A note on the Ricci scalar of six dimensional manifold with SU(2) structure

0.09 MB·

by  Gautier Solard

A note on the Ricci scalar of six dimensional manifold with SU(2)
structure
Gautier Solard
Physics Department, Universit`a di Milano-Bicocca, Piazza della Scienza 3,20100 Milano, ITALY
INFN, sezione di Milano-Bicocca, Milano, ITALY
[email protected]
Abstract
Taking [1] as an inspiration, we study the intrinsic torsion of a SU(2) structure manifold in
six dimensions to give a formula for the Ricci scalar in terms of torsion classes. The derivation is
founded on the SU(3) result coming from the aforementioned paper.
6
1 Introduction
0
2
Generalized geometry [2,3] showed the need to understand better SU(3)×SU(3) manifolds and in
b
e particular SU(2) manifolds. Some early works [4–9] looked into the matter of compactification on
F these manifolds. The Ricci scalar of the internal manifold can give precious insights on a particular

9 solution, for example on the presence of sources. Thus, a better understanding of the Ricci scalar is
1 important and in particular its expression in terms of the SU(2) torsion classes in six dimensions (it

has already been done for five dimensional manifolds [10]).
]
h
This work has been inspired by [1] where the authors give the expression of the Ricci scalar in
t
- terms of the SU(3) structure torsion classes on six dimensional manifold. Then we do something
p
e similar in spirit to what was done in [11] namely using the SU(3) result to derive the SU(2) result.
h In section 1, we review briefly six dimensional manifolds with SU(3) and SU(2) structure. After
[
expressing the SU(3) torsion classes in terms of the SU(2) torsion classes in section 2, we use one of

