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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Green-Schwarz mechanism} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory} [[!include string theory - contents]] \hypertarget{physics}{}\paragraph*{{Physics}}\label{physics} [[!include physicscontents]] \hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology} [[!include differential cohomology - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{the_higher_magnetic_charge_anomaly}{The higher magnetic charge anomaly}\dotfill \pageref*{the_higher_magnetic_charge_anomaly} \linebreak \noindent\hyperlink{YMactionfunctional}{The higher abelian Yang-Mills action functional\ldots{}}\dotfill \pageref*{YMactionfunctional} \linebreak \noindent\hyperlink{WithElectricCharge}{\ldots{} with electric charge \ldots{}}\dotfill \pageref*{WithElectricCharge} \linebreak \noindent\hyperlink{_and_with_magnetic_charge}{\ldots{} and with magnetic charge.}\dotfill \pageref*{_and_with_magnetic_charge} \linebreak \noindent\hyperlink{the_anomaly_line_bundle}{The anomaly line bundle}\dotfill \pageref*{the_anomaly_line_bundle} \linebreak \noindent\hyperlink{the_greenschwarz_mechanism}{The Green-Schwarz mechanism}\dotfill \pageref*{the_greenschwarz_mechanism} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{heterotic_supergravity}{Heterotic supergravity}\dotfill \pageref*{heterotic_supergravity} \linebreak \noindent\hyperlink{axions_and_the_strong_cp_problem_in_heterotic_supergravity}{Axions and the strong CP problem in heterotic supergravity}\dotfill \pageref*{axions_and_the_strong_cp_problem_in_heterotic_supergravity} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{for__supergravity}{For $10 D$ supergravity}\dotfill \pageref*{for__supergravity} \linebreak \noindent\hyperlink{in_axion_physics}{In axion physics}\dotfill \pageref*{in_axion_physics} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A \textbf{Green-Schwarz mechanism} (named after [[Michael Green]] and [[John Schwarz]]) is a modification of an [[action functional]] of a [[quantum field theory]] involving higher [[gauge field]]s that makes a [[quantum anomaly]] of the original action functional disappear. More in detail: \begin{itemize}% \item An [[action functional]] in [[path integral|path integral quantization]] is said to be [[quantum anomaly|anomalous]] if it is only locally identified with a function on the configuration space of fields, but is globally instead a section of a line [[bundle]] (usually equipped with [[connection on a bundle|connection]]). \item Given two anomalous action functionals in this sense, it may happen that while the two corresponding line bundles on configuration space are each nontrivial, their [[tensor product]] becomes trivializable. In this case one can consider the non-anomalous action function given by the sum of the two anomalous action functionals. This is what is called \textbf{anomaly cancellation} of one piece of an action functional against another. \item The two main sources of examples for action functionals that are anomalous are \begin{itemize}% \item the standard [[fermionic action functional]] (see there for details) for chiral fermions \begin{itemize}% \item this is a section of the [[Pfaffian line bundle]] for the [[Dirac operator]] \end{itemize} \item the action functional for [[differential cohomology|differential cocycles]] (higher connection, higher [[gauge theory]]) in the presence of [[electric charge|electric]] and [[magnetic charge]]s: \begin{itemize}% \item this is a [[section]] of the [[transgression]] of the [[line bundle]] arising as the [[cup product]] of the [[electric charge|electric]] and [[magnetic charge|magnetic differential cocycles]]. \end{itemize} \end{itemize} \end{itemize} A \textbf{Green--Schwarz mechanism} is the addition of an action functional for higher differential