New mathematical connections concerning string theory 2

New mathematical connections concerning string theory: II

Michele Nardelli

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.

Nella presente tesi, vengono evidenziate ulteriori connessioni trovate tra alcuni settori della teoria di stringa ed il modello di Palumbo.

Ricordiamo che tale modello 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, (correlata all’azione di stringa bosonica) costituita a sua volta da insiemi parziali di onde, definite Fi (correlate all’azione di

superstringa). Vengono evidenziate le connessioni trovate tra il modello di Palumbo e: 1) le D-stringhe, 2) la corrispondenza gauge/gravità e la dualità stringa aperta/chiusa, 3) la connessione trovata tra alcune equazioni della tesi di Durr “On a Gauge and Conformal Invariant Nonlinear Spinor Theory” e le azioni Dirac-Born-Infeld per una D3-brana e quelle che sono alla base della congettura di dualità Het/T4 −IIA/K3.

Vengono inoltre descritte ulteriori connessioni trovate tra altre formule legate alla funzione zeta di Riemann ed alcune soluzioni in cosmologia di stringa e teoria di campo di stringa.

Infine, vengono studiate alcune equazioni differenziali che descrivono configurazioni con singolarità nude e le connessioni matematiche trovate tra singolarità nude ed alcuni teoremi applicati a soluzioni di problemi al contorno per equazioni differenziali riguardanti insiemi aperti. Di tali equazioni differenziali, definite in insiemi aperti, sono state studiate anche le condizioni al contorno alla frontiera di tali insiemi.

1.Mathematical connections between Palumbo’s model and some equations concerning D-term strings.[1]-[2]

It is known that string theories admit various BPS-saturated string-like objects in the effective 4d theory. These are D_1+q-branes wrapped on some q-cycle. We shall refer to these objects as effective

D

₁-strings, or D-strings for short. Thus, we conjecture that the string theory D-strings (that is, wrapped D1+q-branes) are

seen as D-terms strings in 4d supergravity. Since according to the conjecture D_1+q branes are D-term strings, it immediately follows that the energy of the

D

_3+q

−

D

_3+q-system must be seen from the point of view of the 4d supergravity as D-term energy.

The supergravity model is defined by one scalar field φ, charged under U(1), with K=

φ

∗

φ

and superpotential W=0, so that we reproduce the supergravity version of the cosmic string in the critical Einstein-Higgs-Abelian gauge field model. This model can be also viewed as a D-term inflation model. In such case, the bosonic part of the supergravity action is reduced to

D P bos M R F F V L e− =− −∂ ∂ ∗ − µν µν − µ µ

φ

φ

4 1 ˆ ˆ 2 1 2 1

, (1.1) where D-term potential is defined by

2 2 1

D

VD = D= g

ξ

−g

φ

∗

φ

. (1.2) Here W_µ is an abelian gauge field,

µ ν ν µ µν W W F ≡∂ −∂ , ∂ˆ_µ

φ

≡

(

∂_µ −igW_µ

)

φ

. (1.3) The energy of the string is:

(

)( )



+











+

+

+

∂

∂

=

∫

g

drd

∗

F

F

D

M

P

R

string

2

2

1

4

1

ˆ

ˆ

det

2 2 µν µν µ µ

φ

φ

θ

µ

(

0

)

2 det det _=∞

∫

₌

∫

− +MP d

θ

hK r d

θ

hK r , (1.4)

where K is the Gaussian curvature at the boundaries (on which the metric is h). These boundaries are at

∞ =

r and r =0. Further, for the metric ds2 =−dt2 +dz2 +dr2 +C2

( )

r d

θ

2, (1.5) we have

( )

r

C

g

=

det

,

det

g

R

=

2

C

''

, dethK =−C' (1.6)

Eq.(1.4) can be rewritten by using the Bogomol’nyi method as follows

( )

(

)

[

]

∫

+       + ∂ ± ∂ = − 12 2 2 1 2 1 ˆ ˆ _{iC F D} r C drd r string

θ

φ

φ

m

µ

θ + 2

∫

[

∂

(

±

)

∂

]

− 2

∫

_=∞+ 2

∫

₌₀ ' ' ' B _rB _{P r P r} r P drd C A A M d C M d C M

θ

_θ m _θ

θ

θ

, (1.7)

Where we have used the explicit form of the metric (1.5). The energy of the string, can be also defined as:

ξ

π

θ

µ

string drd detgT 2 n 0 0 = =

∫

, (1.8) where

(

)

[

]

[

B

]

r B r P r iC F D M A A T

φ

_θ

φ

± ∂ _θ −∂_θ       + ∂ ± ∂ = − 2 2 12 2 1 0 0 2 1 ˆ ˆ m _{. (1.9)}

The definition of the energy of the string that we are using in (1.4), which is valid for time independent configurations, is

K

h

M

L

R

M

g

E

M P matter P M

∫

∫



_

+

∂









−

=

det

2

det

2 2 . (1.10)

Now we see that the term

_











−

_matter P

L

R

M

2

2

produced in addition to two BPS bounds in (1.9) also a term

(

)

[

B

]

r B

r

C

±

A

θ

∂

θ

A

∂

_'

m

_{. (Note that the BPS state is a state that is invariant under a nontrivial subalgebra of} the full supersymmetry algebra. Such states always carry conserved charges, and the supersymmetry algebra determines the mass of the state exactly in terms of its charges). Due to the gravitino BPS bound

( )

B

A r

C =± _θ

− '

1 , the surface term ∂rAθB in

0 0

T is cancelled by the Einstein term

g

R

. This is not

surprising since the Einstein equation of motion must be satisfied due to vanishing gravitino transformations. The remaining term in the energy, the Gibbons-Hawking K surface term, give the non-vanishing contribution to the energy of the string which is directly related to the deficit angle ∆ , where

M

_P2

∆

=

µ

_string

.

