New mathematical connections concerning string theory 1

New mathematical connections concerning string theory

Michele Nardelli Presented to

Dipartimento di Matematica ed Applicazioni “R.Caccioppoli”

Università degli Studi di Napoli “Federico II”- Polo delle Scienze e delle Tecnologie Monte S.Angelo, via Cintia (Fuorigrotta), 80126 Napoli, Italy

Riassunto.

Scopo del presente lavoro è quello di descrivere le relazioni trovate tra il modello di Palumbo sull’origine e l’evoluzione dell’Universo e la teoria di stringa. Il modello di Palumbo è sintetizzato dalla relazione

∫

∞

=

0 i i

dF

F

F

, dove F rappresenta l’energia iniziale del Big Bang, ossia, l’esplosione del buco nero dal

quale si originò l’universo. Dal Big Bang, si sprigionarono tutte le onde immaginabili di F . Al pari delle radiazioni elettromagnetiche, le quali consistono di una successione continua di insiemi di onde, anche le radiazioni F sono costituite da insiemi parziali di onde, definite F_i. Dopo aver descritto le azioni di stringa bosonica e quelle di superstringa, si evidenziano le connessioni trovate tra queste ed il modello di Palumbo. Vengono evidenziate, inoltre, le connessioni trovate tra le azioni delle brane di Dirichlet, precisamente le D3 e D9-brane ed il modello di Palumbo. Anche per alcune azioni di stringa inerenti il modello cosmologico di pre Big-Bang, vengono evidenziate delle connessioni con il modello di Palumbo. Infine, vengono descritte le relazioni trovate tra alcune soluzioni solitoniche in teoria di campo di stringa ed alcune equazioni correlate alla funzione zeta di Riemann. Viene evidenziato, quindi, come anche per queste ultime è possibile la connessione con il modello di Palumbo.

1.Bosonic String.

A one-dimensional object will sweep out a two dimensional world-sheet, which can be described in terms of two parameters, µ(

τ

,

σ

)

X . The simplest invariant action, the Nambu-Goto action, is proportional to the

area of the world-sheet. We define the induced metric h_ab where indices a, b,... run over values (

τ

,

σ

):

.

µ µ

X

X

h

_ab

=

∂

_a

∂

_b (a) Then the Nambu-Goto action is

=

∫

M NG NG

d

d

L

S

τ

σ

_,(b) , ) det ( ' 2 1 1/2 ab NG h L =− −

πα

(c) where M denotes the world-sheet. The constant

α

', which has units of

spacetime-length-squared, is Regge slope. The tension Tof the string is related to the Regge slope by

' 2

1

πα

=

T . We can simplify the Nambu-Goto action by introducing an independent world-sheet metric

) , (

τ

σ

γ

ab . We will take

γ

abto have Lorentzian signature

(

−,

+

)

. The action is

(

)

∫

− ∂ ∂ − = M b a ab p X d d X X S , ' 4 1 ] , [

τ

σ

γ

1/2

γ

µ _µ

πα

γ

(d) where

γ

=det

γ

_ab.This is the

generalization with local world-sheet supersymmetry. To see the equivalence to S_NG, use the equation of motion obtained by varying the metric,

(

)

∫













−

−

−

=

M cd cd ab ab ab p

X

d

d

h

h

S

,

2

1

'

4

1

]

,

[

τ

σ

γ

1/2

δγ

γ

γ

πα

γ

δ

γ (e) where hab is again the induced metric

(a). We have used the general relation for the variation of a determinant,

δ

γ

=

γγ

ab

δγ

_ab

=

−

γγ

_ab

δγ

ab

.

Then

δ

_γS_p =0 implies . 2 1 cd cd ab ab h

h =

γ

γ

Dividing this equation by the square root of minus its

determinant gives

h

ab

( )

−

h

−1/2

=

γ

ab

(

−

γ

)

−1/2

,

so that

γ

ab is proportional to the induced metric. This in turn can be used to eliminate

γ

_ab from the action,

( )

∫

− = − → [ ]. ' 2 1 ] , [X d d h 1/2 S X S_p

τ

σ

_NG

πα

γ

(f)

The condition that the world-sheet theory be Weyl-invariant is:

0

=

=

=

β

β

Φ

β

µν µν B G (1.1) , where: ), ( 4 ' ' 2 '

α

α

α

'2

α

β

λω ν µλω ν µ µν µν R H H O G + − ∇ ∇ + = ), ( ' 2 '

_α

_α

'2

α

β

ω _ωµν ωµν ω µν H H O B + Φ ∇ + ∇ − = ). ( 24 ' ' 2 ' 6 26 _{2 '2}

α

α

α

α

β

µνλ µνλ ω ω H H O D + − Φ ∇ Φ ∇ + Φ ∇ − − = Φ The equation G

=

0

µν

β

resembles Einstein’s equation with source terms from the antisymmetric tensor field

and the dilaton. (Note that the dilaton is the massless scalar with gravitational-strength couplings, found in

all perturbative string theories). The equation B

=

0

µν

β

is the antisymmetric tensor generalization of

Maxwell’s equation, determining the divergence of the field strength. The field equations (1.1) can be derived from the spacetime action:

S =

∫

dDx 2 0 2 1

κ

(-G) + − − Φ − ' 3 ) 26 ( 2 [ 2 2 / 1

α

D e 4 ( ')] 12 1

α

µ µ µνλ µνλH O H R− + ∂ Φ∂ Φ+ (1.2)

The normalization constant

κ

₀ is not determined by the field equations and has no physical significance

since it can be changed by a redefinition of Φ. One can verify that:

∫

− Φ Φ − − Φ + + − − = )( 4 )] 2 1 2 ( [ ) ( ' 2 1 1/2 2 2 0

β

β

δ

δ

β

δ

β

δ

α

κ

δ

ω ω µν µν µν µν µν µν G B G D G G B G e G x d S (1.3)

This is the effective action governing the low energy spacetime fields. It is often useful to make a field redefinition of the form:

), ( )] ( 2 exp[ ) ( ~ x G x x

G_µν =

ω

_µν (1.4), which is a spacetime Weyl transformation. The Ricci scalar

constructed from G~_µν is:

] ) 1 )( 2 ( ) 1 ( 2 )[ 2 exp( ~

_ω

2

_ω

_ω

µ

_ω

µ ∂ ∂ − − − ∇ − − − = R D D D R (1.5)

For the special case D=2, this is the Weyl transformation g'1/2R' = g1/2(R−2∇2

ω

).

