### ?

## A finite analog of the AGT relation I: finite W-algebras and quasimaps' spaces

Communications in Mathematical Physics. 2011. Vol. 308. No. 2. P. 457-478.

Recently Alday, Gaiotto and Tachikawa proposed a conjecture relating 4-dimensional super-symmetric gauge theory for a gauge group G with certain 2-dimensional conformal field theory. This conjecture implies the existence of certain structures on the (equivariant) intersection cohomology of the Uhlenbeck partial compactification of the moduli space of framed G-bundles on P^2. More precisely, it predicts the existence of an action of the corresponding W-algebra on the above cohomology, satisfying certain properties.
We propose a "finite analog" of the (above corollary of the) AGT conjecture.

Braverman A., Michael Finkelberg, Nakajima H., Instanton moduli spaces and W-algebras / Cornell University. Series math "arxiv.org". 2014.

We describe the (equivariant) intersection cohomology of certain moduli spaces ("framed Uhlenbeck spaces") together with some structures on them (such as e.g.\ the Poincar\'e pairing) in terms of representation theory of some vertex operator algebras ("W-algebras"). ...

Added: January 30, 2015

Brav C. I., Thomas H., Mathematische Annalen 2011 Vol. 351 No. 4 P. 1005-1017

We establish faithfulness of braid group actions generated by twists along an ADE configuration of 22-spherical objects in a derived category. Our major tool is the Garside structure on braid groups of type ADE. This faithfulness result provides the missing ingredient in Bridgeland's description of a space of stability conditions associated to a Kleinian singularity. ...

Added: September 29, 2014

Braverman A., Dobrovolska G., Michael Finkelberg, Gaiotto-Witten superpotential and Whittaker D-modules on monopoles / Cornell University. Series math "arxiv.org". 2014.

Let G be an almost simple simply connected group over complex numbers. For a positive element α of the coroot lattice of G let Z^α denote the space of based maps from the projective line to the flag variety of G of degree α. This space is known to be isomorphic to the space of ...

Added: February 3, 2015

Michael Finkelberg, Schechtman V., Microlocal approach to Lusztig's symmetries / Cornell University. Series math "arxiv.org". 2014.

We reformulate the De Concini -- Toledano Laredo conjecture about the monodromy of the Casimir connection in terms of a relation between the Lusztig's symmetries of quantum group modules and the monodromy in the vanishing cycles of factorizable sheaves. ...

Added: January 30, 2015

Bezrukavnikov R., Finkelberg M. V., Wreath Macdonald polynomials and categorical McKay correspondence (with Appendices by Ivan Losev, Vadim Vologodsky) / Cornell University. Series math "arxiv.org". 2012. No. 1208.3696.

Mark Haiman has reduced Macdonald positivity conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjectures where the symmetric group is replaced by the wreath product $S_n\ltimes (Z/r Z)^n$. He has proven the original conjecture by establishing the geometric statement about the Hilbert ...

Added: February 6, 2013

Alexander I. Efimov, Derived categories of Grassmannians over integers and modular representation theory / Cornell University. Series math "arxiv.org". 2014.

In this paper we study the derived categories of coherent sheaves on Grassmannians Gr(k,n), defined over the ring of integers. We prove that the category D^b(Gr(k,n)) has a semi-orthogonal decomposition, with components being full subcategories of the derived category of representations of GL_k. This in particular implies existence of a full exceptional collection, which is ...

Added: February 2, 2015

Cruz Morales J. A., Galkin S., Upper Bounds for Mutations of Potentials / Cornell University. Series math "arxiv.org". 2013. No. 1301.4541.

In this note we provide a new, algebraic proof of the excessive Laurent phenomenon for mutations of potentials (in the sense of [Galkin S., Usnich A., Preprint IPMU 10-0100, 2010]) by introducing to this theory the analogue of the upper bounds from [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52]. ...