The “SuperSwirl” is a static, supersymmetric, codimension-two configuration for a nonlinear sigma model, in the context of six dimensional gauged supergravity.

The energy per unit four dimensional volume of the superswirl turns out to diverge, due to the contributions from the boundaries. This energy can be computed from

(

)

(

)

∫

_       − + + − + = − ∗ 2 2 2 2 2 1 ' 8 1 4 1 1 4 1 0 0

φ

φ

φ

φ

θ

ε

ϕ ϕ g e F F e D D R g drd mn mn m m

(

)

− + = =

∫

∫

− + d

θ

hK _{r r} d

θ

hK _{r r} 2 1

, (1.11) where K is the extrinsic curvature of the surfaces r=constant, whose metric is h. In this case these surfaces are the “boundaries” at

r

_±. This energy can be expressed in a Bogomol’nyi type form as follows:

(

)

(

)

               − + + − + =

∫

2 2 2 2 2 1 2 ' 1 1 2 1 0 0

φ

φ

φ

φ

θ

ε

ϕ ϕ θ g e f e iD rD r drd r

(

∫

∫

)

− +− + d rB' r d rB' r 2 1

θ

θ

. (1.12)

From this expression is clear that the supersymmetry constraints

(

2

)

1 2 ' 0

φ

ϕ − − = − e g f and

D

z

φ

=

0

, 0 = ∗

φ

z

D in terms of the

(

r

,

θ

)

coordinates, imply the vanishing of the first two terms of the energy. Thus the energy is given entirely by the last two terms. These are given by

− +      + − +       + − − = r r _r r r _r

ψ

ψ

ψ

ψ

π

ψ

ψ

ψ

ψ

π

ε

' 1 ' ' 1 ' . (1.13)

Hence, we have that the energy (per unit volume) is infinite, since it is proportional to the boundary terms computed at the singular points. This system should have boundary source terms that cover the singularities. These should regularise the latter, rendering the total energy finite. This new solution constitutes a new class of supersymmetric vacua for 6D chiral gauged supergravity, with possible implications for a deeper understanding of the theory itself, in particular its origin from higher dimensional supergravities or string theories.

We note that the equations (1.11) and (1.12) are related at the equations (1.4) and (1.7), above mentioned. Further, these equations can be related to Palumbo’s model, precisely at the D-brane actions, thus with Fi.

We take the equation of coupling of a D-brane to NS-NS closed string fields and the equation of the Born-Infeld form for the gauge action applies by T-duality to the type I theory. For parallelism Palumbo’s model
string theory, we have:

(

)

[

]

{

}

∫

− + + = − 26 −Φ 1/2 25 d

ξ

Tr e det Gab Bab 2

πα

'Fab

µ

(

)

{

[

(

)

]

}

∫

∫

∞

⇒

+

−

−

=

0 2 / 1 10 2 2

det

2

'

'

2

1

µν µν

πα

η

πα

g

_YM

d

xTr

F

(

)( )

_











+

+

+

∂

∂

⇒

∗ ∞

∫

∫

g

drd

F

F

D

M

P

R

2

2

1

4

1

ˆ

ˆ

det

2 2 0 µν µν µ µ

φ

φ

θ

+ +

(

∫

=∞ −

∫

=0

)

⇒ 2 det det r r P d hK d hK M

θ

θ

(

)

(

)

∫

∫

_       − + + − + ⇒ − ∗ ∞ 2 2 2 2 2 0 1 ' 8 1 4 1 1 4 1 0 0

ϕ

ϕ

φ

φ

θ

ϕ ϕ g e F F e D D R g drd mn mn m m + +

(

)

− + = =

∫

∫

d

θ

hK _{r r} − d

θ

hK _{r r} 2 1 . (1.14)

Here, we see that also the energy of the D-strings can be related at the Palumbo’s model.

2.Mathematical connections between Palumbo’s model and some equations concerning gauge/gravity correspondence and open/closed string duality.[3]

With regard to gauge/gravity relations for the gauge theory living on fractional D3 and wrapped D5 branes using supergravity calculations, we have that since also the fractional D3 branes are D5 branes wrapped on a vanishing 2-cycle located at the orbifold fixed point, we can start from the world-volume action of a D5 brane, that is given by:

WZW BI S

S

S = + , (2.1) where the Born-Infeld action

S

BI reads as:

(

)

∫

−

(

+

+

)

−

=

− IJ IJ IJ s BI

d

e

G

B

F

g

S

det

2

'

'

2

'

1

6 5

ξ

πα

α

π

α

φ

, (2.2) while the Wess-Zumino-Witten

action S_WZW is given by:

(

)

∫ ∑

     ∧ = + 6 2 ' 2 5 ' 2 ' 1 V n B F n s WZW C e g S πα

α

π

α

. (2.3) Hence, we have:

(

)

(

)

























∧

+

+

+

−

−

=

∫

−

∫ ∑

+ 6 2 ' 2 6 5

det

2

'

'

2

'

1

V n B F n IJ IJ IJ s

e

C

F

B

G

e

d

g

S

ξ

φ

πα

πα

α

π

α

. (2.4)

We divide the six-dimensional world-volume into four flat directions in which the gauge theory lives and two directions on which the brane is wrapped. Let us denote them with the indices I,J=(α,β;A,B) where α and β denote the flat four-dimensional ones and A e B the wrapped ones. We assume the supergravity fields to be independent from the coordinates α, β. We also assume that the determinant in eq.(2.2) factorizes into a product of two determinants, one corresponding to the four-dimensional flat directions where the gauge theory lives and the other one corresponding to the wrapped ones where we have only the metric and the NS-NS two-form field. By expanding the first determinant and keeping only the quadratic term in the gauge field we obtain:

(

)

(

)

(

)

(

AB AB

)

a a s BI

d

e

G

G

G

F

F

G

B

g

S

=

−

∫

−

−

det

det

+

8

'

2

'

2

'

1

2 ₆ 5 2 αβ γδ βδ αγ αβ φ

ξ

πα

α

π

α

, (2.5) where we have included a factor 1/2 coming from the normalization of the gauge group generators

[

]

2 ab b a T T Tr =

δ

.