Let

ω

=2(Φ₀ −Φ)/(D−2) and define

Φ

~

=

Φ

−

Φ

₀ (1.6), which has vanishing expectation value. The action becomes: S=

∫

∂ − − − + − − − Φ − −Φ − µ µνλ µνλ

α

κ

2 4 ~ 12 1 ~ ' 3 ) 26 ( 2 [ ) ~ ( 2 1 ~_{/( 2)} 8 ) 2 /( ~ 4 2 / 1 2 D H H e R e D G X dD D D )], ' ( ~ ~ ~

α

µ O + Φ ∂

Φ (1.7). In terms of

G

~

µν , the gravitational Lagrangian density takes the standard Hilbert

form (−G~)1/2R~/2

κ

2. The constant 0

0 Φ

=

κ

e

nature has the value 18 1 2 / 1 2 / 1 ) 10 43 , 2 ( ) 8 ( ) 8 ( = = − = x GeV M G p N

π

π

κ

. Commonly, G_µν is called the “sigma

model metric” or “string metric”, and G~_µν the “Einstein metric”. The (1.7) becomes :

∫

∂ Φ∂ Φ+ − − − + − − − = Φ − −Φ − )] ' ( ~ ~ ~ 2 4 ~ 12 1 ~ ' 3 ) 26 ( 2 [ ) ~ ( 8 2 1 ~_{/( 2)} 8 ) 2 /( ~ 4 2 / 1

_α

α

π

µ µ µνλ µνλ O D H H e R e D G X d G S D D D N (1.8)

The amplitudes of string, that corresponding at the classical terms in the effective action, would be obtained in field theory from the spacetime action

S

+

S

T, where S is the action (1.2) for the massless fields, and where

∫

− ∂ ∂ − − = 26 1/2 −2Φ 2 ' 4 ( ) ( 2 1 T T T G e G x d S_T

α

ν µ µν

) (1.9), is the action for the closed string tachyon T. (Note that, in string theory, the tachyon is a particle with a negative mass-squared, signifying an instability of

the vacuum). We take the more general case of D= d+1 spacetime dimensions with xd periodic. Since we

are in field theory we leave D arbitrary. Parameterize the metric as 2 2

₍

µ

₎

µ ν µ µν

dx

dx

G

dx

A

dx

G

dx

dx

G

ds

d dd N M D MN

=

+

+

=

(1.10).

We designate the full D-dimensional metric by G_MND ; the Ricci scalar for the metric (1.10) is:

µν µν σ σ σ F F e e e R R d 2 2 4 1 2 ∇ − − = −

(1.11), where R is constructed from GMN and Rd from Gµν. The

graviton-dilaton action (1.2) becomes:

∫

− + ∇ Φ∇ Φ = = − Φ ) 4 ( ) ( 2 1 1/2 2 2 0 1 µ µ

κ

d x G e R S D =

∫

− − Φ+ − ∂ Φ∂ + ∂ Φ∂ Φ− ) 4 1 4 4 ( ) ( 1/2 2 2 2 0 µν µν σ µ µ µ µ σ

_σ

κ

π

F F e R e G x d R d d d hence:

∫

− −∂ ∂ + ∂ Φ ∂ Φ − = − Φ ) 4 1 4 ( ) ( 1/2 2 2 2 0 1 µν µν σ µ µ µ µ

σ

σ

κ

π

F F e R e G x d R S d _d d _{d d d (1.12) giving kinetic}

terms for all the massless fields. Here G_ddenotes the determinant of G_µν, and in the last line we have introduced the effective d-dimensional dilaton, Φd =Φ−

σ

/2.

The antisymmetric tensor also gives rise to a gauge symmetry by a generalization of the Kaluza-Klein mechanism. Separating B_MN into B_µν and

A

_µ'

=

B

_dµ

, the gauge parameter

ζ

_M separates into a

d-dimensional antisymmetric tensor transformation

ζ

_µ and an ordinary gauge invariance

ζ

_d(Note that the

action

∫

+ ∂ ∂ + Φ = M a b ab ab X R X X X B i X G g g d S [( ( ) ( )) ' ( )] ' 4 1 2

_σ

_ε

_α

πα

ν µ µν µν

σ , where the field Bµν( X)

is the antisymmetric tensor, and the dilaton involves both Φ and the diagonal part of G_µν, is invariant under

) ( ) ( ) (X X X B_{µν µ}

ζ

_{ν ν}

ζ

_µ

δ

=∂ −∂ which adds a total derivative to the Lagrangian density. This is a

generalization of the electromagnetic gauge transformation to a potential with two antisymmetric indices. The gauge parameter

ζ

_M, above mentioned, is defined in this relation). The gauge field is B_dµ and the field

strength H_dµν . The antisymmetric tensor action becomes:

∫

− Φ

∫

− Φ − + − − = − − = ( ) (~ ~ 3 ) 12 ) ( 24 1 _{1/2 2 2} 2 0 2 2 / 1 2 0 2 µν µν σ µνλ µνλ

κ

π

κ

d d d d MNL MNL D D H H e H H e G x d R H H e G x d S d

(1.13). Here, we have defined H~_µνλ =(∂_µB_νλ −A_µH_dνλ)+cyclic permutations. The term proportional to

the vector potential arises from the inverse metric GMN. It is known as a “Chern-Simons term”, this

The relevant terms from the spacetime action (1.7) concerning an D-brane are:

∫

− − ∇ Φ∇ Φ = ~~ ~) 6 1 ~ ( ~ 2 1 26 2 µ µ

κ

d X G R

S (1.14). Recall that the tilde denotes the Einstein metric.

The tilded dilaton has been shifted so that its expectation value is zero. In terms of the same variables, the relevant terms from the D-brane action

[

]

1/2 1 ) ' 2 det( _{ab ab ab} p p p T d e G B F S =− +

ξ

−Φ − + +

πα

∫

, (with T_p is a constant and

)) ( ( ) (

ξ

ξ

ξ

ξ

µν ν µ X G X X Gab _{a b} ∂ ∂ ∂ ∂ = , ( ) ( (

ξ

))

ξ

ξ

ξ

µν ν µ X B X X Bab _{a b} ∂ ∂ ∂ ∂

= are the induced metric and

antisymmetric tensor on the brane) are:

(

)

∫

−      Φ − − = +1 ~ 1/2 det ~ 12 11 exp _ab p p p G p d S

τ

ξ

. We have defined

=

_T

_e

−Φ0 p p

τ

; this is the physical tension

of the Dp-brane when the background value of the dilaton is Φ₀. To obtain the propagator we expand the

spacetime action to second order in h_µν = G_µν −

η

_µν and to second order in Φ~. Also , we need to choose a gauge for the gravitational calculation. The simplest gauge for perturbative calculation is

0 2 1 ˆ ˆ = ∂ − ∂ ≡ µν ν µµ µ ν h h

F , where a hat indicates that an index has been raised with the flat space metric

µν

η

.

