Table Of ContentLocal heterotic geometry in holomorphic
coordinates
Guy Bonneau Galliano Valent
∗ ∗
4
9
9
1
n Abstract
a
J
In the same spirit as done for N=2 and N=4 supersymmetric non-linear σ models in
0
1 2 space-time dimensions by Zumino and Alvarez- Gaum´e and Freedman, we analyse
the (2,0) and (4,0) heterotic geometry in holomorphic coordinates. We study the
2
properties of the torsion tensor and give the conditions under which (2,0) geometry
v
3 is conformally equivalent to a (2,2) one. Using additional isometries, we show that
0 it is difficult to equip a manifold with a closed torsion tensor, but for the real 4 di-
0
mensional case where we exhibit new examples. We show that, contrarily to Callan,
1
0 Harvey and Strominger ’s claim for real 4 dimensional manifolds, (4,0) heterotic
4 geometry is not necessarily conformally equivalent to a (4,4) Ka¨hler Ricci flat geom-
9
etry. We rather prove that, whatever the real dimension be, they are special quasi
/
h
Ricci flat spaces, and we exemplify our results on Eguchi-Hanson and Taub-NUT
t
- metrics with torsion.
p
e
h
:
v
i
X
r
a
PAR/LPTHE/93-56 December 1993
∗
Laboratoire de Physique Th´eorique et des Hautes Energies, Unit´e associ´ee au CNRS URA 280, Uni-
versit´e Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05.
1
2 G. Bonneau, G. Valent
1 Introduction
In 1979, B. Zumino [1] proved the importance of complex geometry for the study of
supersymmetric σ-models. In particular, using holomorphic coordinates, he showed that
N = 2 supersymmetry in two space-time dimensions requires the bosonic part of the
Lagrangian density to be built on a K¨ahler manifold. The work of Alvarez-Gaum´e and
Freedman has generalised this approach to N = 4 supersymmetry which was shown to
require hyperk¨ahler manifolds [2]. The heterotic supersymmetries lead to generalisations
of the aforementioned geometries : in the (2,0) [3, 4] and in the (4,0) cases [5, 6, 7, 8],
it was shown that one needs target manifolds with torsion. It is the aim of the present
work to analyse the properties of such manifolds equipped with torsion using holomorphic
coordinates.
In section 2, we recall the necessary and sufficient conditions on the target space metric
and torsion tensors for (2,0) heterotic supersymmetry and express them in holomorphic
coordinates. Various geometrical objects are then given in such coordinates, in the same
spirit as done by Alvarez-Gaum´e and Freedman [9] and Hull [4]. A torsion potential b
µν
and a vector V (dual of the torsion tensor in the special case of 4 real dimensions) are
µ
then introduced and used to express the relevant geometrical objects.
We then analyse the necessary and sufficient conditions under which, given a complex
Riemannian manifold of real dimension 2N, a (2,0) heterotic geometry (metric g +
µν
torsion) and a (2,2) one (K¨ahler metric gˆ , no torsion) have conformally related metrics :
µν
gˆ = e−2fg .
µν µν
This appears to be a very restrictive condition as the conformal factor has to be a real
function, harmonic with respect to the K¨ahler metric laplacian, whatever the dimension
of the manifold be:
∆ˆe2f = 0 .
In order to describe geometries leading to one-loop finite non-linear σ models in two space
time dimensions, following Friedan [10] in the torsionless case, Friedling and Van de Ven
[11] introduced “ generalised quasi Ricci flat” metrics through 1 :
W and χ such that Ric = W
µ µ (µν) (µ ν)
∃ ∇
1
Ric = Tρ W + χ . (1)
[µν] 2 µν ρ ∇[µ ν]
We then express these conditions in holomorphic coordinates and also discuss the condi-
tions that restrict the holonomy group from U(N) to SU(N) : indeed, it was argued by Hull
[4] that such a requirement is necessary for off-shell one-loop finiteness of (2,0) non-linear
σ models in two space time dimensions. We show that the metric has to be a special
“generalized quasi Ricci flat” one 2.
