Prize Scholar: Nivedita

Am I am a physicist? Am I a mathematician? (Is this the real life? Is this just fantasy?) At times, it feels like I am walking a tightrope between two cliffs. Thurston, in his essay, 'On Proof and Progress in Mathematics', suggests that the role of mathematicians is to advance human understanding of mathematics by finding and formalising new structures and proving theorems about them that pave the way for newer structures. Physicists, on the other hand, pursue the path of advancement of human understanding of the universe. A fundamental question which is set deeply in all human cognitive pursuits is why things are the way they are and what is the best 'language' to describe them in order to unravel the structure underneath.

I work with modern mathematical structures that describe aspects of quantum field theory (QFT). QFT lies at the heart of modern theoretical physics, describing systems and offering stunning experimental accuracy in both condensed matter and high-energy physics. In the last half-century, it has also given us remarkable conjectures and surprising non-trivial connections between seemingly unrelated fields of mathematics. Despite being around for nearly a century now, the mathematical foundations of QFT remain largely mysterious.

Two primary schools putting QFT on firm mathematical footing are Algebraic Quantum Field Theory (AQFT) and Functorial Quantum Field Theory (FQFT). These are analogous to the Heisenberg and Schrödinger pictures of quantum mechanics, where the former focuses on observables and the algebraic properties thereof, and the latter describes the (time) evolution of state spaces. These two approaches have been studied independently until very recently when attempts are being made to relate the two. My work is based on a special subclass of quantum field theories, the conformal field theories (CFTs). Two-dimensional CFTs describe critical phenomena in various condensed matter systems, and they also form the building blocks of string theory. The aim is to give a construction of a fully extended Segal chiral CFT (a functor defined on the category of conformal manifolds) from the algebraic data of a conformal net. This work aims to understand if the two formalisms of QFT are indeed equivalent to each other in this case.

I am also working on giving a rigorous mathematical basis to the idea that a d-dimensional QFT has a monoidal (d-1)-category of symmetries. Global and higher categorical symmetries are being discovered by physicists in QFTs, and my project aims to present that such structures are expected to exist where symmetries are generalised to topological interfaces between QFTs. In the past decade, using higher categories has proven to be an essential ingredient in studying quantum field theories from a mathematical perspective. Studying QFT in the language of higher category theory beautifully and naturally incorporates many subtle concepts like generalised symmetries, topological defects, and locality, to name a few.

Here is a link to my website: https://www.n-nivedita.com/home