Now we compute the one-loop vacuum amplitude of an open string stretching between a fractional D3 brane of the orbifold C2/ Z₂ dressed with a background SU(N) gauge field on its world-volume and a stack of N ordinary fractional D3 branes. The free energy of an open string stretched between a dressed D3 brane and a stack of N D3 branes located at a distance y in the plane

(

x4, x5

)

that is orthogonal to both the world-volume of the D3 branes and the four-dimensional space on which the orbifolds acts, is given by:

( ) ( )

∫

∞ − −



≡

+











−

−











 +

=

0 2 0

1

1

2

o h o e L GSO G F R NS

P

e

Z

Z

h

e

Tr

d

N

Z

s bc πτ

τ

τ

, (2.6)

where F_s is the space-time fermion number, G_bc is the ghost number and the GSO projector is given by:

( )

( )

2 1 1G F GSO P = − + − βγ

, (2.7) with G_βγ being the superghost number:

(

)

∑

∞ = − − + − = 2 / 1 m m m m m Gβγ

γ

β

β

γ

,

∑

(

)

∞ = − − + − − = 1 0 0 m m m m m Gβγ

γ

β

γ

β

β

γ

(2.8) respectively in the NS

and in the R sector. F is the world-sheet fermion number defined by

∑

∞ = − ⋅ − = 2 / 1 1 t t t

F

ψ

ψ

(2.9) in the NS sector and by

( )

₋1F _=Γ11

( )

₋1FR, Γ11 ≡Γ0Γ1...Γ9,

∑

∞ = − ⋅ = 1 n n n R

F

ψ

ψ

(2.10) in the R sector. The superscript o in Eq.(2.6) stands for “open” because we are computing the annulus diagram in the open string channel. We have:

(

)

∫

(

)

∫

(

) (

) (

)

∞ − − Θ Θ × + − − = 0 1 1 4 1 ' 2 4 2 2 sin sin ˆ det ' 8 2

τ

τ

ν

τ

τ

ν

πν

πν

τ

τ

η

α

π

πτ πα τ i i i i e f e d F x d N Z g f g f y o e

×

[

f

₃4

(

e

−πτ

) (

Θ

₃

i

ν

_f

τ

i

τ

) (

Θ

₃

i

ν

_g

τ

i

τ

)

−

f

₄4

(

e

−πτ

) (

Θ

₄

i

ν

_f

τ

i

τ

) (

Θ

₄

i

ν

_g

τ

i

τ

)

−

f

₂4

(

e

−πτ

) (

Θ

₂

i

ν

_f

τ

i

τ

) (

Θ

₂

i

ν

_g

τ

i

τ

)

]

, (2.11) and

(

)

∫

(

)

∫

( )

(

) (

)

∞ −         Θ Θ Θ × + − − = 0 1 1 2 2 ' 2 4 2 2 0 0 sin sin 4 ˆ det ' 8 2

τ

τ

ν

τ

τ

ν

τ

πν

πν

τ

τ

η

α

π

πα τ i i i i i e d F x d N Z g f g f y h

×

[

Θ

2₄

( )

0

i

τ

Θ

₃

(

i

ν

_f

τ

i

τ

) (

Θ

₃

i

ν

_g

τ

i

τ

)

−

Θ

₃2

( )

0

i

τ

Θ

₄

(

i

ν

_f

τ

i

τ

) (

Θ

₄

i

ν

_g

τ

i

τ

)

]

∫

∫

∞ − − 0 ' 2 4 2 2 ~ 32 πα τ αβ αβ

τ

τ

π

y a a e d F xF d iN , (2.12) where F_αβ

ε

_αβδγFδγ 2 1 ~ = .

The three terms in Eq.(2.11) come respectively from the NS, NS

( )

−1 F and R sectors, while the contribution from the R

( )

−1F sector vanishes. In Eq.(2.12) the three terms come respectively from the NS, NS

( )

−1 F

and R

( )

−1F sectors, while the R contribution vanishes because the projector h annihilates the Ramond vacuum.

The above computation can also be performed in the closed string channel where Z_ec and Z_hc are now given by the tree level closed string amplitude between two untwisted and two twisted boundary states respectively: ( )

∫

∞ + −

=

0

3

;

3

2

'

_{0 0} U L L t U c e

dt

D

F

e

D

N

Z

α

π

π (2.13) and

∫

( ) ∞ + −

=

0

3

;

3

2

'

_{0 0} T L L t T c h

dt

D

F

e

D

N

Z

α

π

π (2.14)

where

D

3

;

F

>

is the boundary state dressed with the gauge field F. Hence, we have:

(

)

∫

(

)

∫

(

) (

)

( )

∞ − − Θ Θ + − = 0 4 1 1 1 ' 2 3 4 2 2 sin sin ˆ det ' 8 2 t g f g f t y c e e f it it e t dt F x d N Z πα _π

ν

ν

πν

πν

η

α

π

{

f

( ) (

e

πt

ν

_f

it

) (

ν

_g

it

)

f

( ) (

e

πt

ν

_f

it

) (

ν

_g

it

)

2 2 4 2 3 3 4 3

Θ

Θ

−

Θ

Θ

×

− −

f

( ) (

e

πt

ν

_f

it

) (

ν

_g

it

)

}

4 4 4 4

Θ

Θ

−

− (2.15) and

(

)

∫

(

)

∫

( )

(

) (

)

∞ − Θ Θ Θ + − = 0 1 1 4 2 ' 2 4 2 2 ₀ sin sin 4 ˆ det ' 8 2 it it it e t dt F x d N Z g f g f t y c h

ν

ν

πν

πν

η

α

π

πα

{

( )

it

(

ν

_f

it

) (

ν

_g

it

)

2

( )

it

₂

(

ν

_f

it

) (

₂

ν

_g

it

)

}

3 3 3 2 2

0

Θ

Θ

−

Θ

0

Θ

Θ

Θ

×

∫

∫

−

−

t y a a

e

t

dt

F

xF

d

iN

4 _{2 '} 2 2

~

32

πα αβ αβ

π

. (2.16)

The three terms in Eq.(2.15) respectively come from the NS-NS, R-R and NS-NS

( )

−1 F sectors, while those in Eq.(2.16) from the NS-NS, R-R and R-R

( )

−1F sectors. In particular, the twisted odd R-R

( )

−1 F spin structure gets a nonvanishing contribution only from the zero modes.

