A Geometric Realization of <span class="mathjax-formula">$sl(6,\mathbb {C})$</span>

A Geometric Realization of sl(6; C)

GIOVANNIGAIFFI(*) - MICHELEGRASSI(*)

ABSTRACT- Given an orientable weakly self-dual manifoldXof rank two, we build a geometric realization of the Lie algebrasl(6;C) as a naturally defined algebra LC of endomorphisms of the space of differential forms of X. We provide an explicit description of Serre generators in terms of natural generators ofLC. This construction gives a bundle onXwhich is related to the search for a natural Gauge theory onX. 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 geometric counterpart to the Mirror Symmetry phenomenon, and possibly a geometric language in which to formulate a physical theory interpolating between differents-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 one has that two mirror dual Calabi-Yaus should posses (in some limiting sense) semi-flat special lagrangian torus fibrations f :M!B,^f : ^M!Bwhich 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 hyperkahler, and the special lagrangian tori can be viewed as complex submanifolds (with re- spect to a rotated complex structure), so that the methods of complex al- gebraic geometry can be put to work.

(*) Indirizzo degli A.: Dipartimento di Matematica ``L. Tonelli'', Largo Ponte- corvo, 5 - 56127 Pisa.

E-mail: [email protected] [email protected]

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 spaceXˆM_BM. In the hyperkaÈhler case^ this space is a complex manifold, while in the general case (for example for Mirror Symmetry for Calabi-Yau threefolds) it is just a real manifold of (real) dimension 3dimC(M).

BACKGROUND. The notion of (Weakly) self-dual manifold (cf. [G2]) was conceived in the first place to isolate the geometric aspects of theXabove which are needed to obtain Mirror Symmetry betweenMandM. We re-^ produce here the definition for the reader, while referring to [G2] and [G3]

for all the remarks, examples and observations:

DEFINITION1.1. Aweakly self-dual manifold(WSD manifold for brev- ity) is given by a smooth manifold X, together with two smooth 2-forms v1;v2;a Riemannian metric and a third smooth2-formvD(the dualizing form) on it, which satisfy the following conditions:

1) dv₁ˆdv₂ˆdv_Dˆ0and the distributionv⁰₁‡v⁰₂ is integrable.

2) For all p2X there exists an orthogonal basis dx₁;. . .;dx_m, dy¹₁;. . .;dy¹_m;dy²₁;. . .;dy²_m, dz₁;. . .;dz_c;dw₁;. . .;dw_cof T_pX such that the dx1;. . .;dxm;dy¹₁;. . .;dy¹_m;dy²₁;. . .;dy²_m are orthonormal and

(v₁)_pˆX^m

iˆ1

dx_i^dy¹_i; (v₂)_pˆX^m

iˆ1

dx_i^dy²_i;

(v_D)_pˆX^m

iˆ1

dy¹_i^dy²_i‡X^c

iˆ1

dz_i^dw_i:

Any orthogonal basis of T_pX dual to a basis of1-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 formsv₁;v₂;v_D are covariant constant with respect to the Levi-Civita connection, we speak of 2-KaÈhler manifolds. An example of these comes from mirror symmetry for abelian varieties.

REMARK1.2. The formvDis symplectic once restricted tov⁰₁‡v⁰₂. We have therefore thatv^dim(X)_D ^m6ˆ0.

DEFINITION 1.3. 1) A WSD manifold is nondegenerate if dim(v⁰₁\v⁰₂)_pˆ0 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 v⁰₁‡v⁰₂ have volume one (with respect to the volume form induced by the metric)

Using Self dual manifolds, you can give a first naõÈve geometric defini- tion of Mirror Symmetry as follows:

Two Calabi-Yaumanifolds with B-field (M;B_M) and (M;^ BM^) are mirror dual if there is a Self-dual manifold X together with surjections p:X!M andp^:X!M such that:^

a) p(v_M)ˆv1,^p(vM^)ˆv2.

b) The leaves ofv^?₁ are the fibres of^p.

c) The leaves ofv^?₂ are the fibres ofp.

d) The induced B-fields on M andM are the ones given.^

Here make their first appearence the B-fieldsB_M andB_M^, which are flat unitary gerbes onMandM^ 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.

In the paper [G3] there is a toric construction of a two-dimensional family X^m_k1;k2 of WSD manifolds of rankmand (real) dimension 3m‡2 (see Defi- nition 3.11 on page 11 of that paper). The construction is inspired by Delzant's method of constructing toric projective manifolds (see [Gu]). As the real parametersk1;k2vary, these WSD manifolds interpolate between physically significant asymptotic limits, as described in the following. To these mani- folds one can apply the constructions of the present paper whenmˆ2. In this case the resulting degenerate WSD manifolds have dimension 6‡2.

PHYSICAL MOTIVATION. One of the reasons to introduce SD manifolds however was to get rid of special lagrangian fibrations, which are so dif- ficult to construct, and to be able to attack the problem of Mirror Sym- metry also when these fibrations 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 Ca- labi-Yaus, but via a more sophisticated procedure, which involves a Gro- mov-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 nor-

malized Gromov-Hausdorff sense to the correct limits of the mirror dual Calabi-Yaus. The picture, taken from [G3], is the following:

whereM_AandM_Bare the large KaÈhler and large complex structure limits ofMandM^ respectively. To be precise, the manifolds which come out of the costruction of [G3] are (degenerate) Weakly self-dual manifolds or rankm and dimension 3m‡2 form1.

