The global anomalies of (2,0) superconformal field theories in six dimensions

TL;DR

The paper derives the global gravitational anomalies of six-dimensional $(2,0)$ theories, beginning with the $A$-type realization on stacks of M5-branes and then proposing a unified anomaly formula compatible with the $D$- and $E$-type cases. It demonstrates that the global anomaly comprises additional terms beyond the local $I_8$-based contribution, which can be interpreted as Hopf-Wess-Zumino terms on the Coulomb branch, and shows how these terms arise from topological modes of the M-theory C-field. The author provides explicit anomaly expressions for stacks of M5-branes, for the center-of-mass tensor multiplet, and for the interacting $(2,0)$ theory, together with integrality checks and a discussion of conformal blocks and ADE generalization. The work clarifies the role of differential cocycles and Wu classes in global anomaly cancellation and outlines a framework for an ADE-compatible anomaly formula with implications for dimensional reductions and dualities.

Abstract

We compute the global gauge and gravitational anomalies of the A-type (2,0) superconformal quantum field theories in six dimensions, and conjecture a formula valid for the D- and E-type theories. We show that the anomaly contains terms that do not contribute to the local anomaly but that are crucial for the consistency of the global anomaly. A side result is an intuitive picture for the appearance of Hopf-Wess-Zumino terms on the Coulomb branch of the (2,0) theories.

Figures (2)

• Figure 1: This figure illustrates the argument showing that the value of ${\rm An}_{\mathfrak{F}}$ on twisted doubles depends only on the gluing map $\phi$. We start by picking two manifolds $U$ and $U'$ bounded by $M$. On the top left, the twisted double $U_\phi$ is constructed by gluing two copies of $U$, one of them with its orientation reversed, with the help of the map $\phi$. On the top right, the same construction starting from $U'$, with the opposite orientation, yielding $\bar{U}'_\phi$. By rearranging the pieces, we obtain the second line. Then, noticing that the two twists cancel in the second gluing on the second line, we obtain on the third line $V = U \cup_{\rm id} \bar{U}'$ and $\bar{V}$. This pair of manifolds is bordant to the empty manifold, showing that ${\rm An}_{\mathfrak{F}}$ was zero all along and implying that ${\rm An}_{\mathfrak{F}}(U_\phi) = {\rm An}_{\mathfrak{F}}(U'_\phi)$. In terms of the line bundle $\mathscr{L}$, this translates into the fact that the transition functions do not depend on the sections used to compute them.
• Figure 2: A pictorial representation of the arguments in this section. The three pictures represent a fiber over a point of $M$. On the left, the setup used to compute the anomaly due to a set of non-intersecting M5-branes (black dots). Tubular neighborhoods (grayed out) are cut out and there is an anomaly inflow from the M-theory Chern-Simons term in the bulk (in white). This inflow cancels exactly the sum of the anomalies of the isolated M5-branes. In the middle, the setup presented in Section \ref{['SecIdComp']} in order to compute the anomaly of a stack of M5-branes on its Coulomb branch. A single tubular neighborhood of $M$ is cut out and includes all the M5-branes. Again, there is an anomaly inflow due to the M-theory Chern-Simons term in the bulk. On the right, the difference between the anomaly inflow contributions can be attributed to the M-theory Chern-Simons term integrated over the region $N$, represented in white.

Theorems & Definitions (2)

• Proposition A.1
• proof