cocycles with [[magnetic charge]]s such that their [[quantum anomaly]] cancels a given [[Pfaffian line bundle]]: so it is a choice of by itself ill-defined action functional for higher gauge theory that cancels the ill-definedness of an action functional for chiral fermions. In the more strict and original sense of the word, \emph{the} Green--Schwarz mechanism is the application of this procedure in the theory called [[heterotic string theory|heterotic supergravity]]: there it so happens that the Pfaffian line bundle of the fermionic action has as [[Chern class]] the [[transgression]] of a degree-12 class in [[ordinary differential cohomology]] that factorizes as $I_8 \wedge I_4$. Since heterotic supergravity contains a higher gauge field that couples to [[string theory|strings]], this is precisely of the form $J_{electric} \wedge J_{magnetic}$ that the anomaly for the corresponding higher gauge theory in the presence of magnetic charges gives rise to. So the original Green--Schwarz anomaly cancellation mechanism consist of modifying the ``naive'' action functional for heterotic supergravity by adding the contribution that corresponds to adding a magnetic current of the form \begin{displaymath} j_B := I_4 \,. \end{displaymath} \hypertarget{the_higher_magnetic_charge_anomaly}{}\subsection*{{The higher magnetic charge anomaly}}\label{the_higher_magnetic_charge_anomaly} \hypertarget{YMactionfunctional}{}\subsubsection*{{The higher abelian Yang-Mills action functional\ldots{}}}\label{YMactionfunctional} Consider on some [[spacetime]] $X$ a [[gauge field]] $[\nabla] \in H_{diff}^{n+1}(X)$ modeled in [[ordinary differential cohomology]] in degree $n+1$: a [[circle n-bundle with connection]]. For instance \begin{itemize}% \item in degree $n=1$ this is an [[electromagnetic field]]; \item in degree $n=2$ this is a [[Kalb-Ramond field]]; \item in degree $n = 3$ a [[supergravity C-field]]. \end{itemize} Canonically associated to the [[gauge field]] is its [[field strength]]: the [[curvature]] [[differential form]] \begin{displaymath} F_\nabla \in \Omega^{n+1}_{cl}(X) \,. \end{displaymath} The abelian [[Yang-Mills action functional]] for our gauge field (the [[action functional]] of higher order [[electromagnetism]]) is the [[function]] \begin{displaymath} \exp(i S_{YM}(-)) : H_{diff}^{n+1}(X) \to \mathbb{C} \end{displaymath} that sends the field $\nabla$ \begin{displaymath} \exp(i S_{YM}(-)) : [\nabla] \mapsto \exp(-i \int_X F_\nabla \wedge \star F_\nabla) \end{displaymath} to the [[exponential]] of the [[integral]] over the [[spacetime]] $X$ of the [[differential form]] obtained as the [[wedge product]] of the [[curvature form]] with its image under the [[Hodge star operator]] correspondonding to the [[pseudo-Riemannian manifold|pseudo-Riemannian metric]] on $X$. The fact that this map [[function]] defined in terms of [[cocycle]]s is a well defined function on [[cohomology]] means that the [[action functional]] is [[gauge transformation|gauge invariant]]. At this point this is just the trivial statement that under a [[gauge transformation]] \begin{displaymath} \nabla \stackrel{g}{\to} \nabla' \end{displaymath} the [[curvature]] invariant: $F_\nabla = F_{\nabla'}$. \hypertarget{WithElectricCharge}{}\subsubsection*{{\ldots{} with electric charge \ldots{}}}\label{WithElectricCharge} The \hyperlink{YMactionfunctional}{above} [[action functional]] describes the dynamics of the [[gauge field]] all by itself, with no interactions with other fields or with [[relativistic particle|fundamental particle]]s/[[brane|fundamental brane]]s. A distribution of $n$-[[electric charge]] on $X$ is modeled itself by a [[cocycle]] $\hat j_E$ in [[ordinary differential cohomology]] in degree $dim X - n$ \begin{displaymath} [\hat j_E] \in H_{diff}^{dim X - n}(X) \,. \end{displaymath} The [[curvature]] $j_E$ of $\hat j_E$ is the [[electric