The point of view of [G3] is very different from the current one in the main literature on mathematical Mirror Symmetry: instead of considering the fibre productMBM^ (when it exists) as a device for proving Mirror Sym- metry 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 thes-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 as-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 interpolate between thes-models associated to the Calabi-Yau's, and as a byproduct prove Mirror Symmetry for them. In the present paper we per- form a first step in this direction, constructing natural bundles on rank two WSD manifolds. In view of [G3], this is potentially relevant to Mirror Sym- metry for quartic hypersurfaces inP<sup>3</sup>, i.e. K3 surfaces. 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 is 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 mani-

folds, and we found that in this case the corresponding natural bundle is formed by complex Lie superalgebras. We were able to find a geome- trically 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 in- teresting gauge theories.

THE CONSTRUCTION OFLC. 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 Weakly Self-dual manifolds. We also considered only orientable nondegenerate Weakly Self-dual manifolds of rank two (hence of dimen- sion 6): one can immediately verify that the relations among the resulting generators of the algebraL_C remain unchanged with respect to the de- generate case.

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 mathematically, so we hope that our results could have some useful geo- metric 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 (which is a geometric restate- ment of Theorem 5.11):

The Lie algebrasl(6;C)acts via canonical operators (depending only on the geometric structure) on the smooth differential forms of any ori- entable WSD manifold of rank2.

This action generalizes naturally the action of sl(2;C) on smooth dif- ferential forms of any almost KaÈhler manifold, and is induced by a bundle action on the exterior power of the cotangent bundle.

Recall that a WSD manifold is a Riemannian manifold with three

``compatible'' closed differential forms. We will build a Lie algebra of pointwise operators on complex differential forms onX, as smooth sections of a bundle of Lie algebras of operators on the complexified cotangent bundle ofX. To start, one can define the following operators:

DEFINITION1.4. Forf2V_CX,

L₀(f)ˆv_D^f; L₁(f)ˆ v₂^f; L₂(f)ˆv₁^f:

One can notice immediately the strong resemblance of the operators above with the Lefschetz operator of KaÈhler geometry. Indeed, one can elaborate on this similarity, and use the metric to define the adjoints L_jˆL_j (using a pointwise procedure, as in the almost KaÈhler case).

Simply using theL_jand theL_j, one can show that the algebra generated is isomorphic toSL(4;C) ([G2]). However, there are other natural differ- ential forms on a WSD manifold (which do not have a counterpart in the KaÈhler case), namely the volume forms of the distributionsv^?₁,v^?₂,v^?_D of vectors which contract to zero with the formsv₁;v₂andv_Drespectively. 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 calledLthe algebra generated by theLj;Vj

and their adjoints, andLCits complexification. To studyLCwe introduced an operatorJ, 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 should not be surprising; moreover the holonomy of a WSD manifold in which all v₁;v₂;v_Dare invariant is actually always included in the group generated byJ). One checks that all the operators introduced commute with it:

8j[L_j;J]ˆ[L_j;J]ˆ[V_j;J]ˆ[A_j;J]ˆ0

and therefore one can try to decomposeLT_CXwith respect toJand then use Schur'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-KaÈhler manifolds) the operators above are all covariant constant with respect to the metric connection, and define an action on the cohomology of Xmuch in the same way as in the KaÈhler setting the operatorsLandLdo (due to Hodge-type identities). We do not 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 algebraL_Cinside a copy of the Clifford algebraCl_6;6.

Using this Clifford algebra one can identify ``degree two'' or ``quadratic'' operators (in a way similar to the ones involved in the Spinor re- presentations on standard Spin manifolds) and among these theSO(2;R)- invariant ones. A posteriori, it turns out that the operators ofL_C hJiare all the J-invariant operators of ``degree two'', and this strengthens the rationale in our selection of natural operators.

As a last step one finds that insideLTXthere is anSO(2;R)-isotypical component 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 defining representation). Using the bound on the di- mension ofL_Cobtained computing ``quadratic'' invariants, one then shows that the representation on this isotypical component is faithful. This pro- vides as a byproduct a method for giving presentation 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_pXˆW0W1W2 where the W_j are three mutually orthogonal canonical distributions defined as:

W0ˆ ff2T_pX jf^v²₁ ˆf^v²₂ˆ0g;

W₁ˆ ff2T_pXjf^v²₁ˆf^v²_Dˆ0g;

W2ˆ ff2T_pXjf^v²₂ˆf^v²_Dˆ0g:

The WSD structure also determines canonical pairwise linear identifi- cations amongW0;W1andW2, so that one can also writeT_pXˆW0RR³ or more simply

T_pXˆW_RR³ whereW ˆW₀W₁W₂.

