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Andrew Dancer

Publications and source records attributed to Andrew Dancer.

At least 19 recordsLinked to original sources
arXiv2025

Intersection cohomology of Popov-Vinberg varieties

The Popov-Vinberg variety of a simply connected, split, semisimple algebraic group $G$ is a singular affine variety that contains the basic affine space $G/U$ as a Zariski open subset. It is defined as the spectrum of the ring of functions on $G/U$, and can also be identified with the universal symplectic implosion for the maximal compact subgroup of $G$. We provide a recursive procedure for computing the intersection cohomology of this variety, with an emphasis on the case where $G = \operatorname{SL}_n$.

math.AG↗
arXiv2024

Complex Symplectic Contractions and 3d Mirrors

We propose magnetic quivers for the complex-symplectic contraction spaces, which are related to implosions and have a natural interpretation in terms of the Moore-Tachikawa category. We use 3-d mirrors to provide computational checks.

hep-th↗
arXiv2024

Implosion, Contraction and Moore-Tachikawa

We give a survey of the implosion construction, extending some of its aspects relating to hypertoric geometry from type $A$ to a general reductive group, and interpret it in the context of the Moore-Tachikawa category. We use these ideas to discuss how the contraction construction in symplectic geometry can be generalised to the hyperk\"ahler or complex symplectic situation.

math.SG↗
arXiv2021

Partial Implosions and Quivers

We propose magnetic quivers for partial implosion spaces. Such partial implosions involve a choice of parabolic subgroup, with the Borel subgroup corresponding to the standard implosion. In the subregular case we test the conjecture by verifying that reduction by the Levi group gives the appropriate nilpotent orbit closure. In the case of a parabolic corresponding to a hook diagram we are also able to carry out this verification provided we work at nonzero Fayet-Iliopoulos parameters.

hep-th↗
arXiv2021

Orthosymplectic Implosions

We propose quivers for Coulomb branch constructions of universal implosions for orthogonal and symplectic groups, extending the work on special unitary groups in arXiv:2004.09620. The quivers are unitary-orthosymplectic as opposed to the purely unitary quivers in the A-type case. Where possible we check our proposals using Hilbert series techniques.

hep-th↗
arXiv2020

Symplectic duality and implosions

We discuss symplectic and hyperk\"ahler implosion and present candidates for the symplectic duals of the universal hyperk\"ahler implosion for various groups.

math.SG↗
arXiv2018

Hypertoric manifolds of infinite topological type

We analyse properties of hypertoric manifolds of infinite topological type, including their topology and complex structures. We show that our manifolds have the homotopy type of an infinite union of compact toric varieties. We also discuss hypertoric analogues of the periodic Ooguri-Vafa spaces.

math.SG↗
arXiv2016

Hypertoric manifolds and hyperK\"ahler moment maps

We discuss various aspects of moment map geometry in symplectic and hyperK\"ahler geometry. In particular, we classify complete hyperK\"ahler manifolds of dimension $4n$ with a tri-Hamiltonian action of a torus of dimension $n$, without any assumption on the finiteness of the Betti numbers. As a result we find that the hyperK\"ahler moment in these cases has connected fibres, a property that is true for symplectic moment maps, and is surjective. New examples of hypertoric manifolds of infinite topological type are produced. We provide examples of non-Abelian tri-Hamiltonian group actions of connected groups on complete hyperK\"ahler manifolds such that the hyperK\"ahler moment map is not surjective and has some fibres that are not connected. We also discuss relationships to symplectic cuts, hyperK\"ahler modifications and implosion constructions.

math.DG↗
arXiv2015

A multiplicative analogue of complex symplectic implosion

We introduce a multiplicative version of complex-symplectic implosion in the case of $SL(n, \C)$. The universal multiplicative implosion for $SL(n, \C)$ is an affine variety and can be viewed as a nonreductive geometric invariant theory quotient. It carries a torus action. and reductions by this action give the Steinberg fibres of $SL(n, \C)$. We also explain how the real symplectic group-valued universal implosion introduced by Hurtubise, Jeffrey and Sjamaar may be identified inside this space.

