aa r X i v : . [ m a t h . S G ] M a y A MONOTONE LAGRANGIAN CASEBOOK
JACK SMITH
Abstract.
We present an array of new calculations in Lagrangian Floer theory which demonstrateobservations relating to symplectic reduction, grading periodicity, and the closed–open map. Wealso illustrate Perutz’s symplectic Gysin sequence and the quilt theory of Wehrheim and Woodward. Introduction
Given a monotone Lagrangian submanifold L of a symplectic manifold X , a fundamental invariantis its Floer cohomology. This describes the endomorphisms of L in the montone Fukaya categoryof X , and its non-vanishing indicates that L cannot be displaced from itself by a Hamiltonianflow, but in general it is difficult to calculate. In this note we make some simple observationsabout its properties and exploit them to study specific examples. In addition, we give some explicitcomputations in p CP q and CP ˆ CP which demonstrate Perutz’s symplectic Gysin sequence [29]and Wehrheim–Woodward’s quilt theory [37], for which there are few concrete calculations in theliterature. Throughout we pay particular attention to relative spin structures, and chase throughthe signs on which the answers are subtly dependent.1.1. Setup.
We assume that X is compact or tame (convex or geometrically bounded) at infinity,and that L is closed, connected and monotone, meaning that the Maslov index and area morphisms µ : π p X, L q Ñ Z and ω : π p X, L q Ñ R are positively proportional. We also require that the minimal Maslov number N L P Z ą Y t8u ,meaning the positive generator of µ p π p X, L qq , is at least 2. If the characteristic of the coefficientfield K is not 2 then we assume that L is orientable (which automatically makes N L even and henceat least 2) and relatively spin, and endowed with a choice of relative spin structure s as in Section 2.2.Associated to s is a background class b in H p X ; Z { q . The choice of s orients the moduli spaces ofpseudoholomorphic discs with boundary on L , whose counts are the key ingredients in Floer theory. L may also be equipped with a flat line bundle L over K .We denote the triple p L, s, L q by L and call it a monotone Lagrangian brane . It has a Floercohomology algebra HF ˚ p L , L ; Λ q over the Laurent polynomial ring Λ “ K r T ˘ s , in which the‘Novikov variable’ T has degree N L . The construction and basic properties of this algebra aredescribed in [7], where it is called the Lagrangian quantum homology of L (in contrast to [7] we usecohomological grading). This is our main object of study.We summarise some of its properties in Section 2, but for now it suffices to point out the following.There is a unital Λ-algebra homomorphism, the (length-zero) closed–open string map CO : QH ˚ p X, b ; Λ q Ñ HF ˚ p L , L ; Λ q , from the quantum cohomology of X (with background class b ), and this induces the ‘quantummodule action’ of QH ˚ on HF ˚ . There is also a multiplicative spectral sequence E “ H ˚ p L ; K q b K Λ ùñ HF ˚ p L , L ; Λ q , originally due to Oh [28]. We say L is wide if we have HF ˚ p L , L ; Λ q – H ˚ p L ; K q b K Λ as gradedΛ-modules, and narrow if HF ˚ p L , L ; Λ q “ Main results.
The bulk of the paper comprises worked examples, in which we compute HF ˚ p L , L ; Λ q as a Λ-algebra or -module for various Lagrangians L . After the brief Floer the-ory review in Section 2, each section focuses on a different technique, and can be read independentlyof the others. The only exception is that the examples studied in Section 7 are defined in Section 6.In Section 3, for each sequence k , . . . , k r of positive integers we construct a Lagrangian embeddingof the flag variety F p d , . . . , d r q in the monotone product of Grassmannians Gr p k , n qˆ¨ ¨ ¨ˆ Gr p k r , n q ,where d j : “ k ` ¨ ¨ ¨ ` k j and n : “ d r . We equip it with a specific choice of relative spin structureand the trivial flat line bundle, and combining CO with knowledge of the quantum cohomology ofGrassmannians we compute Theorem 1 (Theorem 3.9) . The Floer cohomology algebra of this Lagrangian is Λ r c , , . . . , c ,k , c , , . . . , c r,k r s N r ˚ j “ p ` c j, ` ¨ ¨ ¨ ` c j,k j q “ ` T, where ˚ denotes the Floer product, c j,k has degree k , and the Novikov variable T has degree n . The importance of the relatively spin condition becomes manifest in this calculation: if one triesto use a background class b which does not admit a relative spin structure then one obtains relationswhich are only consistent in characteristic 2.This Lagrangian is constructed by symplectic reduction using a Hamiltonian action of U p k q ˆ¨ ¨ ¨ ˆ U p k r q on C n , and along the way we prove the following monotonicity transfer property forgeneral symplectic reductions. Theorem 2 (Proposition 3.2) . Suppose X is a symplectic manifold carrying a Hamiltonian actionof a compact connected Lie group K , such that K acts freely on the zero set µ ´ p q of the momentmap. If L Ă X is a K -invariant monotone Lagrangian submanifold contained in µ ´ p q then L { K is monotone in X {{ K . In particular, X {{ K is (spherically) monotone. This allows one to deduce monotonicity of the reduced manifold X {{ K or Lagrangian L { K frommonotonicity of L , which may be much simpler topologically.In Section 4 we study periodicity in the grading of HF ˚ using the discrete Fourier transform.This gives constraints on the minimal Maslov number N L for certain Lagrangians: Theorem 3 (Theorem 4.8) . If H ˚ p L ; K q is an exterior algebra on generators of degree k ´ ď¨ ¨ ¨ ď k r ´ r , and the quantum cohomology of X contains an invertible element of even degree q ď k r then N L is at most k r . In particular, if L is a monotone torus and QH ˚ p X q contains aninvertible element of degree then N L “ . By a more careful analysis we prove the following dichotomy for Lagrangian embeddings of PSU p n q of high minimal Maslov index. Theorem 4 (Theorem 4.11) . Suppose L is a Lagrangian embedding of PSU p n q in a closed mono-tone symplectic manifold X , whose quantum cohomology contains an invertible element of degree .Assume n ě and N L ě n . Then we have N L “ n and either:(i) K has prime characteristic p and n is a power of p , in which case L is wide for all linebundles L and all relative spin structures.(ii) Otherwise L is narrow over K for all such L and s . This completes partial computations [25, Theorem 8, Proposition 27], [20, Example 7.2.3] of afamily of Lagrangian PSU p n q ’s in CP n ´ .Section 5 begins with the following easy result. Theorem 5 (Proposition 5.2) . If E Ñ X is a complex vector bundle whose restriction to L istrivial, and j ă N L { , then we have CO p c j p E qq “ , where c j p E q is the j th Chern class of E . Although this result is not especially noteworthy in itself, it is surprisingly useful, and we demon-strate this by considering a family of Lagrangians L Ă Gr p n ´ k, n q ˆ CP kn ´ diffeomorphic tothe projective Stiefel manifold parametrising projective k -frames in C n . These appear in [39] andspecialise to the above family of PSU p n q ’s in the case k “ n . We show MONOTONE LAGRANGIAN CASEBOOK 3
Theorem 6 (Theorem 5.3) . If p denotes char K and p r its greatest power dividing n (interpreted as if p “ ) then either: k ď p r , in which case L is wide for all choices of relative spin structure andflat line bundle; or k ą p r , in which case L is narrow for all such choices. The wideness result was known by Zapolsky but the narrowness part is new.Finally, in Sections 6 and 7 we study a monotone Lagrangian embedding of SO p q in p CP q andof the lens space L p , q in CP ˆ CP . The former was studied in [33, Section 5], where we showedthat it is narrow unless char K is 3 or 5, depending on the choice of relative spin structure. We alsoshowed that it is wide in the char K “ Theorem 7 (Theorem 6.4, Section 7.2) . The Lagrangian SO p q in p CP q is also wide in the char K “ case. We give two proofs of this result. In Section 6.2 we use Perutz’s Gysin sequence, viewing theLagrangian as a circle bundle over the final CP factor. This expresses HF ˚ as the cone on aquantum version of ‘cupping with the Euler class’ H ˚ p CP q Ñ H ˚ p CP q . In Section 7.2 meanwhile,we use quilt theory (summarised in Section 7.1) to relate L to the Chekanov torus in CP ˆ CP .The Floer theory of the latter is well-understood, and using a result of Wehrheim and Woodward[37, Theorem 6.3.1] we transfer this knowledge to L . This requires computing a certain inducedrelative spin structure, which we do by an indirect method, and the conditions on char K emerge.We apply the same methods to the Lagrangian lens space to obtain Theorem 8 (Theorem 6.9, Section 7.3) . The Lagrangian L p , q in CP ˆ CP is wide when char K is or , depending on the choice of relative spin structure, and is narrow otherwise. Acknowledgements.
Much of this work was carried out whilst I was an EPSRC-funded PhDstudent in Cambridge, and I am grateful to my supervisor, Ivan Smith, for all of his help over my timethere (one small part of which was the suggestion of applying the Gysin sequence to the examplesin Section 6). I also thank Tim Perutz, Nick Sheridan, Brunella Torricelli, Chris Woodward, andFrol Zapolsky for helpful discussions. I am supported by EPSRC grant [EP/P02095X/1].2.
Floer theory prerequisites
Monotonicity.
Recall that a Lagrangian L Ă p
X, ω q is monotone if there exists a positivereal number λ L such that ω “ λ L µ as homomorphisms π p X, L q Ñ R , where ω is the area and µ the Maslov index. Similarly X is monotone if there exists a positive λ X such that ω “ λ X c p X q as homomorphisms π p X q Ñ R . Assuming π p X q ‰
0, which will always be the case in ourexamples, monotonicity of L implies montonicity of X with λ L “ λ X ; we call this common valuethe monotonicity constant of X or L . This is because µ restricts to 2 c p X q on π p X q . By the longexact sequence of the pair p X, L q , if X is monotone and π p L q is torsion then L is also monotone.2.2. Relative spin structures.
A relative spin structure on a Lagrangian orients the modulispaces of holomorphic discs which are counted by the Floer differential. The mechanics of this donot concern us, but we will need to manipulate relative spin structures and how they change signs.First we recap the basic definitions. Suppose V is an orientable real vector bundle V over a space M . We shall assume rank V ě
2, since ranks 0 and 1 require special treatment but are rather trivial.The second Stiefel–Whitney class w p V q is the obstruction to lifting the GL ` -frame bundle of V to a principal GL „` -bundle, where GL ` is the orientation-preserving subgroup of GL p rank V, R q andGL „` is its unique connected double cover. V is spin if and only if w p V q “
0, and in this case a spinstructure is a choice of such a lift of the frame bundle. The set of spin structures forms a torsor for H p M ; Z { q . If M is an orientable manifold and V “ T M then we talk simply of w p M q and spinstructures on M . Definition 2.1 ([22, pp. 675–676], as reformulated in [36, Remark 3.1.3, Proposition 3.1.5(b)]) . Anorientable submanifold M of a manifold N is relatively spin if there exists a class b in H p N ; Z { q with b | M “ w p M q . In this case, a relative spin structure on M comprises a choice of b (thebackground class) and an equivalence class of ˇCech 1-cochain describing a principal GL „` -bundle, JACK SMITH with cocycle condition twisted by b , which lifts the frame bundle. Here we are viewing the coefficientgroup Z { b as the deck group of the covering GL „` Ñ GL ` . The set of relative spin structuresforms a torsor for H p N, M ; Z { q . For an orientable vector bundle B on N , a relative spin structurewith background class w p B q is equivalent to a spin structure on B | M ‘ T M . We will usually beinterested in the case where N is symplectic and M Lagrangian. {{ The construction of orientations on holomorphic disc moduli spaces is described in [22, Chapter8]. All we need to know is how changing relative spin structure affects these orientations.