It is useful to write Eq.(2.12) in a more convenient way. Using the notation for the Θ-functions

( ) ∑

∞ −∞ =               +       + +       +

=













Θ

n b a n t a n i

e

t

b

a

2 2 2 2 1 2 2 ν π

ν

, (2.17) and the identity

( )

( )

( )

i i i i i i i i b a ab b a

g

h

g

b

h

a

'

1

1

1

2

1

4 1 4 1 1 0 ,

ν

ν

∏

∏

∑

= = = + +













−

−

Θ

−

=













+

+

Θ

−

, (2.18) with 0 2 1 = = =g g h_i ; g₃ = g− ₄ =1;

ν

₁ =i

ν

_f

τ

;

ν

₂ =i

ν

_g

τ

;

ν

₃ =

ν

₄ =0

(

ν

ν

)

τ

ν

ν

g f i − = − = 2 ' '_{1 2} ;

ν

=

ν

= i

(

ν

_g +

ν

_f

)

τ

2 '

'_{3 4} , we can rewrite Eq.(2.12) as follows:

(

)

∫

(

)

∫

( )

(

) (

)

∞ −         Θ Θ Θ + − − = 0 1 1 2 2 ' 2 4 2 2 ₀ sin sin 4 ˆ det ' 8 2 2

τ

τ

ν

τ

τ

ν

τ

πν

πν

τ

τ

η

α

π

πα τ i i i i i e d F x d N Z g f g f y o h

_











+

Θ













−

Θ













−

Θ

×

i

ν

g

ν

f

τ

i

τ

i

ν

f

ν

g

τ

i

τ

i

ν

f

ν

g

τ

i

τ

2

2

2

2 2 1 1 . (2.19)

By expanding the previous equation up to the second order in F and using the following relations

( )

( )

t

( )

t

e

f

e

f

it

=

−π −π

Θ

2 4 , 3 , 2 1 4 , 3 , 2

0

;

( )

_{( )}

t e f it _π ν

_πν

ν

₋ → =− Θ ₃ 1 1 0_2sin

lim (2.20) together with

π

ν

_f ≅−i f and

π

ν

g ≅−g , we get:

(

)

∫

−

∫

∞ − = 0 ' 2 4 2 2 ~ 32 πα τ αβ αβ αβ αβ

τ

τ

π

y a a a a o h e d F iF F F x d N Z , (2.21) which reduces to

( )

_          − Λ     − →

∫

∫

∞ Λ − 2 2 ' 1 ' 2 2 2 4 8 1 4 1 ) ( α πα τ αβ αβ

τ

τ

π

y YM a a o h e d N g F xF d F Z

∫

∫

∞ Λ −     − 2 2 ' 1 ' 2 4 2 ~ 32 1 α πα τ αβ αβ

τ

τ

π

y a a e d F F x d iN . (2.22)

In the closed string channel we get instead:

        − Λ     − →

∫

∫

Λ − 2 2 ' 0 ' 2 2 2 4 8 ) ( 1 4 1 ) ( α πα αβ αβ

π

t y YM a a c h e t dt N g F xF d F Z

∫

∫

Λ −     − 2 2 ' 0 ' 2 4 2 ~ 32 1 αβ α _πα αβ

π

t y a a e t dt F xF d iN . (2.23)

Now we study the one-loop vacuum amplitude of an open string stretching between a stack of

(

I

=

1

,...,

4

)

N

_I branes of type I and a D3 fractional brane, with a background SU(N) gauge field turned-on

on its world-volume. Due to the structure of the orbifold C3/(Z₂×Z₂), this amplitude is the sum of four

terms:

∑

= + = 3 1 i h e Z _i Z Z , where Z_e and i h

Z are obtained in the open [closed] channel by multiplying Eq.s (2.11) and (2.12) [Eq.s (2.15) and (2.16)] by an extra 1/2 factor due to the orbifold projection. In the open string channel, _ho i

Z

is:

( )

(

) (

)

∫

∫

−

+

∞ −

_Θ

_Θ

_Θ

=

0 1 1 2 2 ' 2 4 2 2

0

sin

2

sin

2

)

ˆ

det(

)

'

8

(

2

)

(

2

τ

τ

ν

τ

τ

ν

τ

πν

πν

τ

τ

η

α

π

πα τ

i

i

i

i

i

e

d

F

x

d

N

f

Z

g f g f y i o h i i

×

{

Θ

₃2

( )

0

i

τ

Θ

₄

(

i

ν

_f

τ

i

τ

) (

Θ

₄

i

ν

_g

τ

i

τ

)

−

Θ

₄2

( )

0

i

τ

Θ

₃

(

i

ν

_f

τ

i

τ

) (

Θ

₃

i

ν

_g

τ

i

τ

)

}

( )

∫

∫

− − 4 2 ' 2 2 ~ 64 πα τ αβ αβ

τ

τ

π

i y a a i e d F xF d N if . (2.24)

