algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
D=2 Yang-Mills theory (D=2 YM theory) studies the Yang-Mills equations over a base manifold with dimension . 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 Yang-Mills theory leads to the “‘t Hooft model” also called D=2 QCD.
Consider:
a Lie group,
a connection,
its curvature.
Chern-Weil theory implies that the first Chern class of the gauge bundle is
The complex Hopf fibration is a principal -bundle over , which encodes the charge quantization of the magnetic charge of a magnetic monopole in three dimensions (Dirac monopole) using:
Given an , the corresponding principal bundle is given by pullback of the universal principal bundle along the composition of the canonical inclusion and the map induced by .
See also the references at D=2 QCD.
The original reference:
Further developments:
Ninoslav E. Bralić, Exact computation of loop averages in two-dimensional Yang-Mills theory, Phys. Rev. D 22 12 (1980) 3090–3103 [doi:10.1103/PhysRevD.22.3090]
V. A. Kazakov, I. K. Kostov, Non-linear strings in two-dimensional U(∞) gauge theory, Nuclear Physics B 176 1 (1980) 199–215 [doi:10.1016/0550-3213(80)90072-3]
Leonard Gross, Christopher King, Ambar Sengupta, Two dimensional Yang-Mills theory via stochastic differential equations, Annals of Physics 194:1 (1989) 65–112 [doi:10.1016/0003-4916(89)90032-8]
B. Ye. Rusakov, Loop averages and partition functions in gauge theory on two-dimensional manifolds, Modern Physics Letters A 5 9 (1990) 693–703 [doi:10.1142/S0217732390000780]
Dana S. Fine, Quantum Yang-Mills on the two-sphere, Communications in Mathematical Physics 134 (1990) 273–292 [doi:10.1007/BF02097703]
Dana S. Fine, Quantum Yang-Mills on a Riemann surface, Communications in Mathematical Physics 140 (1991) 321–338. [doi:10.1007/BF02099502]
Ambar Sengupta, The Yang-Mills measure for , Journal of Functional Analysis 108 2 (1992) 231–273 [doi;10.1016/0022-1236(92)90025-E]
Ambar Sengupta, Quantum Gauge Theory on Compact Surfaces, Annals of Physics 221 1 (1993) 17–52 [doi:10.1006/aphy.1993.1002]
Matthias Blau, George Thompson, Quantum Yang-Mills theory on arbitrary surfaces, International Journal of Modern Physics A 7 16 (1992) 3781–3806 [doi:10.1142/S0217751X9200168X]
Edward Witten. On quantum gauge theories in two dimensions, Commun.Math. Phys. 141 (1991) 153–209 [doi:10.1007/BF02100009]
Edward Witten, Two-dimensional gauge theories revisited, Journal of Geometry and Physics 9 4 (1992) 303–368 [arXiv:hep-th/9204083, doi:10.1016/0393-0440(92)90034-X90034-X)]
Stefan Cordes, Gregory Moore, Sanjaye Ramgoolam. Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories (1994) [hep-th/9411210]
On D=2 QCD
Gerard ’t Hooft, A Two-Dimensional Model For Mesons, Nucl. Phys. B 75 (1974) 461-470 [doi:10.1016/0550-3213(74)90088-1]
Michael Douglas, Keke Li, Matthias Staudacher, Generalized Two-Dimensional QCD, Nucl.Phys. B 420 (1994) 118-140 [arXiv:hep-th/9401062, doi:10.1016/0550-3213(94)90377-8]
Emanuel Katz, Gustavo Marques Tavares, Yiming Xu, A solution of 2D QCD at Finite using a conformal basis [arxiv:1405.6727]
and its relation to string theory, the “Gross-Taylor model”:
David Gross. Two Dimensional QCD as a String Theory, Nuclear Physics B 400 1–3 (1993) 161-180 [hep-th/9212149, doi:10.1016/0550-3213(93)90402-B]
David Gross, Washington Taylor, Two Dimensional QCD is a String Theory, Nucl. Phys. B 400 (1993) 181-210 [arxiv:hep-th/9301068, doi:10.1016/0550-3213(93)90403-C]
David Gross, Washington Taylor, Twists and Wilson loops in the string theory of two-dimensional QCD, Nucl. Phys. B 403 (1993) 395-452 [hep-th/9303046, doi:10.1016/0550-3213(93)90042-N]
and the Hořava model:
Petr Hořava, Topological Strings and QCD in Two Dimensions (1993) [hep-th/9311156, spire]
Petr Hořava. Topological Rigid String Theory and Two Dimensional QCD, Nucl.Phys. B 463 (1996) 238-286 [hep-th/9507060, doi:10.1016/0550-3213(96)00036-3 ]
In the context of generalized global symmetries:
Mendel Nguyen, Yuya Tanizaki, Mithat Ünsal. Non-invertible 1-form symmetry and Casimir scaling in 2d Yang-Mills theory (2021) [arXiv:2104.01824]
Tony Pantev, Eric Sharpe. Decomposition and the Gross-Taylor string theory (2023) [arXiv:2307.08729]
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.