Lemma 2.2 (de Silva [17, Theorem Q], Cho [14, Theorem 6.4], Fukaya–Oh–Ohta–Ono [22, Propo-sition 8.1.16]) . If two relative spin structures differ by a class ε P H p X, L ; Z { q then the associatedorientations on the moduli space of discs in class A P H p X, L ; Z q differ by p´ q x ε,A y . (cid:3) Similarly, the effect of a background class b on quantum cohomology is to modify counts of rationalcurves in class A P H p X ; Z q by p´ q x b,A y .2.3. Floer theory as a deformation of Morse theory.
Given a monotone Lagrangian L Ă X as in Section 1.1, its Floer theory can be regarded as a deformation of Morse theory in the followingsense. Biran–Cornea [7] construct a ‘pearl’ model for HF ˚ whose underlying chain complex is theMorse complex of a Morse function on L , tensored with the ring Λ. The differential, product, andclosed–open map only involve non-negative powers of T , so respect the filtration of the complexby T -exponent, and at the associated graded level (i.e. the T terms) they coincide with the cor-responding classical operations: the Morse differential, the cup product, and the Morse restrictionmap H ˚ p X q Ñ H ˚ p L q (recall that H ˚ p X ; Λ q is canonically identified with H ˚ p X ; K q b K Λ as aΛ-module).The spectral sequence induced by this filtration is precisely the Oh spectral sequence, and the r thpage differential encodes the T r terms. In particular, it maps the surviving elements of H ˚ p L ; K q b Λ to H ˚` ´ rN L p L ; K q b Λ. Combining grading considerations in this spectral sequence with themultiplicative structure, one obtains the following well-known result of Biran–Cornea.
Proposition 2.3 ([7, Proposition 6.1.1]) . If H ˚ p L ; K q is generated as an algebra by H ď m p L ; K q with m ď N L ´ then L is either wide or narrow, and only the former can occur if the inequalityis strict. (cid:3) Morse cocycles of index ă N L are automatically Floer cocycles, and we obtain a ‘PSS’ map H ă N L p L ; K q Ñ HF ˚ p L , L ; Λ q . This intertwines CO with the restriction map H ă N L p X ; K q Ñ H ă N L p L ; K q Symplectic reduction
Our first observations relate to symplectic reduction. After establishing some useful results aboutmonotonicity and relative spin structures, including Theorem 2, we apply them to a family ofLagrangian flag varieties, and prove Theorem 1.3.1.
Hamiltonian actions.
Recall that an action of a Lie group K on X is Hamiltonian , withmoment map µ : X Ñ k ˚ (not to be confused with the Maslov index!), if: the action preserves ω ; µ intertwines the K -action on X with the coadjoint action on k ˚ ; and µ generates the action in thesense that for all ξ in k we have x d µ, ξ y “ ω p´ , V ξ q , where V ξ is the vector field describing the action of ξ . If K is compact and acts freely on µ ´ p q ,which implies that 0 is a regular value of µ , then the symplectic reduction X {{ K is the quotient µ ´ p q{ K equipped with the unique symplectic form ω X {{ K whose pullback to µ ´ p q coincides with ω | µ ´ p q . If L is a Lagrangian submanifold of X contained in µ ´ p q , and is preserved setwise by the K -action, then L { K defines a Lagrangian in X {{ K . We shall always assume that K is connected, soin particular it acts in an orientation-preserving way.We remark for later use that MONOTONE LAGRANGIAN CASEBOOK 5
Lemma 3.1 ([13, Proposition 1.3]) . K -orbits contained in µ ´ p q are isotropic. (cid:3) To do Floer theory with L { K we would like to understand when it is monotone and relativelyspin, and these are the subjects of the next two subsections.3.2. Monotonicity for reductions.
Our goal is to relate monotonicity of symplectic reductionsto monotonicity upstairs:
Proposition 3.2.
Suppose X is a symplectic manifold carrying a Hamiltonian action of a compactconnected Lie group K , which acts freely on µ ´ p q . If L Ă X is a K -invariant monotone Lagrangiansubmanifold contained in µ ´ p q then L { K is monotone in X {{ K with λ L { K “ λ L .Remark . Any connected K -invariant Lagrangian automatically lies in µ ´ p q , possibly aftershifting µ by a fixed point of the coadjoint action—see [15, Lemma 4.1]. The monotonicity constantsappearing in the statement λ L { K “ λ L may not be uniquely determined, in which case we meanthat there exist choices for which equality holds. {{ Proof.
First we claim that p ˚ : π p µ ´ p q , L q Ñ π p X {{ K, L { K q is an isomorphism, where p denotesthe quotient-by- K map. We will prove this by repeated application of the five lemma, so to beginlet Z denote µ ´ p q , let K be a K -orbit inside L , and for j ě π j p K q π j p L q π j p L, K q π j ´ p K q π j ´ p L q π j p K q π j p L q π j p L { K q π j ´ p K q π j ´ p L q p ˚ The unlabelled vertical maps are the obvious isomorphisms, the top row is from the long exactsequence of the pair, and the bottom row is from the long exact sequence of the fibration. Notethat a priori π p L, K q is only a pointed set, but the K -action on L makes it into a group so that p ˚ is a homomorphism; the multiplication on K and connectedness of L ensure that π p K q “ π p K q and π p L q “ t point u are also groups, but we don’t need this. The squares all commute (theonly one which requires a little thought is the third of the four) so the five lemma shows that p ˚ : π j p L, K q Ñ π j p L { K q is an isomorphism for all j ě
1. Similarly we have isomorphisms p ˚ : π j p Z, K q Ñ π j p Z { K “ X {{ K q for all j ě π p L, K q π p Z, K q π p Z, L q π p L, K q π p Z, K q π p L { K q π p X {{ K q π p X {{ K, L { K q π p L { K q π p X {{ K q p ˚ p ˚ p ˚ p ˚ p ˚ The top row is from the long exact sequence of the triple p Z, L, K q whilst the bottom row is from thelong exact sequence of the pair. By the previous paragraph the vertical maps are all isomorphisms,except possibly the middle one, so applying the five lemma again we see that the middle map p ˚ : π p Z, L q Ñ π p X {{ K, L { K q is also an isomorphism, as claimed.Our strategy now is to understand monotonicity of L { K by lifting discs by p p ˚ q ´ , so take anarbitrary class β in π p X {{ K, L { K q and choose a disc u : p D, B D q Ñ p Z, L q with r p ˝ u s “ β . Since p ˚ ω X {{ K “ ω X | Z we immediately have that u and β have equal areas. We claim that they have equalMaslov indices, i.e. that the bundle pairs p u ˚ T X, u | ˚B D T L q and p u ˚ p ˚ T p X {{ K q , u | ˚B D p ˚ T p L { K qq haveequal indices, and we shall do this by exhibiting the latter pair as a quotient of the former by atrivial sub-pair.So fix an ω -compatible almost complex structure J on X . For any non-zero ξ P k we have xp J V ξ q { d µ, ξ y “ ω p J V ξ , V ξ q ą , and hence p J V ξ q { d µ ‰
0, so the subbundle J p k ¨ Z q of T X | Z is sent fibrewise injectively to k ˚ by d µ and thus provides a complementary subbundle to T Z . This means that k ¨ Z is a subbundle of T Z
JACK SMITH which is complementary to E Z : “ T Z X J p T Z q , giving T X | Z “ p k ¨ Z q ‘ J p k ¨ Z q ‘ E Z as real vector bundles over Z . Letting E k denote the sum of the first two terms, we obtain a splitting T X | Z “ E k ‘ E Z as complex vector bundles.Now let F k denote the totally real subbundle k ¨ L “ T L X E k | L of E k | L , and F Z the projection of T L onto E Z | L with kernel F k . The short exact sequence of pairs0 Ñ p E k , F k q Ñ p T X | Z , T L q Ñ p E Z , F Z q Ñ µ p u ˚ T X, u | ˚B D T L q “ µ p u ˚ E k , u | ˚B D F k q ` µ p u ˚ E Z , u | ˚B D F Z q . The first term on the right-hand side vanishes, since the bundle pair is trivialised by the action of k ,and we are left to show that the second term is µ p u ˚ p ˚ T p X {{ K q , u | ˚B D p ˚ T p L { K qq . To see that thisis indeed the case, note that E Z projects isomorphically onto p ˚ T p X {{ K q “ T X | Z { E k and that thisprojection identifies F Z with p ˚ T p L { K q . Moreover, the complex structure on E Z is compatible with p ˚ ω X {{ K “ ω | T Z . We conclude that u and β have equal indices and so if L is monotone then L { K is monotone with the same monotonicity constant. (cid:3) Relative spin structures for reductions.
We would like to understand relative spin struc-tures on reduced Lagrangians L { K Ă X {{ K in the setting of Proposition 3.2, assuming also that L is orientable. These often (for example, if the K -action extends to a larger group K and L isthe zero set of the K -moment map; see Lemma 3.13) have the property that their normal bundleextends to X {{ K , and in this case we can apply Lemma 3.4.
Given a vector bundle V on N whose restriction to M is identified with the normalbundle to M in N , M carries a natural relative spin structure with background class w p V q ` w p N q .Proof. We need to show that
T M ‘ V | M ‘ T N | M , i.e. p T N ‘ T N q| M , carries a natural spin structure.But the double of any orientable rank k vector bundle has a ntural spin structure since its structuregroup naturally reduces to the block diagonal subgroup GL p k, R q ` ã Ñ GL p k, R q ` , and this lifts toGL p k, R q „` [36, Proposition 3.1.6(b)]. (cid:3) To apply this one needs to compute w p V q , and taking M “ L { K Ă N “ X {{ K and assuming V pulls back to a spin bundle on Z “ µ ´ p q , this can be done as follows. Fix a spin structure on p ˚ V , where p : Z Ñ X {{ K is the projection, and for a loop γ P π p K q define ε p γ q P Z { γ on p ˚ V lifts to the GL „` -bundle determined by the spin structure, and 0 otherwise.Doing this for all classes in π p K q we obtain an element ε of Hom p π p K q , Z { q “ H p K ; Z { q . Lemma 3.5.
