Anti-symplectic Involution Research Articles

Symplectic splitting of Hamiltonian structures and reduced monodromy matrices

In Hamiltonian systems monodromy and reduced monodromy matrices of periodic orbits are symplectic. A symmetry is a symplectic or anti-symplectic involution which leaves the Hamiltonian invariant. A natural class to study are periodic orbits which are invariant under such involutions. Depending on whether the orbit is invariant under a symplectic or anti-symplectic symmetry, its monodromy matrix satisfies special properties. In the symplectic case, the monodromy matrix admits a symplectic decomposition, depending on the symplectic involution's fixed point set. In the anti-symplectic case, the monodromy matrix is of special symmetric form, which was constructed by Frauenfelder–Moreno. Such monodromy matrices are very beneficial in applied problems. The goal of this paper is to design a general construction behind the reduced monodromy's symplectic splitting, by using the notion of Hamiltonian manifolds, and to give a general framework for monodromy and reduced monodromy matrices of invariant periodic orbits in Hamiltonian systems with symmetries. In addition, we show an application related to contact forms, especially to the spatial circular restricted three-body problem for energy level sets of contact type.

The linear symmetries of Hill’s lunar problem

A symmetry of a Hamiltonian system is a symplectic or anti-symplectic involution which leaves the Hamiltonian invariant. For the planar and spatial Hill lunar problem, four resp. eight linear symmetries are well-known. Algebraically, the planar ones form a Klein four-group {mathbb {Z}}_2 times {mathbb {Z}}_2 and the spatial ones form the group {mathbb {Z}}_2 times {mathbb {Z}}_2 times {mathbb {Z}}_2. We prove that there are no other linear symmetries. Remarkably, in Hill’s system the spatial linear symmetries determine already the planar linear symmetries.

Cubic fourfolds with an involution

There are three types of involutions on a cubic fourfold; two of anti-symplectic type, and one symplectic. Here we show that cubics with involutions exhibit the full range of behaviour in relation to rationality conjectures. Namely, we show a general cubic fourfold with a symplectic involution has no associatedK3K3surface and is conjecturely irrational. In contrast, a cubic fourfold with a particular anti-symplectic involution has an associatedK3K3, and is in fact rational. We show such a cubic is contained in the intersection of all non-empty Hassett divisors; we call such a cubic Hassett maximal. We study the algebraic and transcendental lattices for cubics with an involution both lattice theoretically and geometrically.

Degeneration of natural Lagrangians and Prymian integrable systems

Starting from an anti-symplectic involution on a K3 surface, one can consider a natural Lagrangian subvariety inside the moduli space of sheaves over the K3. One can also construct a Prymian integrable system following a construction of Markushevich–Tikhomirov, extended by Arbarello–Saccà–Ferretti, Matteini and Sawon–Shen. In this article we address a question of Sawon, showing that these integrable systems and their associated natural Lagrangians degenerate, respectively, into fix loci of involutions considered by Heller–Schaposnik, García-Prada–Wilkin and Basu–García-Prada. Along the way we find interesting results such as the proof that the Donagi–Ein–Lazarsfeld degeneration is a degeneration of symplectic varieties, a generalization of this degeneration, originally described for K3 surfaces, to the case of an arbitrary smooth projective surface, and a description of the behaviour of certain involutions under this degeneration.

On the Duality of F-Theory and the CHL String in Seven Dimensions

We show that the duality between F-theory and the CHL string in seven dimensions defines algebraic correspondences between K3 surfaces polarized by the rank-ten lattices $H \oplus N$ and $H\oplus E_8(-2)$. In the special case when the F-theory admits an additional anti-symplectic involution or, equivalently, the CHL string admits a symplectic one, both moduli spaces coincide. In this case, we derive an explicit parametrization for the F-theory compactifications dual to the CHL string, using an auxiliary genus-one curve, based on a construction given by Andr\'e Weil.

Remarks on the systoles of symmetric convex hypersurfaces and symplectic capacities

In this note we study the systoles of convex hypersurfaces in $${\mathbb {R}}^{2n}$$ invariant under an anti-symplectic involution. We investigate a uniform upper bound of the ratio between the systole and the symmetric systole of the hypersurfaces using symplectic capacities from Floer theory. We discuss various concrete examples in which the ratio can be understood explicitly.

Algebraic cycles and Lehn–Lehn–Sorger–van Straten eightfolds

AbstractThis article is about Lehn–Lehn–Sorger–van Straten eightfolds$Z$and their anti-symplectic involution$\iota$. When$Z$is birational to the Hilbert scheme of points on a K3 surface, we give an explicit formula for the action of$\iota$on the Chow group of$0$-cycles of$Z$. The formula is in agreement with the Bloch–Beilinson conjectures and has some non-trivial consequences for the Chow ring of the quotient.

