Theories of Class F and Anomalies

TL;DR

The paper develops Theories of Class F by compactifying the 6d $(2,0)$ theory on nontrivial genus-$g$ curve fibrations over 4d spacetime, producing 4d theories with duality defects and space-time varying couplings that realize $SL(2,\mathbb{Z})$ monodromies.A central technical tool is pushing forward the 8-form anomaly polynomial $I_8$ of the 6d theory along the fiber to obtain the 6d-to-4d anomaly polynomial $I_6$, with explicit universal ($\mathbb{L}$-universal) terms and defect-induced corrections, and complementary field-theoretic checks in the torus case via a $U(1)_D$ anomaly in 4d $\mathcal{N}=4$ SYM.The work provides explicit results for torus fibrations and higher-genus fibrations, including Kodaira-classified duality defects, their 2d remnants on compactifications to strings, and a broad network of extensions to non-supersymmetric, $(1,0)$ conformal matter, and $(\mathcal{N}=1)$ theories, highlighting the role of spacetime-dependent couplings in modifying anomaly structures.Overall, the paper establishes a robust framework to compute and interpret anomaly polynomials in Class F theories, connects to F-theory/D3-brane pictures, and opens avenues to incorporate punctures and more general fibrations.

Abstract

We consider the 6d (2,0) theory on a fibration by genus g curves, and dimensionally reduce along the fiber to 4d theories with duality defects. This generalizes class S theories, for which the fibration is trivial. The non-trivial fibration in the present setup implies that the gauge couplings of the 4d theory, which are encoded in the complex structures of the curve, vary and can undergo S-duality transformations. These monodromies occur around 2d loci in space-time, the duality defects, above which the fiber is singular. The key role that the fibration plays here motivates refering to this setup as theories of class F. In the simplest instance this gives rise to 4d N=4 Super-Yang-Mills with space-time dependent coupling that undergoes SL(2, Z) monodromies. We determine the anomaly polynomial for these theories by pushing forward the anomaly polynomial of the 6d (2,0) theory along the fiber. This gives rise to corrections to the anomaly polynomials of 4d N=4 SYM and theories of class S. For the torus case, this analysis is complemented with a field theoretic derivation of a U(1) anomaly in 4d N=4 SYM. The corresponding anomaly polynomial is tested against known expressions of anomalies for wrapped D3-branes with varying coupling, which are known field theoretically and from holography. Extensions of the construction to 4d N = 0 and 1, and 2d theories with varying coupling, are also discussed.

Figures (1)

• Figure 1: Theories of Class F: A sketch of the structure of the 6d space-time for the $(2,0)$ theory, which upon reduction along the fibration $\pi$ gives rise to theories of class F, for $T^2$ and $C_{3,1}$, respectively, which are 4d theories on $M_4$ with varying coupling. Singularities in the fiber result in monodromies of the couplings of the 4d theory, which take values in the mapping class group $\text{MCG}_{g,n}$. On the left hand side this is $SL(2, \mathbb{Z})$. In the class F theories the singular loci correspond to duality defects (shown in red), which are real codimension two in $M_4$.