nLab gauged WZW model

Idea

The gauged WZW model is a field theory (physics) which combines the WZW model with gauge theory: given a (simple) Lie group $G$ and a subgroup $H \hookrightarrow G \times G$, the corresponding gauged WZW model is a 2-dimensional prequantum field theory on some worldvolume $\Sigma_2$ whose fields are pairs consisting of a smooth function $\Sigma_2 \to G$ and a Lie algebra valued 1-form $A \in \Omega^1(\Sigma_2, \mathfrak{h})$, with values in the Lie algebra of $H$.

Where the Lagrangian/action functional of the ordinary WZW model is the sum/product of a standard kinetic action and the surface holonomy of a circle 2-bundle with connection whose curvature 3-form is the canonical 3-form $\langle \theta \wedge [\theta \wedge \theta]\rangle \in \Omega^3(G)^G$, so the action functional of the gauged WZW model is that obtained by refining this circle 2-bundle to the $H$-equivariant differential cohomology of $G$, with curvature 3-form in equivariant de Rham cohomology.

Definition

The Chevalley-Eilenberg algebra of the Lie algebra $\mathfrak{g}$ is naturally identified with the sub-algebra of left invariant differential forms on $G$:

$$CE(\mathfrak{g}) \simeq \Omega^\bullet(G)^G \,.$$

The ordinary WZW model is given by the basic circle 2-bundle with connection on $G$ whose curvature 3-form is

$$H = \langle \theta \wedge [\theta \wedge \theta]\rangle \in \Omega^3(G)^G \,.$$

Now for $H \hookrightarrow G$ a subgroup, write

$$CE(\mathfrak{g}//\mathfrak{h}) \coloneqq \Omega^\bullet(G, \mathfrak{h}^\ast[1])^G$$

for the corresponding dg-algebra of (say) the Cartan model for equivariant de Rham cohomology on $G$. There is a canonical projection

$$CE(\mathfrak{g}//\mathfrak{h}) \to CE(\mathfrak{g}) \,.$$

A curvature 3-form for the gauged WZW model is a 3-cocycle

$$\tilde H \in CE^3(\mathfrak{g}//\mathfrak{h})$$

in this equivariant de Rham cohomology which lifts $H \coloneqq \langle \theta \wedge [\theta \wedge \theta]\rangle$ through this projection.

One finds (Witten 92, appendix) that in terms of the degree-2 generators $\{F^a\}$ of the Cartan model (see there) with respect to some basis $\{t_a\}$ of $\mathfrak{g}$, these lifts are of the form (Witten 92, (A.14))

$$\tilde H = H + \lambda_a \wedge F^a$$

where $\lambda_a \in \Omega^1(G)$ is given by (in matrix Lie algebra notation)

$$\lambda_a = \left\langle t_a^l \cdot (d g)g^{-1} + t_a^r \cdot g^{-1} d g \right\rangle$$

and exist precisely if (Witten 92, (A.16)) for all pairs of basis elements

$$\langle t_a^l \cdot t_b^l - t_a^r \cdot t_b^r \rangle = 0 \,.$$

This condition had originally been seen as a anomaly cancellation-condition of the gauged WZW model. A systematic discussion of these obstructions in equivariant de Rham cohomology is in (Figueroa-O’Farrill-Stanciu 94).

Now by ∞-Wess-Zumino-Witten theory, the corresponding WZW model has as target the smooth groupoid $\tilde G//H$ such that maps into it are locally a map $g$ into $G$ together with 1-form potentials $A^a$ for the $F^a$, and the WZW term is locally a 2-form built from $d g$ and $A^a$ such that its curvature 3-form is $\tilde H$. This is the gauged WZW model (Witten 92, (A.16)).

Properties

Partition function in (equivariant) elliptic cohomology

The partition function of the gauged WZW model as an elliptic genus is considered in (Henningsonn 94, (8)). When done properly this should give elements in equivariant elliptic cohomology, hence an equivariant elliptic genus.

Related concepts

• coset WZW model

• gauged linear sigma model?

References

The WZW term of QCD chiral perturbation theory

The gauged WZW term of chiral perturbation theory/quantum hadrodynamics which reproduces the chiral anomaly of QCD in the effective field theory of mesons and Skyrmions:

General

The original articles:

• Julius Wess, Bruno Zumino, Consequences of anomalous Ward identities, Phys. Lett. B 37 (1971) 95-97 (spire:67330, doi:10.1016/0370-2693(71)90582-X)

• Edward Witten, Global aspects of current algebra, Nuclear Physics B Volume 223, Issue 2, 22 August 1983, Pages 422-432 (doi:10.1016/0550-3213(83)90063-9)

See also:

• O. Kaymakcalan, S. Rajeev, J. Schechter, Nonabelian Anomaly and Vector Meson Decays, Phys. Rev. D 30 (1984) 594 (spire:194756)

