Showing papers on "Symplectic representation published in 2019"

Symplectic resolutions of Quiver varieties and character varieties

Abstract: In this article, we consider Nakajima quiver varieties from the point of view of symplectic algebraic geometry. Namely, we consider the question of when a quiver variety admits a projective symplectic resolution. A complete answer to this question is given. We also show that the smooth locus of a quiver variety coincides with the locus of $\theta$-canonically stable points, generalizing a result of Le Bruyn. An interesting consequence of our results is that not all symplectic resolutions of quiver varieties appear to come from variation of GIT. In the final part of the article, we consider the $G$-character variety of a compact Riemann surface of genus $g > 0$, when $G$ is $\mathrm{SL}(n,\mathbb{C})$ or $\mathrm{GL}(n,\mathbb{C})$. We show that these varieties admit symplectic singularities. When the genus $g$ is greater than one, we show that the singularities are terminal and locally factorial. As a consequence, these character varieties do not admit symplectic resolutions.

UNCHAINING SURGERYAND TOPOLOGY of SYMPLECTIC 4-MANIFOLDS

Abstract: We study a symplectic surgery operation we call unchaining, which effectively reduces the second Betti number and the symplectic Kodaira dimension at the same time. Using unchaining, we give novel constructions of symplectic Calabi-Yau surfaces from complex surfaces of general type, as well as from rational and ruled surfaces via the natural inverse of this operation. Combining the unchaining surgery with others, which all correspond to certain monodromy substitutions for Lefschetz pencils, we provide further applications, such as a complete resolution of a conjecture of Stipsicz on the existence of exceptional sections in Lefschetz fibrations, new constructions of exotic symplectic 4-manifolds, and inequivalent pencils of the same genera and the same number of base points on families of symplectic 4-manifolds. Meanwhile, we give a handy criterion for determining from the monodromy of a pencil whether its total space is spin or not.

Homology of the family of hyperelliptic curves

Abstract: Homology of braid groups and Artin groups can be related to the study of spaces of curves. We completely calculate the integral homology of the family of smooth curves of genus g with one boundary component, that are double coverings of the disk ramified over n = 2g+1 points. The main part of such homology is described by the homology of the braid group with coefficients in a symplectic representation, namely the braid group Brn acts on the first homology group of a genus g surface via Dehn twists. Our computations show that such groups have only 2-torsion. We also investigate stabilization properties and provide Poincare series, for both unstable and stable homology.

Partitioning Pauli Operators: in Theory and in Practice

Abstract: Measuring the expectation value of Pauli operators on prepared quantum states is a fundamental task in the variational quantum eigensolver. Simultaneously measuring sets of operators allows for fewer measurements and an overall speedup of the measurement process. In this thesis, we look both at the task of partitioning all Pauli operators of a fixed length and of partitioning a random subset of these Pauli operators. We first show how Singer cycles can be used to optimally partition the set of all Pauli operators, giving some insight to the structure underlying many constructions of mutually unbiased bases. Thereafter, we show how graph coloring algorithms promise to provide speedups linear with respect to the lengths of the operators over currently-implemented techniques in the measurement step of the variational quantum eigensolver.

Braid groups, mapping class groups and their homology with twisted coefficients

Abstract: We consider the Birman-Hilden inclusion $\varphi\colon\mathfrak{Br}_{2g+1}\to\Gamma_{g,1}$ of the braid group into the mapping class group of an orientable surface with boundary, and prove that $\varphi$ is stably trivial in homology with twisted coefficients in the symplectic representation $H_1(\Sigma_{g,1})$ of the mapping class group; this generalises a result of Song and Tillmann regarding homology with constant coefficients. Furthermore we show that the stable homology of the braid group with coefficients in $\varphi^*(H_1(\Sigma_{g,1}))$ has only $4$-torsion.