[Paper Review] Mirrors, Functoriality, and Derived Geometry

Yuri I. Manin|arXiv (Cornell University)|Aug 9, 2017

Homotopy and Cohomology in Algebraic Topology38 references3 citations

TL;DR

This paper proposes a functorial approach to quantum cohomology using derived deformation theory and $L_π$-algebras, framing quantum cohomology as a derived deformation of cohomology rings via Gromov-Witten invariants. The key contribution is the introduction of $F_\infty$-structures on extended deformation spaces, generalizing Frobenius manifolds and providing a functorial framework for mirror symmetry.

ABSTRACT

In this survey, I suggest to approach the problem of functorial properties of quantum cohomology by drawing lessons from several versions of Mirror duality involving deformation spaces.

Motivation & Objective

• To address the mystery of functorial properties in quantum cohomology by embedding it within deformation theory frameworks.
• To investigate how mirror symmetry constructions—especially those involving deformation spaces—exhibit rich functoriality.
• To extend classical deformation theory using derived and enriched structures, particularly $L_\infty$-algebras and $F_\infty$-manifolds.
• To establish a bridge between quantum cohomology and Frobenius manifold structures through derived geometric tools.
• To provide a conceptual framework for understanding self-referentiality in the modular operad of stable curves via derived deformation functors.

Proposed method

• Uses controlling $L_\infty$-algebras to formalize deformation functors, generalizing DGLA-based deformation theory.
• Applies extended deformation functors to model quantum cohomology as a formal deformation of $H^*(V)$ via Novikov rings $K_q$.
• Constructs the quantum product via third derivatives of a potential $\Phi$ encoding genus-zero Gromov-Witten invariants.
• Introduces $F_\infty$-structures on formal dg manifolds as a derived generalization of Frobenius manifolds.
• Defines $F_\infty$-identity through cohomology classes of polybrackets $[[\mu_\bullet, \mu_\bullet]]$ in the $C_\infty$-algebra framework.
• Establishes that Hochschild cohomology and singular cohomology of compact spaces naturally carry $F_\infty$-structures on their formal dg resolutions.

Experimental results

Research questions

• RQ1 How can functorial properties of quantum cohomology be systematically understood through deformation-theoretic methods?
• RQ2 What is the role of $L_\infty$-algebras and their extended versions in encoding quantum cohomology as a deformation of cohomology rings?
• RQ3 How do $F_\infty$-structures generalize Frobenius manifolds and provide a derived framework for mirror symmetry?
• RQ4 In what sense is the modular operad of stable curves self-referential, and how does derived geometry resolve this?
• RQ5 What is the precise relationship between quantum cohomology, Gromov-Witten invariants, and the $F_\infty$-structure on the formal dg manifold of Hochschild cohomology?

Key findings

• Quantum cohomology $H^*_q(V)$ is realized as a formal deformation of $H^*(V)$ via a potential $\Phi$ whose third derivatives define the quantum product.
• The quantum product satisfies the $F$-identity in the derived setting, generalizing the Frobenius identity to $F_\infty$-structures.
• The formal dg manifold associated with Hochschild cohomology of an associative algebra naturally carries an $F_\infty$-structure.
• The formal dg manifold of singular cohomology of a compact topological space is also an $F_\infty$-manifold.
• The $F_\infty$-identity is encoded in the vanishing of the cohomology class $[[\mu_\bullet, \mu_\bullet]]$ of polybrackets on the $C_\infty$-algebra of tangent sheaves.
• Compatibility of flat structures with $\circ$-products implies the $F$-identity, establishing a deep link between flatness and Frobenius properties in the derived setting.