nLab
Green-Schwarz mechanism

Idea

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

    • this is a section of the Pfaffian line bundle for the Dirac operator

  • the action functional for differential cocycles (higher connection, higher gauge theory) in the presence of electric and magnetic charges:

    • this is a section of the transgression of the line bundle arising as the cup product of the electric and magnetic differential cocycles.

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 $J_{\mathrm{electric}}\wedge J_{\mathrm{magnetic}}$ 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

$$j_B:=I_4.$$j_B := I_4 \,.

The higher magnetic charge anomaly

The higher abelian Yang-Mills action functional…

Consider on some spacetime $X$ a gauge field $[\nabla ]\in H_{\mathrm{diff}}^{n+1}(X)$ 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

$$F_{\nabla }\in {\Omega }_{\mathrm{cl}}^{n+1}(X).$$F_\nabla \in \Omega^{n+1}_{cl}(X) \,.

The abelian Yang-Mills action functional for our gauge field (the action functional of higher order electromagnetism) is the function

$$\exp (i{S}_{\mathrm{YM}}(-)):H_{\mathrm{diff}}^{n+1}(X)\to ℂ$$\exp(i S_{YM}(-)) : H_{diff}^{n+1}(X) \to \mathbb{C}

that sends the field $\nabla$

$$\exp (i{S}_{\mathrm{YM}}(-)):[\nabla ]\mapsto \exp (-i{\int }_X{F}_{\nabla }\wedge \star F_{\nabla })$$\exp(i S_{YM}(-)) : [\nabla] \mapsto \exp(-i \int_X F_\nabla \wedge \star F_\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

$$\nabla \stackrel{g}{\to }\nabla \prime$$\nabla \stackrel{g}{\to} \nabla'

the curvature invariant: $F_{\nabla }=F_{\nabla \prime }$.

… with electric charge …

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 $\dim X-n$

$$[{\hat{j}}_E]\in H_{\mathrm{diff}}^{\dim X-d}(X).$$[\hat j_E] \in H_{diff}^{dim X - d}(X) \,.

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

$$2\pi i{\int }_X(-):H_{\mathrm{diff}}^{\dim X}(X)\to H_{\mathrm{diff}}^0(*)=U(1)$$2 \pi i\int_X (-) : H_{diff}^{dim X}(X) \to H_{diff}^0(*) = U(1)

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

$$\exp (i{S}_{\mathrm{YM}}(-)+i{S}_{\mathrm{el}}(-)):H_{\mathrm{diff}}^{n+1}(X)\times H_{\mathrm{diff}}^{\dim X-n}(X)\to ℂ$$\exp(i S_{YM}(-) + i S_{el}(-)) : H^{n+1}_{diff}(X) \times H^{dim X - n}_{diff}(X) \to \mathbb{C}

given by

$$([\nabla ],[{\hat{j}}_E])\mapsto \exp (i{\int }_X{F}_{\nabla }\wedge \star F_{\nabla })\exp (2\pi i{\int }_X{\hat{j}}_E\cdot \nabla ).$$([\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) \,.

… and with magnetic charge.

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

$$d{F}_{\nabla }=j_B,$$d F_\nabla = j_B \,,

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 $C_{\mathrm{diff}}^{n+1}(X{)}_{{\hat{j}}_B}$ for the ∞-groupoid of twisted cocycles for this fixed twist. The crucial point is now the following:

the above expression

$$\exp (i{S}_{\mathrm{el}}(\nabla ,{\hat{j}}_E)):\exp (2\pi i{\int }_X{\hat{j}}_E\cdot \nabla )$$\exp(i S_{el}(\nabla, \hat j_E)) : \exp(2 \pi i \int_X \hat j_E \cdot \nabla)

for the electric coupling can still be given sense, even with $\nabla \in C_{\mathrm{diff}}^{n+1}(X{)}_{{\hat{j}}_B}$, 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

$$\begin{array}{ccc}& & {\mathrm{Anom}}_{{\hat{j}}_B}\\ & {}^{\exp (S_{\mathrm{el}})}\nearrow & \downarrow \\ \mathrm{Conf}& =& \mathrm{Conf}\end{array}$$\array{ && Anom_{\hat j_B} \\ & {}^{\mathllap{\exp(S_{el})}}\nearrow & \downarrow \\ Conf &=& Conf }

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.