The class ε P H p K ; Z { q transgresses to w p V q P H p X {{ K ; Z { q in the Serre spectralsequence for K ã Ñ Z ։ X {{ K .Proof. Consider the two fibrations K ã Ñ Z ։ X {{ K and B p Z { q ã Ñ B GL „` ։ B GL ` . The bundle V is classified by a map ϕ : X {{ K Ñ B GL ` between their base spaces, and the choice of spin structureon p ˚ V gives a lift r ϕ to the total spaces. Restricting to the fibres yields a map K Ñ B p Z { q , i.e. aclass ε in H p K ; Z { q , and we claim that ε “ ε . To see this, it suffices to prove that looped mapsΩ ε, Ω ε : Ω K Ñ Z { A -spaces. But both maps can be described as the restriction to Ω K of the map Ω X Ñ GL „` describing the monodromy of our bundle with respect to some connection.Finally we need to show that ε transgresses to w p V q . For this we deloop the fibrations to obtaina diagram up to homotopy Z X {{ K BKB GL „` B GL ` B Z { r ϕ πϕ Bε w MONOTONE LAGRANGIAN CASEBOOK 7
By definition, w p V q is the composite w ˝ ϕ “ Bε ˝ π , where π is the classifying map X { K Ñ BK as shown. The class ε transgresses to Bε in the Serre spectral sequence for K ã Ñ EK ։ BK .Pulling back by the map π gives our fibration K ã Ñ X ։ X {{ K , so by naturality of the Serrespectral sequence we see that ε transgresses to w p V q in the spectral sequence for the latter. (cid:3) Remark . Changing the spin structure on p ˚ V by a class δ in H p Z ; Z { q changes ε by thepullback of δ to K . However, the image of this pullback is precisely the kernel of the map π ˚ : H p BK ; Z { q – H p K ; Z { q Ñ H p X {{ K ; Z { q , so the transgression w p V q is unaffected. {{ Worked example: flag varieties.
Consider the space X “ C n equipped with the standardsymplectic form. Fix a tuple of positive integers k , . . . , k r which sum to n (with r ě K be the block diagonal subgroup U p k q ˆ ¨ ¨ ¨ ˆ U p k r q of U p n q . We shall define a Hamiltonian K -action on X and a Lagrangian L in the zero set of the moment map, apply the above results toshow that L { K has well-defined Floer theory in X {{ K , and use the closed–open map to compute HF ˚ p L { K, L { K q as a ring from knowledge of the quantum cohomology of X {{ K .The K -action will actually be defined as the restriction of a Hamiltonian U p n q -action, so wediscuss that first. To construct this action we view elements of C n as n ˆ n matrices w , so thatU p n q acts by left multiplication. This action is Hamiltonian with moment map µ U p n q : X Ñ u p n q ˚ given by(1) x µ U p n q p w q , A y “ i p A q ´ i p w : Aw q “ x i p ww : ´ I q , A y for all w P C n and A P u p n q , where I is the n ˆ n identity matrix and x A , A y “ Tr p A : A q is theusual inner product on u p n q (note that iww : and iI both lie in this space); the Tr p A q term controlsthe normalisation of the reduced symplectic form. Therefore µ ´ p n q p q comprises those w such that ww : “ I , so is exactly U p n q Ă C n . By Lemma 3.1 it is isotropic, and its dimension is n “ dim C X ,so it is Lagrangian. This will be our L . Lemma 3.7. L is orientable and monotone.Proof. L is actually parallelisable since it’s a Lie group. To prove monotonicity consider the holomor-phic disc u p z q in X with boundary on L given by the diagonal matrix with diagonal entries z, , . . . , u generates π p L q , so from the long exact sequence u generates π p X, L q . It hasindex 2 ą π ą (cid:3) Now restrict this action to K . The resulting K -action is Hamiltonian and its moment map µ isgiven by projecting µ U p n q under u p n q ˚ Ñ k ˚ “ ` u p k q ‘ ¨ ¨ ¨ ‘ u p k r q ˘ ˚ . This is the Hamiltonian action we shall study. The space Z “ µ ´ p q comprises those w such that i p ww : ´ I q is orthogonal to k , which amounts to each row having norm 1, the first k rows beingpairwise orthogonal, and similarly for the next k , and so on. Note that K acts freely on Z , and L lies in Z and is K -invariant, so we are in the setting of Proposition 3.2, which tells us that Lemma 3.8. L { K is monotone. (cid:3) Our goal is the following result
Theorem 3.9.
Equipping L { K with a specific choice of relative spin structure (defined below) andwith the trivial line bundle to give a brane p L { K q , we have HF ˚ pp L { K q , p L { K q ; Λ q “ Λ r c , , . . . , c ,k , c , , . . . , c r,k r s N r ˚ j “ p ` c j, ` ¨ ¨ ¨ ` c j,k j q “ ` T, where ˚ is the Floer product, each c j,k has degree k , and the Novikov variable T has degree n . JACK SMITH
Remark . We’ll see shortly that L { K is simply connected so in fact the only flat line bundle itadmits is the trivial one. {{ The first step is to understand the topology of L { K . Letting V j p w q Ă C n denote the span of thefirst d j : “ k ` ¨ ¨ ¨ ` k j rows of a matrix w in L , we obtain a diffeomorphism L { K Ñ flag variety F p d , . . . , d r q given by w ÞÑ “ V p w q Ă V p w q Ă ¨ ¨ ¨ Ă V r p w q “ C n . We have tautological bundles E , . . . , E r of ranks k , . . . , k r , with fibres V r p w q{ V r ´ p w q respectively. Lemma 3.11. L { K “ F p d , . . . , d r q is simply connected and H ˚ p L { K ; Z q is the polynomial algebraby the Chern classes of the tautological bundles E , . . . , E r modulo the relations c p E q ! ¨ ¨ ¨ ! c p E r q “ arising from the fact that their sum is trivial.Proof. The first claim follows from the long exact sequence in homotopy groups for the fibration K “ U p k q ˆ ¨ ¨ ¨ ˆ U p k r q ã Ñ L “ U p n q ։ L { K “ F p d , . . . , d r q . The second, meanwhile, comes from the Serre spectral sequence for the fibrationU p n q ã Ñ U p n q ˆ K EK ։ BK, whose total space is homotopy equivalent to L { K . Explicitly, the cohomology of the base is thepolynomial algebra on the Chern classes of the E j , whilst the generators x , x , . . . , x n ´ for thecohomology of the fibre transgress to the Chern classes of E ‘ ¨ ¨ ¨ ‘ E r . (cid:3) The symplectic reduction X {{ K has a similar description as the productGr p k , n q ˆ ¨ ¨ ¨ ˆ Gr p k r , n q of Grassmannians. The cohomology of each Gr p k j , n q is the polynomial algebra on the Chern classesof its tautological bundle E j (of rank k j ) and of the quotient F j “ C n { E j (of rank n ´ k j ), modulothe relation c p E j q ! c p F j q “
1. These bundles E j restrict to the E j above on L { K , so our useof the same notation is justified; if we wish to distinguish them we will denote them by E j p X {{ K q and E j p L { K q . The tangent bundle T Gr p k j , n q is naturally identified with E _ j b F j —the fibre of thelatter over the point of Gr p k j , n q corresponding to a rank k j subspace V Ă C n comprises linear maps V Ñ C n { V , and hence infinitesimal deformations of V —so by considering Chern roots we obtain c p Gr p k j , n qq “ k j c p F j q ´ p n ´ k j q c p E j q . We have just seen that H p Gr p k j , n q ; Z q is free of rank 1,generated by c p E j q “ ´ c p F j q , so we deduce c p Gr p k j , n qq “ ´ nc p E j q . Lemma 3.12.
The minimal Maslov number N L { K is n .Proof. Since L { K is simply connected (by Lemma 3.11) any disc in π p X {{ K, L { K q lifts to a spherein π p X {{ K q , and the Maslov index becomes twice the Chern number. We just saw that c p X {{ K q “´ n ř j c p E j q , so it suffices to show that there is a sphere which pairs to ˘ c p E j q . To do this,pick k j ` v , . . . , v k j ` in C n and consider the sphere u : r z s P CP ÞÑ @ v , . . . , v k j ´ , v k j ` zv k j ` D P Gr p k j , n q . The bundle u ˚ E j is O ‘ k j ´ CP ‘ O CP p´ q , so xr u s , c p E j qy “ ´
1, as needed. (cid:3)
Lemma 3.13. L { K carries a natural relative spin structure with background class b : “ ř j k j c p E j q .Proof. To construct the relative spin structure we apply Lemma 3.4. The normal bundle to L in X is naturally identified with the trivial bundle with fibre u p n q ˚ , whilst the normal bundle to Z is identified with the trivial bundle with fibre k ˚ . Therefore the normal bundle to L in Z is thetrivial bundle with fibre Φ “ p u p n q{ k q ˚ . This extends K -equivariantly to all of Z (the K -action iscoadjoint) and thus descends to a vector bundle V on X {{ K . By Lemma 3.4 L { K carries a naturalrelative spin structure with background class w p X {{ K q ` w p V q . MONOTONE LAGRANGIAN CASEBOOK 9
We compute w p V q as in Lemma 3.5, equipping p ˚ V with the spin structure induced by itstrivialisation as Z ˆ Φ. The class ε is precisely the pushforward on π given by the coadjoint actionmap K Ñ GL ` , and π p K q is freely generated by the loops γ , . . . , γ r defined as follows: γ j p t q isdiagonal with diagonal entries 1 , . . . , e it , . . . ,
1, where e it occurs in the d j th position. The coadjointaction of γ j loops n ´ k j times the generator of π p GL ` q , so we conclude that ε transgresses to r ÿ j “ ´p n ´ k j q c p E j q . Hence w p X {{ K q ` w p V q “ ´ n ř j c p E j q ´ ř j p n ´ k j q c p E j q “ ř j k j c p E j q in H p X {{ K ; Z { q . (cid:3) We are almost ready to prove Theorem 3.9. The final input we need is
Lemma 3.14.
The quantum cohomology QH ˚ p X {{ K, b ; Λ q is the polynomial algebra over Λ on theChern classes of the E j and F j , modulo the relations c p E j q ˚ c p F j q “ ` T , where T is the Novikovvariable of degree n .Proof. We already saw that without the T term this gives the classical cohomology. Witten [38,Section 3.2] showed that with zero background class the quantum correction is p´ q k j T (Witten’srelation is in terms of the duals of these bundles so comes with a different sign), and we are left toshow that turning on the class b modifies this sign to `
1. To prove this, recall from Section 2.2 thata background class b changes the count of curves in class A by a factor of p´ q x b,A y . In our case,the curves contributing to c p E j q ˚ c p F j q lie on the Gr p k j , n q factor and have Chern number n (fordegree reasons) and thus pair to ´ c p E j q and to zero with all other c p E k q . They thereforepair to k j mod 2 with b , giving precisely the required factor. (cid:3) Remark . If k “ p k, n q is CP n ´ and E is O CP n ´ p´ q with Chern class 1 ´ H , where H is the hyperplane class. The relation c p E q ! c p F q “ c p F q “ ` H ` ¨ ¨ ¨ ` H n ´ ,then Witten’s relation reduces to H ˚ n “ T : the familiar description of QH ˚ p CP n ´ q . {{ Proof of Theorem 3.9.
