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## Coulomb branches of 3d N = 4 quiver gauge theories and slices in the affine Grassmannian

Advances in Theoretical and Mathematical Physics. 2019. Vol. 23. No. 1. P. 75-166.

This is a companion paper of [Part II]. We study Coulomb branches

of unframed and framed quiver gauge theories of type ADE. In the

unframed case they are isomorphic to the moduli space of based ra-

tional maps from P^1 to the flag variety. In the framed case they are

slices in the affine Grassmannian and their generalization. In the

appendix, written jointly with Joel Kamnitzer, Ryosuke Kodera,

Ben Webster, and Alex Weekes, we identify the quantized Coulomb

branch with the truncated shifted Yangian.

Braverman A., Michael Finkelberg, Nakajima H., Advances in Theoretical and Mathematical Physics 2019 Vol. 23 No. 2 P. 253-344

We consider the
morphism from the variety of triples introduced in our previous paper to the
affine Grassmannian. The direct image of the dualizing complex is a
ring object in the equivariant derived category on the affine Grassmannian (equivariant derived Satake category). We show that various constructions in our previous paper work for an arbitrary commutative
ring object.
The second purpose of this ...

Added: November 12, 2019

Braverman A., Michael Finkelberg, Nakajima H., Advances in Theoretical and Mathematical Physics 2018 Vol. 22 No. 5 P. 1071-1147

Consider the 3-dimensional N = 4 supersymmetric gauge theory associated with a compact Lie group Gc and its quaternionic representation M. Physicists study its Coulomb branch, which is a noncompact hyper-K¨ahler manifold with an SU(2)-action, possibly with singularities. We give a mathematical definition of the Coulomb branch as an affine algebraic variety with C×-action when ...

Added: May 3, 2019

Гончаров Е. А., Finkelberg M. V., Функциональный анализ и его приложения 2019 Т. 53 № 4 С. 3-13

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 ...

Added: November 28, 2019

Braverman A., Etingof P., Michael Finkelberg, Annales Scientifiques de l'Ecole Normale Superieure 2020 Vol. 53 No. 5 P. 1249-1312

We show that the partially spherical cyclotomic rational Cherednik algebra (obtained from the full rational Cherednik algebra by averaging out the cyclotomic part of the underlying
reflection group) has four other descriptions: (1) as a subalgebra of the degenerate DAHA of type A
given by generators; (2) as an algebra given by generators and relations; (3) as ...

Added: December 8, 2020

Michael Finkelberg, Feigin E., Reineke M., Kyoto Journal of Mathematics 2017 Vol. 57 No. 2 P. 445-474

We study the connection between the affine degenerate Grassmannians in type A, quiver Grassmannians for one vertex loop quivers and affine Schubert varieties. We give an explicit description of the degenerate affine Grassmannian of type GL(n) and identify it with semi-infinite orbit closure of type A_{2n-1}. We show that principal quiver Grassmannians for the ...

Added: May 10, 2017

Krylov V., Functional Analysis and Its Applications 2018 Vol. 52 No. 2 P. 113-133

Let $G$ be a connected reductive algebraic group over $\mathbb{C}$. Let $\Lambda^{+}_{G}$ be the monoid of dominant weights of $G$. We construct the integrable crystals $\mathbf{B}^{G}(\lambda),\ \lambda\in\Lambda^{+}_{G}$, using the geometry of generalized transversal slices in the affine Grassmannian of the Langlands dual group. We construct the tensor product maps $\mathbf{p}_{\lambda_{1},\lambda_{2}}\colon \mathbf{B}^{G}(\lambda_{1}) \otimes \mathbf{B}^{G}(\lambda_{2}) \rightarrow \mathbf{B}^{G}(\lambda_{1}+\lambda_{2})\cup\{0\}$ ...

Added: September 11, 2018

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

Braverman A., Michael Finkelberg, Moscow Mathematical Journal 2013 Vol. 13 No. 2 P. 233-265

This is the third paper in a series which describes a conjectural analogue of the affine Grassmannian for affine Kac-Moody groups (also known as the double affine Grassmannian). The present paper is dedicated to the description of the conjectural analogue of the convolution diagram for the double affine Grassmannian and affine zastava. ...

Added: September 18, 2013

Braverman A., Michael Finkelberg, Ginzburg V. et al., Compositio Mathematica 2021 Vol. 157 No. 8 P. 1724-1765

We construct a mirabolic analogue of the geometric Satake equivalence. We also prove an equivalence that relates representations of a supergroup to the category of GL(N − 1, C[[t]])-equivariant perverse sheaves on the affine Grassmannian of GLN . We explain how our equivalences fit into a more general framework of conjectures due to Gaiotto and ...

Added: July 22, 2021

Michael Finkelberg, Tsymbaliuk A., Arnold Mathematical Journal 2019 Vol. 5 No. 2-3 P. 197-283

We define an integral form of shifted quantum affine algebras of type A and construct
Poincaré–Birkhoff–Witt–Drinfeld bases for them. When the shift is trivial, our integral
form coincides with the RTT integral form. We prove that these integral forms are
closed with respect to the coproduct and shift homomorphisms. We prove that the
homomorphism from our integral form to ...

