We consistently incorporate Yang Mills matter fields into string corrected
(deformed), D=10, N=1 Supergravity. We solve the Bianchi identities within the
framework of the modified beta function favored constraints to second order in
the string slope parameter $\g$ also including the Yang Mills fields. In the
torsion, curvature and H sectors we find that a consistent solution is readily
obtained with a Yang Mills modified supercurrent $A_{abc}$. We find a solution
in the F sector following our previously developed method.Comment: 11 pages, LaTeX file, PACS number: 04.65.+

Bellucci, S.

O'Reilly, D.

Physics Letters B

English

arXiv

AbstractWe consistently incorporate Yang–Mills matter fields into string corrected (deformed) D=10, N=1 supergravity. We solve the Bianchi identities within the framework of the modified beta function favored constraints to second order in the string slope parameter γ also including the Yang–Mills fields. In the torsion, curvature and H sectors we find that a consistent solution is readily obtained with a Yang–Mills modified supercurrent Aabc. We find a solution in the F sector following our previously developed method

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Inclusion of Yang–Mills fields in string corrected supergravity 

Physics Letters B 666 (2008) 193–198Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Inclusion of Yang–Mills ﬁelds in string corrected supergravity
S. Bellucci a,∗, D. O’Reilly b
a INFN-Laboratori Nazionali di Frascati, Via E. Fermi 40, 00044 Frascati, Italy
b Department of Mathematics, The Graduate School and University Center, 365 Fifth Avenue, New York, NY 10016-4309, USA
a r t i c l e i n f o a b s t r a c t
Article history:
Received 27 May 2008
Received in revised form 30 June 2008
Accepted 3 July 2008
Available online 5 July 2008
Editor: T. Yanagida
PACS:
04.65.+e
We consistently incorporate Yang–Mills matter ﬁelds into string corrected (deformed) D = 10, N = 1
supergravity. We solve the Bianchi identities within the framework of the modiﬁed beta function favored
constraints to second order in the string slope parameter γ also including the Yang–Mills ﬁelds. In the
torsion, curvature and H sectors we ﬁnd that a consistent solution is readily obtained with a Yang–
Mills modiﬁed supercurrent Aabc . We ﬁnd a solution in the F sector following our previously developed
method.
© 2008 Elsevier B.V. All rights reserved.
1. Introduction
String corrections to quantum ﬁeld theories are believed to contribute Gauss–Bonnet terms to the action. This invariant has been
studied in many different scenarios, for example Gauss–Bonnet modiﬁed cosmology [1]. These terms are introduced by hand into the
quantum gravity models. However it is also known that Gauss–Bonnet terms occur naturally in the context of string corrected gravity [2].
The leading order corrections in terms of the string tension parameter, γ contain these invariants. Hence we are motivated by this and by
other reasons to study string deformed supergravity, with D = 10, N = 1 as the low energy limit of string theory. Here we wish to include
Yang–Mills ﬁelds in the non-minimal theory. As several of the terms are extremely lengthy we avoid writing them explicitly. In a future
work we will explore the simpliﬁed bosonic case.
Some years ago a scenario was developed to construct a manifestly supersymmetric theory of string corrected supergravity [2–6].
This involved incorporating the Lorentz Chern–Simons superform, XLL into the geometry of D = 10, N = 1 supergravity. It was initially
successful at ﬁrst order in γ [2]. It ran into diﬃculties and controversy at second order [7]. It was suggested that at second order the
torsion Tαβ g should be modiﬁed to include the so-called X tensor [3]. A search for the X tensor ensued and a candidate was proposed in
[8] which was shown to allow for the solution of the Bianchi identities in the H sector and also the torsion and curvature sectors.
In this Letter we show that a simple modiﬁcation of the X tensor Ansatz allows also for the inclusion of matter ﬁelds in the H torsion
and curvature sectors. We show that the assumption F (2)αβ = 0 does not allow for a solution. We propose a candidate that does allow for
a solution. Prior to ﬁnding the X tensor also by way of an Ansatz, in [8], the search for a second order solution consisted of systematically
studying the table of irreducible representations [9]. However the task proved formidable because of the number of unknown quantities.
