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On some equations concerning quantum electrodynamics coupled to quantum gravity, the
gravitational contributions to the gauge couplings and quantum effects in the theory of
gravitation: mathematical connections with some sector of String Theory and Number Theory
Michele Nardelli 2,1
1 Dipartimento di Scienze della Terra
Universit degli Studi di Napoli Federico II, Largo S. Marcellino, 10
80138 Napoli, Italy
2 Dipartimento di Matematica ed Applicazioni R. Caccioppoli
Universit degli Studi di Napoli Federico II Polo delle Scienze e delle Tecnologie
Monte S. Angelo, Via Cintia (Fuorigrotta), 80126 Napoli, Italy
Abstract
This paper is principally a review, a thesis, of principal results obtained from various authoritative
theoretical physicists and mathematicians in some sectors of theoretical physics and mathematics.
In this paper in the Section 1, we have described some equations concerning the quantum
electrodynamics coupled to quantum gravity. In the Section 2, we have described some equationsconcerning the gravitational contributions to the running of gauge couplings. In the Section 3, we
have described some equations concerning some quantum effects in the theory of gravitation. In the
Section 4, we have described some equations concerning the supersymmetric Yang-Mills theory
applied in string theory and some lemmas and equations concerning various gauge fields in any
non-trivial quantum field theory for the pure Yang-Mills Lagrangian. Furthermore, in conclusion, in
the Section 5, we have described various possible mathematical connections between the argument
above mentioned and some sectors of Number Theory and String Theory, principally with some
equations concerning the Ramanujans modular equations that are related to the physical vibrations
of the bosonic strings and of the superstrings, some Ramanujans identities concerning and the
zeta strings.
1. On some equations concerning the quantum electrodynamics coupled to quantumgravity. [1]
The key equations that govern the behaviour of the coupling constants in quantum field theory are
the renormalisation group Callan-Symanzik equations. If we let g denote a generic coupling
constant, then the value of g at energy scale E, the running coupling constant ( )Eg , is determinedby
( )( )gE
dE
EdgE ,= , (1)
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where ( )gE, is the renormalisation group -function. Asymptotic freedom is signaled by
( ) 0Eg as E , requiring 0
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As with the Green function (4) the divergent part of (7) comes from the 0 limit of the proper
time integral. The divergent part of L is
divp ( )
++=
2
21
2
0
44
2 ln2
1
32
1ccc EtrEtrEEtrEExdL . (9)
The lower limit on the proper time integration can be kept as 0= and the divergent part of the
effective action L contains a simple pole as 0 given by
divp ( ) = 24
216
1xtrEdL
. (10)
The general form of ij is
( ) ( ) ( )iji
j
i
j
i
j CBA ++=
(11)
for coefficients ( )ijA , ( )ijB
and ( )ijC that depend on the spacetime coordinates through the
background field. Normal coordinates are introduced at 'x with yxx += ' and all of the
coefficients in (11) are expanded about 0=y . This gives
( ) ( ) ( )
=
+=1
...0 ...1
1
n
i
j
i
j
i
jn
nyyAAA
(12)
with similar expansions for ( )ijB and ( )ijC . The Green function is Fourier expanded as usual,
( )( )
( )
= pGpedxxGi
j
yipn
n
i
j2
1', , (13)
except that the Fourier coefficient ( )pGij can also have a dependence on the origin of the coordinate
system 'x that is not indicated explicitly. If
( ) ( ) ( ) ( ) ...210 +++= pGpGpGpGi
j
i
j
i
j
i
j (14)
where ( )pG ijr
is of order rp 2 as p it is easy to see that to calculate the pole part of ( )xxG ij ,
as 4n only terms up to and including ( )pG ij2
are needed.
The gravity and gauge field contributions result in
tr ( )++++
++++=
222222
1 381232
1
32
1
8
3
2
1
8
3
8
1
4
3
8
3 vvFE . (15)
The overall result for the quadratically divergent part of the complete one-loop effective action (8)
that involves 2F is
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( )
++++=
2422
2
221
32
1
32
1
8
3
2
1
8
3
8
5
4
3
8
3
32Fxd
Ecquad
. (16)
( )
++++=
2422
2
221
32
1
32
1
8
3
2
1
8
3
8
5
4
3
8
3
216
1Fxd
Ecquad
. (16b)
If 0 , 0 , 1 are taken to obtain the gauge condition independent result, the non-zero
result
( )=
24
2
221
128Fxd
Ecquad
(17)
is found.
The net result for the divergent part of the effective action that involves 2F and therefore
contributes to charge renormalization is
divp ( )( )
+
=
242
2
22
2
2
2
221 ln
48ln
256
3
128FxdE
eE
Ecc
c
. (18)
divp ( )( )
+
=
242
2
22
2
2
2
221 ln
3ln
16
3
816
1FxdE
eE
Ecc
c
. (18b)
From this the renormalization group function in (1) that governs the running electric charge to be
calculated to be
( ) eEeeE
+=
23
3212, 2
2
2
2
3
. (19)
The first term on the right hand side of (19) is that present in the absence of gravity and results in
the electric charge increasing with the energy. The second term on the right hand side of (19)
represents the correction due to quantum gravity. For pure gravity with no cosmological constant, or
for small cosmological constant , the quantum gravity contribution to the renormalization group
-function is negative and therefore tends to result in asymptotic freedom.
2. On some equations concerning the gravitational contributions to the running of gaugecouplings. [2]
The action of Einstein-Yang-Mills theory is
=
aaggRgxdS
FF
4
112
4 (20)
where R is Ricci scalar and aF is the Yang-Mills fields strength = ,igF .
It is hard to quantize this lagrangian because of gravity-parts non-linearity and minus-dimension
coupling constant G 16= . Usually, one expands the metric tensor around a background metric
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++=
2
2
22
222
2
2
1 1ln16
1
c
sw
s
csc
My
MM
. (25)
Putting 1 and2 in the following equation
=
122
3g , (26)
we obtain the gravitational corrections to the gauge function
++
=
2
2
02
2
2
22 1ln16 c
sw
s
cs
My
Mg
(26b)
In general, the total function of gauge field theories including the gravitational effects may bewritten as follows
++
+=
2
2
02
2
2
223
021ln
1616
1
c
sw
s
cs
My
Mggb
. (27)
The interesting feature of gauge theory interactions is the possible gauge couplings unification at
ultra-high energy scale when the gravitational effects are absent. Where the running of gauge
coupling in the Model Standard Super Symmetric (MSSM) without gravitational contributions is
known to be
( ) ( )
M
Mee ln10
3311+=
; ( ) ( )
M
Mww ln2
111+=
; ( ) ( )
M
Mss ln2
311=
(28)
with experimental input at ZM
( ) 05.097.581 = Ze M ; ( ) 05.061.291
=
ZwM ; ( ) 22.047.81 = Zs M (29)
We note that these values are equal to the following values connected with the aurea ratio:
38,12461180 + 20,56230590 = 58,68691770; 29,12461180; 8,49844719.