The functions fi(N) introduced in Eq. (2.24) depend on the number of the different kinds of fractional

branes

N

I and their explicit expressions are: 4 3 2 1 1(N ) N N N N f _I = + − − , f₂(N_I)=N₁−N₂ +N₃ −N₄, f₃(N_I)=N₁−N₂ −N₃ +N₄ (2.25) Let us now extract in both channels the quadratic terms in the gauge field F. In the open sector, we get:

                − Λ     − →

∫

∑

∫

= ∞ Λ − 3 1 _{1/( ' )} ' 2 2 2 4 2 2 16 ) ( ) ( 1 4 1 ) ( i y i YM a a o h i e d N f g F xF d F Z α πα τ αβ αβ

τ

τ

π

(

)

∑

∫

∫

= ∞ Λ −     − 3 1 _{1/ '} ' 2 4 2 2 2 2 ) ( ~ 32 1 i y i a a i e d N f F xF d i α πα τ αβ αβ

τ

τ

π

, (2.26)

while in the closed string channel we obtain:

                − Λ     − →

∫

∑

∫

= Λ − 3 1 ' 0 ' 2 2 2 4 2 2 16 ) ( ) ( 1 4 1 ) ( i t y i YM a a c h i e t dt N f g F xF d F Z α πα αβ αβ

π

∫

∑

∫

= Λ −     − 3 1 ' 0 ' 2 4 2 2 2 2 ) ( ~ 32 1 i t y i a a i e t dt N f F xF d i α πα αβ αβ

π

, (2.27) where the divergent contribution is due

to the massless states in both channels.

Now we consider the validity of the gauge/gravity correspondence in the 26-dimensional bosonic string and we consider it in the orbifold Cδ/2/ Z₂ with

δ

<22. We consider the one-loop vacuum amplitude of an open string stretching between a D3 brane dressed with a background gauge field and a system on N undressed D3 branes. It is given by:

( )

o h o e L G

Z

Z

e

h

e

Tr

d

N

Z

bc

_≡

₊













−











 +

=

∫

∞ − 0 2 0

1

2

πτ

τ

τ

, (2.28) where L₀ includes the ghost and the matter contribution. By performing the explicit calculation of the one-loop vacuum amplitude one gets:

(

)

∫

(

)

∫

∞ − + − − = 0 ' 2 4 2 2 2 ˆ det ' 8 πα τ

τ

τ

η

α

π

y o e e d F x d N Z

(

)

(

) (

ν

τ

τ

) (

ν

τ

τ

)

πν

πν

πτ ν ν πτ i i i i e f e g f g f g f 1 1 18 1 sin sin 2 2 2 Θ Θ × ₋ + (2.29) and

(

)

(

)

(

)

(

) (

)

∫

∞

∫

+ −         Θ Θ + − − = 0 1 1 ' 2 4 2 2 sin sin 2 ˆ det ' 8 2 2 2

τ

τν

τ

τν

πν

πν

τ

τ

η

α

π

ν ν πτ πα τ i i i i e e d F x d N Z g f g f y o h g f

[

( )

]

( δ)

[

( )

]

δ δ − − −

×₂2 f₁ k 18 f₂ k _{, (2.30) where the power 18 is obtained from d-8 for the value of the critical} dimension d=26. The previous expressions can also be rewritten in the closed string channel and one gets:

(

)

∫

(

)

∫

( ) (

) (

)

∞ − − Θ Θ + − = 0 1 1 18 1 ' 2 11 4 2 2 sin sin 2 ˆ det ' 8 2 it it e f e t dt F x d N Z g f t g f t y c e

ν

ν

πν

πν

η

α

π

π

πα _{(2.31) for the untwisted}

sector and

(

)

∫

(

)

∫

(

) (

)

∞ − −         Θ Θ + − = 0 1 1 ' 2 2 / 11 4 2 2 sin sin 2 ˆ det ' 8 2 it it e t dt F x d N Z g f g f t y c h

ν

ν

πν

πν

η

α

π

πα δ

×2δ/2

[

f₁(q)

]

−(18−δ)

[

f₄(q)

]

−δ (2.32) for the twisted sector.

Also these equations can be related with the Palumbo’s model. For example, we take the equation of Scherck-Schwarz theory, the equation of heterotic string action and the equation of the one-loop vacuum amplitude of an open string stretching between a D3 brane dressed with a background gauge field and a system of N undressed D3 branes, in bosonic string theory (2.29-2.30), we have:

( )

∫

∫

∫

∞ − Φ ∞

=













−

−

Φ

∂

Φ

∂

+

−

→

=

0 2 2 2 10 2 10 2 3 2 10 2 10 0

~

2

1

4

2

1

F

Tr

g

H

R

e

G

x

d

F

dF

F

i i ν µ µ

κ

κ

=

∫

_− −

(

)

( )

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

(

)

−

(

+

)

× Θ Θ + − → ₋ + ∞ −

∫

∫

) ( ) ( ) ( sin sin 2 ˆ det ' 8 1 1 18 1 ) ( 0 ' 2 4 2 2 2 2 2

τ

τ

ν

τ

τ

ν

πν

πν

τ

τ

η

α

π

πτ ν ν πτ πα τ i i i i e f e e d F x d N g f g f y f g

∫

∫

∞ + − ×         Θ Θ + − − 0 1 1 ) ( ' 2 4 2 2 ( ) ( ) sin sin 2 ) ˆ det( ) ' 8 ( 2 2 2

τ

τν

τ

τν

πν

πν

τ

τ

η

α

π

ν ν πτ πα τ i i i i e e d F x d N g f g f y f g

[

]

δ

[

]

δ δ − − − ×22 _f1(_k) (18 ) _f2(_k) _{. (2.33)}

3. Mathematical connections between linear subcanonical spinor theory in third order formalism, Dirac-Born-Infeld action, Duality Het/T4 −IIA/K3 and Palumbo’s Model.[4]

Linear subcanonical spinor theory in third order formalism.