2
the formulas of [1] to express the Ricci scalar in section 3.
v
6
1
1 Review of SU(3) and SU(2) structures
7
4
0 1.1 SU(3) structure
.
1
0 Asixdimensionalmanifoldis saidtobeof SU(3)structureifitadmits aglobally definedrealtwo-form
6 J and a globally defined complex decomposable three-form Ω verifying :
1
v: 3i
J ∧Ω =0 J ∧J ∧J = Ω∧Ω (1.1)
i 4
X
r We can quantify the failure of these forms to be closed using the following torsion classes :
a
dJ =3/2Im(W¯ Ω)+W ∧J +W (1.2)
1 4 3
dΩ =W J2+W ∧J +W¯ ∧Ω (1.3)
1 2 5
with W a complex scalar, W a complex primitive (ie. J∧J∧W = 0) (1,1) form, W a real primitive
1 2 2 3
(ie. J ∧W = 0) (2,1)+(1,2) form, W a real one form and W a complex (1,0) form. Constraints
3 4 5
on these torsion classes put constraints on the manifold itself. For example, if all the torsion classes
vanish, the manifold is Calabi-Yau. If W = W = 0, then the manifold is complex etc.
1 2
1.2 SU(2) structure
A six-dimensional manifold is said to be of SU(2) structure if it admits a globally defined complex
one-form K = K1+iK2, a real two form j and a complex two form ω verifying :
ι K =0 ι K =2 (1.4)
K K
1
j ∧ω =0 ι j = ι ω =0 j ∧j = ω∧ω (1.5)
K K
2
One can then define a triplet of real two forms Ja with J1 = j, J2 = Re(ω) and J3 = Im(ω). This
triplet verifies :
ι K =0 ι K =2 (1.6)
K K
ι Ja =0 Ja∧Jb =δaJ1∧J1 (1.7)
K b
As in the SU(3) structure case, one can define torsion classes as follows1 :
dKi =mi Ja+mi K1∧K2+µ˜i ∧Kj +µi (1.8)
a 0 j
dJa =n0Ki∧Ja+ǫabcnb Ki∧Jc+νa ∧Ki+ν˜a+ν˜a∧K1∧K2 (1.9)
i i i 3 1
mia, mi0, nai, n0i are 16 real scalars. µ˜ij are 4 real one-forms verifying ιKkµ˜ij = 0. µi, νai are 8
real two-forms verifying : µi ∧Jb = νai ∧Jb = 0 and ιKkµi = ιKkνai = 0. ν˜3a are 3 real three-forms
verifying ιKkν˜3a = 0 (and thus ν˜3a ∧Jb = 0). Finally ν˜1a are three real one-forms verifying ιKkν˜1a but
they are not independant from one another. Indeed, from considering d(Ja∧Jb = δaJ1∧J1), one can
b
see that ν˜2 and ν˜3 can be expressed in terms of ν˜1. Nevertheless we will continue to use the three ν˜a
1 1 1 1
in order to simplify the expressions.
To summarize the independant real torsion classes are : 16 scalars, 5 one-forms, 8 two-forms and
3 three-forms.
2 Going from SU(2) to SU(3)
From a SU(2) structure, one can define a family of SU(3) structure. Let R be a SO(3) matrix and
define2 j = R1 Ja and ω = (R2 +iR3 )Ja. Then Jˆ= j +K1∧K2 and Ωˆ = K∧ω define a SU(3)
c a c a a c c
structure. By computing dJˆ and dΩˆ using (1.8) and (1.9) and comparing with (1.2) and (1.3), one
obtains the expressions of the SU(3) torsion classes in terms of the SU(2) torsions classes :
1
W = C (m+ +na) (2.1)
1 3 a a +
Xa
1 j −2K1∧K2
W = C 2(2m+ −na)( c )+2iνa +∗(ν˜a∧K)−νˆa∧K
2 2 a(cid:20) a + 3 + 3 (cid:21)
Xa
1
+ (δi −iǫ )∗(µ˜+∧Kj ∧ω ) (2.2)
2 j ij i c
Xi,j
1stricto sensu, the torsion classes should be defined with respect to j and ω rather than the Ja but the expressions
are simpler and more ”covariant” using theJa
2Notethat jc and ωc together with K definea SU(2) structure.
1 1
W = ǫ (mj −nb )R1 Kj ∧Ja− ǫ (R1 R1 −δ )(mj −nb )Kk ∧Ja
3 2 abc b j c 2 jk a b ab b j
aX,b,c,j aX,b,j,k
1 1
− ǫ Kj ∧µk + R1 Kj ∧νa − ν˜a∧Ja+ R1 K1∧K2∧ν˜a
jk a j 6 1 2 a 1
Xj,k Xj,a Xa Xa
1 1 1
+ (K1∧K2−j )∧ µ˜i + R1 ν˜a+ R1 νˆa∧K1∧K2 (2.3)
2 c i 2 a 3 2 a
Xi Xa Xa
1 1 1
W = n0 Kj + ǫ mj R1 Kk + R1 ν˜a+ µ˜i − R1 νˆa (2.4)
4 j jk a a 2 a 1 2 i 2 a
Xj jX,k,a Xa Xi Xa
K 1
W = (n0 +im− −iR1 na)+ (δi +iǫ ) µ˜i +i∗ µ˜i ∧K1∧K2∧j
5 2 − 0 a − 4 j ij j j c
Xi,j (cid:2) (cid:0) (cid:1)(cid:3)
1
− C ∗[(∗ν˜a)∧ω ] (2.5)
a 3 c
4
Xa
with C =R2 +iR3 , φ +iφ = φ , φ −iφ = φ , νˆa = ∗[(∗ν˜a)∧j ].
a a a 1 2 + 1 2 − 3 c
3 Ricci scalar
In [1], the authors gave the expression of the Ricci scalar in terms of the SU(3) torsion classes3 :
15 1 1
R = |W |2− |W |2− |W |2−|W |2+4δW +2δW +8 < W ,W > (3.1)
6 1 2 3 4 5R 4 4 5R
2 2 2
Thanks to our expressions of the SU(3) torsion classes in terms of the SU(2) torsion classes (2.1)-
(2.5), we can give the Ricci scalar in terms of the SU(2) torsion classes :
1 1
R =− (mi )2+3 (n0)2+4 ǫ n0mj +4 mi na − |µi|2− |νa|2 (3.2)
6 a i ij i 0 a i 2 2 i
Xa,i Xi Xi,j Xa,i Xi Xa,i
1 1 3
− (ǫ ǫ +ǫ ǫ )∗(µ˜i ∧∗(µ˜k))− |ν˜a|2− |ν˜a|2− ∗ µ˜i ∧(∗ν˜a)∧Ja
2 ik jl il jk j l 6 1 2 3 i 3
iX,j,k,l Xa Xa Xa,i (cid:2) (cid:3)
1 1
+ ∗ (∗µ˜i )∧∗ ν˜a∧Ja∧Ki∧Kj + (1−δa)∗ ∗((∗ν˜a)∧Ja)∧(∗ν˜b)∧Jb
3 j 1 2 b 3 3
Xi,j,a (cid:2) (cid:0) (cid:1)(cid:3) Xa,b h i
−2 ǫ δ(mi Kj)+4 δ(n0Ki)+2 δ(µ˜i )− ∗ (δν˜a)∧Ja∧K1∧K2
ij 0 i i 3
Xi,j Xi Xi Xa (cid:0) (cid:1)
+ R1aδ(ν˜1a)− ǫabcǫij(2mib+nbi)ncj + ǫij(2mia+4nai)n0j −4 miami0
Xa bX,c,i,j Xi,j Xi

