Dan Freed, Harvard mathematician and director of the Center of Mathematical Sciences and Applications (CMSA), discusses the nature of quantum field theory, the impact of AI on mathematics, and his career spanning geometry, topology, and physics.
Teaching philosophy and the transition to research
Freed emphasizes that research requires living with confusion and uncertainty, unlike classroom learning where students understand nearly everything.
He tells students: “If you’re not confused, you’re not working,” and designs open-ended problems to bridge the gap to research.
Example: in group theory, have students explore subgroups of a group of order eight, discover patterns, and run experiments — now enhanced by AI.
Freed encourages AI use but insists on a solid mathematical foundation first; he assigns one AI problem per week in his differential geometry course while keeping other problems AI-free.
Harvard’s math department and CMSA are actively adapting: hosting summits on PhD education and math research in the age of AI, running an “AI for the Working Mathematician” seminar, and embracing the paradigm shift rather than resisting it.
The AI explosion in mathematics (summer–fall 2024)
May 2024: AI solved the Erdős unit distance conjecture by constructing a counterexample using a tower of number fields.
Summer 2024: Solutions to the Jacobian conjecture (disproved) and complex structure on S⁶ appeared.
Early August: OpenAI released solutions to 10 significant problems across diverse areas.
Early September: OpenAI released a purported solution to the Navier-Stokes millennium problem.
Early October: OpenAI released ~475 AI-generated papers, each claiming a proof or counterexample to well-known problems.
The community faces a massive verification task; Harvard’s CMSA is convening mathematicians, government, and industry to navigate the transition.
Freed notes two AI success modes: (1) pushing construction ideas from literature further than humans pursued, and (2) connecting distant mathematical areas where no single expert knows both.
On Yang–Mills mass gap: less well-formulated than Hodge conjecture; recent OpenAI progress on 2D sigma models used cross-area techniques, but full 4D problem remains open.
What is quantum field theory? Axiom systems and the “elephant”
Freed uses the real numbers as analogy: “complete ordered field” is a hard-won three-word characterization that doesn’t reflect our decimal experience but enables powerful theorems.
QFT lacks a similarly crisp, unique axiom system. Two 1950s–60s axiom systems for relativistic QFT (Wightman axioms; Haag–Kastler) are analytic and don’t capture all QFTs.
1980s brought geometric examples from string theory and study of QFT in other dimensions — “the proverbial elephant”: different people see different parts (trunk, tail) but never the whole.
Graeme Segal’s late-1980s axiom system for Wick-rotated (Euclidean) QFT, initially for 2D conformal field theories, proved surprisingly powerful across topology, symplectic geometry, algebraic geometry, algebra, and number theory.
Two streams now: geometric/topological (Segal-style) and analytic (Wightman/Haag–Kastler). Future task: merge them into a unified picture.
Axiom system vs. definition: largely interchangeable in this context; both start from undefined terms, add definitions, axioms, then build theorems.
Anomalies as features, not bugs
Quantum mechanics is fundamentally projective geometry: pure states are lines in a Hilbert space (projective space), not vectors.
Quantization requires linearizing the projective system; the obstruction to linearization is an invertible field theory — what physicists call an anomaly.
In the 1980s anomalies were seen as bugs to cancel (e.g., Green–Schwarz mechanism). Freed’s framework shows anomalies are themselves QFTs with locality, making them computable invariants.
‘t Hooft anomaly matching: when flowing from UV to IR, the anomaly/projectivity must match, constraining the low-energy theory — a powerful feature for studying physical systems.
Anomaly is a bug during construction (need to linearize to continue quantizing) but a feature of the finished projective theory.
Determinant line bundles and the Atiyah–Singer index theorem (mid-1980s)
Freed’s 1980s paper with Jean-Michel Bismut grew from the anomaly–index theorem connection.
Isadore Singer (Freed’s advisor) saw early that mathematicians must engage directly with QFT; the Atiyah–Singer index theorem (1963) suddenly appeared in fermionic anomaly computations.
Dan Quillen studied determinant line bundles on Riemann surfaces for Arakelov theory; Bismut brought probabilistic proofs of index theorems.
Freed and Bismut developed the geometry of determinant line bundles in a general setting applicable to physics: connection, curvature, holonomy, and variations (holomorphic families, Riemann surfaces, topological aspects).
Key insight: kernel and cokernel dimensions can jump in a family, but their difference (the index) is constant — a topological invariant.
Chern–Simons theory: the first exact QFT mathematicians could touch
Witten’s 1988 paper “Quantum Field Theory and the Jones Polynomial” originated from a dinner with Segal, Atiyah, and Witten in Swansea.
Before Chern–Simons, mathematicians had only perturbation theory (“asymptotic to what?”) and no exact non-perturbative examples.
Chern–Simons gave both an asymptotic expansion (using classical invariants like Chern–Simons invariant and Reidemeister torsion) and an exact topological answer (via Reshetikhin–Turaev’s algebraic construction).
The asymptotic expansion remains unproved rigorously; the two mathematical languages (perturbative vs. exact) are very different — a measure of how much physics intuition remains unabsorbed by mathematics.
Non-invertible symmetries / global categorical symmetries
Freed is part of a Simons Collaboration on “Global Categorical Symmetries” — a name he finds oxymoronic (“non-invertible symmetry” contradicts the geometric notion of symmetry as reversible).
Better framing: algebras acting on quantum theories. Algebras have a zero element that kills everything — hard to call that a symmetry.
