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\section*{parameterized WZW model}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{wesszuminowitten_theory}{}\paragraph*{{$\infty$-Wess-Zumino-Witten theory}}\label{wesszuminowitten_theory}

[[!include infinity-Wess-Zumino-Witten theory - contents]]

\hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory}

[[!include functorial quantum field theory - contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{the_heterotic_string}{The heterotic string}\dotfill \pageref*{the_heterotic_string} \linebreak
\noindent\hyperlink{the_greenschwarz_super_branes_on_curved_super_spacetime}{The Green-Schwarz super $p$-branes on curved super spacetime}\dotfill \pageref*{the_greenschwarz_super_branes_on_curved_super_spacetime} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea}

Where a \emph{[[WZW model]]} is a [[sigma model]] [[quantum field theory]] whose [[target space]] is a [[group]] $G$, a \emph{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 (\hyperlink{GatesSiegel88}{Gates-Siegel 88}, \hyperlink{GKKS91}{Gates-Ketov-Kozenko-Solovev 91}, \hyperlink{DistlerSharpe10}{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 ([[schreiber:∞-Wess-Zumino-Witten theory|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]]

\begin{displaymath}
\mathbf{L}_{WZW}
  \;\colon\;
  G \longrightarrow 
  \mathbf{B}^n U(1)_{conn}
\end{displaymath}
(whose [[curvature]] is given by the global [[Maurer-Cartan form]] on $G$).

If one has a $G$-[[principal bundle]]

\begin{displaymath}
\itexarray{
    G &\hookrightarrow& X
    \\
    && \downarrow
    \\
    && B
  }
\end{displaymath}
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]]

\begin{displaymath}
\mathbf{L}_{WZW}^X \;\colon\; X \longrightarrow \mathbf{B}^n U(1)_{conn}
\end{displaymath}
on the total space $X$ of this bundle. This hence yields what one may think of as a coherent collection of [[WZW models]] \emph{parameterized} over base space $B$.

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

\begin{displaymath}
Heis_G(\mathbf{L}_{WZW})\simeq String(G)
\end{displaymath}
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 \hyperlink{cwzw}{cwzw} for details of this claim).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{the_heterotic_string}{}\subsubsection*{{The heterotic string}}\label{the_heterotic_string}

The [[heterotic string]] [[worldsheet]] [[theory (physics)|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 circle 3-bundle|universal Chern-Simons 3-connection]] of $Spin$ with that of $E_8 \times E_8$.

By (\hyperlink{FiorenzaRogersSchreiber13}{Fiorenza-Rogers-Schreiber 13, section 2.6.1}) we find in this case that the [[Heisenberg 2-group]] is the [[string 2-group]]

\begin{displaymath}
Heis(\mathbf{L}_{WZW^{het}})
  \simeq
  String(G)
  \,.
\end{displaymath}
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 (\hyperlink{DistlerSharpe10}{Distler-Sharpe 10, section 7}).

\hypertarget{the_greenschwarz_super_branes_on_curved_super_spacetime}{}\subsubsection*{{The Green-Schwarz super $p$-branes on curved super spacetime}}\label{the_greenschwarz_super_branes_on_curved_super_spacetime}

By \hyperlink{FiorenzaSatiSchreiber13}{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]]/[[schreiber:The brane bouquet|the brane bouqet]] are [[schreiber:∞-Wess-Zumino-Witten theory]] [[sigma-model]]

\begin{displaymath}
\mathbf{L}_{WZW^{brane}}
  \;\colon\;
  \mathbb{R}^{d-1,1;N} \longrightarrow \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn}
  \,,
\end{displaymath}
where $\Gamma$ is the group of [[periods]] of the defining [[super L-∞ algebra]] [[L-∞ algebra cohomology|L-∞ cocycle]]

\begin{displaymath}
\mathbf{B} sIso_N(d-1,1) \longrightarrow \mathbf{B}^{p+2}(\mathbb{R}/\Gamma)
  \,.
\end{displaymath}
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 (\hyperlink{FiorenzaSatiSchreiber13}{Fiorenza-Sati-Schreiber 13})).

