Chern-Weil Global Symmetries and How Quantum Gravity Avoids Them

TL;DR

The paper introduces Chern-Weil global symmetries as currents built from products of gauge field strengths and shows that quantum gravity eliminates such stable global symmetries by either gauging or breaking them. Through field theory, string theory, and AdS/CFT analyses, it demonstrates that gauging CW currents via Chern-Simons terms and dissolving brane charges on higher-dimensional branes are recurring mechanisms ensuring consistency with quantum gravity. The work connects familiar phenomena—axions, axion monodromy, Chern-Simons couplings, worldvolume degrees of freedom, brane ending/dissolution, and holographic anomalies—under a single CW-symmetry lens, and discusses implications for axion physics and boundary CFTs. It argues that the absence of CW global symmetries may be a deep organizing principle in quantum gravity with potential phenomenological consequences and guidance for future model building.

Abstract

We draw attention to a class of generalized global symmetries, which we call "Chern-Weil global symmetries," that arise ubiquitously in gauge theories. The Noether currents of these Chern-Weil global symmetries are given by wedge products of gauge field strengths, such as $F_2 \wedge H_3$ and $\text{tr}(F_2^2)$, and their conservation follows from Bianchi identities. As a result, they are not easy to break. However, it is widely believed that exact global symmetries are not allowed in a consistent theory of quantum gravity. As a result, any Chern-Weil global symmetry in a low-energy effective field theory must be either broken or gauged when the theory is coupled to gravity. In this paper, we explore the processes by which Chern-Weil symmetries may be broken or gauged in effective field theory and string theory. We will see that many familiar phenomena in string theory, such as axions, Chern-Simons terms, worldvolume degrees of freedom, and branes ending on or dissolving in other branes, can be interpreted as consequences of the absence of Chern-Weil symmetries in quantum gravity, suggesting that they might be general features of quantum gravity. We further discuss implications of breaking and gauging Chern-Weil symmetries for particle phenomenology and for boundary CFTs of AdS bulk theories. Chern-Weil global symmetries thus offer a unified framework for understanding many familiar aspects of quantum field theory and quantum gravity.

Figures (5)

• Figure 1: The multi-branched potential of axion monodromy. Each branch of the potential is labeled by a distinct integer value of the discrete flux parameter $f_0$. The theory is invariant under the gauge transformation $\phi \rightarrow \phi +2\pi$, $f_0 \rightarrow f_0 + m$.
• Figure 2: Creating a gravitational soliton with nonzero $\mathop{\rm tr}\nolimits(R^2)$ charge on $\mathbb{R}^4$ via the connected sum construction. We take a manifold with nonzero $\int \mathop{\rm tr}\nolimits(R^2)$, such as K3, and glue it to $\mathbb{R}^4$ via a small tube. To a far away observer, this looks like a "particle" with nonzero gravitational Chern-Weil charge. This picture should be understood as a constant time snapshot, and can be generalized to other dimensions and gravitational solitons. See McNamara:2019rup for an extended discussion.
• Figure 3: Illustration of the process that converts between a Yang-Mills instanton localized on a stack of D$p$-branes (depicted at left) and a D$(p-4)$-brane that can move freely into the bulk (depicted at right). The instanton solution has a size modulus, and small instantons are indistinguishable from lower-dimensional D-branes. As a result, instanton charge and D$(p-4)$-brane charge are not independently conserved, avoiding a global symmetry.
• Figure 4: The left panel depicts a D6-brane with dissolved D4-brane charge, represented as a blurry core. The D4 charge is measured by the flux on a linking $S^4$, represented by the dotted sphere. This cycle does not uplift to an $S^4$ in eleven dimensions because the circle fibration over $S^4$ has no global section, as it is the double suspension of the Hopf fibration. In the right panel, we now have a spherical D6-brane. The $S^4$ no longer intersects the D6 worldvolume, so the fibration is trivial over $S^4$ and the cycle has an 11d uplift. This $S^4$ is nontrivial in homology (since it intersects the 2-cycle $\omega_2$ involving the M-theory circle and a radial direction, as shown in the figure).
• Figure 5: Schematic representation of the compactification space $M_3$ (see also Cremonesi:2015bldApruzzi:2017nck). It looks like an American football, with stacks of D8-branes at particular positions wrapping the angular $S^2$. A stack of $f_I$ branes provides an $SU(f_I)$ symmetry at low energies.