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A 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 fields that makes a quantum anomaly of the original action functional disappear.
More in detail:
An action functional in path integral quantization is said to be 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).
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 anomaly cancellation of one piece of an action functional against another.
The two main sources of examples for action functionals that are anomalous are
the standard fermionic action functional? (see there for details) for chiral fermions
the action functional for differential cocycles (higher connection, higher gauge theory) in the presence of electric and magnetic charges:
A Green–Schwarz mechanism is the addition of an action functional for higher differential cocycles with magnetic charges 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, the Green–Schwarz mechanism is the application of this procedure in the theory called 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 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
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
in degree $n=1$ this is an electromagnetic field;
in degree $n=2$ this is a Kalb-Ramond field;
in degree $n = 3$ a supergravity C-field.
Canonically associated to the gauge field is its field strength: the curvature differential form
The abelian Yang-Mills action functional for our gauge field (the action functional of higher order electromagnetism) is the function
that sends the field $\nabla$
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 metric on $X$.
The fact that this map function defined in terms of cocycles is a well defined function on cohomology means that the action functional is gauge invariant. At this point this is just the trivial statement that under a gauge transformation
the curvature invariant: $F_\nabla = F_{\nabla'}$.
The above action functional describes the dynamics of the gauge field all by itself, with no interactions with other fields or with fundamental particles/fundamental branes.
A distribution of $n$-electric charge on $X$ is modeled itself by a cocycle $\hat j_E$ in ordinary differential cohomology in degree $dim X - n$
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 charts $U \subset X$ given by the integral $\int_X A_U \wedge j_E$, where $A_U$ is the local connection $n$-form of the gauge fiedl $\nabla$.
Globally, this contribution is given by the push-forward
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
given by
We now consider one more additional term in the action functional, one that desribes 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 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 kinemtics 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
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 cocycles in ordinary differential cohomology.
One fines that instead, more one has to model $\nabla$ not as a circle n-bundle with connection, but as an $n$-twisted bundle with connection, where the twisgt 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 above expression
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
on configuration space. The characteristic class of this line bundle – its Chern class – is hence the magnetic anomaly in higher gauge theory.
In the next section we formalize properly the notion of this line bundle on configuration space.
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
whose
objects are field configurations on $X$;
morphisms are gauge transformations;
2-morphisms are gauge transformations of gauge transformation,
and so on.
Moreover this cocycle ∞-groupoid is not just a discrete ∞-groupoid but it naturally has smooth structure : it is naturally a smooth ∞-groupoid: an ∞-stack over the category SmoothMfd. We shall write
for this smooth $\infty$-groupoid of configuration of the physical system – defined as the internal hom in terms of the closed monoidal structure on the (∞,1)-topos 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 here; 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
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 ∞-groupoids, and accordingly all the differential geometric structures in cohesive $(\infty,1)$-topos are available. In particular we may speak of line bundle with connection on $Conf$, gevin for instance by morphisms
in Smooth∞Grpd.
We say
the underlying class in ordinary cohomology
is the anomaly of the system of higher electromagnetism couple to electic and magnetic charge;
its curvature 2-form
is the differential anomaly.
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 $X$:
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
We can now state the 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 $Conf$
and equipped with an action functional
that is a section of an anomaly line bundle $Anom_{rest}$
such that the curvature 2-form of $Anom_{tot}$ happens to be of the form
for some $I_{n+2} \in \Omega^{n+2}_{cl}(X)$ and $I_{dim X - n} \in \Omega^{dim X - n}(X)$.
Then the 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
This means by the above that the new action functional is now a section
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
This is the anomaly-free action functional after the Green-Schwarz mechanism has been applied.
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
a Spin-principal bundle with connection $\hat \omega$ – the spin connection (the graviton);
a unitary group-principal bundle with connection $\hat A$ (the Yang-Mills field);
a circle 2-bundle with connection $\hat B$ – the Kalb-Ramond field.
sections of the associated spinor bundles: the fermions $\psi$ (gravitino, gaugino, dilatino).
The path integral over the fermionic part of the action
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
with
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
where the ellipses indicate decomposable curvature characteristic forms.
Therefore in this case the Green-Schwarz mechanism consists of
adding to the system 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
adding to the system string electric charge $j_E \in \Omega^{10 - 2}(X)$ .
This means that to the action functional is added the factor
which is locally on $U \hookrightarrow X$ given in the exponent by the integral
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 twisted differential string structure on $X$. See there for more details.
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 (Svrcek-Witten 06):
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. … the string compactifications always generate PQ symmetries, often spontaneously broken at the string scale and producing axions, but sometimes unbroken.(Svrcek-Witten 06, pages 3-4)
In heterotic string theory KK-compactified to 4d, the 4d B-field, dualized, serves as the axion field, called the “model independent axion” (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
(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):
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 :
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 of motion
hence the usual relation or electro-magnetic duality, expressing what used to be the Lagrange multiplier now as the magentic dual field to the twisted B-field.
Plugging this algebraic equation of motion back into the above action functional for $H$ gives
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 eta invariant)
Review, broader context and further discussion is given in
Discussion higher gauge theory:
Hisham Sati, Urs Schreiber, Jim Stasheff, Twisted differential String and Fivebrane structures, Commun. Math. Phys. 315 (2012), 169-213 (arXiv:0910.4001)
Clay Cordova, Thomas Dumitrescu, Kenneth Intriligator, Exploring 2-Group Global Symmetries (arXiv:1802.04790)
An account of historical developments is in section 7 of
The full formula for the differential form data including the fermionic contributions is in
and references given there.
Discussion relating to axions is in