Several examples of metrics with special isometries are then constructed in Section 3.
Despite the large isometry groups considered, which enforce conformal equivalence with
1 A d=ef 1(A A ) and A d=ef 1(A + A ). Moreover is the ordinary
λ[µν]ρ 2 λµνρ − λνµρ λ(µν)ρ 2 λµνρ λνµρ ∇µ
covariantderivative with the symmetric Christoffel connection.
2 χ = W where W is related to the trace of the torsion tensor.
µ µ µ
−
complex heterotic geometry 3
the K¨ahler torsionless case, (2,0) supersymmetry is not sufficient to define uniquely the
metrics and several examples of supplementary properties are analysed. In particular, we
generalise the (torsionless) metrics of LeBrun [12] which have a vanishing scalar curvature,
to target spaces with torsion and obtain as a special case an easy derivation of Eguchi-
Hanson metric with torsion 3 directly in holomorphic coordinates. Unfortunately, despite
their conformal equivalence with regular K¨ahler ones, all the metrics we obtained are
singular.
Section4isdevotedto(4,0)heteroticgeometry. Thenecessary andsufficient conditions
for (4,0) supersymmetry are given (as shown in [7], they are slightly less restrictive than
often asserted ( see for example [8])), and we prove that in any 4N dimensional case, the
metric has to be a special “ generalised quasi Ricci flat” one where the vector V depends
µ
on a single complex function F. We also show that scalar flatness is obtained on general
grounds only for 4 real dimensions.
We then analyse the possible conformal equivalence of such metrics with torsionless hy-
perk¨ahler ones involved in (4,4) supersymmetry and show that, even in 4 real dimensions,
contrarily to Callan et al.’s claim [14], (4,0) world sheet supersymmetry does not imply
that the corresponding metric is conformally equivalent to a Ricci flat K¨ahler one.
Section 5 is then devoted to a detailed study of Eguchi-Hanson and Taub-NUT metrics
with torsion [13] in holomorphic coordinates. While Eguchi-Hanson with torsion is indeed
conformally equivalent to its torsionless counterpart, this fails to be true for Taub-NUT
with torsion. To conclude some remarks are offered in Section 6.
2 (2,0) heterotic geometry in complex coordinates
2.1 Generalities and notations
As explained in the introduction, we consider 2N real dimensional complex Riemannian
manifolds with torsion and assume that the metric is hermitian with respect to the covari-
antly constant complex structure J and that the torsion tensor three form is closed. These
conditions are necessary and sufficient to build a (2,0) supersymmetric σ model with a
Wess-Zumino term [3]. All these hypothesis write :
- distance :
dτ2 = g dxµdxν, g = g (2)
µν µν νµ
- torsion : the torsion tensor T = g Tσ is fully skew-symmetric and its associated
µνρ µσ νρ
three form is closed
1
T = T dxµ dxν dxρ, dT = 0 (3)
µνρ
3! ∧ ∧
- Jν is an almost complex structure :
µ
JνJρ = δρ (4)
µ ν − µ
- integrability condition on Jν to be a complex structure :
µ
Nρ Jλ(∂ Jρ ∂ Jρ) (µ ν) = 0 (5)
µν ≡ µ λ ν − ν λ − ↔
3 first obtained by Delduc and Valent [13] .
