Morse functions and Real Lagrangian Thimbles on Adjoint Orbits
aa r X i v : . [ m a t h . S G ] A ug MORSE FUNCTIONS AND REAL LAGRANGIANTHIMBLES ON ADJOINT ORBITS
ELIZABETH GASPARIM
AND
LUIZ A. B. SAN MARTIN
Abstract.
We compare Lagrangian thimbles for the potential of aLandau–Ginzburg model to the Morse theory of its real part. We ex-plore Landau–Ginzburg models defined using Lie theory, constructingtheir real Lagrangian thimbles explicitly and comparing them to thestable and unstable manifolds of the real gradient flow.
Contents
1. Real and complex Morse functions 12. Symplectic Lefschetz fibrations 33. The gradient field of of Re f H
54. Lagrangian vanishing cycles 95. Real Lagrangian thimbles 126. The potential and graphs 157. Minimal semisimple orbits 198. Acknowledgements 24References 241.
Real and complex Morse functions
Given a real manifold M , a smooth function f : M → R is called a Morsefunction if it has only nondegenerate critical points. Recall that a criticalpoint p of f is nondegenerate if the Hessian matrix ∂ f∂x i ∂x j ( p ) is nonsingular.Nondegenerate critical points are isolated, and the lemma of Morse tells usthat, on a neighborhood of such a critical point, the Morse function can bewritten in local coordinates as f = f ( p ) − x − · · · − x λ + x λ +1 + · · · + x n . The integer λ is called the index of f at p . Morse theory tells us how torecover the topology of a compact manifold M from the data of indicesof critical points of f . In fact, M has the homotopy type of a finite CW-complex with one cell of dimension λ for each critical point of index λ , see[M].Given a complex manifold M , a complex function f : M → C is calleda Landau–Ginzburg model and the function f is called the superpotential .If in addition df is surjective outside a finite number of points and f isa holomorphic Morse function, that is, it has only nondenegerate criticalpoints, then f is called a Topological Lefschetz fibration . On a neighborhood of a critical point, a Lefschetz fibration can be written in local coordinatesas f = f ( p ) + z + · · · + z n . Note that changing the coordinate z j to iz j takes z j to − z j so that itis meaningless to talk about indices of critical points in the complex case.Furthermore, observe that if the complex manifold M is compact, thenany holomorphic function f : M → C is constant. Hence, this type ofLandau–Ginzburg model is interesting only in the case when M is non-compact. For compact manifolds the natural concept of Landau–Ginzburgmodel is f : M → P .In algebraic geometry functions f : M → P give rise to so-called pen-cils, which are studied extensively within the context of Picard–Lefschetztheory. Fibers of such pencils intersect in the base locus, and a topologicalLefschetz fibration can be obtained by blowing up this base locus. If M b is a regular fibre contained in a small neighborhood of a singular fibre M o then there is a retraction M b → M o which induces a surjection in homology H ∗ ( M b ) → H ∗ ( M o ). The classes in the kernel of this surjection are called vanishing cycles and are important objects of study in Hodge theory, see[PS]. The fundamental theorem of Picard–Lefschetz theory describes theintersection theory of vanishing cycles in the case when M is a projective va-riety. In particular, for compact M this fundamental theorem implies thateach critical point of f has a corresponding vanishing cycle. However, inthe noncompact case existence of a vanishing cycle corresponding to eachcritical point is not guaranteed, see [S].If ( M, ω ) is a symplectic manifold, then a topological Lefschetz fibration f : M → C is called a Symplectic Lefschetz fibration provided the symplecticform ω is nondegenerate on the fibre M x for all x in the sense that: ι. M x is a symplectic submanifold of M for each regular value x , and ιι. for each critical point p the symplectic form ω p is non degenerateover the tangent cone of M f ( p ) at p .For any symplectic fibration there exists a natural connection obtainedby taking the symplectic orthogonal to the fibre. If o is a critical value of f and b is a regular value contained in a neighborhood of o , then considera path λ : [0 , → C from λ (0) = b to λ (1) = o . Given a vanishing cycle α ⊂ M b we can use the connection to parallel transport the cycle α along λ all the way to the corresponding critical point p . For each t ∈ [0 ,
1] weobtain a cycle α t ⊂ M λ ( t ) so that α = α and α = p . The object tracedby the cycle α on its way to p is topologically a closed disc D = {∪ t ∈ [0 , α t } with boundary ∂D = α and is called a thimble .Vanishing cycles live naturally in the middle homology of the regular fi-bre, hence dim R α = dim C M b . Thus, it makes sense ask whether α is aLagrangian submanifold of M b , that is, if ω vanishes on α , and in the affir-mative case α is called a Lagrangian vanishing cycle . If the correspondingthimble is a Lagrangian submanifold of M it is then called a Lagrangianthimble . Lagrangian thimbles are the objects that generate the so-called
Fukaya–Seidel category of the fibration, and they are our main objects ofstudy in this paper.
ORSE FUNCTIONS AND REAL LAGRANGIAN THIMBLES ON ADJOINT ORBITS 3
We explore symplectic Lefschetz fibrations on semisimple adjoint orbits,recalling the construction of the complex superpotential in section 2 anddescribing the gradient vector field of its real part in section 3. We thenconstruct Lagrangian vanishing cycles in section 4 and Lagrangian thimblesof a preferred type which we name real Lagrangian thimbles (definition 18)obtained using the Morse theory of the real part of the superpotential.Profiting from the knowledge of Lagrangian submanifolds of the adjoint or-bits described in [GGSM2] and existence of Lagrangian submanifolds insidetheir compactifications described in [GSMV] we have existence of Lagrangiansubmanifolds V passing through any critical value c of the superpotential f H = f + if and containing a real sphere that is a vanishing cycle for f constructed in section 4. Then, considering the restriction of the real part g = f | V to the Lagrangian submanifold V of O ( H ) we are able to findout explicitly the desired real thimbles: Theorem (17) . Take c near f ( x ) = g ( x ) . We have that g − [ c, g ( x )] = f − [ c, f ( x )] ∩ V in the negative definite case, or g − [ g ( x ) , c ] = f − [ f ( x ) , c ] ∩ V in the positive definite caseis homeomorphic to a closed ball in R dim V . This ball is a Lagrangian thimble.
We provide examples that illustrate the behaviour of the superpotentialover Lagrangian submanifolds obtained from graphs in section 6. Finally,exploring the graph of Γ( R w ) of the right translation by the principal in-volution of the Weyl group, we explicitly describe examples of the relationbetween the Morse theory of the real part and the real Lagrangian thimblesof the superpotential, concluding this work with: Theorem (29) . The stable and unstable manifolds of grad ( Re f H ) at thecritical point [ e j ] are open in the graph Γ( m ± j ◦ R w ) . The real Lagrangianthimbles are closed balls contained in the graph Γ( m ± j ◦ R w ) . The relation between real thimbles in symplectic Lefschetz fibrations andstable and unstable manifolds of the gradient flow is part of the folkloreof the subject and is presumably well known to experts. Nevertheless, wewere unable to find such relation explained in detail anywhere in the liter-ature, and we believe that the explicit constructions given here are usefulillustrations of the construction of Lagrangians.2.