(Note that an D-brane is a dynamical object on which strings can end. The term is a contraction of “Dirichlet brane”. The coordinates of the attached strings satisfy Dirichlet boundary conditions in the directions normal to the brane and Neumann conditions in the directions tangent to the brane. The mass or tension of a D-brane is intermediate between that of an ordinary quantum or a fundamental string and that of “soliton”. The soliton is a state whose classical limit is a smooth, localized and topologically nontrivial classical field configuration; this includes particle states, which are localized in all directions, as well as extended objects. The low energy fluctations of D-branes are described by supersymmetric gauge theory, which is non-Abelian for coincident branes). Expanding the action to second order and adding a gauge-fixing term −F_νFνˆ /4

κ

2, the spacetime action becomes:

∫

∂ ∂ − ∂ ∂ + ∂ Φ∂ Φ − = ~ ~) 3 2 2 1 ( 8 1 26 ˆ ˆˆ ˆ ˆ ˆ ˆ 2 µ µ λ λ µ ν ν µ λ ν µ νλ µ

κ

d X h h h h S (1.15).

2.The oscillation modes of string corresponding to the graviton.

In the paper “Dual Models for Nonhadrons” of J. Scherk and John H. Schwarz of California Institute of Technology, published in April 1974, (2) the possibility of describing particles other than adrons (leptons, photons, gauge bosons, gravitons, etc.) by a dual model is explored. The Virasoro-Shapiro model is studied, interpreting the massless spin-two state of the model as a graviton.

Both the 26-dimensional Veneziano model (VM) and the 10-dimensional meson-fermion model (MFM) have a massless “photon”, the nonplanar Virasoro-Shapiro model (VSM) has a massless “graviton”, and the MFM has a massless “lepton”. The VM and the MFM have both been studied and found to yield the Yang-Mills theory of massless self-interacting vector mesons needed to describe electromagnetism and weak interactions.

2.1.Virasoro-Shapiro Model.

The most general action for massless gravitons and scalars with general coordinate invariance, involving no more than two derivatives, is:

( )

( )

( )

∫

_    ∂ ∂ − − =

φ

φ

φ

φ

π

φ

µ ν µν 3 2 1 4 2 1 16 1 f g Rf G f g x d S , (2.1)

where the

f

₁ are functions, analytic at the origin, subject to the normalization constraints

1 ) 0 ( ) 0 ( ₃ 2 = f =

The first term in the equation (2.1) contains no derivatives, while the second and third each contain two derivatives. The form of the action in the equation (2.1) can be simplified by performing a Weyl transformation gµν =g'µν f₂

( )

φ

−1 (2.2).

Under this transformation

g

and R become:

g

=

g

'

f

₂

( )

φ

−2 (2.3) and

( )

( )

( )

( )

φ

( )

φ

( )

φ

( )

φ

φ

φ

φ

α α α α 2 2 2 2 2 2 2

2

2

'

3

'

f

f

f

f

f

D

f

f

R

R

_

+

∂

∂









 ∂

−

=

, (2.4)

where D_α represents a covariant derivative. Then, the action becomes:

( )

( )

[

]

( )

( )

( )

( )

∫





























+

∂

∂

−

−

=

2 2 2 2 3 2 2 1 4

'

16

3

2

1

16

1

φ

φ

π

φ

φ

φ

φ

π

φ

φ

ν µ µν

f

f

G

f

f

g

R

G

f

f

g

x

d

S

, where setting

( )

( )

( )

[

]

2 2 1 1

φ

φ

φ

f f k = and

( )

( )

( )

( )

( )

2 2 2 2 3 2 ' 16 3       + =

φ

φ

π

φ

φ

φ

f f G f f k , we have:

( )

( )

∫

_    ∂ ∂ − − =

φ

φ

φ

π

φ

µν _{µ ν} 2 1 4 2 1 16 1 k g R G k g x d S (2.5).

The action can be simplified further by a transformation of

φ

itself. Setting

φ

=

φ

[

ψ

( )

x

]

, where the function

( )

ψ

φ

satisfies the differential equation

( )

φ

' 2k₂

( )

φ

=1, gives an action of the form in the equation (2.5) with

( )

φ

2

k

replaced by one and

k

₁

( )

φ

by a new function

d

( )

φ

. Therefore the first order terms in

α

' are completely determined and it only remains to evaluate the zeroth order terms. The absence of constant and linear terms implies that in tree approximation the multigraviton amplitudes must be precisely those of general relativity, as given by the Hilbert-Einstein action (the second term in the equation (2.5)). The vanishing of

d

( )

φ

means that there is no cosmological term in the multigraviton interactions, whereas the vanishing of

d

'

( )

0

, implies that multigraviton amplitudes cannot contain scalar poles. To first order in the zero-slope expansion the action become:

∫

_    ∂ ∂ + − =

φ

φ

π

µ ν µν g R G g x d S 2 1 16 1 4 (2.6).

There is a TSS (1 tensor and two scalars) interaction in which the graviton couples to the energy momentum tensor of the

φ

field. Furthermore, all pure scalar self-interactions terms vanish identically. The

graviton+gauge boson interactions have to be both Yang-Mills invariant and generally covariant. Therefore the unique action for these fields to first order in

α

' is:

[

]

∫

−

∫

− = µν ρσ µσ µρ

π

G d x gR d x gg g TrG G S 4 4 8 1 16 1

(2.7). (Note that “Tr” denote the trace of matrix square).

2.2.Interaction of the Scalar Field with Matter.

