Aspects of Mirror Symmetry in Chiba 2019
Dates: 10 and 11 July 2019
Venue: Department of Science Bldg. 1, 1st Floor, 107 / 3rd Floor, 320 (11th morning)
Chiba University [ access | campus map ]
Speakers
Yu-Wei Fan (UC Berkeley)
Hansol Hong (Yonsei)
Doğancan Karabaş (IBS)
Kazushi Kobayashi (Chiba)
Yat-Hin Suen (IBS)
Program
10 July:
13:00-14:30 Hong: Enlarging local mirrors from Floer theory of Lagrangians 1
15:00-16:30 Suen: Reconstruction of $T_{¥mathbb{P}_2}$ via tropical Lagrangian multi-section
16:40-18:10 Kobayashi: The bijectivity of mirror functors on tori
11 July:
10:30-12:00 Free discussion
13:00-14:30 Hong: Enlarging local mirrors from Floer theory of Lagrangians 2
15:00-16:30 Fan: Systolic inequality for K3 surfaces via stability conditions
16:40-18:10 Karabaş: Wrapped Fukaya category of some rational homology balls via microlocal sheaf theory
Title and Abstract
Hansol Hong (Yonsei)
Title: Enlarging local mirrors from Floer theory of Lagrangians
Abstract: Maurer-Cartan solutions in Lagrangian Floer complex of a fixed Lagrangian $L$ give rise to a local mirror that encodes a mirror geometry around $L$. Furthermore, such a construction naturally induces a functor from the Fukaya category to the mirror matrix factorization category (or derived categories of coherent sheaves) of the local mirror. However, the functor can not detect the objects in the Fukaya category that do not intersect $L$, which shows the locality of the construction. To remedy, one can consider Maurer--Cartan deformation in noncommutative directions. In this case, the corresponding mirror typically appears as a quiver algebra. In the first talk I will explain general features of mirror spaces arising in this way, and exhibit a few interesting examples as applications.
Another way to enlarge the above local mirrors is to consider several different Lagrangians and their Maurer--Cartan spaces. The condition that two Lagrangians define an isomorphic object in the Fukaya category induces a gluing map between associated mirror charts. The method applies to recover the well-known wall-crossing formula for generic singular torus fibers. Starting from a general gluing scheme, I shall investigate typical situations of wall-crossing in various dimensions with an application to $2$-plane Grassmanians.
Yu-Wei Fan (UC Berkeley)
Title: Systolic inequality for K3 surfaces via stability conditions
Abstract: We propose a natural generalization of Loewner's torus systolic inequality from the perspective of Calabi--Yau geometry. We ask whether the square of the minimum volume of special Lagrangians in a Calabi--Yau manifold is bounded above by the total volume of the Calabi--Yau. We introduce the categorical analogues of systole and volume in terms of Bridgeland stability conditions, which enables us to formulate the mirror question under mirror symmetry. Finally, we give an affirmative answer to the mirror question for K3 surfaces of Picard rank one.
Doğancan Karabaş (IBS)
Title: Wrapped Fukaya category of some rational homology balls via microlocal sheaf theory
Abstract: It is shown by Kashiwara and Schapira (1980s) that for every constructible sheaf on a smooth manifold, one can construct a closed conic Lagrangian subset of its cotangent bundle, called the microsupport of?the sheaf. This eventually led to the equivalence of the category of constructible sheaves on a manifold and the Fukaya category of its cotangent bundle by the work of Nadler and Zaslow (2006), and Ganatra, Pardon, and Shende (2018) for partially wrapped Fukaya categories. One can try to generalise this and conjecture that Fukaya category of a Weinstein manifold can be given by constructible (microlocal) sheaves associated with its skeleton. In this talk, I will explain these concepts and con firm the conjecture for a family of Weinstein manifolds which are certain quotients of An-Milnor fibres. I will outline the computation of their wrapped Fukaya categories and microlocal sheaves on their skeleta, called pinwheels.
Kazushi Kobayashi (Chiba)
Title: The bijectivity of mirror functors on tori
Abstract: By the SYZ construction, a mirror pair $(X, X^¥wedge)$ of a complex torus $X$ and a mirror partner $X^¥wedge$ of the complex torus $X$ is described as the special Lagrangian torus fibrations $X¥to B$ and $X^¥wedge¥to B$ on the same base space $B$. Then, by the SYZ transform, we can construct a simple projectively flat bundle on $X$ from each affine Lagrangian multi section of $X^¥wedge¥to B$ with a unitary local system along it. However, there are non-unique choices of transition functions of it, and this fact actually causes difficulties when we try to construct a functor between the symplectic geometric category and the complex geometric category. In the present paper, by solving this problem, we prove that there exists a bijection between the set of the isomorphism classes of their objects.
Yat-Hin Suen (IBS)
Reconstruction of $T_{¥mathbb{P}^2}$ via tropical Lagrangian multi-section
Abstract: In this talk, I am going to talk about the reconstruction problem of the holomorphic tangent bundle $T_{¥mathbb{P}^2}$ of the complex projective plane. I will introduce the notion of tropical Lagrangian multi-section and cook up one from a family of Hermitian metrics defined on $T_{¥mathbb{P}^2}$. Then I will perform the reconstruction of $T_{¥mathbb{P}^2}$ from this tropical Lagrangian multi-section. Walling-crossing phenomenon will occur in the reconstruction process.
Organizers
Masahiro Futaki (Chiba)
Hiroshige Kajiura (Chiba)
Atsushi Kanazawa (Kyoto)
Supported by:
JSPS Grant-in-aid (C) 18K03269 (Masahiro Futaki)
JSPS Grant-in-aid (C) 18K03293 (Hiroshige Kajiura)
Contact: futaki at math.s.chiba-u.ac.jp