Originated during COVID (2020): Michele Del Zotto assembled mathematicians and physicists on Zoom; the collaboration formed and won funding remotely.
With Greg Moore and Constantin Teleman, Freed developed a framework phrasing these as algebras in the Segal axiom system / topological field theory context.
In quantum mechanics (low-dimensional case), reduces to operator algebras commuting with the Hamiltonian — an old von Neumann idea, now extended.
Extended locality and unitarity
Segal’s axioms: QFT as a linear representation of a bordism category (encoding locality) into a category of vector spaces.
Two bedrock principles: locality (in the domain) and unitarity (a property of the representation, like for group representations).
Basic Segal axioms allow only codimension-1 cuts; extended locality allows cutting further (surfaces → lines → points).
Reflection positivity (Wick-rotated unitarity) is understood in the basic case but hard to incorporate in the extended framework.
Freed and Hopkins (∼10 years ago) incorporated unitarity in the extended invertible case to classify phases of matter — a topological formula that gave wrong answers without unitarity.
Recent progress on extended unitarity for non-invertible theories in topological settings.
Topologically twisted theories and “quantum” vs. “classical”
Topologically twisted supersymmetric theories are not unitary; Freed often drops “quantum” and says “topological field theory.”
Seiberg’s view: invertible field theories = classical; non-invertible = quantum. Freed is unsure — other sources of invertible theories (e.g., from index theory) may not deserve “classical.”
The distinction may be semantic; the mathematical definition “invertible” is precise.
Does the Standard Model satisfy Segal’s axioms?
Segal axioms live on compact Riemannian manifolds (Wick rotation: Minkowski → Euclidean → curved).
One can contemplate putting the Standard Model in this framework, but no one has proved even a free field theory satisfies the axioms, let alone interacting Yang–Mills.
Constructing pure Yang–Mills on all Riemannian 4-manifolds would imply controlling long-distance behavior (mass gap) — equivalent to solving the millennium problem.
Segal axioms provide a beautiful mathematical window but remain far from the physical theories of interest.
Charge quantization and generalized differential cohomology
(Referenced but not detailed in this conversation; Freed has a paper on this topic.)
Exotic ℝ⁴ and physics
Dimension 4 is unique: 1,2,3 have unique smooth structures; ≥5 have finitely many on compact manifolds; 4 has countably many on compact manifolds and uncountably many on ℝ⁴ (exotic ℝ⁴s).
Discovered via Freedman (topological 4-manifolds) + Donaldson (smooth structures); Bob Gompf (Freed’s former student) constructed uncountably many exotic ℝ⁴s.
Physics relevance unclear: in quantum gravity one might sum over smooth structures, but no cogent argument links exotic ℝ⁴s specifically to 3+1 spacetime.
Quantization: not a functor, but a family of procedures
Quantization (classical → quantum) fails to be a functor: no universal well-defined correspondence.
Planck’s constant as a parameter: in Chern–Simons, it’s 1/integer — no continuous classical limit, only discrete asymptotic expansion.
In 2D CFT with central charge < 1, theories are discrete with no classical limit parameter.
Abelian Chern–Simons: classical data maps to quantum data infinitely-to-one (many classical systems → same quantum system).
Over time, different quantization methods (geometric, canonical, path integral) fit into a more coherent picture.
Freed’s early 1990s work: interpreted geometric quantization as a path integral integrating vector spaces (not numbers), and in 1D integrated linear categories — fitting canonical and path integral quantization together in simple cases.
Fully local topological field theories and physical predictions
Freed, Teleman, Scheimbauer (2024 paper): used the cobordism hypothesis (full locality) to construct Reshetikhin–Turaev / Chern–Simons theories from algebraic data.
Full locality unlikely to change physical predictions; physicists already assume it. The mathematics is hard for honest QFTs with infinite degrees of freedom, but in topological/invertible models one can explore which theories extend and which don’t.
”Second quantization” terminology
Likely a misnomer: first quantization → one-particle states; second quantization → Fock space / multi-particle states from a field.
The classical system is the field; quantizing it once gives the multi-particle picture. “Second quantization” is a stepping stone, not a literal second application of a quantization functor.
Advice for students in the AI era
Core skill: finding your own problems, not just solving posed ones. Breaking problems into digestible pieces.
In settled fields (e.g., PDE), foundations exist; in physics/QFT, the problem is “what is a QFT?” — a different kind of problem.
If AI solves well-posed problems, human value shifts to: formulating new conceptualizations, asking important questions, communicating them.
Freedom to err is essential: Singer (Freed’s advisor) valued freedom to explore, err, create. Without it, no progress — like needing freedom to ask “stupid” questions.
Example: Atiyah asking “Why is the Â-genus of a spin manifold an integer?” — seemed stupid (already proved), but led to the Dirac operator and index theory.
Naive questions often reach the heart of why something works.
Current projects and outlook
With Hopkins and Moore: orienting moduli spaces of instantons (Donaldson theory) using invertible field theory structures; extending to spin structures and Floer theory.
With Teleman and collaborators: defects in topological field theory (submanifolds with coupled sub-theories) — an underdeveloped part of the mathematical picture.
AI: wants to use it interactively to formulate larger research directions, break them down, and assemble teams — not just solve standing problems.
Hopeful but acknowledges personal loss: the “charmed life” of slow, deep mathematical work is changing. Young people should not be discouraged — radical change brings new opportunities and new forms of mathematics.
