[Paper Review] Quantum Gravity Bounds on N=1 Effective Theories in Four Dimensions

Luca Martucci, Nicolo Risso+1 more|arXiv (Cornell University)|Oct 19, 2022

Black Holes and Theoretical Physics4 citations

TL;DR

This paper derives quantum gravity constraints on four-dimensional N=1 effective field theories (EFTs) by analyzing EFT strings—axionic, half-BPS defects whose consistency as weakly coupled nonlinear sigma models imposes positivity bounds and quantization conditions on axion couplings to gauge and gravitational sectors. The key result is a universal bound on gauge group rank in terms of gravitational couplings, which rules out otherwise consistent supergravity theories as viable quantum gravity completions.

ABSTRACT

We propose quantum gravitational constraints on effective four-dimensional theories with N=1 supersymmetry. These Swampland constraints arise by demanding consistency of the worldsheet theory of a class of axionic, or EFT, strings whose existence follows from the Completeness Conjecture of quantum gravity. Modulo certain assumptions, we derive positivity bounds and quantization conditions for the axionic couplings to the gauge and gravitational sector at the two- and four-derivative level, respectively. We furthermore obtain general bounds on the rank of the gauge sector in terms of the gravitational couplings to the axions. We exemplify how these bounds rule out otherwise consistent effective supergravity theories as theories of quantum gravity. Our derivations of the quantum gravity bounds are tested and further motivated in concrete string theoretic settings. In particular, this leads to a sharper version of the bound on the gauge group rank in F-theory on elliptic four-folds with a smooth base, which improves the known geometrical Kodaira bounds. We furthermore provide a detailed derivation of the EFT string constraints in heterotic string compactifications including higher derivative corrections to the effective action and apply the bounds to M-theory compactifications on $G_2$ manifolds.

Motivation & Objective

• To derive quantum gravity constraints on N=1 supersymmetric effective field theories in four dimensions using axionic strings as probes.
• To identify conditions under which EFT strings—arising from the Completeness Conjecture—can be treated as weakly coupled nonlinear sigma models (NLSMs) with controlled backreaction.
• To translate worldsheet anomaly matching into consistency conditions on bulk couplings, particularly for axion-gauge and axion-gravity interactions.
• To derive universal bounds on the rank of the gauge sector in terms of gravitational couplings to axions, improving over known geometric bounds in F-theory.
• To test and motivate these bounds in concrete string compactifications, including F-theory, heterotic, and M-theory on G2 manifolds.

Proposed method

• Apply the Completeness Conjecture of quantum gravity to postulate the existence of axionic EFT strings charged under U(1) and gravitational symmetries.
• Use anomaly inflow to compute worldsheet anomalies of EFT strings, requiring their cancellation for UV consistency.
• Derive positivity bounds on axion couplings at two-derivative level (e.g., $\partial_\mu \phi \partial^\mu \phi$ terms) via anomaly matching.
• Derive quantization conditions for axion couplings to gravity at four-derivative level (e.g., $R^2$ terms), enforcing consistency with quantum gravity.
• Construct a weakly coupled NLSM description of the EFT string worldsheet, valid near the core due to enhanced shift symmetries.
• Test bounds in F-theory on elliptic four-folds, heterotic compactifications with higher-derivative corrections, and M-theory on G2 manifolds.

Experimental results

Research questions

• RQ1 What quantum gravity constraints arise from the consistency of EFT strings in N=1 supersymmetric 4D EFTs?
• RQ2 How do axionic couplings to gauge and gravitational sectors constrain the allowed structure of the effective action?
• RQ3 What is the maximal allowed rank of the gauge group in terms of gravitational couplings to axions?
• RQ4 How do these bounds improve upon known geometric bounds in F-theory compactifications?
• RQ5 Can the derived bounds rule out otherwise anomaly-free supergravity theories as quantum gravity completions?

Key findings

• Positivity bounds are derived for axion couplings to the gauge and gravitational sectors at the two-derivative level, ensuring consistency with quantum gravity.
• Quantization conditions are established for axion couplings to gravity at the four-derivative level, constraining the curvature-squared terms.
• A universal upper bound on the gauge group rank is derived in terms of gravitational couplings to axions, which improves the known Kodaira bound in F-theory on smooth elliptic four-folds.
• The bounds rule out otherwise consistent N=1 supergravity theories as quantum gravity completions, such as those with unbounded gauge rank.
• The constraints are microscopically verified in F-theory compactifications on elliptic four-folds with smooth bases, where the bounds match explicit flux and divisor counting.
• In M-theory compactifications on G2 manifolds, the consistency of EFT strings is linked to the positivity of M5-instanton actions, requiring $s^0 > 0$ and $s^0 - p_a s^a + \sum_k s^k > 0$.