Let us now come back to the canonical operatorsL_j mentioned in the introduction:

DEFINITION1.4 Forf2V_CX,

L₀(f)ˆv_D^f; L₁(f)ˆ v₂^f; L₂(f)ˆv₁^f:

We now choose a (non-canonical) orthonormal basis g₁;g₂ for W₀, and this together with the standard identifications of the Wj

determines an orthonormal basis for T_pX, whi ch we wri te as fv_ijˆg_ie_j jiˆ1;2; jˆ0;1;2g. We remark that the v_ij are an adapted coframe for the WSD structure, and therefore we have the explicit expressions:

v1ˆv10^v11‡v20^v21; v2ˆv10^v12‡v20^v22; v_Dˆv11^v12‡v21^v22:

A different choice of theg₁;g₂would be related to the previous one by an element inO(2;R) or, taking into account the orientability ofXmentioned in the Introduction, an element ofSO(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 operatorJ:

DEFINITION 2.1. The operator J2EndR(V(X)) is induced by its pointwise action on theLT_pX for varying p2X, defined in terms of the standard basis v_ijas

J(v_1j)ˆv_2j; J(v_2j)ˆ v_1j for j2 f0;1;2g and J(v^w)ˆJ(v)^w‡v^J(w)for v;w2LT_pX.

REMARK 2.2. As J commutes with itself, it is well defined, in- dependently of the choice of an oriented adapted basis.

Using the chosen (orthonormal) basis, one can define corresponding (non canonical) wedge and contraction operators:

DEFINITION 2.3. Let i2 f1;2g and j2 f0;1;2g. The operators E_ij and I_ij are respectively the wedge and the contraction operator with the form v_ij on V

TX (defined using the given basis); we use the notation

@

@v_ij to indicate the element of T_pX dual to v_ij2T_pX:

E_ij(f)ˆv_ij^f; I_ij(f)ˆ @

@v_ij*f:

PROPOSITION2.4. The operators E_ij;I_ijsatisfy the following relations:

8i;j;k;l E_ijE_klˆ E_klE_ij; I_ijI_klˆ I_klI_ij; 8i;j E_ijI_ij‡I_ijE_ijˆId;

8(i;j)6ˆ(k;l) EijIklˆ IklEij; 8i;j E_ijˆI_ij; I_ijˆE_ij whereis adjunction with respect to the metric.

PROOF. The proof is a simple direct verification, which we omit. p It is then immediate to verify that:

PROPOSITION2.5. J can be expressed as JˆX²

jˆ0

E2jI1j E1jI2j on the wholeV

T_pX. From this expression and the previous proposition one obtains that Jˆ J, i.e. for every p the Lie algebra generated by J is a subalgebra of o(V

T_pX) isomorphic to so(2;R)R. Moreover, the ex- ponential images inside Aut_R(V(X))of the operators of type tJ for t2R form a group isomorphic toSO(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 expres- sions for the pointwise action of the other canonical operators, the volume formsV_j:

DEFINITION2.6. Forf2V T_pX,

V₀(f)ˆE₁₀E₂₀(f); V₁(f)ˆE₁₁E₂₁(f); V₂(f)ˆE₁₂E₂₂(f):

Remember however that the operatorsV_jdo not depend on the choice of a basis, as they are simply multiplication by the volume forms of the spacesW_j. We use thev_ijalso as a orthonormal basis for the complexified space T_pRC (with respect to the induced hermitian inner product). We in- dicate with the same symbolsVj the complexified operators acting on the spacesV

CT_pX.

The riemannian metric induces a Riemannian metric onT_pXand on the spaceV

T_pX.

DEFINITION2.7. Forj2 f0;1;2g

LjˆL_j; AjˆV_j:

By construction the canonical operatorsLj;Vj;Lj;AjonV

T_pXare the pointwise restrictions of corresponding global operators on smooth dif- ferential forms, which we indicate with the same symbols: forj2 f0;1;2g,

L_j;V_j;L_j;A_j:V(X)!V(X):

Summing up:

DEFINITION 2.8. The -Lie algebra L is the -Lie subalgebra of End_R…V(X)†generated by the operators

fL_j;V_j;L_j;A_j jfor jˆ0;1;2g:

The operator on L is induced by the adjoint with respect to the Rie- mannian metric. The -Lie algebra LC is L C, and is in a natural way a -Lie subalgebra of End_C V_C(X)

. The operator on L_C is in- duced by the adjoint with respect to the induced Hermitian metric.

The canonical splittingT_pXˆW₀W₁W₂together with the cano- nical identifications W0W1W2 induce an action of the symmetric group S3, which propagates to V

TX and to its C¹ sections. At every point, the action can be written explicitly in terms of the basis as

s(v_ij)ˆv_is(j):

The induced action on endomorphisms via conjugation,s(f)ˆsfs ¹, preservesLC. Indeed, one can check directly using the basisvij at every point that fors2 S3

s(V_j)ˆV_s(j); s(L_j)ˆe(s)L_s(j):

Since S₃ acts on L_C by conjugation with unitary operators, its action commutes with adjunction (theoperator), and therefore

s(Aj)ˆAs(j); s(Lj)ˆe(s)Ls(j):

Moreover, one also has that s(J)ˆJ which means that the action ofS3

commutes with that ofso(2;R).

3. The action ofso(2,R).

When one deals with mirror simmetry for 2-KaÈhler manifolds (see the Introduction), the WSD manifolds which arise have the property that the formsv1;v2andvDare covariant constant with respect to the metric. In this case, the maximal possible holonomy of the WSD manifold X is in- cluded in theso(2;R) generated by the operatorJ. We will show now thatJ commutes with L_C. Our proof will be strictly algebraic, so that the com- mutativity betweenso(2;R) andL_C will hold also on WSD manifolds for which the holonomy is more general.