math.SG↗
arXiv2014

Symplectic and hyperkahler implosion

We review the quiver descriptions of symplectic and hyperk\"ahler implosion in the case of SU(n) actions. We give quiver descriptions of symplectic implosion for other classical groups, and discuss some of the issues involved in obtaining a similar description for hyperk\"ahler implosion.

math.SG↗
arXiv2013

Twistor spaces for hyperkaehler implosions

We study the geometry of the twistor space of the universal hyperkaehler implosion Q for SU(n). Using the description of Q as a hyperkaehler quiver variety, we construct a holomorphic map from the twistor space Z_Q of Q to a complex vector bundle over P^1, and an associated map of Q to the affine space R of the bundle's holomorphic sections. The map from Q to R is shown to be injective and equivariant for the action of SU(n) x T^{n-1} x SU(2). Both maps, from Q and from Z_Q, are described in detail for n=2 and n=3. We explain how the maps are built from the fundamental irreducible representations of SU(n) and the hypertoric variety associated to the hyperplane arrangement given by the root planes in the Lie algebra of the maximal torus. This indicates that the constructions might extend to universal hyperkaehler implosions for other compact groups.

math.SG↗
arXiv2013

Implosions and hypertoric geometry

The geometry of the universal hyperKaehler implosion for SU(n) is explored. In particular, we show that the universal hyperKaehler implosion naturally contains a hypertoric variety described in terms of quivers. Furthermore, we discuss a gauge theoretic approach to hyperKaehler implosion.

math.SG↗
arXiv2012

Implosion for hyperkahler manifolds

We introduce an analogue in hyperkahler geometry of the symplectic implosion, in the case of SU(n) actions. Our space is a stratified hyperkahler space which can be defined in terms of quiver diagrams. It also has a description as a non-reductive geometric invariant theory quotient.

math.SG↗
arXiv2010

Non-Abelian Cut Constructions and Hyperk\"ahler Modifications

We discuss a general framework for cutting constructions and reinterpret in this setting the work on non-Abelian symplectic cuts by Weitsman. We then introduce two analogous non-Abelian modification constructions for hyperk\"ahler manifolds: one modifies the topology significantly, the other gives metric deformations. We highlight ways in which the geometry of moment maps for non-Abelian hyperk\"ahler actions differs from the Abelian case and from the non-Abelian symplectic case.

math.DG↗
arXiv2009

Classifying superpotentials: three summands case

We give an overview of our earlier classification results in [DW4] and [DW6] for superpotentials of scalar curvature type of the cohomogeneity one Ricci-flat equations. We then give an account of the classification in the case where the isotropy representation of the principal orbit consists of exactly three distinct irreducible real summands--the leftover case from [DW6].

math.DG↗
arXiv2007

Classification of superpotentials

We extend our previous classification of superpotentials of ``scalar curvature type" for the cohomogeneity one Ricci-flat equations. We now consider the case not covered in our previous paper, i.e., when some weight vector of the superpotential lies outside (a scaled translate of) the convex hull of the weight vectors associated with the scalar curvature function of the principal orbit. In this situation we show that either the isotropy representation has at most 3 irreducible summands or the first order subsystem associated to the superpotential is of the same form as the Calabi-Yau condition for submersion type metrics on complex line bundles over a Fano Kähler-Einstein product.

math.DG↗
arXiv2005

Modifying hyperkaehler manifolds with circle symmetry

A construction is introduced for modifying hyperkaehler manifolds with tri-Hamiltonian circle action, that in favourable situations increases the second Betti number by one. This is based on the symplectic cut construction of Lerman. In 4 or 8 dimensions the construction may be interpreted as adding a D6-brane. A number of examples are given and generalisations to three-Sasaki and hypersymplectic geometry discussed.

math.DG↗