nLab parameterized WZW model

Idea

Where a WZW model is a sigma model quantum field theory whose target space is a group $G$, a parameterization of such a model is a sigma-model (subject usually to some constraints) whose target space now is the total space of a $G$-principal bundle such that the action functional when restricted to fields that take values only in any one fiber, reduces to the given un-parameterized model.

Parameterized WZW models have been argued to provide a more geometric and more complete description of the current algebra-sector of heterotic string backgrounds than the traditional construction in terms of worldsheet fermions (Gates-Siegel 88, Gates-Ketov-Kozenko-Solovev 91, Distler-Sharpe 10, section 7). In particular the Green-Schwarz anomaly of the heterotic string finds a natural interpretation as the obstruction to parameterizing the given WZW term over the given principal bundle:

A (higher) WZW model is an $n$-dimensional sigma-model field theory whose target space is a group $G$ and whose interaction-part of the action functional is the higher holonomy of a circle n-bundle with connection

$$\mathbf{L}_{WZW} \;\colon\; G \longrightarrow \mathbf{B}^n U(1)_{conn}$$

(whose curvature is given by the global Maurer-Cartan form on $G$).

If one has a $G$-principal bundle

$$\array{ G &\hookrightarrow& X \\ && \downarrow \\ && B }$$

over some base space $B$, then a lift of the structure group to the Heisenberg n-group $Heis_G(\mathbf{L}_{WZW})$ of $\mathbf{L}_{WZW}$ regarded as a prequantum n-bundle, is the structure necessary and sufficient for the fiber-wise copies of $G$ and $\mathbf{L}_{WZW}$ glue fiberwise to a single circle n-bundle with connection

$$\mathbf{L}_{WZW}^X \;\colon\; X \longrightarrow \mathbf{B}^n U(1)_{conn}$$

on the total space $X$ of this bundle. This hence yields what one may think of as a coherent collection of WZW models parameterized over base space $B$.

For the case that $G$ is a compact semisimple Lie group and $\mathbf{L}_{WZW}$ its canonical WZW term (the 2-connection on the string 2-group), then

$$Heis_G(\mathbf{L}_{WZW})\simeq String(G)$$

and hence the obstruction to the existence of a parameterization is precisely a string structure, recovering the traditional statement of the Green-Schwarz anomaly (see cwzw for details of this claim).

Examples

The heterotic string

The heterotic string worldsheet theory is a sigma model on spacetime $X$ combined with a chiral $G$-WZW model for $G = Spin \times E_8 \times E_8$ or similar and with the WZW term being the difference between the differentially twisted looping of the universal Chern-Simons 3-connection of $Spin$ with that of $E_8 \times E_8$.

By (Fiorenza-Rogers-Schreiber 13, section 2.6.1) we find in this case that the Heisenberg 2-group is the string 2-group

$$Heis(\mathbf{L}_{WZW^{het}}) \simeq String(G) \,.$$

It follows thus from the above discussion that a consistent heterotic sigma model requires that the underlying $G$-principal bundle on spacetime admits a string structure. This is the famous Green-Schwarz anomaly cancellation condition, re-expressed as a consistency condition for parameterized WZW models.

On the level of action functionals in codimension 0 this observation is due to (Distler-Sharpe 10, section 7).

The Green-Schwarz super $p$-branes on curved super spacetime

By Fiorenza-Sati-Schreiber 13 the super-branes of string theory/M-theory on super Minkowski spacetime $\mathbb{R}^{d-1,1;N}$ classified by the brane scan/the brane bouqet are ∞-Wess-Zumino-Witten theory sigma-model

$$\mathbf{L}_{WZW^{brane}} \;\colon\; \mathbb{R}^{d-1,1;N} \longrightarrow \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn} \,,$$

where $\Gamma$ is the group of periods of the defining super L-∞ algebra L-∞ cocycle

$$\mathbf{B} sIso_N(d-1,1) \longrightarrow \mathbf{B}^{p+2}(\mathbb{R}/\Gamma) \,.$$

A necessary structure globalizing this to a curved super-spacetime $X$ is a G-structure on $X$ for $G = \mathbf{QuantMorph}(\mathbf{L}_{WZW^{brane}}^{inf})$ the restriction of the WZW term to the infinitesimal disk.

Given this then the Green-Schwarz action functional refines to a local prequantum field theory datum $\mathbf{L}_{WZW^{brane}}$ globally defined on $X$.

(For branes on which other branes may end, such as the D-branes and the M5-brane, here super Minkowski spacetime is replaced by a extended Minkowski super spacetime, as discussed in (Fiorenza-Sati-Schreiber 13)).

More details are in (cwzw).

Related concepts

• gauged WZW model

• definite globalization of WZW term

• higher Cartan geometry

References

Parameterized WZW models as sigma models for the heterotic string originate in

• Jim Gates, Warren Siegel, Leftons, Rightons, Nonlinear $\sigma$-Models, and Superstrings, Phys.Lett. B206 (1988) 631 (spire)

• Jim Gates, Strings, superstrings, and two-dimensional lagrangian field theory, pp. 140-184 in Z. Haba, J. Sobczyk (eds.) Functional integration, geometry, and strings, proceedings of the XXV Winter School of Theoretical Physics, Karpacz, Poland (Feb. 1989), , Birkhäuser, 1989.

• Jim Gates, S. Ketov, S. Kozenko, O. Solovev, Lagrangian chiral coset construction of heterotic string theories in $(1,0)$ superspace, Nucl.Phys. B362 (1991) 199-231 (spire)

Discussion relating this to equivariant elliptic genera is in section 7 of

• Jacques Distler, Eric Sharpe, Heterotic compactifications with principal bundles for general groups and general levels, Adv. Theor. Math. Phys. 14:335-398, 2010 (arXiv:hep-th/0701244)

Review of this includes

• Eric Sharpe, Recent developments in heterotic compactifications, in Eric Sharpe, Arthur Greenspoon, Advances in String Theory: The First Sowers Workshop in Theoretical Physics, AMS 2008 (pdf slides (39-49))

The corresponding elliptic genera had been considered in

• Matthew Ando, The sigma orientation for analytic circle-equivariant elliptic cohomology, Geom. Topol. 7 (2003) 91-153 (arXiv:math/0201092)

and with more emphasis on equivariant elliptic cohomology in

• Matthew Ando, Equivariant elliptic cohomology and the Fibered WZW models of Distler and Sharpe, talk 2007 (lecture notes pdf)

The discussion of the relevant Heisenberg n-group theory is in

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, Higher geometric prequantum theory, 2013 (arXiv:1304.0236)

with more details in

• Urs Schreiber, Obstruction theory for parameterized higher WZW terms

• Urs Schreiber, section “definite forms” in differential cohomology in a cohesive topos

General ∞-Wess-Zumino-Witten theory is set up in section 6 of

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields, 2013 (arXiv:1308.5264)