Point-like bounding chains in open Gromov–Witten theory

We present a solution to the problem of defining genus zero open Gromov–Witten invariants with boundary constraints for a Lagrangian submanifold of arbitrary dimension. Previously, such invariants were known only in dimensions 2 and 3 from the work of Welschinger. Our approach does not require the Lagrangian to be fixed by an anti-symplectic involution, but can use such an involution, if present, to obtain stronger results. Also, non-trivial invariants are defined for broader classes of interior constraints and Lagrangian submanifolds than previously possible even in the presence of an anti-symplectic involution. The invariants of the present work specialize to invariants of Welschinger, Fukaya, and Georgieva in many instances. The main obstacle to defining open Gromov–Witten invariants with boundary constraints in arbitrary dimension is the bubbling of J-holomorphic disks. Unlike in low dimensions or for interior constraints, disk bubbles do not cancel in pairs by anti-symplectic involution symmetry. Rather, we use the technique of bounding chains introduced in Fukaya–Oh–Ohta–Ono’s work on Lagrangian Floer theory to cancel disk bubbling. At the same time and independently, gauge equivalence classes of bounding chains play the role of boundary constraints, in place of the cohomology classes that usually serve as constraints in Gromov–Witten theory. A crucial step in our construction is to identify a canonical up to gauge equivalence family of “point-like” bounding chains, which specialize in dimensions 2 and 3 to the point constraints considered by Welschinger.

On the Cohomology Groups of Real Lagrangians in Calabi–Yau Threefolds

The quintic threefold X is the most studied Calabi-Yau 3-fold in the mathematics literature. In this article, using Čech-to-derived spectral sequences, we investigate the mod 2 and integral cohomology groups of a real Lagrangian , obtained as the fixed locus of an anti-symplectic involution in the mirror to X. We show that is the disjoint union of a 3-sphere and a rational homology sphere. Analyzing the mod 2 cohomology further, we deduce a correspondence between the mod 2 Betti numbers of and certain counts of integral points on the base of a singular torus fibration on X. By work of Batyrev, this identifies the mod 2 Betti numbers of with certain Hodge numbers of X. Furthermore, we show that the integral cohomology groups of are 2-primary for , we conjecture that this holds in much greater generality.

ON THE CHOW RING OF CERTAIN LEHN–LEHN–SORGER–VAN STRATEN EIGHTFOLDS

AbstractWe consider a 10-dimensional family of Lehn–Lehn–Sorger–van Straten hyperkähler eightfolds, which have a non-symplectic automorphism of order 3. Using the theory of finite-dimensional motives, we show that the action of this automorphism on the Chow group of 0-cycles is as predicted by the Bloch–Beilinson conjectures. We prove a similar statement for the anti-symplectic involution on varieties in this family. This has interesting consequences for the intersection product of the Chow ring of these varieties.

The Chekanov torus in $S^2 \times S^2$ is not real

We prove that the count of Maslov index 2 $J$-holomorphic discs passing through a generic point of a real Lagrangian submanifold in a closed spherically monotone symplectic manifold must be even. As a corollary, we exhibit a genuine real symplectic phenomenon in terms of involutions, namely that the Chekanov torus $\mathbb{T}_{\text{Chek}}$ in $S^2\times S^2$, which is a monotone Lagrangian torus not Hamiltonian isotopic to the Clifford torus $\mathbb{T}_{\text{Clif}}$, can be seen as the fixed point set of a smooth involution, but not of an antisymplectic involution.

Infinitely many non‐isotopic real symplectic forms on S2×S2

Let $(S^2,\omega)$ be a symplectic sphere, and let $\tau \colon S^2 \to S^2$ be an anti-symplectic involution of $(S^2,\omega)$. We consider the product $(S^2,\omega) \times (S^2,\omega)$ endowed with the anti-symplectic involution $\tau \times \tau$, and study the space of monotone anti-invariant symplectic forms on this four-manifold. We show that this space is disconnected. In addition, during the course of the proof, we produce a diffeomorphism of the grassmannian (2,4) which induces the identity map on all homology and homotopy groups, but which is not homotopic to the identity.

Zero-cycles on double EPW sextics

The Chow rings of hyperKähler varieties are conjectured to have a particularly rich structure. In this paper, we focus on the locally complete family of double EPW sextics and establish some properties of their Chow rings. First, we prove a Beauville–Voisin type theorem for zero-cycles on double EPW sextics; precisely, we show that the codimension-4 part of the subring of the Chow ring of a double EPW sextic generated by divisors, the Chern classes and codimension-2 cycles invariant under the anti-symplectic covering involution has rank one. Second, for double EPW sextics birational to the Hilbert square of a K3 surface, we show that the action of the anti-symplectic involution on the Chow group of zero-cycles commutes with the Fourier decomposition of Shen–Vial.

$K3$ surfaces with involution and analytic torsion

This is the abstruct of the revised paper. We study the equivariant analytic torsion for K3 surfaces with an anti-symplectic involution with the invariant lattice M (such a surface is called a 2-elementary K3 surface of type M in this paper), and show that it (together with the analytic torsion of the fixed curves) can be identified with the automorphic form on the moduli space characterizing the discriminant locus. Three lattices A_1, II_{1,1}(2), II_{1,9}(2) are of particular interest, because they consist of the building blocks of 2-elementary lattices. An explicit formula is given for them. In particular, if M is twice the Enriques lattice, the automorphic form coincides with Borcherds's Phi-function which confirms an observation by Jorgenson-Todorov and Harvey-Moore. Some other examples are shown to be related to Borcherds's product and generalized Kac-Moody algebras.