Indeed, we have:
( ) ( ) ( ) 12461180,38370820393,12361803399,109016994,1137/77/35 =+=+ ;
( )[ ] 56230590,20385410197,637/28 = ;
( ) ( ) ( ) ( ) =++=++ 323606798,061803399,285410197,637/217/147/28
12461180,29370820394,9 = ;
( ) ( ) ( ) ( ) ( +++=+++ 055728089,0145898033,061803399,237/637/427/287/14
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) 49844719,83832815729,23013155617,0 =+ .
As usual, 61803399,12
15
+= , i.e. the value of the aurea ratio.
3. On various equations concerning some quantum effects in the theory of gravitation[3]
The general spherical static metric is given by
( ) ( ) 222122 ++= drdrrhdtrfds , (30)
where ( )rf and ( )rh are arbitrary functions of the coordinate r. The angular part of the metric isdiagonal and given by
( )
=
+=
=
2
1
2
1
222 sind
i
d
ij
jidd . (31)
Consider an interesting classical scalar field ( )x living in a spacetime with the metric (30). Thisfield has an action given by:
=
=2
2
n
n
n
dgxdS . (32)
The n are a set of arbitrary coupling constants. In particular,2
2 m gives the mass of a weakly
coupled excitation of this field. We will expand in the eigenmodes of the free, classical wave
equation such that
( ) ( )= xadx ppp , (33)
pp m 22
= . (34)
A sufficiently large set of quantum numbers p label the eigenbasis. The abstract formal expression
pd simply represents an appropriate measure over the modes under which
( ) ( ) ( ) = yxyxdd
ppp ,
and
( ) ( ) = pqqpd xxgxd .
Now we want to consider the energy density, , of a massless scalar field in a infinitely large
hypercubic blackbody cavity at temperature T. Consider a real scalar field ( )txI , , where I issome kind of p -dimensional polarization index representing p internal degrees of freedom (for
example, in a well-chosen gauge, the transverse polarization of an Abelian vector field behavesessentially like an internal index on a scalar field with 2= dp ). Further assume that the field is
sufficiently weakly coupled that each polarization component can be treated as an independent field
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obeying an action similar to eq. (32) with all interaction coefficients higher than 22 M set equal
to zero. Then each field component obeys the classical equation of motion
( ) ( )xMx II 22 = . (35)
The solutions to equation (35) may be expressed as a sum over modes labelled by a wave vector ak
obeying22
Mk = :a
a
I
k
a
a
I
k
I
kxkBxkA cossin += , (36)
for arbitrary real coefficients IkA andI
kB . We take the state to be labelled by the 1d spatial
components ofak and fix the frequency of each mode by
i
ik kkMk +=22
0
2 . We now confine
the field to live in a cubic box of side length L by demanding Dirichlet boundary conditions at
Lxi ,0= for 1...1 = di . This demands 0=I
kB and
ii mL
k
= , (37)
whereim is a spatial vector whose components are non-negative integers. The total energy in the
hypercube is given by
=
=
=
=p
I m m
mIk
d
nU1 0 01 1
... , (38)
where we understand that k is given by
=
+=1
1
2
2
22
d
i
ik mL
M
. (39)
The set of integers mIn defining the quantum state must obey the appropriate statistics for the field
I . We have described a pure quantum state of the theory. At a finite temperature T and zero
chemical potential, the system will be in a mixed state governed by the partition function
{ }
=mI
k
n
eQ
, (40)
where T/1 . This can be evaluated to give
( )
=
=
=0 01 1
1ln...lnm md
kepQ , (41)
where 1= for bosons and 1= for fermions. The overall factor ofp occurs because the energy
is independent of p , so each polarization mode contributes equally. The average occupation
number of a given momentum mode in the thermal state is then given by
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=
=
==
p
I k
mImke
pQnn
1
ln1
. (42)
The total energy in the hypercube can now be found by combining the expressions (38) and (42), or
by
=
=
=
=
==
=
0 0 0 01 1 1 1
......lnm m m m
kmk
d d
ke
pnQU
. (43)
We should pass from a state labelling in terms of quantum numbers im to a labelling in terms of
physical momenta ik , with a mode density determined by the differential limit of equation (37).
The sums over im then become integrals over ik as
( )
k
e
pLkdU k
d
dd
L
1
11
2
, (44)
where k is understood asi
ikkM +2 . The factors of 2 in the denominator of the measure arise
because the integrals over the ik run over both positive and negative values, whereas the im were
only summed over non-negative values. Equation (44) scales properly with the volume, so that even
in the infinite volume limit we can define the spectral energy density over ik modes. Using the
spherical symmetry of the infinite volume limit and defining ii
i kkkk == , eq. (44) becomes
( )( )
+
+
== 0
222
1
2
22
2 kM
d
d
d
e
kMkdk
SVolpV
U
. (45)
This defines the spectral energy density over the magnitude of the spatial momentum, via
( ) kdkuk , as
( )( )
( )
+=
+
22
222
1
2
2 kM
d
d
d
k
e
kMkSVolpku . (46)
Similarly, we can define the spectral energy density over the frequency as
( )( )
( )
( )( )
=
e
MSVolpu
d
d
d 2/3222
1
2
2, (47)
where runs over [ ],M .The total energy density can now be evaluated using either eq. (46) or eq. (47) match. Simple
analytic results can be found for the case 0=M , which will also apply when MT >> . In this case,
k= and equations (46) and (47) match. They give
( )( )
= 0
1
1
2
2 x
d
d
dd
e
x
dx
SVol
pT . (48)
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The integral can be evaluated by pulling the exponential into the numerator, performing a Taylor
series in xe , doing the integral, and resumming the Taylor series. The result for the general definite
integral is given by
( )
( )( )
=
=
b
a
d
n
bnan
dx
d
n
ebeadde
dxx 1
0
2/11
!!12
21
, (49)
where ( )s is the Riemann zeta function, which can be defined for real 1>s as
( )
=
=1
1
nsn
s . (50)
This series arises in the evaluation of the integral with the 1= , for bosons. For fermions, 1= ,
the corresponding series is
( ) ( )
=
1
1 11n
s
n
nsR . (51)
This series can be evaluated using
( ) ( ) ( )sssssss
2
2...