DEFINITION3.1. Given n2Z, we indicate with Vnthe one dimensional complex representation ofSO(2;R)S¹ R=Zgiven by the character:

u!e^2p{nu:

PROPOSITION3.2. Under the SO(2;R)representation induced by the operator J, for any p2X:

1)The spaceV₁

(T_CXp)splits as V³₁M

V₁³: 2)The whole spaceV

(T_CX_p)splits according to the following picture:

V₀

(T_CXp) ˆ V0

V₁

(T_CXp) ˆ V³₁ L

V₁³ V₂

(T_CXp) ˆ V³₂ L

V₀⁹ L V₂³ V₃

(T_CXp) ˆ V 3 L

V⁹₁ L

V₁⁹ L V3

V₄

(T_CXp) ˆ V³₂ L

V₀⁹ L V₂³ V₅

(T_CXp) ˆ V³₁ L

V₁³ V₆

(T_CXp) ˆ V0

PROOF. 1) The spaceT_CX_pis a direct sum of the threeW_j, and each one of these is the standard two dimensional real representation ofso(2;R). We therefore diagonalize the representation introducing a new basis for each W_j ˆ hv_1j;v_2ji:

w_j ˆv_1j‡{v_2j; w_j ˆv_1j {v_2j: From the definition ofJ, one has then for everyj2 f0;1;2g

J(wj)ˆ {wj; J(wj)ˆ{wj: Therefore one has for everyj2 f0;1;2g

hw_ji V ₁; hw_ji V₁:

2) To prove the general case, we use the fact that the operatorJdetermines an almost complex structure on the manifoldX, compatible with the metric.

From this, following standard arguments, the complex differential forms and also the elements ofV

T_CXyfor anyy2Ycan be divided according to their type:

^

T_CXyˆ^dimXM

nˆ0

M

p‡qˆn

^^p;q T_CXy:

In the notation adopted in the proof of the first statement, one has

^^p;q

T_CX_yˆ hw_i1^ ^w_ip^w_j1^ ^w_jq ji₁;. . .;j_q 2 f0;1;2gi:

From the definition of the action ofJone has therefore that for anyp;q

^^p;q

T_CX_yV_{q p}^k withkˆ 3

p 3

q from which the second statement of the proposition can

be esily deduced. p

THEOREM3.3. The operators L_j;V_j for j2 f0;1;2gcommute with the generator J ofso(2;R).

PROOF. We prove the statements by a direct computation using the basisvij; moreover, using the action ofS3(which permutes theLj;Vj and fixesJ), it is enough to prove the commutativity forL0andV0. It useful to rewritev₀ (and hence L₀ which is wedge with v₀) in terms of the basis generated by thew_j:

v₀ˆv₁₁^v₁₂‡v₂₁^v₂₂ˆ1

2…w₁^w₂ w₂^w₁† and then:

[J;L₀](w_i1^ ^w_ip^w_j1^ ^w_jq)ˆ J 1

2…w1^w2 w2^w1†

^ wi1^ ^wip^wj1^ ^wjq

‡ 1

2…w1^w2 w2^w1†

^J wi1^ ^wip^wj1^ ^wjq

1

2…w1^w2 w2^w1† ^J(w_i1^ ^w_ip^w_j1^ ^w_jq):

Therefore the result follows from the fact that J(1

2…w1^w2 w2^w1† ˆ0

asw_jandw_k have opposite weight with respect toJfor anyj;k.

Similarly, [J;V0]ˆ0 follows from the fact that for anya V₀(a)ˆv₁₀^v₂₀^aˆ{

2 w₀^w₀^a

p From the previous theorem one obtains the following corollary, which holds on any WSD manifold (not necessarily 2-KaÈhler):

COROLLARY3.4. The algebraLCcommutes with the action ofso(2;R) induced by J.

PROOF. We already know that [J;L_j]ˆ[J;V_j]ˆ0 forj2 f0;1;2g. The corresponding commutation relations for the adjoint generatorsLj;Aj of LCfollow from the fact thatJˆ J, as noticed in Proposition 2.5. p REMARK3.5. From Schur's lemma it follows that the columns of the diagram of Proposition3.2are preserved by the action ofL_C.

4. An irreducible representation ofL_C.

Looking at the table in Proposition 3.2 we notice that the second column from the left is a representation ofLC (by Remark 3.5) of dimension 6:

VV⁶₂ ˆ hw0^w1; w0^w2; w1^w2; w0^w1^w2^w0; w0^w1^w2^w1; w0^w1^w2^w2i:

In this section we will compute explicitely this representation.