4 G. Bonneau, G. Valent
- hermiticity : the metric is hermitian with respect to the complex structure
g Jλ + Jλg = 0 (6)
µλ ν µ λν
which is equivalent to the statement that J d=ef Jσg is skew symmetric and therefore
µν µ σν
locally defines a two-form :
1
ω = J dxµ dxν (7)
µν
2 ∧
- covariant constancy of Jν :
µ
D Jρ ∂ Jρ +Γρ Jλ Γλ Jρ = 0 (8)
µ ν ≡ µ ν λµ ν − νµ λ
(Theconnection withtorsionis taken asΓρ = γρ 1Tρ where γρ isthe usualsymmetric
µν µν−2 µν µν
Christoffel connection). As a consequence of equ. (8), the integrability condition (5)
reduces to an algebraic constraint on the torsion :
Nρ Tρ JλJσTρ + (JλTσ JλTσ )Jρ = 0 (9)
µν ≡ µν − µ ν λσ µ λν − ν λµ σ
( compare with equ.(21) of [8] ) and then one obtains :
1 1
dω = −2Tµ′νρJµµ′dxµ ∧dxν ∧dxρ = 2Tµ′ν′ρ′Jµµ′Jνν′Jρρ′dxµ ∧dxν ∧dxρ (10)
In the absence of torsion, equations (2,4,6,8) define a K¨ahler manifold.
In the following, due to (4,5), one can choose a coordinate system where J is diagonal
and constant. The real coordinates xµ being split into the complex ones zi and z¯i, one
has:
J¯i = Ji = 0 , Ji = J¯i = iδi J = J = 0 , J = J = ig . (11)
j ¯j j − ¯j j ⇔ ij ¯i¯j j¯i − ¯ij j¯i
n o n o
2.2 The geometrical objects in complex coordinates
In these complex coordinates, equation (6) gives :
g = g = 0
ij ¯i¯j
The distance becomes :
dτ2 = 2g dzidz¯j (12)
i¯j
and the complex structure two-form
ω = ig dzi dz¯j (13)
i¯j
∧
The covariant constancy of J then implies :
(8) Γi = Γi = Γ¯i = Γ¯i = 0 (14)
⇒ ¯jk ¯jk¯ jk jk¯
complex heterotic geometry 5
Using the well known expression of the symmetric connection γρ corresponding to the
µν
hermitian metric g 4 :
µν
1
γi = gi¯l [g + g ]
jk 2 ¯lk,j ¯lj,k
1
γi = γi = gi¯l [g g ]
¯jk k¯j 2 ¯lk,¯j − ¯jk,¯l
γi = 0 (15)
¯jk¯
and conjugate expressions for γ¯i , equations (14) give :
µν
T = T = 0
ijk ¯i¯jk¯
T = g g
ijk¯ ik¯,j − jk¯,i
Γi = γi 1Ti = gi¯lg
jk jk − 2 jk k¯l,j
Γi = Ti = Ti = gi¯l[g g ] = 2γi (16)
jk¯ − jk¯ k¯j j¯l,k¯ − jk¯,¯l jk¯
Of course, the algebraic constraint (9) is identically satisfied.
The closedness of the torsion, equ. (3), then writes :
[T T ]+[T T ] = 0 (17)
ijk¯,¯l − ij¯l,k¯ k¯¯li,j − k¯¯lj,i
or equivalently :
g = g
i[k¯,¯l]j j[k¯,¯l]i
Equation(17) can also be written :
D T + D T = TνT 2Tν T where ν (n,n¯) . (18)
[¯l k¯]ij [j i]k¯¯l ij νk¯¯l − i[k¯ ¯l]νj ≡
The closedness of the torsion may also be solved through the introduction of a skew-
symmetric torsion potential b ([3]) :
µν
T = b + b + b
µνρ νρ,µ ρµ,ν µν,ρ
or, in complex coordinates
T = b b + b
ijk¯ jk¯,i − ik¯,j ij,k¯
Equation (16) then implies the existence of a vector potential (χ ,χ ) and of a “gauge”
i ¯i
freedom (v ,v ) such that [3],[8] :
i ¯i
b = v v
ij j,i i,j
−
b + g = χ v
i¯j i¯j ¯j,i i,¯j
−
Due to equation (14), the Riemann tensor defined by
Rµ = ∂ Γµ + Γµ Γλ (ρ σ) (19)
νρσ ρ νσ λρ νσ − ↔
4 As usual, a comma indicates a derivative with respect to the coordinate.
6 G. Bonneau, G. Valent
has many vanishing components in holomorphic coordinates
Ri = R¯i = 0 . (20)
¯jκλ jκλ
The surviving ones are
Ri = gin¯[∂ T 2Tm¯ g ]
jkl − j kln¯ − n¯[k l]m¯,j
Ri = Ri = ∂ Γi D Ti + Tn¯Ti
jk¯l − j¯lk − ¯l jk − k j¯l ¯lk jn¯
Ri = gin¯[∂ T 2Tmg ] (21)
jk¯¯l n¯ k¯¯lj − j[k¯ ¯l]m,n¯
and similar expressions for R¯i .