6
2
4
2
2
2=+++= R , (52)
so that
( ) ( )sss
=
2
21R , (53)
which is the origin of this factor in eq. (49). So, eq. (48) becomes
( )( )( )
( )
d
ddd
Td
ddp
2/12
2/1
2
12
!1
2
21
=
. (54)
Furthermore, we can rewrite the eq. (48) also as follows:
( )( )
=
=
0
1
1
2
2
x
d
d
dd
e
xdx
SVolpT
( ) ( )( )
( )
d
ddd
Td
ddp
2/12
2/1
2
12
!1
2
21
. (54b)
Before, we have considered the energy density, , of a massless scalar field in a infinitely large
hypercubic cavity at temperature T. We now want to calculate the energy flux, , emitted from a
blackbody with this same temperature. For the flux calculation, instead of 2dSVol we will
encounter
( ) =
222
2cos dddd
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( ) ( )( )( )
( )( ) ==
=
=
=
2
0 0 0
2/
0
32/23
2
122321
2
1
2
22
2sincos... d
dd
i
iidddSVol
ddd
dddd
( ) ( ) ( )22
2
22
1
21
== dd SVol
d
d
dBVol
, (55)
where ( )x is the step function and nB is the n-dimensional unit ball: the compact subspace of nR
bounded by1nS (for example, 3B is the unit 3-ball of volume 3/4 bounded by the sphere 2S of
area 4 ). The fact that the expression for the energy density becomes that for the flux when
( )2dSVol is replaced by ( )2dBVol makes physical sense, since nB is the projection of nS onto nR .We are left with the relationship of flux to energy density as
( )
( )( )
2
2
2
22
2
1
=
=d
d
SVol
BVol
dd
d
. (56)
Thus, the d-dimensional Stefan-Boltzmann law is given by
( )( )( )
( )
d
ddd
Td
d
ddp
2/2
2/1
2
222
!1
2
21
=
. (57)
4. On some equations concerning the supersymmetric Yang-Mills theory applied in string
theory and some lemmas and equations concerning various gauge fields in any non-trivial
quantum field theory for the pure Yang-Mills Lagrangian. [4] [5]
The fields of the minimal 2=N supersymmetric Yang-Mills theory are the following: a gauge field
mA , fermionsi
and i& transforming as ( )2/1,0,2/1 and ( )2/1,2/1,0 under
( ) ( ) ( )IRL SUSUSU 222 , and a complex scalar B - all in the adjoint representation of the gaugegroup. Covariant derivatives are defined by
( )+= mmm iAD , (58)
and the Yang-Mills field strength is
[ ]nmmnnmmn AAAAF ,+= . (59)
The supersymmetry generators transform as ( ) ( )2/1,2/1,02/1,0,2/1 ; introducing infinitesimal
parameters i and i& , furthermore [ ]BBD ,= . The minimal Lagrangian is
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[ ] [ ] [ ]
+
=
M
ii
ijji
ij
m
m
i
m
m
i
mn
mn Bi
Bi
BBBDBDDiFFxTrde
L
&
&&
&,
2,
2,
2
1
4
11 242
(60)
Here Tr is an invariant quadratic form on the Lie algebra which for ( )NSUG = we can
conveniently take to be the trace in the N dimensional representation.It is possible to realize a mass term for the 1=N matter multiplet (which consists of B and2&
= ) by adding to the Lagrangian a term of the form ( ) { },...1QI + , where 1Q is the charge
corresponding to the 1 transformation. Furthermore, 1Q is the only essential symmetry. We
obtain:
( ) { } ( ) +=++=M M
BBmTrmxdemmxTrdLQILL 22
1,... 42
22
224
1 &&
&
&
&&
, (61)
with
mnklklmnm 22&&= . (62)
The mass is proportional to the holomorphic two-form . The 1=N gauge multiplet, consisting of
the gauge field mA and the gluinoi
= , remains massless.
With regard the 16 -amplitude in the type IIB description, the classical action for the operator ( )16
in the5
5 SAdS supergravity action is
[ ] ( ) ( )( ) ( ) =
+=
16
1
0
5
0
0
004165
0
00
412
12 1
;,1
5
0
p
pppSs
ig
xJxxxxKtdxdVgeJSs
. (63)
This result is in agrees with the following expression obtained in the Yang-Mills calculation:
( )( )[ ]
( )( ) =
+
+
+=
16
1
00
0
42
0
2
0
4
088
5
0
00
4
8
8
16
1122
p
A
p
A
p
ig
YMp
pp
pp
p
p
YM
YM xxxx
dddxdegxG
&
&
(64)
i.e. the correlation function in the super Yang-Mills description.
The low energy effective action for type IIB superstring theory in ten dimensions includes the
interaction (in the string frame)
( ) ( ) ( ) ( ) ( )
++++
464222104 ...69
2462323 RDeeegxdlS s
, (65)
where the involve contributions form D-instantons.
We consider the perturbative contributions to the46
RD interaction. The sum of the contributions to
the four graviton amplitude at tree level and at one loop in type II string theory compactified on 2T
is proportional to ( ) ( ) ( )( ) ( ) ( )
4
222
2222
2 24/14/14/1
4/4/4/R
+
+++
I
ultlsl
ultlsleV
sss
sss , (66)
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where 2V is the volume of2
T in the string frame, uts ,, are the Mandelstam variables, and I is
obtained from the one loop amplitude. The amplitude is the same for type IIA and type IIB string
theories. Now I is given by
( )
= F ,22
2
FZd
I lat (67)
where F is the fundamental domain of ( )ZSL ,2 , and 2/2 = ddd . In the above expression, the
lattice factor latZ which depends on the moduli is given by
( ) ( )( )
=
ZMatA
lat AUU
TAiTVZ
,22
2
22
22
11det2exp
. (68)
Expanding eq. (67) to sixth order in the momenta, we get that
( )[ ]213336
3
IIutsl
I s +++= , (69)
where
( ) ( ) ( ) =
=
Li
ilat
dZ
dI
F T
3
1
323121
2
2
2
2
2
1 ;;ln;ln4
, (70)
and
=2I ( )[ ]
=
Li
ilat
dZ
d
F T
3
1
3
21
2
2
2
2
2
;ln
. (71)
We have also that
( ) ( ) ( ) ++=
=L
IIIEZd
IZSL
latF
3
3
2
1
1
1
,2
32
2
2
1
3
,4
4, (72)
where 211
1, II , and 31
I are the contributions from the zero orbit, the non-degenerate orbits and the
degenerate orbits of ( )ZSL ,2 respectively. In the eq. (72), we have used the expression
( ) ( ) ( ) ( ) ( ) 12121
2
2122/5
2/5
0,0 2
12
3
2
2
3
2
,2
3 254
362,
+
+= mimmm
ZSLemmK
m
mE
. (73)
Thence, we can rewrite the eq. (72) also as follows:
( )
=
LlatZ
dI
F2
2
2
1
3
4
4 ( ) ( ) ( ) 121
21
2
2122/5
2/5
0,0 2
12
3
2
2
3
2 254
362
+
+ mimmm
emmKm
m
. (74)
The contribution from the zero orbit gives
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( ) ( ) =
=L
ZSLE
dVI
F
0,,2
32
2
2
2
1
1 . (75)
The contribution from the non-degenerate orbits gives
( ) ( )
>
++
=
=
00,0
2,2
32
2
212
2
1
2
22
2
,2
pjk
pUjkU
TiTkp
ZSLeE
ddVI
( ) ( ) ( )
=0,0
2
22/5
2/5,2
3212,2
kp
ipkTZSLepkTK
k
pUUET
. (76)
Furthermore, the contribution from the degenerate orbits gives
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
+
+=
=
2/1
2/1 0 0,0,
,2
32
2
3
23
,2
32
2
212
3
1 ,
4
5362
2,
2
22
2
pj
ZSLpUj
U
T
ZSLUUE
T
TeEd
dVI
.