It was eventually shown that the form given in Eq. (33) of the ﬁrst reference of [8] for T (2)αβ g , satisfying the torsion identity at dimension
one half, thereafter allowed ultimately for a consistent solution in the torsion and curvature sectors.
In Ref. [8], we found a solution to D = 10, N = 1 supergravity to second order in the string slope parameter with the modiﬁed tensor
GADG = HADG + γ Q ADG . (1)
Here we extend this as follows:
GADG = HADG + γ Q ADG + βY ADG . (2)
Here β is the Yang–Mills coupling constant. Matter ﬁelds will not have consequences for the appearance of the Gauss–Bonnet term
however, but we wish to construct a complete model.
* Corresponding author.
E-mail addresses: bellucci@lnf.infn.it (S. Bellucci), doreilly@gc.cuny.edu, dunboyne@vzavenue.net (D. O’Reilly).0370-2693/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.physletb.2008.07.004
194 S. Bellucci, D. O’Reilly / Physics Letters B 666 (2008) 193–1982. The solution
The Bianchi identities in superspace are as follows:[[∇[A,∇B},∇C)}= 0. (3)
Here we have extended the commutator in [8] to include the Yang–Mills ﬁeld strengths F IAB
[∇A,∇B} = T ABC∇C + 1
2
RAB
deMed + i F IABtI . (4)
The t I are the generators of the Yang–Mills gauge group. For notation convenience we drop the group index I . For notational brevity we
also write
RABde = R(0) ABde + R(1) ABde + R(2) ABde + · · · (5)
and
T AD
G = T (0) ADG + T (1) ADG + T (2) ADG + · · · . (6)
Where the numerical superscript refers to the order of the quantity. A quantity which re-occurs is the following
Ω(1) gef = L(1)gef − 14 A
(1)
gef (7)
and its spinor derivative which we denote simply as
Ω(1)γ gef = ∇γ
[
L(1)gef − 14 A
(1)
gef
]
. (8)
We evaluate this in Appendix A. Ref. [2] began with the conventional constraints as follows:
iσb
αβ Tαβ
a = 16δab, σabcdeαβ Tαβa = 0, σaαβ Tαβa = 0,
Tα[bc] = 0, Tabc = 16 T [abc], Taα
β = 1
48
σaαλσ
pqrβλApqr . (9)
Using these constraints then led to the ﬁrst order solution. That is G(1) ABC , T (1) ABC , and R(1) ABde were found, along with additional
modiﬁed constraints. In [8] we found that no modiﬁcation to the condition Hαβγ = 0 was necessary. We continue with this assumption.
We have the H sector Bianchi identities with Yang–Mills ﬁelds as follows [2]:
1
6
∇(α|H |βγ δ) − 14 T(αβ|
MHM|γ δ) = −γ4 R(αβ|ef R |γ δ)
ef − β
4
F(αβ| I F |γ δ) I , (10)
1
2
∇(α|H |βγ )d − ∇dHαβγ − 12 T(αβ|
MHM|γ )d + 12 Td(α|
MHM|βγ ) = −γ R(αβ|ef R |γ )def − β F(αβ| I F |γ )d I , (11)
∇(α|Hβ)cd + ∇[c|H |d]αβ − TαβMHMcd − TcdMHMαβ − T(α|[c|MHM|d]|β)
= −γ [2Rαβef Rcdef + R(α|[c|ef R |d]β)ef ]− β[2Fαβ I Fcd I + F(α|[c| I F |d]β) I]. (12)
We must also solve the torsions and curvatures
T(αβ|λT |γ )λd − T(αβ| g T |γ )gd − ∇(α|Tβγ )d = 0, (13)
T(αβ|λT |γ )λδ − T(αβ| g T |γ )g δ − ∇(α|T |βγ )δ − 14 R
(2)
(αβ|deσ de |γ )δ = 0, (14)
T(αβ|λR |γ )λde − T(αβ|λR |γ )λde − ∇(α|R |βγ )de = 0. (15)
To this list we now add the equations arising form the F sector
∇[A|F |BC) − T M [AB|FM|C) = 0. (16)
In our previous work the supercurrent A(1)gef was given by
A(1)gef = +iγ σgef 	τ Tmn	 Tmnτ . (17)
To begin with we modify A(1)gef to include matter ﬁelds as in [2].