Symplectic Lefschetz fibrations
In this section we summarize the construction of symplectic Lefschetz fi-brations on adjoint orbits, their compactifications and Lagrangian subman-ifolds discussed in [GGSM1, GGSM2, BGGSM].Let G be a complex semisimple Lie group with Lie algebra g and denoteby h X, Y i := tr(ad( X ) , ad( Y )) the Cartan–Killing form of g . Fix a Cartansubalgebra h ⊂ g and a real compact form u of g . Associated to thesesubalgebras are the subgroups T = h exp h i = exp h and U = h exp u i = exp u .Denote by τ the conjugation associated to u which is defined by τ ( X ) = X if X ∈ u and τ ( Y ) = − Y if Y ∈ i u , that is, if Z = X + iY ∈ g with X, Y ∈ u ORSE FUNCTIONS AND REAL LAGRANGIAN THIMBLES ON ADJOINT ORBITS 4 then τ ( X + iY ) = X − iY . In this case we can define the Hermitian form H τ : g × g → C as H τ ( X, Y ) := −h X, τ Y i . (1)If we write the real and imaginary parts of H τ as H τ ( X, Y ) = (
X, Y ) + i Ω( X, Y ) , it is well known that the real part ( · , · ) is an inner product and the imaginarypart Ω is a symplectic form on g . Indeed, we have0 = i H ( X, X ) = H ( iX, X ) = i Ω ( iX, X ) , that is, Ω ( iX, X ) = 0 for all X ∈ g , which shows that Ω is nondegenerate.Moreover, d Ω = 0 because Ω is a constant bilinear form. The fact thatΩ ( iX, X ) = 0 for all X ∈ g guarantees that the restriction of Ω to anycomplex subspace of g is also nondegenerate.We denote by O ( H ) the adjoint orbit Ad G ( H ) of H ∈ g . Denote by h ∗ the dual vector space of h and by Π the set of all roots associated to theCartan subalgebra h . An element H ∈ h is called regular if α ( H ) = 0 forall α ∈ Π. As the restriction of Cartan–Killing form to h is nondegenerate,the map ϕ : h → h ∗ defined by ϕ ( X ) = h X, ·i is a linear isomorphism. Wedenote by h R the real subspace of h generated by ϕ − (Π). The pullback ofthe symplectic form Ω by the inclusion O ( H ) ֒ → g defines a symplectic formon O ( H ). With this choice of symplectic form, we have our constructionof Symplectic Lefschetz fibrations via Lie theory as follows: Theorem 1. [GGSM1, Thm. 2.2]
Let h be the Cartan subalgebra of acomplex semisimple Lie algebra g . Given H ∈ h and H ∈ h R with H aregular element. The height function f H : O ( H ) → C defined by f H ( x ) = h H, x i x ∈ O ( H ) has a finite number (= |W| / |W H | ) of isolated singularities and gives O ( H ) the structure of a symplectic Lefschetz fibration. Here W = Nor G ( h ) / Cen G ( h ) denotes the Weyl group. In the languageused in Homological Mirror Symmetry the pair ( O ( H ) , f H ) is called aLandau–Ginzburg model with superpotential f H . See [BBGGSM] for a dis-cussion of the mirror of O ( H ) in the case of sl (2 , C ).Given a regular element H ∈ g , consider the set Θ of simple roots thathave H in their kernel. Let p Θ be the parabolic subalgebra determined byΘ, with corresponding parabolic subgroup P Θ . The quotient F Θ := G/P Θ by the parabolic subgroup is the flag manifold determined by H . Anotherregular element in g will correspond to the same flag manifold if it is annihi-lated by the same set of roots Θ, so for questions regarding the isomorphismwith T ∗ F Θ we denote the regular element by H Θ instead of H .The adjoint orbit of a regular element H Θ is isomorphic to the cotan-gent bundle of the flag manifold F Θ [GGSM2, Thm. 2.1]. The isomorphism ι : O ( H Θ ) → T ∗ F Θ is obtained observing that O ( H Θ ) = [ k ∈ K Ad ( k ) (cid:16) H Θ + n +Θ (cid:17) , ORSE FUNCTIONS AND REAL LAGRANGIAN THIMBLES ON ADJOINT ORBITS 5 then taking for each X ∈ n +Θ , the correspondence:Ad ( k ) ( H Θ + X )
7→ h
Ad ( k ) X, ·i where Ad ( k ) n − Θ is identified with the tangent space T kb Θ F Θ , where b Θ is theorigin of the flag.Let µ be the moment map of the action a : G × T ∗ F Θ → T ∗ F Θ . Then µ : T ∗ F Θ → Ad ( G ) H Θ is the inverse of the map ι : Ad ( G ) H Θ → T ∗ F Θ , andsatisfies µ ∗ ω = Ω , where Ω is the canonical symplectic form of T ∗ F Θ and ω the (real) Kirillov–Kostant–Souriau form on Ad ( G ) H Θ .We compactify the total space of T ∗ F Θ to the trivial product F Θ × F Θ ∗ as: O ( H Θ ) ∼ → T ∗ F Θ ֒ → T ∗ F Θ = F Θ × F Θ ∗ . [BGGSM, Thm. 5.3] showed how to extend the potential f H to the com-pactification in the case of minimal adjoint orbits.Let w be the principal involution of the Weyl group W , that is, theelement of highest length as a product of simple roots. The right action R w : F H → F H ∗ is anti-symplectic with respect to the K¨ahler forms on F H and F H ∗ given by the Borel metric and canonical complex structures. Weuse graphs of anti-symplectic maps to construct Lagrangian submanifolds of F H and F H ∗ , these graphs will be used in the construction of Lagrangianthimbles of O ( H ) in section 5. Notation . We will denote by Γ( f ) the graph of a map f . Remark . Γ( R w ), that is, the graph of R w the right translation by theprincipal involution of W , is the orbit of K by the diagonal action. Thisorbit is the zero section of T ∗ F H under the identification with O ( H ) ≈ G · ( H , − H ). Therefore, Γ ( R w ) is a real Lagrangian submanifold of theproduct. 3. The gradient field of of Re f H The field Z ( x ) = [ x, [ τ x, H ]] is defined over the whole algebra g and istangent to the adjoint orbits, since the tangent space to Ad ( G ) x at x is theimage of ad ( x ). Assume here that both H and H are regular and belongto the Weyl chamber h + R . The field Z is gradient, not with respect to theinner product coming from g (the real part of H ), but with respect to theRiemannian metric m on the adjoint orbit O ( H ), which does not extendnaturally to g .The metric m is defined as follows: the tangent space T x O ( H ) is theimage of ad ( x ), which is the sum of the eigenspaces associated to the nonzeroeigenvalues of x . This happens because ad ( x ) is conjugate to ad ( H ) (theformula ad ( φx ) = φ ◦ ad ( x ) ◦ φ − holds true for any automorphism φ ∈ Aut ( g ), in particular for φ = Ad ( g ), g ∈ G ). Now, ad ( H ) is diagonalizableand its image is the sum of the root spaces, which are the eigenspaces ofthe nonzero eigenvalues of ad ( H ) (since H is regular). By conjugation thesame is true for ad ( x ), x ∈ O ( H ). As a consequence, the restriction ofad ( x ) to its image is an invertible linear transformation. ORSE FUNCTIONS AND REAL LAGRANGIAN THIMBLES ON ADJOINT ORBITS 6
Taking this into account, define m x ( u, v ) = (cid:16) ad ( x ) − u, ad ( x ) − v (cid:17) , where ( · , · ) is the inner product given by the real part of H ( · , · ), and theinverse of ad ( x ) is just the inverse of its restriction to the tangent space.The form m x ( · , · ) is a well defined Riemannian metric on O ( H ). Remark . The realification g R of g is a real semisimple Lie algebra. ItsCartan–Killing form h· , ·i R is given by h· , ·i R = 2 ℜh· , ·i (see [SM]). Conse-quently, the inner product ( · , · ) is given by( X, Y ) = − h X, τ Y i R where τ is conjugation with respect to u , which is a linear transformation of g R (over R ).Returning to the field Z ( x ), define the height function h H : O ( H ) → R by h H ( x ) = ( x, H ) . Given A ∈ g , the tangent vector [ A, x ] is given by[
A, x ] = ddt | t =0 Ad (cid:16) e tA (cid:17) x. Therefore, ( dh H ) x ([ A, x ]) = ddt | t =0 (cid:16) Ad (cid:16) e tA (cid:17) x, H (cid:17) = ([ A, x ] , H ) . (2)On one hand, m x ([ A, x ] , Z ( x )) = − m x (ad ( x ) A, ad ( x ) [ τ x, H ])= − ( A, [ τ x, H ]) , by definition of m x . On the other hand, by lemma 6 below,( A, [ τ x, H ]) = ( A, ad ( τ x ) H ) = − (ad ( x ) A, H ) .Thus, m x ([ A, x ] , Z ( x )) = (ad ( x ) A, H ) = − ([ A, x ] , H ) . Combining this with (2) we arrive at( dh H ) x ([ A, x ]) = − m x ([ A, x ] , Z ( x )) . In conclusion:
Proposition 5. Z ( x ) = − grad h H with respect to the metric m x . Lemma 6.
