6d SCFTs and U(1) Flavour Symmetries

TL;DR

The paper shows that in 6d N=(1,0) theories, abelian gauge symmetries cannot survive decoupling gravity and instead become global flavour symmetries with nonzero 't Hooft anomalies, while ABJ anomalies are canceled. It provides a geometric and supergravity-based argument that the height pairing associated with rational sections in F-theory is non-contractible, forcing the corresponding U(1) couplings to vanish in the gravity-decoupled limit. The authors then analyze abelian flavour symmetries in 6d SCFTs, giving a field-theoretic interpretation as anomaly-free linear combinations of diagonal U(1)s within maximal non-abelian flavour groups and illustrating this with explicit global and local F-theory models across T=1, T=2 bases and conformal matter. They propose a general local rule for identifying the surviving abelian flavour symmetries from the hypermultiplet content and anomaly constraints, highlighting the role of the Shioda map and height pairing in connecting geometry to low-energy symmetries. The results illuminate the structure of 6d SCFTs and their tensor-branch dynamics, with implications for discrete symmetries and potential extensions to lower dimensions.

Abstract

We study the behaviour of abelian gauge symmetries in six-dimensional N=(1,0) theories upon decoupling gravity and investigate abelian flavour symmetries in the context of 6d N=(1,0) SCFTs. From a supergravity perspective, the anomaly cancellation mechanism implies that abelian gauge symmetries can only survive as global symmetries as gravity is decoupled. The flavour symmetries obtained in this way are shown to be free of ABJ anomalies, and their 't Hooft anomaly polynomial in the decoupling limit is obtained explicitly. In an F-theory realisation the decoupling of abelian gauge symmetries implies that a mathematical object known as the height pairing of a rational section is not contractible as a curve on the base of an elliptic Calabi-Yau threefold. We prove this prediction from supergravity by making use of the properties of the Mordell-Weil group of rational sections. In the second part of this paper we study the appearance of abelian flavour symmetries in 6d N=(1,0) SCFTs. We elucidate both the geometric origin of such flavour symmetries in F-theory and their field theoretic interpretation in terms of suitable linear combinations of geometrically massive U(1)s. Our general results are illustrated in various explicit examples.

Figures (1)

• Figure 1: The two-dimensional fan for the toric surface $\widehat{\mathbb{P}^2_{132}}$. To each ray is associated a homogeneous coordinate indicated by the subscript.