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Some Landau--Ginzburg Models Viewed As Rational Maps

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Some Landau--Ginzburg Models Viewed As Rational Maps SOME LANDAU–GINZBURG MODELS VIEWED AS RATIONAL MAPS E. BALLICO, E. GASPARIM, L. GRAMA, AND L. SAN MARTIN ABSTRACT. [GGSM2] showed that height functions give adjoint orbits of semisimple Lie algebras the structure of symplectic Lefschetz fibra- tions (superpotential of the LG model in the language of mirror sym- metry). We describe how to extend the superpotential to compactifi- cations. Our results explore the geometry of the adjoint orbit from 2 points of view: algebraic geometry and Lie theory. CONTENTS 1. Motivationandstatementofresults 1 2. SymplecticLefschetzfibrationsonadjointorbits 4 3. Adjointorbitsasalgebraicvarieties 5 4. Severalincarnationsoftheorbit 6 5. Topologyon Oµ 8 6. Thepotentialviewedasarationalmap 9 7. Algebraiccompactificationsandtheconjectureof[BG] 11 8. Lie theoretical compactification and the Segre embedding 12 References 16 1. MOTIVATION AND STATEMENT OF RESULTS The data W : X → C of a manifold together with a complex function is commonly known in the literature as a Landau–Ginzburg model; W is called the superpotential. Such LG models are fundamental ingredients to the study of questions motivatedby the Homological Mirror Symmetry conjecture of Kontsevich [Ko]. When the superpotential is defined over arXiv:1601.05119v1 [math.AG] 19 Jan 2016 a symplectic manifold, this involves the construction of a Fukaya–Seidel category. The objects of this category are associated to the singularities of W (Lagrangian vanishing cycles), see [Se]. These questions motivated us to construct examples of LG models and to study their symplectic geom- etry [GGSM1], [GGSM2]. Here we explore these examples from different points of view, namely, we are interested in the algebraic geometry and Lie theory of the smooth variety X . Part of the homological mirror symmetry conjecture describes a dual- ity between algebraic varieties and symplectic LG models. Subsequently Clarke [Cl] showed that the conjecture can be interpreted in further gen- erality in such a way that both sides are LG models. According to whether 1 2 E. BALLICO, E. GASPARIM, L. GRAMA, AND L. SAN MARTIN X is considered with a Kähler structure and algebraic potential or whether X is considered with a symplectic structure and holomorphic potential these are called B-side LG model, or A-side LG model, see [AAK] or [KKP]. In such terminology, the work of [GGSM1] provides a large family of ex- amples of A-side LG models. Here we look at these examples from the point of view of B-side LG models. Our constructions are carried out using Lie theory. Let g be a complex semisimple Lie algebra with Lie group G, and h the Cartan subalgebra. Consider the adjoint orbit O(H0) of an element H0 ∈ h, that is, O(H0) := {Ad(g )H0, g ∈ G}. Let H ∈ h be a regular element, and 〈.,.〉 the Cartan–Killing form. The main result of [GGSM1] shows that the height function fH : O(H0) → C X 7→ 〈X , H〉 gives the orbit the structure of a symplectic Lefschetz fibration; thus cor- responding to an A-side LG model. We show here that this height func- tion can also be interpreted as a rational map on a projective compactifi- cation of O(H0); hence corresponding to a B-side LG model. In thiswork we restrict ourselvestothe class of adjointorbits which are diffeomorphic to cotangent bundles of projective spaces Pn. This is the simplest case of semisimple orbit, yet already presenting somewhat sur- prising features. A harmonious combination of Lie theory and algebraic geometryhappens naturallyin this context, for example, we shall see that Lie theory provides rather efficient methods to identify the Segre embed- ding of a compactification of O(H0). This method of carrying along Lie theory together with algebraic geometry is arguably where the core value of our contributionlies. We put forth the idea that there is much to profit from applying Lie theoretical methods to algebraic geometric problems. This work is a first instance of what we propose as a long term program. Certainly such combinations of the 2 areas have appeared in the litera- ture in other contexts, the particularly new features of our contribution are the applications to the study Lefschetz fibrations and LG models. Our results go as follows. We take G = SL(n + 1,C) and consider the adjoint orbit passing through µ = Diag(n,−1,...,−1), which we denote by ∗ n Oµ. The diffeomorphism type is then Oµ ∼dif T P . In Section 2 we recall the main result of [GGSM1] showing that the heightfunction with respect to a regular element gives the adjoint orbit the structure of a symplectic Lefschetz fibration. We then describe the orbit and the regular fibers of fH as affine varieties, and consider fibrewise compactifications. In sec- tion 3 for the case of sl(2,C) we obtain: Theorem. 