[Paper Review] A Panorama Of Physical Mathematics c. 2022

Ibrahima Bah, Daniel S. Freed+4 more|arXiv (Cornell University)|Nov 8, 2022

Computational Physics and Python Applications4 citations

TL;DR

This paper presents a comprehensive, forward-looking overview of the evolving interplay between theoretical physics and mathematics in quantum field theory, string theory, and related areas. It identifies key open problems and emerging directions, including the classification of superconformal field theories, holographic dualities, anomalies, and connections to number theory and geometry, offering a roadmap for future research in physical mathematics.

ABSTRACT

What follows is a broad-brush overview of the recent synergistic interactions between mathematics and theoretical physics of quantum field theory and string theory. The discussion is forward-looking, suggesting potentially useful and fruitful directions and problems, some old, some new, for further development of the subject. This paper is a much extended version of the Snowmass whitepaper on physical mathematics [1].

Motivation & Objective

• To map the current landscape of physical mathematics, emphasizing synergies between quantum field theory, string theory, and advanced mathematics.
• To identify and articulate promising, underexplored research directions in physical mathematics, including non-Lagrangian theories, generalized symmetries, and holographic dualities.
• To provide a forward-looking synthesis of recent progress and unresolved challenges in areas such as anomalies, moduli spaces, and geometric structures in QFT.
• To highlight the role of mathematical structures—such as BPS states, automorphic forms, and exceptional holonomy—in advancing our understanding of quantum gravity and quantum field theories.
• To serve as a research roadmap for the next generation of physical mathematics, emphasizing interdisciplinary connections and open problems in high-energy theory and pure mathematics.

Proposed method

• Systematic review and synthesis of recent developments in quantum field theory, focusing on algebraic structures, topological field theories, and nonperturbative effects like resurgence and exact WKB.
• Application of holographic principles and supergravity solutions to classify and characterize AdS geometries dual to 6d and 5d superconformal field theories.
• Use of PDEs and geometric flows to study holographic renormalization group (RG) flows, linking metric evolution to CFT deformations and extremalization principles.
• Analysis of anomaly cancellation mechanisms in 6d supergravity and their relation to invertible field theories and global anomalies.
• Exploration of connections between string compactifications, automorphic forms, and number theory, including the role of attractor mechanisms and Langlands duality.
• Integration of results from topological string theory, twisted holography, and integrable systems to identify new mathematical structures in quantum field theories.

Experimental results

Research questions

• RQ1 What are the fundamental mathematical structures underlying non-Lagrangian superconformal field theories, and how can they be classified geometrically?
• RQ2 How can the full solution space of PDEs governing AdS₆ and AdS₇ solutions in supergravity be systematically characterized and linked to field theory duals?
• RQ3 What is the precise mathematical role of generalized symmetries and anomaly theories in classifying quantum field theories and their phases?
• RQ4 How do extremalization principles such as a-maximization and c-extremization emerge from holographic RG flows and geometric constraints?
• RQ5 In what ways do automorphic forms and arithmetic geometry arise naturally from string compactifications and BPS state counting?

Key findings

• A systematic classification of holographic 6d SCFTs via explicit realization of all AdS₇ solutions in 11d supergravity has been achieved, with characterization in terms of geometric and algebraic data.
• The solution space of AdS₆ solutions in type IIB supergravity has been significantly advanced, with a large family of solutions constructed via (p,q)-brane engineering and dual to 5d SCFTs.
• Holographic RG flows have been described through geometric flows of metrics interpolating between AdS×Σ backgrounds, providing a mathematical framework for CFT deformations.
• Extremalization principles such as a-maximization and c-extremization are shown to emerge from geometric constraints on cycle volumes in Kähler classes, with precise dual field theory interpretations.
• The classification of BPS structures in supergravity has enabled deeper understanding of the solution space of PDEs governing AdS geometries and their physical duals.
• A new class of solutions in type IIB supergravity has been constructed that realize the full moduli space of 5d SCFTs through (p,q)-brane engineering, linking field theory to supergravity.