Slodowy Slices Research Articles

Singularities of nilpotent Slodowy slices and collapsing levels of W-algebras

Abstract We develop techniques to construct isomorphisms between simple affine W-algebras and affine vertex algebras at admissible levels. We then apply these techniques to obtain many new, and conjecturally all, admissible collapsing levels for affine W-algebras. In short, if a simple affine W-algebra at a given level is equal to its affine vertex subalgebra generated by the centraliser of an ${\mathfrak {sl}}_2$ -triple associated with the underlying nilpotent orbit, then that level is said to be collapsing. Collapsing levels are important both in representation theory and in theoretical physics. Our approach relies on two fundamental invariants of vertex algebras. The first one is the associated variety, which, in the context of admissible level simple affine W-algebras, leads to the Poisson varieties known as nilpotent Slodowy slices. We exploit the singularities of these varieties to detect possible collapsing levels. The second invariant is the asymptotic datum. We prove a general result asserting that, under appropriate hypotheses, equality of asymptotic data implies isomorphism at the level of vertex algebras. Then we use this to give a sufficient criterion, of combinatorial nature, for an admissible level to be collapsing. Our methods also allow us to study isomorphisms between quotients of W-algebras and extensions of simple affine vertex algebras at admissible levels. Based on such examples, we are led to formulate a general conjecture: for any finite extension of vertex algebras, the induced morphism between associated Poisson varieties is dominant.

Universal filtered quantizations of nilpotent Slodowy slices

Every conic symplectic singularity admits a universal Poisson deformation and a universal filtered quantization, thanks to the work of Losev and Namikawa. We begin this paper by showing that every such variety admits a universal equivariant Poisson deformation and a universal equivariant quantization with respect to a reductive group acting on it by \mathbb{C}^\times -equivariant Poisson automorphisms. We go on to study these definitions in the context of nilpotent Slodowy slices. First, we give a complete description of the cases in which the finite W -algebra is a universal filtered quantization of the slice, building on the work of Lehn–Namikawa–Sorger. This leads to a near-complete classification of the filtered quantizations of nilpotent Slodowy slices. The subregular slices in non-simply laced Lie algebras are especially interesting: with some minor restrictions on Dynkin type, we prove that the finite W -algebra is a universal equivariant quantization with respect to the Dynkin automorphisms coming from the unfolding of the Dynkin diagram. This can be seen as a non-commutative analogue of Slodowy's theorem. Finally, we apply this result to give a presentation of the subregular finite W -algebra of type \mathsf{B} as a quotient of a shifted Yangian.

A new family of isolated symplectic singularities with trivial local fundamental group

AbstractWe construct a new infinite family of four‐dimensional isolated symplectic singularities with trivial local fundamental group, answering a question of Beauville raised in 2000. Three constructions are presented for this family: (1) as singularities in blowups of the quotient of by the dihedral group of order , (2) as singular points of Calogero–Moser spaces associated with dihedral groups of order at equal parameters, and (3) as singularities of a certain Slodowy slice in the ‐fold cover of the nilpotent cone in .

Arc Spaces and Chiral Symplectic Cores

We introduce the notion of chiral symplectic cores in a vertex Poisson variety, which can be viewed as analogs of symplectic leaves in Poisson varieties. As an application we show that any quasi-lisse vertex algebra is a quantization of the arc space of its associated variety, in the sense that its reduced singular support coincides with the reduced arc space of its associated variety. We also show that the coordinate ring of the arc space of Slodowy slices is free over its vertex Poisson center, and the latter coincides with the vertex Poisson center of the coordinate ring of the arc space of the dual of the corresponding simple Lie algebra.

