Gauge theory techniques in quantum cohomology (Q2752200)

This paper deals with the computation of the flat sections of the tangent bundle to a complex vector space, carrying a flat connection. This is a general question arising from a work of Givental [\textit{A. Givental}, Equivariant Gromov-Witten invariants, Int. Math. Res. Not. 1996, No. 13, 613-663 (1996; Zbl 0881.55006)], where equivariant Gromov-Witten invariants are used to obtain flat sections of the Dubrovin connection on the tangent bundle to the even cohomology of a quintic hypersurface in the projective space \(\mathbb{P}^3\).NEWLINENEWLINENEWLINELet \(H\) be a complex vector space with an associative, and commutative product denoted by \(*\). On \(TH\) the Dubrovin connection is defined by: \(\nabla_Y X=dX (Y)+ iY*X\), with connection 1-form \(\omega(Y) (X)=iY*X\). The de Rham cohomology of \(H\), computed with respect to the exterior derivative coupled to \(\nabla\), vanishes and since \(H\) is contractible, the moduli space (under gauge transformations) of flat connections reduces to a point. Thus \(\nabla\) is globally gauge equivalent to the trivial connection \(d\), given by \(d_YX=dX(Y)\), and finding flat sections is equivalent to solving: \(g^{-1}dg= \omega\) for \(g: \mathbb{C}^m \to\text{GL}(m,\mathbb{C})\), \(m\) being the dimension of \(H\). In coordinates such equation reads as a PDE system.NEWLINENEWLINENEWLINEThe authors compute the gauge transformation mentioned and the corresponding flat sections, using systematically the Frobenius integrability theorem. They find a basis of the flat sections for the Dubrovin connection for the small quantum product, a deformation in the \(H^2(M;\mathbb{C})\) directions of the cup product on the even cohomology \(H=H^{\text{ev}}(M;\mathbb{C})\), considering \(M\) as a Fano variety. Then, they compute flat sections for \(\mathbb{P}^m\) and in this case the calculation reduces to exponentials of ordinary matrices, whereas, in general, it involves exponentiating an infinite matrix with matrices as entries. They recover all the Gromov-Witten invariants. The paper contains also some comparisons with the result obtained by \textit{R. Pandharipande} [Rational curves on hypersurfaces (after A. Givental), Séminaire Bourbaki, Volume 1997/98. Exposés 835-849. Paris: Soc. Math. de France, Astérisque 252, 307-340, Exp. No. 848 (1998; Zbl 0932.14029)].NEWLINENEWLINENEWLINEFinally, there are some comments for the big quantum product and for the quantum cohomology coupled to gravity in genus zero, which will be treated in a future work.NEWLINENEWLINEFor the entire collection see [Zbl 0966.00024].