14,895 research outputs found
Crystal melting on toric surfaces
We study the relationship between the statistical mechanics of crystal
melting and instanton counting in N=4 supersymmetric U(1) gauge theory on toric
surfaces. We argue that, in contrast to their six-dimensional cousins, the two
problems are related but not identical. We develop a vertex formalism for the
crystal partition function, which calculates a generating function for the
dimension 0 and 1 subschemes of the toric surface, and describe the
modifications required to obtain the corresponding gauge theory partition
function.Comment: 30 pages; v2: references adde
Counting and computing regions of -decomposition: algebro-geometric approach
New methods for -decomposition analysis are presented. They are based on
topology of real algebraic varieties and computational real algebraic geometry.
The estimate of number of root invariant regions for polynomial parametric
families of polynomial and matrices is given. For the case of two parametric
family more sharp estimate is proven. Theoretic results are supported by
various numerical simulations that show higher precision of presented methods
with respect to traditional ones. The presented methods are inherently global
and could be applied for studying -decomposition for the space of parameters
as a whole instead of some prescribed regions. For symbolic computations the
Maple v.14 software and its package RegularChains are used.Comment: 16 pages, 8 figure
Convexity and Zariski decomposition structure
This is the first part of our work on Zariski decomposition structures, where
we study Zariski decompositions using Legendre-Fenchel type transforms. In this
way we define a Zariski decomposition for curve classes. This decomposition
enables us to develop the theory of the volume function for curves defined by
the second named author, yielding some fundamental positivity results for curve
classes. For varieties with special structures, the Zariski decomposition for
curve classes admits an interesting geometric interpretation.Comment: 38 pages; revised half of the previous arXiv:1507.04316 [math.AG
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