The Alexander polynomial as an intersection of two cycles in a symmetric power

We consider a braid β which acts on a punctured plane. Then we construct a local system on this plane and find a homology cycle D in its symmetric power, such that D ⋅ β(D) coincides with the Alexander polynomial of the plait closure of β.

THREE-POINT FUNCTIONS ON THE SPHERE OF CALABI-YAU d-FOLDS

Using mirror symmetry in Calabi-Yau manifolds M, we study three-point functions of A(M) model operators on the genus 0 Riemann surface in cases of one-parameter families of d-folds realized as Fermat type hypersurfaces embedded in weighted projective spaces and a two-parameter family of d-folds embedded in a weighted projective space Pd+1 [2,2,2,...,2,2,1,1] (2 (d + 1)). These three-point functions [Formula: see text] are expanded by indeterminates [Formula: see text] associated with a set of Kähler coordinates {tl}, and their expansion coefficients count the number of maps with a definite degree which map each of the three-points 0, 1 and ∞ on the world sheet on some homology cycle of M associated with a cohomology element. From these analyses, we can read the fusion structure of Calabi-Yau A(M) model operators. In our cases they constitute a subring of a total quantum cohomology ring of the A(M) model operators. In fact we switch off all perturbation operators on the topological theories except for marginal ones associated with Kähler forms of M. For that reason, the charge conservation of operators turns out to be a classical one. Furthermore, because their first Chern classes c1 vanish, their topological selection rules do not depend on the degree of maps (in particular, a nilpotent property of operators [Formula: see text] is satisfied). Then these fusion couplings {κl} are represented as some series adding up all degrees of maps.

Generalized characters of the level-one Wess–Zumino–Witten model for simply laced groups on Riemann surfaces

The generalized characters of the level-one Al, Dl, and El Wess–Zumino–Witten models on the higher genus Riemann surfaces are obtained from their behaviors under the pinching limit of the zero-homology cycles. It is important in the construction of the higher genus characters that these models have fusion rules of the same type as the rational Gaussian model. The two-point correlators are also obtained by pinching the nonzero-homology cycle.