Magnetic lattices for orthosymplectic quivers

For any gauge theory, there may be a subgroup of the gauge group which acts trivially on the matter content. While many physical observables are not sensitive to this fact, the choice of the precise gauge group becomes crucial when the magnetic lattice of the theory is considered. This question is addressed in the context of Coulomb branches for 3d mathcal{N} = 4 quiver gauge theories, which are moduli spaces of dressed monopole operators. We compute the Coulomb branch Hilbert series of many unitary-orthosymplectic quivers for different choices of gauge groups, including diagonal quotients of the product gauge group of individual factors, where the quotient is by a trivially acting subgroup. Choosing different such diagonal groups results in distinct Coulomb branches, related as orbifolds. Examples include nilpotent orbit closures of the exceptional E-type algebras and magnetic quivers that arise from brane physics. This includes Higgs branches of theories with 8 supercharges in dimensions 4, 5, and 6. A crucial ingredient in the calculation of exact refined Hilbert series is the alternative construction of unframed magnetic quivers from resolved Slodowy slices, whose Hilbert series can be derived from Hall-Littlewood polynomials.

Four-dimensional conical symplectic hypersurfaces

We show that every indecomposable conical symplectic hypersurface of dimension four is isomorphic to the known one, namely, the Slodowy slice Xn which is transversal to the nilpotent orbit of Jordan type [2n−2,1,1] in the nilpotent cone of sp2n for some n≥2. In the appendix written by Yoshinori Namikawa, conical symplectic varieties of dimension two are classified by using contact Fano orbifolds.

An Application of Spherical Geometry to Hyperkähler Slices

AbstractThis work is concerned with Bielawski’s hyperkähler slices in the cotangent bundles of homogeneous affine varieties. One can associate such a slice with the data of a complex semisimple Lie group $G$, a reductive subgroup $H\subseteq G$, and a Slodowy slice $S\subseteq \mathfrak{g}:=\text{Lie}(G)$, defining it to be the hyperkähler quotient of $T^{\ast }(G/H)\times (G\times S)$ by a maximal compact subgroup of $G$. This hyperkähler slice is empty in some of the most elementary cases (e.g., when $S$ is regular and $(G,H)=(\text{SL}_{n+1},\text{GL}_{n})$, $n\geqslant 3$), prompting us to seek necessary and sufficient conditions for non-emptiness.We give a spherical-geometric characterization of the non-empty hyperkähler slices that arise when $S=S_{\text{reg}}$ is a regular Slodowy slice, proving that non-emptiness is equivalent to the so-called $\mathfrak{a}$-regularity of $(G,H)$. This $\mathfrak{a}$-regularity condition is formulated in several equivalent ways, one being a concrete condition on the rank and complexity of $G/H$. We also provide a classification of the $\mathfrak{a}$-regular pairs $(G,H)$ in which $H$ is a reductive spherical subgroup. Our arguments make essential use of Knop’s results on moment map images and Losev’s algorithm for computing Cartan spaces.

Quiver theories and Hilbert series of classical Slodowy intersections

We build on previous studies of the Higgs and Coulomb branches of SUSY quiver theories having 8 supercharges, including 3dN=4, and Classical gauge groups. The vacuum moduli spaces of many such theories can be parameterised by pairs of nilpotent orbits of Classical Lie algebras; they are transverse to one orbit and intersect the closure of the second. We refer to these transverse spaces as “Slodowy intersections”. They embrace reduced single instanton moduli spaces, nilpotent orbits, Kraft-Procesi transitions and Slodowy slices, as well as other types. We show how quiver subtractions, between multi-flavoured unitary or ortho-symplectic quivers, can be used to find a complete set of Higgs branch constructions for the Slodowy intersections of any Classical group. We discuss the relationships between the Higgs and Coulomb branches of these quivers and Tσρ theories in the context of 3d mirror symmetry, including problematic aspects of Coulomb branch constructions from ortho-symplectic quivers. We review Coulomb and Higgs branch constructions for a subset of Slodowy intersections from multi-flavoured Dynkin diagram quivers. We tabulate Hilbert series and Highest Weight Generating functions for Slodowy intersections of Classical algebras up to rank 4. The results are confirmed by direct calculation of Hilbert series from a localisation formula for normal Slodowy intersections that is related to the Hall Littlewood polynomials.