Added: November 14, 2019

Michael Finkelberg, Krylov V., Mirkovic I., Journal of Topology 2020 Vol. 13 No. 2 P. 683-729

Let G be a reductive complex algebraic group. We fix a pair of opposite Borel subgroups
and consider the corresponding semi-infinite orbits in the affine Grassmannian Gr G . We prove
Simon Schieder’s conjecture identifying his bialgebra formed by the top compactly supported
cohomology of the intersections of opposite semi-infinite orbits with U (n ∨ ) (the universal
enveloping ...

Added: March 19, 2020

Finkelberg Michael, Fujita R., Representation Theory 2021 Vol. 25 P. 67-89

The convolution ring of loop rotation equivariant K-homology of the affine Grassmannian of GL(n) was identified with
a quantum unipotent cell of the loop group of SL(2) by Cautis and Williams. We identify the basis formed by
the classes of irreducible equivariant perverse coherent sheaves with the dual
canonical basis of the quantum unipotent cell. ...

Added: January 29, 2021

Braverman A., Michael Finkelberg, Nakajima H., Advances in Theoretical and Mathematical Physics 2021 Vol. 25 No. 4 P. 957-993

This is the third companion paper of [Part II]. When a gauge theory has
a flavor symmetry group, we construct a partial resolution of the Coulomb branch as a
variant of the definition. We identify the partial resolution with a partial resolution of
a generalized slice in the affine Grassmannian, Hilbert scheme of points, and resolved
Cherkis bow variety ...

Added: April 13, 2022

Finkelberg M. V., Braverman A., / Cornell University. Series arXiv "math". 2018.

In arXiv:1807.09038 we formulated a conjecture describing the derived category D-mod(Gr_GL(n)) of (all) D-modules on the affine Grassmannian of the group GL(n) as the category of ind-coherent sheaves on a certain stack (it is explained in loc. cit. that this conjecture "follows" naturally from some heuristic arguments involving 3-dimensional quantum field theory). In this paper we prove a ...

Added: December 3, 2018

Braverman A., Michael Finkelberg, Nakajima H., Journal fur die reine und angewandte Mathematik, Germany 2022 Vol. 2022 No. 787 P. 45-78

We deduce the Kazhdan–Lusztig conjecture on the multiplicities of simple modules over a simple complex Lie algebra in Verma modules in category O from the equivari-ant geometric Satake correspondence and the analysis of torus fixed points in zastava spaces. We make similar speculations for the affine Lie algebras and W-algebras. ...

Added: June 10, 2022

Braverman A., Michael Finkelberg, Travkin R., Communications in Number Theory and Physics 2022 Vol. 16 No. 4 P. 695-732

We prove an equivalence relating representations of a degenerate orthosymplectic supergroup with the category of SO(N − 1, C[[t]])-equivariant perverse sheaves on the affine Grassmannian of SON . We explain how this equivalence fits into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh. ...

Added: October 22, 2022

Finkelberg M. V., Goncharov E., / Cornell University. Series arXiv "math". 2019.

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 ...

Added: June 9, 2019

Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183-189

The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...

Added: January 28, 2020

Borzykh D., ЛЕНАНД, 2021

Книга представляет собой экспресс-курс по теории вероятностей в контексте начального курса эконометрики. В курсе в максимально доступной форме изложен тот минимум, который необходим для осознанного изучения начального курса эконометрики. Данная книга может не только помочь ликвидировать пробелы в знаниях по теории вероятностей, но и позволить в первом приближении выучить предмет «с нуля». При этом, благодаря доступности изложения и небольшому объему книги, ...

Added: February 20, 2021

В. Л. Попов, Математические заметки 2017 Т. 102 № 1 С. 72-80

Мы доказываем, что аффинно-треугольные подгруппы являются борелевскими подгруппами групп Кремоны. ...

Added: May 3, 2017

Красноярск : ИВМ СО РАН, 2013

Труды Пятой Международной конференции «Системный анализ и информационные технологии» САИТ-2013 (19–25 сентября 2013 г., г.Красноярск, Россия): ...

Added: November 18, 2013

Min Namkung, Younghun K., Scientific Reports 2018 Vol. 8 No. 1 P. 16915-1-16915-18

Sequential state discrimination is a strategy for quantum state discrimination of a sender’s quantum
states when N receivers are separately located. In this report, we propose optical designs that can
perform sequential state discrimination of two coherent states. For this purpose, we consider not
only binary phase-shifting-key (BPSK) signals but also general coherent states, with arbitrary prior
probabilities. Since ...

Added: November 16, 2020

Grines V., Gurevich E., Pochinka O., Russian Mathematical Surveys 2017 Vol. 71 No. 6 P. 1146-1148

In the paper a Palis problem on finding sufficient conditions on embedding of Morse-Smale diffeomorphisms in topological flow is discussed. ...

Added: May 17, 2017

Okounkov A., Aganagic M., Moscow Mathematical Journal 2017 Vol. 17 No. 4 P. 565-600

We associate an explicit equivalent descendent insertion to any relative insertion in quantum K-theory of Nakajima varieties.
This also serves as an explicit formula for off-shell Bethe eigenfunctions for general quantum loop algebras associated to quivers and gives the general integral solution to the corresponding quantum Knizhnik Zamolodchikov and dynamical q-difference equations. ...

Added: October 25, 2018