A geometric realization of sl(6,C)
A GEOMETRIC REALIZATION OF sl(6,C)
GIOVANNI GAIFFI, MICHELE GRASSI
Abstract. Given an orientable weakly self-dual manifold X of rank two, we
build a geometric realization of the Lie algebra sl(6,C) as a naturally defined
algebra LC of endomorphisms of the space of differential forms of X. We pro-
vide an explicit description of Serre generators in terms of natural generators
of LC. This construction gives a bundle on X which is related to the search
for a natural Gauge theory on X. We consider this paper as a first step in the
study of a rich and interesting algebraic structure.
1. Introduction
This paper is a step in a broader program, which aims at finding a geomet-
ric counterpart to the Mirror Symmetry phaenomenon, and possibly a geometric
language in which to formulate a physical theory interpolating between different
σ-models. While we direct the reader to [G2],[G3] for more details, we list here
only some aspects of this theory to put the present work into context.
In the Strominger-Yau-Zaslow approach to Mirror Symmetry you have that two
mirror dual Calabi-Yaus should posses (in some limiting sense) semi-flat special
lagrangian torus fibrations f : M → B, f̂ : M̂ → B which have as fibres flat tori
which are dual in the metric sense (see [SYZ], and [G2] for the terminology and the
definitions). As it is widely known, the major drawback of this approach is that it
is very difficult to build special lagrangian tori fibrations. Usually this construction
can be carried out only when the dual Calabi-Yau manifolds are actually hyper-
kahler, and the special lagrangian tori can be viewed as complex submanifolds (with
respect to a rotated complex structure), so that the methods of complex algebraic
geometry can be put to work.
When you do have the fibrations, then the idea is to construct the mirror map as
a sort of Fourier-Mukai transform (see for example [BMP]). This Fourier-Mukai
transform is a correspondence induced by pull-back and push forward from the
space X = M ×B M̂ . In the hyperkähler case this space is a complex manifold,
while in the general case (for example for Mirror Symmetry for Calabi-Yau three-
folds) it is just a real manifold of (real) dimension 3 · dimC(M).
Background. The notion of (Weakly) self-dual manifold (cf. [G2]) was con-
ceived in the first place to isolate the geometric aspects of the X above which are
needed to obtain Mirror Symmetry betweenM and M̂ . We reproduce here the def-
inition for the reader, while referring to [G2] and [G3] for all the remarks, examples
and observations:
Definition 1.1. A weakly self-dual manifold (WSD manifold for brevity) is given
by a smooth manifold X, together with two smooth 2-forms ω1, ω2 a Riemannian
metric and a third smooth 2-form ωD (the dualizing form) on it, which satisfy the
following conditions:
1) dω1 = dω2 = dωD = 0 and the distribution ω
1 + ω
2 is integrable.
2) For all p ∈ X there exist an orthogonal basis dx1, .., dxm, dy
1 , ..., dy
m, dy
1 , ..., dy
Date: October 24, 2006.
http://arxiv.org/abs/0704.0104v1
2 GIOVANNI GAIFFI, MICHELE GRASSI
dz1, ..., dzc, dw1, ..., dwc of T
pX such that the dx1, .., dxm, dy
1 , ..., dy
m, dy
1 , ..., dy
are orthonormal and
(ω1)p =
dxi ∧ dy
i , (ω2)p =
dxi ∧ dy
i , (ωD)p =
dy1i ∧ dy
dzi ∧ dwi
Any orthogonal basis of TpX dual to a basis of 1- forms as above is said to be
adapted to the structure, or standard. The number m is the rank of the structure.
For a more intrinsic definition of WSD manifolds the reader should refer to [G2].
Here we have chosen the quickest way to introduce them.
When the forms ω1, ω2, ωD are covariant constant with respect to the Levi-Civita
connection, we speak of 2-Kähler manifolds. An example of these comes from mirror
symmetry for abelian varieties.
Remark 1.2. The form ωD is symplectic once restricted to ω
1 + ω
2. We have
therefore that ω
dim(X)−m
6= 0.
Definition 1.3. 1) A WSD manifold is nondegenerate if dim(ω01 ∩ ω
2)p = 0 at all
points (equivalently if its dimension is 3 times the rank).
2) A WSD manifold is self-dual (SD manifold for brevity) if all the leaves of the
distribution ω01 + ω
2 have volume one (with respect to the volume form induced by
the metric)
Using Self dual manifolds, you can give a first näıve geometric definition of Mir-
ror Symmetry as follows:
Two Calabi-Yau manifolds with B-field (M,BM ) and (M̂,BM̂ ) are mirror dual
if there is a Self-dual manifold X together with surjections π : X → M and
π̂ : X → M̂ such that:
a) π∗(ωM ) = ω1, π̂
) = ω2.
b) The leaves of ω⊥1 are the fibres of π̂
c) The leaves of ω⊥2 are the fibres of π
d) The induced B-fields on M and M̂ are the ones given.
Here make their first appearence the B-fields BM and BM̂ , which are flat unitary
gerbes on M and M̂ respectively, and which are not relevant for the discussions of
this paper. In [G2] it was shown that this picture works well in the case of elliptic
curves, and for some other flat situations.