A quantum Mirković-Vybornov isomorphism

We present a quantization of an isomorphism of Mirković and Vybornov which relates the intersection of a Slodowy slice and a nilpotent orbit closure in g l N \mathfrak {gl}_N to a slice between spherical Schubert varieties in the affine Grassmannian of P G L n PGL_n (with weights encoded by the Jordan types of the nilpotent orbits). A quantization of the former variety is provided by a parabolic W-algebra and of the latter by a truncated shifted Yangian. Building on earlier work of Brundan and Kleshchev, we define an explicit isomorphism between these non-commutative algebras and show that its classical limit is a variation of the original isomorphism of Mirković and Vybornov. As a corollary, we deduce that the W-algebra is free as a left (or right) module over its Gelfand-Tsetlin subalgebra, as conjectured by Futorny, Molev, and Ovsienko.

Resolutions With Conical Slices and Descent for the Brauer Group Classes of Certain Central Reductions of Differential Operators in Characteristicp

Abstract“Even more so is the word ‘crystalline’, a glacial and impersonal concept of his which disdains viewing existence from a single portion of time and space” Eileen Myles, “The Importance of Being Iceland”For a smooth variety $X$ over an algebraically closed field of characteristic $p$ to a differential 1-form $\alpha $ on the Frobenius twist $X^{\textrm{(1)}}$ one can associate an Azumaya algebra ${{\mathcal{D}}}_{X,\alpha }$, defined as a certain central reduction of the algebra ${{\mathcal{D}}}_X$ of “crystalline differential operators” on $X$. For a resolution of singularities $\pi :X\to Y$ of an affine variety $Y$, we study for which $\alpha $ the class $[{{\mathcal{D}}}_{X,\alpha }]$ in the Brauer group $\textrm{Br}(X^{\textrm{(1)}})$ descends to $Y^{\textrm{(1)}}$. In the case when $X$ is symplectic, this question is related to Fedosov quantizations in characteristic $p$ and the construction of noncommutative resolutions of $Y$. We prove that the classes $[{{\mathcal{D}}}_{X,\alpha }]$ descend étale locally for all $\alpha $ if ${{\mathcal{O}}}_Y\widetilde{\rightarrow }\pi _\ast{{\mathcal{O}}}_X$ and $R^{1}\pi _*\mathcal O_X = R^2\pi _*\mathcal O_X =0$. We also define a certain class of resolutions, which we call resolutions with conical slices, and prove that for a general reduction of a resolution with conical slices in characteristic $0$ to an algebraically closed field of characteristic $p$ classes $[{{\mathcal{D}}}_{X,\alpha }]$ descend to $Y^{\textrm{(1)}}$ globally for all $\alpha $. Finally we give some examples; in particular, we show that Slodowy slices, Nakajima quiver varieties, and hypertoric varieties are resolutions with conical slices.

Coulomb Branch of a Multiloop Quiver Gauge Theory

We compute the Coulomb branch of a multiloop quiver gauge theory for the quiver with a single vertex, r loops, one-dimensional framing, and dim V = 2. We identify it with a Slodowy slice in the nilpotent cone of the symplectic Lie algebra of rank r. Hence it possesses a symplectic resolution with 2r fixed points with respect to a Hamiltonian torus action. We also identify its flavor deformation with a base change of the full Slodowy slice.

Spaltenstein varieties of pure dimension

We show that Spaltenstein varieties of classical groups are pure dimensional when the Jordan-type of the nilpotent element involved is an even or odd partition. We further show that they are Lagrangian in the partial resolutions of the associated nilpotent Slodowy slices, from which their dimensions are known to be one half of the dimension of the partial resolution minus the dimension of the nilpotent orbit. The results are then extended to the σ \sigma -quiver-variety setting.

