nLab Green-Schwarz mechanism



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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:

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 8I 4I_8 \wedge I_4. Since heterotic supergravity contains a higher gauge field that couples to strings, this is precisely of the form J electricJ magneticJ_{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

j BI 4. j_B \,\coloneqq\, I_4 \,.

The higher magnetic charge anomaly

The higher abelian Yang-Mills action functional…

Consider on some spacetime XX a gauge field []H diff n+1(X)[\nabla] \in H_{diff}^{n+1}(X) modeled in ordinary differential cohomology in degree n+1n+1: a circle n-bundle with connection. For instance

Canonically associated to the gauge field is its field strength: the curvature differential form

F Ω 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(iS YM()):H diff n+1(X) \exp\big( \mathrm{i} S_{YM}(-) \big) \;\colon\; H_{diff}^{n+1}(X) \to \mathbb{C}

that sends the field \nabla

exp(iS YM()):[]exp(i XF F ) \exp\big( \mathrm{i} S_{YM}(-)\big) \;\colon\; [\nabla] \mapsto \exp\big( - \mathrm{i} \textstyle{\int}_X F_\nabla \wedge \star F_\nabla \big)

to the exponential of the integral over the spacetime XX 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 XX.

The fact that this map 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

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

the curvature is invariant (due to the higher gauge group being abelian): F =F F_\nabla = F_{\nabla'}.

… 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 nn-electric charge on XX is modeled itself by a cocycle j^ E\hat j_E in ordinary differential cohomology in degree dimXndim X - n

[j^ E]H diff dimXn(X), [\hat j_E] \;\in\; H_{diff}^{dim X - n}(X) \,,

so that the curvature j Ej_E of j^ E\hat j_E is the electric current form. The action functional that encodes the Lorentz force of the gauge field exerted on this electric charge distribution is locally on coordinate charts UXU \subset X given by the integral XA Uj E\int_X A_U \wedge j_E, where A UA_U is the local connection nn-form of the gauge field \nabla.

Globally, this contribution is given by the push-forward

2πi X():H flat dimX+1(X)H flat 1(*)=U(1). 2 \pi \mathrm{i} \textstyle{\int}_X (-) \;\colon\; H_{flat}^{dim X + 1}(X) \to H_{flat}^1(*) \;=\; U(1) \,.

of the cup product j^ E\hat j_E \cdot \nabla in ordinary differential cohomology.

In total then, the action functional of higher abelian Yang-Mills theory in the presence of electric charge is the function

exp(iS YM()+iS el()):H diff n+1(X)×H diff dimXn(X) \exp\big( \mathrm{i} S_{YM}(-) + \mathrm{i} S_{el}(-) \big) \;\;\colon\;\; H^{n+1}_{diff}(X) \times H^{dim X - n}_{diff}(X) \longrightarrow \mathbb{C}

given by

([],[j^ E])exp(i XF F )exp(2πi Xj^ E). \big( [\nabla], [\hat j_E] \big) \;\mapsto\; \exp\big( \mathrm{i} \textstyle{\int}_X F_\nabla \wedge \star F_\nabla \big) \, \exp\big( 2 \pi \mathrm{i} \int_X \hat j_E \cdot \nabla \big) \,.

[cf. Freed 2002 (2.17)]

… 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 nn-magnetic charge on XX, in addition to the interaction with the distribution of electric charge described above.

The magnetic charge distribution itself is also modeled as a cocycle j^ B\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

dF =j B, \mathrm{d} F_\nabla \;=\; j_B \,,

where j Bj_B is the magnetic charge distribution. If F F_\nabla is the curvature of a circle n-bundle with connection, then necessarily dF =0d 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 finds instead that one has to model \nabla not as a circle n-bundle with connection, but as an nn-twisted bundle with connection, where the twist is j^ B\hat j_B.

We shall write C diff n+1(X) j^ BC^{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

exp(iS el(,j^ E)):exp(2πi Xj^ E) \exp\big( \mathrm{i} S_{el}(\nabla, \hat j_E) \big) \;\colon\; \exp\big( 2 \pi \mathrm{i} \textstyle{\int}_X \hat j_E \cdot \nabla \big)

for the electric coupling can still be given sense, even with C diff n+1(X) j^ B\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.

