arXiv subjects

Duiliu-Emanuel Diaconescu

Publications and source records attributed to Duiliu-Emanuel Diaconescu.

At least 19 recordsLinked to original sources

arXiv2026

Nilpotent cohomological Hall algebras of surfaces

This paper develops a framework for systematically studying cohomological "Hecke operators" associated with modifications of coherent sheaves on a smooth surface $X$ along a fixed proper curve $Z \subset X$ (possibly singular and reducible), using the theory of cohomological Hall algebras. More precisely, we construct a moduli stack of coherent sheaves $\mathbf{Coh}(\widehat{X}_Z)$ on $X$ with set-theoretic support $Z$ and we prove that its reduced is an Artin stack locally of finite type. This provides a vast generalization of the global nilpotent cone. Subsequently, we develop the needed background to define the (motivic, $T$-equivariant) cohomological Hall algebra $\mathbf{HA}^{T}_{X,Z}$ of the moduli stack of coherent sheaves on $X$ with set-theoretic support on $Z$, in the setting of a general motivic formalism $\mathbf{D}$ in the sense of Khan. The algebra $\mathbf{HA}^{\mathbf{D}, A}_{X,Z}$ is functorial with respect to closed immersions $Z' \subset Z$ and transformations of the motivic formalism $\mathbf{D}$, and only depends on the formal neighborhood $\widehat{X}_Z$ of $Z$ in $X$. In the companion paper arXiv:2603.03386, we use the nilpotent COHA $\mathbf{HA}^{T}_{X,Z}$ to answer a question previously raised in arXiv:2004.13685 about the precise relationship between the COHA of a minimal resolution of a Kleinian singularity and the corresponding preprojective COHA.

math.AG↗

arXiv2026

Cohomological Hall algebras of one-dimensional sheaves on surfaces and Yangians

This paper provides the first algebraic characterization of an algebra of cohomological Hecke operators associated with modifications of coherent sheaves on a smooth surface $X$ along a fixed proper curve $Z \subset X$ (possibly singular and reducible), establishing a direct connection with Yangians. It is based on the theory of equivariant nilpotent cohomological Hall algebras $\mathbf{HA}^T_{X,Z}$, developed by the same authors. More precisely, let $X$ be a resolution of a Kleinian singularity (for example, $X = T^\ast\mathbb{P}^1$) and let $Z$ be the exceptional divisor. One of the main results of this paper is an explicit isomorphism $\mathbf{HA}^T_{X,Z} \simeq \mathbb{Y}^+_\infty$, where $\mathbb{Y}^+_\infty$ is a completed, nonstandard, positive half of the affine Yangian $\mathbb{Y}(\mathfrak{g})$ of the corresponding affine ADE Lie algebra $\mathfrak{g}$. Furthermore, the generators of $\mathbf{HA}^T_{X,Z}$--given by fundamental classes of substacks of zero-dimensional sheaves and of pushforwards of line bundles on $Z$--are expressed explicitly in terms of Yangian generators. Our main tools, which may be of independent interest, are: (i) a `continuity' theorem describing the behavior of cohomological Hall algebras of objects in the heart of $t$-structures $τ_n$ when the sequence $(τ_n)_n$ converges, in an appropriate sense, to a fixed $t$-structure $τ_\infty$; (ii) the definition of a multi-parameter Yangian $\mathbb{Y}_Q$ for an arbitrary quiver $Q$, given by generators and relations; (iii) a theorem relating the algebraic action of the braid group $B_Q$ on the Yangian $\mathbb{Y}_Q$ to the action of $B_Q$ on the equivariant 2-dimensional cohomological Hall algebra $\mathbf{HA}^T_Q$ of $Q$, where the latter can be described in terms of derived reflection functors of the bounded derived category of modules over the preprojective algebra of $Q$.

math.AG↗

arXiv2025

Cohomological Hall algebras, their categorification, and their representations via torsion pairs