The Chern classes of the E j p L { K q lie in degree ă N L (by Lemma 3.12) andtherefore define Floer cohomology classes c j,k via the PSS map. Since the classical versions generate H ˚ p L { K ; K q as a K -algebra with respect to the cup product (Lemma 3.11), the Floer versionsgenerate HF ˚ “ HF ˚ pp L { K q , p L { K q ; Λ q as a Λ-algebra with respect to the Floer product.We also have the Chern classes of the E j p X {{ K q in QH ˚ “ QH ˚ p X {{ K, b ; Λ q , and using again thefact that they lie in degree ă N L we see that their images under CO coincide with their classicalrestrictions to L { K via PSS. In other words, for each j we have CO p c p E j p X {{ K qqq “ c p E j p L { K qq P HF ˚ . This also forces the c j,k to commute with respect to the Floer product, since QH ˚ is commutative.For each j , the complement F j restricts to E p L { K q ‘ ¨ ¨ ¨ ‘ { E j p L { K q ‘ ¨ ¨ ¨ ‘ E r p L { K q on L { K , so its total Chern class (which lies in degree ă N L ) satisfies CO p c p F j qq “ ˚ l ‰ j p ` c l, ` ¨ ¨ ¨ ` c l,k l q . The relation c p E j q ˚ c p F j q “ ` T in QH ˚ , and the fact that CO is a ring map, then gives(2) r ˚ j “ p ` c j, ` ¨ ¨ ¨ ` c j,k j q “ ` T in HF ˚ . Therefore HF ˚ is a quotient of the algebra claimed in Theorem 3.9. To show that thisquotient is by zero, and thus prove the theorem, it suffices to show that HF ˚ and A have the samedimension over K in each degree, i.e. that p L { K q is wide, and this follows from Proposition 2.3. (cid:3) We end by noting that if we had tried to run this argument without the background class b thenthe right-hand side of (2) would have been 1 ` p´ q k j T , and varying j we would have obtainedrelations which are inconsistent outside characteristic 2 unless the k j all have the same parity.The most general background class that makes them consistent is of the form b ` δ ř j c p E j q for δ P Z {
2, which makes the right-hand side into 1 ` p´ q δ T , and these two choices correspond to thetwo different relative spin structures on L { K : recall that relative spin structures form a torsor for H p X {{ K, L { K ; Z { q , and in our case the long exact sequence of the pair tells us that this group isprecisely the kernel of H p X {{ K ; Z { q Ñ H p L { K ; Z { q since L { K is simply connected; this kernelis the span of ř j c p E j q , so there is a unique relative spin structure which differs from that inLemma 3.13, and the two background classes differ by ř j c p E j q .4. The Floer–Poincar´e polynomial
In this section we explore consequences of grading periodicity.4.1.
Periodicity and the Poincar´e polynomial.
So far we have worked with the Z -graded Λ-algebra HF ˚ p L , L ; Λ q , but by setting the Novikov variable T to 1 we can turn it into a Z { N L -graded K -algebra which we denote by HF ˚ p L , L ; K q . Definition 4.1.
For an integer q , say L is q -periodic if HF ˚ p L , L ; K q – HF ˚` q p L , L ; K q as Z { N L -graded vector spaces. {{ L is tautologically N L -periodic, but it often turns out to be q -periodic for some proper divisor q of N L , and this can impose strong restrictions (see for example the work of Seidel [31] and Biran–Cornea [7, 8]). We shall introduce a simple tool, closely related to the discrete Fourier transform,which allows us to extract new information. Remark . Sources of q -periodicity include: ‚ The quantum module action of invertible elements of degree q in the Z { N L -graded quantumcohomology of QH ˚ p X ; K q (obtained by setting T “ QH ˚ p X ; Λ q ), as in [7, Corollary6.2.1]. For example, monotone Lagrangians in compact toric varieties are 2-periodic sincethe toric divisiors are quantum invertible (McDuff–Tolman [26, Section 5.1] exhibit them aselements in the image of the Seidel representation [30] π p Ham p X qq Ñ QH ˚ p X ; K q ˆ , which in general provides a rich source of invertibles). Similarly, monotone Lagrangiansin quadrics of even complex dimension are 2-periodic by [32, Lemma 4.3]. Note that if ourLagrangian is equipped with a relative spin structure with background class b then we shouldwork with QH ˚ p X, b ; K q . ‚ An isomorphism between the shift functor r q s and the identity functor on the Fukaya categoryof X , as in [31, Section 3]. For example, on CP n the path of symplectomorphisms p ϕ t : r z : z : ¨ ¨ ¨ : z n s ÞÑ r e πit z : z : ¨ ¨ ¨ : z n sq t Pr , s gives a Hamiltonian isotopy between the identity and r´ s . In fact, the Seidel representationsends this path (which is actually a closed loop) to the hyperplane class, which provides 2-periodicity via the quantum module action. ‚ The Floer–Gysin sequence of Biran–Khanevsky [9, Corollary 1.3], in the case q “ K “
2, although this restriction can probably be lifted). Here one assumes that X embeds as a codimension 2 symplectic submanifold of some symplectic manifold p M, ω M q ,so that X is Poincar´e dual to a positive multiple of r ω M s and M z X is subcritical. A monotoneLagrangian in X lifts to a Lagrangian circle bundle in M z X , and the self-Floer cohomologyof the latter is the cone over ‘multiplication by the Euler class’ on the self-Floer cohomologyof the former. Subcriticality ensures that the circle bundle must have vanishing self-Floercohomology, so multiplication by the Euler class is an isomorphism of degree 2. {{ MONOTONE LAGRANGIAN CASEBOOK 11
The main idea is to encode the degreewise dimensions of HF ˚ in a generating function. Recallthat the Poincar´e polynomial of L over K is P p S q “ ÿ j P Z dim K H j p L ; K q ¨ S j P Z r S s . This is multiplicative under tensor product decompositions of H ˚ p L ; K q as a graded vector space. Definition 4.3.
The
Floer–Poincar´e polynomial of L is the generating function P F p S q “ ÿ j P Z { N L dim K HF j p L , L ; K q ¨ S j P Z r S s{p S N L ´ q of the ‘Floer–Betti numbers’. Note that this is only defined modulo S N L ´ {{ Its key property for us is
Proposition 4.4. If L is q -periodic for some proper divisor q of N L , then P F p S q is divisible by S q ` S q ` ¨ ¨ ¨ ` S N L . In particular, the total dimension P F p q of HF ˚ p L , L ; K q is divisible by N L { q , whilst P F p ζ q must vanish for any N L th root of unity ζ that isn’t also a q th root of unity.Proof. Assuming q -periodicity we have P F p S q “ q ÿ j “ N L { q ÿ k “ dim K HF j ` kq p L , L ; K q ¨ S j ` kq “ q ÿ j “ dim K HF j p L , L ; K q ¨ S j N L { q ÿ k “ S kq , and the inner sum is the claimed factor. When S “ N L { q . Meanwhile, when S q ‰ S q p S N L ´ q{p S q ´ q , so vanishes if S is also an N L th root of unity. (cid:3) Remark . For ζ “ e ´ πi { N L the sequence P F p q , P F p ζ q , . . . , P F p ζ N L ´ q is the discrete Fouriertransform of the sequence of Floer–Betti numbers. {{ Minimal Maslov constraints.
A simple consequence for monotone tori is
Proposition 4.6.
A monotone Lagrangian torus L Ă X which is -periodic has minimal Maslovnumber . In particular this applies to any monotone Lagrangian torus if the Z { N X -graded quantumcohomology contains an invertible element of degree .Proof. Suppose for contradiction that L is a 2-periodic monotone Lagrangian n -torus of minimalMaslov number N L ą
2, and equip it with an arbitrary spin structure and flat line bundle over K .By Proposition 2.3 L is wide, so its Floer–Poincar´e polynomial is the reduction mod p S N L ´ q ofits ordinary Poincar´e polynomial, p ` S q n . Applying P F p ζ q “
0, where ζ “ e πi { N L , we see that1 ` ζ “
0. Since N L ą L cannot exist. (cid:3) Remark . Using his theory of Floer (co)homology on the universal cover, Damian [16, Theorem1.6] proved that N L “ L in the product of CP n (for n ě
1) with an arbitrary symplectic manifold W . Fukaya [21, Theorem 14.1] obtained a similarresult, without monotonicity, for aspherical spin Lagrangians in CP n and other uniruled symplecticmanifolds. {{ In fact, the same argument generalises to give
Theorem 4.8.
Suppose L Ă X is a (closed, connected) monotone Lagrangian, whose cohomology H ˚ p L ; K q is an exterior algebra on generators of degree k ´ ď ¨ ¨ ¨ ď k r ´ r , and that either char K “ or that L is orientable and relatively spin (with background class b ). If QH ˚ p X, b ; K q contains an invertible element of even degree q ď k r then N L is at most k r . Proof.
Let L be the brane obtained by equipping L with an arbitrary relative spin structure (ifchar K ‰
2) and the trivial line bundle. If N L ą k r then in particular N L ě HF ˚ p L , L ; K q is defined, and from Proposition 2.3 we get P F p S q “ r ź j “ p ` S k j ´ q mod p S N L ´ q . We claim that this is impossible, assuming the existence of the invertible element h of degree q .Well, after reducing the grading of QH ˚ modulo N L , some power of h constitutes an invertibleelement of degree q , where q : “ gcd p q, N L q is the smallest element of the subgroup of Z { N L generated by q . Proposition 4.4 then gives P F p ζ q “ ζ “ e πi { N L , and N L { q | r . The formermeans that 1 ` ζ k j ´ “ j , and hence that N L “ p k j ´ q , and the latter then yields q “ k j ´ p k j ´ q . Since N L is even, our assumption that q is even means that q is also even.We must therefore have q “ p k j ´ q “ N L , but this contradicts the fact that q ď q ď k r ă N L .We conclude that if the element h exists then N L must be at most 2 k r . (cid:3) Remark . The hypotheses are satisfied for monotone Lagrangian embeddings of compact con-nected Lie groups K and their quotients by finite subgroups Γ, taking K “ Q . Hopf [24, Satz1] showed that the rational cohomology algebras of such K are exterior algebras on odd degreegenerators, and the same holds for finite quotients by showing that the quotient map induces anisomorphism on rational homology (an inverse is provided by sending a simplex in K { Γ to the av-erage of its | Γ | lifts to K : this is clearly a right inverse, and to see that it’s a left inverse note thatif we project a cycle σ in K and then lift we obtain1 | Γ | ÿ γ P Γ γ ¨ σ and this is homotopic to σ by homotoping each γ back to the identity). All such K { Γ are orientableand spin since they are parallelised by the infinitesimal action of k . {{ Worked example:
PSU p n q . We now apply these ideas to the following family of examples.For an integer n ě p n q acts by left multiplication on the space of n ˆ n complexmatrices, and projectivising gives a Hamiltonian PSU p n q -action on X “ CP n ´ (with the Fubini–Study form). The action on the identity matrix is free, and its orbit L gives a Lagrangian embeddingof PSU p n q . This is the projectivisation of the Lagrangian U p n q from Section 3.4. Its fundamentalgroup is Z { n , so since X is monotone with minimal Chern number n we see that L is monotoneand has minimal Maslov number divisible by 2 n . It is parallelisable (it’s a Lie group) and thereforeorientable and spin.This family was originally discovered by Amarzaya and Ohnita [1] and later rediscovered byChiang [13, Section 4], and we denote it by L AOC . Iriyeh [25, Theorem 8, Proposition 27] provedthat L AOC is wide over a field of characteristic 2 if n is a power of 2 and narrow otherwise, andwhen n is 3 or 5 L AOC is non-narrow over Z . Later, Evans and Lekili [20, Example 7.2.3] showedthat L AOC is wide over a field of characteristic p when n is a power of p , and narrow if n is notdivisible by p . Their arguments work with any relative spin structure and flat line bundle. Recentlythis family has also been studied by Torricelli [35].We complete these partial computations by showing Theorem 4.10. If n is not a prime power then L AOC is narrow for any choice of coefficient field,relative spin structure, and flat line bundle L . We shall actually deduce this from the following more general result
Theorem 4.11. If L is a -periodic Lagrangian embedding of PSU p n q in a closed monotone sym-plectic manifold X , with n ě and N L ě n , then N L “ n and either:(i) K has prime characteristic p and n is a power of p , in which case L is wide for all linebundles L and all relative spin structures.(ii) Otherwise L is narrow over K for all such L and s . MONOTONE LAGRANGIAN CASEBOOK 13
Remark . By the same arguments as for L AOC , any such L inherits monotonicity from X and isorientable and spin. If the 2-periodicity depends on the choice of background class for X (e.g. if theperiodicity comes from an invertible element in QH ˚ p X, b ; K q that only exists for a specific choiceof b ) then the result only applies to relative spin structures with this background class. {{ This immediately proves Theorem 4.10, since for all n we have QH ˚ p CP n , b ; K q “ K r H s{p H n ` ´ p´ q x b, r line sy q , where H is the hyperplane class which is a degree 2 invertible. Here r line s is the homology class ofthe curve which contributes to the quantum product H n ˚ H .Before proving Theorem 4.11 we need Lemma 4.13 ([10, Th´eor`eme 11.4], [4, Corollary 4.2]) . If char K “ p ą and p r is the greatestpower of p dividing n then we have an isomorphism of graded algebras H ˚ p PSU p n q ; K q – Λ p x , x , . . . , p x p r ´ , . . . , x n ´ q b R r y s{p y p r q , where the Λ denotes the exterior algebra over K generated by elements x j ´ of degree j ´ ; exceptin the case p “ and r “ where the relation x “ should be replaced by x “ y . If char K “ then as graded algebras H ˚ p PSU p n q ; K q – Λ p x , x , . . . , x n ´ q . (cid:3) We can now give the proof.