Added: May 27, 2013

Braverman A., Michael Finkelberg, Twisted zastava and q-Whittaker functions / Cornell University. Series math "arxiv.org". 2014.

In this note, we extend the results of arxiv:1111.2266 and arxiv:1203.1583 to the non simply laced case. To this end we introduce and study the twisted zastava spaces. ...

Added: February 5, 2015

Brav C. I., International Mathematics Research Notices 2009 No. 8 P. 1355-1387

Kirillov has described a McKay correspondence for finite subgroups of that associates to each “height” function an affine Dynkin quiver together with a derived equivalence between equivariant sheaves on and representations of this quiver. The equivalences for different height functions are then related by reflection functors for quiver representations. The main goal of this article ...

Added: September 29, 2014

Efimov A., Homotopy finiteness of some DG categories from algebraic geometry / Cornell University. Series math "arxiv.org". 2013.

In this paper, we show that bounded derived categories of coherent sheaves (considered as DG categories) on separated schemes of finite type over a field of characteristic zero are homotopically finitely presented. This confirms a conjecture of Kontsevich. The proof uses categorical resolution of singularities of Kuznetsov and Lunts, which is based on the ordinary ...

Added: October 31, 2013

Rybakov S., On classification of groups of points on abelian varieties over finite fields / Cornell University. Series math "arxiv.org". 2014.

A k-isogeny class of abelian varieties over a finite field k is uniquely determined by the Weil polynomial f of any variety from this class. When we consider classification problems concerning abelian varieties inside an isogeny class, the classification can be given in terms of the corresponding Weil polynomial. In this paper we improve our ...

Added: January 21, 2014

Bershtein M., Gavrylenko P., Marshakov A., Twist-field representations of W-algebras, exact conformal blocks and character identities / . 2017. No. 1705.00957.

We study twist-field representations of the W-algebras and generalize the construction of the corresponding vertex operators to D- and B-series. We demonstrate how the computation of characters of such representations leads to the nontrivial identities involving lattice theta-functions. We propose a construction of their exact conformal blocks, which for D-series express them in terms of ...

Added: May 4, 2017

Positselski L., Efimov A., Coherent analogues of matrix factorizations and relative singularity categories / Cornell University. Series math "arxiv.org". 2013. No. arXiv:1102.0261.

We define the triangulated category of relative singularities of a closed subscheme in a scheme. When the closed subscheme is a Cartier divisor, we consider matrix factorizations of the related section of a line bundle, and their analogues with locally free sheaves replaced by coherent ones. The appropriate exotic derived category of coherent matrix factorizations ...

Added: December 22, 2013

Michael Finkelberg, Leonid Rybnikov, Quantization of Drinfeld Zastava in type C / Cornell University. Series math "arxiv.org". 2013.

Drinfeld zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of an affine Lie algebra g^. In case g is the symplectic Lie algebra spN, we introduce an affine, reduced, irreducible, normal quiver variety Z which maps to the zastava space isomorphically in characteristic ...

Added: December 27, 2013

Michael Finkelberg, Kamnitzer J., Pham K. et al., Advances in Mathematics 2018 Vol. 327 P. 349-389

We study a coproduct in type A quantum open Toda lattice
in terms of a coproduct in the shifted Yangian of sl2. At
the classical level this corresponds to the multiplication of
scattering matrices of euclidean SU(2) monopoles. We also
study coproducts for shifted Yangians for any simply-laced
Lie algebra. ...

Added: February 21, 2018

Lev Soukhanov, On the phenomena of constant curvature in the diffusion-orthogonal polynomials / Cornell University. Series math "arxiv.org". 2014.

We consider the systems of diffusion-orthogonal polynomials, defined in the
work [1] of D. Bakry, S. Orevkov and M. Zani and (particularly) explain why
these systems with boundary of maximal possible degree should always come from
the group, generated by reflections. Our proof works for the dimensions $2$ (on
which this phenomena was discovered) and $3$, and fails in ...

