[Paper Review] Computing the quantum cohomology of primitive classes

Xiaowen Hu|arXiv (Cornell University)|Jan 15, 2015

Algebraic Geometry and Number Theory33 references4 citations

TL;DR

This paper uses deformation invariance and $S_n$-symmetry of Gromov-Witten invariants, combined with classical invariant theory, to reduce the WDVV equation via monodromy groups, enabling reconstruction of quantum cohomology for primitive classes in smooth complete intersections. It establishes a reconstruction theorem for cubic hypersurfaces and odd-dimensional complete intersections of two quadrics, and fully describes the cohomology ring of the Fano variety of lines on smooth cubic hypersurfaces.

ABSTRACT

For smooth complete intersections in the projective spaces, we make use of the deformation invariance and $S_n$-symmetry of the Gromov-Witten invariants and results in classical invariant theory to study the symmetric reduction of the WDVV equation by the monodromy groups. We discuss to what extent the quantum cohomology involving primitive cohomology classes can be determined. In particular, we obtain a reconstruction theorem for cubic hypersurfaces and odd dimensional complete intersection of two quadrics. By the way, we obtain a complete description of the cohomology ring of the Fano variety of lines on smooth cubic hypersurfaces.

Motivation & Objective

• To determine the quantum cohomology of primitive cohomology classes in smooth complete intersections in projective spaces.
• To understand the extent to which quantum cohomology can be reconstructed using symmetry and deformation invariance of Gromov-Witten invariants.
• To apply classical invariant theory to the symmetric reduction of the WDVV equation under monodromy group actions.
• To provide a complete description of the cohomology ring of the Fano variety of lines on smooth cubic hypersurfaces.

Proposed method

• Leveraging deformation invariance of Gromov-Witten invariants to stabilize quantum product structures.
• Utilizing $S_n$-symmetry of invariants to constrain the form of the quantum cohomology algebra.
• Applying results from classical invariant theory to analyze symmetric reductions of the WDVV equation.
• Reducing the WDVV equation via monodromy group actions to simplify the quantum product relations.
• Focusing on primitive cohomology classes to isolate essential quantum corrections.
• Using the symmetric structure of the invariants to derive recursive constraints on the quantum cup product.

Experimental results

Research questions

• RQ1 To what extent can the quantum cohomology of primitive classes in smooth complete intersections be reconstructed from symmetry and invariance properties?
• RQ2 How does the monodromy group act on the WDVV equation, and what symmetric reductions emerge from this action?
• RQ3 What is the structure of the cohomology ring of the Fano variety of lines on a smooth cubic hypersurface?
• RQ4 Can a reconstruction theorem be established for cubic hypersurfaces using symmetric reduction of the WDVV equation?
• RQ5 What role does $S_n$-symmetry play in simplifying the quantum cohomology relations for complete intersections?

Key findings

• A reconstruction theorem is established for cubic hypersurfaces, showing that their quantum cohomology for primitive classes is fully determined by symmetric constraints and deformation invariance.
• The quantum cohomology of odd-dimensional complete intersections of two quadrics is also fully reconstructible using the same method.
• The cohomology ring of the Fano variety of lines on a smooth cubic hypersurface is completely described, including its Betti numbers and multiplicative structure.
• The symmetric reduction of the WDVV equation under monodromy groups leads to a finite system of constraints that determine the quantum product.
• The method successfully isolates the quantum corrections to primitive cohomology classes by exploiting $S_n$-symmetry and classical invariant theory.
• The results demonstrate that the quantum cohomology of these spaces is determined by their geometric symmetries and topological invariants.