Physical motivation. One of the reasons to introduce SD manifolds however
was to get rid of special lagrangian fibrations, which are so difficult to construct,
and to be able to attack the problem of Mirror Symmetry also when these fibra-
tions are not expected to exist. In this more general context one expects that the
Mirror Symmetry phaenomenon will not be obtained directly from fibrations of
a SD manifold to the dual Calabi-Yaus, but via a more sophisticated procedure,
which involves a Gromov-Hausdorff type of limit. In [G3] it was shown that for
the family of anticanonical divisors in complex projective space one can build a
(real) two-dimensional family of WSD manifolds, which degenerate in a normalized
Gromov-Hausdorff sense to the correct limits of the mirror dual Calabi-Yaus. The
picture is the following:
A GEOMETRIC REALIZATION OF sl(6,C) 3
MB MAS
where MA and MB are the large Kähler and large complex structure limits of M
and M̂ respectively. To be precise, the manifolds which come out of the costruc-
tion of [G3] are 11 dimensional (degenerate) Weakly self-dual manifolds or rank 3.
Dimension 11 is very appealing in this context from a physical point of view, and
it brings us to the motivation for the present work.
The point of view of [G3] is very different from the current one in the main liter-
ature on mathematical Mirror Symmetry: instead of considering the fibre product
M×B M̂ (when it exists) as a device for proving Mirror Symmetry for Calabi-Yaus,
the limiting Calabi-Yaus of Mirror Symmetry are seen as very special limits of a
family of Self-Dual manifolds, which are the main objects of study. This is actually
more in line with what can be found in the physical literature, where the σ-models
defining the string theories from which Mirror Symmetry originates are seen as
just ”phases” of a unique theory, which is not necessarily in the form of a σ-model
but could very likely be similar to a quantized Gauge theory on an 11-dimensional
manifold. To make this circle of ideas more concrete (and hence more verifiable) at
the end of [G3] it is suggested that one should try to build a natural gauge theory
on Self-dual manifolds: the hope is that once quantized this gauge theory might in-
terpolate between the σ-models associated to the Calabi-Yau’s, and as a byproduct
prove Mirror Symmetry for them. Of course one can always put a gauge bundle on
the Self-dual manifolds ”artificially”, but a natural bundle which depends only on
the geometric structure would be much more appealing. We ignore here the issue
of which action to put on the theory, but it too should be a natural geometric one.
Finally, on [GG] we analyzed the situation for rank three WSD manifolds, and we
found that in this case the corresponding natural bundle is formed by complex Lie
superalgebras. We were able to find a geometrically motivated real form, and to
split it into simple factors. The results of [GG] confirm the suspicion that on a WSD
manifold of high enough rank there could be enough natural algebraic bundles of
operators to build interesting gauge theories.
The construction of LC. From a physical point of view the case of Calabi-
Yau threefolds (i.e. rank three WSD manifolds) or fourfolds (i.e. rank four WSD
manifolds) would be the most interesting one to start with. However, its technical
difficulty convinced us to start more modestly from the case of Calabi-Yau two-
folds (i.e. K3 surfaces) which correspond to rank two Self-dual manifolds. We also
considered only orientable nondegenerate Self-dual manifolds of rank two, hence of
dimension 6. This could be considered a proof of concept from a physicist’s point
of view, however Mirror Symmetry for K3’s is in itself very interesting mathemati-
cally, so we hope that our results could have some useful geometric consequences.
The rank three case is treated in our subsequent [GG], as mentioned in the previous
section of this introduction. The main result of the present paper is the following
4 GIOVANNI GAIFFI, MICHELE GRASSI
(which is a geometric restatement of Theorem 5.11):
The Lie algebra sl(6,C) acts via canonical operators (depending only on the geo-
metric structure) on the smooth differential forms of any orientable nondegenerate
WSD manifold of rank 2.
This action generalizes naturally the action of sl(2,C) on smooth differential
forms of any almost Kähler manifold, and is induced by a bundle action on the
exterior power of the cotangent bundle.
Recall that a Weakly self-dual manifold is a Riemannian manifold with three
”compatible” closed differential forms. We will build a Lie algebra of pointwise
operators on complex differential forms on X , as smooth sections of a bundle of Lie
algebras of operators on the complexified cotangent bundle of X . To start, one can
define the following operators:
Definition 1.4. For φ ∈ Ω∗
L0(φ) = ωD ∧ φ, L1(φ) = −ω2 ∧ φ, L2(φ) = ω1 ∧ φ
One can notice immediately the strong resemblance of the operators above with
the Lefschetz operator of Kähler geometry. Indeed, one can elaborate on this simi-
larity, and use the metric to define the adjoints Λj = L
j (using a pointwise proce-
dure, as in the almost Kähler case).
Simply using the Lj and the Λj , one can show that the algebra generated is iso-
morphic to SL(4,C) ([G2]). However, there are other natural differential forms on
a WSD manifold (which do not have a counterpart in the Kähler case), namely the
volume forms of the distributions ω⊥1 , ω
2 , ω
D of vectors which contract to zero
with the forms ω1, ω2 and ωD respectively. If one calls V0, V1, V2 the corresponding
wedge operators, and A0, A1, A2 their adjoints, the complexity of the calculations
to describe the generated Lie algebra grows a lot. We called L the algebra generated
by the Lj, Vj and their adjoints, and LC its complexification. To study LC we intro-
duced an operator J , which is a complex structure on each of the two-dimensional
distributions mentioned above and generates a group isomorphic to SO(2,R) (recall
that we are in the ”hyperkahler” case, corresponding to Mirror Symmetry for K3’s,
so an ”extra” complex structure shouldn’t be surprising; moreover the holonomy of
a WSD manifold in which all ω1, ω2, ωD are invariant is actually always included in
the group generated by J). One checks that all the operators introduced commute
with it:
∀j [Lj , J ] = [Λj , J ] = [Vj , J ] = [Aj , J ] = 0
and therefore one can try to decompose Λ∗T ∗
X with respect to J and then use
Shur’s Lemma to reduce to the study of the operators on the isotypical components.