Using the above described basis, it is not difficult to compute the ma- trices by hand:

PROPOSITION4.1. Indicating withbthe ordered basis for V indicated above, the matrices for the (restrictions to V of) the generators ofL_Care the following:

Mb(L0) ˆ

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

12 0 0 0 0 0 0 ¹₂ 0 0 0 0 0

BB BB BB

@

1 CC CC CC A

; Mb(L0) ˆ

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

0 BB BB BB

@

1 CC CC CC A

;

M_b(L₁) ˆ

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

12 0 0 0 0 0

0 0 0 0 0 0

0 0 ¹₂ 0 0 0 0

BB BB BB

@

1 CC CC CC A

; M_b(L₁) ˆ

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

0 BB BB BB

@

1 CC CC CC A

;

M_b(L₂) ˆ

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ¹₂ 0 0 0 0 0 0 ¹₂ 0 0 0 0 0 0 0 0 0 0

BB BB BB

@

1 CC CC CC A

; M_b(L₂) ˆ

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 0

BB BB BB

@

1 CC CC CC A

;

M_b(V₀) ˆ

0 0 0 0 0 0 0 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

BB BB BB

@

1 CC CC CC A

; M_b(A₀) ˆ

0 0 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 0 0

0 BB BB BB

@

1 CC CC CC A

;

M_b(V₁) ˆ

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 ₂^{ 0 0 0 0

0 0 0 0 0 0

0 BB BB BB

@

1 CC CC CC A

; M_b(A₁) ˆ

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 0 0 0 0 0 0 0 0

BB BB BB

@

1 CC CC CC A

;

Mb(V2) ˆ

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

2{ 0 0 0 0 0 0

BB BB BB

@

1 CC CC CC A

; Mb(A2) ˆ

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

0 0 0 0 0 0

0 BB BB BB

@

1 CC CC CC A :

PROOF. Direct computation using the basis generated by thewj. p COROLLARY4.2. The algebra generated by the restriction ofL_Cto V is isomorphic tosl(6;C), with V its natural representation.

One can sum up the computations above in the following theorem:

THEOREM4.3. 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 Kˆ f0g, and therefore the re- presentationV is faithful andLCsl(6;C).

5. Quadratic invariants.

We begin by showing that the action of Lie algebraLC is induced by a (non-canonical) Clifford algebra representation. We use for simplicity the canonical identification TX_pTX_p without further comment, so that if fv_ijg is a basis for T_pX, then @

@v_ij

is the corresponding dual basis for T_pX.

DEFINITION5.1. For p2X, the Clifford algebraCpis CpˆCl(TpXT_pX;q)

with the quadratic form q induced by the metric 8i;j;h;k hv_ij;v_hki ˆ0;

8i;j;h;k @

@v_ij; @

@v_hk

ˆ0;

8(i;j)6ˆ(h;k) v_ij; @

@vhk

ˆ0;

8i;j v_ij; @

@v_ij

ˆ 1 2:

REMARK5.2. The Clifford algebrasCpfor varying p define a Clifford bundleC 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 TpXT_pX!R. 2

PROPOSITION5.3. The Clifford algebraC_phas a canonical representa- tionr_ponV

T_pX, induced by the operators E_ij and I_ijvia the map r_p(v_ij)ˆE_ij; r_p @

@v_ij

ˆI_ij: PROOF. The Clifford relations

fc‡cfˆ 2hf;ci

are precisely the content of Proposition 2.4. The representation is canonical, even if the operatorsEijandIijare not, because it can be defined in a basis independent way as

r_p(v)(a)ˆv^a; r_p @

@v ˆ @

@v*a

p Abusing slightly the notation, we will identifyC_pwith its (faithful) im- age insideEnd_R V

T_pX

, and we will omit any reference to the mapr_p. Actually, as the representation above is a real analogue of the Spinor re- presentation, it is easy to check that the mapr_pis an isomorphism of as- sociative algebras. One then has:

DEFINITION5.4. The linear subspaceC²_pofC_pis the image of the natural mapV₂

(TpXT_pX)! Cp. The linear subspaceC⁰_p ofCp is the subspace generated by1.

Recall thatC²_pis a Lie subalgebra ofC_p(with the commutator bracket).

PROPOSITION5.5. The Lie algebraLpand the operator J sit insideC²_p for all p2X.

PROOF. The operatorsL_j, theL_j, theV_jand theA_jlie insideC²_p C⁰_pby Proposition 2.4 and the fact thatv1;v;vDlie inV₂

T_pX. The operatorJlies insideC²_p C⁰_p by Proposition 2.5. By definition the elementsC²_p are com- mutators, and therefore have trace zero in any representation, and hence also in ther_p. Moreover, again by inspection all the generators ofL_phave trace zero once represented viar_p(they are nilpotent), and therefore they must lie insideC²_p. The operatorJ is in the Lie algebra of the isometry group, and therefore it too has trace zero and hence sits insideC²_p. AsC²_pis closed under the commutator bracket ofCp, and this commutator coincides with the composition bracket of operators, we have the conclusion. p REMARK5.6. Giving degree1to the operators Eijand degree 1to the operators I_ij, we induce aZ-degree onCp. This degree coincides with the degree of the operators induced from the grading on the forms from V

TX.

REMARK5.7. For any p2X, the Clifford algebra Cp is isomorphic to Cl6;6, as the metric u sed to define it has signatu re (6;6). The

previous proposition therefore shows that Lp is a Lie subalgebra of Cl²_6;6spin(6;6)ˆso(6;6), generated by smooth global sections of the Clifford bundle C.

The operator J acts on all of C_p by adjunction with respect to the commutator bracket, and sends its quadratic part C²_p to itself from Pro- position 5.5.