A(1)gef −→ +iσgef 	τ
[
γ Tmn	 Tmn
τ + βλ	λτ ]= A(γ )gef + A(β)gef . (18)
Here we use the notation where the superscript γ or β is self evident. This has the effect of splitting the Bianchi identities however
we still have to be careful of cross terms. In order to be cautious we will examine all contributions in the H sector, in particular that
for H(2)aβd , for thoroughness (see Appendix A). In all of the Bianchi identities we encounter the spinor derivative ∇α A(1)abc . We now
show that a key equation which led to the previous second order modiﬁed beta function favored (β F F ) solution is still valid but with the
modiﬁed A(1)abc . The spinor derivative of A(1)abc will now contain the contributions due to the spinor derivative of λτ at zeroth order.
We have the following modiﬁcations from [2]:
S. Bellucci, D. O’Reilly / Physics Letters B 666 (2008) 193–198 195∇γ T (0)ef δ = −14σ
mn
γ
δR(0)ef mn + T (0)ef λT (0)γ λδ, (19)
∇γ λ(0)δ = −1
4
σmnγ
δ Fmn + λλT (0)γ λδ. (20)
The fundamental equation which enable a solution to be found with higher order β F F constraints, now with the modiﬁed supercurrent,
A(1)pqr is still given by
T (0)(αβ|λσ pqref |γ )λA(1)pqr H(0)def − σ pqref (αβ|H(0)def ∇|γ )A(1)pqr = −24σ g (αβ|H(0)def
[
Ω(1)|γ )gef
]
. (21)
In the H sector we will have the following quantities
HABC = H(γ γ ) ABC + H(ββ) ABC + H(βγ ) ABC . (22)
We will also adopt the modiﬁed torsion
T (2)αβ
g = − iγ
6
σ pqref αβH
(0)d
ef A
(1)
pqr . (23)
We then ﬁnd no change in the form of the H sector results.
H¯(2)αβd = H(γ β)αβd = σαβ g
[−iγ H(0)def A(β)gef ]− σ pqref αβ
[
iγ
12
H(0)def A
(β)
pqr
]
. (24)
Eq. (21) contains the second order contribution of the spinor derivative ∇γ Hαβg . Term by term we notice no order β2 contributions
occur. We also seek terms of the form O (γ β) and note that none appear other than those resulting from the modiﬁcation to Aabc . We
also ﬁnd H(2)gγ d as in [8], but we write it in a different way
+ i
2
σ(αβ|g H(2)g|γ )d = − γ12σ
γ |
(αβ|σ pqr g f |γ )λA(1)pqr Tdf λ + iσ g (αβ|
{
4γ
[∇|γ )H(0)def ]L(1)gef + 4γ H(0)def ∇|γ )L(1)gef
− γ
2
[∇|γ )H(0)def ]A(1)gef − γ2 H(0)def ∇|γ )A(1)gef − 2γ H(0) gef R(1)|γ )def − 2γ L(1) gef R(0)|γ )def
+ γ
4
A(1)gef R
(0)|γ )def + 2γ∇|γ )
[
H(0)d
ef H(0)gef
]Order(1)}
. (25)
The symmetrized result can be extracted as in [2]. We list it in Appendix A. We also believe that it can be simpliﬁed further. We also
have, along with (33) of the ﬁrst reference of [8],
iσ g (αβ|T (2)|γ )gδ = +2iγ σ g (αβ|T (0)ef δΩ(1)|γ )gef , (26)
+iσ g (αβ|T (2)|γ )gd = +4iγ σ g (αβ|H(0)def
[
Ω(1)|γ )gef
]+ γ
6
