The n-Category Café

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 $\psi$ with the standard contribution $$\propto \int_X \bar\psi D_\phi \psi$$ to the action functional, where $D_\phi$ 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 $\phi$ an element in a Pfaffian line. These glue to the Pfaffian line bundle $$\array{ Pfaff \\ \downarrow \\ conf_{bos} }$$ over $conf_{bos}$. 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 $e^{-S}$ is a section in the tensor bundle $$\array{ Charge \otimes Pfaff \\ \downarrow & \uparrow^{e^{-S}} \\ conf_{bos} } \,.$$

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 $e^{-S}$ with a complex function requires a choice of isomorphism $$Charge \otimes Pfaff \stackrel{\simeq}{\to} conf_{bos} \times \mathbb{C}$$ of line bundles with connection, with the trivial bundle on the right carrying the trivial connection.

One hence says that the curvature 2-form $curv(Charge \otimes Pfaff)$ of the anomaly bundle is the local anomaly, while its holonomy group $Hol(Charge \otimes Pfaff)$ 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 $(d=10$)-dimensional target space. Therefore their magnetic duals are $(d-(n+1)-2) = 5$-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 $$curv(Pfaff) = \int_X I_4 \wedge I_8 \in \Omega^2_{closed}(conf_{bos})$$ where the integrand 12-form happens to factor into a 4-form $I_4$ and an 8-form $I_8$. Here the integral is to be understood in the sense discussed last time. More explicitly, $I_4$ and $I_8$ depend on the bosonic fields given by the Levi-Civita connection $\omega$ on the Spin bundle of $X$ and a connection $A$ on a complex vector bundle on $X$ as $$I_4 = \frac{1}{2}p_1(F_\omega) - ch_2(F_A)$$ $$I_8 = \frac{1}{48} p_2(F_\omega) - ch_4(F_A) + decomposable characteristic forms \,.$$

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 $j_B$ and electric string current measured by an 8-form $j_E$ is of just the same form $$curv(Charge) = \int_X j_B \wedge j_E \,.$$

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

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 $$d H_3 = 0$$ 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 $j_B$ is the curvature 4-form: $$d H_3 = j_B \;\;= I_4 = \frac{1}{2}p_1(F_\omega) - ch_2(F_A) \,.$$ For fixed $\hat j_B$, a point in the new $conf_{bos}$ now specifies one such twisted 2-bundle for $\hat j_B$ 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 $B$ 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 $$\int_X B \wedge j_E = \int_X B \wedge I_8 \,,$$ encoding the coupling of the electric charge distribution $j_E$ to the electric background field $B$. 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 $conf_{bos}$.

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 $$d H_3 = \frac{1}{2}p_1(F_\omega) - ch_2(F_A) \,,$$ 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.