[cf. Freed 2002 (2.29)]

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 diff dimXn(X)×H diff n+1(X)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

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


Moreover, this cocycle ∞-groupoid is not just a discrete ∞-groupoid but it naturally has smooth structure making it a smooth ∞-groupoid: an ∞-stack over the category SmoothMfd. We shall write

Conf[X,(B nU(1)×B dimXn1U(1)) conn]SmoothGrpd Conf \;\coloneqq\; \Big[ X ,\, \big( \mathbf{B}^{n}U(1) \times \mathbf{B}^{dim X - n-1}U(1) \big)_{conn} \Big] \;\in\; SmoothGrpd_{\infty}

for this smooth \infty-groupoid of configurations of the physical system – defined as the internal hom in terms of the closed monoidal structure on the (∞,1)-topos Smooth∞Grpd of XSmoothMfdSmoothGrpdX \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 UU \in SmoothMfd the UU-parameterized smooth families of field configurations, gauge transformations, etc. form the ∞-groupoid

Conf(U)C diff dimXn(U×X)×C diff n+1(U×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 XX with the parameter space UU. (See for instance Lie integration and connection on an ∞-bundle for details on how differential forms on U×XU \times X encode UU-families of forms on XX).

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 (,1)(\infty,1)-topos are available. In particular we may speak of line bundles with connection on ConfConf, given for instance by morphisms

Anom j^ B:ConfBU(1) conn. Anom_{\hat j_B} \;\colon\; Conf \to \mathbf{B} U(1)_{conn} \,.

in Smooth∞Grpd.

We say that:

  • the underlying class in ordinary cohomology

    [Anom j^ B]H 2(Conf,) [Anom_{\hat j_B}] \in H^2(Conf, \mathbb{Z})

    is the anomaly of the system of higher electromagnetism coupled to electric and magnetic charge;

  • its curvature 2-form

    Curv Anom j^ B:ConfΩ flat 2 Curv_{Anom_{\hat j_B}} \;\colon\; Conf \to \Omega^2_{flat}

    is the differential anomaly.

One finds that this curvature 2-form is given by the fiber integration of the wedge product of the electric current (n+1)(n+1)-form with the magnetic charge dimXndim X - n-form over XX:

Curv Anom j^ B= Xj Ej B. Curv_{Anom_{\hat j_B}} \,=\, \textstyle{\int}_X \, j_E \wedge j_B \,.

This means that for every parameter space UU \in SmoothMfd and every morphism ϕ:UConf\phi : U \to Conf – which corresponds by the nature of the ∞-stack ConfConf to a field configuration (,j^ E)C diff n+1(U×X)×C diff dimXn(U×X)(\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 UU yields the ordinary differential form Xj Ej B\int_X j_E \wedge j_B which is the image of (,j^ E)(\nabla, \hat j_E) under the fiber integration map

X():Ω (U×X)Ω (U). \textstyle{\int}_X(-) \;\colon\; \Omega^\bullet(U \times X) \to \Omega^\bullet(U) \,.

The Green-Schwarz mechanism

We can now state the Green-Schwarz mechanism itself.

Let Conf^\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 ConfConf

Conf^=Conf rest×Conf \hat Conf \,=\, Conf_{rest} \times Conf

and equipped with an action functional

exp(iS rest()+iS el()):Conf^Anom rest \exp\big( \mathrm{i} S_{rest}(-) + \mathrm{i} S_{el}(-) \big) \;\colon\; \hat Conf \to Anom_{rest}

that is a section of an anomaly line bundle Anom restAnom_{rest}

Anom rest exp(S tot) Conf rest = Conf rest \array{ && Anom_{rest} \\ & {}^{\mathllap{\exp(S_{tot})}}\nearrow & \downarrow \\ Conf_{rest} &=& Conf_{rest} }

such that the curvature 2-form of Anom totAnom_{tot} happens to be of the form

CurvAnom ref= XI n+2j el, Curv{Anom_{ref}} \,=\, \textstyle{\int}_X I_{n+2} \wedge j_{el} \,,

for some I n+2Ω cl n+2(X)I_{n+2} \in \Omega^{n+2}_{cl}(X) (understood, as explained above, as forms in Ω cl n+2(X×U)\Omega^{n+2}_{cl}(X\times U)) and j elΩ dimXn(X)j_{el} \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 j^ B\hat j_B with

[j^ B]=[Anom rest] [\hat j_B] \,=\, - [Anom_{rest}]
j B=I n+2. j_B \,=\, - I_{n+2} \,.