σ g (αβ|σ pqre g|γ )φ A(1)pqr T (0)deφ, (27)
Rαβde = − iγ12σ
pqref
αβ A
(1)
pqr Ref de. (28)
3. F sector Bianchi identities
We have seen that in the H and torsion sectors the required minimal Bianchi identities have been fully satisﬁed by simply modifying
the supercurrent as in Eq. (28). The fundamental identity (21) is then used to solve in each case. This is the identity that enables such
solutions to be obtained in the modiﬁed β function favored set of constraints. Here we show that it can also be used to solve in the F
sector coupled with an Ansatz. To begin with we consider the following Bianchi identities:
T(αβ|λF |γ )λ − T(αβ| g F |γ )g − ∇(α|F |βγ ) = 0, (29)
∇α Fab − ∇[a|Fα|b] − Tα[a|M FM|b] − TabM FM α = 0. (30)
We have from [2] to ﬁrst order,
F (0)αβ = F (1)αβ = F (1)αd = 0, (31)
F (0)αd = −iσdαφλφ, (32)
∇α F (0)ef = iσ[e|αφ∇| f ]λφ − 2iσ gαφH(0) gef . (33)
Eq. (29) is not satisﬁed by F (2)αβ = 0. We propose the following candidate and show that it works.
F (2)αβ = − iγ
12
σ pqref αβ A
(1)
pqr Fef . (34)
At second order equation (29) becomes
T (0)(αβ|λF (2)|γ )λ − ∇(α|F (2)|βγ ) − iσ g (αβ|F (0)|γ )g + i γ6 σ
pqref
(αβ|A(1)pqr H(0)gef
[−iσg|γ )φλφ]= 0 (35)
or, more clearly
196 S. Bellucci, D. O’Reilly / Physics Letters B 666 (2008) 193–198T (0)(αβ|λ
[
− iγ
12
σ pqref |γ )λFef A(1)pqr
]
+ iγ
12
σ pqref (αβ|Fef ∇|γ )A(1)pqr − iγ12σ
pqref
(αβ|A(1)pqr∇|γ )Fef
− iσ g (αβ|F (0)|γ )g + i γ6 σ
pqref
(αβ|A(1)pqr H(0)gef
[−iσg|γ )φλφ]= 0. (36)
Using (21) generates two solvable terms and we also set up a cancellation. We also must use
σ pqref (αβ|σe|γ )φ = −σ pqref φ(α|σe|βγ ). (37)
Hence, we obtain
−iσ g (αβ|F (2)|γ )g + 2iγ σ g (αβ|Ω(1)|γ )gef F (0)ef − γ6 σ
g
(αβ|σ pqrg f |γ )φ A(1)pqr∇ f λφ + iγ12σ
pqref
(αβ|A(1)pqr
[−2iσ g |γ )φH(0) gef ]
− i γ
6
σ pqref (αβ|A(1)pqr H(0)gef
[−iσg|γ )φλφ]= 0. (38)
The last two terms conveniently cancel. We ﬁnd
iσ g (αβ|F (2)|γ )g = +2iγ σ g(αβ|Ω(1)|γ )gef F (0)ef −
γ
6
σ g (αβ|σ pqrg f |γ )φ A(1)pqr∇ f λφ (39)
or
F (2)γ g = +2γΩ(1)γ gef F (0)ef + i γ6 σ
pqr
g
f
γ φ A
(1)
pqr∇ f λφ. (40)
Finally for completeness, we consider the derivatives, ∇α F (2)bc and ∇(α|F (2)|β)g . We have
∇(α|F |β)d − Tαβ g F gd − TαβλFλd + T(α|d g F g|β) + T(α|dλFλ|β) = 0 (41)
and
∇α Fbc = ∇[b|Fα|c] + Tα[b| g F g|c] + Tα[b|λFλ|c] − Tbc g F gα + TbcλFλα. (42)
Substitution of (31), (32), (33), (34) and (40), as well as other results quoted in [2] and
T (2)αβ
λ = − iγ
12
σ pqref αβ A
(1)
pqr T
(0)λ
ef (43)
give, respectively, for (41) and (42),
−iσd(α|φ∇|β)λφ |(2) + 2γ∇(α|Ω(1)|β)def F ef + 4iγΩ(1)(α|def σe|β)φ∇ f λφ − 4iγΩ(1)(α|def σg|β)φH(0)gef