Before to write the action, is useful to recall that the scalar field couples in all orders to the trace of the stress

tensor of matter, as predicted by the Brans-Dicke theory. The parameter

ω

in the Brans-Dicke theory, which

fixes the relative strength of scalar and tensor couplings with matter, is determined by the dual models. The action, for the Brans-Dicke theory takes the form:

(

)

(

)

∫

_− − ∂ ∂ − ∂ ∂ − − − _ =

φ

φ

ψ

φ

ψ

φ

π

µ ν µν ν µ µν c m c g g G R g x d S exp 2 2 1 exp 2 1 2 1 16 2 2 4 , (2.8) where

.

2

3

16

ω

+

=

G

c

ψ

is a scalar field that has been introduced to represent matter. One can easily show

_µν

ω

π

φ

G

T

2

3

4

+

=

(2.9), where T_µν is the canonical matter stress tensor obtained by varying g_µν. In this

theory the exchange of a graviton and a scalar between two

ψ

particles is given by:

      + +       − _{µµ νν µµ νν} µν µν

ω

π

π

' 2 3 4 ' 2 1 ' 8 1 2 T T G T T T T G q . (2.10)

One sees that in the dual model, the last two terms of the equation (2.10) cancel exactly, which would

correspond to

ω

=−1 in the Brans-Dicke model. Then, if

ω

π

2

3

16

+

=

G

c

, for

ω

=−1 we have c= 16

π

G and (2.8) become:

(

)

(

)

∫

_− − ∂ ∂ − ∂ ∂ − − _ =

φ

φ

ψ

ψ

π

φ

ψ

π

φ

π

µ ν µν ν µ µν G m G g g G R g x d S exp2 16 2 1 16 exp 2 1 2 1 16 2 2 4 (2.11).

In conclusion, the action representing the VM to first order in

α

' contains photon, graviton, and scalar fields. It is described by the equation:

(

)

( )

∫

_− − − ∂ ∂ _ =

φ

φ

φ

π

µ ν µν ρσ µν νσ µρ g f G G Tr g g G R g x d S 2 1 8 1 16 26 _{(2.12), where}

( )

φ

=

1

+

k

φ

+

...,

f

and k 8

π

G 6 5

= . If the scalar couplings are given by the Brans-Dicke theory, then

( )

φ

f

should be

exp

( )

k

φ

.

3.Superstrings.

The superstring action is obtained introducing a supersymmetry on the world-sheet that connect the spacetime coordinates µ

(

τ

,

σ

)

,

X that are bosonic fields on the world-sheet, at a fermionic partner

(

τ

,

σ

)

.

µ α

Ψ The index

µ

denote that the fermionic coordinate changes as a vector, whose components are

spinors on the world-sheet. The theory obtained is defined “superstring theory”, and the corresponding action is:

[

]

∫

      − ∂ Γ Γ − ∇ Γ − ∂ ∂ − − = , 4 ' 4 1 µν ν ν µ ν µ ν µ

_ψ

_ψ

_χ

_ψ

_χ

_ψ

_η

γ

γ

τ

σ

πα

b b a b a a a b a ab i X i i X X d d S (3.1)

where

χ

_a is a Majorana gravitino and, as

−

γ

γ

ab

,

is a Lagrange multiplier without dynamic. The action

can be simplified selecting the equivalent of bosonic conformal gauge, defined superconformal gauge,

,

φ

η

γ

_ab = _abe

χ

a =Γa

ζ

, (3.2) and to make us the identity of bidimensionals gamma matrices

, 0 = Γ Γ Γ b a

a the action become

(

)

∫

∂ ∂ − Γ ∂ − = , ' 4 1 µν ν µ ν µ

_ψ

_ψ

_η

η

τ

σ

πα

a a b a ab i X X d d

S (3.3) the action of D scalar fields and D

fermionic fields free. We have two preserved currents, the energy-momentum tensor, obtained of the change of metric on the world-sheet

(

)

0 2 1 ' 2 1 4 ' 1 =       ∂ Γ + ∂ ∂ +       ∂ Γ + ∂ Γ + ∂ ∂ − = µ _µ µ _µ

η

µ _µ

ψ

µ

ψ

_µ

α

ψ

ψ

α

X X i X X T c _c ab a b b a b a ab , (3.4)

and the supercurrent, obtained by varying the gravitino,

. 0 ' 2 1 = ∂ Γ Γ =

_ψ

µ _µ

α

X Ja b a _b (3.5)

The equations (3.4) and (3.5) are constraints of superstring theory. These are called “super-Virasoro” constraints.

a)Type IIA superstring.

The action of type IIA superstring is:

, CS R NS IIA S S S S = + + (3.6)

(

)

∫

      − Φ ∂ Φ ∂ + − = − Φ , 2 1 4 2 1 2 3 2 2 / 1 10 2 10 H R e G x d SNS µ µ

κ

(3.7)

(

)

∫

      ₊ − − = ~ , 4 1 2 4 2 2 2 / 1 10 2 10 F F G x d S_R

κ

(3.8)

∫

∧ ∧ − = . 4 1 4 4 2 2 10 F F B S_CS

κ

(3.9)

We have regrouped terms according to whether the fields are in the NS-NS or R-R sector of the string theory; the Chern-Simons action contains both.

[The NS-NS (Neveu-Schwarz) states in type I and type II superstring theories, are the bosonic closed string states whose left- and right-moving parts are bosonic. The R-R (Ramond-Ramond) states in type I and type II superstring theories, are the bosonic closed string states whose left- and right-moving parts are fermionic. The Chern-Simons term, is a term in the action which involves p-form potentials as well as field strengths. Such a term is gauge-invariant as a consequence of the Bianchi identity and/or the modification of the p-form gauge transformation].

b)Type IIB superstring.

The action of type IIB superstring is:

, CS R NS IIB S S S S = + + (3.10)

(

)

∫













−

Φ

∂

Φ

∂

+

−

=

− Φ

,

2

1

4

2

1

2 3 2 2 / 1 10 2 10

H

R

e

G

x

d

k

S

_{NS µ} µ (3.11)

(

)

∫













+

+

−

−

=

~

,

2

1

~

4

1

2 5 2 3 2 1 2 / 1 10 2 10

F

F

F

G

x

d

S

_R

κ

(3.12)

∫

∧ ∧ − = , 4 1 3 3 4 2 10 F H C S_CS

κ

(3.13) where

F

~

₃

=

F

₃

−

C

O

∧

H

₃

,

(3.14) and . 2 1 2 1 ~ 3 2 3 2 5 5 F C H B F F = − ∧ + ∧ (3.15) c)Type I superstring.

