On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator.

*(English)*Zbl 0974.14040From 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 \(x\in C\).

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 |