We will show that the space ofJ-invariants insideC²_p(the ``quadratic''J- invariants) coincides withL_C. To describe it explicitely, let us introduce the following notation:

DEFINITION5.8.

Ewj ˆE_1j‡{E_2j; E_wj ˆE_1j {E_2j; Iwj ˆI_1j {I_2j; I_wj ˆI_1j‡{I_2j:

LEMMA5.9. The adjoint action of the operator J on Ewj;Iwj;E_wj;I_wjis:

[J;Ewj]ˆ {Ewj; [J;Iwj]ˆ{Iwj; [J;E_wj]ˆ{E_wj; [J;I_wj]ˆ {I_wj:

PROOF. It is enough to consider the corresponding J-weights of

the w_j;w_j. p

PROPOSITION5.10. The following36operators provide a linear basis for the quadratic J-invariants:

(1) [Ew0;E_w1];[Ew0;E_w2];[Ew1;E_w2];[Ew1;E_w0];[Ew2;E_w0];[Ew2;E_w1];

(2) [I_w0;I_w1];[I_w0;I_w2];[I_w1;I_w2];[I_w1;I_w0];[I_w2;I_w0];[I_w2;I_w1];

(3) [Ew0;E_w0];[Ew1;E_w1];[Ew2;E_w2];

(4) [I_w0;I_w0];[I_w1;I_w1];[I_w2;I_w2];

(5) [E_w0;I_w1];[E_w0;I_w2];[E_w1;I_w0];[E_w1;I_w2];[E_w2;I_w0];[E_w2;I_w1];

(6) [E_w0;I_w1];[E_w0;I_w2];[E_w1;I_w0];[E_w1;I_w2];[E_w2;I_w0];[E_w2;I_w1];

(7) [E_w0;I_w0];[E_w1;I_w1];[E_w2;I_w2];[E_w0;I_w0];[E_w1;I_w1];[E_w2;I_w2]:

PROOF. TheJ-weight of a bracket ofJ-homogeneous operators is the sum of the respective weights. The quadratic ``monomials'' (with respect to the bracket) in theEwj;Iwj;E_wj;I_wjare allJ-homogeneous, and therefore to find a basis ofJ-invariant quadratic operators it is enough to identify theJ- invariant quadratic monomials. To be J-invariant means simply to have weight zero, and the computation of theJ-weight of the quadratic mono-

nials follows immediately from those of Ewj;Iwj;E_wj;I_wj, which are re-

spectively {; {; {; {. p

We end this section with the following:

THEOREM5.11. In the exact sequence of Theorem4.3 the kernel K is equal tof0g. The algebraLCis therefore isomorphic tosl(6;C).

PROOF. SinceL_Cis included in the Lie algebra of quadratic invariants, it is enough to show thatJ62 L_C, as from this and the previous proposition it follows thatdim_C(L_C)35. AsL_Cmaps surjectively tosl(6;C) which has dimension 35, the kernel must be zero. When restricted to the sub- representationV, the generators ofLChave all trace zero by inspection of their matrices. However, by definition of V, J restricted to it is multi- plication by 2{, and has therefore trace equal to 12{. p COROLLARY5.12. The Lie algebraLC hJiequals the Lie algebra of quadratic invariants insideC²_p.

6. A geometric presentation of Serre generators.

In this section, to gain a better geometric understanding of the re- presentationL_Cofsl(6;C), we explore in greater detail its relation to the geometric structure of a WSD manifold. In particular, we give a pre- sentation of a natural choice of Cartan subalgebra and Serre generators in terms on the geometric generatorsLj;Lj;Vj;Aj.

TheLj operators are similar in nature to the Lefschetz operators of a KaÈhler manifold. This analogy is what provided the initial interest in the algebraic structure of L_C. Similarly to the corresponding standard con- struction of a representation ofsl(2;C), we define

DEFINITION6.1. For j2 f0;1;2g

H_jˆ[L_j;L_j]:

These operators are self-adjoint, as L_j ˆL_j by definition. As in the context of KaÈhlerian geometry, for every jthe algebra hL_j;L_j;H_jiturns out to be a copy ofsl(2;C). Moreover, the following proposition shows that the operators Hj are semisimple on the whole algebraLC, and therefore generate a toral subalgebra ofLC:

PROPOSITION 6.2. The geometric operators Hj generate a toral sub- algebra ofLC, and the following relations hold: for j6ˆk2 f0;1;2g

(1) [Hj;Lj]ˆ2Lj; [Hj;Lj]ˆ 2Lj; (2) [H_j;L_k]ˆL_k; [H_j;L_k]ˆ L_k; (3) [H_j;V_j]ˆ0; [H_j;A_j]ˆ0;

(4) [H_j;V_k]ˆ2V_k; [H_j;A_k]ˆ 2A_k:

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 ofLCtoV. p

The whole algebra L_C splits into a direct sum of weight spaces with respect tohH0;H1;H2i, as this subalgebra is toral. The weight ofL0 with respect to the basis dual toH0;H1;H2is:

a_L0 ˆ (a_L0(H0);a_L0(H1);a_L0(H2)) ˆ (2;1;1):

