light of the orbifold blow-up:

as:

thus allowing us to define the crucial exceptional metric:

Since the full Type-IIB Calabi-Yau superpotential is given by:

where the Kähler Type-IIB orientifold moduli is:

with:

and the volume of the divisor, , is:

with:

thus we now have the ingredients to write the modular exceptional field theory action as:

with the exceptional Ricci scalar:

the kinetic part:

and the gauge term:

and the 10+3-D Chern-Simons topological term:

where the potential has the form:

This is a theory dynamically equivalent to 11-D SUGRA and Type-IIB under covariantized U-duality group-action. However, the gauged kinetic terms corresponding to the gauge form appears only topologically in:

Hence, the EoM for the field is given by:

Since exceptional field theory based on the modular group uses a dimensionally extended spacetime to 12-D that fully covariantizes supergravity under the U-duality symmetry groups of M-theory, homological mirror symmetry entails there ought to be a deep internal symmetry induced between M-theory and F-theory upon dimensional-reduction to Type-IIB SUGRA which in the formalism, taking the 6/8 Klebanov-Witten limit, is defined by the action:

We are now in a position to explore this EFT-duality between M-theory and F-theory, noting that it is a duality that is rich in …

The post M-Theory/Type-IIB Duality, Brane-Dynamics and EFT appeared first on George Shiber.

]]>with:

and:

where the Chern-Simons-topological Lagrangian has covariant variational form:

with:

and the Yang-Mills field equation for the covariant field strength form is:

Thus, we can derive the Chern-Simons-type topological action:

with:

and:

and the covariant curvature form and holomorphic curvature form are, respectively:

and:

where the Ramond-Ramond gauge-coupling sector is given by the action:

and the Ramond-Ramond term being:

thus giving us the Type-IIB Calabi-Yau three-fold superpotential:

Before we can see the duality relations between M-theory and F-theory elliptic fibrational Standard-Model constructions, note that the topologically mixed Yang-Mills action:

where the corresponding Chern-Simons action is:

with the Ramond-Ramond coupling-term:

has variational action:

with:

Now, since 11-D SUGRA on a torus is equivalent to Type-IIB string-theory on a circle, the action of the modular group on the Type-IIB axio-dilaton allows us to take the zero limit of:

and by mirror symmetry, we get a Type-IIA dimensional uplift to M-theory, given that in the Einstein frame, the Type-IIB bosonic SUGRA action is:

with:

This is one aspect of appreciating, via the exceptional field theory (EFT) modular group action, the F-theory/M-theory duality. The essence of EFT is thus a deeper double duality relating M-theory/Type-IIA and F-theory/Type-IIB. Key is the role of U-duality in the modular holomorphic action on the Neveu-Schwarz sector of Type-IIB. Our generalized diffeomorphisms, generated by a vector , act fully locally on yielding the Lie derivative that differs from the classic Lie derivative by a Calabi-Yau induced -tensor and is implicitly defined by the transformation rules for a generalized vector:

The associated diffeomorphism algebra has an exceptional field bracket:

with closure condition:

The action diffeomorphism-symmetries are parametrized by vector bundles over the metaplectic space and take the form:

with:

where the gauge vector transforms as:

The corresponding generalized exceptional scalar metric has the following property:

which decomposes in light of the orbifold blow-up:

as:

thus allowing us to define the crucial exceptional metric:

Since the full Type-IIB Calabi-Yau superpotential is given by:

where the Kähler Type-IIB orientifold moduli is:

with:

and the volume of the divisor, , is:

with:

thus we now have the ingredients to write the modular exceptional field theory action as:

with the exceptional Ricci scalar:

the kinetic part:

and the gauge term:

and the 10+3-D Chern-Simons topological term:

where the potential has the form:

This is a theory dynamically equivalent to 11-D SUGRA and Type-IIB under covariantized U-duality group-action. However, the gauged kinetic terms corresponding to the gauge form appears only topologically in:

Hence, the EoM for the field is given by:

Since exceptional field theory based on the modular group uses a dimensionally extended spacetime to 12-D that fully covariantizes supergravity under the U-duality symmetry groups of M-theory, homological mirror symmetry entails there ought to be a deep internal symmetry induced between M-theory and F-theory upon dimensional-reduction to Type-IIB SUGRA which in the formalism, taking the 6/8 Klebanov-Witten limit, is defined by the action:

We are now in a position to explore this EFT-duality between M-theory and F-theory, noting that it is a duality that is rich in …

The post M-Theory/Type-IIB Duality, Brane-Dynamics and EFT 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.

]]>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.

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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 …

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

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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 …

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

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