Consider the inner product ( · , · ) := ℜH ( · , · ) , then • the conjugation τ is an isometry for this inner product, and (ad ( X ) Y, Z ) = − ( Y, ad ( τ X ) Z ) . • if τ X = X , that is, if X ∈ u , then ad ( X ) is antisymmetric for ( · , · ) , • if τ Y = − Y , that is, Y ∈ i u , then ad ( Y ) is symmetric for ( · , · ) . ORSE FUNCTIONS AND REAL LAGRANGIAN THIMBLES ON ADJOINT ORBITS 7
Proof. If X ∈ g then H (ad ( X ) Y, Z ) = −H ( Y, ad ( τ X ) Z ), and it followsthat the same relation holds true for the inner product ( · , · ) . In fact, H ( τ X, Y ) = −h τ X, τ Y i = −h τ Y, τ X i = H ( τ Y, X ) = H ( X, τ Y ) , which means that ( τ X, Y ) = ( X, τ Y ). For the second item: H ([ X, Y ] , Z ) = −h [ X, Y ] , τ Z i = h Y, [ X, τ Z ] i = h Y, τ [ τ X, Z ] i = −H ( Y, [ τ X, Z ]) . (cid:3) Remark . ( Z as a field on g ) We show that considered on the entire vectorspace g the vector field Z ( x ) = [ x, [ τ x, H ]] is not gradient with respectto ( · , · ). Take the differential form α x ( v ) = ( v, Z ( x )). Then dα ( v, w ) = vα ( w ) − wα ( v ) − α [ v, w ], where the last term vanishes if v and w areregarded as constant vector fields on g . The expression for Z then gives( dα ) x ( v, w ) =( w, [ v, [ τ x, H ]]) + ( w, [ x, [ τ v, H ]]) − ( v, [ w, [ τ x, H ]]) − ( v, [ x, [ τ w, H ]]) . Evaluating this expression on x = H ∈ h , we obtain( dα ) x ( v, w ) = ( w, [ H , [ τ v, H ]]) − ( v, [ H , [ τ w, H ]])= ( w, ad ( H ) ad ( H ) τ v ) − ( v, ad ( H ) ad ( H ) τ w ) . We have ( w, ad ( H ) ad ( H ) τ v ) = (cid:16) τ w, τ τ − ad ( H ) ad ( H ) τ v (cid:17) = ( τ w, ad ( τ H ) ad ( τ H ) v )= (ad ( τ H ) ad ( τ H ) τ w, v ) = (ad ( τ H ) ad ( τ H ) τ w, v )where the last equality comes from the fact that ad ( H ) commutes withad ( H ). Therefore,( dα ) x ( v, w ) = (ad ( τ H ) ad ( τ H ) τ w, v ) − (ad ( H ) ad ( H ) τ w, v ) . But, setting τ H = − H and τ ( H ) = H , then the right hand side becomes − H ) ad ( H ) τ w, v ) which does not vanish identically on v, w . Thus, dα = 0, implying that the vector field is not gradient on g .We return to the study of the singularities of the gradient field Z on theorbit O ( H ). We have verified that the set of such singularities is O ( H ) ∩ h ,which is the orbit of H ∈ h by the Weyl group. We now recall the proofthat these singularities are nondegenerate. To see this, let x = wH be oneof the singularities. Then the differential of Z at x is given by dZ x ( v ) = [ v, [ τ x, H ]] + [ x, [ τ v, H ]] = [ x, [ τ v, H ]]= − ad ( x ) ad ( H ) ( τ v ) . The tangent space to O ( H ) at x is T x O ( H ) = X α ∈ Π g α = X α> ( g α ⊕ g − α ) . ORSE FUNCTIONS AND REAL LAGRANGIAN THIMBLES ON ADJOINT ORBITS 8 If v = P α ∈ Π a α X α then τ v = − P α ∈ Π a α X − α . Consequently, dZ x ( v ) = ad ( x ) ad ( H ) X α ∈ Π a α X − α = X α ∈ Π a α α ( x ) α ( H ) X − α . In particular, let α be a positive root. Then, g α + g − α (regarded as a real vector space) is invariant by dZ x . Furthermore, with respect to the basis { X α , X − α , iX α , iX − α } , the restriction of dZ x to this subspace is given bythe matrix α ( x ) α ( H ) − − , which has eigenvalues ± α ( x ) α ( H ) with associated eigenspaces V − α ( x ) α ( H ) = span R { X α − X − α , i ( X α + X − α ) } = ( g α + g − α ) ∩ u ,V α ( x ) α ( H ) = span R { X α + X − α , i ( X α − X − α ) } = ( g α + g − α ) ∩ i u . Therefore, T x O ( H ) = P α ∈ Π g α decomposes into T x O ( H ) = V + x ⊕ V − x ,where V + x (unstable space) is the sum of eigenspaces with positive eigen-values and V − x (stable space) is where dZ x has negative eigenvalues. Thedimension of T x O ( H ) over R is 2 | Π | , whereas dim R V ± = | Π | . Proposition 8.
The subspaces V + x and V − x are Lagrangian with respect tothe symplectic form Ω = ℑH .Proof. V α ( x ) α ( H ) and V − α ( x ) α ( H ) are isotropic subspaces, since they are con-tained in either u or i u and both are subspaces where the Hermitian form H takes real values. On the other hand, if α = β are positive roots, then g α + g − α is orthogonal to g β + g − β with respect to the Cartan–Killing form,and since these subspaces are τ -invariant they are also orthogonal with re-spect to H . Therefore, H assumes real values on V + x and on V − x as well,hence these subspaces are isotropic, and by dimension count they are La-grangian. (cid:3) The subspaces V + x and V − x are the tangent subspaces to the unstableand stable submanifolds of Z with respect to the fixed point x . Thesesubmanifolds are denoted by V + x and V − x , respectively.We will investigate the stable manifold of Z for the case x = H . If α > α ( H ), α ( H ) and α ( H ) α ( H ) are all positive. It follows that V + H = i u ∩ X α> ( g α + g − α ) , V − H = u ∩ X α> ( g α + g − α ) . Observe that u ∩ P α> ( g α + g − α ) is a real vector space with basis { A α = X α − X − α , iS α = i ( X α + X − α ) : α > } . Moreover, for H ∈ h and α > • [ H, A α ] = α ( H ) ( X α + X − α ). • [ H, S α ] = α ( H ) i ( X α − X − α ). ORSE FUNCTIONS AND REAL LAGRANGIAN THIMBLES ON ADJOINT ORBITS 9 • h A α , S α i = 0, and h A α , A α i = h A α , A α i = 2 since, h X α , X − α i = 1. • If β = α then h A α , A β i = h S α , S β i = h A α , S β i = h S α , A β i = 0. Lemma 9.
For x = z + y with y ∈ u and z ∈ h + R , we have that H τ ( Z ( x ) , y ) is real and negative.Proof. We have τ x = y − z . Thus, Z ( x ) = [ y + z, [ y − z, H ]] = [ y + z, [ y, H ]].Consequently,( Z ( x ) , y ) = ([ y + z, [ y, H ]] , y ) = ([ y, H ] , [ z, y ])= − ([ H, y ] , [ z, y ]) .Set y = P α> ( a α S α + b α A α ) with a α , b α ∈ R . Then, by the above relations[ H, y ] = X α> α ( H ) ( a α ( X α − X − α ) + b α i ( X α + X − α )) , and similarly with z in place of H . Still using the above relations, we obtain h [ H, y ]] , [ z, y ] i = 2 X α> α ( H ) α ( H ) (cid:16) a α + b α (cid:17) which is > α ( H ) , α ( H ) > a α , b α ∈ R . This finishes theproof, since ( Z ( x ) , y ) = − ([ H, y ] , [ z, y ]). (cid:3) Lagrangian vanishing cycles
We construct Lagrangian spheres inside regular fibres, which are our can-didates for vanishing cycles. The correct dimension of the desired spheres is n − n is thecomplex dimension of the adjoint orbit, and the real dimension of the flag F Θ where Θ = Θ ( H ) = { α ∈ Σ : α ( H ) = 0 } . The number of Lagrangianspheres to be found equals |W| , that is, the number of singularities.Here we assume that H ∈ cl a + and that H ∈ a + , hence H is regular.Recall that the symplectic form Ω on the orbit O ( H ) is the restriction ofthe imaginary part of the Hermitian form of g H τ ( X, Y ) = −h X, τ Y i . On the other hand, the real part is the inner product defined by B τ ( X, Y ) = − Re h X, τ Y i = − h X, τ Y i R , where h· , ·i R is the Cartan–Killing form of the realification of g . Thus, H τ ( X, Y ) = B τ ( X, Y ) + i Ω (
X, Y )and the equality Ω (
X, Y ) = B τ ( X, iY ) holds since H τ ( X, Y ) is Hermitian.We can search for an isotropic submanifold by taking a subspace V ⊂ g where H τ takes real values, and then check whether the intersection V ∩ g is indeed a submanifold.Two examples of subspaces where H τ takes real values are: i) the compactreal form u , where H τ is negative definite and ii) the symmetric part i u ,where H τ is positive definite.The intersection u ∩ O ( H ) is empty because the eigenvalues of ad ( X ),for X ∈ O ( H ) are real whereas those of ad ( Y ), for Y ∈ u are imaginary.The latter happens because ad ( Y ) is anti-symmetric with respect to the ORSE FUNCTIONS AND REAL LAGRANGIAN THIMBLES ON ADJOINT ORBITS10