(77)
Now we want to show two lemmas and equations concerning the gauge fields as described in the
Jormakkas paper Solutions to Yang-Mills equations [5]. Thence, in the next Section, we
describe some possible mathematical connections between some equations concerning this
interesting argument and some equations concerning the Ramanujans modular equations that are
related to the physical vibrations of the bosonic strings and of the superstrings, and some
Ramanujans identities concerning .
Lemma 1
Let the gauge field satisfy
( ) = =
3
1
22
321
, ,,,0 jj
ecsxxxA aRa
(78)
where j and c are as in the following expressions:
jjj ir +=
, Rjj
, ; 3211 2xxx+=
; 3212 2xxx=
; 313 2
1
xx+=
;
021 == ; 032
1x= ; ) ) )jjjjj ivurh ,, += (79)
20 =c ; 121 == cc ; 03 =c ; 20 =e ; 121 == ee ; 03 =e , (80)
and Rsa , . Then
( )( )
=
3
2/3
222
321
3 1
2,,,0
ka
a
k csxxxAxd . (81)
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The expression (79) describes a localized gauge field, gauge boson, which moves in the 21,xx
direction with the speed of light as a function of 0x . We select a concrete case that gives easy
calculations. Let ( ) 22 jjf = and extend it to ( )22
jj rrh = . The real and imaginary parts of
iecd += are described from (80). We evaluate the gauge potential at 00 =x and take the real
part.
We change the variables to 321 ,, yyy
32116
1
3
23 xxxy = ; 322
3
1
3
2xxy = ; 33 2xy = (82)
Then
=
++=3
1
2
3
2
2
2
1
2
j
j yyy (83)
As 2y and 3y are not functions of 1x we can change the order of integration
( ) ( ) ( ) ( )
===
= ++ ==
21
223
22
221
223
22
23
1
223
1
22 22
1
1
22
2
1
1
233
3
1 yyyyyyyedyxededxxedxedxed j
jj j
( ) ( )
+
=1
2 223
123
22
2
yyxed (84)
As 3y
is not a function of 2x we can change the order of integration
( ) ( )
+21
223
22
2 22
1
1
2
3
1 yyyedyxed
( ) ( ) ==
+
12 22
3
123
22
2
yyxed
( )( )
( )( )
===
22
223
222
223
2 22
1
23
122
1
23
1
222
3
3
122
3
1 yyyyedyedxedxedx
( )( ) ( )( ) ( ) ( ) 323
3
2
3
2
222
1
2
3
3
122
2
1
2
3
3
122
2
3
3
1 23
223
2
=== yy
edyedx . (85)
Thus
3
2
3
23 1
2
22
=
jxed
=
3
323 1
2
22
jxed
( )( )
=
3
2/3
222
321
3 1
2,,,0
ka
a
k csxxxAxd . (86)
Thence, we can conclude that the eq. (81) is true.
We can rewrite the eq. (85) also as follows:
( ) ( ) + 21
223
22
2 22
1
1
2
3
1 yyy edyxed ( ) ( ) ==
+ 12 223
123222 yyxed ( ) ( ) =3
23
2221
23
3
1
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( ) ( )( )6
321
2
2
1
4822
3
123
22
2
=
=
+ yy
xed . (86b)
Lemma 2
Let the gauge field satisfy
( ) = =
3
1
22
321
, ,,,0 jj
ecsxxxA aRa
(87)
where j and c are as in (79), (80) and Rsa , . Then
= a aBsd 2
3
3
161
RL (88)
where in Minkowskis metric at 00 = x , we have 0=B . In the negative definite metric
( )
=
1000
0100
0010
0001
,g , (88b)
we have that 43
2
3
13++=B .
From (79) and (82) follows that
32112
1
6
1
3
1yyy += ; 3212
2
1
6
1
3
1yyy = ; 213
3
2
3
1yy = . (89)
For Minkowskis metric
( ) 3263152142332222110 yyByyByyByByByBP +++++== (90)
where 0=kB for all k. For the metric (88b)
( ) 3263152142
33
2
22
2
1132
2
3
2
2
2
1 444 yyByyByyByByByBP +++++=++= (91)
where
3
131 =B ;
3
22 =B ; 43 =B ; 2
3
44 =B ;
6
45 =B ;
3
46 =B . (92)
We do the integration with generic parameters jB . Then
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( ) ( )( )
( ) ( )
++++
=23
22
21
223
22
21
22
2
1
32
2
1
3yyy
xedPxed
)( 3263152142
33
2
22
2
11 yyByyByyByByByB +++++ .
