Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLagrangian torus fibrations and homological mirror symmetry for the conifold
131
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We discuss homological mirror symmetry for the conifold from the point of view of the Strominger-Yau-Zaslow conjecture.
rate research
Read More
This is an expository article on the A-side of Kontsevichs Homological Mirror Symmetry conjecture. We give first a self-contained study of $A_infty$-categories and their homological algebra, and later restrict to Fukaya categories, with particular emphasis on the basics of the underlying Floer theory, and the geometric features therein. Finally, we place the theory in the context of mirror symmetry, building towards its main predictions.
We prove homological mirror symmetry for the universal centralizer $J_G$ (a.k.a Toda space), associated to any complex semisimple Lie group $G$. The A-side is a partially wrapped Fukaya category on $J_G$, and the B-side is the category of coherent sheaves on the categorical quotient of a dual maximal torus by the Weyl group action (with some modification if $G$ has a nontrivial center). This is the first and the main part of a two-part series, dealing with $G$ of adjoint type. The general case will be proved in the forthcoming second part [Jin2].
The main result of the present paper concerns finiteness properties of Floer theoretic invariants on affine log Calabi-Yau varieties $X$. Namely, we show that: (a) the degree zero symplectic cohomology $SH^0(X)$ is finitely generated and is a filtered deformation of a certain algebra defined combinatorially in terms of a compactifying divisor $mathbf{D}.$ (b) For any Lagrangian branes $L_0, L_1$, the wrapped Floer groups $WF^*(L_0,L_1)$ are finitely generated modules over $SH^0(X).$ We then describe applications of this result to mirror symmetry, the first of which is an ``automatic generation criterion for the wrapped Fukaya category $mathcal{W}(X)$. We also show that, in the case where $X$ is maximally degenerate and admits a ``homological section, $mathcal{W}(X)$ gives a categorical crepant resolution of the potentially singular variety $operatorname{Spec}(SH^0(X))$. This provides a link between the intrinsic mirror symmetry program of Gross and Siebert and the categorical birational geometry program initiated by Bondal-Orlov and Kuznetsov.
We define a local Riemann-Roch number for an open symplectic manifold when a complete integrable system without Bohr-Sommerfeld fiber is provided on its end. In particular when a structure of a singular Lagrangian fibration is given on a closed symplectic manifold, its Riemann-Roch number is described as the sum of the number of nonsingular Bohr-Sommerfeld fibers and a contribution of the singular fibers. A key step of the proof is formally explained as a version of Wittens deformation applied to a Hilbert bundle.
We describe supersymmetric A-branes and B-branes in open N=(2,2) dynamically gauged nonlinear sigma models (GNLSM), placing emphasis on toric manifold target spaces. For a subset of toric manifolds, these equivariant branes have a mirror description as branes in gauged Landau-Ginzburg models with neutral matter. We then study correlation functions in the topological A-twisted version of the GNLSM, and identify their values with open Hamiltonian Gromov-Witten invariants. Supersymmetry breaking can occur in the A-twisted GNLSM due to nonperturbative open symplectic vortices, and we canonically BRST quantize the mirror theory to analyze this phenomenon.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
