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Quantization via Mirror Symmetry

Sergei Gukov

TL;DR

The paper reframes geometric quantization as an A-model problem on a complexified phase space $Y$, with the Hilbert space realized as morphisms between branes, and then translates this into a B-model computation on the mirror $\widetilde Y$ via Kontsevich mirror symmetry. It develops the role of coisotropic and Lagrangian branes, especially the canonical coisotropic brane $\mathcal B_{cc}$ and a Lagrangian brane on $M$, and shows how hyperholomorphic/$({\rm B},{\rm B},{\rm B})$ branes organize the mirror description. Through a sequence of concrete examples—in particular toy models, $K3$ surfaces, and Chern–Simons theory—the text demonstrates how the Verlinde formula emerges as an Ext-dimension computation on the mirror, with the rank and Chern-character data tied to SYZ-fibre volumes and $SU(2)$-invariance. The construction provides a geometric, brane-based pathway to understanding quantization in representation theory and gauge theory, revealing a deep link between quantum dimensions and mirror-symmetric geometry. Overall, the work integrates A-/B-model techniques, hyperkähler geometry, and Langlands duality to derive Verlinde-type dimensions from brane data.” (All mathematical objects are denoted with explicit $...$ delimiters as needed.)

Abstract

When combined with mirror symmetry, the A-model approach to quantization leads to a fairly simple and tractable problem. The most interesting part of the problem then becomes finding the mirror of the coisotropic brane. We illustrate how it can be addressed in a number of interesting examples related to representation theory and gauge theory, in which mirror geometry is naturally associated with the Langlands dual group. Hyperholomorphic sheaves and (B,B,B) branes play an important role in the B-model approach to quantization.

Quantization via Mirror Symmetry

TL;DR

The paper reframes geometric quantization as an A-model problem on a complexified phase space , with the Hilbert space realized as morphisms between branes, and then translates this into a B-model computation on the mirror via Kontsevich mirror symmetry. It develops the role of coisotropic and Lagrangian branes, especially the canonical coisotropic brane and a Lagrangian brane on , and shows how hyperholomorphic/ branes organize the mirror description. Through a sequence of concrete examples—in particular toy models, surfaces, and Chern–Simons theory—the text demonstrates how the Verlinde formula emerges as an Ext-dimension computation on the mirror, with the rank and Chern-character data tied to SYZ-fibre volumes and -invariance. The construction provides a geometric, brane-based pathway to understanding quantization in representation theory and gauge theory, revealing a deep link between quantum dimensions and mirror-symmetric geometry. Overall, the work integrates A-/B-model techniques, hyperkähler geometry, and Langlands duality to derive Verlinde-type dimensions from brane data.” (All mathematical objects are denoted with explicit delimiters as needed.)

Abstract

When combined with mirror symmetry, the A-model approach to quantization leads to a fairly simple and tractable problem. The most interesting part of the problem then becomes finding the mirror of the coisotropic brane. We illustrate how it can be addressed in a number of interesting examples related to representation theory and gauge theory, in which mirror geometry is naturally associated with the Langlands dual group. Hyperholomorphic sheaves and (B,B,B) branes play an important role in the B-model approach to quantization.

Paper Structure

This paper contains 15 sections, 167 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Quantization is an art
  3. Geometric quantization
  4. Deformation quantization
  5. Brane quantization
  6. When does quantization exist?
  7. $B$-model approach to quantization
  8. $(B,B,B)$ branes and hyperholomorphic bundles
  9. Toy model
  10. $(B,A,A)$ and $(B,B,B)$ branes on $K3$
  11. What became of $\hbar$
  12. Quantization of Chern-Simons theory
  13. Mirror of the Lagrangian brane ${\mathcal{B}}'$
  14. Mirror of the coisotropic $(B,A,A)$ brane ${\mathcal{B}}_{cc}$
  15. The Verlinde formula via mirror symmetry

Figures (1)

  • Figure 1: Singularities of $\widetilde{Y}$: $a)$ four simple singularities of type $A_1$ when $\alpha = 0$, $b)$ one singularity of type $D_4$ when $\alpha = \frac{1}{2}$, and $c)$ three $A_1$ singularities for all other values $0 < \alpha < \frac{1}{2}$.

Theorems & Definitions (2)

  • Conjecture 3.1
  • Conjecture 4.1: Rigid