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].

scholarly journals Gromov–Witten Theory of Toric Birational Transformations

2019 ◽  
Vol 2020 (20) ◽  
pp. 7037-7072
Author(s):  
Pedro Acosta ◽  
Mark Shoemaker
Keyword(s):  
Asymptotic Expansion ◽  
Linear Transformation ◽  
Invariant Theory ◽  
Geometric Invariant Theory ◽  
Complete Intersections ◽  
Geometric Invariant ◽  
Genus Zero ◽  
Birational Transformations ◽  
Witten Theory ◽  
Wall Crossing

Abstract We investigate the effect of a general toric wall crossing on genus zero Gromov–Witten theory. Given two complete toric orbifolds $X_{+}$ and $X_{-}$ related by wall crossing under variation of geometric invariant theory quotients, we prove that their respective $I$-functions are related by linear transformation and asymptotic expansion. We use this comparison to deduce a similar result for birational complete intersections in $X_{+}$ and $X_{-}$. This extends the work of the previous authors in [2] to the case of complete intersections in toric varieties and generalizes some of the results of Coates–Iritani–Jiang [15] on the crepant transformation conjecture to the setting of non-zero discrepancy.

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.

Branched Coverings and Minimal Free Resolution for Infinite-Dimensional Complex Spaces

2003 ◽  
Vol 10 (1) ◽  
pp. 37-43
Author(s):  
E. Ballico
Keyword(s):  
Projective Space ◽  
Projective Spaces ◽  
Free Resolution ◽  
Complete Intersections ◽  
Vanishing Theorem ◽  
Sheaf Cohomology ◽  
Minimal Free Resolution ◽  
Branched Coverings ◽  
Infinite Dimensional ◽  
Complex Spaces

Abstract We consider the vanishing problem for higher cohomology groups on certain infinite-dimensional complex spaces: good branched coverings of suitable projective spaces and subvarieties with a finite free resolution in a projective space P(V ) (e.g. complete intersections or cones over finitedimensional projective spaces). In the former case we obtain the vanishing result for H 1. In the latter case the corresponding results are only conditional for sheaf cohomology because we do not have the corresponding vanishing theorem for P(V ).

scholarly journals Local BPS Invariants: Enumerative Aspects and Wall-Crossing

2018 ◽  
Vol 2020 (17) ◽  
pp. 5450-5475 ◽  
Author(s):  
Jinwon Choi ◽  
Michel van Garrel ◽  
Sheldon Katz ◽  
Nobuyoshi Takahashi
Keyword(s):  
Projective Space ◽  
Euler Characteristic ◽  
Moduli Spaces ◽  
Del Pezzo Surfaces ◽  
Arithmetic Genus ◽  
One Dimensional ◽  
Wall Crossing ◽  
Bps Invariants ◽  
Del Pezzo ◽  
Dimensional Projective Space

Abstract We study the BPS invariants for local del Pezzo surfaces, which can be obtained as the signed Euler characteristic of the moduli spaces of stable one-dimensional sheaves on the surface $S$. We calculate the Poincaré polynomials of the moduli spaces for the curve classes $\beta $ having arithmetic genus at most 2. We formulate a conjecture that these Poincaré polynomials are divisible by the Poincaré polynomials of $((-K_S).\beta -1)$-dimensional projective space. This conjecture motivates the upcoming work on log BPS numbers [8].

scholarly journals Hyperbolicity related problems for complete intersection varieties

2013 ◽  
Vol 150 (3) ◽  
pp. 369-395 ◽  
Author(s):  
Damian Brotbek
Keyword(s):  
Projective Space ◽  
Existence Theorem ◽  
Complete Intersection ◽  
Cotangent Bundle ◽  
Projective Variety ◽  
Complete Intersections ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Complex Projective ◽  
Intersection Surface

AbstractIn this paper we examine different problems regarding complete intersection varieties of high multidegree in a smooth complex projective variety. First we prove an existence theorem for jet differential equations that generalizes a theorem of Diverio. Then we show how one can deduce hyperbolicity for generic complete intersections of high multidegree and high codimension from the known results on hypersurfaces. Finally, motivated by a conjecture of Debarre, we focus on the positivity of the cotangent bundle of complete intersections, and prove some results towards this conjecture; among other things, we prove that a generic complete intersection surface of high multidegree in a projective space of dimension at least four has an ample cotangent bundle.

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.

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 LIOUVILLE THEORY AND SPECIAL COADJOINT VIRASORO ORBITS

1995 ◽  
Vol 10 (06) ◽  
pp. 785-799 ◽  
Author(s):  
A. GORSKY ◽  
A. JOHANSEN
Keyword(s):  
Quantum Mechanics ◽  
Type Theory ◽  
Vector Fields ◽  
Liouville Theory ◽  
Hamiltonian Reduction ◽  
Coadjoint Orbits ◽  
Point Particles ◽  
Witten Theory ◽  
Wrong Sign

We describe the Hamiltonian reduction of the coadjoint Kac–Moody orbits to the Virasoro coadjoint orbits explicitly in terms of the Lagrangian approach for the Wess–Zumino–Novikov–Witten theory. While a relation of the coadjoint Virasoro orbit Diff S1/ SL (2, R) to the Liouville theory has already been studied, we analyze the role of special coadjoint Virasoro orbits Diff [Formula: see text]corresponding to stabilizers generated by the vector fields with double zeros. The orbits with stabilizers with single zeros do not appear in the model. We find an interpretation of zeros xi of the vector field of stabilizer [Formula: see text] and additional parameters qi, i = 1, …, n, in terms of quantum mechanics for n-point particles on the circle. We argue that the special orbits are generated by insertions of "wrong sign" Liouville exponential into the path integral. The additional parmeters qi are naturally interpreted as accessory parameters for the uniformization map. Summing up the contributions of the special Virasoro orbits we get the integrable sinh–Gordon type theory.

scholarly journals Wall-crossing in genus zero Landau–Ginzburg theory

2017 ◽  
Vol 2017 (733) ◽  
Author(s):  
Dustin Ross ◽  
Yongbin Ruan
Keyword(s):  
Moduli Spaces ◽  
Homogeneous Polynomial ◽  
Genus Zero ◽  
Quantum Invariants ◽  
Rational Parameter ◽  
The Family ◽  
Wall Crossing

AbstractWe study a family of moduli spaces and corresponding quantum invariants introduced recently by Fan–Jarvis–Ruan. The family has a wall-and-chamber structure relative to a positive rational parameter ϵ. For a Fermat quasi-homogeneous polynomial

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.

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