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April 27, 2008

Charges and Twisted Bundles, IV: Anomaly Cancellation

Posted by Urs Schreiber

Last time # I had talked about how the presence of electric and magnetic charges makes the would-be action functional of (bosonic, abelian, possibly higher) gauge theory a section of a potentially nontrivial line bundle Charge conf bos \array{ Charge \\ \downarrow \\ conf_{bos} } with connection on the space of fields, here called conf bosconf_{bos}. This time I talk about how this “anomaly cancels” against another anomly caused by spinorial fields: the Pfaffian line bundle.

In the presence of further fields on top of the (abelian, here) gauge fields (i.e. the (higher) connections on (higher) line bundle) there may be other nontrivial line bundles on configuration space such that the action functional is a section of the tensor product of all of them.

In particular, if there are also fermionic fields ψ\psi with the standard contribution Xψ¯D ϕψ \propto \int_X \bar\psi D_\phi \psi to the action functional, where D ϕD_\phi is a Dirac operator depending on the bosonic fields, the path integral over them is taken to compute the “determinant” of the Dirac operator, which is, for each ϕ\phi an element in a Pfaffian line. These glue to the Pfaffian line bundle Pfaff conf bos \array{ Pfaff \\ \downarrow \\ conf_{bos} } over conf bosconf_{bos}. Hence also the fermionic contribution to the action functional may be anomalous in that this line bundle may be nontrivial.

But the full action functional e Se^{-S} is a section in the tensor bundle ChargePfaff e S conf bos. \array{ Charge \otimes Pfaff \\ \downarrow & \uparrow^{e^{-S}} \\ conf_{bos} } \,.

Anomaly cancellation hence occurs when this tensor product bundle is trivializable. In fact, all line bundles occuring here are line bundles with connection, and the consistent interpretation of the section e Se^{-S} with a complex function requires a choice of isomorphism ChargePfaffconf bos× Charge \otimes Pfaff \stackrel{\simeq}{\to} conf_{bos} \times \mathbb{C} of line bundles with connection, with the trivial bundle on the right carrying the trivial connection.

One hence says that the curvature 2-form curv(ChargePfaff)curv(Charge \otimes Pfaff) of the anomaly bundle is the local anomaly, while its holonomy group Hol(ChargePfaff)Hol(Charge \otimes Pfaff) is the global anomaly. The famous Green-Schwarz anomaly cancellation mechanism is the construction of a suitable charge anomaly line bundle such that it cancels a given Pfaffian anomaly line bundle:

the supergravity theory wich is the effective target space theory of the heterotic string is, in its usual formulation, a 2-gauge theory for “electrically charged” strings (2-particles) propagating on a (d=10(d=10)-dimensional target space. Therefore their magnetic duals are (d(n+1)2)=5(d-(n+1)-2) = 5-branes (6-particles). The fermionic fields in the theory (called the dilatino. the gravitino and the gaugino) can be computed to produce an anomaly line bundle whose curvature 2-form has the form curv(Pfaff)= XI 4I 8Ω closed 2(conf bos) curv(Pfaff) = \int_X I_4 \wedge I_8 \in \Omega^2_{closed}(conf_{bos}) where the integrand 12-form happens to factor into a 4-form I 4I_4 and an 8-form I 8I_8. Here the integral is to be understood in the sense discussed last time. More explicitly, I 4I_4 and I 8I_8 depend on the bosonic fields given by the Levi-Civita connection ω\omega on the Spin bundle of XX and a connection AA on a complex vector bundle on XX as I 4=12p 1(F ω)ch 2(F A) I_4 = \frac{1}{2}p_1(F_\omega) - ch_2(F_A) I 8=148p 2(F ω)ch 4(F A)+decomposablecharacteristicforms. I_8 = \frac{1}{48} p_2(F_\omega) - ch_4(F_A) + decomposable characteristic forms \,.

As also discussed last time, the curvature of the charge anomaly line bundle in the presence of 5-brane magnetic current measured by a 4-form j Bj_B and electric string current measured by an 8-form j Ej_E is of just the same form curv(Charge)= Xj Bj E. curv(Charge) = \int_X j_B \wedge j_E \,.

So here it is obvious how the (local) anomaly is to be cancelled: we identify the magenetic current with I 4I_4 and the electric current with I 8I_8. j B=I 4 j_B = I_4 and electric current with I 8I_8 j E=I 8. j_E = I_8 \,.

Once this identification has been done, the precise setup needed for anomaly cancellation can be derived by simply matching with the general formulas listed last time:

first we need to change the configuration space of the theory: the electric field which used to be a line 2-bundle with connection – the Kalb-Ramond field – whose curvature 3-form was necessarily closed dH 3=0 d H_3 = 0 has to be taken now into a twisted line 2-bundle with connection, which constitutes a “section” of the twisting line 3-bundle for which j Bj_B is the curvature 4-form: dH 3=j B=I 4=12p 1(F ω)ch 2(F A). d H_3 = j_B \;\;= I_4 = \frac{1}{2}p_1(F_\omega) - ch_2(F_A) \,. For fixed j^ B\hat j_B, a point in the new conf bosconf_{bos} now specifies one such twisted 2-bundle for j^ B\hat j_B being the twist.

Having changed the configuration space, we next modify the action functional. We need to find the term that needs to be added to the original anomalous (due to the fermions) action functional such that the charge anomaly enters the game. Denoting by BB the local connection 2-form of the electric 2-bundle that the string couples to (the Kalb-Ramond field) this was the term that locally reads XBj E= XBI 8, \int_X B \wedge j_E = \int_X B \wedge I_8 \,, encoding the coupling of the electric charge distribution j Ej_E to the electric background field BB. We add this term (or rather its proper interpretation in terms of push-forward of differential cocycles) to the former action function and interpret the result as a new action functional on the new conf bosconf_{bos}.

Doing so produces, by construction, an anomaly free action functional: the charges have cancelled the fermionic anomaly. This is the Green-Schwarz anomaly cancellation mechanism.

Remarkably, this anomaly cancellation has also a different interpretation: from the point of view not of the target space theory, but of the worldsheet theory of the electric string, the equation dH 3=12p 1(F ω)ch 2(F A), d H_3 = \frac{1}{2}p_1(F_\omega) - ch_2(F_A) \,, lifted properly to an identity in differential cohomology, says that the virtual difference of the Spin-lift of the tangent bundle minus the gauge bundle have string structure. This is the precise analog for a string of the condition that the tangent bundle on the target of a 1-partcile needs Spin-structure.

Posted at April 27, 2008 6:17 PM UTC

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Re: Charges and Twisted Bundles, IV: Anomaly Cancellation

In response to public demand here in Lisbon I have finally taken the time to write out the story of the Green-Schwarz mechanism on the nnLab: here.

Posted by: Urs Schreiber on February 12, 2011 12:07 AM | Permalink | Reply to this

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