Musings

The constraints of anomaly-cancellation are rather severe, in $4k+2$ dimensions.

First of all, the gauge group can be decomposed as $$G = G_H \times G_S \times G_r$$ where $G_H$ is a compact subgroup of the isometry group of the hypermultiplet manifold¹, $G_S$ is a group under which the hypermultiplets are neutral, and $G_r = Sp(1)$ or $U(1)$, in the case of gauged supergravity, where the R-symmetry group, or a $U(1)$ subgroup thereof, is gauged. To kill the $Tr(R^4)$ term in $I_8$, we must have

where $n_T$ is the number of tensor multiplets.

We also need the $Tr(F^4_i)$ term to vanish, for each simple factor of the gauge group. For $G_S$, this means that it must be a product of simple factors with no quartic Casimir ($SU(2)$, $SU(3)$, $G_2$, $F_4$, $E_6$, $E_7$, or $E_8$). If $G_H$ contains $Sp(N)$ ($N\gt 1$), $SU(N)$ ($N\gt 3$) or $SO(N)$ ($N\gt 4$) factors, we must arrange for the contributions of the hypermultiplets to cancel the contributions of the vectors. Having done that, $I_8$ may be written as $$I_8 = C^{A B} \rho_A \rho_B$$ where $\rho = (Tr(R^2), Tr(F_1^2), \dots , Tr(F_n^2))$. If the quadratic form, $C^{A B}$, has rank $r\leq n_T +1$ and signature $(1,r-1)$ (for its nonzero eigenvalues), then the anomaly can be cancelled by a (generalized) Green-Schwarz mechanism.

Even with $n_T=1$, there are lots of theories, including two infinite series discovered by Schwarz². Kumar and Taylor argue, correctly I think, that these infinite series are pathological. The gauge kinetic terms (related by supersymmetry to the coefficients in the anomaly polynomial) are not positive-definite. Many of the theories that remain can, indeed, be embedded in string theory.

Wth less than three simple factors in the gauge group, there are still plenty of theories with no (known) string construction. There’s the $SU(24)\times SO(8)$ model, mentioned in their paper. But there are plenty of others in Avramis and Kehagias. For instance, there are models with $G=SO(29)$, and $G=SO(30)$ with³, respectively, 21 and 22 fundamental hypermultiplets, and the appropriate number of neutral hypermultiplets to satisfy (1).

And Avramis and Kehagias make no pretence that their list of anomaly-free models is complete, even for $n_T=1$, let alone for $n_T\gt 1$. (Note that the set of 6D theories which can be embedded in string theory include the low-energy limit of the $E_8\times E_8$ heterotic string, compactified on K3, which has branches of its moduli space with $n_T\gt 1$. So it seems particularly artificial, in this context, to restrict to $n_T=1$.)

In the physically-interesting case of 4 dimensions, the absence of gravitational anomalies means that the anomaly-cancellation does not impose much of a constraint at all on the possible gauge groups and matter content. It’s not clear at all why the 6D case (even if it were true that all anomaly-free 6D supersymmetric theories could be UV-completed in string theory) would be a good guide to 4D.