The action of type I superstring is:

O C I S S S = + (3.16)

(

)

(

)

∫

− + ∂ Φ∂ Φ − = − Φ ~ ], 2 1 4 [ 2 1 2 3 2 2 / 1 10 2 10 F R e G x d S_{C µ} µ

κ

(3.17)

(

)

( )

∫

−Φ − − = . 2 1 2 2 2 / 1 10 2 10 F Tr e G x d g S_{O v} (3.18)

The open string potential and field strength are written as matrix-valued forms

A

₁ and

F

₂, which are in the

vector representation as indicated by the subscript on the trace. Here ~ _{2 3},

10 2 10 2 3

ω

κ

g dC F = − (3.19) and

ω

₃ is

the Chern-Simons 3-form .

3 2 1 1 1 1 1 3       ∧ ∧ − ∧ =Trv A dA i A A A

ω

(3.20)

Under an a ordinary gauge transformation

δ

A

₁

=

d

λ

−

i

[

A

₁

,

λ

],

the Chern-Simons form transforms as

(

₁

)

.

3 dTrv

λ

dA

δω

= (3.21) Thus it must be that

(

₁

)

.

2 10 2 10 2 Tr dA g C

κ

_v

λ

δ

= (3.22)

Hence the equation (3.17) becomes:

(

)

[

(

)

]

2 1 1 1 1 1 2 10 2 10 2 2 2 / 1 10 2 10

3

2

2

1

4

2

1

∫













∧

∧

−

∧

−

−

Φ

∂

Φ

∂

+

−

=

− Φ

A

A

A

i

dA

A

Tr

g

dC

R

e

G

x

d

S

_C

κ

_v

κ

µ µ (3.23) d)Heterotic strings.

The heterotic strings have the same supersymmetry as the type I string and so we expect the same action.

However, in the absence of open strings or R-R fields the dilaton dependence should be e−2Φ throughout:

(

)

[

( )

]

∫

− + ∂ Φ∂ Φ− − = − Φ . ~ 2 1 4 2 1 2 2 2 10 2 10 2 3 2 2 / 1 10 2 10 F Tr g H R e G x d S_het

κ

_v

κ

µ µ (3.24) Here , ~ 3 2 10 2 10 2 3

ω

κ

g dB H = − 2

(

1

)

10 2 10 2 Tr dA g B v

λ

κ

δ

= (3.25) are the same as in the type I string, with the form

renamed to reflect the fact that it is from the NS sector. 4.D-brane actions.

The coupling of a D-brane to NS-NS closed string fields is the same Dirac-Born-Infeld action as in the bosonic string,

(

)

[

]

{

}

∫

− + + − = _p p+1 −Φ det _{ab ab} 2 ' _ab 1/2 , Dp d Tr e G B F S

µ

ξ

πα

(4.1)

where G_ab and B_ab are the components of the spacetime NS-NS fields parallel to the brane and F_ab is the gauge field living on the brane.

The gravitational coupling is 2

( )

2 7 2 '4

2 1

α

π

κ

= g . (4.2) Expanding the action (4.1) gives

the coupling of the Yang-Mills theory on the Dp-brane,

(

2 '

)

( )

2 ' . 1 2 ( 3)/2 2 2 − − = = p p p Dp g g

π

α

τ

πα

(4.3)

The Born-Infeld form for the gauge action applies by T-duality to the type I theory, is:

(

)

∫

{

[

−

(

+

)

]

}

−

=

det

2

'

,

'

2

1

10 1/2 2 2

η

µν

πα

µν

πα

g

d

xTr

F

S

YM

(4.4) where for the relations (4.2) and (4.3) we

have the type I relation 2

( )

2 7/2 '

2

α

π

κ

= YM g (type I) (4.5)

Another low energy action with many applications is that for a Dp-brane and Dp’-brane. There are three kinds of light strings: p-p, p-p’, and p’-p’, with ends on the respective D-branes. We will consider explicitly the case p=5 and p’=9. The massless content of the 5-9 spectrum amounts to half of a hypermultiplet. The other half comes from strings of opposite orientation, 9-5. The action is fully determined by supersymmetry and the charges; we write the bosonic part:

(

)

∫

∫

∫

∑













+

−

−

−

=

=

.

2

'

'

4

1

4

1

3 1 2 2 5 6 6 2 5 10 2 9 A j A ij i D MN MN D MN MN D

g

D

D

x

d

F

xF

d

g

F

xF

d

g

S

χ

ι µ

χ

χ

ι

σ

χ

µ (4.6)

The integrals run respectively over the 9-brane and the 5-brane, with M =0,...,9,

µ

=0,...,5, and

. 9 ,..., 6 =

m The covariant derivative is D_µ =∂_µ +iA_µ −iA'_µ with A_µ and A'_µ the 9-brane and 5-brane

gauge fields. The field

χ

_i is a doublet describing the hypermultiplet scalars. The 5-9 strings have one

endpoint on each D-brane so

χ

carries charges +1 and -1 under the respective symmetries. The gauge

couplings gDp were given in equation (4.3).

5.Mathematics concerning the parallelism between Palumbo’s model and string theory.

We take now the relationship of Palumbo’s model:

∫

∞

=

0 i i

dF

F

F

(5.1) Here F denotes the initial energy

present at the Big Bang explosion (the explosion of initial black hole) and F_i all the partial waves belonging

to F (12)-(13). For the parallelism found between Palumbo’s model and string theory, F is the oscillation

mode of a bosonic string having mass equal to zero (graviton) and F_i are the oscillation modes of supersymmetric strings. Then, we have from the equations (d) and (3.1):

∫

∞

=

0 i i

dF

F

F

->

∫

∫

∫

∞

−

−

=

∂

∂

−

−

M b a ab

d

d

X

X

d

d

0

4

'

1

'

4

1

γ

τ

σ

πα

γ

γ

σ

τ

πα

µ µ

γ

µ ν

ψ

µ

ψ

ν

χ

ψ

µ ν

ψ

ν _

η

_µν            − ∂ Γ Γ − ∇ Γ − ∂ ∂ b b a b a a a b a ab X i X i i X X 4 (5.2)

This equation for the equations (1.15) and (3.24) can be defined also

(

)

