arXiv subjects

Robert Silversmith

Publications and source records attributed to Robert Silversmith.

4 recordsLinked to original sources

arXiv2016

Gromov-Witten invariants of $\mathrm{Sym}^d\mathbb{P}^r$

We give a graph-sum algorithm that expresses any genus-$g$ Gromov-Witten invariant of the symmetric product orbifold $\mathrm{Sym}^d\mathbb{P}^r:=[(\mathbb{P}^r)^d/S_d]$ in terms of "Hurwitz-Hodge integrals" -- integrals over (compactified) Hurwitz spaces. We apply the algorithm to prove a partial mirror theorem for $\mathrm{Sym}^d\mathbb{P}^r$ in genus zero. The theorem states that a generating function of Gromov-Witten invariants of $\mathrm{Sym}^d\mathbb{P}^r$ is equal to an explicit power series $I_{\mathrm{Sym}^d\mathbb{P}^r},$ conditional upon a conjectural combinatorial identity. This is a first step towards proving Ruan's Crepant Resolution Conjecture for the resolution $\mathrm{Hilb}^{(d)}(\mathbb{P}^2)$ of the coarse moduli space of $\mathrm{Sym}^d\mathbb{P}^2.$

math.AG↗

arXiv2016

Gromov-Witten theory of toroidal orbifolds and GIT wall-crossing

Toroidal 3-orbifolds $(S^1)^6/G$, for $G$ a finite group, were some of the earliest examples of Calabi-Yau 3-orbifolds to be studied in string theory. While much mathematical progress towards the predictions of string theory has been made in the meantime, most of it has dealt with hypersurfaces in toric varieties. As a result, very little is known about curve-counting theories on toroidal orbifolds. In this paper, we initiate a program to study mirror symmetry and the Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence for toroidal orbifolds. We focus on the simplest example $[E^3/μ_3],$ where $E\subseteq\mathbb{P}^2$ is the elliptic curve $\mathbb{V}(x_0^3+x_1^3+x_2^3).$ We study this orbifold from the point of GIT wall-crossing using the gauged linear sigma model, a collection of moduli spaces generalizing spaces of stable maps. Our main result is a mirror symmetry theorem that applies simultaneously to the different GIT chambers. Using this, we analyze wall-crossing behavior to obtain an LG/CY correspondence relating the genus-zero Gromov-Witten invariants of $[E^3/μ_3]$ to generalized Fan-Jarvis-Ruan-Witten invariants.

math.AG↗

arXiv2012

Duality properties of indicatrices of knots

The bridge index and superbridge index of a knot are important invariants in knot theory. We define the bridge map of a knot conformation, which is closely related to these two invariants, and interpret it in terms of the tangent indicatrix of the knot conformation. Using the concepts of dual and derivative curves of spherical curves as introduced by Arnold, we show that the graph of the bridge map is the union of the binormal indicatrix, its antipodal curve, and some number of great circles. Similarly, we define the inflection map of a knot conformation, interpret it in terms of the binormal indicatrix, and express its graph in terms of the tangent indicatrix. This duality relationship is also studied for another dual pair of curves, the normal and Darboux indicatrices of a knot conformation. The analogous concepts are defined and results are derived for stick knots.

math.GT↗

arXiv2010

A Midsummer Knot's Dream

In this paper, we introduce playing games on shadows of knots. We demonstrate two novel games, namely, To Knot or Not to Knot and Much Ado about Knotting. We also discuss winning strategies for these games on certain families of knot shadows. Finally, we suggest variations of these games for further study.

math.GT↗