# Research

#### 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

**arXiv:2307.07339**

**Lagrangian multiform for cyclotomic Gaudin models**

Vincent Caudrelier, Anup Anand Singh, and Benoît Vicedo

*Preprint*, May 2024

**arXiv:2405.12837**

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 long*ish*-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

**arXiv:2009.07623**

#### 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.*