scholarly journals Instantons and vortices on noncommutative toric varieties

2014 ◽  
Vol 26 (09) ◽  
pp. 1430008 ◽  
Author(s):  
Lucio S. Cirio ◽  
Giovanni Landi ◽  
Richard J. Szabo
Keyword(s):  
Moduli Spaces ◽  
Toric Varieties ◽  
Partition Functions ◽  
Toric Geometry ◽  
Monoidal Categories ◽  
Klein Quadric ◽  
Noncommutative Algebraic Geometry ◽  
Fundamental Matter ◽  
Braided Monoidal Categories ◽  
Noncommutative Gauge Theories

We elaborate on the quantization of toric varieties by combining techniques from toric geometry, isospectral deformations and noncommutative geometry in braided monoidal categories, and the construction of instantons thereon by combining methods from noncommutative algebraic geometry and a quantized twistor theory. We classify the real structures on a toric noncommutative deformation of the Klein quadric and use this to derive a new noncommutative four-sphere which is the unique deformation compatible with the noncommutative twistor correspondence. We extend the computation of equivariant instanton partition functions to noncommutative gauge theories with both adjoint and fundamental matter fields, finding agreement with the classical results in all instances. We construct moduli spaces of noncommutative vortices from the moduli of invariant instantons, and derive corresponding equivariant partition functions which also agree with those of the classical limit.

scholarly journals Quasitriangular Hopf group algebras and braided monoidal categories

2014 ◽  
Vol 64 (4) ◽  
pp. 893-909
Author(s):  
Shiyin Zhao ◽  
Jing Wang ◽  
Hui-Xiang Chen
Keyword(s):  
Group Algebras ◽  
Monoidal Categories ◽  
Braided Monoidal Categories

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 On toric geometry and K-stability of Fano varieties

2021 ◽  
Vol 8 (19) ◽  
pp. 548-577
Author(s):  
Anne-Sophie Kaloghiros ◽  
Andrea Petracci
Keyword(s):  
Moduli Spaces ◽  
Deformation Theory ◽  
The Other ◽  
Fano Varieties ◽  
Toric Geometry ◽  
Content Type ◽  
Closed Point ◽  
Toric Fano Varieties

We present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. First, we present two examples of K-polystable toric Fano 3 3 -fold with obstructed deformations. In one case, the K-moduli spaces and stacks are reducible near the closed point associated to the toric Fano 3 3 -fold, while in the other they are non-reduced near the closed point associated to the toric Fano 3 3 -fold. Second, we study K-stability of the general members of two deformation families of smooth Fano 3 3 -folds by building degenerations to K-polystable toric Fano 3 3 -folds.

scholarly journals Counting curves on surfaces

2017 ◽  
Vol 28 (02) ◽  
pp. 1750012
Author(s):  
Norman Do ◽  
Musashi A. Koyama ◽  
Daniel V. Mathews
Keyword(s):  
Moduli Spaces ◽  
Symplectic Geometry ◽  
Differential Forms ◽  
Combinatorial Problem ◽  
Partition Functions ◽  
Low Dimensional Topology ◽  
Moduli Spaces Of Curves ◽  
Finite Set ◽  
Curves On Surfaces ◽  
Low Dimensional

We consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology: for a compact surface [Formula: see text], with a finite set of points [Formula: see text] fixed on its boundary, how many configurations of disjoint arcs are there on [Formula: see text] whose boundary is [Formula: see text]? We find that this enumerative problem, counting curves on surfaces, has a rich structure. We show that such curve counts obey an effective recursion, in the general spirit of topological recursion, and exhibit quasi-polynomial behavior. This “elementary curve-counting” is in fact related to a more advanced notion of “curve-counting” from algebraic geometry or symplectic geometry. The asymptotics of this enumerative problem are closely related to the asymptotics of volumes of moduli spaces of curves, and the quasi-polynomials governing the enumerative problem encode intersection numbers on moduli spaces. Among several other results, we show that generating functions and differential forms for these curve counts exhibit structure that is reminiscent of the mathematical physics of free energies, partition functions and quantum curves.

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