Convexity of Charged Operators in CFTs with Multiple Abelian Symmetries

TL;DR

This paper explores convexity properties of charged operators in conformal field theories with multiple Abelian symmetries, proposing constraints on their charge spectrum related to the Weak Gravity Conjecture and analyzing dimensional differences.

Contribution

It extends the convexity constraints to multi-dimensional charge spaces and provides proofs and examples in two and higher dimensions, refining the conjecture accordingly.

Findings

In 2D CFTs, the sub-lattice index can be parametrically large, but is bounded by current levels.

In higher dimensions, the sub-lattice index generated by BPS operators can be large, but no parametric delay in convexity is observed.

The convexity conjecture requires modification in 2D to incorporate current levels, which can be proven.

Abstract

Motivated by the Weak Gravity Conjecture in the context of holography in AdS, it has been proposed that operators charged under global symmetries in CFTs, in three dimensions or higher, should satisfy certain convexity properties on their spectrum. A key element of this proposal is the charge at which convexity must appear, which was proposed to never be parametrically large. In this paper, we develop this constraint in the context of multiple Abelian global symmetries. We propose the statement that the convex directions in the multi-dimensional charge space should generate a sub-lattice of the total lattice of charged operators, such that the index of this sub-lattice cannot be made parametrically large. In the special case of two-dimensional CFTs, the index can be made parametrically large, which we prove by an explicit example. However, we also prove that in two dimensions there always exist convex directions generating a sub-lattice with an index bounded by the current levels of the global symmetry. Therefore, in two dimensions, the conjecture should be slightly modified to account for the current levels, and then it can be proven. In more than two dimensions, we show that the index of the sub-lattice generated by marginally convex charge vectors associated to BPS operators only, can be made parametrically large. However, we do not find evidence for parametric delay in convexity once all operators are considered.