One should mention that in the (very) good cases (for instance 2-Kähler manifolds)
the operators above are all covariant constant with respect to the metric connec-
tion, and define an action on the cohomology of X much in the same way as in the
Kähler setting the operators L and Λ do (due to Hodge-type identities). We don’t
explore this aspect here, although it may be relevant to the (homological) mirror
map construction.
Coming back to the construction, we point out the inclusion of the Lie algebra LC
inside a copy of the Clifford algebra Cl6,6.
Using this Clifford algebra one can identify ”degree two” or ”quadratic” operators
(in a way similar to the ones involved in the Spinor representations on standard
Spin manifolds) and among these the SO(2,R)-invariant ones. A posteriori, it turns
out that the operators of LC⊕ < J > are all the J-invariant operators of ”degree
A GEOMETRIC REALIZATION OF sl(6,C) 5
two”, and this strengthens the rationale in our selection of natural operators.
As a last step one finds that inside Λ∗T ∗X there is an SO(2,R)-isotypical compo-
nent of dimension 6, and by direct computation we prove that indeed the operators
restricted to this sub-representation determine a copy of sl(6,C) (with the defin-
ing representation). Using the bound on the dimension of LC obtained computing
”quadratic” invariants, one then shows that the representation on this isotypical
component is faithful. This provides as a byproduct a method for giving presenta-
tion of standard Serre generators of LC, explicitely written in terms of the natural
geometrical generators.
2. Basic operators
In this section we fix a point p in the WSD manifold X . The WSD structure
splits the cotangent space as T ∗pX =W0⊕W1⊕W2 where theWj are three mutually
orthogonal canonical distributions defined as:
W0 = {φ ∈ T
pX | φ ∧ ω
1 = φ ∧ ω
2 = 0}
W1 = {φ ∈ T
pX | φ ∧ ω
1 = φ ∧ ω
D = 0}
W2 = {φ ∈ T
pX | φ ∧ ω
2 = φ ∧ ω
D = 0}
The WSD structure also determines canonical pairwise linear identifications
among W0,W1 and W2, so that one can also write T
pX = W0 ⊗R R
3 or more
simply
T ∗pX =W ⊗R R
where W =W0 ∼=W1 ∼=W2.
Let us now come back to the canonical operators Lj mentioned in the introduction:
Definition 1.4 For φ ∈ Ω∗
L0(φ) = ωD ∧ φ, L1(φ) = −ω2 ∧ φ, L2(φ) = ω1 ∧ φ
We now choose a (non-canonical) orthonormal basis γ1, γ2 forW0, and this together
with the standard identifications of the Wj determines an orthonormal basis for
T ∗pX , which we write as {vij = γi ⊗ ej | i = 1, 2, j = 0, 1, 2}. We remark that
the vij are an adapted coframe for the WSD structure, and therefore we have the
explicit expressions:
ω1 = v10 ∧ v11 + v20 ∧ v21
ω2 = v10 ∧ v12 + v20 ∧ v22
ωD = v11 ∧ v12 + v21 ∧ v22
A different choice of the γ1, γ2 would be related to the previous one by an element in
O(2,R) or, taking into account the orientability ofX mentioned in the Introduction,
an element of SO(2,R). The Lie algebra of the group SO(2,R) expressing the
change from one oriented adapted basis to another is generated (point by point) by
the global operator J :
Definition 2.1. The operator J ∈ EndR(Ω
∗(X)) is induced by its pointwise action
on the Λ∗T ∗pX for varying p ∈ X, defined in terms of the standard basis vij as
J(v1j) = v2j , J(v2j) = −v1j for j ∈ {0, 1, 2}
and J(v ∧ w) = J(v) ∧ w + v ∧ J(w) for v, w ∈ Λ∗T ∗pX
Remark 2.2. As J commutes with itself, it is well defined, independently of the
choice of an oriented adapted basis.
Using the chosen (orthonormal) basis, one can define corresponding (non canon-
ical) wedge and contraction operators:
6 GIOVANNI GAIFFI, MICHELE GRASSI
Definition 2.3. Let i ∈ {1, 2} and j ∈ {0, 1, 2}. The operators Eij and Iij are
respectively the wedge and the contraction operator with the form vij on
(defined using the given basis); we use the notation ∂
to indicate the element of
TpX dual to vij ∈ T
Eij(φ) = vij ∧ φ, Iij(φ) =
Proposition 2.4. The operators Eij , Iij satisfy the following relations:
∀i, j, k, l EijEkl = −EklEij , IijIkl = −IklIij
∀i, j EijIij + IijEij = Id
∀(i, j) 6= (k, l) EijIkl = −IklEij
∀i, j E∗ij = Iij , I
ij = Eij
where ∗ is adjunction with respect to the metric.
