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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.
The Chevalley-Eilenberg algebra of the Lie algebra $\mathfrak{g}$ is naturally identified with the sub-algebra of left invariant differential forms on $G$:
The ordinary WZW model is given by the basic circle 2-bundle with connection on $G$ whose curvature 3-form is
Now for $H \hookrightarrow G$ a subgroup, write
for the corresponding dg-algebra of (say) the Cartan model for equivariant de Rham cohomology on $G$. There is a canonical projection
A curvature 3-form for the gauged WZW model is a 3-cocycle
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))
where $\lambda_a \in \Omega^1(G)$ is given by (in matrix Lie algebra notation)
and exist precisely if (Witten 92, (A.16)) for all pairs of basis elements
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)).
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.
gauged linear sigma model?
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
This is expanded on in
José Figueroa-O'Farrill, S Stanciu, Gauged Wess-Zumino terms and Equivariant Cohomology, Phys.Lett. B341 (1994) 153-159 (arXiv:hep-th/9407196)
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
which further generalizes the discussion to non-compact Lie groups.
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
Emphasis of the special case of abelian gauging in Section 2 of
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