∫

∫

∞

∫

− Φ

−

=













Φ

∂

Φ

∂

+

∂

∂

−

∂

∂

−

0 2 2 / 1 10 2 10 ˆ ˆ ˆ ˆ ˆ ˆ ˆ 26 2

2

1

~

~

3

2

2

1

8

1

e

G

x

d

h

h

h

h

X

d

κ

κ

µ µ λ λ µ ν ν µ λ ν µ νλ µ

( )













−

−

Φ

∂

Φ

∂

+

2 2 2 10 2 10 2 3

~

2

1

4

Tr

F

g

H

R

_µ µ

κ

_v (5.3)

and for the equation (2.12) of Scherck-Schwarz theory, we have:

(

)

( )

(

)

∫

∫

∞

∫

− Φ

−

=









∂

∂

−

−

−

−

0 2 2 / 1 10 2 10 26

2

1

2

1

8

1

16

G

g

g

Tr

G

G

f

g

d

x

G

e

R

g

x

d

κ

φ

φ

φ

π

µ ν µν ρσ µν νσ µρ

( )













−

−

Φ

∂

Φ

∂

+

2 2 2 10 2 10 2 3

~

2

1

4

Tr

F

g

k

H

R

_µ µ _ν (5.4), where the sign minus indicates the expansion force: i.e.

the Einstein cosmological constant.

With regard the D-branes the equation that is related to F_i, thus to the supersymmetric action is (see eq.4.4)

(

)

∫

{

[

−

(

+

)

]

}

−

=

10 1/2 2 2

det

2

'

'

2

1

µν µν

πα

η

πα

g

d

xTr

F

S

YM

, while the equation related to F, thus to the

bosonic action is (see eq.4.1)

(

)

[

]

{

}

∫

− + + − = 26 −Φ 1/2 25 25 det ab ab 2 ' ab D d Tr e G B F

S

µ

ξ

πα

. Thus, for parallelism Palumbo’s model ->

string theory, we have:

(

)

[

]

{

}

(

)

{

[

(

)

]

}

∫

−

+

+

=

∫

−

∫

−

+

−

∞ Φ − 10 1/2 2 2 0 2 / 1 26 25

det

2

'

'

2

1

'

2

det

η

_µν

πα

_µν

πα

πα

ξ

µ

d

xTr

F

g

F

B

G

e

Tr

d

YM ab ab ab (5.5)

6.Further correlations between Palumbo’s model and D-brane actions. a) First order action for type IIB Dirichlet 3-brane.

Now we want to discuss a new first order action for type IIB Dirichlet 3-brane (3). Its form is inspired by the superfield equations of motions obtained from the generalized action principle. The action involves auxiliary symmetric spin tensor fields.

The original D-brane action is of Dirac-Born-Infeld (DBI) type:

(

)

∫

− + − = + 0 , det 1 M mn mn p DBI d g F

I

ξ

(6.1) where

g

_mn

=

∂

_m

X

m

η

_mn

∂

_m

X

n is the induced metric, and the

field Fmn are the components of the 2-form field strength nm,

n m F d d dA F = =

ξ

∧

ξ

m. m A d A=

ξ

(6.2)

The moving frame action for the bosonic Dp-brane in flat D=10 space-time reads

(

)

∫

+ + + + = 1 1 1 0 1 p M p p D L L S (6.3)

(

)

a a

(

ab ab

)

a a p E E F p L p p + − ∧ ∧ + = + ...

ε

det

η

! 1 1 ... 0 1 ₀ 0 _(6.4)     ∧ − ∧ = − + ab a b p p Q dA E E F L 2 1 1 1

1 (6.5) In (6.4) Qp−1 is the Lagrange Multiplier

(

p

−1

)

−

form,

( ) ( )

ξ

a

ξ

m m a

u

dX

E

=

(6.6) is the pull-back of the

(

p

−

1

)

components of a target space vielbein form

(

m

=

0

,...,

9

;

a

=

0

,...,

9

;

a

=

0

,...,

p

;

i

=

1

,...,

9

−

p

)

(

a i

)

a m m a

E

E

u

dX

E

=

=

,

(6.7) which is related to the holonomic vielbein dXm by a Lorentz rotation

parametrized by the matrix

( )

u

_ma

=

(

u

_ma

,

u

_mi

)

∈

SO

( )

1

,

9

⇔

u

_ma

η

mn

u

_nb

=

η

ab (6.8) The moving frame action for strings and for type I p-branes can be written as

(

)

∫

∧ ∧ + = + ... . ! 1 1 ... ' 0 0 1 p p p a a a a M I E E p

S

ε

(6.9) However, its original form was the first order with respect

to the X variable

(

)

∫

      ∧ ∧ ∧ + − ∧ ∧ ∧ = + p p p p p a a a a a a a a a a M I e e e p e e E p S ... ... ' 0 1 0 0 1 0 1 ... ! 1 1 ... ! 1

ε

ε

(6.10) where

( )

ξ

ξ

a m m a e d

e = is the (auxiliary) world volume vielbein 1-form field. For the case of a type IIB 3-brane, an

adequate choice of the matrix m is given by

( ) ( )

(

~

)

. 1 F F Tr F z F z m_ab =

δ

_ab +

σ

_a

σ

b (6.11) In the trace of

(6.11) F F, are spinor representations for the selfdual and the anti-selfdual part of the tensor F_ab

(

~

)

, 4 αβ βα αβ

σ

a

σ

b ab F i F F = =

(

~

)

, 4 βα α β β

α&& &_&

σ

a

σ

b && ab F i F F = =− (6.12)

,

2

2

~

β α β α β α β α β β α α β β α α

F

σ

σ

δ

F

δ

F

F

_&&

≡

_ab a_{& b}&

=

_&&

+

_&&

∗

F

_αβ_α&β&

=

2

δ

_αβ

F

_α&β&

−

2

δ

_α&β&

F

_αβ

,

(6.13)

and the scalar factors z and its complex conjugate z are expressed through the F_ab tensor as

(

)

, det 8 1 2 1       + − + + = i F F F z

ε

abcd _{ab cd}

η

(6.14)

The action for the D3-brane can be represented as =

∫

+

(

+

)

,

1 4 ' 0 4 ' 3 13 L L S_{D M} (6.15) where

( )

(

)

      ∧ ∧ ∧ − ∧ ∧ ∧ + − = − abcd d c b a abcd d c b a a a ab ab F E m e e e e e e e m L

η

ε

ε

! 4 * 3 1 ! 3 * 3 1 det det ' 1 ' ' 0 4

(6.16) can be obtained from (6.10) if one replaces ea by ba b

m

e , includes an overall multiplier

(

+

F

)

( )

, det ' ' ' ' ' ' ' 'm m m m m abcd d d c c b b a a abcd

ε

ε

= det

( )

' ' ' ', ' 1 ' ' ' abcd a a d d c c b b abcdm m m mm

ε

ε

− =

The second term in the functional (6.15)

(

_{( )}

)

,

2 1 2 2 2 1 4       ∧ − − ∧ = − ab a b F E E B dA e Q L φ (6.17) corresponds to (6.5), but with the dependence on dilaton

φ

=

φ

(

X

( )

ξ

)

and NS-NS 2-form background field

( )

X B dX dX B = m ∧ n _nm 2 1 2 restored.

