arXiv subjects

Andrei Neguţ

Publications and source records attributed to Andrei Neguţ.

52 records · Page 3Linked to original sources

arXiv2021

Toward AGT for parabolic sheaves

We construct explicit elements $W_{ij}^k$ in (a completion of) the shifted quantum toroidal algebra of type $A$, and show that these elements act by 0 on the $K$-theory of moduli spaces of parabolic sheaves. We expect that the quotient of the shifted quantum toroidal algebra by the ideal generated by the elements $W_{ij}^k$ will be related to $q$-deformed $W$-algebras of type $A$ for arbitrary nilpotent, which would imply a $q$-deformed version of the AGT correspondence between gauge theory with surface operators and conformal field theory.

math.AG↗

arXiv2021

The R-matrix of the quantum toroidal algebra

We consider the R-matrix of the quantum toroidal algebra of type gl_1, both abstractly and in Fock space representations. We provide a survey of a certain point of view on this object which involves the elliptic Hall and shuffle algebras, and show how to obtain certain explicit formulas.

math.QA↗

arXiv2020

Affine Laumon spaces and a conjecture of Kuznetsov

We prove a conjecture of Kuznetsov stating that the equivariant K-theory of affine Laumon spaces is the universal Verma module for the quantum affine algebra U_q(gl_n^). We do so by reinterpreting the action of the quantum toroidal algebra U_q(gl_n^^) on the K-theory from [14] in terms of the shuffle algebra studied in [12], which constructs an embedding of U_q(gl_n^) into U_q(gl_n^^)

math.AG↗

arXiv2020

The Chow of $S^{[n]}$ and the universal subscheme

We prove that any element in the Chow ring of the Hilbert scheme $\text{Hilb}_n$ of $n$ points on a smooth surface $S$ is a universal class, i.e. the pushforward of a polynomial in the Chern classes of the universal subscheme on $\text{Hilb}_n \times S^k$ for some $k \in \mathbb{N}$, with coefficients pulled back from the Chow of $S^k$.

math.AG↗

arXiv2020

The universal sheaf as an operator

We compute the universal sheaf of moduli spaces M of sheaves on a surface S, as an operator {Symmetric Polynomials} $\rightarrow$ K(M), thus generalizing the viewpoint of Carlsson-Nekrasov-Okounkov to arbitrary rank and general smooth surfaces.

math.AG↗

arXiv2020

Quantum toroidal and shuffle algebras

In this paper, we prove that the quantum toroidal algebra of gl_n is isomorphic to the double shuffle algebra of Feigin and Odesskii for the cyclic quiver. The shuffle algebra viewpoint will allow us to prove a factorization formula for the universal R-matrix of the quantum toroidal algebra.

math.RT↗

arXiv2020

Lehn's formula in Chow and Conjectures of Beauville and Voisin

The Beauville-Voisin conjecture for a hyperkähler manifold X states that the subring of the Chow ring A^*(X) generated by divisor classes and Chern characters of the tangent bundle injects into the cohomology ring of X. We prove a weak version of this conjecture when X is the Hilbert scheme of points on a K3 surface, for the subring generated by divisor classes and tautological classes. This in particular implies the weak splitting conjecture of Beauville for these geometries. In the process, we extend Lehn's formula and the Li-Qin-Wang W_{1+infinity} algebra action from cohomology to Chow groups, for the Hilbert scheme of an arbitrary smooth projective surface S

math.AG↗

arXiv2020

Deformed W-algebras in type A for rectangular nilpotent

For any natural numbers n,r, we construct an algebra P_n^r via generators and quadratic relations, and show that it deforms the W-algebra of gl_{nr} with respect to a nilpotent with Jordan block decomposition r+...+r. We introduce a surjective morphism from half of the quantum toroidal algebra of gl_n to P_n^r, and show that the action of the quantum toroidal algebra on the K-theory groups of the moduli spaces of parabolic sheaves factors through P_n^r.

math.RT↗

arXiv2018

Affine Laumon spaces and integrable systems

In this paper we formalize and prove a conjecture of Braverman concerning integrals of the Chern polynomial of the tangent bundle to affine Laumon spaces. This provides the computation of the Nekrasov partition function of N = 2 gauge theory with adjoint matter on C^2 in the Omega background, in the presence of a full surface operator insertion

math.AG↗

arXiv2017

The q-AGT-W relations via shuffle algebras

We construct the action of the q-deformed W-algebra on its level r representation geometrically, using the moduli space of U(r) instantons on the plane and the double shuffle algebra. We give an explicit LDU decomposition for the action of W-algebra currents in the fixed point basis of the level r representation, and prove a relation between the Carlsson-Okounkov Ext operator and intertwiners for the deformed W-algebra. We interpret this result as a q-deformed version of the AGT-W relations.