Proof of Theorem 4.11.
The equality N L “ n follows directly from Theorem 4.8 with K “ Q .Let p “ char K (prime or zero), and suppose first that n is a power of p . By Lemma 4.13 H ˚ p PSU p n q ; K q is generated as a K -algebra by elements of degree at most 2 n ´ n “ p “ x “ y ), so by Proposition 2.3 we see that (i) holds. This is how Evansand Lekili proved wideness in [20].Now suppose that n is not a power of p . This time H ˚ p PSU p n q ; K q is generated as a K -algebraby elements of degree ď n ´
1, so for any given L and s Proposition 2.3 tells us that L is eithernarrow or wide. We need to rule out the latter, so suppose for contradiction that L is wide.Assume that p is prime and let p r be its greatest power dividing n . We then have(3) P F p S q “ p ` S ` ¨ ¨ ¨ ` S p r ´ q ź j “ ,..., x p r ,...,n p ` S j ´ q mod S N L ´ , and from Proposition 4.4 we get that p r ¨ n ´ is divisible by N L { “ n and that P F p ζ q “ n th root of unity ζ . Setting S “ ζ in (3), the first bracketed term cannot vanish, so weobtain 2 j ´ “ n for some j in t , . . . , p p r , . . . , n u . Thus n is odd, and the condition n | p r ¨ n ´ forces n to divide p r . Since p r is a proper factor of n , this is impossible. We are left to deal with p “ p r is interpreted as 1. (cid:3) Remark . Iriyeh’s proof [25, pp. 260–261] that L AOC is narrow in characteristic 2 when n is evenbut not a power of 2 is similar in spirit. He uses 2-periodicity in the Z { n -grading, plus Poincar´eduality, to deduce that if n is even then HF ˚ p L AOC , L
AOC q has the same rank in every degree. If L AOC were non-narrow, and hence wide, then this would mean that 2 n divides the sum of the Z { L AOC . This sum is a power of 2, so n itself must be a power of 2. {{ Trivial vector bundles
Restricting Chern classes.
Recall from Section 2.3 that for a monotone Lagrangian brane L the PSS map H ă N L p L ; K q Ñ HF ˚ p L , L ; Λ q intertwines the closed–open map CO : QH ˚ p X ; Λ q Ñ HF ˚ p L , L ; Λ q , with the classical restriction map H ˚ p X ; K q Ñ H ˚ p L ; K q on classes of degree ă N L . In particular Lemma 5.1. If α P H ď N L p X ; K q restricts to in H ˚ p L ; K q then CO p α q “ . (cid:3) An obvious corollary of this is the following observation.
Proposition 5.2.
Suppose L Ă X is a monotone Lagrangian brane and E Ñ X is a complex vectorbundle whose restriction to L is trivial. For all j with ď j ă N L { we have CO p c j p E qq “ , where c j p E q is the j th Chern class of E . (cid:3) The point of making this statement separately is that many algebraic varieties X carry naturalvector bundles whose Chern classes generate large parts of the cohomology (e.g. as in Section 3.4).Moreover, these classes can be easily manipulated using exact sequences and the splitting principle.5.2. Worked example: projective Stiefel manifolds.
We now apply this technique to the fol-lowing family of examples, introduced to the author by Frol Zapolsky (who proved the wideness partof Theorem 5.3); these spaces appear in his work [39] constructing quasi-morphisms on contactomor-phism groups. Fix positive integers n and k with n ě , k . We view CP kn ´ as the projectivisationof the space of k ˆ n matrices and let PSU p k q act by left multiplication in the obvious way. Thisaction is Hamiltonian, with moment map x µ pr w sq , A y “ ´ i w : Aw Tr w : w for all A in su p k q and all k ˆ n matrices Z , and the symplectic reduction at the zero level is the(complex) Grassmannian Gr p k, n q . The set µ ´ p q embeds as a Lagrangian L Ă X “ Gr p k, n q ´ ˆ CP kn ´ “ Gr p n ´ k, n q ˆ CP kn ´ , where ´ denotes reversal of the sign of the symplectic form, and is diffeomorphic to the (complex)projective Stiefel manifold, i.e. the quotient of the Stiefel manifold V parametrising unitary k -framesin C n by the obvious action of U p q . The case k “ n gives the family L AOC from Section 4.3, whilst k “ Ă p CP n ´ q ´ ˆ CP n ´ . We shall show Theorem 5.3. If p denotes char K and p r its greatest power dividing n (interpreted as if p “ )then either: k ď p r , in which case L is wide for all choices of relative spin structure and flat linebundle; or k ą p r , in which case L is narrow for all such choices. Note that this is consistent with Theorem 4.11 when k “ n , and with the fact that ∆ is alwayswide when k “ HF ˚ p ∆ , ∆ q – QH ˚ p CP n ´ q ). The k “ n case behaves slightly differently from theothers (the Lagrangian is not simply connected, for example), so since we have already dealt with itby other means we henceforth exclude it. The first task is to establish the basic properties of theseLagrangians: Lemma 5.4. L is monotone and orientable, with N L “ n .Proof. First consider the Stiefel manifold V . Projecting a unitary frame to its first entry realises V as a fibration over S n ´ . Projecting the fibre to its second entry then realises is as a fibrationover S n ´ , whose fibre is a fibration over S n ´ , and so on, until we reach fibre S n ´ k ` . Byiterating the long exact sequence in homotopy groups we see that V is simply connected. The longexact sequence in homotopy groups for the fibration U p q ã Ñ V ։ L then shows that L is simplyconnected. This proves that L is orientable and (from the long exact sequence in homotopy for thepair p X, L q ) that it is monotone if and only X is monotone, with N L given by twice the minimalChern number N X . To see that N X “ n , recall from Lemma 3.12 that Gr p k, n q has minimal Chernnumber n , and we know that CP kn ´ has minimal Chern number kn (in fact, this is the special caseGr p , kn q ), so their product has minimal Chern number gcd p n, kn q “ n .It remains to prove monotonicity, and since L is simply connected it suffices to show that CP kn ´ and Gr p k, n q are monotone with the same monotonicity constant. In fact, we claim that CP kn ´ ˆ CP p n ´ k q n ´ (equipped with the sum of appropriately scaled Fubini–Study forms) and its PSU p k q ˆ PSU p n ´ k q -reduction Gr p k, n q ˆ Gr p n ´ k, n q are monotone with the same monotonicity constant.For this note that the Hamiltonian U p n q -action on the space C n of n ˆ n matrices, with momentmap (1), restricts to Hamiltonian actions of both U p q ˆ U p q and U p k q ˆ U p n ´ k q —in each casethe first factor acts on the first k rows and the second factor acts on the remaining n ´ k rows. The MONOTONE LAGRANGIAN CASEBOOK 15 reductions are CP kn ´ ˆ CP p n ´ k q n ´ and Gr p k, n qˆ Gr p n ´ k, n q , and the symplectic form on the lattercomes from the symplectic reduction of the former by the residual action of PSU p k q ˆ PSU p n ´ k q .The claim then follows from Proposition 3.2, by considering the monotone Lagrangian U p n q in C n from Section 3.4: this Lagrangian is monotone (Lemma 3.7), so both reductions are monotone withthe same monotonicity constant. (cid:3) Lemma 5.5. L is relatively spin.Proof. Let pr and pr be the projections from X “ Gr p k, n q ´ ˆ CP kn ´ onto its two factors. Weclaim that in fact L carries a natural relative spin structure with background class pr ˚ w p CP kn ´ q ,i.e. that pr ˚ T CP kn ´ | L ‘ T L carries a natural spin structure. Fix a compatible almost complexstructure J , and apply the argument from the proof of Proposition 3.2 (with X replaced by CP kn ´ and Z by L ) to see that pr ˚ T CP kn ´ | L “ J p k ¨ L q ‘ T L . Thus pr ˚ T CP kn ´ | L ‘ T L is the doubledbundle
T L ‘ T L plus the trivial summand J p k ¨ L q . The former has a natural spin structure fromthe doubling construction of Lemma 3.4, whilst the latter has a natural spin structure from itstrivialisation, completing the proof. (cid:3) Remark . Similarly L has a natural spin structure with background class pr ˚ w p Gr p k, n qq ; nowpr ˚ T Gr p k, n q| L ‘ T L is a quotient of the doubled bundle
T L ‘ T L by the trivial bundle k ¨ L . Ingeneral L need not be (absolutely) spin: take k “ n odd for example. {{ We are now ready for
Proof of Theorem 5.3.
