[Paper Review] The Lagrangian Cubic Equation

Paul Biran, Cedric Membrez|arXiv (Cornell University)|Jun 23, 2014

Geometric and Algebraic TopologyMathematics35 references1 citations

TL;DR

This paper establishes a universal cubic relation in the quantum homology ring of a monotone symplectic manifold involving the homology class of a Lagrangian sphere, proving that the quantum product of the class with itself satisfies a specific cubic equation depending on the dimension and Chern number. The key result is a Lagrangian cubic equation governing the quantum product of the class with itself, linking it to Gromov-Witten invariants and Floer-theoretic invariants, with applications to Dehn twists and enumerative geometry in the monotone case.

ABSTRACT

Let $M$ be a closed symplectic manifold and $L \subset M$ a Lagrangian submanifold. Denote by $[L]$ the homology class induced by $L$ viewed as a class in the quantum homology of $M$. The present paper is concerned with properties and identities involving the class $[L]$ in the quantum homology ring. We also study the relations between these identities and invariants of $L$ coming from Lagrangian Floer theory. We pay special attention to the case when $L$ is a Lagrangian sphere.

Motivation & Objective

• To establish a universal cubic equation satisfied by the quantum homology class of a Lagrangian sphere in a monotone symplectic manifold.
• To explore the interplay between quantum homology, Lagrangian Floer homology, and Gromov-Witten invariants for Lagrangian submanifolds.
• To characterize the structure of the quantum homology ring for Lagrangian spheres, particularly in the case where the minimal Maslov number divides the dimension.
• To provide a framework for computing invariants such as the coefficient γS in the cubic relation using Floer-theoretic and enumerative geometric methods.

Proposed method

• Use of Lagrangian Floer homology to derive the cubic relation for the quantum product of the homology class [S] of a Lagrangian sphere S.
• Application of the quantum product ∗ on the quantum homology ring QH(M; Z[q]) with degree |q| = −2.
• Computation of the cubic equation [S]∗3 = γS[S]qn under conditions on the dimension n and the minimal Chern number CM.
• Use of the Dehn twist along S to show that σL = 0 in the case of even-dimensional spheres, leading to a simplified cubic equation.
• Employment of spectral sequences from the pearl complex to compute quantum homology of Lagrangian submanifolds, particularly for Q-homology spheres.
• Expression of the coefficient σL in terms of genus 0 Gromov-Witten invariants over classes A with ⟨c1, A⟩ = n/2.

Experimental results

Research questions

• RQ1 What universal cubic relation does the quantum homology class [S] of a Lagrangian sphere satisfy in the quantum homology ring of a monotone symplectic manifold?
• RQ2 How does the coefficient γS in the cubic relation [S]∗3 = γS[S]qn depend on the dimension n and the minimal Chern number CM?
• RQ3 In what cases does the cubic equation reduce to a quadratic or vanish identically, and what topological or geometric conditions enforce this?
• RQ4 How can the coefficient σL in the generalized Lagrangian cubic equation be expressed in terms of Gromov-Witten invariants?
• RQ5 What role does the Dehn twist play in forcing σL = 0, and how does this relate to the symmetry of the quantum product?

Key findings

• For an odd-dimensional Lagrangian sphere S, the quantum product satisfies [S]∗[S] = 0, indicating a quadratic vanishing in the quantum homology ring.
• For even-dimensional S, if CM divides n, then [S]∗3 = γS[S]qn with γS ∈ Z, and γS is divisible by 4 if 2CM does not divide n.
• When CM does not divide n, the cubic relation vanishes: [S]∗3 = 0.
• The coefficient σL in the generalized Lagrangian cubic equation is expressible as a sum of genus 0 Gromov-Witten invariants over classes A with ⟨c1, A⟩ = n/2.
• In the case of a Lagrangian sphere with χ = 2 and 2CM|n, the Dehn twist symmetry forces σL = 0, reducing the cubic to [S]∗3 = χ²τL[S]qn with τL ∈ 1/4Z.
• For even-dimensional Lagrangian Q-homology spheres, the quantum homology QH∗(L; ΛQ) is isomorphic to H∗(L; Q)⊗ΛQ if NL ∤ n+1 or [L] ≠ 0, otherwise it may vanish or be isomorphic depending on the count of Maslov index n+1 disks.