Cartan–Killing form of u , see lemma 6. On the other hand, the intersection i u ∩ O ( H ) is the flag F Θ itself, since it is the orbit of the compact group U = exp u .Therefore, F Θ is an isotropic submanifold, in fact Lagrangian, and anysubmanifold of F Θ is isotropic as well. Moreover, the function f H ( x ) = h H, x i takes real values on F Θ = i u ∩ O ( H ). Since by hypothesis H isregular, it follows that the restriction f Θ H of f H to F Θ is a Morse func-tion. The origin H is a singularity and the hypothesis that H ∈ cl a + implies that H is an attractor (with negative definite Hessian). Therefore,the levels (cid:16) f Θ H (cid:17) − (cid:16) f Θ H ( x ) (cid:17) of f Θ H around H are codimension 1 spheres in F Θ . These levels are isotropic submanifolds. Clearly (cid:16) f Θ H (cid:17) − (cid:16) f Θ H ( x ) (cid:17) ⊂ ( f H ) − ( f H ( x )) and since dim ( f H ) − ( f H ( x )) = dim F Θ −
2, it follows thatfor x around H the spheres (cid:16) f Θ H (cid:17) − (cid:16) f Θ H ( x ) (cid:17) are Lagrangian cycles at thelevels ( f H ) − ( f H ( x )) of the Lefschetz fibrations.We can now carry out the analogous construction around other criticalpoints wH , w ∈ W . We use the following notation( ι ) Given a root α >
0, let u α = ( g α ⊕ g − α ) ∩ u and i u α = ( g α ⊕ g − α ) ∩ i u . Taking a Weyl basis X β ∈ g β , with β a root, these subspaces aregenerated by: • u α = span R { A α = X α − X − α , iS α = i ( X α + X − α ) } and • i u α = span R { iA α = i ( X α − X − α ) , S α = X α + X − α } .( ιι ) For w ∈ W , let Π w = Π + ∩ w − Π − be the set of positive roots thatare taken to negative roots by w .( ιιι ) For w ∈ W define the real vector subspace V w = h R ⊕ X α ∈ Π w u α ⊕ X α ∈ Π + \ Π w i u α . When w = 1 the subspace V = i u , since Π = ∅ . The subspaces V w ,1 = w ∈ W , will replace i u in the constructions of spheres around thecritical points wH . Lemma 10. H τ and the Cartan–Killing form h· , ·i take real values in V w .Proof. Both H τ and h· , ·i are real in each of the components of V w (posi-tive definite in h R and P α ∈ Π + \ Π w i u α and negative definite in P α ∈ Π w u α ).Moreover h R , u α and u β are orthogonal with respect to H τ and to h· , ·i if α = β . (cid:3) Consequently, the restriction of the imaginary part Ω of H τ to V w vanishesidentically. On the orbit O ( H ) we define a distribution ∆ w ( x ) ⊂ T x O ( H )by ∆ w ( x ) = V w ∩ T x O ( H ) . By lemma 10, the subspaces ∆ w ( x ) are isotropic with respect to the sym-plectic form Ω (restricted to the orbit). The goal is to prove that thisdistribution is integrable (at least around the singularity wH ). Once this ORSE FUNCTIONS AND REAL LAGRANGIAN THIMBLES ON ADJOINT ORBITS11 is accomplished, the integral submanifold passing through wH will be a La-grangian submanifold (for Ω). Consequently, a ball around the singularity wH , inside the integral submanifold will be our candidate to a Lagrangianthimble. Remark . A priori a distribution obtained by intersecting a fixed subspacewith the tangent spaces of an embedded submanifold (such as our distribu-tion ∆ w ) might not even be continuous (that is, admit local parametriza-tions by continuous fields). As an example, consider the case of the circle S = { x ∈ R : | x | = 1 } . The horizontal line { ( t,
0) : t ∈ R } contains thetangent space at (0 ,
1) however, it intersects in dimension zero the tangentspaces of points near (0 , wH (or any other singularity) the distribution is∆ w ( wH ) = X α ∈ Π w u α ⊕ X α ∈ Π + \ Π w i u α . This is due to the fact that the tangent space at wH is given by T wH O ( H ) = X α ∈ Π g α which intersects V w at u α and i u α , showing thatdim R ∆ w ( wH ) = 12 dim R O ( H ) . Hence, ∆ w ( wH ) is a Lagrangian subspace. It follows that dim R ∆ w ( wH ) ≥ dim ∆ w ( x ), x ∈ O ( H ), since the subspaces ∆ w ( x ) are isotropic for Ω.We will parametrize ∆ w around wH by Hamiltonian fields.Given X ∈ g define the real height function (with respect to the innerproduct B τ ) f X : O ( H ) → R by f X ( x ) = B τ ( X, x ) . Denote by ham f X the Hamiltonian field of f X with respect to Ω and bygrad f X its gradient with respect to B τ (both Ω and B τ are restricted to O ( H )). By definition, if v ∈ T x O ( H ) then( df X ) x ( v ) = Ω ( v, ham f X ( x )) = B τ ( v, grad f X ( x )) . The formula Ω (
X, Y ) = B τ ( X, iY ), guaranties thatΩ ( v, ham f X ( x )) = B τ ( v, i ham f X ( x )) = B τ ( v, grad f X ( x )) . Since this equality holds for all v ∈ T x O ( H ) it follows thatham f X ( x ) = − i grad f X ( x ) ,for all x ∈ O ( H ).A basis for ∆ w ( wH ) is given by ham f X ( iH ) with X belonging to { iA α , S α : α ∈ Π w } ∪ { A α , iS α : α ∈ Π + \ Π w } where A α = X α − X − α and S α = X α + X − α .Moreover, these Hamiltonian fields are tangent to the distribution ∆ w . ORSE FUNCTIONS AND REAL LAGRANGIAN THIMBLES ON ADJOINT ORBITS12 Real Lagrangian thimbles
Let u be the Lie algebra of K . We first we recall the following result. Proposition 12. [GGSM2, Prop 6.2]
The tangent space to
Γ ( k ◦ R w ) at ( x, y ) = ( x, k ◦ R w ( x )) is given by { ( A, Ad ( k ) A ) ∼ ( x, k ◦ R w ( x )) : A ∈ u } where ( A, Ad ( k ) A ) ∼ is the vector field on F H × F H ∗ = F ( H ,H ∗ ) induced by ( A, Ad ( k ) A ) ∈ u × u . For Lefschetz fibrations on an adjoint orbit O ( H ) we can obtain La-grangian submanifolds as graphs of symplectic maps on the correspondingflag. The idea of our construction is based on the following generalities. Let f : N → C be a Lefschetz fibration where the total space N is a Hermitianmanifold with Hermitian metric h , complex structure J , and K¨ahler formΩ. Set f = f + if and let V be a Lagrangian submanifold which containsa critical point x of f . Let g = g + ig be the restriction of f to V . Wedefine following gradient vector fields F = grad f , F = grad f , G = grad g , and G = grad g . Since f is a holomorphic function, df ( J v ) = idf ( v ) for all v ∈ T N . Thismeans df ( J v ) + idf ( J v ) = idf ( v ) − df ( v ) , hence df ( v ) = − df ( J v ). That is, h ( F , v ) = − h ( F , J v ) = h ( J F , v )which shows that F = J F F = J F . From these equalities it follows that x is a critical point of f if and only if x is a critical point of both f and f . The Hessians of f and f at thecritical point x are related as follows. If A and B are vector fields, thenHess f ( A, B ) =
BAf = Bh ( F , A )= Bh ( F , J A ) = B ( J A ) f = Hess f ( J A, B ) . If f has isolated critical points, then both f and f are Morse functions.The relation between the Hessians shows that at every critical point of f thenumber of positive eigenvalues of the Hessian equals the number of negativeeigenvalues. In fact, if Hess f ( A, A ) > f ( J A, J A ) = Hess f ( − A, J A ) = − Hess f ( J A, A ) = − Hess f ( A, A )hence the positive definite and negative definite parts have the same dimen-sion.To obtain the relation between F i and G i we observe that, since V is a La-grangian submanifold, the tangent space to N at a point y ∈ V decomposesinto T y N = T y V ⊕ J T y V y ∈ V, as J T y V is the orthogonal complement with respect to the Hermitian metric M ( · , · ), of T y V . Indeed, if u, v ∈ T y V then M ( u, J v ) = − Ω ( u, v ) = 0
ORSE FUNCTIONS AND REAL LAGRANGIAN THIMBLES ON ADJOINT ORBITS13 therefore the subspaces T y V and J T y V are orthogonal and have the samedimension, thus are complementary. Consequently, the following relationbetween F i and G i holds on points of V . Proposition 13. If y ∈ V then F ( y ) = G ( y ) − J G ( y ) and F ( y ) = G ( y ) − J G ( y ) .Proof. For the case of F take the decomposition F ( y ) = u + J v ∈ T y V ⊕ J T y V . Since g is the restriction of f to V , it follows that ( df ) y ( w ) =( dg ) y ( w ) if w ∈ T y V . Therefore, for w ∈ T y V ( dg ) y ( w ) = ( df ) y ( w ) = M ( F ( y ) , w )= M ( u + v, w ) = M ( u, w )and we see that u = G ( y ). Now take J w ∈ J T y V . So,( df ) y ( J w ) = − ( df ) y ( w ) = − ( dg ) y ( w )= − M ( G ( y ) , w )and M ( F ( y ) , J w ) = − M ( G ( y ) , w ), that is, M ( v, w ) = M ( J v, J w ) = − M ( G ( y ) , w ) .Since w is arbitrary, it follows that v = − G ( y ). Thus, for y ∈ VF = − J F = − J ( G − J G ) = G − J G . (cid:3) The expressions from proposition 13 show that if G = 0, then F = G and, consequently F is tangent to V . It follows that Corollary 14.