1
+ ǫ ǫ ∗(νb ∧∗νc )+ ǫ ∗(νa ∧∗µj)+ ∗(µ˜i ∧ν˜a∧Ki∧Kj)
2 abc ij i j ij i j 3
bX,c,i,j Xi,j Xi,j
− ∗ µ˜i ∧µ˜i ∧Kj ∧Kk ∧Ja +2 ǫ δ((mi +na)Kj)
j k ij a i
Xi,j,k h i Xi,j
1
− ∗ ν˜a∧(∗ν˜b)− ν˜b∧(∗ν˜a) ∧Jb − δ[∗(µ˜i ∧Ki ∧Kj ∧Ja)]
(cid:20)(cid:18) 1 3 3 1 3 (cid:19) (cid:21) j
Xb Xi,j

3Fortwoforms, ω1 andω2,onedefines<ω1,ω2 >=∗[∗(ω1)∧ω2]and |ω1|2 =<ω1,ω1 >,thecodifferential isdefined
as δω1=∗(d(∗ω1))
The curvature of the manifold doesn’t depend on the specific SU(2) structure we choose and so
shouldn’t depend on the matrix R. To solve this puzzle, one has to remember that nowhere we
requested that d2 = 0 on the forms defining the SU(2) structure. And indeed, one can show :
1
R6|R1a part = ∗ R1a2 ǫa,b,cd(d(Jb))∧Jc− d(d(Ki))∧Ki∧Ja = 0 (3.3)
Xa Xb,c Xi
 
so that the Ricci scalar is independent of the matrix R.
Conclusion
Using the link between SU(3) and SU(2) structures, we were able to express the Ricci scalar in terms
of the SU(2) structure torsion classes. Unfortunately, applying the same technique to the Ricci tensor
has been unfruitful so far but is still under investigation. Another way to derive this formula could
have been to use the result of [12] where they give the expression of the Ricci scalar in term of the
pure spinors4. Indeed, one could have expressed the pure spinors in terms of the SU(2)-forms and
derived the result.
Acknowledgement
The author would like to thank Hagen Triendl who participated in the project at an earlier stage.
The author is supported in part by INFN and by the European Research Council under the Euro-
pean Union’s Seventh Framework Program (FP/2007-2013) ERC Grant Agreement n.307286 (XD-
STRING).
References
[1] L. Bedulli and L. Vezzoni, The Ricci tensor of SU(3)-manifolds, Journal of Geometry and
Physics 57 (Mar., 2007) 1125–1146 [math/0606786].
[2] N. Hitchin, Generalized Calabi-Yau manifolds, Quart.J.Math.Oxford Ser. 54 (2003) 281–308
[math/0209099].
[3] M. Gualtieri, Generalized complex geometry, math/0401221.
[4] J. P. Gauntlett, D. Martelli and D. Waldram, Superstrings with intrinsic torsion, Phys.Rev.
D69 (2004) 086002 [hep-th/0302158].
[5] G. Dall’Agata and N. Prezas, N = 1 geometries for M theory and type IIA strings with fluxes,
Phys.Rev. D69 (2004) 066004 [hep-th/0311146].
[6] G. Dall’Agata, On supersymmetric solutions of type IIB supergravity with general fluxes, Nucl.
Phys. B695 (2004) 243–266 [hep-th/0403220].
[7] M. Grana, R. Minasian, M. Petrini and A. Tomasiello, Generalized structures of N=1 vacua,
JHEP 0511 (2005) 020 [hep-th/0505212].
[8] J. Bovy, D. Lust and D. Tsimpis, N = 1,2 supersymmetric vacua of IIA supergravity and SU(2)
structures, JHEP 08 (2005) 056 [hep-th/0506160].
4The authorthanksDavid Andriot for bringing this formula to his attention
[9] H. Triendl and J. Louis, Type II compactifications on manifolds with SU(2) x SU(2) structure,
JHEP 07 (2009) 080 [0904.2993].
[10] L. Bedulli and L. Vezzoni, Torsion of SU(2)-structures and Ricci curvature in dimension 5,
ArXiv Mathematics e-prints (Feb., 2007) [math/0702790].
[11] S. Chiossi and S. Salamon, The intrinsic torsion of SU(3) and G 2 structures, ArXiv
Mathematics e-prints (Feb., 2002) [math/0202282].
[12] D. Lust, F. Marchesano, L. Martucci and D. Tsimpis, Generalized non-supersymmetric flux
vacua, JHEP 11 (2008) 021 [0807.4540].