Physics:Arnold–Givental conjecture

The Arnold–Givental conjecture, named after Vladimir Arnold and Alexander Givental, is a statement on Lagrangian submanifolds. It gives a lower bound in terms of the Betti numbers of a Lagrangian submanifold L on the number of intersection points of L with another Lagrangian submanifold which is obtained from L by Hamiltonian isotopy, and which intersects L transversally.

Statement

Let [math]\displaystyle{ (M, \omega) }[/math] be a compact [math]\displaystyle{ 2n }[/math]-dimensional symplectic manifold. An anti-symplectic involution is a diffeomorphism [math]\displaystyle{ \tau: M \to M }[/math] such that [math]\displaystyle{ \tau^* \omega = -\omega }[/math]. The fixed point set [math]\displaystyle{ L \subset M }[/math] of [math]\displaystyle{ \tau }[/math] is necessarily a Lagrangian submanifold.

Let [math]\displaystyle{ H_t\in C^\infty(M), 0 \leq t \leq 1 }[/math] be a smooth family of Hamiltonian functions on [math]\displaystyle{ M }[/math] which generates a 1-parameter family of Hamiltonian diffeomorphisms [math]\displaystyle{ \varphi_t: M \to M }[/math]. The Arnold–Givental conjecture says, suppose [math]\displaystyle{ \varphi_1(L) }[/math] intersects transversely with [math]\displaystyle{ L }[/math], then

[math]\displaystyle{ \# (\varphi_1(L) \cap L) \geq \sum_{i=0}^n {\rm dim} H_*(L; {\mathbb Z}_2). }[/math]

Status

The Arnold–Givental conjecture has been proved for certain special cases.

Givental proved it for the case when [math]\displaystyle{ (M, L) = (\mathbb{CP}^n, \mathbb{RP}^n) }[/math].

Yong-Geun Oh proved it for real forms of compact Hermitian spaces with suitable assumptions on the Maslov indices.

Lazzarini proved it for negative monotone case under suitable assumptions on the minimal Maslov number.

Kenji Fukaya, Yong-Geun Oh, Ohta, and Ono proved for the case when [math]\displaystyle{ (M, \omega) }[/math] is semi-positive.

Frauenfelder proved it for the situation when [math]\displaystyle{ (M, \omega) }[/math] is a certain symplectic reduction, using gauged Floer theory.

See also

• Arnold conjecture

References

Bibliography

• Frauenfelder, Urs (2004), "The Arnold–Givental conjecture and moment Floer homology", International Mathematics Research Notices 2004 (42): 2179–2269, doi:10.1155/S1073792804133941 .
• Fukaya, Kenji; Oh, Yong-Geun; Ohta, Hiroshi; Ono, Kaoru (2009), Lagrangian intersection Floer theory - anomaly and obstruction, International Press, ISBN 978-0-8218-5253-8 
• Givental, A. B. (1989a), "Periodic maps in symplectic topology", Funktsional. Anal. I Prilozhen 23 (4): 37–52, https://www.mathnet.ru/eng/faa/v23/i4/p37 
  • Givental, A. B. (1989b), "Periodic maps in symplectic topology (translation from Funkts. Anal. Prilozh. 23, No. 4, 37-52 (1989))", Functional Analysis and Its Applications 23 (4): 287–300, doi:10.1007/BF01078943 
• "Floer cohomology and Arnol'd-Givental's conjecture of [on Lagrangian intersections"], Comptes Rendus de l'Académie des Sciences 315 (3): 309–314, 1992, https://gallica.bnf.fr/ark:/12148/bpt6k5470708m/f315.item?lang=FR# .
• Oh, Yong-Geun (1995), "Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks, III: Arnold-Givental Conjecture", The Floer Memorial Volume, pp. 555–573, doi:10.1007/978-3-0348-9217-9_23, ISBN 978-3-0348-9948-2