In this paper we provide a systematic way of producing representations of cohomological, K-theoretical and categorified Hall algebras, and study the output of our construction in several cases. We thus recover and categorify in a unified framework the action of the COHA of a quiver on the cohomology of Nakajima quiver variety, the action of the COHA of zero-dimensional sheaves on the the cohomology of Hilbert schemes of points and moduli spaces of Gieseker-stable sheaves on smooth surfaces, recovering the constructions of Neguţ and DeHority. We also obtain new examples, associated to Pandharipande-Thomas stable pairs. Along the way, we explain carefully under which conditions one can associate to a pair $(\mathscr{C},τ)$ consisting of a stable $\infty$-category with a t-structure a COHA. This requires a careful analysis and extension of Khan's theory of motivic Borel-Moore homology to the non quasi-compact setting, and it allows to produce new examples of COHAs arising from Bridgeland's stability conditions. The representations that we construct take an extra categorical input: that of a torsion pair $(\mathscr{T},\mathscr{F})$ on the heart $\mathscr{C}^\heartsuit$ of $τ$. Under favorable conditions, the homology of the moduli stack associated to $\mathscr{T}$ acquires a Hall multiplication, that acts both on the left and on the right on the homology of the moduli stack associated to $\mathscr{F}$. The left action generalizes and categorifies Nakajima's positive operators, while the right action corresponds to negative operators.

math.AG↗

arXiv2025

Coulomb branch localization, quasimaps, and surface counting in Calabi--Yau fourfolds

We present a string theoretic approach to surface counting in local Calabi--Yau fourfolds via supersymmetric localization in topologically twisted four-dimensional gauge theories. This approach is based on a spectral correspondence between PT1-stable pairs on local fourfolds and twisted quasimaps with fixed two-dimensional domain associated to the ADHM quiver, or, equivalently, ADHM sheaves. For local toric fourfolds, we derive a conjectural residue formula for the K-theoretic quasimap partition function via Coulomb branch localization. As a result, in this case, we obtain a conjectural prescription fixing all usual sign ambiguities in the equivariant computation of such invariants. We present some explicit computations for local P2, extending the results available in the literature, and describe the formalism in general. This is the first instance of Coulomb branch localization for a quasimap theory in the context of four-dimensional gauge theories.

math.AG↗

arXiv2025

Flops and Hilbert schemes of space curve singularities

Using pagoda flop transitions between smooth projective threefolds, a relation is derived between the Euler numbers of moduli spaces of stable pairs which are scheme-theoretically supported on a fixed singular space curve and Euler numbers of Flag Hilbert schemes associated to a plane curve singularity. When the space curve singularity is locally complete intersection, one obtains a relation between the latter and Euler numbers of Hilbert schemes of the space curve singularity. It is also shown that this relation yields explicit results for a class of torus-invariant locally complete intersection singularities.

math.AG↗

arXiv2023

McKay correspondence, cohomological Hall algebras and categorification

Let $π\colon Y\to X$ denote the canonical resolution of the two dimensional Kleinian singularity $X$ of type ADE. In the present paper, we establish isomorphisms between the cohomological and K-theoretical Hall algebras of $ω$-semistable properly supported sheaves on $Y$ with fixed slope $μ$ and $ζ$-semistable finite-dimensional representations of the preprojective algebra of affine type ADE of slope zero respectively, under some conditions on $ζ$ depending on the polarization $ω$ and $μ$. These isomorphisms are induced by the derived McKay correspondence. In addition, they are interpreted as decategorified versions of a monoidal equivalence between the corresponding categorified Hall algebras. In the type A case, we provide finer descriptions of the cohomological, K-theoretical and categorified Hall algebra of $ω$-semistable properly supported sheaves on $Y$ with fixed slope $μ$: for example, in the cohomological case, the algebra can be given in terms of Yangians of finite type ADE Dynkin diagrams.