The superfield (embedding) equations for type II super-D3-brane include the symmetric spin tensor (super)fields h_αβ , h_α&β& . They are expressed in terms of anti-selfdual and selfdual components F_αβ and F_α&β&

of the antisymmetric tensor Fab by

( )

, 1 _αβ αβ F F z h =

( )

, 1 β α β

α&& F&&

F z

h = (6.18) where the functions z, z

are defined in eq.(6.14). The expression (6.11) for the matrix

m

simplifies when written in the terms of these spin tensors

(

~

)

, 2 1 h h Sp

m_ab =

δ

_ab +

σ

_a

σ

b ⇔

m

_αβ_αβ_&&

≡

m

_ab

σ

_αa_α&

σ

~

_bββ&

=

2

(

δ

_αβ

δ

_α&β&

+

h

_αβ

h

_α&β&

)

.

(6.19)

From eqs.(6.14) we have

z

+

z

=

1

+

−

det

(

η

+

F

)

,

ab cd

abcd F F i z z

ε

8 = − (6.20) whereas from

eqs.(6.12) and (6.18) we obtain

, * 2 1 4 2 2 2 2 h z h z F F F F i ba ab cd ab abcd − = ≡

ε

, 2 1 2 2 2 2 h z h z F F_ab ba =− − (6.21) with the

abbreviations

h

2

≡

h

_αβ

h

αβ

,

h2 ≡h_α&β&hα&β&. (6.22) These relations and the identity

(

)

2 8 1 2 1 1 det       − − ≡ + − ab cd abcd ba abF F F F F

ε

η

can be used to write the product of z with z (6.14) as

(

)

(

)

2

(

)

. 8 1 det 1 4 2 2 2 2 2 2 h z h z z z F F F z z abcd _{ab cd} = + + +      + + − + =

η

ε

(6.23)

From the second eq.(6.20) together with (6.21) and replacing the first eq.(6.20) by (6.23) we arrive at

(

)

, 2 1 2 2 2 2 h z h z z z− = −

(

)

., 2 1 2zz z2h2 z2h2 z z+ = − + (6.24)

The sum and the difference of these equations are homogeneous in z and z, respectively. Thus for

nonvanishing z we can extract a system of linear equations for z and z

, 2 1 1=z− h2z , 2 1

1=z− h2z (6.25) with the solution ,

4 1 1 2 1 1 2 2 2 h h h z − + = . 4 1 1 2 1 1 2 2 2 h h h z − + = (6.26)

Substituting (6.26) into (6.18) we obtain the expression for the selfdual and anti-selfdual parts (6.12) of Fab and, hence, for the whole tensor Fab from (6.13). The expression for the DBI-like square root can be obtained directly from eqs.(6.26) and (6.20):

(

+

F

)

−

det

η

= 4 1 2 1 2 1 2 2 2 2 h h h h −       +       + (6.27)

The inverse matrix m−1 and the determinant

det

( )

m

are

(

)

( )

, det 1 ~ 2 1 1 m h h Sp m ba _ab _a b       − = −

σ

σ

δ

(6.28)

( )

. 4 1 det 2 2 2       − = h h m (6.29)

Eq.(6.28) can be obtained directly from the spinor representation for the matrix

m

(6.19) while the simplest

way to obtain (6.29) is to use a special gauge, where only two of the components of the tensor Fab

,

01 = f+

F F34 = f₋ (6.30) are nonvanishing. Substituting (6.27), (6.28), (6.29), into (6.15), (6.16), (6.17)

we arrive at

(

14'

)

, ' 0 4 ' 3 13 L L SD =

∫

_M+ + (6.31) , 4 1 2 1 ~ 2 1 ! 3 * 3 2 1 2 1 2 2 ' ' ' 2 2 ' 0 4       ∧ ∧ ∧       − − ∧ ∧ ∧       −       +       + = a b c d _abcd abcd d c a a a a a e e e e h h e e e h h E h h L

δ

σ

σ

α

ε

ε

β α α α β β β

α & && & (6.32) ( )

(

)

                      + +       + ∧ − −       − ∧ =Q h h dA B Eαα E_ββ

δ

_αβ h h_αβ

δ

_αβ h h_αβ L 2 1 2 1 4 1 4 1 2 2 2 2 2 ' 2 ' 1 4 & & & & & & _{, (6.33)}

where, in addiction to (6.22) the bispinor representation for the vielbein indices Eαα& =Eα

σ

~aαα& are used. We recall that h_αβ =h_βα, h_α&β& =h_β&α& are auxiliary symmetric spin tensor fields replacing F_ab. The functional (6.31), (6.32), (6.33) is the result for the first order action for the type IIB D3-brane.

b) A supersymmetric action functional describing the interaction of the fundamental superstring with the D=10, type IIB Dirichlet super-9-brane (4).

The geometric action for the super-D9-brane in flat D=10, type IIB superspace is

(

)

∫

=

∫

+ + = WZ M M L L L L S 10 ₁₀ 10 ₁₀0 1_{10 10} (6.34), where

L

=

Π

+

F

∧

η

10 0 10 (6.35) with

(

)

, det mn Fmn F ≡ − + +

η

η

( )

10! ... , 1 _{1 10} 10 1... 10 m m m m Π ∧ ∧Π ≡ Π∧

_ε

(6.36) . 2 1 2 8 1 10       Π ∧ Π − − ∧ = nm n m F B dA Q L (6.37)

( )

x A dx

A= m _m is the gauge field inherent to the Dirichlet branes,

B

₂ represents the NS-NS gauge field with

flat superspace value 2

=

Π

∧

(

Θ

1

Θ

1

−

Θ

2

Θ

2

)

+

Θ

1

Θ

1

∧

Θ

2 m

Θ

2

m m m m

d

d

d

d

i

B

σ

σ

σ

σ

(6.38) and field strength

H

3

=

dB

2

=

i

Π

∧

(

d

Θ

1 m

∧

d

Θ

1

−

d

Θ

2 m

∧

d

Θ

2

)

.

m

_σ

_σ

(6.39) The Wess-Zumino Lagrangian form is the same as the one appearing in the standard formulation

10 10 2

_C

e

L

WZ

=

F

∧

, C n C2n 5 0 = ⊕ = , n n F F n e =⊕ = 2∧ 5 0 ! 1 2 _{, (6.40)}

where the formal sum of the RR superforms C =C0 +C2 +... and of the external powers of the 2-form

2

2

dA

B

F

≡

−

(6.41) is used and ₁₀ means the restriction to the 10-superform input.

