Lagrangian multiforms, integrability, and quantisation

Much of my current work is rooted in the \(r\)-matrix approach to integrable systems and the theory of Lagrangian multiforms. Together with my doctoral supervisor Dr. Vincent Caudrelier and our collaborators, I am 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. Incorporating ingredients and ideas from the Hamiltonian framework for integrability – in particular, the theory of Lie dialgebras – we have achieved this for a large class of finite-dimensional integrable systems. At the moment, I am working towards incorporating affine models into our framework. A related longish-term goal is to use our construction together with the path integral formalism to quantise integrable field 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.

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.

Preprints and publications

  1. arXiv
    Lagrangian multiform for cyclotomic Gaudin models
    Vincent Caudrelier, Anup Anand Singh, and Benoît Vicedo
    Preprint, May 2024
  2. LMP
    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
  3. CQG
    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