Quantum cohomology
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| Quantum cohomology | |
|---|---|
| Name | Quantum cohomology |
| Field | Algebraic geometry, Symplectic geometry, Mathematical physics |
| Introduced | 1990s |
| Key concepts | Gromov–Witten invariants, Frobenius manifold, Novikov ring, Quantum product |
| Notable people | Maxim Kontsevich, Mikhail Gromov, Alexander Givental, Yuri Manin, Edward Witten |
Quantum cohomology is a mathematical theory combining methods from Algebraic geometry, Symplectic geometry, and Mathematical physics to enrich classical cohomology rings of Manifolds by incorporating counts of holomorphic curves. Originating from ideas in String theory, Topological field theory, and work of researchers such as Edward Witten and Maxim Kontsevich, the subject provides a deformation of the cup product using enumerative invariants and has deep connections to Mirror symmetry, Frobenius manifold theory, and the study of moduli spaces.
Introduction
Quantum cohomology augments the classical cohomology ring of a compact Kähler manifold or projective Variety by a family of products parameterized by elements of a Novikov ring or formal parameters arising from curve classes. Early milestones include predictions from Mirror symmetry conjectures relating counts on Calabi–Yau Manifolds to period integrals, and rigorous foundations established through compactness results of Gromov compactness and virtual fundamental cycle techniques developed by groups including those around Mikhail Gromov, Yuri Manin, and Alexander Givental. Connections to mathematical structures such as Frobenius manifolds and representations of Virasoro algebra emerged through work motivated by Enumerative geometry and Topological quantum field theory.
Mathematical foundations
The formal setup requires an oriented compact Symplectic manifold or projective Algebraic variety X with a chosen class in H2(X; Z). One builds a deformation of H*(X; Λ) over a Novikov ring Λ capturing energy of holomorphic maps, using compactified moduli spaces of stable maps modeled on the Deligne–Mumford compactification of Moduli space of curves and analytic foundations from Elliptic operator theory and Fredholm theory. Foundational analytical tools involve the study of pseudo-holomorphic curves from Gromov’s work on compactness in symplectic topology, and algebraic foundations rely on techniques developed in the context of Chow varietys and Intersection theory by authors in the tradition of Grothendieck, Jean-Pierre Serre, and Alexander Grothendieck’s school. Virtual cycle constructions by teams associated with Kontsevich and later formalizations by groups including those influenced by Kuranishi methods yield the necessary enumerative invariants.
Gromov–Witten invariants
Gromov–Witten invariants count (virtually) holomorphic stable maps from marked nodal curves into X and are indexed by genus, number of marked points, and curve class in H2(X; Z). Their definition uses the virtual fundamental class of the moduli space of stable maps introduced in formalisms influenced by Behrend and Fantechi, and their properties satisfy axioms paralleling those from Topological field theory and Witten’s axioms for two-dimensional gravity. Gromov–Witten theory exhibits structures such as the WDVV equations named after work related to Dijkgraaf and Verlinde, and they form the structure constants for the quantum product via three-point, genus-zero invariants. Techniques for computation draw upon localization methods related to the Atiyah–Bott fixed-point theorem, degenerations as in work of Jun Li, and mirror formulas conjectured by Candelas and verified by methods of contending authors such as Givental and Lian.
Quantum product and ring structure
The quantum product is a graded, associative deformation of the cup product on H*(X) whose structure constants are three-point, genus-zero Gromov–Witten invariants. Associativity is equivalent to the WDVV equations and endows H*(X; Λ) with the structure of a Frobenius algebra and often a Frobenius manifold after inclusion of parameter spaces as in constructions by Dubrovin. Under suitable convergence hypotheses one obtains a quantum connection whose flatness relates to isomonodromic deformations studied in contexts involving Jimbo and Miwa in integrable systems. The quantum cohomology ring sometimes admits presentation analogous to classical presentations, with deformation parameters corresponding to effective curve classes; examples include presentations for flag varieties related to Schubert calculus and relations to Hecke algebra representations studied in works influenced by Bernstein.
Examples and computations
Key computable classes of examples include projective spaces, Grassmannians, toric varieties, and Fano varieties. For projective space P^n explicit formulas for Gromov–Witten invariants and the small quantum ring were computed in early work by authors such as Kontsevich and Ruan, while quantum Schubert calculus for Grassmannians links to combinatorial structures explored by Bertram and later by proponents of Buch’s rules. Toric mirror theorems by Givental and collaborators enable computations for toric varieties and semi-Fano hypersurfaces, and examples of Calabi–Yau threefolds feature prominently in predictions associated with Candelas’s quintic calculations. Computational tools include localization via Equivariant cohomology, degeneration formulas, and quantum Lefschetz principles advanced by researchers including Coates and Ruan.
Relations to mirror symmetry and enumerative geometry
Quantum cohomology sits at the heart of the mathematical formulation of mirror symmetry, connecting A-model enumerative invariants on a symplectic manifold to B-model deformation theory on a mirror complex manifold, as conjectured by Strominger–Yau–Zaslow and developed by Kontsevich’s homological mirror symmetry program. Enumerative predictions such as counts of rational curves on Calabi–Yau threefolds were inaugurated by Candelas’s work and later formalized using period computations and variation of Hodge structure as studied by Griffiths and Deligne. Mirror theorems by Givental and proofs by Lian, Liu, and Yau unify these perspectives and link to broader contexts including Derived category methods and Homological algebra techniques championed by proponents like Seidel.
Applications and extensions
Applications span enumerative predictions in Enumerative geometry, structural results in Symplectic topology such as existence of pseudo-holomorphic curves via Gromov–Witten invariants, and links to integrable systems and representation theory through quantum connection and mirror symmetry. Extensions include quantum K-theory developed by authors including Givental and Lee, relative Gromov–Witten theory for pairs as in work of Jun Li, orbifold quantum cohomology from studies by Chen and Ruan, and logarithmic Gromov–Witten theory pursued by collaborators influenced by Gross and Siebert. Ongoing research connects quantum cohomology to areas such as noncommutative geometry, categorical mirror symmetry, and enumerative problems arising in modern problems influenced by institutions like Institute for Advanced Study and consortia across universities and research centers.