(93)
As 2y and 3y are not functions of 1x we can change the order of integration and change the
integration parameter 1x to 1y
( ) ( )( )
( ) ( )
++++
=23
22
21
223
22
21
22
2
1
32
2
1
3yyy
xedPxed
)( 3263152142
33
2
22
2
11 yyByyByyByByByB +++++ =
( ) ( ) ( )( )
( )
++++=
+ 2122
122
322
22
2
1
1326
2
33
2
22
22
1
2
111
22
1
2yyyy
edxyyByByBeydxBxed
( )( )
=
++
212
22
1
113524
y
eydxyByB
( ) ( ) ( ) ( ) ( ) ++++=
+ 21221223222 221
1326
2
33
2
22
2212
111
2212
3
1 yyyy edyyyByByBeydyBxed
( )( )
=
++
212
22
1
113524
y
eydyyByB
( ) ( )
( )( )
+++=
+
2
12
2
12
3
1326
2
33
2
2231
22
1
223
22
2
yyByByBBxedyy
. (94)
As 3y is not a function of 2x we can change the order of integration and change the integration
parameter 2x to 2y :
( ) ( )( )
( ) ( )
++++
=23
22
21
223
22
21
22
2
1
32
2
1
3yyy
xedPxed
)( 3263152142
33
2
22
2
11 yyByyByyByByByB +++++ =
( ) ( )
( )( ) =
+++=
+
2
12
2
12
3
1326
2
33
2
2231
22
1
223
22
2
yyByByBBxedyy
( )
( )
( ) ( )
++=
2222
222
32
22
1
2
222
22
1
231
22
1
32
1
2
12
2
3
3
1 yyyeydyBedyBedx
( ) ( )
=
++
2222
22
2
2
1
2236
2
2
1
22332
1
2
1 yyeydyyBedyyB
( )
( ) ( )=
++=
2
12
2
1
2
12
2
1
2
12
2
12
2
3
3
1 2333231
22
1
3
23
2
yBBBedxy
( )
( ) ( ) ( )=
++=
2
2
334241
22
1
32
1
2
1
2
12
2
1
2
3
3
1 232
yBBBedyy
( ) ( )( ) ( )
( )32152
3
53212
3
2
1
2
12
2
1
2
3
3
1BBBBBB ++=++=
. (95)
We can rewrite this equation also as follows:
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( ) ( )( )
( )253
2
326315214
2
33
2
22
2
11
22
1
3
16
81
4
23
22
21
2
=
+++++
++
yyByyByyByByByBxedyyy
. (95b)
Thence, we can conclude that the eq. (88) is true. Indeed, we have that:
( ) ( )( )
( ) ( )
++++
=23
22
21
223
22
21
22
2
1
32
2
1
3yyy
xedPxed
=+++++ )( 3263152142
33
2
22
2
11 yyByyByyByByByB
( )( )3215
2
3
2
1BBB ++=
==
a
aBsd2
33
16
1
R
L . (96)
5. Ramanujans equations, zeta strings and mathematical connections
Now we describe some mathematical connections with some sectors of String Theory and Number
Theory, principally with some equations concerning the Ramanujans modular equations that are
related to the physical vibrations of the bosonic strings and of the superstrings, the Ramanujans
identities concerning and the zeta strings.
3.1Ramanujans equations [6] [7]With regard the Ramanujans modular functions, we note that the number 8, and thence the
numbers2864 = and 8232
2= , are connected with the modes that correspond to the physical
vibrations of a superstring by the following Ramanujan function:
( )
++
+
=
4
2710
4
21110log
'
142
'
cosh
'cos
log4
3
18
2
'
'4
0
'
2
2
wtitwe
dxex
txw
anti
w
wt
wx
. (97)
Furthermore, with regard the number 24 (12 = 24 / 2 and 32 = 24 + 8) this is related to the physical
vibrations of the bosonic strings by the following Ramanujan function:
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19
( )
++
+
=
42710
421110log
'
142
'
cosh
'cos
log4
24
2
'
'4
0
'
2
2
wtitwe
dxex
txw
anti
w
wt
wx
. (98)
It is well-known that the series of Fibonaccis numbers exhibits a fractal character, where the forms
repeat their similarity starting from the reduction factor /1 = 0,618033 = 2
15
(Peitgen et al.
1986). Such a factor appears also in the famous fractal Ramanujan identity (Hardy 1927):
++
+=
==
q
t
dt
tf
tf
qR
0 5/45/1
5
)(
)(
5
1exp2
531
5)(
2
15/1618033,0
, (99)
and
++
+=
q
t
dt
tf
tfqR
0 5/45/1
5
)(
)(
5
1exp
2
531
5)(
20
32
, (100)
where 215 +=
.
Furthermore, we remember that arises also from the following identities (Ramanujans paper:Modular equations and approximations to Quarterly Journal of Mathematics, 45 (1914), 350-
372.):
( )( )
++=
2
13352log
130
12
, (100a) and
++
+=
4
2710
4
21110log
142
24
.
(100b)
From (100b), we have that
++
+=
4
2710
4
21110log
14224
. (100c)
Let ( )qu denote the Rogers-Ramanujan continued fraction, defined by the following equation
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20
( ) ,...1111
::325/1
++++==
qqqqquu 1
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21
Using (109) in (107), we find that
( )( )
( ) ( ) ( ) =+= dqvuqdqd
dqq
dqqqqq
dq
q
q/log8
5
840
5
8 5/1535
5
( ) ( ) ( ) ( ) ( ) +
+== k
kkdqqqqvudqqqq 1
1log5
24log5
840/log840
5353
, (111)
where (104) has been employed. We note that we can rewrite the eq. (111) also as follows:
( )( )
( ) ( ) +
+=k
kkdqqqq
q
dq
q
q
1
1log
5
24log
5
840
5
8 535
5
. (112)
Multiplying both sides for64
5, we obtain the following identical expression:
( )( )
( ) ( ) +
+=k
kkdqqqq
q
dq
q
q
1
1log
8
13log
8
1
8
125
8
1 535
5
, (112b)
or
( )( )
( ) ( )
+
+=
k
kkdqqqq
q
dq
q
q
1
1log3log25
8
1
8
1 535
5
. (112c)
In the Ramanujans notebook part IV in the Section Integrals are examined various results on
integrals appearing in the 100 pages at the end of the second notebook, and in the 33 pages of the
third notebook. Here, we have showed some integrals that can be related with some argumentsabove described.