Recall that p denotes char K and p r the greatest power of p dividing n (takento be 1 if p “ k ą p r . We need to show that L is narrow for all choices ofrelative spin structure and flat line bundle.Let pr and pr be as in the proof of Lemma 5.5, let E be the tautological bundle over Gr p k, n q (of rank k ) and let F be the quotient C n { E . Consider the bundles E p q : “ pr ˚ E b pr ˚ O CP kn ´ p q and F p q : “ pr ˚ F b pr ˚ O CP kn ´ p q . The short exact sequence 0 Ñ E p q Ñ pr ˚ O CP kn ´ p q ‘ n Ñ F p q Ñ c p E p qq ! c p F p qq “ p ` H q n in classical cohomology, where H is the pullback of thehyperplane class from CP kn ´ , and the same then holds in quantum cohomology in degrees ă n .In particular, we have(4) j ÿ l “ c l p E p qq ˚ c j ´ l p F p qq “ ˆ nj ˙ H ˚ j in QH ˚ for j “ , . . . , n ´ E p q| L is trivial so by Proposition 5.2 we have CO p c j p E p qqq “ ď j ă n (usingthe fact that N L “ n ). Applying this to CO of (4), we obtain CO p c j p F p qqq “ ˆ nj ˙ CO p H q ˚ j for j “ , . . . , n ´
1, and since F p q has rank n ´ k we conclude that both sides vanish for j “ n ´ k ` , . . . , n ´
1. Setting j “ n ´ p r we get ˆ nn ´ p r ˙ CO p H q ˚ p n ´ p r q “ HF ˚ . The left-hand side is invertible since ` nn ´ p r ˘ “ ` np r ˘ is coprime to p “ char K and H is invertible in QH ˚ , so the only possibility is that HF ˚ “
0, i.e. that L is narrow as claimed.It remains (for the case k ą p r ) to show that E p q| L is trivial. To see that this is the case notethat the fibre of E p q over a point p V Ă C n , l Ă C kn q P X , where V is a subspace of rank k and l is a line, comprises linear maps l Ñ V . If p V, l q lies in L then there exists a k ˆ n matrix A withorthonormal rows such that l is the span of A and V is the span of the rows of A . We therefore have k natural maps l Ñ V given by projecting an element λ ¨ A of l to each of its k rows. These mapsdefine k sections of E p q| L which provide a trivialising frame, completing the proof of narrownessfor k ą p r .Now assume k ď p r . We claim that H ˚ p L ; K q is generated as an algebra in degrees ă n ´ N ą n ´ k such that ` nN ˘ ı p is n . (cid:3) Remark . The narrowness result could have been proved by periodicity considerations as inSection 4.3, and conversely the results there on L AOC (the k “ n case above) could have beenproved using these Chern class arguments. It is interesting to note that whilst the former techniquerequires full knowledge of the Betti numbers of L , the latter relies on much softer calculations butis more dependent on the geometry of the ambient manifold X . {{ The symplectic Gysin sequence
The exact triangle.
In this section we illustrate the symplectic Gysin sequence by filling in amissing computation from [33] and studying a related example. There are two distinct approaches tothis theory in the literature, using different methods but leading to similar results: the Lagrangiancircle bundle construction and Floer–Gysin sequence of Biran [5], Biran–Cieliebak [6] and Biran–Khanevsky [9], and Perutz’s symplectic Gysin sequence associated to a spherically fibred coisotropicsubmanifold [29]. We shall follow the latter because Perutz explicitly deals with coefficient rings ofcharacteristic other than 2. Strictly Perutz works with a Novikov variable that can have arbitraryreal exponents, see [29, Notation 1.5], but the monotonicity hypotheses mean that this is not strictlynecessary and we can restrict to integer exponents as we have been using.The setup is as follows. M and N are closed symplectic manifolds and L is a Lagrangian sub-manifold of X : “ M ´ ˆ N (recalling that ´ denotes reversal of the sign of the symplectic form)such that the projections pr M and pr N to M and N respectively have the following properties: pr M embeds L in M ; and pr N exhibits L as an oriented S k -bundle over N . As usual we assume that L is monotone with minimal Maslov number N L at least 2. Perutz shows Theorem 6.1 ([29, Theorem 6.2, Addendum 1.6]) . If N L “ k ` and L is equipped with (thetrivial flat line bundle and) a relative spin structure whose background class is pulled back from b M P H p M ; Z { q then there is an exact triangle of QH ˚ p N ; Λ q -modules QH ˚´p k ` q p N ; Λ q QH ˚ p N ; Λ q HF ˚ p L , L ; Λ q . p e ˚ r s The horizontal arrow is quantum product with p e “ e ` νT , where e is the Euler class of the orientedsphere bundle L Ñ N and ν is the signed count of index k ` discs through a point x of L whichsend a second boundary marked point to a global angular chain . The latter is a chain on L whichintersects a generic fibre of pr N | L in a single point and whose boundary is the union of the fibresover a chain in the base representing the Poincar´e dual of the Euler class. The QH ˚ p N ; Λ q -actionon HF ˚ p L , L ; Λ q is by pulling back to QH ˚ p X ; Λ q and using the closed–open map.Remark . In [29] Perutz denotes L by p V , and works with Hamiltonian Floer cohomology HF ˚ p µ q of a symplectomorphism µ of N . We take µ “ id N so that HF ˚ p µ q becomes QH ˚ p N q . His argumentall goes through for background classes of the form pr ˚ M b M ` pr ˚ N b N , with b N P H p N ; Z { q as longas QH ˚ p N ; Λ q , is deformed to QH ˚ p N, b N ; Λ q . {{ Worked example I: SO p q . In this subsection we revisit a monotone Lagrangian studied in[33]. There we showed that it is narrow except possibly when char K “ K “
5, dependingon the choice of relative spin structure, and that it is wide in the char K “ K “ MONOTONE LAGRANGIAN CASEBOOK 17
Take M “ p CP ˆ CP q ´ and N “ CP , with each CP given the Fubini–Study form, so that X “ p CP q . Let L be the zero set of the moment map for the standard SO p q -action by rotation onthe three CP “ S factors. This comprises ordered triples of points on S which form the vertices ofan equilateral triangle on a great circle, so is precisely the lift of the Chiang Lagrangian [13, Section2] in CP under the branched cover p CP q Ñ Sym CP – CP . Such a triangle is determined bytwo of its vertices, so pr M embeds L in M , and the projection pr N to the third vertex exhibits L as an orientable circle bundle over N , so we are in the setup of the Gysin sequence. Note that L is monotone (since X is monotone and π p L q is torsion), has N L ě p q -orbit), and carries a standard spin structure defined by the trivialisation of its tangentbundle coming from the infinitesimal SO p q -action. Equip L with the trivial line bundle and anarbitrary relative spin structure to give a brane L .This Lagrangian is the ‘ N “
3’ case of the main family of examples in [33] and the computationsof [33, Sections 5.2–5.3] show the following. Given a point p x , x , x q in L there is a holomorphicdisc u : p D, B D q Ñ p X, L q defined by u p z q “ p x , RM z R ´ x , RM z R ´ x q where R is any rotation of CP sending to x and M z is the map C Ñ C given by multiplicationby z . This meets the SO p q -invariant divisor Z “ tpr z s , r z s , r z sq P X : r z s “ r z su , which is Poincar´e dual to the class H ` H ( H j denotes the pullback of the hyperplane class from the j th CP factor), and the count of this disc computes CO p H ` H q “ ˘ T ¨ L . Moreover, this signis positive for the standard spin structure. Similarly there are discs u and u where the roles of thethree factors are interchanged, and these meet divisors Z and Z and compute CO p H ` H q and CO p H ` H q . Up to reparametrisation, these are the only three index 2 holomorphic discs through p x , x , x q (strictly they are the ‘axial’ index 2 discs, but by [19, Corollary 3.10] all holomorphicindex 2 discs are axial), and their classes A , A and A freely generate H p X, L ; Z q Since relative spin structures form a torsor for H p X, L ; Z { q , and we have a distinguishedchoice—namely the standard spin structure—we can label each relative spin structure by a class ε P H p X, L ; Z { q . Letting ε j “ p´ q x ε,A j y the above results can then be written as(5) CO p H ` H ` H ´ H j q “ ε j T ¨ L . Lemma 6.3 ([33, Theorem 5.4.5], ‘ N “ . If HF ˚ p L , L ; Λ q is non-zero then either char K “ and the ε j are all equal, or char K “ and the ε j are not all equal.Proof. By taking linear combinations of the relations (5) we obtain(6) CO p H q “ p ε ` ε ´ ε q T ¨ L . In QH ˚ p X, ε ; Λ q we have H “ ˘ T , where the sign is determined by pairing the background class(which is the image of ε in H p X ; Z { q ) with the class of a line on the third CP factor, and sincethis line intersects Z and Z once each, but not Z , the sign is exactly ε ε . Squaring (6), wethus have 4 ε ε ¨ L “ p ε ` ε ´ ε q T ¨ L . If HF ˚ ‰ “ p ε ε ` ε ε ` ε ε q in K . If the ε j coincide then the right-hand side is 6, so char K must be 3; otherwise the right-handside is ´ K must be 5. (cid:3) In [33, Theorem 5.7.3] we used the symmetric group action that permutes the three CP factorsto show that L is wide when char K “ ε j are equal. We can now prove the main result Theorem 6.4. L is wide in the cases allowed by Lemma 6.3, i.e. when char K “ and the ε j areall equal or when char K “ and the ε j are not. Proof.
