nLab
gauged WZW model

Contents

Context

\infty-Wess-Zumino-Witten theory

Quantum field theory

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

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

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 θ[θθ]Ω 3(G) G\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 HH-equivariant differential cohomology of GG, with curvature 3-form in equivariant de Rham cohomology.

Definition

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

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

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

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

Now for HGH \hookrightarrow G a subgroup, write

CE(𝔤//𝔥)Ω (G,𝔥 *[1]) G 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 GG. There is a canonical projection

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

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

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

in this equivariant de Rham cohomology which lifts Hθ[θθ]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}\{F^a\} of the Cartan model (see there) with respect to some basis {t a}\{t_a\} of 𝔤\mathfrak{g}, these lifts are of the form (Witten 92, (A.14))

H˜=H+λ aF a \tilde H = H + \lambda_a \wedge F^a

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

λ a=t a l(dg)g 1+t a rg 1dg \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

t a lt b lt a rt b r=0. \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 G˜//H\tilde G//H such that maps into it are locally a map gg into GG together with 1-form potentials A aA^a for the F aF^a, and the WZW term is locally a 2-form built from dgd g and A aA^a such that its curvature 3-form is H˜\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.

References

The original articles are

  • Edward Witten, Nucl. Phys. B223 (1983) 433.

  • Edward Witten, Commun. Math. Phys. 92 (1984) 455.

  • 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

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

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

See also

  • Edward Witten, The NN 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

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

Last revised on June 11, 2019 at 11:05:54. See the history of this page for a list of all contributions to it.