( )( )( ) ( )( )
=
+
++=
01
0 442
2
442
2
2
2
414
414
8
12
kxk
na
xae
xdxa
kae
ka
adnne
; (113)
( )( ) ( )
= +++=
+0 122442
2 1
44
1
414
kx
kaaa
axae
xdxa
. (114)
Multiplying both sides for2 , we obtain the equivalent expression:
( )( ) ( )
= +++=
+
01
22
232
442
22 1
44414
kx
kaaa
axae
xdxa
. (114b)
Let 0n . Then
( )( ) ( )( )
( ) ( ) ( ){ }( ) ( ){ }
=
+
++
+=
+0 0
12
2/12cosh12
12cos12
4coscosh
2sin
k
nkk
kk
nke
xxx
dxnx
. (115)
Also here, multiplying both sides for 2 , we obtain the equivalent expression:
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22
( )( ) ( )( )
( ) ( ) ( ){ }( ) ( ){ }
=
+
++
+=
+0 0
122
32
2/12cosh12
12cos12
4coscosh
2sin
k
nkk
kk
nke
xxx
dxnx
. (115b)
Now we analyze the following integral:
=
++
=1
0
2
15
2
411log
:
dxx
x
I ; (116)
Let ( ) 2/411 xu ++= , so that uux = 2 . Then integrating by parts, setting vu /1= and using the
following expression ( )( )
=z
dww
wzLi
02
1log, Cz , and employing the value
=
2
15log
102
152
2
2
Li , we find that
( )( ) ( )( )( )( )
+ + +
=
+=
=
=
2/15
1
2/15
1
2/15
1
2
2
1logloglog12
logdu
u
u
u
udu
u
uuduu
uu
uI
( )( )
+
=
+
+=
2/15
1
2 log1log
2
15log
2
1dv
v
vv
( )156102
15log
2
11
2
15
2
15log
2
1 222222
2 =+=
++
+= LiLi . (117)
Thence, we obtain the following equation:
++
=1
0
2
411log
dxx
x
I ( )( )
1512
log 22/15
1 2
=
=
+
duuuu
u. (118)
Multiplying both sides for 4
15, we obtain the equivalent expression:
++
=1
0
2
411log
4
15dx
x
x
I ( )( )
4154
1512
log
4
15 322/15
1 2
==
=
+
duuuu
u;
++
=1
0
2
411log
4
15dx
x
x
I ( )( )
412
log
4
15 32/15
1 2
=
=
+
duuuu
u. (118b)
In the Ramanujans paper: Some definite integrals connected with Gausss sums (Ramanujan,1915 Messanger of Mathematics, XLIV, 75-85), are described the following integrals:
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23
=
20
23 3
4
1
16
1
sinh
cos
dx
x
xx , (119)
=0
23
16
1
sinh
sin
dx
x
xx , (120)
+=
022
53
2
1
64
1
1
2cos
dx
e
xxx
, (121)
+=
022
2 551
256
1
1
2cos
dx
e
xxx
. (122)
Now, we sum (119) and (120) obtaining:
+0
23
sinh
cosdx
x
xx
+=+
=
0 22
23
16
1
16
3
64
1
16
13
4
1
16
1
sinh
sin
dx
x
xx .
Now we multiply both the sides for3 and obtain:
+0
233
sinh
cos
( dxx
x
x
+=+=+=02
3233
2
3323
3416
1
1616
3
641616
3
64)sinh
sin
dxx
x
x ,
thence
+=
+
2
3
0 0
23
233 3
416
1
sinh
sin
sinh
cos
dx
x
xxdx
x
xx . (123)
Now, we sum (121) and (122) obtaining:
+
0 2 1
2cosdx
e
xx
x
+++=
0 222
2
256
5
256
5
256
1
64
5
64
3
128
1
1
2cos
dxe
xx
x
.
Now we multiply both the sides for 3 and obtain the following equivalent equation:
+++=
+
8
5
8
5
8
1
2
5
2
3
42
1
16
1
1
2cos
1
2cos 2323
0 0 2
2
2
3dx
e
xxdx
e
xxxx
. (124)
In the work of Ramanujan, [i.e. the modular functions,] the number 24 (8 x 3) appears repeatedly.
This is an example of what mathematicians call magic numbers, which continually appear where we
least expect them, for reasons that no one understands. Ramanujans function also appears in stringtheory. Modular functions are used in the mathematical analysis of Riemann surfaces. Riemann
surface theory is relevant to describing the behavior of strings as they move through space-time.
When strings move they maintain a kind of symmetry called "conformal invariance". Conformal
invariance (including "scale invariance") is related to the fact that points on the surface of a string's
world sheet need not be evaluated in a particular order. As long as all points on the surface are taken
into account in any consistent way, the physics should not change. Equations of how strings must
behave when moving involve the Ramanujan function. When a string moves in space-time by
splitting and recombining a large number of mathematical identities must be satisfied. These are the
identities of Ramanujan's modular function. The KSV loop diagrams of interacting strings can be
described using modular functions. The "Ramanujan function" (an elliptic modular function that
satisfies the need for "conformal symmetry") has 24 "modes" that correspond to the physical
vibrations of a bosonic string. When the Ramanujan function is generalized, 24 is replaced by 8 (8 +
2 = 10) for fermionic strings.
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24
With regard the Section 1, we have the following mathematical connections between the eqs. (16b)
and (18b) and the eqs. (112c), (123) and (124):
( )
++++=
2422
2
221
32
1
32
1
8
3
2
1
8
3
8
5
4
3
8
3
48
1
Fxd
Ecquad
( )( )
( ) ( )
+
+=
k
kkdqqqq
q
dq
q
q
1
1log3log25
8
1
8
1 535
5
; (125)
( )
++++=
2422
2
221
32
1
32
1
8
3
2
1
8
3
8
5
4
3
8
3
216
1Fxd
Ecquad
+=
+
2
3
0 0
23
233 3
416
1
sinh
sin
sinh
cos
dx
x
xxdx
x
xx ; (126)
( )
++++=
2422
2
221
32
1
32
1
8
3
2
1
8
3
8
5
4
3
8
3
216
1Fxd
Ecquad
+++=
+
8
5
8
5
8
1
2
5
2
3
42
1
16
1
1
2cos
1
2cos 2323
0 0 2
2
2
3dx
e
xxdx
e
xxxx
; (127)
divp ( )( )
+
=
242
2
22
2
2
2
221 ln
3ln
16
3
82
1
8
1FxdE
eE
Ecc
c
( )
( )( )
( )
+
+=
k
kkdqqqq
q
dq
q
q
1
1log3log25
8
1
8
1 535
5
; (128)
divp ( )( )
+
=
242
2
22
2
2
2
221 ln
3ln
16
3
816
1FxdE
eE
Ecc
c
+=
+
2
3
0 0
23
233 3
416
1
sinh
sin
sinh
cos
dx
x
xxdx
x
xx ; (129)
divp ( )( )
+
=
242
2
22
2
2
2
221 ln
3ln
16
3
816
1FxdE
eE
Ecc
c
+++=
+
8
5
8
5
8
1
2
5
2
3
42
1
16
1
1
2cos
1
2cos 2323
0 0 2
2
2
3dx
e
xxdx
e
xxxx
. (130)
With regard the Section 2, we have the following mathematical connections between the eqs. (24)
and (27) and the eqs. (123) and (124):
( ) ( ) ( ) ( ) ( ) ( ) ( )
+++=+ 2
2
2
22 02
1162 k
R
q
RR
qkp
ed
LR IgkkgIgqqgIVdxGigT MM
+ 22 pR
Igppg M
+=
+
2
3
0 0
23
233 3
416
1
sinh
sin
sinh
cos
dx
x
xxdx
x
xx ; (131)
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25
( ) ( ) ( ) ( ) ( ) ( ) ( )
+++=+ 2
2
2
22 02
1162 k
R
q
RR
qkp
ed
LR IgkkgIgqqgIVdxGigT MM
( ) ( )]+ 22 pRIgppg M
+++=
+
85
85
81
25
23
421
161
1
2cos
1
2cos 2323
0 0 2
2
2
3dx
e
xxdxe
xxxx
; (132)
++
+=
2
2
02
2
2
223
021ln
1616
1
c
sw
s
cs
My
Mggb
+=
+
2
3
0 0
23
233 3
416
1
sinh
sin
sinh
cos
dx
x
xxdx
x
xx ; (133)
+++= 2
2
02
2
2
2
2302
1ln1616
1c
sw
s
cs
MyMggb
+++=
+
8
5
8
5
8
1
2
5
2
3
42
1
16
1
1
2cos
1
2cos 2323
0 0 2
2
2
3dx
e
xxdx
e
xxxx
. (134)
Thence, mathematical connections with the Ramanujans modular equations that are related to the
physical vibrations of the superstrings and with the Ramanujans identities concerning .
