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parameterized WZW model

Context

\infty-Wess-Zumino-Witten theory

Models

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Quantum field theory

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Contents

Idea

Where a WZW model is a sigma model quantum field theory whose target space is a group GG, 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 GG-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 nn-dimensional sigma-model field theory whose target space is a group GG and whose interaction-part of the action functional is the higher holonomy of a circle n-bundle with connection

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

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

If one has a GG-principal bundle

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

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

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

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

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

Heis G(L WZW)String(G) 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 XX combined with a chiral GG-WZW model for G=Spin×E 8×E 8G = 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 SpinSpin with that of E 8×E 8E_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(L WZW het)String(G). 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 GG-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 pp-branes on curved super spacetime

By Fiorenza-Sati-Schreiber 13 the super-branes of string theory/M-theory on super Minkowski spacetime d1,1;N\mathbb{R}^{d-1,1;N} classified by the brane scan/the brane bouqet are ∞-Wess-Zumino-Witten theory sigma-model

L WZW brane: d1,1;NB p+1(/Γ) conn, \mathbf{L}_{WZW^{brane}} \;\colon\; \mathbb{R}^{d-1,1;N} \longrightarrow \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn} \,,

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

BsIso N(d1,1)B p+2(/Γ). \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 XX is a G-structure on XX for G=QuantMorph(L WZW brane inf)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 L WZW brane\mathbf{L}_{WZW^{brane}} globally defined on XX.

(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).

References

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

  • Jim Gates, Warren Siegel, Leftons, Rightons, Nonlinear σ\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)(1,0) superspace, Nucl.Phys. B362 (1991) 199-231 (spire)

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

Review of this includes

The corresponding elliptic genera had been considered in

and with more emphasis on equivariant elliptic cohomology in

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

with more details in

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

Last revised on February 19, 2015 at 20:30:42. See the history of this page for a list of all contributions to it.