¯jκλ
The Ricci tensor
Ric d=ef Rσ (22)
µν µνσ
then writes
Ric = D Tk + TkTl
ij i kj ij lk
Ric = ∂ ∂ logdet g + D Tk TkT¯l (23)
i¯j i ¯j k k k i¯j − i¯l ¯jk
(for vanishing torsion, one recovers the usual results [9]) .
We introduce the vector :
1 V = Tk = T¯k
V = JνJλTρ i ki − k¯i (24)
µ 2 µ ρ νλ ⇔ V¯i = Tk¯k¯¯i = −Tkk¯i !
and obtain, by contraction of the closedness relation (18),
(D V + D V ) + (D Tk + D Tk¯) 2(TkV + Tk¯V + TkT¯l ) = 0 (25)
i ¯j ¯j i k i¯j k¯ ¯ji − i¯j k ¯ji k¯ i¯l ¯jk
and
gi¯j(D V + D V ) = gi¯j(TkT¯l + 2V V ) (26)
i ¯j ¯j i ¯li k¯j i ¯j
Moreover, as a consequence of equation (16), we find :
D T = D T
[¯l k¯]ij [j i]k¯¯l
which gives by contraction
D Tk D Tk¯ + D V D V = 0 . (27)
k i¯j − k¯ ¯ji i ¯j − ¯j i
Using these equations, the Ricci tensor may be expressed in holomorphic coordinates as :
Ric = D V +2∂ V
ij j i [i j]
1
Ric = D V + ∂ (∂ logdet g 2V ) + ∂ (∂ logdet g 2V ) . (28)
i¯j ¯j i 2 i ¯j k k − ¯j ¯j i k k − i
h i
Finally, the scalar curvature
R d=ef gµνRic
µν
writes :
1
R = 2gi¯j ∂ ∂ logdet g + 2V V (D V + D V ) (29)
i ¯j k k i ¯j − 2 i ¯j ¯j i
(cid:20) (cid:21)
complex heterotic geometry 7
2.3 Generalised Quasi Ricci flat metrics
For further use, we recall that in [10], Friedan defines “quasi Ricci flat metrics ” through
(no torsion)
V such that Ric = D V + D V
µ µν µ ν ν µ
∃
Such geometries lead to one-loop finiteness for the D=2 non-linear σ models built on such
target space metrics. They have been studied by Bonneau and Delduc and some explicit
expressions have been obtained [15]. Here, in the presence of torsion, the Ricci tensor is no
longer symmetric and, following Friedling andvandeVen [11], we define “generalised quasi
Ricci flatness ” by the same one loop finiteness requirement. This leads to the definition :
W and χ such that Ric = W
µ µ (µν) (µ ν)
∃ ∇
1
Ric = Tρ W + χ (30)
[µν] 2 µν ρ ∇[µ ν]
In this (2,0) heterotic geometry, conditions (30) simplify to :
Ric = D W +∂ (W +χ)
ij j i [i j]
Ric = D W +∂ (W +χ) . (31)
i¯j ¯j i [i ¯j]
2.4 SU(N) holonomy and generalised quasi Ricci flatness
Let us recall that the holonomy group of 2N dimensional (with or without torsion) man-
ifolds with a covariantly constant complex structure is a subgroup of U(N). Moreover,
particular cases where the holonomy is SU(N) play a special role : in the absence of tor-
sion, this means Ricci flatness ; in the present case, the vanishing of the U(1) part of the
Riemann curvature
C = JσRρ
µν ρ σµν
writes :
C = 2∂ Γ = 0 (32)
µν [µ ν]
where Γ = JνΓρ , (a priori not a vector), is found in holomorphic coordinates as 5[4] :
µ ρ νµ
Γ = i[∂ logdet g 2V ] , Γ = i[∂ logdet g 2V ] . (33)
i i i ¯i ¯i ¯i
k k− − k k−
Using the tensor C , equation (28) writes
µν
i i
Ric = D V + C , Ric = D V + C (34)
ij j i ij i¯j ¯j i i¯j
2 2
Then SU(N) holonomy C = 0 , leads to a special case of “generalised quasi Ricci
µν
⇔
flatness” (31) with
W = V , χ = V .