With regard the Section 4, we have the following mathematical connections between the eqs. (85)
and (95b), that derive from the (95), and the eqs. (114b), (115b), (118b), (123) and (124):
( ) ( )
+21
223
22
2 22
1
1
2
3
1 yyyedyxed
( ) ( ) ==
+
12 22
3
123
22
2
yyxed
( )( )
( )( )
===
22
223
222
223
2 22
1
23
122
1
23
1
222
3
3
122
3
1 yyyyedyedxedxedx
( )( ) ( )( ) ( ) ( ) 323
3
2
3
2
222
1
2
3
3
122
2
1
2
3
3
122
2
3
3
1 23
223
2
=== yy
edyedx
( ) ( )( )
=
=
+
6
321
2
2
1
4
822
3
123
22
2
yyxed
( )( ) ( )
= +++=
+
01
22
232
442
22 1
44414
kx
kaaa
axae
xdxa
; (135)
( ) ( )
+21
223
22
2 22
1
1
2
3
1 yyyedyxed
( ) ( ) ==
+
12 22
3
123
22
2
yyxed
( )( )
( )( )
===
22
223
222
223
2 22
1
23
122
1
23
1
222
3
3
122
3
1 yyyyedyedxedxedx
( )( ) ( )( ) ( ) ( )3
2
3
3
2
3
2
2221
23
3122
21
23
3122
23
31
2
3
22
3
2
=== yy edyedx
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26
( ) ( )( )
=
=
+
6
321
2
2
1
4822
3
123
22
2
yyxed
( )( ) ( )( )
( ) ( ) ( ){ }( ) ( ){ }
=
+
++
+=
+
00
122
32
2/12cosh12
12cos12
4coscosh
2sin
k
nkk
kk
nke
xxx
dxnx
; (136)
( ) ( )
+21
223
22
2 22
1
1
2
3
1 yyyedyxed
( ) ( ) ==
+
12 22
3
123
22
2
yyxed
( )( )
( )( )
===
22
223
222
223
2 22
1
23
122
1
23
1
222
3
3
122
3
1 yyyyedyedxedxedx
( )( ) ( )( ) ( ) ( ) 323
3
2
3
2
222
1
2
3
3
122
2
1
2
3
3
122
2
3
3
1 23
223
2
=== yy
edyedx
( ) ( )( )
=
=
+
6
321
2
2
1
4
822
3
123
22
2
yyxed
++
=1
0
2
411log
4
15dx
x
x
I ( )( )
412
log
4
15 32/15
1 2
=
=
+
duuuu
u; (137)
( ) ( )
+21
223
22
2 22
1
1
2
3
1 yyyedyxed
( ) ( ) ==
+
12 22
3
123
22
2
yyxed
( )( )
( )( )
===
22
223
222
223
2 22
1
23
122
1
23
1
222
3
3
122
3
1 yyyyedyedxedxedx
( )( ) ( )( ) ( ) ( ) 323
3
2
3
2
222
1
2
3
3
122
2
1
2
3
3
122
2
3
3
1 23
223
2
=== yy
edyedx
( ) ( )( )
=
=
+
6
321
2
2
1
4822
3
123
22
2
yyxed
+=
+
2
3
0 0
23
233 3
416
1
sinh
sin
sinh
cos
dx
x
xxdx
x
xx ; (138)
( ) ( )
+21
22
3
2
2
2 2
2
1
1
2
3
1 yyyedyxed
( )
( ) ==
+1
2
223
123
2
2
2
yy
xed
( )( )
( )( )
===
22
223
222
223
2 22
1
23
122
1
23
1
222
3
3
122
3
1 yyyyedyedxedxedx
( )( ) ( )( ) ( ) ( ) 323
3
2
3
2
222
1
2
3
3
122
2
1
2
3
3
122
2
3
3
1 23
223
2
=== yy
edyedx
( ) ( )( )
=
=
+
6
321
2
2
1
4822
3
123
22
2
yyxed
+++=
+
8
5
8
5
8
1
2
5
2
3
42
1
16
1
1
2cos
1
2cos 2323
0 0 2
2
2
3
dxe
xxdxe
xxxx . (139)
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27
( ) ( )( )
( )=
+++++
++
25
32
326315214
2
33
2
22
2
11
22
1
3
16
81
4
23
22
21
2
yyByyByyByByByBxed
yyy
( )( ) ( )
= +++=
+
01
22
232
442
22 1
44414
kx
kaaa
axae
xdxa
; (140)
( ) ( )( )
( )=
+++++
++
25
32
326315214
2
33
2
22
2
11
22
1
3
16
81
4
23
22
21
2
yyByyByyByByByBxed
yyy
( )( ) ( )( )
( ) ( ) ( ){ }( ) ( ){ }
=
+
++
+=
+
00
122
32
2/12cosh12
12cos12
4coscosh
2sin
k
nkk
kk
nke
xxx
dxnx
; (141)
( ) ( )( )
( )=
+++++
++
25
32
326315214
2
33
2
22
2
11
22
1
3
16
81
4
23
22
21
2
yyByyByyByByByBxed
yyy
++
=1
0
2
411log
4
15dx
x
x
I ( )( )
412
log
4
15 32/15
1 2
=
=
+
duuuu
u; (142)
( ) ( )( )
( )=
+++++
++
25
32
326315214
2
33
2
22
2
11
22
1
3
16
81
4
23
22
21
2
yyByyByyByByByBxed
yyy
+=
+
2
3
0 0
23
233 3
416
1
sinh
sin
sinh
cos
dx
x
xxdx
x
xx ; (143)
( ) ( )( )
( )=
+++++
++
25
32
326315214
2
33
2
22
2
11
22
1
3
16
81
4
23
22
21
2
yyByyByyByByByBxed
yyy
+++=
+
8
5
8
5
8
1
2
5
2
3
42
1
16
1
1
2cos
1
2cos 2323
0 0 2
2
2
3dx
e
xxdx
e
xxxx
. (144)
Thence, mathematical connections with the Ramanujans identities concerning , precisely4
3.
