### ?

## Coulomb branches of 3-dimensional gauge theories and related structures

Ch. 1. P. 1-52.

Braverman A., Michael Finkelberg

These are (somewhat informal) lecture notes for the CIME summer

school “Geometric Representation Theory and Gauge Theory” in June 2018. In

these notes we review the constructions and results of Braverman et al. (Adv Theor

Math Phys 22(5):1017–1147, 2018; Adv Theor Math Phys 23(1):75–166, 2019;

Adv Theor Math Phys 23(2):253–344, 2019) where a mathematical definition of

Coulomb branches of 3d N = 4 quantum gauge theories (of cotangent type)

is given, and also present a framework for studying some further mathematical

structures (e.g. categories of line operators in the corresponding topologically

twisted theories) related to these theories.

### In book

Braverman A., Michael Finkelberg, Negut A., Oblomkov A. Vol. 2248. , Switzerland : Springer, 2019

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, 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., 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, Negut A. et al., Switzerland : Springer, 2019

In the last 30 years a new pattern of interaction between mathematics and physics
emerged, in which the latter catalyzed the creation of new mathematical theories.
Most notable examples of this kind of interaction can be found in the theory of
moduli spaces. In algebraic geometry the theory of moduli spaces goes back at
least to Riemann, but they ...

Added: December 24, 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, 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

Braverman A., Michael Finkelberg, , in : Representation Theory and Algebraic Geometry. : Switzerland : Birkhauser/Springer, 2022. P. 133-149.

-- ...

Added: June 20, 2022

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, 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

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

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

Braverman A., Michael Finkelberg, Nakajima H., 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 rational maps from P^1 to the flag variety. In the framed case they are
slices in the affine Grassmannian and their generalization. In ...

Added: September 28, 2019

Dumanski I., Feigin E., Finkelberg M. V., / Cornell University. Series math "arxiv.org". 2020. No. 2003.12930.

We compute the spaces of sections of powers of the determinant line bundle on the spherical Schubert subvarieties of the Beilinson- Drinfeld affine Grassmannians. The answer is given in terms of global Demazure modules of the current Lie algebra. ...

Added: April 2, 2020

Braverman A., Finkelberg M. V., Nakajima H., / Cornell University. Series arXiv "math". 2016.

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 rational maps from ℂP1 to the flag variety. In the framed case they are slices in the affine Grassmannian and their generalization. In the appendix, written jointly with ...

Added: April 21, 2016

Bershtein M., Vargulevich A., Journal of Mathematical Physics 2022 Vol. 63 No. 6

It was conjectured by Belavin et al. [J. High Energy Phys. 2013(3), 35] that bosonization of a singular vector (in the Neveu–Schwarz sector) of the 𝒩=1N=1 super analog of the Virasoro algebra can be identified with the Uglov symmetric function. In this paper, we prove this conjecture. We also extend this result to the Ramond sector of the 𝒩=1N=1 super-Virasoro algebra. ...

Added: August 18, 2022

Krylov V., Perunov I., Advances in Mathematics 2021 Vol. 392 No. 3 Article 108034

Let $G$ be a connected reductive complex algebraic group with a maximal torus $T$. We denote by $\La$ the coweight lattice of $T$.
Let $\La^+ \subset \La$ be the submonoid of dominant coweights. For $\la \in \La^+,\,\mu \in \La,\,\mu \leqslant \la$, in "Coulomb branches of
3d $\mathcal{N}=4$ quiver gauge theories and slices
in the ...

Added: January 10, 2022

Loktev S., Kato S., / Cornell University. Series arXiv "math". 2017. No. 1712.03508.

We construct a filtration on integrable highest weight module of an affine Lie algebra whose adjoint graded quotient is a direct sum of global Weyl modules. We show that the graded multiplicity of each Weyl module there is given by a corresponding level-restricted Kostka polynomial. This leads to an interpretation of level-restricted Kostka polynomials as ...

Added: December 11, 2017

Losiakov V., Тютин И. В., Теоретическая и математическая физика 2019 Т. 199 № 1 С. 142-153

In the framework of general relativity theory, we consider a space-time whose metric depends on only one coordinate and time. We choose a gauge class such that all the constraint conditions of the theory of gravity as a gauge theory are explicitly solved and construct a Hamiltonian depending only on dynamical physical variables (gravitons). We ...

Added: October 31, 2019

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

Marchuk N., М. : Издательская группа URSS, 2023

В книге изучаются релятивистские уравнения теории поля, в частности рассматриваются свойства ковариантности и симметрии уравнений Дирака—Максвелла и Дирака—Янга—Миллса. Вводится ряд новых систем уравнений, называемых модельными уравнениями теории поля. Эти системы уравнений воспроизводят основные свойства стандартных систем уравнений теории поля. В то же время модельные уравнения имеют ряд отличий от стандартных уравнений теории поля, в частности обладают новой внутренней симметрией ...

Added: January 3, 2024