In this post, part one, I will discuss the importance of Gromov-Witten invariants, and for an excellent read, Bernd Siebert’s SGWI is a must. For X a compact almost Kähler manifold of complex dimension D, let be the moduli orbifold of degree d stable holomorphic maps to X of genus g where d takes values in the lattice . By compactness, is equipped with a rational coefficient virtual fundamental cycle of complex dimension
and the total descendent potential of is
with the genus g descendent potential
with being the powers of the 1-st Chern class of the universal cotangent line bundle over corresponding to the i-th marked point, and are pull-backs by the evaluation map at the i-th symplectic-marked point of the cohomology classes and is the representative of d in the semigroup ring of the semigroup of degrees of holomorphic curves in X. The genus g Gromov-Witten potential of X is defined as the restriction
According to Witten’s conjecture, which Kontsevich proved, the total descendent potential coincides with the tau-function of the KdV-hierarchy satisfying the string equation
Let the Witten-Kontsevich tau-function be denoted by and let be an N-dimensional vector space equipped with a non-degenerate symmetric bilinear form. With the space of Laurent polynomials in one indeterminate z with vector coefficients from , introduce a symplectic bilinear form in by
and let correspond to the decomposition
of the Laurent polynomials into polynomial and polar parts, and deeply, note that the subspaces are Lagrangian. One then quantizes infinitesimal symplectic transformations on to order ≤ 2 linear differential operators . In a Darboux coordinate system compatible with our decomposition one has
holding and satisfying
and for any other pairs of quadratic Darboux monomials.
Let us move now to the Witten-Kontsevich Tau-function: let
It follows that
and thus shows that the operators are infinitesimal symplectic transformations on . However, is conjugate to , with , therefore commute as and by the Fourier transform, as the vector fields on the line. So, the Poisson brackets satisfy and hence we have a representation of the Lie algebra of vector fields on the line to the Lie algebra of quadratic hamiltonians on .
In the 1-dimensional with the standard inner product, using the Darboux coordinate system
on , one gets
We then get
unless m, n = ±1, in which case:
Thus the operators form a representation of the Lie algebra of vector fields on the line.
Proposition The Witten-Kontsevich tau-function is annihilated by the operators , m = −1, 0, 1, 2, …; and thus is completely characterized by this property – up to a scalar factor
- Hodge integrals: Let denote the Hodge bundle over the moduli space . Then analytically, the fiber of over the point represented by a stable map is the complex space of dimension dual to
and so is the pull-back of by the contraction map
of the Hodge bundle over the Deligne-Mumford space. It is a beautiful theorem that even components of the Chern character vanish. Hence, let us define the total Hodge potential of as an extension of the total descendent potential depending on the …