A single mathematical framework—conformal field theory—is being used to calculate the behaviour of quantum gravity and exotic quantum materials that standard methods cannot touch. These theories describe strongly coupled systems, where particles interact so intensely that the usual mathematical tools (Feynman diagrams) break down. The researcher is combining three advanced techniques—the conformal bootstrap, supersymmetric localisation, and harmonic analysis—to solve problems that have resisted decades of effort. In the first strand, the goal is to compute graviton scattering in string and M-theory to all orders, and to extend a recent derivation of the AdS/CFT correspondence from negative to positive spacetime curvature—the kind relevant to cosmology. In the second strand, the aim is to determine whether certain 2+1 dimensional quantum chromodynamics theories are conformal, and to compute critical exponents that describe phase transitions in magnetic materials and fractional quantum Hall systems. This is fundamental science. If successful, it will not produce a new battery or a faster computer next year. But it would provide exact predictions for experiments on quantum materials, and could reveal whether the holographic principle—the idea that a universe’s physics can be encoded on its boundary—holds in our own universe. Past work on conformal field theory has already reshaped our understanding of phase transitions and black holes.
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My proposal will explore non-perturbative aspects of conformal field theories (CFTs) with applications to both high energy and condensed matter systems. In condensed matter, CFTs describe quantum materials that are the target of current and future experiments. In high energy, CFTs provide the only known non-perturbative description of quantum gravity via the famous AdS/CFT duality. These CFTs are often strongly coupled, however, so they cannot be studied using standard perturbative tools such as Feynman diagrams. My plan is to combine cutting edge non-perturbative methods such as the conformal bootstrap, supersymmetric localization, and harmonic analysis to answer long standing questions in strongly coupled physics. This proposal is divided into two related strands: Strand I. Non-perturbatively study quantum gravity via the dual CFT. For string and M-theory, the goals are to compute graviton scattering to all orders in the Planck length expansion, and study black hole states that appear in this scattering. For the simpler case of higher spin gravity, the goals are to extend my recent derivation of AdS/CFT, which applies to negative spacetime curvature, to the cosmologically relevant case of positive spacetime curvature, and to connect to string/M-theory AdS/CFT. The outputs of this strand will realize the dream of the holographic principle by computing exact physical observables in quantum gravity. Strand II. Study quantum chromodynamics in 2+1 dimensions as an emergent description of algebraic spin liquids, deconfined criticality, and the transition between fractional quantum hall states. The goals are to determine when these theories are conformal, compute critical exponents, verify recently proposed dualities, and find new dualities. The outputs of this strand will predict physical observables that can guide ongoing and future experiments. Since these CFTs are dual to higher spin gravity, the output of strand I will also inform the research of strand II.
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