We concentrate our attention on the investigation of the simplest possible nonlinear spinor theory, namely a theory for a self-coupled 2-component Weyl spinor field

ψ

(x) which obeys the nonlinear field equation

⋅

∂

( )

+

'

:

(

∗

)

:

(

)

=

0

x

g

x

i

σ

ψ

σ

ψ

ψ

σ

µ

ψ

µ (3.1).

This is essentially the Heisenberg nonlinear spinor equation in the form as given by Durr. An invariance of this spinor equation under dilatations requires to assume the spinor field to have the subcanonical dimension dim

ψ

=1/2 (3.2)

The linear theory corresponding to this subcanonical spinor theory is the third order Weyl equation −i(

σ

⋅∂)∂2

ψ

(x)=0 (3.3)

or the set of first order equations

i

σ

⋅

∂

ψ

=

ψ

ˆ

i

σ

⋅

∂

ψ

ˆ

=

ψ

ˆˆ

i

σ

⋅

∂

ψ

ˆˆ =

0

(3.4)

This linear theory could be shown to be invariant under the full 15-parameter conformal group. The transition back to the nonlinear theory will be essentially performed by the requirement of phase-gauge invariance of the theory, which demands the replacement

∂_µ →∇_µ =∂_µ +igR_µ (3.5) in the Lagrangian, where R_µ is identified with the bilinear form

R

µ

(

x

)

:

ψ

σ

µ

ψ

:

(

x

)

∗

−

=

(3.6)

Now we shortly review the linear subcanonical spinor theory in the third order derivative formalism and explicitly consider its solutions. These solutions span a quantum mechanical state space with indefinite metric.

We consider the free massless third order derivative theory for a 2-component Weyl spinor field with the field equation

−i(

σ

⋅∂)∂2

ψ

(x)=0 (3.7) which can be formally derived from the Lagrangian density

=

[

ψ

∗(

σ

⋅∂)∂

ψ

−((

σ

⋅∂)∂

ψ

)∗

ψ

]

2 2 2 3 i L (3.8)

This theory is invariant under the full 15-parameter conformal group if we require the Weyl spinor field to transform according to an irreducible representation with mass dimension

2 1 dim

ψ

= (3.9)

Quantization of the spinor field is achieved by the requirement that the anticommutator of pseudo-hermitian conjugate fields is connected with an invariant solution of (3.7) which vanishes for space-like distances, and a normalization which is fixed by the normalization of the Lagrangian density (3.8). One obtains

∫

− ⋅ ∗ ⋅ = ⋅ − =             −       ipx e p p p d i x x x x x 2 2 4 4 2 0 ) ( ) 2 ( ) ( ) ( 2 1 ) ( 2 1 2 , 2

σ

π

δ

ε

π

σ

ψ

ψ

=−

∫

d p ⋅p (p ) '(p )e−ip⋅x ) 2 ( 1 _{4 0 2} 3

σ

ε

δ

π

(3.10) where ( ) ( ) 2 1 _{0 2} x x

δ

ε

π

is the invariant function of a massless field. The integrand in the momentum integral

(3.10) has the form

) ( ) ( 1 ) )( )( ( 1 ) (p2 2 p p p p0 p 2 p0 p p r r r r ⋅ + ⋅ − = ⋅ ⋅ ⋅ = ⋅

σ

σ

σ

σ

σ

σ

(3.11)

which indicates that there exists a double pole for positive chirality states (positive-energy positive-helicity or negative-energy negative-helicity states)

p

0

=

σ

r

⋅

p

r

=

p

r

h

(3.12)

(

h

=

σ

r

⋅

p

r

/

p

r

=

helicity

), and a single pole for negative chirality states (positive-energy negative-helicity or negative-energy positive-helicity states)

p

0

=

−

σ

r

⋅

p

r

=

−

p

r

h

(3.13)

both with zero mass. The field operator ψ(x) will contain annihilation operators for a massless right-handed good and bad ghost, ag and ab, and an annihilation operator an for an ordinary massless left-handed state

similar to the neutrino, and also the creation operators b_g,b_b,b_n for the corresponding “antiparticles”. It is convenient to use the pseudo-hermitian operators

∗ =

η

x

η

−1 b

b (3.14)

constructed with the metric tensor η in the quantum mechanical state space, because in a theory with indefinite metric the pseudo-hermitian conjugation takes over the role of the hermitian conjugation in a theory with positive definite metric. In the 1-particle sector of the quantum mechanical state space the metric tensor η has the form





















=

=

−

1

0

0

0

0

1

0

1

0

1

η

η

(3.15)

b

_b∗

=

b

_gx

;

b

_g∗

=

b

_bx

; b_n∗ =b_nx

(3.16) For the creation and annihilation operators we have the anticommutation rules

( ) ( )

{

}

ij j b i g

p

a

p

p

p

a

(

r

),

∗

(

r

'

)

=

δ

(

r

−

r

'

)

δ

;

{

a

_b( )i

(

p

r

),

a

( )_gj∗

(

p

r

'

)

}

=

δ

(

p

r

−

p

r

'

)

δ

_ij ; ( ) ( )

{

}

ij j n i n

p

a

p

p

p

a

(

r

),

∗

(

r

'

)

=

δ

(

r

−

r

'

)

δ

(3.17)

and similar anticommutation rules for the b-operators. All other anticommutators are zero. The superscript (i) refers to the spin degree of freedom. The Weyl spinor field ψ(x) can be expanded in terms of these operators

∑∫

( )

( )

( )

( )

]

(

( )

)

= + −           +       + − = 2 , 1 1 3 2 / 3 ₂ 2 1 ) 2 ( ) 2 ( 1 ) ( i i i g i b p i pt a p h p a p p d x _α α

π

ψ

r r r r r + ( )

( ) ( )

(

−

)

]