current]] form. The [[action functional]] that encodes the [[force]] of the [[gauge field]] exerted on this electric charge distribution is locally on [[coordinate chart]]s $U \subset X$ given by the [[integral]] $\int_X A_U \wedge j_E$, where $A_U$ is the local [[connection on an infinity-bundle|connection]] $n$-form of the gauge field $\nabla$. Globally, this contribution is given by the push-forward \begin{displaymath} 2 \pi i\int_X (-) : H_{diff}^{dim X}(X) \to H_{diff}^0(*) = U(1) \end{displaymath} of the [[cup product]] $\hat j_E \cdot \nabla$ in [[ordinary differential cohomology]]. In total the [[action functional]] of higher abelian [[Yang-Mills theory]] in the presence of [[electric charge]] is the [[function]] \begin{displaymath} \exp(i S_{YM}(-) + i S_{el}(-)) : H^{n+1}_{diff}(X) \times H^{dim X - n}_{diff}(X) \to \mathbb{C} \end{displaymath} given by \begin{displaymath} ([\nabla], [\hat j_E]) \mapsto \exp(i \int_X F_\nabla \wedge \star F_\nabla) \exp(2 \pi i \int_X \hat j_E \cdot \nabla) \,. \end{displaymath} \hypertarget{_and_with_magnetic_charge}{}\subsubsection*{{\ldots{} and with magnetic charge.}}\label{_and_with_magnetic_charge} We now consider one more additional term in the [[action functional]], one that describes moreover the interaction of our [[gauge field]] with a distribution of $n$-[[magnetic charge]] on $X$, in addition to the interaction with the distribution of [[electric charge]] described \hyperlink{WithElectricCharge}{above}. The [[magnetic charge]] distribution itself is also modeled as a [[cocycle]] $\hat j_B$ in [[ordinary differential cohomology]]. As opposed to the [[electric charge]] it is however not part of the dynamics but of the kinematics of the system: it does not manifestly show up in the integral expression for the [[action functional]], but does modify the nature of the configuration space that this action functional is defined on. Namely the kinematic higher [[Maxwell equations]] is a condition of the form \begin{displaymath} d F_\nabla = j_B \,, \end{displaymath} where $j_B$ is the [[magnetic charge]] distribution. If $F_\nabla$ is the [[curvature]] of a [[circle n-bundle with connection]], then necessarily $d F_\nabla = 0$. Therefore the system of higher electromagnetism in the presence of magnetic charge cannot be modeled any more by [[cocycle]]s in [[ordinary differential cohomology]]. One finds instead that one has to model $\nabla$ not as a [[circle n-bundle with connection]], but as an $n$-[[twisted bundle]] with connection, where the [[twisted cohomology|twist]] is $\hat j_B$. We shall write $C^{n+1}_{diff}(X)_{\hat j_B}$ for the [[∞-groupoid]] of twisted cocycles for this fixed twist. The crucial point is now the following: the \hyperlink{WithElectricCharge}{above} expression \begin{displaymath} \exp(i S_{el}(\nabla, \hat j_E)) : \exp(2 \pi i \int_X \hat j_E \cdot \nabla) \end{displaymath} for the electric coupling can still be given sense, even with $\nabla \in C^{n+1}_{diff}(X)_{\hat j_B}$, but it no longer has the interpretation of a [[circle group]]-valued [[function]]. Rather, it has now the interpretation of a [[section]] of a [[line bundle]] \begin{displaymath} \itexarray{ && Anom_{\hat j_B} \\ & {}^{\mathllap{\exp(S_{el})}}\nearrow & \downarrow \\ Conf &=& Conf } \end{displaymath} on configuration space. The [[characteristic class]] of this line bundle -- its [[Chern class]] -- is hence the \emph{magnetic anomaly} in higher gauge theory. In the next section we formalize properly the notion of this line bundle on configuration space. \hypertarget{the_anomaly_line_bundle}{}\subsubsection*{{The anomaly line bundle}}\label{the_anomaly_line_bundle} In order to formalize this we have to refine the formalization of the structure of the configuration space. So far we had regarded the [[set]] $H^{dim X - n}_{diff}(X) \times H_{diff}^{n+1}(X)$ of gauge equivalence classes of field configurations. This is the set of connected components of the full [[cocycle]] [[∞-groupoid]] \begin{displaymath} C^{dim X - n}_{diff}(X) \times C_{diff}^{n+1}(X) \in \infty Grpd \end{displaymath} whose \begin{itemize}% \item [[object]]s are field configurations on $X$; \item [[morphism]]s are [[gauge transformation]]s; \item [[2-morphism]]s are gauge transformations of gauge transformation, \item and so on. \end{itemize} Moreover this cocycle [[∞-groupoid]] is not just a [[discrete ∞-groupoid]] but it naturally has \emph{smooth structure} : it is naturally a [[smooth ∞-groupoid]]: an [[∞-stack]] over the [[category]] [[SmoothMfd]]. We shall write \begin{displaymath} Conf := [X,(\mathbf{B}^{n}U(1) \times \mathbf{B}^{dim X - n-1}U(1))_{conn}] \in Smooth\infty Grpd \end{displaymath} for this smooth $\infty$-groupoid of configuration of the physical system -- defined as the [[internal hom]] in terms of the [[Smooth∞Grpd]] of $X \in SmoothMfd \hookrightarrow Smooth\infty Grpd$ into the target object of the higher [[gauge theory]], (this object is discussed in detail ; it is presented under the [[Dold-Kan correspondence]] by the [[Deligne complex]] of sheaves on [[CartSp]]). This smooth structure is characterized by saying that for any $U \in$ [[SmoothMfd]] the $U$-parameterized smooth families of field configurations, gauge transformations, etc. form the [[∞-groupoid]] \begin{displaymath} Conf(U) \simeq C^{dim X - n}_{diff}(U \times X) \times C_{diff}^{n+1}(U \times X ) \end{displaymath} of gauge fields on the [[product]] of [[spacetime]] $X$ with the parameter space $U$. (See for instance [[Lie integration]] and [[connection on an ∞-bundle]] for details on how differential forms on $U \times X$ encode $U$-families of forms on $X$). This way the configuration space of higher electromagnetism in the presence of electric and magnetic charge is naturally incarnated as an object in the [[cohesive (∞,1)-topos]] of [[smooth ∞-groupoid]]s, and accordingly all the in cohesive $(\infty,1)$-topos are available. In particular we may speak of [[circle n-bundle with connection|line bundle with connection]] on $Conf$, given for instance by morphisms \begin{displaymath} Anom_{\hat j_B} : Conf \to \mathbf{B} U(1)_{conn} \end{displaymath} in [[Smooth∞Grpd]]. We say \begin{itemize}% \item the underlying class in [[ordinary cohomology]] \begin{displaymath} [Anom_{\hat j_B}] \in H^1(Conf, U(1)) \end{displaymath} is \textbf{the anomaly} of the system of higher electromagnetism coupled to electric and magnetic charge; \item its [[curvature]] 2-form \begin{displaymath} Curv_{Anom_{\hat j_B}} : Conf \to \mathbf{\flat}_{dR} \mathbf{B}^2 \mathbb{R} \end{displaymath} is the \textbf{differential anomaly}. \end{itemize} One finds that this curvature 2-form is given by the [[fiber integration]] of the wedge product of the [[electric charge]] $(n+1)$-form with the [[magnetic charge]] $dim X - n$-form over $X$: \begin{displaymath} Curv_{Anom_{\hat j_B}} = \int_X j_E \wedge j_B \,. \end{displaymath} This means that for every parameter space $U \in$ [[SmoothMfd]] and every morphism $\phi : U \to Conf$ -- which corresponds by the nature of the [[∞-stack]] $Conf$ to a field configuration $(\nabla, \hat j_E) \in C^{n+1}_{diff}(U \times X) \times C^{dim X - n}_{diff}(U \times X)$ -- the pullback of this differential form to $U$ yields the ordinary differential form $\int_X j_E \wedge j_B$ in the image of $(\nabla, \hat j_E)$ under the [[fiber integration]] map \begin{displaymath} \int_X(-) : \Omega^\bullet(U \times X) \to \Omega^\bullet(U) \,. \end{displaymath} \hypertarget{the_greenschwarz_mechanism}{}\subsubsection*{{The Green-Schwarz mechanism}}\label{the_greenschwarz_mechanism} We can now state the \textbf{Green-Schwarz mechanism} itself. Let $\hat Conf \in$ [[Smooth∞Grpd]] be the configuration space