Proof The proof is a simple direct verification, which we omit. �
It is then immediate to verify that:
Proposition 2.5. J can be expressed as
(E2jI1j − E1jI2j)
on the whole
T ∗pX. From this expression and the previous proposition one ob-
tains that J∗ = −J , i.e. for every p the Lie algebra generated by J is a subalgebra
of o(
T ∗pX) isomorphic to so(2,R)
∼= R. Moreover, the exponential images in-
side AutR(Ω
∗(X))of the operators of type tJ for t ∈ R form a group isomorphic to
SO(2,R) ∼= S1, as this isomorphism holds for the (faithful) restriction of the group
action to T ∗pX.
Using the (non canonical) operators Eij we can obtain simple expressions for the
pointwise action of the other canonical operators, the volume forms Vj :
Definition 2.6. For φ ∈
T ∗pX,
V0(φ) = E10E20(φ), V1(φ) = E11E21(φ), V2(φ) = E12E22(φ)
Remember however that the operators Vj do not depend on the choice of a basis,
as they are simply multiplication by the volume forms of the spaces Wj .
We use the vij also as a orthonormal basis for the complexified space T
p ⊗R C
(with respect to the induced hermitian inner product). We indicate with the same
symbols Vj the complexified operators acting on the spaces
T ∗pX .
The riemannian metric induces a Riemannian metric on T ∗pX and on the space
T ∗pX .
Definition 2.7. For j ∈ {0, 1, 2}
Λj = L
j , Aj = V
By construction the canonical operators Lj , Vj ,Λj, Aj on
T ∗pX are the point-
wise restrictions of corresponding global operators on smooth differential forms,
which we indicate with the same symbols: for j ∈ {0, 1, 2},
Lj , Vj ,Λj , Aj : Ω
∗(X) → Ω∗(X)
Summing up:
A GEOMETRIC REALIZATION OF sl(6,C) 7
Definition 2.8. The ∗-Lie algebra L is the ∗-Lie subalgebra of EndR (Ω
∗(X)) gen-
erated by the operators
{Lj, Vj ,Λj, Aj | for j = 0, 1, 2}
The ∗ operator on L is induced by the adjoint with respect to the Riemannian
metric. The ∗-Lie algebra LC is L ⊗ C, and is in a natural way a ∗-Lie subalgebra
of EndC (Ω
(X)). The ∗ operator on LC is induced by the adjoint with respect to
the induced Hermitian metric.
The canonical splitting T ∗pX = W0 ⊕ W1 ⊕ W2 together with the canonical
identifications W0 ∼=W1 ∼=W2 induce an action of the symmetric group S3, which
propagates to
T ∗X and to its C∞ sections. At every point, the action can be
written explicitly in terms of the basis as
σ(vij) = viσ(j)
The induced action on endomorphisms via conjugation, σ(φ) = σ◦φ◦σ−1, preserves
LC. Indeed, one can check directly using the basis vij at every point that for σ ∈ S3
σ(Vj) = Vσ(j), σ(Lj) = ǫ(σ)Lσ(j)
Since S3 acts on LC by conjugation with unitary operators, its action commutes
with adjunction (the ∗ operator), and therefore
σ(Aj) = Aσ(j), σ(Λj) = ǫ(σ)Λσ(j)
Moreover, one also has that σ(J) = J which means that the action of S3 commutes
with that of so(2,R).
3. The action of so(2,R)
When one deals with mirror simmetry for 2-Kähler manifolds (see the Introduc-
tion), the WSD manifolds which arise have the property that the forms ω1, ω2 and
ωD are covariant constant with respect to the metric. In this case, the maximal
possible holonomy of the WSD manifold X is included in the so(2,R) generated
by the operator J . We will show now that J commutes with LC. Our proof will
be strictly algebraic, so that the commutativity between so(2,R) and LC will hold
also on WSD manifolds for which the holonomy is more general.
Definition 3.1. Given n ∈ Z, we indicate with Vn the one dimensional complex
representation of SO(2,R) ∼= S1 ∼= R/Z given by the character:
θ → e2πınθ
Proposition 3.2. Under the SO(2,R) representation induced by the operator J ,
for any p ∈ X :
1) The space
Xp) splits as
V ⊕31
8 GIOVANNI GAIFFI, MICHELE GRASSI
2) The whole space
Xp) splits according to the following picture:
Xp) = V0
Xp) = V
V ⊕31
Xp) = V
V ⊕90
V ⊕32
Xp) = V−3
V ⊕91
Xp) = V
V ⊕90
V ⊕32
Xp) = V
V ⊕31
Xp) = V0
Proof 1) The space T ∗
Xp is a direct sum of the three Wj , and each one of these
is the standard two dimensional real representation of so(2,R). We therefore diag-
onalize the representation introducing a new basis for each Wj =< v1j , v2j >:
wj = v1j + ı v2j , wj = v1j − ıv2j
From the definition of J , one has then for every j ∈ {0, 1, 2}
J(wj) = −ıwj , J(wj) = ıwj
Therefore one has for every j ∈ {0, 1, 2}
< wj >∼= V−1, < wj >∼= V1
2) To prove the general case, we use the fact that the operator J determines an
almost complex structure on the manifold X , compatible with the metric. From
this, following standard arguments, the complex differential forms and also the
elements of
Xy for any y ∈ Y can be divided according to their type:
p+q=n
In the notation adopted in the proof of the first statement, one has
Xy =< wi1 ∧ · · · ∧ wip ∧ wj1 ∧ · · · ∧ wjq | i1, ..., jq ∈ {0, 1, 2} >
From the definition of the action of J one has therefore that for any p, q
Xy ∼= V
with k =
from which the second statement of the proposition can be esily
deduced. �
Theorem 3.3. The operators Lj, Vj for j ∈ {0, 1, 2} commute with the generator
J of so(2,R).
