scholarly journals Homological Mirror Symmetry and Tropical Geometry

2014 ◽  
Keyword(s):  
Mirror Symmetry ◽  
Tropical Geometry ◽  
Homological Mirror Symmetry

scholarly journals Homological mirror symmetry for higher-dimensional pairs of pants

2020 ◽  
Vol 156 (7) ◽  
pp. 1310-1347
Author(s):  
Yankı Lekili ◽  
Alexander Polishchuk
Keyword(s):  
Mirror Symmetry ◽  
Derived Category ◽  
Affine Variety ◽  
Fukaya Category ◽  
Homological Mirror Symmetry ◽  
Endomorphism Algebra ◽  
Generating Set ◽  
Fukaya Categories ◽  
Higher Dimensional ◽  
Abelian Covers

Using Auroux’s description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of $k+1$ generic hyperplanes in $\mathbb{CP}^{n}$, for $k\geqslant n$, with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of $n+2$ generic hyperplanes in $\mathbb{C}P^{n}$ ($n$-dimensional pair of pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of the derived category of the singular affine variety $x_{1}x_{2}\ldots x_{n+1}=0$. By localizing, we deduce that the (fully) wrapped Fukaya category of the $n$-dimensional pair of pants is equivalent to the derived category of $x_{1}x_{2}\ldots x_{n+1}=0$. We also prove similar equivalences for finite abelian covers of the $n$-dimensional pair of pants.

scholarly journals Moduli Space in Homological Mirror Symmetry

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Matsuo Sato
Keyword(s):  
Elliptic Curve ◽  
Configuration Space ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Mirror Symmetry ◽  
Feynman Diagrams ◽  
Holomorphic Curves ◽  
Homological Mirror Symmetry

We prove that the moduli space of the pseudo holomorphic curves in the A-model on a symplectic torus is homeomorphic to a moduli space of Feynman diagrams in the configuration space of the morphisms in the B-model on the corresponding elliptic curve. These moduli spaces determine the A∞ structure of the both models.

scholarly journals Double Solids, Categories and Non-Rationality

2013 ◽  
Vol 57 (1) ◽  
pp. 145-173 ◽  
Author(s):  
Atanas Iliev ◽  
Ludmil Katzarkov ◽  
Victor Przyjalkowski
Keyword(s):  
Category Theory ◽  
Mirror Symmetry ◽  
Homology Group ◽  
Hochschild Homology ◽  
Singular Points ◽  
Double Covering ◽  
Homological Mirror Symmetry ◽  
New Approach ◽  
The Third

AbstractThis paper suggests a new approach to questions of rationality of 3-folds based on category theory. Following work by Ballard et al., we enhance constructions of Kuznetsov by introducing Noether–Lefschetz spectra: an interplay between Orlov spectra and Hochschild homology. The main goal of this paper is to suggest a series of interesting examples where the above techniques might apply. We start by constructing a sextic double solid X with 35 nodes and torsion in H3(X, ℤ). This is a novelty: after the classical example of Artin and Mumford, this is the second example of a Fano 3-fold with a torsion in the third integer homology group. In particular, X is non-rational. We consider other examples as well: V10 with 10 singular points, and the double covering of a quadric ramified in an octic with 20 nodal singular points. After analysing the geometry of their Landau–Ginzburg models, we suggest a general non-rationality picture based on homological mirror symmetry and category theory.

scholarly journals Homological mirror symmetry for punctured spheres

2013 ◽  
Vol 26 (4) ◽  
pp. 1051-1083 ◽  
Author(s):  
Mohammed Abouzaid ◽  
Denis Auroux ◽  
Alexander I. Efimov ◽  
Ludmil Katzarkov ◽  
Dmitri Orlov
Keyword(s):  
Mirror Symmetry ◽  
Homological Mirror Symmetry

scholarly journals Self-organized criticality and pattern emergence through the lens of tropical geometry

2018 ◽  
Vol 115 (35) ◽  
pp. E8135-E8142 ◽  
Author(s):  
N. Kalinin ◽  
A. Guzmán-Sáenz ◽  
Y. Prieto ◽  
M. Shkolnikov ◽  
V. Kalinina ◽  
...  
Keyword(s):  
Mirror Symmetry ◽  
Tropical Geometry ◽  
Scaling Limit ◽  
Reaction Diffusion ◽  
Auction Theory ◽  
Discrete Models ◽  
Self Organized ◽  
Reaction Diffusion Model ◽  
Pure Mathematics ◽  
Self Organized Criticality

Tropical geometry, an established field in pure mathematics, is a place where string theory, mirror symmetry, computational algebra, auction theory, and so forth meet and influence one another. In this paper, we report on our discovery of a tropical model with self-organized criticality (SOC) behavior. Our model is continuous, in contrast to all known models of SOC, and is a certain scaling limit of the sandpile model, the first and archetypical model of SOC. We describe how our model is related to pattern formation and proportional growth phenomena and discuss the dichotomy between continuous and discrete models in several contexts. Our aim in this context is to present an idealized tropical toy model (cf. Turing reaction-diffusion model), requiring further investigation.

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