math.AG↗

arXiv2022

Affine Laumon spaces and iterated W-algebras

A family of vertex algebras whose universal Verma modules coincide with the cohomology of affine Laumon spaces is found. This result is based on an explicit expression for the generating function of Poincare polynomials of these spaces. There is a variant of quantum Hamiltonian reduction that realizes vertex algebras which we call iterated W-algebras and our main conjecture is that the vertex algebras associated to the affine Laumon spaces are subalgebras of iterated W-algebras.

math.QA↗

arXiv2018

Atiyah class and sheaf counting on local Calabi Yau fourfolds

We discuss Donaldson-Thomas (DT) invariants of torsion sheaves with 2 dimensional support on a smooth projective surface in an ambient non-compact Calabi Yau fourfold given by the total space of a rank 2 bundle on the surface. We prove that in certain cases, when the rank 2 bundle is chosen appropriately, the universal truncated Atiyah class of these codimension 2 sheaves reduces to one, defined over the moduli space of such sheaves realized as torsion codimension 1 sheaves in a noncompact divisor (threefold) embedded in the ambient fourfold. Such reduction property of universal Atiyah class enables us to relate our fourfold DT theory to a reduced DT theory of a threefold and subsequently then to the moduli spaces of sheaves on the base surface using results in arXiv:1701.08899 and arXiv:1701.08902 and . We finally make predictions about modularity of such fourfold invariants when the base surface is an elliptic K3.

math.AG↗

arXiv2018

Twisted spectral correspondence and torus knots

Cohomological invariants of twisted wild character varieties as constructed by Boalch and Yamakawa are derived from enumerative Calabi-Yau geometry and refined Chern-Simons invariants of torus knots. Generalizing the untwisted case, the present approach is based on a spectral correspondence for meromorphic Higgs bundles with fixed conjugacy classes at the marked points. This construction is carried out for twisted wild character varieties associated to (l, kl-1) torus knots, providing a colored generalization of existing results of Hausel, Mereb and Wong as well as Shende, Treumann and Zaslow.

hep-th↗

arXiv2017

Local curves, wild character varieties, and degenerations

Conjectural results for cohomological invariants of wild character varieties are obtained by counting curves in degenerate Calabi-Yau threefolds. A conjectural formula for E-polynomials is derived from the Gromov-Witten theory of local Calabi-Yau threefolds with normal crossing singularities. A refinement is also conjectured, generalizing existing results of Hausel, Mereb and Wong as well as recent joint work of Donagi, Pantev and the author for weighted Poincaré polynomials of wild character varieties.

math.AG↗

arXiv2017

BPS states, torus links and wild character varieties

A string theoretic framework is constructed relating the cohomology of wild character varieties to refined stable pair theory and torus link invariants. Explicit conjectural formulas are derived for wild character varieties with a unique irregular point on the projective line. For this case the string theoretic construction leads to a conjectural colored generalization of existing results of Hausel, Mereb and Wong as well as Shende, Treumann and Zaslow.

hep-th↗

arXiv2016

Vertical D4-D2-D0 bound states on K3 fibrations and modularity

An explicit formula is derived for the generating function of vertical D4-D2-D0 bound states on smooth K3 fibered Calabi-Yau threefolds, generalizing previous results of Gholampour and Sheshmani. It is also shown that this formula satisfies strong modularity properties, as predicted by string theory. This leads to a new construction of vector valued modular forms which exhibits some of the features of a generalized Hecke transform.

hep-th↗

arXiv2015

Vertical sheaves and Fourier-Mukai transform on elliptic Calabi-Yau threefolds

This paper studies the action of the Fourier-Mukai transform on moduli spaces of vertical torsion sheaves on elliptic Calabi-Yau threefolds in Weierstrass form. Moduli stacks of semistable one dimensional sheaves on such threefolds are identified with open and closed substacksof moduli stacks of vertical semistable two dimensional sheaves on their Fourier-Mukai duals. In particular, this yields explicit conjectural results for Donaldson-Thomas invariants of vertical two dimensional sheaves on K3-fibered elliptic Calabi-Yau threefolds.

