# zbMATH — the first resource for mathematics

On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator. (English) Zbl 0974.14040
From the introduction: Quantum multiplications on the cohomology of symplectic manifolds were first proposed by the physicist Vafa, based on Witten’s topological sigma models. Y. Ruan and G. Tian [J. Differ. Geom. 42, No. 2, 259-367 (1995; Zbl 0860.58005)] gave a mathematical construction of quantum multiplications on cohomology groups of positive symplectic manifolds (chapter 1). The definition uses certain symplectic invariants, called Gromov-Witten (GW-) invariants, that were previously defined by Ruan for (semi-) positive symplectic manifolds. A large class of such manifolds is provided by Fano manifolds (complex manifolds with ample anti-canonical bundle). Examples are low degree complete intersections and compact complex homogeneous spaces like Grassmann manifolds. If $$M$$ is a Fano (or positive symplectic) manifold the quantum cohomology $$QH^*_{[\omega]}(M)$$ is just the cohomology space $$H^*(M,\mathbb{C})$$ with a (non-homogeneous) associative, graded commutative multiplication, the quantum multiplication. This multiplication depends on the choice of a (complexified) Kähler class $$[\omega]$$ on $$M$$. Its homogeneous part (the “weak coupling limit” $$\lambda\cdot[\omega]$$, $$\lambda \to\infty)$$ is the usual cup product.
In this note we observe that quantum cohomology rings have a nice description in terms of generators and relations: If $$H^*(M,\mathbb{C}) =\mathbb{C}[X_1, \dots,X_N]/(f_1, \dots,f_k)$$ is a presentation of the cohomology ring (for simplicity we assume $$\deg X_i$$ even for the moment) then $$QH^*_{[\omega]} (M)=\mathbb{C}[X_1, \dots,X_N]/ (f_1^{[\omega]}, \dots, f_k^{[\omega]})$$ , where $$f_1^{[\omega]}, \dots,f_k^{[\omega]}$$ are just the polynomials $$f_1,\dots,f_k$$ evaluated in the quantum ring associated to $$[\omega]$$ (Theorem 2.2). The $$f_i^{[\omega]}$$ are real-analytic in $$[\omega]$$ and thus have a natural analytic extension to $$H^{1,1}(M)$$, and the quantum cohomology rings fit together into a flat analytic family over $$H^{1,1}(M)$$.
As an application of this observation we compute the quantum cohomology of the Grassmannians. The calculations for $$G(k,n)$$ reduce to the single quantum product $$c_k\wedge_Qs_{n-k}$$ of the top non-vanishing Chern respectively Segre class of the tautological $$k$$-bundle (see chapter 3). We see in chapter 4 that formulas of this type occur whenever the cohomology ring has a presentation as complete intersection. In particular, we prove as corollary 4.6:
Vafa-Intriligator formula: For any Kähler class $$[\omega]$$ there is a finite set $$C\subset\mathbb{C}^n$$ and non-zero constants $$a_x$$, $$x\in C$$, such that, for any $$F\in\mathbb{C} [X_1,\dots, X_k],$$ $\langle F\rangle_g^{[\omega]} =\sum_{x\in C}(a_x)^{g-1} \cdot F(x).$ Here $$\langle F\rangle_g^{[\omega]}$$ is the genus $$g$$ GW-invariant associated to $$F$$. In fact, there is a polynomial $$W^{[\omega]}$$ in the $$X_i$$ having $$C$$ as its set of critical points with $$a_x$$ being (up to sign) the determinants of the Hessians at $$x\in C$$.

##### MSC:
 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 14J45 Fano varieties 14N10 Enumerative problems (combinatorial problems) in algebraic geometry
Full Text: