Integrability, gauge theory, and quantisation

Much of my ongoing work is rooted in three major themes: the \(r\)-matrix approach to integrable systems, the theory of Lagrangian multiforms, and the gauge-theoretic origins of integrable structures. Together with my doctoral supervisors and our collaborators, I am currently developing a framework for systematically constructing Lagrangian multiforms, objects that provide a variational description of integrable hierarchies through a generalised action and a variational principle. In the following two papers, we used the theory of Lie dialgebras to achieve this goal for a large class of finite-dimensional integrable systems:

Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems
Vincent Caudrelier, Marta Dell’Atti, and Anup Anand Singh
Letters in Mathematical Physics, Feb 2024

Lagrangian multiform for cyclotomic Gaudin models
Vincent Caudrelier, Anup Anand Singh, and Benoît Vicedo
Preprint, May 2024

At present, we are attempting to construct a Lagrangian multiform for the Hitchin integrable system using gauge-theoretic ideas. The Hitchin system is related to vector bundles on Riemann surfaces and unifies many interesting integrable models – its multiform description will therefore be an important addition to the theory of Lagrangian multiforms. A related longish-term goal is to use our construction together with the path integral formalism to quantise integrable theories in a covariant manner.

Causal set theory

As a visiting scholar at the Raman Research Institute, Bengaluru, I worked on causal set theory, an approach to quantum gravity that postulates that spacetime is fundamentally discrete and replaces the spacetime continuum with locally finite posets. If one were to randomly pick a poset from the space of all finite posets, it would far more likely be non-manifold-like. This poses a challenge to causal set theory, since one expects continuum-like dynamics to arise in its classical limit. A result from my collaboration with Dr. Abhishek Mathur and Prof. Sumati Surya helped establish that certain classes of non-manifold-like causal sets are suppressed in the causal set path sum despite being more typical that manifold-like ones, making the emergence of manifold-like behavior possible without any restrictions.

Entropy and the link action in the causal set path-sum
Abhishek Mathur, Anup Anand Singh, and Sumati Surya
Classical and Quantum Gravity, Dec 2020

Black holes, information scrambling, and quantum chaos

For my MS thesis work done under the supervision of Prof. Spenta Wadia at the International Centre for Theoretical Sciences (ICTS-TIFR), Bengaluru, I tried to better understand the scrambling of information by black holes using ideas from holography, quantum information and chaotic dynamics. I focused, in particular, on the Sachdev-Ye-Kitaev (SYK) model, a strongly coupled, quantum many-body system that is maximally chaotic, nearly conformally invariant, and exactly solvable, properties that make it interesting from the point of view of holography. A new result from my MS project was a derivation of the effective action of a charged version of the SYK model achieved by reducing the original theory of Majorana fermions to a theory of bilocal fields.

Chaos in field theory and gravity
Anup Anand Singh
Supervisor: Prof. Spenta Wadia
MS Thesis, May 2018

You can find a consolidated list of my preprints and publications here.