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

exp(iS rest()+iS el()):Conf rest×ConfAnom restAnom j^ B \exp\big( \mathrm{i} S_{rest}(-) + \mathrm{i} S_{el}(-) \big) \,\colon\, Conf_{rest} \times Conf \to Anom_{rest} \otimes Anom_{\hat j_B}

of the tensor product of the two anomaly line bundles. The first Chern class of the tensor product is the sum of the two 1st Chern-classes, hence by definition of j Bj_B they cancel, so that Anom restAnom j^ BAnom_{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

exp(iS rest()+iS el()):Conf rest×ConfU(1). \exp\big( \mathrm{i} S_{rest}(-) + \mathrm{i} S_{el}(-)\big) \;\colon\; Conf_{rest} \times Conf \to U(1) \,.

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


Heterotic supergravity

The original work of Green-Schwarz concerned anomaly cancellation in the effective supergravity theory on a dimX=10dim X = 10-dimensional target spacetime in heterotic string theory.

The configurations of this theory are given by

The path integral over the fermionic part of the action

exp(iS ferm()):(ω,A,B,ψ)exp(i Xψ¯D ω,Aψ) \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 (A^,ω^)(\hat A, \hat \omega), a section of a Pfaffian line bundle, whose curvature form turns out to be

curv Pfaff= XI 4I 8 curv_{Pfaff} = \int_X I_4 \wedge I_8


I 4=12p 1(F ω)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 SpinSpin-principal bundle and second Chern class of the unitary group-principal bundle


I 8=148p 2(F ω)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 BΩ 2+2j_B \in \Omega^{2+2} given by I 4I_4.

    This means that the Kalb-Ramond field B^\hat B becomes a twisted field whose field strength HH 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 EΩ 102(X)j_E \in \Omega^{10 - 2}(X) .

    This means that to the action functional is added the factor

exp(i XB^I^ 8) \exp(i \int_X \hat B \cdot \hat I_8 )

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

UB UI 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 XX. See there for more details.

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 (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 HH locally has in addition to the exact differential dBd B also a fundamental 3-form contribution, so that

H=dB+C H = d B + C

(locally). Moreover, the differential dHd 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):

dH=tr(F F ). d H = tr \left(F_\nabla \wedge F_\nabla\right) \,.

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 :

S= XHHkinetic actionfor B-field+ Xa(dHtr(F F ))Green-Schwarz constraint. 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) } } \,.

Now by the usual argument, one says that instead of varying by aa 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 aa into the action functional.

Now since we are dealing with a twisted B-field, with free 3-form component CC, we actually vary with respect to CC. This yields the Euler-Lagrange equation of motion

da=H, d a = \star H \,,

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 HH gives

S˜= Xdadakinetic actionfor axion field+ Xatr(F F )axionicinteraction. \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)}} \,.



The original articles are

Rederivation using the elliptic genus/Witten genus (then: “character-valued partition function”):


Review (with an eye towards KK-compactification to 6d, see also at D=6 N=(1,0) SCFT):

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

An account of historical developments is in section 7 of

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=10D = 10 N=1N= 1 supergravity theories, Physics Letters B 277 (1992) (pdf)

and references given there.

Discussion relating to axions:

Higher gauge theory of the Green-Schwarz mechanism

Discussion of higher gauge theory modeling the Green-Schwarz mechanisms for anomaly cancellation in heterotic string theory, on M5-branes, and in related systems in terms of some kind of nonabelian differential cohomology (ordered by arXiv time-stamp):

Last revised on February 13, 2024 at 12:19:30. See the history of this page for a list of all contributions to it.