3.2Zeta Strings [8]The exact tree-level Lagrangian for effective scalar field which describes open p-adic stringtachyon is
++
=
+
12
2
2 1
1
2
1
1
1 pp
pp
p
p
g
L
, (145)
wherep
is any prime number,
22+= t
is the D-dimensional dAlambertian and we adoptmetric with signature ( )++ ... . Now, we want to show a model which incorporates the p-adicstring Lagrangians in a restricted adelic way. Let us take the following Lagrangian
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28
+
++=
==
1 1 1 1
1222 1
1
2
111
n n n n
n
nnnn
ngn
nCL
LL
. (146)
Recall that the Riemann zeta function is defined as
( )
==1 1
11
n pss
pns
, is += , 1> . (147)
Employing usual expansion for the logarithmic function and definition (147) we can rewrite (146)
in the form
( )
++
= 1ln
22
112
gL
, (148)
where1= 2220
2kkkr
, (149)
where( ) ( ) ( )dxxek ikx
=
~
is the Fourier transform of ( )x .
Dynamics of this field
is encoded in the (pseudo)differential form of the Riemann zeta function.When the dAlambertian is an argument of the Riemann zeta function we shall call such string a
zeta string. Consequently, the above is an open scalar zeta string. The equation of motion for
the zeta string is
( )( ) +> =
=
2
2
220 1
~
22
1
2 kkixk
Ddkk
ker
(150)
which has an evident solution 0= .
For the case of time dependent spatially homogeneous solutions, we have the following equation ofmotion
( )( )
( )( )
( )tt
dkkk
etk
tikt
=
=
+>
1
~
22
1
200
2
0
2
2
0
0
. (151)
With regard the open and closed scalar zeta strings, the equations of motion are
( )( )
( )
=
=
n
n
nn
ixk
Ddkk
ke
1
2
12 ~
22
1
2
, (152)
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29
( )( )
( )( )
( )
( )
+
+
+=
=
1
11
2
12
112
1~
42
1
4
2
n
n
nn
nixk
Dn
nndkk
ke
, (153)
and one can easily see trivial solution 0== .
With regard the Section 3, we have the following mathematical connection between the eq. (54b)
and the eq. (150):
( )( ) +> =
=
2
2
220 1
~
22
1
2 kkixk
Ddkk
ker
( )( )
=
=
0
1
1
2
2
x
d
d
dd
e
xdx
SVolpT
( )( )( )
( )
d
ddd
Td
ddp
2/12
2/1
2
12
!1
2
21
. (154)
With regard the Section 4, we have the following mathematical connections between the eqs. (74)
and (77) and the eqs. (150):
( )
=
LlatZ
dI
F2
2
2
1
3
4
4 ( ) ( ) ( ) +
+
12121
2
2122/5
2/5
0,0 2
12
3
2
2
3
2 254
362
mim
mm
emmKm
m
( )
( ) +> =
=
2
2
220 1
~
22
1
2 kkixk
Ddkk
ker
; (155)
( ) ( ) ( ) ( ) ( ) ( )( ) ( )
+=
=
+
2/1
2/1 00,0,
,2
32
2
3
23
,2
32
2
212
3
1 ,4
5362
2,
2
22
2
pj
ZSLpUj
U
T
ZSLUUE
TTeE
ddVI
( )
( ) +> =
=
2
2
220 1
~
22
1
2 kkixk
Ddkk
ker
. (156)
In conclusion, multiplying for 6
8
1 the eq. (77), we obtain the following equivalent equation:
( )( )
( )
( )
( )( )
( ) ( )
+
+=
=
2/1
2/1 00,0,
,2
322
3
2
3,2
322
2
12
63
1 ,4
53
624,8
12
22
2
pj
ZSLpUj
U
T
ZSL
UUETTeE
d
dVI
(157)
This equation can be related with the expression (138) and with the eq. (150), giving the following
interesting complete mathematical connections:
( ) ( ) ( ) ( ) ( ) ( )( ) ( )
+=
=
+
2/1
2/1 00,0,
,2
32
2
3
2
3,2
32
2
212
63
1 ,4
5362
4,
8
12
22
2
pj
ZSLpUj
U
T
ZSLUUE
TTeE
ddVI
( ) ( )
+21
223
22
2 22
1
1
2
3
1 yyyedyxed
( ) ( ) ==
+
12 22
3
123
22
2
yyxed
( )( )
( )( )
===
22
223
222
223
2 22
1
23
122
1
23
1
222
3
3
122
3
1 yyyyedyedxedxedx
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( )( ) ( )( ) ( ) ( ) 323
3
2
3
2
222
1
2
3
3
122
2
1
2
3
3
122
2
3
3
1 23
223
2
=== yy
edyedx
( ) ( )( )
=
=
+
6
321
2
2
1
4822
3
123
22
2
yyxed
+=
+
2
3
0 0
23
233 3
416
1
sinh
sin
sinh
cos
dx
x
xxdx
x
xx
( )( ) +> =
=
2
2
220 1
~
22
1
2 kkixk
Ddkk
ker
. (158)
Acknowledgments
I would like to thank Prof. Branko Dragovich of Institute of Physics of Belgrade (Serbia) for his
availability and friendship. Ild like to thank Prof. Jorma Jormakka of Department of Military
Technology, National Defence University, Helsinki, for his availability. In conclusion, Ild like to
thank also Dr. Pasquale Cutolo for his important and useful work of review and his kind advices.
References
[1] David J. Toms Quantum gravitational contributions to quantum electrodynamics arXiv:
1010.0793v1 [hep-th] 5 Oct 2010
[2] Yong Tang and Yue-Liang Wu Gravitational Contributions to the Running of Gauge
Couplings arXiv:0807.0331v2 [hep-ph] 10 Dec 2008
[3] Sean Patrick Robinson Two Quantum Effects in the Theory of Gravitation Massachusetts
Institute of Technology June 2005
[4] Edward Witten Supersymmetric Yang-Mills Theory on a four-manifold arXiv:hep-th/
9403195v1 31 Mar 1994;
Massimo Bianchi and Stefano Kovacs Yang-Mills Instantons vs. type IIB D-instantons
arXiv:hep-th/9811060v2 27 Nov 1998;Anirbau Basu The
46RD term in type IIB string theory on
2T and U-duality arXiv:0712.
1252v1 [hep-th] 8 Dec 2007
[5] Jorma Jormakka Solutions to Yang-Mills Equations arXiv:1011.3962v2 [math.GM]
15 Nov 2010 - http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.3962v2.pdf
[6] Bruce C. Berndt, Heng Huat Chan, Sen-Shan Huang Incomplete Elliptic Integrals in
Ramanujans Lost Notebook - American Mathematical Society, Providence, RI, 2000, pp. 79-
126.
[7] Bruce C. Berndt Ramanujans Notebook Part IV Springer-Verlag (1994).