+ iγ
6
σ pqrd
f
(α|φ
(∇|β)A(1)pqr)[∇ f λφ]+ iγ6 σ pqrd f (α|φ A(1)pqr
[∇|β)∇ f λφ]− iσ gαβ F (2)gd + γ12σ pqref αβσdλφ A(1)pqr Tef λλφ
− 2γ T (0)αβλΩ(1)λdef F (0)ef − i γ6 T
(0)
αβ
λσ pqrd
f
λ f A
(1)
pqr∇ f λφ − iσg(α|φλφT (2)|β)d g = 0, (44)
∇α Fbc = 2γ∇[b|Ω(1)α|c]ef F ef − i γ6 σ
pqr [b| f αφ A(1)pqr∇|c]∇ f λφ − i γ6 σ
pqr [b| f αφ
[∇|c]A(1)pqr]∇ f λφ
+ T (2)α[b|g F (0)g|c] − iT (2)α[b|λσ|b]λφλφ + 2γ T (0)gbcΩαgef F ef + i γ6 σ
pqr
g
f
λφ A
(1)
pqr T
(2)
bc
g∇ f λ f − iγgαφλφT (2)bc g
− i γ
12
σ pqref λα A
(1)
pqr F
(0)
ef T
(0)
bc
λ. (45)
4. Modiﬁed commutator
Finally, we note how the commutator (4) is modiﬁed. We have
[∇α,∇β} = − iγ
12
σ pqref αβ A
(1)
pqr
[
Tef
γ ∇γ + 2H(0)ef g∇g + 12 R
(0)
ef
mnMmn + i F (0)ef I t I
]
. (46)
But
2H(0)ef
g = −T (0)ef g, (47)
and
R(0)ef
mnMmn = −1
8
R(0)ef λ
δσmnδ
λ (48)
so we get the modiﬁed geometry proportional to σ 5 as follows:
[∇α,∇β}|(2) = − iγ
12
σ pqref αβ A
(1)
pqr
[
T (0)ef
γ ∇γ − T (0)ef g∇g − 116 R
(0)
ef λ
δσmnδ
λMmn − i F (0)ef I t I
]
. (49)
A point conceptually important can be stressed here by observing that the latter expression agrees in its structure with the conjecture
in Ref. [5] for supersymmetric Yang–Mills couplings.
S. Bellucci, D. O’Reilly / Physics Letters B 666 (2008) 193–198 1975. Conclusion
The geometrical methods currently known as deformations [3] and the constraints often referred to as beta function favored con-
straints [10–12] allowed for the determination of the most general higher derivative Yang–Mills action to the third order, which is globally
supersymmetric and Lorentz covariant in D = 10 spacetime (see e.g. [13]), a result which is important for topologically nontrivial gauge
conﬁgurations of the vector ﬁeld, such as, for instance, in the case of compactiﬁed string theories on manifolds with topologically nontriv-
ial properties. Building upon our previous works on the subject, we have been able here to provide a consistent inclusion of Yang–Mills
matter ﬁelds into string corrected (deformed) D = 10, N = 1 supergravity. Our solution to the Bianchi identities, obtained within the
framework of the modiﬁed beta function favored constraints, holds to the second order in the string slope parameter γ and includes
also the Yang–Mills ﬁelds. We obtained as well a consistent solution in the torsion, curvature and H sectors with a Yang–Mills modiﬁed
supercurrent Aabc . Following a technique we developed in earlier papers, we also found a solution in the F sector and gave an explicit
formula for the modiﬁcation induced in the commutator expression.
