[Paper Review] A Brief Summary Of Global Anomaly Cancellation In Six-Dimensional Supergravity

Samuel Monnier, Gregory W. Moore|arXiv (Cornell University)|Aug 3, 2018

Black Holes and Theoretical Physics4 citations

TL;DR

This paper establishes a necessary and sufficient condition for global anomaly cancellation in six-dimensional $(1,0)$ supergravity theories using a 7-dimensional spin topological field theory. It shows that global anomalies cancel precisely when the partition function of this topological field theory on closed 7-manifolds is trivial, extending the Green-Schwarz mechanism to include global and torsion effects via differential cohomology and refined anomaly coefficients.

ABSTRACT

This is a short summary of a talk at Strings 2018. See also arXiv:1808.01334.

Motivation & Objective

• To identify new consistency conditions for low-energy six-dimensional supergravity theories that arise from quantum gravity constraints.
• To systematically address global anomaly cancellation, which had not been fully analyzed before in the context of 6D supergravity.
• To extend the Green-Schwarz mechanism beyond perturbative anomalies to include global and torsion anomalies via differential cohomology.
• To clarify the role of the global structure of gauge groups—especially the connected component and torsion in cohomology—in anomaly cancellation.
• To provide a mathematically rigorous framework for the Green-Schwarz counterterm in topologically nontrivial spacetimes.

Proposed method

• The authors model the $B$-field using generalized cohomology $H^*( ext{BG}; Λ)$, lifting the anomaly 4-form $Y$ to a differential cocycle $\check{Y}$ to capture torsion components invisible in the fieldstrength.
• They define a 7-dimensional spin topological field theory via the partition function $Z_{\text{top}}: \Omega^\text{Spin}_7(BG) \to U(1)$, whose triviality is shown to be necessary and sufficient for global anomaly cancellation.
• The anomaly coefficient $b \in H^4(BG; \Lambda)$ is decomposed into free and torsion parts, with the torsion component $b_T \in \text{Tors}(H^4(BG; \Lambda))$ playing a key role in global anomaly cancellation.
• The Green-Schwarz counterterm $\Psi_{\text{CT}}$ is constructed as a section of a line bundle over the moduli space $\mathfrak{B}$, using differential cohomology to ensure locality and topological invariance.
• The method incorporates characteristic classes: $p_1$ for gravity, $c_2^i$ for simple gauge groups, and $c_1^I c_1^J$ for abelian $U(1)$ factors, with coefficients $a, b_i, b_{IJ}$.
• The framework generalizes the standard Green-Schwarz mechanism by ensuring independence from extensions of $B$-fields into 7-manifolds, using the condition that $\frac{1}{2}\int_{\mathcal{U}_7} dBY$ is well-defined modulo 1 only when the topological field theory is trivial.

Experimental results

Research questions

• RQ1 What is the complete set of conditions required to cancel both local and global anomalies in six-dimensional supergravity?
• RQ2 How do torsion classes in $H^4(BG; \Lambda)$ affect the consistency of anomaly cancellation in 6D supergravity?
• RQ3 Can the Green-Schwarz mechanism be extended to nontrivial topological spacetimes using differential cohomology?
• RQ4 What is the precise mathematical structure that encodes global anomaly cancellation in 6D supergravity?
• RQ5 Under what conditions on the gauge group and representation data is the 7-dimensional topological field theory trivial?

Key findings

• Global anomaly cancellation in 6D supergravity is equivalent to the triviality of a 7-dimensional spin topological field theory defined by the partition function $Z_{\text{top}}: \Omega^\text{Spin}_7(BG) \to U(1)$.
• The torsion component $b_T \in \text{Tors}(H^4(BG; \Lambda))$ of the anomaly coefficient $b$ is a new physical degree of freedom that must be specified in the data of a consistent supergravity theory.
• The condition $\bar{b} \in 2H^4(BG_1; \Lambda)$ is necessary for anomaly cancellation, where $G_1$ is the simply-connected cover of the semi-simple part of $G$.
• When $\Omega^\text{Spin}_7(BG) = 0$, all anomalies—including global ones—are canceled by the Green-Schwarz mechanism, implying that such theories are anomaly-free if the standard conditions are met.
• The refined Green-Schwarz counterterm $\Psi_{\text{CT}}$ is constructed using differential cohomology and is independent of Wu structures, ensuring consistency in nontrivial topology.
• The framework verifies that F-theory compactifications satisfy the necessary anomaly cancellation conditions, including the correct global structure of the gauge group.