The equations of motions from the geometric action (6.34)-(6.37) split into the algebraic ones obtained from the variation of auxiliary fields Q8 and Fmn

, 2 1 2 2 nm n m F B dA

F ≡ − ≡ Π ∧Π (6.42)

Q

₈

=

Π

_nm∧8

η

+

F

(

η

+

F

)

−1nm (6.43) and the dynamical ones

0 7 8 8 + = −D WZ dL dQ , (6.44) Π∧_m9

η

+F

(

η

+F

)

−1mn

σ

_nµν ∧

(

dΘ2ν −dΘ1ρh_ρν

)

=0. (6.45) Turning back to the fermionic equations (6.45), let us note that after decomposition

, 1 2 ν ν ρ ρ ν m m h d

dΘ − Θ =Π Ψ (6.46) where Ψ_mν =∇_mΘ2ν −∇_mΘ1ρhν_ρ (6.47) and ∇_m defined by

m m m m

dx

(

)

1 0, 10 = + Ψ + Π − ∧ −mn m k F F i

η

σ

ν

η

µν ⇔

(

)

0

.

1

=

+

Ψ

m − mn k

η

F

σ

ν µν (6.48)

In order to represent the kinetic term

∫

_M+ L ≡

∫

_M+ Eˆ++ ∧Eˆ−− 2 1 ˆ 1 1 1

1 ₀ as an integral over the D9-brane world

volume too, we have to introduce of the harmonic fields in the whole 10-dimensional space or in the D9-brane world volume

( )

x

(

u

( )

x

u

( )

x

u

( )

x

)

u

_ma

≡

_m++

,

_m−−

,

_mi ∈

SO

( )

1

,

9

and =

(

+ I−

)

q I q v v vα_{µ µ} , _µ& ∈

Spin

( )

1

,

9

Such a “lifting” of the harmonics to the super-D9-brane worldvolume creates the fields of an auxiliary ten dimensional

SO

( )

1

,

9

/

(

SO

( )

1

,

1

*

SO

( )

8

)

“sigma model”. The only restriction for these new fields is that they should coincide with the “stringy” harmonics on the worldsheet:

++

(

( )

ξ

)

ˆ

++

( )

ξ

,

=

m m

x

u

u

ˆ

−−

(

( )

ξ

)

ˆ

−−

( )

ξ

,

=

m m

x

u

u

u

_mi

(

x

( )

ξ

)

=

u

ˆ

_mi

( )

ξ

(6.49)

( )

(

ξ

)

α

( )

ξ

µ α µ x v v = ˆ : _µ+

(

( )

ξ

)

ˆ_µ+

( )

ξ

, = I q I q x v v _µ−

(

( )

ξ

)

= _µI−

( )

ξ

q I q x v v _& ˆ _& (6.50)

In this manner we arrive at the full supersymmetric action describing the coupled system of the open fundamental superstring interacting with the super-D9-brane:

(

)

∫

+ ∧ + ∧ = = L J L dJ A S _M10 10 8 IIB 8 =

∫

(

)

_      ∧ +       Π ∧ Π − − ∧ + + − Π∧10 _{8 2 10} 2 1 det 10 F Q dA B F e C F mn n m mn mn M

η

+

∫

+

∫

∧      − ∧ ∧ ++ −− A dJ B E E J _M M10 8 2 10 8 2 1 (6.51)

Also the equations (6.31)-(6.33) and (6.51) can be connected at the Palumbo’s model, precisely at the Fi, thus at the superstring action. Then, taking the expression obtained from parallelism between Palumbo’s model and string theory and putting, in the right hand, type IIB action, from equations (2.12) and (3.10)-(3.13) we have:

(

)

( )

∫

_− − − ∂ ∂ _= −

φ

φ

φ

π

µ ν µν ρσ µν νσ µρ g f G G Tr g g G R g x d 2 1 8 1 16 26 =



−











+

+

−

−













−

Φ

∂

Φ

∂

+

−

∫

∫

∫

− Φ ∞ 2 5 2 3 2 1 10 2 10 2 3 2 10 2 10 0

~

2

1

~

4

1

2

1

4

2

1

F

F

F

G

x

d

H

R

e

G

x

d

κ

κ

µ µ

∫

∧ ∧ − _{2 4 3 3} 10 4 1 F H C

κ

⇒ ⇒ _   + ∧ ∧ ∧       −       +       +

∫

+ _abcd d c a a a a a a M E h h e e e h h

ε

σ

σ

δ

α β α α α β β β & & & & ' ' ' 2 2 ~ 2 1 ! 3 * 3 2 1 2 1 3 1

]



(

_{( )}

)

  ∧ − −       − ∧ + ∧ ∧ ∧       − −

ε

αα _ββ& & E E B dA h h Q e e e e h h abcd d c b a 4 1 4 1 4 1 2 1 2 2 2 ' 2 2 2

]

              + +       + _αβ _αβ _αβ β α

δ

δ

h h h h 2 1 2 1 2 2 & & & & (6.52) and

(

)

( )

_=   ∂ ∂ − − − −

∫

φ

φ

φ

π

µ ν µν ρσ µν νσ µρ g f G G Tr g g G R g x d 2 1 8 1 16 26 =

∫

∫

∫



−











+

+

−

−













−

Φ

∂

Φ

∂

+

−

− Φ ∞ 2 5 2 3 2 1 10 2 10 2 3 2 10 2 10 0

~

2

1

~

4

1

2

1

4

2

1

F

F

F

G

x

d

H

R

e

G

x

d

κ

κ

µ µ