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 $H_{\mathrm{diff}}^{\dim X-n}(X)\times H_{\mathrm{diff}}^{n+1}(X)$ of gauge equivalence classes of field configurations. This is the set of connected components of the full cocycle ∞-groupoid

$$C_{\mathrm{diff}}^{\dim X-n}(X)\times C_{\mathrm{diff}}^{n+1}(X)\in \mathrm{\infty }\mathrm{Grpd}$$C^{dim X - n}_{diff}(X) \times C_{diff}^{n+1}(X) \in \infty Grpd

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

$$\mathrm{Conf}:=[X,(B^{n}U(1)\times B^{\dim X-n-1}U(1){)}_{\mathrm{conn}}]\in \mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}$$Conf := [X,(\mathbf{B}^{n}U(1) \times \mathbf{B}^{dim X - n-1}U(1))_{conn}] \in Smooth\infty Grpd

for this smooth $\mathrm{\infty }$-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 $X\in \mathrm{SmoothMfd}\hookrightarrow \mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}$ 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

$$\mathrm{Conf}(U)\simeq C_{\mathrm{diff}}^{\dim X-n}(U\times X)\times C_{\mathrm{diff}}^{n+1}(U\times X)$$Conf(U) \simeq C^{dim X - n}_{diff}(U \times X) \times C_{diff}^{n+1}(U \times X )

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 $(\mathrm{\infty },1)$-topos are available. In particular we may speak of line bundle with connection on $\mathrm{Conf}$, gevin for instance by morphisms

$${\mathrm{Anom}}_{{\hat{j}}_B}:\mathrm{Conf}\to BU(1{)}_{\mathrm{conn}}$$Anom_{\hat j_B} : Conf \to \mathbf{B} U(1)_{conn}

in Smooth∞Grpd.

We say

• the underlying class in ordinary cohomology

  $$[{\mathrm{Anom}}_{{\hat{j}}_B}]\in H^1(\mathrm{Conf},U(1))$$[Anom_{\hat j_B}] \in H^1(Conf, U(1))

  is the anomaly of the system of higher electromagnetism couple to electic and magnetic charge;

• its curvature 2-form

  $${\mathrm{Curv}}_{{\mathrm{Anom}}_{{\hat{j}}_B}}:\mathrm{Conf}\to {♭}_{\mathrm{dR}}{B}^2ℝ$$Curv_{Anom_{\hat j_B}} : Conf \to \mathbf{\flat}_{dR} \mathbf{B}^2 \mathbb{R}

  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 $X$:

$${\mathrm{Curv}}_{{\mathrm{Anom}}_{{\hat{j}}_B}}={\int }_X{j}_E\wedge j_B.$$Curv_{Anom_{\hat j_B}} = \int_X j_E \wedge j_B \,.

This means that for every parameter space $U\in$ SmoothMfd and every morphism $\varphi :U\to \mathrm{Conf}$ – which corresponds by the nature of the ∞-stack $\mathrm{Conf}$ to a field configuration $(\nabla ,{\hat{j}}_E)\in C_{\mathrm{diff}}^{n+1}(U\times X)\times C_{\mathrm{diff}}^{\dim X-n}(U\times X)$ – 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

$${\int }_X(-):{\Omega }^{•}(U\times X)\to {\Omega }^{•}(U).$$\int_X(-) : \Omega^\bullet(U \times X) \to \Omega^\bullet(U) \,.

The Green-Schwarz mechanism

We can now state the Green-Schwarz mechanism itself.

Let $\widehat{\mathrm{Conf}}\in$ 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 $\mathrm{Conf}$

$$\widehat{\mathrm{Conf}}={\mathrm{Conf}}_{\mathrm{rest}}\times \mathrm{Conf}$$\hat Conf = Conf_{rest} \times Conf

and equipped with an action functional

$$\exp (i{S}_{\mathrm{rest}}(-)+i{S}_{\mathrm{el}}(-)):\widehat{\mathrm{Conf}}\to {\mathrm{Anom}}_{\mathrm{rest}}$$\exp(i S_{rest}(-) + i S_{el}(-)) : \hat Conf \to Anom_{rest}

that is a section of an anomaly line bundle ${\mathrm{Anom}}_{\mathrm{rest}}$

$$\begin{array}{ccc}& & {\mathrm{Anom}}_{\mathrm{rest}}\\ & {}^{\exp (S_{\mathrm{tot}})}\nearrow & \downarrow \\ {\mathrm{Conf}}_{\mathrm{rest}}& =& {\mathrm{Conf}}_{\mathrm{rest}}\end{array}$$\array{ && Anom_{rest} \\ & {}^{\mathllap{\exp(S_{tot})}}\nearrow & \downarrow \\ Conf_{rest} &=& Conf_{rest} }

