# Integrable harmonic map equations

Constructing integrable harmonic map equations from 4d Chern-Simons

As part of their work on deriving integrable systems from 4d Chern-Simons theory, Costello and Yamazaki (Costello & Yamazaki, 2019) give a physical construction of a class of two-dimensional classical integrable field theories from the data of an algebraic curve \(C\), a one-form \(\omega\) on \(C\), and a reductive group \(G\). Fixing the input data, the target space of this integrable field theory is a certain open substack of \(\text{Bun}_G(C)\) which I will denote \(\text{Bun}_G^0(C)\). The existence of this integrable field theory implies a number of remarkable and novel algebro-geometric structures on \(\text{Bun}_G^0(C)\).

In the paper (Derryberry, 2021) I gave an algebro-geometric construction of these expected geometric structures, and proved that they satisfied the properties expected from the work of Costello and Yamazaki.

Specifically, the structures that I construct on \(\text{Bun}_G^0(C)\) are, firstly, an algebraic metric and an algebraic 3-form. These are what are needed to define a classical \(\sigma\)-model with target \(\text{Bun}_G^0(C)\). Further, for each \(z \in C\), I find *two* algebraic connections on the evaluation bundle \(P_z\) over \(\text{Bun}_G^0(C)\). From the \(\sigma\)-model perspective, these two connections are what is needed to define the Lax matrix for a specific harmonic map equation (the equations of motion for the \(\sigma\)-model). The primary result of this paper is therefore that the class of harmonic map equations which arise in this manner are all integrable.

Future directions for this project include:

- The construction of explicit solutions to the harmonic map equations.
- Leveraging integrability to understand the topology of the space of harmonic maps (cf (Uhlenbeck, 1989)).
- Proving a conjecture of Costello on the form of the 1-loop beta function for the \(\sigma\)-model (the statement and partial results can be found in (Derryberry, 2021)).

#### References:

- Gauge Theory And Integrability, IIIPreprint, arXiv: 1908.02289, 2019
- Lax formulation for harmonic maps to a moduli of bundlesPreprint, arXiv: 2106.09781, 2021
- Harmonic maps into Lie groups: classical solutions of the chiral model
*Journal of Differential Geometry*, 1989