−( − ⋅ )− x p t p i i i n p h p e a r r r r r α

b

_b( )i

( )

p

i

p

t

b

_g( )i

( )

p

(

h

( )

p

)

_i

b

_n( )i

( ) ( )

p

(

h

p

)

_i

e

i(pt px) r r r

r

r

r

r

r

r

_{+ − ⋅} − ∗ + ∗ ∗













−

























−

−

−

2

_{α α}

2

1

(3.18)

with the helicity projection operators

( )

(

p

p

)

p

p

h

r

r

v

r

r

⋅

±

=

±

σ

2

1

(3.19)

The expansion for

ψ

_α∗

( )

x is given by the pseudo-hermitian expression of (3.18). With (3.18) we deduce for the anticommutator (3.10) on the basis of the anticommutator rules (3.17)

( )

( )

(

)

]

( )

[

{

∫

− + + =             −       − − ⋅ + − ∗ i pt px e h t p i h p p d x x _r r rr r 1 2 2 1 2 1 2 , 2 2 3 3

π

ψ

ψ

[

h

(

i pt

)

h

]

e i(pt px) r r r r _{+ − ⋅} + − − − + 1 2

( )

∫





(

)

(

)

(

)

(

)



+

−

+

+

+

−

=

₊ − _{− 0} −1 0 0 1 0 0 0 4 3

2

1

p

p

dp

d

p

p

h

p

p

dp

d

p

p

h

p

d

δ

r

r

δ

r

r

π

+

( )

2pr −2

[

h₋

δ

(

p₀ − rp

)

+h₊

δ

(

p₀ + pr

)

]

}

e−ip⋅x

( )

∫

(

) (

) (

) (

)

⋅ − − +

















−

+

+

+

−

=

ipx

e

p

p

p

p

h

p

p

p

p

h

p

d

i

r

r

r

r

0 2 0 0 2 0 4 4

2

π

(3.20)

i.e. the correct expression (3.10).

The situation in the state space is less pathological if we generalize the third order spinor theory (3.7) to include a mass, i.e.

−i

(

σ

⋅∂

)

(

∂2 +m2

)

ψ

( )

x =0 (3.21)

In this case, of course, the symmetry under dilatation and special conformal transformation will be broken. The anticommutator then has the form

[

(

)

( )

]

( )

∫

(

−

)

= ⋅ = ∆ − ∆ ∂ ⋅ =             −       ∗ −ip⋅x e m p p p p d i x m x m x x 2 2 2 4 4 2 2 2 0 ; ; 1 2 , 2

σ

π

σ

ψ

ψ

( )

∫

( ) (

[

) ( )

]

⋅ − − − ⋅ = ipx e p m p p p p d m 2 2 2 0 4 2 3 2 1

δ

δ

ε

σ

π

. (3.22)

From this we deduce that

ψ

( )

x

now annihilates positive norm states of mass m, containing positive and negative chirality components, and negative norm zero states with zero mass and positive chirality. The Weyl spinor field has the expansion

( )

( )

∑∫

=

[

(

)

] [

]

× + ⋅ + + = 2 , 1 2 / 1 3 2 / 3 4 2 1 i i p p p E m E p m E p d m x _α α

σ

π

ψ

r r

×

[

a

_m( )i

( )

p

r

exp

[

−

i

(

E

_p

t

−

p

r

⋅

x

r

)

]

−

b

_m( )i∗

( )

p

r

exp

[

i

(

E

_p

t

−

p

r

⋅

x

r

)

]

]

+

+

( )

h

_i

[

a

( )i

( )

r

p

[

−

i

(

p

r

t

−

p

r

⋅

x

r

)

]

−

b

( )i∗

( )

p

r

[

i

(

p

r

t

−

p

r

⋅

x

r

)

]

]

− − + α

exp

exp

(3.23) with

E

_p

=

(

p

r

2

+

m

2

)

1/2 (3.24)

The annihilation and creation operators obey the anticommutation rules

{

}

ij j m i m

p

a

p

p

p

a

()

(

r

),

( )∗

(

r

'

)

=

δ

(

r

−

r

'

)

δ

{

a

(i)

(

p

r

),

a

(j)∗

(

r

p

'

)

}

=

−

δ

(

p

r

−

p

r

'

)

δ

_ij − − (3.25)

and similar anticommutators for the b(i) . All other anticommutators vanish. The negative sign in the second anticommutator of (3.25) indicates that a∗− creates a negative norm state. It is possible verify easily that the

expansion (3.23) leads back to (3.22):

( )

∫

_

[

(

(

)

)

]

[

(

)

]

   − ⋅ − + ⋅ − − ⋅ + =             −       ∗ x p t E i x p t E i E p E p d m x x p p p p r r r r r r exp exp 2 2 1 2 , 2 3 2 3

σ

π

ψ

ψ

−

h

+

[

(

−

i

(

p

t

−

p

⋅

x

)

)

]

+

[

i

(

p

t

−

p

⋅

x

)

]

}

=

r

r

r

r

r

r

exp

exp

( )

∫

[

(

) ( )

]

⋅ − − − ⋅ = ipx e p m p p p p d m 2 2 2 0 4 2 3 ( ) 2 1

δ

δ

ε

σ

π

. (3.26)

3.1 Born-Infeld action and D-brane actions.[5]

Born and Infeld realized the final version of their non-linear electrodynamics through a manifestly covariant action. In modern language this can be expressed by saying that the world-volume theory of the brane is described by the action

_{( )}

( )

∫

−

(

+

)

−

=

+ µν µν

σ

π

g

d

G

F

S

p s p p

det

2

1

₁ (3.27)

where F is the world-volume electromagnetic field strength, measured in units in which 2

πα

'=1 . G is the induced metric on the brane

G

_µν

=

η

_mn

∂

_µ

X

m

∂

_ν

X

n (3.28) Thence, we have from (3.27):

_{( )}

( )

∫

−

(

∂

∂

+

)

−

=

+ µν ν µ

η

σ

π

g

d

X

X

F

S

mn m n p s p p

det

2

1

₁ (3.29).