Corrections and streamlining of the computations:

• Chou Kuang-chao, Guo Han-ying, Wu Ke, Song Xing-kang, On the gauge invariance and anomaly-free condition of the Wess-Zumino-Witten effective action, Physics Letters B Volume 134, Issues 1–2, 5 January 1984, Pages 67-69 (doi:10.1016/0370-2693(84)90986-9))

• H. Kawai, S.-H. H. Tye, Chiral anomalies, effective lagrangians and differential geometry, Physics Letters B Volume 140, Issues 5–6, 14 June 1984, Pages 403-407 (doi:10.1016/0370-2693(84)90780-9)

• J. L. Mañes, Differential geometric construction of the gauged Wess-Zumino action, Nuclear Physics B Volume 250, Issues 1–4, 1985, Pages 369-384 (doi:10.1016/0550-3213(85)90487-0)

• Tomáš Brauner, Helena Kolešová, Gauged Wess-Zumino terms for a general coset space, Nuclear Physics B Volume 945, August 2019, 114676 (doi:10.1016/j.nuclphysb.2019.114676)

See also

• Yasunori Lee, Kantaro Ohmori, Yuji Tachikawa, Revisiting Wess-Zumino-Witten terms (arXiv:2009.00033)

Interpretation as Skyrmion/baryon current:

• Jeffrey Goldstone, Frank Wilczek, Fractional Quantum Numbers on Solitons, Phys. Rev. Lett. 47, 986 (1981) (doi:10.1103/PhysRevLett.47.986)

• Edward Witten, Current algebra, baryons, and quark confinement, Nuclear Physics B Volume 223, Issue 2, 22 August 1983, Pages 433-444 (doi:10.1016/0550-3213(83)90064-0)

• Gregory Adkins, Chiara Nappi, Stabilization of Chiral Solitons via Vector Mesons, Phys. Lett. 137B (1984) 251-256 (spire:194727, doi:10.1016/0370-2693(84)90239-9)

  (beware that the two copies of the text at these two sources differ!)

• Mannque Rho et al., Introduction, In: Mannque Rho et al. (eds.) The Multifaceted Skyrmion, World Scientific 2016 (doi:10.1142/9710)

Concrete form for $N$-flavor quantum hadrodynamics in 2d:

• C. R. Lee, H. C. Yen, A Derivation of The Wess-Zumino-Witten Action from Chiral Anomaly Using Homotopy Operators, Chinese Journal of Physics, Vol 23 No. 1 (1985) (spire:16389, pdf)

Concrete form for 2 flavors in 4d:

• Masashi Wakamatsu, On the electromagnetic hadron current derived from the gauged Wess-Zumino-Witten action, (arXiv:1108.1236, spire:922302)

Including light vector mesons

Concrete form for 2-flavor quantum hadrodynamics in 4d with light vector mesons included (omega-meson and rho-meson):

• Ulf-G. Meissner, Ismail Zahed, equation (6) in: Skyrmions in the Presence of Vector Mesons, Phys. Rev. Lett. 56, 1035 (1986) (doi:10.1103/PhysRevLett.56.1035)

• Ulf-G. Meissner, Norbert Kaiser, Wolfram Weise, equation (2.18) in: Nucleons as skyrme solitons with vector mesons: Electromagnetic and axial properties, Nuclear Physics A Volume 466, Issues 3–4, 11–18 May 1987, Pages 685-723 (doi:10.1016/0375-9474(87)90463-5)

• Ulf-G. Meissner, equation (2.45) in: Low-energy hadron physics from effective chiral Lagrangians with vector mesons, Physics Reports Volume 161, Issues 5–6, May 1988, Pages 213-361 (doi:10.1016/0370-1573(88)90090-7)

• Roland Kaiser, equation (12) in: Anomalies and WZW-term of two-flavour QCD, Phys. Rev. D63:076010, 2001 (arXiv:hep-ph/0011377, spire:537600)

Including heavy scalar mesons

Including heavy scalar mesons:

specifically kaons:

• Curtis Callan, Igor Klebanov, equation (4.1) in: Bound-state approach to strangeness in the Skyrme model, Nuclear Physics B Volume 262, Issue 2, 16 December 1985, Pages 365-382 (doi10.1016/0550-3213(85)90292-5)

• Igor Klebanov, equation (99) of: Strangeness in the Skyrme model, in: D. Vauthrin, F. Lenz, J. W. Negele, Hadrons and Hadronic Matter, Plenum Press 1989 (doi:10.1007/978-1-4684-1336-6)

• N. N. Scoccola, D. P. Min, H. Nadeau, Mannque Rho, equation (2.20) in: The strangeness problem: An $SU(3)$ skyrmion with vector mesons, Nuclear Physics A Volume 505, Issues 3–4, 25 December 1989, Pages 497-524 (doi:10.1016/0375-9474(89)90029-8)

specifically D-mesons:

