nLab D=2 Yang-Mills theory

Contents

Context

Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Differential cohomology

Contents

Idea

D=2 Yang-Mills theory (D=2 YM theory) studies the Yang-Mills equations over a base manifold with dimension D=2D=2. This special case allows the definition of the Yang-Mills measure.

As with ordinary 4d Yang-Mills theory being the basis of quantum chromodynamics (QCD), so coupling suitable fermion physics (“quarks”) to D=2D=2 Yang-Mills theory leads to the “‘t Hooft model” also called D=2 QCD.

Basics

Consider:

  • GG a Lie group,

  • BB an orientable Riemannian 2-manifold,

  • EBE\twoheadrightarrow B a principal G G -bundle,

  • AΩ Ad 1(E,𝔤)Ω 1(B,Ad(E))A\in\Omega_{\operatorname{Ad}}^1(E,\mathfrak{g})\cong\Omega^1(B,\operatorname{Ad}(E)) a connection,

  • F Ad AA=dA+[AA]Ω Ad 2(E,𝔤)Ω 2(B,Ad(E))F_A \coloneqq \mathrm{d}_A A=\mathrm{d}A+[A\wedge A]\in\Omega_{\operatorname{Ad}}^2(E,\mathfrak{g})\cong\Omega^2(B,\operatorname{Ad}(E)) its curvature.

Chern-Weil theory implies that the first Chern class of the gauge bundle is

(1)c 1(E),[B]=c 1(Ad(E)),[B]=i2π Btr(F A), \big\langle c_1(E),[B] \big\rangle \,=\, \big\langle c_1\big(\operatorname{Ad}(E)\big), [B] \big\rangle \,=\, -\frac{\mathrm{i}}{2\pi} \int_B \operatorname{tr}(F_A) \;\in\; \mathbb{Z} \,,

Application on the 2-sphere

The complex Hopf fibration is a principal U ( 1 ) U(1) -bundle over S 2 S^2 , which encodes the charge quantization of the magnetic charge of a magnetic monopole in three dimensions (Dirac monopole) using:

Prin U(1)(S 2)[S 2,BU(1)]=π 2(BU(1))π 1(U(1))π 1S 1. \operatorname{Prin}_{\operatorname{U}(1)}(S^2) \;\cong\; \big[ S^2,\operatorname{BU}(1) \big] \;=\; \pi_2\big( \operatorname{BU}(1) \big) \;\cong\; \pi_1\big( \operatorname{U}(1) \big) \;\cong\; \pi_1 S^1 \;\cong\; \mathbb{Z} \,.

Given an mm\in\mathbb{Z}, the corresponding principal bundle is given by pullback of the universal principal bundle EU(1)BU(1)EU(1)\twoheadrightarrow BU(1) along the composition of the canonical inclusion S 2P 1P BU(1)S^2\cong\mathbb{C}P^1\hookrightarrow\mathbb{C}P^\infty\cong BU(1) and the map BU(1)BU(1)BU(1)\rightarrow BU(1) induced by U(1)U(1),zz m\mathrm{U}(1)\rightarrow \mathrm{U}(1),z\mapsto z^m.

References

See also the references at D=2 QCD.

The original reference:

Further developments:

On D=2 QCD

and its relation to string theory, the “Gross-Taylor model”:

and the Hořava model:

In the context of generalized global symmetries:

In the more general context of volume-dependent theories:

Discussion of lattice 2d Yang-Mills theory via derived algebraic geometry and prefactorization algebras:

Last revised on September 12, 2024 at 07:20:54. See the history of this page for a list of all contributions to it.