of a physical system that contains among its fields higher abelian gauge theory with [[electric charge]] with configuration space $Conf$ \begin{displaymath} \hat Conf = Conf_{rest} \times Conf \end{displaymath} and equipped with an [[action functional]] \begin{displaymath} \exp(i S_{rest}(-) + i S_{el}(-)) : \hat Conf \to Anom_{rest} \end{displaymath} that is a [[section]] of an anomaly line bundle $Anom_{rest}$ \begin{displaymath} \itexarray{ && Anom_{rest} \\ & {}^{\mathllap{\exp(S_{tot})}}\nearrow & \downarrow \\ Conf_{rest} &=& Conf_{rest} } \end{displaymath} such that the [[curvature]] 2-form of $Anom_{tot}$ happens to be of the form \begin{displaymath} Curv{Anom_{ref}} = \int_X I_{n+2} \wedge I_{(dim X - n)} \,, \end{displaymath} for some $I_{n+2} \in \Omega^{n+2}_{cl}(X)$ and $I_{dim X - n} \in \Omega^{dim X - n}(X)$. Then the \textbf{Green-Schwarz mechanism} is the map that changes this physical system by adding magnetic charge to it, given by a cocycle $\hat j_B$ with \begin{displaymath} [\hat j_B] = - [Anom_{rest}] \end{displaymath} \begin{displaymath} j_B = - I_{dim X - n} \,. \end{displaymath} This means by the above that the new [[action functional]] is now a section \begin{displaymath} \exp(i S_{rest}(-) + i S_{el}(-)) : Conf_{rest} \times Conf \to Anom_{rest} \otimes Anom_{\hat j_B} \end{displaymath} of the [[tensor product]] of the two anomaly line bundles. The [[Chern class]] of the tensor product is the sum of the two Chern-classes, hence by definition of $j_B$ they cancel, so that $Anom_{rest} \otimes Anom_{\hat j_B}$ is trivializatable as a line bundle with connection. A choice of such trivialization identifies the section then with an ordinary function \begin{displaymath} \exp(i S_{rest}(-) + i S_{el}(-)) : Conf_{rest} \times Conf \to U(1) \,. \end{displaymath} This is the anomaly-free action functional after the Green-Schwarz mechanism has been applied. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{heterotic_supergravity}{}\subsubsection*{{Heterotic supergravity}}\label{heterotic_supergravity} The original work of Green-Schwarz concerned anomaly cancellation in the effective [[supergravity]] theory on a $dim X = 10$-dimensional target [[spacetime]] in [[heterotic string theory]]. The configurations of this theory are given by \begin{itemize}% \item a [[Spin]]-[[principal bundle]] [[connection on a bundle|with connection]] $\hat \omega$ -- the [[spin connection]] (the [[graviton]]); \item a [[unitary group]]-[[principal bundle]] [[connection on a bundle|with connection]] $\hat A$ (the [[Yang-Mills field]]); \item a [[circle n-bundle with connection|circle 2-bundle with connection]] $\hat B$ -- the [[Kalb-Ramond field]]. \item [[section]]s of the [[associated bundle|associated]] [[spinor bundle]]s: the fermions $\psi$ ([[gravitino]], [[gaugino]], [[dilatino]]). \end{itemize} The [[path integral]] over the fermionic part of the action \begin{displaymath} \exp(i S_{ferm}(-)) : (\omega, A, B, \psi) \mapsto \exp(i \int_X \bar \psi D_{\omega,A} \psi) \end{displaymath} is an anomalous action functional on the configuration space of the remaining bosonic fields $(\hat A, \hat \omega)$, a [[section]] of a [[Pfaffian line bundle]], whose curvature form turns out to be \begin{displaymath} curv_{Pfaff} = \int_X I_4 \wedge I_8 \end{displaymath} with \begin{displaymath} I_4 = \frac{1}{2} p_1(F_\omega) - ch_2(F_A) \end{displaymath} the difference between the (image in [[de Rham cohomology]] of the) first fractional [[Pontryagin class]] of the $Spin$-principal bundle and second [[Chern class]] of the [[unitary group]]-[[principal bundle]] and \begin{displaymath} I_8 = \frac{1}{48} p_2(F_\omega) - ch_4(F_A) + ... \end{displaymath} where the ellipses indicate decomposable [[curvature characteristic form]]s. Therefore in this case the Green-Schwarz mechanism consists of \begin{enumerate}% \item