Quantization via mirror symmetry
type: Item
instance of: scholarly article
title: Quantization via mirror symmetry
zbMATH Open document ID: 1281.18002
author: Sergei Gukov
published in: Japanese Journal of Mathematics. 3rd Series
publication date: 2013-01-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1011.2218
full work available at URL: CaltechAUTHORS:20120113-091504097
review text: This is a write-up of the author's \(8\)th Takagi lectures on brane quantization and mirror symmetry delivered at Kyoto University in 2010. The style is descriptive and expository with focus on ideas and concepts rather than technicalities, and computations are mostly sketched. The brane quantization of a symplectic manifold \((M,\omega)\) is a new approach developed by the author and \textit{E. Witten} in [Adv. Theor. Math. Phys. 13, No. 5, 1445--1518 (2009; Zbl 1247.81378)]. It requires a choice of complexification \(Y\) of \(M\) with a non-degenerate holomorphic \(2\)-form \(\Omega\) that restricts to \(\omega\) on \(M\), and a unitary line bundle over \(Y\) that restricts to the prequantum line bundle of geometric quantization, with a connection of curvature \(\operatorname{Re}\Omega\). The Hilbert space of states is then given by the space of morphisms in the Fukaya category of \(Y\) (open strings) \(\mathcal{H}=\Hom(\mathcal{B}_{cc},\mathcal{B}')\), where \(\mathcal{B}'\) is a Lagrangian object (\(A\)-brane) supported on \(M\), and \(\mathcal{B}_{cc}\) is a canonical coisotropic object filling the entire \(Y\). Two classes of examples are considered in which the brane quantization can be carried out explicitly. The first are coadjoint orbits of a real Lie group \(G\) with the Kirillov-Kostant-Souriau form, the setting of the geometric quantization and the orbit method of the representation theory. A natural complexification is provided by switching to the complexified group \(G_{\mathbb{C}}\) (e.g. \(\mathrm{SL}(2,\mathbb{C})\) for \(\mathrm{SU}(2)\) and \(\mathrm{SL}(2,\mathbb{R})\)). Brane quantization offers new insights into representation theory explaining, for instance, why some unitary representations do not correspond to orbits (e.g. the complementary series representations of \(\mathrm{SL}(2,\mathbb{R})\)), and vice versa (e.g. the minimal orbits of \(\mathrm{SO}(p,q)\) with \(p,q\geq4\)). It also leads to new conditions for \((M,\omega)\) to be quant…
reviewed by: Sergiy Koshkin
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 14J33
Mathematics Subject Classification ID: 53D37
Mathematics Subject Classification ID: 81S10
Mathematics Subject Classification ID: 81T70
zbMATH DE Number: 6132929
zbMATH Keywords: symplectic manifold
zbMATH Keywords: brane quantization
zbMATH Keywords: prequantum line bundle
zbMATH Keywords: Fukaya category
zbMATH Keywords: SYZ fibration
zbMATH Keywords: \((B,B,B)\) branes
zbMATH Keywords: Chern-Simons gauge theory
zbMATH Keywords: Higgs \(G\)-bundles
zbMATH Keywords: Hitchin moduli
zbMATH Keywords: Verlinde formula
MaRDI profile type: Publication
OpenAlex ID: W2120082977
arXiv ID: 1011.2218
cites work: The Yang-Mills equations over Riemann surfaces
cites work: The index of elliptic operators. I
cites work: Deformation theory and quantization. I: Deformations of symplectic structures
cites work: Vector bundles on curves and generalized theta functions: recent results and open problems
cites work: Spectral curves and the generalised theta divisor.
cites work: Q4002484
cites work: Topological reduction of \(4\)D SYM to \(2\)D\(\sigma\)-models
cites work: Surface group representations and \(U(p,q)\)-Higgs bundles.
cites work: Q4656911
cites work: The McKay correspondence as an equivalence of derived categories
cites work: Topological--anti-topological fusion.
cites work: Hyper-Kähler Calabi metrics, \(L^2\) harmonic forms, resolved M2-branes, and AdS\(_4\)/CFT\(_3\) correspondence
cites work: Q4265278
cites work: Geometric endoscopy and mirror symmetry
cites work: Higgs bundles for the non-compact dual of the unitary group
cites work: The modular group action on real \(SL(2)\)-characters of a one-holed torus
cites work: Surface Operators and Knot Homologies
cites work: Branes and quantization
cites work: Q3597344
cites work: Rigid surface operators
cites work: Mirror symmetry, Langlands duality, and the Hitchin system
cites work: The Self-Duality Equations on a Riemann Surface
cites work: Q4434432
cites work: Kleinian singularities, derived categories and Hall algebras
cites work: Remarks on A-branes, mirror symmetry, and the Fukaya category
cites work: Electric-magnetic duality and the geometric Langlands program
cites work: Homological Algebra of Mirror Symmetry
cites work: Symplectic structure of the moduli space of sheaves on an abelian or K 3 surface
cites work: Fukaya categories as categorical Morse homology
cites work: Constructible sheaves and the Fukaya category
cites work: Quantification géométrique.
cites work: Mirror symmetry is \(T\)-duality
cites work: Fusion rules and modular transformations in 2D conformal field theory
cites work: Quantum field theory and the Jones polynomial
cites work: On quantum gauge theories in two dimensions
cites work: Geometric Langlands duality and the equations of Nahm and Bogomolny
cites work: Q3108747
DOI: 10.1007/S11537-011-1033-2
Recommended article: Q4871005
Recommended article: Q4883401
Recommended article: Q4549216
Recommended article: Second-quantized mirror symmetry
Recommended article: Q2739528
Recommended article: Quantized mirror curves and resummed WKB
Recommended article: Feynman integrals and mirror symmetry
Recommended article: Renormalization method and mirror symmetry
Recommended article: Quantum symmetries and marginal deformations
Recommended article: Q5482628
description: scientific article