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Panagiotis Dimakis

Publications and source records attributed to Panagiotis Dimakis.

12 recordsLinked to original sources
arXiv2026

The conformal limit for Nakajima quiver varieties

Inspired by Gaiotto's conformal limit construction for Higgs bundles we define and study a conformal limit construction for Nakajima quiver varieties. We prove that the conformal limit is indeed a limit of a one parameter family of points inside a specified quiver variety and that it gives a biholomorphic map between holomorphic Lagrangian submanifolds foliating two different quiver varieties. In the last part of the paper we discuss the analog of Simpson's conjecture on the completeness of these holomorphic Lagrangian submanifolds.

math.AG↗
arXiv2026

Lagrangian correspondences for moduli spaces of Higgs bundles and holomorphic connections

On a compact connected Riemann surface $C$ of genus at least $2$, we construct Lagrangian correspondences between moduli spaces of rank-$n$ Higgs bundles (respectively, holomorphic connections) and the Hilbert schemes of points on $T^\ast C$ (respectively, the twisted cotangent bundles of $C$). Central to these constructions are Higgs bundles (respectively, holomorphic connections) which are transversal to line subbundles of the underlying bundles: these naturally induce divisors on $C$ together with auxiliary parameters, namely lifts to divisors on spectral curves for Higgs bundles and residue parameters of apparent singularities for holomorphic connections. We discuss the evidence showing that the Dolbeault geometric Langlands correspondence is generically realized by these Lagrangian correspondences; we expect that the de Rham geometric Langlands correspondence can be realized by their quantization, following Drinfeld's construction of Hecke eigensheaves. We also discuss the relations of our constructions to various topics, including reductions of Kapustin-Witten equations, the conformal limit, separation of variables, and degenerate fields in conformal field theories.

math.AG↗
arXiv2024

Asymptotic geometry at infinity of quiver varieties

Using an approach developed by Melrose to study the geometry at infinity of the Nakajima metric on the reduced Hilbert scheme of points on $\mathbb{C}^2$, we show that the Nakajima metric on a quiver variety is quasi-asymptotically conical (QAC) whenever its defining parameters satisfy an appropriate genericity assumption. As such, it is of bounded geometry and of maximal volume growth. Being QAC is one of two main ingredients allowing us to use the work of Kottke and the second author to compute its reduced $L^2$-cohomology and prove the Vafa-Witten conjecture. The other is a vanishing theorem in $L^2$-cohomology for exact wedge $3$-Sasakian metrics generalizing a result of Galicki and Salamon for closed $3$-Sasakian manifolds.

math.DG↗
arXiv2024

On a conjecture of Simpson

On a compact Riemann surface $\Sigma$ of genus $g\ge 2$, equipped with a complex vector bundle $E$ of rank $2$ and degree zero let $M_H$ be the moduli space of Higgs bundles. $M_H$ admits a $\mathbb C^{\star}$-action and to each stable $\mathbb C^{\star}$-fixed point $[(\bar\partial_0,\Phi_0)]$ is associated a holomorphic Lagrangian submanifold $W^1(\bar\partial_0,\Phi_0)$ inside the de Rham moduli space $M_{dR}$ of complex flat connections. In this note we prove a conjecture of Simpson stating that $W^1(\bar\partial_0,\Phi_0)$ is closed inside $M_{dR}$.

math.DG↗
arXiv2024

The moduli space of solutions to the Extended Bogomolny equations on $\Sigma \times \mathbb{R_+}$

We study moduli spaces of solutions to the extended Bogomolny equations on $\Sigma \times \mathbb{R_{+,y}}$ with gauge group $\operatorname{SL}(2,\mathbb{C})$ satisfying the generalized Nahm pole boundary condition as $y\to 0$ and limiting to complex flat connections as $y\to \infty$. Refining the Kobayashi-Hitchin correspondence of \cite{MH2}, we identify these moduli spaces with certain holomorphic lagrangian sub-manifolds inside the moduli space of Higgs bundles.