Added: September 19, 2014

Romaskevich O. L., L'Enseignement Mathématique 2014

We consider 3 -periodic orbits in an elliptic billiard. Numerical experiments conducted by Dan Reznik have shown that the locus of the centers of inscribed circles of the corresponding triangles is an ellipse. We prove this fact by the complexification of the problem coupled with the complex law of reflection. ...

Added: December 25, 2014

Galkin S., Shinder E., Exceptional collections of line bundles on the Beauville surface / Cornell University. Series math "arxiv.org". 2012. No. 1210.3339.

We construct quasi-phantom admissible subcategories in the derived category of coherent sheaves on the Beauville surface S. These quasi-phantoms subcategories appear as right orthogonals to subcategories generated by exceptional collections of maximal possible length 4 on S. We prove that there are exactly 6 exceptional collections consisting of line bundles (up to a twist) and these collections ...

Added: September 14, 2013

Feigin E., Makhlin I., Vertices of FFLV polytopes / Cornell University. Series math "arxiv.org". 2016. No. arXiv:1604.08844 .

FFLV polytopes describe monomial bases in irreducible representations of sln. We study various sets of vertices of FFLV polytopes. First, we prove the locality of set of vertices with respect to the type A Dynkin diagram. Second, we describe all the permutation vertices. Third, we describe all the simple vertices and prove that their number ...

Added: May 6, 2016

Positselski L., Contraherent cosheaves / Cornell University. Series math "arxiv.org". 2014. No. 1209.2995.

Contraherent cosheaves are globalizations of cotorsion (or similar) modules over commutative rings obtained by gluing together over a scheme. The category of contraherent cosheaves over a scheme is a Quillen exact category with exact functors of infinite product. Over a quasi-compact semi-separated scheme or a Noetherian scheme of finite Krull dimension (in a different version ...

Added: February 6, 2013

Fedor Bogomolov, De Oliveira B., Local structure of closed symmetric 2-differentials / Cornell University. Series math "arxiv.org". 2014.

In the authors's previous work on symmetric differentials and their
connection to the topological properties of the ambient manifold, a class of
symmetric differentials was introduced: closed symmetric differentials
([BoDeO11] and [BoDeO13]). In this article we give a description of the local
structure of closed symmetric 2-differentials on complex surfaces, with an
emphasis towards the local decompositions as products of ...

Added: November 21, 2014

Ivan Cheltsov, Park J., Won J., Cylinders in singular del Pezzo surfaces / Cornell University. Series math "arxiv.org". 2013.

For each del Pezzo surface $S$ with du Val singularities, we determine
whether it admits a $(-K_S)$-polar cylinder or not. If it allows one, then we
present an effective divisor $D$ that is $\mathbb{Q}$-linearly equivalent to
$-K_S$ and such that the open set $S\setminus\mathrm{Supp}(D)$ is a cylinder.
As a corollary, we classify all the del Pezzo surfaces with du ...

Added: December 27, 2013

Victor Kulikov, Shustin E., Duality of planar and spacial curves: new insight / Cornell University. Series math "arxiv.org". 2014.

We study the geometry of equiclassical strata of the discriminant in the space of plane curves of a given degree, which are families of curves of given degree, genus and class (degree of the dual curve). Our main observation is that the use of duality transformation leads to a series of new sufficient conditions for ...

Added: February 2, 2015

Feigin E., Makedonskyi I., Weyl modules for osp(1,2) and nonsymmetric Macdonald polynomials / Cornell University. Series arXiv "math". 2015. No. 1507.01362.

The main goal of our paper is to establish a connection between the Weyl modules of the current Lie superalgebras (twisted and untwisted) attached to osp(1,2) and the nonsymmetric Macdonald polynomials of types $A_2^2$ and ${A_2}^{2\dagger}$ . We compute the dimensions and construct bases of the Weyl modules. We also derive explicit formulas for the ...

Added: July 8, 2015