If the imaginary part is constant on the Lagrangian subva-riety V , then grad f is tangent to V . Consequently, we obtain the following method of constructing stable andunstable manifolds of grad f at a critical point x (in the case of Morsefunctions). Proposition 15.
Let V be a Lagrangian submanifold that contains a criticalpoint x of the function f = f + if that defines a Lefschetz fibration. Supposethat f is constant on V and that the restriction of the Hessian Hess ( f ) ( x ) tothe tangent subspace T x V is negative definite (respectively positive definite).Then, the stable (respectively unstable) manifold of g in V is an open subsetof the stable (respectively unstable) manifold of f .Proof. The Hessian of g is the restriction to T x V of the Hessian of f . Thehypothesis guaranties that the fixed point x is an attractor (respectivelyrepeller) of G = grad g . Consequently, in the negative definite case, thestable manifold of G is and open subset V that contains x . In this openset F coincides with G , since by hypothesis f is constant on V , that is, G = 0. Therefore, the stable manifold of G is contained in the stablemanifold of F . A similar argument handles the positive definite case. (cid:3) Since the levels of a Morse function in the neighborhood of an attractingor repelling singularity are spheres (by the Morse lemma), this propositionhas the following consequence.
ORSE FUNCTIONS AND REAL LAGRANGIAN THIMBLES ON ADJOINT ORBITS14
Corollary 16.
In the setup of proposition 15 consider a level g − { c } = f − ( c ) ∩ V with c near g ( x ) = f ( x ) such that c < g ( x ) in the negativedefinite case and c > g ( x ) in the positive definite case. Then, g − { c } is asphere of dimension dim V − . The sphere g − { c } in this corollary is a Lagrangian submanifold of thelevel f − { c } (since in proposition 15 we took the hypothesis that g is con-stant).The next goal is to construct a Lagrangian thimble having as boundary thesphere g − { c } contained in the Lagrangian submanifold V . For this observethat for any y ∈ N , the symplectic orthogonal of the fibre Φ y = f − { f ( y ) } is generated by F = grad f and J grad f = − grad f = − F . In fact, F ( y ) is the metric orthogonal of T y Φ y since it is gradient. However, Φ y isa complex submanifold, thus Ω ( F ( y ) , v ) = M ( F ( y ) , J v ) = 0 if v ∈ T y Φ y .It follows thatΩ ( J F ( y ) , v ) = M ( J F ( y ) , J v ) = M ( F ( y ) , v ) = 0if v ∈ T y Φ y , which shows that F ( y ) and J F ( y ) generate the symplecticorthogonal to Φ y .Consequently we obtain the following Lagrangian thimble for f . Theorem 17.
In the setup of proposition 15 take c near f ( x ) = g ( x ) .We have that g − [ c, g ( x )] = f − [ c, f ( x )] ∩ V in the negative definite case, or g − [ g ( x ) , c ] = f − [ f ( x ) , c ] ∩ V in the positive definite caseis homeomorphic to a closed ball in R dim V . This ball is a Lagrangian thimble.Proof.
In the negative definite case g − [ c, g ( x )] is the Lagrangian thimbleobtained by parallel transport of the Lagrangian sphere g − { c } along theline segment [ c, g ( x )] ⊂ R . In fact, if s ∈ [ c, g ( x )] and z ∈ g − { s } thenthe horizontal lift of the vector d/dt is a multiple of F ( z ). This happensbecause the horizontal lift is a vector W = aF ( z ) + bJ F ( z ), a, b ∈ R ,which satisfies df z ( W ) = ( df ) z ( W ) + i ( df ) z ( W ) = d/dt , thus, df z ( W ) isreal and therefore coincides with ( df ) z ( W ). This implies ( df ) z ( W ) = 0,so, 0 = M ( F , W ) = − M ( J F ( z ) , aF ( z ) + bJ F ( z ))= − bM ( J F ( z ) , J F ( z ))consequently, b = 0. In the negative definite case we have the coefficient a >
0, since f grows in the direction of F .Therefore, the parallel transport of a point of g − { c } along the segment[ c, g ( x )] follows the trajectories of F (reparametrized). Such trajectoriesconverge to x , thus, the union of parallel transports of s ∈ [ c, g ( x )] is theball g − [ c, g ( x )] = f − [ c, f ( x )] ∩ V .The same argument works in the positive definite case, with − F in placeof F . (cid:3) Definition 18.
A Lagrangian thimble inside a stable or unstable of sub-manifold constructed as in theorem 17 is called a real Lagrangian thimble ,since it is obtained by lifting of a real horizontal curve.
ORSE FUNCTIONS AND REAL LAGRANGIAN THIMBLES ON ADJOINT ORBITS15 The potential and graphs
The goal in this section is to analyze the behavior of the potential givenby the height function f H ( x ) = h x, H i on Lagrangian graphs. The casesof interest here are the graphs of the composites m ◦ R w with m in thetorus T = exp ( i h R ). Such graphs all pass through the critical points of f H . In fact, in the product F H × F H ∗ these critical points are given by( wH , ww H ∗ ) = ( wH , − wH ). Since m ◦ R w ( wH ) = Ad ( m ) ( ww H ∗ ) = ww H ∗ we see that these pairs belong to Γ ( m ◦ R w ).The Hessian of f H at a critical point is calculated considering everythingfrom the point of view of the adjoint orbit O ( H ) = Ad ( G ) H . In this casethe field e A induced by A ∈ g is linear e A = ad ( A ). Therefore e Af H ( x ) = h [ A, x ] , H i and the second derivative is e B e Af H ( x ) = h [ A, [ B, x ]] , H i . Thus,if x = wH is a critical point, thenHess ( f H ) (cid:16) e A ( x ) , e B ( x ) (cid:17) = −h [ B, wH ] , [ A, H ] i = −h [ wH , B ] , [ H, A ] i . (3)The goal now is to find the restriction of this Hessian to the tangentspaces to the graphs Γ ( m ◦ R w ), m ∈ T at the critical points. These tan-gent spaces were described in proposition 12 using the realization of thehomogenous space as an orbit inside the product F H × F H ∗ = F ( H ,H ∗ ).Such description must be translated to the viewpoint where the homoge-neous space is the adjoint orbit O ( H ) = Ad ( G ) H . This translationwill be made in the next proposition. First recall that from the point ofview of the open orbit G · ( H , − H ) ⊂ F H × F H ∗ the critical points are( wH , ww H ∗ ) = ( wH , − wH ), w ∈ W . Proposition 19.