The full list is:

aL0ˆ(2;1;1); aL0 ˆ aL0; aL1ˆ(1;2;1); aL1 ˆ aL1; a_L2ˆ(1;1;2); a_L2 ˆ a_L2; a_V0 ˆ(0;2;2); a_A0ˆ a_V0; a_V1 ˆ(2;0;2); a_A1ˆ a_V1; aV2 ˆ(2;2;0); a_A2ˆ aV2:

To find a natural geometric expression for two ad-semisimple elements which complete hH0;H1;H2ito a Cartan subalgebra we look at the gen- eratorsV_jandA_j. However, it turns out that the natural candidates [V_j;A_j] already lie in the algebrahH₀;H₁;H₂i. We instead build the new operators by ``subtracting'' from theV_jtheir weighta_Vj:

DEFINITION6.3. We define:

S0ˆ{[[[V0;L1];L2];L0];

S₁ˆ{[[[V₁;L₂];L₀];L₁];

S2ˆ{[[[V2;L0];L1];L2] and denote byHthe Lie algebra (overC):

H ˆ hH₀;H₁;H₂;S₀;S₁;S₂i:

The coefficients{which appear in the formulas above are dictated by the fact that with this choice the (diagonal) matrices of theSjrestricted to V have integer entries.

PROPOSITION6.4. The algebraHis a Cartan subalgebra ofL_C. More precisely, the following are the diagonals of the operators H₀;. . .;S₂ once restricted to V

H0: 1 01 01 1 0 BB BB BB

@ 1 CC CC CC A

; H1: 1 01 10 1 0 BB BB BB

@ 1 CC CC CC A

; H2: 0

11 11 0 0 BB BB BB

@ 1 CC CC CC A

; S0: 1 10 01 1 0 BB BB BB

@ 1 CC CC CC A

; S1: 1 01 01 1 0 BB BB BB

@ 1 CC CC CC A

; S2: 0 11 11 0 0 BB BB BB

@ 1 CC CC CC A :

PROOF. The computation of the matrices above shows that, once restricted toV, the algebraHspans the space of diagonal matrices of trace

zero in the given basis. p

REMARK 6.5. The computation above shows also that operators S0;S1;S2safisfy the relation

S0‡S1‡S2ˆ0:

Even if from the previous proposition we know thatHis maximal toral insideL_C, the natural geometric generatorsL_j;L_jare not eigenvectors for the adjoint action of theSk. At this point however it is possible to single out in natural geometric terms operators ofLCwhich have ``pure'' weight with respect to the algebraHand which contain in their linear span theL_j;L_j:

DEFINITION6.6. For j2 f0;1;2g

L1j ˆ 2Lj‡[Sj;Lj]; L2j ˆ2Lj‡[Sj;Lj];

L_1jˆ 2L_j [S_j;L_j]; L_2j ˆ2L_j [S_j;L_j]:

PROPOSITION 6.7. Indicating with e^h_k the 66 matrix with a 1 in position k (row) and h (column) and zero otherwise, the matrices of the operators L_ijandL_ijrestricted on V are:

L₁₀ˆ2e²₆; L₁₁ˆ 2e¹₄; L₁₂ ˆ 2e³₅; L20ˆ 2e¹₅; L21ˆ 2e³₆; L22 ˆ2e²₄; L10ˆ8e⁶₂; L11ˆ 8e⁴₁; L12ˆ 8e⁵₃; L₂₀ˆ 8e⁵₁; L₂₁ˆ 8e⁶₃; L₂₂ˆ8e⁴₂:

COROLLARY6.8. We have the following relations for the operators of LCrestricted to V:

[H_k;L_ij]ˆ(1‡d_kj)L_ij; [H_k;L_ij]ˆ (1‡d_kj)L_ij; [Sk;Lij]ˆ( 1)^i‡1(1 3dkj)Lij; [Sk;Lij]ˆ( 1)ⁱ(1 3dkj)Lij;

[S_k;V_j]ˆ0; [S_k;A_j]ˆ0:

Guided by all the explicit computations of the action on the isotypical componentVˆV⁶₂ made up to this point, we now define in terms of the natural geometric operators a set of Serre generators for the algebraLC.

DEFINITION6.9.

e1ˆ1

4[L20;A1]; f1ˆ1

4[V1;L20];

e₂ˆ1

4[L₂₂;A₀]; f₂ˆ1

4[V₀;L₂₂];

e3ˆV0; f3ˆA0; e₄ˆ1

4[L₁₂;A₀]; f₄ˆ1

4[V₀;L₁₂];

e₅ˆ1

4[L₁₀;A₁]; f₅ˆ1

4[V₁;L₁₀]:

Moreover, for all i2 f1;. . .;5gwe defineh_iˆ[e_i;f_i].

As thee_ihave by construction associated matrixeⁱ_i‡1once restricted to V and thef_iare their respective adjoints, one gets:

PROPOSITION6.10. The operatorsei;fj;hk satisfy the Serre relations forsl(6;C)and theh_ispan the Cartan subalgebraH:

h1ˆ1

2…H1 H2 S1 S2†;

h2ˆ1

2…H0 H1‡S2†;

h₃ˆ1

2… H₀‡H₁‡H₂†;

h₄ˆ1

2…H₀ H₁ S₂†;

h₅ˆ1

2…H₁ H₂‡S₁‡S₂†:

It would be interesting as a last remark to identify in the list of quadratic invariants the geometric operatorsLij;Lij;Vj;Aj, the algebraH and theso(2;R) generatorJ. To do this one could of course use the explicit matrices for the quadratic 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 TXˆW0W1W2 induces naturally a multi- degree onV

T_CX with values inZ³, which we indicate withmdeg. This follows from the equation

^ⁿ

T_CX M

p‡q‡rˆn

^^p

W0C … † ^^q

W1C … † ^^r

W2C … †:

We notice furthermore that the (complexified) decomposition above is preserved by the operatorJ, and thereforemdegcommutes with the action ofso(2;R).

