In A Nutshell

I am primarily interested in the relations between integrability and string theory, and the consequences thereof, such as dualities involving integrable systems and gauge theories. The main tool connecting integrability with string theory that I am studying is a four-dimensional analog of Chern-Simons (CS) theory. This theory can be related to integrable spin chains [CWY18a, CWY18b] and integrable Quantum Field Theories (QFTs) [CY19]. 4d CS appears in string theory as sub-sectors of certain D-brane world-volume theories [CY20, IMRY21]. In this context, many known dualities involving integrable systems can be seen as various applications of string dualities. Examples of known dualities that arise this way include Bethe/Gauge correspondence [CY20, IMRY21], fermionic duality of super spin chains [IMRY21], particular examples of topological holography [IMZ20], geometric Langlands correspondence [AT90], etc.

Why Study Integrability?

The main idea behind integrability is the ability to find exact solutions to a dynamical problem. Roughly speaking, integrable systems are those where given initial conditions we can exactly determine the trajectory of evolution. This problem is at the heart of theoretical physics and is generally a hard one. In the realm of QFTs, instances, where we can compute something of interest with exact precision, are few and far between. Quantum field theory is our most successful mathematical framework so far for building physical models of the observable universe, ranging from subatomic particles to cosmology, and all sorts of matter in between. However, most of the time, we have to content ourselves with approximate computations. Given the far-reaching consequences of predictions from QFTs, it is therefore of exceptional interest to find integrable substructures hidden in them.

Why String Theory?

Arguably, the biggest and the most well-understood class of integrable quantum systems consists of integrable spin chains. These are quantum mechanical systems with a large number of conserved charges. In its simplest incarnation, known as the Heisenberg model, an integrable spin chain models a simple one-dimensional magnet [Bax82]. Tools from string theory can be used to relate integrable spin chains to certain sub-sectors of various QFTs. It is by now part of the canon that D-branes in string theory can be used to create large families of QFTs (review [GK98]). A relatively new idea is that integrable spin chains can also be created in string theory using D-branes [CY20, IMRY21]. String dualities can then be used to relate the spin chains to certain sub-sectors of different QFTs. Dualities abound in string theory, giving drastically different descriptions of the same underlying physical systems. Therefore, being able to embed spin chains in string theory opens the door to finding integrable sub-sectors of numerous QFTs. In addition, brane realizations of integrable systems can provide valuable geometric intuition behind elaborate algebraic constructions related to quantum algebras and their representations.