We shall apply Theorem 6.1. The map p b M , b N q ÞÑ pr ˚ M b M ` pr ˚ N b N is an isomorphism H p M ; Z { q‘ H p N ; Z { q Ñ H p X ; Z { q , so Remark 6.2 allows us to take any relative spin structureon L , and the computation in the proof of Lemma 6.3 shows that p´ q x b N , r CP sy “ ε ε , so the exacttriangle we obtain isΛ r H , T ˘ s{p H ´ ε ε T q Λ r H , T ˘ s{p H ´ ε ε T q HF ˚ p L , L ; Λ q . p e ˚ r s The class p e is the sum of the Euler class 2 H with νT , where ν counts holomorphic index 2 discsthrough a generic point x of L , each weighted by the intersection of its boundary with a globalangular chain.One can explicitly construct a global angular chain and compute the value of ν , but in fact wecan use Lemma 6.3 to save us the trouble. With respect to the basis 1 , H of QH ˚ p CP , b N ; Λ q as aΛ-module, the map p e ˚ has matrix ˆ νT ε ε T νT ˙ , with determinant p ν ´ ε ε q T . In particular, HF ˚ ‰ ν ´ ε ε vanishes in K .For all integers ν and for all ε j P t˘ u , the quantity ν ´ ε ε is never ˘
1, so we see that thereis always some characteristic in which HF ˚ non-zero. By Lemma 6.3 we then conclude that L isnon-narrow in characteristic 3 when the ε j are all equal and in characteristic 5 when they are not.In each case, H ˚ p L ; K q has rank 2 so L is automatically wide (there is only one potentially non-zerodifferential in the Oh spectral sequence and non-narrowness means this differential is zero). (cid:3) Remark . In the wide cases we know that HF ˚ is the cone on multiplication by 2 H ` νT so 2 H acts as ´ νT . Hence CO p H q “ ´ νT ¨ L , so by (6) we get that ν “ ε ´ ε ´ ε in K . This agreeswith the explicit calculation of ν over Z , by counting discs meeting the global angular chain. {{ Worked example II: L p , q . We now consider the following closely-related example. Take M “ p CP q ´ and N “ CP , each equipped with an appropriate multiple of the Fubini–Studyform so that the product X “ M ´ ˆ N is monotone. Take the Hamiltonian SO p q -action on X “ p Sym CP q ˆ CP which rotates the CP ’s, and let L be the zero set of the moment map. Thisis an SO p q -orbit comprising triples p x , x , x q of points on the sphere, with x and x unordered,which form the vertices of an isosceles triangle with apex at x of a specific angle. The stabiliser ofsuch a configuration is the group of order 2 generated by the rotation through angle π about x , sothe orbit is diffeomorphic to the lens space L p , q . As before, it is monotone, orientable (hence has N L ě L with the trivial flat line bundle andan arbitrary relative spin structure to give a brane L , our goal is to compute the characteristics inwhich L is wide. Remark . We can explicitly calculate the moment map and apex angle following the conventionsof [34, Sections 3.1–3.2] but with the symplectic form on CP n scaled by n `
1; this ensures that acomplex line has area p n ` q π and Chern number n `
1, which gives monotonicity. In detail, wetake x , y as the basis for the standard representation of SU p q and consider the bases x n , x n ´ y, x n ´ y , . . . , y n and x n , dˆ n ˙ x n ´ y, dˆ n ˙ x n ´ y , . . . , y n for its n th symmetric power. We call the corresponding homogeneous coordinates on CP n standard and unitary coordinates respectively. If z is the vector of unitary coordinates on CP n then themoment map µ for the SU p q -action is x µ pr z sq , ξ y “ p n ` q i z : ϕ p ξ q zz : z for all ξ in su p q , where ϕ p ξ q is the matrix for the the infinitesimal action in unitary coordinates. MONOTONE LAGRANGIAN CASEBOOK 19
In our case we take w “ p w , w , w q and z “ p z , z q as unitary coordinates on CP and CP respectively, and see that the moment map satisfies x µ pr w s , r z sq , ` i ´ i ˘ y “ | w | ´ | w | k w k ` | z | ´ | z | k z k . The isosceles triangle with apex 0 and ‘base’ vertices ˘ λ corresponds to w “ p´ λ , , q and z “p , q , and if this lies in µ ´ p q then 3 1 ´ λ ` λ ` “ . This yields λ “ {
2, and hence the apex angle is arccos p ´ ? q « ˝ using [34, Equation (9)]. {{ Again there are three holomorphic index 2 discs through each point of L . The analogues of u and u meet the SO p q -invariant divisor comprising triples of points on CP , the first two unordered,such that (at least) one of the first two points coincides with the third; this is given by tpr ax ` bxy ` cy s , r dx ` ey sq : ae ` be p´ d q ` c p´ d q “ u so its Poincar´e dual is H ` H , where H and H are the hyperplane classes on CP and CP respectively. The analogue of u meets the invariant divisor comprising triples of points where theunordered pair coincide, given by tpr ax ` bxy ` cy s , r dx ` ey sq : b ´ ac “ u , Poincar´e dual to 2 H . Let A , A and A denote the homology classes of these discs. Lemma 6.7. A and A form a basis for H p X, L ; Z q .Proof. The long exact sequence of the pair gives a short exact sequence0 Ñ H p X ; Z q – Z Ñ H p X, L ; Z q Ñ H p L ; Z q – Z { Ñ , whilst intersecting with the two divisors above gives a map θ : H p X, L ; Z q Ñ Z . The latter sends A and A to p , q and p , q respectively, so it suffices to show it’s injective. Since H p X ; Z q hasindex 4 as a subgroup of H p X, L ; Z q , we have that θ is injective if and only if θ | H p X ; Z q is injectiveand θ p H p X ; Z qq has index 4 in θ p H p X, L ; Z qq , and this is what we shall prove.To show these two properties, note that lines on the CP and CP factors form a basis for H p X, Z q ,and are sent by θ to p , q and p , q respectively. Thus θ | H p X ; Z q is injective and has cokernel Z { θ itself is surjective we see that θ p H p X ; Z qq has index 4 in θ p H p X, L ; Z qq , so we’re done. (cid:3) As before, we introduce signs ε and ε to parametrise the relative spin structure, and now thethree discs compute (by [33, Theorem 3.5.3]) that CO p H ` H q “ ε T ¨ L and CO p H q “ ε T ¨ L . The relations in quantum cohomology, meanwhile, become H “ ε T and H “ T . Lemma 6.8. L is narrow unless: char K “ and ε “ ε ; or char K “ and ε “ ´ ε .Proof. Cubing the equality CO p H q “ ε T ¨ L gives 8 ε T ¨ L “ ε T ¨ L , so if HF ˚ ‰ ε must be equal to ε in K . (cid:3) Since we are in the setting of the Gysin sequence we can use it to prove
Theorem 6.9. L is wide when char K “ and ε “ ε or when char K “ and ε “ ´ ε .Proof. We argue as in Theorem 6.4. The exact triangle is nowΛ r H , T ˘ s{p H ´ T q Λ r H , T ˘ s{p H ´ T q HF ˚ p L , L ; Λ q p e ˚ r s with p e “ H ` νT , and the determinant of the map p e ˚ is p ν ´ q T . Since ν ´
16 is never ˘
1, asbefore there is always some characteristic in which HF ˚ non-zero. By Lemma 6.8 we deduce that L is non-narrow in characteristic 7 when ε “ ε and in characteristic 3 when ε “ ´ ε . Again, ineach case H ˚ p L ; K q has rank 2 so L is automatically wide. (cid:3) Quilt theory and the Chekanov tori
Lagrangian correspondences.
We now turn to the quilt theory of Wehrheim and Woodward,set out in [37] and subsequent papers by the same authors and by Ma’u-Wehrheim–Woodward (thistheory is actually also the basis of Perutz’s Gysin sequence). We shall use their composition theoremto relate the Lagrangians studied in Sections 6.2 and 6.3 to the Chekanov tori in CP ˆ CP and CP respectively.Recall that given symplectic manifolds p X j , ω j q , a Lagrangian correspondence from X j ´ to X j isa Lagrangian submanifold L p j ´ q j of X ´ j ´ ˆ X j , where as above X ´ j ´ is shorthand for p X j ´ , ´ ω j ´ q .These generalise both ordinary Lagrangians in X j , when X j ´ is a point, and symplectomorphismsfrom X j ´ to X j “ X j ´ , when L p j ´ q j is the graph. The composition of correspondences L p j ´ q j and L j p j ` q , written L p j ´ q j ˝ L j p j ` q , is the subset(7) π p j ´ qp j ` q ` p L p j ´ q j ˆ L j p j ` q q X p X ´ j ´ ˆ ∆ X j ˆ X j ` q ˘ Ă X ´ j ´ ˆ X j ` , where π p j ´ qp j ` q is the projection X ´ j ´ ˆ X j ˆ X ´ j ˆ X j ` Ñ X ´ j ´ ˆ X j ` and ∆ X j is the diagonal in X ´ j ˆ X j . The correspondence is said to be embedded if the intersectionin (7) is transverse and the restriction of π p j ´ qp j ` q to this intersection is an embedding, in whichcase it is a Lagrangian correspondence from X j ´ to X j ` .Under appropriate hypotheses, Wehrheim–Woodward define a ‘quilted’ Floer cohomology forcycles of Lagrangian correspondences X to X to . . . to X r ` “ X , and prove that it is invariantunder replacing consecutive correspondences by their composition when it is embedded. Moreover,when r “ X is a point, so the cycle of correspondences is just a pair of Lagrangians in X , theirtheory reproduces the ordinary Lagrangian intersection Floer cohomology of the two Lagrangians.For us the important result is: Theorem 7.1 ([37, Theorem 6.3.1]) . Suppose we have a Lagrangian correspondence L from X to X and a Lagrangian L in X such that the composition L : “ L ˝ L is embedded. Assumemoreover that all of these manifolds are closed, oriented and monotone, with the same monotonicityconstant, and that π p X ˆ X q is torsion. If HF ˚ p L , L q ‰ then HF ˚ p L , L q ‰ . We have been deliberately vague about the coefficients here: as stated the result only appliesin characteristic 2, and to move outside this setting we need the orientations constructed in [36].First we fix relative spin structures on L and L with background classes b ` w p X q ` b and b ` w p X q for some b j P H p X j ; Z { q . This induces a relative spin structure on L with backgroundclass b and it is with respect to these relative spin structures that Theorem 7.1 holds. Moreover,for these relative spin structures we have [37, Equation (24)](8) w p L q ` w p L q ` w p L q “ , where w denotes the signed count of index 2 discs through a generic point of L . Remark . The proof of Theorem 7.1 shows that HF ˚ p L , L q is isomorphic to HF ˚ p L sh01 , L ˆ L q in X ´ ˆ X , where L sh01 denotes L with background class shifted by w p X ˆ X q in the sense of[36, Remark 5.1.8]. This shift reverses the sign of the count of index 2 discs, so the differential on CF ˚ p L sh01 , L ˆ L q squares to w p L ˆ L q ´ w p L sh01 q “ w p L q ` w p L q ` w p L q “ . {{ MONOTONE LAGRANGIAN CASEBOOK 21
