nLab D=2 Yang-Mills theory

Idea

D=2 Yang-Mills theory (2D YM) is Yang-Mills theory (e.g. the study of the Yang-Mills equations) over base manifolds of dimension $D=2$ (2-manifolds/surfaces).

In this low-dimensional special case the path integral measure exists: known as the Yang-Mills measure (cf. Sengupta 1992, Pickrell 1996).

While the Lagrangian density of Yang-Mills theories generally depends on a background metric, in 2D it actually only depends on the induced volume form. Accordingly, 2D YM is “generally covariant” under area-preserving diffeomorphisms (cf. Witten 1992, Pickrell 1996).

Just as ordinary 4d Yang-Mills theory is the backbone of quantum chromodynamics (QCD), so coupling suitable fermion fields (“quarks”) to $D=2$ Yang-Mills theory leads to 2D QCD known as the ‘t Hooft model.

Basics

Consider:

• $G$ a Lie group,

• $B$ an orientable Riemannian 2-manifold,

• $E\twoheadrightarrow B$ a principal $G$-bundle,

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

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

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

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

Application on the 2-sphere

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

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

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

Related concepts

On ordinary Yang-Mills theory (YM):

• D=2 YM, D=3 YM, D=4 YM, D=5 YM, D=6 YM, D=7 YM, D=8 YM

• Maxwell theory/electromagnetism (U(1) YM), Donaldson theory (SU(2) YM), quantum chromodynamics (SU(3) YM)

• Yang-Mills equation, linearized Yang-Mills equation, Yang-Mills instanton, Yang-Mills field, stable Yang-Mills connection, Yang-Mills moduli space, Yang-Mills flow, F-Yang-Mills equation, Bi-Yang-Mills equation

• Uhlenbeck's singularity theorem, Uhlenbeck's compactness theorem

On variants of Yang-Mills theory and on super Yang-Mills theory (SYM):

• Yang-Mills-Higgs equations, stable Yang-Mills-Higgs pair, Yang-Mills-Higgs flow

• Einstein-Yang-Mills theory, Einstein-Yang-Mills-Dirac theory, Einstein-Yang-Mills-Dirac-Higgs theory

• 3D Yang-Mills-Chern-Simons theory

• 3d superconformal gauge field theory: D=3 N=1 SYM, D=3 N=2 SYM, D=3 N=4 SYM

• 4d superconformal gauge field theory: D=4 N=1 SYM, D=4 N=2 SYM, D=4 N=4 SYM

• D=5 SYM, D=6 SYM, D=7 SYM, D=10 SYM

• topological Yang-Mills theory, topologically twisted D=4 super Yang-Mills theory

• D=2 QCD

References

See also the references at D=2 QCD.

The original reference:

• Alexander A. Migdal, Recursion equations in gauge field theories, Zh. Eksp. Teor. Fiz. 69 (1975) 810–822 (Sov. Phys. JETP 42, 413–418) [pdf, pdf spire:100330]

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 N. 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 $U(N)$ 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]

• Edward Witten: On quantum gauge theories in two dimensions, Commun. Math. Phys. 141 (1991) 153–209 [doi:10.1007/BF02100009, euclid:cmp/1104248198]

• 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)]

• 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]

• Stefan Cordes, Gregory Moore, Sanjaye Ramgoolam: Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories (1994) [hep-th/9411210]

On rigorous construction of the path integral measure invariant under area-preserving diffeomorphisms:

• Ambar N. Sengupta, The Yang-Mills measure for $S^2$, Journal of Functional Analysis 108 2 (1992) 231–273 [doi;10.1016/0022-1236(92)90025-E]

• Ambar N. Sengupta: Quantum Gauge Theory on Compact Surfaces, Annals of Physics 221 1 (1993) 17–52 [doi:10.1006/aphy.1993.1002]

• Ambar N. Sengupta: Quantum Gauge Theory on Compact Surfaces, Memoirs of the AMS 126 (1997) [ISBN:978-1-4704-0185-6]

• Doug Pickrell: On $YM_2$ measures and area-preserving diffeomorphisms, Journal of Geometry and Physics 19 4 (1996) 315-367 [doi:10.1016/0393-0440(95)00034-8]

• Doug Pickrell: On the Action of the Group of Diffeomorphisms of a Surface on Sections of the Determinant Line Bundle, Pacific Journal of Mathematics 193 1 (2000) 177-199 [doi:10.2140/pjm.2000.193.177, pdf]

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 $N$ 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:

• Richard Wedeen. Volume-Dependent Field Theories (2024). [arXiv:2402.06691]

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

• Marco Benini, Tomás Fernández, Alexander Schenkel: Derived algebraic geometry of 2d lattice Yang-Mills theory [arXiv:2409.06873]