Let m ∈ T = exp ( i h R ) and consider Γ ( m ◦ R w ) as aLagrangian submanifold of O ( H ) = Ad ( G ) · H . Then the tangent space to Γ ( m ◦ R w ) at the critical point wH , w ∈ W , is generated by the vectors (1) e X α ( wH ) − Ad ( m ) e X − α ( wH ) = [ X α , wH ] − [Ad ( m ) X − α , wH ] with α ( wH ) < and (2) f iX α ( wH )+Ad ( m ) f iX − α ( wH ) = i [ X α , wH ]+ i [Ad ( m ) X − α , wH ] with α ( wH ) < .Proof. By proposition 12 the tangent space to Γ ( m ◦ R w ) at the criticalpoint ( wH , − wH ) (seen as the orbit in the product) is generated by( A, Ad ( m ) A ) ∼ ( wH , − wH ) = (cid:16) e A ( wH ) , Ad ( m ) e A ( − wH ) (cid:17) with A ∈ u .The real compact form u is generated by i h R , A α = X α − X − α and Z α = i ( X α + X − α ) with α running through all roots. The field inducedby an element of i h R vanishes at the critical point wH hence it suffices toconsider the fields induced by A α and Z α .Choose a root α such that α ( wH ) <
0. Then, in F H , e A α ( wH ) = e X α ( wH ) and e Z α ( wH ) = f iX α ( wH ) since e X − α ( wH ) = 0.On the other hand, Ad ( m ) A α = Ad ( m ) X α − Ad ( m ) X − α and Ad ( m ) Z α =Ad ( m ) iX α + Ad ( m ) iX − α given that both Ad ( m ) X ± α and Ad ( m ) iX ± α belong to g ± α since Ad ( m ) g ± α = g ± α (because m ∈ T ). ORSE FUNCTIONS AND REAL LAGRANGIAN THIMBLES ON ADJOINT ORBITS16
Taking now the induced field on F H ∗ and using the fact that α ( wH ) < m ) e X α ( − wH ) = 0 on F H ∗ (since α ( − wH ) > m ) e A α ( − wH ) = − Ad ( m ) e X − α ( − wH ) and Ad ( m ) e Z α ( − wH ) = i Ad ( m ) e X − α ( − wH ).Now, the isomorphism between G · ( H , − H ) and O ( H ) takes a fieldinduced by an element of u to an induced field. Moreover, the isomorphismassociates ( wH , − wH ) ∈ F H × F H ∗ to wH ∈ O ( H ). This way, the iso-morphism takes (cid:16) e A ( wH ) , Ad ( m ) e A ( − wH ) (cid:17) to e A ( wH )+Ad ( m ) e A ( wH )(where the former e · means the field induced on F H and F H ∗ whereas thelatter the one induced on O ( H )).Hence the tangent space at the critical point wH ∈ O ( H ) is generatedby e X α ( wH ) − Ad ( m ) e X − α ( wH ) and f iX α ( wH )+ i Ad ( m ) e X − α ( wH ). (cid:3) The generators of the tangent space at Γ ( m ◦ R w ) of the previous propo-sition can also be described in the following simpler manner. Take H ∈ h R such that m = e iH . Then, Ad ( m ) X α = e iα ( H ) X α . This way, the vectorfields that provide the generators at wH become • e X α − Ad ( m ) e X − α = e X α − e − iα ( H ) e X − α with α ( wH ) < • f iX α + Ad ( m ) f iX − α = f iX α + e − iα ( H ) f iX − α with α ( wH ) < f H ) at wH using formula (3). Theelements of g which define the generating fields belong to g α ⊕ g − α . Hencethe Hessian vanishes at a pair of generators coming from distinct roots, since,with respect to the Cartan-Killing form, g ± α is orthogonal to g ± β if β = ± α .For the fields provided by a root α with α ( wH ) <
0, we obtain (in wH ): • Hess ( f H ) (cid:16) e X α − e − iα ( H ) e X − α , e X α − e − iα ( H ) e X − α (cid:17) = −h [ wH , X α − e − iα ( H ) X − α ] , [ H, X α − e − iα ( H ) X − α ] i .The second term equals −h α ( wH ) X α + e − iα ( H ) α ( wH ) X − α , α ( H ) X α + e − iα ( H ) α ( H ) X − α i . Now, h X α , X α i = h X − α , X − α i = 0 and h X α , X − α i = 1 (Weyl basis)therefore, the Hessian becomes − α ( wH ) α ( H ) e − iα ( H ) . • Hess ( f H ) (cid:16) f iX α + e − iα ( H ) f iX − α , f iX α + e − iα ( H ) f iX − α (cid:17) = −h [ wH , iX α + e − iα ( H ) iX − α ] , [ H, iX α + e − iα ( H ) iX − α ] i . That is, h α ( wH ) X α − e − iα ( H ) α ( wH ) X − α , α ( H ) X α − e − iα ( H ) α ( H ) X − α i . Thus the Hessian equals − α ( wH ) α ( H ) e − iα ( H ) . • Hess ( f H ) (cid:16) e X α − e − iα ( H ) e X − α , f iX α + e − iα ( H ) f iX − α (cid:17) = −h [ wH , X α − e − iα ( H ) X − α ] , [ H, iX α + e − iα ( H ) iX − α ] i . ORSE FUNCTIONS AND REAL LAGRANGIAN THIMBLES ON ADJOINT ORBITS17
That is, − i h α ( wH ) X α + e − iα ( H ) α ( wH ) X − α , α ( H ) X α − e − iα ( H ) α ( H ) X − α i = 0 . Summing up,
Proposition 20.
The Hessian of f H restricted to the tangent space T wH (cid:16) Γ (cid:16) e iH ◦ R w (cid:17)(cid:17) is diagonalizable in the basis { (cid:16) e X α − e − iα ( H ) e X − α (cid:17) ( wH ) , (cid:16) f iX α + e − iα ( H ) f iX − α (cid:17) ( wH ) : α ( wH ) < } . The diagonal elements are given by − α ( wH ) α ( H ) e − iα ( H ) . For example, the orbit of the compact group (the zero section in the iden-tification with the cotangent bundle) is Γ ( R w ). If w = 1 and H = 0, then − α ( wH ) α ( H ) e − iα ( H ) = − α ( H ) α ( H ) which is < α ( H ) < α < α ( H ) <
0. That is, the Hessian isnegative definite, which was to be expected given that the critical point H is a maximum of Re f H on the zero section.We now consider graphs in the compactification F H µ × F ∗ H µ The isomor-phism between the open orbit in F H µ × F H ∗ µ (diagonal action) and the orbit G · ( v ⊗ ε ) of v ⊗ ε ∈ V ⊗ V ∗ (representation of G ) leads to a conve-nient description of the intersection of graphs of anti-holomorphic functions F H µ → F H ∗ µ with the open orbit.We return to the anti-holomorphic functions m ◦ R w : F H µ → F H ∗ µ with m ∈ T , the maximal torus. The submanifold determined by graph ( R w ) in R w on F H µ × F H ∗ µ is the orbit of the compact group K through ( v , ε ). Thisorbit stays inside G · ( v , ε ) and is identified with the K -orbit of v ⊗ ε in V ⊗ V ∗ (by equivariance). The isomorphism with the adjoint orbit Ad ( G ) H µ associates this K -orbit inside V ⊗ V ∗ with the intersection i u ∩ Ad( G ) H µ (the Hermitian matrices in the case of sl ( n + 1 , C ) or else the zero sectionof T ∗ F H µ ). This set is formed by the elements v ⊗ ε ∈ G · ( v ⊗ ε ) suchthat ker ε = v ⊥ (with respect to the K -invariant Hermitian form ( · , · ) µ ),since u ∈ K is an isometry of ( · , · ) µ and ker ε = v ⊥ . The converse is trueas well: if v ⊗ ε ∈ G · ( v ⊗ ε ) and ker ε = v ⊥ then v ⊗ ε ∈ Γ ( R w ). In fact,if ker ε = v ⊥ and X ∈ u then ρ µ ( X ) is anti-Hermitian, thus ( ρ µ ( X ) v, v ) µ ispurely imaginary and since ker ε = v ⊥ , then ε ( ρ µ ( X ) v ) is purely imaginaryas well. Therefore, h M ( v ⊗ ε ) , X i = ε ( ρ µ ( X ) v ) is imaginary for arbitrary X ∈ u , which implies that M ( v ⊗ ε ) ∈ i u .Summing up, we obtain the following description of Γ ( R w ) regarded asa subset of G · ( v ⊗ ε ). Consider Φ − (Γ ( R w )) ⊂ G · ( v ⊗ ε ), which,abusing notation, we also denote by Γ ( R w ). We have: Proposition 21.
Γ ( R w ) = { v ⊗ ε ∈ G · ( v ⊗ ε ) : ker ε = v ⊥ } . Consider now the graph of m ◦ R w : F H µ → F H ∗ µ with m ∈ T . In generalΓ ( m ◦ R w ) ⊂ F H µ × F H ∗ µ is not contained in the open orbit and, conse-quently, intercepts this orbit in a noncompact subset. In either case, take ORSE FUNCTIONS AND REAL LAGRANGIAN THIMBLES ON ADJOINT ORBITS18 the subgroup U m = { (cid:16) u, mum − (cid:17) ∈ U × U : u ∈ U } . The graph Γ ( m ◦ R w ) is the orbit of K m through ( v , ε ). This happensbecause, if x = u · v ∈ F H µ then R w ( x ) = u · ε therefore( x, m ◦ R w ( x )) = ( x, m · uε ) . This means that Γ ( m ◦ R w ) is formed by elements of the form ( x, my ) with( x, y ) ∈ Γ ( R w ), that is,Γ ( m ◦ R w ) = m (Γ ( R w ))where m ( x, y ) = ( y, mx ). Passing to the realization inside V ⊗ V ∗ we obtaina geometric realization of Φ − (Γ ( m ◦ R w )), also denoted by Γ ( m ◦ R w ): Proposition 22.