Hessenberg varieties, Slodowy slices, and integrable systems

This work is intended to contextualize and enhance certain well-studied relationships between Hessenberg varieties and the Toda lattice, thereby building on the results of Kostant, Peterson, and others. One such relationship is the fact that every Lagrangian leaf in the Toda lattice is compactified by a suitable choice of Hessenberg variety. It is then natural to imagine the Toda lattice as extending to an appropriate union of Hessenberg varieties. We fix a simply-connected complex semisimple linear algebraic group $G$ and restrict our attention to a particular family of Hessenberg varieties, a family that includes the Peterson variety and all Toda leaf compactifications. The total space of this family, $X(H_0)$, is shown to be a Poisson variety with a completely integrable system defined in terms of Mishchenko--Fomenko polynomials. This leads to a natural embedding of completely integrable systems from the Toda lattice to $X(H_0)$. We also show $X(H_0)$ to have an open dense symplectic leaf isomorphic to $G/Z \times S_{\text{reg}}$, where $Z$ is the centre of $G$ and $S_{\text{reg}}$ is a regular Slodowy slice in the Lie algebra of $G$. This allows us to invoke results about integrable systems on $G\times S_{\text{reg}}$, as developed by Rayan and the second author. Lastly, we witness some implications of our work for the geometry of regular Hessenberg varieties.

Quiver varieties and symmetric pairs

We study fixed-point loci of Nakajima varieties under symplectomorphisms and their antisymplectic cousins, which are compositions of a diagram isomorphism, a reflection functor, and a transpose defined by certain bilinear forms. These subvarieties provide a natural home for geometric representation theory of symmetric pairs. In particular, the cohomology of a Steinberg-type variety of the symplectic fixed-point subvarieties is conjecturally related to the universal enveloping algebra of the subalgebra in a symmetric pair. The latter symplectic subvarieties are further used to geometrically construct an action of a twisted Yangian on a torus equivariant cohomology of Nakajima varieties. In the typeAAcase, these subvarieties provide a quiver model for partial Springer resolutions of nilpotent Slodowy slices of classical groups and associated symmetric spaces, which leads to a rectangular symmetry and a refinement of Kraft–Procesi row/column removal reductions.

Aspects of Calabi-Yau Integrable and Hitchin Systems

In the present notes we explain the relationship between Calabi-Yau integrable systems and Hitchin systems based on work by Diaconescu-Donagi-Pantev and the author. Besides a review of these integrable systems, we highlight related topics, for example variations of Hodge structures, cameral curves and Slodowy slices, along the way.

Abstract integrable systems on hyperkähler manifolds arising from Slodowy slices

We study holomorphic integrable systems on the hyperk\"ahler manifold $G\times S_{\text{reg}}$, where $G$ is a complex semisimple Lie group and $S_{\text{reg}}$ is the Slodowy slice determined by a regular $\mathfrak{sl}_2(\mathbb{C})$-triple. Our main result is that this manifold carries a canonical \textit{abstract integrable system}, a foliation-theoretic notion recently introduced by Fernandes, Laurent-Gengoux, and Vanhaecke. We also construct traditional integrable systems on $G\times S_{\text{reg}}$, some of which are completely integrable and fundamentally based on Mishchenko and Fomenko's argument shift approach.

Quiver theories and formulae for Slodowy slices of classical algebras

We utilise SUSY quiver gauge theories to compute properties of Slodowy slices; these are spaces transverse to the nilpotent orbits of a Lie algebra g. We analyse classes of quiver theories, with Classical gauge and flavour groups, whose Higgs branch Hilbert series are the intersections between Slodowy slices and the nilpotent cone S∩N of g. We calculate refined Hilbert series for Classical algebras up to rank 4 (and A5), and find descriptions of their representation matrix generators as algebraic varieties encoding the relations of the chiral ring. We also analyse a class of dual quiver theories, whose Coulomb branches are intersections S∩N; such dual quiver theories exist for the Slodowy slices of A algebras, but are limited to a subset of the Slodowy slices of BCD algebras. The analysis opens new questions about the extent of 3d mirror symmetry within the class of SCFTs known as Tσρ(G) theories. We also give simple group theoretic formulae for the Hilbert series of Slodowy slices; these draw directly on the SU(2) embedding into G of the associated nilpotent orbit, and the Hilbert series of the nilpotent cone.

