scholarly journals Quantum (non-commutative) toric geometry: Foundations

2021 ◽  
Vol 391 ◽  
pp. 107945
Author(s):  
Ludmil Katzarkov ◽  
Ernesto Lupercio ◽  
Laurent Meersseman ◽  
Alberto Verjovsky
Keyword(s):  
Toric Geometry

scholarly journals Branes at Bbb C4/Γ singularity from toric geometry

1999 ◽  
Vol 1999 (04) ◽  
pp. 012-012 ◽  
Author(s):  
Changhyun Ahn ◽  
Hoil Kim
Keyword(s):  
Toric Geometry

Toric Geometry ofSL2(ℂ) Free Group Character Varieties from Outer Space

2018 ◽  
Vol 70 (2) ◽  
pp. 354-399 ◽  
Author(s):  
Christopher Manon
Keyword(s):  
Free Group ◽  
Outer Space ◽  
Integrable Hamiltonian System ◽  
Outer Automorphism ◽  
Character Variety ◽  
Toric Geometry ◽  
Character Varieties ◽  
Simplicial Space ◽  
Proper Map ◽  
Divisorial Valuations

AbstractCuller and Vogtmann defined a simplicial spaceO(g), calledouter space, to study the outer automorphism group of the free groupFg. Using representation theoretic methods, we give an embedding ofO(g) into the analytification of X(Fg,SL2(ℂ)), theSL2(ℂ) character variety ofFg, reproving a result of Morgan and Shalen. Then we show that every pointvcontained in a maximal cell ofO(g) defines a flat degeneration of X(Fg,SL2(ℂ)) to a toric varietyX(PΓ). We relate X(Fg,SL2(ℂ)) andX(v) topologically by showing that there is a surjective, continuous, proper map Ξv:X(Fg,SL2(ℂ)) →X(v). We then show that this map is a symplectomorphism on a dense open subset of X(Fg, SL2(ℂ)) with respect to natural symplectic structures on X(Fg, SL2(ℂ)) andX(v). In this way, we construct an integrable Hamiltonian system in X(Fg, SL2(ℂ)) for each point in a maximal cell ofO(g), and we show that eachvdefines a topological decomposition of X(Fg, SL2(ℂ)) derived from the decomposition ofX(PΓ) by its torus orbits. Finally, we show that the valuations coming from the closure of a maximal cell inO(g) all arise as divisorial valuations built from an associated projective compactification of X(Fg, SL2(ℂ)).

scholarly journals Computational Tools for Cohomology of Toric Varieties

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Ralph Blumenhagen ◽  
Benjamin Jurke ◽  
Thorsten Rahn
Keyword(s):  
Software Package ◽  
Computational Algorithm ◽  
Toric Varieties ◽  
Dual Pair ◽  
Target Space ◽  
Toric Geometry ◽  
Computational Tools ◽  
String Compactifications ◽  
String Models

Novel nonstandard techniques for the computation of cohomology classes on toric varieties are summarized. After an introduction of the basic definitions and properties of toric geometry, we discuss a specific computational algorithm for the determination of the dimension of line-bundle-valued cohomology groups on toric varieties. Applications to the computation of chiral massless matter spectra in string compactifications are discussed, and using the software packagecohomCalg, its utility is highlighted on a new target space dual pair of(0,2)heterotic string models.

scholarly journals Probing non-toric geometry with rotating membranes

2010 ◽  
Vol 838 (1-2) ◽  
pp. 238-252 ◽  
Author(s):  
Jung Hun Lee ◽  
Sunchang Kim ◽  
Jongwook Kim ◽  
Nakwoo Kim
Keyword(s):  
Toric Geometry

scholarly journals Toric geometry and the dual of ℐ-extremization

2019 ◽  
Vol 2019 (6) ◽  
Author(s):  
Jerome P. Gauntlett ◽  
Dario Martelli ◽  
James Sparks
Keyword(s):  
Toric Geometry

scholarly journals Secant Cumulants and Toric Geometry

2014 ◽  
Author(s):  
M. Micha ek ◽  
L. Oeding ◽  
P. Zwiernik
Keyword(s):  
Toric Geometry

scholarly journals Toric Geometry of Spin(7)-Manifolds

2019 ◽  
Author(s):  
Thomas Bruun Madsen ◽  
Andrew Swann
Keyword(s):  
Symmetry Group ◽  
Orbit Space ◽  
Moment Map ◽  
Toric Geometry ◽  
Hamiltonian Action ◽  
Trivalent Graph ◽  
Dense Set

Abstract We study $ \operatorname{Spin}(7) $-manifolds with an effective multi-Hamiltonian action of a four-torus. On an open dense set, we provide a Gibbons–Hawking type ansatz that describes such geometries in terms of a symmetric $ 4\times 4 $-matrix of functions. This description leads to the 1st known $ \operatorname{Spin}(7) $-manifolds with a rank $ 4 $ symmetry group and full holonomy. We also show that the multi-moment map exhibits the full orbit space topologically as a smooth four-manifold, containing a trivalent graph in $ \mathbb{R}^4 $ as the image of the set of the special orbits.

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