(…)

specifically B-mesons:

• Mannque Rho, D. O. Riska, N. N. Scoccola, above (2.1) in: The energy levels of the heavy flavour baryons in the topological soliton model, Zeitschrift für Physik A Hadrons and Nuclei volume 341, pages343–352 (1992) (doi:10.1007/BF01283544)

Including heavy vector mesons

Inclusion of heavy vector mesons:

specifically K*-mesons:

• S. Ozaki, H. Nagahiro, Atsushi Hosaka, Equations (3) and (9) in: Magnetic interaction induced by the anomaly in kaon-photoproductions, Physics Letters B Volume 665, Issue 4, 24 July 2008, Pages 178-181 (arXiv:0710.5581, doi:10.1016/j.physletb.2008.06.020)

Including electroweak interactions

Including electroweak fields:

• J. Bijnens, G. Ecker, A. Picha, The chiral anomaly in non-leptonic weak interactions, Physics Letters B Volume 286, Issues 3–4, 30 July 1992, Pages 341-347 (doi:10.1016/0370-2693(92)91785-8)

• Gerhard Ecker, Helmut Neufeld, Antonio Pich, Non-leptonic kaon decays and the chiral anomaly, Nuclear Physics B Volume 413, Issues 1–2, 31 January 1994, Pages 321-352 (doi:10.1016/0550-3213(94)90623-8)

Discussion for the full standard model of particle physics:

• Jeffrey Harvey, Christopher T. Hill, Richard J. Hill, Standard Model Gauging of the WZW Term: Anomalies, Global Currents and pseudo-Chern-Simons Interactions, Phys. Rev. D77:085017, 2008 (arXiv:0712.1230)

General gauged WZW models

The original articles are

• Edward Witten, Nonabelian bosonization in two dimensions, Commun. Math. Phys. 92 (1984) 455 (euclid:cmp/1103940923)

• Krzysztof Gawedzki, A. Kupiainen, G/H conformal field theory from gauged WZW model Phys. Lett. 215B, 119 (1988);

• Krzysztof Gawedzki, A. Kupiainen, Coset construction from functional integrals, Nucl. Phys. B 320 (FS), 649 (1989)

• Krzysztof Gawedzki, in From Functional Integration, Geometry and Strings, ed. by Z. Haba and J. Sobczyk (Birkhaeuser, 1989).

The (curvature of the)gauged WZW term was recognized/described as a cocycle in equivariant de Rham cohomology is in the appendix of

• Edward Witten, On holomorphic factorization of WZW and coset models, Comm. Math. Phys. Volume 144, Number 1 (1992), 189-212. (EUCLID)

This is expanded on in

• José Figueroa-O'Farrill, Sonia Stanciu, Gauged Wess-Zumino terms and Equivariant Cohomology, Phys. Lett. B 341 (1994) 153-159 [arXiv:hep-th/9407196, doi:10.1016/0370-2693(94)90304-2]

• José de Azcárraga, J. C. Perez Bueno, On the general structure of gauged Wess-Zumino-Witten terms (arXiv:hep-th/9802192)

A quick review of this class of 3-cocycles in equivariant de Rham cohomology is also in section 4.1 of

• Hugo Garcia-Compean, Pablo Paniagua, Bernardo Uribe, Equivariant extensions of differential forms for non-compact Lie groups (arXiv:1304.3226)

which further generalizes the discussion to non-compact Lie groups.

On boundary conditions (BCFT/D-branes) for the gauged WZW model via parafermions:

• Juan Maldacena, Gregory Moore, Nathan Seiberg, Geometrical interpretation of D-branes in gauged WZW models, JHEP 0107:046 (2001) $[$arXiv:hep-th/0105038, doi:10.1088/1126-6708/2001/07/046$]$

See also:

• Edward Witten, The $N$ matrix model and gauged WZW models, Nuclear Physics B Volume 371, Issues 1–2, 2 March 1992, Pages 191–245

• Stephen-wei Chung, S.-H. Henry Tye, Chiral Gauged WZW Theories and Coset Models in Conformal Field Theory, Phys. Rev. D47:4546-4566,1993 (arXiv:hep-th/9202002)

• Konstadinos Sfetsos, Gauged WZW models and Non-abelian duality, Phys.Rev. D50 (1994) 2784-2798 (arXiv:hep-th/9402031)

• Elias Kiritsis, Duality in gauged WZW models (pdf)

The partition function/elliptic genus of the SU(2)/U(1) gauged WZW model is considered in

• Måns Henningson, N=2 gauged WZW models and the elliptic genus, Nucl.Phys. B413 (1994) 73-83 (arXiv:hep-th/9307040)

Emphasis of the special case of abelian gauging in Section 2 of

• Stefan Förste, Deformations of WZW models, Class. Quant. Grav. 21 (2004) S1517-1522 (arXiv:hep-th/0312202)