AN EQUIVARIANT DESCRIPTION OF CERTAIN HOLOMORPHIC SYMPLECTIC VARIETIES

Varieties of the form$G\times S_{\!\text{reg}}$, where$G$is a complex semisimple group and$S_{\!\text{reg}}$is a regular Slodowy slice in the Lie algebra of$G$, arise naturally in hyperkähler geometry, theoretical physics and the theory of abstract integrable systems. Crooks and Rayan [‘Abstract integrable systems on hyperkähler manifolds arising from Slodowy slices’,Math. Res. Let., to appear] use a Hamiltonian$G$-action to endow$G\times S_{\!\text{reg}}$with a canonical abstract integrable system. To understand examples of abstract integrable systems arising from Hamiltonian$G$-actions, we consider a holomorphic symplectic variety$X$carrying an abstract integrable system induced by a Hamiltonian$G$-action. Under certain hypotheses, we show that there must exist a$G$-equivariant variety isomorphism$X\cong G\times S_{\!\text{reg}}$.

Poisson traces, D-modules, and symplectic resolutions

We survey the theory of Poisson traces (or zeroth Poisson homology) developed by the authors in a series of recent papers. The goal is to understand this subtle invariant of (singular) Poisson varieties, conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations. The main technique is the study of a canonical D-module on the variety. In the case the variety has finitely many symplectic leaves (such as for symplectic singularities and Hamiltonian reductions of symplectic vector spaces by reductive groups), the D-module is holonomic, and hence, the space of Poisson traces is finite-dimensional. As an application, there are finitely many irreducible finite-dimensional representations of every quantization of the variety. Conjecturally, the D-module is the pushforward of the canonical D-module under every symplectic resolution of singularities, which implies that the space of Poisson traces is dual to the top cohomology of the resolution. We explain many examples where the conjecture is proved, such as symmetric powers of du Val singularities and symplectic surfaces and Slodowy slices in the nilpotent cone of a semisimple Lie algebra. We compute the D-module in the case of surfaces with isolated singularities and show it is not always semisimple. We also explain generalizations to arbitrary Lie algebras of vector fields, connections to the Bernstein–Sato polynomial, relations to two-variable special polynomials such as Kostka polynomials and Tutte polynomials, and a conjectural relationship with deformations of symplectic resolutions. In the appendix we give a brief recollection of the theory of D-modules on singular varieties that we require.

Filtrations on Springer fiber cohomology and Kostka polynomials

We prove a conjecture which expresses the bigraded Poisson-de Rham homology of the nilpotent cone of a semisimple Lie algebra in terms of the generalized (one-variable) Kostka polynomials, via a formula suggested by Lusztig. This allows us to construct a canonical family of filtrations on the flag variety cohomology, and hence on irreducible representations of the Weyl group, whose Hilbert series are given by the generalized Kostka polynomials. We deduce consequences for the cohomology of all Springer fibers. In particular, this computes the grading on the zeroth Poisson homology of all classical finite W-algebras, as well as the filtration on the zeroth Hochschild homology of all quantum finite W-algebras, and we generalize to all homology degrees. As a consequence, we deduce a conjecture of Proudfoot on symplectic duality, relating in type A the Poisson homology of Slodowy slices to the intersection cohomology of nilpotent orbit closures. In the last section, we give an analogue of our main theorem in the setting of mirabolic D-modules.