scholarly journals The Complex Orthogonal Gelfand–Zeitlin System

2019 ◽  
Author(s):  
Mark Colarusso ◽  
Sam Evens
Keyword(s):  
Integrable System ◽  
Algebraic Groups ◽  
Moment Map ◽  
Linear Case ◽  
General Linear ◽  
Regular Elements ◽  
Strongly Regular ◽  
Adjoint Orbits ◽  
The Moment ◽  
Orthogonal Case

Abstract In this paper, we use the theory of algebraic groups to prove a number of new and fundamental results about the orthogonal Gelfand–Zeitlin system. We show that the moment map (orthogonal Kostant–Wallach map) is surjective and simplify criteria of Kostant and Wallach for an element to be strongly regular. We further prove the integrability of the orthogonal Gelfand–Zeitlin system on regular adjoint orbits and describe the generic flows of the integrable system. We also study the nilfibre of the moment map and show that in contrast to the general linear case it contains no strongly regular elements. This extends results of Kostant, Wallach, and Colarusso from the general linear case to the orthogonal case.

scholarly journals On collective complete integrability according to the method of Thimm

1983 ◽  
Vol 3 (2) ◽  
pp. 219-230 ◽  
Author(s):  
Victor Guillemin ◽  
Shlomo Sternberg
Keyword(s):  
Integrable System ◽  
Symplectic Manifold ◽  
Lie Group ◽  
Geodesic Flow ◽  
Necessary Conditions ◽  
Complete Integrability ◽  
Moment Map ◽  
The Real ◽  
Completely Integrable System ◽  
The Moment

AbstractLet G be a Lie group acting in Hamiltonian fashion on a symplectic manifold M with moment map Φ:M → g*. A function of the form ƒ∘Φ where ƒ is a function on g* is called ‘collective’. We obtain necessary conditions on the G action for there to exist enough Poisson commuting functions on g* so that the corresponding collective functions on M form a completely integrable system. For the case G = O(n) or U(n) these conditions are sufficient. This explains Thimm's proof [17] of the complete integrability of the geodesic flow on the real and complex grassmanians. We also discuss related questions in the geometry of the moment map.

Kählerian potentials and convexity properties of the moment map

1996 ◽  
Vol 126 (1) ◽  
pp. 65-84 ◽  
Author(s):  
Peter Heinzner ◽  
Alan Huckleberry
Keyword(s):  
Moment Map ◽  
Convexity Properties ◽  
The Moment

scholarly journals Bootstrapping Coulomb and Higgs branch operators

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Aleix Gimenez-Grau ◽  
Pedro Liendo
Keyword(s):  
Recent Work ◽  
Moment Map ◽  
Mixed System ◽  
Flavor Symmetry ◽  
Coulomb Branch ◽  
Conformal Bootstrap ◽  
General Constraints ◽  
Superconformal Theories ◽  
The Moment

Abstract We apply the numerical conformal bootstrap to correlators of Coulomb and Higgs branch operators in 4d$$ \mathcal{N} $$ N = 2 superconformal theories. We start by revisiting previous results on single correlators of Coulomb branch operators. In particular, we present improved bounds on OPE coefficients for some selected Argyres-Douglas models, and compare them to recent work where the same cofficients were obtained in the limit of large r charge. There is solid agreement between all the approaches. The improved bounds can be used to extract an approximate spectrum of the Argyres-Douglas models, which can then be used as a guide in order to corner these theories to numerical islands in the space of conformal dimensions. When there is a flavor symmetry present, we complement the analysis by including mixed correlators of Coulomb branch operators and the moment map, a Higgs branch operator which sits in the same multiplet as the flavor current. After calculating the relevant superconformal blocks we apply the numerical machinery to the mixed system. We put general constraints on CFT data appearing in the new channels, with particular emphasis on the simplest Argyres-Douglas model with non-trivial flavor symmetry.

Inverse complements and strongly unit regular elements

2021 ◽  
pp. 2250190
Author(s):  
Umashankara Kelathaya ◽  
Savitha Varkady ◽  
Manjunatha Prasad Karantha
Keyword(s):  
Partial Order ◽  
Regular Element ◽  
Generalized Inverse ◽  
Generalized Inverses ◽  
Order Unit ◽  
Regular Elements ◽  
Strongly Regular ◽  
Minus Partial Order

In this paper, the notion of “strongly unit regular element”, for which every reflexive generalized inverse is associated with an inverse complement, is introduced. Noting that every strongly unit regular element is unit regular, some characterizations of unit regular elements are obtained in terms of inverse complements and with the help of minus partial order. Unit generalized inverses of given unit regular element are characterized as sum of reflexive generalized inverses and the generators of its annihilators. Surprisingly, it has been observed that the class of strongly regular elements and unit regular elements are the same. Also, several classes of generalized inverses are characterized in terms of inverse complements.

scholarly journals Brick Manifolds and Toric Varieties of Brick Polytopes

10.37236/5038 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Laura Escobar
Keyword(s):  
Toric Variety ◽  
Toric Varieties ◽  
Moment Map ◽  
Twisted Product ◽  
General Fiber ◽  
Moment Polytope ◽  
The Moment ◽  
Subword Complex ◽  
Do So

Bott-Samelson varieties are a twisted product of $\mathbb{C}\mathbb{P}^1$'s with a map into $G/B$. These varieties are mostly studied in the case in which the map into $G/B$ is birational to the image; however in this paper we study a fiber of this map when it is not birational. We prove that in some cases the general fiber, which we christen a brick manifold, is a toric variety. In order to do so we use the moment map of a Bott-Samelson variety to translate this problem into one in terms of the "subword complexes" of Knutson and Miller. Pilaud and Stump realized certain subword complexes as the dual of the boundary of a polytope which generalizes the brick polytope defined by Pilaud and Santos. For a nice family of words, the brick polytope is the generalized associahedron realized by Hohlweg, Lange and Thomas. These stories connect in a nice way: we show that the moment polytope of the brick manifold is the brick polytope. In particular, we give a nice description of the toric variety of the associahedron. We give each brick manifold a stratification dual to the subword complex. In addition, we relate brick manifolds to Brion's resolutions of Richardon varieties.

The moment map

1994 ◽  
pp. 144-189
Author(s):  
David Mumford ◽  
John Fogarty ◽  
Frances Kirwan
Keyword(s):  
Moment Map ◽  
The Moment
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