The action is invariant under arbitrary diffeomorphysms of the world-volume. One way of fixing this freedom is to adopt the so-called “static gauge” for which the world-volume coordinates are equated with the first p+1 space-time coordinates:

Xµ ≡

σ

µ,

µ

=0,1,...,p

. (3.30)

This “static gauge” description is most convenient if the brane is indeed positioned along those directions. The rest of the coordinates become world-volume fields

Xm ≡

φ

m,m= p+1,...,9. (3.31) The Born-Infeld action becomes

_{( )}

( )

∫

−

(

+

∂

∂

+

)

−

=

+

σ

η

_{µν µ}

φ

_ν

φ

_µν

π

g

d

F

S

p i i s p p

det

2

1

'

1 . (3.32)

Note that this is in some sense a modification of pure Born-Infeld: it has extra scalar fields

φ

and that the action (3.27) can be also write as:

=

−

∫

d

x

−

Det

(

G

_µν

+

F

_µν

)

g

S

p 4

1

with gp

( )

gs 3 2

π

= , hence:

( )

∫

−

(

+

)

−

=

µν µν

π

g

d

x

Det

G

F

S

s 4 3

2

1

. (3.33)

The action for a Dp-brane comes in two parts, the Dirac-Born-Infeld part, and the Wess-Zumino part. These are

SDBI =−

µ

p

∫

dp+

ζ

e−φ −det

(

g_αβ + f_αβ

)

1

, (3.34)

where f =2

πα

'F−B is a U(1) field strength (the world volume gauge field therefore transforms as

'

2

/

πα

λ

δ

A

=

B under a SUGRA gauge transformation

δ

B

2

=

d

λ

B), and

=

∫

∧⊕q q f

p

WZ e C

S

µ

, (3.35)

where the integral projects onto p+1 forms. The D-brane charge is

=

1

/(

2

)

p

'

(p+1)/2

p

π

α

µ

. The coordinates

α

ζ

are the embedding coordinates of the D-brane. Note that the spacetime fields are pulled back to the world volume. Hence, we have

=− ₊

∫

+ − − + + ₊

∫

∧⊕q q f p p p p p d e g f e C S _{( 1)/2} 1 _{( 1)/2} ' ) 2 ( 1 ) det( ' ) 2 ( 1

α

π

ζ

α

π

αβ αβ φ . (3.36)

With regard to string corrections, the most important corrections are those to the D7-brane action because they give an induced D3-brane charge and tension. There are also corrections to the DBI action that are responsible for modifying the tension of wrapped D7-branes. Considering the bosonic part only, the DBI action becomes

∫

_

(





−

+

−

−

=

₊ + − αβγδ αβγδ φ

πα

ζ

α

π

192

(

)

(

)

)

'

2

(

1

)

det(

'

)

2

(

1

2 1 2 / ) 1 ( T T p p p DBI

d

e

g

f

R

R

S

ab

)

]

b a b a N b a N T T R R R R R R _ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ( ) 2 ) ( ) ( ) ( 2 − + − _αβ αβ _αβ αβ (3.37)

up to Ο(

α

')2. There is an additional contribution at this order with an undetermined coefficient, but it vanishes on-shell, so it does not affect S-matrix elements or dispersion relations. Here, a ˆˆ,b are normal bundle indices in an orthonormal basis with vielbein

ξ

aˆ.

3.2 Duality type I-SO(32).[6]

In these theories, the action is fixed from the supersymmetry. The heterotic action contain the fields

φ

µν µν, B ,

G and

A

_µa; the type I G_µν and

φ

from the closed sector (NS)2, B_µν from the closed sector

2 )

(R and

A

_µa from the open sector. In the Einstein frame for the two actions, we have

∫

      − − ∂ ∂ − − = − − _{µνρ σεζ} φ εζ ρσ µν ρσ µν φ ρσ µν ν µ µν

_φ

_φ

π

d x g R g g g e trF F g g g e H H SH 10 4 2 7 12 1 4 1 8 1 ) 2 ( 1 , (3.38)

∫

      − − ∂ ∂ − − = _{µνρ σεζ} φ εζ ρσ µν ρσ µν φ ρσ µν ν µ µν

φ

φ

π

d x g R g g g e trF F g g g e H H SI 10 4 2 7 12 1 4 1 8 1 ) 2 ( 1 , (3.39) where F_µν =∂_µA_ν −∂_νA_µ + 2

[

A_µ,A_ν

]

,

[

,

]

. 3 2 2 1 cicl A A A F A Tr B H _+       − − ∂ = _{µ νρ µ νρ µ ν ρ} µνρ

These two actions are obtained each other identifying among them the fields corresponding of the two different theories and putting

φ

H =−

φ

I; the change of sign in dilaton connected the perturbative aspect of Type I with that non-perturbative of heterotic and vice versa.

3.3 Duality Het/T4 −IIA/K3[6].

With regard to duality Het/T4 −IIA/K3, the heterotic relation contain, metric, antisymmetric tensor, dilaton, 10+6+64=80 scalars and 8+16=24 vectors; with M∈O(4,20), t

M M = we can write

∫

_

(

)





∂

∂

+

∂

∂

−

−

=

d

x

g

R

g

g

Tr

ML

ML

S

_{µ ν} µν ν µ µν

_φ

_φ

π

2

8

1

)

2

(

1

₆ 3 µν ρσ _{µρ νσ} φ µν ρσ εζ _{µρε νσζ}

]

φ H H g g g e F LML F g g e ab b a − − − − 12 1 ) ( 4 1 ₂ , (3.40) where Hµνρ =∂µBνρ + AµaL_abFνρb +cicl 2 1 .