Geometry
Enumerative Geometry: Kontsevich's Recursion and Gromov-Witten Invariants
Цели урока
- Understand the curve-counting problem and Gromov-Witten numbers N_d
- Master Kontsevich's recursion for computing N_d
- Know the moduli spaces of stable maps
- Understand quantum cohomology as a deformation of classical cohomology
Предварительные знания
- Algebraic Curves and Bezout's Theorem
- Homology overview
- Projective Geometry P^2
How did physicists compute N_3 = 317,206,375 before mathematicians, and what does this say about the relationship between physics and mathematics?
- String theory: rational curve counts N_d arise as correlation functions of topological string theory - math serving physics
- Mirror symmetry: N_3 = 317,206,375 curves of degree 3 on the quintic Calabi-Yau was predicted by physicists before a rigorous proof
- Combinatorics: Kontsevich's recursion computes N_d in polynomial time - a concrete algorithm from abstract geometry
- Algebraic topology: quantum cohomology connects algebraic geometry to integrable systems and representation theory
From Steiner to Kontsevich
Jakob Steiner (1848) posed the question: how many conics are tangent to 5 given conics? The answer (supposedly 7776) was wrong due to multiplicity issues. Schubert developed characteristic classes in the 19th century. Witten in 1990-91 predicted curve counts from string theory. Kontsevich in 1994 proved the recursion, unifying algebraic geometry, physics, and combinatorics - Fields Medal 1998.
Classical Enumerative Geometry
Through 2 points passes 1 line. Through 5 points - 1 conic. Through 8 points - 12 cubics. Through 11 points - 620 quartics. This is not coincidence - it is deep structure. How many rational curves of degree d pass through 3d-1 points in general position in P^2?
Kontsevich's recursion is one of the most beautiful results of 1990s mathematics. Behind it lies a deep structure: quantum cohomology, axiomatized as an operad. Formally, the N_d are integrals over moduli spaces of stable maps.
How many rational curves of degree 3 in P^2 pass through 8 general points?
N_3 = 12 rational cubics through 8 general points in P^2. Kontsevich's recursion computes all N_d automatically from N_1=1.
Gromov-Witten Invariants and Moduli Spaces
Mathematicians could count rational curves through finitely many points. Physicists from quantum string theory derived the same numbers - by completely different means. Kontsevich in 1994 explained why these approaches agree: Gromov-Witten invariants are correlation functions of topological string theory.
Connection to Mirror Symmetry
Mirror symmetry computes GW numbers
Candelas-de la Ossa formula (1991): the quintic Calabi-Yau threefold contains 2875 lines (d=1), 609250 conics (d=2), 317206375 degree-3 curves. These numbers are computed not geometrically but via the mirror manifold: instead of a difficult integral over mapping spaces - a simple period calculation. This is the main practical result of mirror symmetry.
What are Gromov-Witten invariants geometrically?
GW invariants are integrals over moduli spaces M_{g,n}(X, beta). They count (with weights) algebraic curves of genus g in class beta with n marked points.
Quantum Cohomology
The classical cohomology ring H*(X) uses the cup product. Quantum cohomology QH*(X) deforms it: the product includes corrections from rational curves. Each rational curve in X contributes through GW invariants.
Quantum cohomology turns out to be richer than classical: it connects to integrable systems, representation theory, and mirror symmetry. The operad of commutative algebras deforms into an operad encoding all GW invariants.
The GW axioms formulated by Kontsevich and Manin give a rigorous mathematical foundation for string theory computations.
How does quantum cohomology differ from classical cohomology?
In QH*(X) the cup product alpha cup beta is replaced by the quantum product alpha * beta including sums over rational curves. At q=0 the quantum product reduces to classical.
Connections to Other Topics
Enumerative geometry uses topology, physics, and combinatorics.
- Algebraic Curves: Bezout's Theorem and Genus — Enumerative problems are built from intersections of algebraic curves
- Mirror Symmetry — Mirror symmetry computed the number of rational curves on the quintic Calabi-Yau threefold
- Derived Algebraic Geometry: Introduction — Kontsevich's virtual fundamental class is constructed in derived geometry
Итоги
- N_d = number of rational degree-d curves through 3d-1 points in P^2
- N_1=1, N_2=1, N_3=12, N_4=620, N_5=87304
- Kontsevich's recursion computes all N_d from N_1=1 in polynomial time
- GW invariants = integrals over [M_{g,n}(X, beta)]^{vir}
- Quantum cohomology QH*(X) deforms the classical ring by rational curve contributions
Вопросы для размышления
- Why do exactly 3d-1 point conditions determine a rational curve of degree d, and why not fewer?
- How does Kontsevich's recursion express N_d via N_{d1} and N_{d2} with d1+d2=d - what is the geometric meaning of this splitting?
- What does the quantum correction H^{n+1} = q in the quantum cohomology of P^n mean geometrically?