Superadditivity at Large Charge (2503.16603v1)

Abstract: The weak gravity conjecture has been invoked to conjecture that the dimensions of charged operators in a CFT should obey a superadditivity relation (sometimes referred to as convexity). In this paper, we study superadditivity of the operator spectrum in theories expanded about the semi-classical saddle point that dominates correlators of large charge operators. We explore this in two contexts. The first is a model with two scalar fields that carry different charges, at a non-trivial Wilson-Fisher fixed point. A careful analysis of the semi-classics for this two field model demonstrates that 'quantum' violations of superadditivity (those not forbidden by the conjecture) persist in the large charge regime. We then turn to study the general properties of CFTs at large charge as bottom-up EFTs. By a trial and error procedure we come up with a seemingly consistent family of examples violating the conjecture. In so doing the presence of a genuine dilaton field appears necessary. On the one hand our result demonstrates that the superadditivity conjecture cannot be proven purely on the basis of a bottom-up analysis. On the other hand, the need for a dilaton, with the corresponding infinite fine tuning, indicates the conjecture-violating EFTs are unlikely to be UV completable.

Analysis of Superadditivity in Conformal Field Theory at Large Charge

The paper in question investigates the conjectured superadditivity property of the spectrum of charged operators within Conformal Field Theories (CFTs), particularly when examined in the large charge regime. The context for this paper arises from the weak gravity conjecture (WGC), which implies certain convexity conditions on CFT operator dimensions as a function of charge. The superadditivity conjecture suggests that for certain discrete operator charges $q$, the operator dimension $\Delta(q)$ should satisfy the inequality $$\Delta(n_1 q_0 + n_2 q_0) \geq \Delta(n_1 q_0) + \Delta(n_2 q_0)$$. This property is analogized from gravitational theories and has significant implications for CFTs in higher-dimensional quantum field theories.

Primary Findings and Methodology

The paper explores superadditivity in two primary contexts: a specific two-field Wilson-Fisher model and a more general analysis based on the Effective Field Theory (EFT) framework at large charge. The Wilson-Fisher model employs two scalar fields with varied charges and is analyzed using semi-classical methods at large charge, revealing insights into when and why superadditivity might be violated due to quantum fluctuations. Quantum corrections, captured in the large charge expansion, highlight non-trivial violations at small charges if the minimal charge $q_0$ is not pragmatically selected.

In parallel, the authors develop a bottom-up EFT analysis to investigate if additional low-energy dynamics could lead to a violation of the conjecture. They examine the potential for additional light degrees of freedom, such as a dilatonic field, to alter the expected superadditive behavior. Their systematic trial-and-error approach confirms that simple Goldstone modes within the EFT maintain superadditivity, but introducing a genuine dilaton field could, under certain conditions, lead to violations, suggesting potential swampland theories not realized in full CFTs.

Implications

The findings impact our understanding of operator dimensions in CFTs – if superadditivity holds across all CFTs, it places stringent conditions on possible UV completions of quantum field theories, implicating fundamental aspects of quantum gravity and the landscape of effective theories. Superadditivity, thus, could play a vital role in constraining the configurations of possible UV-complete theories, aligning with tenets of the swampland program.

Future Directions

Further exploration is warranted in several areas touched upon by the paper:

1. Broadening Models: Testing a wider class of models beyond the specific constructions analyzed could reveal deeper geometric properties of the CFT landscape.
2. Supersymmetric Theories: Extending the analysis to include supersymmetric theories might reveal more robust regimes where certain non-trivial factions of charge lead to definite superadditive or subadditive scenarios.
3. Extensions and Applications: Considering applications of superadditivity in adjacent areas like black hole thermodynamics or the AdS/CFT correspondence might provide a fertile ground for theoretical developments.

In sum, the paper proposes a rigorous examination of superadditivity at large charges in CFTs and introduces significant methodological tools to further the theoretical understanding, with implications stretching across various domains within high energy physics. These exploratory results invite further scrutiny and indicate rich, underlying structures in the fabric of quantum field theories and their associated symmetries.