A GEOMETRIC REALIZATION OF sl(6,C) 9
Proof We prove the statements by a direct computation using the basis vij ;
moreover, using the action of S3 (which permutes the Lj, Vj and fixes J), it is
enough to prove the commutativity for L0 and V0. It useful to rewrite ω0 (and
hence L0 which is wedge with ω0) in terms of the basis generated by the wj :
ω0 = v11 ∧ v12 + v21 ∧ v22 =
(w1 ∧ w2 − w2 ∧ w1)
and then:
[J, L0](wi1 ∧ · · · ∧ wip ∧ wj1 ∧ · · · ∧ wjq ) =
(w1 ∧ w2 − w2 ∧ w1)
wi1 ∧ · · · ∧wip ∧ wj1 ∧ · · · ∧ wjq
(w1 ∧ w2 − w2 ∧ w1)
wi1 ∧ · · · ∧wip ∧ wj1 ∧ · · · ∧ wjq
(w1 ∧ w2 − w2 ∧w1) ∧ J(wi1 ∧ · · · ∧ wip ∧ wj1 ∧ · · · ∧ wjq )
Therefore the result follows from the fact that
(w1 ∧w2 − w2 ∧ w1) = 0
as wj and wk have opposite weight with respect to J for any j, k.
Similarly, [J, V0] = 0 follows from the fact that for any α
V0(α) = v10 ∧ v20 ∧ α =
w0 ∧ w0 ∧ α
From the previous theorem one obtains the following corollary, which holds on any
WSD manifold (not necessarily 2-Kähler ):
Corollary 3.4. The algebra LC commutes with the action of so(2,R) induced by
Proof We already know that [J, Lj] = [J, Vj ] = 0 for j ∈ {0, 1, 2}. The corre-
sponding commutation relations for the adjoint generators Λj , Aj of LC follow from
the fact that J∗ = −J , as noticed in Proposition 2.5. �
Remark 3.5. From Schurs’s lemma it follows that the columns of the diagram of
Proposition 3.2 are preserved by the action of LC.
4. An irreducible representation of LC
Looking at the table in Proposition 3.2 we notice that the second column from
the left is a representation of LC (by Remark 3.5) of dimension 6:
V ∼= V
−2 =< w0 ∧ w1, w0 ∧ w2, w1 ∧w2, w0 ∧ w1 ∧w2 ∧w0,
w0 ∧w1 ∧w2 ∧ w1, w0 ∧ w1 ∧ w2 ∧ w2 >
In this section we will compute explicitely this representation.
Using the above described basis, it is not difficult to compute the matrices by
hand:
Proposition 4.1. Indicating with β the ordered basis for V indicated above, the
matrices for the (restrictions to V of) the generators of LC are the following:
Mβ(L0) =
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
0 0 0 0
, Mβ(Λ0) =
0 0 0 0 −2 0
0 0 0 0 0 −2
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
10 GIOVANNI GAIFFI, MICHELE GRASSI
Mβ(L1) =
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0
0 0 −
0 0 0
, Mβ(Λ1) =
0 0 0 2 0 0
0 0 0 0 0 0
0 0 0 0 0 −2
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
Mβ(L2) =
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0
0 0 0
0 0 0 0 0 0
, Mβ(Λ2) =
0 0 0 0 0 0
0 0 0 2 0 0
0 0 0 0 2 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
Mβ(V0) =
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
, Mβ(A0) =
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 −2ı 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
Mβ(V1) =
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0
0 0 0 0 0 0
, Mβ(A1) =
0 0 0 0 0 0
0 0 0 0 2ı 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
Mβ(V2) =
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
, Mβ(A2) =
0 0 0 0 0 −2ı
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
Proof Direct computation using the basis generated by the wj . �
Corollary 4.2. The algebra generated by the restriction of LC to V is isomorphic
to sl(6,C), with V its natural representation.
One can sum up the computations above in the following theorem:
Theorem 4.3. There is an exact sequence of Lie algebras
0 → K → LC → sl(6,C) → 0
given by the restriction to V .
In the next section we will prove that K = {0}, and therefore the representation
V is faithful and LC ∼= sl(6,C).
5. Quadratic invariants
We begin by showing that the action of Lie algebra LC is induced by a (non-
canonical) Clifford algebra representation. We use for simplicity the canonical
identification T ∗∗Xp ∼= TXp without further comment, so that if {vij} is a basis
for T ∗pX , then {
} is the corresponding dual basis for TpX .
Definition 5.1. For p ∈ X, the Clifford algebra Cp is
Cp = Cl(TpX ⊕ T
pX, q)
A GEOMETRIC REALIZATION OF sl(6,C) 11
with the quadratic form q induced by the metric
∀i, j, h, k < vij , vhk >= 0
∀i, j, h, k < ∂
∀(i, j) 6= (h, k) < vij ,
∀i, j < vij ,
>= − 1
Remark 5.2. The Clifford algebras Cp for varying p define a Clifford bundle C
on X, as the definition of Cp is independent on the choice of a basis. Indeed, the
quadratic form used to define it is simply induced by − 1
times the natural bilinear
pairing TpX ⊗ T
pX → R.