math.AG↗

arXiv2013

Parabolic refined invariants and Macdonald polynomials

A string theoretic derivation is given for the conjecture of Hausel, Letellier, and Rodriguez-Villegas on the cohomology of character varieties with marked points. Their formula is identified with a refined BPS expansion in the stable pair theory of a local root stack, generalizing previous work of the first two authors in collaboration with G. Pan. Haiman's geometric construction for Macdonald polynomials is shown to emerge naturally in this context via geometric engineering. In particular this yields a new conjectural relation between Macdonald polynomials and refined local orbifold curve counting invariants. The string theoretic approach also leads to a new spectral cover construction for parabolic Higgs bundles in terms of holomorphic symplectic orbifolds.

hep-th↗

arXiv2013

Geometric engineering of (framed) BPS states

BPS quivers for N=2 SU(N) gauge theories are derived via geometric engineering from derived categories of toric Calabi-Yau threefolds. While the outcome is in agreement of previous low energy constructions, the geometric approach leads to several new results. An absence of walls conjecture is formulated for all values of N, relating the field theory BPS spectrum to large radius D-brane bound states. Supporting evidence is presented as explicit computations of BPS degeneracies in some examples. These computations also prove the existence of BPS states of arbitrarily high spin and infinitely many marginal stability walls at weak coupling. Moreover, framed quiver models for framed BPS states are naturally derived from this formalism, as well as a mathematical formulation of framed and unframed BPS degeneracies in terms of motivic and cohomological Donaldson-Thomas invariants. We verify the conjectured absence of BPS states with "exotic" SU(2)_R quantum numbers using motivic DT invariants. This application is based in particular on a complete recursive algorithm which determine the unframed BPS spectrum at any point on the Coulomb branch in terms of noncommutative Donaldson-Thomas invariants for framed quiver representations.

hep-th↗

arXiv2012

Moduli of ADHM Sheaves and Local Donaldson-Thomas Theory

The ADHM construction establishes a one-to-one correspondence between framed torsion free sheaves on the projective plane and stable framed representations of a quiver with relations in the category of complex vector spaces. This paper studies the geometry of moduli spaces of representations of the same quiver with relations in the abelian category of coherent sheaves on a smooth complex projective curve $X$. In particular it is proven that this moduli space is virtually smooth and related byrelative Beilinson spectral sequence to the curve counting construction via stable pairs of Pandharipande and Thomas. This yields a new conjectural construction for the local Donaldson-Thomas theory of curves as well as a natural higher rank generalization.

math.AG↗

arXiv2011

Chamber Structure and Wallcrossing in The ADHM Theory of Curves I

ADHM invariants are equivariant virtual invariants of moduli spaces of twisted cyclic representations of the ADHM quiver in the abelian category of coherent sheaves of a smooth complex projective curve X. The goal of the present paper is to present a generalization of this construction employing a more general stability condition which depends on a real parameter. This yields a chamber structure in the ADHM theory of curves, residual ADHM invariants being defined by equivariant virtual integration in each chamber. Wallcrossing results and applications to local stable pair invariants will be presented in the second part of this work.

math.AG↗

arXiv2011

Chamber Structure and Wallcrossing in the ADHM Theory of Curves II

This is the second part of a project concerning variation of stability and chamber structure for ADHM invariants of curves. Wallcrossing formulas for such invariants are derived using the theory of stack function Ringel-Hall algebras constructed by Joyce and the theory of generalized Donaldson-Thomas invariants of Joyce and Song. Some applications are presented, including strong rationality for local stable pair invariants of higher genus curves and comparison with wallcrossing formulas of Kontsevich and Soibelman, and the halo formula of Denef and Moore.

math.AG↗