Involutions, obstructions and mirror symmetry

Consider a Maslov zero Lagrangian submanifold diffeomorphic to a Lie group on which an anti-symplectic involution acts by the inverse map of the group. We show that the Fukaya A∞ endomorphism algebra of such a Lagrangian is quasi-isomorphic to its de Rham cohomology tensored with the Novikov field. In particular, it is unobstructed, formal, and its Floer and de Rham cohomologies coincide. Our result implies that the smooth fibers of a large class of singular Lagrangian fibrations are unobstructed and their Floer and de Rham cohomologies coincide. This is a step in the SYZ and family Floer cohomology approaches to mirror symmetry.More generally, our result continues to hold if the Lagrangian has cohomology the free graded algebra on a graded vector space V concentrated in odd degree, and the anti-symplectic involution acts on the cohomology of the Lagrangian by the induced map of negative the identity on V. It suffices for the Maslov class to vanish modulo 4.

Antisymplectic involution and Floer cohomology

The main purpose of the present paper is a study of orientations of the moduli spaces of pseudo-holomorphic discs with boundary lying on a \emph{real} Lagrangian submanifold, i.e., the fixed point set of an anti-symplectic involutions $\tau$ on a symplectic manifold. We introduce the notion of $\tau$-relatively spin structure for an anti-symplectic involution $\tau$, and study how the orientations on the moduli space behave under the involution $\tau$. We also apply this to the study of Lagrangian Floer theory of real Lagrangian submanifolds. In particular, we study unobstructedness of the $\tau$-fixed point set of symplectic manifolds and in particular prove its unobstructedness in the case of Calabi-Yau manifolds. And we also do explicit calculation of Floer cohomology of $\R P^{2n+1}$ over $\Lambda_{0,nov}^{\Z}$ which provides an example whose Floer cohomology is not isomorphic to its classical cohomology. We study Floer cohomology of the diagonal of the square of a symplectic manifold, which leads to a rigorous construction of the quantum Massey product of symplectic manifold in complete generality.

Symmetric periodic orbits and uniruled real Liouville domains

A real Liouville domain is a Liouville domain with an exact anti-symplectic involution. The authors call a real Liouville domain uniruled if there exists an invariant finite energy plane through every real point. Asymptotically, an invariant finite energy plane converges to a symmetric periodic orbit. In this note, they work out a criterion which guarantees uniruledness for real Liouville domains.

Open Gromov–Witten disk invariants in the presence of an anti-symplectic involution

For a symplectic manifold with an anti-symplectic involution having non-empty fixed locus, we construct a model of the moduli space of real sphere maps out of moduli spaces of decorated disk maps and give an explicit expression for its first Stiefel–Whitney class. As a corollary, we obtain a large number of examples, which include all odd-dimensional projective spaces and many complete intersections, for which many types of real moduli spaces are orientable. For these manifolds, we define open Gromov–Witten invariants with no restriction on the dimension of the manifolds or the type of the constraints if there are no boundary marked points; a WDVV-type recursion obtained in a sequel computes these invariants for many real symplectic manifolds. If there are boundary marked points, we define the invariants under some restrictions on the allowed boundary constraints, even though the moduli spaces are not orientable in these cases.

Counting genus-zero real curves in symplectic manifolds

There are two types of $J$-holomorphic spheres in a symplectic manifold invariant under an anti-symplectic involution: those that have a fixed point locus and those that do not. The former are described by moduli spaces of $J$-holomorphic disks, which are well studied in the literature. In this paper, we first study moduli spaces describing the latter and then combine the two types of moduli spaces to get a well-defined theory of counting real curves of genus 0. We use equivariant localization to show that these invariants (unlike the disk invariants) are essentially the same for the two (standard) involutions on $\mathbb{P}^{4n-1}$.

The moduli space of maps with crosscaps: the relative signs of the natural automorphisms

Just as a symmetric surface with separating fixed locus halves into two oriented bordered surfaces, an arbitrary symmetric surface halves into two oriented symmetric half-surfaces, i.e. surfaces with crosscaps. Motivated in part by the string theory view of real Gromov-Witten invariants, we previously introduced moduli spaces of maps from surfaces with crosscaps, developed the relevant Fredholm theory, and resolved the orientability problem in this setting. In this paper, we determine the relative signs of the automorphisms of these moduli spaces induced by interchanges of boundary components of the domain and by the anti-symplectic involution on the target manifold, without any global assumptions on the latter. As immediate applications, we describe sufficient conditions for the moduli spaces of real genus 1 maps and for real maps with separating fixed locus to be orientable; we treat the general genus 2+ case in a separate paper. Our sign computations also lead to an extension of recent Floer-theoretic applications of anti-symplectic involutions and to a related reformulation of these results in a more natural way.