113 research outputs found
Twisted Gromov-Witten r-spin potential and Givental's quantization
The universal curve p:C->\Mbar over the moduli space \Mbar of stable r-spin
maps to a target K\"ahler manifold X carries a universal spinor bundle L->C.
Therefore the moduli space \Mbar itself carries a natural K-theory class Rp_*L.
We introduce a twisted r-spin Gromov-Witten potential of X enriched with
Chern characters of Rp_*L. We show that the twisted potential can be
reconstructed from the ordinary r-spin Gromov-Witten potential of X via an
operator that assumes a particularly simple form in Givental's quantization
formalism.Comment: 25 pages, 3 figure
Changes of variables in ELSV-type formulas
In [5] I.P. Goulden, D.M. Jackson, and R. Vakil formulated a conjecture
relating certain Hurwitz numbers (enumerating ramified coverings of the sphere)
to the intersection theory on a conjectural Picard variety. We are going to use
their formula to study the intersection theory on this variety (if it is ever
to be constructed) by methods close to those of M. Kazarian and S. Lando in
[7]. In particular, we prove a Witten-Kontsevich-type theorem relating the
intersection theory and integrable hierarchies.
We also extend the results of [7] to include the Hodge integrals over the
moduli spaces, involving one lambda-class.Comment: 25 pages. Final versio
Tautological relations via r-spin structures
Relations among tautological classes on the moduli space of stable curves are
obtained via the study of Witten's r-spin theory for higher r. In order to
calculate the quantum product, a new formula relating the r-spin correlators in
genus 0 to the representation theory of sl2 is proven. The Givental-Teleman
classification of CohFTs is used at two special semisimple points of the
associated Frobenius manifold. At the first semisimple point, the R-matrix is
exactly solved in terms of hypergeometric series. As a result, an explicit
formula for Witten's r-spin class is obtained (along with tautological
relations in higher degrees). As an application, the r=4 relations are used to
bound the Betti numbers of the tautological ring of the moduli of nonsingular
curves. At the second semisimple point, the form of the R-matrix implies a
polynomiality property in r of Witten's r-spin class.
In the Appendix (with F. Janda), a conjecture relating the r=0 limit of
Witten's r-spin class to the class of the moduli space of holomorphic
differentials is presented.Comment: Corrected powers of phi in the analysis of the second shift. Appendix
on the moduli of holomorphic differentials by F. Janda, R. Pandharipande, A.
Pixton, and D.Zvonkine. Final versio
On double Hurwitz numbers with completed cycles
In this paper, we collect a number of facts about double Hurwitz numbers,
where the simple branch points are replaced by their more general analogues ---
completed (r+1)-cycles. In particular, we give a geometric interpretation of
these generalised Hurwitz numbers and derive a cut-and-join operator for
completed (r+1)-cycles. We also prove a strong piecewise polynomiality property
in the sense of Goulden-Jackson-Vakil. In addition, we propose a conjectural
ELSV/GJV-type formula, that is, an expression in terms of some intrinsic
combinatorial constants that might be related to the intersection theory of
some analogues of the moduli space of curves. The structure of these
conjectural "intersection numbers" is discussed in detail.Comment: 31 page
- …