i i i i
−
As a consequence, and as will be exemplified in subsection (3.3.2), SU(N) holonomy is a
more restrictive requirement that the sole “generalised quasi Ricci flatness” of Friedling
and van de Ven.
5 We use equations (21,16,24).
8 G. Bonneau, G. Valent
2.5 (2,0) versus (2,2) geometries
Given a complex Riemannian manifold - Jν being the complex structure - equipped with
µ
a metric g and a torsion tensor T satisfying equations (2,3,4,5,6,8), we address the
µν µνρ
following question :
“does there exist a conformal transformation that transforms this (2,0) geometry into
a (2,2) one ”, i.e. that relates the metric g and the torsion tensor T to a K¨ahler metric
µν µνρ
gˆ (no torsion) through :
µν
gˆ = e−2fg (35)
µν µν
(the positivity of the distance requires the reality of f ).
The K¨ahler condition writes in complex coordinates :
gˆ = gˆ , gˆ = gˆ
i¯j,k k¯j,i i¯j,k¯ ik¯,¯j
which leads, using equation (16), to
T = g g = 2(g ∂ f g ∂ f)
ijk¯ ik¯,j − jk¯,i ik¯ j − jk¯ i
T = g g = 2(g ∂ f g ∂ f) (36)
¯i¯jk ¯ik,¯j ¯jk,¯i ¯ik ¯j ¯jk ¯i
− −
The vector V is then a gradient :
µ
V = Tk = 2(N 1)∂ f ,V = Tk¯ = 2(N 1)∂ f . (37)
i ki − i ¯i k¯¯i − ¯i
This is a very restrictive condition as it means that the whole torsion tensor depends on
a single real function f. Notice that equation (36) may be rewritten as :
1 1
T = g V g V , T = g V g V (38)
ijk¯ (N 1) ik¯ j − jk¯ i ¯i¯jk (N 1) ¯ik ¯j − ¯jk ¯i
− h i − h i
The closedness relation (26) writes
N 2
D Vi + D V¯j = 2 − V Vi where Vi = gi¯jV ,V¯j = gi¯jV . (39)
i ¯j N 1 i ¯j i
−
Equation (27) gives D Vi = D V¯i and, for N = 2, D V = D V . As a consequence,
i ¯i i ¯j ¯j i
6
the conformal factor e2f satisfies :
∆ˆe2f d=ef 2gˆi¯jˆ ∂ e2f = 2gˆi¯j∂ ∂ e2f = 4gˆi¯je2f ∂ ∂ f +2∂ f∂ f
i ¯j i ¯j i ¯j i ¯j
∇
1 h i
= 4e4fgi¯j D ∂ f V ∂ f + V ∂ f
" i ¯j − i ¯j (N 1) i ¯j #
−
2e4f N 2
= D Vi − V Vi = 0 . (40)
i i
N 1 − N 1
− (cid:20) − (cid:21)
and we have also
8(N 2)
∆f d=ef 2gi¯jD ∂ f = − V 2. (41)
i ¯j (N 1)2k k
−
Notice that using (13, 36) one has
dω = 2ω df. (42)
∧
Finally, the scalar curvatures are related by :
2N(N 2)
R[g,T] = e−2f Rˆ[gˆ,Tˆ 0]+ − V 2 (43)
" ≡ (N 1)2 k kgˆ#
−
complex heterotic geometry 9
3 (2,0) heterotic geometry in 2 complex dimensions
3.1 The torsion tensor
In that special case T has as many components as the vector V and from definition (24)
ijk¯ i
one obtains :
T = g V g V (44)
ijk¯ ik¯ j − jk¯ i
This relation is a duality one, first written in real coordinates in [13]
T = ǫ Vσ
µνρ µνρσ
The closedness relation (26), when compared to equation (27) gives
D Vi = D V¯j = 0 (45)
i ¯j
and the scalar curvature reduces to :
R = 2gi¯j(∂ ∂ logdet g + 2V V ) (46)
i ¯j i ¯j
k k
3.2 From (2,2) to (2,0) supersymmetry through a conformal
rescaling of the metric ?