Γ ( m ◦ R w ) = { v ⊗ ρ ∗ µ ( m ) ε ∈ G · ( v ⊗ ε ) : ker ε = v ⊥ } . Now we have the setup to prove that f H is real on Γ ( m ◦ R w ). Thisis essential to obtain real Lagrangian thimbles. With the realization of G/Z µ as an orbit in V ⊗ V ∗ the proof that f H is real greatly simplifies.Actually, this is not only true for elements m ∈ T , but for more generallinear transformations of V (or more precisely, of V ∗ ).Before stating the result, observe that the function f H is a priori definedon the orbit G · ( v ⊗ ε ) and is given by f H ( v ⊗ ε ) = ε ( ρ µ ( H ) v ). Fromthis expression we see that f H extends to a linear functional of V ⊗ V ∗ , thatis, it is defined on points outside the orbit G · ( v ⊗ ε ) as well. Proposition 23.
Let D : V → V be a linear transformation that is diago-nalizable on a basis adapted to the root subspaces and consider the set D (Γ ( R w )) = { v ⊗ D ∗ ε ∈ V ⊗ V ∗ : ker ε = v ⊥ } where D ∗ ε = ε ◦ D . Suppose that D has real eigenvalues. Then, f H assumesreal values on D (Γ ( R w )) .Proof. If v ⊗ Dε ∈ D (Γ ( R w )) then f H ( v ⊗ D ∗ ε ) = ε ( Dρ µ ( H ) v ) = tr (( v ⊗ ε ) Dρ µ ( H )) . On a basis adapted to the root subspaces, Dρ µ ( H ) is diagonal with realeigenvalues. If this basis is orthonormal, then v ⊗ ε has a Hermitian matrix,and therefore real diagonal entries. Hence, the last term of the equalityabove is real, and f H ( v ⊗ D ∗ ε ) is real as well. (cid:3) Corollary 24. If m ∈ T satisfies m = 1 then f H is real on Γ ( m ◦ R w ) .Proof. In fact, if m = 1 then the eigenvalues of m are ± m ∈ T , ρ µ ( m ) = ± id on the root spaces. (cid:3) Further properties of Lagrangian submanifolds inside products of flagsand their intersection numbers are described in [GSMV].
ORSE FUNCTIONS AND REAL LAGRANGIAN THIMBLES ON ADJOINT ORBITS19 Minimal semisimple orbits
We now focus our attention on the case of minimal semisimple orbits,by considering the orbits of sl ( n + 1 , C ) of smallest dimension. The corre-sponding flag manifolds are F H = P n and F H ∗ = Gr n ( n + 1) = P n ∗ for H = (cid:16) n − n × n (cid:17) H ∗ = (cid:16) n × n − n (cid:17) . These are dual to each other, so itsuffices to consider the case of P n .For minimal flags it is possible to describe real Lagrangian thimbles of f H for all singularities via the graphs graph( m ◦ R w ) with m ∈ T . This happensbecause for each of these singularities there are elements m ∈ T such thatHess ( f H ) restricted to Γ( m ◦ R w ) is either positive definite or negativedefinite. Together with the fact (proved below) that the imaginary partof f H is constant over the corresponding graphs, we obtain the stable andunstable manifolds of grad (Re f H ) and consequently also real Lagrangianthimbles.Our general construction specializes to this situation as follows: • The diagonal action of Sl( n + 1 , C ) on F H × F H ∗ = P n × P n ∗ has2 orbits. An open dense one formed by pairs of transversal vectors( V, W ) ∈ P n × P n ∗ with V ∩ W = { } ; and another orbit formed byvectors ( V, W ) ∈ P n × P n ∗ with V ⊂ W . • The diffeomorphism between the open orbit and the adjoint orbit O ( H ) associates to a pair ( V, W ) ∈ P n × P n ∗ with V ∩ W = { } thelinear transformation T : C n +1 → C n +1 with T v = nv if v ∈ V and T v = − v if v ∈ W . • The map R w : P n → P n ∗ associates to a subspace of dimension 1of C n +1 its orthogonal complement with respect to the canonicalHermitian form of C n +1 . • W is the group of permutation of n + 1 elements. • The set of critical points of the potential in P n (orbit of the Weylgroup at the origin) has n + 1 elements which are the subspaces [ e j ], j = 1 , . . . , n + 1, generated by the vectors of the canonical basis of C n +1 . The origin is given by [ e ], so that under the identificationsthis origin gets identified to H ; whereas [ e j ] gets identified to wH for any permutation w such that w (1) = j . • The roots are α ij with i = j , with corresponding eigenspaces gener-ated by the elementary matrices X α ij = X ij (with 1 in the position ij and 0 elsewhere). • The roots α with α ( H ) < α j with 2 ≤ j ≤ n + 1and consequently the tangent space at the origin is identified withthe space of column matrices (cid:16) ∗ n × n (cid:17) . • The tangent spaces at the other critical points are obtained viapermutation: let w be the permutation such that w (1) = j , then α ( wH ) < α = α ij with i = j . Thus, the tangentspace at [ e j ] ≈ wH is formed by matrices whose nonzero entriesbelong to the j -th column and that have a zero on entry jj .Assume now, once and for all that n is even, hence we are working in sl ( n + 1) with matrices that have an odd number of diagonal entries. ORSE FUNCTIONS AND REAL LAGRANGIAN THIMBLES ON ADJOINT ORBITS20
Definition 25.
Given a singularity [ e j ] define m ± j ∈ T = exp ( i h R ) asfollows: m ± j = ∓ ( − j diag(1 , . . . , , ± j , − , . . . , − j ).The fact that the number of diagonal entries is odd, guaranties that inall cases det m ± j = 1 and, therefore, m ± j indeed belongs to T . (Although inthe first and last cases this is also true for even n .) Proposition 26.
