22 research outputs found
Higher-genus wall-crossing in the gauged linear sigma model
We introduce a technique for proving all-genus wall-crossing formulas in the
gauged linear sigma model as the stability parameter varies, without assuming
factorization properties of the virtual class. We implement this technique
explicitly for the hybrid model, which generalizes our previous work to the
Landau--Ginzburg phase.Comment: 42 pages, 2 figures. v2: Added an appendix (by Yang Zhou) proving the
n=0 case of the main theorem; additional substantial changes to Sections 2.5
(broad vanishing) and 4.3 (localization contributions
What If?: Mathematics, Creative Writing, and Play
Mathematics can inform creative writing by suggesting structures for it to follow, as well as by providing the imaginative impetus for common rules to be broken. In a workshop co-taught by the author, a class of sixth-grade students explored this interplay as they produced fractal-inspired poetry and geometry-inspired fiction. This article describes the form and results of the workshop in the context of a broader discussion of the influence of mathematics upon literature
Open -spin theory II: The analogue of Witten's conjecture for -spin disks
We conclude the construction of -spin theory in genus zero for Riemann
surfaces with boundary. In particular, we define open -spin intersection
numbers, and we prove that their generating function is closely related to the
wave function of the th Gelfand--Dickey integrable hierarchy. This provides
an analogue of Witten's -spin conjecture in the open setting and a first
step toward the construction of an open version of Fan--Jarvis--Ruan--Witten
theory. As an unexpected consequence, we establish a mysterious relationship
between open -spin theory and an extension of Witten's closed theory.Comment: The more foundational parts of the previous version, v3, were moved
to the article arXiv:2003.01082. These include the description of objects,
constructions of moduli spaces and bundles and proofs of orientations
theorems. The name of the paper and the abstract were changed accordingl
Topological recursion relations from Pixton's formula
For any genus g \leq 26, and for n \leq 3 in all genus, we prove that every
degree-g polynomial in the psi-classes on Mbar_{g,n} can be expressed as a sum
of tautological classes supported on the boundary with no kappa-classes. Such
equations, which we refer to as topological recursion relations, can be used to
deduce universal equations for the Gromov-Witten invariants of any target.Comment: 17 page
Powers of the theta divisor and relations in the tautological ring
We show that the vanishing of the -st power of the theta divisor on
the universal abelian variety implies, by pulling back along a
collection of Abel--Jacobi maps, the vanishing results in the tautological ring
of of Looijenga, Ionel, Graber--Vakil, and
Faber--Pandharipande. We also show that Pixton's double ramification cycle
relations, which generalize the theta vanishing relations and were recently
proved by the first and third authors, imply Theorem~ of Graber and
Vakil, and we provide an explicit algorithm for expressing any tautological
class on of sufficiently high codimension as a
boundary class.Comment: 29 page