adding to the system \emph{fivebrane [[magnetic charge]]} $j_B \in \Omega^{2+2}$ given by $I_4$. This means that the Kalb-Ramond field $\hat B$ becomes a twisted field whose [[field strength]] $H$ is no longer closed, but satisfies the kinematical [[Maxwell equation]] \begin{displaymath} d H = I_4 \,. \end{displaymath} \item adding to the system \emph{string [[electric charge]]} $j_E \in \Omega^{10 - 2}(X)$ . This means that to the [[action functional]] is added the factor \end{enumerate} \begin{displaymath} \exp(i \int_X \hat B \cdot \hat I_8 ) \end{displaymath} which is locally on $U \hookrightarrow X$ given in the exponent by the integral \begin{displaymath} \int_U B_U \wedge I_8 \,. \end{displaymath} The nature of the field configuration obtained this way -- spin connection with twist of th Kalb-Ramond field by the Pontryagin class -- may be understood conciesely as constituting a [[differential string structure|twisted differential string structure]] on $X$. See there for more details. \hypertarget{axions_and_the_strong_cp_problem_in_heterotic_supergravity}{}\subsubsection*{{Axions and the strong CP problem in heterotic supergravity}}\label{axions_and_the_strong_cp_problem_in_heterotic_supergravity} The Green-Schwarz anomaly cancellation mechanism naturally makes the twisted [[B-field]] in [[heterotic string theory]] behave like the [[axion]] with the correct potential to serve as the [[theta angle]] and serve as the solution to the [[strong CP problem]] (\hyperlink{SvrcekWitten06}{Svrcek-Witten 06}): \begin{quote}% An obvious question about the axion hypothesis is how natural it really is. Why introduce a global PQ ``symmetry'' if it is not actually a symmetry? What is the sense in constraining a theory so that the classical Lagrangian possesses a certain symmetry if the symmetry is actually anomalous? It could be argued that the best evidence that PQ ``symmetries'' are natural comes from string theory, which produces them without any contrivance. \ldots{} the string compactifications always generate PQ symmetries, often spontaneously broken at the string scale and producing axions, but sometimes unbroken.(\hyperlink{SvrcekWitten06}{Svrcek-Witten 06, pages 3-4}) \end{quote} In [[heterotic string theory]] [[KK-compactification|KK-compactified]] to 4d, the 4d [[B-field]], dualized, serves as the axion field, called the ``model independent axion'' (\hyperlink{SvrcekWitten06}{Svrcek-Witten 06, starting p. 15}). This works as follows: By the [[Green-Schwarz anomaly cancellation]] mechanism, then [[B-field]] in [[heterotic string theory]] is a twisted 2-form field, whose [[field strength]] $H$ locally has in addition to the exact differential $d B$ also a fundamental 3-form contribution, so that \begin{displaymath} H = d B + C \end{displaymath} (locally). Moreover, the differential $d H$ is constrained to be the Pontryagin 4-form of the gauge potential $\nabla$ (minus that of the [[Riemann curvature]], but let's suppress this notationally for the present purpose): \begin{displaymath} d H = tr \left(F_\nabla \wedge F_\nabla\right) \,. \end{displaymath} Now suppose [[KK-compactification]] to 4d has been taken care of, then this constraint may be implemented in the [[equations of motion]] by adding it to the [[action functional]], multiplied with a [[Lagrange multiplier]] : \begin{displaymath} S = \underset{ \text{kinetic action} \atop \text{for B-field} }{ \underbrace{\int_X H \wedge \star H} } + \underset{ \text{Green-Schwarz constraint} }{ \underbrace{ \int_X a \left( d H - tr(F_\nabla \wedge F_\nabla) \right) } } \,. \end{displaymath} Now by the usual argument, one says that instead of varying by $a$ and thus implementing the [[Green-Schwarz anomaly cancellation]] constraint, it is equivalent to fist vary with respect to the other fields, and then insert the resulting equations in terms of $a$ into the action functional. Now since we are dealing with a twisted [[B-field]], with free 3-form component $C$, we actually vary with respect to $C$. This yields the [[Euler-Lagrange equation]] [[equation of motion|of motion]] \begin{displaymath} d a = \star H \,, \end{displaymath} hence the usual relation or [[electro-magnetic duality]], expressing what used to be the [[Lagrange multiplier]] now as the magentic dual [[field (physics)|field]] to the twisted [[B-field]]. Plugging this algebraic [[equation of motion]] back into the above [[action functional]] for $H$ gives \begin{displaymath} \tilde S = \underset{\text{kinetic action} \atop \text{for axion field}}{\underbrace{\int_X d a \wedge \star d a}} + \underset{\text{axionic} \atop \text{interaction}}{\underbrace{\int_X a \, tr(F_\nabla \wedge F_\nabla)}} \,. \end{displaymath} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[Freed-Witten anomaly]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} A clear and precise account of what the relevant anomalies are and what the Green-Schwarz mechanism is to cancel them is given in (see also the relevant bits at \emph{[[eta invariant]]}) \begin{itemize}% \item [[Edward Witten]], section 2.2 of \emph{World-Sheet Corrections Via D-Instantons}, JHEP 0002:030,2000 (\href{http://arxiv.org/abs/hep-th/9907041}{arXiv:9907041}) \end{itemize} Review, broader context and further discussion is given in \begin{itemize}% \item [[Dan Freed]], \emph{[[Dirac charge quantization and generalized differential cohomology]]} (\href{http://arxiv.org/abs/hep-th/0011220}{arXiv:hep-th/0011220}) \end{itemize} Discussion [[higher gauge theory]]: \begin{itemize}% \item [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{Twisted differential String and Fivebrane structures}, Commun. Math. Phys. 315 (2012), 169-213 (\href{https://arxiv.org/abs/0910.4001}{arXiv:0910.4001}) \item [[Clay Cordova]], [[Thomas Dumitrescu]], [[Kenneth Intriligator]], \emph{Exploring 2-Group Global Symmetries} (\href{https://arxiv.org/abs/1802.04790}{arXiv:1802.04790}) \end{itemize} \hypertarget{for__supergravity}{}\subsubsection*{{For $10 D$ supergravity}}\label{for__supergravity} An account of historical developments is in section 7 of \begin{itemize}% \item [[John Schwarz]], \emph{The Early Years of String Theory: A Personal Perspective} in \emph{\href{books+about+string+theory#TheBirthOfStringTheory}{The birth of string theory}} (\href{http://arxiv.org/abs/0708.1917}{arXiv:0708.1917}) \end{itemize} The full formula for the differential form data including the fermionic contributions is in \begin{itemize}% \item L. Bonora, M. Bregola, R. D'Auria, P. Fr\'e{} K. Lechner, P. Pasti, I. Pesando, M. Raciti, F. Riva, M. Tonin and D. Zanon, \emph{Some remarks on the supersymmetrization of the Lorentz Chern-Simons form in $D = 10$ $N= 1$ supergravity theories}, Physics Letters B 277 (1992) ([[BonoraSuperGS.pdf:file]]) \end{itemize} and references given there. \hypertarget{in_axion_physics}{}\subsubsection*{{In axion physics}}\label{in_axion_physics} Discussion relating to [[axions]] is in \begin{itemize}% \item Peter Svrcek, [[Edward Witten]], \emph{Axions In String Theory}, JHEP 0606:051,2006 (\href{http://arxiv.org/abs/hep-th/0605206}{arXiv:hep-th/0605206}) \end{itemize} [[!redirects Green-Schwarz mechanisms]] [[!redirects Green Schwarz mechanism]] [[!redirects Green-Schwarz mechanism]] [[!redirects Green?Schwarz mechanism]] [[!redirects Green--Schwarz mechanism]] [[!redirects Green Schwartz mechanism]] [[!redirects Green-Schwartz mechanism]] [[!redirects Green?Schwartz mechanism]] [[!redirects Green--Schwartz mechanism]] [[!redirects Green-Schwarz anomaly cancellation]] [[!redirects Green-Schwarz quantum anomaly cancellation]] [[!redirects Green-Schwarz anomaly]] [[!redirects Green–Schwarz mechanism]] [[!redirects Green–Schwarz mechanisms]] \end{document}