Seiberg-Witten invariants, the topological degree and wall crossing formula

2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Maciej Starostka
Keyword(s):  
Simple Proof ◽  
Topological Degree ◽  
Witten Invariant ◽  
Homotopy Invariance ◽  
Finite Dimensional ◽  
Wall Crossing ◽  
Wall Crossing Formula

AbstractFollowing S. Bauer and M. Furuta we investigate finite dimensional approximations of a monopole map in the case b 1 = 0. We define a certain topological degree which is exactly equal to the Seiberg-Witten invariant. Using homotopy invariance of the topological degree a simple proof of the wall crossing formula is derived.

scholarly journals A wall-crossing formula for degrees of Real central projections

2014 ◽  
Vol 25 (04) ◽  
pp. 1450038 ◽  
Author(s):  
Christian Okonek ◽  
Andrei Teleman
Keyword(s):  
Algebraic Geometry ◽  
Projective Space ◽  
Pole Placement ◽  
Real Projective Space ◽  
Central Projections ◽  
Wall Crossing ◽  
Subspace Problem ◽  
Wall Crossing Formula ◽  
Real Subspace ◽  
Crucial Assumption

The main result is a wall-crossing formula for central projections defined on submanifolds of a Real projective space. Our formula gives the jump of the degree of such a projection when the center of the projection varies. The fact that the degree depends on the projection is a new phenomenon, specific to Real algebraic geometry. We illustrate this phenomenon in many interesting situations. The crucial assumption on the class of maps we consider is relative orientability, a condition which allows us to define a ℤ-valued degree map in a coherent way. We end the article with several examples, e.g. the pole placement map associated with a quotient, the Wronski map, and a new version of the Real subspace problem.

On the Moduli Space of a Spherical Polygonal Linkage

1999 ◽  
Vol 42 (3) ◽  
pp. 307-320 ◽  
Author(s):  
Michael Kapovich ◽  
John J. Millson
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Level Sets ◽  
Morse Function ◽  
Great Circle ◽  
Winding Number ◽  
Wall Crossing ◽  
Polygonal Linkage ◽  
Spherical Linkages ◽  
Wall Crossing Formula

AbstractWe give a “wall-crossing” formula for computing the topology of the moduli space of a closed n-gon linkage on 𝕊2. We do this by determining the Morse theory of the function ρn on the moduli space of n-gon linkages which is given by the length of the last side—the length of the last side is allowed to vary, the first (n − 1) side-lengths are fixed. We obtain a Morse function on the (n − 2)-torus with level sets moduli spaces of n-gon linkages. The critical points of ρn are the linkages which are contained in a great circle. We give a formula for the signature of the Hessian of ρn at such a linkage in terms of the number of back-tracks and the winding number. We use our formula to determine the moduli spaces of all regular pentagonal spherical linkages.

scholarly journals An extention of Nomizu’s Theorem –A user’s guide–

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Hisashi Kasuya
Keyword(s):  
Lie Group ◽  
Simple Proof ◽  
Graded Algebra ◽  
De Rham Cohomology ◽  
Differential Graded Algebra ◽  
Finite Dimensional ◽  
De Rham ◽  
Simply Connected ◽  
Complex Valued ◽  
Solvable Lie Group

AbstractFor a simply connected solvable Lie group G with a lattice Γ, the author constructed an explicit finite-dimensional differential graded algebra A*Γ which computes the complex valued de Rham cohomology H*(Γ\G, C) of the solvmanifold Γ\G. In this note, we give a quick introduction to the construction of such A*Γ including a simple proof of H*(A*Γ) ≅ H*(Γ\G, C).

The wall-crossing behavior for Bridgeland’s stability conditions on abelian and K3 surfaces

2018 ◽  
Vol 2018 (735) ◽  
pp. 1-107 ◽  
Author(s):  
Hiroki Minamide ◽  
Shintarou Yanagida ◽  
Kōta Yoshioka
Keyword(s):  
Stability Condition ◽  
Derived Category ◽  
The Other ◽  
Abelian Surface ◽  
Stability Conditions ◽  
Abelian Surfaces ◽  
Coherent Sheaves ◽  
Fundamental Properties ◽  
Wall Crossing ◽  
Wall Crossing Formula

AbstractThe wall-crossing behavior for Bridgeland’s stability conditions on the derived category of coherent sheaves on K3 or abelian surface is studied. We introduce two types of walls. One is called the wall for categories, where thet-structure encoded by stability condition is changed. The other is the wall for stabilities, where stable objects with prescribed Mukai vector may get destabilized. Some fundamental properties of walls and chambers are studied, including the behavior under Fourier–Mukai transforms. A wall-crossing formula of the counting of stable objects will also be derived. As an application, we will explain previous results on the birational maps induced by Fourier–Mukai transforms on abelian surfaces. These transformations turns out to coincide with crossing walls of certain property.

scholarly journals General wall crossing formula

1995 ◽  
Vol 2 (6) ◽  
pp. 797-810 ◽  
Author(s):  
T. J. Li ◽  
A. Liu
Keyword(s):  
Wall Crossing ◽  
Wall Crossing Formula

scholarly journals Wall-crossing in genus-zero hybrid theory

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Emily Clader ◽  
Dustin Ross
Keyword(s):  
Projective Space ◽  
Hybrid Model ◽  
Type Theory ◽  
Singularity Theory ◽  
Complete Intersections ◽  
Genus Zero ◽  
Witten Theory ◽  
Wall Crossing ◽  
Wall Crossing Formula ◽  
Hybrid Theory

Abstract The hybrid model is the Landau–Ginzburg-type theory that is expected, via the Landau–Ginzburg/ Calabi–Yau correspondence, to match the Gromov–Witten theory of a complete intersection in weighted projective space. We prove a wall-crossing formula exhibiting the dependence of the genus-zero hybrid model on its stability parameter, generalizing the work of [21] for quantum singularity theory and paralleling the work of Ciocan-Fontanine–Kim [7] for quasimaps. This completes the proof of the genus-zero Landau– Ginzburg/Calabi–Yau correspondence for complete intersections of hypersurfaces of the same degree, as well as the proof of the all-genus hybrid wall-crossing [11].

The Topological Degree of A-Proper Maps

1971 ◽  
Vol 23 (3) ◽  
pp. 403-412 ◽  
Author(s):  
H. Ship Fah Wong
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Topological Degree ◽  
Fixed Point Theorems ◽  
Brouwer Degree ◽  
Finite Dimensional ◽  
Essential Idea ◽  
Proper Maps ◽  
Finite Dimensional Case

Recently several fixed-point theorems have been proved for new classes of non-compact maps between Banach spaces. First, Petryshyn [15] generalized the concept of compact and quasi-compact maps when he introduced the P-compact maps and proved a fixed-point theorem for this class of maps. Then in [6], de Figueiredo defined the notion of G-operator to unify his own work on fixed-point theorems and that of Petryshyn. He also proved that the class of G-operators was a fairly large one.We notice the following facts: (i) The essential idea in the above cases is that if certain finite-dimensional “approximations” of the map have fixed points, then the map has a fixed point; (ii) One of the tools used in proving fixed-point theorems in the finite-dimensional case is the Brouwer degree and its generalization to maps of the type Identity + Compact in [8].

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