scholarly journals T-duality and homological mirror symmetry for toric varieties

2012 ◽  
Vol 229 (3) ◽  
pp. 1873-1911 ◽  
Author(s):  
Bohan Fang ◽  
Chiu-Chu Melissa Liu ◽  
David Treumann ◽  
Eric Zaslow
Keyword(s):  
Mirror Symmetry ◽  
Toric Varieties ◽  
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 Quantum Cohomology and Closed-String Mirror Symmetry for Toric Varieties

2020 ◽  
Vol 71 (2) ◽  
pp. 395-438
Author(s):  
Jack Smith
Keyword(s):  
Toric Variety ◽  
Mirror Symmetry ◽  
Complex Structure ◽  
Toric Varieties ◽  
Quantum Cohomology ◽  
Closed String ◽  
Generalized Jacobian ◽  
Novikov Ring

Abstract We give a short new computation of the quantum cohomology of an arbitrary smooth (semiprojective) toric variety $X$, by showing directly that the Kodaira–Spencer map of Fukaya–Oh–Ohta–Ono defines an isomorphism onto a suitable Jacobian ring. In contrast to previous results of this kind, $X$ need not be compact. The proof is based on the purely algebraic fact that a class of generalized Jacobian rings associated to $X$ are free as modules over the Novikov ring. When $X$ is monotone the presentation we obtain is completely explicit, using only well-known computations with the standard complex structure.

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
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