math.RT↗

arXiv2016

Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology

We construct a categorification of the maximal commutative subalgebra of the type $A$ Hecke algebra. Specifically, we propose a monoidal functor from the (symmetric) monoidal category of coherent sheaves on the flag Hilbert scheme to the (non-symmetric) monoidal category of Soergel bimodules. The adjoint of this functor allows one to match the Hochschild homology of any braid with the Euler characteristic of a sheaf on the flag Hilbert scheme. The categorified Jones-Wenzl projectors studied by Abel, Elias and Hogancamp are idempotents in the category of Soergel bimodules, and they correspond to the renormalized Koszul complexes of the torus fixed points on the flag Hilbert scheme. As a consequence, we conjecture that the endomorphism algebras of the categorified projectors correspond to the dg algebras of functions on affine charts of the flag Hilbert schemes. We define a family of differentials $d_{N}$ on these dg algebras and conjecture that their homology matches that of the $\mathfrak{gl}_N$ projectors, generalizing earlier conjectures of the first and third authors with Oblomkov and Shende.

math.GT↗

arXiv2016

Exts and the AGT relations

We prove the connection between the Nekrasov partition function of N=2 super-symmetric U(2) gauge theory with adjoint matter and conformal blocks for the Virasoro algebra, as predicted by the Alday-Gaiotto-Tachikawa relations. Mathematically, this is achieved by relating the Carlsson-Okounkov Ext vector bundle on the moduli space of rank 2 sheaves with Liouville vertex operators.Our approach is geometric in nature, and uses a new method for intersection-theoretic computations of the Ext operator.

hep-th↗

arXiv2015

Infinitesimal change of stable basis

The purpose of this note is to study the Maulik-Okounkov $K-$theoretic stable basis for the Hilbert scheme of points on the plane, which depends on a "slope" $m \in \mathbb{R}$. When $m = \frac ab$ is rational, we study the change of stable matrix from slope $m-\varepsilon$ to $m+\varepsilon$ for small $\varepsilon>0$, and conjecture that it is related to the Leclerc-Thibon conjugation in the $q-$Fock space for $U_q\widehat{\mathfrak{gl}}_b$. This is part of a wide framework of connections involving derived categories of quantized Hilbert schemes, modules for rational Cherednik algebras and Hecke algebras at roots of unity.

math.AG↗

arXiv2015

Quantum Algebras and Cyclic Quiver Varieties

The purpose of this thesis is to present certain viewpoints on the geometric representation theory of Nakajima cyclic quiver varieties, in relation to the Maulik-Okounkov stable basis. Our main technical tool is the shuffle algebra, which arises as the K-theoretic Hall algebra of the double cyclic quiver. We prove the isomorphism between the shuffle algebra and the quantum toroidal algebra U_{q,t}(sl_n^^), and identify the quotients of Verma modules for the shuffle algebra with the K-theory groups of Nakajima cyclic quiver varieties, which were studied by Nakajima and Varagnolo-Vasserot. The shuffle algebra viewpoint allows us to construct the universal R-matrix of the quantum toroidal algebra U_{q,t}(sl_n^^), and to factor it in terms of pieces that arise from subalgebras isomorphic to quantum affine groups U_{q}(gl_m^), for various m. This factorization generalizes constructions of Khoroshkin-Tolstoy to the toroidal case, and matches the factorization that Maulik-Okounkov produce via the stable basis in the K-theory of Nakajima quiver varieties. We connect the two pictures by computing formulas for the root generators of U_{q,t}(sl_n^^) acting on the stable basis, which provide a wide extension of Murnaghan-Nakayama and Pieri type rules from combinatorics.

math.RT↗

arXiv2015

The m/n Pieri rule

The Pieri rule is an important theorem which explains how the operators e_k of multiplication by elementary symmetric functions act in the basis of Schur functions s_lambda. In this paper, for any rational number m/n, we study the relationship between the rational version e_k^{m/n} of the operators (given by the elliptic Hall algebra) and the "rational" version s_lambda^{m/n} of the basis (given by the Maulik-Okounkov stable basis construction)

math.CO↗

arXiv2015

Refined knot invariants and Hilbert schemes

We consider the construction of refined Chern-Simons torus knot invariants by M. Aganagic and S. Shakirov from the DAHA viewpoint of I. Cherednik. We give a proof of Cherednik's conjecture on the stabilization of superpolynomials, and then use the results of O. Schiffmann and E. Vasserot to relate knot invariants to the Hilbert scheme of points on the plane. Then we use the methods of the second author to compute these invariants explicitly in the uncolored case. We also propose a conjecture relating these constructions to the rational Cherednik algebra, as in the work of the first author, A. Oblomkov, J. Rasmussen and V. Shende. Among the combinatorial consequences of this work is a statement of the m/n shuffle conjecture.

math.RT↗