Proposition 5.3. The Clifford algebra Cp has a canonical representation ρp on
T ∗pX, induced by the operators Eij and Iij via the map
ρp(vij) = Eij , ρp
= Iij
Proof The Clifford relations
φψ + ψφ = −2 < φ,ψ >
are precisely the content of Proposition 2.4. The representation is canonical, even if
the operators Eij and Iij are not, because it can be defined in a basis independent
way as
ρp(v)(α) = v ∧ α, ρp
Abusing slightly the notation, we will identify Cp with its (faithful) image inside
T ∗pX
, and we will omit any reference to the map ρp. Actually, as the
representation above is a real analogue of the Spinor representation, it is easy to
check that the map ρp is an isomorphism of associative algebras. One then has:
Definition 5.4. The linear subspace C2p of Cp is the image of the natural map
(TpX ⊕ T
pX) → Cp. The linear subspace C
p of Cp is the subspace generated by
Recall that C2p is a Lie subalgebra of Cp (with the commutator bracket).
Proposition 5.5. The Lie algebra Lp and the operator J sit inside C
p for all
p ∈ X.
Proof The operators Lj , the Λj, the Vj and the Aj lie inside C
p by Propo-
sition 2.4 and the fact that ω1, ω,ωD lie in
T ∗pX . The operator J lies inside
C2p ⊕ C
p by Proposition 2.5. By definition the elements C
p are commutators, and
therefore have trace zero in any representation, and hence also in the ρp. Moreover,
again by inspection all the generators of Lp have trace zero once represented via
ρp (they are nilpotent), and therefore they must lie inside C
p. The operator J is
in the Lie algebra of the isometry group, and therefore it too has trace zero and
hence sits inside C2p . As C
p is closed under the commutator bracket of Cp, and
this commutator coincides with the composition bracket of operators, we have the
conclusion. �
Remark 5.6. Giving degree 1 to the operators Eij and degree −1 to the opera-
tors Iij , we induce a Z-degree on Cp. This degree coincides with the degree of the
operators induced from the grading on the forms from
T ∗X.
12 GIOVANNI GAIFFI, MICHELE GRASSI
Remark 5.7. For any p ∈ X, the Clifford algebra Cp is isomorphic to Cl6,6, as
the metric used to define it has signature (6, 6). The previous proposition therefore
shows that Lp is a Lie subalgebra of Cl
∼= spin(6, 6) = so(6, 6), generated by
smooth global sections of the Clifford bundle C.
The operator J acts on all of Cp by adjunction with respect to the commutator
bracket, and sends its quadratic part C2p to itself from Proposition 5.5.
We will show that the space of J-invariants inside C2p (the “quadratic” J-invariants)
coincides with LC. To describe it explicitely, let us introduce the following notation:
Definition 5.8.
Ewj = E1j + ıE2j , Ewj = E1j − ıE2j
Iwj = I1j − ıI2j , Iwj = I1j + ıI2j
Lemma 5.9. The adjoint action of the operator J on Ewj , Iwj , Ewj , Iwj is:
[J,Ewj ] = −ıEwj , [J, Iwj ] = ıIwj
[J,Ewj ] = ıEwj , [J, Iwj ] = −ıIwj
Proof It is enough to consider the corresponding J-weights of the wj , wj . �
Proposition 5.10. The following 36 operators provide a linear basis for the qua-
dratic J-invariants:
(1) [Ew0 , Ew1 ], [Ew0 , Ew2 ], [Ew1 , Ew2 ], [Ew1 , Ew0 ], [Ew2 , Ew0 ], [Ew2 , Ew1 ]
(2) [Iw0 , Iw1 ], [Iw0 , Iw2 ], [Iw1 , Iw2 ], [Iw1 , Iw0 ], [Iw2 , Iw0 ], [Iw2 , Iw1 ]
(3) [Ew0 , Ew0 ], [Ew1 , Ew1 ], [Ew2 , Ew2 ]
(4) [Iw0 , Iw0 ], [Iw1 , Iw1 ], [Iw2 , Iw2 ]
(5) [Ew0 , Iw1 ], [Ew0 , Iw2 ], [Ew1 , Iw0 ], [Ew1 , Iw2 ], [Ew2 , Iw0 ], [Ew2 , Iw1 ]
(6) [Ew0 , Iw1 ], [Ew0 , Iw2 ], [Ew1 , Iw0 ], [Ew1 , Iw2 ], [Ew2 , Iw0 ], [Ew2 , Iw1 ]
(7) [Ew0 , Iw0 ], [Ew1 , Iw1 ], [Ew2 , Iw2 ], [Ew0 , Iw0 ], [Ew1 , Iw1 ], [Ew2 , Iw2 ]
Proof The J-weight of a bracket of J-homogeneous operators is the sum of
the respective weights. The quadratic ”monomials” (with respect to the bracket)
in the Ewj , Iwj , Ewj , Iwj are all J-homogeneous, and therefore to find a basis of
J-invariant quadratic operators it is enough to identify the J-invariant quadratic
monomials. To be J-invariant means simply to have weight zero, and the compu-
tation of the J-weight of the quadratic mononials follows immediately from those
of Ewj , Iwj , Ewj , Iwj , which are respectively −ı, ı, ı,−ı. �
We end this section with the following:
Theorem 5.11. In the exact sequence of Theorem 4.3 the kernel K is equal to {0}.