PROPOSITION6.11. The operators L_j;V_j;L_j;A_j;H_j;S_j are mdeg-homo- geneous, with multi-degrees:

mdeg(L0)ˆ(0;1;1); mdeg(L1)ˆ(1;0;1); mdeg(L2)ˆ(1;1;0);

mdeg(L0)ˆ(0; 1; 1); mdeg(L1)ˆ( 1;0; 1); mdeg(L2)ˆ( 1; 1;0);

mdeg(V0)ˆ(2;0;0); mdeg(V1)ˆ(0;2;0); mdeg(V2)ˆ(0;0;2);

mdeg(A0)ˆ( 2;0;0); mdeg(A1)ˆ(0; 2;0); mdeg(A2)ˆ(0;0; 2);

mdeg(H0)ˆ(0;0;0); mdeg(H1)ˆ(0;0;0); mdeg(H2)ˆ(0;0;0);

mdeg(S0)ˆ(0;0;0); mdeg(S1)ˆ(0;0;0); mdeg(S2)ˆ(0;0;0):

PROOF. The values for mdeg for theLj and theVjfollow immediately from mdeg of the corresponding forms and the dual (contraction) operators have opposite value of mdeg. The remaing values can be computed using the additivity of mdeg with respect to the bracket. p

PROPOSITION6.12. Letfj;k;lg ˆ f0;1;2g. Then:

Span L_1j;L_2j

ˆ Span [E_wk;E_wl];[E_wl;E_wk]

; Span L_1j;L_2j

ˆ Span [I_wk;I_wl];[I_wl;I_wk]

; Span Vj

ˆ Span[Ewj;E_wj]

; Span A_j

ˆ Span[I_wj;I_wj]

; H Span J… † ˆ M²

mˆ0

Span [Ewm;Iwm];[E_wm;I_wm] :

PROOF. The mdeg of the Lij is the same of the corresponding Lj, and similarly for their adjoints. The mdegs of the quadratic monomials are immediately computed as they are the sum of those of their com- ponents. For example, mdeg(E_w0)ˆmdeg(E_w0)ˆ(1;0;0), mdeg(E_w1)ˆ

ˆmdeg(E_w1)ˆ(0;1;0) and therefore mdeg([E_w0;E_w1])ˆ(1;1;0), equal

to that of L12 andL22. p

REFERENCES

[B] V. BATYREV, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom.,3(1994), pp. 493-535.

[BMP] U. BRUZZO- G. MARELLI- F. PIOLI,A Fourier transform for sheaves on real tori Part II. Relative theory, J. of Geometry and Phy.,41(2002), pp.

312-329.

[CDGP] P. CANDELAS- X.C. DE LAOSSA - P.S. GREEN - L. PARKES, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nucl.

Phys.,B359(1991), pp. 21-74.

[GG] G. GAIFFI- M. GRASSI,A natural Lie superalgebra bundle on rank three WSD manifolds, J. of Geom. and Phys.,59(2009), 207-220.

[G1] M. GRASSI, Polysymplectic spaces, s-KaÈhler manifolds and lagrangian fibrations, math.DG/0006154 (2000).

[G2] M. GRASSI,Mirror symmetry and self-dual manifolds, math.DG/0202016 (2002).

[G3] M. GRASSI, Self-dual manifolds and mirror symmetry for the quintic threefold, Asian J. Math.,9(2005), pp. 79-102.

[GP] B.R. GREENE- M.R. PLESSER,Duality in Calabi-Yau moduli space, Nucl.

Phys.,B338(1990), pp. 15-37.

[GVW] B.R. GREENE - C. VAFA - N.P. WARNER, Calabi-Yaumanifolds and renormalization group flows, Nucl. Phys.,B324(1989), pp. 371-390.

[Gr] M. GROMOV, Metric structures for Riemannian and non-Riemannian spaces, BirkhaÈuser P.M.,152(Boston, 1999).

[GW] M. GROSS - P.M.H. WILSON, Large Complex Structure limits of K3 surfaces, math.DG/0008018 (2001).

[Gu] V. GUILLEMIN,Moment maps and combinatorial invariants of Hamilto- nianTⁿ-spaces, P. M. BirkhaÈuser,122(1994).

[M] A. MCINROY,Orbifold mirror symmetry for complex tori, preprint.

[KS] M. KONTSEVICH - Y. SOIBELMAN, Homological mirror symmetry and torus fibrations, math.SG/0011041 (2001).

[SYZ] A. STROMINGER- S.T. YAU- E. ZASLOW,Mirror Symmetry is T-Duality, Nucl. Phys.,B479(1996), pp. 243-259.

Manoscritto pervenuto in redazione l'11 luglio 2007.