such that the curvature 2-form of ${\mathrm{Anom}}_{\mathrm{tot}}$ happens to be of the form

$$\mathrm{Curv}{\mathrm{Anom}}_{\mathrm{ref}}={\int }_X{I}_{n+2}\wedge I_{(\dim X-n)},$$Curv{Anom_{ref}} = \int_X I_{n+2} \wedge I_{(dim X - n)} \,,

for some $I_{n+2}\in {\Omega }_{\mathrm{cl}}^{n+2}(X)$ 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

$$[{\hat{j}}_B]=-[{\mathrm{Anom}}_{\mathrm{rest}}]$$[\hat j_B] = - [Anom_{rest}]

$$j_B=-I_{\dim X-n}.$$j_B = - I_{dim X - n} \,.

This means by the above that the new action functional is now a section

$$\exp (i{S}_{\mathrm{rest}}(-)+i{S}_{\mathrm{el}}(-)):{\mathrm{Conf}}_{\mathrm{rest}}\times \mathrm{Conf}\to {\mathrm{Anom}}_{\mathrm{rest}}\otimes {\mathrm{Anom}}_{{\hat{j}}_B}$$\exp(i S_{rest}(-) + i S_{el}(-)) : Conf_{rest} \times Conf \to Anom_{rest} \otimes Anom_{\hat j_B}

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 ${\mathrm{Anom}}_{\mathrm{rest}}\otimes {\mathrm{Anom}}_{{\hat{j}}_B}$ is trivializatable as a line bundle with connection.

A choice of such trivialization identifies the section then with an ordinary function

$$\exp (i{S}_{\mathrm{rest}}(-)+i{S}_{\mathrm{el}}(-)):{\mathrm{Conf}}_{\mathrm{rest}}\times \mathrm{Conf}\to U(1).$$\exp(i S_{rest}(-) + i S_{el}(-)) : Conf_{rest} \times Conf \to U(1) \,.

This is the anomaly-free action functional after the Green-Schwarz mechanism has been applied.

Examples

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

• 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

$$\exp (i{S}_{\mathrm{ferm}}(-)):(\omega ,A,B,\psi )\mapsto \exp (i{\int }_X\overline{\psi }{D}_{\omega ,A}\psi )$$\exp(i S_{ferm}(-)) : (\omega, A, B, \psi) \mapsto \exp(i \int_X \bar \psi D_{\omega,A} \psi)

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

$${\mathrm{curv}}_{\mathrm{Pfaff}}={\int }_X{I}_4\wedge I_8$$curv_{Pfaff} = \int_X I_4 \wedge I_8

with

$$I_4=\frac{1}{2}{p}_1(F_{\omega })-{\mathrm{ch}}_2(F_A)$$I_4 = \frac{1}{2} p_1(F_\omega) - ch_2(F_A)

the difference between the (image in de Rham cohomology of the) first fractional Pontryagin class of the $\mathrm{Spin}$-principal bundle and second Chern class of the unitary group-principal bundle

and

$$I_8=\frac{1}{48}{p}_2(F_{\omega })-{\mathrm{ch}}_4(F_A)+...$$I_8 = \frac{1}{48} p_2(F_\omega) - ch_4(F_A) + ...

where the ellipses indicate decomposable curvature characteristic forms.

Therefore in this case the Green-Schwarz mechanism consists of

1. 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

   $$dH=I_4.$$d H = I_4 \,.

2. 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

$$\exp (i{\int }_X\hat{B}\cdot {\hat{I}}_8)$$\exp(i \int_X \hat B \cdot \hat I_8 )

which is locally on $U\hookrightarrow X$ given in the exponent by the integral

$${\int }_U{B}_U\wedge I_8.$$\int_U B_U \wedge I_8 \,.

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.

References

A clear and precise account of what anomalies are and what the Green-Schwarz mechanism is to cancel them is given in

• Dan Freed, Dirac charge quantization and generalized differential cohomology (arXiv:hep-th/0011220)

For the moment, see there for further references.

For $10D$ supergravity

An account of historical developments is in section 7 of

• John Schwarz, The Early Years of String Theory: A Personal Perspective in The birth of string theory (arXiv:0708.1917)

The full formula for the differential form data including the fermionic contributions is in

• L. Bonora, M. Bregola, R. D’Auria, P. Fré K. Lechner, P. Pasti, I. Pesando, M. Raciti, F. Riva, M. Tonin and D. Zanon, Some remarks on the supersymmetrization of the Lorentz Chern-Simons form in $D=10$ $N=1$ supergravity theories (pdf)

and references given there.