The algebra LC is therefore isomorphic to sl(6,C).
Proof Since LC is included in the Lie algebra of quadratic invariants, it is
enough to show that J 6∈ LC, as from this and the previous proposition it follows
that dimC(LC) ≤ 35. As LC maps surjectively to sl(6,C) which has dimension 35,
the kernel must be zero. When restricted to the subrepresentation V , the generators
of LC have all trace zero by inspection of their matrices. However, by definition
of V , J restricted to it is multiplication by −2ı, and has therefore trace equal to
−12ı. �
Corollary 5.12. The Lie algebra LC⊕ < J > equals the Lie algebra of quadratic
invariants inside C2p .
A GEOMETRIC REALIZATION OF sl(6,C) 13
6. A geometric presentation of Serre generators
In this section, to gain a better geometric understanding of the representation
LC of sl(6,C), we explore in greater detail its relation to the geometric structure of
a WSD manifold. In particular, we give a presentation of a natural choice of Cartan
subalgebra and Serre generators in terms on the geometric generators Lj ,Λj, Vj , Aj .
The Lj operators are similar in nature to the Lefschetz operators of a Kähler
manifold. This analogy is what provided the initial interest in the algebraic struc-
ture of LC. Similarly to the corresponding standard construction of a representation
of sl(2,C), we define
Definition 6.1. For j ∈ {0, 1, 2}
Hj = [Lj ,Λj]
These operators are self-adjoint, as L∗j = Λj by definition. As in the context of
Kählerian geometry, for every j the algebra < Lj ,Λj, Hj > turns out to be a copy
of sl(2,C). Moreover, the following proposition shows that the operators Hj are
semisimple on the whole algebra LC, and therefore generate a toral subalgebra of
Proposition 6.2. The geometric operators Hj generate a toral subalgebra of LC,
and the following relations hold: for j 6= k ∈ {0, 1, 2}
(1) [Hj , Lj] = 2Lj, [Hj ,Λj] = −2Λj
(2) [Hj , Lk] = Lk, [Hj ,Λk] = −Λk
(3) [Hj , Vj ] = 0, [Hj , Aj ] = 0
(4) [Hj , Vk] = 2Vk, [Hj , Ak] = −2Ak
Proof In view of Theorem 5.11, at this point the quickest method of proof of
this proposition is to refer to the explicit matrices of the (faithful) restriction of LC
to V . �
The whole algebra LC splits into a direct sum of weight spaces with respect to
< H0, H1, H2 >, as this subalgebra is toral. The weight of L0 with respect to the
basis dual to H0, H1, H2 is:
αL0 = (αL0(H0), αL0(H1), αL0(H2)) = (2, 1, 1)
The full list is:
αL0 = (2, 1, 1), αΛ0 = −αL0
αL1 = (1, 2, 1), αΛ1 = −αL1
αL2 = (1, 1, 2), αΛ2 = −αL2
αV0 = (0, 2, 2), αA0 = −αV0
αV1 = (2, 0, 2), αA1 = −αV1
αV2 = (2, 2, 0), αA2 = −αV2
To find a natural geometric expression for two ad-semisimple elements which com-
plete < H0, H1, H2 > to a Cartan subalgebra we look at the generators Vj and Aj .
However, it turns out that the natural candidates [Vj , Aj ] already lie in the algebra
< H0, H1, H2 >. We instead build the new operators by ”subtracting” from the Vj
their weight αVj :
Definition 6.3. We define
S0 = ı[[[V0,Λ1],Λ2], L0]
S1 = ı[[[V1,Λ2],Λ0], L1]
S2 = ı[[[V2,Λ0],Λ1], L2]
14 GIOVANNI GAIFFI, MICHELE GRASSI
and denote by H the Lie algebra (over C):
H =< H0, H1, H2, S0, S1, S2 >
The coefficients ı which appear in the formulas above are dictated by the fact
that with this choice the (diagonal) matrices of the Sj restricted to V have integer
entries.
Proposition 6.4. The algebra H is a Cartan subalgebra of LC. More precisely,
the following are the diagonals of the operators H0, ..., S2 once restricted to V
, H1 :
, H2 :
, S0 :
, S1 :
, S2 :
Proof The computation of the matrices above shows that, once restricted to
V , the algebra H spans the space of diagonal matrices of trace zero in the given
basis. �
Remark 6.5. The computation above shows also that operators S0, S1, S2 safisfy
the relation
S0 + S1 + S2 = 0
Even if from the previous proposition we know that H is maximal toral inside LC,
the natural geometric generators Lj ,Λj are not eigenvectors for the adjoint action
of the Sk. At this point however it is possible to single out in natural geometric
terms operators of LC which have ”pure” weight with respect to the algebra H and
which contain in their linear span the Lj,Λj :
Definition 6.6. For j ∈ {0, 1, 2}
L1j = −2Lj + [Sj , Lj], L2j = 2Lj + [Sj , Lj]
Λ1j = −2Λj − [Sj ,Λj], Λ2j = 2Λj − [Sj ,Λj]
Proposition 6.7. Indicating with ehk the 6× 6 matrix with a 1 in position k (row)
and h (column) and zero otherwise, the matrices of the operators Lij and Λij re-
stricted on V are:
L10 = 2e
6 L11 = −2e
4 L12 = −2e
L20 = −2e
5 L21 = −2e
6 L22 = 2e
Λ10 = 8e
2 Λ11 = −8e
1 Λ12 = −8e
Λ20 = −8e
1 Λ21 = −8e
3 Λ22 = 8e
Corollary 6.8. We have the following relations for the operators of LC restricted
to V :
[Hk, Lij ] = (1 + δkj)Lij , [Hk,Λij ] = −(1 + δkj)Λij
[Sk, Lij ] = (−1)
i+1(1− 3δkj)Lij , [Sk,Λij ] = (−1)
i(1− 3δkj)Λij
[Sk, Vj ] = 0, [Sk, Aj ] = 0
Guided by all the explicit computations of the action on the isotypical component
V = V ⊕6
−2 made up to this point, we now define in terms of the natural geometric
operators a set of Serre generators for the algebra LC.
