Unpicking the Links Between Knots and Quantum Theory

Benjamin Skuse

When a researcher collects a Nobel Prize in Physics, Chemistry, or Physiology/Medicine it is often a reward for one astonishing breakthrough or a series of breakthroughs many years before that have had a lasting impact on a field or society. With no Nobel Prize for mathematics, the Fields Medal is one of two awards (alongside the Abel Prize) that carry a similar level of prestige. Yet this award is not given just for outstanding achievements that have already been made. Given that Fields Medals are only handed out to mathematicians under the age of 40, its other purpose is to highlight mathematicians’ potential for making significant contributions in the future.

Undergraduate Promise in Knot Theory

John Pardon, of Stony Brook University in New York, is one of four recipients of this year’s Fields Medal (alongside Yu Deng, Jacob Tsimerman and Hong Wang). The unassuming American already has significant achievements to his name. For instance, his first original contribution to mathematics came when he was a Princeton University undergraduate, where he disproved a 1983 conjecture by renowned geometer Mikhael Gromov (2009 Abel Prize).

Gromov’s conjecture was to do with knots, a mathematical topic that includes real-world rope knots that we all know; described in three-dimensional (3D) space mathematically as smooth closed curves. Gromov’s work related to a specific property of knots known as a distortion, otherwise known as how tangled a knot is geometrically. He surmised that there must be some universal upper bound on distortion for a class of knots known as torus knots. In other words, there must be a limit to how tangled a torus knot can be, even for the most complex of knots. In work published in 2011, Pardon proved this was not true, finding certain infinitely repeating knot families whose geometric tangling was inherently limitless.

In the intervening years, Pardon has made significant achievements in many highly technical areas. His advances across symplectic geometry particularly have been groundbreaking. However, instead of providing a superficial glimpse into a number of different areas that have piqued Pardon’s interest and his subsequent achievements in them, what is perhaps more insightful is to focus on just one of these interests – knots – and how he is now and might in the future apply his mathematical talents to vexing unsolved problems in this area.

Picking Apart One of Pardon’s Research Interests

Though by all accounts a quiet and reserved mathematician who rarely makes his thoughts known, a strong hint of Pardon’s focus in this area was given in a rare interview for SCGP News shortly after he joined the Simons Center for Geometry and Physics (SCGP) at Stony Brook in 2022. “I’ve always been fascinated by quantum invariants of 3-manifolds arising from Witten’s path integral reformulation of the Jones polynomial based on the Chern–Simons functional,” he said.

Now, that sentence contains a litany of terms – none of which are ‘knot ’ – that need some unpacking. A ‘3-manifold’ is the 3D analogue of a 2D surface of a sphere, torus, or any other surface with more holes. Continuing the 2D analogy, the number of holes of a surface is an example of an invariant: a property of a shape or space that remains the same even when other properties change. A ‘quantum invariant’ can be loosely thought of as invariants constructed using ideas originating in quantum theory.

Digging deeper into the weeds, the ‘Jones polynomial’ is an example of a knot and link (a disjoint union of several knots) invariant. Invented in the 1980s by Sir Vaughan Jones (1990 Fields Medal), it is an algebraic rule that assigns a polynomial equation to any knot/link, acting as a mathematical fingerprint, often distinguishing different knots, although distinct knots can sometimes share the same Jones polynomial.

How the ‘Chern–Simons functional’ relates to the Jones polynomial in Pardon’s sentence is far from obvious on face value. This is because Chern–Simons theories are now often regarded as physical quantum field theories that describe how forces and fields behave in a given space; even though they were introduced as mathematical concepts by Shiing-Shen Chern and James Simons (latterly of Simons Foundation fame) in 1974. To make sense of this, a little history might help.

In the late 1980s, theoretical physicist Edward Witten (1990 Fields Medal; the first physicist to receive this honour) started thinking about a problem posed to quantum field theorists by mathematician Sir Michael Atiyah (1966 Fields Medal, 2004 Abel Prize): Why did definitions of the Jones polynomial and its generalizations always involve looking in some way at a 2D projection or slicing of a knot when these objects were inherently 3D invariants? In other words, it didn’t feel right that defining a fundamental property of a 3D object should rely on drawing it in 2D.

Detailed in his seminal 1989 paper, Witten took a unique approach to the problem. Rather than studying knots directly, he considered special observables – known as Wilson loops – associated with knots in Chern–Simons quantum field theory. Using the path integral formalism introduced by Richard Feynman (1965 Nobel Prize in Physics) – a way of averaging together every possible quantum state the field could take – he argued that their expectation values reproduce the Jones polynomial. What Witten had shown was that the Jones polynomial from knot theory was a natural outcome of 3D quantum field theory, where the physical observables are topological invariants of the spacetime in which the theory lives.

The Intersection of Abstract Mathematics and Fundamental Theoretical Physics

So, when Pardon said he was interested in “quantum invariants of 3-manifolds arising from Witten’s path integral reformulation of the Jones polynomial based on the Chern–Simons functional,” one interpretation of this is that he was going to explore how the intrinsic topology of 3D (and higher-dimensional) spaces can be understood through invariants derived from physical theories.

This interest has already borne fruit. Calabi–Yau 3-folds (named after Eugenio Calabi and 1983 Fields Medallist Shing-Tung Yau) are manifolds that crop up in superstring theory as candidates for extra dimensions proposed to be curled up in our universe. Associated with them are slightly more complicated invariants than numbers of holes. This set of ‘curve-counting’ invariants can be thought of as answering how many distinct geometric curves of a given type can fit inside the 6D Calabi–Yau space (though they do not literally count embedded curves).

Gromov and Witten proposed a way of counting these curves dynamically derived from string theory using certain invariants, whereas Simon Donaldson (1986 Fields Medal) and Richard Thomas proposed a static method derived from gauge theory using different invariants. In 2023, Pardon established the longstanding MNOP correspondence – proposed by Davesh Maulik, Nikita Nekrasov, Andrei Okounkov (2006 Fields Medal), and Rahul Pandharipande in the 2000s – that the two methods are actually equivalent.

Given how the MNOP correspondence is inspired by gauge/string duality, and how long the correspondence remained unproven, it is no surprise then that, as his citation states, Pardon’s MNOP conjecture proof was a key achievement being recognized with the Fields Medal.

What Pardon will do next in this area could have even greater consequences. Expanding curve-counting techniques beyond standard Calabi–Yau spaces to broader classes of manifolds and developing these methods even further to make them highly generalizable would be significant advances. These could deepen mathematical understanding of enumerative geometry, symplectic topology, and quantum field theory, while potentially providing new tools for studying conjectural links between geometry and theoretical physics. With Pardon still only 37, there will hopefully be many years to see what exciting contributions he will make from his unwavering interest in knots and the hidden topology of space.

The post Unpicking the Links Between Knots and Quantum Theory originally appeared on the HLFF SciLogs blog.