Worked example I: CP ˆ CP . Consider the Lagrangian SO p q -oribt L Ă p CP q fromSection 6.2. Our aim is to reprove Theorem 6.4 using Theorem 7.1. To do this we view L as aLagrangian correspondence L from X “ p CP ˆ CP q ´ to X “ CP , and consider its composition L with the Clifford torus (equatorial circle) L in X . This is equivalent to performing symplecticreduction at the equatorial level set for the S -action on the third CP factor by rotation about thevertical axis. This composition is embedded, and L is precisely the monotone Chekanov torus T Ch in X as presented by Entov–Polterovich [18, Example 1.22]. It consists of ordered triples of pointson the sphere which form the vertices of an equilateral triangle on a great circle, such that the thirdpoint is constrained to the equator. Remark . This torus was first discovered by Chekanov in R [11], and appears in both CP and CP ˆ CP in many Hamiltonian isotopic guises. Comparisons between various different constructionsare given by Gadbled [23] and Oakley–Usher [27]. {{ The hypotheses of Theorem 7.1 are satisfied so after choosing appropriate relative spin structuresthe non-vanishing of HF ˚ p T Ch , T Ch q implies the non-vanishing of HF ˚ p L, L q . By [7, Proposition6.1.4], the former is equivalent to the vanishing of the homology class swept by the boundaries of theindex 2 discs through a generic point of T Ch , and these discs were explicitly computed (for a specificregular complex structure) by Chekanov–Schlenk [12, Lemma 5.2]. There are exactly five such discs,in classes D , S ´ D ´ D , S ´ D , S ´ D and S ´ D ` D in H p CP ˆ CP , T Ch q , where S and S are the classes of the spheres in each factor and D and D are discs whose boundaries forma basis for H p T Ch ; Z q . We shall show that the relative spin structures and signs work out to giveTheorem 6.4.First equip L “ L with the standard spin structure, so its three index 2 discs all count positively,and equip L with the trivial spin structure. By the paragraph after Theorem 7.1 this induces arelative spin structure on L “ T Ch with background class 0, satisfying (by (8)) w p T Ch q “ ´ w p L q ´ w p L q “ ´ . This means the five discs computed by Chekanov–Schlenk must all count with negative signs, so thesum of their boundaries is 3 B D . We deduce that in this case L is non-narrow (hence wide) whenchar K “ L to the one with ε “ ε “ ´ ε “
1, withbackground class H ` H , recalling that H j represents the hyperplane class on the j th factor of X “ p CP q . This reverses the sign of the u disc, so w p L q becomes 1 ` ´ “
1. The inducedrelative spin structure on T Ch then has background class H ` H and satisfies w p T Ch q “ ´ w p L q ´ w p L q “ ´ . Let δ P H p CP ˆ CP , T Ch ; Z { q describe the difference between this relative spin structure andthe one in the previous paragraph, with respect to which all discs counted negatively. Since thebackground class is H ` H , we can write δ as S _ ` S _ ` δ D _ ` δ D _ for some δ j P Z {
2, where S _ , S _ , D _ , D _ is the basis of H p CP ˆ CP , T Ch ; Z { q dual to the basis S , S , D , D of H p CP ˆ CP , T Ch ; Z { q . We then have ´ “ w p T Ch q “ ´p´ q δ ` p´ q δ ` δ ` p´ q δ ` p´ q δ ` p´ q δ ` δ “ p´ q δ p ` p´ q δ q , where the five terms correspond to the signs attached to the five discs in the order listed above, andwe conclude that p´ q δ “ ´ p´ q δ “
1. The sum of the boundaries is thus 5 B D , so L isnon-narrow (hence wide) when char K “ ε “ ε “ ε “ ´ ε “ ε “ ´ ε “ ´ L can be obtained from the four alreadyconsidered by permuting the CP factors. Worked example II: CP . We can do the same thing for the Lagrangian lens space L in CP ˆ CP from Section 6.3, composing with the equator on the CP factor to give the Chekanovtorus in CP (again see the papers of Gadbled [23] and Oakley–Usher [27] for equivalences of variousdefinitions). Using the methods of Chekanov–Schlenk [12], Auroux [3, Proposition 5.8] computedthat again this torus bounds five index 2 holomorphic discs through a generic point, this time inclasses D , S ´ D ´ D , S ´ D , S ´ D , S ´ D ` D , where S is the class of a line on CP and D and D are classes of discs whose boundaries form a basis of H p T Ch ; Z q .When L is equipped with the standard spin structure and the equator with the trivial spinstructure, we obtain w p T Ch q “ ´ B D . Thus L is wide when char K “ L to the one with ε “ ´ ε “
1. This has zero back-ground class and gives w p L q “
1, so the induced relative spin structure on T Ch has zero backgroundclass and w p T Ch q “ ´
3. The difference between this relative spin structure on T Ch and the previousone is thus of the form δ D _ ` δ D _ with ´ “ w p T Ch q “ ´p´ q δ ´ p´ q δ ´ ´ ´ p´ q δ . We deduce that p´ q δ “ ´ p´ q δ “
1, so the sum of the boundaries is 9 B D and L is widewhen char K “ References [1] A. Amarzaya and Y. Ohnita, “Hamiltonian stability of certain minimal Lagrangian submanifolds in complexprojective spaces,”
Tohoku Math. J. (2) no. 4, (2003) 583–610. http://projecteuclid.org/euclid.tmj/1113247132 .[2] L. Astey, S. Gitler, E. Micha, and G. Pastor, “Cohomology of complex projective Stiefel manifolds,” Canad. J.Math. no. 5, (1999) 897–914. https://doi.org/10.4153/CJM-1999-039-2 .[3] D. Auroux, “Mirror symmetry and T -duality in the complement of an anticanonical divisor,” J. G¨okova Geom.Topol. GGT (2007) 51–91.[4] P. F. Baum and W. Browder, “The cohomology of quotients of classical groups,” Topology (1965) 305–336.[5] P. Biran, “Lagrangian non-intersections,” Geom. Funct. Anal. no. 2, (2006) 279–326. http://dx.doi.org/10.1007/s00039-006-0560-0 .[6] P. Biran and K. Cieliebak, “Symplectic topology on subcritical manifolds,” Comment. Math. Helv. no. 4, (2001) 712–753. http://dx.doi.org/10.1007/s00014-001-8326-7 .[7] P. Biran and O. Cornea, “Quantum structures for Lagrangian submanifolds,” arXiv:0708.4221v1 [math.SG] .[8] P. Biran and O. Cornea, “Rigidity and uniruling for Lagrangian submanifolds,” Geom. Topol. no. 5, (2009) 2881–2989. http://dx.doi.org/10.2140/gt.2009.13.2881 .[9] P. Biran and M. Khanevsky, “A Floer-Gysin exact sequence for Lagrangian submanifolds,” Comment. Math. Helv. no. 4, (2013) 899–952. http://dx.doi.org/10.4171/CMH/307 .[10] A. Borel, “Sur l’homologie et la cohomologie des groupes de Lie compacts connexes,” Amer. J. Math. (1954)273–342.[11] Y. V. Chekanov, “Lagrangian tori in a symplectic vector space and global symplectomorphisms,” Math. Z. no. 4, (1996) 547–559. http://dx.doi.org/10.1007/PL00004278 .[12] Y. Chekanov and F. Schlenk, “Notes on monotone Lagrangian twist tori,”
Electron. Res. Announc. Math. Sci. (2010) 104–121. http://dx.doi.org/10.3934/era.2010.17.104 .[13] R. Chiang, “New Lagrangian submanifolds of CP n ,” Int. Math. Res. Not. no. 45, (2004) 2437–2441. http://dx.doi.org/10.1155/S1073792804133102 .[14] C.-H. Cho, “Holomorphic discs, spin structures, and Floer cohomology of the Clifford torus,”
Int. Math. Res. Not. no. 35, (2004) 1803–1843. http://dx.doi.org/10.1155/S1073792804132716 .[15] K. Cieliebak, A. R. Gaio, and D. A. Salamon, “ J -holomorphic curves, moment maps, and invariants ofHamiltonian group actions,” Internat. Math. Res. Notices no. 16, (2000) 831–882. http://dx.doi.org/10.1155/S1073792800000453 .[16] M. Damian, “Floer homology on the universal cover, Audin’s conjecture and other constraints on Lagrangiansubmanifolds,”
Comment. Math. Helv. no. 2, (2012) 433–462. http://dx.doi.org/10.4171/CMH/259 .[17] V. de Silva, Products in Symplectic Floer Homology of Lagrangian Intersection . PhD thesis, University ofOxford, 1999.[18] M. Entov and L. Polterovich, “Rigid subsets of symplectic manifolds,”
Compos. Math. no. 3, (2009) 773–826. http://dx.doi.org/10.1112/S0010437X0900400X .[19] J. D. Evans and Y. Lekili, “Floer cohomology of the Chiang Lagrangian,”
Selecta Math. (N.S.) no. 4, (2015) 1361–1404. http://dx.doi.org/10.1007/s00029-014-0171-9 . MONOTONE LAGRANGIAN CASEBOOK 23 [20] J. D. Evans and Y. Lekili, “Generating the Fukaya categories of Hamiltonian G -manifolds,” J. Amer. Math. Soc. no. 1, (2019) 119–162. https://doi.org/10.1090/jams/909 .[21] K. Fukaya, “Application of Floer homology of Langrangian submanifolds to symplectic topology,” in Morsetheoretic methods in nonlinear analysis and in symplectic topology , vol. 217 of
NATO Sci. Ser. II Math. Phys.Chem. , pp. 231–276. Springer, Dordrecht, 2006. http://dx.doi.org/10.1007/1-4020-4266-3_06 .[22] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono,
Lagrangian intersection Floer theory: anomaly and obstruction.Two-volume set , vol. 46 of
AMS/IP Studies in Advanced Mathematics . American Mathematical Society,Providence, RI; International Press, Somerville, MA, 2009.[23] A. Gadbled, “On exotic monotone Lagrangian tori in CP and S ˆ S ,” J. Symplectic Geom. no. 3, (2013)343–361. http://projecteuclid.org/euclid.jsg/1384282840 .[24] H. Hopf, “ ¨Uber die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen,” Ann. of Math. (2) (1941) 22–52. https://doi.org/10.2307/1968985 .[25] H. Iriyeh, “Symplectic topology of Lagrangian submanifolds of C P n with intermediate minimal Maslovnumbers,” Adv. Geom. no. 2, (2017) 247–264. http://dx.doi.org/10.1515/advgeom-2017-0005 .[26] D. McDuff and S. Tolman, “Topological properties of Hamiltonian circle actions,” IMRP Int. Math. Res. Pap. (2006) 72826, 1–77.[27] J. Oakley and M. Usher, “On certain Lagrangian submanifolds of S ˆ S and C P n ,” Algebr. Geom. Topol. no. 1, (2016) 149–209. http://dx.doi.org/10.2140/agt.2016.16.149 .[28] Y.-G. Oh, “Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings,” Internat. Math. Res. Notices no. 7, (1996) 305–346. http://dx.doi.org/10.1155/S1073792896000219 .[29] T. Perutz, “A symplectic Gysin sequence,” arXiv:0807.1863v1 [math.SG] .[30] P. Seidel, “ π of symplectic automorphism groups and invertibles in quantum homology rings,” Geom. Funct. Anal. no. 6, (1997) 1046–1095. http://dx.doi.org/10.1007/s000390050037 .[31] P. Seidel, “Graded Lagrangian submanifolds,” Bull. Soc. Math. France no. 1, (2000) 103–149. .[32] I. Smith, “Floer cohomology and pencils of quadrics,”
Invent. Math. no. 1, (2012) 149–250. http://dx.doi.org/10.1007/s00222-011-0364-1 .[33] J. Smith, “Discrete and continuous symmetries in monotone Floer theory,” arXiv:1703.05343v3 [math.SG] .[34] J. Smith, “Floer cohomology of Platonic Lagrangians,” arXiv:1510.08031v3 [math.SG] .[35] B. Torricelli, “A survey on the Lagrangian SU p n q{p Z { n q in CP n ´ ,” Master’s thesis, University College London,2016, unpublished.[36] K. Wehrheim and C. T. Woodward, “Orientations for pseudoholomorphic quilts,” arXiv:1503.07803v6 [math.SG] .[37] K. Wehrheim and C. T. Woodward, “Quilted Floer cohomology,” Geom. Topol. no. 2, (2010) 833–902. http://dx.doi.org/10.2140/gt.2010.14.833 .[38] E. Witten, “The Verlinde algebra and the cohomology of the Grassmannian,” in Geometry, topology, & physics ,Conf. Proc. Lecture Notes Geom. Topology, IV, pp. 357–422. Int. Press, Cambridge, MA, 1995.[39] F. Zapolsky, “Quasi-morphisms on contactomorphism groups and Grassmannians of 2-planes,” arXiv:1902.02403 [math.SG] . Department of Mathematics, University College London, Gower Street, London, WC1E 6BT
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