When one compares equations (44) and (37,38), one sees that in 2 complex dimensions one
gets conformal equivalence between (2,0) and (2,2) geometries, if and only if the vector V
is the gradient of a real function :
V = 2∂ f , V = 2∂ f (47)
j j ¯j ¯j
whith the constraints
∆ˆe2f = ∆f = 0 (48)
and, in that case, the scalar curvatures are proportional :
R[g,T] = e−2fRˆ[gˆ,Tˆ 0] (49)
≡
We now construct somes examples of (2,0) heterotic metrics with particular isometries.
3.3 Special cases with linear U(N) isometry
3.3.1 Generalities
Let us consider the special family of hermitian metrics g (i,¯j = 1,..,N) with linear U(N)
i¯j
symmetry
N
g = A(s)δ + B(s)z¯izj , s = z¯izi (50)
i¯j ij
i=1
X
10 G. Bonneau, G. Valent
Defining C(s) = A(s)+sB(s) we have
1 B(s)
gi¯j = δ ziz¯j (51)
ij
A(s) − "A(s)C(s)#
and det g = A(s)C(s). From (16), the torsion tensor is found to be 6 :
k k
T = [A˙(s) B(s)](z¯jδ z¯iδ ) (52)
ijk¯ − ik − jk
and its closedness gives
d
[A˙(s) B(s)]δ[jzk]z¯l + [A˙(s) B(s)]δ[jδk] (i l) = 0 (53)
ds − i − i l − ↔
For N 3, this implies A˙(s) B(s) = 0 which means that the metric is K¨ahler and that
≥ −
no torsion can be put on a manifold with such isometries. For N = 1, the torsion tensor
identically vanishes whereas for N = 2 (2 complex dimensions), equation (53) gives
d L
[s2(A˙(s) B(s))] = 0 A˙(s) B(s) = (54)
ds − ⇒ − 2s2
In such a case, the vector V writes
i
A˙(s) B(s)
V = Tk = − z¯i d=ef ∂ logγ(L,s) (55)
i ki " A(s) # i
Due to the reality of the functions A(s) and B(s) and consequently of γ(L,s), the looked-
for metric is conformally equivalent to a K¨ahler one through the conformal transformation
gˆ = γ(L,s)g (compare equations (47) and (55)). We then rescale the looked-for functions
A(s) and C(s) according to :
A(s) = γ(L,s)η(s) , C(s) = γ(L,s)µ(s) (56)
where, as a consequence of equations (54,55) we have :
d(sη(s)) dγ L
µ(s) = , s = . (57)
ds ds 2sη
In this U(2) case, it is convenient to use the coordinates
z1 = √scos θei(φ+2ψ) , z2 = √ssin θei(φ−2ψ)
2 2
to write the distance (12) :
C(s)
dτ2 = (ds)2 +2sA(s)(η2 +η2)+2sC(s)η2 (58)
2s 1 2 3
where the expressions for the η are given in subsection (5.1).
i
6 A dot indicates a derivative with respect to the variable s.