math.DG↗
arXiv2022

Model knot solutions for the twisted Bogomolny equations

In this paper we prove existence for model knot solutions to the dimensionally reduced twisted Kapustin-Witten equations on $\mathbb{R}^3_+$ for any twisting parameter $t\in(0,\infty)$. We start with the explicit solutions for $t = 1$ derived in \cite{WFivebranes} and perform a continuity argument in $t$. This corroborates a prediction of Gaiotto and Witten \cite[][p. 961]{GW}.

math-ph↗
arXiv2020

Equivariant K-theory and Resolution II: Non-Abelian actions

The smooth action of a compact Lie group on a compact manifold can be resolved to an iterated space, as made explicit by Pierre Albin and the second author. On the resolution the lifted action has fixed isotropy type corresponding to the open stratum and also in an iterated sense, with connecting equivariant fibrations over the boundary hypersurfaces covering the resolutions of the other strata. This structure descends to a resolution of the quotient as a stratified space. For an Abelian group action the equivariant K-theory can then be described in terms of bundles over the bases `dressed' by the representations of the isotropy types with morphisms covering the connecting maps. A similar model is given here covering the non-Abelian case. Now the reduced objects are torsion-twisted bundles over finite covers of the bases, corresponding to the projective action of the normalizers on the representations of the isotropy groups, again with morphisms over all the boundaries. This leads to a closely related iterated deRham model for equivariant cohomology and, now with values in forms twisted by flat bundles of representation rings over the bases, for delocalized equivariant cohomology. We show, as envisioned by Baum, Brylinksi and MacPherson, that the usual equivariant Chern character, mapping to equivariant cohomology, factors through a natural Chern character from equivariant K-theory to delocalized equivariant cohomology with the latter giving an Atiyah-Hirzebruch isomorphism.

math.KT↗
arXiv2019

Compactification of semi-simple Lie groups

We discuss the `hd-compactification' of a semi-simple Lie group to a manifold with corners; it is the real analog of the wonderful compactification of deConcini and Procesi. There is a 1-1 correspondence between the boundary faces of the compactification and conjugacy classes of parabolic subgroups with the boundary face fibering over two copies of the corresponding flag variety with fiber modeled on the (compactification of the) reductive part. On the hd-compactification Harish-Chandra's Schwartz space is identified with a space of conormal functions of rapid-logarithmic decay relative to square-integrable functions.

math.DG↗
arXiv2018

Compactification of SL(2)

We discuss `hd-compactifications' of $\SL(2,\bbK)$ for $\bbK=\bbC$ or $\bbR.$ These are compact manifolds with boundary on which both the Schwartz and the Harish-Chandra Schwartz spaces are shown to be relatively standard spaces of conormal functions relative to the boundary. Closure under convolution and other module properties are shown to follow from the structure of appropriate generalized product spaces and the functorial properties of conormal functions and smooth maps between manifolds with corners. It is anticipated that a similar approach applies to general real reductive Lie groups, with the additional complications for $\SL(n,\bbK)$ being essentially combinatorial.

math.GR↗
arXiv2018

Equivariant K-theory and Resolution I: Abelian actions

The smooth action of a compact Lie group on a compact manifold can be resolved to an iterated space, as made explicit by Pierre Albin and the second author. On the resolution the lifted action has fixed isotropy type, in an iterated sense, with connecting fibrations and this structure descends to a resolution of the quotient. For an abelian group action the equivariant K-theory can then be described in terms of bundles over the base with morphisms covering the connecting maps. A similar model is given, in terms of appropriately twisted deRham forms over the base as an iterated space, for delocalized equivariant cohomology in the sense of Baum, Brylinski and MacPherson. This approach allows a direct proof of their equivariant version of the Atiyah-Hirzebruch isomorphism.

math.AT↗
arXiv2017

Combinatorial Wall-Crossing and the Mullineux Involution

In this paper, we define the combinatorial wall-crossing transformation and the generalized column regularization on partitions and prove that a certain composition of these two transformations has the same effect on the one-row partition $(n)$. As corollaries we explicitly describe the quotients of the partitions which arise in this process. We also prove that the one-row partition is the unique partition that stays regular at any step of the wall-crossing transformation.

math.CO↗