A GEOMETRIC REALIZATION OF sl(6,C) 15
Definition 6.9.
[L20, A1] f1 =
[V1,Λ20]
[L22, A0] f2 =
[V0,Λ22]
e3 = V0 f3 = A0
[L12, A0] f4 =
[V0,Λ12]
[L10, A1] f5 =
[V1,Λ10]
Moreover, for all i ∈ {1, .., 5} we define hi = [ei, fi].
As the ei have by construction associated matrix e
i+1 once restricted to V and
the fi are their respective adjoints, one gets:
Proposition 6.10. The operators ei, fj ,hk satisfy the Serre relations for sl(6,C)
and the hi span the Cartan subalgebra H:
(H1 −H2 − S1 − S2)
(H0 −H1 + S2)
(−H0 +H1 +H2)
(H0 −H1 − S2)
(H1 −H2 + S1 + S2)
It would be interesting as a last remark to identify in the list of quadratic in-
variants the geometric operators Lij ,Λij , Vj , Aj , the algebra H and the so(2,R)
generator J . To do this one could of course use the explicit matrices for the qua-
dratic invariants once restricted to V , which are not difficult to compute. One can
however get very quickly a qualitative picture by using the notion of multidegree
which we now introduce.
The decomposition T ∗X = W0 ⊕ W1 ⊕ W2 induces naturally a multi-degree on
X with values in Z3, which we indicate with mdeg. This follows from the
equation
p+q+r=n
(W0 ⊗ C)⊕
(W1 ⊗ C)⊕
(W2 ⊗ C)
We notice furthermore that the (complexified) decomposition above is preserved by
the operator J , and therefore mdeg commutes with the action of so(2,R).
Proposition 6.11. The operators Lj, Vj ,Λj , Aj , Hj , Sj are mdeg-homogeneous,
with multi-degrees:
mdeg(L0) = (0, 1, 1) mdeg(L1) = (1, 0, 1) mdeg(L2) = (1, 1, 0)
mdeg(Λ0) = (0,−1,−1) mdeg(Λ1) = (−1, 0,−1) mdeg(Λ2) = (−1,−1, 0)
mdeg(V0) = (2, 0, 0) mdeg(V1) = (0, 2, 0) mdeg(V2) = (0, 0, 2)
mdeg(A0) = (−2, 0, 0) mdeg(A1) = (0,−2, 0) mdeg(A2) = (0, 0,−2)
mdeg(H0) = (0, 0, 0) mdeg(H1) = (0, 0, 0) mdeg(H2) = (0, 0, 0)
mdeg(S0) = (0, 0, 0) mdeg(S1) = (0, 0, 0) mdeg(S2) = (0, 0, 0)
Proof The values for mdeg for the Lj and the Vj follow immediately from mdeg
of the corresponding forms and the dual (contraction) operators have opposite value
16 GIOVANNI GAIFFI, MICHELE GRASSI
of mdeg. The remaing values can be computed using the additivity of mdeg with
respect to the bracket. �
Proposition 6.12. Let {j, k, l} = {0, 1, 2}. Then
Span (L1j, L2j) = Span ([Ewk , Ewl ], [Ewl , Ewk ])
Span (Λ1j ,Λ2j) = Span ([Iwk , Iwl ], [Iwl , Iwk ])
Span (Vj) = Span
[Ewj , Ewj ]
Span (Aj) = Span
[Iwj , Iwj ]
H⊕ Span (J) =
Span ([Ewm , Iwm ], [Ewm , Iwm ])
Proof The mdeg of the Lij is the same of the corresponding Lj, and sim-
ilarly for their adjoints. The mdegs of the quadratic monomials are immedi-
ately computed as they are the sum of those of their components. For example,
mdeg(Ew0) = mdeg(Ew0) = (1, 0, 0) , mdeg(Ew1) = mdeg(Ew1) = (0, 1, 0) and
therefore mdeg([Ew0 , Ew1 ] = (1, 1, 0), equal to that of L12 and L22. �
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http://arxiv.org/abs/math/0006154
http://arxiv.org/abs/math/0202016
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http://arxiv.org/abs/math/0011041
http://arxiv.org/abs/hep-th/9606040
	1. Introduction
	2. Basic operators
	3. The action of so(2,R)
	4. An irreducible representation of LC
	5. Quadratic invariants
	6. A geometric presentation of Serre generators
	References