3.1 Let X = O(1,−1), H = Diag(1,−1) and fH : X → C. Then fH admits a fiberwise compactification with fibres isomorphic to P1. SOME LANDAU–GINZBURG MODELS VIEWED AS RATIONAL MAPS 3 In section 4 we present several ways to interpret the adjoint orbit, thus illustrating the interactions between Lie theory and algebraic geometry. In section 5 we describe (Zariski) open charts for Oµ in terms of Bruhat cells. Corollary. 5.4 The domains of the parametrizations D j corresponding to the Bruhat cells are open and dense in Oµ. The orbit Oµ is not compact, thus we must choose a compactication. Once again we are faced with the decision whether to use Lie theory or algebraic geometry. Recently in [GGSM2] a holomorphic open and dense ∗ ∗ embbeding of Oµ into F × F was constructed. Here, F and F represent ∗ a flag manifold and its dual flag, chosen such that O(H0) ∼dif T F is a diffeomorphism. The immediate task that then follows is to extend the potential fH to this compactification. Such an extensioncan not be made holomorphically,as explained in lemmas 6.1 and 6.2. We then proceed to extend the potential as a rational function. We consider here Oµ as the n n ∗ n adjoint orbit of e1 ⊗ ε1 in C × (C ) . Set V = C . Theorem. 6.3 The rational function on V ⊗ V ∗ that coincides with the potential fH on O (v0 ⊗ ε0) is given by tr Aρµ (H) RH (A) = ¡ tr(A) ¢ for A ∈ V ⊗V ∗ = End(V ). On the algebraic geometric side, [BCG] compactified the affine variety X = Oµ to a projective variety X by homogenising its defining ideal. Then the question begging to be asked is whether their algebraiccompactifica- tionagrees with our Lie theoreticalone. Using methodsof computational algebraic geometry and a Macaulay2 algorithm [BG] identified a projec- tive embbeding of X as the Segreembbeding for thecases of sl(n+1) with n < 10, and conjectured that the result holds true for all n. We provide an affirmative answer to this question, in particular concluding that the Lie theoretical compactification does coincide with the algebraic geometric one for the case of Oµ. n n∗ Theorem. 7.1 The embbeding Oµ ,→ P × P obtained by Lie theoretical methods agrees with the Segre embbeding obtained algebro-geometrically by homogenisation of the ideal cutting out the orbit Oµ as an affine variety in sl(n + 1). Remark 1.1. Observe that the algebraic geometric method will in general produce singular compactifications, see [BCG, Sec. 6], whereas that the Lie theoretical method always embeds the orbit into a product of smooth flag manifolds. We conclude the paper by presenting in section 7.1 the expressions of the Segre embbeding and the rational potential RH first for the case n = 4 E. BALLICO, E. GASPARIM, L. GRAMA, AND L. SAN MARTIN ∗ 2 3, with µ = (2,−1,−1) hence Oµ ≈ T P , and finally in general for µ = ∗ n (n,−1,...,−1) when Oµ ≈ T P . 2. SYMPLECTIC LEFSCHETZ FIBRATIONS ON ADJOINT ORBITS Let g be a complex semisimple Lie algebra with Cartan–Killing form 〈X ,Y 〉= tr(ad(X )ad(Y )) ∈ C, and G a connected Lie group with Lie alge- bra g . Let H0 ∈ h. The adjoint orbit of H0 is defined as O (H0) = AdG · H0 = {Ad(g )H0 ∈ g : g ∈ G}. Fix a Cartan subalgebra h ⊂ g and a compact real form u of g. Associ- ated to these subalgebras there are the subgroups T =〈exph〉= exph and U =〈expu〉= expu. Denote by τ the conjugation associated to u, defined by τ(X ) = X if X ∈ u and τ(Y ) =−Y if Y ∈ iu. Hence if Z = X +iY ∈ g with X ,Y ∈ u then τ(X + iY ) = X − iY . In this case, Hτ : g × g → C defined by (2.1) Hτ (X ,Y ) =−〈X ,τY 〉 is a Hermitian form on g (see [SM, lemma 12.17]). We write the real and imaginary parts of H as H (X ,Y ) = (X ,Y ) + iΩ(X ,Y ) X ,Y ∈ g. The real part (·,·) is an inner product and the imaginary part of Ω is a symplectic form on g. Indeed, we have 0 6= iH (X , X ) = H (iX , X ) = iΩ(iX , X ), that is, Ω(iX , X ) 6= 0 for all X ∈ g, which shows that Ω is nondegenerate. Moreover, dΩ = 0 because Ω is a constant bilinear form. The fact that Ω(iX , X ) 6= 0 for all X ∈ g guarantees that the restriction of Ω to any com- plex subspace of g is also nondegenerate. Now, the tangent spaces to O (H0) are complex vector subspaces of g. Therefore, the pullback of Ω by the inclusion O (H0) ,→ g defines a sym- plectic form on O (H0). With this choice of symplectic form, the main result of [GGSM1] says: Theorem [GGSM1, Thm. 2.2] Let h be the Cartan subalgebra of a com- plex semisimple Lie algebra.
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