Acknowledgement
This work of has been supported in part by the European Community Human Potential Program under contract MRTN-CT-2004-005104
“Constituents, fundamental forces and symmetries of the universe”.
Appendix A
For the sake of extra caution as the Bianchi identity for example for Hαbd is long, we can write out the full version as follows:
1
2
∇(α|H |βγ )d(γ γ ;ββ;γ β) − ∇dHαβγ Order(2) − 12 T
(0)
(αβ|λH(γ γ )λ|γ )d − 12 T
(0)
(αβ|λH(γ β)λ|γ )d − 12 T
(0)
(αβ|λH(ββ)λ|γ )d
− 1
2
T (γ )(αβ|λH(β)λ|γ )d − 12 T
(γ )
(αβ|λH(γ )λ|γ )d − 12 T
(β)
(αβ|λH(β)λ|γ )d − 12 T
(β)
(αβ|λH(γ )λ|γ )d − 12 T
(γ γ )
(αβ|λH(0)λ|γ )d
− 1
2
T (γ β)(αβ|λH(0)λ|γ )d − 12 T
(ββ)
(αβ|λH(0)λ|γ )d − 12 T
(0)
(αβ|g H(γ γ ) g|γ )d − 12 T
(0)
(αβ|g H(γ β) g|γ )d − 12 T
(0)
(αβ| g H(ββ)g|γ )d
− 1
2
T (γ )(αβ|g H(β) g|γ )d − 12 T
(β)
(αβ| g H(γ )g|γ )d + 12 T
(γ γ )
d(α|g H(0) g|βγ ) + 12 T
(γ β)
d(α| g H(0) g|βγ ) + 12 T
(ββ)
d(α|g H(0) g|βγ )
+ 1
2
T (γ )d(α|g H(γ ) g|βγ ) + 12 T
(γ )
d(α| g H(β)g|βγ ) + 12 T
(β)
d(α|g H(γ ) g|βγ ) + 12 T
(β)
d(α| g H(β)g|βγ ) + 12 T
(0)
d(α| g H(γ γ )g|βγ )
+ 1
2
T (0)d(α|g H(γ β) g|βγ ) + 12 T
(0)
d(α|g H(ββ)g|βγ )
+ γ [R(γ )(αβ|ef R(0)|γ )def + R(β)(αβ|ef R(0)|γ )def + R(0)(αβ|ef R(β)|γ )def + R(0)(αβ|ef R(γ )|γ )def ]
+ β[F (γ )(αβ|F (0)|γ )d + F (β)(αβ|F (0)|γ )d + F (0)(αβ|F (β)|γ )d + F (0)(αβ|F (γ )|γ )d]= 0. (A.1)
Applying the ﬁrst order constraints and allowing HABC terms to drop out leaves the H¯ ABC contributions.
1
2
∇(α|H |βγ )d(ββ;γ β) − 12 T
(0)
(αβ|λH(γ β)λ|γ )d − 12 T
(0)
(αβ|λH(ββ)λ|γ )d − 12 T
(γ β)
(αβ|λH(0)λ|γ )d − 12 T
(γ β)
(αβ|λH(0)λ|γ )d
− 1
2
T (0)(αβ|g H(γ β) g|γ )d − 12 T
(0)
(αβ|g H(ββ)g|γ )d + 12 T
(γ β)
d(α|g H(0) g|βγ ) + 12 T
(ββ)
d(α|g H(0) g|βγ )
+ 1
2
T (γ γ )d(α|g H(0) g|βγ ) + γ
[
R(β)(αβ|ef R(0)|γ )def + R(0)(αβ|ef R(β)|γ )def
]= 0. (A.2)
We therefore ﬁnd as before that no order β2 terms exist. Also there is no need to make modiﬁcations other than the adjustment of
Aabc in the H sector.
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Inclusion of Yang-Mills Fields in String Corrected Supergravity

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