More details are in (\hyperlink{cwzw}{cwzw}).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[gauged WZW model]]


\item [[definite globalization of WZW term]]


\item [[higher Cartan geometry]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

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

\begin{itemize}%
\item [[Jim Gates]], [[Warren Siegel]], \emph{Leftons, Rightons, Nonlinear $\sigma$-Models, and Superstrings}, Phys.Lett. B206 (1988) 631 (\href{https://inspirehep.net/record/251286/}{spire})


\item [[Jim Gates]], \emph{Strings, superstrings, and two-dimensional lagrangian field theory}, pp. 140-184 in Z. Haba, J. Sobczyk (eds.) \emph{Functional integration, geometry, and strings}, proceedings of the XXV Winter School of Theoretical Physics, Karpacz, Poland (Feb. 1989), , Birkh\"a{}user, 1989.


\item [[Jim Gates]], S. Ketov, S. Kozenko, O. Solovev, \emph{Lagrangian chiral coset construction of heterotic string theories in $(1,0)$ superspace}, Nucl.Phys. B362 (1991) 199-231 (\href{http://inspirehep.net/record/314337/?ln=en}{spire})



\end{itemize}
Discussion relating this to equivariant [[elliptic genera]] is in section 7 of

\begin{itemize}%
\item [[Jacques Distler]], [[Eric Sharpe]], \emph{Heterotic compactifications with principal bundles for general groups and general levels}, Adv. Theor. Math. Phys. 14:335-398, 2010 (\href{http://arxiv.org/abs/hep-th/0701244}{arXiv:hep-th/0701244})

\end{itemize}
Review of this includes

\begin{itemize}%
\item [[Eric Sharpe]], \emph{Recent developments in heterotic compactifications}, in [[Eric Sharpe]], [[Arthur Greenspoon]], \emph{\href{http://www.ams.org/bookstore-getitem/item=AMSIP-44}{Advances in String Theory: The First Sowers Workshop in Theoretical Physics}}, AMS 2008 (\href{http://www.phys.vt.edu/sowers/talks/sharpe-sowers.pdf}{pdf slides (39-49)})

\end{itemize}
The corresponding [[elliptic genera]] had been considered in

\begin{itemize}%
\item [[Matthew Ando]], \emph{The sigma orientation for analytic circle-equivariant elliptic cohomology}, Geom. Topol. 7 (2003) 91-153 (\href{http://arxiv.org/abs/math/0201092}{arXiv:math/0201092})

\end{itemize}
and with more emphasis on [[equivariant elliptic cohomology]] in

\begin{itemize}%
\item [[Matthew Ando]], \emph{Equivariant elliptic cohomology and the Fibered WZW models of Distler and Sharpe}, \href{http://www.math.ucsb.edu/~drm/GTPseminar/2007-fall.php}{talk 2007} (\href{http://www.math.ucsb.edu/~drm/GTPseminar/notes/20071026-ando/20071026-malmendier.pdf}{lecture notes pdf})

\end{itemize}
The discussion of the relevant [[Heisenberg n-group]] theory is in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:Higher geometric prequantum theory]]}, 2013 (\href{http://arxiv.org/abs/1304.0236}{arXiv:1304.0236})

\end{itemize}
with more details in

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:Obstruction theory for parameterized higher WZW terms]]}


\item [[Urs Schreiber]], section ``definite forms'' in \emph{[[schreiber:differential cohomology in a cohesive topos]]}



\end{itemize}
General [[schreiber:∞-Wess-Zumino-Witten theory]] is set up in section 6 of

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]}, 2013 (\href{http://arxiv.org/abs/1308.5264}{arXiv:1308.5264})

\end{itemize}
[[!redirects parameterized WZW models]] [[!redirects parameterized Wess-Zumino-Witten model]] [[!redirects parameterized Wess-Zumino-Witten models]]

[[!redirects parametrized WZW model]] [[!redirects parametrized WZW models]] [[!redirects parametrized Wess-Zumino-Witten model]]

[[!redirects parametrized Wess-Zumino-Witten models]]

[[!redirects fibered WZW model]] [[!redirects fibered WZW models]] [[!redirects fibered Wess-Zumino-Witten model]] [[!redirects fibered Wess-Zumino-Witten models]]

[[!redirects parameterized WZW term]] [[!redirects parameterized WZW terms]]

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[[!redirects definite parameterizations of higher WZW terms]]



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