Consider the singularity [ e j ] ≈ wH . The restriction of Hess( f H ) to the tangent space T [ e j ] Γ( m ◦ R w ) is positive definite if m = m + j and negative definite if m = m − j .Proof. By proposition 20 the restriction of Hess( f H ) is diagonal in the ba-sis given by roots α ( wH ) <
0. The diagonal entries associated to the 2dimensional subspace corresponding to the root α are: − α ( wH ) α ( H ) e − iα ( H ) and α ( wH ) < H is such that m = exp iH .As mentioned earlier, if wH ≈ [ e j ] then the roots α such that α ( wH ) < α kj with k = j . Also, recall that from the start H was takenin the positive Weyl chamber h + R . Therefore, for these roots we have • α kj ( H ) > k < j , since α kj >
0, and • α kj ( H ) < k > j , since α kj < m = exp iH = diag { ε , . . . , ε n } , then e − iα kj ( H ) = ε k ε j . Hence, • for m + j = exp iH , we have e − iα kj ( H ) = (cid:26) k < j − k > j and • for m − j = exp iH , we have e − iα kj ( H ) = (cid:26) − k < j k > j. Combining the signs of e − iα kj ( H ) and of α kj ( H ) we see that the HessianHess ( f H ) is positive definite on T [ e j ] Γ( m + j ◦ R w ) and negative definite on T [ e j ] Γ( m − j ◦ R w ). (cid:3) The goal now is to show that the imaginary part of f H is constant onΓ( m ± j ◦ R w ). These graphs intercept the zero section Γ( R w ) where f H isreal. So, we wish to show that f H is real on Γ( m ± j ◦ R w ).For a transversal pair ( V, W ) ∈ P n × P n ∗ denote by Φ ( V, W ) the lineartransformation in O ( H ) corresponding to the pair. As mentioned earlier,Φ ( V, W ) v = nv if v ∈ V and Φ ( V, W ) w = − w then w ∈ W . To showthat f H is real on the graph Γ( m ± j ◦ R w ) we will prove that the diagonal ofΦ ( V, W ) has real entries.The following calculations work for any m ∈ T such that m = 1. So,fix once and for all m ∈ T such that m = 1. Take [ u ] ∈ P n . Then R w [ u ] = [ u ] ⊥ and Φ (cid:16) [ u ] , [ u ] ⊥ (cid:17) is a Hermitian matrix whose diagonal entries ORSE FUNCTIONS AND REAL LAGRANGIAN THIMBLES ON ADJOINT ORBITS21 are real. Since, m ◦ R w [ u ] = m [ u ] ⊥ , we ought to show that Φ (cid:16) [ u ] , m [ u ] ⊥ (cid:17) has real diagonal entries. Here assume that [ u ] and m [ u ] ⊥ are transversal,that is ( u, mu ) = 0.Suppose | u | = 1 and take an orthonormal basis { v , . . . , v n } of [ u ] ⊥ with respect to the canonical Hermitian form ( · , · ) of C n +1 . The basis { u, v , . . . , v n } is orthonormal in C n +1 and since m ∈ SU ( n ) the basis β = { mu, mv , . . . , mv n } is orthonormal as well, whereas the basis γ = { u, mv , . . . , mv n } is not orthonormal. By the definition of Φ the matrix ofΦ (cid:16) [ u ] , m [ u ] ⊥ (cid:17) on the basis γ is given by h Φ (cid:16) [ u ] , m [ u ] ⊥ (cid:17)i γ = diag { n, − , . . . , − } . The matrices for the change of basis between β and γ are[ I ] γβ = ( u, mu ) 0 · · · u, mv ) 1 . . . ...... ... . . . 0( u, mv n ) 0 · · · with inverse [ I ] βγ = / ( u, mu ) 0 · · · − ( u, mv ) / ( u, mu ) 1 . . . ...... ... . . . 0 − ( u, mv n ) / ( u, mu ) 0 · · · . Therefore, h Φ (cid:16) [ u ] , m [ u ] ⊥ (cid:17)i β = [ I ] γβ h Φ (cid:16) [ u ] , m [ u ] ⊥ (cid:17)i γ [ I ] βγ is given by h Φ (cid:16) [ u ] , m [ u ] ⊥ (cid:17)i β = n · · · ν ( u, mv ) − ν ( u, mv n ) 0 · · · − , where we have set ν := ( n + 1) / ( u, mu ).We now claim that the diagonal elements of Φ (cid:16) [ u ] , m [ u ] ⊥ (cid:17) are given by (cid:16) Φ (cid:16) [ u ] , m [ u ] ⊥ (cid:17) e j , e j (cid:17) where e j is an element of the canonical basis. To seethis, take an arbitrary x ∈ C n +1 and write h Φ (cid:16) [ u ] , m [ u ] ⊥ (cid:17)i β = A + B with A = n · · · ν ( u, mv ) 0 . . . ...... ... . . . 0 ν ( u, mv n ) 0 · · · ORSE FUNCTIONS AND REAL LAGRANGIAN THIMBLES ON ADJOINT ORBITS22 and B = · · · − · · · − . In coordinates x = ( x, mu ) mu + ( x, mv ) mv + · · · + ( x, mv n ) mv n . So,[ Ax ] β = n · · · ν ( u, mv ) 0 . . . ...... ... . . . 0 ν ( u, mv n ) 0 · · · ( x, mu )( x, mv )...( x, mv n ) = n ( x, mu ) ν ( u, mv ) ( x, mu )... ν ( u, mv n ) ( x, mu ) and [ Bx ] β = · · · − · · · − ( x, mu )( x, mv )...( x, mv n ) = − ( x, mv )... − ( x, mv n ) . Since β is an orthonormal basis, we have that (cid:16) Φ (cid:16) [ u ] , m [ u ] ⊥ (cid:17) x, x (cid:17) is givenby the sum of • ( Ax, x ) = n | ( x, mu ) | + n ( u,mu ) ( x, mu ) P nj =1 ( u, mv j ) ( x, mv j ) and • ( Bx, x ) = − P nj =1 | ( x, mv j ) | .In this sum, the only part that is not evidently real is n ( u, mu ) ( x, mu ) n X j =1 ( u, mv j ) ( x, mv j ) . To analyze this part, start up observing that ( u, mu ) ∈ R since m is anisometry of ( · , · ). Thus, ( u, mu ) = ( mu, u ) = ( u, mu ). So, we ought to verifythat ( x, mu ) n X j =1 ( u, mv j ) ( x, mv j ) ∈ R (4)whenever x is an element of the canonical basis. This works specifically foran element x of the canonical basis, because in this case mx = ± x , thatis, x belongs to an eigenspace of m . The sum in (4) can be rewritten as P nj =1 ( mu, v j ) ( mx, v j ), because m ∈ SU( n ). It can also be expressed as n X j =1 ( mu, ( mx, v j ) v j ) = mu, n X j =1 ( mx, v j ) v j . The second factor is the orthogonal projection proj ( mx ) of mx over [ v , . . . , v n ] =[ u ] ⊥ . Since ( mu, proj ( mx )) = (proj ( mu ) , mx ) it then follows that the sumin (4) is ( mx, u ) n X j =1 ( mu, v j ) v j , mx . ORSE FUNCTIONS AND REAL LAGRANGIAN THIMBLES ON ADJOINT ORBITS23
Finally, since mx = ± x (as it happens for the elements of the canonicalbasis) the previous expression becomes( x, u ) n X j =1 ( mu, v j ) v j , x . (5)We can now prove this expression is real. Lemma 27.
Expression (5) is real.Proof.
Let E ± be the eigenspaces associated to the eigenvalues ± m (since m = 1), and write u = u + + u − ∈ E + ⊕ E − (that is, u + = 1 / u + mu )and u + = 1 / u − mu ). Then, for each index j , 0 = ( u, v j ) = (cid:0) u + , v j (cid:1) +( u − , v j ), that is, (cid:0) u − , v j (cid:1) = − (cid:16) u + , v j (cid:17) . It follows that ( mu, v j ) = (cid:0) u + − u − , v j (cid:1) = 2 (cid:0) u + , v j (cid:1) = − u − , v j ). Suppose,for example, that x ∈ E + . Then, ( x, u ) = (cid:0) x, u + (cid:1) , since E + is orthogonal to E − , given that m is unitary. Thus (5) can be rewritten as n X j =1 ( x, u ) ( mu, v j ) ( v j , x ) = 2 n X j =1 (cid:16) x, u + (cid:17) (cid:16) u + , v j (cid:17) ( v j , x )= 2 n X j =1 (cid:16) x, u + (cid:17) (cid:16) u + , ( x, v j ) v j (cid:17) = 2 (cid:16) x, u + (cid:17) n X j =1 (cid:16) u + , ( x, v j ) v j (cid:17) = 2 (cid:16) x, u + (cid:17) u + , n X j =1 ( x, v j ) v j . The sum inside the Hermitian form is n X j =1 ( x, v j ) v j = ( x, u ) u + n X j =1 ( x, v j ) v j − ( x, u ) u = x − ( x, u ) u since { u, v , . . . , v n } is an orthonormal basis. So, the last term is given by2 (cid:16) x, u + (cid:17) (cid:16) u + , x (cid:17) − (cid:16) x, u + (cid:17) (cid:16) u + , ( x, u ) u (cid:17) =2 (cid:16) x, u + (cid:17) (cid:16) u + , x (cid:17) − (cid:16) x, u + (cid:17) ( u, x ) (cid:16) u + , u (cid:17) . To see that this is real observe that the first term of the right hand side is | (cid:0) x, u + (cid:1) | . As for the second term, ( u, x ) = (cid:0) u + , x (cid:1) is (cid:0) u + , u (cid:1) = (cid:0) u + , u + (cid:1) ,so, the second term is 2 | (cid:0) x, u + (cid:1) | (cid:0) u + , u + (cid:1) which is also real. (cid:3) Summing up, we have obtained:
Proposition 28. If m ∈ T with m = 1 and V ∈ P n is such that V doesnot belong to m ◦ R w ( V ) = mV ⊥ then the matrix of Φ (
V, m ◦ R w V ) hasreal diagonal entries (in the canonical basis). In conclusion:
ORSE FUNCTIONS AND REAL LAGRANGIAN THIMBLES ON ADJOINT ORBITS24
Theorem 29.
Let m ± j be as in definition 25. Then, the stable and unstablemanifolds of grad ( Re f H ) at the critical point [ e j ] are open in the graph Γ( m ± j ◦ R w ) . The real Lagrangian thimbles are closed balls contained in thegraph Γ( m ± j ◦ R w ) . Acknowledgements
Gasparim was partially supported by a Simons Associateship Grant of theAbdus Salam International Centre for Theoretical Physics and by the Vice-Rector´ıa de Investigaci´on y Desarrollo Tecnol´ogico of Universidad Cat´olicadel Norte, Chile. San Martin was partially supported by CNPq grant no.303755/09-1 and Fapesp grant no. 2018/13481. We thank F. Valencia forpointing out a few corrections needed on an early version of the text.
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